Understanding the Energy Balance of Transition Region Structures Observed by IRIS in Non-Equilibrium Emission

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RICE UNIVERSITY

Understanding the energy balance of Transition Region structures observed by IRIS in non-equilibrium emission

Shah Mohammad Bahauddin

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE

Doctor of Philosophy

APPROVED, THESIS COMMITTEE

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

Dr. David Alexander OBE Director, Rice Space Institute Professor of Physics and Astronomy

Dr. Laurence Yeung Assistant Professor, Department of Earth, Environmental and Planetary Sciences

HOUSTON, TEXAS March 2019

Shah Mohammad Bahauddin

2019

ABSTRACT

Understanding the energy balance of Transition Region structures observed by IRIS in non-equilibrium emission

by Shah Mohammad Bahauddin

The corona, the outer atmosphere of the Sun, is a multi-million degree plasma, nearly three orders magnitude hotter than the visible surface. The exact mechanism by which the corona is heated is still the subject of debate, but possibilities include magnetic reconnection and magnetohydrodynamic waves. Studying the thin boundary layer connecting the cooler chromosphere to the hotter corona, named the

TR, is an important step toward understanding mass and energy transport from the chromosphere to the corona. Thus, spectral emissions from the cool (< 1 MK) loop- like structures in this region are in need of extensive study and analysis. Because observations lack sufficient spatial resolution, this type of structure was called the

“unresolved fine structure”, which is now considered resolved by the Interface Region

Imaging Spectrograph (IRIS). In the active TR of the Sun, IRIS has observed loop-like structures with intermittent brightenings which are thought to originate from impulsive heating. In this thesis, the author present evidence of magnetic field line braiding and reconnection mediated brightenings of TR loops using IRIS slit-jaw images and spectral data, complemented by the EUV channels of the Atmospheric

Imaging Assembly (AIA) of the Solar Dynamics Observatory (SDO). The set of observables used to characterize the brightenings consists of diagnostics of

temperature, density, line broadening, and Doppler-shift on a pixel-by-pixel basis.

The characterization scheme is extended by accumulating time dependent differential emission measure (DEM) distributions to define the nature of the spatial heating profile and frequency. A field-aligned hydrodynamic simulation and a forward modeling code, designed to generate synthetic observations from numerical experiments for comparison with real data, are employed. Non-equilibrium ionization is included in the computation of synthetic spectra. In addition, the relatively high-density TR plasma requires the inclusion of density-dependent dielectronic recombination rates to calculate the ion populations and the emission line intensities. We show that the observations and the numerical experiments are consistent with reconnection mediated impulsive heating at the braiding sites of multi-stranded TR loops. The combination of observation and numerical analysis will provide the building blocks of time-dependent 3D models of these loops and their contribution to active region emission which will, in turn, help us to understand the energy balance of these structures and may shed light on the long standing coronal heating problem: “Why is the Sun’s corona so much hotter than the surface?”.

Acknowledgments

Foremost, I would like to express my sincere gratitude to my advisor Prof.

Stephen J. Bradshaw for the continuous support of my graduate study and research, in addition to his patience, motivation, and guidance. Besides my advisor, I would like to thank the rest of my thesis committee: Prof. David Alexander and Prof. Laurence

Yeung for their encouragement, insightful comments and questions. I thank my colleague and friend in the Bradshaw Solar Physics group: Will T. Barnes for the stimulating discussions, for the sleepless nights we were working together and for all the fun we had in the last four years. Also, I thank my friends and teachers at the

University of Dhaka. In particular, I am grateful to the Late Prof. Zahid Hasan

Mahmood for believing in me during my time in college. My sincere thanks also go to

Prof Atiqur Rahman Ahad, Mr. Mahbubul Hoq and Prof Biduyt Kumar Bhodro, who were selfless and kindhearted toward me and were sources of inspiration and knowledge. I would also like to take this opportunity to express gratitude to the Quazi family (Fairuz Zaman Chowdhury, Mohiuzzaman Quazi and P Zaman Quazi) without whom I would not be able to take care of my family and study. Last but not the least,

I would like to thank my family: my parents Md Quamruz Zaman and Salma Zaman, and my sister Raahima Zannat (with Budim) for their love and care throughout my life. I am also grateful to my wonderful wife Mahbuba Khan (Mouree) for her love and grace, and my son Ehaan M Muhtadeen for making me realize how precious life truly is.

Contents

Acknowledgments ...... iv

Contents ...... v

List of Figures ...... vii

List of Tables ...... xv

Nomenclature ...... xvii

Introduction ...... 1

The Sun and Its Atmosphere ...... 8 2.1. Origin and Characteristics ...... 8 2.2. Composition and Structure ...... 12 2.3. The Solar Atmosphere ...... 15 2.4. Magnetism and Solar Activity ...... 18

Coronal EUV and X-Ray Radiation ...... 22 3.1. Spectral Lines in the Coronal Model Approximation ...... 22 3.2. Line Broadening Mechanisms ...... 25 3.2.1. Natural Broadening ...... 26 3.2.2. Collisional Broadening ...... 26 3.2.3. Doppler Broadening ...... 27 3.2.4. Non-thermal Line Broadening ...... 29 3.2.5. Line Broadening due to the Limitation of Instrument ...... 30 3.3. Line Formation by Charge State Transitions ...... 31 3.4. Calculation of Ion Population ...... 38 3.5. The Optically Thin Radiative Plasma ...... 41 3.6. Effect of Abundance ...... 43

Hydrodynamic Modeling of Solar Loops ...... 46 4.1. The Hydrodynamic Model ...... 46 4.2. The Hydrodynamic and Radiative Emission (HYDRAD) Code ...... 52 4.3. Forward Modeling Methods ...... 56

IRIS Observations of Transition Region Loops...... 62

5.1. Pre-IRIS Era and Theoretical Models ...... 62 5.2. The IRIS Mission ...... 66 5.2.1. Instrument Overview ...... 69 5.2.2. IRIS Science Results ...... 71 5.3. The Dynamic Transition Region of Solar Active Regions ...... 77 5.4. Non-equilibrium Ionization ...... 79 5.5. IRIS observations of TR loops ...... 83 5.6. Doppler Shift and Line Broadening ...... 90 5.7. Density Diagnostics ...... 91 5.8. Co-aligning the IRIS data with AIA ...... 95 5.9. Differential Emission Measure Analysis ...... 96 5.10. TR Loops in AR 12396 observed on 6th August 2015 ...... 100

On the Origin of Transition Region loop Brightenings ...... 111 6.1. Case I: Footpoint Heating ...... 115 6.2. Case II: Uniform Heating ...... 125 6.3. Case III: Heating by Loop Braiding ...... 131 6.4. Case IV: Heating in Underlying Atmosphere ...... 152 6.4.1. Magneto-acoustic Shocks from Chromosphere ...... 152 6.4.2. Self-Absorption Lines from Chromospheric Reconnection ...... 167

Conclusion and Future Work...... 180 7.1. Magnetic Reconnection and Ion Heating Mechanisms ...... 182 7.2. Non-Maxwellian Velocity Distributions ...... 183 7.3. Multi-wavelength Multi-Instrument Data-driven Analysis ...... 186

References ...... 189

List of Figures

Figure 1: Illustrations depicting the structure and atmosphere of the Sun. Image Courtesy: tapir.caltech.edu ...... 17

Figure 2: The daily adjusted measurements of the solar flux density (10-22 W m-2 Hz-1), from 1947 till the end of January 2018. The green dots mark the time of the official minima of the different sunspot cycles. The orange dots highlight the time period when the flux drops below or is equal to 70 SFU. (Image Courtesy: Solar-Terristrial Center of Excellence, Royal Observatory of Belgium) ...... 20

Figure 3: Radiative Recombination: Ion capturing a free electron into a lower energy, bound state, releasing excess electron energy by emission of a photon...... 32

Figure 4: Free electron is captured by an ion without the emission of a photon, thus giving rise to a doubly exited state. This is the first step to initiate dielectronic recombination...... 34

Figure 5: While staying in the doubly excited state, the inner electron can radiatively decay directly to its initial state and the resultant energy release can free the captured electron with its original energy. This is called autoionization...... 35

Figure 6: Alternatively, spontaneous radiative transition can take place by the inner electron, leading both electrons remained bound to the ion and emitting a photon. This process is known as dielectronic recombination...... 36

Figure 7: Cartoon geometry of a coronal loop with a representation of the numerical grid used to disretesized the domain: the building block of the HYDRAD computational grid is the cell. The cell contains information of its own characteristics as well as the physical properties of the plasma...... 54

Figure 8: Schematic representation of the process by which the emission from each grid cell is binned into the appropriate detector pixel (Bradshaw and Klimchuck 2011)...... 60

Figure 9: Solar atmosphere models developed since the 1950s (from Schrijver 2001)...... 64

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Figure 10 : The variation of the Doppler shift versus temperature at disk center (Peter and Judge 1999). Blue and red shifts (or upward and downward, respectively) motions are represented with positive and negative velocity values, respectively. The solid line demonstrates a by-eye fit to the Doppler shifts of the lines studied by Peter and Judge (1999)...... 65

Figure 11: Left: IRIS data search and collection suite of Lockheed Martin Solar and Astrophysics Laboratory (http://iris.lmsal.com/search/). Right: Example loop candidate seen by IRIS in an active region on in 20th July 2016 at 05:23:09 UTC...... 85

Figure 12: Left: Fitting Si IV line in two Lorentzian components. The corresponding pixel is given in the image at Right. The red marked pixel has much stronger intensity than the balck marked one and exhibited complex broadend line with multiple peaks and Doppler shifts...... 91

Figure 13: O IV ratio generated with CHIANTI 8. Black line T = 105.15 and Red line T = 104.4 (Young 2015)...... 92

Figure 14: Candidate loop 1: (Top Row) 1st 2 columns represent intensity vs density in spatio-temporal coordinate. The following columns represent intensity vs Doppler shift for double component Si IV (denoted by I and II) and single components O IV and S IV lines observed by IRIS. (Bottom Row) 1st 2 columns represent intensity vs Si/O ratio and the latter ones represent intensity vs non-thermal line broadenings for Si IV, O IV and S IV lines. Here, strong bi-directional flow in Si IV line, large non-thermal line broadening in Si IV and S IV and large Si/O peak ratio were observed...... 103

Figure 15: Candidate loop 2: Top Row: the number density was maximum at the loop top when it was brightened. At the same time, the Si IV 1403 Å exhibited compound peaks (I and II) indicating strong bidirectional flow (≈ 100 kms-1). Large Doppler shift was found at S IV (≈ 65 km/s) in contrast to O IV (≈ 25 km/s). Bottom Row: peak to peak ratio of Si IV and O IV exhibited abnormally large ratio which helped to constrains numerical models. Although large non-thermal component in Si IV and S IV line were observed, no significant non-thermal component was observed in O IV...... 104

Figure 16: Pixel-by-Pixel analysis on the candidate loop 3: High density, Strong bi-directional flow in Si IV lines along with large Doppler shift in S IV lines were found at the brightenings like previous candidates. However, the bidirectional flow was stronger into downward direction in contrast to the ix previous observations (where it was nearly equal). Again, the O IV exhibited relatively small Doppler shift. A large non-thermal line broadening in Si IV and S IV were observed in contrast to O IV. Following the trend of previous candidates, large Si/O peak ratio were observed...... 105

Figure 17: Pixel-by-Pixel analysis on the candidate loop 4: High density, Strong bi-directional flow in Si IV lines along with large Doppler shift in S IV lines were found at the brightenings. Here, the bidirectional flow was predominantly upward unlike candidate 3. The O IV exhibited relatively small Doppler shift and a large non-thermal line broadening in Si IV and S IV were observed in contrast to O IV. Following the trend of previous candidates, large Si/O peak ratio were observed (twice as strong as loop 2 and 3)...... 106

Figure 18: Pixel-by-Pixel analysis on the candidate loop 5: High density, Strong bi-directional flow in Si IV lines along with large Doppler shift in S IV lines were found at the brightenings. Here, the bidirectional flow was predominantly upward similar to candidate 4. The O IV exhibited relatively small Doppler shift and a large non-thermal line broadening in Si IV and S IV were observed in contrast to O IV. Following the trend of previous candidates, large Si/O peak ratio were observed (as strong as loop 2 and 3)...... 107

Figure 19: Pixel-by-Pixel analysis on the candidate loop 6. High density, Strong bi-directional flow in Si IV lines along with large Doppler shift in S IV lines were found at the brightenings. Here, the strength of both upward and downward flow was nearly equal (similar to loop 2). The O IV exhibited relatively small Doppler shift and a large non-thermal line broadening in Si IV and S IV were observed in contrast to O IV. Following the trend of previous candidates, large Si/O peak ratio were observed (as strong as loop 2, 3 and 5)...... 108

Figure 20: Pixel-by-Pixel analysis on the candidate loop 7. High density, Strong bi-directional flow in Si IV lines along with large Doppler shift in S IV lines were found at the brightenings. Again, the bidirectional flow was predominantly upward similar to loop 4 and 5. The O IV exhibited relatively small Doppler shift and a large non-thermal line broadening in Si IV and S IV were observed in contrast to O IV. The loop displayed abnormally large Si/O peak ratio (3–4 times as strong as loop 2, 3, 5 and 6)...... 109

Figure 21: Pixel-by-Pixel analysis on the candidate loop 8. High density, Strong bi-directional flow in Si IV lines along with large Doppler shift in S IV lines were found at the brightenings. Here, the strength of both upward and x downward flow was nearly equal (similar to loop 2 and 6). The O IV exhibited relatively small Doppler shift and a large non-thermal line broadening in Si IV and S IV were observed in contrast to O IV. Following the trend of previous candidates, large Si/O peak ratio were observed (as strong as loop 2, 3, 5 and 6)...... 110

Figure 22: Temporal evolution of electron and ion temperature (top) and electron density and velocity (bottom) along the loop axis of a 10 Mm TR loop. The heating rate of the loop is 0.02 erg/cm3/s and a scale height of sH = 1.25 Mm positioned at s0 = 1.25 Mm from both footpoint of the loop...... 120

Figure 23: Evolution of candidate loop 1 seen by IRIS. The time step is approximately 26 seconds spanning over 154 seconds in total (1 – 6). The diagnostics of this loop is given in Figure 12...... 134

Figure 24: Top: A TR loop brightened by magnetic braiding: The 1600Å SDO/AIA passband can see the brightening but cannot resolve its origin. The IRIS 1400Å can resolve that the brightening might be due to the heating of the loop. By introducing the intensity-band filtered unsharp masking technique, one can incorporate the information for the image gradient of the original IRIS image pixel-by-pixel and the origin of the loop brightening can be revealed as braiding of multiple strands in a single loop. Bottom: Spectral information obtained from IRIS for 3 positions: 1 and 3 where braiding causes heating and brightening; 2 where there is no indication of magnetic braiding and reconnection and no brightening is seen...... 136

Figure 25: Top: The light curves of candidate loop 1 are shown for both the IRIS 1400 channel as well as six AIA EUV channels (131 Å, 171 Å, 193 Å, 211 Å, 336 Å and 94 Å). The IRIS 1400 channel appears at least 20 seconds earlier than the AIA EUV channels. Middle: Temporal evolution of the IRIS spectrum and the DEM distribution derived from AIA channels. A significant hot shoulder appears momentarily at the same time the IRIS event occurs and decays. Bottom: The temporal evolution of the slope of the coolward DEM and the integrated DEM (Logarithmic) above 14 MK: A 2 – 3 order of magnitude increase in emission measure is seen when the IRIS events occur. At the same time, a remarkable sharp drop in the coolward slope of the DEM is noticed.140

Figure 26: Top: The light curves of candidate loop 2 are shown for both the IRIS 1400 channel as well as six AIA EUV channels (131 Å, 171 Å, 193 Å, 211 Å, 336 Å and 94 Å). Middle: Temporal evolution of the IRIS spectrum and the DEM distribution derived from AIA channels following a similar pattern to xi candidate 1. Bottom: The temporal evolution of the slope of the coolward DEM and the integrated DEM (Logarithmic) above 14 MK; again showing a similar pattern of evolution...... 141

Figure 27 : Temporal heating profile of one of the loops that was applied to generate the synthetic observation in HYDRAD...... 144

Figure 28: HYDRAD simulation of temperature (top), density and velocity (bottom). At t = 190 seconds, a short, impulsive heating of 0.3 erg/cm3/s was applied for 4 seconds emulating the possible reconnection mechanism. The resulting large temperature difference between electrons and ions is seen. Due to the expansion of plasma by heating, a strong flow of 50 km/s was observed in HYDRAD originating near the apex of the loop. This 50 km/s can be assumed due to the expansion of plasma in all direction as the braiding and reconnection heated apex of the loop. By adding this velocity with the observed non-thermal line broadening (59 – 78 km/s), the LOS doppler shift should correspond over 100 km/s (as observed)...... 146

Figure 29: Observables derived from the synthetic spectrum simulated by the Forward Modeling code: (Top) Normalized IRIS pixel intensity. (Bottom) Synthetic Si/O peak ratio. The results from non-equilibrium (NE) ionization including the density dependent dielectronic recombination rates produce stronger pixel intensity and larger Si/O peak ratio. This quantitatively agrees better with the observation. Here, dielectronic recombination rates are shown for both LD = low density limit and DD = density-dependent rates...... 147

Figure 30: Differences in intensity spectra between the coronal low-density limit and the TR high-density limit accounting for density dependent dielectronic recombination rates for both equilibrium (1st row) and non- equilibrium (2nd row) ionization...... 151

Figure 31: (A)–(C) 흀 − 풕 diagram for Si IV, C II, and Mg II at the IRIS slit located on a sunspot. (D)–(F) Same as (A)–(C) but for a shorter time range (Tian et al. 2014). Notice the correlation between shock occurrence (swing from red to blue) in the Mg II h line concurrently with Si IV and C II lines...... 154

Figure 32: 흀 − 풕 diagram of IRIS Si IV (left) and Mg II h (right) line (De Pontieu et al. 2015). The first pair of diagrams are for a location at a plage. The second pair are observations of a coronal hole region. Overplotted in red and white is the evolution of the non-thermal line broadening as derived from a single Gaussian fit to the Si IV line. The non-thermal line broadening scale is shown xii on the top axis. Notice the correlation between shock occurrence (swing from red to blue) in the Mg II h line and the increased line broadening and brightness in Si IV...... 155

Figure 33: 흀 − 풕 diagrams for the Mg II h (left) and the Si IV lines (middle and right) from numerical simulations (De Pontieu et al. 2015). The first two plots assume ionization equilibrium. The right plot assumes non-equilibrium ionization for Si IV. Overplotted in red or white is the evolution of the non- thermal line broadening as derived from a single Gaussian fit to the Si IV line. Notice the correlation between shock occurrence (swing from red to blue) in the Mg II h line and the increased line broadening and brightness in Si IV. .. 157

Figure 34: 흀 − 풕 diagram of candidate Loop 1. In the first pair of diagrams intensity is integrated along x–axis of the IRIS image. The second pair of diagrams have intensity integrated along both x and y–axis of the IRIS image. In all cases, the integrated intensity is normalized by the pixel array...... 159

Figure 35: 흀 − 풕 diagram of candidate Loop 2. In the first pair of diagrams intensity is integrated along the x–axis of the IRIS image. The second pair of diagrams have intensity integrated along both x and y–axis of the IRIS image. In all cases, the integrated intensity is normalized by the pixel array...... 160

Figure 36: 흀 − 풕 diagram of candidate Loop 3. In the first pair of diagrams intensity is integrated along the x–axis of the IRIS image. The second pair of diagrams have intensity integrated along both x and y–axis of the IRIS image. In all cases, the integrated intensity is normalized by the pixel array...... 161

Figure 37: 흀 − 풕 diagram of candidate Loop 4. In the first pair of diagrams intensity is integrated along the x–axis of the IRIS image. The second pair of diagrams have intensity integrated along both x and y–axis of the IRIS image. In all cases, the integrated intensity is normalized by the pixel array...... 162

Figure 38: 흀 − 풕 diagram of candidate Loop 5. In the first pair of diagrams intensity is integrated along the x–axis of the IRIS image. The second pair of diagrams have intensity integrated along both x and y–axis of the IRIS image. In all cases, the integrated intensity is normalized by the pixel array...... 163

Figure 39: 흀 − 풕 diagram of candidate Loop 6. In the first pair of diagrams intensity is integrated along the x–axis of the IRIS image. The second pair of diagrams have intensity integrated along both x and y–axis of the IRIS image. In all cases, the integrated intensity is normalized by the pixel array...... 164 xiii

Figure 40: 흀 − 풕 diagram of candidate Loop 7. In the first pair of diagrams intensity is integrated along the x–axis of the IRIS image. The second pair of diagrams have intensity integrated along both x and y–axis of the IRIS image. In all cases, the integrated intensity is normalized by the pixel array...... 165

Figure 41: 흀 − 풕 diagram of candidate Loop 8. In the first pair of diagrams intensity is integrated along the x–axis of the IRIS image. The second pair of diagrams have intensity integrated along both x and y–axis of the IRIS image. In all cases, the integrated intensity is normalized by the pixel array...... 166

Figure 42: Spectra covering Si IV (top) and C II (bottom) lines as a function of space and time in three IRIS pixels (Yan et al 2015). Each panel shows the time evolution of the line profiles over 3 minutes starting at 09:21:16. Different colors represent different times (in second) beginning from blue and ending at red...... 169

Figure 43: Cartoon illustrating a possible scenario for the formation of the self-absorption features in Si IV (Yan et al. 2015) due to reconnection in the deep chromosphere...... 170

Figure 44: Single pixel array analysis (averaged over y-axis) and spatially- averaged analysis (averaged over x–y area) of candidate Loop 1 insearch of self-absorption lines...... 172

Figure 45: Single pixel array analysis (averaged over y-axis) and spatially- averaged analysis (averaged over x–y area) of candidate Loop 2 insearch of self-absorption lines...... 173

Figure 46: Single pixel array analysis (averaged over y-axis) and spatially- averaged analysis (averaged over x–y area) of candidate Loop 3 insearch of self-absorption lines...... 174

Figure 47: Single pixel array analysis (averaged over y-axis) and spatially- averaged analysis (averaged over x–y area) of candidate Loop 4 insearch of self-absorption lines...... 175

Figure 48: Single pixel array analysis (averaged over y-axis) and spatially- averaged analysis (averaged over x–y area) of candidate Loop 5 in search of self-absorption lines...... 176 xiv

Figure 49: Single pixel array analysis (averaged over y-axis) and spatially- averaged analysis (averaged over x–y area) of candidate Loop 6 insearch of self-absorption lines...... 177

Figure 50: Single pixel array analysis (averaged over y-axis) and spatially- averaged analysis (averaged over x–y area) of candidate Loop 7 insearch of self-absorption lines...... 178

Figure 51: Single pixel array analysis (averaged over y-axis) and spatially- averaged analysis (averaged over x–y area) of candidate Loop 8 insearch of self-absorption lines...... 179

List of Tables

Table 1: IRIS Spectrograph Channels in FUV (C II and Si IV) and NUV (Mg II) bands...... 70

Table 2: IRIS Slit-Jaw Image Channels. Filter positions can be transmitting (T) or reflecting/mirrors (M) ...... 70

Table 3: Candidate observations (1–4) in active regions in search of the TR loops. The highlighted entry is the best scan having more than 10 loops evolving inside the vicinity of raster scan...... 86

Table 4: Candidate observations (5–8) in active regions in search of the TR loops. The highlighted entry is the best scan having more than 10 loops evolving inside the vicinity of raster scan...... 87

Table 5: Candidate observations (9–12) in active regions in search of the TR loops. The highlighted entry is the best scan having more than 10 loops evolving inside the vicinity of raster scan...... 88

Table 6: Observed transitions by IRIS Si IV 1403 Å used for density diagnostics...... 93

Table 7: Tabulated physical properties extracted from IRIS observations for 8 candidate TR loops emerged in AR 12396 on 6th August 2015...... 102

Table 8: Parameter space in TNE cycles for the observed TR loop brightenings...... 117

Table 9: Parameter space of TNE cycles for the observed TR loop brightenings...... 123

Table 10: Parameter space of uniform heating in search for the observed TR loop brightenings (temporal profile of the heating is chosen to be triangular)...... 125

Table 11: Comparison of synthetic and observed loops for uniform heating...... 131

Table 12: Comparison between observation and modeling under different assumptions of ionization states and dielectronic recombination rates...... 148

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Table 13: Comparison between observation and prediction for loop heating due to magnetic field braiding and reconnection...... 151

Nomenclature

TR Transition Region

IRIS Interface Region Imaging Spectrograph

SDO Solar Dynamics Observatory

AIA Atmospheric Imaging Assembly

TRACE TR and Coronal Explorer

SOHO Solar and Heliospheric Observatory

SUMER Solar Ultraviolet Measurements of Emitted

Radiation

CDS Coronal Diagnostic Spectrometer

MDI Michelson Doppler Imager

EIT Extreme ultraviolet Imaging Telescope

EIS EUV imaging spectrograph

NE Non-equilibrium

Chapter 1

Introduction

The corona (Latin, 'crown') is an aura of plasma that surrounds the Sun and other celestial bodies. The high temperature of the Sun’s corona gives it unusual spectral features, which led 19th century’s scientists to suggest that it contained a mysterious element, namely "Coronium". It was not until the late 1930’s that the astronomers realized these spectral features arise due to highly ionized iron (Fe-XIV)

(Grotrian 1939, Dere 2009). The highly charged states of iron indicate plasma at a temperature of more than a million Kelvin, much hotter than the surface of the Sun

(5800 K). The physical origin of this high temperature is still under debate. The thin region with strong temperature and density gradients that is sandwiched between the chromosphere and the corona is known as the TR (TR). The amount of power required to heat the solar corona can be estimated by summing the coronal losses by radiation and by thermal conduction toward the chromosphere through the TR. The required amount is estimated to be about 1 kW/m2 or 0.0025% of the amount of light

2 energy that escapes the Sun. Possible heating theories (Priest et al. 1998) that allow for such a large heat flux are Alfven wave heating (De Pontieu et al. 2007, Cirtain et al. 2007, McIntosh et al. 2011), magnetic reconnection (Parker 1965, Schrijver et al.

1998, Parker 1988) and magnetic swirls (Wedemeyer-Böhm 2012). Although each theory has its own merits and limitations, none can universally explain the physical origin of the extreme coronal temperatures.

The question of how the upper atmosphere of the Sun is heated to million- degree temperatures, “the Coronal Heating Problem”, remains one of the greatest unsolved problems in space science (Ulmschneider 1998, Malara and Velli 2001).

Although considerable progress has been made in recent years due to the advancement of instrumentation and computational power, a detailed and comprehensive understanding is still lacking (De Moortel et al. 2015, Klimchuck

2015). Proposed coronal heating mechanisms are usually divided into two broad classes: AC heating and DC heating. AC heating corresponds to a periodic rise and fall in the deposited energy, which resembles the expected characteristics of wave heating. One example is the dissipation of propagating waves that originate from below the surface or are driven by motions at the surface of convection cells. As the wave propagates, the velocity amplitude of the wave increases with height to conserve momentum and when the wave reaches the lower corona, the energy dissipates via some turbulent or repetitive dissipation processes (McIntosh et al.

2011, van der Holst et al. 2014). The reasoning behind this idea arises from the observed density stratification of the atmosphere (McIntosh et al. 2011). On the other hand, DC heating involves the gradual storage of energy to be released in a burst of

3 heating following some triggering mechanism and/or instability. One favorable example is the gradual storage of energy in the magnetic field and then sudden release triggered by some form of reconnection. The sudden release converts the magnetic energy into heat energy (Parker 1988, Schrijver et al. 1998). The central idea is based on the assumption that while emission from large active regions appears to be in a steady-state, there exist many individual, small-scale heating pulses with high frequency. With large enough frequency and ubiquity, if each individual impulse is capable of releasing energy of order 1024 erg into the corona, then they can readily supply enough energy needed to maintain its observed temperature (Parker 1988,

Cargill 1994, Cargill and Klimchuck 2004). The required energy to be released per event is about one billionth the energy of a full-scale flare and, these impulsive events were named “nanoflares” by Parker.

However, to be a successful candidate, a mechanism must be able to deliver energy at a high enough rate in the dense region of the atmosphere. Although having a thickness of few hundred kilometers and being optically thin, the turbulent interface region contains more mass than does all the rest of the corona and heliosphere.

Moreover, given how much material is there, the chromosphere requires a heating rate at least ten times greater than that of the corona itself. With the development of modeling and instrumentation, the author now know that the heating is highly time- dependent and varies widely in spatial scales (Litwin 1993, Klimchuck 2006). To grasp the underlying mechanism, the author turns his attention to the bright, curving structures that appear as arcs above the solar surface known as magnetic loops. These loops hold possible observational signatures of dynamic heating mechanisms in the

4 solar atmosphere. Recent observations suggest that coronal heating is impulsive, and a single coronal loop contains hundreds or more individual, unresolved strands that are heated quasi-independently (De Moortel et al. 2015, Klimchuck 2015, Reale

2010). Thus, it is necessary to understand and model such structures to extract the relevant physics. The emission from the TR cannot be fully explained by a simple thin boundary layer in loops connecting the chromosphere and the corona (Klimchuck

2015, Warren and Winebarger 2000). As this moving interface region covers a wide range of heights above the Sun’s surface, the temperatures of this region can vary dramatically from 5000 K to almost a million degrees Kelvins, but there are also enormous density contrasts, with certain areas up to a million times denser than others. As a highly dynamic region constantly in motion, it is a formidable task to profile the temperatures of any structure in this region with respect to position. And to make observational inference even more challenging, a wide range of temperatures can occur at similar heights in this region, with different swaths of material propelled upward and downward in response to the release of magnetic energy, as well as various types of plasma waves.

It has been argued the majority of TR emission is dominated by dynamic, low- lying loops with peak temperatures < 1 MK (Feldman 1983). This argument led to an explorer mission that would perform small-scale analysis of the dynamic network of low-lying loops which would reveal mechanisms powering this region and those above. To explore this hypothesis, NASA launched its solar observation satellite the

Interface Region Imaging Spectrograph (IRIS) on 28 June 2013 (De Pontieu et al.

2014). The IRIS mission aimed to make use of high-resolution images, data and

5 advanced computer models to unravel how matter, light, and energy move from the

Sun’s 6000 K surface to its million K corona. To achieve this, IRIS observes the lowest part of the Sun's atmosphere, the chromosphere and the TR, at the relevant wavelengths. The observations are coordinated by the Lockheed Martin Solar and

Astrophysics Laboratory (LMSAL) and NASA's Ames Research Center in a manner such that IRIS can tease apart what is happening in greater detail than ever before.

Indeed, recent images from IRIS confirmed this new interpretation of a dynamic TR full of low lying, previously unresolved structures (Hansteen 2014). In this research, the author intends to explore the dynamic emission signatures observed by IRIS and study the characteristics and evolution of TR loops to understand how they are powered. It is the author’s belief this study will lead to answers at the core of several outstanding questions about the Sun's atmosphere, such as how it creates giant explosions like solar flares or coronal mass ejections (CMEs), and in particular material in the corona reaches millions of degrees, several thousand times hotter than the surface of the Sun itself.

The science goals of this thesis are to: (1) understand the ion statistics and energy balance of these structures, which dominate the TR emission, as they evolve; and (2) determine the properties of the energy deposition required to produce such low-lying TR loops. The first study reveals how non-equilibrium ionization impacts standard diagnostic analysis techniques and its importance in the numerical modeling of TR loops. The second goal provides insight into how much energy is required to heat these loops relative to the total energy required to sustain the active region emission and the mechanism responsible for delivering it.

6 In chapter 2, the basic structures of the solar atmosphere are discussed accompanied by the introduction of ideas and terms that have become common to the community of solar astrophysicists but might not be familiar to a reader outside of the discipline. In addition, it is only fair to introduce the star to the reader along with all its wonderful characteristics with which we live every day. Chapter 3 comprises brief reviews of theoretical and observational aspects of the radiative emission from the solar atmosphere. On the theoretical side: the basic atomic processes that lead to the formation of spectral lines are introduced; the ionization rates, the recombination rates and the ion populations used by the author are discussed; and an optically-thin radiative emission model employed by the author for use in subsequent work is described. On the observational side: the explanations of spectral line broadening mechanisms are described for various physical processes of plasma. The hydrodynamic modeling of solar loops comprises Chapter 4: the chapter begins with a review of the hydrodynamic equations and the physical processes they describe; then an introduction to the computational hydrodynamic code, HYDRAD, is given, which was employed by the author for the purpose of coupling the hydrodynamics with the atomic processes, finally a brief description of the forward modeling code used to generate synthetic observations is discussed. In essence, Chapter 4 is a review of the previous original work done by the author’s supervisor, Dr. Stephen J Bradshaw

(Bradshaw 2003).

Chapter 5 starts with the conventional view of the TR from the beginning of the space era and progresses to observations until the 2013, the year IRIS was launched. Next, a brief review of the IRIS instrument is presented and the science

7 results it brought to the community. Then the author starts to discuss his original contributions where the techniques of spectroscopic diagnostics (Doppler shifts and line broadening, density diagnostics), co-aligning IRIS data with AIA, and differential emission measure analysis for determining the physical properties of the TR loops and associated brightenings are explored in detail. The set of observations carried out by IRIS and analyzed by the author are also described in this chapter along with their physical implications.

Chapter 6 heavily concentrates on the numerical simulation and forward modeling efforts in order to interpret the IRIS observations. Different heating processes are discussed in detail and the most probable candidate is identified, supported with observational and numerical evidence. The analysis is methodically performed in a such a manner that a large parameter space is covered and bias for a particular process is avoided during the numerical experimentation. The author concludes the thesis with Chapter 7 and the scope of extensions to this work. These include a detailed investigation of the ion heating processes that might play an important role hinted by IRIS observations; the inclusion of non-Maxwellian distributions (such as the 휅-distribution) in the prediction and analysis of spectroscopic data; and the assimilation of multi-wavelength, multi-instrument data to the existing analysis to impose further constraints on the numerical modeling in order to obtain more reliable interpretation of the observational data.

Chapter 2

The Sun and Its Atmosphere

2.1. Origin and Characteristics

In the center of the solar system lies the Sun as the brightest star in the Earth’s sky. The Sun, as a sphere of hot plasma with internal convective motion, maintains a magnetic field through the dynamo process (Lang 2000). It is a G-type main sequence star that accounts for approximately 99.86 percent of the solar system mass and estimated to be brighter than about 85 percent of the stars in the Milky Way

(Woolfson 2000 and Lada 2006). The formation and evolution of the Sun began approximately 4.6 billion years ago with the gravitational collapse of a fragment of a giant molecular cloud (Bonanno et al. 2002 and Connelly et al. 2012). The collapse is believed to be triggered by shockwaves from one or more supernovae nearby. Most

of the collapsing mass was collected in the center, forming the Sun, while the rest flattened into a protoplanetary disk out of which the planets, moons, asteroids, and other small Solar System bodies formed. This theory is called ‘nebular hypothesis’ and it is conjectured that the giant nebular cloud was more than 60 light years across, while the fragments were roughly 4 light years across which had a mass just over that of the Sun. The composition of the cloud fragment was hydrogen (along with helium and trace amounts of lithium produced by Big Bang nucleosynthesis) forming about

98% of its mass. The remaining 2% of the mass consisted of heavier elements that were created by nucleosynthesis in earlier generations of stars (Hansen et al. 2012).

The origin and age of the Sun can be estimated by examining the elements found in meteorites, which can trace the first solid material to form in the pre-solar nebula (Bouvier and Meenakshi 2010). Studies of ancient meteorites reveal traces of stable daughter nuclei of short-lived isotopes, such as Fe-60, that can only form in exploding, short-lived stars. This indicates that one or more supernovae occurred near the Sun while it was forming. A shock wave from a supernova may have triggered the formation of the Sun by creating relatively dense regions within the cloud, causing these regions to collapse. Because only massive, short-lived stars produce supernovae, the Sun must have formed in a large star-forming region that produced massive stars. Studies of the structure of the Kuiper belt and of anomalous materials within it suggest that the Sun was formed within a cluster of between 1,000 and

10,000 stars with a diameter of between 6.5 and 19.5 light years (Zeilik and Gregory

1998). This cluster began to break apart a few hundred million years after formation.

At its dying moments, the pre-solar nebula spun faster as it collapsed preserving the conservation of angular momentum, and as the material within the nebula condensed, the atoms within it began to collide with increasing frequency, converting their kinetic energy into heat. As the center collected most of the mass, it became increasingly hotter than the surrounding disc. Over about 100,000 years, the competing forces of gravity, gas pressure, magnetic fields, and rotation caused the contracting nebula to flatten into a spinning protoplanetary disc with a diameter of about 200 AU and form a hot, dense protostar at the center.

At this point in its evolution, the Sun is thought to have been a T-Tauri star

(Caffe et al 1987). Such stars are observed to be accompanied by discs of pre- planetary matter (Momose et al. 2003). These discs are observed by Hubble to be extended over several hundred AU and spectral analysis found materials reaching a surface temperature of about a thousand Kelvins at most. Within 50 million years of its birth, the temperature and pressure at the core of the Sun became so great that its hydrogen began to fuse, creating an internal source of energy that countered gravitational contraction until hydrostatic equilibrium was achieved. Thus, the Sun entered into the prime phase of its life, known as the main sequence. Main-sequence

stars derive energy from the fusion of hydrogen into helium in their cores (Zeilik and

Gregory 1998). The Sun remains a main-sequence star today.

As a sphere of plasma, the Sun does not have a definite boundary, but its density decreases with increasing height above the photosphere. For the purpose of measurement, the Sun's radius is considered to be the distance from its center to the edge of the photosphere, the visible surface of the Sun (where the plasma becomes optically thin). By this measure, the Sun is a near-perfect sphere having its polar diameter differing from its equatorial diameter by only 10 km (Lang 2000). The Sun is observed to be rotated faster at its equator than at its poles (Lang 2000). In a frame of reference defined by the stars, the rotational period is approximately 25.6 days at the equator and 33.5 days at the poles. Viewed from Earth as it orbits the Sun, the apparent rotational period of the Sun at its equator is about 28 days. The tidal effect of the planets does not significantly affect the shape and rotation of the Sun.

The Sun is a G2V star, with G2 indicating its surface temperature of approximately 5800 K and V that it is a main-sequence star (Bouvier and Meenakshi

2010). The Sun is by far the brightest object in the Earth's sky, about 13 billion times brighter than the next brightest star, Sirius. The average luminance of the Sun is about

1.88 G cd/m2. The solar constant is the amount of power that the Sun deposits per unit area that is directly exposed to sunlight and is equal to approximately 1,368

W/m2 at the distance of Earth from the Sun. The incident sunlight on the surface of

Earth is attenuated by Earth's atmosphere and the energy of the sunlight supports almost all life on Earth and drives Earth's climate and weather.

2.2. Composition and Structure

The Sun is composed primarily of hydrogen and helium. Currently in the Sun's life, they account for 74.9% and 23.8% of the mass of the Sun in the photosphere, respectively (Lodders 2003). All heavier elements, called metals in astronomy, account for less than 2% of the mass, with 1% oxygen, 0.3% carbon, 0.2% neon, and

0.2% iron being the most abundant (Hansen et al. 2012). This chemical composition was inherited from the nebula fragment out of which it formed. The hydrogen and most of the helium in the Sun would have been produced by Big Bang nucleosynthesis in the first 20 minutes of the universe, and the heavier elements were produced by previous generations of stars before the Sun was formed and spread into the interstellar medium during the final stages of stellar life and by supernovae. The Sun has a higher abundance of elements heavier than hydrogen and helium than the older stars and it is believed that this high metallicity played a crucial role in the development of its planetary system (Lineweaver 2001).

The core of the Sun extends from the center to about 20–25% of the solar radius. It has a density of up to 150 g/cm3 and a temperature of close to 15.7 million

Kelvins (K) (García et al. 2007). By contrast, the Sun's surface temperature is

approximately 5,800 K. The core is the only region in the Sun that produces an appreciable amount of thermal energy through fusion; 99% of the power is generated within 24% of the Sun's radius, and by 30% of the radius fusion stops nearly entirely.

The remainder of the Sun is heated by this energy as it is transferred outwards, by radiation and convection, finally to the solar photosphere where it escapes into space as sunlight or the kinetic energy of particles (Zirker 2002).

The Sun releases energy at the mass–energy conversion rate of 4.26 million metric tons per second (which requires 600 metric megatons of hydrogen). The fusion rate in the core is in a self-correcting equilibrium: a slightly higher rate of fusion would cause the core to heat up more and expand slightly against the weight of the outer layers, reducing the temperature and hence the fusion rate and correcting the perturbation; and a slightly lower rate would cause the core to cool and shrink slightly, increasing the temperature and increasing the fusion rate and again reverting it to its present rate.

From the core out to about 70% of solar radii, thermal radiation is the primary means of energy transfer. The temperature drops from approximately 7 million to 2 million Kelvins with increasing distance from the core. This temperature gradient is less than the value to drive convection (sub-adiabatic) and thus the transfer of energy can only be facilitated by photon absorption by electrons and re-emission. The density drops a hundred-fold (from 20 g/cm3 to 0.2 g/cm3) from the core to the top

of the radiative zone. The radiative zone and the convective zone are separated by a transition layer known as the tachocline (Spiegel and Zahn 1992). This is a region where the sharp change between the uniform rotation of the radiative zone and the differential rotation of the convection zone results in a large shear between the two successive horizontal layers as they slide past one another. Presently, it is hypothesized that a magnetic dynamo operates at the tachocline and generates the solar magnetic field.

The Sun's convection zone extends from 70% of the solar radii to the bottom of the photosphere. In this layer, the solar plasma is neither dense enough nor hot enough to transfer the heat energy outward via radiation. Instead, the density of the plasma is low enough to allow convective currents to develop and move the Sun's energy outward towards its surface. Material heated at the tachocline absorbs heat and expands, thereby reducing its density and allowing it to rise. As a result, an orderly motion of the mass develops into thermal cells that carry the majority of the heat outward. Once the material diffusively and radiatively cools just beneath the photospheric surface, its density increases, and it sinks to the base of the convection zone, where it again picks up heat from the top of the radiative zone and the convective cycle continues. At the ceiling of the convective zone, the temperature drops to 5800 K and the density to only 0.2 g/m3.

The visible surface of the Sun, the photosphere, is the layer below which the

Sun becomes opaque to visible light. Above the photosphere visible sunlight is free to propagate into space, and almost all of its energy escapes the Sun entirely (Gibson

1973). The change in opacity is due to the decreasing amount of H− ions, which absorb visible light easily. The photosphere is tens to hundreds of kilometers thick and is slightly less opaque than air on Earth. The spectrum of the sunlight traced back from the optically thin photosphere mimics the spectrum of a black-body radiating at

5800 K, interspersed with atomic absorption lines from the tenuous layers above the photosphere. Thus, the temperature of the photosphere is estimated to be 5800 K and it has a particle density of ~1023 m−3. Because of this temperature, the photosphere is not fully ionized, having an ionization of about 3% while leaving almost all of the hydrogen in its neutral form. The coolest layer of the Sun is a temperature minimum region extending to about 500 km above the photosphere, and has a temperature of about 4100 K. This part of the Sun is cool enough to allow the existence of simple molecules such as carbon monoxide and water, which can be detected via their absorption spectra (Solanki et al. 1994).

2.3. The Solar Atmosphere

During a total solar eclipse, when the disk of the Sun is covered by that of the

Moon, parts of the Sun's surrounding atmosphere can be seen. It is composed of four

distinct parts: the chromosphere, the TR, the corona and the heliosphere. All these four layers are much hotter than the surface of the Sun (Lang 2000) and shown in

Figure 1 (except the TR due to its minute thickness). Above the temperature minimum layer lies a layer about 2,000 km thick, dominated by a spectrum of emission and absorption lines and is called the chromosphere. The temperature of the chromosphere increases gradually with altitude, up to around 20,000 K near the ceiling. In the upper part of the chromosphere helium becomes partially ionized.

Above the chromosphere, there exists a thin TR, spanning over a few hundred kilometers. Here, the temperature rises rapidly from around 20,000 K in the upper chromosphere to coronal temperatures of order 1 MK. The temperature increase facilitates the full ionization of helium in the TR, which significantly reduces radiative cooling of the plasma. The TR does not occur at a well-defined altitude. Rather, it forms around chromospheric features such as spicules and filaments, and is in constant, chaotic motion. The TR is only observable from space by instruments sensitive to the extreme ultraviolet portion of the spectrum (Lang 2000).

The corona is the final and extended layer of the Sun’s atmosphere. The low corona, near the surface of the Sun, has a particle density around 1015 m−3 to 1016 m−3

(Hansteen et al. 1997). The average temperature of the corona and solar wind is a couple of million Kelvins; however, in the hottest regions it can reach 8 million to 20 million Kelvins. Although no complete theory yet exists to account for the

temperature of the corona, at least some of its heat is conjectured to be from nanoflare and Alfvén wave dissipation. A flow of plasma outward from the corona into interplanetary space is called the solar wind.

Figure 1: Illustrations depicting the structure and atmosphere of the Sun. Image Courtesy: tapir.caltech.edu

The heliosphere, the tenuous outermost atmosphere of the Sun, is filled with solar wind plasma. This layer of the Sun is defined to begin at the distance where the flow of the solar wind becomes super-Alfvénic—that is, where the flow becomes faster than the speed of Alfvén waves (Parker 1958, Emslie and Miller 2003), at approximately 20 solar radii (0.1 AU). Turbulence and dynamic forces in the heliosphere cannot affect the solar corona within, because the information can only

travel at the speed of Alfvén waves. The solar wind travels outward continuously through the heliosphere, forming the solar magnetic field into a spiral shape, until it impacts the heliopause more than 50 AU from the Sun. In December 2004, the

Voyager 1 probe passed through a shock front that is thought to be part of the heliopause (Stone et al. 2005). In late 2012, Voyager 1 recorded a marked increase in cosmic ray collisions and a sharp drop in lower energy particles from the solar wind

(Stone et al. 2013), which suggested that the probe had passed through the heliopause and entered the interstellar medium.

2.4. Magnetism and Solar Activity

The Sun has a magnetic field that varies across the surface of the Sun. Its polar field is 1–2 gauss, whereas the field is typically 3000 gauss in sunspots (Lang 2000).

The magnetic field also varies in time and location. The quasi-periodic 11-year solar cycle is the most visible spatial variation in which the appearance of the concentrated magnetic field lines oscillates due to the exchange of energy between toroidal and poloidal solar magnetic fields (Lang 2000, Charbonneau 2014). At solar-cycle maximum, the external poloidal dipolar magnetic field is near its dynamo-cycle minimum strength, but an internal toroidal quadrupolar field, generated through differential rotation within the tachocline, is near its maximum strength. At this point in the dynamo cycle, buoyant upwelling within the convective zone forces emergence

of toroidal magnetic field through the photosphere, giving rise to pairs of sunspots, roughly aligned east–west and having footpoints with opposite magnetic polarities.

During the solar cycle's declining phase, energy shifts from the internal toroidal magnetic field to the external poloidal field, and sunspots diminish in number and size. At solar-cycle minimum, the poloidal field is at its maximum strength while the toroidal field rests at its minimum and thus sunspots are relatively rare. With the rise of the next 11-year sunspot cycle, differential rotation shifts magnetic energy back from the poloidal to the toroidal field, but with a polarity that is opposite to the previous cycle. The process carries on continuously, and in an idealized, simplified scenario, each 11-year sunspot cycle corresponds to a change, in the overall polarity of the Sun's large-scale magnetic field. The phenomenon of the alternating magnetic polarity of sunspot pairs is known as the Hale cycle (Hale et al. 1919). Figure 2 shows the variation of solar flux density for last 70 years. In roughly each 11 years, the flux density maximizes indicating the solar sunspot cycle.

Figure 2: The daily adjusted measurements of the solar flux density (10-22 W m- 2 Hz-1), from 1947 till the end of January 2018. The green dots mark the time of the official minima of the different sunspot cycles. The orange dots highlight the time period when the flux drops below or is equal to 70 SFU. (Image Courtesy: Solar-Terristrial Center of Excellence, Royal Observatory of Belgium)

Sunspots are visible as dark patches on the Sun's photosphere and correspond to concentrations of magnetic field where the convective transport of heat is inhibited from the solar interior to the surface (Lang 2000). As a result, sunspots are slightly cooler than the surrounding photosphere, and appear relatively darker. At a typical solar minimum, few sunspots are visible and appear only at high solar latitudes. As the solar cycle progresses towards its maximum, sunspots tend form closer to the

solar equator covering as large as tens of thousands of kilometers across. High velocity solar wind, solar flares and coronal-mass ejections are all found to occur at sunspot groups (Zirker 2002). Both coronal-mass ejections and high-speed streams of solar wind carry plasma and magnetic fields outward into the Solar System.

The solar magnetic field extends well beyond the Sun itself. The electrically conducting solar wind plasma carries the Sun's magnetic field into space, forming what is called the interplanetary magnetic field. The outward-flowing solar wind stretches the interplanetary magnetic field outward, forcing it into a roughly radial structure. For a simple dipolar solar magnetic field, with opposite hemispherical polarities on either side of the solar magnetic equator, a thin current sheet is formed in the solar wind. At great distances, the rotation of the Sun twists the dipolar magnetic field and corresponding current sheet into an Archimedean spiral structure called the Parker spiral (Thomas and Smith 1980). The magnetic field exerted by the

Sun leads to many effects that are collectively called solar activity. The effects of solar activity on Earth include auroras at moderate to high latitudes and the disruption of radio communications and electric power (Lang 2000). Solar activity is thought to have played a large role in the formation and evolution of the Solar System.

Chapter 3

Coronal EUV and X-Ray Radiation

3.1. Spectral Lines in the Coronal Model Approximation

A telescope observing the Sun’s corona is illuminated from three primary sources, from the same volume of space. The first one is K-corona (K for kontinuierlich, "continuous" in German), created by sunlight scattering off free electrons. Lines of K-corona is spread by Doppler broadening of the reflected photospheric absorption greatly as they become completely obscure, giving the spectral appearance of a continuum with no absorption lines (House et al. 1981).

Next, the F-corona (F for Fraunhofer) is created by sunlight bouncing off dust particles and is observable by telescope as it contains the Fraunhofer absorption lines extending to very high elongation angles from the Sun, where it is called the zodiacal light (Kaiser 1970). Finally, the E-corona is due to spectral emission lines (E for emission) produced by ions of charged elements that are present in the coronal plasma. Depending on the physical mechanisms at the region of observation, it may

be observed in broad or forbidden line, or hot spectral emission lines and is the main source of information about the corona's composition.

The fact that the structure of the corona is complex and quite dynamic was first proved when high-resolution X-ray photograph arrived from the Skylab in 1973

(MacQueen 1974), and then later by Yohkoh (Acton et al. 1992) and the other following space instruments such as TRACE, SDO and Hinode (Schrijver et al. 1999 and Del Zanna et al. 2011). At that time, solar astronomers had a grasp on the nature of the Sun’s stratified atmosphere and were able to generally define the coronal plasma. Under this definition, the solar corona is a high temperature (> 106 K), dilute

(< 1010 cm-3) plasma. It is mainly comprised of hydrogen (H) and helium (He), which are fully ionized at coronal temperatures, and so the plasma is treated as fully ionized despite the existence of trace quantities of partially ionized heavier elements.

The corona is optically-thin to visible, UV and X-ray radiation. Photons at these wavelengths pass through unhindered by opacity effects: absorption, re-emission and scattering. In the non-flaring corona, most radiation arises due to electron transitions within the ions, called bound-bound transitions. The intensity of a spectral line due to a bound-bound transition from a column of plasma, with volume V and cross- sectional area A can be written as,

1 퐼(휆푗,푖) = ∫ 푃(휆푗,푖)푑푉 4휋퐴 푉

Where, 푃(휆푗,푖) is the emissivity of the spectral line that arises from an atom of element X, charge state +m (lost m electrons) and in bound/excited state j emitting a photon of energy Δ퐸푗,푖 = ℎ푐⁄휆푗,푖 in order to arrive at a lower energy state i. The emissivity 푃(휆푗,푖) is the product of three physical parameters: the number density of ions of charge +m in excited state j, the Einstein spontaneous emission coefficient and the energy of the emitted photon. Here, the number density can be written in terms of a series of ratios that can be observed and calculated theoretically,

푁 (푋+푚) +푚 +푚 푗 푁(푋 ) 푁(푋) 푁(퐻) 푁푗(푋 ) = +푚 푁푒 푁(푋 ) 푁(푋) 푁(퐻) 푁푒

Where, 푁(푋+푚) is the total population of ions of charge +m, 푁(푋) is the total element abundance, 푁(퐻) is the abundance of hydrogen and 푁푒 is the number density of electrons in the column. As described above, the Sun’s corona is an optically thin layer of plasma and this approximation allows one to assume that the majority of excitations and de-excitations occur through collisional excitation and spontaneous radiative decay from the ground state (g) of each ion present. Thus, one can assume that changes in the energy level populations of the emitting ions take place in much shorter timescale than changes in the charge state do. In other words, the processes

푁 (푋+푚) that determine the excitation state ( 푗 ) are decoupled from those that 푁(푋+푚)

푁(푋+푚) determine the charge state ( ); this simplification is known as the coronal 푁(푋)

model approximation. Under the coronal model approximation, the expression for the emissivity in terms of the electron collisional excitation rate coefficient from the

푒 ground state g (퐶푔,푗) becomes,

+푚 푁(푋 ) 푁(푋) 푁(퐻) 푒 2 푃(휆푗,푔) = 퐶푔,푗Δ퐸푗,푔푁푒 푁(푋) 푁(퐻) 푁푒

In the above equation, the rate at which collisional transitions occur depends on the interaction cross-section presented to incident particles by the target and on the flux of incident particles. Since particle interactions are mostly through collisions, it is instinctive to assume that the distribution is a collisionally relaxed Maxwellian

(although this is not true for many dynamic events).

3.2. Line Broadening Mechanisms

A spectral line is a fingerprint that identifies the atoms, elements or molecules present in the plasma and the nature of the transitions that take place within them.

Ideally, the radiative transitions should give rise to line profiles that are infinitely sharp, or mathematically, delta-functions. However, atomic levels are not infinitely sharp, nor are the lines that connect the levels and their states. Thus, a spectral line can extend over a range of wavelengths, and its center may be shifted from its rest wavelength. Many physical effects can affect the shape and the position of the line and

the author would like to discuss some of the most relevant line broadening mechanisms that occur in TR and coronal plasma.

3.2.1. Natural Broadening

The uncertainty principle dictates that the spread in energy Δ퐸 of an atomic state and the lifetime Δ푡 of that state must satisfy Δ퐸Δ푡~ℏ. In other words, a finite lifetime of a state must have an energy uncertainty and thus a minimum width of atomic level will be measured by the observer. Since, the spontaneous decay of an atomic state n proceeds at a rate, 훾 = ∑푛′ 퐴푛푛′, where the sum is all states n’ of lower

′ energy and 퐴푛푛′ is the decay rate for the transition 푛 → 푛 ; therefore, the line profile would encapsulate a decaying sinusoid electric field and results in an unshifted

Lorentz profile (Rybicki and Lightman 2008),

훾/4휋2 휙(휈) = 2 2 (휈 − 휈0) + (훾/4휋)

3.2.2. Collisional Broadening

The collision of other atoms within the plasma can interrupt the emitting atom of interest during its emission process. For example, if the emitting atom suffers collisions with other particles while it is emitting, then the phase of the emitted radiation can be altered suddenly and consequently shorten the characteristic time for the process, increasing the uncertainty in the energy emitted (as occurs in natural

broadening). Generally, the duration of collision is much shorter than the lifetime of the emission process and strongly dependent on the temperature and the density of the plasma. If the collisions occur with frequency 휈푐표푙, then the broadening effect can be described by a Lorentzian profile (Rybicki and Lightman 2008),

Γ/4휋2 휙(휈) = 2 2 (휈 − 휈0) + (Γ/4휋)

where, Γ = 훾 + 2휈푐표푙

3.2.3. Doppler Broadening

Among all the physical mechanisms that can cause broadening due to local effects, perhaps the simplest one for line broadening is the Doppler effect. In this scenario, an atom is in thermal motion so that the frequency of emission (or absorption) in its own frame of reference corresponds to a different frequency in the frame of reference of the observer. As each atom suffers from its own shifts of frequency, the net effect is to spread the line out, while keeping the total strength of radiation constant. The change in frequency associated with an atom with velocity component 푣푧 along the line of sight (say, z axis) can be written by,

휈 푣 휈 − 휈 = 0 푧 0 푐

Where 휈0 is the frequency in the atom’s frame of reference. The atoms in a plasma emitting radiation would have a distribution of velocities. Each photon emitted will be "red"- or "blue"-shifted by the Doppler effect depending on the velocity of the atom relative to the observer. Thus, in thermal equilibrium, the number of atoms having velocities in the range 푣푧 to 푣푧 + 푑푣푧 would be proportional to the Maxwellian distribution,

푚 푣2 exp (− 푎푡표푚 푧 ) 푑푣 2푘푇 푧

Therefore, the strength of the emission in the frequency range 휈 to 휈 + 푑휈 is proportional to

2 2 푚푎푡표푚푐 (휈 − 휈0) exp (− 2 ) 푑휈 2휈0 푘푇

1 2 2 with the line profile function, 휙(휈) = 푒−(휈−휈0) /(Δ휈퐷) . Here, the Doppler Δ휈퐷√휋 width Δ휈퐷 is defined by,

휈0 2푘푇 Δ휈퐷 = √ 푐 푚푎푡표푚

The expression of the effective width can be extended in a similar manner for additional physical mechanisms. In almost all cases, a radiating atom in a plasma shows both a Lorentzian profile and a Gaussian profile. The combination of thermal

broadening with the natural or collisionally broadened line profile is called the Voigt profile which is effectively the convolution of the Lorentz profile with a Gaussian profile and has no simple analytic form. However, the Lorentzian broadening is generally negligible and dominates at the wings, while the center of the line is dominated by the Gaussian profile.

3.2.4. Non-thermal Line Broadening

Other processes may contribute to line broadening such that the ion temperature may be overestimated if one assumes that thermal broadening is the dominant component to the line width. Such broadenings are often named as

'turbulent' broadening - an ad hoc description of nonthermal motions. The physical reasons for such broadening are usually non-thermal processes e.g. magnetic reconnection, Alfvén wave heating etc. In the absence of a firm physical model for the additional sources of velocity dispersion, it is customary to adopt a gaussian line-of- sight velocity distribution for the sake of convenience. As mentioned in the previous section, the root-mean-square (rms) line-of-sight (1-D) thermal Doppler velocity dispersion arising from a Maxwellian velocity distribution is

휈0 2푘푇 Δ휈퐷 = √ 푐 푚푎푡표푚

The turbulent motion experienced by the plasma due to various non-thermal velocity fields can be characterized by assuming the line-of-sight turbulent velocity distribution as gaussian, with an RMS value 휉; then the total broadening is obtained simply by adding the thermal and turbulent widths in quadrature

휈 2푘푇 0 2 Δ휈퐷 = √ + 휉 푐 푚푎푡표푚 keeping the form of the gaussian component of the emission profile unchanged. In principle, since every element abundant in that atmosphere should share the same non-thermal broadening but will undergo different (mass-dependent) thermal broadening, it should be possible to separate the two components by careful analysis.

However, the various physical processes occurring simultaneously at different timescales and length-scales make such calculation extremely difficult in practice.

3.2.5. Line Broadening due to the Limitation of Instrument

In spectroscopy, a continuous physical quantity e.g. intensity vs wavelength is recorded. The corresponding data is sampled electronically as discrete points.

However, the observed distribution of photon intensity over wavelength is limited by the diffraction limit of the instrument and the digital system that discretizes the data.

Moreover, various external factors such as axial divergence of the incident and diffracted beams, the configuration of slits in the diffractometer and any

misalignment during calibration and the extreme conditions of launch can introduce additional uncertainties. Thus, the approximation of an optically-thin spectral line must take account of an excess broadening term in the Gaussian profile, known as instrumental width. Under such assumptions, the intensity per unit wavelength contains an additional broadening term 휎퐼 which encapsulates all the contribution of line broadening due to the instrument itself,

2 퐼 (휆 − 휆0) 퐼휆 = 푒푥푝 [− ] √2휋휎 2휎2

2 2 휈0 2푘푇 2 2 Where, 휎 = 2 ( + 휉 ) + 휎퐼 2푐 푚푎푡표푚

3.3. Line Formation by Charge State Transitions

As an ion of an element can change its state by undergoing the transition of an electron between its energy levels, it can also change its state by processes where the ion gains or loses an electron: electrons can be freed from their bound states or free electrons can be captured into a bound state of an ion. The process is called a bound- free transition or a free-bound transition, respectively. Bound-free transitions are also called ionization while free-bound transitions are called recombination. The rates of ionization and recombination governs the charge state of the ions in a plasma.

The energy needed for the charge state transition may come from an absorbed

photon, or from collisions with particles. If the energy to initiate ionization comes from another excited electron within the ion itself, then the process is known as autoionization.

Generally, the recombination mechanism has two components: radiative recombination and dielectronic recombination. The former one occurs when an ion can capture a free electron into a lower energy, bound state, and the excess electron energy is released by the emission of a photon (shown in Figure 3). This can be expressed as,

+푧+1 +푧 ℎ푐 푋 + 푒 ⇒ 푋 ′ + 푖 푖 휆

Figure 3: Radiative Recombination: Ion capturing a free electron into a lower energy, bound state, releasing excess electron energy by emission of a photon.

The latter was first recognized by Massey and Bates (1942) and its importance in coronal plasma was first demonstrated by Burgess (1964, 1965, 1977). In this process, a free electron is captured by an ion without the emission of photon, thus giving rise to a doubly exited state and then a resonant photon is emitted so that the ion can reach a stable singly excited state. It is a two-step process that can potentially increase the efficiency for electrons and ions to recombine in a plasma. Using detailed balance arguments and the correspondence principle, Burgess derived formulas for the rate of the process and calculated the recombination coefficients for iron and calcium in the corona. The revised ionization balance curves derived by Burgess agreed better with the observations found from the Doppler widths of coronal lines.

The dielectronic recombination can be divided into two steps: A free electron collides with an ion of element 푋 having an ionization state 푧 + 1 at an energy just below the threshold for excitation of a resonance transition but remains itself captured in an outer excited state 푛푙. Thus, a doubly excited state is formed with the excited core shown in Figure 4.

푧+1 − 푧 푋푖 + 푒 ↔ 푋푗,푛푙

Figure 4: Free electron is captured by an ion without the emission of a photon, thus giving rise to a doubly exited state. This is the first step to initiate dielectronic recombination.

At this point, the inner electron can radiatively decay directly to its initial state

푗 → 푖 and the resultant energy release can free the captured electron with its original energy. This means that the process leads to the ionization of the ion itself, an example of autoionization. Autoionization can also be initiated by the inner electron but decaying to another excited state than the initial one. In this case, the ion would be autoionized, but the captured electron would be released with less than its original energy (Figure 5).

푧 푧+1 − 푋푗,푛푙 → 푋푘 + 푒

Alternatively, spontaneous radiative transition can take place by the inner electron, but to a state having a photon emitted of energy less than the ionization energy of level 푛푙. Note from Figure 6, both electrons remain bound to the ion in this transition emitting a photon. Although it is a stabilizing transition, the singly excited state of the ion eventually cascades down to the ground state completing dielectronic recombination.

ℎ푐 푋푧 → 푋푧 + 푗,푛푙 푚,푛푙 휆

The overall effect of these multiple pathways of the process obtained by summing over all possible states of the inner excited electron as well as the one in the state 푛푙 give an increase of the total recombination rate in the plasma. Hence, previous calculations before Burgess led to estimates of the temperature of the corona lower than measured, since those calculations assumed that the dominant recombination mechanism in the solar corona was radiative recombination. The inclusion of dielectronic recombination leads to a higher rate of recombination which can only be balanced by introducing an increased rate of ionization and, the inferred temperature calculated from the study would be substantially increased. This led to a higher estimate for the coronal temperature which agreed with the observed

spectral line profiles and radio observations of the time indicating temperatures of

1.5 – 2.5 × 106 K.

Burgess presented formulae for the dielectronic recombination rate coefficient for a given initial and intermediate state and also the total rate coefficient for a given initial state summed over all intermediate states, culminating in the well- known Burgess formula for dielectronic recombination for coronal ions.

Studies showed that the dielectronic recombination rate coefficients exhibit a significant dependence on the density of the plasma. Such dependence severely affects the predictions of the amount of a particular ion and the temperature at which the peak population occurs in the case of density stratification. One of the most used atomic databases for numerical modeling and simulations in the solar physics community is CHIANTI (Landi et al. 2013). This freely available database provides a comprehensive set of spectral line energies for different elements, their corresponding transition probabilities and other data such as element abundances and ionization balances which are used for the calculation of the radiative loss function. However, CHIANTI and similar atomic databases generally assumes ionization and recombination rates in the low-density limit. Although this assumption performs well in coronal plasma, it may not return realistic results in the exploration of denser atmosphere, such as the TR. The substantial density of the TR can suppress the dielectronic recombination rate by opening additional pathways to ionization and

thus preventing the recombination process from being fully completed. Bradshaw &

Mason 2003, Polito et al. 2016 and Bradshaw & Testa 2019 reported that such quenching of dielectronic recombination shifts the ion population to lower temperatures than predicted in the low-density limit (coronal).

While Burgess’ work led to a plethora of published literature calculating dielectronic recombination rates in the low-density (coronal) limit, there was a scarcity of published recombination rates that take density-dependence into account.

Since the TR observed by IRIS is essentially a steep gradient of dense atmosphere, the dielectronic recombination rate inevitably varies within a short scale height and this must be taken account of in calculations of emission spectrum. Thus, the author was provided the best of the available density dependent dielectronic recombination rates from the ADAS (Atomic Data and Analysis Structure) database to incorporate in

HYDRAD’s ionization and radiation calculations (Summers and O’Mullane 2002).

Although the ADAS database is not open-source and it was designed specifically for applications involving fusion research, the rates it provides apply equally well to TR plasma observed by IRIS.

3.4. Calculation of Ion Population

Ion population fraction (IPF) calculations are essential to understand the radiative spectrum emitted by hot coronal plasma. In order to successfully calculate

the IPF of each individual ion belonging to an element, a radiation model needs to possess the full set of ionization and recombination rates. Here, the term population fraction means the fraction of all ions belonging to an element that are in a particular charge state. As an example, if the population fraction of O IV is 0.3, then 30% of the ions of oxygen that are present in the plasma are in the form of O IV. The IPF is strongly dependent on temperature and density; the latter relation becomes particularly significant when dielectronic recombination rates are taken account. The ionization balance of an element is defined as the distribution among such charge states of the ions belonging to the element of interest. In order to calculate the full ionization balance of the plasma, one needs to calculate the individual balance of each element present in that plasma. Under thermal equilibrium, the calculation of ionization balance states that the temperature at which the ionization and recombination rates of an ion are equal corresponds to the temperature at which its population fraction reaches the maximum. Moreover, the steady-state coronal plasma assumption ensures that every ionization is balanced by a corresponding recombination, which means, all of the processes responsible for populating a particular charge state are balanced by all of those responsible for depopulating it.

However, the term “steady-state” puts an important constraint on the timescales of temperature and density variation: the temperature and density slowly follows the evolution of charge state during the time the plasma evolve to a new charge state; in other words, the timescales for temperature and density variation are far greater than

the characteristic timescales of ionization and recombination (for coronal plasma, the timescales are generally on the order of 100 seconds). It is understandable that this condition would not be met in various dynamic events where the plasma is definitely out of equilibrium, Especially, when it comes to the solar TR, where the temperature and density gradient is steep and small scale events take place on significantly short timescales, the physical mechanism must be scrutinized in the light of a non- equilibrium ionization balance. In such cases, where the heating timescales of the plasma is much shorter than the timescales of the atomic processes, an ion of relatively low charge state can be present at much higher temperatures than the range within which it exists in equilibrium. The time-scales of excitations and radiative decays are much smaller and so these processes follow the changing temperature much more closely than the ionization and recombination processes.

Since an ion will generally emit more strongly with increasing temperature, the radiative emission from the plasma could be significantly enhanced compared with the emission one may expect in equilibrium. Consequently, the nature of the radiative emission from the TR may potentially be altered as a result of the populations of its constituent ions existing at temperatures far from their equilibrium formation temperatures. A large amount of the research work carried out and presented here by the author is devoted to investigating the observational and theoretical consequences of non-equilibrium ion populations due to transient processes occurring in the solar TR.

Under the assumption that the processes of excitation and radiative decay can be de-coupled from ionization and recombination processes, the equation of ionization balance from which one can determine the ion population fractions is written as

휕푋 휕 푖 + (푋 푣) = 푛(퐼 푋 + 푅 푋 − 퐼 푋 − 푅 푋 ) 휕푡 휕푠 푖 푖−1 푖−1 푖 푖+1 푖 푖 푖−1 푖

here 푋푖 denotes the population fraction (normalized to 1) of ion stage 푖 (in the ground state configuration) of element 푋; the coefficients 퐼푖 and 푅푖 are the ionization and recombination rates from/to ion stage 푖; and n is the electron number density of the plasma (푠 and 푡 being the spatial and temporal coordinate, respectively). This equation represents the full time-dependent equation of ionization balance for each element 푋. The author employed the HYDRAD code to calculate the ion population balance of the TR plasma as a function of spatial location and time.

3.5. The Optically Thin Radiative Plasma

Assuming a steady-state optically-thin corona, the total amount of radiative energy lost per unit volume and per unit time through the emission of spectral lines can be expressed as:

2 퐸푅(푛, 푇) = 푛 Λ(푛, 푇)

where 푛 and 푇 are the number density of electrons and the plasma temperature, and Λ(푛, 푇) is the optically-thin radiative loss function, sometimes called the total emissivity. Efforts have been undertaken to calculate the radiative loss function since the 1970s based upon the available atomic data (Raymond et al. 1976,

Landini & Monsignori Fossi 1990, Landi & Landini 1998, 1999). With the advent of improved models and observations, atomic data are recalculated, and the radiative loss functions were revised and updated in order.

The radiative loss function depends upon the abundance of the constituent plasma elements, the ionization balance of the elements, the atomic data (such as, transition energies and wavelengths) and the transition probabilities (Einstein coefficients) used to determine the level populations for the emitting ions. It is generally expressed in terms of the ion emissivity which is, for a single transition between energy levels 푖 and 푗, defined as:

휀푖,푗 = Δ퐸푛푗퐴푗,푖

where ∆E is the difference between the energy levels 푖 and 푗; 푛푗 is a measure of the proportion of the population of the emitting ion in the state 푗; and 퐴푗,푖 is the

Einstein coefficient for the transition. The total ion emissivity is obtained by summing over all of the transitions that may occur within the ion.

The radiative loss function (or simply emissivity) for a single ion is then:

푍+1 ( ) Λ푋푖 = 0.83 × 퐴푏 푋 × ∑ 푋푖 × 휀푖,푗 푖=1

where 0.83 is the proton-electron ratio; 퐴푏(푋) is the abundance of element 푋 relative to hydrogen; 푋푖 is the population fraction of ion 푖 of element 푋; and 휀푖 is the total ion emissivity.

3.6. Effect of Abundance

The radiative loss function for optically thin plasma has the term 퐴푏(푋), which is the abundance of element 푋 relative to hydrogen and depends on the chemical composition of the region of interest in the solar atmosphere. As for an example, the element abundances in the photosphere can differ substantially from those in the corona. The differences arise due to the structure of magnetic field and the temperature and density of the plasma. Generally, the elements with low first ionization potentials (FIP) have their abundances varied most since these elements are most susceptible to ionize from neutral by collisionally removing a single electron.

One such example is Iron. Due to its low FIP, it is found to be four times more abundant in open field regions of the corona than it is in the photosphere, and in closed field regions it can be twice as abundant in the corona as in the photosphere.

In addition, the element abundance can differ substantially in different regions of similar height scale: a young, newly emerged active region in the corona can be found

to have photospheric abundances instead of coronal abundances. Last but not the least, element abundance can vary between individual features of the same region of interest, such as flares vs. sunspots. Thus, it is important to consider the implications of element abundance in the calculation of spectra obtained by hydrodynamic modeling.

Obtaining element abundance data is extremely delicate and difficult work.

Without it, we are essentially blind when it comes to interpreting the data provided by the observatories, since everything that is known about different layers of the solar atmosphere comes from the radiation that they emit. Thus, the importance of having accurate element abundances for individual layers of the solar atmosphere cannot be overstated. From the early efforts of Russell (1929), Suess & Urey (1956) and

Goldberg, Müller & Aller (1960) to the more recent works of Anders & Grevesse

(1989), Grevesse & Sauval (1998), Lodders (2003, 2009), Asplund (2005, 2009) and

Grevesse et al. 2007 compilations of the solar system abundances have found extremely wide-ranging use in solar astronomy and space physics. Generally, two independent processes are followed by researchers in determining the abundances of solar atmosphere: mass spectroscopy of meteorites and 3D time-dependent hydrodynamic modeling predicting the ratio of different spectral lines. Although mass spectroscopy in terrestrial laboratories offers direct measurement of abundance of almost every element and isotope with remarkable precision, the

meteorite itself can suffer from depletion of volatile elements to various degrees including the ones that are abundant in the solar atmosphere such as helium, carbon and oxygen. Hence, it is not advisable to depend on the abundance analysis of the meteorites alone. The solar spectrum must be interpreted using realistic hydrodynamic models of the solar atmosphere and spectrum formation processes. In case of the hydrodynamic modeling, the element abundance is not directly observed but inferred. Hence the process lacks the precision and accuracy of atomic and molecular data that are found in the meteoritic measurements. Thus, the two processes are often performed in conjunction to complement each other. In addition to these, analysis of the spectra from local stellar neighborhood, data from helioseismic oscillations and examining the decay channels of neutrinos originated from solar core can provide tighter constraints in order to obtain the most realistic abundance set possible.

Since, the science goal of this project is to study the low-lying transition region loops of newly emerged active regions, the element abundances the author choose for the solar TR plasma was retained from the latest photospheric abundance derived by

Asplund (2009).

Chapter 4

Hydrodynamic Modeling of Solar Loops

4.1. The Hydrodynamic Model

As already seen, the TR harbors fine-scale structures that are dynamic and heated impulsively, in addition to the footpoints of hotter overlying structures. The complexity of the region is even more compounded by the sharp gradient of temperature and density (temperatures rising from below 104 K to above 106 K and corresponding densities from above 1010 to below 108 cm−3). Thus, it is strenuous and often intractable to develop numerical models that can accurately encapsulate any structure in this region. A modeler has to devise an adequate simulation environment that can accommodate a variety of physical processes such as convective transport, thermal conduction, impulsive heating, and non-equilibrium ion populations, all of which may or may not couple together and produce a range of phenomena including propagating waves and shocks.

The above processes occur over small spatial and fast temporal scales in the

TR, making it even more challenging to develop a numerical model that can accurately handle the coupling of these processes across all of the relevant orders of magnitude.

No matter how complex the nature of the problem, the first step to building a model is expressing the system in terms of the fluid equations of hydrodynamics. Although a magnetohydrodynamic (MHD) treatment is suggested to capture the complexities associated to the magnetic field of the plasma, it only adds a significant degree of numerical complexity that is otherwise unnecessary when the plasma remains confined to the field during the time of its evolution. In addition, the ratio of the thermal pressure of the confined plasma to the magnetic pressure 훽 ≪ 1, meaning the magnetic pressure far exceeds the plasma pressure and it is safe to assume that the magnetic flux tube remains rigid only to confine the plasma, and mostly unaffected by the dynamical processes occurring within. Next, the set of hydrodynamic fluid equations needs to be representative for the physical system in order to capture the processes accurately. Secondly, a numerical scheme must be developed which can solve the aforementioned set of equations efficiently. The numerical method has to be able to solve within a sufficient degree of accuracy such that quantifiable physical properties can be extracted from the model. However, if the chosen set of equations are not a suitable representation of the problem, the model will fail to generate realistic results despite the accuracy of the method itself. Finally,

the method needs to be implemented by an efficient algorithm and tested by different cases in order to insure consistency and reliability.

In order to understand the hydrodynamic and radiative code employed, a brief description of the hydrodynamic model equations is given below. Under the single fluid assumption, the 1D hydrodynamic equations are written in terms of density 휌, momentum 휌푣, energy 퐸 and gravitational acceleration 𝑔 as

휕휌 휕 + (휌휈) = 0 휕푡 휕푠

휕 휕 휕푃 (휌휈) + (휌휈2) = − – 휌𝑔 휕푡 휕푠 휕푠

휕퐸 휕 휕퐹 + ([퐸 + 푃]휈) = − 푐 − 퐸 – 휌휈𝑔 + 퐸 휕푡 휕푠 휕푠 푅 퐻

The term 퐹퐶 corresponds to heat flux, [퐸 + 푃] is the enthalpy and [퐸 + 푃]휈 is then the enthalpy flux, where 푃 is the pressure and expressed as 2푘퐵푛푇. 퐸퐻 is the heating rate per unit volume delivered to the system. A component of 퐸퐻 can be considered as background heating and required to maintain the structure at its current temperature. Next, the rate of energy loss due to radiation is expressed by 퐸푅.

As discussed in the last chapter, 퐸푅 is calculated by summing the values of the optically-thin radiative loss function for all of the elements present in the plasma,

2 퐸푅 = 푛 ∑ Λ푋 푋

where, the optically-thin radiative loss function for a single element 푋 can be written as the product of the proton/electron ratio, the abundance of element 푋 relative to hydrogen, the sum over all charge states of the fractional population of 푋 and the total ion emissivity 휀푖 for each ion 푖.

The fluid equations above describe a plasma comprised of ions and electrons, with a common temperature 푇, density 휌 and bulk flow velocity 푣. It is assumed here that the plasma is neutral meaning, 푛 = 푛푒 = 푛푖 where, 푛 = 휌/푚푖. Under this condition, one can exclude any possibility of the development of strong, large-scale electric fields ( 휈 = 휈푒 = 휈푖). Assuming that there is no energy exchange between the species (푇 = 푇푒 = 푇푖), the species can be treated as they have relaxed into a state of

Boltzmann equilibrium having near Maxwellian distributions. The final equation required to close the system is the expression of the total energy of the plasma per unit volume

1 퐸 = 3푘 푛푇 + (푚 푛)휈2 퐵 2 푖

Energetic events in the TR (and broadly, the solar atmosphere) involve electron and ion interactions. Different particle species in the plasma are expected to undergo physical processes at different spatial and temporal scales (due to cyclotron radius and frequency). Thus, it is intuitive that any heating mechanism will resonate in favor of one particle species over another. For example, in a given scenario

electrons can undergo direct heating and will exchange energy with the ions via collisions until the temperatures of the ions equilibrate with electrons. Now, if the heating timescale is much longer than the electron-ion collision time scale, then the ions will evolve in temperature equilibrium and a single-fluid treatment is acceptable.

However, observational evidences indicate that heating in the Sun’s atmosphere takes place on relatively short timescales and it is possible that the ions may have a very different temperature to the electrons, such that the single-fluid treatment is no longer satisfactory. In such a case, the parameter of interest is the electron-ion collision frequency that has to be incorporated in the hydrodynamic treatment,

3 4 − 4√2휋푒 푘퐵푇푒 2 휈푒푖 = 푛 ( ) ln Λ, where Λ is the Coulomb logarithm. 3푚푒푚푖 푚푒

In order to account for the possibility that (푇푒 ≠ 푇푖) the single-fluid energy equation must be split into two components; one for electrons and one for ions.

However, this naturally questions the validity of charge neutrality and the possibility of developing large electric fields. This predicament can be circumvented by assuming that any electric fields that would arise are quickly neutralized by fast- moving free electrons, thus restoring charge neutrality (푛푒 = 푛푖) and that the

푚 electrons move in bulk with the ions due to the ion’s greater inertia( 푒 ≪ 1 ). 푚푖

Therefore, one can maintain a single mass conservation equation and a single

momentum conservation equation. Finally, two new energy equations can be written.

One for the electrons and one for the ions,

휕퐸 휕 휕푃 휕퐹푐 푘 푛휈 푒 + ([퐸 + 푃 ]휈) = 휈 푒 − 푒 + 퐵 푒푖 (푇 − 푇 ) − 퐸 휕푡 휕푠 푒 푒 휕푠 휕푠 (훾 − 1) 푖 푒 푅

휕퐸 휕 휕푃 휕퐹푐 푘 푛휈 푖 + ([퐸 + 푃 ]휈) = − 휈 푒 − 푖 + 퐵 푒푖 (푇 − 푇 ) – 휌휈𝑔 휕푡 휕푠 푖 푖 휕푠 휕푠 (훾 − 1) 푒 푖

For the above equation, the subscript 푒 and 푖 denotes an electron and an ion quantity, respectively. The solution of the two-fluid hydrodynamic equations provides the plasma temperature, number density and flow velocity. These physical properties are fed to the system of time dependent ionization balance equations discussed in 3.4 with the ionization and recombination rates prepared for the appropriate atmosphere,

휕푋 휕 푖 + (푋 푣) = 푛(퐼 푋 + 푅 푋 − 퐼 푋 − 푅 푋 ) 휕푡 휕푠 푖 푖−1 푖−1 푖 푖+1 푖 푖 푖−1 푖

Solving the above equation yields new values of population fractions 푋푖 for each element. The new values of 푋푖 are then used to calculate the total radiative loss rate which is fed back into the system of fluid equations. The fluid equations of hydrodynamics are then solved again to find new sets of temperature, number density and flow velocity which would be again given to the ionization balance equation. The process is repeated for the duration of the run. Under this scheme, the

two systems of equations are coupled to each other via the radiative loss term of the energy equation. In the present study, the time-dependent ionization balance equation is given density-dependent dielectronic recombination rates in order to model the TR accurately.

4.2. The Hydrodynamic and Radiative Emission (HYDRAD) Code

In the previous section, the process of solving the systems of hydrodynamic equation and the ionization balance equations is discussed and it was shown how the two systems are coupled to each other via the radiative loss term of the energy equation. At first glance, the hydrodynamic equations pose a system of three coupled equations having only three variables of physical interest: temperature (푇푒, 푇푖), number density (푛) and bulk flow velocity (푣), closed by the expressions for the total energy and thermal pressure. To solve these equations, one has to iteratively solve the system of time-dependent ionization balance equations in conjunction. In order to estimate the complexity of the problem let us first draw attention to the TR loops observed in this work: firstly, it is expected that these low-lying loops would be dynamic and short-lived, meaning a full time-dependent form of the balance equation should be needed. Secondly, the density-dependent dielectronic recombination rates must be fetched from the ADAS database and used in the system of balance equations according to the local temperature and density. This means if a standard treatment of

the optically thin radiative emission from the TR includes even 3 elements (O, Si and

S: whose spectral emission is observed by IRIS Si IV channel), then this would require

41 ions to be included in the system of ionization balance equations that needs to be iterated and updated over time and location. Therefore, the number of coupled equations to be solved increases from 4 to 45, and thus becomes extremely difficult to handle. Additionally, the calculation of the ion populations of any element as a function of temperature, given the ionization and recombination rates itself is a challenging task. To achieve this, a numerical scheme is to be employed at each discrete temperature value that would search for the ion stage that has the largest population by comparing the ionization and recombination rates. However, the steep gradients of temperature and density in the TR coupled with the non-linear thermal conduction term results the variation of the ionization and recombination rates over several orders of magnitude and thus the ionization balance equation can quickly become a stiff differential equation which usually requires an implicit scheme to be solved (Bradshaw 2003). To successful handle the above-mentioned complications, the author used the HYDRAD: HYDrodynamics and RADiative emission model code developed and maintained by Dr. Stephen J Bradshaw (Bradshaw 2009, Bradshaw &

Cargill 2013).

HYDRAD employs finite-difference method by discretizing the computational grid in a linear array of cells in order to represent the loop system (shown in Figure

7). It solves the time-dependent equations for the evolution of mass, momentum, and energy for multi-fluid plasma (electrons, ions and neutrals) for arbitrary loop geometry. In addition to solving the systems of hydrodynamic and balance equations, it accounts for the optically-thick radiation in the lower atmosphere transitioning to optically-thin radiation (lines and continuum) in the overlying atmosphere. The code has been written by Dr. Bradshaw in C++ to take advantage of its memory management capabilities ensuring maximum efficiency. The code ensures that the

spatial density of grid cells increases wherever needed, while keeping the total number of cells manageable. Thus, HYDRAD adopts adaptive mesh refinement that is capable of very fine resolution (up to meter scales on current hardware). This feature is particularly necessary for the investigation of TR loops, where the temperature gradient is very steep and spatial resolution is of paramount importance in order to capture the dynamic response of the atmosphere to heating.

HYDRAD treats optical depth effects in the lower solar atmosphere by adopting the VAL (Vernazza, Avrett, & Loeser 1981) model C chromosphere and the prescription for optically-thick radiative cooling and heating developed by Carlsson

& Leenaarts (2012). Under this prescription, empirical formulas for the radiative cooling and heating for the chromosphere are derived from the calculation of time- dependent radiative transfer and snapshots generated by 2D MHD simulations. At the footpoints of TR loops, where the plasma is partially ionized, the ionization and recombination of hydrogen is included in the energy equation solved by HYDRAD, as is the contribution of H I to energy transport by thermal conduction (Orrall & Zirker

1961), thus ensuring the simulation of more realistic footpoints buried inside the chromosphere.

4.3. Forward Modeling Methods

Since the mean free path of photons becomes effectively infinite above the solar photosphere, the corona and the TR are typically considered to be optically thin.

As a result, observations are 2D projections of a 3D structure along the line-of-sight

(LOS). This makes interpreting observations quite difficult, since locating the precise position of an individual structure along the LOS is not possible. One exception is stereoscopy with the STEREO space mission (e.g., Marsh et al., 2009; Verwichte et al.,

2009; Aschwanden, 2011), where one uses multiple vantage points to infer the 3D structure of the observations.

Since information in at least one dimension is spatially integrated (or smoothed) during observations, the comparison of simulation data to observations is non-trivial. In addition, hydrodynamic and magnetohydrodynamic simulations produce physical properties of the plasma (such as density, temperature etc.), while observations only show either line profiles or the intensities of spectra integrated over space and time. This leads to a challenge since similar spectral information can originate from different physical properties of plasma and thus, careful analysis is necessary to infer the process that is responsible for the observed spectra. To allow direct comparison, a hydrodynamic model should be accompanied with a conversion method that can create synthetic observations equivalent to an observing instrument.

Such a method is called forward modeling.

In solar coronal physics, several tools are available for forward modeling.

Perhaps the most prominent one is FORWARD, which computes EUV emission and polarimetric signals from a given coronal model (Gibson, 2014; Gibson et al., 2016).

Besides that, there is the GX_SIMULATOR (Nita et al., 2015), which computes radio and X-ray emission and is mainly aimed at forward modeling flares. Then there is the

FOMO tool which was mainly motivated by the desire to perform forward modeling of coronal wave models. The FOMO code (Antolin and Van Doorsselaere 2013, Van

Doorsselaere et al. 2016) was also used for modeling the emission from propagating slow waves and periodic upflow. Lately FOMO was extended to compute gyrosynchrotron emission and the modeling of standing kink waves in coronal loops

(Antolin et al. 2014).

In this thesis, the author investigated TR loops and associated brightenings.

Such structures are generally thought to be composed of many individual strands. The thickness of individual strands can be measured by looking at the transverse length to the magnetic field direction. However, the scale lengths of the thickness are far smaller than any current observing instrument can resolve. Some indirect estimate is needed in order to find how many sub-resolution strands exist in a single resolution element. Oftentimes modeling individual strand is not computationally economic, although many modern codes (such as EBTEL and HYDRAD) can perform such heavy operations depending upon the desired detail of the system. A more efficient

approach is to run the model for a single strand for a complete heating and cooling cycle; then writing out the loop state at regular intervals (snapshots) in time. Now each snapshot holds information on the state of the strand and by combining these snapshots and assuming that each snapshot corresponds to a different strand, one can effectively build up a bundle of strands into a monolithic loop. To understand the above procedure by an example, let us assume that a strand is simulated undergoing a complete cycle of heating and cooling in 1000 s and the state of the strand in every second is recorded. Thus, the modeler obtains 1001 (0–1000 s) unique temperature, density and bulk velocity profiles, which can be assumed to represent 1001 different strands comprising the full loop. In other words, it is inherently assumed that the observed loop is composed of many strands, each representing one of every (or at least many) possible state encountered during the heating and cooling of a single strand. However, such an assumption is only true when there exists a statistically significant number of strands and each of them evolves in a similar manner.

Moreover, the lifetime of the loop should also be long enough such that a few of the strands can undergo a complete cycle of heating and cooling.

The forward modeling process employed in this work was developed by

Bradshaw and Klimchuck (2011). The code accepts output from HYDRAD: the field- aligned evolution of temperatures, pressures and number densities, the bulk flow velocity and the ion populations as a function of position and time. Figure 8

demonstrates how the code projects the loop from one footpoint to another onto a linear array of detector pixels. Since the loop length is user-defined, this process is done by the simulation environment. However, the diameter of the loop needs to be set by the user depending on the instrument employed for observation. In this work, the author set it to be 0.33 arcsecond or about 230 km (approximately the size of a single IRIS pixel). This diameter of the bundle is chosen so that the predicted intensities can easily be scaled to arbitrary line-of-sight depths observed by IRIS.

Taking the above example of 1001 elementary strands comprising the loop, then each strand has a cross-section of 41500 푘푚2/1001 ≈ 41.5 푘푚2 with a diameter of about

7 km. Now, it is evident that a loop can be observed in any part of the solar disk and, depending on the location and tilt of the loop with respect to the observing instrument, the emission and brightness can vary. Thus, the author selected IRIS observations targeted near the center of the disk. Under this condition, the problem of loop projection and binning becomes a more straightforward one: the loop can be assumed to be located at disk center and oriented perpendicularly to the solar surface, with the plane of the loop aligned in the east–west direction. This facilitates binning the spatial emission profile along a single row of detector pixels (Figure 8).

From this orientation, it is evident that the curvature of the loop leads to an increase in the line-of-sight depth of the emitting plasma toward the footpoints, increasing their brightness.

Figure 8: Schematic representation of the process by which the emission from each grid cell is binned into the appropriate detector pixel (Bradshaw and Klimchuck 2011).

Numerically, the forward model calculates the intensity of a line under isothermal condition (since each grid cell is associated with a single electron temperature) and expressed as a data number count rate (DN pixel−1s−1) which is given by (Bradshaw and Klimchuck 2011)

1 푁(푋) 푁(퐻) 퐼(휆, 푛, 푇) = 퐼푅퐹(휆) × 푋푖 × 휖(휆, 푛, 푇) × 〈퐸푀〉 4휋 × ℎ푐/휆 푁(퐻) 푁푒

where ℎ푐/휆 is the photon energy, 퐼푅퐹(휆) is the instrument response

푁(퐻) 푁(푋) function, = 0.83 is the proton:electron ratio, is the abundance (relative to 푁푒 푁(퐻)

hydrogen) of element X, 푋푖 is the population fraction of charge state i of element X,

휖(휆, 푛, 푇) is the emissivity of the line and 〈퐸푀〉 = 퐸푀푝푖푥푒푙/퐴푝푖푥푒푙 is the spatially

2 averaged column emission measure in the pixel, where 퐸푀푝푖푥푒푙 = 푛푒푑푉 is the emission measure in the observing pixel and 퐴푝푖푥푒푙 is the pixel area. The instrument response function, 퐼푅퐹(휆), is the product of the effective area, plate scale, and gain of the instrument. The above equation calculates the emission intensity for each grid cell in the numerical plane. The contribution from a number of cells is added for each pixel. This procedure is realized by projecting each grid cell onto a single row of detector pixels (Figure 8) and establishing what proportion of the emission from the grid cell falls onto each pixel. This is repeated for the spectral lines of the IRIS Si IV,

O IV and S IV ions, for all of the grid cells of the strand, and for every strand comprising the loop. The resultant emission from that spectrum is then a synthetic IRIS observation of the loop, having contributions from the many sub-resolution strands at different stages of evolution. Direct comparisons between numerical and observational data can be made for any IRIS channel by integrating the synthetic emission over the appropriate cadence of the event.

Chapter 5

IRIS Observations of Transition Region Loops

5.1. Pre-IRIS Era and Theoretical Models

During the 1950s and 1960s, the details of the structure of the atmosphere of the Sun were still unknown. It was possible to perform ground-based observations of the photosphere and chromosphere, but for the hotter TR and corona, the emission lines were mainly located in the range of near and far ultra-violet (UV:100 to 2000 Å) and became accessible only after the advent of the space era. Space missions, such as

NASA’s Skylab (1973-1974), started providing a more detailed view of the coronal structures (such as active region loops and coronal holes; Aschwanden et al. 2001).

The mass flows in the TR were observed for the first time by Doschek et al. (1976) using The Naval Research Laboratory (NRL) normal incidence spectrograph on

Skylab and later on by the NASA Solar Maximum Mission (1980-1989) and a few other rocket experiments (e.g., sounding rocket flight 27.107 US of 1987, Hassler et al.

1991). By the 1980s modelers were trying to address the large-scale magnetic structures and the presence of quasi-steady flows in the stratified solar atmosphere.

During the 1990s, the very successful SOHO (Domingo et al. 1995) and TRACE

(Handy et al. 1999) space missions revealed the much more complicated and fine structures of the atmosphere of the Sun. Many types of mass flows (ejection of spicular and flare materials, siphon flows, draining of coronal material, flows in the individual loops, etc.) and magnetohydrodynamic transversal plasma waves were detected. Later on, in the 2000s, Hinode (Kosugi et al. 2007), and SDO (Pesnell et al.

2011) revealed even more detailed structures and dynamics. The classical layers of the atmosphere of the Sun (photosphere, chromosphere, TR, and corona) were distinguished more by temperature than by geometry and later the corona and the

TR by the role of the dominant energy flux (Bradshaw and Cargill 2005). The right panel of Figure 9 shows a sketch of this complex scenario. For instance, the height of the chromosphere in areas with open magnetic field lines (like coronal holes) could be higher than the height of the TR in closed magnetic field regions (like quiet Sun or active region, Tian et al. 2010). It appears that there are different heating mechanisms operating over the wide energy ranges (alternatively the same mechanism depositing different amounts of energy) in different coronal structures (such as quiet Sun, coronal holes, loops in active regions) although which one dominates at a given location and time is still unknown (Walsh and Ireland 2003; Klimchuk 2006). By the

end of beginning of this century, the dynamic and stratified nature of the TR was revealed to solar astronomers. Thus, continuous attempts to include this region in order to correctly model the heating of the solar atmosphere and its response to energy deposition were initiated.

Figure 9: Solar atmosphere models developed since the 1950s (from Schrijver 2001).

During the observation of dynamic nature of the TR, an average net red shift or downward motion was detected for emission lines (e.g., C IV, O IV, and N V) forming at TR temperatures in the quiet Sun around disk center since the launch of

SO82B/Skylab spectrograph (Doschek et al. 1976). Later on, other UV spectrometers

(HRTS, SMM/UVSP, LASP, and SoHO/SUMER) observed and confirmed this

phenomenon. In general, the amount of the red shift increases as an observer goes toward hotter lines (starting at chromospheric temperatures), and reaches maximum velocities of 10 to 15 kms-1 in the quiet Sun at temperatures of about 0.2 MK. As the temperature increase further, the Doppler shift decreases. However, determining the rest wavelength of the lines formed at high temperatures is technically difficult and associated with large uncertainties. Figure 10, from Peter and Judge (1999), shows the Doppler shift with respect to line formation temperature.

The prevalent average red shift observed in TR lines suggested that such a strong mass flux downward would be enough to empty the whole corona within an hour (Kjeldseth-Moe 2003) and therefore there should be some mechanism(s) that has to counter this huge mass flux, since structures observed in this region persist much longer. Consequently, researchers introduced physical models to explain such average net red shifts of the TR lines such as siphon flows, the return of spicule material, upward propagating waves, downward propagating acoustic waves, nanoflare models, and so on. However, none of the proposed explanations appear to be complete enough without further high-resolution observations that can more finely resolve the TR; thus, the predictions made by the models are yet to be able to explain all of its features. As the emission from the TR cannot be fully explained by contemporary models, it has been argued the majority of “TR emission” is dominated by dynamic, low-lying magnetic structures that remain unresolved and require multi- wavelength spectroscopic imaging at high spatial resolution and temporal cadence.

5.2. The IRIS Mission

The Interface Region Imaging Spectrograph (IRIS), also called Explorer 94, is a NASA solar observation satellite (De Pontieu 2014). The mission was funded through the Small Explorer (SMEX) program to investigate the physical conditions of the solar limb, particularly the chromosphere and the TR of the Sun. The spacecraft

consists of a satellite bus and spectrometer built by the Lockheed Martin Solar and

Astrophysics Laboratory (LMSAL), and a telescope provided by the Smithsonian

Astrophysical Observatory. IRIS is operated by LMSAL and NASA's Ames Research

Center. NASA announced on 19 June 2009 that IRIS was selected from six Small

Explorer mission candidates for further study, along with the Gravity and Extreme

Magnetism (GEMS) space observatory (Jahoda 2010).

Despite the importance of the TR for solar activity, the heating of the corona, and the genesis of the solar wind, this interface region remains poorly understood. It has been observed that the atmosphere is highly structured and non-uniform, reflecting rich magnetic structure and activity and, thus, must be observed over a wide spectral range (from the visible to the EUV). As a result, it presents a challenging target for observers and modelers alike. In the TR, magnetic fields and plasma compete for dominance as one goes from the photosphere (T < 6,000 K, β > 1) through the chromosphere to the corona (T > 1 MK, β < 1). The region is extremely narrow in height with a steep temperature gradient and experiences a decrease in the electron density by three orders of magnitude. In addition, the plasma undergoes transitions from partially ionized in the chromosphere to fully ionized in the corona and shows evidence of supersonic and super-Alfvénic motions. Along with the TR itself, the upper chromosphere offers further challenges as it is partially opaque, with non-local thermodynamic equilibrium (non-LTE) effects dominating the radiative transfer, so

that interpreting the radiation, and determining the local energy balance and ionization state, is non-intuitive and requires advanced computer models. This has been confirmed by observations with Hinode and ground-based telescopes.

Thus, it is natural that the emission from the TR cannot be fully explained by a simple thin boundary layer, and it has been argued that the TR is composed of two main components: the foot-points of large, hot coronal loops (Testa et al. 2016); and short, low-lying loops that are highly dynamic with short lifetimes (< minutes). The majority of the TR emission is dominated by the latter one with peak temperatures <

1 MK. Because of the lack of spatial resolution previously available to observations, these structures were called the “unresolved fine structure” (Feldman 1983) and believed to store an abundance of clues to the formation and dynamics of the TR and perhaps the mechanism of energy release at the adjacent layers. Recent improvements in the spatial and temporal resolution of observations provided by the

Interface Region Imaging Spectrograph (IRIS) have made it possible to study these structures in detail. The spectral ranges that IRIS observes have previously been studied at lower resolution using rockets, balloons, and satellites. IRIS succeeds previous state-of-the-art solar instrumentation, such as TRACE, HMI (Scherrer et al.

2012) and SDO/AIA (Lemen et al. 2011), as the principal instrument to study the TR plasma by exploiting its novel, high-throughput, and high-resolution instrumentation, supported by numerical simulations.

As the first mission designed to simultaneously observe images of the chromosphere and TR at multiple temperatures and provide spectroscopic information at fine spatial scales and fast temporal cadence, the mission's science objectives are to answer the following questions (De Pontieu 2014):

• Which types of non-thermal energy dominate in the chromosphere and

beyond?

• How does the chromosphere regulate mass and energy supply to the

corona and heliosphere?

• How do magnetic flux and matter rise through the lower atmosphere

and what role does flux emergence play in flares and mass ejections?

5.2.1. Instrument Overview

The IRIS instrument consists of a Cassegrain telescope with a 19-cm primary mirror and an active secondary mirror with a focus mechanism. Spectrally, the telescope feeds a far-UV (FUV: from 1332 to 1407 Å) and a near-UV (NUV: from 2783 to 2835 Å) band which pass through a spectrograph system to record spectral data.

The system is divided into two major components in order to obtain both image and spectral data simultaneously. (1) Incoming light rays pass through a slit that is 0.33 arcseconds wide and 175 arcseconds long and are dispersed onto either an FUV or an

NUV grating. The FUV grating is followed by two CCDs, whereas light from the NUV

grating is collected by a separate CCD. (2) The area around the slit is a highly reflective surface and the reflected light rays are filtered and captured by a fourth CCD to produce an image of the scene around the slit. The spectrograph and image channels used by IRIS are tabulated below IN Table 1 and Table 2:

Dispersion Pixel Temp. (Log10 Band Wavelength (Å) (mÅ/pixel) (") T)

FUV 1 1331.7–1358.4 12.98 0.1663 3.7 – 7.0

FUV 2 1389.0–1407.0 12.72 0.1663 3.7 – 5.2

NUV 2782.7–2835.1 25.46 0.1664 3.7 – 4.2 Table 1: IRIS Spectrograph Channels in FUV (C II and Si IV) and NUV (Mg II) bands.

Filter Temp. (Log10 Bandpass Center (Å) Width (Å) Pixel (“) Wheel T)

C II 31 M 1330 55 0.1656 3.7 – 7.0

Mg II h/k 61 T 2796 4 0.1679 3.7 – 4.2

Si IV 91 M 1400 55 0.1656 3.7 – 5.2

Mg II wing 121 T 2832 4 0.1679 3.7 – 3.8 Table 2: IRIS Slit-Jaw Image Channels. Filter positions can be transmitting (T) or reflecting/mirrors (M)

To meet the objectives and the science goals of the research presented here, the author first identified IRIS datasets for active regions located near disk center,

that included the 1400 Å (Si IV) slit-jaw images captured at better than 10 s cadence.

For density diagnostics and forward modeling, the author extracted and analyzed the

FUV 2: O IV spectrum and Si IV intensity for each pixel, in every time step, highlighted in the Table 1.

5.2.2. IRIS Science Results

IRIS achieved first light on 17 July 2013. On 31 October 2013, calibrated IRIS data and images were released on the project website. Data collected from the IRIS spacecraft has shown that the interface region of the Sun is significantly more complex than previously realized. This includes features described as solar heat bombs, high-speed plasma jets, nanoflares, and mini-tornadoes (discussed below).

These features are an important step in understanding the transfer of heat to the corona. A brief highlight of a few early science results from IRIS’s maiden observations is discussed chronologically hereafter:

• Hansteen et al. (2014) successfully exploited the high spatial and temporal

resolution of IRIS to reveal structures remarkably similar to those postulated

as the unresolved fine structures. Images of the lower TR at the solar limb with

the IRIS slit-jaw camera in the Si IV 1400 Å and in the C II 1330 Å filter

invariably showed bright low-lying loops or loop segments in quiet Sun

regions. In addition to these bright structures, a much fainter component

forms a background that extends up to 10 arcsec above the limb. The

background includes a large number of linear structures, with properties

similar to the well-known spicules observed from the ground in the Hα 656.3-

nm line.

• Peter et al. (2014) reported hot explosions or bomb-like events having

bidirectional flows in the cool TR (< 0.1 MK) in an emerging active region. They

believed the explosions were triggered by the emergence of undulating

magnetic field lines, and the resulting U-shaped loop getting dragged down.

Being squeezed together, the magnetic field reconnected, and plasma got

heated and accelerated deep in the atmosphere. The bidirectional outflow

from the reconnection region caused observed double-humped line profiles of

Si IV, C II, and Mg II, whereas cool material above caused observed absorption

lines.

• Tian et al. (2014) observed the prevalence of intermittent small-scale jets with

speeds of 80 to 250 kilometers per second from the narrow bright network

lanes of the interface region. These jets had lifetimes of 20 to 80 seconds and

widths of ≤300 kilometers. They originated from small-scale bright regions,

often preceded by footpoint brightenings and accompanied by transverse

waves with amplitudes of ~20 kms-1. Many jets reached temperatures of at

least ~105 Kelvin and constitute an important element of the TR structures.

They were likely an intermittent but persistent source of mass and energy for

the solar wind.

• De Pontieu et al. (2014) presented high-resolution (0.33-arc second)

observations revealing an upper-chromosphere and TR that are replete with

twist or torsional motions on sub–arc second scales, occurring in active

regions, quiet Sun regions, and coronal holes alike. The team coordinated

observations with the Swedish 1-meter Solar Telescope (SST) to quantify

these twisting motions and their association with rapid heating to TR

temperatures and beyond. The observations indicated that strong

photospheric vortical flows and the resulting twist in flux tubes, as well as

dissipation of torsional Alfvén waves generated by very small-scale

photospheric vortices might be responsible for heating the low solar

atmosphere.

• Tian et al. (2014) reported first results of sunspot oscillations from

observations by the IRIS. The study found a positive correlation between the

maximum velocity and deceleration, a result that is consistent with state-of-

the-art MHD simulations of upward propagating magneto-acoustic shock

waves.

• Testa et al. (2014) presented evidence of nonthermal particles in coronal loops

heated impulsively by nanoflares. The authors witnessed rapid variability

(~20 to 60 seconds) of intensity and velocity on small spatial scales (≲500

kilometers) at the footpoints of hot and dynamic coronal loops. The

observations were consistent with numerical simulations of heating by beams

of nonthermal electrons, which are generated in small impulsive (≲30

seconds) heating events called “coronal nanoflares.” The accelerated electrons

deposited a sizable fraction of their energy (≲1025 erg) in the chromosphere

and TR.

• Pereira et al. (2014) studied IRIS observations of solar spicules in quiet-Sun

regions. The high-resolution observations allowed the authors to follow the

thermal evolution of type II spicules and finally confirm the fading of Ca II H

spicules appeared to be caused by rapid heating to higher temperatures,

persisting in that high-temperature state for more than 500 s. These are the

type II spicules that are very different from type I spicules (much lower

velocities). Spectroheliograms from spectral rasters also confirmed that these

quiet-Sun spicules originate in bushes from the magnetic network. Their

results suggested that type II spicules were the site of vigorous heating,

reinforcing earlier work.

• Okamoto et al. (2015) reported high spatial, temporal, and spectral resolution

observations of a solar prominence that showed a compelling signature of so-

called resonant absorption, a long-hypothesized mechanism to efficiently

convert and dissipate transverse wave energy into heat. Aside from coherence

in the transverse direction, their observations showed telltale phase

differences around 180° between transverse motions in the plane-of-sky and

line-of-sight velocities of the oscillating fine structures, and also suggested

significant heating from chromospheric to higher temperatures. Comparison

with advanced numerical simulations supported scenarios in which

transverse oscillations trigger a Kelvin–Helmholtz instability (KHI) at the

boundaries of oscillating threads via resonant absorption. This instability led

to numerous thin current sheets in which wave energy was dissipated and

plasma was heated. The results provided direct evidence for wave-related

heating in action, one of the candidate coronal heating mechanisms.

• Vissers et al. (2015) published their observations on Ellerman bomb-type

events and their properties in the ultraviolet lines sampled by the Interface

Region Imaging Spectrograph (IRIS), using simultaneous imaging

spectroscopy in Hα with the Swedish 1-m Solar Telescope (SST) and

ultraviolet images from the Solar Dynamics Observatory. The IRIS spectra

strengthened their view that Ellerman bombs marked reconnection between

bipolar kilogauss fluxtubes and resulted in bi-directional jets located within

the solar photosphere and shielded by overlying chromospheric fibrils in the

cores of strong lines. The spectra also suggested that the reconnecting

photospheric gas underneath was heated sufficiently to momentarily reach

stages of ionization normally assigned to the TR and the corona.

• Dudík et al. (2016) investigated the occurrence of slipping magnetic

reconnection, chromospheric evaporation, and coronal loop dynamics in the

2014 September 10 X-class flare. Their IRIS observation confirmed the

dynamics of both the flare and outlying coronal loops to be consistent with the

predictions of the standard solar flare model in three dimensions.

• Martínez-Sykora et al. (2017) examined the origin of solar spicules and the

generation of Alfvénic waves using IRIS and Swedish 1-m Solar Telescope

observations. They found strong LOS blueshifts in the Ca II 8542 Å (middle

chromosphere), Mg II h 2803 Å (upper chromosphere), Si IV 1403 Å (TR), and

Fe IX 171 Å (corona) lines. This suggested that spicule assisted energy and

mass transfer should be present and may play a substantial role in energizing

the corona.

• Tian et al. (2018) showed evidence of magnetic reconnection in sunspots using

coaligned observations from the 1.6 m Goode Solar Telescope and IRIS. They

detected fine-scale jets at light bridges produced by magnetic reconnection.

The IRIS spectra revealed the characteristic bidirectional flows (up to 200

kms−1) and the heating of weakly ionized plasma to 80,000 K.

5.3. The Dynamic Transition Region of Solar Active Regions

Although the traditional definition of the TR is that it is a layer of the Sun's atmosphere between the chromosphere and corona, this prevailing definition is far from reality. Rather, the solar TR is enormously dynamic with large contrasts in density and temperature occurring throughout. To understand its nature, let us start with the temperature gradients that clearly exist at the base of the solar corona. It is then obvious that an important component of the energy balance is the heat flux carried by this temperature gradient. The simplest version of the energy equation in the hydrostatic limit then becomes,

푑퐹 퐸 − 퐸 − 퐶 = 0 퐻 푅 푑푠

Where, 퐸퐻 is the heating rate per unit volume, 퐸푅 is the rate of energy loss due

푑퐹 to radiation and the term 퐶 represents the redistribution of energy by thermal 푑푠 conduction with 푠 being the magnetic field aligned coordinate. By writing,

푑퐹 퐸 − 퐸 = 퐶 퐻 푅 푑푠

We can see that the heat flux either makes up the deficit between heating and radiation (퐸푅 > 퐸퐻) or it carries off the excess heat (퐸퐻 > 퐸푅) from the local volume.

푑퐹 In other words, if 퐶 < 0, then the amount of energy transported into the volume by 푑푠

thermal conduction is greater than the amount leaving the volume. Therefore, the energy lost from the heat flux must have been deposited into the volume to help

푑퐹 replace the energy lost by radiation. On the other hand, if 퐶 > 0, then the amount of 푑푠 energy transported into the volume by thermal conduction is less than the amount leaving the volume. For realistic values of temperature and density at the base of the

4 11 -3 corona (e. g. 푇0 = 2 × 10 K and 푛0 = 10 cm ), 퐸푅 ≫ 퐸퐻. The radiation at the base of the loop is strong enough that heating alone in this region cannot resupply the energy lost. Therefore, it can only come from thermal conduction. Since the temperature at the base of the loop is so low then the only way for enough energy to be transported into that region, to help the heating balance the energy lost by radiation, is for the temperature gradient to steepen dramatically. Evidently, this process leads to strong temperature increases over short spatial scales, forming the

TR where temperatures typically rise from chromospheric values (2 × 104 K) to coronal values (> 106 K) on spatial scales of a few hundred to a couple of thousand kilometers.

In the TR, thermal conduction acts as an energy source. The spatial location where it switches from source to sink (at the corona) provides an explicit way to define the position of the interface between the corona and the TR. In the hydrostatic assumption, an active region loop exhibits heat flux transitions from a source to a sink at temperature 푇0 = 0.6푇푀푎푥 (Bradshaw & Mason 2003). Given that the TR and the

corona are separated by this point (defined by the different behavior of the heat flux and, therefore, by fundamental differences in the energy balance), it is incorrect to think of TR as a layer of plasma temperature below 106 K and coronal plasma having temperature above 106 K. Thus, a distinction between the TR and the corona based on a single temperature or density alone is rather erroneous, but a distinction based on a temperature (or density) extracted from a physical consideration of the energy balance is more or less representative to the actual scenario (Bradshaw and Cargill

2010). This statement is also proven as FUV observations showed that the TR harbors various multi-thermal events such as heat bombs, small loops and brightenings, high- speed plasma jets, and flare like events.

5.4. Non-equilibrium Ionization

The emission from the constituent ions of a plasma depends strongly upon how far they exist from their temperature of formation in equilibrium. The energetic heating that takes place during solar flares can heat the plasma to high temperatures far more quickly than the population of ions can respond by forming higher charge states. A rapid flow, quickly transporting ions across strong temperature gradients could have the same effect. A population of ions emitting in plasma conditions far from local thermodynamic equilibrium will have a major effect upon: the observed spectral line emission from which all information about the properties of the plasma

is derived; and the energy balance of the plasma, all of which are of importance for modeling purposes. The consequences of a non-equilibrium population of emitting ions for plasma diagnostics and modeling the energy balance of the solar TR has long been recognized. Before the advent of modern, high-resolution instrumentation, it was realized that the dynamic events and steep gradients already observed could have an important effect upon the ionization balance of the TR plasma, and thus upon the spectral line intensities and the overall magnitude of the radiative emission.

Observations of such dynamic events in the TR can be traced back to the

1980’s when Porter et al. (1987) reported loop brightenings in SMM observations,

Dere et al. (1989) recorded EUV brightenings at TR temperatures using the Naval

Research Laboratory’s High-Resolution Telescope and Spectrograph (NRL HRTS) and lately, Perez et al. (1999a,b) observed EUV brightenings in the C IV and O VI lines formed at the TR by SOHO/SUMER. Madjarska & Doyle (2002) also carried out SOHO-

SUMER observations of explosive events but they used the Lyman 6 chromospheric line and Si VI TR line and found evidence of a propagating disturbance passing through the chromosphere to the TR and releasing energy into the low-lying, cool TR plasma. In modern times, it was predicted that brightenings of loop-like structures in the TR should appear in EUV images due to the presence of TR lines in the 94, 131,

171, 193, 211 and 335 Å EUV passbands of AIA (Warren and Winebarger 2000,

Gudiksen et al. 2011). Subsequently, this claim was verified by Martínez-Sykora et al.

(2011) who used the Bifrost code to assess the contribution of the non-dominant spectral lines to SDO/AIA images. The observational signature of TR emission in the

193 Å EUV channel was finally confirmed by Winebarger et al. (2013) studying rapidly evolving, low-lying loops observed during the Hi-C rocket flight. The observed loop diameters were less than 800 km and projected lengths were less than 7.5 Mm.

These loops appeared to evolve simultaneously in all the AIA channels. In addition, there was no indication that the loops appeared in hotter passbands (e. g., 211 Å) before the cooler ones (e. g., 171 Å); hence the loops did not appear to be cooling through coronal temperatures. These clues all supported the interpretation of these loops as cool structures. The most likely temperature was found to be 푇 ∼ 105.4±0.1

K in all cases. Despite their low temperatures and relatively short scale heights, these loops have a rather high density due to their low-lying nature. The rapid temporal evolution is believed to occur from a combination of heating on short timescales and the efficient radiative cooling of the dense plasma. MHD simulations (Winebarger et al. 2013) showed that such loops occur naturally because of rapid magnetic energy release at chromospheric/low coronal heights. The simulated loops tended to occur in a magnetic field configuration where their footpoints are separated by only a few

Mm with the loop apex forming just a few Mm above the photosphere.

More recent work by Olluri et al. 2013(a,b) solved the full time-dependent rate equations for an oxygen atom in the three-dimensional numerical model of the solar

atmosphere generated by the Bifrost code in order to construct synthetic intensity maps and study the emission. The O IV lines of interest were within the wavelength range of IRIS and they found that predicted electron densities using O IV diagnostics can be up to an order of magnitude higher when NE effects are accounted for. Doyle et al. (2013) considered Si IV ions at TR temperatures to be out-of-equilibrium and found that the ratio of Si IV 1394 Å to O IV 1401 Å can vary by up to a factor of 4 due to non-equilibrium ionization. Dudík et al. (2013) investigated the formation of the O

IV, Si IV, and S IV lines observable in the second FUV channel of IRIS. They demonstrated that the Si IV line can predominantly be formed at very low temperatures of 104 K even for weakly non-Maxwellian situations, thus a significant amount of Si IV emission can be emitted out-of-equilibrium. Martínez et al. (2016) analyzed the properties of non-equilibrium ionization of silicon and oxygen ions in five solar targets (quiet Sun; coronal hole; plage; quiescent active region, AR; and flaring AR) observed by IRIS. Their observed emission and the intensity of line ratios could only be reproduced when the 2D self-consistent radiative MHD simulation they used took account of nonequilibrium ionization, thus concluding that these lines were formed out of equilibrium.

Most recently, Bradshaw & Testa (2019) investigated atomic processes leading to the formation of emission lines within the IRIS wavelength range at temperatures near 105 K (TR temperatures). They reported significant impacts on

spectroscopic diagnostic measurements of quantities associated with the plasma that emission lines from S IV and O IV provide. By examining nanoflare-based coronal heating to determine what the detectable signatures are in TR emission, they undertook a detailed comparison between predictions from numerical experiments and several sets of observations to ascertain when non-equilibrium ionization and/or density-dependent atomic processes become important. Their findings included that non-equilibrium ionization significantly affected the line intensity ratio of S/O and the density dependence of atomic rate coefficients became significant when the ion population was out of equilibrium. Each of these contemporary studies indicates that the TR plasma is out-of-equilibrium and the density dependent rate coefficients should be used when modeling (or predicting).

5.5. IRIS observations of TR loops

The following sections of this chapter comprise detailed discussions of the diagnostic techniques performed by the author, and the resultant images and spectroscopic data that were collected. These data are compared with numerical experiments in Chapter 6.

The IRIS spectrograph collects light from the telescope at far-UV (FUV: from

1332 Å to 1407 Å) and a near-UV (NUV: from 2783 to 2835 Å) bands. A list of candidate observations was first identified from the Lockheed Martin Solar and

Astrophysics Laboratory (LMSAL) datasets of IRIS active region observations. Those shown in Table 3 – 5 had the following necessary characteristics:

• The targeted active region appeared near the center of the disk. Under this

condition, the loops could be assumed to be oriented perpendicularly to the

solar surface. Thus, the co-aligning with AIA was straightforward and

comparing the forward modeled spectra with observations would be free of

effects that may arise from any angular projection.

• The observed active region had distinguishable bright loops having loop length

less than 10 Mm that did not extend into the high temperature corona.

• The spectrograph raster could scan through the loop at the same time the loop

was evolving. This facilitated a large number of datapoints during different

phases in the loop’s lifetime.

• A dense small-step raster scan was available to sample the maximum possible

pixels of the loop to ensure that there were enough datapoints to

observationally distinguish different parts of the loops (e.g. footpoints and

apex).

• The cadence of the observation was fast enough (< 10 s) to capture the loop

dynamics since the loops were short-lived (lifetime < 500 s) and dynamic.

• Data could be collected for both Si IV and Mg II channels to run the necessary

diagnostics.

Overview Location Raster Slit-Jaw Image Example AR 12396 FOV: 33"x119" x,y: -543",-353" FOV: 119"x119" Dense 96-step raster Steps: 96x0.35" Max FOV: Si IV: 17s, 639 images 2015-08-05 Step Cadence: 5.6s 152"x119" Mg II: 17s, 639 images 08:01:17-10:59:14 Raster Cad: 534s, 20 rasters

AR 12396 FOV: 14"x119" x,y: -357",-369" FOV: 120"x119" Large 8-step raster Steps: 8x2" Max FOV: Si IV: 36s, 136 images 2015-08-06 Step Cadence: 9.1s 134"x119" Mg II: 36s, 136 images 15:19:21-16:41:59 Raster Cad: 73s, 68 rasters

AR 12396 FOV: 14"x119" x,y: -254",-367" FOV: 119"x119" Large 8-step raster Steps: 8x2" Max FOV: Si IV: 38s, 332 images 2015-08-06 Step Cadence: 9.5s 133"x119" Mg II: 38s, 332 images 23:29:51-03:00:14 +1d Raster Cad: 75s, 167 rasters

AR 12396 FOV: 14"x119" x,y: 296",-385" FOV: 119"x119" Dense 96-step raster Steps: 8x2" Max FOV: Si IV: 38s, 757 images 2015-08-09 Step Cadence: 9.4s 133"x119" Mg II: 38s, 757 images 19:49:51-03:48:35 +1d Raster Cad: 75s, 380 rasters

Table 3: Candidate observations (1–4) in active regions in search of the TR loops. The highlighted entry is the best scan having more than 10 loops evolving inside the vicinity of raster scan.

Overview Location Raster Slit-Jaw Image Example AR 12041 FOV: 11"x122" x,y: 81",-276" FOV: 122"x122" Dense 32-step raster Steps: 32x0.35" Max FOV: Si IV: 9s, 400 images 2015-08-18 Step Cadence: 1.1s 132"x122" Mg II: 9s, 400 images 05:46:44-06:47:41 Raster Cad: 36s, 100 rasters

AR 12403 FOV: 141"x174" x,y: 290",-337" FOV: 167"x174" Dense 400-step raster Steps: 400x0.35" Max FOV: Si IV: 37s, 100 images 2015-08-25 Step Cadence: 9.4s 307"x174" Mg II: 37s, 100 images 05:10:09-06:12:38 Raster Cad: 3748s, 1 raster

AR 12454 FOV: 112"x119" x,y: 825",88" FOV: 119"x119" Dense 320-step raster Steps: 320x0.35" Max FOV: Si IV: 38s, 80 images 2015-11-23 Step Cadence: 9.4s 231"x119" Mg II: 38s, 80 images 10:15:20-11:05:20 Raster Cad: 3001s, 1 raster

AR FOV: 112"x119" x,y: -664",-173" FOV: 119"x119" Dense 320-step raster Steps: 320x0.35" Max FOV: Si IV: 38s, 80 images 2016-01-20 Step Cadence: 9.6s 231"x119" Mg II: 38s, 80 images 10:37:21-11:28:19 Raster Cad: 3057s, 1 raster

Table 4: Candidate observations (5–8) in active regions in search of the TR loops. The highlighted entry is the best scan having more than 10 loops evolving inside the vicinity of raster scan.

Overview Location Raster Slit-Jaw Image Example AR FOV: 112"x119" x,y: -577",-167" FOV: 119"x119" Dense 320-step raster Steps: 320x0.35" Max FOV: Si IV: 38s, 80 images 2016-01-20 Step Cadence: 9.6s 231"x119" Mg II: 38s, 80 images 23:37:21-00:28:19 +1d Raster Cad: 3057s, 1 raster

AR FOV: 112"x119" x,y: -347",-145" FOV: 119"x119" Dense 320-step raster Steps: 320x0.35" Max FOV: Si IV: 38s, 80 images 2016-01-22 Step Cadence: 9.6s 231"x119" Mg II: 38s, 80 images 04:52:21-05:43:19 Raster Cad: 3057s, 1 raster

AR 12567 FOV: 141"x175" x,y: 458",11" FOV: 167"x175" Dense 400-step raster Steps: 400x0.35" Max FOV: Si IV: 37s, 100 images 2016-07-20 Step Cadence: 9.2s 307"x175" Mg II: 37s, 100 images 05:09:11-06:10:33 Raster Cad: 3,683s, 1 rasters

AR 12641 FOV: 112"x119" x,y: -25",367" FOV: 120"x119" Dense 320-step raster Steps: 320x0.35" Max FOV: Si IV: 21s, 320 images 2017-03-03 Step Cadence: 5.2s 233"x119" Mg II: 21s, 320 images 07:42:17-09:32:25 Raster Cad: 1,652s, 4 rasters

Table 5: Candidate observations (9–12) in active regions in search of the TR loops. The highlighted entry is the best scan having more than 10 loops evolving inside the vicinity of raster scan.

Among all the datasets, the best scan was taken on 2015 August 06 from

15:19:21 to 16:41:59 UTC with an FOV of 14"x 119" (highlighted in Table 3). The 0.33" x 119" slit was stepped 8 times with 2" steps and a step cadence of 9.1 seconds.

However, this observation consisted of the repetition of 68 such frames making the raster cadence for each loop 73 seconds. Thus, the total number of raster scans was

544. The slit-jaw images (SJI) had an FOV of 120"x119" with a cadence of 36 seconds.

Since the SJI cadence was four times longer than the raster cadence there were 136 total slit-jaw images. All of the data were calibrated to level 2 by including dark current, flat-field, and geometric correction, and subsequently transformed to Level

3 data by building data cubes of spatial and temporal sequenced spectra.

The selection of this particular loop was based on two criteria: (1) the brightening was small enough to be considered a TR loop event (< 10 Mm) and (2) the intensity profile of the pixels was distinguishable above the noise threshold.

However, since the raster scans through the entire image, it covered the loop only for a fraction of its evolution. Next, the author extracted the spectral data from the raster file. To determine line intensities, Doppler shifts and broadenings, the author fitted the Si IV FUV line using a multiple-peak fitting algorithm with a background baseline.

The baseline was first estimated within multiple shifted windows of width 200 separation units. Then, the author adjusted the baseline for the corresponding spectrum by regressing the varying baseline to the window points using a spline

approximation. Once the fitted spectrum was retrieved, the author extracted the O IV

1399.766 Å, O IV 1401.157 Å, Si IV 1402.77 Å and S IV 1404.808 Å peaks.

5.6. Doppler Shift and Line Broadening

We extracted observational parameters of physical significance: the peak ratio of O IV 1401.157 Å and Si IV 1402.77 Å lines; and the Doppler shift and line width of

O IV 1401.157 Å, Si IV 1402.77 Å and S IV 1404.808 Å lines. The Si IV 1402.77 Å line did not exhibit single Gaussian or Lorentzian form, thus the author adopted a bi- component Lorentzian fit (since it fitted with minimum mean squared error) for this line (Figure 12) and then measured the width and Doppler shift for each component.

From the line width of the measured lines, the author calculated the non-thermal broadening with the equation,

2 2 ∆푣푁푇 = √∆푣 − ∆푣푇ℎ − ∆푣푖푛푠푡

2푘퐵푇 Where, 푣푇ℎ = √ is the maximum thermal line broadening at the peak 푚푖표푛

−1 formation temperature of 0.1 MK in the TR and 푣푖푛푠푡 = 3.9 푘푚푠 . For IRIS, the non- thermal line broadening can be caused by several different processes, including for example fine scale unresolved flows, waves, turbulence (De Pontieu et al. 2015 and

Testa et al. 2016).

Note, a reliable component analysis could not be done for S IV bulk flow due to its weak signal strength and the presence of blending lines (Table 6); thus, a single component analysis is performed.

5.7. Density Diagnostics

One of the most accurate ways to calculate the plasma density is by the exploitation of the intensity ratio of spectral lines of a single element of same charge number but a different dependence on the electron number density. Ideally, this approach removes the uncertainties associated with the ion fractional abundance and

chemical abundance. However, the lines should have a similar temperature dependence, so that their ratio remains independent on the plasma temperature. To achieve this, the existence of metastable levels is required within that ion which can decay radiatively via forbidden transitions. Generally, the intensity of an allowed

2 transition is proportional to the square of electron density 푛푒, whereas for forbidden

훽 transitions, the intensity is proportional to 푛푒 with 1 < 훽 < 2 (this is because the radiative decay rate for forbidden transition is so small that collisional de-excitation can become an important de-populating mechanism). Thus, based on the relative importance of collisional and radiative de-excitation, density diagnostics can be performed by either the ratio of two forbidden transitions, two allowed transitions or an allowed and a forbidden transition.

Figure 13: O IV ratio generated with CHIANTI 8. Black line T = 105.15 and Red line T = 104.4 (Young 2015).

The best density-sensitive line ratios observed by IRIS is given by the spin- forbidden O IV transitions at 1399.766 Å and 1401.157 Å included in the Si IV 1403 Å spectral window (Figure 13). These lines are particularly suitable for density measurements as their ratio is largely independent of the electron temperature, and only weakly dependent on the electron distribution (Dudík et al. 2013). The other advantage is that the lines are close in wavelength, minimizing any calibration effects.

Hence, in this work, the author employed the atomic database CHIANTI to calculate the theoretical intensity ratio of this pair of O IV lines near 1400 Å as a function of density. Then the observations were interpolated with this theoretical model and the density was derived for each pixel, at each time step. For the current study, the observed lines in this spectral window is presented in detail in Table 6.

Transition Wavelength (Å) Remarks

Si IV 3s 2S1/2 ↔ 3p 2P1/2 1402.77 Strongest

O IV 2s22p 2P1/2 ↔ 2s2p2 4P3/2 1397.198 Weak

O IV 2s22p 2P1/2 ↔ 2s2p2 4P1/2 1399.766 Average

O IV 2s22p 2P3/2 ↔ 2s2p2 4P5/2 1401.157 Strong

O IV 2s22p 2P3/2 ↔ 2s2p2 4P3/2 1404.779 Blended with S IV

S IV 3s23p 2P1/2 ↔ 3s3p2 4P3/2 1398.040 Very Weak

S IV 3s23p 2P1/2 ↔ 3s3p2 4P1/2 1404.808 Blended with O IV Table 6: Observed transitions by IRIS Si IV 1403 Å used for density diagnostics.

In the line intensities shown in Table 6, the O IV multiplets are generally considered to be significantly stronger than the S IV multiplets in the solar TR.

However, the O IV 1397.198 Å line and the S IV 1398.04 Å line are very weak in most solar events and thus can be neglected during observation. Note that, there is a S IV

(at 1404.85 Å) blend in the O IV 1404.779 Å line leaving the O IV 1399.766 Å and O IV

1401.157 Å as the only unblended IRIS lines fit for density diagnostics. Although, the allowed transition of Si IV 1402.77 Å line has a strong intensity and the line ratios between Si IV 1402.77 Å and O IV 1401.157 Å can be used to calculate electron densities (Cheng at al. 1981, Peter et al. 2014, Doschek et al. 2016), the validity of using the this ratio has been disputed for the facts that the ratio gives very high densities compared to the more reliable ones obtained from the O IV ratios alone

(Hayes and Shine, 1987) and that Si IV and O IV are formed at quite different temperatures for equilibrium ionization (Judge 2015) or in the presence of non-

Maxwellian velocity distributions (Dudík et al. 2013). In addition, there exists ambiguity in the element abundances of Si and O in TR region with the possibility that the abundance might vary depending on the height scale of the observed structure.

Finally, Si IV and O IV show drastically different responses to transient heating leading to non-equilibrium ionization (shown in the later part of this thesis) because of their different formation processes, and taking into account the effect of TR high- densities on the ionization balance causes the fractional ion abundances of O IV and

Si IV to depart significantly from the low density approximation (Polito et al. 2016,

Young et al. 2018).

The author presents the density diagnostics of the 8 candidate loops in spatial and temporal coordinates in the 1st rows of the 2nd columns of Figures 14 – 21.

5.8. Co-aligning the IRIS data with AIA

The slit-jaw images provided by IRIS have higher spatial resolution (0.33") and potentially faster cadence (as low as 9s) than the AIA data (0.6" and 12s). In addition, IRIS images have a smaller FOV (175 arcsec x 175 arcsec) and are often rotated with respect to AIA (41 arcmin x 41 arcmin along the detector axis). Taking all these into consideration, the author carefully aligned the IRIS data to the AIA 1600

Å channel and the AIA 1600 Å channel to the other AIA EUV channels (94 Å, 131 Å,

171 Å, 193 Å, 211 Å and 335 Å) using the solar limb. Since our required spatial scales were extremely small, the author revised the coalignments by a trial and error method and excluded any unrelated bright pixels present in the region of interest.

Finally, the intensity of pixels from the SJI image of the IRIS 1402 Å channel and the corresponding AIA EUV channels were spatially integrated for each time in order to obtain the light curve. Level 2 data for IRIS slit-jaw and Level 1.5 data for AIA EUV images were used for analysis since they are corrected for dark-current and flat-field, and solar rotation as well as internal co-alignment drifts.

5.9. Differential Emission Measure Analysis

Recall from the coronal model approximation in chapter 3.1:

+푚 푁(푋 ) 푁(푋) 푁(퐻) 푒 2 푃(휆푗,푔) = 퐶푔,푗Δ퐸푗,푔푁푒 푁(푋) 푁(퐻) 푁푒

푁(푋+푚) Here the quantity 퐶푒 = 퐺(푇, 휆 ) is called the contribution function 푁(푋) 푔,푗 푗,푔 and it is strongly peaked in temperature, being related as it is to the ion population fraction. In terms of the contribution function, the intensity can be written as,

1 푁(푋) 푁(퐻) 2 퐼(휆푗,푔) = ∫ 퐺(푇, 휆푗,푔)훥퐸푗,푔푁푒 푑푉 4휋퐴 푉 푁(퐻) 푁푒

The quantity, 푁2푑푉 = 퐸푀(푇) is a measure of the amount of plasma ∫푉 푒 emitting at a particular temperature and is called the emission measure. Any observation of the solar atmosphere with temperature sensitive spectral lines needs to be modeled and verified in terms of the emission measure since it holds crucial information about the temperature distribution along the LOS of the emitting plasma.

However, calculating the spectral line intensity from the emission measure (EM) is only possible if the observed plasma is close to isothermal. For our current regions of interest, that obviously was not the case. Thus, the author used a quantity called the differential emission measure (DEM), which is related to the amount of material in the temperature interval [푇, 푇 + Δ푇] emitting the observed radiation and to the

temperature gradient along the LOS. The latter relationship also means that the DEM can be related to the heat flux and may thus help us to resolve the question of which mechanisms heat different solar phenomena: whether the candidate loops were heated by a train of nanoflares originated from magnetic reconnection of multi- braided loop strands (e.g. Parker 1988); or through micro-flares (Fletcher et al. 2011;

Hannah et al. 2011) due to energetic particles, or other mechanisms such as waves.

To reliably answer these questions, one needs to know not only the uncertainties on the emission for a given temperature, but also the uncertainties on the temperature itself, i.e. the temperature resolution.

2 The DEM 휙(푇) = 푁푒 푑푅/푑푇 can be incorporated in the equation of spectral intensity as,

1 푁(푋) 푁(퐻) 퐼(휆푗,푔) = 훥퐸푗,푔 ∫ 퐺(푇, 휆푗,푔)휙(푇)푑푇 4휋 푁(퐻) 푁푒 푇

In general, one has to measure many spectral lines across a wide temperature range (in intervals Δ푇) in order to find a full DEM. Reliable 퐺(푇, 휆푗,푔) functions must be found for each line and an inversion procedure applied to the above equation to extract 휙(푇). Although it may seem straightforward, the calculation of the DEM is often found to be non-trivial and cumbersome: the uncertainties associated with the observations such as counting statistics, background and instrumental errors compounds the difficulty in the determination of the DEM and results in an ill-posed

inverse problem (Tikhonov 1963; Bertero et al. 1985; Craig & Brown 1986; Schmitt et al. 1996). Any direct attempts to solve the inversion normally leads to the amplification of the uncertainties, and hence unphysical solutions.

To reconstruct the DEM additional constraints must be added and numerous approaches have been developed to solve this problem. The simplest of these is described in the previous section: assume that all the emission is at a single temperature (isothermal) with 휉(푇) ∝ 훿(푇 − 푇0) where 훿(푥) is the Dirac delta function. Then, dividing the observed density by the response function and plotting this as a function of temperature, the intersection points of the different curves (EM loci curves) for different lines gives the isothermal temperature and an upper bound to the emission measure (e.g. Schmelz et al. 2011). Although this is a simple and computationally fast method, it does require the isothermal assumption and if the

DEM is multithermal this method will produce erroneous results. Another approach is to forward fit a chosen form of DEM while minimizing the differences in observable space. This has been implemented for a discretized spline model DEM (Monsignori

Fossi & Landini 1992; Brosius et al. 1996; Parenti et al. 2000) and more recently using the IDL mpfit routine with SDO/AIA and Hinode/XRT (Weber et al. 2004; Golub et al.

2004). An iterative forward fitting approach has been developed with multiple

Gaussian model DEMs using the IDL POWELL routine (Aschwanden & Boerner 2011).

To estimate the error in the DEM with these methods a Monte Carlo approach is

adopted, producing multiple realizations within a given noise range. Although these approaches calculate parameters for the model DEM with reasonable precision, they are computationally slow in almost all cases, and not practical for fine-resolution spatio-temporal analysis.

Regularized inversion methods introduce an additional “smoothness” to constrain the amplification of the uncertainties, allowing a stable inversion to recover the DEM solution (e.g. Craig 1977; Craig & Brown 1986). This was demonstrated to have promise for solar observations by Craig (1977) and subsequently tested on simulated data by Fossi & Landini (1991). Several forms of regularized inversion – truncated (or zeroth-order) Singular Value Decomposition (SVD), second-order regularization and maximum entropy regularization – have also been tested using simulated EUV spectral line emission (Judge et al. 1997). Although they determined that these approaches were superior to the other methods described above, they found several problems with the regularized inversion: the smoothness criterion used may become unphysical; the solutions are highly sensitive to uncertainties in the kernel (the Kernel K is the matrix representing the spectral line contribution function and comes from system of linear equations to be solved for the DEM, see

Hannah and Kontar 2012); and the return of negative solutions. In 2012, Hannah and

Kontar presented a regularization method, which resolved some of these problems and robustly recovered the underlying DEM with estimated errors. By applying

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Generalized Singular Value Decomposition, they decreased the computational burden of the method and showed that it can naturally provide an estimate to the temperature resolution as well. This method has been implemented and exhibited robust results when applied to solar data for the inversion of RHESSI (Lin et al. 2003)

X-ray spectra to reveal their source electron distribution (Kontar et al. 2004) and active region observations of SDO/AIA and Hinode/EIS. The author adopted this regularized inversion technique for the calculation of the DEM in each pixel for the

AIA dataset used here. An IDL routine can be found in GitHub to efficiently implement this technique: /ianan/demreg and the details of the method are described in Hannah

& Kontar (2012).

5.10. TR Loops in AR 12396 observed on 6th August 2015

The observations presented in Figures 14 – 15 concentrate on a pixel-by-pixel analysis of loop brightenings and the evolution of their spectra. The IRIS 1400 Å channel was the primary means for this analysis since its temperature sensitivity is close to the region starting from the upper chromosphere to the lower corona and its range contains O IV lines readily available for density diagnostics. The spatio- temporal evolution of the density sensitive O IV ratio (1399.766 and 1401.157 Å) showed that the bright pixels of the loop had average number densities of 1011 cm-3, while the darker pixels of the loops had an average number density close to 1013 cm−3,

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meaning plasma observed in these pixels was closer to chromospheric conditions. In contrast to O IV, the Si IV emission exhibited enormous differences between the dark and bright regions of the loops; in fact, the Si IV spectrum at the bright loop cores was fundamentally different to any other spectrum from the surrounding active region.

The Si IV 1403 Å spectrum appeared to show compound peaks with large line broadenings (∆λ → 3Å ≈ 400 km/s). Thus, the author decomposed the Si IV 1403 Å profiles and found at least two components. One component displayed a blue-shift and the other a red-shift of the line centroid, indicating a strong bidirectional flow with a maximum speed of 100 km/s toward and away from the observer. As the observed active region appeared close to the center of the solar disk, this can be interpreted as a predominantly radial bidirectional flow. In addition to strong

Doppler shifts, a number of the observed loops showed that each component of the Si

IV bidirectional flow had a non-thermal component as large as 60 km/s. Such strong

Doppler shifts with broad non-thermal components were also observed in the S IV

1404 Å line profile. Compared to Si IV and S IV, the line profile of O IV ion appeared to not contain any significant non-thermal component and the maximum Doppler shift was no more than 25 km/s. To further investigate the differences between the heavier and lighter ions, the ratio of the peaks of Si IV 1403 Å and O IV 1401 Å was plotted for each pixel in Figures 14 – 21. A remarkably large Si/O peak ratio is observed at the bright pixels, indicating a clear correlation between the Si/O peak ratio and intensity. Table 7 shows the pixel-averaged values of the apex and footpoint

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density, Si IV, S IV and O IV Doppler shift, and non-thermal line broadening at the brightened loop top and the Si/O peak ratio at the brightenings.

Si/O 푛 푛 Δ푣 Δ푣 Δ푣 Δ푣 Δ푣푁푇 Δ푣푁푇 Δ푣푁푇 푓표표푡 푎푝푒푥 푆푖 퐼 푆푖 퐼퐼 푂 푆 푆푖 푂 푆 peak 10x cm-3 10x cm-3 (km/s) (km/s) (km/s) (km/s) (km/s) (km/s) (km/s) ratio 1 12.88 10.49 +83 −40 27 60 58 0 90 40

2 12.98 12.5 +80 −100 25 62 69 0 66 19

3 13 11.9 +20 −100 26 34 60 4 73 10

4 13 11.75 +100 −36 27 55 71 5 92 38

5 12.79 11.5 +100 −39 22 46 79 4 91 16

6 12.65 12.33 +84 −100 25 30 59 0 65 22

7 12.84 11.71 +91 −44 25 60 78 0 87 71

8 12.74 10.3 +78 −90 26 61 65 0 78 26 Table 7: Tabulated physical properties extracted from IRIS observations for 8 candidate TR loops emerged in AR 12396 on 6th August 2015. 103

Figure 16: Pixel-by-Pixel analysis on the candidate loop 3: High density, Strong bi-directional flow in Si IV lines along with large Doppler shift in S IV lines were found at the brightenings like previous candidates. However, the bidirectional flow was stronger into downward direction in contrast to the previous observations (where it was nearly equal). Again, the O IV exhibited relatively small Doppler shift. A large non-thermal line broadening in Si IV and S IV were observed in contrast to O IV. Following the trend of previous candidates, large Si/O peak ratio were observed. 106

Figure 21: Pixel-by-Pixel analysis on the candidate loop 8. High density, Strong bi-directional flow in Si IV lines along with large Doppler shift in S IV lines were found at the brightenings. Here, the strength of both upward and downward flow was nearly equal (similar to loop 2 and 6). The O IV exhibited relatively small Doppler shift and a large non- thermal line broadening in Si IV and S IV were observed in contrast to O IV. Following the trend of previous candidates, large Si/O peak ratio were observed (as strong as loop 2, 3, 5 and 6).

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Chapter 6

On the Origin of Transition Region loop Brightenings

The dramatically increased temporal resolution of the current generation of observing instruments has made it clear that besides large-scale events, such as solar flares, there also exist smaller-scale dynamic events which plausibly play a large role in heating the solar corona. Consequently, both the spatial and the temporal dependence of the heat input are required to help determine the actual heating mechanism, and the introduction of a time-dependent component to the heating mechanism was inevitable in the development of hydrodynamic models. As discussed earlier, such heating events can be sufficiently fast that the emitting ions may not be able to respond to the changing plasma temperature and are outcompeted by the heating timescale. Thus, any self-consistent analysis of observational data and the

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treatment of radiation in modeling studies required the radiative emission from non- equilibrium ion populations to be accounted for.

The heating mechanisms which have been proposed so far can be broadly categorized as either AC or DC processes. It has been confidently predicted that waves and MHD waves, in particular, should be an important component of the dynamic solar atmosphere, given the nature of its magnetic structuring. The dissipation of wave energy is an AC heating mechanism (McIntosh et al. 2011, van der Holst et al.

2014). Alternatively, magnetic field lines can be twisted and stressed over relatively long periods of time, and this gradual build-up of energy can result in a sudden release due to the process of magnetic reconnection, which occurs as the magnetic field reconfigures itself into a state of lower energy. The transfer of energy from stressed magnetic fields into heat is an example of a DC heating mechanism (Schrijver et al.

1998, Parker 1988). In the paradigm of DC mechanisms, the central idea is the conjecture of nanoflares and the nanoflare storm. In his breakthrough paper, Parker

(1988) noted that UV and X-ray observations of the solar atmosphere showed that emission from small magnetic bipole structures was intermittent and impulsive, whereas the emission from larger bipole structures was steadier and appeared to be the sum of a large number of individual impulses. The individual impulses were determined to be the basic unit of energy release and termed nanoflares because their energies were on the order of one-billionth the energy of a typical flare. These

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observations led Parker (1988) to suggest that the solar corona is heated to such high temperatures by a swarm of nanoflares. In the nanoflare model of Parker (1988), the spatial-scale of a nanoflare was estimated to be on the order of 103 km and its typical energy release on the order of 1024 erg. In reality, it is likely that these events exhibit a range of spatial-scales and characteristic energies. The basic nanoflare mechanism is the twisting and braiding of magnetic field lines in the solar atmosphere, driven by turbulent motions in the photosphere where the field lines are anchored (Peter

2004). The turbulent motions arise due to the convection zone beneath the solar surface. Eventually, the magnetic field lines become so stressed (as a result of the twisting and braiding) that strong current sheets are formed at the tangential discontinuities between them, resulting in magnetic reconnection and the dissipation of large amounts of magnetic energy as heat.

Since nanoflares are such small-scale events, both spatially and energetically, none have yet been directly observed. In fact, it may be necessary to look for indirect evidence of their existence (Cargill 1994) and their ability to provide a sufficient amount of energy to satisfy the heating requirements of the corona. Until now, a surfeit of modeling and simulation work has been produced in support of the nanoflare heating, but no direct observational evidence has been found. However,

SOHO-SUMER, SOHO-CDS, TRACE, SDO/AIA, SDO/HINODE and IRIS have provided tantalizing evidence in support of nanoflares.

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Though it may not be possible to directly observe nanoflares, there is ample observational evidence for transient brightenings that are associated with heating mechanisms that might hold key information on the heating of the corona. Before we dive into that, let us first define the order of energies that the author considers as nanoflares: it is the heating events which have characteristic energies of 1024 erg, or less. Here, the preferred location of heating is not known, though there is observational evidence to suggest that it may occur in the chromosphere, the TR and the corona. Thus, the author experimented by varying heating parameter values, both spatially and temporally, in the numerical code and investigated the origin in detail.

Since Skylab, loops have been recognized as a vital ingredient in coronal structure and energetics. It is not unreasonable to imagine that the corona and the TR are entirely composed of resolved and unresolved loops with varying lengths, temperatures, heating rates, and activity levels. The rest of this chapter is dedicated to discovering the origin of the energetics and the dynamics of the cool TR loops observed by IRIS. To understand the dynamics of these TR loops, three-dimensional magneto-hydrodynamic simulations would be extremely useful, but they are limited in spatial scale and computationally extremely demanding, and thus cannot adequately address the complexity present in a multi-stranded loop. A more feasible approach is to solve the field-aligned hydrodynamic equations along many sub- resolution strands using a spatially and temporally-dependent heating function to

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emulate the properties of the underlying mechanism. The author followed this recipe and the system of equations appropriate to this treatment is solved using HYDRAD:

The HYDrodynamics and RADiative emission model. Next, the forward modeling code developed by Prof. Stephen Bradshaw (discussed in Chapter 4.3) was employed that can produce synthetic spectra and broad/narrow band emission along the field lines based on the output from HYDRAD, for a given observing instrument.

In order to model with fidelity the observed brightenings at the loop apex, one must satisfy several observational criteria: 1) the density profile showed a dense plasma concentration (1010 to 1011 cm-3) at the top of the loop; 2) the bright apex of the loop exhibited a bulk flow of 100 km/s where there also existed a non-thermal velocity component of more than 40 km/s; and 3) the Si/O peak ratio in the Si 1400

Å line was of order 10. These three critical constraints will act as yardsticks to measure the success of the numerical model in simulating the observed events.

6.1. Case I: Footpoint Heating

Although various mechanisms of coronal loop heating have been proposed independent of the detailed process of energy release, there are both numerical and observational evidence that loops are predominantly heated at the footpoints

(Bingert and Hardi 2001, Aschwanden 2006) . Thus, the author initiated the numerical calculations with footpoint heating and shows later in this chapter that

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accumulation of cool, condensed mass and subsequently periodic loop-top brightening are possible in these short, cool TR loops. The author studied the evolution of these loops, discussed dynamic solutions and calculated the time- dependent emission of TR lines arising from this model to determine whether the observed spectrum can emerge from such a heating mechanism.

To realize this scheme, the author adopted one of the popular choices of expression to describe the spatial distribution of heating along the loop,

2 (푠 − 푠0) 퐸퐻(푠) = 퐸퐻0exp [− 2 ] 2푠퐻

In the above formulation, the heating has a Gaussian distribution of spatial width 푠퐻 along the loop, centered at the location 푠0. The maximum volumetric heating rate is 퐸퐻0 which varies spatially following the above equation. The heating becomes spatially uniform as 푠퐻 → ∞. The equation is flexible enough to localize the heating and place it anywhere along the loop. For experimenting with footpoint heating, the spatial width was varied between 0.5 and 1.50 Mm in the simulations presented below. Here, the loop length was assumed to be 10 Mm and the simulation was run for 10000 seconds. The spatial width variations were combined with the variation in the heating rate and spatial location of the maximum heating. The full parameter space is given below in Table 8. In all cases, the heating was time-independent.

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Maximum Spatial Location Volumetric Heating Spatial Width (Mm) 3 (Mm) Rate (erg/cm /s) 풔푯 풔ퟎ 푬푯ퟎ 0.005 1.0 0.75 2.00

0.010 1.25 1.00 2.25

0.015 1.50 1.25 2.50

0.020 1.75 1.50 2.75

0.025 2.0 1.75 3.00 Table 8: Parameter space in TNE cycles for the observed TR loop brightenings.

Thus, the parameter space explored 5 × 5 × 10 = 250 loops in order to find a candidate to match the loop observed by IRIS. In all of these cases, the maximum volumetric heating rate was kept fixed at 0.002 erg/cm3/s and the spatial location of the heating function was set s0 = 1.5 Mm above the chromosphere. The simulated loops had foot-points of 2.2 Mm buried in the chromosphere (total length of the loop

10 + 2.2 × 2 = 14.4 Mm) with a foot-point temperature of 2 × 104 K. The foot-point density was selected to be 1012 cm-3 and the loop assumed to be initially isothermal.

This condition was convenient for keeping the initial temperature low in short, dense loops; otherwise they can start off too hot and become numerically intractable. As discussed in Chapter 4, an equilibrium radiation treatment was employed which takes account of the optically thick photosphere and chromosphere.

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In the case of foot-point heating described above, the localized heating can lead to thermal non-equilibrium (TNE), a phenomenon that occurs in the solar atmosphere when the heating is highly stratified (Müller 2003, 2004 and Froment

2018). The localization of the heating produces chromospheric evaporative upflows that supply the coronal structure with dense and hot material. A thermal runaway is eventually triggered when the radiative losses overcome the limited heating at coronal heights. Condensations are formed locally in the corona and fall down to the loop footpoints along the magnetic field lines. Furthermore, if the heating is steady or quasi-steady, i.e. with a high heating frequency compared to the typical cooling time, this phenomenon can be cyclic. Such a system has no existing thermal equilibrium and will undergo evaporation and condensation cycles, hence the name thermal non- equilibrium cycles. The evolution of one of the simulated loops that goes through periodic thermal non-equilibrium cycles is described in the following text. If the author define the TNE period (the time when thermal non-equilibrium initiates condensation at the loop apex to the beginning of next condensation) by Δ푡 (in case

Δ푡 of Figure 22, Δ푡 = 3700 seconds), then during the first 푡 = + 1000 = 2200 s the 3 hot TR loop is formed and then starts to cools from lower coronal temperature

6 5.5 (푇푡표푝(푡 = 0) = 10 K) to upper TR temperature (≈ 10 K), while the density stratification remains roughly constant and a low amplitude acoustic wave continues to bounce in between the footpoints. Then, there is a sharp transition: The

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temperature at the loop apex is no longer the maximum loop temperature, and this dip in the temperature stratification amplifies rapidly. At the same time, the developing pressure gradient drives a slow flow towards the cooling loop apex with a velocity of 푣 ≈ 10 − 20 푘푚푠−1 at 푡 = 2400 푠. At this time, a clump of cool (104

K) material with rapidly increasing mass content forms at the loop-top. This clump

(or condensation region) eventually starts moving slowly towards one loop leg and is accelerated to 푣 ≈ 5 − 15 푘푚푠−1 before draining into the chromosphere at 푡 =

5250 푠. As a result, a weak rebound shock forms, followed by a phase of chromospheric evaporation which refills the evacuated loop with plasma. This upflow decreases with time from 푣 ≈ 10 푘푚푠−1 to 푣(푡 = Δ푡 푠) ≈ 0. In the mean time, the apex temperature of the loop reaches to its previous maximum temperature

6 (푇푚푎푥,푡표푝 = 10 K at 푡 = Δ푡 푠). The whole process is then repeated. During the simulation run, the author introduced an asymmetry of 5% between the deposited energy at each leg which proved suﬃcient to dictate the draining direction: the condensation moves to the side on which less energy is supplied.

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In the above scenario, the formation of the central dip of the temperature stratification resulted from the concentration of heating near the footpoints of the loop (in other words, from insuﬃcient heating at the top to balance radiation). In order to better understand the evolution of the loop, let us consider the energy balance at the loop apex. The relevant terms for this are the mechanical energy supply

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푄푚 and the radiative losses 퐿푟푎푑. As the density in the coronal part of the loop (loop apex) increases, the mechanical heating per particle (푄푚/푛푒) decreases (here, the ion density, 푛푖표푛 equals approximately the electron density, 푛푒). At the same time, the radiative losses per particle, (퐿푟푎푑/푛푒) increase as the temperature drops to TR temperature, which is predominantly due to the temperature dependence of the radiative losses. Now the time-dependence of the total energy balance at the apex is dominated by the interaction between the increase of radiative losses and the increase of density; as a result of this interplay, the energy supply at the loop top

Δ푡 becomes negative at t = s, which results in cooling and the developing dip in the 3 temperature profile. The simultaneous decrease of the gas pressure initiates a symmetric flow towards the center of the loop, so that more and more mass is advected and a condensation region is formed. Once the temperature dip has formed as a consequence of the described loss of equilibrium, a thermal instability sets in

2 because 퐿푟푎푑 ∝ 푛푒. Since, the simulated loop is of semicircular shape, the configuration with a condensation region located at the very center of the loop is gravitationally unstable. Therefore, the slightest perturbation forces the condensation region to move downward in either direction, where it experiences increasing acceleration.

From the search in parameter space it was evident that the heating scale length strongly dictated the periodicity of the TNE cycle. Thus, the author found it to

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be necessary to discuss its role in the loop evolution and under what circumstances the TNE cycle is observed. Let us consider the limiting cases first: for short heating scale lengths of 푠퐻 ≤ 1.0 푀푚, the loop cooled as not enough energy was deposited in the upper part of the loop to balance the radiative and conductive losses. In this case, the temperature in the loop fell during the first 2Δ푡 s to roughly 104 K and stayed at that level for the remainder of the model run. On the other hand, for longer heating scale lengths with 푠퐻 ≥ 2 푀푚, the energy deposition at the loop center was large enough to sustain a stable loop against radiative and conductive losses and an average

6.1 loop temperature of 10 K (for 푠퐻 = 2 푀푚) was reached and maintained.

The regime in between, with intermediate heating scale lengths of 1 푀푚 <

푠퐻 < 2 푀푚, showed the cyclic behavior described above. In these cases, the loop exhibited a dynamic behavior, triggered by the onset of thermal instability. For the cases of shorter heating scale lengths, 푠퐻 = 1.25 푀푚 and 푠퐻 = 1.5 푀푚, the formation of a condensation region followed qualitatively in the same way. However, as the heating was more strongly concentrated towards the footpoints of the loop, the net energy supply per particle at the loop top started to decrease at an earlier point in time so that the maximum loop temperature attained was lower, namely 푇푚푎푥 =

5 5 5 × 10 K for 푠퐻 = 1.25 푀푚, and 푇푚푎푥 = 6 × 10 K for 퐻푚 = 1.5 푀푚 compared to

5 푇푚푎푥 = 8 × 10 K for 푠퐻 = 1.75 푀푚. Due to the strong radiative losses at high temperature (and might also have been denser due to the high temperature), these

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loops also cooled faster, so that the period of the condensation cycle was shorter than

3 3 for 푠퐻 = 1.75 푀푚 (Δ푡 = 6.1 × 10 푠): Δ푡 = 3.7 × 10 푠 for 푠퐻 = 1.25 푀푚 and Δ푡 =

3 5 4.4 × 10 푠 for 푠퐻 = 1.5 푀푚. In the lower temperature regime (2 × 10 K < 푇 <

6 × 105 K), the cooling rate was found to be very similar for all three cases, which leads us to the conclusion that the increased period for the heating scale length of

푠퐻 = 1.75 푀푚 was mostly due to the longer duration of loop reheating and loop cooling in the high temperature regime (푇 > 6 × 105 K). Table 9 summarizes the relevant parameters for diﬀerent heating scale lengths.

Spatial Width (Mm) TNE Periodicity (s) Type of Loop 풔푯 횫풕

1.0 - Catastrophic Cooling

1.25 3.7 × 103 TNE

1.50 4.4 × 103 TNE

1.75 6.5 × 103 TNE

Condensation then 2.00 - Thermally Stable

2.50 - Thermally Stable Table 9: Parameter space of TNE cycles for the observed TR loop brightenings.

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For all loops which formed a condensation region, the minimum mean

3.7 4 temperature was very similar to the lower/mid TR (푇푚푖푛 = 10 − 10 K), thus recreating the low-lying TR loops observed by IRIS.

Varying the maximum volumetric heating rate only recreated the above scenario in a similar pattern, with the exception that the TNE periodicity was either increased if the heating rate was increased or went down if the heating rate was decreased. This was intuitive since increasing heating rate meant more energy was deposited in the upper part of the loop to balance the radiative and conductive losses and thus lengthened the periodicity of the cycle and vice versa. On the other hand, altering the spatial location (s0) qualitatively yielded similar result except a heating function closer to the loop apex resulted in a thermally stable loop as expected.

Within the exhaustive parameter space, there were cases found where the density and temperature of the loops would coincide with the IRIS observation, but the author could not locate any cases where the high bi-directional velocity (along with the non-thermal component) of 100 km/s were reproduced in numerical experiments. The maximum velocity of the mass at the loop apex was no more than

20 km/s which was far less than the observed velocity. More importantly, the TNE cycles of these simulated low-lying loops were no less than couple of thousands of seconds while the observed loops had lifetimes of less than a thousand seconds. Thus,

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the author reached to the conclusion that the observed loop brightenings were not mediated by TNE cycles.

6.2. Case II: Uniform Heating

In the next phase of numerical experiments, the author turned his attention to the possibility of a uniform heating mechanism that might be responsible for the observed loop brightening. Recall the expression that was adopted to describe the spatial distribution of heating along the loop,

2 (푠 − 푠0) 퐸퐻(푠) = 퐸퐻0exp [− 2 ] 2푠퐻

Maximum Spatial Location Spatial Width Duration of Volumetric Heating (Mm) (Mm) Heating (s) Rate (erg/cm3/s) 풔ퟎ 풔푯 ∆풕 푬푯ퟎ (횫 = step size) 0.01 – 0.1, Δ=0.01 0.1 – 1.0, Δ=0.1 1.0 100 10 – 100, Δ=10 1 – 3, Δ=0.25 Table 10: Parameter space of uniform heating in search for the observed TR loop brightenings (temporal profile of the heating is chosen to be triangular).

In the above formulation, the heating became spatially uniform as 푠퐻 → ∞.

Once again, the author ran an exhaustive search in the parameter space given in Table

10 and could generate a condensation at loop apex when an exceedingly high

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volumetric heating rate was applied. As a result of large volumetric heating, chromospheric ablation was generated with opposing flows following the magnetic field lines. In the two-fluid model (electrons and ions), strong compressive heating at the interface region where the two flows meet led to the conversion of ion kinetic energy (neglecting electron kinetic energy due to their low mass) to ion thermal energy. Thus, the ion temperature was raised above the electron temperature. Given this induced temperature difference at the interface, there followed a collisional exchange of energy between the heated ions and the cooler electrons. The electrons received energy from the ions in order to equilibrate the temperatures of the particle species. At the same time, the electrons lost energy through radiation, which increased in proportion to the square of the density as the plasma at the interface was compressed by the interacting flows. At the beginning, the ion temperature was above the electron temperature and acted as a thermal reservoir for the electrons to draw thermal energy. Now, in order for the loop to be thermally stable, the rate of energy gain by the electrons from the ions, via collisions, had to exceed the rate of energy loss by the electrons due to radiation. However, as the density of the apex increased with

2 the opposing flows, the radiation loss increased (∝ 푛푒) and thus, a strong flow (above some threshold) from the opposing footpoints led to a cooling event and the formation of a condensation. During the simulation run, there was an asymmetry of

1% in the heating rate between the footpoints to dictate the draining direction.

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In order to establish the above argument quantitatively, the author, with the help of his supervisor Dr. Stephen J Bradshaw, derived a hierarchy of timescales that must be satisfied to avoid the formation of a condensation whenever two oppositely directed flows interact with one another. The derivation begins by considering the physical nature of this interaction: to successfully heat the electrons, thereby avoiding cooling and the formation of a condensation, two conditions concerning the rate of energy transfer must be satisfied:

1. The rate of compressive heating of the ions must exceed the rate of energy

loss to the electrons by collisions. This maintains the ion temperature

above the electron temperature and ensures that the ion fluid can continue

to act as a thermal reservoir for the electron fluid to draw upon.

2. The rate of energy gain by the electrons from the ions, via collisions, must

exceed the rate of energy loss by the electrons due to radiation. This allows

the electron temperature to increase (or be maintained) and avoids

cooling/condensation formation.

The relevant hydrodynamic energy equations for the two-particle species that will allow us to explore the consequences of these conditions are:

휕퐸 휕 푘 푖 + [(퐸 + 푃 )푣] = − 퐵 푛휈 (푇 − 푇 ) 휕푡 휕푥 푖 푖 훾 − 1 푒푖 푖 푒

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휕퐸 푘 푒 = 퐵 푛휈 (푇 − 푇 ) − 푛2휒푇 α 휕푡 훾 − 1 푒푖 푖 푒 푒

Using the first one of the above equations one can see that the first condition

(above) is satisfied when:

휕휈 푘 −(퐸 + 푃 ) > 퐵 푛휈 (푇 − 푇 ) 푖 푖 휕푥 훾 − 1 푒푖 푖 푒

휕휈 Where < 0 in the case of compressive flow and 푇 > 푇 . Here only the 휕푥 푖 푒 compressive part of the enthalpy transport term is considered. The second condition is satisfied when,

푘 퐵 푛휈 (푇 − 푇 ) > 푛2휒푇 α 훾 − 1 푒푖 푖 푒 푒

Let us consider the first condition. To order of magnitude:

푣 푘 (퐸 + 푃 ) > 퐵 푛휈 (푇 − 푇 ) 푖 푖 푙 훾 − 1 푒푖 푖 푒

where 푙 is the characteristic width of the compressed region of the fluid (this could be a few times the smallest resolved length scale/grid cell length in a numerical simulation). Substituting for the energy and the pressure:

5 1 푣 푘 ( 푘 푛푇 + 푚 푛푣2) > 퐵 푛휈 (푇 − 푇 ) 2 퐵 푖 2 푖 푙 훾 − 1 푒푖 푖 푒

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Here, the kinetic energy term on the left-hand side of the above equation is neglected since it is only the thermal energy that is collisionally exchanged between the particle species.

훾 푣 푘 ( 푘 푛푇 ) > 퐵 푛휈 (푇 − 푇 ) 훾 − 1 퐵 푖 푙 훾 − 1 푒푖 푖 푒

푣 휈 (푇 − 푇 ) > 푒푖 푖 푒 푙 훾푇푖

The left- and right-hand sides of equation (6) can then simply be inverted to find the relevant timescales,

푙 훾푇 < 푖 푣 휈푒푖 (푇푖 − 푇푒)

The left-hand side of the equation above is the timescale for the compressive heating (휏푐표푚푝) of the ions and the right-hand side is the timescale for the collisional energy loss (휏푐표푙푙) from the ions. The inequality represented by this equation must be satisfied for the ion fluid to continue acting as a thermal reservoir for the electron fluid.

Now let us consider the second condition,

푘 퐵 푛휈 (푇 − 푇 ) > 푛2휒푇 α 훾 − 1 푒푖 푖 푒 푒

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Or more compactly inverted to,

훾푇푖 푘퐵훾푇푖 < α 휈푒푖(푇푖 − 푇푒) (훾 − 1)푛χ푇푒

The left-hand side of the equation above is the timescale for energy gain by the electrons (the same as the timescale for energy loss by the ions) and the right-hand side is the timescale for the radiative energy loss (휏푟푎푑) from the electrons.

The final hierarchy of timescales (휏푐표푚푝 < 휏푐표푙푙 < 휏푟푎푑) that must be satisfied in order to prevent cooling in the region where the oppositely directed flows interact and, consequently, to avoid the formation of a condensation is then:

푙 훾푇푖 푘퐵훾푇푖 < < α 휈 휈푒푖(푇푖 − 푇푒) (훾 − 1)푛χ푇푒

In the limit 푇푖 ≫ 푇푒, the above equation reduces to:

푙 훾 푘퐵훾푇푖 < < α 휈 휈푒푖 (훾 − 1)푛χ푇푒

For a large electron density resulting from a colliding flow at the apex, n can decrease 휏푟푎푑 substantially and thus cooling and the subsequent formation of a condensation becomes unavoidable.

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Predicted Observed

Density (cm-3) 11 – 12.5 11 – 13

Velocity (km/s) 5 – 20 40 – 100

Peak Apex Intensity (DN) > 10000 < 4000 Table 11: Comparison of synthetic and observed loops for uniform heating.

However, the numerical experiments showed that although such a heating rate could produce a dense, cool mass at the loop top, it could only produce a maximum velocity of the mass no more than 20 km/s which was again far less than the observed velocity (Table 11). To make matters worse, the resulting synthetic intensity derived from the forward model was two orders of magnitude higher than the observed intensities. Evidently, this is due to the exceedingly high volumetric heating rate that uniformly heated the condensation region of high densities and the resulting emission was excessively larger than the observed emission. Hence, the author reached the conclusion that the observed loop brightenings were not due to any uniform heating mechanism.

6.3. Case III: Heating by Loop Braiding

As described in 5.10, in the active TR IRIS has observed loop-like structures with intermittent brightenings which are thought to originate from impulsive heating. These intermittent brightenings are associated with strong non-thermal line

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broadenings which have been interpreted in the published literature as a signature of nanoflares (Testa 2014), turbulence (De Pontieu 2014), plasmoid instabilities

(Innes 2015), and magneto-acoustic shocks (De Pontieu 2015). Most, if not all, of these transient heating mechanisms satisfy the conditions for non-equilibrium ionization. Hence, the author ran an exhaustive search in the parameter space of both footpoint heating and uniform heating of the loop. But the fact that none of the models could simulate all the features of the observed loop(s) told us that the model had to be built bottom up. In other words, the simulations had to become more complex than the previous sets of assumptions and guided closely by observation. To achieve this, lets draw our attention to the Si IV and O IV line (shown in Table 7 and Figure 14 –

21). The following observations were ubiquitous in all eight loops:

• The loops are brightened near the summit (apex) and the lifetime of the

brightenings were 100 – 300 seconds.

• From the density diagnostics of O IV, the brightened pixels exhibited large

densities (1010.5 – 1013 cm-3) than their surroundings.

• At the IRIS pixels where brightenings occur (presumably at the apex), large

broadenings in Si IV line were observed. In addition, the Si IV lines were

neither Gaussian nor Lorentzian, rather a complex shape.

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• By decomposing the Si IV lines, at least two components were found, one blue

shifted and other red shifted.

• Large Non-thermal (NT) line broadenings were also observed in Si IV (and S

IV) lines at the brightened apex in contrast to O IV lines.

The first three characteristics suggested that there were possibly condensations or accumulations of mass in all of these loops. The following three characteristics suggested that there must be some sort of heating mechanism(s) that caused an explosive event which led to bidirectional flow. The heating had a non- thermal component and the effect of heating left the brightening evolving for at least

100 – 300 seconds (For example, see Figure 14 – 21, 23). At first glance these brightenings might seem similar to bomb type events originated deep inside the chromosphere (reported by Peter et al. 2014); however, the fact that the brightenings did not appear isolated in Figure 23, rather they appeared on top of the loops and evolved along the field lines stood them apart. Since, the brightenings were evolving along the field lines and it is well known that magnetic field lines can release bursts of energies (Parker 1988), the natural course of action was to investigate whether energy released by magnetic fields can play role in the loop brightenings.

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Figure 23: Evolution of candidate loop 1 seen by IRIS. The time step is approximately 26 seconds spanning over 154 seconds in total (1 – 6). The diagnostics of this loop is given in Figure 14.

In this section, the author simulated a scenario where heating by magnetic field line braiding and reconnection can be realized and whether such heating can explain the observed brightenings of TR loops in IRIS slit-jaw images and spectral data. If the synthetic spectra agree with the observables, then it would indicate indirect evidence of nanoflare heating and would require further verification by the

EUV channels of the Atmospheric Imaging Assembly (AIA) of the Solar Dynamics

Observatory (SDO). The characterization scheme would then be extended by accumulating time-dependent differential emission measure (DEM) distributions to define the nature of the spatial heating profile and frequency.

First of all, the rapidly evolving, low-lying TR loop brightenings observed by

IRIS are remarkably consistent with previous Hi-C observations of reconnection

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mediated heating of coronal loops (Winebarger 2013). Thus, the author carefully analyzed the IRIS images in search of possible signatures of braiding or overlapping flux tubes. To achieve this, the density distribution of the intensity of the pixels was estimated by realizing a skewed, unimodal histogram of five disjoint categories (or bins). From this, five Gaussian filters were constructed for intensity bands where each filter had the maximum response at the mean of the corresponding bin and a 2σ standard deviation corresponding to the width of the bin. Next, each filter was normalized in reverse so that the dimmer intensity could become more pronounced and comparable to the brightest one. At the end, a new image was reconstructed by overlapping five filtered image frames using a spatial-persistent brightening method

(in each pixel, the maximum intensity among the frames is selected). The filters were applied to each image frame in a similar manner at every timestep.

The new filtered image of each frame was resized by a factor of 5 increment and processed by unsharp masking through applying two step Gaussian Blur. Since, the new image was resized by a factor of 5, thus the standard deviation of the

Gaussian lowpass filter for unsharp masking was set to 5. The strength of the sharpening effect for an edge pixel was set to 3. This value could be increased to introduce sharper contrast, however, an excessive large value for this parameter may create undesirable artifacts in the resulting image.

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Figure 24 shows information unveiled by IRIS on the geometries and the evolution of the loops. Intensity-band filtered unsharp masking revealed the above suspicion that even the simplest bright loop region was composed of multiple strands and their interaction could arise heating. This strongly supports the long-believed idea that bright loops should occur naturally as a result of rapid magnetic energy release (in this case due to the reconnection of braided magnetic field lines). The observed low-lying loops and the associated brightenings occurred in a magnetic field configuration where their footpoints were separated by less than a few pixels (as in our observations) but their apices overlap along the LOS. From this observation, the author came to the conclusion that the rapid temporal evolution of the brightenings at the braiding sites occured by a combination of heating on short timescales and the efficient radiative cooling of the dense plasma.

Next, the presence of strong radial bidirectional flows and a significant non- thermal component in the Si IV and S IV lines (with no such component in O IV lines) provided evidence of heating of heavier ion species. In addition, the image intensity- gradient resolved data of Figure 24 clearly shows that this selective ion heating was taking place at the pixel locations where there is a strong concentration and overlap of sub-resolution strands while simultaneously the Doppler data shows strong radial bidirectional flows for Si IV and S IV (but relative weak for O IV) in the same pixel locations at sub-arcsecond resolution. This represents evidence that heating took

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place at braiding sites where reconnection is a plausible mechanism. Here the power of IRIS data is demonstrated as it captured spectral information in image pixels which was not possible in Hi-C observations (since it was an imaging instrument). As stated above, the spectral observations showed that the lighter O IV ion exhibited no line broadening which reveals clues about the nature of heating mechanism and how ions of different elements were responding to the heating. Based on this observation, the author conjecture that ion cyclotron turbulence might be responsible for the enhanced (non-thermal) broadening: Since the observed broadening was possibly associated with the magnetic reconnection then strong field-aligned currents might exist, which could drive ion cyclotron waves in the plasma. Depending on the ratio of electron and ion temperature, and the ratio of number densities of the different species, each ion species in the multi-ion plasma must exceed its own critical drift velocity to trigger the instability and undergo heating by turbulence (Kindel and

Kennel 1971, Satyanarayana 1985, Forme 1993). In favorable conditions, heavier ions have smaller drift velocities and thus shorter onset times for the instability, and they experience stronger heating.

One important question is whether loop brightenings occur and evolve simultaneously in the IRIS slit-jaw and the energetically stronger EUV lines of AIA images. The time-lag between the light curves at a target location obtained in spectral bands that sample different temperature plasmas reveals this information and tells

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us at a given time whether the observer is witnessing a heating phase or a cooling one

(Viall 2012, 2015; Lionello 2016). Figure 25 and 26 (top) shows two representative loops whose light curves are shown for both the IRIS 1400 channel as well as six AIA

EUV channels (131, 171, 193, 211, 336 and 94 Å). In both examples, the IRIS 1400 channel appeared at least 20 seconds earlier than AIA EUV channels, confirming heating events were being detected. The light curves of the AIA EUV channels peaked concurrently, although the plots showed miniscule delays among them which, in reality, fell inside the temporal limits of the instrument.

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Figure 26: Top: The light curves of candidate loop 2 are shown for both the IRIS 1400 channel as well as six AIA EUV channels (131 Å, 171 Å, 193 Å, 211 Å, 336 Å and 94 Å). Middle: Temporal evolution of the IRIS spectrum and the DEM distribution derived from AIA channels following a similar pattern to candidate 1. Bottom: The temporal evolution of the slope of the coolward DEM and the integrated DEM (Logarithmic) above 14 MK; again showing a similar pattern of evolution.

The properties that are expected of hot, non-flaring plasmas due to impulsive heating in active regions have been investigated using two-fluid hydrodynamic

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models and various metrics proposed to infer properties of the heating mechanism

(Cargill 1994, 1995; Cargill and Klimchuck 2004; Barnes 2016). Among these metrics, the most significant two are: (1) the shallow slope of the DEM distribution coolward of the peak (Cargill and Klimchuck 2004) and (2) the presence of a significant hot shoulder adjacent to the peak emission in the DEM distribution (Cargill 1994, 1995;

Barnes et al. 2016). Because of the impulsive nature of nanoflare heating, the individual strands that comprises the loop undergo many heating and cooling cycles.

Thus, at any given time, there exists small strands emitting at different stages of heating and cooling and the resulting plasma emits at all temperatures from the TR

(105 K) to the corona (107 K). Thus, the DEM distribution is broadened with a hot component of 10 MK or more and the above metrics arise. The heating frequency can be revealed by analyzing the first metric while the second metric might contain the

“smoking gun” evidence for the impulsive heating (e.g. nanoflares). In the present case, the DEM was calculated from the coaligned SDO/AIA channels for two representative loops with respect to time and temperature in Figure 25 and 26

(middle). In both cases, a significant hot shoulder in the DEM distribution appeared momentarily immediately after the IRIS events occurred and decayed quickly, conforming with the first metric. In Figure 25 and 26 (bottom), the coolward DEM slope and the integrated DEM above 14 MK are plotted with respect to time. Although such high temperature EUV observations and the subsequent DEM calculation contain instrument uncertainties, remarkable correlation was witnessed between the

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decrease of coolward DEM slope and the increase in emission measure from the hot component of the DEM at fine temporal resolution (less than a minute). The author proposes this momentary extended tail of “hot shoulder” in the DEM to be the first detection of faint, high-temperature emission due to nanoflare heating in the non- flaring active TR.

Recall the first observational constraint was: the density profile showed a dense plasma concentration (1010 to 1011 cm-3) at the top of the loop. In the HYDRAD runs, the first criterion was achieved by gently heating the footpoint of the loops with a train of intermittent nanoflares. Once a condensation of the observed plasma density was reached, a new train of impulsive heating was introduced at the condensation. This satisfies the second condition by producing a strong velocity component near the heating site, emulating the observed non-thermal velocity shown in Figures 14 – 21. For the third constraint to be satisfied, one must carefully consider the ionization states and the population distribution when predicting the synthetic spectrum. The temporal heating profile that was applied to of one of the loops to generate the synthetic observation is given below in Figure 27:

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Figure 27 : Temporal heating profile of one of the loops that was applied to generate the synthetic observation in HYDRAD.

The simulated loop had a length of 10 Mm along with each footpoint having

2.2 Mm buried in the chromosphere resulting the total length to be 14.4 Mm. The spatial profile of the heating was a Gaussian distribution spread across the field line.

In the above example, the footpoints of the loop was first heated by 0.018 erg/cm3/s having a broad Gaussian with 푠0 = 5.8 Mm and 푠퐻 = 1 Mm so that a condensation can be formed at the apex of the loop. Once, a condensation is formed, the apex was heated

3 by 0.3 erg/cm /s having a narrow 푠퐻 = 0.2 Mm. Here, an asymmetry of 10% in the heating magnitude between the footpoints were introduced. In addition, the apex heating was not applied at 퐿/2, rather a spatial asymmetry of 3% were introduced

(푠0 = 퐿/2 + 0.03퐿) in order to recreate a realistic image of loop brightening. These spatial asymmetries dictated the direction of the fall of mass concentration. To obtain

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draining at both foot-points of the loop, as seen in observation, the author simulated multiple loops together with a slightly different 푠0 = 퐿/2 ± 0.03퐿 and let them evolve independently. The resulting loop dynamics showed mass concentration predominantly draining through one foot-point while a relatively weak flow was found at the other foot-point consistent with the observations.

The evolution of loop temperature, density and velocity along the loop axis is shown in Figure 28. During the first 50 seconds the loop evolved in a steady state with

5 10 −3 a temperature below 10 K and a condensation of 푛푎푝푒푥 = 10 푐푚 . At earlier times, the spatio-temporal evolution of ∆푇 = 푇푒 − 푇푖 showed the signature of gentle

3 footpoint heating (0.018 erg/cm /s with 푠0 = 5.8 Mm and 푠퐻 = 1 Mm) of the strand.

Once a steady condensation was achieved at 푡 = 190 푠, a short, isolated, nanoflare of

0.3 erg/cm3/s was applied for 4 seconds to emulate possible energy release by reconnection. The heating increased the temperature of the surrounding plasma to coronal temperatures, and a large temperature difference between electrons and ions arose. When the observed non-thermal line-broadening was inserted into the forward modeling code, then the synthetic spectrum reproduced both the magnitude of intensity and the observed radial bi-directional flow of 100 km/s. By initiating a train of nanoflares, one can sustain the bright non-thermal velocity component for the observed period of time. Once the heating was turned off, the hot mass content fell either side of the loop to the chromosphere.

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The evolution of the intensity and Si/O peak ratio at the apex, synthesized by the forward modeling code, was also presented in Figure 29. Here, multiple strands, having slightly different but similar apex heating profiles, were combined to produce a synthetic spectrum. Significant differences in both the magnitude and form of the temporal profiles of the observables were predicted for different ionization statistics.

The Si IV emission calculated for non-equilibrium ionization generated a stronger emission profile, while exhibiting a large Si/O peak ratio. The results for non- equilibrium ionization were in quantitatively better agreement with the corresponding observation in terms of intensity, density and Si/O peak ratio (Table

12).

Density Velocity Intensity Si/O Peak −3 퐿표𝑔 푛푒(푐푚 ) (푘푚/푠) (퐷푁/푠) Ratio Observation from 10.5 – 13 83 – 100 180 – 700 16 – 70 IRIS Equilibrium 10.5 – 11 >100 68 2 Low Density Non-Equilibrium 10.5 – 11 >100 80 3 Low Density Equilibrium 9.8 – 10.5 >100 200 14 Density dependent Non-Equilibrium 10 – 10.7 >100 400 40 Density Dependent Table 12: Comparison between observation and modeling under different assumptions of ionization states and dielectronic recombination rates.

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The reasons for the stronger Si emission are due to the timescale of heating (in this case, 0.3 erg/cm3/s for 4 seconds) and the adoption of density-dependent dielectronic recombination rates. The short heating timescale drove the ionization state out-of-equilibrium and favored an enhanced population of Si IV. During the cooling phase, the Si IV population persisted for an extended period due to the density-dependent quenching of dielectronic recombination and thus leading to stronger emission. The differences between the low-density limit and the high- density limit accounting for the density dependent dielectronic recombination rates for both equilibrium (1st row) and non-equilibrium (2nd row) ionization are shown below in Figure 30. Note that, since most of the density diagnostic techniques assume that the emitting plasma is thermal and in ionization equilibrium, which clearly is not this case. Thus, the author extracted the synthetic spectra from the forward modeled intensity profile and stored the O IV line ratio as synthetic observations. This synthetic observation is interpolated with the theoretical intensity ratio obtained by

CHIANTI and the synthetic density diagnostics was derived for each pixel, at each time step. Thus, the author preserved consistency in density diagnostics between real observation and hydrodynamic simulation.

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Accounting for the density dependent dielectronic recombination rates:

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Unlike the other heating mechanism, this assumption of nanoflare mediated heating led HYDRAD to generate synthetic observations from numerical experiments that significantly corresponded to the real data. To achieve this, Non-equilibrium ionization had to be included in the computation of synthetic spectra. In addition, the relatively high-density TR plasma required the inclusion of density-dependent dielectronic recombination rates to calculate the ion populations and the emission line intensities. The author showed that the observations and the numerical experiments are consistent with reconnection mediated impulsive heating at the braiding sites of multi-stranded TR loops (Table 13).

Observation from IRIS Predicted

High Density at Apex 1010.5 – 1013 cm-3 1010 – 1011 cm-3

Large Si IV/O IV ratio 16 – 70 40

Bidirectional Flow 40 – 100 km/s 100 km/s

Apex Intensity 180 – 700 DN/s 400 DN/s

Time of Loop Evolution 120 – 400 seconds 350 seconds Table 13: Comparison between observation and prediction for loop heating due to magnetic field braiding and reconnection.

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6.4. Case IV: Heating in Underlying Atmosphere

The previous section presents compelling evidence that the braiding of magnetic fields and reconnection is the source of heating in the TR. However, the author intends to investigate the physical location of the reconnection sites, since reconnection is ubiquitous in both the chromosphere and the TR. Recent observations of the solar chromosphere have suggested that reconnection is occurring there and driving localized, transient outflows on a number of different length scales. The largest class of such outflows is called “chromospheric jets”

(Shibata et al. 2007). These are surges of plasma, observed in Hα and Ca II, with typical lifetimes of 200–1000 s, lengths of 5 Mm, and velocities at their base of 10 kms-

1. A smaller class of these outflows are called “spicules” (Sterling 2000), and are mainly observed on the solar limb (Sekse et al. 2012). Spicules have lifetimes of 10–

600 s, lengths of up to 1 Mm and velocities of 20–150 kms-1. The pervasive existence of such impulsive events indicates that chromospheric heating sources have the capacity to produce the observed IRIS spectra and must be critically examined.

6.4.1. Magneto-acoustic Shocks from Chromosphere

Hansteen et al. (2006) and De Pontieu et al. (2007a) demonstrated that some

TR loop brightenings are formed by chromospheric shocks that occur when waves caused by convective flows or global oscillations leak into the chromosphere along

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inclined magnetic field lines. Both of them employed advanced numerical radiative magneto-hydrodynamical simulations to validate their claim. Such a mechanism was previously suggested by Michalitsanos (1973); Bel & Leroy (1977); Suematsu (1990) and De Pontieu et al. (2004, 2005). This work was later supported Heggland et al.

(2007, 2011) who studied shockwave driven jets in more detail and found that correlations between velocity and deceleration of the jets in their numerical simulations matched well with TR loop brightenings, which supported the view that

TR loops are driven by magneto-acoustic shocks. Later Martínez-Sykora et al. (2009) studied spicules in a 3D numerical simulation that covered the solar atmosphere from the upper layers of the convection zone to the lower corona. These simulated spicules described parabolic trajectories similar to TR loops and were found to be driven by a variety of mechanisms that include p-modes, collapsing granules, and magnetic reconnection in the lower chromosphere etc.

In a seminal paper, Langangen 2008 studied TR loops spectroscopically and identified their characteristic shock signature in the spectral evolution (휆푡– diagrams): a sudden (strong) blueshift of the line, followed by a gradual (linear) shift of the line to redshifts of typically 15 푘푚푠−1. Similar transverse motion of TR loops in

Hα was studied by Koza et al. (2007) and a rebound shock mechanism for fibrils and loops was presented by Sterling & Hollweg (1989). In 2014, Tian et al. and later De

Pontieu et al. 2015 reported spectral observations of the solar TR and corona showing

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the broadening of spectral lines beyond what is expected from thermal and instrumental broadening. These results are shown in Figure 31 and 32.

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Figure 32: 흀 − 풕 diagram of IRIS Si IV (left) and Mg II h (right) line (De Pontieu et al. 2015). The first pair of diagrams are for a location at a plage. The second pair are observations of a coronal hole region. Overplotted in red and white is the evolution of the non-thermal line broadening as derived from a single Gaussian fit to the Si IV line. The non-thermal line broadening scale is shown on the top axis. Notice the correlation between shock occurrence (swing from red to blue) in the Mg II h line and the increased line broadening and brightness in Si IV.

The remaining non-thermal broadening was significant (5 − 30 푘푚푠−1) and correlated with intensity. They study spectra of the Si IV 1403 Å line obtained at high resolution with the IRIS and found that the non-thermal line broadening which, in most regions, remains at about 20 푘푚푠−1. Their detailed comparison with IRIS chromospheric observations showed that, in regions where the LOS was more

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parallel to the field, magneto-acoustic shocks driven from below impacted the TR and could lead to significant non-thermal line broadening. The authors were able to recreate the observed scenario in MHD simulations. In addition, they showed that the correlation could be caused by the shocks, but only if non-equilibrium ionization was taken into account. As shown in Figure 33 the MHD simulated shocks led to increased nonthermal line broadening in the TR at the time of the shock passage. However, this was more evident in the simulation assuming non-equilibrium ionization: the broadening was increased by 2 − 10푘푚푠−1 during shock passage and by several kms−1 throughout the simulation. More importantly, the ionization equilibrium simulation did not reproduce the correlation between non-thermal line broadening and the logarithm of the intensity. That correlation only appeared when non- equilibrium ionization is included in the simulations. This phenomenon was explained by stating the fact that non-equilibrium ionization led to the presence of

Si3+ ions over a much wider range of temperatures than under ionization equilibrium

(also Olluri et al. 2013). In such a case, the line formation region more easily captures both the pre- and post-shock environment. This naturally leads to a larger range of velocities along the LOS of the optically thin Si IV lines and thus non-thermal line broadening. Since the observations were made in Si IV channels, the sensitivity to the shocks in the Si IV emission would correlate the intensity with the non-thermal line broadening.

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In order to extract evidence of such magneto-acoustic shock mediated heating, the author collected IRIS data for the Mg II -h and -k chromospheric lines in parallel to the Si IV TR line. These resonance lines are the 3푠 − 3푝 transitions to the ground state of singly ionized magnesium from upper levels that are close in energy. De

Pontieu et al. (2015) showed characteristic line shifts and broadening in these IRIS channels of chromospheric lines. They also showed that the line shifts and the broadenings were concurrent with the Si IV channels confirming the heating from the underlying chromospheric shock.

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However, in case of the TR loop brightenings observed by the author of this thesis, no such characteristics of chromospheric shock assisted heating were observed as shown in Figure 34 – 41. Thus, it is evident that the observed loop brightenings were not due to magneto-acoustic shocks originating from the underlying chromosphere.

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6.4.2. Self-Absorption Lines from Chromospheric Reconnection

In general, the emission originating in the TR as well as from the corona itself is considered to be optically thin for emission lines observed on the Sun. Mostly, the low density of the plasma in the upper atmosphere allows each emitted photon to escape without further scattering. There are some exceptions, though. In particular, strong resonance lines have been reported to show effects of optical depth through the analysis of line ratios. The resonance lines of Li-like ions (such as C IV at 1548 and

1550 Å) should have a line ratio of about 2 under optically thin conditions (according to their Einstein A coefficients). However, when observed toward the limb of the Sun and a longer LOS, an increased non-thermal width and a reduced line ratio indicate that optical depth effects become important (Mariska 1992; Peter 1999) and may hold information of heating activities in the underlying atmosphere.

If such self-absorption features could be found, they would give important information (e. g., spatial location of heating) on the loop-like structures of the solar atmosphere. Concerning the TR emission, the existence or absence of self-absorption features may provide an estimate of depth of the thermal structure and how it interacts in different part of the atmosphere along the LOS. Traditionally, it was often suggested that the TR is energetically disconnected; that is, its physical state derives from local processes, such as short-range electric currents, magnetic reconnection, small jets etc. However, past observations of explosive events (Dere et al. 1989; Innes

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et al. 1997), blinkers (Harrison et al. 1999; Peter & Brković 2003), MHD shocks

(Ryutova & Tarbell 2003) and recent IRIS observation of pockets of hot gas in the cool chromosphere questioned this paradigm (Peter et al. 2014), along with the observation of self-absorption features confirmed that the TR is energetically connected to its chromospheric and coronal boundaries.

Yan et al. 2015 presented observations from IRIS showing self-absorption features in a TR line. In recent observations, such tilt and dips of spectral lines had been attributed to torsional or rotational motions (Curdt & Tian 2011; Curdt et al.

2012; De Pontieu et al. 2014a; Li et al. 2014). However, the velocities Yan et al. 2015 observed (Figure 42) were exceeding 100 km s−1, sometimes up to 250 km s−1; hence, significantly larger than reported before. The fact that these velocities were observed to be close to or even exceeding the Alfvén speed questions the interpretation of a torsional motion in their observations. Instead, Yan et al. 2015 provided evidence that the outflow had the nature of a bidirectional jet. The interpretation for the existence of bidirectional jets in TR lines is either explosive events (e.g., Dere et al. 1991; Innes et al. 1997) or an indication of more recently reported Ellerman-bomb-type events

(Peter et al. 2014). It is assumed that strong reconnection outflows are taking place in the underlying atmosphere (e.g. the chromosphere) and leading to the strong brightening and broadening of the Si IV line profile along with the presence of chromospheric absorption lines and self-absorption features in TR lines.

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In Figure 43, a cartoon summary of a possible scenario is given from Yan et al.

2015. At a height where the middle chromosphere would be located, a reconnection event gives rise to the increase in intensity of Si IV through heating of the previously cool chromospheric plasma. At the same time the multiple flows created by the reconnection cause a very strong broadening (Figure 43 – top). Above the location of the heating and the origin of the strong Si IV emission the pre-existing upper atmosphere is still present. The thin upper chromospheric layer above will give rise to the TR absorption lines such as Ni II and Fe II (Figure 43 – middle). Yan et al. 2015

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presented chromospheric absorption features and the high density derived from line ratios as strong arguments to place the origin of the strong Si IV emission in the middle chromosphere in the first place. Now, the TR above the event contains a sufficient amount of Si IV that can cause the self-absorption of Si IV lines too. The existence of the self-absorption lines was found in the 1400 Å slit-jaw-images of Yan et al. (2015) verifying the possible scenario.

Figure 43: Cartoon illustrating a possible scenario for the formation of the self- absorption features in Si IV (Yan et al. 2015) due to reconnection in the deep chromosphere.

Inspired by these investigations, the author of the thesis inspected the observed IRIS data pixel-by-pixel to see whether there existed any Si IV self- absorption lines (S I 1401.515 Å and Fe II 1401.774 Å, 1403.101 Å, 1403.255 Å from

Peter et al. 2014) that can be tied to a chromospheric reconnection mechanism.

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However, both single pixel array analysis and spatially-averaged analysis of the loop candidates did not return any significant self-absorption features (Figure 44 – 51) except for loop candidate 7, for which the self-absorption features were not particularly strong. This suggested that the reconnection mediated heating, that might be responsible for the TR loop brightening under investigation, was not occurring in the deep chromosphere. 172

Figure 44: Single pixel array analysis (averaged over y-axis) and spatially-averaged analysis (averaged over x–y area) of candidate Loop 1 insearch of self-absorption lines. 173

Figure 45: Single pixel array analysis (averaged over y-axis) and spatially-averaged analysis (averaged over x–y area) of candidate Loop 2 insearch of self-absorption lines. 174

Figure 46: Single pixel array analysis (averaged over y-axis) and spatially-averaged analysis (averaged over x–y area) of candidate Loop 3 insearch of self-absorption lines. 175

Figure 47: Single pixel array analysis (averaged over y-axis) and spatially-averaged analysis (averaged over x–y area) of candidate Loop 4 insearch of self-absorption lines. 176

Figure 48: Single pixel array analysis (averaged over y-axis) and spatially-averaged analysis (averaged over x–y area) of candidate Loop 5 in search of self-absorption lines. 177

Figure 49: Single pixel array analysis (averaged over y-axis) and spatially-averaged analysis (averaged over x–y area) of candidate Loop 6 insearch of self-absorption lines. 178

Figure 50: Single pixel array analysis (averaged over y-axis) and spatially-averaged analysis (averaged over x–y area) of candidate Loop 7 insearch of self-absorption lines. 179

Figure 51: Single pixel array analysis (averaged over y-axis) and spatially-averaged analysis (averaged over x–y area) of candidate Loop 8 insearch of self-absorption lines.

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Chapter 7

Conclusion and Future Work

A set of observations that provided evidence of impulsive heating by braiding and magnetic reconnection is discussed, although subject to the usual instrument limitations. Nonetheless, the co-aligned observations at unprecedented spatiotemporal scale were sufficiently convincing to make a plausible claim that the

TR is powered by magnetic reconnection causing trains of nanoflares. Although there have been a plethora of differential emission measure studies of coronal loops, the author believes that this analysis on the TR loops provides a compelling reason to consider these transient, low-lying structures as forensic evidence in search of the intermittent heat sources. Under non-equilibrium ionization conditions, the numerical experiments described above found that the treatment of atomic processes, particularly, density-dependence in dielectronic recombination rates

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influenced the intensity peaks of the emitting ions dramatically and must be accounted for. More importantly, the author is the first to identify the high Si IV/O IV ratio in the IRIS data as a diagnostic tool to characterize the ionization population of these impulsive heating. The analysis offers quantifiable and distinct constraints in order to model TR plasma out-of-equilibrium.

The study on how the interface region of Sun’s atmosphere contributes to mass and energy transport from the chromosphere to the corona is crucial to the scientific return of the IRIS mission. Since this thesis focuses on the analysis and interpretation of IRIS data to reveal the contribution of the equilibrium and non- equilibrium ionization in the fine-structures lying at TR, this study evidently maximizes the scientific return in full. Furthermore, the research addresses one of the science objectives of the Solar and Space Physics Decadal Survey: “Determine the origins of the Sun’s activity and predict the variations in the space environment” (The

2013-2022 Decadal Survey in Solar and Space Physics). Finally, because the ionization of plasma is a fundamental process in many areas of astrophysics and cosmology, it also covers the Science Objective 4: “Discover and characterize fundamental processes that occur both within the heliosphere and throughout the universe” of the Solar and Space Physics Decadal Survey. Hence, the author believes this research will mark an enduring impact on space science society for days to come.

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Lastly, understanding our own star better will help deepen our insight into atmospheres at distant stars as well.

7.1. Magnetic Reconnection and Ion Heating Mechanisms

When reconnection occurs along with large aspect-ratio current sheets, structures that appear to look like current sheets with embedded plasmoids have been seen, for example, behind coronal mass ejections. However, in case of nanoflares the one the author investigated, small-scale (∼5000 km), short-lived (1–5 minutes) reconnection sites identified by broad non-Gaussian TR line profiles, it is difficult to imaging and disentangling the dynamics of the reconnection process. Here, ions of various species participate in a competing environment where various shocks and acceleration take place. However, the information of the ion dynamics is embedded in the spatiotemporal spectroscopic emission. In this thesis, the author presented an idea that links the broadening of Si IV along with a bi-directional flow of heavier ions with the possible indication of ion-cyclotron instability at the reconnection site. Such observation needs to be extended to multi-species multi-charge state analysis and quantified by numerical experiments satisfying all the observed constraints. As for an example, it was thought that much of the acceleration and heating in reconnection occurs, as proposed by Petschek (Petschek 1964), along with shocks attached to the reconnection region, so that most of the plasma at the reconnection site moves with

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the Alfvén speed. In the realm of theory, there has been a clear consensus that the

Petschek mechanism does not hold unless the resistivity of the plasma is enhanced locally at the X-point. Despite this theoretical reservation about the Petschek theory, there has been a general tendency to continue comparing observations with the theory. However, recent efforts showed that it is possible that the breakup of current sheets into magnetic islands can be taken place independent of Lundquist number leading to fast reconnection mechanisms. Such a mechanism, also known as plasmoid instability (Innes 2015), should exhibit distinct non-Gaussian line profile corresponding to density and velocity structures that can be employed to characterize the nature of the mechanism. The author intends to pursue such an investigation in the future.

7.2. Non-Maxwellian Velocity Distributions

It has long been known that a high-energy tail to the electron velocity distribution is likely to play a significant role in the plasma physics of the solar TR.

Shoub (1982, 1983) solved the Landau-Fokker-Planck equation, which allows for the buildup of an electric field in an inhomogeneous plasma and showed that the solutions for the electron velocity distributions in the TR could deviate from

Maxwellian.

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MacNeice (1986) investigated the deviations from Maxwellian of the distribution function and the effect upon the plasma temperature and density for a compact flare simulation and found a difference between the estimates of the temperature were only about 3% and the differences in the densities only 1%. The steep gradients encountered during solar flares did not cause a substantial departure from a Maxwellian velocity distribution because the high densities ensured that the plasma remained collisional to an excellent approximation.

The consequences of non-Maxwellian velocity distribution in the TR with regard to theory and observations are two-fold (Holman et al. 1992, 2011). First of all, the Spitzer-Harm formulation of the heat flux employed in almost all hydrodynamic models to-date needs to be revised because it was derived under the assumption that departures from Maxwellian are small. A non-Maxwellian tail to the electron velocity distribution could contribute significantly to the heat flux (a higher moment of the distribution function). The second important consequence of a non-

Maxwellian tail to the electron velocity distribution affects the collisional ionization rates and may significantly alter the ion populations in the plasma and the emission spectrum. In equilibrium the ionization threshold for most ions is in the tail of the energy distribution function, so a non-Maxwellian component to the distribution could make a substantial contribution to the ionization rate. Therefore, ions could be formed at temperatures much lower than those suggested by a Maxwellian

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distribution and would compound the tendency already noted by the author for non- equilibrium effects to enhance the populations of higher ionization stages. The collisional excitation rates for some transition may also be affected, which would further alter the intensities of the corresponding emission lines, with consequences for spectroscopically derived observational results and estimates of the overall radiative emission employed by theoretical models.

Most recently, Dzifcáková and Kulinová (2011) proved the occurrence of the non-thermal κ-distribution in the solar TR and diagnose its parameters. They found that the intensity ratios of Si III lines observed by SUMER in 1100–1320 Å region do not correspond to the line ratios computed under the assumption of the Maxwellian electron distribution. They computed a set of synthetic Si iii spectra for the electron

κ-distributions with different values of the parameter κ. We had to include the radiation field in our calculations to explain the observed line ratios. The inclusion of

κ-distribution facilitates the authors to able to explain the observed Si III line spectrum in the TR. The degree of non-thermality increases with the activity of the solar region, it is lower for CH and higher for the AR. After the launch of IRIS, Dudík et al. (2014) investigated the formation of the TR O IV and Si IV lines observable by

IRIS for both Maxwellian and non-Maxwellian conditions characterized by a κ- distribution exhibiting a high-energy tail. They found that the Si IV lines are formed at lower temperatures than the O IV lines for all κ. In addition, it was observed that in

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non-Maxwellian situations with lower κ, the contribution functions are shifted to lower temperatures. Combined with the slope of the differential emission measure, it was found to be possible for the Si IV lines to be formed at very different regions of the solar TR than the O IV lines; possibly close to the solar chromosphere. By combining these investigations that have previously been carried out, it is intended to incorporate a method of calculating a more suitable distribution function for the electrons in the solar TR, which will take into account its non-Maxwellian nature.

7.3. Multi-wavelength Multi-Instrument Data-driven Analysis

Since IRIS acquires approximately eight Gigabytes of data every day (De

Pontieu 2014), the most efficient way to utilize the full potential of IRIS observational data to unveil the subtle trends and patterns in structures of interest would be to employ a feature-detecting, parameter extraction algorithm. Such a comprehensive search founded on state-of-the-art machine learning algorithms will ensure consistency and establish (or otherwise) the universality of any heating properties that would be identified. Hence, it is expected to reveal information concerning the variety of processes occurring in these structures and provide constraints on the parameter space the heating properties must occupy.

Last but not the least, along with IRIS and SDO/AIA, data from EIS, HINODE, and SDO/HMI along with ground-based telescopes on chromospheric absorption

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lines are needed to be assimilated into the models. In addition, extensive use of the forthcoming explorer missions namely the Parker Solar Probe and the Solar Orbiter should be employed to characterize the energetic particles of these regions. Such a multi-wavelength, multi-instrument analysis will facilitate the constraining of numerical model. Once, given an estimate of the energy budget of the underlying physical processes of the loops is assembled, the study may include a calculation of the reduced χ2 parameter, which accounts for the number of degrees of freedom and can be used to derive a probability giving a quantitative measure for the goodness-of- fit. Such an extension of study will provide an extremely robust indication of whether the contemporary models are compatible with the physics and will validate the basis for our understanding of the energy balance in these structures.

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References

"Solar and Space Physics: A Science for a Technological Society", The 2013-2022

Decadal Survey in Solar and Space Physics, Space Studies Board, Division on

Engineering & Physical Sciences, August 2012.

Acton, L., et al. "The Yohkoh mission for high-energy solar physics." Science 258.5082

(1992): 618-625.

Anders, Edward, and Nicolas Grevesse. "Abundances of the elements: Meteoritic and solar." Geochimica et Cosmochimica acta 53.1 (1989): 197-214.

Antolin, Patrick, and Tom Van Doorsselaere. "Line-of-sight geometrical and instrumental resolution effects on intensity perturbations by sausage modes."

Astronomy & Astrophysics 555 (2013): A74.

Antolin, Patrick, Takaaki Yokoyama, and Tom Van Doorsselaere. "Fine strand-like structure in the solar corona from magnetohydrodynamic transverse oscillations."

The Astrophysical Journal Letters 787.2 (2014): L22.

Aschwanden, Markus J. "Solar stereoscopy and tomography." Living Reviews in Solar

Physics 8.1 (2011a): 5.

190

Aschwanden, Markus J., and Paul Boerner. "Solar corona loop studies with the

Atmospheric Imaging Assembly. I. Cross-sectional temperature structure." The

Astrophysical Journal 732.2 (2011b): 81.

Aschwanden, Markus J., Arthur I. Poland, and Douglas M. Rabin. "The new solar corona." Annual Review of Astronomy and Astrophysics 39.1 (2001): 175-210.

Aschwanden, Markus. Physics of the solar corona: an introduction with problems and solutions. Springer Science & Business Media, 2006.

Asplund, Martin, et al. "The chemical composition of the Sun." Annual review of astronomy and astrophysics 47 (2009): 481-522.

Asplund, Martin. "New light on stellar abundance analyses: departures from LTE and homogeneity." Annu. Rev. Astron. Astrophys. 43 (2005): 481-530.

Bel, N., and B. Leroy. "Analytical study of magnetoacoustic gravity waves." Astronomy and Astrophysics 55 (1977): 239.

Bertero, M., C. De Mol, and El R. Pike. "Linear inverse problems with discrete data. I.

General formulation and singular system analysis." Inverse problems 1.4 (1985): 301.

Bingert, Sven, and Hardi Peter. "Intermittent heating in the solar corona employing a

3D MHD model." Astronomy & Astrophysics 530 (2011): A112.

191

Bonanno, A., H. Schlattl, and L. Paterno. "The age of the Sun and the relativistic corrections in the EOS." Astronomy & Astrophysics 390.3 (2002): 1115-1118.

Bouvier, Audrey, and Meenakshi Wadhwa. "The age of the Solar System redefined by the oldest Pb–Pb age of a meteoritic inclusion." Nature geoscience 3.9 (2010): 637.

Bradshaw, S. J. "A numerical tool for the calculation of non-equilibrium ionisation states in the solar corona and other astrophysical plasma environments." Astronomy

& Astrophysics 502.1 (2009): 409-418.

Bradshaw, S. J., and H. E. Mason. "A self-consistent treatment of radiation in coronal loop modeling." Astronomy & Astrophysics 401.2 (2003b): 699-709.

Bradshaw, S. J., and P. J. Cargill. "A new enthalpy-based approach to the transition region in an impulsively heated corona." The Astrophysical Journal Letters 710.1

(2010): L39.

Bradshaw, S. J., and P. J. Cargill. "The cooling of coronal plasmas-II. Properties of the radiative phase." Astronomy & Astrophysics 437.1 (2005): 311-317.

Bradshaw, Stephen J. A Coupled Hydrodynamic and Optically-Thin Radiative

Emission Model for the Solar Atmosphere. 2003a. Cambridge U, PhD dissertation.

192

Bradshaw, Stephen J., and James A. Klimchuk. "What dominates the coronal emission spectrum during the cycle of impulsive heating and cooling?." The Astrophysical

Journal Supplement Series 194.2 (2011): 26.

Bradshaw, Stephen J., and Paola Testa. "Quantifying the Influence of Key Physical

Processes on the Formation of Emission Lines Observed by IRIS: I. Non-Equilibrium

Ionization and Density-Dependent Rates." arXiv preprint arXiv:1901.03935 (2019).

Bradshaw, Stephen J., and Peter J. Cargill. "The influence of numerical resolution on coronal density in hydrodynamic models of impulsive heating." The Astrophysical

Journal 770.1 (2013): 12.

Brković, A., and Hardi Peter. "Relation of TR blinkers to the low chromosphere."

Astronomy & Astrophysics 406.1 (2003): 363-371.

Brosius, Jeffrey W., et al. "Measuring active and quiet-Sun coronal plasma properties with extreme-ultraviolet spectra from SERTS." The Astrophysical Journal Supplement

Series 106 (1996): 143.

Burgess, A., et al. "Cross-sections for ionization of positive ions by electron impact."

Monthly Notices of the Royal Astronomical Society 179.2 (1977): 275-292.

193

Burgess, Alan. "A General Formula for the Estimation of dielectronic Recombination

Co-Efficients in Low-Density Plasmas." The Astrophysical Journal 141 (1965): 1588-

1590.

Burgess, Alan. "Delectronic Recombination and the Temperature of the Solar Corona."

The Astrophysical Journal 139 (1964): 776-780.

Caffe, M. W., et al. "Evidence in meteorites for an active early Sun." The Astrophysical

Journal 313 (1987): L31-L35.

Cargill, Peter J. "Some implications of the nanoflare concept." The Astrophysical

Journal 422 (1994): 381-393.

Cargill, Peter J., and James A. Klimchuk. "Nanoflare heating of the corona revisited."

The Astrophysical Journal 605.2 (2004): 911.

Cargill, Peter J., John T. Mariska, and Spiro K. Antiochos. "Cooling of solar flares plasmas. 1: Theoretical considerations." The Astrophysical Journal 439 (1995): 1034-

1043.

Carlsson, Mats, and Jorrit Leenaarts. "Approximations for radiative cooling and heating in the solar chromosphere." Astronomy & Astrophysics 539 (2012): A39.

Charbonneau, Paul. "Solar dynamo theory." Annual Review of Astronomy and

Astrophysics 52 (2014): 251-290.

194

Cheng, C-C., et al. "Spatial and temporal structures of impulsive bursts from solar flares observed in UV and hard X-rays." The Astrophysical Journal 248 (1981): L39-

L43.

Cirtain, J. W., et al. "Evidence for Alfvén waves in solar X-ray jets." Science 318.5856

(2007): 1580-1582.

Connelly, James N., et al. "The absolute chronology and thermal processing of solids in the solar protoplanetary disk." Science 338.6107 (2012): 651-655.

Craig, I. J. D. "The use of a-priori information in the derivation of temperature structures from X-ray spectra." Astronomy and Astrophysics 61 (1977): 575-590.

Craig, Ian JD, and John C. Brown. "Inverse problems in astronomy: a guide to inversion strategies for remotely sensed data." Research supported by SERC. Bristol, England and Boston, MA, Adam Hilger, Ltd., 1986, 159 p. (1986).

Curdt, Werner, and Hui Tian. "Spectroscopic evidence for helicity in explosive events."

Astronomy & Astrophysics 532 (2011): L9.

Curdt, Werner, Hui Tian, and Suguru Kamio. "Explosive events: swirling TR jets." Solar

Physics 280.2 (2012): 417-424.

De Moortel, Ineke, and Philippa Browning. "Recent advances in coronal heating." Phil.

Trans. R. Soc. A 373.2042 (2015): 20140269.

195

De Pontieu, B., et al. "Chromospheric Alfvénic waves strong enough to power the solar wind." Science 318.5856 (2007): 1574-1577.

De Pontieu, B., R. Erdélyi, and I. De Moortel. "How to channel photospheric oscillations into the corona." The Astrophysical Journal Letters 624.1 (2005): L61.

De Pontieu, Bart, et al. "High-resolution observations and modeling of dynamic fibrils." The Astrophysical Journal 655.1 (2007): 624.

De Pontieu, Bart, et al. "On the prevalence of small-scale twist in the solar chromosphere and TR." Science 346.6207 (2014): 1255732.

De Pontieu, Bart, et al. "The interface region imaging spectrograph (IRIS)." Solar

Physics 289.7 (2014): 2733-2779.

De Pontieu, Bart, et al. "Why is non-thermal line broadening of spectral lines in the lower TR of the Sun independent of spatial resolution?." The Astrophysical Journal

Letters 799.1 (2015): L12.

De Pontieu, Bart, Robert Erdélyi, and Stewart P. James. "Solar chromospheric spicules from the leakage of photospheric oscillations and flows." Nature 430.6999 (2004):

536.

Del Zanna, G., B. O’Dwyer, and H. E. Mason. "SDO AIA and Hinode EIS observations of

“warm” loops." Astronomy & Astrophysics 535 (2011): A46.

196

Dere, K. P., et al. "CHIANTI–an atomic database for emission lines-IX. Ionization rates, recombination rates, ionization equilibria for the elements hydrogen through zinc and updated atomic data." Astronomy & Astrophysics 498.3 (2009): 915-929.

Dere, K. P., et al. "Explosive events and magnetic reconnection in the solar atmosphere." Journal of Geophysical Research: Space Physics 96.A6 (1991): 9399-

9407.

Dere, K. P., J-DF Bartoe, and G. E. Brueckner. "Explosive events in the solar transition zone." Solar physics 123.1 (1989): 41-68.

Domingo, V., B. Fleck, and Arthur I. Poland. "The SOHO mission: an overview." Solar

Physics 162.1-2 (1995): 1-37.

Doschek, G. A., H. P. Warren, and P. R. Young. "The electron density in explosive transition region events observed by IRIS." The Astrophysical Journal 832.1 (2016):

77.

Doschek, G. A., U. Feldman, and J. D. Bohlin. "Doppler wavelength shifts of transition zone lines measured in SKYLAB solar spectra." The Astrophysical Journal 205 (1976):

L177-L180.

197

Doyle, J. G., et al. "Diagnosing transient ionization in dynamic events." Astronomy &

Astrophysics 557 (2013): L9.

Dudík, Jaroslav, et al. "Slipping magnetic reconnection, chromospheric evaporation, implosion, and precursors in the 2014 September 10 X1. 6-Class Solar Flare." The

Astrophysical Journal 823.1 (2016): 41.

Dudík, Jaroslav, et al. "Solar TR lines observed by the Interface Region Imaging

Spectrograph: Diagnostics for the O IV and Si IV lines." The Astrophysical Journal

Letters 780.1 (2013): L12.

Dzifčáková, E., and A. Kulinová. "Diagnostics of the κ-distribution using Si III lines in the solar TR." Astronomy & Astrophysics 531 (2011): A122.

Emslie, A. G., and J. A. Miller. "Particle acceleration." Dynamic Sun (2003): 262-287.

Feldman, U. "On the unresolved fine structures of the solar atmosphere in the 30,000-

200,000 K temperature region." The Astrophysical Journal 275 (1983): 367-373.

Fletcher, Lyndsay, et al. "An observational overview of solar flares." Space science reviews 159.1-4 (2011): 19.

Forme, F. R. E., et al. "Effects of current driven instabilities on the ion and electron temperatures in the topside ionosphere." Journal of atmospheric and terrestrial physics 55.4-5 (1993): 647-666.

198

Fossi, BC Monsignori, and M. Landini. "Models for inner corona parameters."

Advances in Space Research 11.1 (1991): 281-292.

Froment, Clara, et al. "On the Occurrence of Thermal Nonequilibrium in Coronal

Loops." The Astrophysical Journal 855.1 (2018): 52.

García, Rafael A., et al. "Tracking solar gravity modes: the dynamics of the solar core."

Science 316.5831 (2007): 1591-1593.

Gibson, E. G. "The Quiet Sun, NASA SP-303." National Aeronautics and Space

Administration, Washington, DC (1973).

Gibson, Sarah E., et al. "FORWARD: A toolset for multiwavelength coronal magnetometry." Frontiers in Astronomy and Space Sciences 3 (2016): 8.

Gibson, Sarah. "Data-model comparison using FORWARD and CoMP." Proceedings of the International Astronomical Union 10.S305 (2014): 245-250.

Goldberg, Leo, Edith A. Muller, and Lawrence H. Aller. "The Abundances of the

Elements in the Solar Atmosphere." The Astrophysical Journal Supplement Series 5

(1960): 1.

Golub, L., et al. "The Solar-B Mission and the Forefront of Solar Physics (ASP Conf. Ser.

325), ed." T. Sakurai & T. Sekii (San Francisco, CA: ASP) 217 (2004).

199

Grevesse, N., and A. J. Sauval. "Standard solar composition." Space Science Reviews

85.1-2 (1998): 161-174.

Grevesse, Nicolas, Martin Asplund, and A. J. Sauval. "The solar chemical composition."

Space Science Reviews 130.1-4 (2007): 105-114.

Grotrian, W. "On the question of the significance of the lines in the spectrum of the solar corona." Naturwissenschaften 27.214 (1939): 1900-1975.

Gudiksen, Boris V., et al. "The stellar atmosphere simulation code Bifrost-Code description and validation." Astronomy & Astrophysics 531 (2011): A154.

Hale, G. E. et al., "The Magnetic Polarity of Sun- Spots", Astrophysical Journal, 49

(1919): pp. 153.

Handy, B. N., et al. "The TR and coronal explorer." Solar Physics 187.2 (1999): 229-

260.

Hannah, I. G., et al. "Microflares and the statistics of X-ray flares." Space science reviews 159.1-4 (2011): 263.

Hannah, Iain G., and Eduard P. Kontar. "Differential emission measures from the regularized inversion of Hinode and SDO data." Astronomy & Astrophysics 539

(2012): A146.

200

Hansen, Carl J., Steven D. Kawaler, and Virginia Trimble. Stellar interiors: physical principles, structure, and evolution. Springer Science & Business Media, 2012.

Hansteen, V. H., E. Leer, and T. E. Holzer. "The role of helium in the outer solar atmosphere." The Astrophysical Journal 482.1 (1997): 498.

Hansteen, V., et al. "The unresolved fine structure resolved: IRIS observations of the solar TR." Science 346.6207 (2014): 1255757.

Hansteen, Viggo H., et al. "Dynamic fibrils are driven by magnetoacoustic shocks." The

Astrophysical Journal Letters 647.1 (2006): L73.

Harrison, R. A., et al. "A study of extreme ultraviolet blinker activity." Astronomy and

Astrophysics 351 (1999): 1115-1132.

Hassler, D. M., G. J. Rottman, and F. Q. Orrall. "Absolute velocity measurements in the solar TR and corona." Advances in Space Research 11.1 (1991): 141-145.

Hayes, Marion, and R. A. Shine. "SMM observations of Si iv and O iv bursts in solar active regions." The Astrophysical Journal 312 (1987): 943-954.

Heggland, L., et al. "Wave propagation and jet formation in the chromosphere." The

Astrophysical Journal 743.2 (2011): 142.

201

Heggland, Lars, B. De Pontieu, and V. H. Hansteen. "Numerical simulations of shock wave-driven chromospheric jets." The Astrophysical Journal 666.2 (2007): 1277.

Holman, Gordon D., and Stephen G. Benka. "A hybrid thermal/nonthermal model for the energetic emissions from solar flares." The Astrophysical Journal 400 (1992):

L79-L82.

Holman, G. D. et al., Space Sci. Rev., 159, 107 (2011).

House, L. L., et al. "Studies of the corona with the Solar Maximum Mission coronagraph/polarimeter." The Astrophysical Journal 244 (1981): L117-L121.

Innes, D. E., et al. "Bi-directional plasma jets produced by magnetic reconnection on the Sun." Nature 386.6627 (1997): 811.

Innes, D. E., et al. "IRIS Si IV line profiles: an indication for the plasmoid instability during small-scale magnetic reconnection on the Sun." The Astrophysical Journal

813.2 (2015): 86.

Jahoda, Keith. "The gravity and extreme magnetism small explorer." Space Telescopes and Instrumentation 2010: Ultraviolet to Gamma Ray. Vol. 7732. International

Society for Optics and Photonics, 2010.

202

Judge, P. G., Veronika Hubeny, and John C. Brown. "Fundamental limitations of emission-line spectra as diagnostics of plasma temperature and density structure."

The Astrophysical Journal 475.1 (1997): 275.

Judge, Philip G. "Uv spectra, bombs, and the solar atmosphere." The Astrophysical

Journal 808.2 (2015): 116.

Kaiser, C. B. "The thermal emission of the F corona." The Astrophysical Journal 159

(1970): 77.

Kindel, J. Mo, and C. F. Kennel. "Topside current instabilities." Journal of Geophysical

Research 76.13 (1971): 3055-3078.

Kjeldseth-Moe, O. "The solar TR." Dynamic Sun (2003): 196-216.

Klimchuk, James A. "Key aspects of coronal heating." Phil. Trans. R. Soc. A 373.2042

(2015): 20140256.

Klimchuk, James A. "On solving the coronal heating problem." Solar Physics 234.1

(2006): 41-77.

Kontar, Eduard P., et al. "Generalized regularization techniques with constraints for the analysis of solar bremsstrahlung X-ray spectra." Solar Physics 225.2 (2004): 293-

309.

203

Kosugi, Takeo, et al. "The Hinode (Solar-B) mission: an overview." The Hinode

Mission. Springer, New York, NY, 2007. 5-19.

Koza, Julius, et al. "Temporal variations in fibril orientation." arXiv preprint astro- ph/0703733 (2007).

Lada, Charles J. "Stellar multiplicity and the initial mass function: most stars are single." The Astrophysical Journal Letters 640.1 (2006): L63.

Landi, E., and M. Landini. "Radiative losses of optically thin coronal plasmas."

Astronomy and Astrophysics 347 (1999): 401-408.

Landi, E., and M. Landini. "The Arcetri spectral code for thin plasmas." Astronomy and

Astrophysics Supplement Series 133.3 (1998): 411-425.

Landi, E., et al. "CHIANTI—An atomic database for emission lines. XIII. Soft X-ray improvements and other changes." The Astrophysical Journal 763.2 (2013): 86.

Landini, M. A. S. S. I. M. O., and BRUNELLA C. Monsignori Fossi. "The X-UV spectrum of thin plasmas." Astronomy and Astrophysics Supplement Series 82 (1990): 229-

260.

Lang, Kenneth R. "The Sun from space." Astrophysics and Space Science 273.1 (2000):

1-6.

204

Lemen, James R., et al. "The atmospheric imaging assembly (AIA) on the solar dynamics observatory (SDO)." The Solar Dynamics Observatory. Springer, New York,

NY, 2011. 17-40.

Lin, R. P., et al. "The Reuven Ramaty high-energy solar spectroscopic imager

(RHESSI)." The Reuven Ramaty High-Energy Solar Spectroscopic Imager (RHESSI).

Springer, Dordrecht, 2003. 3-32.

Lineweaver, Charles H. "An estimate of the age distribution of terrestrial planets in the universe: quantifying metallicity as a selection effect." Icarus 151.2 (2001): 307-

313.

Lionello, Roberto, et al. "Can large time delays observed in light curves of coronal loops be explained in impulsive heating?." The Astrophysical Journal 818.2 (2016):

129.

Litwin, Christof, and Robert Rosner. "On the structure of solar and stellar coronae-

Loops and loop heat transport." The Astrophysical Journal 412 (1993): 375-385.

Lodders, K., H. Palme, and H-P. Gail. "4.4 Abundances of the elements in the Solar

System." Solar system. Springer, Berlin, Heidelberg, 2009. 712-770.

Lodders, Katharina. "Solar system abundances and condensation temperatures of the elements." The Astrophysical Journal 591.2 (2003): 1220.

205

Lodders, Katharina. "Solar system abundances and condensation temperatures of the elements." The Astrophysical Journal 591.2 (2003): 1220.

MacNeice, Peter. "A numerical hydrodynamic model of a heated coronal loop." Solar physics 103.1 (1986): 47-66.

MacQueen, R. M., et al. "The outer solar corona as observed from Skylab: Preliminary results." The Astrophysical Journal 187 (1974): L85.

Madjarska, M. S., and J. G. Doyle. "Temporal evolution of different temperature plasma during explosive events." Astronomy & Astrophysics 382.1 (2002): 319-327.

Malara, F., and M. Velli. "Observations and Models of Coronal Heating." Recent

Insights into the Physics of the Sun and Heliosphere: Highlights from SOHO and Other

Space Missions. Vol. 203. 2001.

Mariska, John T. The solar TR. Vol. 23. Cambridge University Press, 1992.

Marsh, M. S., Robert William Walsh, and S. Plunkett. "Three-dimensional coronal slow modes: toward three-dimensional seismology." The Astrophysical Journal 697.2

(2009): 1674.

Martínez-Sykora, Juan, et al. "Forward modeling of emission in Solar Dynamics

Observatory/Atmospheric Imaging Assembly passbands from dynamic three- dimensional simulations." The Astrophysical Journal 743.1 (2011): 23.

206

Martínez-Sykora, Juan, et al. "On the generation of solar spicules and Alfvénic waves."

Science 356.6344 (2017): 1269-1272.

Martínez-Sykora, Juan, et al. "Spicule-like structures observed in three-dimensional realistic magnetohydrodynamic simulations." The Astrophysical Journal 701.2

(2009): 1569.

Martínez-Sykora, Juan, et al. "Time dependent nonequilibrium ionization of TR lines observed with IRIS." The Astrophysical Journal 817.1 (2016): 46.

Massey, H. S. W., and D. R. Bates. "The properties of neutral and ionized atomic oxygen and their influence on the upper atmosphere." Reports on Progress in Physics 9.1

(1942): 62.

McIntosh, Scott W., et al. "Alfvénic waves with sufficient energy to power the quiet solar corona and fast solar wind." Nature 475.7357 (2011): 477-480.

Michalitsanos, A. G. "The five-minute period oscillation in magnetically active regions." Solar Physics 30.1 (1973): 47-61.

Momose, Munetake, et al. "Investigation of the Physical Properties of Protoplanetary

Disks around T Tauri Stars by a High-resolution Imaging Survey at lambda= 2 mm."

The Proceedings of the IAU 8th Asian-Pacific Regional Meeting, Volume 1. Vol. 289.

2003.

207

Monsignori-Fossi, B. C., and M. Landini. "The differential emission measure and the composition of the solar corona." Memorie della Societa Astronomica Italiana 63

(1992): 767-772.

Müller, D. A. N., H. Peter, and V. H. Hansteen. "Dynamics of solar coronal loops-II.

Catastrophic cooling and high-speed downflows." Astronomy & Astrophysics 424.1

(2004): 289-300.

Müller, D. A. N., V. H. Hansteen, and H. Peter. "Dynamics of solar coronal loops-I.

Condensation in cool loops and its effect on TR lines." Astronomy & Astrophysics

411.3 (2003): 605-613.

Nita, Gelu M., et al. "Three-dimensional radio and X-ray modeling and data analysis software: Revealing flare complexity." The Astrophysical Journal 799.2 (2015): 236.

Okamoto, Takenori J., et al. "Resonant absorption of transverse oscillations and associated heating in a solar prominence. I. Observational aspects." The Astrophysical

Journal 809.1 (2015): 71.

Olluri, K., B. V. Gudiksen, and V. H. Hansteen. "Non-equilibrium ionization effects on the density line ratio diagnostics of O IV." The Astrophysical Journal 767.1 (2013a):

43.

208

Olluri, K., B. V. Gudiksen, and V. H. Hansteen. "Non-equilibrium ionization in the

Bifrost stellar atmosphere code." The Astronomical Journal 145.3 (2013b): 72.

Orrall, F. Ql, and J. B. Zirker. "The Coefficient of Thermal Conductivity in the Sun's

Atmosphere." The Astrophysical Journal 134 (1961): 63.

Parenti, S., Bromage, B. J. I., Poletto, G., et al. 2000, A&A, 363, 800

Parker, E. N. "Dynamical theory of the solar wind." Space Science Reviews 4.5-6

(1965): 666-708.

Parker, E. N. "Nanoflares and the solar X-ray corona." The Astrophysical Journal 330

(1988): 474-479.

Parker, E. N. "Nanoflares and the solar X-ray corona." The Astrophysical Journal 330

(1988): 474-479.

Parker, Eugene N. "Dynamics of the interplanetary gas and magnetic fields." The

Astrophysical Journal 128 (1958): 664.

Pereira, Tiago Mendes Domingos, et al. "An Interface Region Imaging Spectrograph first view on solar spicules." The Astrophysical Journal Letters 792.1 (2014): L15.

Pérez, M. E., et al. "Explosive events in the solar atmosphere." Astronomy and

Astrophysics 342 (1999a): 279-284.

209

Pérez, M. E., et al. "Temporal variability in the electron density at the solar TR."

Astronomy and Astrophysics 351 (1999b): 1139-1148.

Pesnell, W. Dean, B. J Thompson, and P. C. Chamberlin. "The solar dynamics observatory (SDO)." The Solar Dynamics Observatory. Springer, New York, NY, 2011.

3-15.

Peter, H. "Analysis of transition-region emission-line profiles from full-disk scans of the Sun using the SUMER instrument on SOHO." The Astrophysical Journal 516.1

(1999): 490.

Peter, H., et al. "Hot explosions in the cool atmosphere of the Sun." Science 346.6207

(2014): 1255726.

Peter, Hardi, and P. G. Judge. "On the Doppler shifts of solar ultraviolet emission lines."

The Astrophysical Journal 522.2 (1999): 1148.

Peter, Hardi, Boris V. Gudiksen, and Åke Nordlund. "Coronal heating through braiding of magnetic field lines." The Astrophysical Journal Letters 617.1 (2004): L85.

Petschek, H. E. "The physics of solar flares." AAS-NASA Symp.. Vol. 425. NASA, 1964.

Polito, V., et al. "Density diagnostics derived from the O iv and S iv intercombination lines observed by IRIS." Astronomy & Astrophysics 594 (2016): A64.

210

Porter, J. G., et al. "Microflares in the solar magnetic network." The Astrophysical

Journal 323 (1987): 380-390.

Priest, E. R., et al. "Nature of the heating mechanism for the diffuse solar corona."

Nature 393.6685 (1998): 545-547.

Raymond, John Charles, Donald P. Cox, and Barham Woodford Smith. "Radiative cooling of a low-density plasma." The Astrophysical Journal 204 (1976): 290-292.

Reale, Fabio. "Coronal loops: Observations and modeling of confined plasma." Living

Reviews in Solar Physics 7.1 (2010): 1-78.

Russell, Henry Norris. "On the composition of the Sun's atmosphere." The

Astrophysical Journal 70 (1929): 11.

Rybicki, George B., and Alan P. Lightman. Radiative processes in astrophysics. John

Wiley & Sons, 2008.

Ryutova, Margarita, and Theodore Tarbell. "MHD shocks and the origin of the solar transition region." Physical review letters 90.19 (2003): 191101.

Satyanarayana, P., et al. "Theory of the current‐driven ion cyclotron instability in the bottomside ionosphere." Journal of Geophysical Research: Space Physics 90.A12

(1985): 12209-12218.

211

Scherrer, Philip Hanby, et al. "The helioseismic and magnetic imager (HMI) investigation for the solar dynamics observatory (SDO)." Solar Physics 275.1-2

(2012): 207-227.

Schmelz, J. T., et al. "Isothermal and multithermal analysis of coronal loops observed with AIA." The Astrophysical Journal 731.1 (2011): 49.

Schmitt, J. H. M. M., et al. "The Extreme-Ultraviolet Spectrum of the Nearby K Dwarf ε

Eridani." The Astrophysical Journal 457 (1996): 882.

Schrijver, C. J. "The Coronae of the Sun and Solar-type Stars (CD-ROM Directory: contribs/schrijv)." 11th Cambridge Workshop on Cool Stars, Stellar Systems and the

Sun. Vol. 223. 2001.

Schrijver, Carolus J., et al. "A new view of the solar outer atmosphere by the TR and coronal explorer." Solar Physics 187.2 (1999): 261-302.

Schrijver, Carolus J., et al. "Large-scale coronal heating by the small-scale magnetic field of the Sun." Nature 394.6689 (1998): 152-154.

Sekse, Dan Henrik, L. Rouppe van der Voort, and Bart De Pontieu. "Statistical properties of the disk counterparts of type II spicules from simultaneous observations of rapid blueshifted excursions in Ca II 8542 and Hα." The Astrophysical

Journal 752.2 (2012): 108.

212

Shibata, Kazunari, et al. "Chromospheric anemone jets as evidence of ubiquitous reconnection." Science 318.5856 (2007): 1591-1594.

Shoub, E. C., Invalidity of local thermodynamic equilibrium for electrons in the solar

TR. I. Fokker-Planck results. The Astrophysical Journal 266 (1983): 339

Shoub, Edward C. Invalidity of local thermodynamic equilibrium for electrons in the solar transition region. I. Fokker-Planck results. No. SUIPR-930. STANFORD UNIV CA

INST FOR PLASMA RESEARCH, 1982.

Shoub, Edward C. Invalidity of Local Thermodynamic Equilibrium for Electrons in the

Solar Transition Region. II. Analysis of a Linear BGK Model. No. SU-IPR-946.

STANFORD UNIV CA INST FOR PLASMA RESEARCH, 1982.

Solanki, S. K., W. Livingston, and T. Ayres. "New light on the heart of darkness of the solar chromosphere." Science 263.5143 (1994): 64-66.

Spiegel, E. A., and J-P. Zahn. "The solar tachocline." Astronomy and Astrophysics 265

(1992): 106-114.

Sterling, Alphonse C., and Joseph V. Hollweg. "A rebound shock mechanism for solar fibrils." The Astrophysical Journal 343 (1989): 985-993.

Stone, E. C., et al. "Voyager 1 explores the termination shock region and the heliosheath beyond." Science 309.5743 (2005): 2017-2020.

213

Stone, E. C., et al. "Voyager 1 observes low-energy galactic cosmic rays in a region depleted of heliospheric ions." Science 341.6142 (2013): 150-153.

Suess, Hans E., and Harold C. Urey. "Abundances of the elements." Reviews of Modern

Physics 28.1 (1956): 53.

Summers, Hugh P., and M. G. O’Mullane. "The atomic data and analysis structure." AIP

Conference Proceedings. Vol. 543. No. 1. AIP, 2000.

Testa, Paola, Bart De Pontieu, and Viggo Hansteen. "High spatial resolution Fe XII observations of solar active regions." The Astrophysical Journal 827.2 (2016): 99.

Testa, Paola, et al. "Evidence of nonthermal particles in coronal loops heated impulsively by nanoflares." Science 346.6207 (2014): 1255724.

Thomas, Barry T., and Edward J. Smith. "The Parker spiral configuration of the interplanetary magnetic field between 1 and 8.5 AU." Journal of Geophysical

Research: Space Physics 85.A12 (1980): 6861-6867.

Tian, H., et al. "High-resolution observations of the shock wave behavior for sunspot oscillations with the interface region imaging spectrograph." The Astrophysical

Journal 786.2 (2014): 137.

Tian, H., et al. "Prevalence of small-scale jets from the networks of the solar TR and chromosphere." Science 346.6207 (2014): 1255711.

214

Tian, Hui, et al. "Frequently occurring reconnection jets from sunspot light bridges."

The Astrophysical Journal 854.2 (2018): 92.

Tian, Hui, et al. "New views on the emission and structure of the solar TR." New

Astronomy Reviews 54.1-2 (2010): 13-30.

Tikhonov, A. N., Soviet Math. Dokl., 4 (1963): 1035

Ulmschneider, Peter. "Heating of chromospheres and coronae." Space Solar Physics.

Springer Berlin Heidelberg, 1998. 77-106. van der Holst, Bart, et al. "Alfvén wave solar model (AWSoM): coronal heating." The

Astrophysical Journal 782.2 (2014): 81.

Van Doorsselaere, Tom, et al. "Forward modeling of EUV and gyrosynchrotron emission from coronal plasmas with FoMo." Frontiers in Astronomy and Space

Sciences 3 (2016): 4.

Vernazza, J. Eo, E. Ho Avrett, and R. Loeser. "Structure of the solar chromosphere. III-

Models of the EUV brightness components of the quiet-Sun." The Astrophysical

Journal Supplement Series 45 (1981): 635-725.

Verwichte, E., et al. "Seismology of a large solar coronal loop from EUVI/STEREO observations of its transverse oscillation." The Astrophysical Journal 698.1 (2009):

397.

215

Viall, Nicholeen M., and James A. Klimchuk. "Evidence for widespread cooling in an active region observed with the SDO Atmospheric Imaging Assembly." The

Astrophysical Journal 753.1 (2012): 35.

Viall, Nicholeen M., and James A. Klimchuk. "The TR response to a coronal nanoflare: forward modeling and observations in SDO/AIA." The Astrophysical Journal 799.1

(2015): 58.

Vissers, Gregal Joan Maria, et al. "Ellerman bombs at high resolution. III. Simultaneous observations with IRIS and SST." The Astrophysical Journal 812.1 (2015): 11.

Walsh, R. W., and J. Ireland. "The heating of the solar corona." The Astronomy and

Astrophysics Review 12.1 (2003): 1-41.

Warren, Harry P., and Amy R. Winebarger. "Small scale structure in the solar TR." The

Astrophysical Journal Letters 535.1 (2000): L63.

Weber, M. A., et al. "Multi-Wavelength Investigations of Solar Activity (IAU Symp.

223), ed." AV Stepanov, EE Benevolenskaya, & AG Kosovichev (Cambridge: Cambridge

Univ. Press) 223 (2004): 321.

Wedemeyer-Böhm, Sven, et al. "Magnetic tornadoes as energy channels into the solar corona." Nature 486.7404 (2012): 505-508.

216

Winebarger, Amy R., et al. "Detecting nanoflare heating events in subarcsecond inter- moss loops using Hi-C." The Astrophysical Journal 771.1 (2013): 21

Woolfson, Michael M. The origin and evolution of the solar system. CRC Press, 2000.

Yan, Limei, et al. "Self-absorption in the solar TR." The Astrophysical Journal 811.1

(2015): 48.

Young, P. R., et al. "A Si IV/O IV electron density diagnostic for the analysis of IRIS solar spectra." The Astrophysical Journal 857.1 (2018): 5.

Young, Peter R. "Report on the O IV and S IV lines observed by IRIS." arXiv preprint arXiv:1509.05011 (2015).

Zeilik, Michael, and Stephen Gregory. "Introductory Astronomy & Astrophysics fourth edition. New York, NY: Thomson Learning." (1998): 233.

Zirker, Jack B. Journey from the Center of the Sun. Princeton University Press. 2002.

217