Diagnosing the Frequency of Energy Deposition in the Magnetically

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RICE UNIVERSITY

Diagnosing the Frequency of Energy Deposition in

the Magnetically-closed Solar Corona

b y

Will Barnes

A THESIS SUBMITTED

I N PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE

Doctor of Philosophy

APPROVED, THESIS COMMITTEE:

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Dr. Stephen Bradshaw, Chair Associate Professor of Physics and Astronomy

^ v -Dr. David Alexander, OBE Professor of Physics and Astronomy

)r. Maarten V. de Hoop Simons Chair and Professor of Computational and Applied Mathematics and Earth Science

HOUSTON, TEXAS

MAY 2019 ABSTRACT

Diagnosing the Frequency of Energy Deposition in the

Magnetically-Closed Solar Corona by

Will Barnes

The solar corona, the outermost layer of the Sun’s atmosphere, is heated to temperatures in excess of one million Kelvin, nearly three orders of magnitude greater than the surface of the Sun. While it is generally agreed that the continually stressed coronal magnetic field plays a role in producing these million-degree temperatures, the exact mechanism responsible for transporting this stored energy to the coronal plasma is yet unknown. Nanoflares, small-scale bursts of energy, have long been proposed as a candidate for heating the non-flaring corona, especially in areas of high magnetic activity. However, a direct detection of heating by nanoflares has proved difficult and as such, properties of this proposed heating mechanism remain largely unconstrained. In this thesis, I use a hydrodynamic model of the coronal plasma combined with a sophisticated forward modeling approach and machine learning classification techniques to predict signatures of nanoflare heating and compare these predictions to real observational data. In particular, the focus of this work is constraining the frequency with which nanoflares occur on a given magnetic field line in non-flaring active regions. First, I give an introduction to the structure of the solar atmosphere and coronal heating, discuss the hydrodynamics of coronal loops, and provide an overview of the important emission mechanisms in a high-temperature, optically-thin plasma. Then, I describe the forward modeling pipeline for predicting time-dependent, multi-wavelength emission over an entire active region. Next, I use a hydrodynamic model of a single coronal loop to predict signatures of “very hot” plasma produced by nanoflares and find that several effects are likely to affect the observability of this direct signature of nanoflare heating. Then, I use the forward modeling code described above to simulate time-dependent, multi-wavelength AIA emission from active region NOAA 1158 for a range of nanoflare frequencies and find that signatures of the heating frequency persist in multiple observables. Finally, I use these predicted diagnostics to train a random forest classifier and apply this model to real AIA observations of NOAA 1158. Altogether, this thesis represents a critical step in systematically constraining the frequency of energy deposition in active regions. Acknowledgements

This work would not have been possible, or at the very least far less enjoyable, without the help and support of my supervisors, colleagues, friends and family. First, I would like to thank my thesis committee members, Dr. David Alexander and Dr. Maarten de Hoop, for agreeing to serve on my committee and reading a first draft of this work. I would especially like to thank Dr. Alexander for his advice and guidance, both career- and research-related, during my time as a graduate student and for helping me navigate the field of solar physics. As a graduate student, it has been my great pleasure to work with and be advised by Dr. Stephen Bradshaw. I have benefited immensely from his vast knowledge of field-aligned hydrodynamics and atomic physics as well as his careful and measured approach to research. Most importantly, he has taught me how to be an independent researcher and I am extremely grateful for his mentorship and friendship during my time at Rice. I also owe a special debt of gratitude to my undergraduate research advisor, Dr. Lorin Matthews (Baylor University), for teaching me about the microphysics of astrophysical dusty plasmas and for inspiring me to go to graduate school. I am grateful for her patience and kindness as a mentor early in my physics education. During my brief time in the solar physics community, I have been fortunate to collaborate with several talented and accomplished researchers. I am extremely grateful to Professor Peter Cargill (Imperial College London, University of St An- drews) for sharing his unparalleled knowledge of coronal loop physics and for v

his patience in guiding me through the writing and publication of two papers early in my graduate career, the first of which comprises Chapter 5 of this thesis. Additionally, I would like to thank Dr. Nicholeen Viall (NASA Goddard Space Flight Center) for lending her observational expertise and detailed knowledge of the temperature sensitivity of the AIA passbands and for providing helpful comments and revisions on Chapter 6 and Chapter 7 of this thesis. I would also like to thank Dr. Jim Klimchuk, Dr. Harry Warren, Dr. Jeffrey Reep, Dr. Jack Ireland, and Dr. Ken Dere. I am extremely indebted to the members of the SunPy community for volunteer- ing their time and effort to build a sustainable software ecosystem for solar physics. In particular, I would like to thank Dr. Stuart Mumford for his tireless and often thankless efforts to continually improve and develop SunPy and for always having the answer to any question related to Python or solar coordinate systems. I am very grateful to the many people at Rice and in Houston who made my time as a graduate student all the more enjoyable. Many thanks go to Dan, Kong, Joe, Nathan, Loah, Brandon, Laura, Alison, Alex, and Shah for hearing my complaints at lunch, sharing more than a few beers at Valhalla, and making graduate school bearable and, on occasion, fun. I would especially like to thank Joe and Mitch for their friendship and support over the last decade, both in Waco and in Houston. I would like to thank my parents, Mark and Terri, for their financial, emotional, and physical support throughout my entire life, across multiple states and even a few continents. I would also like to thank my siblings, Jessie and Wesley, for always being willing to remind me that I am not that smart. I owe special thanks to my in-laws, Jim and Susan, as well as to my siblings-in-law, Tara and Michael, for the many rounds of disc golf and even more free meals; and Tamara and Mike for making me feel welcome when I first moved to Houston and for continuing to support me as I prepare to leave. vi

Lastly and most importantly, I am forever grateful to my wife Morgan, to whom this thesis is dedicated. Her ever-present optimism, constant encouragement, and unmatched love of dogs have made life all the more enjoyable, even in the face of looming deadlines. Without her unconditional love and support, I would not have made it to graduate school let alone ﬁnished this thesis. The Sun is a miasma Of incandescent plasma The Sun’s not simply made out of gas No, no, no

—“Why Does the Sun Really Shine” They Might Be Giants Table of contents

List of ﬁgures xiii

List of tables xxx

Nomenclature xxxii

1 Introduction 1 1.1 The Structure of the Solar Atmosphere ...... 2 1.1.1 Interior ...... 3 1.1.2 Photosphere ...... 5 1.1.3 Chromosphere ...... 7 1.1.4 Transition Region ...... 7 1.1.5 Corona ...... 8 1.1.6 The Solar Wind ...... 9 1.2 The Solar Magnetic Field ...... 10 1.2.1 Origin of the Magnetic Field and Flux Emergence ...... 11 1.2.2 Observations ...... 14 1.2.3 Field Extrapolation ...... 15 1.2.4 Reconnection ...... 17 1.3 Heating in the Solar Corona ...... 20 1.3.1 Waves versus Reconnection ...... 21 Table of contents ix

1.3.2 Nanoﬂare Heating ...... 23 1.4 Thesis Outline ...... 26 1.5 Use of Data and Software ...... 28

2 The Physics of Coronal Loops 29 2.1 Hydrostatics ...... 32 2.1.1 Equations of Hydrostatic Equilibrium ...... 32 2.1.2 The Isothermal Limit ...... 36 2.1.3 Scaling Laws ...... 39 2.1.4 Numerical Solutions ...... 44 2.2 Hydrodynamics ...... 46 2.2.1 Equations of Field-aligned Hydrodynamics ...... 47 2.2.2 The Heating, Cooling, and Draining Cycle of Coronal Loops . 50 2.2.3 The HYDRAD Model ...... 54 2.2.4 The EBTEL Model ...... 55

3 Emission Mechanisms and Diagnostics of Coronal Heating 67 3.1 The CHIANTI Atomic Database ...... 68 3.2 Spectral Line Formation ...... 69 3.2.1 Collisional Excitation of Atomic Levels ...... 70 3.2.2 Level Populations ...... 75 3.2.3 Processes which Affect the Ion Charge State ...... 76 3.2.4 The Charge State in Equilibrium ...... 82 3.2.5 Non-Equilibrium Ionization ...... 84 3.3 Continuum Emission ...... 87 3.3.1 Free-free Emission ...... 87 3.3.2 Free-bound Emission ...... 88 3.4 Temperature Sensitivity of the AIA Passbands ...... 90 Table of contents x

3.5 The Differential Emission Measure Distribution ...... 94 3.5.1 The Emission Measure Slope ...... 96 3.5.2 Determining the DEM from Observations ...... 98 3.6 Time-Lag Analysis ...... 104 3.6.1 Cross-Correlation ...... 105 3.6.2 Time Lag between AIA Channel Pairs ...... 107

4 synthesizAR: A Framework for Modeling Optically-thin Emission 111 4.1 Building the Magnetic Skeleton ...... 112 4.1.1 Potential Field Extrapolation ...... 113 4.1.2 Tracing Magnetic Field Lines ...... 116 4.1.3 Aside: Coordinate Systems in Solar Physics ...... 117 4.2 Field-Aligned Modeling ...... 122 4.3 Atomic Physics ...... 124 4.4 Instrument Effects ...... 126 4.4.1 Constructing the Virtual Observer ...... 126 4.4.2 Projecting Along the LOS ...... 127

5 Inferring Heating Properties of “Hot” Plasmas in Active Region Cores 133 5.1 Introduction ...... 133 5.2 Summary of Relevant Physics ...... 138 5.2.1 Heat Flux Limiters ...... 140 5.2.2 Two-fluid Modeling ...... 140 5.2.3 Ionization Non-Equilibrium ...... 142 5.3 Results ...... 143 5.3.1 Single-fluid Parameter Variations ...... 143 5.3.2 Two-fluid Effects ...... 149 5.3.3 Ionization Non-Equilibrium ...... 155 Table of contents xi

5.4 Discussion ...... 157

6 Predicting Diagnostics for Nanoﬂares of Varying Frequency 160 6.1 Introduction ...... 160 6.2 Modeling ...... 165 6.2.1 Magnetic Field Extrapolation ...... 165 6.2.2 Hydrodynamic Modeling ...... 168 6.2.3 Heating Model ...... 169 6.2.4 Forward Modeling ...... 172 6.3 Results ...... 177 6.3.1 Intensities ...... 177 6.3.2 Emission Measure Slopes ...... 180 6.3.3 Time Lags ...... 184 6.4 Discussion ...... 189 6.5 Summary ...... 192

7 Mapping the Heating Frequency in Active Region NOAA 11158 195 7.1 Introduction ...... 195 7.2 Observations and Analysis ...... 199 7.2.1 Emission Measure Slopes ...... 201 7.2.2 Time Lags ...... 203 7.3 Classiﬁcation Model ...... 207 7.3.1 Data Preparation and Model Parameters ...... 209 7.3.2 Different Feature Combinations ...... 211 7.3.3 Feature Importance ...... 215 7.4 Discussion ...... 217 7.5 Conclusions and Summary ...... 219 Table of contents xii

8 Conclusions and Future Work 222 8.1 Conclusions ...... 222 8.2 Future Work ...... 225 8.2.1 Nanoﬂare Storms on Bundles of Strands ...... 225 8.2.2 Thermal Non-Equilibrium ...... 229

References 233

Appendix A ﬁasco: A Python Interface to the CHIANTI Atomic Database 250 A.1 Parsing Data ...... 251

A.2 The Ion Class ...... 252 A.3 The Element Class ...... 254 A.4 Working with Multiple Ions ...... 254

Appendix B An Implicit Method for Computing Non-Equilibrium Charge States 256 List of ﬁgures

1.1 Schematic of the solar interior. In the core and radiative zone, radiation is the dominant energy transfer mechanism while convection, the cyclic rise of hot gas to the surface and subsequent infall of cooled gas, dominates in the convection zone. Adapted from Figure 11.2 of Carroll & Ostlie (2007)...... 3 1.2 The three branches of the proton-proton nucleosynthesis reaction. The ppI branching ratio is 69% and the ppII branching ratio is 99.7%. Adapted from Figure 10.8 in Carroll & Ostlie (2007)...... 4 1.3 Temperature (blue, left axis) and density (orange, right axis) of the

solar atmosphere as a function of height, h, above the solar surface. These profiles are based on the semi-empirical models of McWhirter et al. (1975) and Vernazza et al. (1981). The data points show the exact values from the models and the smooth lines are first-order spline fits to the data. The dotted black lines denote the different regions of the solar atmosphere...... 6 List of figures xiv

1.4 The layers of the Sun’s atmosphere revealed in multiple wavelengths at approximately 19:00 UTC on 2010 December 1. The top left panel shows the photosphere, the top right panel shows the chromosphere, and the bottom left panel shows the EUV corona all imaged by SDO/AIA. The bottom right panel shows the hot X-ray corona as

observed by Hinode/XRT. All data are courtesy of the AIA and XRT instrument teams...... 8 1.5 The LOS magnetic field strength at the photosphere (i.e. a magnetogram) as measured by the HMI instrument on the SDO spacecraft at 19:00 UTC 2010 December 1. Red indicates positive polarity (out of the page) and blue indicates negative polarity (into the page). The colorbar is on a semi-log scale from 750 G to +750 G...... 15 1.6 Cartoon illustrating the breaking and reconnecting of magnetic field lines. When two regions of oppositely directed field (red and blue, left panel) are brought together (middle panel), a discontinuity devel-

ops in the narrow diffusion region (shaded gray box). Non-ideal effects cause the field lines to “reconnect” (right panel), thus changing the topology of the magnetic field. Adapted from Figure 6.1 of Priest (2014)...... 18 List of figures xv

1.7 Illustration of the nanoflare heating scenario of Parker (1988). The flux tubes have been straightened out such that both the top and bottom gray surfaces correspond to the photosphere. Flux tubes in an initially uniform field (a) are braided when their footpoints are shuffled by the underlying convective motions of the photosphere (b). At some critical angle between the braided flux tubes, they reconnect (c), dissipate their stored energy into the plasma, and relax back to some lower energy state (d). The axes in panel (a) indicate the directions parallel and perpendicular to the magnetic field. Adapted from Figures 5 and 7 of Klimchuk (2015)...... 24

2.1 Left: A simple empirical model of b as a function of height, h, above the solar surface as calculated by Equation 2 and Equation 3 of Gary (2001) for the magnetic ﬁeld above a sunspot (blue) and a plage region (orange). The dotted lines indicate the tops of the photosphere, chromosphere and corona. The dashed line denotes b = 1. Adapted

from Figure 3 of Gary (2001). Right: An arcade of loops extending into the corona observed off the solar limb by the 171 Å EUV band of the TRACE satellite on 6 November 1999. Adapted from Figure 11 of Reale (2010)...... 30 2.2 Radiative loss as a function of temperature for an optically thin plasma. The blue line shows the true value of L computed by CHI- ANTI using the abundances of Feldman et al. (1992) and assuming a

9 3 constant density of 10 cm . The orange line shows the Raymond- Klimchuk (RK) power-law approximation given in Klimchuk et al. (2008) and the green line shows the power-law approximation of Rosner et al. (1978, RTV)...... 35 List of ﬁgures xvi

2.3 Density as a function of ﬁeld-aligned coordinate, s, for an isothermal ﬂux tube assuming a vertical (dashed) and semi-circular (solid) ge-

ometry for a half length of L = 500 Mm and a footpoint density of

10 3 n0 = 10 cm . The different colors correspond to different temperatures, T, as denoted in the legend...... 38

2.4 Loop apex temperature, Tmax, as a function of pressure, p, calculated from Equation 2.17 for several different values of the loop half-length,

L. Adapted from Figure 9 of Rosner et al. (1978)...... 41 2.5 Temperature (left) and density (right) as a function of s for a full semi- circular loop of length 2L = 80 Mm heated unformly (blue) and at

the apex with lH = 10 Mm (orange). An isothermal chromosphere of depth 5 Mm is attached to each footpoint. The footpoint temperature

is T = 2 104 K and the footpoint density is n = 1011 cm 3 though 0 ⇥ 0 the chromospheric density is much higher...... 45 2.6 A cartoon illustration of the heating and cooling cycle of an impul-

sively heated coronal loop. The loop has a half-length of L and is assumed to be symmetric about the apex. The red arrows denote energy injected by heating (a) and energy transported by thermal conduction (b). The blue arrows denote energy lost by radiation (d and e). The thick green arrows indicate the bulk transport of material in the loop (c, d, and e). The gray arrows denote the order in which the cycle proceeds...... 51 2.7 Energy loss and gain mechanisms arising from a nanoﬂare with t = 200 s and electron heating only. The various curves correspond to the terms in the EBTEL two-ﬂuid electron energy equation, Equation 2.62: electron and ion thermal conduction, radiation, binary Coulomb

interactions, and yTR. The loop parameters are as in Section 5.3.... 62 List of ﬁgures xvii

3.1 Effective collision strength, U, as a function of Te for 100 selected

transitions in Fe XII. U was interpolated to Te using fit coefficients provided by the CHIANTI atomic database and computed using the method of Burgess & Tully (1992)...... 74 3.2 Level population of the first five levels of O II as a function of electron

6 density, ne, at Te = 10 K. Note that the ground state is the most

abundant for all ne. The level population is normalized to the total

number of O II ions such that Âj Nj = 1. Adapted from Figure 4.3 of Phillips et al. (2008)...... 77 3.3 Ionization (blue) and recombination (orange) rates as a function of

electron temperature, Te, for Fe XVI. The constituent rates are denoted by dashed and dot-dashed lines. Note that the recombination rate

dominates at low Te while the ionization rate dominates at high Te, as expected...... 80 3.4 Ion population fractions for every ionization state of Fe as a function

of Te. The population fractions were computed assuming ionization equilibrium using Equation 3.31. Note that increasingly higher ionization states become populated with increasing electron temperature and vice versa...... 84 3.5 Equilibrium (dashed) and non-equilibrium (solid) population frac-

tions as a function of time, t, for Fe X through Fe XV. The time-

dependent temperature proﬁle, Te, is shown on the right axis in 9 3 black. The density is held constant at ne = 10 cm for the entire simulation interval...... 86 List of ﬁgures xviii

3.6 Free-free emission summed over all ions of Fe as a function of wavelength. The different curves correspond to 1 MK (blue), 10 MK (or-

ange), and 100 MK (green). The factor neni is not included here such

that Pff has no density dependence...... 89 3.7 AIA wavelength response functions for the six primary EUV channels. For each channel, the response function is shown at 10 Å of ± the nominal wavelength. Each response function is normalized to

the maximum value of Rc over this interval...... 92 3.8 Temperature response functions for the six EUV channels of AIA listed in Table 3.1 as computed by Equation 3.40. Together, these six channels provide observational coverage over the temperature range

3 105 T 2 107 K...... 93 ⇥ . . ⇥ 3.9 Left panel: Two-ﬂuid EBTEL simulations of the electron temperature,

Te, for a loop heated by a single nanoflare at t = 0s(blue) and a loop heated by 10 nanoflares every 300 s (orange) for a total simulation time of 5 103 s. In both cases, the loop length is 40 Mm, the dura- ⇥ tion of each nanoflare is 200 s, and the total energy deposited in the

3 electrons in the loop is 10 erg cm . Right panel: EM(Te) for the two loops shown in the left panel. The dashed lines denote the power-law

ﬁt, Ta, to each distribution over the interval 1.25 MK T 4 MK e  e  and the emission measure slopes are shown in the legend. EM(Te) is 2 approximated by binning Te,i, weighted by ni L, at each timestep ti in 4 8.5 temperature bins between 10 K to 10 K with width 0.05 in log Te and then time-averaging over the whole simulation. Note that both distributions peak at approximately the same temperature...... 96 List of ﬁgures xix

3.10 An example of the regularized inversion method of Hannah & Kontar

(2012) for a simple model DEM(Te) and simulated AIA observations.

The dashed, gray line is the true DEM(Te), a single Gaussian pulse 6.5 centered at 10 K, and the black error bars in Te and DEM(Te) de-

note the regularized solution. The true DEM(Te) has a total emission measure of 3.76 1022 cm 5 and spread of s = 0.15 in log T . The ⇥ e colored curves as given in the legend are the emission measure loci curves for each AIA EUV channel. The lower panel shows the residuals between the true and recovered intensities for each channel. Adapted from Figure 3 of Hannah & Kontar (2012)...... 103

3.11 The left panel shows two Gaussian signals f1 and f2 with peaks at 0.25 s (blue) and 0.75 s (orange), respectively. The right panel shows

the cross-correlation (t) between f and f as a function of the C12 1 2 offset t. The dotted black line denotes t = 0 s. Note that (t) C12 peaks at t = 0.5 s, the separation in t between the peaks of f1 and f2. 106 3.12 Top panel: Simulated light curves (normalized to the peak value) for the six EUV channels of AIA (left axis) and electron temperature (black line, right axis) for a loop of half-length 40 Mm cooling from 5 MK to 0.2 MK. The hydrodynamic evolution of the loop was ⇡ ⇡ simulated using the EBTEL model and the six light curves were

computed using Te and ne from the simulation. Bottom panel: Cross- correlation as a function of temporal shift, t, computed from the light curves shown in the top panel for six selected channel pairs. The dotted black line indicates a temporal shift of t = 0s. The dotted lines and dots at the peak of each curve denote the time lag for that channel pair...... 108 List of ﬁgures xx

4.1 HMI LOS magnetogram observed on 2019 January 24 14:00:22 UTC. The full-disk observation is shown on the left and the zoomed-in view of NOAA 12733 is shown on the right. In both panels, the colorbar is on a symlog scale from 750 G to 750 G. Note that at the time of this observation, the active region was close to the center of the disk...... 113

4.2 A slice through the center of the extrapolated volume of Bx along the

y-axis. The black streamlines indicate the Bx and Bz components of the ﬁeld at this slice. The colorbar is on a log scale and ranges from 750 G to 750 G...... 116 4.3 LOS magnetogram of NOAA 12733 as observed by SDO/HMI. All 200 traced ﬁeld lines are overlaid in black...... 118 4.4 Helioprojective coordinate system (black) overlaid on a Stonyhurst heliographic coordinate system (blue) as viewed by an observer at (0°, 20°, 1 AU). The spacing between the lines of HGS longitude and latitude is 10° and the spacing between the lines of HPC longi-

tude and latitude is 25000...... 120 4.5 Temperature (left) and density (right) as a function of ﬁeld-aligned

coordinate, s, normalized to the loop length, L, for all 200 strands in the model active region as determined by the scaling laws of Martens (2010)...... 124 4.6 Coordinates for a sample of ten strands binned into a histogram defined by the HPC frame of the specified observer. The actual coordinates of the strands are shown in black and the unweighted binned values are shown in blue. The grid lines denote the edges of the bins. The resolution has been reduced to 20 pixel 1 in both ⇡ 00 coordinates for illustrative purposes...... 128 List of figures xxi

1 1 4.7 Predicted intensities, in DN pixel s , for active region NOAA 12733 as observed by four out of the six AIA EUV channels: 94 Å, 131 Å, 211 Å and 335 Å. The colorbar is on an arcsinh scale from 0 to the maximum intensity in that channel and the color tables are the standard AIA color tables as deﬁned in SunPy. The coordinate frame of each map is a helioprojective coordinate system deﬁned by an observer at the location of the SDO satellite on 2019 January 24 14:00:22 UTC...... 131

5.1 Left: Temperature (upper panel) and density (lower panel) profiles for a loop with 2L = 80 Mm. Each heating profile is triangular in shape with a steady background heating of H = 3.5 10 5 erg cm 3 s 1. bg ⇥ The duration of the heating pulse is varied according to t = 20 s, 40 s, 200 s, 500 s, with each value of t indicated by a different color, as shown in the right panel. The total energy injected into the loop is fixed at

3 10 erg cm . Note that time is shown on a log scale to emphasize the behavior of the heating phase. Right: Corresponding EM(T) for each pulse duration t. The relevant parameters and associated colors are

shown in the legend. EM(T) is calculated according to the procedure outlined in the beginning of Section 5.3. In all panels, the solid (dotted) lines show the corresponding EBTEL (HYDRAD) results (see Section 5.3.1)...... 144 List of ﬁgures xxii

5.2 EM(T) calculated from the single-fluid EBTEL model when only pure Spitzer conduction is used and when a flux limiter is imposed according to Section 5.2.1. In the free-streaming limit, five differ-

ent values of f are considered (see legend). The pulse duration is t = 200 s. All other parameters are the same as those discussed in

Section 5.3.1. Note that EM(T) is only shown for T > Tpeak as the cool side of EM(T) is unaffected by the choice of f ...... 148 5.3 Two-ﬂuid EBTEL simulations for t = 20 s, 40 s, 200 s, 500 s in which

only the electrons are heated. Left: Electron temperature (upper panel), ion temperature (middle panel), and density (lower panel).

Right: Corresponding EM(T) calculated according to Section 5.3. The pulse durations and associated colors for all panels are shown in the legend. All parameters are the same as those discussed in Section 5.3.1. In all panels, the solid (dotted) lines show the corresponding EBTEL (HYDRAD) results...... 150 5.4 Pressure (left axis) and density (right axis) as a function of temperature for the t = 200 s case. All parameters are the same as those

discussed in Section 5.3.1. The single-ﬂuid pressure p and density n are denoted by the solid blue and orange lines, respectively. The

electron pressure, pe, ion pressure, pi, and two-ﬂuid total pressure,

pe + pi, are denoted by the dashed, dotted, and dot-dashed blue lines respectively. The two-ﬂuid density is represented by the dashed orange line. Pressure, density, and temperature are all shown on a log scale...... 151 List of ﬁgures xxiii

5.5 Two-ﬂuid EBTEL simulations for t = 20 s, 40 s, 200 s, 500 s in which

only the ions are heated. Left: Electron temperature (upper panel), ion temperature (middle panel), and density (lower panel). Right: Corresponding EM(T) calculated according to Section 5.3. The pulse durations and associated colors for all panels are shown in the legend. All parameters are the same as those discussed in Section 5.3.1. In all panels, the solid (dotted) lines show the corresponding EBTEL (HYDRAD) results...... 153

5.6 Teff (red) for pulse durations of 20 s (top panel) and 500 s (bottom panel) for the single-ﬂuid case (solid) as well as the cases where only

the electrons (dashed) or only the ions (dot-dashed) are heated. T(t) proﬁles (i.e. assuming ionization equilibrium) for t = 20 s (blue lines) and t = 500 s (brown lines) for all three heating scenarios are repeated here for comparison purposes...... 156

5.7 EM(Teff) (red) for pulse durations of 20 s (top panel) and 500 s (bottom panel) for the single-ﬂuid (solid), electron heating (dashed),

and ion heating (dot-dashed) cases. EM(T) (i.e. assuming ionization equilibrium) for t = 20 s (blue lines) and t = 500 s (brown lines) for all three heating scenarios are repeated here for comparison purposes.

Note that in both panels EM(T) is only shown for log T > log Tpeak..157 List of ﬁgures xxiv

6.1 Active region NOAA 1158 on 12 February 2011 15:32:42 UTC as observed by HMI (left) and the 171 Å channel of AIA (right). The gridlines show the heliographic longitude and latitude. The left panel shows the LOS magnetogram and the colorbar range is 750 G ± on a symmetrical log scale. In the right panel, 500 out of the total 5000 field lines are overlaid in white and the red and blue contours show the HMI LOS magnetogram at the +5% (red) and 5% (blue) levels...... 166 6.2 Distribution of footpoint-to-footpoint lengths (in Mm) of the 5000 field lines traced from the field extrapolation computed from the magnetogram of NOAA 1158...... 167 6.3 Heating rate (top), electron temperature (middle), and density (bottom) as a function of time for the three heating scenarios for a single strand. The colors denote the heating frequency as defined in the

legend. The strand has a half length of L/2 40 Mm and a mean ⇡ ﬁeld strength of B¯ 30 G...... 170 ⇡ 6.4 SSW temperature response functions (solid black) and effective temperature response functions for the elements in Table 6.2 (dashed black) for all six EUV AIA channels. The colored, dashed curves, as indicated in the legend, denote the contributions of the individual elements to the total response. For this calculation, I have assumed

15 3 equilibrium ionization and a constant pressure of 10 K cm . The time-varying degradation of the instrument is not included...... 175

6.5 n T phase-space orbits for a single strand for the ﬁrst three heating scenarios in Table 6.1. The black line indicates a constant pressure of

15 3 10 K cm ...... 176 List of ﬁgures xxv

1 1 6.6 Snapshots of intensity, in DN pixel s , across the whole active region at t = 15 103 s. The rows correspond to the three different ⇥ heating frequencies and the columns are the six EUV channels of AIA. In each column, the colorbar is on a square root scale and is normalized between zero and the maximum intensity in the low- frequency case. The color tables are the standard AIA color tables as implemented in SunPy (SunPy Community et al., 2015)...... 178

6.7 Maps of the emission measure slope, a, in each pixel of the active region for the high- (left), intermediate- (center), and low-frequency

(right) cases. The EM(Te) is computed using time-averaged intensities from the six AIA EUV channels using the method of Hannah

a & Kontar (2012). The EM(Te) in each pixel is then ﬁt to T over the temperature range 8 105 K T < T . Any pixels with r2 < 0.75 ⇥  peak are masked and colored white...... 181

6.8 Distribution of emission measure slopes, a, for every pixel in the simulated active region for the high-, intermediate-, and low-frequency heating scenarios as shown in Figure 6.7. The histogram bins are determined using the Freedman Diaconis estimator (Freedman & Diaconis, 1981) as implemented in the Numpy package for array computation in Python (Oliphant, 2006) and each histogram is normalized such that the area under the histogram is equal to 1...... 183 List of ﬁgures xxvi

6.9 Time lag maps for three different channel pairs for all ﬁve of the heating models described in Table 6.1. The value of each pixel indicates the temporal offset, in s, which maximizes the cross-correlation (see Equation 3.55). The rows indicate the different channel pairs and the columns indicate the varying heating scenarios. The range of the colorbar is 5000 s. If max < 0.1, the pixel is masked and ± CAB colored white...... 185 6.10 Same as Figure 6.9 except each pixel shows the maximum cross- correlation, max ...... 188 CAB 6.11 Histograms of time lag values across the whole active region. The rows indicate the different channel pairs and the columns indicate the different heating models. Colors are used to denote the various heating models. The black dashed line denotes zero time lag. The bin range is 104 s and the bin width is 60 s. As with the time-lag ± maps, time lags corresponding to max < 0.1 are excluded. . . . . 190 CAB

7.1 Active region NOAA 1158 as observed by AIA on 2011 February 12 15:32 UTC in the six EUV channels of interest. The data have been processed to level-1.5, aligned to the image at 2011 February 12 15:33:45 UTC, and cropped to the area surrounding NOAA 1158.

1 1 The intensities are in units of DN pixel s . In each image, the colorbar is on a square root scale and is normalized between zero and the maximum intensity. The color tables are the standard AIA color tables as implemented in SunPy...... 200 List of ﬁgures xxvii

7.2 Map of emission measure slope, a, in each pixel of active region

NOAA 1158. The EM(Te) is computed from the observed AIA intensities in the six EUV channels time-averaged over the 12 h observing

a window. The EM(Te) in each pixel is then ﬁt to T over the temperature interval 8 105 K T < T . Any pixels with r2 < 0.75 are ⇥  peak masked and colored white...... 202 7.3 Distribution of emission measure slopes from Figure 7.2 (black) and from Chapter 6 (blue, orange, green). In each case, the bins are determined using the Freedman Diaconis estimator (Freedman & Diaconis, 1981) as implemented in the Numpy package for array computation in Python (Oliphant, 2006). Each histogram is normalized such that the area under the histogram is equal to 1...... 203 7.4 Time-lag maps of active region NOAA 1158 for all 15 channel pairs. The value of each pixel indicates the temporal offset, in s, which maximizes the cross-correlation (see Section 3.6.1). The range of the colorbar is 5000 s. If max < 0.1, the pixel is masked and ± CAB colored white. Each map has been cropped to emphasize the core of the active region such that the bottom left corner and top right corner of each image correspond to ( 440 , 380 ) and ( 185 , 125 ), 00 00 00 00 respectively...... 205 7.5 Same as Figure 7.4, but instead of the time lag, the maximum value of the cross-correlation, max , is shown in each pixel for each CAB channel pair...... 206 List of ﬁgures xxviii

7.6 Classification probability for each pixel in the observed active region. The rows denote the different cases in Table 7.1 and the columns correspond to the different heating frequency classes. If any of the 31 features is not valid in a particular pixel, the pixel is masked and colored white. Note that summing over all heating probabilities in each row gives 1 in every pixel...... 212 7.7 Predicted heating frequency classification in each pixel of NOAA 1158 for each of the cases in Table 7.1. The classification is determined by which heating frequency class has the highest mean probability over all trees in the random forest. Each pixel is colored blue, orange, or green depending on whether the most likely heating frequency is high, intermediate, or low, respectively. If any of the 31 features is not valid in a particular pixel, the pixel is masked and colored white. 213

8.1 Time-lag maps produced by the bundle heating model as simulated from a ﬁeld extrapolation of active region NOAA 1158. A sample of four channel pairs are shown here: 94-335, 335-171, 211-193, and 171-131 Å. The value of each pixel indicates the temporal offset, in s, which maximizes the cross-correlation (see Equation 3.55). The range of the colorbar is 5000 s. If max < 0.1, the pixel is masked and ± CAB colored white...... 228

8.2 Heating input as a function of ﬁeld-aligned coordinate, s, for simulating TNE using the HYDRAD code. The full-length of the strand is 120 Mm. The total heating proﬁle is a combination of two Gaussian

heating proﬁles of width 10 Mm at the two footpoints, s = 5 Mm and s = 115 Mm. The heating rate of both pulses is 5 10 3 erg cm 3 s 1. ⇥ The heating is turned on at t = 0sand is kept constant for the entire simulation...... 230 List of ﬁgures xxix

3 8.3 Electron temperature, in MK (top), and density, in cm (bottom), as a function of ﬁeld-aligned coordinate, s, and time, t as simulated by the HYDRAD code for a semi-circular loop of full-length 2L = 120 Mm. The time-independent heating function is localized to the

footpoints and is shown as a function of s in Figure 8.2...... 231 List of tables

2.1 Comparison between HYDRAD (H) and EBTEL (E) with c1 = 2 and

c1 given by Equation 2.65, for n < neq. The ﬁrst three columns show the full loop length, heating pulse duration, and maximum heating

rate. The last three columns show nmax for the three models. Only

nmax is shown as Tmax is relatively insensitive to the value of c1. The ﬁrst two rows correspond to the t = 200, 500 s cases considered in Chapter 5. The next four rows are the four cases shown in Table 2 of Cargill et al. (2012a). The last two rows are cases 6 and 11 from Table 1 of Bradshaw & Cargill (2013)...... 65

3.1 Primary ions observed by the six AIA EUV channels of interest. Adapted from Table 1 of Lemen et al. (2012)...... 91 3.2 Summary of observational and modeling studies that have used the

emission measure slope, a, as a diagnostic for the underlying energy

deposition. The approximate range of observed slopes is 2 . a . 5. Adapted from Table 3 of Bradshaw et al. (2012)...... 99

6.1 All three heating models plus the two single-event control models. In the single-event models, the energy ﬂux is not constrained by Equation 6.3...... 172 List of tables xxxi

6.2 Elements included in the calculation of Equation 6.5. For each element, all ions for which CHIANTI provides sufﬁcient data for computing the emissivity are included...... 174

6.3 sI/I¯ as deﬁned by Equation 11 of (Guarrasi et al., 2010) computed on a single image at t = 15 103 s for each channel and heating ⇥ frequency. A larger value denotes a greater degree of contrast. . . . . 179

7.1 The four different combinations of emission measure slope, time lag, and maximum cross-correlation. The third column lists the total number of features used in the classiﬁcation. The fourth column

gives the misclassification error as evaluated on Xtest, Ytest. The fifth, sixth, and seventh columns show the percentage of pixels labeled as high-, intermediate-, and low-frequency heating, respectively. . . . . 210 7.2 Ten most important features as determined by the random forest classifier in case C. The second column shows the variable importance as computed by Equation 7.2 and the third column, s, is the standard deviation of the feature importance over all trees in the random forest. The second column is normalized such that the most important feature is equal to 1...... 216

A.1 The ﬁrst 5 rows of the energy level ﬁle for Fe XVI, fe_16.elvlc. Additional metadata and units are available in the metadata of the

Table object...... 252 Nomenclature

Roman Symbols c speed of light in a vacuum g gravitational acceleration at the solar surface h Planck constant kB Boltzmann constant me electron mass mi ion mass ne electron density

R radius of the Sun, 6.957 1010 cm ⇡ ⇥

Te electron temperature

Ti ion temperature

Greek Symbols l wavelength n photon frequency tAB time lag between signal A and signal B Nomenclature xxxiii

Subscripts k ionization stage

Other Symbols

0 arcminute

00 arcsecond

cross-correlation C

F Fourier transform

Acronyms / Abbreviations

AIA Atmospheric Imaging Assembly

AR active region

AU astronomical unit

DEM differential emission measure

DN digital number, equivalent to counts

EBTEL Enthalpy-Based Thermal Evolution of Loops model

EUV extreme ultraviolet

FFT fast Fourier transform

HMI Helioseismic Magnetic Imager

HYDRAD Hydrodynamics and Radiation code

IDL Interactive Data Language

LOS line-of-sight Nomenclature xxxiv

MHD magnetohydrodynamics

NEI non-equilibrium ionization

NOAA National Oceanic and Atmospheric Administration

PFSS potential ﬁeld source surface

SDO Solar Dynamics Observatory

SSW SolarSoftware, a suite of IDL tools for analysis of solar data

TR transition region Chapter 1

Introduction

For the last ﬁve billion years, the Sun has provided the light by which humans observe the world around them and the heat to save the planet from the frigid temperatures of interplanetary space. While energy from the Sun is critical to sustaining life on Earth, space weather, driven by magnetized material ejected from the solar atmosphere, threatens modern technological infrastructure. Additionally, due to its proximity, the Sun also provides astronomers an exclusive and unique look into how stars behave via continuous, high-resolution observations at wavelengths across the entire electromagnetic spectrum. The structure and complexity of the Sun is partially revealed to the naked eye during a total solar eclipse. Solar eclipses have been observed and recorded for thousands of years, with reported sightings dating back to the fourteenth century BC (Golub & Pasachoff, 2010). Chinese rock drawings from the Han dynasty (approximately 1900 years ago) appear to show the moon completely obscuring the Sun. Most recently, the “Great American Eclipse” captured the attention of millions from Oregon to South Carolina as it diagonally traversed the United States on 21 August 2017, offering a breathtaking view of the otherwise-invisible outermost layer of the Sun’s atmosphere: the solar corona.

1 1.1 The Structure of the Solar Atmosphere 2

Despite being observed for thousands of years, the true mystery of the corona was not realized until the early twentieth century. Enabled by a modern understanding of atomic structure, analysis of spectroscopic eclipse observations revealed the temperature of the corona to be in excess of one million kelvin, many orders of magnitude hotter than the solar surface. This question of what exactly causes these unexpectedly high temperatures, dubbed the “coronal heating problem,” remains unanswered and has occupied solar astronomers for nearly eighty years. This thesis addresses the question of how energy is deposited into magnetically- active regions of the solar corona and, in particular, whether observations, combined with sophisticated models, can be used to constrain properties of the energy deposition. This chapter serves as a brief introduction to the astrophysics of the Sun and its dynamic and highly-complex atmosphere. In Section 1.1, I give a brief description of the interior of the Sun and the layers of the solar atmosphere. Section 1.2 describes the magnetic ﬁeld of the Sun and Section 1.3 discusses the coronal heating problem. Section 1.4 provides an outline of the remainder of this thesis and in Section 1.5,I provide a few comments on the use of software and data in this work.

1.1 The Structure of the Solar Atmosphere

The Sun is a main-sequence G2 type star and its current age is 4.6 109 yr. It ⇡ ⇥ has a mass of M = 1.99 1033 g and a radius of R = 6.955 1010 cm (Priest, ⇥ ⇥ 2014). The Sun emits primarily as a blackbody in the visible and the infrared bands of the electromagnetic spectrum and the effective temperature of the surface

is Teff = 5777 K (Carroll & Ostlie, 2007). However, as will be discussed in later sections, observations at shorter wavelengths show that the temperature structure of the solar atmosphere is far more complicated. In the following sections, I discuss the structure of the stellar interior (Section 1.1.1) and then give a brief description of each 1.1 The Structure of the Solar Atmosphere 3

layer of the solar atmosphere: the photosphere (Section 1.1.2), the chromosphere (Section 1.1.3), the transition region (Section 1.1.4), the corona (Section 1.1.5), and the solar wind (Section 1.1.6).

1.1.1 Interior

1R R Radiative Zone 0.714 Convection Zone

R 0.3

Core

Figure 1.1 Schematic of the solar interior. In the core and radiative zone, radiation is the dominant energy transfer mechanism while convection, the cyclic rise of hot gas to the surface and subsequent infall of cooled gas, dominates in the convection zone. Adapted from Figure 11.2 of Carroll & Ostlie (2007).

The interior of the Sun cannot be directly observed because it is opaque to radiation. All knowledge of its structure must be inferred from detailed stellar structure calculations or through helioseismology, the study of global oscillations (Priest, 2014). The solar interior can be divided into three distinct layers: the core, the radiative zone, and the convection zone. This is illustrated in Figure 1.1.

The core of the Sun is very hot ( 1.57 107 K) and dense ( 9 1025 cm 3) ⇡ ⇥ ⇡ ⇥ (Bahcall et al., 2001; Carroll & Ostlie, 2007) and, as denoted in Figure 1.1, extends 1.1 The Structure of the Solar Atmosphere 4

1H + 1H 2H + e+ + n 1 1 !1 e 2H + 1H 3H + g 1 1 !2

3He + 3He 4He + 21H 2 2 !2 1 3He + 4He 7Be + g (ppI) 2 2 ! 4

7Be + 1H 8B + g 7 7 4 1 !5 4Be + e 3Li + ne 8 8 + ! B Be + e + ne 7Li + 1H 24He 5 !4 3 1 ! 2 8Be 4He (ppII) 4 !2 (ppIII)

Figure 1.2 The three branches of the proton-proton nucleosynthesis reaction. The ppI branching ratio is 69% and the ppII branching ratio is 99.7%. Adapted from Figure 10.8 in Carroll & Ostlie (2007).

radially over r . 0.3R . The primary mechanism of energy production in the 1 4 core is the fusion of 1H into 2He via a reaction called the proton-proton chains or 4 + the “pp” chains. Besides 2He, the pp chain also produces positrons (e ), weakly-

interacting electron neutrinos (ne), and photons (g). The full reaction chain is shown in Figure 1.2. These reactions can only occur at very high densities such as those found in the core as quantum tunneling is required to overcome the Coulomb

1 barrier between the two 1H atoms. The energy produced in this reaction is carried away by both the neutrinos and the photons. Only a small fraction is contained in the weakly-interacting neutrinos which travel practically uninhibited out of the interior while most of the energy is contained in the photons. These photons thus undergo a “random walk” from the core to the surface as the they are continually absorbed and reemitted isotropically. Moving radially outward from the core, the density drops thereby inhibiting

the pp chain reaction, and the temperature decreases such that dT/dr < 0. This decrease in temperature causes a decrease in radiation pressure with increasing r, leading to the slow upward diffusion of photons produced by the pp chain reactions 1.1 The Structure of the Solar Atmosphere 5

in the core. This region, which extends over 0.3R r 0.714R , is often referred   to as the radiative zone (Carroll & Ostlie, 2007). Here and in the core, radiation is the dominant energy transport mechanism.

As the temperature gradient continues to steepen with increasing r, the opacity increases, inhibiting energy transport by radiation. In the region 0.714R < r  1R , the so-called convection zone (see Figure 1.1), hot, buoyant mass elements carry excess energy outward while cool mass elements fall inward and the cycle repeats such that energy is continually transported to the surface. In particular, convection becomes the dominant transport mechanism over radiation when the

actual temperature gradient becomes greater than the adiabatic temperature gradient. If this is the case, the temperature gradient is said to be “super adiabatic” (see Section 10.4 of Carroll & Ostlie, 2007) such that rising parcels of gas exchange heat with the surrounding medium.

1.1.2 Photosphere

Just above the convection zone lies the photosphere, the lowest layer of the solar atmosphere and often considered the “surface” of the Sun. The photosphere is a dense, relatively thin layer and can be observed in visible light. Here, the optical

depth, t, or the transparency of the plasma, is t . 1 in the visible band such that photons at these wavelengths can escape without being absorbed and reemitted. Thus, most of the Sun’s emission in the visible spectrum originates in the photosphere (Priest, 2014). The photosphere extends to approximately 500 km above the surface and corresponds to a minimum in the temperature as a function of height,

T 4400 K. This is illustrated in Figure 1.3. The deﬁnition of the base of the min ⇡ photosphere is a bit more arbitrary, but is usually said to be 100 km below the ⇡ point where t 1 for photons of wavelength 5 103 Å(Carroll & Ostlie, 2007). ⇠ ⇥ 1.1 The Structure of the Solar Atmosphere 6

Photosphere Transition Region 106 1017

Chromosphere 1015

10 Corona ]

13 3 ] T 10 K [

n cm [ T

1011 n

104 109

0 1 2 3 4 5 h [Mm]

Figure 1.3 Temperature (blue, left axis) and density (orange, right axis) of the solar atmosphere as a function of height, h, above the solar surface. These profiles are based on the semi-empirical models of McWhirter et al. (1975) and Vernazza et al. (1981). The data points show the exact values from the models and the smooth lines are first-order spline fits to the data. The dotted black lines denote the different regions of the solar atmosphere.

The top left panel of Figure 1.4 shows an observation of the photosphere at 4500 Å by the Atmospheric Imaging Assembly instrument (AIA, Lemen et al., 2012) on the Solar Dynamics Observatory spacecraft (SDO, Pesnell et al., 2012). The image appears relatively smooth as the Sun emits primarily as a blackbody in the visible and infrared wavelengths (Carroll & Ostlie, 2007). Note the appearance of a dark

sunspot in the upper-right quadrant of the image. Sunspots correspond to areas of intense magnetic activity and appear dark due to a localized inhibition of energy transport by the solar magnetic ﬁeld (Priest, 2014). 1.1 The Structure of the Solar Atmosphere 7

1.1.3 Chromosphere

Above the photosphere lies the chromosphere which has a depth of 1600 km. ⇡ Moving upward through the chromosphere from the temperature minimum at the top of the photosphere, the temperature increases at ﬁrst gradually and then more rapidly to many times 104 K (see Figure 1.3). At the same time, the density falls off very quickly. Though not visible to the naked eye, the chromosphere is highly structured. Spicules, tall columns of gas that extend high into the solar atmosphere (De Pontieu et al., 2011), are primarily visible off the solar disk in Ha and originate in the photosphere as do ﬁlaments, spicules observed on-disk, and plage, bright regions surrounding sunspots. The top right panel Figure 1.4 shows the chromosphere as observed by the 304 Å channel of SDO/AIA. Compared to the underlying photosphere, the chromosphere is much more highly structured. Note that the intensity enhancement in the top right quadrant is spatially coincident with the sunspot in the photosphere in the top left panel.

1.1.4 Transition Region

The extremely thin transition region (TR) sits between the chromosphere and the corona and primarily emits emission in the extreme ultraviolet (EUV) portion of the electromagnetic spectrum. It is only a few hundred km thick, but is characterized by very steep temperature gradients as the temperature increases over an order of magnitude, from a few 104 K to well above 105 K, as illustrated in Figure 1.3. The density also continues to rapidly decrease in the TR. The upper boundary of the TR is not static and is more properly deﬁned in terms of the role of thermal conduction in the TR energy balance. This is discussed in more detail in Section 2.1.1. 1.1 The Structure of the Solar Atmosphere 8

SDO/AIA 4500 A˚ SDO/AIA 304 A˚

2010/01/12 2010/01/12 SDO/AIA 171 A˚ Hinode/XRT Open/Al mesh

2010/01/12 2010/01/12

Figure 1.4 The layers of the Sun’s atmosphere revealed in multiple wavelengths at approximately 19:00 UTC on 2010 December 1. The top left panel shows the photosphere, the top right panel shows the chromosphere, and the bottom left panel shows the EUV corona all imaged by SDO/AIA. The bottom right panel shows the hot X-ray corona as observed by Hinode/XRT. All data are courtesy of the AIA and XRT instrument teams.

1.1.5 Corona

The outermost layer of the solar atmosphere is the corona (Latin, “crown”) and begins a little over 2 103 km above the surface. The corona is very hot ( 106 K) ⇥ & 9 3 and diffuse (. 10 cm ) and is characterized primarily by optically-thin emission in the X-ray and EUV bands such that photons are not reemitted or absorbed once 1.1 The Structure of the Solar Atmosphere 9

they are produced in the corona. The corona is only visible to the naked eye during an eclipse. The corona is highly-structured by the complex solar magnetic ﬁeld because of the relative strength of the magnetic ﬁeld compared to the gas pressure (see Section 1.2.1 and Chapter 2). The bottom panels of Figure 1.4 show the corona at 8 105 K as imaged by the 171 Å channel of SDO/AIA (left panel) and at 107 K ⇡ ⇥ ⇡ as imaged by the X-ray Telescope (XRT Golub et al., 2007) on the Hinode satellite (Kosugi et al., 2007) (right panel). Note how drastically the appearance of the Sun changes moving upward from the photosphere (top left) to the chromosphere (top right) and to the EUV and X-ray corona (bottom). The dynamics of these EUV- and X-ray-bright structures in the corona is the primary focus of this thesis.

1.1.6 The Solar Wind

Above the corona, the solar atmosphere transitions to the solar wind. The existence of the solar wind was ﬁrst predicted by Parker (1958) based on the relatively simple idea that a hot, hydrostatic corona should be expanding given the pressure measured in interplanetary space. Its existence was later conﬁrmed experimentally

by Neugebauer & Snyder (1962) using in-situ measurements from the Mariner-2 spacecraft (Golub & Pasachoff, 2010). The solar wind has two components: a fast solar wind with velocity 800 km s 1 and a slow solar wind with velocity ⇡ 400 km s 1 (Golub & Pasachoff, 2010). While it is generally agreed that the fast ⇡ wind originates from cool, dark coronal holes, where the solar magnetic ﬁeld is open to the interplanetary magnetic ﬁeld rather than closing back at the surface, the origin of the slow wind is much less certain. 1.2 The Solar Magnetic Field 10

1.2 The Solar Magnetic Field

Like the Earth, the Sun possesses an intrinsic magnetic field. Near the polar regions, the solar magnetic field is approximately dipolar, but closer to the equator, the field is highly nonuniform and dynamic. In areas of intense magnetic activity, the magnetic field strength can reach a few 103 G1 while in more “quiet” regions, it is much lower, 0.1 G to 0.5 G (Aschwanden, 2006). Interestingly, global magnetic activity on the Sun varies on a 11 yr cycle in which the dipolar field also reverses ⇡ (Golub & Pasachoff, 2010). The exact physical mechanism responsible for this cyclic variability and the generation of the intrinsic field are not well understood. The solar magnetic field extends high into the atmosphere and dominates the structuring and dynamics in the hot, tenuous corona. Because of the high temperatures that characterize both the solar atmosphere and interior, much of the gas that makes up the Sun is ionized; that is, each atom has been stripped of at least one of its electrons. This means that the Sun is filled by a sea of charged particles

called a plasma. Because these particles are charged, the dynamics of the plasma are strongly influenced by solar magnetic field and vice versa such that the solar plasma and magnetic field represent a coupled system. In general, the coupled dynamics of the solar plasma and magnetic field are described by the equations of

magnetohydrodynamics (MHD),

d r + r v = 0, (1.1) dt r· d 1 r v = j B p, (1.2) dt c ⇥ r c j = B, (1.3) 4p r⇥ ∂ hc2 B = (v B)+ 2B, (1.4) ∂t r⇥ ⇥ 4p r 1For reference, the Earth’s magnetic ﬁeld at its surface has an average value of 0.5 G (Finlay et al., 2010). ⇡ 1.2 The Solar Magnetic Field 11

B = 0, (1.5) r· rg d p = F R + Q, (1.6) g 1 dt rg r · ✓ ◆ where r is the mass density, v is the bulk ﬂow velocity, j is the current density, B is

the magnetic field, p is the thermal pressure, h is the magnetic diffusivity, g = 5/3 is the ratio of specific heats, q is the heat flux, R is the radiative loss term, H is heating due to Ohmic and viscous dissipation, and d/dt ∂/∂t + v (Priest, ⌘ ·r 2014). Solving the MHD equations for the time-dependent, vector magnetic field in three-dimensions is an extremely challenging problem that requires sophisticated numerical codes and significant computational resources.

1.2.1 Origin of the Magnetic Field and Flux Emergence

The generation and emergence of the complex magnetic ﬁeld at the surface is due primarily to two mechanisms in the solar interior: differential rotation and convection. Because the Sun is not a rigid body, the rotation rate of the solar plasma varies latitudinally as well as radially where the radial dependence is also latitude dependent. For example, at a latitude of 60°, the solar interior seems to rotate faster than the surface while the opposite is true at the equator (Thompson et al., 1996). At

r 0.6R near the base of the convection zone (see Figure 1.1) the rotation rates at ⇠ all latitudes converge (Golub & Pasachoff, 2010) such that the radiative zone rotates rigidly compared to the outer layers. This region of convergence is often called the tachocline (Aschwanden, 2006). This radial and latitudinal dependence of the rotation rate is important to the generation of the magnetic ﬁeld because the ﬁeld is “frozen-in” to the plasma.

Another way of stating this is that the magnetic ﬂux, F = dS B, through a S · R 1.2 The Solar Magnetic Field 12

surface S does not change in time along the path of a ﬂuid element such that,

d F = 0. (1.7) dt

As proof of this, consider a small change in ﬂux dF as the ﬂuid element travels

through two surfaces S and S0 in time dt with velocity v. dF can be expressed as,

dF = dS B(t + dt) dS B(t). ZS0 · ZS ·

Using the divergent-free condition of B (Equation 1.5) combined with Stoke’s

theorem and the fact that dS = d` dtv gives, ⇥

dF = dS (B(t + dt) B(t)) dt d` v B(t + dt), S · · ⇥ Z IC dF B(t + dt) = dS (v B(t + dt)) , dt S · dt r⇥ ⇥ Z ✓ ◆ where is the curve that encloses S. Taking the limit dt 0 and using the ideal C ! MHD induction equation (Equation 1.4 with h = 0) gives Equation 1.7. Under the condition of Equation 1.7, a magnetized plasma drags the magnetic field with along with it. Following the treatment of Golub & Pasachoff (2010), for an initially straight magnetic field line oriented along the solar rotation axis, the plasma frozen to the field line will undergo differential rotation such that the field line will be stretched perpendicular to the original orientation. As the Sun continues to rotate, the field is continually “wound up” and the perpendicular component increases. This amplification of effect of an initially dipolar field is purely a consequence of differential rotation and the assumption of flux freezing (Equation 1.7) and provides a qualitative picture for the generation of the intrinsic magnetic field of the Sun. A more quantitative approach requires solving the non-linear dynamo equations (see Section 4.3.3 of Golub & Pasachoff, 2010). 1.2 The Solar Magnetic Field 13

As the frozen-in magnetic field in the solar interior becomes continually de- formed due to differential rotation, these twisted magnetic field lines are carried upward through the convection zone to the surface due to magnetic buoyancy. First proposed by Parker (1955), this effect occurs when the internal pressure of plasma plus the magnetic pressure of the frozen-in field cannot balance the ambient pressure, thus forcing the field line upward. This twisted and amplified field is carried up through the photosphere, forming dipolar loop-like structures that extend high above the surface and into the chromosphere, transition region, and corona.

This phenomenon, called ﬂux emergence, leads to the formation of active regions, areas of intense magnetic activity characterized by densely-packed closed magnetic structures. Active regions are manifested as cool, dark sunspots in the photosphere due to the reduced internal pressure required to bring the ﬂux to the surface. However, in the corona, active regions appear as bright loop-like structures in the EUV and X-ray bands. This is because the hot, diffuse corona is a “low-b” plasma, where,

8pp b , (1.8) ⌘ B2

is the ratio between the gas or thermal pressure (p) and the magnetic pressure (B2/8p). Because the magnetic pressure dominates over the thermal pressure such that b 1, the plasma is confined by the magnetic field and traces out the complex ⌧ and twisted field lines that emerged from the interior. The multi-wavelength appearance of an active region can be seen in the top right quadrants of each panel in Figure 1.4. 1.2 The Solar Magnetic Field 14

1.2.2 Observations

Knowledge of the magnetic field in the corona is crucial to understanding the dynamics and heating (see Section 1.3) of the coronal plasma. However, direct measurements of the coronal vector magnetic field have proved challenging2 because the optical and infrared coronal lines sensitive to the field are relatively faint compared to the much brighter solar disk (Judge et al., 2001). As such, it is common practice to measure the line-of-sight (LOS) component of the photospheric magnetic field and then treat this measurement as a lower boundary condition for a model of the coronal field (see Section 1.2.3). The photospheric field can be measured using

the Zeeman effect wherein the magnetic ﬁeld “breaks” the degeneracy of the atomic energy levels with respect to the total angular momentum operator. The resulting level splitting can be approximated by,

13 2 Dl 5 10 Bl , (1.9) ⇡ ⇥ 0

where B is the ﬁeld strength and l0 is the wavelength when B = 0 (Phillips et al., 2008). In order for this splitting to be resolved compared to the instrument or

thermal width, B must be sufficiently strong and l0 sufficiently long such that measurements of the coronal field (B 102 G) can typically only be made at far ⇠ visible or infrared wavelengths. Many ground-based telescopes (e.g. GONG, Mt. Wilson, Howard, 1976) and space-based instruments (e.g. SOHO/MDI, Scherrer et al., 1995) provide high- quality measurements of the LOS photospheric magnetic field. Figure 1.5 shows a

full-disk magnetogram produced from measurements by the Helioseismic Magnetic Imager (HMI, Hoeksema et al., 2014) on SDO which measures the Fe I 6173.34 Å

2The Daniel K. Inouye Telescope (Elmore et al., 2014), expected to see ﬁrst light in 2020, will provide vastly improved measurements of the coronal magnetic ﬁeld via high-resolution spectropo- larimetry. 1.2 The Solar Magnetic Field 15

Figure 1.5 The LOS magnetic ﬁeld strength at the photosphere (i.e. a magnetogram) as measured by the HMI instrument on the SDO spacecraft at 19:00 UTC 2010 December 1. Red indicates positive polarity (out of the page) and blue indicates negative polarity (into the page). The colorbar is on a semi-log scale from 750 G to +750 G.

absorption line. Note that the region of enhanced adjacent positive and negative polarities seen in the top right quadrant is spatially coincident with the sunspot and bright loops seen in the photospheric, EUV, and X-ray images in Figure 1.4.

1.2.3 Field Extrapolation

In the absence of direct measurements of the coronal magnetic field, magnetic field extrapolation techniques provide useful and efficient approximations of the three-dimensional vector magnetic field in the corona given a LOS photospheric magnetogram. Following the treatment in Priest (2014, Chapter 3), Equation 1.2, the ideal MHD momentum equation, in magnetohydrostatic balance can be written as,

1 0 = j B p. (1.10) c ⇥ r 1.2 The Solar Magnetic Field 16

In a low-b plasma (see Equation 1.8), the second term on the right-hand side can often be neglected such that Equation 1.10 becomes,

j B = 0. (1.11) ⇥

Equation 1.11 is the so-called force-free condition. Combined with Equation 1.3, Ampére’s law, this implies that,

B = aB, (1.12) r⇥ where, in general, the scalar a may be some function of position r. In the case of a = 0, j = 0 (from Equation 1.3) and the magnetic ﬁeld is said to

be current-free or potential. Equation 1.12 implies that B is also curl-free such that it can be expressed as,

B = f, (1.13) r where f is some scalar potential. Combining this expression with the requirement from Maxwell’s equations that the magnetic ﬁeld must always be divergence-free (Equation 1.5) gives Laplace’s equation,

2f = 0. (1.14) r

If the normal component of the magnetic field is specified on the lower boundary (e.g. from a photospheric LOS magnetogram), the solution within a closed volume is unique (Priest, 2014). Several methods have been developed to solve Equation 1.14 for the coronal magnetic field. The potential field source surface (PFSS) model of Schatten et al. (1969) solves Equation 1.14 for the global corona given a synoptic photospheric magnetogram as the lower boundary input and under the assumption that the field 1.2 The Solar Magnetic Field 17

is purely radial at some “source surface,” typically 2.5R . Additionally, the Green’s function method of Schmidt (1964) can be used to efficiently determine the potential magnetic field on the scale of a single active region on a Cartesian grid given a LOS magnetogram. Section 4.1.1 will describe the latter method in detail. While the work presented in this thesis will only make use of photospheric LOS magnetogram data, there exist many techniques for computing field extrapolations from vector magnetograms as well (see review by Welsch & Fisher, 2016). The potential field represents the lowest possible energy state of the magnetic field and is likely to be an appropriate approximation provided the magnetic energy dominates over the thermal energy (b < 1) and the field has had sufficient time to relax to the lowest energetic state (Priest, 2014). Thus, a field with a non-zero

current is in a higher energy state than a potential field. From Equation 1.3, if j = 0 6 then a = 0. Provided Equation 1.11 holds, solutions to Equation 1.12 represent 6 non-potential force-free fields and in general are much more difficult to compute than potential field solutions. If a is constant, the solution is a linear force-free field,

but if a is a function of a position r, the magnetic ﬁeld is said to be non-linear force- free. See Wiegelmann & Sakurai (2012) for a comprehensive review of force-free magnetic ﬁelds in solar physics as well as Schrijver et al. (2008) for a comparison of several non-linear force-free models.

1.2.4 Reconnection

After the magnetic ﬁeld is forced into the solar atmosphere by the buoyant motion

of the convection zone, it remains rooted in the photosphere, whether it is open (ﬂux tube extends radially outward, possibly connecting with the interplanetary

magnetic field) or closed (both ends attached to the solar surface). Because the field is frozen into the photospheric plasma (Equation 1.7), the turbulent motion of the photosphere deforms and stresses the overlying field, leading to the storage of 1.2 The Solar Magnetic Field 18

magnetic energy. This motion can eventually lead to a topological restructuring of the magnetic ﬁeld as it relaxes from a stressed to an equilibrium state, a process

commonly referred to as magnetic reconnection.

Figure 1.6 Cartoon illustrating the breaking and reconnecting of magnetic field lines. When two regions of oppositely directed field (red and blue, left panel) are brought together (middle panel), a discontinuity develops in the narrow diffusion region (shaded gray box). Non-ideal effects cause the field lines to “reconnect” (right panel), thus changing the topology of the magnetic field. Adapted from Figure 6.1 of Priest (2014).

Magnetic reconnection is thought to be a dominant process in a variety of space and astrophysical plasma environments, including Earth’s magnetosphere and accretion disks. Reconnection is observed in laboratory experiments like the tokamak and the reversed field pinch (Priest & Forbes, 2000) and is the primary driver of some proposed coronal heating mechanisms (see Section 1.3.2). Solar flares, brightenings across the entire electromagnetic spectrum that produce 1033 erg of ⇠ energy, are also thought to be triggered by magnetic reconnection. The basic idea behind reconnection is illustrated in Figure 1.6. When two oppositely-directed field lines are brought together in a conducting fluid, a tangen- tial discontinuity develops between them with current-carrying plasma squeezed into this area of discontinuity. Because the field lines are frozen into the plasma, a large magnetic gradient develops at the discontinuity and a current sheet forms.

Because of these large gradients, the resistivity in this diffusion region (the gray 1.2 The Solar Magnetic Field 19

box in Figure 1.6) becomes very high, allowing the magnetic field lines to diffuse through the plasma, reconnect, and relax to a topologically different, but more energetically favorable state. As the field lines reconnect and are pushed out of the end of the diffusion region by the enhanced pressure (right panel of Figure 1.6), the current sheet diffuses away and the plasma is heated by Ohmic dissipation of the stored magnetic energy. Reconnection is thus a non-ideal process as it violates the “frozen-in” flux requirement (Equation 1.7) and allows for the conversion of stored magnetic energy to kinetic and thermal energy via dissipation (Priest, 2014; Priest & Forbes, 2000). While the idea behind reconnection was introduced by Dungey (1953), the first complete theory was proposed by Sweet (1958) and further developed by Parker (1957, 1963). In the so-called Sweet-Parker model, a diffusion region of length

2L and width 2` is defined between two anti-parallel fields. The field lines are

carried into the diffusion region with speed vi = h/`, where h is the magnetic diffusivity. Using Equation 1.5, Equation 1.1, and Equation 1.2, the inﬂow velocity,

or reconnection rate, can be rewritten as vi = vAi/pRmi, where vAi is the Alfvén

speed and Rmi = LvAi/h is the magnetic Reynolds number (Priest & Forbes, 2000). The Sweet-Parker model predicts a reconnection rate far too slow to properly account for the energy release timescales observed in ﬂares. In an effort to remedy the slow reconnection in the Sweet-Parker mechanism, Petschek (1964) suggested that magnetoacoustic shocks could provide an additional acceleration mechanism for the reconnection rate. Additionally, he proposed a smaller diffusion region, further shortening the reconnection timescale. For many years following Petschek’s work, it was thought that the problem of fast reconnection was solved. Today, however, thanks in part to increased computing power that makes three-dimensional and kinetic simulations possible, reconnection is now understood to be a far more subtle 1.3 Heating in the Solar Corona 20

mechanism than previously thought, with the Petschek and Sweet-Parker models as only special cases (Priest & Forbes, 2000).

1.3 Heating in the Solar Corona

The mystery of the anomalously-high temperatures in the Sun’s outer atmosphere, the so-called “coronal heating problem”, is a central question in the field of solar astrophysics. The discovery of the >106 K corona was made over the course of nearly fifty years through a combination of eclipse observations and laboratory experiments. Spectroscopic measurements during the 1869 solar eclipse by Charles Young and William Harkness yielded a surprising result: an unknown “coronal green line” (Golub & Pasachoff, 2010). Because it could not be associated with any known element, the line was initially labeled as a new element, “coronium”. Later examinations of the coronal spectrum revealed several more unidentifiable spectral lines, including a “red line” and a “yellow line”. Grotrian (1939) showed that these lines correspond to forbidden transitions in Fe XIV at 5303 Å (“green”), Fe X at 6374 Å (“red”), and Ca XV at 5694 Å (“yellow”) (from Table 2.1 of Golub & Pasachoff, 2010). Edlén (1943) later identified four additional coronal lines, Fe X, Fe XI, Ca XII, and Ca XIII. The presence of these high ionization states implies coronal temperatures in excess of 106 K and explained the high gas pressure needed to support extended corona seen in eclipse observations (Golub & Pasachoff, 2010). Because of their work in coronal spectroscopy, Grotrian (1939) and Edlén (1943) are generally credited with the discovery of the million-degree solar corona. How- ever, some (see Peter & Dwivedi, 2014) have argued that early coronal spectro- scopists did not imply a million-degree corona and that it was Alfvén (1941) who first proposed a hot corona, even arguing that it was the interaction between the 1.3 Heating in the Solar Corona 21

magnetic ﬁeld and the charged particles in the solar atmosphere that lead to heating in the corona.

1.3.1 Waves versus Reconnection

Since the discovery of the million-degree corona, a number of mechanisms have been proposed as candidates for heating the coronal plasma. Historically, these physical mechanisms have been divided into two categories: AC, mechanisms that rely on waves to transfer energy from the lower atmosphere into the corona and

DC, mechanisms that involve dissipation of energy stored in the stressed magnetic field. More rigorously, the AC- or DC-type heating classification depends on the timescale of the stressing motion: if it is longer than a characteristic crossing time of the coronal structure, it is classified as DC heating. If the timescale is shorter than a crossing time, it is classified as AC heating. Any viable heating mechanism must be able to explain the energy source of the heating, how the energy is converted to heat, how the plasma responds, and any observational signatures from the resulting plasma emission (see Figure 1 of Klimchuk, 2006).

AC Heating

A variety of wave modes have been observed in both the open and closed corona, including acoustic waves, Alfvén waves, and slow- and fast-mode MHD waves (Aschwanden, 2006), but the simple existence of these waves is not enough to make them a viable candidate for coronal heating. They must be able to propagate into the corona with an adequate amount of energy and then efficiently dissipate this energy in order to heat the coronal plasma. For example, acoustic waves are capable of carrying enough energy to heat the corona, but are almost entirely reflected by the steep density gradients in the TR. Alfvén waves may also be capable of carrying enough energy to heat the corona, but specific frequencies are required 1.3 Heating in the Solar Corona 22

to avoid reﬂection at the TR. These particular modes have also been found to be non-dissipative under coronal conditions, making it hard for them to actually deposit their energy even if they make it to the corona (Klimchuk, 2006). Wave modes generated in the corona would of course overcome the problem of crossing the TR boundary. However, while oscillations in the corona have been observed (De Moortel et al., 2002a,b), the properties of these waves have not been measured precisely enough to say whether or not they are capable of heating the corona (Klimchuk, 2006).

DC Heating

The footpoints of magnetic field lines rooted in the photosphere are subject to the turbulent velocity field of the underlying convection zone and thus undergo a random walk across the surface. The turbulent motion of the footpoints causes the overlying field to become twisted and braided, leading to a highly stressed magnetic field above the solar surface. If the stressing and subsequent buildup of magnetic energy is sufficiently slow, the field can evolve through a series of equilibrium stages and store energy above the potential level of the field. However, if the stressing happens quickly such that the field does not have time to reach equilibrium, the stored energy is likely to be dissipated by reconnection, heating the plasma, and allowing the field to relax to a near-potential state (Priest, 2014). Gold (1964) first proposed the idea of a twisted field created by photospheric footpoint motions, suggesting that it was the relaxation of the stressed field to its equilibrium (or force-free) state that provided the energy release mechanism needed to power flares. Later, Parker (1972) proposed that closely-packed flux tubes in the corona will lead to a braided and twisted field and that current sheets will form at the boundaries between braided field lines. Dissipation of these current sheets then provides sufficient energy to power the corona (Parker, 1983a,b). Recent 1.3 Heating in the Solar Corona 23

observations by Cirtain et al. (2013) using the High-resolution Coronal Imager (Hi- C) sounding rocket provide possible evidence of braiding of the coronal magnetic ﬁeld.

1.3.2 Nanoﬂare Heating

Parker (1988) suggests that the observed X-ray corona is due to the superposition

of many nanoflares, small-scale reconnections in the twisted and braided coronal magnetic field that release an amount of energy equal to 10 9 of that of a typical ⇡ solar flare. Current sheets develop at discontinuities between adjacent flux tubes and when these discontinuities reach some critical value, the field reconnects and the current sheet dissipates, heating the plasma. Based on the hard X-ray observations of Lin et al. (1984) and the EUV observations of Brueckner & Bartoe (1983), Parker estimates that each reconnection releases 1024 erg on average though the amount could be as high as 1027 erg. A simplified cartoon version of this process is illustrated in Figure 1.7. Following Parker (1988), the nanoflare energy can be estimated as follows. Consider the simplified geometry shown in Figure 1.7 in which a flux tube extends

from s = 0 to s = 2L, where both surfaces correspond to the photosphere and the flux tube has been straightened. Let the footpoint at s = 2L be fixed and the footpoint at s = 0 move randomly across the surface with velocity v. The angle q between the vertical and the displaced flux tube as a function of time t is,

vt tan q(t) , (1.15) ⇡ 2L 1.3 Heating in the Solar Corona 24

(a) (b) ⎢⎢ L s=2

s=0

⏊ (c) (d)

Figure 1.7 Illustration of the nanoflare heating scenario of Parker (1988). The flux tubes have been straightened out such that both the top and bottom gray surfaces correspond to the photosphere. Flux tubes in an initially uniform field (a) are braided when their footpoints are shuffled by the underlying convective motions of the photosphere (b). At some critical angle between the braided flux tubes, they reconnect (c), dissipate their stored energy into the plasma, and relax back to some lower energy state (d). The axes in panel (a) indicate the directions parallel and perpendicular to the magnetic field. Adapted from Figures 5 and 7 of Klimchuk (2015). provided the angle is small (i.e. q(t) < 1). If B is the vertical component of the k field, the resulting perpendicular component can be expressed as,

B vt B = B tan q(t) k . (1.16) ? k ⇡ 2L

The Poynting ﬂux associated with the work done by the footpoint motion on the ﬁeld is given by,

2 2 2 1 B v B v t F = B B v = k tan q(t) k . (1.17) 4p k ? · ? 4p ⇡ 4p(2L) 1.3 Heating in the Solar Corona 25

7 2 1 For typical active region values of F = 10 erg cm s (Withbroe & Noyes, 1977), 1 B = 100 G, and v = 0.5 km s , q 14°. Once q, the angle between B and B , k ⇡ ? k reaches this critical value, the energy is rapidly dissipated by reconnection and the

field relaxes back to its equilibrium state, destroying B . Note that the absence of ? dissipation and reconnection would result in an infinite build-up of stress in the field (Klimchuk, 2015). The energy associated with each of these discontinuities in the field can be written as, B2 # = `2DL , (1.18) 8p? where ` is the length of each random step and DL is the length of winding along the neighboring flux tube. Assuming the lifetime of each step is 500 s, ` = 250 km ⇡ and DL = `/ tan q = 103 km. Thus, the free energy associated with the winding of the field is # 6 1024 erg, in approximate agreement with observations. Parker ⇡ ⇥ (1988) notes that the energy of the resulting nanoflare will be, on average, less than this value. Much progress has been made in understanding the role of nanoflares in heating the solar corona though a definitive detection has yet to be made. Early modeling efforts by Cargill (1994) and Cargill & Klimchuk (2004) showed that nanoflares lead to a broad distribution of temperatures and can produce “very hot” plasma at temperatures in excess of 8 106 K, the so-called “smoking gun” of nanoflare heat- ⇥ ing. While spectroscopic observations of “warm” plasma, 1 MK to 2 MK, provide compelling, indirect evidence of nanoflares (Viall & Klimchuk, 2012; Warren et al., 2011, 2012, e.g.), a direct detection of > 8 MK with current instruments is not likely (Winebarger et al., 2012). However, recent results from two sounding rockets, the Extreme Ultraviolet Normal Incidence Spectrograph (EUNIS, Brosius et al., 2014) and the Focusing Optics X-ray Solar Imager (FOXSI, Ishikawa et al., 2017), provide compelling evidence of this very hot plasma. 1.4 Thesis Outline 26

Though the original nanoflare concept of Parker (1988) was explicitly tied to reconnection, the modern definition of a nanoflare is far more general. Through- out the remainder of this thesis, I adopt the definition of Klimchuk (2015) that a nanoflare is “an impulsive energy release on a small cross-field spatial scale without regard to physical mechanism”. Nanoflares may be caused by waves or by reconnection as both have been shown to be impulsive (Klimchuk, 2006, 2015). Nanoflare events may occur frequently or infrequently on a given flux tube and their energy spectrum may be broad though it is likely to favor lower energy events (Hudson, 1991). Rather than probing a specific physical mechanism, the work in this thesis is focused on constraining the properties of the heating, regardless of the underlying driver.

1.4 Thesis Outline

The primary focus of this thesis is the heating of the solar corona via impulsive nanoflare heating. Understanding the energy deposition in the corona via EUV and X-ray observations is made difficult by several mitigating factors, including the optically-thin nature of the upper solar atmosphere, inadequate spectral, temporal, and spatial resolution of current observing instruments, and uncertainties in the atomic physics. A large volume of high-quality observations combined with detailed forward models are needed to adequately constrain the frequency with which energy is dissipated in the coronal plasma. In Chapter 2, I discuss the field-aligned physics of coronal loops and detail both hydrostatic and hydrodynamic approaches to modeling the thermal structure of these loops. In Chapter 3, I outline the dominant mechanisms for producing spectral line and continuum emission in the corona. Additionally, I discuss the two primary 1.4 Thesis Outline 27

observables used in this thesis for diagnosing the properties of the heating: the emission measure distribution and the time lag. Chapter 4 provides a detailed explanation by example of the synthesizAR forward modeling code developed for predicting optically-thin emission from an active region using an ensemble of many loop models. The next three chapters make up the primary research findings of this thesis. Chapter 5 examines signatures of nanoflare heating in the “hot” component of the differential emission measure distribution. In particular, I use the EBTEL model (Section 2.2.4) to examine the extent to which nanoflare duration, heat flux limiting, ion heating, and non-equilibrium ionization affect the observability of this very hot plasma. In Chapter 6, I predict the time-dependent, multi-wavelength AIA intensities from active region NOAA 1158 for three different nanoflare heating frequencies using the forward modeling code described in Chapter 4 combined with the EBTEL model. From these predicted intensities, I compute the emission measure slope and time lag diagnostics in each pixel of the active region. In Chapter 7, I compute the emission measure slope and time lag from real AIA observations of active region NOAA 1158. I then train a random forest classifier using the predicted diagnostics from Chapter 6 and use it to “map” the heating frequency across the entire active region based on the classification of the observed diagnostics. Finally, Chapter 8 summarizes the research findings of this thesis and suggests several topics for future work. 1.5 Use of Data and Software 28

1.5 Use of Data and Software

This thesis makes use of observational data from the Helioseismic Magnetic Imager as well as the Atmospheric Imaging Assembly, both aboard the Solar Dynamics Observatory spacecraft. Data from both of these instruments are kindly made publicly available by the respective instrument teams via the Joint Science Operations Center (JSOC, Couvidat et al., 2016) operated by Stanford University. All atomic data used in this work are from version 8 of the CHIANTI atomic database (see Section 3.1). CHIANTI is a collaborative project involving George Mason Univer- sity, the University of Michigan (USA), University of Cambridge (UK), and NASA Goddard Space Flight Center (USA). The work presented in this thesis makes use of the greater scientific Python ecosystem, including Astropy for unit-aware computation and astronomy-specific functionality (The Astropy Collaboration et al., 2018), Dask for parallel and distributed computing (Rocklin, 2015), matplotlib (Hunter, 2007) and seaborn (Waskom et al., 2018) for visualization, NumPy for array computation (Oliphant, 2006), PlasmaPy, a community-developed open source core Python package for plasma physics (PlasmaPy Community et al., 2018), scikit-learn for machine learning (Pe- dregosa et al., 2011), scipy for general purpose scientific computing (e.g. interpola- tion, curve fitting, special functions Jones et al., 2001), and SunPy, an open-source and free community-developed solar data analysis package written in Python (SunPy Community et al., 2015). Additionally, this work makes use of SolarSoftware (SSW, Freeland & Handy, 1998), a common programming and data analysis environment for solar physics written in the proprietary Interactive Data Language (IDL).

This thesis was typeset using LATEX and the PythonTEX package (Poore, 2015). With the exception of Figure 1.7 and Figure 2.6, all ﬁgures are created inline in the

document using PythonTEX and matplotlib. Chapter 2

The Physics of Coronal Loops

While parts of the solar magnetic ﬁeld may “open” to form the solar wind, many ﬁeld lines close back at the photosphere to form arch-like structures extending high

into the tenuous corona called coronal loops, the primary building blocks of the highly-structured solar corona (Reale, 2010). As can be seen in the left panel of Figure 2.1, b 1 (see Equation 1.8) in the corona such that the plasma is strongly ⌧ confined by the magnetic field resulting in negligible cross-field motion. Conduction of heat is also severely inhibited perpendicular to the field-aligned direction such that loops are thermally isolated. As a result, hot and radiating coronal plasma traces out the enormously complicated solar magnetic field. The right panel of Figure 2.1 shows several distinct loop structures observed off the solar limb by the Transition Region and Coronal Explorer satellite (TRACE, Handy et al., 1999). Because of the relatively high temperatures of the corona ( 105 K to 107 K), ⇠ coronal loops are observed primarily at EUV and X-ray wavelengths. Additionally, they are long-lived structures, observable for many hours and sometimes even days. While most coronal loop temperatures exceed 105 K, loop plasma can span a range of temperatures and densities, due to both the complexity of the underlying field and the fact that individual loops are thermally isolated. Loops are often

29 30

categorized based on their thermal properties: cool, 0.1 MK to 1 MK, warm, 1 MK to 1.5 MK, and hot, 2 MK (Reale, 2010). Whether these thermal categories actually represent distinct classes of loops or if they are all just transient states of the same type of loop is an open question.

104

103

102

101 [Mm] h

100

1 10 sunspot plage 2 10 3 0 10 10

Figure 2.1 Left: A simple empirical model of b as a function of height, h, above the solar surface as calculated by Equation 2 and Equation 3 of Gary (2001) for the magnetic ﬁeld above a sunspot (blue) and a plage region (orange). The dotted lines indicate the tops of the photosphere, chromosphere and corona. The dashed line denotes b = 1. Adapted from Figure 3 of Gary (2001). Right: An arcade of loops extending into the corona observed off the solar limb by the 171 Å EUV band of the TRACE satellite on 6 November 1999. Adapted from Figure 11 of Reale (2010).

The first evidence of magnetically confined loop structures came from soft X- ray observations on rocket missions in the late 1960s as reported by Vaiana et al. (1968). Analysis of data from these early missions also allowed for classification of distinct topological features on the solar surface by Vaiana et al. (1973). These early findings provided the first look at the X-ray-bright, highly-structured corona. Later, observations from the Orbiting Solar Observatory IV, equipped with a grazing- 31

incidence X-ray telescope, and the X-ray telescope aboard the Skylab space station (Krieger et al., 1972; Reale, 2010) allowed for more accurate determinations of loop lifetimes and better comparisons between observations and loop models (Rosner et al., 1978). Modern EUV observing instruments such as TRACE, the EUV

Imaging Spectrometer (EIS, Culhane et al., 2007) on Hinode and most recently the AIA telescope on SDO have revealed dynamic, multi-thermal structures that are incompatible with the model of a simple hydrostatic atmosphere (e.g. Del Zanna, 2008; Viall & Klimchuk, 2012; Warren et al., 2002; Winebarger et al., 2002) Additionally, it is now apparent that loop structures are multi-stranded (Gomez

et al., 1993), where a sub-resolution strand is the smallest loop for which the cross- section is isothermal (Bradshaw et al., 2012). The fundamental limit of the fine- structuring of the corona is unknown though recent sounding rocket flights have sought to address this question (Aschwanden & Peter, 2017; Cirtain et al., 2013). This natural discretization of the corona by the magnetic field means that the

dynamics of the plasma can be modeled in the field-aligned direction as an ensemble of individual isolated atmospheres rather than having to solve the equations of MHD (Equations 1.1 to 1.6) in three dimensions. Furthermore, because these models greatly simplify the geometry of the system, they are efficient, relatively easy to interpret, and capable of resolving the spatial and length scales needed to accurately model the dynamics in the corona and TR (Bradshaw & Cargill, 2013). Notably, many modern MHD codes used for modeling the corona do not satisfy the latter two criteria. In this chapter, I provide a detailed discussion of the field-aligned physics and modeling of coronal loops. In Section 2.1, I discuss the case of hydrostatic equilibrium and various methods for solving the energy and pressure balance equations in order to determine the thermal structure of a loop for some energy supplied by coronal heating. Then, in Section 2.2, I discuss the equations of field- 2.1 Hydrostatics 32

aligned hydrodynamics (Section 2.2.1) and the physics of the loop heating, cooling and draining cycle (Section 2.2.2). Additionally, I provide a detailed description of the EBTEL model in Section 2.2.4 which will be used extensively in this thesis to efﬁciently model the response of coronal loops to impulsive heating.

2.1 Hydrostatics

2.1.1 Equations of Hydrostatic Equilibrium

For a single strand in hydrostatic equilibrium with uniform cross-sectional area, the

equations of pressure and energy balance as a function of s, the coordinate parallel to the magnetic ﬁeld, are given by,

d Q = n2L(T)+ F , (2.1) ds c d p = m ng, (2.2) ds i

p = 2kBnT, (2.3)

where Q is the heating rate, L(T) is the radiative loss term, Fc is the conductive ﬂux,

p is the thermal pressure, n is the number density, T is the temperature, mi is the 2 R dr average ion mass, g = g is the gravitational acceleration along the ﬁeld r ds line, r is the radial distance⇣ from⌘ the center of the Sun, and g is the gravitational acceleration at the solar surface (r = R ). To determine p and T as a function of s along the strand, one must solve Equa- tion 2.1 and Equation 2.2 given some predeﬁned heating rate Q, which is, in general, a function of the loop coordinate and is often assumed to be proportional to powers

of n and T (Priest, 2014). Both of these equations are subject to closure by the equation of state given in Equation 2.3. 2.1 Hydrostatics 33

Equation 2.2 says that the downward gravitational pull on the plasma is balanced by the gradient of the thermal pressure. Equation 2.1 says that the energy lost by radiation and thermal conduction in the corona must be balanced by coronal heating. I will now discuss the conductive ﬂux and radiative loss terms in more detail. Several possible solution methods for Equation 2.1 and Equation 2.2 are discussed in Sections 2.1.2 to 2.1.4.

Conductive Flux Term

Thermal conduction efficiently transfers energy from regions of high temperature to regions of low temperature in the direction parallel to the magnetic field and opposite to the temperature gradient. Cross-field (i.e. perpendicular to the magnetic field) thermal conduction is severely inhibited due to the low-b nature of the corona such that energy transfer is limited to the field-aligned direction.

It is commonly assumed that the ﬁeld-aligned thermal conduction, Fc, is well- described by expression of Spitzer & Härm (1953),

dT F = k T5/2 , (2.4) c 0 ds where k 10 6 erg cm 1 s 1 K 7/2 is the coefﬁcient of thermal conduction in the 0 ⇠ ﬁeld-aligned direction. In the corona, thermal conduction is an energy sink needed

to balance coronal heating. However, in the TR, where Q = 0, thermal conduction

is an energy source to compensate for losses due to radiation. Thus, Fc is used to define the boundary between the TR and the corona such that the base of the corona is the point at which thermal conduction transitions from an energy source to a sink (Bradshaw & Cargill, 2010a; Vesecky et al., 1979). Note that no energy is lost from the system due to the heat flux; it is only a mechanism for transferring 2.1 Hydrostatics 34 energy between the corona and TR provided the heat flux to the cooler, underlying chromosphere is negligible such that the loop is thermally isolated.

According to Equation 2.4, F • as dT •. However, at low densities, there c ! ds ! may be an insufficient number of particles to support the implied heat flux such that the heat flux saturates at the free-streaming limit (Bradshaw & Cargill, 2006; Bradshaw & Raymond, 2013; Patsourakos & Klimchuk, 2005). Failure to account for this limiting of the heat flux can result in overestimation of cooling due to thermal conduction, particularly in cases where the heating is very impulsive. A more detailed discussion of the free-streaming limit is given in Section 5.2.1. In a rarified, impulsively-heated plasma, the electron mean free path may become large relative to the temperature length scale of the plasma such that the electron distribution becomes non-Maxwellian (Bradshaw & Raymond, 2013). In this case, non-local contributions to the heat flux may become important such that determining the heat flux at any one point along the loop requires integrating over the entire loop. This “non-localization” of the heat flux has been addressed by a number of authors (Karpen & DeVore, 1987; Ljepojevic & MacNeice, 1989; Luciani et al., 1983; West et al., 2008) and is likely to have important consequences for the observability of hot, low-density plasmas.

Radiative Loss Term

Energy from the coronal plasma is also lost to space by radiation. The radiative loss term is proportional to n2 so it is most dominant in areas where the density is high. The amount of energy which the plasma radiates away by spectral line and continuum emission also depends on the temperature through the radiative loss 2.1 Hydrostatics 35

function, L(T), given by,

• L(T)= 4pG + dl Pcont. , (2.5) Â 0Â ji X,k 1 X,k ji Z0 { } @ A where Gji is the contribution function of transition ji in ion k of element X as given cont. by Equation 3.42, PX,k is the total continuum emissivity of ion k in element X (see Section 3.3), and ji is the set of all atomic transitions in ion k of element { } X. Practically speaking, the summations are taken over all elements, ions, and transitions in the CHIANTI atomic database (see Section 3.1). Note that L(T) is

also dependent on the density due to the density dependence of Gji though this dependence is relatively weak compared to that of T.

21 10 CHIANTI RK RTV ] 1 s

3 22 10 erg cm [ L

23 10

104 105 106 107 108 109 T [K]

Figure 2.2 Radiative loss as a function of temperature for an optically thin plasma. The blue line shows the true value of L computed by CHIANTI using the abun- 9 3 dances of Feldman et al. (1992) and assuming a constant density of 10 cm . The orange line shows the Raymond-Klimchuk (RK) power-law approximation given in Klimchuk et al. (2008) and the green line shows the power-law approximation of Rosner et al. (1978, RTV). 2.1 Hydrostatics 36

Calculating L(T) can be computationally expensive as it requires solving Equa- tion 3.18, the level population equation, for each ion over a large temperature range. Rosner et al. (1978) calculate an analytic ﬁt to the radiative loss function of the form,

L(T)=cTa, (2.6) where c and a are piecewise functions in T. The values of a and c are given in Appendix A of Rosner et al. (1978). Klimchuk et al. (2008) provide an improved piecewise fit to L(T) of the same form as Equation 2.6 using updated atomic radiative loss calculations. Additionally, Cargill (2014) provides a minor fix to the fits of Klimchuk et al. (2008) given the results of Reale & Landi (2012) who find that using the power-law fit of Rosner et al. (1978) versus the full CHIANTI radiative loss function gives significantly different cooling behavior in an impulsively heated loop. Figure 2.2 shows a comparison between the full radiative loss function as computed by Equation 2.5 (blue) and the power-law approximations of Klimchuk et al. (2008, orange) and Rosner et al. (1978, green). Note that the upturn at 3 107 K is due ⇡ ⇥ to the increased contribution of free-free continuum emission (see Section 3.3) to the radiative losses at high temperatures.

2.1.2 The Isothermal Limit

I now discuss several methods for solving the hydrostatic energy and pressure equations. First, consider the case of an isothermal, semi-circular loop where

T(s)=T such that dT/ds = 0 and Fc = 0 in the case where heat ﬂux, Fc, is given by Equation 2.4. Equation 2.1 and Equation 2.2 then become,

Q = n2L(T), (2.7) 2 d mig R ps p = p 2 cos , (2.8) ds 2kBT r (s) 2L ⇣ ⌘ 2.1 Hydrostatics 37 where r(s)=R + 2L sin ps and L is the half-length of the loop. Note that in this p 2L case, the heating term is now balanced only by the radiative loss term.

Deﬁning the hydrostatic scale height as lh = 2kBT/mig , making a change of variables x = r(s) such that dx = cos (ps/2L)ds, and separating variables, the pressure equation becomes,

p 1 R2 x 1 dp0 = dx0 2 , p0 p0 lh R x0 Z Z p R2 1 1 ln = , p0 lh x R ✓ ◆ ✓ ◆ 1 (2L/p) sin (ps/2L) p(s)=p0 exp , (2.9) lh 1 +(2L/pR ) sin (ps/2L)  where p0 is the pressure at s = 0. At the apex of the loop, s = L, the pressure is,

2L/p p(L)=p0 exp . lh(1 + 2L/pR ) 

In the limit of a vertical ﬂux tube such that the loop has no curvature, Equa- tion 2.9 becomes,

s h p(s)=p0 exp = p0 exp , (2.10) lh(1 + s/R ) lp   where lp = lh(1 + h/R ) is the pressure scale height. Similarly, from Equation 2.3, the density can be expressed as,

h n(s)=n exp , 0 l  p where n0 = p0/2kBT. Thus, in a vertical, gravitationally stratified flux tube, the pressure and density fall off exponentially with a scale height lp. For short loops (h R ), gravitational stratification plays only a minor role in the pressure (and ⌧ density) structure and l l . In the case of longer loops, gravitational stratifi- p ⇡ h 2.1 Hydrostatics 38

1010

109 ] 3 108 cm [ n

107 1 105 K 2 106 K ⇥ ⇥ 5 105 K 5 106 K ⇥ ⇥ 1 106 K ⇥ 106 0 100 200 300 400 500 s [Mm]

Figure 2.3 Density as a function of ﬁeld-aligned coordinate, s, for an isothermal ﬂux tube assuming a vertical (dashed) and semi-circular (solid) geometry for a half 10 3 length of L = 500 Mm and a footpoint density of n0 = 10 cm . The different colors correspond to different temperatures, T, as denoted in the legend.

cation becomes more important. Additionally, note that lh µ T such that hotter loops are less stratified. Figure 2.3 shows the density as a function of s for a vertical flux tube and half of a semi-circular flux tube for a range of temperatures. As the temperature increases, the gravitational stratification of the density decreases such that hot plasma is likely to be found at higher altitudes while cooler plasma is more likely to be found near the base of the loop. Continuing with the assumption of a vertical flux tube, the heating term can now be expressed as, h Q = n2L(T) exp . (2.11) 0 l /2 ✓ p ◆ Note that the heating falls off twice as fast as the pressure such that in long loops, the heating needed to balance losses from radiation will be concentrated near the footpoints. 2.1 Hydrostatics 39

2.1.3 Scaling Laws

In the non-isothermal case, an analytic solution of Equation 2.1 and Equation 2.2 is

not possible due to the dFc/ds term being nonlinear in T. As a result, a number of so-called scaling laws have been developed to provide analytic and interpretable approximations for the thermal structure of coronal loops. In order to interpret

early space-based coronal observations made with the S-054 SkyLab X-ray telescope, Rosner et al. (1978) derived two scaling laws relating the heating and maximum temperature to the isobaric pressure and loop length. I will now summarize this derivation. Using Equation 2.3 and Equation 2.4, Equation 2.1 can be written as,

F dFc p2 c = . 5/2 2 2 Q k0T dT 4kBT

Separating variables and integrating from the base of the loop yields,

Fc 2 T T k0 p 1/2 5/2 dFc0 Fc0 = 2 dT0 T0 L(T0) k0 dT0 T0 Q, ZFc,0 4kB ZT0 ZT0 where the “0” subscript denotes a quantity at the base of the loop. If the loop is

thermally isolated from the lower atmosphere, Fc,0 = 0 such that no thermal energy is conducted across the lower boundary. Using this assumption to simplify the left-hand side gives,

F2 = f (T) f (T), (2.12) c R H where,

2 T k0 p 1/2 fR(T)= 2 dT0 T0 L(T0), (2.13) 2kB ZT0 T 5/2 fH(T)=2k0 dT0 T0 Q. (2.14) ZT0 2.1 Hydrostatics 40

Note that Equation 2.12 requires that f (T) f (T) to guarantee a physical R H solution for the heat ﬂux. Using Equation 2.4 and integrating once more from the base of the loop yields,

dT 2 k T5/2 = f (T) f (T), 0 ds R H ✓ ◆ dT k T5/2 =(f (T) f (T))1/2, 0 ds R H T 5/2 T0 s s0 = k0 dT0 , (2.15) T0 f (T ) f (T ) Z R 0 H 0 p where s0 = s(T0) is the coordinate at the base of the loop. Using the appropriate boundary conditions, Equation 2.15 can then be used to derive both scaling laws. First, Equation 2.13 can be simpliﬁed by using the piecewise power-law approx-

imation in Equation 2.6. For T > 105 K, a = 1/2 is a suitable approximation (Cargill, 1994) and assuming T T , Equation 2.13 becomes 0

2 k0c0 p fR(T)= 2 T, (2.16) 2kB

18.8 3 1/2 1 where c0 = 10 erg cm K s . At the base of the loop, losses due to radiation dominate over energy supplied by

the heating term such that f (T) f (T). Using this condition and Equation 2.16, R H integrating the right-hand side of Equation 2.15 from the base of the loop (T0) to the

apex (Tmax) gives,

Tmax kB 2k0 2 s s = dT0 T0 , max 0 c p s 0 ZT0

kB 2k0 3 3 smax s0 = (Tmax T0 ), 3ps c0 where smax = s(Tmax) is the coordinate at the apex of the loop where the temperature is maximized. Deﬁning the loop half-length, the distance between the apex 2.1 Hydrostatics 41

108 L = 10 Mm L = 100 Mm L = 1000 Mm ] K [ 107 max T

106 2 1 0 1 2 3 10 10 10 10 10 10 2 p [dyne cm ]

Figure 2.4 Loop apex temperature, Tmax, as a function of pressure, p, calculated from Equation 2.17 for several different values of the loop half-length, L. Adapted from Figure 9 of Rosner et al. (1978).

and the base, as L = s(T ) s(T ), assuming T T since T monotonically max 0 max 0 increases from the base to the apex, and solving for Tmax yields the ﬁrst scaling law,

1/3 Tmax = c1(pL) , (2.17)

1/3 3 c0k0 1/3 1/3 where c1 = 1829 K cm dyne . Figure 2.4 shows the apex kB 2 ⇡ ⇣ q ⌘ temperature, Tmax, as a function of p for several different loop half-lengths. Note that longer loops have higher apex temperatures. If the loop is heated uniformly, the maximum temperature will occur at the apex

such that dT/ds = 0 at s = smax and by Equation 2.4, the heat ﬂux at the apex also vanishes. In this case, plugging Equation 2.14 and Equation 2.16 into Equation 2.12 gives,

fR(Tmax)= fH(Tmax), 2.1 Hydrostatics 42

Tmax k0c0 2 5/2 2 p Tmax = 2k0Q dTT , 2kB ZT0 c0 2 4 7/2 7/2 2 p Tmax = Q(Tmax T0 ). (2.18) 2kB 7

Using the approximation T T , the second term on the right-hand side can be max 0 dropped. Finally, using the ﬁrst scaling law (Equation 2.17) and solving for Q gives the second scaling law of Rosner et al. (1978),

7/6 5/6 Q = c2 p L , (2.19)

7c0 5/2 4/3 1/6 1 where c2 = 2 c1 50863 cm erg s . Note that this scaling law implies 8kB ⇡ that less heat is needed to sustain a long loop compared to a short loop at the same pressure. Serio et al. (1991) extend the work of Rosner et al. (1978) by considering two

additional cases: (1) loops with L greater than the pressure scale height such that the isobaric approximation does not hold; and (2) loops with a local temperature minimum at the top. They considered a more general form of the heating function,

Q(s)=Q exp ( s/l ), where the heating is now stratiﬁed and falls off exponen- 0 H tially over a heating scale height lH. The modiﬁed scaling laws of Serio et al. (1991) are,

2 1 T = c (p L)1/3 exp 0.04L + , (2.20) max 1 0 l l  ✓ H p ◆ 7/6 5/6 1 1 Q = c p L exp 0.5L , (2.21) 0 2 0 l l  ✓ H p ◆ where p0 is the pressure at the base of the loop, Q0 is the heating rate at the

base of the loop, and lp is the pressure scale height as deﬁned in Section 2.1.2. The derivation is largely the same as that of Rosner et al. (1978) except that p is 2.1 Hydrostatics 43

allowed to vary with s according to Equation 2.10 in the limit of no gravitational stratification. Note that the scaling laws of Rosner et al. (1978)(Equations 2.17 and 2.19) are recovered in the limits of constant pressure (l •) and uniform heating p ! (l •). If the heating is sufficiently stratified such that l < L/2, there will H ! H be a local temperature minimum at the apex. In cases where the heating is very

localized near the footpoints such that lH < lp/3, a cool condensation may form at the apex, causing the loop to be Rayleigh-Taylor unstable. See Section 8.2.2 for additional discussion of non-equilibrium in steadily-heated loops. More recently, Martens (2010) derived an analytic expression for the temperature proﬁle along the loop under the assumption of constant pressure p0 and using a heating function of the form, ? b a? Q = CQ p0 T , (2.22)

? ? where CQ is a constant of proportionality and b and a determine the dependence

of Q on p0 and T, respectively. Assuming a radiative loss function of the form of Equation 2.6, Martens solves a dimensionless form of Equation 2.1 and ﬁnds a closed-form expression for the temperature as a function of the ﬁeld-aligned

coordinate, s, 1 T(s)=Tmax br (s/L; l + 1, 1/2) , (2.23) h i 1 where br is the inverse of the regularized incomplete b-function (see Section 6.6 of Abramowitz & Stegun, 1972), l = 11/2+g 1, and g = a from Equation 2.6. 2(2+g+a?) Martens (2010) points out that the solution in Equation 2.23 is overconstrained by the boundary conditions on his dimensionless energy equation such that Equa- tion 2.23 is only valid for speciﬁc sets of parameters. These additional constraints produce two scaling laws of similar form to those of Rosner et al. (1978); Serio et al. 2.1 Hydrostatics 44

(1991). The scaling laws of Martens (2010) are,

11+2g k 1/2 (3 2g)1/2 p L = T 4 0 B(l + 1, 1/2), (2.24) 0 max c 4 + 2g + 2a? ✓ 0 ◆ 2 ? p c0(7/2 + a ) Q = 0 , (2.25) apex (2+g) T (3/2 g) max where B is the complete b-function (see Equation 6.2.1 of Abramowitz & Stegun,

2 1972), c0 = c0/4kB, and Qapex is the heating rate at the apex of the loop (i.e. where

T = Tmax). Equation 2.24 and Equation 2.25 provide a generalization of the scaling laws of Rosner et al. (1978) for non-uniform heating along the loop. Notice that in the case of uniform heating (a? = 0) and g = a = 1/2, Equation 2.24 and Equation 2.25 reduce to Equations 2.17 and 2.19, respectively. In Chapter 4, I use the scaling laws of Martens (2010) to efﬁciently model time-independent emission from an active region composed of many hundreds of loops.

2.1.4 Numerical Solutions

While the analytical expressions outlined in Section 2.1.3 provide useful approximations of the thermal structure of a coronal loop, it is often necessary to solve the full hydrostatic energy (Equation 2.1) and pressure (Equation 2.2) balance equations.

Because of the non-linear dependence on T in the thermal conduction term of the energy equation, Equation 2.1 and Equation 2.2 must be integrated numerically (e.g. with an Euler or Runge-Kutta scheme, see Press et al., 1992, Chapter 16). Numerical solutions of these hydrostatic solutions have been used by many authors (e.g. As- chwanden et al., 2001) to constrain the properties of the heating in the corona by comparing with loop proﬁles derived from observations. Taken together, Equations 2.1, 2.2 and 2.4, closed by Equation 2.3, represent a system of three coupled linear differential equations subject to the boundary 2.1 Hydrostatics 45

conditions,

Fc(s = 0)=0, (2.26)

Fc(s = L)=0, (2.27)

p(s = 0)=p0, (2.28)

T(s = 0)=T0, (2.29)

where p0 and T0 are free parameters. The additional boundary condition on the heat ﬂux is because Q, the energy supplied by coronal heating, is not known such

that there four unknowns: Q, Fc, p, T. Setting the heat ﬂux equal to zero at the boundaries thermally isolates the loop such that no energy leaves the system by thermal conduction.

2.5 1011

2.0

10 ]

] 10 1.5 3 MK [ cm [ T

1.0 n

109 0.5 uniform apex

0 25 50 75 0 25 50 75 s [Mm] s [Mm]

Figure 2.5 Temperature (left) and density (right) as a function of s for a full semi- circular loop of length 2L = 80 Mm heated unformly (blue) and at the apex with lH = 10 Mm (orange). An isothermal chromosphere of depth 5 Mm is attached 4 to each footpoint. The footpoint temperature is T0 = 2 10 K and the footpoint 11 3 ⇥ density is n0 = 10 cm though the chromospheric density is much higher. 2.2 Hydrodynamics 46

If the heating function is assumed to have the form,

s Q(s)=Q exp , 0 l  H

the problem is to determine which value of Q0 satisﬁes Equations 2.26 and 2.27 for

some choice of p0, T0, lH, L using a “shooting” method (see Section 17.1 of Press et al., 1992). Uniform heating corresponds to l •. Typically, Equations 2.1, H ! 2.2 and 2.4 are solved on a numerical grid spanning half a semi-circular loop from the top of the chromosphere to the loop apex and symmetry is assumed about the

apex. If the grid spans the entire loop of length 2L, the second boundary condition

is modiﬁed to Fc(s = 2L)=0 such that heat can be conducted across the apex. Figure 2.5 shows an example hydrostatic solution for a full loop computed using the numerical procedure described above for uniform (blue) and apex (orange) heating.

2.2 Hydrodynamics

Thus far, I have only discussed cases where the loop is in hydrostatic equilibrium such that the downward gravitational pressure is exactly balanced by the thermal pressure and the energy input by coronal heating is exactly balanced by radiation

and conduction to the lower atmosphere. In this case, it was assumed that Q, the energy supplied by coronal heating, was time-independent. However, as discussed

in Section 1.3.2, heating in the corona is likely time-dependent (Q = Q(t)) and impulsive. If Q changes sufﬁciently fast such that thermal conduction and radiation do not have time to immediately balance the energy supplied by coronal heating, the loop is no longer in hydrostatic equilibrium and evolves according to the equations

of ﬁeld-aligned hydrodynamics. 2.2 Hydrodynamics 47

2.2.1 Equations of Field-aligned Hydrodynamics

The two-ﬂuid ﬁeld-aligned hydrodynamic mass, momentum, and energy equations in conservative form, as given in Bradshaw & Cargill (2013, Appendix A), are,

∂r ∂(rv) + =0, (2.30) ∂t ∂s ∂ ∂ ∂ ∂ 4 ∂v (rv)+ (rv2)= (p + p )+ µ + rg, (2.31) ∂t ∂s ∂s e i ∂s 3 i ∂s ✓ ◆ ∂E ∂ ∂p ∂F 1 e + [(E + p )v]=v e ce + k nn (T T ) (2.32) ∂t ∂s e e ∂s ∂s g 1 B ei i e n2L(T )+Q , e e ∂E ∂ ∂p ∂F 1 i + [(E + p )v]= v e ci + k nn (T T ) (2.33) ∂t ∂s i i ∂s ∂s g 1 B ei e i ∂ 4 ∂v + µ v + rvg + Q , ∂s 3 i ∂s i ✓ ◆ where g = 5/3,

p E = e , (2.34) e g 1 p rv2 E = i + , (2.35) i g 1 2

and the set of equations is closed by an equation of state for both the electrons and the ions,

pe = kBnTe, (2.36)

pi = kBnTi. (2.37)

Equations 2.30 to 2.33 describe the mass, momentum, and energy transport of the plasma in the ﬁeld-aligned direction in response to the injection of energy via

coronal heating. Note that the subscripts e and i denote quantities pertaining to either the electron or ion ﬂuids, respectively. Under the current-free and quasi- 2.2 Hydrodynamics 48

neutrality assumptions, ve = vi = v and ne = ni = n, respectively such that there is only one mass equation and one momentum equation. It then follows that

r = m n + m n = n(m + m ) m n since m m . The kinetic energy term in e e i i e i ⇡ i e ⌧ i Equation 2.32 is neglected for this same reason. I will now give a brief a description of each equation and their respective terms.

The Mass Equation

According to Equation 2.30, the equation of mass conservation, changes in the mass

density, r, are due to variations in the mass flux, rv, where v is the bulk flow velocity in the field-aligned direction. An increase in r at a given point is due to an inflow

∂ of mass into the surrounding region ( ∂s (rv) < 0) and any decrease in r is due to an ∂ outﬂow of mass from the region ( ∂s (rv) > 0)(Priest, 2014). Note that Equation 2.30 is equivalent to Equation 1.1 in one dimension (i.e. ∂ ). r!∂s

The Momentum Equation

Equation 2.31 describes the evolution of the momentum of the plasma, rv. The second term on the right-hand side is the viscous contribution to the ion momentum

and µi is the classical coefficient of viscosity as given by Spitzer (1962). The electron viscosity is negligible since m m . Viscous effects are expected to become e ⌧ i important at high temperatures (Bradshaw & Klimchuk, 2011) and, in particular, Peres & Reale (1993) found that inclusion of viscosity affected the formation of shocks in a flaring loop and subsequently influenced the predicted X-ray line profiles for Ca XIX and Fe XXV.

g in the third term on the right-hand side is the ﬁeld-aligned gravitational acceleration and is given in Section 2.1.1. Note that in the limit of zero bulk ﬂow

(v = 0) and electron-ion equilibrium (pe = pi), Equation 2.31 reduces to Equation 2.2, the hydrostatic pressure balance equation. Furthermore, the MHD momentum 2.2 Hydrodynamics 49

equation (Equation 1.2) reduces to Equation 2.31 if the Lorentz force (j B)is ⇥ neglected and the gravitational force and viscous effects are included.

The Energy Equations

Equations 2.32 and 2.33 describe the evolution of the electron and ion energy. The second term on the left-hand side of both equations represents the enthalpy flux and the second term on the right-hand side denotes the energy transported by the heat flux. The form of each heat flux term is the same as that in Equation 2.4 with a modified coefficient of thermal conduction: k 7.8 10 7 erg cm 1 s 1 K 7/2 0,e ⇡ ⇥ for electrons, k 3.2 10 8 erg cm 1 s 1 K 7/2 for ions (Bradshaw & Cargill, 0,i ⇡ ⇥ 2013). Note that this implies that thermal conduction is less efficient at cooling the ions. The fourth term on the right-hand side of Equation 2.32 denotes energy lost to radiation and is the same as in Equation 2.1. Radiation from the ions is considered negligible. The fourth term on the right-hand side of Equation 2.33 is the viscous contribution to the ion energy (see Section 2.2.1) and the fifth term is work done against gravity. The contributions of both of these terms to the electron energy are

negligible because m m . e ⌧ i The electron and ion equations are coupled through the first and third terms on the right-hand sides of Equations 2.32 and 2.33. The first term represents the energy loss or gain as the fluids move through the electric field that maintains quasi-neutrality, given by E = 1 ∂ p . The third term models the exchange of ne ∂s e energy between the electron and ion populations via binary Coulomb collisions

and is attributed to Braginskii (1965). nei is the frequency of electron-ion collisions and is given by, 4 3/2 16pp e 2k T n = B e n ln L, (2.38) ei 3 m m m e i ✓ e ◆ 2.2 Hydrodynamics 50 where e is the electron charge and ln L is the Coulomb logarithm (see both Equation 2.5e and Section 3 of Braginskii, 1965). Though the expression presented here differs by a factor of 2 compared to that of Braginskii, the electron-ion equilibration time is not signiﬁcantly changed by this relatively small numerical factor.

Lastly, Qe and Qi denote the energy injected into the electrons and ions by coronal heating. In general, this is a free parameter and may be a function of

both s and t. While it is often assumed that all of the heating is supplied to the electrons, some heating mechanisms may also preferentially energize the ions. This is discussed in more detail in Chapter 5.

Assuming electron-ion equilibrium (Te = Ti) and adding Equations 2.32 and 2.33 gives the single-ﬂuid hydrodynamic energy equation,

∂E ∂ ∂ ∂ 4 ∂v + [(E + p)v]= F n2L(T)+ µ v + rvg + Q, (2.39) ∂t ∂s ∂s c ∂s 3 i ∂s ✓ ◆ were E = Ee + Ei, p = pe + pi, and Fc = Fc,e + Fc,i. Under the assumption of ∂ hydrostatic equilibrium (v = 0, ∂t = 0), this reduces to the equation of hydrostatic energy balance (Equation 2.1).

2.2.2 The Heating, Cooling, and Draining Cycle of Coronal Loops

If energy supplied by coronal heating cannot be balanced by radiation and thermal conduction in order to maintain hydrostatic equilibrium, the loop undergoes a cycle of heating, cooling, and draining according to the physics in Equations 2.30 to 2.33. This response of the coronal loop plasma to an impulsive release of energy is reasonably well-understood and has been studied extensively, both in the context of ﬂares and nanoﬂares (e.g. Antiochos & Sturrock, 1976, 1978; Bradshaw, 2008; Bradshaw & Cargill, 2005, 2010a,b; Cargill, 1994; Cargill & Klimchuk, 2004; Cargill 2.2 Hydrodynamics 51

(a) corona (e) transition region L

(b)

(d)

(c)

Figure 2.6 A cartoon illustration of the heating and cooling cycle of an impulsively heated coronal loop. The loop has a half-length of L and is assumed to be symmetric about the apex. The red arrows denote energy injected by heating (a) and energy transported by thermal conduction (b). The blue arrows denote energy lost by radiation (d and e). The thick green arrows indicate the bulk transport of material in the loop (c, d, and e). The gray arrows denote the order in which the cycle proceeds.

et al., 1995). Figure 2.6 shows a cartoon of the different phases of the heating, filling, 1 cooling, and draining cycle of a coronal loop. Consider a loop initially in hydrostatic equilibrium that is then (uniformly) heated sufficiently quickly such that radiation and thermal conduction do not have time to immediately balance the supplied energy. For the sake of simplicity, I will assume the single-fluid approximation as given in Equation 2.39 and a semi-circular

loop with half-length L that is symmetric about the apex. This is illustrated in phase (a) of Figure 2.6. This excess heat raises the temperature of the loop and sets

up a strong heat ﬂux toward the lower atmosphere because F µ T5/2 ∂T and T c ∂s monotonically increases with s from the TR up to the apex. This is shown in phase 2.2 Hydrodynamics 52

(b). Once conduction is able to match the coronal heating rate, the plasma begins to cool. At this point, the loop is said to be “underdense” as its density is lower than that expected from hydrostatic equilibrium given the increase in temperature. Both radiation and conduction remove energy from corona and thus cool the plasma, but during this initial phase, conduction is more efﬁcient. This is easily proved by considering the respective timescales of each of these processes. Following Cargill et al. (1995), dropping all but the thermal conduction term in Equation 2.39 gives an approximation of the conductive cooling time,

∂ ∂ E F , ∂t ⇡∂s c 7/2 3kBnT k0T 2 , tC ⇠ L 2 3kBL n tC 5/2 , (2.40) ⇠ k0T

and neglecting all but the radiative loss gives an approximation of the radiative cooling time,

∂ E n2L(T), ∂t ⇡ 3k nT B n2cTa, tR ⇠ 3kB tR a 1 . (2.41) ⇠ cnT

Using a = 1/2, for high temperatures and low densities t t or in other C ⌧ R words conduction is initially more efﬁcient than radiation at removing the excess energy in the corona.

Because energy lost by radiation is proportional to n2, the dense underlying chromosphere and TR are efﬁcient at radiating away the energy conducted downward from the corona. However, if transition region radiative losses cannot keep pace with the coronal conductive ﬂux, this excess energy will heat the chromosphere 2.2 Hydrodynamics 53

and effectively destabilize the pressure balance in Equation 2.2. This increase in temperature causes chromospheric and TR (Bradshaw & Cargill, 2013) material to expand into the corona and increase the coronal density. This process is illustrated in phase (c) of Figure 2.6 and is called chromospheric ablation1. Note that this upﬂow of material provides energy to the corona via an enthalpy ﬂux (Antiochos & Sturrock, 1978).

As material fills the corona, n increases and radiative cooling becomes more efficient. This further lowers the coronal temperature, weakens the downward conductive flux, and thus reduces chromospheric ablation. This is illustrated in panel (d). Once the corona has cooled sufficiently such that the downward conductive flux can be balanced by radiative losses in the TR, chromospheric ablation ceases. At this point, radiation has become just as efficient as thermal conduction such that t t . In contrast to phase (b), the loop is now said to be R ⇠ C “overdense” as the increased density in the loop due to chromospheric ablation is greater than that expected from hydrostatic equilibrium. As the corona continues to cool, the ablated material can no longer support the upward pressure gradient and the plasma falls back down the loop due to gravity (see Equation 2.2). This is shown in phase (e) of Figure 2.6. At this point, radiative cooling is much more efficient compared to conductive cooling such that t t . However, (Bradshaw & Cargill, 2010b) note that the resulting enthalpy R C flux out of the corona from the downflow of material also represents a significant and sometimes dominant loss mechanism during this radiative and enthalpy-driven cooling phase2. Furthermore, Bradshaw (2008) and Bradshaw & Cargill (2010b) show that this enthalpy flux is important in powering the TR against collapse due to runaway radiative cooling.

1Historically, this process is referred to as chromospheric “evaporation.” However, this terminol- ogy is potentially misleading and “ablation” is a more accurate term. 2While this phase of the loop evolution is often referred to as only the “radiative cooling phase,” this nomenclature is incomplete as it neglects an important component of the energy budget. 2.2 Hydrodynamics 54

Provided the time-independent heating that initially supported the loop in hydrostatic equilibrium is still present, the loop will cool and drain until it reaches its equilibrium temperature and density. If the loop again undergoes some time- dependent heating, the cycle will start over. Note that the evolution of plasma is complicated if the loop is re-energized before the cooling and draining cycle is complete (e.g. Barnes et al., 2016a; Cargill, 2014, and see Section 6.2.3).

2.2.3 The HYDRAD Model

Taken together, Equations 2.30 to 2.33 are a set of coupled, non-linear partial differential equations and must be solved numerically. The state-of-the-art Hydrodynamics and Radiation code (HYDRAD, Bradshaw & Cargill, 2013; Bradshaw & Mason, 2003a,b) solves the two-fluid, field-aligned hydrodynamic equations over a full loop of arbitrary geometry using an explicit second-order finite difference scheme on an adaptively-refined grid. HYDRAD runs easily on a variety of platforms, from a laptop computer to a large computing cluster, and has recently been modified to take advantage of multiple threads via the OpenMP library for multithreading (Reep et al., 2019). The code also includes a Java GUI for easily and intuitively configuring the input parameters to the code. A critical feature of HYDRAD is the use of adaptive mesh refinement (AMR) as it ensures that the code adequately resolves steep gradients and discontinuities (e.g. shocks) that can develop along the loop. For example, failure to adequately resolve the TR in cases of very impulsive heating can result in severe underestimation of the coronal density and temperature (Bradshaw & Cargill, 2013). Furthermore,

AMR allows the code to add grid cells as needed by ensuring that r, Ee, and/or

Ei do not vary by more than some predefined tolerance (e.g. 10%) from one grid cell to the next. The maximum level of refinement can be adjusted by the user to increase computational efficiency in cases where high spatial resolution is not 2.2 Hydrodynamics 55

needed. HYDRAD also uses an adaptive time step in order to adequately resolve the thermal conduction timescale, a restriction that can greatly increase compute times for very long loops or very impulsive heating (though see a possible alternative in Johnston et al., 2017). Additionally, HYDRAD (optionally) simultaneously solves the time-dependent ionization and recombination equations (Equation 3.19) for any number of elements. These time-dependent population fractions can then be fed back into the calculation of Equation 2.5 to account for non-equilibrium ionization in the radiative loss function. Additionally, Reep et al. (2019) recently added the ability to solve the H level populations in non-local thermal equilibrium (NLTE) in order to more properly account for optically-thick radiation in the chromosphere. It should be mentioned that a number of codes have been developed to solve the field-aligned hydrodynamic equations including, but not limited to, the NRL Solar Flux Tube Model (Mariska et al., 1989; Warren et al., 2003), RADYN (Allred et al., 2015), Lare1D (Johnston et al., 2017), and the model of Mikićet al. (2013). However, all of these models lack at least one of the following critical features: separate treatment of electron and ion fluids, an adaptively-refined grid to ensure proper spatial resolution, incorporation of non-equilibrium ionization, and/or ease of use.

2.2.4 The EBTEL Model

While ﬁeld-aligned models like HYDRAD are extremely powerful, they are often too computationally expensive to be used in a parameter space exploration, for example, of a range of heating properties. Such studies are critical to constraining properties of the heating in coronal loops. To this end, many zero-dimensional (“0D”) loop models have been developed (e.g. Aschwanden & Tsiklauri, 2009; Cargill, 1994; Fisher & Hawley, 1990; Kopp & Poletto, 1993; Kuin & Martens, 1982), 2.2 Hydrodynamics 56

so-called because they compute the hydrodynamic evolution of a single, spatially- unresolved point rather than an entire loop. Such models are very useful as they incorporate the key physics of the evolution of a coronal loop, but are extremely efficient such that a large parameter space can be explored in a reasonable amount of time. Once a parameter space is sufficiently constrained, a more sophisticated field-aligned model can be used. By far the most advanced and widely-used 0D model is the enthalpy-based thermal evolution of loops model (EBTEL, Klimchuk et al., 2008). Unlike earlier 0D models, EBTEL can include any form of time-dependent heating, allows for cooling by thermal conduction and radiation at all times, and can account for heat flux saturation (see Section 2.1.1). The idea behind the EBTEL model is to equate an enthalpy flux due to ablation or draining with the balance between the heat flux out of the corona and radiative loss rate in the TR. If the TR radiation cannot balance the downward heat flux, this drives an upflow. Conversely, if the TR is radiating away more energy than the heat flux can supply, this drives a downflow. EBTEL was originally developed by Klimchuk et al. (2008) as an improvement to the widely-used cooling model of Cargill (1994). Later, Cargill et al. (2012a) improved the model by including a more sophisticated treatment of the ratio between the radiative losses in the TR and corona. Most recently, Barnes et al. (2016b) generalized the EBTEL model to treat the electron and ion fluids separately. I will now derive the two-fluid EBTEL equations from the field-aligned hydrodynamic equations and then show how these reduce to the original single-fluid EBTEL equations of Cargill et al. (2012a); Klimchuk et al. (2008).

Two-ﬂuid Model

This derivation is adapted directly from Appendix B of Barnes et al. (2016b, Chap- ter 5 of this thesis) and closely follows the derivation of the original EBTEL equations 2.2 Hydrodynamics 57

in Cargill et al. (2012a); Klimchuk et al. (2008). First, substitute the deﬁnitions of the electron and ion energy, Equations 2.34 and 2.35, in the electron and ion energy equations, Equations 2.32 and 2.33, to get expressions for the evolution of the

electron pressure, pe, and ion pressure, pi, respectively,

1 ∂p g ∂ ∂p ∂F 1 e + (p v)=v e ce + k nn (T T ) (2.42) g 1 ∂t g 1 ∂s e ∂s ∂s g 1 B ei i e n2L(T )+Q , e e 1 ∂p g ∂ ∂p ∂F 1 i + (p v)= v e ci + k nn (T T )+Q . (2.43) g 1 ∂t g 1 ∂s i ∂s ∂s g 1 B ei e i i

Note that three terms have been dropped: the kinetic energy component of Ei and the viscous and gravitational terms in the ion energy equation. Following Klim-

chuk et al. (2008), under the assumptions of sub-sonic ﬂows, v < C = 1.5 104T1/2 s ⇥ (= 2.6 107 cm s 1 at T = 3 MK), where C is the sound speed, terms which are ⇥ s second-order (or higher) in v are small such that kinetic energy and viscous terms can be neglected. Furthermore, for loops shorter than a gravitational scale height,

L < l = 5 103T(= 150 Mm for T = 3 MK), the gravitational contribution to the g ⇥ ion energy is negligible. The next step is to integrate each equation over the coronal portion of the loop in order to “zero-dimensionalize” them. Assuming symmetry about the loop apex,

the coronal average of a quantity x x(s, t) is deﬁned as, ⌘

1 1 sa x¯ = dsx= dsx, L ZC L Zs0 where L is the loop half-length, the subscript “a” denotes the apex of the loop, and the subscript “0” denotes the base of the corona where thermal conduction changes from being an energy source in the TR to an energy sink in the corona. For terms 2.2 Hydrodynamics 58

differentiated with respect to s, the coronal average can be expressed as,

∂ ds x = x(s ) x(s ). ∂ a 0 ZC s

Averaging Equations 2.42 and 2.43 over the coronal portion of the loop gives,

L dp¯ g e = (p v) + F + y R + LQ¯ , (2.44) g 1 dt g 1 e 0 ce,0 C C e L dp¯ g i = (p v) + F y + LQ¯ , (2.45) g 1 dt g 1 i 0 ci,0 C i where thermal conduction at the base of the loop, s = s0, is given by,

dT 2 T7/2 F = k T5/2 k a (2.46) c,0 0 ds ⇡7 0 L

(Klimchuk et al., 2008), v(sa)=0 and Fc(sa)=0 due to the assumption of symmetry 2 about the apex, RC = C dsn L(Te) and, R ∂pe kB yC = dsv + ds nnei(Ti Te). (2.47) C ∂s C g 1 Z Z

Similarly, the integrals of x and ∂x/∂s over the TR are deﬁned as,

1 1 s0 x¯ = dsx= dsx, ` ` ZTR Zsc ∂ ds x = x(s ) x(s ), ∂ 0 c ZTR s where ` L is the length of the TR and the “c” subscript denotes the top of the ⌧ chromosphere. Integrating Equations 2.42 and 2.43 over the TR portion of the loop gives,

g (p v) = F + y R , (2.48) g 1 e 0 ce,0 TR TR 2.2 Hydrodynamics 59

g (p v) = F y , (2.49) g 1 i 0 ci,0 TR

Any terms µ ` are neglected because ` L (Klimchuk et al., 2008) and it is assumed ⌧ that the enthalpy ﬂux and heat ﬂux go to zero at the top of the chromosphere.

2 Additionally, RTR = TR dsn L(Te) and R ∂pe kB yTR = dsv + ds nnei(Ti Te). (2.50) TR ∂s TR g 1 Z Z

The second term in this expression is usually small, but is retained for completeness. Substituting Equation 2.48 into Equation 2.44 and Equation 2.49 into Equa- tion 2.45 gives, respectively,

L dp¯ e =y + y (R + R )+LQ¯ , (2.51) g 1 dt TR C C TR e L dp¯ i = (y + y )+LQ¯ . (2.52) g 1 dt C TR i

These equations describe the spatially-averaged evolution of the coronal electron

and ion energy given some heating input Q¯ e and Q¯ i. As with the energy expressions, Equation 2.30 can be similarly integrated over the corona,

∂ ∂ r = (rv), ∂t ∂s ∂ ∂ ds r = ds (rv), ∂ ∂ ZC t ZC s dn¯ L =(nv) . dt 0

Using Equations 2.36 and 2.48, the above equation can be written as

dn¯ (p v) L = e 0 , dt kBTe,0 2.2 Hydrodynamics 60

dn¯ c2(g 1) L = ( Fce,0 RTR + yTR), dt c3gkBT¯e dn¯ c2(g 1) L = ( Fce,0 RTR + yTR), (2.53) dt c3gkBT¯e where c2 = T¯e/Te,a and c3 = Te,0/Te,a. This equation describes the spatially- averaged evolution of the coronal density in response to the heating. The problem

now is to ﬁnd expressions for the integrals yTR and yC.

Recall that yC and yTR are comprised of terms associated with the quasi-neutral electric ﬁeld and temperature equilibration. The integral of the former can be considered as the gain or loss of energy associated with plasma motion through the net electric potential. Using integration by parts, the ﬁrst integral in Equation 2.47 becomes,

∂pe sa dsv =(pev) dvpe = (pev)0 dvpe C ∂s s0 C C Z Z Z (pev)0 p¯e dv = (pev)0 + p¯ev0 0. (2.54) ⇡ ZC ⇡

Thus, Equation 2.47 becomes,

k L y B n¯n (T¯ T¯ ), (2.55) C ⇡ g 1 ei i e where nei = nei(T¯e, n¯).

The next step is to ﬁnd an expression for yTR. Dividing Equation 2.36 by

Equation 2.37, multiplying by v, and using the quasi-neutrality condition (n = ne =

ni) gives, p v T e = e . (2.56) piv Ti 2.2 Hydrodynamics 61

Evaluating this expression at the TR/corona interface, s = s0, and using Equa- tions 2.48 and 2.49 yields,

F + y R ce,0 TR TR = x, (2.57) F y ci,0 TR where x T /T . Solving for y gives, ⌘ e,0 i,0 TR

1 y = (F + R xF ). (2.58) TR 1 + x ce,0 TR ci,0

Finally, the set of two-ﬂuid EBTEL equations can be expressed as,

d g 1 p¯ = (y R (1 + c )) + k n¯n (T¯ T¯ )+(g 1)Q¯ , (2.59) dt e L TR C 1 B ei i e e d g 1 p¯ = y + k n¯n (T¯ T¯ )+(g 1)Q¯ , (2.60) dt i L TR B ei e i i d x c2(g 1) n¯ = (Fce,0 + Fci,0 + c1RC), (2.61) dt 1 + x c3LgkBT¯e

2 where c1 = RTR/RC is discussed more fully in Section 2.2.4, RC = Ln¯ L(T¯e), c 0.9, and c 0.6. 2 ⇡ 3 ⇡ Equations 2.59 to 2.61 are a set of three coupled, non-linear ordinary differential equations. While they still need to be solved numerically, solving these equations is fast and relatively straightforward compared to resolved ﬁeld-aligned codes (e.g. HYDRAD) due to the lack of a spatial grid. The ebtel++ code3 solves the two-ﬂuid EBTEL equations using a Runge-Kutta Cash-Karp method (Press et al., 1992, Section

16.2) given a loop half-length L and a prescribed heating function. ebtel++ is written in the C++ programming language and can use either a static or an adaptive time step. It is very efﬁcient, capable of computing solutions for thousands of loops

3ebtel++ is thoroughly documented and openly developed. The full source code is available here: github.com/rice-solar-physics/ebtelPlusPlus. The original single-ﬂuid EBTEL model of Cargill et al. (2012a); Klimchuk et al. (2008) is implemented in IDL and is available here: github.com/rice-solar- physics/EBTEL 2.2 Hydrodynamics 62

evolving over many hours of simulation time in a few minutes or less. The ebtel++ code is used extensively in both Chapter 5 and Chapter 6.

0.000 ] 1 s 3

0.008 erg cm [

e e thermal conduction ¯ E ion thermal conduction D radiation 0.016 collisions yTR

100 101 102 103 t [s]

Figure 2.7 Energy loss and gain mechanisms arising from a nanoﬂare with t = 200 s and electron heating only. The various curves correspond to the terms in the EBTEL two-ﬂuid electron energy equation, Equation 2.62: electron and ion thermal conduction, radiation, binary Coulomb interactions, and yTR. The loop parameters are as in Section 5.3.

Substituting Equation 2.58 into Equation 2.59, the electron energy evolution equation can be written,

1 dp¯ 1 x x(c + 1)+1 e = F F 1 R (2.62) g 1 dt L(1 + x) ce,0 L(1 + x) ci,0 L(1 + x) C k + B n¯n (T¯ T¯ )+Q¯ , g 1 ei i e e where the ﬁrst two terms on the right-hand side represent the contributions from electron and ion thermal conduction, the third term represents losses from radiation, and the last two terms are as before. Figure 2.7 shows the contribution of each

term, with the exception of the heating term, Q¯ e. As expected, (electron) thermal conduction dominates during the early heating and cooling phase and losses from 2.2 Hydrodynamics 63

radiation takeover in the late draining and cooling stage. Between these two phases, energy exchange between the two species by collisions is important to the evolution

of the electron energy. yTR is included to show its importance in the formation of the two-ﬂuid EBTEL equations.

Single-ﬂuid Model

I now show how the two-fluid EBTEL equations reduce to the original single-fluid EBTEL equations developed by Cargill et al. (2012a); Klimchuk et al. (2008). In the single-fluid limit, n • such that the electron and ion populations are always in ei ! equilibrium, Te = Ti. Adding Equation 2.51 and Equation 2.52 gives,

L d d p¯ + p¯ = R (c + 1)+L(Q¯ + Q¯ ), g 1 dt e dt i C 1 e i ✓ ◆ L d p¯ = R (c + 1)+LQ¯ , g 1 dt C 1 where p¯ = p¯e + p¯i = 2kBn¯T¯ and Q¯ = Q¯ e + Q¯ i. This expression is equivalent to Equation 5.2, the single-ﬂuid EBTEL energy equation.

In the case of Te = Ti, x = 1 and Equation 2.61 becomes,

d c2(g 1) n¯ = (Fc,0 + c1RC), dt 2c3LgkBT¯ where Fce,0 + Fci,0 = Fc,0 because k0 = k0,e + k0,i. Comparison with Equation 5.4 shows that the single-ﬂuid equation is recovered in the limit of inﬁnitely fast binary Coulomb collisions between electrons and ions.

Modiﬁcations to c1 During the Conductive Cooling Phase

This section is adapted directly from Appendix A of Barnes et al. (2016b, Chapter 5

of this thesis). The coefﬁcient c1 is deﬁned as the ratio between the radiative losses 2.2 Hydrodynamics 64

in the TR and corona, RTR c1 = . (2.63) RC

As can be seen in Equations 2.59 to 2.61, c1 plays an important role in the evolution of the loop in response to the energy supplied by heating. Klimchuk et al. (2008)

initially assumed a constant value of c1 = 4 though they found c1 was much larger

at high temperatures. Cargill et al. (2012a) modiﬁed the calculation of c1 to include a correction for gravitational stratiﬁcation and a more sophisticated approach to radiative cooling such that,

eqm c1 , n < neq, = (2.64) c1 8 eqm 2 + rad(( / )2 1) > c1 c1 n neq < 2 , n > neq, 1+(n/neq) > :> where neq was the loop density that would exist for the calculated temperature were eqm the loop to be in static equilibrium (Equation 17 of Cargill et al., 2012a) and c1 rad and c1 are the values of c1 in equilibrium and during the radiative cooling phase, respectively (see Equations 12 and 16 of Cargill et al., 2012a).

In Section 3 of Cargill et al. (2012a) it was assumed that the parameter c1 decreased from its equilibrium value at the time of maximum density, to that com- mensurate with radiative/enthalpy cooling as the loop drained, deﬁned in terms

of the ratio n/neq. In the radiative phase, n > neq while c1 takes on its equilibrium eqm value, c when n < n . Deﬁning D (n n )/n , this gave 1 eq ⌘ EBTEL HYDRAD HYDRAD D . 0.2, acceptable errors in the EBTEL value of n, as shown in Cargill et al. (2012a).

A modiﬁed description of c1 for n < neq is needed for many of the examples discussed in Chapter 5. Speciﬁcally, for intense heating events, the coronal density calculated by the version of EBTEL in Cargill et al. (2012a) is unacceptably high when compared to results from HYDRAD. Quantitatively, it is found that D & 0.3 2.2 Hydrodynamics 65

at nmax. While this may seem to be reasonable for an approximate model, the high EBTEL density is a systematic feature, and requires further investigation. Examination of the HYDRAD results shows that EBTEL signiﬁcantly underesti- mates the TR radiative losses during the heating and conductive cooling phases. At this time, the loop is underdense (e.g. Cargill & Klimchuk, 2004), so that an excess of

the conducted energy goes into evaporating TR material. c1 is modiﬁed as follows

for n < neq, eqm cond 2 2c1 + c1 ((neq/n) 1) c1 = 2 , (2.65) 1 +(neq/n)

as a direct analogy to Equation 2.64 (Equation 18 of Cargill et al., 2012a). In the eqm early phases of an event, n n , so that c ccond. When n = n , c = c . ⌧ eq 1 ⇡ 1 eq 1 1 cond After some experimentation, it is found that c1 = 6 gives reasonable agreement

between EBTEL and HYDRAD. There is no impact on the solution for n > neq.

Table 2.1 Comparison between HYDRAD (H) and EBTEL (E) with c1 = 2 and c1 given by Equation 2.65, for n < neq. The ﬁrst three columns show the full loop length, heating pulse duration, and maximum heating rate. The last three columns show nmax for the three models. Only nmax is shown as Tmax is relatively insensitive to the value of c1. The ﬁrst two rows correspond to the t = 200, 500 s cases considered in Chapter 5. The next four rows are the four cases shown in Table 2 of Cargill et al. (2012a). The last two rows are cases 6 and 11 from Table 1 of Bradshaw & Cargill (2013).

H E E 2L t Q0 nmax nmax nmax, (Eq. 2.65) 3 1 8 3 8 3 8 3 [Mm] [s] [ erg cm s ] [10 cm ] [10 cm ] [10 cm ] 80 200 0.1 37.6 46.0 42.0 80 500 0.04 37.7 44.9 40.0 150 500 0.0015 3.7 4.0 3.7 50 200 0.01 10.7 11.6 10.6 50 200 2 339.0 398.9 357.3 50 200 0.01 15.5 16.5 14.6 40 600 0.8 350.0 458.2 393.2 160 600 0.005 10.0 10.5 9.4

Table 2.1 shows a set of runs carried out to compare the results from HYDRAD eqm and EBTEL with c1 = c1 = 2 (ﬁfth column) and with c1 given by Equation 2.65 2.2 Hydrodynamics 66

(sixth column), when n < neq. Using the modiﬁcation in Equation 2.65 gives, for the more intense heating cases with t 200 s, D 0.1 at n . For the more gentle ⇠ max heating proﬁles of Bradshaw & Cargill (2013); Cargill et al. (2012a) (i.e. rows 3, 4,

6, and 8 of Table 2.1), D . 0.2, conﬁrming that the modiﬁcation proposed here is applicable to a wide range of heating scenarios. For short, intense pulses like the t = 20, 40 s cases addressed in Chapter 5, D > 0.2. The limitations of such cases are addressed in Section 5.3.1. Equation 2.65 is motivated by simplicity while including

the essential physics. Alternative, more complex determinations of c1 have been considered, but involve limitations on how EBTEL can be used both now and in the future. Chapter 3

Emission Mechanisms and Diagnostics of Coronal Heating

Diagnosing the properties of the underlying energy deposition in the corona is nontrivial as measurements are limited to remote sensing data from ground- and space-based instruments. Until the recently launched Parker Solar Probe mission

(Fox et al., 2016), in-situ measurements were limited to the solar wind by satellites at

the L1 point. Thus, inferring the dynamics and energy budget of the coronal plasma necessitates the use of multiple diagnostics computed from the observed coronal emission at multiple wavelengths. In this chapter, I will give an overview of how emission is produced in the corona and discuss observational diagnostics that can provide meaningful insight into how the plasma is heated. Section 3.1 notes the practical importance of the CHIANTI database in modeling and interpreting solar observations. In Section 3.2 and Section 3.3, I review the physics of the formation of spectral lines and continuum radiation in the corona and Section 3.4 provides a detailed explanation of the temperature sensitivity of the AIA passbands. In the last two sections, I discuss two primary diagnostics for inferring properties of the

67 3.1 The CHIANTI Atomic Database 68

energy deposition from observations: the differential emission measure distribution (Section 3.5) and the time lag (Section 3.6).

3.1 The CHIANTI Atomic Database

The CHIANTI atomic database (Del Zanna et al., 2015a; Dere et al., 1997, 2001, 2009; Landi et al., 2012, 2006, 2002, 1999; Landi & Phillips, 2006; Landi & Young, 2009; Landi et al., 2013; Young et al., 2003, 2016; Young & Landi, 2009; Young et al., 1998) is an essential tool for analyzing and modeling spectra of optically-thin astrophysical plasmas such as the solar corona. It is primarily used in the study of the solar atmosphere though it has broader astrophysical applications as well (see Figure 4 of Young et al., 2016). The database provides information on atomic transitions for all ions of over 30 different elements, from hydrogen to zinc. For a given ion, CHIANTI provides wavelengths and energies (among other information) for many thousands of atomic transitions as well as various derived quantities, including the ionization and recombination rates, energy level populations, and spectral line intensities. Additionally, CHIANTI provides multiple measurements of elemental abundances in both the corona and photosphere. Data and routines are also included for computing the free-free and free-bound continuum emission. The CHIANTI project is an international collaboration between the University of Cambridge, the University of Michigan, George Mason University, and NASA Goddard Space Flight Center and is an invaluable asset to the solar physics community. Version 1.0 of the database was released in 1995 and at the time of writing, the current version is 9.0. Users typically interact with the database via the provided IDL routines or the more recently-released ChiantiPy package (Barnes & Dere, 2017; Landi et al., 2012), an interface to CHIANTI implemented in the Python program- 3.2 Spectral Line Formation 69

ming language. All work presented in this thesis makes heavy use of the CHIANTI atomic database via the ﬁasco Python package (see Appendix A).

3.2 Spectral Line Formation

The solar corona is optically thin, meaning that all emitted photons are observed and that these photons are not absorbed or scattered between the emission site and detector. Because these photons travel uninterrupted, they provide a direct signature of the properties of the coronal plasma. The primary mechanism for the formation of spectral emission lines in the solar corona is the spontaneous radiative

decay of an electron in an excited state j to a lower energy state i,

X X + hn , (3.1) k,j ! k,i ji

where Xk is an ion of element X in ionization stage k, nji is the frequency of the

atomic transition, and hnji is the energy of the emitted photon.

The intensity of a spectral line for an atomic transition of wavelength lji = c/nji, where c is the speed of light in a vacuum, is given by,

1 I(l )= dhP(l ), (3.2) ji p ji 4 ZLOS where the integration is taken along the LOS between the observer and the emission

site and P(lji) is the emissivity, or the radiative power per unit volume. The emissivity is given by, hc P(lji)= nX,k,j Aji, (3.3) lji where nX,k,j is the number density of Xk ions in excited state j and Aji is the probability of spontaneous emission, often referred to as the Einstein coefﬁcient (Bradshaw & Raymond, 2013; Del Zanna & Mason, 2018). 3.2 Spectral Line Formation 70

In general, it is quite difﬁcult to determine nX,k,j, the density of ions in excited

state j. As such, we can rewrite nX,k,j as,

nX,k,j nX,k nX nH nX,k,j = ne, nX,k nX nH ne nH = NX,k,j fX,kAb(X) ne, (3.4) ne where ne is the electron density, Ab(X)=nX/nH is the abundance of element X

relative to hydrogen, nH/ne is the ratio of hydrogen ions to electrons (often approximated as n /n 0.83), N = n /n is the population of level j or the H e ⇡ X,k,j X,k,j X,k fraction of Xk ions in excited state j, and fX,k = nX,k/nX is the population fraction of

ion Xk (Del Zanna & Mason, 2018). Plugging Equation 3.4 into Equation 3.3 yields a more convenient expression for the emissivity,

hc P(lji)=0.83 Ab(X) fX,k NX,k,j Ajine. (3.5) lji

Both Ab(X) and Aji, the latter of which depends on the electron temperature, Te,

can be looked up in CHIANTI (see Section 3.1). NX,k,j is a function of both Te and ne and can be computed by assuming the the excitation and de-excitation processes are

in equilibrium. This is discussed in Section 3.2.1 and Section 3.2.2. fX,k is primarily

a function of Te and is discussed in Section 3.2.3 and Section 3.2.4. Thus, for a

given distribution of Te and ne along the LOS, one can compute the intensity of a

transition lji using Equation 3.5 and Equation 3.2.

3.2.1 Collisional Excitation of Atomic Levels

For a photon to be produced by spontaneous radiative decay from excited state j to lower energy state i (Equation 3.1), the ion must ﬁrst be excited into state j. In the solar atmosphere, the most important excitation process is the inelastic collisions 3.2 Spectral Line Formation 71

between ions and free electrons,

X + e(E ) X + e(E ) (3.6) k,i initial ! k,j ﬁnal

where e denotes the free electron and Einitial and Eﬁnal are the initial and ﬁnal energies of the electron, respectively (Phillips et al., 2008). The energies of levels i

and j are Ei and Ej, respectively. If Ei < Ej, Xk has been collisionally excited from a lower to a higher energy state and the free electron has lost an amount of energy equal to the separation between these two levels,

E E = E E . ﬁnal initial i j

Conversely, if Ei > Ej, Xk is collisionally de-excited from i to j and the free electron gains an amount of energy equal to E E . i j In order to understand how energy levels are populated and depopulated by collisions, it is necessary to compute the rate at which collisions occur in a plasma with electron temperature Te for an ion Xk. I will now derive the excitation and de-excitation rate coefﬁcients. This derivation closely follows the treatment in Sections 3.2 and 3.3 of Del Zanna & Mason (2018) as well as Section 4.2 of Phillips et al. (2008). The rate coefﬁcient for collisional excitation is given by,

• e Cij = dvvsij(v) f (v), (3.7) Zv0 where v is the electron velocity, sij(v) is the cross-section for inelastic collisions between the ion and electrons, and f (v) is the velocity distribution function of the electrons. Additionally, v is the threshold velocity such that m v2/2 = E E , 0 e 0 j i where me is the mass of the electron. Any electron with v < v0 will not be able to 3.2 Spectral Line Formation 72

excite the atom from level i to j. It is commonly assumed that the distribution of free electrons in the solar atmosphere is in thermodynamic equilibrium such that it is well-described by a Maxwell-Boltzmann distribution1,

m 3/2 m v2 f (v)=4pv2 e exp e , (3.8) 2pk T 2k T ✓ B e ◆ ✓ B e ◆ where kB is the Boltzmann constant. Substituting Equation 3.8 into Equation 3.7,

m 3/2 • m v2 Ce = 4p e dvv3s (v) exp e , ij 2pk T ij 2k T ✓ B e ◆ Zv0 ✓ B e ◆

2 and making the change of variables E = mev /2 gives,

m 3/2 • 2E E Ce = 4p e dE s (E) exp , ij 2pk T m2 ij k T ✓ B e ◆ ZE0 e ✓ B e ◆ • 8 3/2 E = (k T ) dEEs (E) exp , pm B e ij k T r e ZE0 ✓ B e ◆ 8k T • E E E = B e d s (E) exp , (3.9) pm k T k T ij k T s e ZE0 ✓ B e ◆ B e ✓ B e ◆ where E = mv2/2 = E E is the minimum electron energy required to excite the 0 0 j i ion from i to j. The cross-section for excitation by inelastic collisions can be expressed as,

2 IH sij(E)=pa0Wij(E) , (3.10) wiE where a0 is the Bohr radius, IH is the ionization potential of hydrogen, wi is the

statistical weight of level i, and Wij is the dimensionless collision strength. It should be noted that W is symmetric such that W (E)=W (E ), where E = E E = ij ij ji 0 0 ij E (E E ) is the ﬁnal energy of the electron after it has been scattered. j i 1Observations of non-thermal particles (e.g Dzifˇcáková& Kulinová, 2011) suggest that the distribution of free electrons in the solar corona may be better described by a k-distribution. For more details see Cranmer (2014) or the comprehensive review by Dudík et al. (2017). 3.2 Spectral Line Formation 73

Substituting Equation 3.10 into Equation 3.9,

• e 2 8p 1 1/2 E E C = I a w T d W (E) exp , (3.11) ij H 0 k m i e k T ij k T s B e ZE0 ✓ B e ◆ ✓ B e ◆

exploiting the symmetry of Wij, and making a change of variables to E0 gives,

• e 2 8p 1 1/2 E E C = I a w T d W (E0) exp , ij H 0 k m i e k T ji k T s B e ZE0 ✓ B e ◆ ✓ B e ◆ • 2 8p 1 1/2 E0 E0 + Eij = IH a0 wi Te d Wji(E0) exp , k m 0 k T k T s B e Z ✓ B e ◆ B e !

2 8p 1/2 Uij Eij = I a T exp . (3.12) H 0 k m e w k T s B e i ✓ B e ◆

The term Uij, originally introduced by Seaton (1953), is called the effective collision strength (or alternatively the Maxwellian-averaged collision strength) and is defined as, • E E Uij = d Wji(E) exp , (3.13) 0 k T k T Z ✓ B e ◆ ✓ B e ◆ where E is now the final energy of the electron after the collision. In general, computing cross-sections for excitations by collisions with free- electrons is very difficult and time consuming and requires the use of sophisticated atomic codes which properly account for the energy levels of the target ion and the detailed physics of the interaction between the free electron and target ion (Bautista, 2000; Phillips et al., 2008, Section 4.2.3). Burgess & Tully (1992) computed

fit coefficients to U as a function of Te in terms of compact, dimensionless variables for a large number of atomic transitions. Reduced fit parameters for U are provided in the CHIANTI atomic database using the methods of Burgess & Tully (1992) for all relevant transitions such that effective collision strengths can be efficiently

computed for arbitrary Te. Figure 3.1 shows U as a function of Te for a number of transitions of Fe XII. 3.2 Spectral Line Formation 74

100

1 10 U

2 10

3 10 106 107 Te [K]

Figure 3.1 Effective collision strength, U, as a function of Te for 100 selected transitions in Fe XII. U was interpolated to Te using ﬁt coefﬁcients provided by the CHIANTI atomic database and computed using the method of Burgess & Tully (1992)

The rate coefﬁcient for de-excitation can also be computed using Equation 3.12, the excitation rate coefﬁcient. Under the assumption of thermodynamic equilibrium, the processes of excitation and de-excitation by collisions must be in balance such that

e d nineCij = njneCji, (3.14)

d where Cji is the rate coefﬁcient for collisional de-excitation, and the populations of the two levels are in Boltzmann equilibrium,

n w Eij i = i exp , (3.15) n w k T j j ✓ B e ◆ where wi and wj are the statistical weights of levels i and j, respectively. Combining d Equation 3.14 and Equation 3.15 gives an expression for Cji, the de-excitation rate 3.2 Spectral Line Formation 75

coefﬁcient,

w Eij Cd = i Ce exp , ji w ij k T j ✓ B e ◆

2 8p 1/2 Uij = IH a0 Te . (3.16) skBme wj

3.2.2 Level Populations

In optically-thin, astrophysical plasmas, it is often assumed that the processes which inﬂuence populations of atomic energy levels are decoupled from those processes which inﬂuence the charge state of the atom (see Section 3.2.3). This is because changes in the energy level populations occur much more frequently than changes in the charge state. Another common approximation is that energy levels are populated primarily by collisional excitation and depopulated by spontaneous

radiative decay and that these processes occur primarily between the ground state g and an excited state j. Assuming a steady-state equilibrium between these processes gives,

e nX,k,gneCgj = nX,k,j Ajg, (3.17) where the left-hand side corresponds to processes that populate j and the right-hand side corresponds to processes that depopulate j. Taken together, these assumptions are often referred to as the coronal model approximation (Bradshaw & Raymond, 2013; Del Zanna & Mason, 2018).

The coronal model approximation assumes a two-level system (g and j) in which the only two competing processes are excitation by collisions and spontaneous

radiative decay. However, an ion may have so-called metastable levels where the probability of spontaneous radiative decay is very low such that depopulation by collisional de-excitation is not negligible (Del Zanna & Mason, 2018; Phillips et al., 2008). In this case, the level population calculation must account for transitions 3.2 Spectral Line Formation 76

between excited states such that the two-level approximation is no longer appropriate. Thus, for a multi-level atom, the system of equations required to calculate the population of nj is (temporarily dropping the X, k subscripts),

d e d e Â ni Aij + ne Â niCij + ne Â niCij = nj Â Aji + ne Â Cji + ne Â Cji , (3.18) i>j i>j ij ! where the left-hand side denotes processes which populate level j and the right- hand side denotes processes that depopulate level j (Del Zanna & Mason, 2018).

e d Cij and Cji can be computed from Equations 3.12 and 3.16, respectively. Values

for Aji as a function of Te are available in CHIANTI. Additional processes such as collisional excitation by protons or photoionization by an external radiation ﬁeld

may also inﬂuence nj (see Sections 3.4.1 and 3.4.2 of Del Zanna & Mason, 2018).

In general, the level population nj is a function of both temperature and density. As Equation 3.18 is a system of J coupled equations, where J is the total number of energy levels of the ion, calculating n requires solving a J J matrix equation and j ⇥ can be very computationally expensive, depending on the number of energy levels and relevant atomic transitions. Figure 3.2 shows the level populations of several energy levels of O II as a function of electron energy at 106 K. The resulting relative

level population for level j of charge state k of element X, NX,k,j, can then be used

to compute the emissivity for transition lji (Equation 3.5) and the resulting spectral line intensity (Equation 3.2).

3.2.3 Processes which Affect the Ion Charge State

In addition to the relative populations of each energy level of the ion, one must also

know the population of each charge state of the ion in order to compute the emissivity (Equation 3.5). The relative population fraction of a charge state k of element X,

denoted fX,k = nX,k/nX, is the number of ions of element X in charge state k relative 3.2 Spectral Line Formation 77

100

1 10 j N

2 10 4 2 1: 2s2 2p3 S3/2 4: 2s2 2p3 P1/2 2 2 2: 2s2 2p3 D5/2 5: 2s2 2p3 P3/2 2 3: 2s2 2p3 D3/2 3 10 104 106 108 1010 1012 3 ne [cm ]

Figure 3.2 Level population of the ﬁrst ﬁve levels of O II as a function of electron 6 density, ne, at Te = 10 K. Note that the ground state is the most abundant for all ne. The level population is normalized to the total number of O II ions such that Âj Nj = 1. Adapted from Figure 4.3 of Phillips et al. (2008).

to the total number of ions of element X. Ion charge states are determined by two primary processes: ionization, in which a bound electron is freed by some external perturbation, and recombination, in which a a free electron is captured by the ion.

Thus, the time evolution of the population fraction fk (temporarily dropping the element label) is given by,

d I R I R fk = ne(ak 1 fk 1 + ak+1 fk+1 ak fk ak fk), (3.19) dt

I R where ak and ak are the ionization and recombination rates of charge state k, re-

spectively, and the population fractions are subject to the constraint Âk fk = 1 (Del Zanna & Mason, 2018). In general, the derivative on the left-hand side is

d ∂ ∂ the comoving derivative such that dt = ∂t + v ∂s . Note that ionization from lower charge states and recombination from higher charge states are source terms while 3.2 Spectral Line Formation 78

ionization and recombination out of the current charge state are sinks. Solutions to Equation 3.19 are discussed in Section 3.2.4.

Ionization

In the solar corona, the dominant processes contributing to the total ionization rate

are collisional ionization and excitation-autoionization (Bradshaw & Raymond, 2013). Thus, the total ionization rate can be written as,

aI = aCI + aEA, (3.20) where aCI and aEA are the ionization rates due collisional ionization and excitation- autoionization, respectively.

In the process of collisional ionization, a free electron collides with an ion Xk and frees a bound electron. Following the notation of Bradshaw & Raymond (2013); Mason & Fossi (1994), this can be expressed as,

X + e X + 2e, (3.21) k,i ! k+1,i0

where i0 denotes the ﬁnal energetic state of Xk+1. aCI can be computed in a similar e manner to Cij by integrating the velocity-weighted collision cross-section over a Maxwell-Boltzmann distribution. Using the result from Equation 3.9, the ionization rate due to collisional ionization can be written as,

• CI 8 3/2 E a = (k T ) dEEs (E) exp , (3.22) pm B e CI k T r e ZI ✓ B e ◆ where E is the energy of incident electron, sCI is the collisional ionization cross- section and I is the ionization energy of the initially-bound electron (Del Zanna & 3.2 Spectral Line Formation 79

Mason, 2018). Making the change of variables x =(E I)/k T gives, B e

• CI 8kBTe I x a = exp dxxsCI(kBTex + I)e pm k T 0 s e ✓ B e ◆✓Z (3.23) • I x + dx s (k T x + I)e . k T CI B e B e Z0 ◆

Notice that both integrals have the same form and can be evaluated using Gauss- Laguerre quadrature, • n x dxf(x)e Â wi f (xi), (3.24) Z0 ⇡ i=1 where xi is the zero of the i-th Laguerre polynomial and wi are the associated weights (see Equation 25.4.45 of Abramowitz & Stegun, 1972). Note that in both

terms, evaluating f (xi) requires evaluating the collisional ionization cross-section,

sCI.

As in the case of the collisional excitation cross-section, evaluating sCI is nontrivial. For ions in the hydrogen and helium isoelectronic sequences (i.e. ions with

the same number of electrons as hydrogen or helium), sCI can be calculated using the fitting formula of Fontes et al. (1999). Additionally, Dere (2007) provide fit coefficients for collisional ionization cross-sections for a large number of ions using the method of Burgess & Tully (1992). Fit parameters for both of these methods are

available in CHIANTI and can be used to efﬁciently compute sCI as a function of Te. In the case of excitation-autoionization, if an ion is collisionally excited to a level above the ionization threshold, it can autoionize, resulting in a free electron and an ion in a higher charge state, but lower energy state. This process can be written as,

X + e(E ) X + e(E ) X + e(E )+e0(E), (3.25) k,i0 1 ! k,j 2 ! k+1,i 2

where e0 is the recently freed electron (Phillips et al., 2008). In the ﬁrst step, Xk is

excited from i0 to j and the in the second step, Xk decays from j to i and emits an 3.2 Spectral Line Formation 80

electron e0, producing a higher charge state k + 1. This is only possible provided E E is greater than the ionization threshold of X (Bradshaw & Raymond, 2013). 1 2 k The ionization rate due to excitation-autoionization, aEA, can be computed using an expression analogous to Equation 3.12, but replacing U with the appropriate effective collision strengths for excitation-autoionization. Scaled ﬁt parameters to these collision strengths, as computed by Dere (2007) using the method of Burgess & Tully (1992), are available in CHIANTI. Figure 3.3 shows the total (solid blue), collisional (dashed blue), and excitation-autoionization (dot-dashed blue) ionization rates as a function of temperature for Fe XVI.

9 10

10 10 ] 1 s 3 cm [ aI a 11 CI 10 a aEA aR aRR aDR 12 10 104 105 106 107 108 109 Te [K]

Figure 3.3 Ionization (blue) and recombination (orange) rates as a function of electron temperature, Te, for Fe XVI. The constituent rates are denoted by dashed and dot-dashed lines. Note that the recombination rate dominates at low Te while the ionization rate dominates at high Te, as expected.

Recombination

Along with ionization, recombination, the capture of a free electron by the target ion, is the other primary process which determines the charge state of the ion. 3.2 Spectral Line Formation 81

In an optically-thin plasma, the two dominant processes that contribute to the

total recombination rate are radiative recombination and dielectronic recombination (Bradshaw & Raymond, 2013). As in Equation 3.20, the total recombination rate can be written as, aR = aRR + aDR, (3.26) where aRR and aDR are the recombination rates due to radiative and dielectronic recombination, respectively. In the case of radiative recombination, a free electron is captured into a bound state and a photon is emitted. This process can be expressed as,

X + e(E) X + hn . (3.27) k+1,j ! k,i ji

Note that energy conservation requires that hn = E E + E such that the energy ji j i of the emitted photon must be equal to that of the initial kinetic energy of the

electron plus the energy differential between levels j and i. In general, computing aRR is nontrivial as it requires calculating the photoionization cross-section between

i and j as well as the level population of excited state j (see Equation 4.47 of Phillips et al., 2008). Shull & van Steenberg (1982) calculated aRR for C, N, I, Ne, Mg, Si, S, Ar, Ca, Fe, and Ni using a relatively simple ﬁt function,

h T aRR(T )=A e , (3.28) e 104 ✓ ◆ where A and h are determined by fitting Equation 3.28 to tabulated values of aRR from the literature. Badnell (2006) later improved on this result by computing distorted-wave photoionization cross-sections for all ions up to and including Zn and calculating fit parameters from these results for a more sophisticated analytical fitting function for aRR (see Equation 4 of Verner & Ferland, 1996). The CHIANTI 3.2 Spectral Line Formation 82

database uses the results of both Shull & van Steenberg (1982) and Badnell (2006), as appropriate, to efﬁciently compute aRR as a function of electron temperature. Dielectronic recombination is the inverse process of excitation-autoionization (Equation 3.25) and can be expressed as,

X + e X X + hn . (3.29) k+1,i0 ! k,j ! k,i ji

Note that in the intermediate step of this process, Xk,j is in a doubly-excited state wherein a free electron has been captured and an already-bound electron has been

excited to a higher energy level j. When the ion decays from j to i and a photon is emitted, the captured electron e remains in an excited energy level (Bradshaw & Raymond, 2013). The recombination rate due to dielectronic recombination, aDR, can be evaluated in a similar manner to aRR. Shull & van Steenberg (1982) ﬁt an

DR analytical form for a (Te) to data from the literature and provide ﬁt parameters for the same set of ions as aRR. Improved analytical forms and ﬁt parameters for aDR to results obtained from a distorted-wave approximation calculation are provided in a series of papers by Badnell et al. (2003). As in the case of radiative recombination, the approaches of both Shull & van Steenberg (1982) and Badnell et al. (2003) are used

DR to compute a (Te) in CHIANTI. Figure 3.3 shows the total (solid orange), radiative (dashed orange), and dielectronic (dot-dashed orange) recombination rates as a

function of Te for Fe XVI. Note that dielectronic recombination dominates at high temperatures while radiative recombination is more important at low temperatures

(T 3 104 K). e . ⇥

3.2.4 The Charge State in Equilibrium

As noted in Section 3.2.3, there are many methods for computing the ionization and

recombination rate coefﬁcients, primarily as a function of electron temperature, Te. 3.2 Spectral Line Formation 83

One can then use these rate coefﬁcients to compute the relative population fractions of all ions of a particular element as a function of temperature. When computing these population fractions for a high-temperature, low-density plasma such as the

solar corona, it is often assumed that the charge state of the plasma is in ionization equilibrium or that the processes which populate and depopulate a particular charge state are in balance. Under this assumption, Equation 3.19 becomes (temporarily

dropping the element label X),

I R I R ne(ak 1 fk 1 + ak+1 fk+1 ak fk ak fk)=0. (3.30)

For element X with Z + 1 total charge states, where Z is the atomic number of X, Equation 3.30 represents a system of Z + 1 linear, homogeneous equations and can be expressed in matrix form as,

AF = 0, (3.31)

where F =(f1, f2, ..., fk, ..., fZ+1) is a column vector of length Z + 1, A is a Z + 1 Z + 1 matrix containing all of the ionization and recombination rates, and 0 is ⇥ the zero vector with Z + 1 entries. For a given element X and electron temperature

I R Te, one can compute a and a (e.g. using CHIANTI) for all k to ﬁnd A. To solve Equation 3.31, consider the singular value decomposition (SVD) of A,

A = UWVT, (3.32) where W is a Z + 1 Z + 1 diagonal matrix with singular values (w ) along the ⇥ k diagonal and U and V are square Z + 1 Z + 1 matrices whose columns are or- ⇥ thonormal. Provided A is singular (i.e. det (A)=0), any column of V for which

the corresponding wk = 0 is a solution of Equation 3.31 (Press et al., 1992). 3.2 Spectral Line Formation 84

Figure 3.4 shows the population fractions, fX,k, for every ionization state of Fe as a function of electron temperature, assuming ionization equilibrium. As the electron temperature rises, increasingly higher charge states are populated because the free electrons are more energetic and are capable of releasing more

tightly-bound electrons. Conversely, at lower Te, the free electrons have lower energy and recombine into the outer bound states of the target ions such that the lower charge states become more populated (Bradshaw & Raymond, 2013). Note that the temperature at which each population fraction peaks is the point at which the ionization and recombination rates for that charge state are equal (see Equation 3.19).

1.0 Fe II Fe III Fe IV 0.8 Fe XXV Fe XVII Fe VFe VI Fe VIII 0.6 Fe VII k , Fe IX X f Fe XXVI 0.4 Fe X Fe XVIII Fe XVI Fe XXIV 0.2

104 105 106 107 108 109 Te [K]

Figure 3.4 Ion population fractions for every ionization state of Fe as a function of Te. The population fractions were computed assuming ionization equilibrium using Equation 3.31. Note that increasingly higher ionization states become populated with increasing electron temperature and vice versa.

3.2.5 Non-Equilibrium Ionization

The population fraction, fk, as a function of Te can be accurately determined by solving Equation 3.30 provided that the charge states are in equilibrium with the 3.2 Spectral Line Formation 85

electron temperature of the plasma. This is a reasonable approximation provided that the electron temperature of the plasma changes sufﬁciently slowly as some ﬁnite amount time is required for the charge states to rearrange themselves following a dramatic change in temperature (Bradshaw & Raymond, 2013). The time required

for a charge state k to reach equilibrium following a change in temperature from T0

to T1 in a plasma of density ne can be expressed as,

1 = tCIE I R I R , ne(a (T1) fk 1(T0)+a + (T1) fk+1(T0) a (T1) fk(T0) a (T1) fk(T0)) k 1 k 1 k k (3.33) where tCIE is often called the collisional ionization equilibration (CIE) timescale.

If Te changes from T0 to T1 in a time less than tCIE, then the charge state k is

out of equilibrium and fk must be computed by solving the full time-dependent population fraction equation (Equation 3.19). Smith & Hughes (2010) computed the

maximum tCIE for several different elements and found that for a plasma of density 9 3 ne = 10 cm , the time needed to equilibrate to within 10% of the equilibrium population fractions of Fe was 200 s for T 106 K and approached several e thousand seconds for much higher temperatures (109 K). In a low-density, high-temperature plasma such as the solar corona which can undergo temperature changes on timescales of a few hundred or even tens of seconds (e.g. flares, heating by nanoflares), the temperature implied by the equilibrium charge states may not be representative of the actual electron temperature. As such, correctly accounting for non-equilibrium ionization (NEI) is critical for understanding the radiative losses from the coronal plasma. MacNeice et al. (1984) solved the full time-dependent ionization and recombination equations, including effects due to radiative and collisional excitation and de-excitation, in order to accurately model the spectrum of Ca in a flaring loop heated by an electron beam. Later, Bradshaw (2009) developed a robust, explicit numerical scheme for solving Equation 3.19 3.2 Spectral Line Formation 86

and provided an exhaustive set of test cases for various temperature and density gradients. Many workers (e.g. Bradshaw & Cargill, 2006; Bradshaw & Klimchuk, 2011; Bradshaw & Mason, 2003a; Bradshaw & Testa, 2019; Hansteen, 1993; Reale & Orlando, 2008) have also found that NEI is critically important when attempting to accurately predict spectral intensities of an impulsively heated, low-density plasma. In particular, effects due to non-equilibrium ionization are likely to limit the observability of “very hot” plasma, one of the primary observable signatures of nanoﬂare heating (see Chapter 5).

Fe X Fe XII Fe XIV Fe XI Fe XIII Fe XV 100 107

2 10 ] k , K

6 [

X 10 f e T

4 10

6 5 10 10 0 20 40 60 80 100 t [s]

Figure 3.5 Equilibrium (dashed) and non-equilibrium (solid) population fractions as a function of time, t, for Fe X through Fe XV. The time-dependent temperature proﬁle, Te, is shown on the right axis in black. The density is held constant at 9 3 ne = 10 cm for the entire simulation interval.

Figure 3.5 shows the population fractions of Fe X through Fe XV in equilibrium (dashed) and non-equilibrium (solid). In this simple example, the electron temperature (black) increases linearly from 105 K to 107 K over 50 s and then decreases

5 9 3 linearly back to 10 K over 50 s. The density is held constant at 10 cm for the entire 100 s. Note that for all 6 ions shown, the population fractions are out of 3.3 Continuum Emission 87

equilibrium for the entire 100 s and may differ by many orders of magnitude. In particular, the peaks of the non-equilibrium population fractions lag those of the equilibrium populations as a ﬁnite amount of time is required for the charge state

to form following a change in Te. Additionally, unlike the equilibrium populations, which track the electron temperature exactly, the non-equilibrium populations are

not symmetric about the peak in Te. The equilibrium populations were determined by computing the SVD of the matrix A in Equation 3.31 and the non-equilibrium populations were computed by solving Equation 3.19 using the implicit method described in Appendix B.

3.3 Continuum Emission

3.3.1 Free-free Emission

While the coronal EUV spectrum is dominated by spectral line emission, continuum emission becomes important for wavelengths in the X-ray band (1Åto 100 Å). There are two main types of continuum emission: free-free and and free-bound. Free-free

emission, also known as bremsstrahlung (or “braking radiation”), is produced when an ion interacts with a free electron, reduces the momentum of the free electron, and, by conservation of energy and momentum, produces a photon. This process can be expressed as,

X + e(E ) X + e(E )+hn, (3.34) k 1 ! k 2 where E1 and E2 are the initial and ﬁnal energies of the electron (Del Zanna & Mason, 2018). Bremsstrahlung is the primary mechanism for producing hard X-ray emission in hot ( 107 K) ﬂare plasma. The emission per unit time, per unit volume, and per unit wavelength produced by the free-free process for an electron with a 3.3 Continuum Emission 88 velocity v in Equation 3.34 is given by,

16pe6Z2 ( )= ( ) Pff l, v 3/2 2 2 2 nenig ff l, v , (3.35) 3 c me vl where ni is the number density of the ions and g ff, the free-free Gaunt factor, is a correction factor for the integral over the interaction cross-section (Rybicki & Lightman, 1979). Because the coronal plasma is often assumed to be thermal (see Section 3.2.1), the distribution of electron velocities can be approximated by a Maxwell-Boltzmann distribution. Integrating Equation 3.35 over Equation 3.8 gives the free-free emission produced by a thermal distribution of electrons as a function of temperature,

32pe6 2p 1/2 Z2 hc P (l, Te)= nen exp g , (3.36) ff 3m c3 3k m 2 1/2 i lk T h ffi e ✓ B e ◆ l Te ✓ B e ◆ where g is the velocity-averaged Gaunt factor and is, in general, nontrivial to h ffi calculate (Rybicki & Lightman, 1979). Itoh et al. (2000) provide an analytical ﬁtting formula for the relativistic, velocity-averaged Gaunt factor which can be used to

evaluate g for the conditions 106 T 108.5 K and 4 log (hc/lk T ) 1. h ffi  e   B e  Otherwise, the non-relativistic Gaunt factors of Sutherland (1998) can be used. Figure 3.6 shows the free-free emission summed over all the ions of Fe as a function

of l for three different temperatures: 1 MK, 10 MK and 100 MK. Note that Pff

decays exponentially with increasing l and increases with increasing Te.

3.3.2 Free-bound Emission

In addition to bremsstrahlung, free-bound emission, wherein a free electron is captured by an ion and produces a photon, can also contribute to the continuum. This 3.3 Continuum Emission 89

26 10 1 MK 10 MK 100 MK ] 1

27 10 Å 1 s 3 28 10 erg cm [ ff

P 29 10

10 30 1 0 1 2 3 10 10 10 10 10 l [Å]

Figure 3.6 Free-free emission summed over all ions of Fe as a function of wavelength. The different curves correspond to 1 MK (blue), 10 MK (orange), and 100 MK (green). The factor neni is not included here such that Pff has no density dependence.

mechanism can be expressed as,

X + e(E) X + hn. (3.37) k+1 ! k

The photon produced has energy hn = E + I, where I is the ionization energy of the bound state of the captured electron (Del Zanna & Mason, 2018). The emissivity due to free-bound emission assuming a Maxwellian distribution of electron velocities is given by Equation 12 of Young et al. (2003). The free-bound continuum spectra is characterized by sharp discontinuities at the ionization threshold of the ion because an electron with energy greater than the ionization threshold will not be captured by the ion. Two-photon decay processes in ions in the H and He isoelectronic sequences may also contribute to the total continuum emission (Young et al., 2003). However, compared to bremsstrahlung and free-found emission, the contribution of the 3.4 Temperature Sensitivity of the AIA Passbands 90

two-photon decay process to the total continuum emission is relatively small for

T >1 104 K (Del Zanna & Mason, 2018). The continuum emission due to free-free, e ⇥ free-bound, and two-photon decay processes can all be computed using data and associated software provided by CHIANTI.

3.4 Temperature Sensitivity of the AIA Passbands

One of the primary focuses of this thesis is the prediction and analysis of observations from the Atmospheric Imaging Assembly (AIA) instrument. Thus, before discussing the differential emission measure (Section 3.5) and time lag diagnostics (Section 3.6), it is important to understand the temperature sensitivity of the EUV passbands of the AIA instrument. The AIA instrument onboard the SDO spacecraft is comprised of four dual-channel normal-incidence telescopes which

have continuously observed the full Sun (410 ﬁeld of view) since 29 April 2010 at a cadence of 10 s to 12 s and spatial resolution of 0.6 per pixel (Boerner et al., ⇡ 00 2012; Lemen et al., 2012). In addition to two ultraviolet (UV, 1600 Å and 1700 Å) channels and one visible (4500 Å) channel, AIA has seven EUV channels: 94 Å, 131 Å, 171 Å, 193 Å, 211 Å, 304 Å and 335 Å. Unlike the other EUV channels which

are primarily sensitive to hot (T 105.8 K) lines in high ionization states of Fe, the 304 Å channel is dominated by the much cooler 303.784 Å He II line (see Table 1 of Lemen et al., 2012). This line forms primarily in the chromosphere and is optically-thick, meaning its observed intensity is not well-modeled by Equation 3.2 and data in the CHIANTI database (Boerner et al., 2012; Warren, 2005). Because the work presented here is primarily concerned with the dynamics of hot plasma in active regions, diagnostics will be limited to the six remaining EUV channels: 94 Å, 131 Å, 171 Å, 193 Å, 211 Å and 335 Å. Table 3.1 summarizes the solar features, dominant Fe ions, and characteristic temperature(s) of each channel. 3.4 Temperature Sensitivity of the AIA Passbands 91

Table 3.1 Primary ions observed by the six AIA EUV channels of interest. Adapted from Table 1 of Lemen et al. (2012).

Channel [Å] Primary ion(s) Solar feature Characteristic temperature [K] 94 Fe XVIII flaring corona 106.8 131 Fe VIII, XXI TR, flaring corona 105.6, 107 171 Fe IX quiet corona, upper TR 105.8 193 Fe XII, XXIV corona, hot flare plasma 106.2, 107.3 211 Fe XIV active region 106.3 335 Fe XVI active region 106.4

For any of the aforementioned EUV channels of AIA, the intensity as observed

by a particular channel c can be written as,

pc = dhnHneKc(Te), (3.38) ZLOS where Kc is the temperature response function of channel c and is given by,

• Kc = dl G(l)Rc(l), (3.39) Z0

G(l) is the total contribution function of all ions, and Rc is the wavelength response of channel c (Boerner et al., 2012). Note that Equation 3.38 gives the intensity in

1 1 1 1 5 units of DN pixel s such that Kc has units DN pixel s cm and Rc has units 2 1 1 cm DN photon sr pixel . The wavelength response function incorporates all of the properties of the telescope channel, including the geometrical collecting area, the reflectance of the mirrors, the transmission efficiency of the filters, the quantum efficiency of the CCD, and the plate scale (see Section 2 and Table 2 of Boerner et al., 2012). An additional correction is applied to account for the degradation of the instrument over time. The wavelength response functions for the six EUV channels of interest are shown in Figure 3.7. 3.4 Temperature Sensitivity of the AIA Passbands 92

1.00 94 Å 131 Å 171 Å 0.75

0.50

0.25

0.00 90 100 130 140 170 180 1.00 193 Å 211 Å 335 Å c

R 0.75

0.50 / max c

R 0.25

0.00 190 200 210 220 330 340 l [Å]

Figure 3.7 AIA wavelength response functions for the six primary EUV channels. For each channel, the response function is shown at 10 Å of the nominal wave- ± length. Each response function is normalized to the maximum value of Rc over this interval.

While Rc is only a function of the properties of the instrument, the tempera-

ture sensitivity of each channel, Kc (Equation 3.39), has an explicit dependence on the atomic data through the total contribution function, G(l). G(l) includes components from both line and continuum emission such that Equation 3.39 can be expressed more practically as,

• Gji K = dl Pcont.(l)R (l)+ R (l ), (3.40) c c Â hc/l c ji Z0 ji ji { } where Pcont. is the continuum emissivity (see Section 3.3) and Gji is the contribu-

tion function for an atomic transition lji (Equation 3.42). The sum is taken over ji , the set of all known atomic transitions for the relevant wavelength range of { } channel c. Note that hc/lji is the energy per photon of wavelength lji such that 3 1 Gji/(hc/lji) has units of photon cm s . The set of relevant atomic transitions, 3.4 Temperature Sensitivity of the AIA Passbands 93

ji , can be obtained from the CHIANTI database such that accurately determining { } the temperature sensitivity of the passbands depends critically on knowledge of the relevant atomic transitions in the bandpass of each channel.

24 10 94 Å 193 Å 131 Å 211 Å

] 171 Å 335 Å 5 25 cm 10 1 s 1 26 10 DN pixel [

c 27 10 K

28 10 105 106 107 108 T [K]

Figure 3.8 Temperature response functions for the six EUV channels of AIA listed in Table 3.1 as computed by Equation 3.40. Together, these six channels provide observational coverage over the temperature range 3 105 T 2 107 K. ⇥ . . ⇥

The temperature response functions for the EUV channels of AIA are shown in Figure 3.8. Combined, the six response functions provide temperature coverage

over the range 3 105 T 2 107 K. Note that several of the response functions, ⇥ . . ⇥ particularly the 94 Å and 131 Å channels, are double-peaked in temperature due to the finite width of the wavelength responses. This degeneracy makes associating observations with specific temperatures very difficult. For example, intensity in the 131 Å channel may correspond to very hot, >107 K plasma or much cooler, <106 K plasma. 3.5 The Differential Emission Measure Distribution 94

3.5 The Differential Emission Measure Distribution

Equation 3.2, the intensity for a transition lji, can be alternatively expressed as,

I(lji)= dTe DEM(Te)Gji, (3.41) Z

2 where Gji, the so-called contribution function , is given by,

1 hc fX,k NX,k,j Aji Gji = Ab(X) . (3.42) 4p lji ne

DEM(Te) is the differential emission measure distribution and is deﬁned as,

1 dT DEM(T )=n n e , (3.43) e e H dh ✓ ◆

dTe where dh is the temperature gradient along the LOS. Qualitatively, the DEM(Te)

measures the amount of material along the LOS between temperatures Te and dTe (Withbroe, 1978). Additionally, for a particular temperature interval D, the emission

measure distribution at a particular temperature Te,j is deﬁned as,

Te,j+D/2 EM(Te,j)= dTe DEM(Te), (3.44) T D/2 Z e,j

(Del Zanna & Mason, 2018). For an unresolved point source (e.g. a star other than

the Sun), the integration in Equation 3.2 is often taken over a volume V such that

DEM(Te) and EM(Te) are deﬁned in terms of dV rather than dh. Because of this ambiguity, Equation 3.43 and Equation 3.44 are sometimes referred to as the column differential emission measure and column emission measure, respectively.

2 Several variants of Gji exist in the literature such as dropping the factor of 1/4p, including the nH/ne 0.83 term, or dropping the abundance factor Ab(X). See section 3.1 of Del Zanna & Mason (2018). ⇡ 3.5 The Differential Emission Measure Distribution 95

Because the DEM(Te) (and EM(Te)) describes the distribution of emitting material in temperature space, it is a useful diagnostic for understanding the underlying thermodynamics of the coronal plasma. This is best illustrated by example. The left panel of Figure 3.9 shows the electron temperature, Te, for a 40 Mm loop heated by a single nanoflare at t = 0s(blue) and 10 nanoflares every 300 s (orange) as simulated by the two-fluid EBTEL model (see Section 2.2.4). In both cases, the duration of

3 each nanoflare is 200 s and the total energy deposited in the loop is 10 erg cm . In the single-nanoflare case, the loop is rapidly heated to >10 MK and then allowed to cool, uninterrupted, by thermal conduction and radiation down to its equilibrium temperature, <1 MK. In contrast, the loop heated by 10 nanoflares is never allowed to cool below 3 MK because the time between consecutive nanoflares is shorter than the fundamental cooling time of the loop (see Equations 2.40 and 2.41). Additionally, the loop is only heated to a maximum temperature of just over 4 MK following each event because the energy of each nanoflare is 1/10 of the single nanoflare energy. Thus, the loop heated by a single nanoflare samples a wide range of temperatures while the loop heated by 10 nanoflares is kept within a very narrow range in Te, centered near 4 MK. The right panel of Figure 3.9 shows the corresponding emission measure dis-

tributions, EM(Te), for the two loops. Note that compared to EM(Te) for the loop heated by 10 nanoﬂares, which is narrowly peaked in the range 3 MK to 4 MK, the

EM(Te) for the loop heated by a single nanoﬂare is very wide because the loop samples a much broader range of temperatures. Thus, for infrequently (relative to

the fundamental loop cooling time) heated loops, a broad EM(Te) is expected while

frequently heated loops lead to a more narrow, isothermal EM(Te). 3.5 The Differential Emission Measure Distribution 96

a = 2.0 1028 10 a = 12.4

8 ] 5 ] 27 MK 6 cm 10 [ [ e T

4 EM

2 1026

0 2000 4000 106 107 t [s] Te [K]

Figure 3.9 Left panel: Two-fluid EBTEL simulations of the electron temperature, Te, for a loop heated by a single nanoflare at t = 0s(blue) and a loop heated by 10 nanoflares every 300 s (orange) for a total simulation time of 5 103 s. In both ⇥ cases, the loop length is 40 Mm, the duration of each nanoflare is 200 s, and the total 3 energy deposited in the electrons in the loop is 10 erg cm . Right panel: EM(Te) for the two loops shown in the left panel. The dashed lines denote the power-law fit, Ta, to each distribution over the interval 1.25 MK T 4 MK and the emission e  e  measure slopes are shown in the legend. EM(Te) is approximated by binning Te,i, 2 4 8.5 weighted by ni L, at each timestep ti in temperature bins between 10 K to 10 K with width 0.05 in log Te and then time-averaging over the whole simulation. Note that both distributions peak at approximately the same temperature.

3.5.1 The Emission Measure Slope

Jordan (1975, 1976) found that for T 105 K to 106 K, the observed emission mea- e ⇡ sure distribution could be described by,

a EM(Te) µ Te , (3.45)

where a is some constant. This implies log EM(Te) is linear in log Te over the given temperature range with slope a such that a is often referred to as the emission measure slope. Athay (1966); Jordan (1980) found that a slope of a = 3/2 was most consistent with observations of the quiet Sun. Cargill (1994) simulated loops heated by nanoﬂares using a loop cooling model and found that this scaling also 3.5 The Differential Emission Measure Distribution 97

held between 106 K and 106.4 K with a 4. Generally speaking, Equation 3.45 ⇡ is applicable for temperatures “coolward” of T = argmax EM(T ) down to peak Te e 105 K. ⇡ Using simple scaling laws, one can derive an expression for a for a loop undergoing radiative cooling. The emission measure distribution can be approximated as

EM(T ) n2t (Cargill, 1994), where t is given by Equation 2.41. Assuming e ⇠ rad rad l Te µ n gives, 1/l+1 a EM(Te) µ Te , (3.46) where a controls the temperature-dependence of the radiative loss function (see

Equation 2.6). While the empirical result l 2 of Jakimiec et al. (1992); Serio et al. ⇡ (1991) is often used, Bradshaw & Cargill (2010b) showed that accounting for the

enthalpy flux during radiative cooling leads to l 1 for long loops and l 2 for ⇠ ⇠ short loops. Using a = 1/2 (Cargill, 1994), one finds a = 5/2 for long loops and a = 2 for short loops. The dashed lines in the right panel of Figure 3.9 show Equation 3.45 fit over the temperature range 1.25 MK T 4 MK and the values  e  of the emission measure slope are shown in the legend. Note that a = 2 for the single nanoflare case as expected from the analytical scaling while a = 12.4 for the repeating nanoflare case because the loop is never allowed to undergo full radiative cooling before being reheated. Many workers have used the emission measure slope to infer properties of the

underlying heating from the observed EM(Te). These studies are summarized in

Table 3.2. Notably, Warren et al. (2012) computed EM(Te) distributions for the cores of 15 different active regions using spectroscopic observations of 22 different

lines from Hinode/EIS and narrowband intensities from the 94 Å channel of AIA. Warren et al. found 2 a 4.8 and that shallower slopes coincided with lower   magnetic ﬂux. Additionally, Cargill (2014) systematically addressed the relationship between the frequency of energy deposition and the emission measure slope by 3.5 The Differential Emission Measure Distribution 98

computing a for a range of waiting times between consecutive nanoﬂare events

using the EBTEL loop model. Cargill found that a became large (& 8) for very short waiting times while a 2 for waiting times greater than the characteristic loop ⇡ cooling time, and that a scaling between the nanoﬂare energy and the waiting time was needed to reproduce the range of observed slopes. While the emission measure slope is a relatively simple and easily interpretable diagnostic, one should exercise caution when comparing slopes derived from observations to those from models as uncertainties in the atomic data can lead to conﬁdence intervals of at least 1 to 1.3

on a (Guennou et al., 2013).

The EM(Te) distribution “hotward” of Tpeak provides a potential diagnostic for the nanoﬂare model (Cargill, 1994; Cargill & Klimchuk, 2004) and a similar scaling

b to Equation 3.45, EM(Te) µ Te , has been claimed for Te > Tpeak (e.g. Warren et al., 2012). However, the amount of emission in this temperature range is not well

constrained by current observations (Winebarger et al., 2012). Furthermore, b is very sensitive to the temperature range over which it is calculated such that it is not a robust diagnostic for characterizing the amount of very hot plasma (Barnes et al., 2016a). I will defer a detailed discussion of the “hot” part of the emission measure distribution to Chapter 5.

3.5.2 Determining the DEM from Observations

Though a useful thermal diagnostic, deriving the differential emission measure distribution from observed spectral line or narrowband intensities is very difﬁcult and can yield ambiguous results. Equation 3.41 is an inhomogeneous Fredholm equation of the ﬁrst kind,

b g(t)= dsK(t, s) f (s), (3.47) Za 3.5 The Differential Emission Measure Distribution 99

Table 3.2 Summary of observational and modeling studies that have used the emission measure slope, a, as a diagnostic for the underlying energy deposition. The approximate range of observed slopes is 2 . a . 5. Adapted from Table 3 of Bradshaw et al. (2012).

Reference Type Slope (a) Temperature range [K] Warren et al. (2011) observation 3.26 106–106.6 model 2.17 Winebarger et al. (2011) observation 3.2 106–106.5 Tripathi et al. (2011) observation 2.08–2.47 105.5–106.55 2.05–2.7a b 6 c Mulu-Moore et al. (2011a) model 1.6–2 10 –Tpeak 2–2.3 Warren et al. (2012) observation 1.7–4.5 106–106.6 6 d Schmelz & Pathak (2012) observation 1.91–5.17 10 –Tpeak 6 e Bradshaw et al. (2012) model 0.81–2.56 10 –Tpeak 6 f Reep et al. (2013) model 0.88–4.56 10 –Tpeak 6.6g Cargill (2014) model 2–8 T0–10 Del Zanna et al. (2015b) observationh 4.4 0.4 106–3 106 ± ⇥ 4.6 0.4 ± a DEM(Te) computed from background-subtracted observations. b Intensities were modeled using photospheric (first row) and coronal (second row) abundances. c 6.6 6.8 Tpeak varied from 10 K to 10 K. d 6.3 6.8 Tpeak varied from 10 K to 10 K. e 5.85 7.35 Tpeak varied from 10 K to 10 K. f 6.35 6.65 Tpeak varied from 10 K to 10 K. g 6 6.25 a is computed for 12 different values of T0 between 10 and10 and averaged. h The slope was computed in every pixel of active region NOAA 11193 once when it first appeared (first row) and then again after one rotation (second row). where g(t) is some measured value, K(t, s) is the kernel, and f (s) is the unknown function to be determined (Press et al., 1992). Comparing with Equation 3.41, s and t correspond to the temperature and wavelength or channel, respectively. In principle,

solving Equation 3.47 to ﬁnd f requires inverting the kernel matrix provided K is invertible. However, this is complicated by the fact that information about f (or

in the case of Equation 3.41, DEM(Te)) is lost when f is “smoothed” by the kernel K such that solutions to f derived by inverting K will be extremely sensitive to 3.5 The Differential Emission Measure Distribution 100

uncertainties in the input g (Press et al., 1992). In applied mathematics, there exists a great deal of literature on solution methods for these so-called inverse problems, of which Equation 3.41 is a prime example. In solar astrophysics speciﬁcally, a great amount of effort has been focused on

methods for solving Equation 3.41 for the DEM(Te) ( f ) given a set of observed EUV and/or X-ray intensities (g) using contribution functions or detector responses (K) derived from atomic data. These methods include, but are not limited to, Gaussian, multi-Gaussian, and spline ﬁts via c2-minimization (e.g. Caspi et al., 2014; Guennou et al., 2012; Ryan et al., 2014; Warren et al., 2013), Markov Chain Monte Carlo (MCMC, Kashyap & Drake, 1998), regularized inversion (Hannah & Kontar, 2012; Plowman et al., 2013), sparse basis pursuit (Cheung et al., 2015), and sparse Bayesian inference (Warren et al., 2017). Many of these methods have been designed to work well with speciﬁc types of observations and, due to the lack of a unique solution

to Equation 3.41, tend to all give somewhat different DEM(Te) for the same set of inputs I (see comparison of 15 different methods by Aschwanden et al., 2015). In addition to the mathematical difﬁculties, uncertainties in the atomic data

(e.g. due to line blends, line misidentiﬁcation) make interpreting the DEM(Te) even more ambiguous though there has been recent progress in understanding how these uncertainties propagate through to plasma diagnostics (e.g. Del Zanna et al., 2019; Guennou et al., 2013; Yu et al., 2018). Additionally, ions in the Li and Na isoelectronic sequences are known to exhibit “anomalous behavior” such that when

lines of these ions are used to constrain the DEM(Te), the intensities are consistently overestimated or underestimated by up to a factor of 5 (Burton W. M. et al., 1971; Del Zanna et al., 2002; Dupree, 1972). It has been suggested that this behavior may be due to departures from ionization equilibrium or non-Maxwellian electron distributions (Del Zanna & Mason, 2018). Thus, when selecting observed lines to

constrain the DEM(Te), care should be taken to avoid these anomalous ions as well 3.5 The Differential Emission Measure Distribution 101

as those ions whose contribution functions have a strong density dependence (e.g. Fe IX, Del Zanna & Mason, 2018). It should be noted that several authors (Craig & Brown, 1976; Judge, 2010; Judge et al., 1997, 1995) claim the uncertainties in the atomic data and observations combined with the ill-posed nature of Equation 3.41 prevent a determination of a temperature distribution to within a degree of certainty that is of physical interest.

Emission Measure Loci

Though inverting Equation 3.41 to ﬁnd a full solution for the DEM(Te) is difﬁcult,

an upper bound on EM(Te) can be estimated using the emission measure loci method ﬁrst developed by Veck & Parkinson (1981). For an isothermal plasma at

temperature Tc, Equation 3.41 becomes,

2 I(lji)=Gji(Tc) dhne = Gji(Tc)EM. (3.48) ZLOS

Given this approximation, one can deﬁne a function called the emission measure loci, I(lji) EM(Te)loci = . (3.49) Gji(Te)

EM(Te)loci depends strongly on Te because of the temperature-dependence of the

contribution function, Gji(Te) (Equation 3.42). Note that EM(Tc)loci = EM. By plotting Equation 3.49 as a function of Te for multiple lines and ﬁnding where all

of the emission measure loci curves intersect, one can ﬁnd Tc, the temperature of

the isothermal plasma and EM. This is because for T = Tc, the right-hand side of Equation 3.49 should be equal for an isothermal plasma with uniform density. For a non-isothermal plasma, Equation 3.49 still provides an estimate for the upper

bound on EM(Te), but the EM(Te)loci will not all intersect at a single temperature (Phillips et al., 2008). 3.5 The Differential Emission Measure Distribution 102

Regularized Inversion

Despite the aforementioned difﬁculties, the differential emission measure distribution is a still a practical diagnostic for understanding how energy is deposited in the coronal plasma. The method developed by Hannah & Kontar (2012) allows for

efﬁcient determination of the DEM(Te) from AIA observations (or observed spectral line intensities) by solving Equation 3.41 through regularized inversion wherein a smoothing parameter is introduced to guarantee a well-behaved and unique solution to an otherwise ill-posed problem. In the work presented in this thesis, I will compute the differential emission measure distribution from both predicted and observed AIA intensities using the method of Hannah & Kontar (2012). I will now

describe this approach for computing the DEM(Te) and provide an example. Using the notation of Hannah & Kontar (2012), Equation 3.41 can be written in matrix form as,

g = Kx + dg, (3.50) where g is a vector of intensities for each EUV channel, dg are the associated uncertainties, K is the matrix of temperature response functions3 (see Section 3.4)

for each channel as a function of Te, and x is the differential emission measure as

a function of Te. Recovering x by simply inverting K is not possible due to noise ampliﬁcation from the uncertainties in the data dg. Additionally, this system may be under-determined if the number of temperature bins is greater than the number of detector channels. One common method for dealing with these difﬁculties is to add linear constraints to x in the form of zeroth-order regularization (e.g. Tikhonov, 1963). The regularized least squares problem for Equation 3.50 then becomes,

2 minimize K˜ x g˜ + l L(x x ) 2 , (3.51) k 0 k 3 Note that the contribution function has been replaced by the temperature response function in the kernel as g denotes a narrowband intensity rather than spectral line emission. The temperature response function is effectively the contribution function for the detector channel. 3.5 The Differential Emission Measure Distribution 103

2 where K˜ =(dg) 1K, g˜ =(dg) 1g, x = Â x2 is the `2-norm, L is the constraint k k i i matrix, l is the regularization parameter, and x0 is the initial guess of the solution. Solutions to Equation 3.51 are unique and well-behaved and can be expressed in terms of l. The exact value of l depends on the desired c2 of the solution and may need to be “tweaked” depending on the input data. An additional constraint may be added to ensure the positivity of x.

1024

] 23 1 10 K 5

1022 cm [

1021 DEM 94 Å 193 Å 1020 131 Å 211 Å 171 Å 335 Å 1019 106 107 Te [K] 10

Residuals 10 100 150 200 250 300 Channel [Å]

Figure 3.10 An example of the regularized inversion method of Hannah & Kontar (2012) for a simple model DEM(Te) and simulated AIA observations. The dashed, 6.5 gray line is the true DEM(Te), a single Gaussian pulse centered at 10 K, and the black error bars in Te and DEM(Te) denote the regularized solution. The true DEM(T ) has a total emission measure of 3.76 1022 cm 5 and spread of s = 0.15 e ⇥ in log Te. The colored curves as given in the legend are the emission measure loci curves for each AIA EUV channel. The lower panel shows the residuals between the true and recovered intensities for each channel. Adapted from Figure 3 of Hannah & Kontar (2012).

Figure 3.10 shows a model DEM(Te) consisting of a single Gaussian pulse centered on 106.5 K (dashed gray) and the regularized solution (black error bars) as 3.6 Time-Lag Analysis 104

computed from the model AIA intensities using the method of Hannah & Kontar (2012)4. The emission measure loci curves for each AIA channel are denoted by the colored curves. The lower panel shows the residuals between the true and recovered intensities for each channel. The uncertainties on the model intensities,

dg, are estimated using the error tables provided by the AIA instrument team in

SSW. Note that the regularized solution and the true DEM(Te) deviate most for >107 K and <7 105 K, where the AIA passbands are least sensitive (see Figure 3.8). ⇥ The vertical error bars represent the propagation of the uncertainties in the intensity through the regularized kernel matrix and are computed by taking the standard

deviation of multiple Monte Carlo samples of the regularized solution from g in the range of dg. The horizontal errors represent the best possible temperature resolution, or the temperature bias, of the method and will be larger the greater the degree of regularization. The temperature bias will also be worse if the uncertainties of the observed intensities are large.

3.6 Time-Lag Analysis

Besides the emission measure distribution, additional diagnostics are needed in order to place tighter constraints on coronal heating properties. The time-lag method of Viall & Klimchuk (2012) is a powerful and efﬁcient tool for understanding large- scale cooling patterns across an entire active region. In the following sections, I will outline the time-lag analysis technique and explain why it is useful in discerning the underlying thermodynamic behavior of the plasma from the observed narrowband intensity.

4An IDL implementation of the regularized inversion method of Hannah & Kontar (2012) has kindly been made publicly available by the authors: github.com/ianan/demreg 3.6 Time-Lag Analysis 105

3.6.1 Cross-Correlation

Broadly speaking, the time-lag method involves computing the cross-correlation between pairs of AIA light curves. The cross-correlation measures the similarity

between two signals as a function of the offset between them. Given two signals fA and f that are a function of time t, the cross-correlation, (t), can be written as, B CAB

• AB(t)=(fA ? fB)(t)= dtfA⇤ (t) fB(t + t), (3.52) C • Z where t is the temporal offset between fA and fB and fA⇤ denotes the complex conjugate of f . Alternatively, (t) can be expressed in terms of the convolution, A CAB

(t)= f ( t) f (t), (3.53) CAB A ⇤ B where now denotes the convolution operator. Taking the Fourier transform of ⇤ both sides of Equation 3.53 and using the convolution theorem (Arfken et al., 2013, Section 20.4) gives,

F (t) = F f ( t) f (t) , {CAB } { A ⇤ B } F (t) = F f ( t) F f (t) , {CAB } { A } { B } 1 (t)=F F f ( t) F f (t) , (3.54) CAB { { A } { B }} where F denotes the Fourier transform. Equation 3.54 deﬁnes the cross-correlation

(t) between two signals f and f as a function of temporal offset t in terms CAB A B of the Fourier transform. Deﬁning the cross-correlation in this manner allows one to efﬁciently compute (t) using highly-optimized fast Fourier transform (FFT) CAB algorithms (e.g. the widely-used FFT algorithm developed by Cooley & Tukey,

1965). Note that fA and fB are centered and scaled such that both signals have zero mean and unit standard deviation. Additionally, (t) is scaled by the CAB 3.6 Time-Lag Analysis 106

length of the signal such that the cross-correlation is always between 1 (perfectly anti-correlated) and +1 (perfectly correlated).

1.0 1.0 f1 f2 0.8 0.8 ]

0.6 0.6 12 C arb. unit [ 0.4 0.4 f

0.2 0.2

0.00 0.25 0.50 0.75 1.00 1.0 0.5 0.0 0.5 1.0 t [s] t [s]

Figure 3.11 The left panel shows two Gaussian signals f1 and f2 with peaks at 0.25 s (blue) and 0.75 s (orange), respectively. The right panel shows the cross- correlation (t) between f and f as a function of the offset t. The dotted black C12 1 2 line denotes t = 0 s. Note that (t) peaks at t = 0.5 s, the separation in t C12 between the peaks of f1 and f2.

A simple example of the cross-correlation is illustrated in Figure 3.11 for two

Gaussian pulses, f1 and f2. The right panel shows the two pulses as a function of t, with f1 and f2 peaking at 0.25 s and 0.75 s, respectively. The right panel shows the cross-correlation, (t), as computed by Equation 3.54. (t) peaks at t = 0.5 C12 C12 s, the separation in t between the peaks of f1 and f2. For a more conceptual understanding of (t), consider ﬁxing f and shifting f forward and backward C12 2 1 in t. The degree to which f1 and f2 overlap or the “similarity” between the two signals is analogous to (t). (t) peaks at t = 0.5 s because f and f are most C12 C12 1 2 overlapping when f1 is shifted by 0.5 s relative to its initial position. Throughout the rest of this thesis, I will refer to the value of t which maximizes the cross-correlation 3.6 Time-Lag Analysis 107

as the time lag. The time lag can be deﬁned formally as,

tAB = argmax AB(t). (3.55) t C

In this example, the time lag between f and f is t = 0.5 s. Note that for (t) 1 2 12 C21 (i.e. f and f reversed), the time lag would be t = 0.5 s as f would need to be 1 2 21 2 shifted 0.5 s to the left in order to maximize the similarity between the two signals. Thus, the order of the two signals in Equation 3.54 will determine the sign of the time lag.

3.6.2 Time Lag between AIA Channel Pairs

Because tAB measures the time delay between peaks in different signals and the temperature sensitivity of each AIA detector channel is approximately known (see

Figure 3.8), the time lag between a pair of signals in channels A and B provides a proxy for the loop plasma cooling time between the nominal temperatures of

channels A and B. For the six channels of interest, there are ﬁfteen possible channel pairs. Recall that changing the order of the channels only changes the sign of the time lag such that t = t . By computing the time lag between every possible AB BA channel pair, one can understand how the coronal plasma evolves through the passbands of the instrument and, by extension, how the temperature of the plasma is changing. An example for a set of simulated AIA light curves is shown in Figure 3.12.A loop of half-length 40 Mm cooling from 5 MK down to 0.2 MK is simulated ⇡ ⇡ with the EBTEL model (see Section 2.2.4) and intensities from each of the six AIA channels are computed from Equation 3.38 using the tabulated temperature response functions. The top panel shows the normalized light curve for each channel (left axis) and the electron temperature (right axis, black). Note that the intensity 3.6 Time-Lag Analysis 108

1.0 94 Å 5 131 Å 0.8 171 Å 4 c I 193 Å ] 0.6 211 Å 3 MK [

335 Å e / max T c 0.4 I 2

0.2 1

0.0 0 500 1000 1500 2000 2500 3000 t [s]

1.0 94,171 211,193 0.8 94,211 193,171 335,171 171,131 0.6

AB 0.4 C

0.2

0.0

0.2 2000 1000 0 1000 2000 t [s]

Figure 3.12 Top panel: Simulated light curves (normalized to the peak value) for the six EUV channels of AIA (left axis) and electron temperature (black line, right axis) for a loop of half-length 40 Mm cooling from 5 MK to 0.2 MK. The ⇡ ⇡ hydrodynamic evolution of the loop was simulated using the EBTEL model and the six light curves were computed using Te and ne from the simulation. Bottom panel: Cross-correlation as a function of temporal shift, t, computed from the light curves shown in the top panel for six selected channel pairs. The dotted black line indicates a temporal shift of t = 0s. The dotted lines and dots at the peak of each curve denote the time lag for that channel pair. peaks in successively “cooler” channels as the loop cools into and out of the temperature bandpass of each channel (see Figure 3.8). The bottom panel shows the cross-correlation as a function of temporal offset, t, for six different channel pairs.

The time lag, tAB, for each cross-correlation is denoted by the dotted line and the dot at the peak of the curve. The 94,171 Å pair has the longest time lag, >2 103 s, ⇥ 3.6 Time-Lag Analysis 109

because the intensity of the 94 Å channel, which is most sensitive to 8 MK plasma, ⇠ peaks at t = 0swhile the intensity of the 171 Å channel, which is most sensitive to 1 MK plasma, does not peak until t 2.4 103 s. Conversely, the 171,131 ⇠ ⇡ ⇥ Å channel pair has the shortest time lag because the 171 Å temperature response function and the cool component of the 131 Å temperature response function are heavily overlapping (see Figure 3.8). Note that for a loop which is purely cooling, as is the case in Figure 3.12, all of the time lags are positive. Using the convention of Viall & Klimchuk (2012), the “hotter” channel is ﬁrst and the “cooler” channel is second in the channel pair such

that a positive time lag implies cooling plasma. In other words, if the hot emission precedes the cooler emission, the hotter light curve will have to be shifted in the positive direction in t to maximize (t). However, negative time lags may also CAB be produced by cooling plasma if a channel is sensitive to both hot and cool plasma

(i.e double-peaked in Te, see Figure 3.8). Additionally, note that a zero time lag,

tAB = 0s, does not imply a steady light curve or a steady Te; it only implies that variability in channels A and B is coincident (Viall & Klimchuk, 2015, 2016). The time-lag analysis is a powerful technique and has been applied in a number of observation and modeling studies (e.g. Froment et al., 2017; Lionello et al., 2016; Winebarger et al., 2018, 2016). Using 24 hours of AIA observations of active region NOAA 11082, Viall & Klimchuk (2012) computed time lags in every pixel of the

active region for all fifteen channel pairs. By computing tAB in each pixel, they built up a map of the cooling pattern across the entire active region and found persistent positive time lags in all channel pairs, indicative of cooling plasma. Viall & Klimchuk (2017) extended this same technique to the catalogue of active regions compiled by Warren et al. (2012) and found similar results for all fifteen active regions. Additionally, Bradshaw & Viall (2016) applied the time lag analysis to a set of forward-modeled AIA intensities for a range of different heating models and 3.6 Time-Lag Analysis 110 found that their high- and intermediate-frequency nanoflare simulations were most consistent with observations, suggestive of a range of nanoflare heating frequencies. Throughout this thesis, I will make frequent use of both the emission measure distribution and time-lag analyses. In Chapter 5, I predict the emission measure distribution for several nanoflare heating scenarios in order to determine the observability of EM(Te) for Te > Tpeak. Additionally, in Chapter 6, I predict AIA intensities over a whole active region for a range of nanoflare heating frequencies and compute the emission measure slope and time lag. Then in Chapter 7, I classify slopes and time lags calculated from real observations in terms of the heating frequency using machine learning models trained on the predicted diagnostics. Chapter 4

synthesizAR: A Framework for Modeling Optically-thin Emission

In order to accurately predict observed optically-thin emission from the impulsively- heated coronal plasma, one must properly account for the field-aligned hydrodynamic response to the energy deposition (Chapter 2), the detailed atomic physics that produces the radiation (Chapter 3), and geometric effects due to the integration along the LOS. During the course of my PhD, I have a developed a software framework for modeling optically-thin coronal emission called synthesizAR1. syn- thesizAR includes tools for extrapolating the three-dimensional magnetic field from observed LOS magnetograms, configuring input and reading output from ensem- bles of field-aligned hydrodynamic simulations, and computing projections of the emission along the LOS for arbitrary viewing angles. synthesizAR is written entirely in the widely-used and open-source Python programming language and developed openly on GitHub2. synthesizAR is built

1The name derives from the words “synthesize” and the common abbreviation for active region, “AR” as the code was built to predict, or synthesize, emission from active regions. synthesizAR is also a homophone of synthesizer. 2The entire source code, including installation instructions and links to documentation, can be found at github.com/wtbarnes/synthesizAR

111 4.1 Building the Magnetic Skeleton 112

on top of many of the packages in the mature scientific Python ecosystem, including scipy (Jones et al., 2001), NumPy(Oliphant, 2006), and Astropy (The Astropy Collaboration et al., 2018). In particular, synthesizAR depends heavily on SunPy (SunPy Community et al., 2015) for manipulating imaging data and solar coordinate transformations. The code is fully-documented, including examples, and also includes a test suite that is executed at every code check-in. In this chapter, I give a detailed description of the software using an example workflow, including working code examples throughout, for a simple dipolar active region in hydrostatic equilibrium. In Chapter 6, I use the synthesizAR code coupled with the two-fluid EBTEL code (see Section 2.2.4) to model time-dependent, multi-wavelength emission from an active region for a range of nanoflare heating frequencies. While similar approaches have been used in the past to model optically-thin coronal X-ray and EUV emission (e.g. Allred et al., 2018; Bradshaw & Viall, 2016; Lundquist et al., 2008a,b; Schrijver et al., 2004; Warren & Winebarger, 2006), the synthesizAR package represents the first effort to organize this forward modeling pipeline into an openly-developed and easily-configurable codebase that leverages the high-quality software developed by the greater scientific Python and astronomy communities.

4.1 Building the Magnetic Skeleton

The ﬁrst step in the forward modeling pipeline is to determine the three-dimensional geometry of each magnetic strand in order to construct the magnetic “skeleton” of the model active region. As noted in Chapter 2, a ﬁeld-aligned hydrodynamic model only computes the evolution of the plasma along a single thermally-isolated strand such that the three-dimensional position and orientation of the strand relative to the solar surface must be imposed externally. In this example, I will model the emission 4.1 Building the Magnetic Skeleton 113

from active region NOAA 12733 as observed by SDO/AIA on 2019 January 24. Figure 4.1 shows the LOS magnetogram as observed by SDO/HMI on 2019 January 24 14:00:22 UTC.

102 0 102 00 0 eipoetv Latitude Helioprojective . 300 00 0 . 200 00 0 . 100

100.000 0.0 100.0 00 00 Helioprojective Longitude

Figure 4.1 HMI LOS magnetogram observed on 2019 January 24 14:00:22 UTC. The full-disk observation is shown on the left and the zoomed-in view of NOAA 12733 is shown on the right. In both panels, the colorbar is on a symlog scale from 750 G to 750 G. Note that at the time of this observation, the active region was close to the center of the disk.

4.1.1 Potential Field Extrapolation

As discussed in Section 1.2.3, a potential field extrapolation provides a reasonable approximation of the lowest energy configuration of the coronal magnetic field and can be computed relatively efficiently given an input LOS photospheric magnetogram as the lower boundary. On the scale of a single active region, the curvature of the solar surface can be ignored and the problem reduces to solving Laplace’s

equation (Equation 1.14) on a Cartesian grid, with the z-axis pointed toward the 4.1 Building the Magnetic Skeleton 114

observer, given the boundary conditions,

n f = B (x, y,0), (4.1) ·r z f(r) 0 as r •, (4.2) ! !

where n is the unit vector in the z-direction and Bz(x, y,0) is the z-component of the magnetic ﬁeld at the photosphere given by the observed LOS magnetogram (Sakurai, 1982). Following the method of Schmidt (1964) as outlined by Sakurai (1982), the

Green’s function, G, for this problem must satisfy the conditions,

2 G(r, r0)=0, r

G(r, r0) 0 as r r0 •, ! | |!

n G(r, r0)=0, ·r where r0 =(x0, y0,0) is the position on the z = 0 plane and the gradient is taken with respect to r. Integrating this Green’s function over the boundary weighted by the observed ﬁeld strength gives a solution for f,

f(r)= dx0 dy0 Bz(r0)G(r, r0), (4.3) Z

and the magnetic ﬁeld is computed trivially from Equation 1.13 using a ﬁnite- difference scheme. The Green’s function in Equation 4.3 has the form,

1 G = . (4.4) 2p r r | 0|

Schmidt (1964) suggest an “oblique” correction to G for cases where the active region is far from disk center and n, the surface normal, is not aligned with `, the 4.1 Building the Magnetic Skeleton 115

unit vector pointing toward the observer. In this case, the modiﬁed Green’s function is, 1 n ` (` (n `)) R G = · + ⇥ ⇥ · (4.5) 2p R R(R + ` R)  · where R = r r . Note that if ` = n, Equation 4.5 reduces to Equation 4.4. The 0 second term in Equation 4.5 corresponds to a line of dipoles extending out to z • ! in order to compensate for a reduction in the ﬁrst term due to the projection of n on ` (Sakurai, 1982). The oblique extrapolation method of Schmidt (1964) is implemented in synthe-

sizAR and can easily be used to compute B above an active region, from synthesizAR.extrapolate import PotentialField m_ar_resampled = m_ar.resample([100, 100] * u.pixel) n_z = 100 * u.pixel w_z = 215 * u.Mm extrapolator = PotentialField(m_ar_resampled, w_z, n_z) B_field = extrapolator.extrapolate()

m_ar is a Map object provided by SunPy that includes both the data and associated metadata of the HMI LOS magnetogram cropped to the area around NOAA 12733 as shown in the right panel of Figure 4.1. The magnetogram is resampled to a lower resolution to reduce the computational cost of the ﬁeld extrapolation. The method

call in the last line computes B using the Green’s function in Equation 4.5 and the observed magnetogram and returns an object containing a 100 100 100 array for ⇥ ⇥ each component of B. Figure 4.2 shows a slice along the y-axis through the center

of the extrapolated volume of Bx. Note that both the shape and width of the z-dimension in the above code

example have units attached to them using the units module, denoted by u,in the Astropy package. The units module provides unit-aware versions of scalar and array quantities as well as the appropriate transformations between units. All inputs and outputs in synthesizAR which correspond to physical quantities must have units attached. This requirement helps to avoid simple unit conversion errors 4.1 Building the Magnetic Skeleton 116

200 102 175

150

125 0 [Mm] 100 z

50 102 25

0 100 50 0 50 100 x [Mm]

Figure 4.2 A slice through the center of the extrapolated volume of Bx along the y-axis. The black streamlines indicate the Bx and Bz components of the ﬁeld at this slice. The colorbar is on a log scale and ranges from 750 G to 750 G.

and allows inputs to be more ﬂexible. The following code snippet shows an example of some of the capabilities of the units module, (4e6*u.K).to(u.MK) #convertfromKtoMK 10*u.cm/u.s #unitscanbecombined 1*u.s + 0.5*u.hour #addcompatibleunits

4.1.2 Tracing Magnetic Field Lines

Now that the vector magnetic ﬁeld has been computed from the observed LOS magnetogram, the next step in the pipeline is to trace magnetic ﬁeld lines through the extrapolated volume in order to obtain three-dimensional coordinates for each of the magnetic strands. synthesizAR wraps the low-level streamline tracing capabilities of yt, a Python package for manipulating large, high-dimensional data sets

(Turk et al., 2011). This is encapsulated in a method attached to the PotentialField object instantiated above, 4.1 Building the Magnetic Skeleton 117

strands = extrapolator.trace_fieldlines( B_field, 200,loop_length_range=[20,250]*u.Mm)

This traces 200 field lines through the extrapolated volume and returns a list of field strengths and three-dimensional coordinates along the strand for each of the 200 traced field lines. By default, synthesizAR places seed points for the field line tracing on the lower boundary in areas of relatively high magnetic field strength though this parameter can be adjusted. Additionally, only strands whose footpoints are both connected to the surface (closed field) and whose loop lengths fall within the specified range (as shown above) are kept. The last step is to create the magnetic “skeleton” object from the traced field lines, from synthesizAR import Field noaa12733 = Field(m_ar_resampled, strands)

The Field object holds all of the information about each of the magnetic strands. When the Field object is created, a Loop object is created for each strand. These are accessible through the .loops attribute on the Field object. Each Loop object holds several attributes related to the properties of that strand noaa12733.loops #listofallstrands noaa12733.loops[0] #accessthefirststrand noaa12733.loops[0].coordinates #accessthe3Dpositions noaa12733.loops[0].full_length #footpoint-to-footpointlength noaa12733.loops[0].field_strength #|B|alongthestreamline

This construction provides an intuitive interface for investigating the properties of the active region. Figure 4.3 shows all 200 strands traced from the extrapolated volume overlaid on the observed magnetogram of NOAA 12733.

4.1.3 Aside: Coordinate Systems in Solar Physics

The extrapolated magnetic ﬁeld is represented in a Cartesian coordinate system where the origin is at the center of the active region, the z-axis is normal to the 4.1 Building the Magnetic Skeleton 118

300.000

200.000 Helioprojective Latitude

100.000

100.000 0.0 100.0 00 00 Helioprojective Longitude

Figure 4.3 LOS magnetogram of NOAA 12733 as observed by SDO/HMI. All 200 traced ﬁeld lines are overlaid in black.

surface, and the y-axis points toward solar north. The coordinates in this local active region frame can be deﬁned in terms of the Heliocentric Earth equatorial (HEEQ) Cartesian coordinate system of Hapgood (1992),

AR xHEEQ zlocal xHEEQ 0 1 0 1 0 AR 1 y = Rz(FAR)Ry( QAR) x + y , (4.6) HEEQ local HEEQ B C B C B C B C B C B AR C BzHEEQC BylocalC BzHEEQC B C B C B C @ A @ A @ A where xlocal, ylocal, zlocal are the coordinates in the local active region frame, xHEEQ,

yHEEQ, zHEEQ are the coordinates in the HEEQ frame, Ry,z are the rotation matrices

about the y and z axes, FAR and QAR are the longitude and latitude of the center of AR AR AR the active region, respectively, and xHEEQ, yHEEQ, zHEEQ are the coordinates of the center of the active region in the HEEQ frame. 4.1 Building the Magnetic Skeleton 119

The HEEQ Cartesian coordinate system is defined such that the z-axis is the solar rotation axis and increases towards solar north, and the x-axis points from the center of the Sun to the intersection between the solar equator and the central meridian as seen from Earth. The longitude and latitude in Equation 4.6 are defined in the Stonyhurst heliographic (HGS) coordinate system which can be defined in HEEQ coordinates as,

2 2 2 r = xHEEQ + yHEEQ + zHEEQ, (4.7) q z Q = arctan HEEQ , (4.8) 0 2 2 1 xHEEQ + yHEEQ @yq A F = arctan HEEQ , (4.9) x ✓ HEEQ ◆ where r is the radial distance, measured in units of physical distance, from the center of the Sun and F and Q are the angles of longitude and latitude, respectively, as measured from origin (Thompson, 2006). Note that both the HEEQ and HGS coordinate systems are deﬁned such that they remained ﬁxed with respect to the Earth. Though the HGS frame provides an intuitive way for representing coordinates on the Sun, real data represent projections of the Sun on the celestial sphere such

that the data must be represented in a projected coordinate system deﬁned by the position of the observer (e.g. a satellite, Earth). Following Thompson (2006), the helioprojective (HPC) coordinate system is deﬁned in terms of the longitude and latitude on the celestial sphere centered on the Sun-observer line. The transformation between the HPC frame and the heliocentric cartesian (HCC) coordinate frame is given by,

d = x2 + y2 +(D z)2, (4.10) q 4.1 Building the Magnetic Skeleton 120

Figure 4.4 Helioprojective coordinate system (black) overlaid on a Stonyhurst heliographic coordinate system (blue) as viewed by an observer at (0°, 20°, 1 AU). The spacing between the lines of HGS longitude and latitude is 10° and the spacing between the lines of HPC longitude and latitude is 25000.

x q = arctan , (4.11) x D z ✓ ◆ y q = arcsin , (4.12) y d ⇣ ⌘ where d is the radial distance from the observer, D is the distance between the observer and the center of the Sun, and x, y, z are the coordinates in the HCC frame. The z-axis of the HCC frame is oriented along the Sun-observer line and the y-axis points toward solar north. The transformation between the HGS and HCC frame is given by,

x = r cos Q sin (F F ), (4.13) 0 y = r (sin Q cos B cos Q cos (F F ) sin B ) , (4.14) 0 0 0 4.1 Building the Magnetic Skeleton 121

z = r (sin Q sin B + cos Q cos (F F ) cos B ) , (4.15) 0 0 0

where F0 and B0 are the HGS longitude and latitude of the observer. Note that unlike the HGS frame, the HPC and HCC frames are deﬁned in terms of an ob-

server at a particular location in HGS coordinates, (F0, B0, D ). Figure 4.4 shows a comparison between the HGS frame and a HPC frame deﬁned by an observer at

F0 = 0°, B0 = 20°, and distance of D = 1 AU. After the ﬁeld lines are traced through the extrapolated volume, synthesizAR transforms the coordinates into an HEEQ coordinate frame (using Equation 4.6)

and returns an Astropy SkyCoord object for each strand. These are stored in the .coordinates attributed of each Loop object in the magnetic skeleton. A SkyCoord object stores the numerical values of the coordinates as well as the frame of reference in which the coordinates are deﬁned such that coordinates can easily be represented in different frames. SunPy provides representations of and transformations between many of the common solar coordinate systems as described by Thompson (2006).

For example, to deﬁne a point p on the surface of the Sun at 20° longitude and 20° latitude and then transform it to a HPC coordinate system deﬁned by an observer on Earth,

from sunpy.coordinates import Helioprojective p = SkyCoord(lon=20*u.deg, lat=20*u.deg, radius=const.R_sun, frame='heliographic_stonyhurst') hpc_frame = Helioprojective( observer='Earth',obstime=astropy.time.Time.now()) p_hpc = p.transform_to(hpc_frame)

Representing the coordinates of each strand in this way makes it simple to compute projections of the magnetic skeleton for any viewing angle. For example, in Fig- ure 4.3, the coordinates of each strand are transformed to a HPC coordinate system deﬁned by an observer at the position of the SDO spacecraft on 2019 January 24 14:00:22 UTC. Note that the coordinates of the magnetic strands do not necessarily 4.2 Field-Aligned Modeling 122

have to be determined by a magnetic ﬁeld extrapolation described in Section 4.1.1.

As long as the coordinates can be expressed in a HGS frame using a SkyCoord object, any method (e.g. a non-linear force-free ﬁeld extrapolation, see Section 1.2.3) may be used to model the magnetic skeleton of the active region.

4.2 Field-Aligned Modeling

After constructing the magnetic skeleton of the active region, the next step in the forward modeling pipeline is to determine the thermal structure of each strand. synthesizAR uses a ﬂexible “plug-in” system for specifying how the properties of the strand, such as the loop length and the magnetic ﬁeld strength, are used to map thermodynamic quantities to the structure of the strand. Several different model

interfaces are provided by synthesizAR or an interface can be deﬁned by the user and passed to the framework without having to modify the synthesizAR code itself. For example, the following minimal interface maps a single temperature of 2 MK,

9 3 1 density of 10 cm , and velocity of 0 cm s to every coordinate of every loop in the active region,

class ExampleInterface(object): def load_results(self,loop): t = u.Quantity([0,], 's') T = 2*u.MK * np.ones( t.shape+loop.field_aligned_coordinate.shape) n = 1e9*u.cm**(-3) * np.ones(T.shape) v = 0*u.cm/u.s * np.ones(T.shape) return t, T, T, n, v

While not very useful, this is a valid model interface. An interface is deﬁned as a

Python class with a method called load_results that accepts a Loop object as the single argument. This method then must return the time, electron temperature, ion temperature, density, and velocity as a function of time and coordinate along the ﬁeld line. 4.2 Field-Aligned Modeling 123

In this example workﬂow, the thermal structure of each loop is determined using the scaling laws of Martens (2010, see Section 2.1.3). An interface to the Martens scaling laws is included in the synthesizAR package. To deﬁne this interface and apply it to each strand in the magnetic skeleton,

from synthesizAR.interfaces import MartensInterface martens = MartensInterface() noaa12733.load_loop_simulations(martens, fn)

The maximum temperature of each strand is determined using the scaling laws of Rosner et al. (1978) (see Equation 2.17) and the base temperature of each strand

is fixed at 104 K. fn specifies the name of the output file for the interface model results. Storing these results on disk rather than in memory is advantageous when constructing an active region with many thousands of loops. After applying the model to each strand, the thermodynamic properties, as a function of loop coordinate and time, can be accessed as attributes on each strand in the active region,

noaa12733.loops[0].time noaa12733.loops[0].electron_temperature noaa12733.loops[0].ion_temperature noaa12733.loops[0].density

Figure 4.5 shows the temperature (left panel) and density (right panel) as a function of loop coordinate for all 200 strands in the active region as determined by the Martens scaling laws. Note that it is assumed that the each loop is semi-circular and symmetric about the apex of the strand as the Martens scaling laws only model the thermal structure of half of the loop. synthesizAR also includes an interface to the EBTEL model (see Section 2.2.4) which I will use in Chapter 6 to model the hydrodynamic evolution of many strands in an active region heated by nanoﬂares. 4.3 Atomic Physics 124

2.0

1011 1.5 ] ] 3

MK 10 [ cm

1.0 [ T n

109 0.5

108 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 s/L s/L

Figure 4.5 Temperature (left) and density (right) as a function of ﬁeld-aligned coordinate, s, normalized to the loop length, L, for all 200 strands in the model active region as determined by the scaling laws of Martens (2010).

4.3 Atomic Physics

To correctly model the optically-thin EUV and X-ray emission, synthesizAR uses the CHIANTI atomic database (see Section 3.1) to model the emissivity from all transitions of selected ions for each strand in the magnetic skeleton using Equa-

tion 3.5. The ﬁrst step is to construct an EmissionModel that includes the relevant ions,

from fiasco import Ion T = np.logspace(4, 8, 80) * u.K kws = { 'abundance_filename': 'sun_coronal_1992_feldman' } ions = [ Ion('Fe 8',T,**kws), Ion('Fe 14',T,**kws), Ion('Fe 16',T,**kws), Ion('Fe 18',T,**kws), ] n = np.logspace(8, 11, 15) * u.cm**(-3) from synthesizAR.atomic import EmissionModel em_model = EmissionModel(n, *ions) 4.3 Atomic Physics 125

For this example workﬂow, I model only a few of the dominant ions in the EUV passbands of AIA: Fe VIII, XIV, XVI, and XVIII. In practice, it is better to include as many ions as possible though this may signiﬁcantly increase compute times.

The ions are speciﬁed as Ion objects from the ﬁasco library, a Python interface to the CHIANTI atomic database. Appendix A provides a detailed description of the

fiasco package. The ions are then passed to the EmissionModel, a subclass of the fiasco IonCollection object (see Section A.4). The T and n variables define the grid of temperature and density over which the emissivity is calculated.

After constructing the EmissionModel, the population fraction of each ion speci- ﬁed above is computed for each loop,

em_model.calculate_ionization_fraction(noaa12733, fn_ionfrac)

The .calculate_ionization_fraction method iterates over each strand in the magnetic skeleton and uses the temperature and density as determined by the ﬁeld-aligned model (Section 4.2) to compute the population fraction of each ion as a function of the ﬁeld-aligned coordinate of the strand and stores it in the

ﬁle fn_ionfrac. By default, the population fractions are computed assuming ionization equilibrium (see Section 3.2.4) though additional arguments can be passed to .calculate_ionization_fraction to account for time-dependent, out- of-equilibrium charge states (see Section 3.2.5). Finally, the emissivity is computed for each transition of each ion speciﬁed above,

em_model.calculate_emissivity(fn_emiss)

Calculating the level population using Equation 3.18 is computationally expensive, particular for ions with many known transitions. .calculate_emissivity computes the product Nj Aji for each transition of each ion over the temperature-density grid deﬁned by T and n and stores it as a look-up table in the ﬁle fn_emiss. The 4.4 Instrument Effects 126

emissivity as given by Equation 3.5 can then be efﬁciently calculated for any loop

and transition by looking up the precomputed values for fX,k and Nj Aji.

4.4 Instrument Effects

4.4.1 Constructing the Virtual Observer

Now that the magnetic skeleton has been constructed, the thermal structure calculated, and the emissivities tabulated for the relevant ions, the last step in the synthesizAR pipeline is to account for the observing instrument. As in previous steps, the instrument is deﬁned in an object-oriented manner and then passed in as an argument,

from synthesizAR.instruments import InstrumentSDOAIA aia = InstrumentSDOAIA( [0,1]*u.s, noaa12733.magnetogram.observer_coordinate) from synthesizAR import Observer observer = Observer(noaa12733, [aia],)

The InstrumentSDOAIA object deﬁnes all of the properties of the instrument, including the temporal cadence, the spatial resolution, and the wavelength and temperature response functions for all six of the EUV channels of the AIA instru-

ment (see Section 3.4). When an instance of InstrumentSDOAIA is created, it accepts two arguments: the range of the observing time and the position of the observer. Because the plasma is assumed to be in hydrostatic equilibrium, the active region

only needs to be observed at t = 0s. The position of this observer is speciﬁed in

HGS coordinates as a SkyCoord object (see Section 4.1.3) and in this case is the same as the magnetogram shown in Figure 4.1: the position of the SDO spacecraft 2019 January 24 14:00:22 UTC.

This instrument object is then passed to the Observer object along with the Field object which contains the magnetic skeleton and information about the ther- 4.4 Instrument Effects 127

mal structure along each strand. Notice that the instrument object is passed in as

a list such that multiple virtual instruments can be passed to the same Observer in order to predict observations of the same active region using multiple instruments. Though synthesizAR includes several predefined instruments, as in the case of the field-aligned model interfaces (see Section 4.2), a user can define their

own instrument and pass it to the Observer without modifying the underlying package code. To maximize code reuse and make it easier to deﬁne custom in-

strument classes, synthesizAR also provides an InstrumentBase class from which user-deﬁned instruments can inherit.

4.4.2 Projecting Along the LOS

Next, the wavelength response functions (see Figure 3.7) defined in the instrument class are used to evaluate the integrand of Equation 3.38 for each channel of the instrument and each strand in the magnetic skeleton. The summation in Equa- tion 3.40 is over all available transitions from CHIANTI for the four ions added to the emission model in Section 4.3 and the contribution from the continuum is not included. This quantity is computed as a function of field-aligned coordinate along the strand and time (for time-dependent loop models). To compute the integration along the LOS in Equation 3.38 for each pixel, the coordinates of each strand are first converted to the HPC frame defined by the observer passed to the constructor of the instrument class. Then, the coordinates of all the strands are binned into a two-dimensional histogram where the width of each bin, or pixel, is equal to the angular resolution of the detector and the range of the bins is dependent on the extent of the magnetic strands in the HPC frame defined by the observer. An illustration of the projection of the coordinates into the histogram bins for a few sample strands is shown in Figure 4.6. 4.4 Instrument Effects 128

Figure 4.6 Coordinates for a sample of ten strands binned into a histogram deﬁned by the HPC frame of the speciﬁed observer. The actual coordinates of the strands are shown in black and the unweighted binned values are shown in blue. The grid lines denote the edges of the bins. The resolution has been reduced to 1 ⇡ 20 00 pixel in both coordinates for illustrative purposes.

Each coordinate is weighted by the integrand of Equation 3.38 at that coordinate

multiplied by the differential in the distance along the LOS, Dd (see Equation 4.10), such that Equation 3.38, the intensity in each pixel p is approximated by,

2 Ip,c Â Ddni Kc(Ti, ni), (4.16) ⇡ i { } where c denotes the channel, i denotes the set of all coordinates that fall into { } pixel p, Kc is given by the second term in Equation 3.40, and Ti and ni denote the temperature and density at coordinate i. This is carried out for each channel and for each time step. A Gaussian ﬁlter is applied to each histogram to emulate the point spread function (PSF) of the instrument which accounts for scatter by the mirrors 4.4 Instrument Effects 129

in the telescope, pixelization, and diffraction due to the ﬁlters on each telescope (Grigis et al., 2012). The approximate Gaussian width of the PSF for each of the EUV channels on AIA is given in Table 3 of Grigis et al. (2012). In the synthesizAR pipeline, the above steps are encapsulated in two method calls,

observer.flatten_detector_counts(emission_model=em_model) observer.bin_detector_counts(dir_results)

2 The first line computes n Kc (Equation 4.16) at every coordinate of every strand at each time step and “flattens” it to a single array of dimension n Â n , where t ⇥ s s ns is the total number of coordinates in strand s and nt is the total number of time steps at the cadence of the instrument (12 s for AIA), for each channel. Each strand is interpolated in time to match the cadence of the instrument and interpolated in space along the field-aligned direction such that,

Ds . D min (Dqx, Dqy), (4.17)

where Ds is the spatial resolution of the strand and Dqx,y are the spatial resolution of the instrument in the directions of HPC longitude and latitude. This condition ensures that the image is not “patchy” and is a consequence of the discrete points traced through the extrapolated volume in Section 4.1.2. These results, along with the “ﬂattened” coordinates in the HGS frame, are stored on-disk as an intermediate result. If no emission model is passed to this method, the tabulated temperature response functions interpolated to the temperature at each coordinate are used to

evaluate Kc. The second line in the above code sample computes the weighted histogram according to the procedure above using the interpolated values produced in the ﬁrst line for each time step and each channel. The resulting Gaussian-ﬁltered image is then saved in the Flexible Image Transport System (FITS) format (Wells 4.4 Instrument Effects 130

et al., 1981), the standard file format in astronomy and solar physics for storing image data. Additionally, the header of each FITS file includes a description of the helioprojective frame in which the image is defined using the standard keywords defined by the World Coordinate System (Greisen & Calabretta, 2002). Because synthesizAR follows the same formatting conventions as real observations, data products from synthesizAR are portable and can be treated in the exact same manner as observational data. Figure 4.7 shows the final output of the synthesizAR forward modeling pipeline as applied to NOAA 12733. Each panel shows a map of the intensity across the active region for four out of the six EUV channels of AIA. Since the thermal structure of each strand was modeled using the hydrostatic scaling laws of Martens (2010), only one map is produced per channel. Note that in the top right panel, the intensity is greatest near the base of each loop because the 131 Å channel is most sensitive to cool plasma (Figure 3.8) and the temperature along each strand decreases with decreasing height (Figure 4.5). In the other three channels, the core is much brighter as 94 Å, 211 Å and 335 Å are all sensitive to much warmer plasma. Because the coronal emission is optically-thin, an important consideration when interpreting observations is the angle between the observer and the object of interest as the temperature structure along the LOS will change depending on the viewing angle. Because synthesizAR defines the projection in terms of a HPC frame, chang-

ing the viewing angle is only a matter of specifying a different SkyCoord object for the observer. For example, to view the active region off-limb rather than on-disk, the HGS longitude is changed to F = 90°, new_observer = SkyCoord( lon=-90*u.deg, lat=noaa12733.magnetogram.observer_coordinate.lat, radius=noaa12733.magnetogram.observer_coordinate.radius, frame=HeliographicStonyhurst ) 4.4 Instrument Effects 131

94 A˚ 131 A˚ 00

0 211 A˚ 335 A˚ . 250 00 0 . 200 00 0 . 150 Helioprojective Latitude

50.000 0.0 50.0 00 00 Helioprojective Longitude

1 1 Figure 4.7 Predicted intensities, in DN pixel s , for active region NOAA 12733 as observed by four out of the six AIA EUV channels: 94 Å, 131 Å, 211 Å and 335 Å. The colorbar is on an arcsinh scale from 0 to the maximum intensity in that channel and the color tables are the standard AIA color tables as deﬁned in SunPy. The coordinate frame of each map is a helioprojective coordinate system deﬁned by an observer at the location of the SDO satellite on 2019 January 24 14:00:22 UTC.

Then, this new observer is passed to the instrument object,

new_aia = InstrumentSDOAIA([0,1]*u.s, new_observer)

Note that the mathematics of the projection are entirely encapsulated in the coordinate transformation to the HPC frame deﬁned by the observer. By deﬁning the positions of each strand as coordinate-aware SkyCoord objects, synthesizAR 4.4 Instrument Effects 132 provides an intuitive interface to exploring how the observed emission varies due to the changing temperature structure along the LOS. Chapter 5

Inferring Heating Properties of “Hot” Plasmas in Active Region Cores

In this chapter, I discuss the presence of “very hot” plasma as a possible signature of impulsive nanoﬂare heating in the corona. Using the two-ﬂuid EBTEL model, I simulate the differential emission measure distribution for a range of heating properties and discuss several effects which can limit the observability of high- temperature (8 MK to 10 MK) plasma. While only the evolution of a single loop is considered here, Chapter 6 takes advantage of the forward modeling framework discussed in Chapter 4 to simulate many thousands of loops. This chapter is adapted directly from Barnes et al. (2016b).

5.1 Introduction

Observations of the magnetically-closed solar corona from the Hinode (Kosugi et al., 2007) and Solar Dynamics Observatory (SDO, Pesnell et al., 2012) spacecraft have led, for the ﬁrst time, to quantitative studies of the distribution of coronal plasma as a function of temperature, and preliminary deductions about the heating process

133 5.1 Introduction 134

(see papers in De Moortel & Browning, 2015). The key to this has been the ability to make measurements of the corona over a wide range of temperatures using the EUV Imaging Spectrometer (EIS, Culhane et al., 2007) and X-Ray Telescope (XRT,

Golub et al., 2007) instruments on Hinode, as well as the Atmospheric Imaging Assembly (AIA, Lemen et al., 2012) on SDO. Underpinning this work is the concept of nanoflare heating of the corona. Nanoflares (e.g. Parker, 1988, see Section 1.3.2) are small bursts of energy release, which, despite the implication in their name, have unknown magnitude and duration. While commonly attributed to small-scale magnetic reconnection, nanoflares can occur in other heating scenarios (e.g. Ofman et al., 1998). One example of this approach has been studies of active region core loops (Brad- shaw et al., 2012; Del Zanna et al., 2015b; Reep et al., 2013; Schmelz & Pathak, 2012; Tripathi et al., 2011; Warren et al., 2011, 2012; Winebarger et al., 2011). These are the brightest structures in active regions, spanning the magnetic polarity line, and are observed over a wide range of temperatures. An important result has been the determination of the emission measure distribution as a function of tempera-

ture (EM(T) n2dh) along a line-of-sight (LOS). These workers showed that the ⇠ 6.5 6.6 emission measure peaked at T = Tpeak =10 K to 10 K with EM(Tpeak) of order 27 5 28 5 a 10 cm to 10 cm . Below Tpeak a relation of the form EM(T) µ T was found, with 2 . a . 5, where a is the so-called emission measure slope (see Section 3.5.1). This distribution can be understood by a combination of radiative cooling of the corona to space and an enthalpy flux to the transition region (TR) (e.g. Bradshaw & Cargill, 2010a,b) and and has significant implications for nanoflare heating. Defin- ing low- and high-frequency (LF and HF, respectively) nanoflares by the ratio of

the average time between nanoflares on a magnetic strand or sub-loop ( t ) to h waiti the plasma cooling time from the peak emission measure (tcool), LF nanoflares have t > t and HF nanoflares have t < t . LF nanoflares have a 2 h waiti cool h waiti cool ⇠ 5.1 Introduction 135

- 3 and thus do not account for many of the observations. In fact, Cargill (2014)

argued that these results implied a heating mechanism with t of order 1000 s h waiti to 2000 s between nanoflares, with the value of twait associated with each nanoflare being proportional to its energy. Such intermediate frequency (IF) nanoflares have different energy build-up requirements from the commonly assumed LF scenario (Cargill, 2014). A second outcome of active region studies is the detection of a “hot” non-flaring

coronal component characterized by plasma with T > Tpeak, a long-predicted consequence of nanoﬂare heating (Cargill, 1994, 1995). This has been identiﬁed

from Hinode and SDO data (Reale et al., 2009; Schmelz et al., 2009; Testa & Reale, 2012), and retrospectively from data obtained by the X-Ray Polychrometer (XRP) instrument ﬂown on the Solar Maximum Mission (Del Zanna & Mason, 2014). While characterizing this emission is difﬁcult (e.g. Testa et al., 2011; Winebarger

b et al., 2012), a similar scaling, EM(T) µ T has been claimed (e.g. Warren et al., 2012), with b of order 7 to 10, though Del Zanna & Mason ﬁnd larger values. Warren et al. quote typical errors of 2.5 to 3 on these values due to the very limited data ± available above Tpeak and Winebarger et al. have noted that the paucity of data from Hinode at these temperatures could be missing signiﬁcant quantities of plasma with T > Tpeak. In an effort to diminish uncertainty in this high temperature “blind spot” in

EM(T), Petralia et al. (2014) analyzed an active region core by supplementing EIS spectral observations with broadband AIA and XRT measurements. By using concurrent observations from the 94 Å channel of AIA and the Ti_poly ﬁlter of

6.6 XRT, the Petralia et al. showed that the EM(T) peaked near Tpeak = 10 K and had a weak, hot component. Additionally, Miceli et al. (2012), using the SphinX instrument (Gburek et al., 2011; Sylwester et al., 2008), analyzed full-disk X-ray spectra integrated over 17 days, during which time two prominent active regions 5.1 Introduction 136 were present. Miceli et al. found that a two-temperature model was needed to fit the resulting spectrum, a strong 3 MK component and a much weaker 7 MK component. More recent data has come from rocket flights. The Focusing Optics X-ray Solar Imager (FOXSI, Krucker et al., 2013) first flew in November 2012 and observed an active region. A joint study with EIS and XRT by Ishikawa et al. (2014) suggested

that while hot plasma existed up to 10 MK, the Hinode instruments over-estimated the amount of plasma there. A rocket flight reported by Brosius et al. (2014) identified emission in an Fe XIX line with a peak formation temperature of 106.95 K and reported an emission measure that was 0.59 times the emission formed at 106.2 K. More recently, a pair of rocket flights gave observations from the Amptek X123-SDD soft X-ray spectrometer (Caspi et al., 2015). This provided comprehensive coverage of the 3Åto 60 Å wavelength range. Caspi et al. demonstrated that the emission in this range could be fit by an emission measure with a power-law distribution slope

of roughly b = 6. While all of these observations are very suggestive of nanoﬂare heating, it should also be noted that pixel-averaging, long time averages and/or

inadequate instrument spatial resolution may lead to contamination of the DEM(T) by multiple structures along the LOS. It is desirable to obtain future measurements

of plasma emission at T > Tpeak from a single structure, such as a core active region loop, along the LOS. Several other workers have combined model results with observations in an effort to better elucidate nanoflare signatures. Using a hydrodynamic loop model, Reale et al. (2011) showed that emission from impulsively heated sub-arcsecond strands is finely structured and that this predicted structure can also be found in active region core emission as observed by the 94 Å channel of AIA. Most recently, Tajfirouze et al. (2016b), using a 0D hydrodynamic model, explored a large parameter space in event energy distribution, pulse duration, and number of loops. Using 5.1 Introduction 137

a probabilistic neural network, the authors compared their many forward-modeled light curves to 94 Å AIA observations of a “hot” AR core. They found that the observed light curves were most consistent with a pulse duration of 50 s and a shallow event energy distribution, suggestive of nanoﬂare heating.

While the distributions of temperature and density above Tpeak are likely to be determined by nanoflare heating and conductive cooling, there are several complications arising from the low density and high temperature present there. These are (i) the breakdown of the usual Spitzer description of thermal conduction which leads to slower conductive cooling, (ii) recognition that in cases of heating in a weakly collisional or collisionless plasma, electrons and ions need not have the same temperature since when one is heated preferentially the time for the temperature to equilibrate is longer than the electron conductive cooling time, and (iii) a lack of ionization equilibrium that can underestimate the quantity of the plasma with a given electron temperature. Thus the aim of the this chapter and Barnes et al. (2016a, BCB16 hereafter) is to investigate this high temperature regime from a modeling viewpoint with the aim of obtaining information that can be of use in the interpretation of present and future observations. In this chapter I focus on single-nanoflare simulations in order to build up an understanding of the roles of the different pieces of physics. BCB16 addresses the properties of nanoflare trains. Given the limitations of present observations, these results are in part predictive for a future generation of instruments. Section 5.2 addresses the methodology, including simple outlines of the physics expected from conductive cooling, the preferred heating of different species, and ionization non- equilibrium. Section 5.3 shows results for the single- and two-fluid models, and Section 5.4 provides discussion of the main points of the results. 5.2 Summary of Relevant Physics 138

5.2 Summary of Relevant Physics

First, I consider the situation when a coronal loop (or sub-loop) cools in response

to a nanoflare by the evolution of a single-fluid plasma (Te = Ti) along a magnetic field line. I deal with the case of electron-ion non-equilibrium in Section 5.2.2. The energy equation is,

∂E ∂ ∂F = [v(E + P)] c + Q n2L(T), (5.1) ∂t ∂s ∂s

p v2 5/2 ∂T where v is the velocity, E = g 1 + r 2 , Fc = k0T ∂s is the heat ﬂux, Q is a heating function that includes both steady and time-dependent components,

L(T)=cTa is the radiative loss function in an optically thin plasma (e.g. Klimchuk et al., 2008) and s is a spatial coordinate along the magnetic ﬁeld. In addition the equations of mass and momentum conservation are solved. These equations are

closed by p = 2nkBT, the equation of state. For a given initial state and Q, the plasma evolution can then be followed. In this chapter, two approaches are used to solve Equation 5.1. One uses the HYDRAD code (Bradshaw & Cargill, 2013, see Section 2.2.3) which solves the full field-aligned hydrodynamic two-fluid equations. The second develops further the zero-dimensional Enthalpy Based Thermal Evolution of Loops (EBTEL) approach which solves for average coronal plasma quantities (Cargill et al., 2012a,b, 2015; Klimchuk et al., 2008). In this chapter I compare the HYDRAD and EBTEL results and outline some restrictions that apply to the use of EBTEL when modeling the hot coronal component. However, the value of the EBTEL approach lies in its simplicity and computational speed, and the consequent ability to model the corona as a multiplicity of thin loops for long times, as is done in BCB16. Such calculations remain challenging for field-aligned hydrodynamic models. 5.2 Summary of Relevant Physics 139

The derivation of the single-ﬂuid EBTEL equations can be found in (Cargill et al., 2012a; Klimchuk et al., 2008, and see Section 2.2.4). EBTEL treats the corona and TR as separate regions, matched at the top of the TR by continuity of conductive and enthalpy ﬂuxes. It produces spatially-averaged, time-dependent quantities (e.g.

T¯ (t), n¯(t)) in the corona and can also compute quantities at the loop apex and the corona/TR boundary. The single-ﬂuid EBTEL equations are,

1 dp¯ 1 = Q¯ (R + R ), (5.2) g 1 dt L C TR g (pv) + F + R = 0, (5.3) g 1 0 c,0 TR dn¯ c2(g 1) = (Fc,0 + RTR). (5.4) dt 2c3gLkBT¯

Here an overbar denotes a coronal average, F = (2/7)k T7/2/L is the heat ﬂux c,0 0 a 2 at the top of the TR (see also Section 5.2.1), RC = n¯ L(T¯ )L, is the integrated coronal

radiation, RTR is the integrated TR radiation, and L is the loop half-length. The subscript “0” denotes a quantity at the top of the TR and “a” denotes a quantity at the loop apex. Solving this set of equations requires the speciﬁcation of three parameters: c1 = RTR/RC, c2 = T¯ /Ta, and c3 = T0/Ta. c2 and c3 can be taken as constant, with values of 0.9 and 0.6 respectively. Cargill et al. (2012a) discuss the full

implementation of c1 = c1(Ta, L) and Section 2.2.4 provides a detailed discussion of

the additional corrections applied to c1 in order to ensure better agreement with HYDRAD for impulsive heating scenarios. Equation 5.2 is a statement of energy conservation in the combined corona and TR. Equation 5.3 is the TR energy equation: if the heat ﬂux into the TR is greater (smaller) than its ability to radiate then there is an enthalpy ﬂux into (from) the corona. Equation 5.4 combines Equation 5.3 with that of mass conservation. 5.2 Summary of Relevant Physics 140

5.2.1 Heat Flux Limiters

It is well known that thermal conduction deviates from the familiar Spitzer-Härm formula (Spitzer & Härm, 1953) at high temperatures (e.g. Ljepojevic & MacNe- ice, 1989). There is a ﬁrm upper limit on the heat ﬂux: the free-streaming limit,

Fs =(1/2) fnkBTVe, where Ve is the electron thermal speed and f , a dimensionless constant, is determined from a combination of lab experiments, theory, and numerical models. The free-streaming ﬂux is included in EBTEL and HYDRAD by a simple modiﬁcation (Klimchuk et al., 2008),

FcFs F ,0 = , (5.5) c 2 2 Fc + Fs p where Fc is the Spitzer-Härm heat ﬂux. Smaller values of f limit the heat ﬂux to a greater degree. There is some disagreement on the optimal value of f . Luciani et al. (1983) use f = 0.1 while Karpen & DeVore (1987) use f = 0.53, and Patsourakos & Klimchuk (2005) choose f = 1/6. Unless explicitly stated otherwise, I use f = 1 in order to compare EBTEL results with those of HYDRAD (see appendix of Bradshaw & Cargill, 2013). The main aspect of inclusion of a free-streaming limit is to slow down conductive cooling. Other conduction models (e.g. the non-local model discussed in the coronal context by Ciaravella et al., 1991; Karpen & DeVore, 1987; West et al., 2008) are not considered here since they lead to similar generic results.

5.2.2 Two-ﬂuid Modeling

In some parameter regimes nanoﬂare heating can also induce electron-ion non- equilibrium if the heating timescale is shorter than the electron-ion equilibration timescale. Interactions between electrons and ions in a fully-ionized hydrogen plasma like the solar corona are governed by binary Coulomb collisions. Thus, the

equilibration timescale is tei = 1/nei, where nei is the collision frequency and is 5.2 Summary of Relevant Physics 141

given by, 4 3/2 16pp e 2k T n = B e n ln L,(2.38) ei 3 m m m e i ✓ e ◆ where Te is the electron temperature, me, mi are the electron and ion masses respectively, and ln L is the Coulomb logarithm (see both Equation 2.5e and Section

3 of Braginskii, 1965). For n 109 cm 3 and T 107 K, parameters typical of ⇠ e ⇠ nanoflare heating, t 800 s. Thus, any heating that occurs on a timescale less than ei ⇡ 800 s, such as a nanoflare with a duration of t 100 s, will result in electron-ion  non-equilibrium. While chromospheric evaporation during and after the nanoflare will increase n and thus decrease nei, I argue that during the early heating phase, t t, with 800 s being an upper bound on t . ei ei While it is often assumed that the electrons are the recipients of the prescribed coronal heating function, ion heating in the solar corona should not be discounted since the exact mechanism behind coronal heating is still unknown. For example, ions may be heated via ion-cyclotron wave resonances (e.g. Markovskii & Hollweg, 2004) or magnetic reconnection (e.g. Drake & Swisdak, 2014; Ono et al., 1996). To address this possibility and include effects due to electron-ion non-equilibrium, I apply the EBTEL analysis outlined in Klimchuk et al. (2008) to the two-fluid hydrodynamic equations in the form given in the appendix of Bradshaw & Cargill (2013). Such an approach allows for efficient modeling of a two-component impulsively- heated coronal plasma, and will be used extensively in BCB16. The two-fluid EBTEL equations are derived fully in Section 2.2.4 and are restated here for convenience,

d g 1 p = (y R (1 + c )) + k nn (T T )+(g 1)Q¯ ,(2.59) dt e L TR C 1 B ei i e e d g 1 p = y + k nn (T T )+(g 1)Q¯ ,(2.60) dt i L TR B ei e i i d x c2(g 1) n = (Fce,0 + Fci,0 + c1RC),(2.61) dt 1 + x c3LgkBT¯e 5.2 Summary of Relevant Physics 142

This set of equations is closed by the equations of state pe = kBnTe and pi = kBnTi.

While the notation above is largely self-evident, the additional term yTR, which originates in the need to maintain charge and current neutrality, is deﬁned by Equation 2.58.

5.2.3 Ionization Non-Equilibrium

Ionization non-equilibrium has long been known to be an issue in the interpretation of data from the impulsive phase of flares, and more recently it has been discussed in the context of nanoflares (Bradshaw & Cargill, 2006; Reale & Orlando, 2008). The main issue is that when a tenuous plasma is heated rapidly, it takes a certain time to reach ionization equilibrium so that the ionization states present do not reflect the actual (electron) temperature, assuming that the heating occurs mainly to electrons (see Section 5.2.2 and Section 5.3.2) rather than the heavier ions such as Fe that contribute to the observed radiation. If the heating is sustained, then eventually ionization equilibrium will be reached, and this may occur in moderate to large flares. However, for nanoflares that may last for anywhere between a few seconds and a few minutes, a different scenario arises in which on termination of heating, rapid conductive cooling arises, so that the high ionization states may never be attained. See Section 3.2.5 for further discussion of non-equilibrium charge states. Bradshaw & Cargill (2006), Reale & Orlando (2008) and Bradshaw (2009) have all addressed this point using slightly different approaches, but with similar conclusions, namely that short nanoflares in a low-density plasma are unlikely to be detectable. In this chapter, I develop this work further to assess how the results in the first parts of Section 5.3 are altered. Following these authors, I calculate an “ef-

fective temperature” (Teff) as a proxy for the deviation from ionization equilibrium. This involves taking a time-series of T and n (e.g. from EBTEL), using the method described in Appendix B to calculate the fractional ionization of as many states of 5.3 Results 143

various elements as needed, and then calculating Teff as described in Bradshaw (2009). I consider all ionization states of Fe. The feature that will prove of great relevance in these results is that despite the

different nanoﬂare durations, Teff does not exceed 10 MK. There is also an “over-

shoot” of Teff when it reaches its maximum value: this is saying that collisions are still not strong enough for the adjustment of the ionization state to be instantaneous.

5.3 Results

I now show a series of simulations of a single nanoflare using the zero-dimensional two-fluid hydrodynamic EBTEL model and the HYDRAD code. BCB16 discusses long trains of multiple nanoflares of varying frequency in multiple loops. An important output of all these models is the coronal emission measure. In EBTEL the emission measure for the entire coronal part of the loop is calculated

using the familiar expression EM = n2(2L), where L is the loop half-length. I consider a temperature range of 4.0 log T 8.5 with bin sizes of D log T = 0.02.   ¯ 2 At each time ti, the spatially-averaged coronal temperature T, weighted by n¯ i (2L), where n¯ i is the spatially-averaged number density at ti, is binned in temperature. The emission measure in each bin is then averaged over the entire simulation period.

When measured observationally, EM(T) is a LOS quantity. The emission measure from HYDRAD is calculated using quantities averaged over the upper 90% of the loop which corresponds to the coronal portion of the loop.

5.3.1 Single-ﬂuid Parameter Variations

Varying Pulse Duration

In the ﬁrst set of results I assume the plasma behaves as a single ﬂuid, use a

ﬂux limiter of f = 1, and ignore ionization non-equilibrium. The solid curves in 5.3 Results 144

t = 20 s t = 40 s t = 200 s t = 500 s 28 24 10

] 16 MK [ T 8 ] 5

1026 cm )[

45 T ( ] 3 EM 30 cm 8 1024 10 [ 15 n

0 100 101 102 103 106 107 t [s] T [K]

Figure 5.1 Left: Temperature (upper panel) and density (lower panel) profiles for a loop with 2L = 80 Mm. Each heating profile is triangular in shape with a steady background heating of H = 3.5 10 5 erg cm 3 s 1. The duration of bg ⇥ the heating pulse is varied according to t = 20 s, 40 s, 200 s, 500 s, with each value of t indicated by a different color, as shown in the right panel. The total energy 3 injected into the loop is fixed at 10 erg cm . Note that time is shown on a log scale to emphasize the behavior of the heating phase. Right: Corresponding EM(T) for each pulse duration t. The relevant parameters and associated colors are shown in the legend. EM(T) is calculated according to the procedure outlined in the beginning of Section 5.3. In all panels, the solid (dotted) lines show the corresponding EBTEL (HYDRAD) results (see Section 5.3.1).

Figure 5.1 show average temperature (upper left panel) and density (lower left panel) as a function of time for a single nanoﬂare in a loop with 2L = 80 Mm where the EBTEL approach is used. The heating function takes the form of a triangular pulse for four different pulse durations, t = 20 s, 40 s, 200 s, 500 s, as indicated by the legend. The peak heating rate is varied such that the total energy

3 input is 10 erg cm for all cases. These parameters correspond roughly to bright AR core loops (Warren et al., 2012). In order to ensure that the temperature and density do not become negative, a small background heating of magnitude Hbg = 5.3 Results 145

3.5 10 5 erg cm 3 s 1 is enforced at all times. It can be seen that shorter pulses ⇥ give higher temperatures, as expected. Furthermore, in this early heating phase, 2/7 one would expect the maximum temperature to scale roughly as H0 (where H0 is the peak heating rate); this is approximately what is found. On the other hand, the different pulse durations give approximately the same maximum density, with the shortest pulse reaching its peak value roughly 200 s before the longest. The solid lines in the right panel of Figure 5.1 show the corresponding EBTEL

emission measure distributions, EM(T). The temperature of maximum emission

(Tpeak) and the peak emission measure (EM(Tpeak))are the same in all cases and are consistent with those found in the studies of AR core loops (e.g. Warren et al., 2012). While shorter pulses lead to higher initial temperatures, the shape of the emission

measure below Tpeak is independent of the properties of the heating pulse, indicating that this part of the emission measure distribution cannot provide information about the actual nanoﬂare duration or intensity. All cases show evidence of the

heating phase, namely the bump on EM(T) at log (T)=6.85, 7, 7.2, 7.3. Below these bumps to just above T = T , EM(T) scales as T 5 T 5.5 for all cases, again peak indicating that information about the heating process is lost at these temperatures.

However, detection of emission above Tpeak in a single structure would still be evidence for nanoﬂare heating, though of undetermined duration. For integration over the lifetime of unresolved structures lying transverse to

the LOS, one can write down an expression EM(T) n2t (n, T) which simply ⇠ cool states that what matters for determining EM(T) is how long the plasma spends at any given temperature (e.g. Cargill, 1994; Cargill & Klimchuk, 2004). For an

analytic solution for the cooling, one can formally deﬁne tcool(n, T)=(T/(dT/dt)). In the absence of a formal solution, order of magnitude scalings can be used: the difference with analytic solutions being a numerical factor. To obtain an expression

b EM(T) µ T , one needs to provide a relation between T and n. For conductive 5.3 Results 146

cooling of the corona, one can write t nL2T 5/2, giving EM(T) n3L2T 5/2. cool ⇠ ⇠ In determining the relationship between T and n, two limits are those of constant density and constant pressure. The former gives static conductive cooling (e.g. Antiochos & Sturrock, 1976) and the latter evaporative cooling with constant ther-

mal energy (e.g. Antiochos & Sturrock, 1978), which then lead to b = 5/2, 11/2, respectively. Fitting the EBTEL EM(T) results for t 200 s (see right panel of  Figure 5.1) to T b on 106.8 < T < 107 K yields b 4.5 5 which are more consistent ⇠ with the latter.

HYDRAD Comparison

I now compare EBTEL and HYDRAD results for the different values of t. The dotted lines in all three panels of Figure 5.1 show the corresponding HYDRAD results, where averaging is over the upper 90% of the loop. The background heating in the two codes has been adjusted to ensure that EBTEL and HYDRAD start with the the same initial density since the initial temperature rise will depend on the assumed background density. There is good agreement between the HYDRAD and EBTEL results for t 200 s with the well-documented result that EBTEL gives somewhat higher density maxima than HYDRAD (see Cargill et al., 2012a). For t = 20 s, 40 s, while the peak temperatures are at a level of agreement consistent with previous work (Cargill et al., 2012a), there are notable differences in the initial temperature decay from the maximum in the upper left panel of Figure 5.1 due to the difference in the initial density response. It can be seen that the EBTEL density begins to rise almost immediately following the onset of heating, while there is a lag in the HYDRAD density. This is due to a delay in the upﬂow of material from the TR because a ﬁnite time is required to get material moving up the loop, an effect absent from 0D models. The slower 5.3 Results 147

density rise seen with HYDRAD leads to the faster conductive cooling. Another feature of the short pulses is the very spiky density profile as a function of time. This is a well-known effect, particularly in flare simulations, and is due to pairs of oppositely-directed flows colliding at the loop top, and subsequently bouncing back and forth. As a result of this discrepancy in the density behavior, while the emission measure calculated from the EBTEL model “sees” temperatures well in excess of 10 MK for short pulses, in the HYDRAD model this will not be the case. This is evident from the short pulses in the right panel of Figure 5.1: the emission above 10 MK predicted by EBTEL is not present in the HYDRAD runs, the emission cutting off just above 107 K. For the longer pulses, EBTEL still shows emission at higher temperatures, but the difference with HYDRAD is evident now over a smaller temperature range. Also, the characteristic bumps on the emission measure seen with EBTEL are largely eliminated in the HYDRAD runs. This regime of short heating pulses was not considered in earlier work using EBTEL, and the associated comparisons with field-aligned hydrodynamic codes (Cargill et al., 2012a; Klimchuk et al., 2008), where pulses of order 200 s or greater were considered. Other workers have used short pulses with EBTEL, albeit much less intense (Tajfirouze et al., 2016a,b). Clearly the more gentle the heating profile used, the slower the rise in the EBTEL density, leading to results closer to those found using HYDRAD. Thus it appears that caution is warranted in the use of approximate models for short, intense heating pulses. This restriction only applies to the high temperature regime: as can be seen from Figure 5.1, the emission measure profiles below 106.8 K are not affected. Nonetheless, the absence of emission near 10 MK for short pulses constitutes one of many obstacles to quantifying any hot plasma component due to nanoflares. 5.3 Results 148

Heat Flux Limiter

1028 f =1.00 f =0.53 f =1/6 f =1/10

] f =1/30 5

1026 Spitzer cm )[ T ( EM

1024

4 106 6 106 107 2 107 3 107 ⇥ ⇥ ⇥ ⇥ T [K]

Figure 5.2 EM(T) calculated from the single-fluid EBTEL model when only pure Spitzer conduction is used and when a flux limiter is imposed according to Sec- tion 5.2.1. In the free-streaming limit, five different values of f are considered (see legend). The pulse duration is t = 200 s. All other parameters are the same as those discussed in Section 5.3.1. Note that EM(T) is only shown for T > Tpeak as the cool side of EM(T) is unaffected by the choice of f .

Figure 5.2 shows the effect of using a ﬂux limiter versus Spitzer conduction

on the emission measure distribution. Five different values of f are shown: 1 (Bradshaw & Cargill, 2013, consistent with HYDRAD), 0.53 (Karpen & DeVore, 1987), 1/6 (Patsourakos & Klimchuk, 2005), 0.1 (Luciani et al., 1983), and 1/30. The pulse duration is 200 s and only the EBTEL results are shown. Note that for this pulse length, the HYDRAD results are expected to be similar.

As expected, inclusion of a limiter extends EM(T) to higher temperatures, though this is only notable above 10 MK. As the temperature falls to this value, evaporative upﬂows have increased the coronal density so that the Spitzer description is recovered. Above 10 MK ﬂux limiting gradually becomes important, albeit with 5.3 Results 149

a small emission measure. Using f = 0.53, 1 yield EM(T) that are not discernibly different from that produced by pure Spitzer conduction while f = 1/6, 0.1 extend EM(T) to significantly hotter temperatures. f = 1/30, the most extreme flux limiter, yields an emission well above 107.5 K. Note that for all cases, EM(T) converges to the same value for T 10 MK.  For flux-limited thermal conduction, t LT 1/2 so that the parameter b cool ⇠ lies between 1/2 and 5/2, depending on the assumption about n. For f = 1/30, b = 5/2 is found in Figure 5.2 by fitting EM(T) to T b on 107 K T 107.5 K.   Since the free-streaming limit slows conductive cooling relative to that given by Spitzer, the plasma will spend more time at any given temperature, leading to

smaller values of b. Similar conclusions hold for other conduction models (e.g. the non-local model discussed in the coronal context by Karpen & DeVore, 1987; West et al., 2008) since they all inhibit conduction. While limiting conduction is often regarded as an important process in coronal cooling, these results suggest that for nanoﬂare heating it may not be that important unless extreme values of the limiting parameter are used.

5.3.2 Two-ﬂuid Effects

Electron Heating

I now use the two-ﬂuid model to consider the role of separate electron or ion heating, focusing on cases when only the electrons or ions are heated in order to highlight the essential difference between the two scenarios. Intermediate cases of energy distribution will be considered in subsequent papers. The solid lines in the left panels of Figure 5.3 show the electron temperature (upper panel), ion temperature (middle panel) and density (lower panel) as a function of time from the two-ﬂuid EBTEL model for t = 20 s, 40 s, 200 s, 500 s for electron heating. The dotted lines 5.3 Results 150

t = 20 s t = 40 s t = 200 s t = 500 s 28 24 10 ] 16 MK [

e 8 T

8 ] 5

26 ] 6 10 cm MK

4 )[ [ i T ( T 2 EM ]

3 45 30 1024 cm 8

10 15 [ n 0 100 101 102 103 106 107 t [s] T [K]

Figure 5.3 Two-ﬂuid EBTEL simulations for t = 20 s, 40 s, 200 s, 500 s in which only the electrons are heated. Left: Electron temperature (upper panel), ion temperature (middle panel), and density (lower panel). Right: Corresponding EM(T) calculated according to Section 5.3. The pulse durations and associated colors for all panels are shown in the legend. All parameters are the same as those discussed in Section 5.3.1. In all panels, the solid (dotted) lines show the corresponding EBTEL (HYDRAD) results.

show the corresponding HYDRAD results. The electrons now cool by a combination of thermal conduction and temperature equilibration, the latter becoming signiﬁcant at 150 (450) s for short (long) pulses. The ions thus heat rather slowly, reaching a peak temperature of 5 MK, which overshoots the electron temperature at that time.

The ions then cool via ion thermal conduction and equilibration, with T T after e ⇡ i typically a few hundred seconds.

The solid lines in the right panel of Figure 5.3 show the resulting EM(T). In the case of electron heating and t < 500 s, the emission measure slope over the

temperature interval log Tpeak < log T < 6.8 is considerably steeper compared to the single-ﬂuid case. Recall that in the single-ﬂuid case it is assumed that conduction 5.3 Results 151

is the only relevant cooling mechanism prior to the onset of radiative cooling such

11/2 that under the assumption of constant pressure, EM(T) µ T (see Section 5.3.1). When the electron and ion ﬂuids are treated separately and only the electrons are heated, both of these assumptions break down. Following the onset of conductive

cooling, T T , but the loop has now begun to ﬁll. The equilibration term plays e i the part of a cooling term so long as Te > Ti and is the dominant cooling mechanism for several hundred seconds in between the peak electron temperature and the peak density (see Figure 2.7). Thus, the expression for tcool should include some contribution from the equilibration term in this temperature regime.

101 1010 p pe + pi pe ns pi nt

0 ] 10 2 ] 3 109 cm [ n dyne cm [

p 1 10

2 8 10 10 106 107 T [K]

Figure 5.4 Pressure (left axis) and density (right axis) as a function of temperature for the t = 200 s case. All parameters are the same as those discussed in Sec- tion 5.3.1. The single-fluid pressure p and density n are denoted by the solid blue and orange lines, respectively. The electron pressure, pe, ion pressure, pi, and two- fluid total pressure, pe + pi, are denoted by the dashed, dotted, and dot-dashed blue lines respectively. The two-fluid density is represented by the dashed orange line. Pressure, density, and temperature are all shown on a log scale.

Figure 5.4 shows pressure (blue lines) and density (orange lines) as a function of temperature for the t = 200 s case; both the single-ﬂuid case and the case where 5.3 Results 152

only the electrons are heated are shown. While pe + pi (blue dotted line), the total pressure, like the single-ﬂuid pressure p (blue solid line) is constant over the

6.65 6.8 interval 10 K < T < 10 K, the electron pressure, pe (blue dashed line) is not, 1 meaning n µ Te is not a valid scaling law in the two-fluid, electron-heating case. Comparing the two-fluid density (dashed orange line) and the single-fluid density (solid orange line) easily confirms this. To derive an emission measure slope for the case in which only the electrons are heated, these effects must be accounted for in

the EM(T) n2t (n, T) scaling. Thus, while a power-law b may be calculated by ⇠ cool b fitting the hot part of the EM(T) to T , it is difficult to gain any physical insight from such a fit using the scaling discussed in Section 5.3.1.

Ion Heating

Figure 5.5 shows the electron temperature (upper left panel), ion temperature (middle left panel), density (lower left panel) and the corresponding emission measure (right panel) for t = 20 s, 40 s, 200 s, 500 s when only the ions are heated. The solid lines show the two-ﬂuid EBTEL results while the dotted lines show the corresponding HYDRAD results. Ion heating leads to signiﬁcantly higher temperatures due to the relative weakness of ion thermal conduction, consistent

2/7 with the expected enhancement of (k0,e/k0,i) . The hot ions cool by a combination of weak ion thermal conduction and temperature equilibration. However, because

the Coulomb coupling timescale during the early heating phase (when T T i e and the density is low) is much larger than the ion thermal conduction timescale, by the time the electrons can “see” the ions, they have cooled far below their peak temperature. The peak electron temperature in all cases lies below 10 MK. Because

EM(T) is constructed from the electron temperature, the emission measure never sees T 107 K, with EM(T) being truncated sharply near 106.9 K for all values of t. 5.3 Results 153

t = 20 s t = 40 s t = 200 s t = 500 s 8 1028

] 6

MK 4 [ e

T 2 ] 5

] 150 10 cm

MK 100 )[ [ i T ( T 50

40 EM ] 3

30 1024 cm

8 20 10

[ 10 n 0 100 101 102 103 106 107 t [s] T [K]

Figure 5.5 Two-ﬂuid EBTEL simulations for t = 20 s, 40 s, 200 s, 500 s in which only the ions are heated. Left: Electron temperature (upper panel), ion temperature (middle panel), and density (lower panel). Right: Corresponding EM(T) calculated according to Section 5.3. The pulse durations and associated colors for all panels are shown in the legend. All parameters are the same as those discussed in Section 5.3.1. In all panels, the solid (dotted) lines show the corresponding EBTEL (HYDRAD) results.

The reason for slower equilibration for ion heating can be seen by comparing the density plots in the lower left panels of Figure 5.3 and Figure 5.5. These show that while the peak values of the density are similar for both heating mechanisms, the temporal behavior differs for ion heating with shorts pulses: for these cases, the density takes considerably longer to reach the maximum value. This can be attributed to the relative weakness of ion thermal conduction. Examination of Equation 2.53 and Equation 2.58 shows that an upward enthalpy flux can only be effective for ion heating once temperature equilibration has become significant and an electron heat flux is established. In turn, once the upflow begins, the coronal density increases, making equilibration more effective. Thus, once temperature 5.3 Results 154

equilibration starts to be effective, these processes combine to give a rapid increase in density, as shown. In the case where the heating pulse duration is long, t = 500 s, the difference between the two-fluid and single-fluid emission measure distributions is diminished. Because the electrons are heated slowly, they do not have much time to evolve out of equilibrium with the ions. This in turn heavily dampens the Coulomb exchange term, allowing the two populations to evolve together as a single fluid.

HYDRAD Comparison

The dotted lines in all panels of Figure 5.3 and Figure 5.5 show the corresponding HYDRAD results for both electron and ion heating, respectively. As in Section 5.3.1, the averaging is done over the upper 90% of the loop and the background heating

has been adjusted appropriately. For t 200 s, there is acceptable agreement in n, Te, Ti, and EM(T). For t = 20, 40 s, the upper and lower panels of Figure 5.3 show discrepancies

in Te and n similar to those discussed in Section 5.3.1. The initial decay from the peak electron temperature is noticeably different in the EBTEL runs compared to the corresponding HYDRAD runs, again due to the difference in the initial density response. The discrepancies in the density are exacerbated in the electron heating case (compared to the single-fluid case) since all of the energy is partitioned to the electrons, resulting in a stronger electron heat flux and a subsequently stronger upflow. The right panel of Figure 5.3 shows the effect of this premature rise in the

density on EM(T) for these short pulses: while EBTEL predicts signiﬁcant emission above 10 MK, the emission in the HYDRAD runs cuts off just below 106.9 K.

In the ion heating case, there is acceptable agreement in Te, n, and EM(T). Notably, there is good agreement in n even for the shortest heating pulses, t = 20, 40 s, though EBTEL still overestimates the peak density compared to the HYDRAD 5.3 Results 155

results. Because no heat is supplied to the electrons directly, the electron heating timescale is set by the Coulomb collision frequency (see Equation 2.38), meaning energy is deposited to the electrons over a timescale much longer than 20 s or 40 s.

The resulting slow evolution of Te leads to subsequently weaker upﬂows. Because of the much more gentle rise in density, the electrons are not able to “see” the ions until they have cooled well below 10 MK (see Section 5.3.2). In the middle panel on the left hand side of Figure 5.3, the ion temperature in HYDRAD is signiﬁcantly greater than that of EBTEL in the late heating/early

conductive cooling phase for t = 20 s, 40 s. These spikes in Ti are due to steep velocity gradients that heat the ions through compressive heating and viscosity, two pieces of physics that are not included in EBTEL. Because ion thermal conduction is

comparatively very weak, these sharp features in Ti are not as efﬁciently smoothed

out. For t = 200 s, 500 s, there is good agreement in Ti between the EBTEL and HYDRAD results. Interestingly, the middle panel on the left hand side of Figure 5.5

shows that EBTEL and HYDRAD predict similar behavior for Ti though EBTEL signiﬁcantly overestimates the ion temperature for t = 20 s, 40 s. As before, there is good agreement for t = 200 s, 500 s.

5.3.3 Ionization Non-Equilibrium

The ﬁnal set of results includes the approximate treatment of non-equilibrium ionization, again using the EBTEL approach. The red curves in the top (bottom) panel of Figure 5.6 show Teff for t = 20 (500) s for the single-ﬂuid, electron heating,

and ion heating cases. For comparison, equivalent results for T (single-ﬂuid) and Te

(two-ﬂuid) that assume ionization equilibrium are shown. For all cases, Teff never rises above 10 MK for the short pulse and 8 MK for the long pulse. Thus, for the short pulse, because a sufﬁciently long time is required to ionize the plasma, the hottest electron temperatures are never likely to be detectable. For the longer pulse, 5.3 Results 156

24 single electron

] 16 ion MK [

T 8

10.0

7.5 ]

MK 5.0 [ T 2.5

0.0 100 101 102 103 t [s]

Figure 5.6 Teff (red) for pulse durations of 20 s (top panel) and 500 s (bottom panel) for the single-ﬂuid case (solid) as well as the cases where only the electrons (dashed) or only the ions (dot-dashed) are heated. T(t) proﬁles (i.e. assuming ionization equilibrium) for t = 20 s (blue lines) and t = 500 s (brown lines) for all three heating scenarios are repeated here for comparison purposes.

the slow heating gives the ionization states the opportunity to “catch up”; thus Teff is a reasonable reﬂection of the actual plasma state.

The red curves in Figure 5.7 show the corresponding EM(Teff). The effect of ionization non-equilibrium is to truncate EM around or below 10 MK. The bump on the distribution characteristic of the heating phase is also relocated to lower temperatures. This conﬁrms the earlier comment that, at least for short pulses, the hot electron plasma above 10 MK is undetectable. While the heating signature is

shifted to smaller values of Teff, one has no way of knowing the duration of the pulse that generates it. Thus it seems as if the temperature range Tpeak < T < 10 MK is the optimal one for searching for this hot component as well as direct signatures 5.4 Discussion 157

1028 single electron ion ] 5 1026 cm )[ T ( EM 1024

1028 ] 5 1026 cm )[ T ( EM 1024

4 106 6 106 107 2 107 3 107 ⇥ ⇥ ⇥ ⇥ T [K]

Figure 5.7 EM(Teff) (red) for pulse durations of 20 s (top panel) and 500 s (bottom panel) for the single-ﬂuid (solid), electron heating (dashed), and ion heating (dot- dashed) cases. EM(T) (i.e. assuming ionization equilibrium) for t = 20 s (blue lines) and t = 500 s (brown lines) for all three heating scenarios are repeated here for comparison purposes. Note that in both panels EM(T) is only shown for log T > log Tpeak. of the heating. However, it is difﬁcult to “map” what would be seen in such a state of ionization non-equilibrium back to the real system.

5.4 Discussion

This chapter has begun to address signatures of the so-called “hot” plasma component in the non-ﬂaring corona, especially in active region s, that is perceived 5.4 Discussion 158

as providing essential evidence for the existence of nanoflares. I have used zero- dimensional and field-aligned single- and two-fluid modeling to examine the possible signatures of a single nanoflare occurring in a low-density plasma. This corresponds to the simplest case of so-called “low frequency” (LF) nanoflares, where a coronal loop is heated by many events with the same energy and with a time between events longer than the characteristic cooling time such that the plasma is allowed cool significantly before being re-energized. When an approximate single-fluid model assuming ionization equilibrium is used, the expected signatures of conductive cooling appear in the distribution of plasma as a function of temperature, as described by the emission measure. In particular, short nanoflares with duration < 100 s should have a significant plasma component well above 10 MK, and for longer duration events, significant plasma between the temperature of the maximum emission measure and 10 MK. However, inclusion of several pieces of additional physics modifies this result considerably, in each case making it much less likely that any plasma that > 10 MK can be detected. For short nanoflares, the time taken for conductively-heated chromospheric plasma to move into the coronal part of a loop is sufficiently long that the initial hot coronal plasma cools rapidly, contributing little to the emission measure such that, once the coronal density has increased, its temperature is below 10 MK. This effect is less important for long duration nanoflares. Consideration of separate electron and ion heating shows that, while electron heating leads to similar results to the single fluid case, ion heating results in no emission measure at 10 MK due to the principal electron heating mechanism being a relatively slow collisional process. Finally, relaxing the assumption of ionization equilibrium leads to a truncation of the emission measure below 10 MK, since the time needed to create highly ionized states such as Fe XXI is longer than any relevant cooling time. In all cases the hot plasma, while still in the corona, is effectively “dark”. In addition, characteristic 5.4 Discussion 159

structures in the emission measure profile that are a signature of the heating itself in simple models are all but eliminated. These results suggest that while showing that such a “hot” plasma should exist in principle may not be difficult, characterizing the heating process from its observed properties may be a lot harder. Of course the work presented in this chapter is limited to LF nanoflares. Furthermore, Cargill (2014) showed that the intermediate frequency nanoflare regime does have significant differences, in large part due to the range of densities in which the nanoflares occur. This is addressed fully, along with other parameter variations, in BCB16, though it is difficult to see how a component hotter than 10 MK can be resurrected. Note though that the results of Caspi et al. (2015) pose a challenge to this scenario unless an undetected microflare or small flare occurred during the observations. The observational aspects of this work are addressed more fully in BCB16. However, one can conclude (i) present day observations do not seem capable of making quantitative statements about the “hot” component, though they are highly suggestive of its existence and (ii) future measurements should be concentrated in the temperature regime 106.6 K to 107 K rather than at higher temperatures. The MaGIXS instrument, due to fly in the summer of 2019, is well positioned to do this. Chapter 6

Predicting Diagnostics for Nanoﬂares of Varying Frequency

In this chapter, I use the forward modeling pipeline outlined in Chapter 4 to predict observable signatures of an active region heated by nanoﬂares of varying frequency. In particular, I predict both the emission measure slope (see Section 3.5.1) and time lag (see Section 3.6). In Chapter 7, these simulated diagnostics are used to train a random forest classiﬁer to predict heating frequency in every pixel of active region NOAA 1158. This chapter is adapted directly from Barnes et al. (2019a) which has recently been submitted for publication.

6.1 Introduction

Nanoflares have long been used to explain the observed million-degree temperatures in the non-flaring solar corona. Though originally pertaining to energetic bursts of order 1024 erg resulting from small-scale reconnection (Parker, 1988), the term nanoflare is now synonymous with any impulsive energy release and is not specific to any particular physical mechanism (Klimchuk, 2015). Due to their faint,

160 6.1 Introduction 161

transient nature, direct observations of nanoflares are made difficult by several factors, including inadequate spectral coverage of instruments, the efficiency of thermal conduction, and non-equilibrium ionization (Barnes et al., 2016b; Cargill, 1994; Winebarger et al., 2012). However, recent observations of “very hot” 8 MK to 10 MK plasma, the so-called “smoking gun” of nanoflares, have provided compelling evidence for their existence (e.g. Brosius et al., 2014; Caspi et al., 2015; Ishikawa et al., 2017; Parenti et al., 2017). Critical to understanding the underlying heating mechanism is knowing whether

the corona in non-flaring active regions is heated steadily or impulsively, or, more precisely, at what frequency do nanoflares repeat on a given magnetic strand. In the case of low-frequency nanoflares, the time between consecutive events on a strand is long relative to its characteristic cooling time, giving the strand time to fully cool and drain before it is re-energized. In the high-frequency scenario, the time between events is short relative to the cooling time such that the strand is not allowed to fully cool before being heated again. Steady heating may be regarded as nanoflare heating in the very high-frequency limit.

Before proceeding, I note that a magnetic strand, the fundamental unit of the low-b corona, is a flux tube oriented parallel to the magnetic field that is isothermal in the direction perpendicular to the magnetic field. I make the distinction that

a coronal loop is an observationally-deﬁned feature representing a magnetic ﬁeld- aligned intensity enhancement relative to the surrounding diffuse emission such that a single coronal loop may be composed of many thermally-isolated strands.

Furthermore, I deﬁne the active region core as the area near the center of the active region whose X-ray and EUV emission is dominated by closed loops with both footpoints rooted in the photosphere. In lieu of a direct observable signature of nanoﬂare heating, two parameters in particular have been used to diagnose the heating frequency in active region 6.1 Introduction 162

cores: the emission measure slope and the time lag. These diagnostics provide

indirect signatures of the energy deposition via observations of the plasma cooling by thermal conduction, enthalpy, and radiation. I will now discuss each of these observables in detail.

2 The emission measure distribution, EM(T)= dhne , where ne is the electron density and the integration is taken along the LOS,R is a useful diagnostic for parameterizing the frequency of energy deposition (see Section 3.5 for an extended discussion of the emission measure distribution). Many observational and theo-

retical studies have suggested that the “cool” portion of the EM(Te) (i.e. leftward of the peak, 105.5 T 106.5 K), can be described by EM(T) Ta (Cargill, 1994; . . ⇠ Cargill & Klimchuk, 2004; Jordan, 1976). The so-called emission measure slope, a, is an important diagnostic for assessing how often a single strand may be reheated and has been used by several researchers to interpret active region core observations in terms of both high- and low-frequency heating (see Table 3.2 and references therein).

The “cool” emission measure slope typically falls in the range 2 < a < 5, with shallower slopes indicative of low-frequency heating and steeper slopes associated with high-frequency heating. Many observational studies of active region cores have used the emission measure slope to make conclusions about the heating frequency (e.g. Del Zanna et al., 2015b; Schmelz & Pathak, 2012; Tripathi et al., 2011; Warren et al., 2011, 2012; Winebarger et al., 2011). The emission measure slope is discussed extensively in Section 3.5.1. To better understand observable properties of nanoﬂare heating, several researchers have used hydrodynamic models of coronal loops to examine how the emission measure slope varies with heating frequency (Bradshaw et al., 2012; Mulu- Moore et al., 2011a; Reep et al., 2013). Most recently, Cargill (2014) found that varying the time between consecutive heating events from 250 s (high-frequency heating) to 5000 s (low-frequency heating) could account for the wide observed 6.1 Introduction 163

distribution of emission measure slopes, with higher values of a corresponding to

higher heating frequency due to the EM(Te) distribution becoming increasingly isothermal (see also Barnes et al., 2016a). In addition to the emission measure slope, the time lag analysis of Viall & Klimchuk (2012) has also been used by several workers to understand the frequency

of energy release in active region cores. The time lag is the temporal delay which maximizes the cross-correlation between two time series, and, qualitatively, can be thought of as the amount of time which one signal must be shifted relative to another in order to achieve the best “match” between the two signals. As the plasma cools through the six EUV channels of AIA, the intensity will peak in successively cooler passbands of the instrument according to the sensitivity of each channel in temperature space (Viall & Klimchuk, 2011). Computing the time lag between light curves in different channels provides a proxy for the cooling time between channels and insight into the thermal evolution of the plasma. Calculating the time lag in each pixel of an AIA image can reveal large scale cooling patterns in coronal loops as well as the diffuse emission between loops across an entire active region. Viall & Klimchuk (2012) computed time lags for all possible AIA EUV channel pairs in every pixel of active region NOAA 11082 and found positive time lags across the entire active region core, indicative of cooling plasma. They interpreted these observations as being inconsistent with a steady heating model. Viall & Klimchuk (2017) extended this analysis to the 15 active regions catalogued by Warren et al. (2012) and found overwhelmingly positive time lags, or cooling plasma, in all cases, with only a few isolated instances of negative time lags, or heating plasma. These observations are consistent with an impulsive heating scenario in which little emission is produced during the heating phase because of the time needed to fill the corona by chromospheric evaporation and the efficiency of thermal conduction. Bradshaw & Viall (2016) predicted AIA intensities for a range of nanoflare heating 6.1 Introduction 164

frequencies in a model active region and applied the time-lag analysis to their simulated images. They found that aspects of both high and intermediate frequency nanoflares reproduced the observed time-lag patterns, but neither model could fully account for the observational constraints, suggestive of a range of heating frequencies across the active region. However, Lionello et al. (2016) used a field- aligned hydrodynamic model to compute time lags for several loops in NOAA 11082 and concluded that an impulsive heating model could not account for the long (> 5000 s) time lags calculated from observations by Viall & Klimchuk (2012). Any successful heating model must be able to reproduce the observed distribution of emission measure slopes and time lags. In order to carry out such a test, both advanced forward modeling and sophisticated comparisons to data are required. In this chapter, I carry out a series of nanoflare heating simulations in order to better understand how the frequency of impulsive heating events on a given strand is related to observable properties of the plasma, notably the emission measure slope and the time lag as derived from AIA observations. To do this, I use a combination of magnetic field extrapolations, hydrodynamic models, and atomic data to produce simulated AIA emission which can be treated in the same manner as real observations. I then apply the emission measure and time-lag analysis to this simulated data. Section 6.2 provides detailed descriptions of the forward modeling pipeline and the nanoflare heating model. In Section 6.3, I show the predicted intensities for each heating model and AIA channel (Section 6.3.1), the resulting emission measure slopes (Section 6.3.2) and the time lags (Section 6.3.3). Section 6.4 provides some discussion of the results and Section 6.5 includes a summary and concluding remarks. This chapter serves to describe the forward modeling procedure and lay out the results of the nanoflare simulations for each heating frequency. In Chapter 7,I use machine learning to make detailed comparisons to AIA observations of active 6.2 Modeling 165

region NOAA 1158. I train a random forest classiﬁer using the predicted emission measure slopes and time lags presented here over the entire heating frequency parameter space in order to classify the heating frequency in each pixel of the observed active region. In contrast to past studies which have relied on a single diagnostic, this approach simultaneously accounts for an arbitrarily large number of observables in deciding which model ﬁts the data “best.” The ability to compare models with large quantities of data statistically is crucial for progress in the current era where the amount of solar coronal data is orders of magnitude larger than in the past. Combined, these two chapters demonstrate a novel method for using real and simulated observations to systematically predict heating properties in active region cores.

6.2 Modeling

In order to understand how signatures of the heating frequency are manifested in the emission measure slope and time lag, I predict the emission over the entire active region as observed by AIA for a range of nanoflare heating frequencies. To do this, I have constructed an advanced forward modeling pipeline through a combination of magnetic field extrapolations, field-aligned hydrodynamic simulations, and atomic data. In the following section, I discuss each step of the pipeline. Additionally, this forward modeling code is discussed in greater detail in Chapter 4.

6.2.1 Magnetic Field Extrapolation

I choose active region NOAA 1158, as observed by the Helioseismic Magnetic Imager (HMI, Hoeksema et al., 2014) on 12 February 2011 15:32:42 UTC, from the list of active regions studied by Warren et al. (2012). The LOS magnetogram is shown in the left panel of Figure 6.1. The geometry of active region NOAA 1158 6.2 Modeling 166 00 0 . 100 00 0 . 200 00 0 . 300 Helioprojective Latitude

400.000 300.000 200.000 400.000 300.000 200.000 Helioprojective Longitude

Figure 6.1 Active region NOAA 1158 on 12 February 2011 15:32:42 UTC as observed by HMI (left) and the 171 Å channel of AIA (right). The gridlines show the heliographic longitude and latitude. The left panel shows the LOS magnetogram and the colorbar range is 750 G on a symmetrical log scale. In the right panel, 500 ± out of the total 5000 ﬁeld lines are overlaid in white and the red and blue contours show the HMI LOS magnetogram at the +5% (red) and 5% (blue) levels.

is modeled by computing the three-dimensional magnetic field using the oblique potential field extrapolation method of Schmidt (1964) as outlined in Sakurai (1982, Section 3). The extrapolation technique of Schmidt is well-suited for this purpose due to its simplicity and efficiency though it is only applicable on the scale of an active region. The oblique correction is included to account for the fact that the active region is off of disk-center. The HMI LOS magnetogram provides the lower boundary condition of the vector magnetic field (i.e. Bz(x, y, z = 0)) for the field extrapolation. I crop the magnetogram to an area of 300 -by-300 centered on ( 288.26 , 223.21 ) and 00 00 00 00 resample the image to 100-by-100 pixels to reduce the computational cost of the field extrapolation. The extrapolated field has a dimension of 100 pixels and spatial 6.2 Modeling 167

extent of 0.3R in the z direction such that each component of the vector magnetic ﬁeld, ~B, has dimensions (100, 100, 100).

1200

1000

800

600

Number of Loops 400

200

0 50 100 150 200 250 L [Mm]

Figure 6.2 Distribution of footpoint-to-footpoint lengths (in Mm) of the 5000 ﬁeld lines traced from the ﬁeld extrapolation computed from the magnetogram of NOAA 1158.

After computing the three-dimensional vector ﬁeld from the observed magnetogram, 5 103 ﬁeld lines are traced through the extrapolated volume using the ⇥ streamline tracing functionality in the yt software package (Turk et al., 2011). Only

closed field lines in the range 20 Mm < L < 300 Mm are chosen, where L is the full length of the field line. The right panel of Figure 6.1 shows a subset of the traced field lines overlaid on the observed AIA 171 Å image of NOAA 1158. Contours from the observed HMI LOS magnetogram are shown in red (positive) and blue (negative). A qualitative comparison between the extrapolated field lines and the loops visible in the AIA 171 Å image reveals that the field extrapolation and line tracing adequately capture the three-dimensional geometry of the active region. Figure 6.2 shows the distribution of footpoint-to-footpoint lengths for all of the traced field lines. 6.2 Modeling 168

6.2.2 Hydrodynamic Modeling

Due to the low-b nature of the corona, each field line traced from the field extrapolation can be treated as a thermally-isolated strand. I use the Enthalpy-based Thermal Evolution of Loops model (EBTEL, Cargill et al., 2012a,b; Klimchuk et al., 2008), specifically the two-fluid version of EBTEL (Barnes et al., 2016b), to model the thermodynamic response of each strand. The two-fluid EBTEL code solves the time-dependent, two-fluid hydrodynamic equations spatially-integrated over the corona for the electron pressure and temperature, ion pressure and temperature, and density. The two-fluid EBTEL model accounts for radiative losses in both the transition region and corona, thermal conduction (including flux limiting), and binary Coulomb collisions between electrons and ions. The time-dependent heating input is configurable and can be deposited in the electrons and/or ions. A detailed description of the model and a complete derivation of the two-fluid EBTEL equations can be found in Section 2.2.4. For each of the 5 103 strands, a separate instance of the two-fluid EBTEL ⇥ code is run for 3 104 s of simulation time in order to model the time-dependent, ⇥ spatially-averaged coronal temperature and density. For each simulation, the loop length is determined from the field extrapolation. Flux limiting, with a flux limiter

constant of f = 1, is included in the heat flux calculation (see Eqs. 21 and 22 of Klimchuk et al., 2008, and Section 5.2.1). Additionally, all of the energy is deposited into the electrons. To map the results back to the extrapolated field lines, a single temperature and density is assigned to every point along the strand at each time step. Though EBTEL only computes spatially-averaged quantities in the corona, its efficiency allows for the calculation of time-dependent solutions for many thousands of strands in a matter of minutes. 6.2 Modeling 169

6.2.3 Heating Model

The heating input is parameterized in terms of discrete heating pulses on a single

strand with triangular proﬁles of duration tevent = 200 s. For each event i, there are

two parameters: the peak heating rate qi and the waiting time prior to the event

twait,i. The waiting time is defined such that twait,i is the amount of time between when event i 1 ends and event i begins. Following the approach of Cargill (2014), the waiting time and the event energy are related such that twait,i µ qi. The physical motivation for this scaling is as follows. In the nanoflare model of Parker (1988), random convective motions continually stress the magnetic field rooted in the photosphere, leading to the buildup and eventual release of energy. If the field is stressed for a long amount of time without relaxation, large discontinuities will have time to develop in the field, leading to a dramatic release of energy. Conversely, if the field relaxes quickly, there is not enough time for the field to become sufficiently stressed and the resulting energy release will be relatively small. In this chapter I explore three different heating scenarios: low-, intermediate- , and high-frequency nanoflares. I define the heating frequency in terms of the ratio between the fundamental cooling timescale due to thermal conduction and

radiation, t , and the average waiting time of all events on a given strand, t , cool h waiti

< 1, high frequency, 8 twait > # = h i > 1, intermediate frequency, (6.1) tcool >⇠ <> > 1, low frequency. > > > The heating frequency is parameterized:> in terms of the cooling time rather than an absolute waiting time as t L (see appendix of Cargill, 2014). While a cool ⇠ waiting time of 2000 s might correspond to low-frequency heating for a 20 Mm strand, it would correspond to high-frequency heating in the case of a 150 Mm 6.2 Modeling 170 strand. Parameterizing the heating in this way ensures that all strands in the active region are heated at the same frequency relative to their cooling time. Figure 6.3 shows the heating rate, electron temperature, and density as a function of time, for a single strand, for the three heating scenarios listed above.

High Intermediate Low ]

s 25 3 erg cm 15 3 10 [ 5 Q 8

] 6 MK

[ 4 e

T 2 2 ] 3

cm 1 9 10 [ n 0 0 5000 10000 15000 20000 25000 30000 t [s]

Figure 6.3 Heating rate (top), electron temperature (middle), and density (bottom) as a function of time for the three heating scenarios for a single strand. The colors denote the heating frequency as deﬁned in the legend. The strand has a half length of L/2 40 Mm and a mean ﬁeld strength of B¯ 30 G. ⇡ ⇡

For a single impulsive event i with a triangular temporal proﬁle of duration tevent, the energy density is Ei = teventqi/2. Summing over all events on all strands that comprise the active region gives the total energy ﬂux injected into the active region, Nstrands Nl tevent Âl Âi qi Ll FAR = (6.2) 2 ttotal 6.2 Modeling 171

where ttotal is the total simulation time, Nstrands is the total number of strands comprising the active region, and N =(t + t )/(t + t ) is the total l total h waiti h waiti number of events occurring on each strand over the whole simulation. Note that

the number of events per strand is a function of both # and tcool. For each heating frequency, the total ﬂux into the active region is constrained

7 2 1 by F = 10 erg cm s (Withbroe & Noyes, 1977) such that FAR must satisfy the ⇤ condition, FAR/Nstrands F | ⇤| < d, (6.3) F ⇤ where d 1. For each strand, N events, each with energy E , are chosen from a ⌧ l i power-law distribution with slope 2.5. The upper bound of the distribution is ¯2 ¯ fixed to be Bl /8p, where Bl is the spatially-averaged field strength along the strand l as derived from the field extrapolation. This is the maximum amount of energy made available by the field to heat the strand. The lower bound on the power-law

distribution for Ei is then iteratively adjusted until Equation 6.3 is satisﬁed within

some numerical tolerance. Note that the set of Ei chosen for each strand may not uniquely satisfy Equation 6.3. The field strength derived from the potential field extrapolation is used to constrain the energy input to the EBTEL model for each strand. While the derived potential field is already in its lowest energy state and thus has no energy to give up, the goal here is only to understand how the distribution of field strength may be related to the properties of the heating. In this way, the potential field is used as a proxy for the non-potential component of the coronal field, with the understanding that no quantitative conclusions can be made regarding the amount of available energy or the stability of the field itself. In addition to these three multi-event heating models, I also run two single-event control models. In both control models every strand in the active region is heated ¯2 exactly once by an event with energy Bl /8p. In the first control model, the start 6.2 Modeling 172

Table 6.1 All three heating models plus the two single-event control models. In the single-event models, the energy ﬂux is not constrained by Equation 6.3.

Name # (see Equation 6.1) Energy Constrained? high 0.1 yes intermediate 1 yes low 5 yes cooling 1 event per strand no random 1 event per strand no

time of every event is t = 0 s such that all strands are allowed to cool uninterrupted

4 for ttotal = 10 s. In the second control model, the start time of the event on each strand is chosen from a uniform distribution over the interval [0, 3 104] s, such ⇥ that the heating is likely to be out of phase across all strands. In these two models, the energy has not been constrained according to Equation 6.3 and the total ﬂux into ¯2 the active region is (Âl Bl Ll)/8pttotal. From here on, I will refer to these two models as the “cooling” and “random” models, respectively. All ﬁve heating scenarios are summarized in Table 6.1.

6.2.4 Forward Modeling

Atomic Physics

For an optically-thin, high-temperature, low-density plasma, the radiated power per unit volume, or emissivity, of a transition lij of an electron in ion k of element X is given by, nH P(lij)= Ab(X)Nj(X, k) fX,k AjiDEjine, (6.4) ne where Nj is the fractional energy level population of excited state j, fX,k is the fractional population of ion k, Ab(X) is the abundance of element X relative to hydrogen, n /n 0.83 is the ratio of hydrogen and electron number densities, A H e ⇡ ji is the Einstein coefﬁcient, and DEji = hc/lij is the energy of the emitted photon 6.2 Modeling 173

(see Del Zanna & Mason, 2018; Mason & Fossi, 1994). To compute Equation 6.4,I use version 8.0.6 of the CHIANTI atomic database (Dere et al., 1997; Young et al., 2016) and the abundances of Feldman et al. (1992) as provided by CHIANTI. For

each atomic transition, Aji and lji can be looked up in the database. To ﬁnd Nj, the level-balance equations for ion k must be solved. The relevant excitation and de-excitation processes as provided by CHIANTI (see section 3.3 of Del Zanna & Mason, 2018) are included. See Section 3.2 for a detailed explanation of the emissivity calculation.

The ion population fractions, fX,k, provided by CHIANTI assume ionization equilibrium (i.e. the ionization and recombination rates are always in balance). However, in the rareﬁed solar corona, where the plasma is likely heated impulsively, it is not guaranteed that the ionization timescale is less than the heating timescale such that the ionization state may not be representative of the electron temperature (Bradshaw, 2009; Bradshaw & Cargill, 2006; Reale & Orlando, 2008).

To properly account for this effect, fX,k is computed by solving Equation 3.19, the time-dependent ion population equations, for each element using the ionization and recombination rates provided by CHIANTI. The details of this calculation are provided in Appendix B. See Section 3.2.5 for a more in-depth discussion of non-equilibrium charge states.

Instrument Effects

Equation 6.4 is combined with the wavelength response function of the instrument to model the intensity as it would be observed by AIA,

1 I = dhP(l )R (l ) (6.5) c 4p Â ij c ij ij ZLOS { } 6.2 Modeling 174

Table 6.2 Elements included in the calculation of Equation 6.5. For each element, all ions for which CHIANTI provides sufﬁcient data for computing the emissivity are included.

Element Number of Ions Number of Transitions O 8 11892 Mg 11 31965 Si 13 30047 S 16 33091 Ca 17 42823 Fe 25 553541 Ni 19 83517

where Ic is the intensity for a given pixel in channel c, P(lij) is the emissivity as

given by Equation 6.4, and Rc is the wavelength response function of the instrument for channel c (see Figure 3.7 and Boerner et al., 2012). Additionally, ij is the set of { } all atomic transitions listed in Table 6.2 and the integration is along the LOS. When computing the intensity in each channel of AIA, the wavelength response functions are used directly rather than relying on the temperature response functions computed by SolarSoft (SSW, Freeland & Handy, 1998). As discussed in Section 6.2.4, the assumption of ionization equilibrium is likely to be violated in the impulsive heating cases considered here. Thus, the contributions of each ion to the total channel response must be recomputed using the result of Equation B.4 in place of the equilibrium ion population fractions. Figure 6.4 shows the effective temperature response functions for the six AIA EUV channels compared to those

calculated from aia_get_response.pro in SSW. Even though only a limited number of transitions from the CHIANTI database are included (see Table 6.2), nearly all of the response is recovered in each channel. The high-temperature contributions in the SSW results are due to continuum emission which is not included here. In all cases, the continuum contribution is several orders of magnitude below peak of the channel response. Additionally, the time variation in the wavelength response 6.2 Modeling 175

O Mg Si S Ca Fe Ni

94 Å 131 Å 171 Å 25 10

27 10

29 10 ] 1 193 Å 211 Å 335 Å 25 10 pixel 1 s

5 27 10

DN cm 29

[ 10 c

K 105 106 107 105 106 107 105 106 107 T [K]

Figure 6.4 SSW temperature response functions (solid black) and effective temperature response functions for the elements in Table 6.2 (dashed black) for all six EUV AIA channels. The colored, dashed curves, as indicated in the legend, denote the contributions of the individual elements to the total response. For this calculation, I have assumed equilibrium ionization and a constant pressure of 15 3 10 K cm . The time-varying degradation of the instrument is not included.

functions due to the degradation of the detector is not included (see Section 2.1.6 of Boerner et al., 2012). Furthermore, during the evolution of a strand, the pressure is not constant for any of the heating scenarios as evidenced by Figure 6.5. The black line of

15 3 constant pressure p = 10 K cm shows the default pressure at which the SSW AIA response functions are evaluated. The other lines show the temperature-density phase space evolution for the high-, intermediate-, and low-frequency cases for a single strand, none of which is well described by the assumption of constant pressure. Recomputing and interpolating the emissivity to the temperatures and 6.2 Modeling 176

High Intermediate Low

109 ] 3 cm [ n 108

106 107 T [K]

Figure 6.5 n T phase-space orbits for a single strand for the ﬁrst three heating 15 3 scenarios in Table 6.1. The black line indicates a constant pressure of 10 K cm .

densities as deﬁned by the hydrodynamic simulation ensures that all quantities in Equation 6.4 are evaluated at the correct temperature and density.

Mapping Back to the Magnetic Field

I compute the emissivity according to Equation 6.4 for all of the transitions in Table 6.2 using the temperatures and densities from the hydrodynamic models for all 5 103 strands. I then compute the LOS integral in Equation 6.5 by ﬁrst ⇥ converting the coordinates of each strand to a helioprojective (HPC) coordinate frame (see Thompson, 2006) using the coordinate transformation functionality in Astropy (The Astropy Collaboration et al., 2018) combined with the solar coordinate frames provided by SunPy (SunPy Community et al., 2015). Thus, the simulated active region can easily be projected along any arbitrary LOS simply by changing the location of the observer that deﬁnes the HPC frame. Here, the HPC frame 6.3 Results 177

is deﬁned by an observer at the position of the SDO spacecraft on 12 February 2011 15:32:42 UTC (i.e. the time of the HMI observation of NOAA 1158 shown in Figure 6.1). See Section 4.1.3 for a detailed discussion of solar coordinate systems. Next, I compute a weighted two-dimensional histogram from the transformed coordinates, using the integrand of Equation 6.5 at each coordinate as the weights. The histogram is constructed such that the bin widths are consistent with the spatial resolution of the instrument. For AIA, a single bin, representing a single pixel, has

1 a width of 0.6 00 pixel . Finally, I apply a Gaussian ﬁlter to the resulting histogram to emulate the point spread function of the instrument. This is done for each time step, using a cadence of 10 s, and for each channel. For every heating scenario, this produces approximately 6(3 104)/10 2 104 separate images. This procedure ⇥ ⇡ ⇥ is discussed in greater detail in Section 4.4.

6.3 Results

I forward model time-dependent AIA intensities using the method outlined in Section 6.2.4 for the heating scenarios given in Table 6.1. I discuss the predicted intensities in Section 6.3.1 for all six EUV channels of AIA and all ﬁve heating models. In Section 6.3.2 and Section 6.3.3, I show the results of the emission measure and time lag analyses, respectively, applied to the simulated data. In Chapter 7, these simulated observables are used to train a machine learning classiﬁcation model to understand with which heating scenario the real data are most consistent.

6.3.1 Intensities

I compute the intensities for the 94 Å, 131 Å, 171 Å, 193 Å, 211 Å and 335 Å channels of AIA using the procedure described in Section 6.2.4. The intensity in each pixel of the model active region is computed for a total simulation period of 3 104 s 8.3 h ⇥ ⇡ 6.3 Results 178 with the exception of the cooling case which is only run for 104 s. For the high-, intermediate-, and low-frequency and “random” models, the ﬁrst and last 5 103 s ⇥ of evolution are discarded to avoid any transient effects in the strand evolution associated with the initial conditions and the constraints placed on the energy, respectively. I complete this procedure for each of the ﬁve heating scenarios in Table 6.1.

50 100 300 600 10002000 600 1200 300 600 50 100

High 00

0 ˚ ˚ ˚ ˚ ˚ ˚ . 94A 131A 171A 193A 211A 335A Intermediate 200 00 0 . 300 Low Helioprojective Latitude

300.000 Helioprojective Longitude

1 1 Figure 6.6 Snapshots of intensity, in DN pixel s , across the whole active region at t = 15 103 s. The rows correspond to the three different heating frequencies ⇥ and the columns are the six EUV channels of AIA. In each column, the colorbar is on a square root scale and is normalized between zero and the maximum intensity in the low-frequency case. The color tables are the standard AIA color tables as implemented in SunPy (SunPy Community et al., 2015).

Figure 6.6 shows a snapshot of the intensity at t = 15 103 s for each channel ⇥ and for the high-, intermediate-, and low-frequency nanoflare heating cases. The rows correspond to the different heating scenarios while the columns show the six channels. In each column, the intensities are normalized to the maximum intensity in the low-frequency case and are on a square root scale. In general, I find that in the high-frequency intensity maps, individual loops are difficult to distinguish while in 6.3 Results 179

Table 6.3 sI /I¯ as deﬁned by Equation 11 of (Guarrasi et al., 2010) computed on a single image at t = 15 103 s for each channel and heating frequency. A larger ⇥ value denotes a greater degree of contrast.

Channel [Å] High Intermediate Low 94 3.06 4.63 4.19 131 5.56 3.61 6.13 171 2.79 2.81 3.25 193 2.69 2.80 2.79 211 2.73 2.83 2.84 335 2.63 3.08 3.21

the low-frequency case individual loops appear bright relative to the surrounding emission. This distinguishability or “fuzziness” can be measured quantitatively as sI/I¯, where sI is the standard deviation taken over all pixels and I¯ is the mean intensity (Guarrasi et al., 2010, Equation 11). A larger value of sI/I¯ indicates a greater degree of contrast and vice versa. sI/I¯ for each channel and heating frequency is shown in Table 6.3. With the exception of 131 Å, for every channel, the high-frequency case is the most “fuzzy”. The low-frequency case shows the most contrast in each channel except 94 Å though the margin between the low and intermediate cases is quite small in some cases. Looking at the first two columns of Figure 6.6, I find that the intensity in the 94 Å and 131 Å channels increases as the heating frequency decreases. Both channels are double peaked (see Figure 3.8) and have “hot” ( 7 MK for 94 Å, 12 MK ⇡ ⇡ for 131 Å) and “warm” ( 1 MK for 94 Å, 0.5 MK for 131 Å) components. In ⇡ ⇡ the case of high-frequency heating, less energy is available per event such that few strands are heated to > 4 MK. There is little emission in the 131 Å channel as strands are not often permitted to cool to 0.5 MK either. However, in the low-  and intermediate-frequency cases, several individual bright loops appear in both the 94 Å and 131 Å channels as the heating rate is sufficient to produce “hot” (i.e. 8 MK to 10 MK) loops. Only a few of these loops are visible as the lifetime of this 6.3 Results 180

hot plasma is short due to the efficiency of thermal conduction. In contrast, the faint, diffuse component of the 94 Å emission that is present in all three cases is due to the contribution of the “warm” component. Additionally, I find that the 171 Å channel is dimmer for high-frequency heating as the peak sensitivity of this channel is < 1 MK and in the case of high-frequency heating, strands are rarely allowed to cool below 1 MK. In contrast, the overall intensity in the 193 Å, 211 Å and 335 Å channels is relatively constant over heating frequency as compared to the three previous channels though individual loops do become more visible with decreasing heating frequency. This relative insensitivity is because the temperature response functions of these three channels all peak in between 1.5 MK and 2.5 MK. In the case of high-frequency heating, strands are being sustained near these temperatures while in the low-frequency case, strands are cooling through this temperature range. This is illustrated for a single strand in Figure 6.3. While there are clear differences in the AIA intensities between all three heating frequencies, quantifying these differences is difficult due in part to the multidi- mensional nature of the intensity data. To better understand how observational signatures differ as a function of heating frequency, it is useful to find a reduced representation of the data that retains signatures of the underlying energy deposition. To this end, I compute two common observables: the emission measure slope (Section 6.3.2) and the time lag (Section 6.3.3).

6.3.2 Emission Measure Slopes

As discussed in Section 6.1, the emission measure slope is a useful quantity for understanding how frequently strands are re-energized. I compute emission measure distributions from the forward-modeled intensities using the regularized inversion method of Hannah & Kontar (2012). This method was designed to work with the 6.3 Results 181

narrowband coverage provided by AIA and so is well-suited to the data at hand. The temperature bins are chosen such that the leftmost edge is at 105.5 K and the

rightmost edge at 107.2 K with bin widths of D log T = 0.1. Rather than computing

EM(Te) at each time step, I compute the time-averaged intensity in each pixel of

each channel and compute EM(Te) only once. I compute the uncertainties on the intensities in each channel using the aia_bp_estimate_error.pro procedure in SSW which incorporates uncertainties due to shot noise, read noise, dark subtraction, quantization, photometric calibration, and onboard compression. After comput-

ing EM(Te) in each pixel using the regularized inversion procedure, a ﬁrst-order polynomial is ﬁt to the log-transformed emission measure and temperature bin

centers, log EM a log T, in order to calculate the emission measure slope, a. In 10 ⇠ 10 Chapter 7, I compare the modeled emission measure slopes to those derived from real AIA observations of NOAA 1158 using this same method.

2 3 4 5 6 7

00 High Intermediate Low 0 . 200 00 0 . 300

Helioprojective Latitude 400.000 300.000 Helioprojective Longitude

Figure 6.7 Maps of the emission measure slope, a, in each pixel of the active region for the high- (left), intermediate- (center), and low-frequency (right) cases. The EM(Te) is computed using time-averaged intensities from the six AIA EUV channels using the method of Hannah & Kontar (2012). The EM(Te) in each pixel is then ﬁt to Ta over the temperature range 8 105 K T < T . Any pixels ⇥  peak with r2 < 0.75 are masked and colored white.

Figure 6.7 shows the resulting emission measure slope, a, in each pixel of the forward-modeled active region for the high-, intermediate-, and low-frequency 6.3 Results 182

cases. I fit EM(T ) over bins in the temperature range 8 105 K T < T , e ⇥  peak where Tpeak = argmaxT EM(T) is the temperature at which the emission measure distribution peaks. r2, the correlation coefficient for the first-order polynomial fit, is used to assess the “goodness-of-fit” and pixels with r2 < 0.75 are masked. Looking at the three panels in Figure 6.7, overall a tends to decrease with decreasing frequency, consistent with previous modeling work (see Section 6.1). The low- frequency map (right panel) shows many values close to 2. As the heating frequency increases, the slopes become larger, indicating an increasingly isothermal emission measure distribution. The intermediate-frequency map (center panel) shows predominantly higher slopes, with most pixels in the

range 2 . a . 3.5 while the high-frequency map (left panel) shows much steeper slopes, most a 3.5, and a much broader range of slopes, 3 a 8. Note that . . in the high frequency case, the slope varies considerably across the active region while the distribution of a appears more spatially uniform in the intermediate- and low-frequency cases.

Below T , Cargill (1994) noted that the EM(T ) could be described by EM(T) peak e ⇠ n2t , where t T1 an 1 is the radiative cooling time. Additionally, Bradshaw rad rad ⇠ & Cargill (2010b) found that T n`, with ` 1 for long loops and ` 2 for short ⇠ ⇡ ⇡ loops. Combining these expressions and assuming a = 1/2 (Cargill, 1994) gives a 2 for short loops and a 2.5 for long loops in the case of single nanoflares. Emis- ⇡ ⇡ sion measure slopes produced by low-frequency nanoflares as shown in the right panel of Figure 6.7 are thus consistent with analytical results for single nanoflares. Figure 6.8 shows histograms of the emission measure slopes for the high-, intermediate-, and low-frequency cases. I find that the low-frequency distribution peaks at a 2.3, inside the range expected from analytical results (see above). The ⇡ intermediate- and high-frequency distributions peak at successively higher values, 2.8 and 4.0, respectively. While the low- and intermediate-frequency distri- ⇡ ⇡ 6.3 Results 183

High 1.2 Intermediate Low 1.0

0.8

0.6

0.4

0.2 Number of Pixels (Normalized)

2 3 4 5 6 7 a

Figure 6.8 Distribution of emission measure slopes, a, for every pixel in the simulated active region for the high-, intermediate-, and low-frequency heating scenarios as shown in Figure 6.7. The histogram bins are determined using the Freedman Diaconis estimator (Freedman & Diaconis, 1981) as implemented in the Numpy package for array computation in Python (Oliphant, 2006) and each histogram is normalized such that the area under the histogram is equal to 1.

butions are more narrowly distributed around their peak values, the distribution of slopes in the high-frequency case is relatively broad and has a positive skew towards steeper slopes. Looking at the distribution of slopes across the entire active region in Figure 6.8, I ﬁnd that when the strands are heated infrequently (low frequency) such that each strand is allowed to cool fully prior to the next event, the distribution of slopes “saturates” in the range expected for single nanoﬂares. However, when the strands are reheated often (high frequency), the value of the slope becomes unsaturated and is subject to a range of infrequent cooling times due to the dependence of each waiting time on the power-law heating rate (see Section 6.2.3). These results are

consistent with Cargill (2014) who computed EM(T)=n2L for a single strand for 6.3 Results 184

a range of heating frequencies and found a converged to 2 for low-frequency ⇡ nanoﬂares and increased slowly with increasing heating frequency. The modeled emission measure slopes show that, even when accounting for

the LOS integration, atomic physics, and information lost in the EM(Te) inversion, signatures of the heating frequency still persist in the emission measure slope. However, while this quantity retains information about the frequency of energy deposition, drawing conclusions about the heating based solely on the observed emission measure slope, particularly for a small number of pixels may be misleading. As shown here and in Del Zanna et al. (2015b), the slope may vary signiﬁcantly

across a given active region. Additionally, calculating EM(Te) from observations is non-trivial due to several factors, including the mathematical difﬁculties of the ill- posed inversion (Craig & Brown, 1976; Judge et al., 1997, 1995), uncertainties in the atomic data (Guennou et al., 2013), and insufﬁcient constraints from spectroscopic observations (e.g. Landi & Klimchuk, 2010; Winebarger et al., 2012), among others.

6.3.3 Time Lags

Next, I apply the time lag method of Viall & Klimchuk (2012) to the predicted intensities for each heating scenario in Table 6.1. For each pixel in the active region, I compute the cross-correlation (Equation 3.54) for all pairs of AIA channels (15 in total) and ﬁnd the temporal offset which maximizes the cross-correlation according to Equation 3.55. The details of the cross-correlation and time lag calculation are given in Section 3.6.1. I consider all possible offsets over the interval 6h. Using ± the convention of Viall & Klimchuk (2012), the channel pairs are ordered such that the “hot” channel is listed ﬁrst, meaning that a positive time lag corresponds to variability in the hotter channel followed by variability in the cooler channel. In

other words, a positive time lag indicates cooling plasma. For the 94 Å and 131 Å channels, both of which have a bimodal temperature response function (see Figure 3.8), 6.3 Results 185 the order is determined by the component which is most dominant such that 94 Å is ordered ﬁrst while 131 Å is ordered second. Thus, it is possible for cooling plasma to produce negative time lags in these channel pairs and the ambiguity can be resolved in the context of the time lags in other channel pairs.

Time-Lag Maps

4000 2000 0 2000 4000 94-335 A˚

00 High Intermediate Low Random Cooling 0 . 211-131 A˚ 200

193-171 A˚ Helioprojective Latitude

400.000 200.000 Helioprojective Longitude

Figure 6.9 Time lag maps for three different channel pairs for all ﬁve of the heating models described in Table 6.1. The value of each pixel indicates the temporal offset, in s, which maximizes the cross-correlation (see Equation 3.55). The rows indicate the different channel pairs and the columns indicate the varying heating scenarios. The range of the colorbar is 5000 s. If max < 0.1, the pixel is masked and ± CAB colored white.

Figure 6.9 shows tAB (Equation 3.55) in each pixel of the simulated active region for all heating scenarios listed in Table 6.1 and three selected channel pairs: 94-335, 211-131, and 193-171 Å. Blacks, blues, and greens correspond to negative time lags; reds, oranges, and yellows correspond to positive time lags; and olive green 6.3 Results 186

indicates zero time lag. The range of the colorbar is 5000 s. Note that the heating ± frequency decreases from left to right across each row. If the correlation in a given pixel is too low (max < 0.1), the pixel is masked and colored white. CAB Looking at the first two rows of Figure 6.9, I find that the positive time lags in the 211-131 Å channel pair are significantly longer than those in the 94-335 Å pair. In the temperature range 2.5 < T < 7.3MK (94-335 Å), the dominant cooling mechanism is field-aligned thermal conduction while radiative cooling dominates

in the range 0.6 < T < 2.5MK (211-131 Å). Because thermal conduction is far

more efﬁcient at high temperatures, the plasma spends less time in the [T335, T94]

temperature range than in [T131, T211]. The 193-171 Å time lags for the cooling case fall in the middle as radiative cooling also tends to dominate in this temperature

interval (0.9 < T < 1.5MK), but the separation in temperature space is smaller than the 211-131 Å pair. In all cases, these differences in the magnitude of the positive time lags become more apparent at lower heating frequencies. In the 94-335 Å pair, I find negative time lags in the longest loops near the edge of the active region, inconsistent with the previous assertion that longer loops lead to longer, positive time lags. These loops are rooted in areas of weaker magnetic field (compared to the center) and thus do not have sufficient energy to evolve significantly into the temperature range of the “hot” component of the 94 Å channel (see Section 6.2.3). Thus, cooling from 335 Å to the cooler part of 94 Å dominates the cross-correlation. These negative time lags become more prominent as the heating frequency decreases. The results in the cooling case are consistent with the negative 94-335 Å time lags of similar magnitude observed by Viall & Klimchuk (2017) in this same active region and the tendency for longer field lines to exhibit cooler plasma, though Viall & Klimchuk found far fewer positive 94-335 Å time lags. I also find negative 211-131 Å time lags in the center of the active region for the high-, intermediate-, and low-frequency cases, indicative of plasma cooling 6.3 Results 187

from the hot part of the 131 Å channel through the 211 Å channel. Though they are not shown here, similar negative time lag signatures are present in nearly all of the other 131 Å channel pairs as well. These results are consistent with that of Cadavid et al. (2014) who found that in inter-moss regions of active region NOAA 11250, intensity variations in the 131 Å channel preceded brightenings in all other EUV channels. In the two control cases, I do not find any negative time lags as the cross-correlations in the core are dominated by uninterrupted cooling from 211 Å to the cool part of 131 Å. For the 193-171 Å channel pair, I find very few negative time lags because, unlike the 94 Å and 131 Å channels, the 193 Å and 171 Å channels are strongly peaked about a single temperature. Along with 211 Å, these channels are important for disambiguating the signals in channels with a bimodal temperature response function. These simulated time lags show far fewer zero time lags than the observations of Viall & Klimchuk (2012, 2017) and the modeling work of Bradshaw & Viall (2016) due to the lack of TR emission in the model. As shown by Viall & Klimchuk (2015), TR emission shows near-zero time lags because every layer (or temperature) of the TR responds in unison. However, for the 193-171 Å channel pair, I find that zero time lags still dominate the inner core of the active region for all five heating scenarios, suggesting that the plasma is cooling into, but not through the 171 Å temperature bandpass (Viall & Klimchuk, 2017). This underscores the point that zero time lags do not imply steady heating (Viall & Klimchuk, 2015, 2016).

Cross-Correlation Maps

Figure 6.10 shows the peak cross-correlation value, max , for each selected CAB channel pair. Looking ﬁrst at all three channel pairs, the cross-correlation, on average, increases as the heating frequency decreases. Additionally, the highest 6.3 Results 188

0.0 0.2 0.4 0.6 0.8 1.0

94-335 A˚

00 High Intermediate Low Random Cooling 0 . 211-131 A˚ 200

193-171 A˚ Helioprojective Latitude

400.000 200.000 Helioprojective Longitude

Figure 6.10 Same as Figure 6.9 except each pixel shows the maximum cross- correlation, max . CAB

cross-correlations tend to be in the center of the active region while the lowest tend to be on the outer edge. Comparing Figure 6.10 with the time lags in Figure 6.9 also reveals that negative time lags are correlated with lower peak cross-correlation values. Furthermore, other than the “cooling” scenario, there are large variations from one loop to the next for all heating frequencies such that the spatial coherence of these peak cross-correlation values is low. In Chapter 7, I use the peak cross- correlation value, in addition to the time lag, to classify the heating frequency in each observed pixel.

Histograms

Figure 6.11 shows histograms of time lags for every channel pair and all ﬁve heating scenarios. The time lags are binned between 104 s and 104 s in 60 s bins. Each histogram is colored according the corresponding heating function, consistent with 6.4 Discussion 189

Figure 6.3 and Figure 6.8. The columns are arranged such that heating frequency decreases from left to right. Every channel pair and every heating model is shown in order to demonstrate how the distribution of time lags evolves as the heating frequency varies. Note that as the frequency decreases (from left to right), the number of negative time lags decreases. In the “cooling” case, there are very few negative time lags except for channel pairs which include one or both of the double-peaked channels (94 Å and 131 Å). For those channel pairs which include 94 Å and/or 131 Å, negative time lags are expected, even in the single-nanoﬂare cooling case as the convention of ordering the “hot” channel ﬁrst has been violated such that cooling plasma can lead to negative time lags. For the remaining channel pairs, negative time lags are associated with the heating and cooling cycle being interrupted by repeated events on a given strand.

6.4 Discussion

For all of the heating models, I ﬁnd negative time lags in at least one of the three channel pairs as shown in Figure 6.9. Negative time lags can be used to disam- biguate the temperature sensitivity of the AIA passbands and can be produced in one of two ways: high-frequency heating in which the time lag is dominated by many frequent reheatings or a channel pair in which one channel is bimodal. While intensity in the 131 Å channel can correspond to either 0.4 MK or > 10 MK plasma (see Figure 3.8), negative time lags in the 211-131 Å channel pair provide a possible signature of 10 MK plasma because a negative time lag implies the plasma is cooling from 131 Å to 211 Å. This also holds for the 171-131 and 193-131 Å channel pairs as well while the 94-131 and 335-131 Å channels are more ambiguous due to the ﬁrst channel in the pair being bimodal as well. As noted in Section 6.3.3, the 6.4 Discussion 190

High Intermediate Low Random Cooling 94-335 94-171 94-193 94-131 94-211 335-131 335-193 335-211 335-171 211-131 211-171 211-193 193-171 193-131 103

171-131 10000 0 10000 tAB [s]

Figure 6.11 Histograms of time lag values across the whole active region. The rows indicate the different channel pairs and the columns indicate the different heating models. Colors are used to denote the various heating models. The black dashed line denotes zero time lag. The bin range is 104 s and the bin width is ± 60 s. As with the time-lag maps, time lags corresponding to max < 0.1 are CAB excluded. high-, intermediate-, and low-frequency maps for the 211-131 Å channel pair all show coherent negative time lags in the core. Because these strands are rooted in areas of strong ﬁeld, enough energy is made available by the ﬁeld (see Section 6.2.3) to heat them well into (and likely above) the hot component of the 131 Å passband. Since these strands are relatively short, the density increases rapidly enough for this hot plasma to be visible before it is washed out by thermal conduction. 6.4 Discussion 191

Plasma undergoing pure cooling by radiation and thermal conduction produces a predictable and well-understood time lag signature. However, complicated heating scenarios and LOS geometries are likely to make it more difﬁcult to interpret observed time lag signatures. Consider the case of a single cooling strand such that

the maximum allowed time lag for a given channel pair AB is the amount of time

it takes to cool from TA to TB by thermal conduction and radiation. The “cooling” case in Figure 6.11 may be regarded as the baseline time lag distribution given

that all strands were heated only once at t = 0s. Because the time lag is primarily determined by the cooling phase of the strand, the time lag becomes primarily

a function of the loop length L since tcool µ L. Two effects are likely to increase the decoherence of the baseline time lag distribution: multiple structures evolving out-of-phase along a given LOS (the “random” heating scenario) and multiple reheatings before the end of the cooling and draining cycle on a given strand. Note that multiple polluting structures along the LOS seem to primarily add negative time lags to the distribution (the “random” case) while increasing the frequency of events on a given strand extends the distribution in the positive direction. The latter effect also produces more negative time lags. Since steady heating can be thought of as nanoflare heating in the high-frequency limit, the distribution of time lags is expected to approach a uniform distribution as the heating frequency increases. This is again consistent with the results of Viall & Klimchuk (2016) who found that steadily-heated loops have no preferred time lag. While the model for the energy deposition presented here (see Section 6.2.3) does not assume any specific physical heating mechanism, the parameterization of the heating frequency in Equation 6.1 has an interesting implication in the context of the Parker (1988) nanoflare model. Rearranging Equation 6.1 and recalling that

t µ L gives t µ L, i.e. longer strands have longer absolute waiting times cool h waiti between heating events. Given that longer field lines tend to be rooted in regions of 6.5 Summary 192 weaker magnetic field, this further implies that, where the field is stronger, energy is more quickly dissipated. According to Parker (1988), this dissipation is due to small-scale reconnection of flux tubes that are continually stressed by the convective motion of the underlying photosphere. Thus, in this context, this heating model implies that the reconnection and the underlying driver are more efficient in areas where the field is strongest. Though I have not addressed it here, another possible mechanism for producing time-varying intensity in active regions is thermal non-equilibrium (TNE) wherein condensation cycles driven by highly-stratified, but steady footpoint heating lead to long-period intensity pulsations (Kuin & Martens, 1982). Though originally used to explain coronal rain (Antolin et al., 2010, 2015; Auchère et al., 2018) and prominences (Antiochos & Klimchuk, 1991), several workers (Froment et al., 2017, 2018; Mok et al., 2016; Winebarger et al., 2018, 2016) have recently claimed that TNE can produce time lag signatures similar to those of impulsive heating models, suggesting that observed time lags may be consistent with both impulsive and steady heating. However, it is not yet clear whether TNE is consistent with observed signatures of very hot (8-10 MK) plasma. Detailed comparisons between TNE and nanoflare simulations and observations are desperately needed. See Section 8.2.2 for additional discussion on TNE as a possible model for explaining observed active region variability.

6.5 Summary

I have carried out a series of numerical simulations in an effort to understand how signatures of the nanoﬂare heating frequency are manifested in two observables: the emission measure slope and the time lag. For a given magnetogram observation of the relevant active region (in this case, NOAA 1158), I compute a potential ﬁeld 6.5 Summary 193

extrapolation and trace a large number of field lines through the extrapolated vector field. For each traced field line, an EBTEL hydrodynamic simulation is run and the resulting temperatures and densities, combined with data from CHIANTI and the instrument response function, are used to compute the time-dependent intensity. These intensities are then mapped back to the magnetic skeleton and integrated along the LOS in each pixel to create time-dependent images of the active region. Using the novel and efficient forward modeling pipeline, I produced AIA images for all six EUV channels for 8hof simulation time. From these results, I computed ⇡ both the emission measure slope and the time lag for all possible channel pairs for three different nanoflare heating frequencies, high, intermediate, and low, (see Equation 6.1) in addition to two control models, for a total of five different heating scenarios (see Table 6.1). The results of this study can be summarized in the following points:

1. As the heating frequency decreases, the emission measure slope, a, becomes increasingly shallow, saturating at a 2. As the heating frequency increases, ⇡ the distribution of slopes over the active region is shifted to higher values and broadens.

2. The time lag becomes increasingly spatially coherent with decreasing heating frequency. When strands are allowed to cool without being re-energized, the spatial distribution of time lags is largely determined by the distribution of loop lengths over the active region.

3. The distribution of time lags becomes increasingly broad and approaches a uniform distribution as the heating frequency increases, consistent with the results of Viall & Klimchuk (2016).

4. Negative time lags in channel pairs where the second (“cool”) channel is 131 Å provide a possible diagnostic for 10 MK plasma 6.5 Summary 194

In this chapter, I have used the advanced forward modeling pipeline, as described in Chapter 4, to systematically examine how the emission measure slope and time lag are affected by the nanoﬂare heating frequency. In Chapter 7 I use the model results presented here to train a random forest classiﬁcation model and apply it to emission measure slopes and time lags derived from real AIA observations of NOAA 1158. The 15 channel pairs for the time lag and cross-correlation combined with the emission measure slope represent a 31-dimensional feature space and a single 500-by-500 pixel active region amounts to 2.5 105 sample points. Perform- ⇥ ing an accurate assessment over this amount of data manually or “by eye” is at least impractical and likely impossible. Thus, the application of machine learning to the problem of assessing models in the context of real data is a critical step in understanding the underlying energy deposition in active region cores. Chapter 7

Mapping the Heating Frequency in Active Region NOAA 11158

In this chapter, I use a machine learning classiﬁcation model to classify the heating frequency in each observed pixel of active region NOAA 1158. In particular, I train a random forest classiﬁer using the predicted emission measure slopes, time lags, and cross-correlations from Chapter 6. I then classify each pixel of NOAA 1158 as consistent with high-, intermediate-, or low-frequency heating based on the emission measure slopes, time lags, and cross-correlations computed from the real AIA data. This chapter is adapted directly from Barnes et al. (2019b) which is currently in preparation for publication.

7.1 Introduction

A central problem in the study of the solar corona is whether EUV and soft X-ray observations of active regions imply the plasma is heated steadily or impulsively. Observations of hot plasma by the X-Ray Telescope (XRT, Golub et al., 2007) on the

Hinode spacecraft (Kosugi et al., 2007) indicate that active region cores are heated

195 7.1 Introduction 196

steadily (e.g. Warren et al., 2011; Winebarger et al., 2011). Alternatively, observations of cooler ( 1 MK) plasma have been shown to be more consistent with impulsive ⇠ heating where the plasma is allowed to cool signiﬁcantly (Mulu-Moore et al., 2011b; Ugarte-Urra et al., 2006; Viall & Klimchuk, 2011, 2012; Winebarger et al., 2003, e.g). More recent work (Bradshaw & Viall, 2016; Del Zanna et al., 2015b) suggests that observations of a single active region may be consistent with both steady and impulsive heating, depending on the location within the active region. Collectively, these results suggest that active regions are heated by a range of frequencies. Two often-used diagnostics of the frequency of energy deposition in the coronal plasma are the cool emission measure slope and the time lag, the temporal offset which maximizes the cross-correlation between pairs of imaging channels. The

2 emission measure distribution, EM(T)= dhne , where ne is the electron density and the integration is taken along the LOS,R is well-described by the power-law rela-

tionship EM(T) Ta, for a > 0, over the temperature range 105.5 K T 106.5 K ⇠ . . (Jordan, 1975, 1976). a, the emission measure slope in log log space, parameterizes the width of the emission measure distribution and is a commonly used diagnostic for the heating frequency (e.g. Bradshaw et al., 2012; Del Zanna et al., 2015b; Mulu-Moore et al., 2011a; Reep et al., 2013; Schmelz & Pathak, 2012; Tripathi et al., 2011; Warren et al., 2011; Winebarger et al., 2011). Section 3.5.1 provides a detailed discussion of the emission measure slope. The time-lag analysis of Viall & Klimchuk (2012) provides an additional diagnostic of the heating frequency. Viall & Klimchuk (2011) showed that, as the plasma cools, the intensity will peak in successively cooler passbands of the Atmospheric Imaging Assembly (AIA, Lemen et al., 2012). The temporal offset which maximizes the cross-correlation between these intensities is a proxy for the cooling time of the plasma between these channels. Provided the “hot” channel precedes the “cool” 7.1 Introduction 197

channel, cooling plasma produces a positive time lag. Additional details about the time lag and the calculation of the cross-correlation are given in Section 3.6. Any viable heating model must account for the range of observed emission measure slopes and time lags (Viall & Klimchuk, 2017). However, accurately predicting the distributions of these observables for a given heating model is challenging as several factors are likely to impact these diagnostics, including multiple emitting structures along the LOS and non-equilibrium ionization (e.g. Barnes et al., 2016b). In Chapter 6, I forward modeled emission from active region NOAA 1158 as observed by the six EUV channels of AIA. Using a potential field extrapolation combined with 5 103 separate instances of the Enthalpy-based Thermal Evolution ⇥ of Loops model (EBTEL, Barnes et al., 2016b; Cargill et al., 2012a,b; Klimchuk et al., 2008), I predicted time-dependent intensities in each pixel of the active region for a range of nanoflare heating frequencies. The heating frequency is defined in terms of the dimensionless ratio

< 1, high frequency, 8 twait > # = h i > 1, intermediate frequency, (6.1) tcool >⇠ <> > 1, low frequency, > > > :> where tcool is the fundamental cooling timescale due to thermal conduction and radiation (see Appendix of Cargill, 2014) and t is the average waiting time h waiti between consecutive heating events on a given strand. As in Chapter 6, I deﬁne a

strand to be a ﬂux tube with the largest possible isothermal cross-section and the fundamental unit of the corona while a loop is an intensity enhancement relative to the background diffuse emission and an observationally-deﬁned feature. From the predicted intensities, I computed the emission measure slope as well as the time lag and the maximum cross-correlation for all 15 AIA channel pairs and 7.1 Introduction 198

found that signatures of the heating frequency persist in both the emission measure slope and the time lag. In particular, I found that negative time lags where the “cool” channel is 131 Å provide a possible diagnostic for 10 MK plasma. While such predicted diagnostics are useful in understanding how observables respond to the frequency of energy deposition, systematically assessing real observations in terms of said model results is nontrivial. Attempts to tune model parameters to exactly match a single observation (e.g. a light curve from a single pixel) are not likely to generalize well to other data (“overfitting”). Additionally, purely qualitative comparisons between real data and forward models provide no constraint on the observation with respect to the model inputs, regardless of how sophisticated the simulation may be. Because of the ability to learn non-linear relationships from arbitrary data, machine learning is an excellent tool for systematically comparing observations and simulations for a range of input parameters. Machine learning has a variety of applications in solar physics, from predicting coronal mass ejections (e.g. Bobra & Ilonidis, 2016) to deconvolving magnetograms (Baso & Ramos, 2018). In particular, Tajfirouze et al. (2016b) trained a probabilistic neural network (PNN) on > 105 modeled 94 Å and 335 Å AIA light curves simulated using EBTEL for a large parameter space of heating properties. They found that a sample of observed light curves were most consistent with many frequent events drawn from a power-law distribution with index a = 1.5 though the overall agreement between the best fit and the observation was poor. Combined with predicted observables from sophisticated forward models, systematic comparisons using machine learning methods are well-poised to place strong constraints on heating properties in active regions. In this chapter, I train a random forest to classify the heating frequency in each pixel of active region NOAA 1158 using the predicted emission measure slopes, time lags, and maximum cross-correlations from Chapter 6. Section 7.2 describes 7.2 Observations and Analysis 199

how the full 12 h of multi-wavelength AIA observations are processed as well as how the emission measure slopes (Section 7.2.1) and time lags (Section 7.2.2) are computed. Section 7.3 describes the random forest classiﬁcation model and the data preparation procedure (Section 7.3.1) and Section 7.3.2 and 7.3.3 show the predicted heating frequency in each pixel for several different combinations of features. Finally, I discuss the results of the classiﬁcation model in Section 7.4 and provide some concluding comments in Section 7.5.

7.2 Observations and Analysis

I analyze 12 h of AIA observations of active region NOAA 1158 in six EUV channels, 94 Å, 131 Å, 171 Å, 193 Å, 211 Å and 335 Å, beginning at 2011 February 12 12:00:00 UTC and ending at 2011 February 13 00:00:00 UTC. The active region was chosen from the catalogue of active regions originally compiled by Warren et al. (2012) and was also studied by Viall & Klimchuk (2017). The full-disk, level-1 AIA data in FITS ﬁle format are obtained from the Joint Science Operations Center (JSOC, Couvidat et al., 2016) archive at the full instrument cadence of 12 s and the full

1 spatial resolution of 0.6 00 pixel . This amounts to a total of 21597 images across all six channels and the entire 12 h observing window.

After downloading the data, I apply the aiaprep method, as implemented in SunPy (SunPy Community et al., 2015), to each full-disk image in order to process the level-1 data into level-1.5 data and divide each image by the exposure time. Next, I align each image with the observation at 2011 February 12 15:33:45 UTC (the time of the original observation of NOAA 1158 by Warren et al., 2012) by “derotating” each image using the Snodgrass empirical rotation rate (Snodgrass, 1983). After aligning the images in every channel to a common time, I crop each full-disk image such that the bottom left corner of the image is ( 440 , 375 ) and 00 00 7.2 Observations and Analysis 200 the top right corner is ( 140 , 75 ), where the two coordinates are the longitude 00 00 and latitude, respectively, in the helioprojective coordinate system (see Thompson, 2006, and Section 4.1.3) deﬁned by an observer at the location of the SDO spacecraft on 2011 February 12 15:33:45. Figure 7.1 shows the level-1.5, derotated, and cropped AIA observations of active region NOAA 1158 at 2011 February 12 15:33:45 in all six EUV channels of interest.

8 16 24 80 160 240 1000 2000 3000

94 A˚ 131 A˚ 171 A˚

1500 3000 4500 500 1000 1500 40 80 120

00 ˚ ˚ ˚ 0 193 A 211 A 335 A . 200 00 0 . 300

Helioprojective Latitude 400.000 300.000 200.000 Helioprojective Longitude

Figure 7.1 Active region NOAA 1158 as observed by AIA on 2011 February 12 15:32 UTC in the six EUV channels of interest. The data have been processed to level-1.5, aligned to the image at 2011 February 12 15:33:45 UTC, and cropped to 1 1 the area surrounding NOAA 1158. The intensities are in units of DN pixel s . In each image, the colorbar is on a square root scale and is normalized between zero and the maximum intensity. The color tables are the standard AIA color tables as implemented in SunPy. 7.2 Observations and Analysis 201

7.2.1 Emission Measure Slopes

After prepping, aligning, and cropping all 12 h of AIA data for all six channels, I carry out the same analysis applied to the predicted observations in Chapter 6 in order to compute two diagnostics of the heating: the emission measure slope and the

time lag. First, I compute the emission measure distribution, EM(Te), in each pixel of the active region from the time-averaged intensities from all six channels using the regularized inversion method of Hannah & Kontar (2012). As in Section 6.3.2,

the temperature bins have width D log T = 0.1 and the leftmost and rightmost edges are 105.5 K and 107.2 K, respectively. The uncertainties on the intensities are

estimated using the aia_bp_estimate_error.pro procedure provided by the AIA instrument team in the SolarSoftware package (SSW, Freeland & Handy, 1998).

Figure 7.2 shows the emission measure slope, a, as computed from the observed emission measure distribution in each pixel of active region NOAA 1158. As in

Chapter 6, a is calculated by fitting a first-order polynomial to the log-transformed emission measure and temperature bin centers, log EM a log T. The fit is 10 ⇠ 10 only computed over the temperature range 8 105 K T T , where T = ⇥   peak peak

argmaxT EM(T) is the temperature at which the emission measure distribution peaks. If r2 < 0.75 in any pixel, where r2 is the correlation coefficient for the first-order polynomial fit, the pixel is masked and colored white. The emission measure slope tends to be more steep near the center of the active region and tends to increase from 2.5 to > 5 from the periphery to the inner core ⇠ of the active region. This result is consistent with Del Zanna et al. (2015b) who computed the emission measure slope in each pixel of active region NOAA 1193

and found that a was greatest near the middle of the active region. The exception to this trend is the spatially-coherent structure on the lower edge of the active region which shows emission measure slopes > 5. A few regions on the top edge also show higher emission measure slopes. The patchy appearance in some areas of 7.2 Observations and Analysis 202

2 3 4 5 00 0 . 200 00 0 . Helioprojective Latitude 300

400.000 350.000 300.000 250.000 Helioprojective Longitude

Figure 7.2 Map of emission measure slope, a, in each pixel of active region NOAA 1158. The EM(Te) is computed from the observed AIA intensities in the six EUV channels time-averaged over the 12 h observing window. The EM(Te) in each pixel is then ﬁt to Ta over the temperature interval 8 105 K T < T . Any pixels ⇥  peak with r2 < 0.75 are masked and colored white.

the slope map is due to the different values of Tpeak and the finite width of the temperature bins in the EM(Te). Figure 7.3 shows the distribution of emission measure slopes for every pixel in the active region where r2 0.75. As noted in the legend, the black histogram denotes the slopes computed from the real AIA observations while the blue, orange, and green histograms are the distributions of emission measure slopes computed from the predicted AIA intensities in Chapter 6 for high-, intermediate-, and low- frequency nanoflares, respectively. The mean of the observed distribution of a is 3.49 and the standard deviation is 0.82. The observed distribution of slopes overlaps the distributions of predicted slopes for all three heating scenarios, suggesting that a range of nanoflare heat- 7.2 Observations and Analysis 203

Observations 1.2 High Intermediate 1.0 Low

0.8

0.6

0.4

0.2 Number of Pixels (Normalized)

2 3 4 5 6 7 a

Figure 7.3 Distribution of emission measure slopes from Figure 7.2 (black) and from Chapter 6 (blue, orange, green). In each case, the bins are determined using the Freedman Diaconis estimator (Freedman & Diaconis, 1981) as implemented in the Numpy package for array computation in Python (Oliphant, 2006). Each histogram is normalized such that the area under the histogram is equal to 1.

ing frequencies is operating across the active region. In particular, the observed distribution of slopes overlaps quite strongly with both the intermediate- and

high frequency-slopes. Compared to the simulated distributions of a for low- and intermediate-frequency heating, the observed distribution is wide with a relatively

ﬂat top between a 3 and a 4. The observed distribution is not strongly peaked ⇡ ⇡ about a single value of a.

7.2.2 Time Lags

Next, I apply the time-lag analysis of Viall & Klimchuk (2012) to every pixel in the active region over the entire 12 h observing window at the full temporal and spatial resolution. As in Section 6.3.3, the cross-correlation, , is computed between all CAB possible “hot-cool” pairs, AB, of the six EUV channels of AIA (15 in total) in order 7.2 Observations and Analysis 204

to find the time lag, tAB, the temporal offset which maximizes the cross-correlation, in each pixel of the observed active region. I consider all possible offsets over the interval 6h. Following the convention of Viall & Klimchuk (2012), the channel ± pairs are ordered such that the “hot” channel is listed first, meaning that a positive time lag indicates cooling plasma. The response for each of the six EUV channels of AIA as a function of temperature is shown in Figure 3.8. The details of the cross-correlation and time lag calculations can be found in Section 3.6. Note that Viall & Klimchuk (2017) carried out the time-lag analysis on this same active region, NOAA 1158 (their region 2), as part of a survey of the catalogue of active regions compiled by Warren et al. (2012). This analysis is repeated here to ensure that the observed intensities are treated in the exact same manner as the predicted intensities from Chapter 6. Figure 7.4 shows the time-lag maps of active region NOAA 1158 for all 15 channel pairs. Blacks, blues, and greens indicate negative time lags while reds, oranges, and yellows correspond to positive time lags. Olive green denotes zero time lag. The range of the colorbar is 5000 s. If the maximum cross-correlation in ± a given pixel is too small (max < 0.1), the pixel is masked and colored white. CAB For the majority of the channel pairs, I find persistent positive time lags across most of the active region, indicative of plasma cooling through the AIA passbands. The 94-131, 211-131, 193-171, and 193-131 Å pairs show coherent positive time lags near the edge of the active region, but zero time lag in the center of the active region. On the other hand, the 211-193 and 171-131 Å channel pairs show zero time lag in nearly every pixel of the active region. Both of these channel pairs are strongly overlapping in temperature space (see Figure 3.8) such that their respective peaks in intensity are likely to be close to coincident in time as the plasma cools. Additionally, the 94-335, 94-193, and 94-211 Å pairs all show significant coherent negative time lags. Because the 94 Å channel is bimodal in temperature (see 7.2 Observations and Analysis 205

4000 2000 0 2000 4000

00 94-335 A˚ 94-171 A˚ 94-193 A˚ 94-131 A˚ 94-211 A˚ 0 . 200

335-131 A˚ 335-193 A˚ 335-211 A˚ 335-171 A˚ 211-131 A˚ Helioprojective Latitude

211-171 A˚ 211-193 A˚ 193-171 A˚ 193-131 A˚ 171-131 A˚

400.000 200.000 Helioprojective Longitude

Figure 7.4 Time-lag maps of active region NOAA 1158 for all 15 channel pairs. The value of each pixel indicates the temporal offset, in s, which maximizes the cross-correlation (see Section 3.6.1). The range of the colorbar is 5000 s. If ± max < 0.1, the pixel is masked and colored white. Each map has been cropped CAB to emphasize the core of the active region such that the bottom left corner and top right corner of each image correspond to ( 440 , 380 ) and ( 185 , 125 ), 00 00 00 00 respectively.

Figure 3.8), a negative time lag is indicative of the plasma cooling ﬁrst through the “cool” channel and then through the cooler component of the 94 Å bandpass. The 94-171 and 94-131 pairs show only positive time lags because the 171 Å and 131 Å channels peak at cooler temperatures than the cool component of the 94 Å channel. See Viall & Klimchuk (2017) for a more detailed discussion of the time lag results from NOAA 1158. Note that unlike the predicted time lags in Chapter 6, none of the pairs involving the 131 Å channel, which is also bimodal in temperature, show any coherent negative time lags and, in particular, the inner cores of each 131 Å pair show zero time lag. 7.2 Observations and Analysis 206

0.0 0.2 0.4 0.6 0.8 1.0

94-335 A˚ 94-171 A˚ 94-193 A˚ 94-131 A˚ 94-211 A˚ 00 0 . 335-131 A˚ 335-193 A˚ 335-211 A˚ 335-171 A˚ 211-131 A˚ 200

211-171 A˚ 211-193 A˚ 193-171 A˚ 193-131 A˚ 171-131 A˚ Helioprojective Latitude

400.000 200.000 Helioprojective Longitude

Figure 7.5 Same as Figure 7.4, but instead of the time lag, the maximum value of the cross-correlation, max , is shown in each pixel for each channel pair. CAB

Figure 7.5 shows the maximum cross-correlation, max , in each pixel of the CAB active region. In this figure, none of the pixels are masked. In every channel pair, I find that the maximum cross-correlation is highly structured, indicating that these loops and the surrounding diffuse emission are evolving coherently through the AIA passbands. In most channel pairs, the inner core tends to have the highest cross- correlation while areas near the corners of the images have low cross-correlation. Note that the channel pairs which had the most zero time lags in Figure 7.4, 211-193 and 171-131, show high cross-correlations across the entire active region. This again emphasizes the point that zero time lags do not correspond to steady heating. If the whole active region was heated steadily, each pixel would have a much smaller cross-correlation and no preferred time lag due to the photon noise dominating the variability in each channel (Viall & Klimchuk, 2016). 7.3 Classification Model 207

7.3 Classiﬁcation Model

Rather than manually comparing the observations and simulations using all of the aforementioned diagnostics, I systematically assess the observations of NOAA 1158 in terms of the heating frequency by training a random forest classifier comprised of many decision tress on the predicted observables from Chapter 6. I then use this trained model to classify each observed pixel in terms of high-, intermediate-, or low-frequency heating as defined in Equation 6.1. Unlike more traditional statistical methods, this approach simultaneously considers multiple observables when deciding which frequency best fits the observation. In the parlance of statistical

learning, the heating frequency (low, intermediate, or high) is the class, the emission measure slope, time lag, and cross-correlation are the features, and the pixels are the samples. Following the explanation of James et al. (2013, Chapter 8), a decision tree recursively partitions the feature space of interest into a set of terminal nodes, or leaves, using a top-down, “greedy” approach called recursive binary splitting. At each node in the tree, a feature and an associated split point are chosen to maximize the number of observations of a single class in the resulting nodes. A common

measure of the homogeneity or purity of each node is the Gini index,

Gm = Â pˆmk(1 pˆmk), (7.1) k where k indexes the class, m indexes the node, and pˆmk is the proportion of the observations at node m that belong to class k. Note that as the purity of m increases (i.e. pˆ 0, 1), G decreases (G 0). Alternative measures of node purity may mk ! m m ! also be used (see Section 9.2.3 of Hastie et al., 2009). For every resulting terminal node in the tree, the assigned class is determined by the most commonly occurring class of every observation at that node. 7.3 Classiﬁcation Model 208

Decision trees are commonly used in classification problems because they are computationally efficient and relatively easy to interpret. Unlike many statistical learning techniques, decision trees do not assume any functional mapping between the inputs and outputs such that arbitrary, non-linear relationships can be learned by the model. However, decision trees have two primary weaknesses: (1) they are known to have lower predictive accuracy than other more restrictive classification strategies and (2) they have high variance such that a single tree is not very robust to small changes in the training data (James et al., 2013). While individual decision trees are “weak learners”, combined they give accurate and robust predictions. Random forest classifiers, first developed by Breiman (2001), provide an ensemble statistical learning method for combining many noisy, decorrelated decision trees in order to improve prediction accuracy and robustness. As in the bootstrap-aggregation, or “bagging”, technique developed by Breiman (1996), each tree in the random forest is trained on only a subset of the training data in order to reduce the variance of the model. Additionally, at each node in each tree, a random subset of the total features are considered as candidates for splitting in order to decrease the correlation between trees. A typical rule-of-thumb is to

consider only pp features at each split, where p is the total number of features. This further reduces⌅ ⇧ the variance and prevents a single feature from dominating the decision in every tree. Once each tree in the forest has been built using the training data, an unlabeled observation is classified by traversing each tree in the forest and taking the majority vote of the class at the terminal node of each tree. See Chapter 15 of Hastie et al. (2009) for a detailed discussion of random forests for both classification and regression. 7.3 Classification Model 209

7.3.1 Data Preparation and Model Parameters

To build the classification model, I use the random forest classifier as implemented in the scikit-learn package for machine learning in Python (Pedregosa et al., 2011). Using the predicted emission measure slopes, time lags, and maximum cross- correlations from Chapter 6, I train a single random forest classifier composed of 500 trees, each with a maximum depth of 30. At each node, p31 = 5 possible split candidates are randomly selected from the 31(= 15 time lagsj +k15 cross-correlations + 1 emission measure slope) total features. Note that all of these features are likely to be correlated with one another to some extent. Before training the model, the predicted emission measure slope, time lag, and cross-correlation maps from Chapter 6 for the high-, intermediate-, and low-

frequency heating cases are each ﬂattened into an array of length nxny, where

nx and ny are the dimensions of the predicted images. As before, the pixel is masked if r2 < 0.75 for the emission measure slope and max < 0.1 for the CAB cross-correlation. If a pixel is masked in one frequency case, it is masked in all other frequencies to ensure an equal number of high-, intermediate-, and low-frequency data points. Each ﬂattened array is stacked column-wise in features and row-wise in heating frequency such that all of the simulated data are encapsulated in a single

data matrix X of dimension n p. p = 31 is the total number of features and ⇥ n = 3n n n = 111261 is the total number of pixels for all heating frequencies x y mask minus those pixels which were masked in at least one feature of one frequency. The heating frequency label or class is numerically encoded as 0 (high), 1 (intermediate),

or 2 (low) and similarly stacked to create a single response vector Y of dimension n 1.A2/3 1/3 test-train split is applied to X and Y such that approximately ⇥ 1/3 of the samples are reserved for model evaluation to ensure that the model has

not overﬁt the data. This produces four separate matrices: Xtrain, Ytrain, Xtest, Ytest. The data are not centered to a mean of 0 or scaled to unit standard deviation. By 7.3 Classiﬁcation Model 210

Table 7.1 The four different combinations of emission measure slope, time lag, and maximum cross-correlation. The third column lists the total number of features used in the classification. The fourth column gives the misclassification error as evaluated on Xtest, Ytest. The fifth, sixth, and seventh columns show the percentage of pixels labeled as high-, intermediate-, and low-frequency heating, respectively.

Case Parameters p Error High Inter. Low A a 1 0.30 0.471 0.369 0.159 B t , 30 0.03 0.801 0.136 0.063 AB CAB C a, tAB, AB 31 0.03 0.647 0.321 0.032 D Top 10 features fromC Table 7.2 10 0.05 0.609 0.319 0.072

transforming the data in this manner, I am treating each pixel in the image as an

independent sample with p associated features per sample. The same procedure as described above is applied to the observed emission measure slopes, time lags, and cross-correlations as shown in Figure 7.2, Figure 7.4,

and Figure 7.5, respectively. These results are ﬂattened to a single data matrix X0 of dimension n p, where n = n n n = 46436. The random forest model is 0 ⇥ 0 0x y0 mask0 trained on Xtrain, Ytrain and model performance is evaluated on the “unseen” test set

Xtest, Ytest. The trained model is then applied to X0 in order to predict the heating

frequency in each pixel, Y0. I do not apply any formal hyperparameter tuning or cross-validation procedure, though a manual exploration of the hyperparameters revealed that adding more than 500 trees to the random forest provided only a marginal decrease in the test error while increasing the training time. Similarly, a maximum depth of 30 for each decision tree provides sufficient complexity to each tree as evaluated by the test error while not significantly increasing the computational cost of the training. However, in case A (see Table 7.1), I find that less complex trees (i.e. lower maximum depth) result in a reduction in the misclassification error by 7 8%. 7.3 Classification Model 211

7.3.2 Different Feature Combinations

I apply the train-test-predict procedure described above to all four cases listed in Table 7.1. In case A, the random forest classiﬁer is trained only on the emission mea-

sure slope, a, such that the X and X have dimensions n 1 and n 1, respectively. 0 ⇥ 0 ⇥ In case B, the classiﬁer is trained on the time lags and maximum cross-correlations

for all 15 channel pairs for a total of p = 30 features while in case C, every feature (emission measure slope, 15 time lags, 15 maximum cross-correlations) is used such

that p = 31. I discuss case D in Section 7.3.3. The fourth column in Table 7.1 lists

the misclassification error as evaluated on the test set, Xtest, Ytest and the fifth, sixth, and seventh columns show the fraction of pixels classified as high-, intermediate-, and low-frequency.

After computing the predicted heating frequency for X0, the resulting classiﬁca-

tions, Y0, are mapped back to the corresponding observed pixel locations to create a map of the heating frequency. Figure 7.6 shows the probability that each pixel corresponds to a particular heating frequency. The rows denote the different feature subsets as given in Table 7.1 and the columns correspond to the different heating frequency classes. The class probability, as computed by the scikit-learn package, in each pixel is the mean class probability of all trees in the random forest classifier. The class probability for an individual tree is the proportion of all training samples at the terminal node that belong to that class. Figure 7.7 shows the heating frequency, or class, as predicted by the random forest classifier in each pixel of the observed active region for all four cases in Table 7.1. The predicted class is the one which has the highest mean probability as computed over all trees in the random forest. Each pixel is colored blue, orange, or green depending on whether the class with the highest mean probability is high-, intermediate-, or low-frequency, respectively. 7.3 Classification Model 212

0.0 0.2 0.4 0.6 0.8 1.0

High Intermediate Low B

C 00 0

. D 200 00 0 . 300

Helioprojective Latitude 400.000 350.000 300.000 250.000 Helioprojective Longitude

Figure 7.6 Classiﬁcation probability for each pixel in the observed active region. The rows denote the different cases in Table 7.1 and the columns correspond to the different heating frequency classes. If any of the 31 features is not valid in a particular pixel, the pixel is masked and colored white. Note that summing over all heating probabilities in each row gives 1 in every pixel. 7.3 Classiﬁcation Model 213

High Intermediate Low

A B

C D 00 0 . 200 00 0 . 300 Helioprojective Latitude

400.000 350.000 300.000 250.000 Helioprojective Longitude

Figure 7.7 Predicted heating frequency classiﬁcation in each pixel of NOAA 1158 for each of the cases in Table 7.1. The classiﬁcation is determined by which heating frequency class has the highest mean probability over all trees in the random forest. Each pixel is colored blue, orange, or green depending on whether the most likely heating frequency is high, intermediate, or low, respectively. If any of the 31 features is not valid in a particular pixel, the pixel is masked and colored white.

I ﬁnd that for each combination of features in Table 7.1, high-frequency heating dominates at the center of the active region. This result is consistent with active region core observations of hot, steady emission (Warren et al., 2011, 2010), steep emission measure slopes (e.g. Del Zanna et al., 2015b; Winebarger et al., 2011), and lack of variability in the intensity (e.g. Antiochos et al., 2003) and the velocity 7.3 Classiﬁcation Model 214

(Brooks & Warren, 2009) near the loop footpoints. The frequency classification map for case A is as expected given the observed emission measure slope map in Figure 7.2 and the well-separated distributions of emission measure slopes from the different heating frequencies as shown in Figure 7.3. From the fourth column of Table 7.1, I find that adding more features to the classifier significantly improves the accuracy as computed on the test data set.

However, comparing frequency maps in cases A (p = 1) and C (p = 31), shows that the general pattern of heating frequency across the active region is similar despite the large differences between the misclassification error in case A (0.30) and case C (0.03). Additionally, looking at the seventh column of Table 7.1 and the panels in the third column of Figure 7.6, I find that adding the time lag and maximum cross-correlation features significantly decreases the number of pixels classified as low frequency. Training the classifier on only the emission measure slope versus all of the features has a comparatively small impact on the fraction of pixels classified as intermediate frequency. Interestingly, I find that when relatively shallow trees are used to build the random forest (e.g. a maximum depth of < 10), the misclassification error on the

test data set in case B becomes larger than that of case A, despite pB > pA. If very complex trees (maximum depth > 100) are used in case A, the model overfits the data and the resulting classification becomes very noisy. However, in case B (and C), increasing the maximum depth continually decreases the test error, indicating that the model is not overfitting the data. This seems to suggest that the relationship between the heating frequency and the time lag, as well as the maximum cross- correlation, is much more complex than that of the relationship between the heating frequency and the emission measure slope. 7.3 Classification Model 215

7.3.3 Feature Importance

In addition to the predicted heating frequency, Y0, for a set of features, X0, it is also useful to know which of the p features is most important in deciding to which class each observation (pixel) belongs. One measure of the importance of each feature is the decrease in the Gini index,

M M M DG = m G m,R G m,L G , (7.2) m M m M m,R M m,L ✓ m m ◆ where Gm is the Gini index as given by Equation 7.1, M is the total number of

samples in the tree, Mm is the total number of samples at parent node m, Mm,R(L)

is the total number of samples at the right (left) child node, and Gm,R(L) is the Gini index at the right (left) child node (Sandri & Zuccolotto, 2008). The importance of a particular feature in the random forest classiﬁcation is then determined by summing Equation 7.2 over all nodes which split on that feature for every tree and averaging over all trees (Breiman et al., 1984).

Note that if Gm,R = Gm,L = Gm, DGm = 0 because the split at node m did not improve the discrimination between classes compared to the split at the previous

node. However, if the purity of the left or right node increases such that Gm,R or

Gm,L decreases relative to Gm, DGm > 0 because the split at node m has added information to the classifier by preferentially sorting samples of a single class to either the left or right child node. Table 7.2 shows the ten most important features from case C as determined by Equation 7.2 summed over all nodes in each tree and averaged over all trees. The importance in the second column is normalized such that the most important feature is equal to 1. In case D as listed in the last row of Table 7.1, only these ten features are used to train the random forest classifier and classify each observed pixel. The probability of each heating frequency for case D is shown in the last 7.3 Classification Model 216

Table 7.2 Ten most important features as determined by the random forest classiﬁer in case C. The second column shows the variable importance as computed by Equation 7.2 and the third column, s, is the standard deviation of the feature importance over all trees in the random forest. The second column is normalized such that the most important feature is equal to 1.

Feature Importance s a 1.0000 0.0769 0.4174 0.0917 C211,193 211,171 0.3447 0.0942 C 0.3430 0.0875 C193,171 t211,193 0.1113 0.0109 0.0934 0.0310 C335,171 t171,131 0.0933 0.0139 335,211 0.0777 0.0176 C 0.0739 0.0255 C335,193 t335,171 0.0662 0.0139

row of Figure 7.6 and the map of the most likely heating frequency in each pixel is shown in the bottom-right panel of Figure 7.7. The probability maps in the last row of Figure 7.6 and the frequency map in the bottom right panel of Figure 7.7 reveal approximately the same patterns of heating frequency across the active region as the maps for case C in which all 31 features were included. Additionally, despite using less than 1/3 of the total number of features, the misclassiﬁcation error of case D (0.05) is comparable to case C.

According to Table 7.2, the emission measure slope, a, has the most discriminating power in the random forest classiﬁer. In particular, a is more important than the second most important feature by over a factor of 2 and more important than the most important time lag feature by nearly an order of magnitude. While useful, the feature importance in random forest classiﬁers should be interpreted cautiously, especially in cases where the features are correlated. The time lags, as well as the maximum cross-correlations, in all channel pairs are very strongly correlated. The emission measure slope is also likely correlated with 7.4 Discussion 217

the time lag and cross-correlation though perhaps more weakly so. In particular, Altmann et al. (2010) found that as the number of correlated features in a random forest classiﬁer increased, the individual importance of each feature in the correlated group decreased and that for a very large number of correlated features ( 50), the ⇠ feature importance of each was close to zero. Here, there are at least two groups of 15 strongly correlated features each such that the values shown in Table 7.2 for the time lag and cross-correlation should be regarded as lower bounds on the feature importance. However, the presence of highly-correlated or unimportant features is not expected to affect the robustness or accuracy of the classiﬁer.

7.4 Discussion

As evidenced in Figure 7.6 and Figure 7.7, I find that high-frequency heating is likely to dominate in the core of the active region. Comparing the heating frequency maps in Figure 7.7 for the different cases in Table 7.1, this high-frequency classification seems largely due to the steep observed emission measure slopes in the center of the active region as seen in Figure 7.2. This result is consistent with X-ray observations of hot, steady emission (Warren et al., 2011, 2010; Winebarger et al., 2011) as well as the result of Del Zanna et al. (2015b) who found high values of the emission measure slope in the center of NOAA 11193. Comparing case C in Figure 7.7 with the observed magnetogram of NOAA 1158 shown in Figure 6.1 reveals that the areas of strongest magnetic field are approximately spatially coincident with most of the pixels classified as high-frequency. This suggests that those strands whose footpoints are rooted in areas of large magnetic field strength are heated more frequently. I will explore the relationship between the heating frequency and the underlying magnetic field strength in a future paper (see also Section 8.2.1). 7.4 Discussion 218

The longer loops surrounding the core are consistent with intermediate frequency heating. Notably, the results from the classiﬁer imply that low-frequency heating, as deﬁned by Equation 6.1, is not needed to explain the observed time lags, suggesting that the waiting time on each strand in this active region is likely to be

on the order of or less than tcool. This result is consistent with that of Bradshaw & Viall (2016) who found that intermediate- and high-frequency nanoﬂares both produced time lags consistent with observations while their cooling experiment, similar to the low-frequency nanoﬂares here, showed fundamental disagreements with the observed time lag maps.

After the emission measure slope, a, the next three most important features in the classification are the maximum cross-correlations for the 211-193, 193-171, and 211-171 Å channel pairs. These three channels, 211 Å, 193 Å and 171 Å, peak sequen- tially in temperature at 1.8 MK, 1.6 MK and 0.8 MK, respectively (see Figure 3.8), suggesting that the plasma dynamics in this temperature range, which are dominated by radiative cooling and draining (e.g. Bradshaw & Cargill, 2005, 2010a,b), are coupled to, and indicative of, the frequency at which energy is deposited in the plasma and that thermal conduction has not erased all signatures of the heating. A strand heated by low-frequency nanoflares will be allowed to cool well below 1 MK, producing a strong cross-correlation in these channel pairs, while a strand heated by high-frequency nanoflares will rarely be allowed to cool below the equilibrium temperature such that the cross-correlation, particularly in the 171 Å channel pairs, is likely to be relatively low. This cooling behavior is illustrated for a single strand in Figure 6.3 While the maximum cross-correlation in the 211-193 Å channel pair (see bottom row of Figure 7.5) is very high across the whole active region, the 193-171 Å and 211-171 Å maps (as well as the other 171 Å pairs except for 171-131 Å) show a comparatively low cross-correlation. Combined with the heating frequency maps 7.5 Conclusions and Summary 219

in Figure 7.7, which indicate that the center of the active region is consistent with high-frequency heating, this suggests that many of the loops in the core are kept from cooling much below 1.6 MK. An important caveat to this method for systematic comparison is that the random forest classifier trained on the simulated emission measure slopes, time lags, and maximum cross-correlations cannot provide any assessment of the accuracy of the the forward-modeled results from Chapter 6. The classifier can only say, out of the provided classes (high-, intermediate-, or low-frequency), which type of heating best describes the data. However, given another method for assessing the heating frequency or perhaps some alternative forward-modeling approach, a random forest classifier could be used to compare these two methods. In this way, machine learning also provides a promising strategy for reconciling different modeling approaches.

7.5 Conclusions and Summary

In this chapter, I have used the predicted diagnostics from Chapter 6 to systematically classify each pixel of active region NOAA 1158 in terms of the frequency of energy deposition. In particular, I ﬁrst collect 12 h of full-resolution SDO/AIA observations of NOAA 1158 in six EUV channels: 94 Å, 131 Å, 171 Å, 193 Å, 211 Å and 335 Å. I then co-align each image to a single time such that a given pixel in each image corresponds to approximately the same spatial coordinate and crop the

image to an area of 50000-by-50000 centered on the active region. Next, I compute two diagnostics from these observed intensities: the emission measure slope and the time lag. I time-average the intensities of all six channels and use the method of Hannah & Kontar (2012) to compute the emission measure distribution in each pixel of the active region. I then compute the emission measure 7.5 Conclusions and Summary 220

slope, a, by ﬁtting log EM a log T over the temperature range 8 105 K T < 10 ⇠ 10 ⇥  Tpeak. Next, I apply the time-lag analysis of Viall & Klimchuk (2012) to the full 12 h

of observations of NOAA 1158 and compute the time lag, tAB, and maximum cross- correlation, max , in each pixel of the active region for all possible “hot-cool” CAB pairs of the six EUV channels, 15 in total. Finally, I train a random forest classiﬁer using the predicted emission measure slopes, time lags, and cross-correlations for the three different heating frequencies from Chapter 6. I then use this trained model to classify each observed pixel as consistent with either high-, intermediate-, or low-frequency heating and map the heating frequency across the entire active region. The results from this study can be summarized as follows:

1. The distribution of observed emission measure slopes overlaps with the distributions of predicted emission measure slopes for high-, intermediate-, and low-frequency heating, suggesting a range of heating frequencies across the active region.

2. High-frequency heating dominates in the center of the active region and is coincident with areas of large magnetic ﬁeld strength.

3. Intermediate-frequency heating is more likely in longer loops surrounding the center of the active region. In most pixels, low-frequency heating, as deﬁned in Equation 6.1, is not needed to explain the observed diagnostics

4. The emission measure slope is the strongest predictor of the heating frequency. Radiative cooling and draining around 1 MK to 2 MK, as manifested in the maximum cross-correlation, also appears to be a strong indicator relative to the time lags. However, the feature importance as determined by the classiﬁer should be interpreted carefully. 7.5 Conclusions and Summary 221

I have demonstrated an efﬁcient and powerful technique for constraining the heating frequency in active region cores and, more broadly, for systematically comparing models and observations. Given that the diagnostics here are known to vary with age (e.g. Del Zanna et al., 2015b; Schmelz & Pathak, 2012) and from one active region to the next (Viall & Klimchuk, 2017; Warren et al., 2012), the next step is to apply this methodology to a large sample of active regions to place strong constraints on the frequency of energy deposition in the magnetically-closed corona. Chapter 8

Conclusions and Future Work

8.1 Conclusions

In this thesis, I have closely examined the observable signatures of impulsive nanoflare heating in active region coronal loops with a particular emphasis on connecting model results and observations. More specifically, I have focused on the degree to which observables can be used to diagnose the frequency with which energy is deposited on individual magnetic strands. To this end, I have developed a modu- lar and flexible software pipeline for modeling time-dependent, multi-wavelength optically-thin coronal emission (Chapter 4) as well as a two-fluid modification to the EBTEL loop model (Section 2.2.4). These modeling tools have been applied to a number of different heating scenarios in order to predict observable diagnostics of the heating frequency. The primary research findings from the three studies that comprise this thesis (Chapters 5 to 7) are summarized below. In Chapter 5, I examined the observability of the high-temperature (8 MK to

10 MK) component of the differential emission measure distribution (DEM(Te)), the “smoking gun” of nanoflare heating, using a single loop modeled with EBTEL. In particular, I studied the impact of nanoflare duration t, heat flux limiting, preferen-

222 8.1 Conclusions 223

tial heating of the electrons and ions, the assumption of electron-ion equilibrium,

and non-equilibrium ionization on the hot component of the DEM(Te). In each case, the loop was heated by a single nanoflare, equivalent to low-frequency heating. Additionally, the EBTEL results were compared with spatially-averaged results from the more sophisticated field-aligned model HYDRAD (Section 2.2.3). In the case where electron-ion equilibrium is enforced at all times (i.e. the single- fluid model) and ionization equilibrium is assumed, the expected signatures of conductive cooling appear in the emission measure. For short nanoflares with t < 100 s, there is a significant component well above 10 MK; for longer duration

events (200 s and 500 s) there is signiﬁcant plasma in the temperature range Tpeak < T < 10 MK. However, the inclusion of several pieces of additional physics modify this result considerably, in each case making it much less likely that any plasma that is above 10 MK can be detected. Preferentially heating the electrons leads to similar results to the single-ﬂuid

case. However, when only the ions are heated, the EM(Te) is sharply truncated below 8 MK due to the principal electron heating mechanism being a relatively slow collisional process. In all cases, when the assumption of ionization equilibrium is

relaxed, the EM(Te) is sharply truncated at or below 10 MK, since the time needed to create highly ionized states such as Fe XXI is longer than any relevant cooling time. Taken together, these results suggest that while low-frequency nanoﬂares may

be able to produce a 10 MK component of the EM(T ), observing this hot plasma ⇠ e is likely to be very challenging. Chapter 6 describes a series of numerical simulations carried out in an effort to understand how signatures of the nanoflare heating frequency, defined in terms of the ratio between the waiting time and the loop cooling time, are manifested in the emission measure slope and the time lag. I apply the synthesizAR forward modeling code combined with the two-fluid EBTEL code to model the projected 8.1 Conclusions 224

LOS intensity as observed by all six EUV channels of AIA for many loops in active region NOAA 1158. These steps are carried out for three different nanoﬂare heating frequencies: high, intermediate, and low, (see Equation 6.1) in addition to two control models, for a total of ﬁve different heating scenarios (see Table 6.1). The results of this study can be summarized as follows. As the heating frequency

decreases, the emission measure slope, a, becomes increasingly shallow, saturating at a 2. As the heating frequency increases, the distribution of slopes over the ⇡ active region is shifted to higher values and broadens. Additionally, the time lag becomes increasingly spatially coherent with decreasing heating frequency and when strands are allowed to cool without being re-energized, the spatial distribution of time lags is largely determined by the distribution of loop lengths over the active region. Furthermore, the ﬂattened distribution of time lags becomes increasingly broad and approaches a uniform distribution as the heating frequency increases, consistent with the results of Viall & Klimchuk (2016). Lastly, negative time lags in channel pairs where the second (“cool”) channel is 131 Å provide a possible diagnostic for 10 MK plasma Finally, in Chapter 7, I used the predicted diagnostics from Chapter 6 to systematically classify each pixel of active region NOAA 1158 in terms of the frequency of energy deposition. First, observations of NOAA 1158 in the six EUV channels of AIA were used to compute the emission measure slope over the temperature range

8 105 K T < T as well as the time lag and cross-correlation for all possible ⇥  peak “hot-cool” pairs of the six EUV channels (15 in total). These observables were computed in each pixel of the active region. Then, I trained a random forest classiﬁer using the predicted emission measure slopes, time lags, and cross-correlations from Chapter 6 and used this trained model to classify each observed pixel as consistent with either high-, intermediate-, or low-frequency heating, thus mapping the heating frequency across the entire active region. 8.2 Future Work 225

The results of this study can be summarized as follows. Comparing the model and observed emission measure slopes, the distribution of observed emission measure slopes overlaps with the distributions of predicted emission measure slopes for high-, intermediate-, and low-frequency heating, suggesting a range of heating frequencies across the active region. Based on the results of the classification model, high-frequency heating dominates in the center of active region and is coincident with the areas of large magnetic field strength while intermediate-frequency heating is more likely in longer loops surrounding the center of the active region. In most pixels, low-frequency heating, as defined by Equation 6.1, is not needed to explain the observed diagnostics. Interpreting the trained classifier revealed that the emission measure slope is the strongest predictor of the heating frequency. Radiative cooling and draining around 1 MK to 2 MK as manifested in the maximum cross-correlation also appears to be a strong indicator relative to the time lags. However, the feature importance as determined by the classifier should be interpreted carefully.

8.2 Future Work

The work presented in this thesis has made signiﬁcant progress toward understanding the observable signatures of nanoﬂare heating and placing strong constraints on the frequency of energy deposition in active regions. However, a number of improvements can be made, particularly in the prescription of the heating model in Chapter 6. The following sections outline some possible improvements.

8.2.1 Nanoﬂare Storms on Bundles of Strands

The heating model described in Section 6.2.3 parameterizes the heating frequency

in terms of a simple ratio # = twait/tcool such that # < 1 denotes high-frequency 8.2 Future Work 226

heating, # 1 denotes intermediate-frequency heating, and # > 1 denotes low- ⇠ frequency heating. While useful, this prescription leaves # as an unconstrained free parameter. However, in the context of the nanoflare model of Parker (1988, see Section 1.3.2), the frequency with which field lines reconnect and dissipate their stored energy into the coronal plasma is certainly related to the magnetic field. Thus, the spatial distribution of magnetic flux across an active region provides a potential constraint on the heating frequency. I will now outline a possible method for relating the heating frequency to the observed magnetic field (e.g. from a field extrapolation). First, consider a collection of magnetic strands in an active region, similar to the 5 103 traced field lines in ⇥ Chapter 6, whose footpoints all close back at the solar surface. Next, divide up

the surface into n equal area boxes and bin the strands into these boxes based on

the coordinates of one of their footpoints. This gives Nbundles “bundles” of strands, where a bundle, b, is a collection of strands whose footpoints are in close proximity. Bins with no footpoints are discarded. Rather than heating each strand individually

as in Chapter 6, the strands on each bundle b are heated collectively. This is often referred to as a nanoﬂare “storm” in which each strand in the bundle is heated once in quick succession (Klimchuk, 2015; Mulu-Moore et al., 2011a; Schmelz & Winebarger, 2015) such that the strands evolve nearly in-phase. Qualitatively, this is also similar to so-called avalanche models (e.g. Hood et al., 2016) wherein a single unstable strand in a ﬂux tube causes neighboring strands to also become unstable, leading to a cascade or avalanche of relaxation and heating of the surrounding plasma.

Given a uniform distribution of nanoﬂare storm onset times (0, t ), where U total ttotal is the total simulation time, Nstorms onset times are chosen, where Nstorms is the total number of storms occurring on all bundles in the active region. Each start 8.2 Future Work 227

start time ts is then assigned to a bundle given the probability,

Bb Pb = , Âb Bb where Bb is the average ﬁeld strength of all strands in bundle b,

Â B = s s Bb strands , Nb

strands Bs is the magnetic ﬁeld strength averaged over the corona of strand s and Nb is the total number of strands in b. Thus, bundles rooted in areas of higher ﬁeld strength

are heated much more frequently. Additionally, Nstorms can be chosen by constraining the total energy ﬂux of all bundles over all storms based on observations (e.g.

7 2 1 10 erg cm s from Withbroe & Noyes, 1977, as in Section 6.2.3). This model has two primary advantages compared to the parameterization in Section 6.2.3: (1) the frequency with which each strand is re-energized is chosen based on the observed ﬁeld strength rather than being left as a free parameter and (2) nanoﬂare storms lead to coherent evolution within bundles, consistent with the observed collective behavior of loops made up of many strands (Klimchuk, 2015). As a proof of concept, I have applied this heating model to the collection of 5 103 strands traced from NOAA 1158 in Chapter 6. The magnetogram of NOAA ⇥ 1158 is divided into 50 segments in each direction such that there are 2500 possible

bundles though most are empty. The total simulation time is t = 3 104 s total ⇥ and Nstorms is constrained such that the average ﬂux through the active region is 107 erg cm 2 s 1. The evolution of the plasma along these strands is then ⇡ simulated using the two-ﬂuid EBTEL model (Section 2.2.4) in the same manner as discussed in Section 6.2.2. Figure 8.1 shows the resulting time-lag maps for four different channel pairs as computed from the predicted intensities from an active region heated by the 8.2 Future Work 228

4000 2000 0 2000 4000 94-335 A˚ 335-171 A˚ 00 0 .

150 211-193 A˚ 171-131 A˚ 00 0 . 250 00 0 . 350 Helioprojective Latitude 400.000 300.000 200.000 Helioprojective Longitude

Figure 8.1 Time-lag maps produced by the bundle heating model as simulated from a ﬁeld extrapolation of active region NOAA 1158. A sample of four channel pairs are shown here: 94-335, 335-171, 211-193, and 171-131 Å. The value of each pixel indicates the temporal offset, in s, which maximizes the cross-correlation (see Equation 3.55). The range of the colorbar is 5000 s. If max < 0.1, the pixel is ± CAB masked and colored white.

bundle model. Notably, the time lags in all channel pairs are significantly more spatially coherent than any of the heating scenarios in Chapter 6 (except perhaps the pure cooling model). Qualitatively, this is more similar to the spatial distribution of time lags found in the observations of NOAA 1158 (see Figure 7.4). More detailed comparisons with observations are needed in order to evaluate the viability of tying the heating frequency directly the magnetic field strength. Additional numerical experiments will be performed using the more sophisticated HYDRAD code (Section 2.2.3) to model the field-aligned physics of each strand. 8.2 Future Work 229

8.2.2 Thermal Non-Equilibrium

Underlying all of the work presented in this thesis is the assumption that the observed EUV and X-ray variability in active region emission must be driven by some time-varying source of energy (e.g. bursty reconnection). However, spatially nonuniform, steady heating may still be consistent with observed loop variability.

The phenomenon of thermal non-equilibrium, or TNE, occurs when the energy deposition is at least quasi-steady (relative to a loop cooling time) and highly-localized near the base of the coronal loop. An example of such a heating profile is shown in Figure 8.2. The resulting chromospheric evaporation and increase in density lead to increased radiative losses and if the heating is sufficiently localized to the footpoints, thermal conduction may not be able to balance the enhanced radiative cooling at the apex (Antiochos & Klimchuk, 1991; Kuin & Martens, 1982). A cool condensation will form near the loop apex and then fall down the loop toward the chromosphere as this configuration is Rayleigh-Taylor unstable. Since there exists no stable equilibrium, this drives a continual evaporation-condensation cycle and, consequently, dynamic evolution of the temperature and density structure of the loop. Figure 8.3 shows the temperature (top) and density (bottom) as a function of

ﬁeld-aligned coordinate, s, and time, t, of a semi-circular loop with full-length 2L = 120 Mm subject to the time-independent heating function shown in Figure 8.2. The ﬁeld-aligned hydrodynamic evolution of the loop is modeled using the HYDRAD code (see Section 2.2.3) and the total simulation time is 24 h. While the thermal structure of the loop is initially quite steady, a cool condensation forms near the

apex of the loop at t 3h. and then falls back down the loop over a period of 7h. ⇡ ⇡ The cycle then repeats with another condensation forming just before t = 15 h and again at t 23 h. Note that compared to the dynamics predicted by impulsive ⇡ nanoﬂare heating, the variability timescales associated with TNE are very long. 8.2 Future Work 230

3 10 5 ⇥

4 ] 1 s 3

erg cm 2 [ H E 1

0 0 20 40 60 80 100 s [Mm]

Figure 8.2 Heating input as a function of field-aligned coordinate, s, for simulating TNE using the HYDRAD code. The full-length of the strand is 120 Mm. The total heating profile is a combination of two Gaussian heating profiles of width 10 Mm at the two footpoints, s = 5 Mm and s = 115 Mm. The heating rate of both pulses is 5 10 3 erg cm 3 s 1. The heating is turned on at t = 0sand is kept constant ⇥ for the entire simulation.

Past studies have shown that TNE provides a viable explanation for both prominences (Antiochos & Klimchuk, 1991; Müller et al., 2003) and coronal rain (Antolin et al., 2010, 2015; Auchère et al., 2018). However, recent results from field-aligned models (Mikićet al., 2013) and AIA observations of long-period intensity variations (Auchère et al., 2014, 2016; Froment et al., 2015) suggest that TNE may be important in driving active region variability as well (though see Klimchuk et al., 2010). Sev- eral recent numerical investigations (Lionello et al., 2016; Winebarger et al., 2016) claim the longest observed time lags are not consistent with impulsive heating, but can be explained by TNE. Additionally, Froment et al. (2017) found that observed long-period intensity variations were consistent with synthetic time lags produced by both TNE and nanoflares. Most recently, Winebarger et al. (2018) performed a parameter search over both TNE and impulsive heating models and found that 8.2 Future Work 231

120 3

90 2 60 [Mm]

s 1 30

120 1010

60 [Mm] s 30

0 109 0 5 10 15 20 t [h]

3 Figure 8.3 Electron temperature, in MK (top), and density, in cm (bottom), as a function of ﬁeld-aligned coordinate, s, and time, t as simulated by the HYDRAD code for a semi-circular loop of full-length 2L = 120 Mm. The time-independent heating function is localized to the footpoints and is shown as a function of s in Figure 8.2.

the time lag and ratio between peak intensities in different channels were useful diagnostics in discriminating between these two heating scenarios. While these ﬁndings challenge the idea that observed time-variability in active regions implies exclusively time-dependent heating, it is not yet clear whether TNE can reproduce the entire range of observables. TNE offers an alternative explanation to the observed variability in active region loops. Both the emission measure slope (Section 3.5.1) and the time lag (Section 3.6) provide a possible path forward for discriminating between these two heating scenarios. However, it is not yet clear whether signatures of these heating scenarios in these diagnostics can be meaningfully discerned. Using the synthesizAR forward modeling code (Chapter 4) and the HYDRAD model, I will forward model time- dependent, multi-wavelength intensities for a subset of active regions from the 8.2 Future Work 232

catalogue of Warren et al. (2012) for both an impulsive nanoﬂare heating scenario and a case in which the energy deposition in each loop in the active region is steady and localized near the footpoints (as in Figure 8.2). From these predicted intensities, I will compute the emission measure slope and in order to understand whether a meaningful distinction can be made between these two heating scenarios. Comparisons to observations in the manner outlined in Chapter 7 are important in quantitatively deciding whether an observed pixel is undergoing TNE or being heating impulsively. References

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ﬁasco: A Python Interface to the CHIANTI Atomic Database

As noted in Section 3.1, the CHIANTI atomic database is an invaluable resource in the field of solar physics, both for modeling and interpreting observations. During the course of my graduate work, I developed the fiasco1 package for interacting with CHIANTI in the Python programming language. fiasco provides an intuitive Python interface to every part of the CHIANTI database as well as many calculations for many common derived quantities. The goal of the fiasco package is to provide largely the same functionality as the CHIANTI IDL routines via an object-oriented interface (similar to ChiantiPy, Barnes & Dere, 2017; Landi et al., 2012) as well as improved interoperability with the greater Python ecosystems in astronomy and heliophysics. The following sections will briefly describe the fiasco software for parsing the raw atomic data and interacting with higher-level objects. fiasco is developed openly on GitHub and is fully documented2, including several examples. fiasco also includes a test suite that is run at each code check-in to

1The name of the package derives from the Italian word fiasco, the style of bottle typically used to serve a wine from the Chianti region of Italy. 2The full source is available at github.com/wtbarnes/fiasco and the documentation, which is rebuilt at each code check-in, is available at fiasco.readthedocs.io.

250 A.1 Parsing Data 251 prevent unexpected changes. In the following sections, I provide a brief overview of the package. Additional details can be found in the documentation.

A.1 Parsing Data

For each ion in the CHIANTI database, there are a number of different filetypes representing the different pieces of information attached to each ion (e.g. energy levels, transition wavelengths, thermally averaged collision strengths). While all of these files are in readable plaintext format, the layout of the data in each file is slightly different. Additionally, no units or descriptions of the data are provided in the file. fiasco provides an easy to use file parser that takes the name of any CHIANTI

ﬁlename in the database and parses it into the commonly-used Astropy Table format. As an example, to read the data from the energy level ﬁle for Fe XVI,

from fiasco.io import Parser p = Parser('fe_16.elvlc') table = p.parse()

Table A.1 shows the LATEX representation of the table directly parsed from the ﬁle by ﬁasco, including units. To access the observed energies of each level of Fe XVI and convert them to erg,

e_obs = table['E_obs'] e_obs_erg = (e_obs * const.h * const.c).to(u.erg)

The data in each column is an Astropy u.Quantity and has units attached to it as appropriate. Additional metadata, including descriptions of each of the columns and the information in the footer of the raw data ﬁle can be accessed via the

table.meta attribute. Other ﬁlenames can be parsed in the exact same manner. Because each ﬁle format

(e.g. wgfa, elvlc) is slightly different, a unique parser subclass is implemented for A.2 The Ion Class 252

Table A.1 The ﬁrst 5 rows of the energy level ﬁle for Fe XVI, fe_16.elvlc. Addi- tional metadata and units are available in the metadata of the Table object.

level conﬁg multiplicity LJ Eobs Eth 1 1 cm cm 1 3s 2 S 0.5 0.0 0.0 2 3p 2 P 0.5 277194.188 276436.0 3 3p 2 P 1.5 298143.094 296534.0 4 3d 2 D 1.5 675501.188 676373.0 5 3d 2 D 2.5 678405.875 679712.0

each ﬁletype in the CHIANTI database. There are 23 unique ﬁletypes. Rather than forcing the user to call a different function or instantiate a separate class for each

unique filetype, fiasco uses a factory pattern, a common design pattern in software engineering, to create an instance of a class “on the fly” based on the filename passed to the Parser object. This greatly simplifies the user interface by providing a single entry point for parsing data files. A key advantage of fiasco is that it rebuilds the entire CHIANTI database as a hierarchical data format (HDF5) file. While ChiantiPy and the CHIANTI IDL routines read the raw data files directly, fiasco parses the raw data files once and then writes the entire database to a single HDF5 file. This is more efficient, particularly in the case of large files (e.g. Fe IX and XI with many transitions), as HDF5 allows efficient read access to parts of the database without reading unneeded data into memory.

A.2 The Ion Class

Similar to the ChiantiPy package, ﬁasco provides several useful abstractions of the

database. The ﬁrst is the Ion object. To instantiate an ion of Fe XVI,

from fiasco import Ion T = np.logspace(4, 9, 100) * u.K A.2 The Ion Class 253

fe16 = Ion('Fe 16',temperature=T) #alternatively fe16 = Ion('iron 16',temperature=T)

Note that, just as in the synthesizAR package (see Chapter 4), all inputs corresponding to physical quantities require units. Basic metadata can be accessed as attributes on the ion, fe16.ion_name fe16.element_name fe16.atomic_symbol fe16.abundance

Additionally, information about the energy levels can be accessed by indexing the object directly as a list fe16[0] #thefirstenergylevel,aLevelobject fe16[1].level fe16[1].energy.to(u.eV)

Note that the result of indexing the Ion object is another object corresponding to the energy level within an ion which has several attributes attached to it. Additionally, the raw data ﬁles for each ion can also be accessed via attributes, fe16._elvlc #energylevels fe16._wgfa #transitions

In general, these are reserved for internal use by fiasco such that the user need not worry about naming conventions of CHIANTI file formats. Perhaps most importantly, fiasco provides a simple interface for computing derived quantities from the atomic data attached to each ion. For example, to calculate the total ionization and recombination rates as a function of temperature for Fe XVI, i_rate = fe16.ionization_rate() r_rate = fe16.recombination_rate()

Many other derived quantities are available. A complete list is available in the documentation. A.3 The Element Class 254

A.3 The Element Class

In addition to an individual ion, ﬁasco also provides an abstraction for many ions

of the same element, the Element class. To implement an Element object for iron,

from fiasco import Element fe = Element('iron',temperature=T)

Basic metadata about the element is available via attributes of the class instance,

fe.element_name fe.atomic_number fe.atomic_symbol fe.abundance

Additionally, just as the Ion object could be indexed to retrieve the energy levels, the Element object can be indexed to retrieve the constituent ions,

fe[0] #returnsIonobjectforFe1 fe[15] #returnsIonobjectforFe16

The Element object can also be used to compute quantities not deﬁned for an individual ion such as the equilibrium population fractions (see Section 3.2.4),

ioneq = fe.equilibrium_ionization()

A.4 Working with Multiple Ions

Some quantities such as spectra or radiative loss curves require data from more than one ion and not necessarily of the same element. ﬁasco provides a general

IonCollection object. An ion collection can be created directly,

from fiasco import IonCollection col = IonCollection(Ion('Fe 5',T),Ion('Fe 13',T))

or by literally adding the objects together

col = Ion('Fe 5',T)+ Ion('Fe 13',T) A.4 Working with Multiple Ions 255

The latter is implemented by overriding the addition operator on the Ion and IonCollection classes. This type of notation allows for an intuitive exploration of how adding or removing ions impacts spectra or other quantities like the radiative loss curve (see Figure 2.2).

Note that the Element class is just a subclass of IonCollection where every ion is from the same element. The EmissionModel class (see Section 4.3) in the synthesizAR forward modeling code is also a subclass of IonCollection. Appendix B

An Implicit Method for Computing Non-Equilibrium Charge States

In cases where the electron temperature, Te, changes sufﬁciently slowly such that

dfk the charge state remains in equilibrium with Te, dt = 0 in Equation 3.19 and the population fraction, fk, is determined by a set of linear, homogeneous equations (Equation 3.31). However, when a low-density, optically-thin plasma undergoes a

rapid change in Te, the charge state cannot keep pace with the electron temperature

such that fk and Te are out of equilibrium and dfk/dt cannot be ignored.

Recall that the time evolution of the population fraction fX,k, for an element X and charge state k, is governed by,

d I R I R fk = ne(ak 1 fk 1 + ak+1 fk+1 ak fk ak fk),(3.19) dt

I R where ne is the electron density and ak and ak are the total ionization and recombination rate coefﬁcients for charge state k, respectively (see Section 3.2.3). Following the approaches of Hughes & Helfand (1985); Masai (1984), Equation 3.19 can be

256 257 written more compactly for all k in matrix form as,

d F = AF, (B.1) dt where F =(f , f , ..., f , ..., f ) and A is a Z + 1 Z + 1 block tri-diagonal matrix 1 2 k Z+1 ⇥ containing all of the ionization and recombination rates given by,

aI aR 0...0 1 2 0 I I R R 1 a1 (a2 + a2 ) a3 ... 0 B C B .. .. C B . . C A = n B C . (B.2) e B . . C B . aI (aI + aR) aR . C B k 1 k k k+1 C B . . C B .. .. C B C B C B I R C B 0...0aZ aZ+1C B C @ A Note that the k = 1 and k = Z + 1 sink terms only have one term as recombination is not permitted past the lowest charge state and when the ion is in the highest

charge state (Z + 1), there are no electrons left to ionize. Due to drastic changes in the ionization and recombination rates with temperature, the system of equations in Equation 3.19 is very “stiff,” making explicit schemes very sensitive to the choice of time step (Bradshaw, 2009; MacNeice et al., 1984). Equation B.1 can be solved using an implicit “deferred correction” method (NPL, 1961), as pointed out by MacNeice et al. (1984),

d d d F = F + F + F + (d2), (B.3) l+1 l 2 dt l+1 dt l O ✓ ◆ where F = F(t ), d = t t is the time step, and (d2) denotes all terms that l l l+1 l O are second-order or higher in d. Dropping the (d2) terms and using Equation B.1 O 258

gives an expression for Fl+1,

d F = F + (A F + A F ) , l+1 l 2 l+1 l+1 l l d d F A F = F + A F , l+1 2 l+1 l+1 l 2 l l 1 d d F = I A I + A F , (B.4) l+1 2 l+1 2 l l ✓ ◆ ✓ ◆ where I is the identity matrix. Thus, to solve Equation B.1 for a given ne(t), Te(t),

one need only compute Al at each Te(tl), set F0 to the equilibrium population fractions (as determined from Equation 3.31) and then iteratively compute Equa-

tion B.4 at each tl. An example calculation of the population fractions of Fe in non-equilibrium using Equation B.4 is shown in Figure 3.5. While Equation B.4 is unconditionally stable for all d, implicit schemes only guarantee convergence to the equilibrium solution (i.e. Equation 3.31) such that a poor choice of d may still give inaccurate results for the non-equilibrium, time-dependent solution to Equation B.1 (Bradshaw, 2009). MacNeice et al. (1984) recommend choosing d sufﬁciently small such that the following conditions are satisﬁed for all

k and l,

f f # , (B.5) | k,l+1 k,l| . d # fk,l+1 # 10 r . . 10 r , (B.6) fk,l where #d and #r are control parameters with typical values of 0.1 and 0.6, respectively. Additionally, Masai (1984), as well as Shen et al. (2015), provide an alternate scheme

for choosing the time step a priori based on ne and the eigenvalues of Equation B.2.