Toward Connecting Core-Collapse Supernova Theory with Observations Timothy A

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2014 Toward Connecting Core-Collapse Supernova Theory with Observations Timothy A. Handy

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COLLEGE OF ARTS AND SCIENCES

TOWARD CONNECTING CORE-COLLAPSE SUPERNOVA THEORY WITH

OBSERVATIONS

TIMOTHY A. HANDY

A Dissertation submitted to the Department of Scientiﬁc Computing in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

Degree Awarded: Spring Semester, 2014

Tomasz Plewa Professor Directing Dissertation

Mark Sussman University Representative

Anke Meyer-Baese Committee Member

Gordon Erlebacher Committee Member

Ionel M. Navon Committee Member

The Graduate School has veriﬁed and approved the above-named committee members, and certiﬁes that the dissertation has been approved in accordance with university requirements.

ii To my wife April for her unwavering support through these years of late nights. And to my mother, grandmother, and grandfather: thank you for providing me with the support, wisdom, and guidance required to reach this destination.

iii ACKNOWLEDGMENTS

This work was supported by a great many people, without whom it would not have been successful. First and foremost, a thank you to my advisor, Tomasz Plewa. We’ve traveled through a long, dark tunnel, and as we step out into the light I hope that we can say that we’ve done good work, and that I’ve made a lifelong colleague and friend. Additional thanks to my collaborators: Andrzej Odrzywolek, for laying the foundation this work is built apon; Andrey Zhiglo, for his insights into physical processes; Konstantinos Kifonidis, for his (often humorous) insights, critiques, and suggestions; and R.P. Drake, for both his research and professional support. A sincere thank you the graduate students I have had the pleasure of working with: Andrew Young, for his initial contributions to the LAVAflow and CASAnova projects; Philip Boehner, for his continued development of LAVAflow and CASAnova; and to all of my other student colleagues who have been forced to share in my setbacks and accomplishments due to proximity effects. And finally, many thanks to my committee members who have donated time out of their lives to help better mine. This work has been supported by the NSF grant AST-1109113. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. Simulations were performed in part using the 3Leaf Systems SMP clusters at the Florida State University Research Computing Center, and the Deszno SMP cluster at Jagiellonian University, Cracow, Poland. The software used in this work was developed in part by the DOE Flash Center at the University of Chicago and the Center for Radiative Shock Hydrodynamics at the University of Michigan.

iv TABLE OF CONTENTS

ListofFigures ...... vii ListofTables...... xiv List of Algorithms ...... xv Abstract...... xvi

1 Introduction 1 1.1 Signiﬁcance of the research ...... 2 1.2 Stellar evolution and core-collapse supernovae ...... 3 1.3 Previouswork...... 7 1.4 Currentwork ...... 12

2 Numerical Model 14 2.1 Hydrodynamics...... 14 2.2 Equation of state ...... 15 2.3 Gravity ...... 15 2.4 Numericalgrid ...... 16 2.5 Boundary conditions ...... 16 2.6 Neutrinophysicseﬀects ...... 17 2.7 Tracerparticles...... 18 2.8 Initialconditions ...... 18 2.9 Tuningexplosionenergies ...... 18

3 Results 20 3.1 General properties of explosion models ...... 20 3.2 Convection ...... 37 3.3 Turbulence ...... 50 3.4 Summaryandfuturework...... 61

Appendix A Estimating Numerical Viscosity and Reynolds Number 67

B LAVAflow: Library for Advanced Variable Analysis 70 B.1 LAVAflow feature highlight: Non-grid-aligned variable averaging for orthogonal co- ordinatesystems ...... 71 B.2 LAVAflow feature highlight: Image segmentation ...... 87 B.3 LAVAflow feature highlight: Vector field decomposition ...... 90

C CASAnova: Computational Astrophysics Simulation Archive 93 C.1 Currentcapabilities ...... 94

v D Application of LAVAflow to Studies of Turbulent, Self-magnetized Plasma 96 D.1 Introduction...... 96 D.2 Experimentalscenario ...... 98 D.3 Computationalmodel ...... 100 D.4 Effect of magnetic fields on hydrodynamics ...... 106 D.5 Hydrodynamic evolution of the self-generated case ...... 110 D.6 Discussionandconclusions...... 119

References...... 122 BiographicalSketch ...... 133

vi LIST OF FIGURES

1.1 The onion-like distribution of elements in a star toward the end of its life. The shaded areas represent the interface between two layers where the conditions are right for fusion. The light areas are stable distributions of the elements in the star. The pressure, temperature, and density in these regions are outside of the range required for nuclear burning. Not to scale...... 4

3.1 Explosion energy for models tuned to the energetics of SN 1987A (saturation at ≈ 1.05 1051 erg). Thick curves denote the average over all models in the group; thin × curves denote the respective minimum and maximum values. Note that the fast contracting proto-neutron star models begin unbinding material in roughly half the explosion time of the standard contracting models...... 22

3.2 Comparison of our explosion models with the results of critical neutrino luminosity studies conducted by other groups. The results of individual studies are shown with diﬀerent symbols, while model dimensionality is represented by line style (1D: solid; 2D: dashed; 3D: dash-dotted). (left panel) The dependence of the critical neutrino luminosity on the accretion rate at the time of explosion. We observe a systematic decrease in the required neutrino luminosity as the model dimensionality increases. (right panel) Parameterized electron neutrino luminosity as a function of explosion time. For a given dimensionality, our models tend to explode sooner and require lower neutrinoluminosities...... 23

3.3 Proto-neutron star kick velocity for our SC models, shown with dashed and solid lines, respectively. The outset shows the estimated saturation velocities based on estimates utilizing the results of Scheck et al. (2006)...... 25

3.4 (left panel) Mass in the gain region. The evolution of mass in the gain region is shown with solid lines for SC models (thin solid lines in 2D and thick solid lines in 3D) and dashed lines for FC models. Note that the mass in the gain region is, on average, greater in FC models than in SC models. Also, after the initial transient oscillations, the mass in the gain region stabilizes, indicating that evolution in the gain region has reached a quasi-steady state period. (right panel) Shock accretion rate. The line types are associated with model family and model dimensionality as in the left panel.Note that after the initial period of fast accretion the accretion rates progressively decrease in both families of models. The accretion rates are, on average, greater in SC models than in FC models after the initial transient. See Section 3.1.3 for details...... 26

3.5 Advective timescale for the gain region. The oscillatory behavior of the advective timescale at early times for SC models likely has the same origin as the transient oscillations observed in the gain region mass. Note that the advective times are similar between model realizations in the same family. The diﬀerences in advective times for diﬀerent model realizations are particularly large for 2D SC models that develop a

vii few large-scale nonuniformities in the gain region. On average, the advective times are longer for FC models than for SC models. See Section 3.1.3 for details...... 28

3.6 Heating efficiency for the gain region. The evolution of heating efficiency is shown for SC and FC models with solid and dashed lines, respectively. The results from the 2D SC models are shown with thin solid lines. Note that the heating efficiency in the FC models is always greater than one. This indicates that a strong, immediate neutrino heating is required in order to produce energetic explosions in this case. On the other hand, the heating becomes efficient in the SC models only once the quasi-steady state is established. Again, as in the case of the mass in the gain region and advective times, the evolution of heating efficiency is qualitatively similar between SC and FC families...... 29

3.7 Evolution of the average shock radius. The average shock radius is shown with solid and dashed lines for SC and FC models, respectively, with 2D SC models shown with thin solid lines. Shaded areas denote regions where (texp, rs) pairs for SC and FC model families are located. Note that the FC and SC models explode at similar shock radii but at signiﬁcantly diﬀerent times. The shock expansion rate for FC models is, on average, twice that of the SC models...... 31

3.8 Shock aspect ratio for our models. Thick lines denote the average over all models in the group. Thin lines denote the minimum and maximum values for the particular families. Our results show relatively mild asymmetries (rmax/rmin 1.3). Visual s s ≈ inspection of SASI-producing models from M¨uller et al. (2012) show aspect ratios upwardsof3to4...... 32

3.9 Evolution of the leading coefficients in the spherical harmonic decomposition of the shock radius. Only the results of the decomposition for SC models are shown. The runs of the α1 (left panel) and α2 (right panel) coefficients are presented with solid and dashed lines for 3D and 2D models, respectively. Prior to the onset of convection (t 50 ms), the shock front remains essentially spherically symmetric. The emergence ≈ of low order perturbations around t 80 ms is due to buoyant convective plumes ≈ deforming select sections of the shock front rather than SASI. Note that the degree of shock perturbation is relatively modest compared to some recent results (see, e.g., Müller et al., 2012)...... 34

3.10 Entropy distribution in the post-shock region at the explosion time for select two- dimensional and three-dimensional models. Entropy distribution is shown with pseudocolor maps for models M194J (panel a), M194C (panel b), M187A (panel c), and M187B (panel d). The entropy distribution maps for two-dimensional models (panels (a), (b)) are reﬂected across the symmetry axis. For three-dimensional models, the entropy is shown for a slice through the computational domain at the equatorial plane. In two dimensions models tend to have either dipolar (panel a) or quadrupolar (panel b) ejecta structures which result in an oblate shape of the shock. In three dimensions the ejecta appear to have more regular structure, and the shock is less deformed. See Section 3.1.5 for discussion...... 36

viii 3.11 Distribution of entropy in the 3D SC model M187A. The entropy is shown with pseudocolor maps for a slice through the computational domain at the equatorial plane. The entropy distribution is shown at t = 50, 100, 150, 189, 500, and 1500 ms in panels (a)-(f). The development of convection can be seen as early as t = 50 ms (panel a). The ﬂow structure becomes progressively more complex at intermediate times (panels (b) and (c)) and the explosion is launched at t = 189 ms (panel (d)). Relatively little change in the structure of the gain region takes place at later times (panels (e) and (f))...... 37

3.12 The distribution of entropy in the M187A series of models with varying angular resolution. The entropy is shown for a slice taken at the equatorial plane at t = 100 ms in models with angular resolutions of (a) 3◦, (b) 6◦, (c) 12◦, and (d) 24◦. Note the overall appearance of convection is degraded as the angular resolution decreases, indicating a drop in the eﬃciency of the neutrino-driven convective engine. The shock radius is signiﬁcantly smaller in the coarsest model. See Section 3.2.1 for discussion. 40

3.13 The temporal evolution of the upﬂowing mass in the gain region. The fraction of the total mass inside the gain region with (ur > 0) is shown with dash-dotted, dashed, and solid lines for 2D FC, 2D SC, and 3D SC models, respectively. See Section 3.2.2 fordiscussion...... 41

3.14 The evolution of relative mass fluctuations inside the gain region shortly before and during the quasi-steady state phase in the 3D SC explosion models. Fluctuations are calculated relative to a linear fit of the upflowing mass in the gain region from the beginning of the quasi-steady state phase until the explosion time, where individual curves terminate. (thin solid) M157A; (medium solid) M157B; (thick solid) M157C; (dashed) M157D; (dotted) M157E. See Section 3.2.2 for discussion...... 43

3.15 Estimate of the total solid angle spanned by buoyant, rising material. The onset of convection at t 50 ms results in a rapid increase in the spanned solid angle as the ≈ ﬁrst convective plumes begin to form. During the quasi-steady convective phase the plumes continue to grow. By the time the explosion is launched they cover roughly 2/3 of the total solid angle. See Section 3.2.3 for discussion...... 44

3.16 Evolution of the number of rising bubbles inside the gain region. The number of bubbles is averaged separately for 2D and 3D SC model realizations (shown with dashed and solid lines, respectively). For the clarity of presentation, the vertical scale is limited to N = 25. The maximum average number of bubbles found in 3D is ⟨ ⟩ 200. See Section 3.2.3 for discussion...... 46 ≈ 3.17 Decomposition of the total energy ﬂux and the distribution of the related work source terms in the SC models. Quantities are averaged over time over the period between t = 110 ms and t = 120 ms. The individual curves represent averages taken over all model realizations for a particular model dimension (shown with dashed and solid lines for 2D and 3D models, respectively). The vertical lines denote the approximate locations of the gain radius and the minimum shock radii for the two families. (left

ix panel) Convective ﬂux, FC , and kinetic ﬂux, FK . (right panel) Buoyant (PA) and expansion (PP ) work. See Section 3.2.4 for discussion...... 47

3.18 Lateral kinetic energy spectra for SC models. The spectra are shown with dashed and solid lines for 2D and 3D models, respectively. The data are averaged spatially over a spherical shell 25 km thick centered at 290 km, and over the time period between t = 120 ms and t = 130 ms. Piecewise powerlaw functions are ﬁt separately to spectra for 2D and 3D model families, and are shown with dashed and solid lines for 2D and 3D, respectively. The ﬁt process allows for optimal locations of the break points in the piecewise powerlaw functions. See Section D.5.6 for discussion...... 51

3.19 Distributions of Reynolds stresses in the gain region for the SC models. The data shown in the ﬁgure is obtained by ﬁrst averaging stress distributions for individual model realizations over the period between t = 110 ms and t = 130 ms, and then by calculating mean values based on the time-averages separately for 2D and 3D. The results for 2D and 3D families are shown with thin and thick lines, respectively. The dashed vertical lines at r 110 km, 400 km, and 570 km mark the approximate ≈ positions of the gain radius, minimum shock radius, and maximum shock radius, respectively. Note that the stresses for 2D models rise much more rapidly across the gain radius region than in 3D. See Section 3.3.2 for discussion...... 53

3.20 Distributions of tracer particles in the gain region as a function of particle residency time for SC models. This distributions are shown with dashed and solid lines for 2D and 3D families, respectively, at t = 50 ms, 100 ms, and 200 ms. The data are averaged over all model realizations for a particular model dimension. For legibility, we smooth the curves with a boxcar method using a time window of 5 ms. Note that the scale diﬀers for 3D models. See Section 3.3.3 for discussion...... 56

3.21 Distribution of the speciﬁc internal energy change for tracer particles in the gain region as a function of particle residency time for SC models. The distributions are shown at t = 50, 100, and 200 ms with thick, medium, and thin lines, respectively. The data are averaged over all model realizations for a particular dimension, and are shown separately with dashed and solid lines for 2D and 3D. For legibility, we smoothed the curves with a boxcar method using a time window of 5 ms. See Section 3.3.3 for discussion...... 58

3.22 Distribution of the positive speciﬁc gravitational binding energy for tracer particles in the gain region as a function of particle residency time for SC models. The data are taken at t = 200 ms, and are averaged over all model realizations for a particular model dimension. The results are shown with dashed and solid lines for 2D and 3D, respectively. For legibility, we smooth the curves using a boxcar method with a time window of 5 ms. Note that in 2D, unbound material is composed primarily of matter which entered the gain region prior to the quasi-steady phase. However, in 3D, there are substantial contributions from material which entered the gain region during the period of vigorous convection. See Section 3.3.3 for discussion...... 60

x A.1 Analysis results for models M194J and M187D. (left panel) Estimated numerical kinematic viscosity. (right panel) Estimated Reynolds number as a function of particle separation distance. We may expect our models to reﬂect ﬂows with Reynolds numbers on the order of 10s or 100s...... 68

B.1 Demonstration of the process of identifying clusters of disjoint pixels in an image as outlined in Algorithm 2. The colors black and white represent unused pixels (where no flooding should occur) and unfilled pixels (where flooding has not yet reached), respectively. Gray denotes filled pixels which have been found by the flood fill algorithm. (left panel) Initial unclustered image. (middle panel) Intermediate step during which Algorithm 1 is finding all pixels in a new cluster. (right panel) Fully segmented image after all unfilled pixelshavebeenvisited...... 89

D.1 Proposed experimental setup. (left panel) Two-material target conﬁguration. The dashed circle shows the region we consider for our analysis, the “turbulent core”. (right panel) Schematic for one round of laser driving. Three lasers (a triple) are positioned 120◦apart. Each laser is a precomputed hydrodynamic proﬁle from the CRASH code, with the leading edge of the shock touching the thick solid line. . . . . 99

D.2 Pseudocolor plots of quantities for the prototypical laser drive calculated with CRASH. All plots are colored in log scale. The solid line which forms an envelope over the on 9 all plots indicates our cutoﬀ region for mapping (pcut = 3 10 Pa). (a) Density [g − − × cm 3]. (b) Pressure [Pa]. (c) Velocity magnitude [cm s 1]...... 105

D.3 RMS-Mach number for the hydrodynamic, pre-existing magnetic field, and self-generating magnetic field cases. The Mach number behaves similarly for all magnetization cases, indicating that the magnetic field is likely too low to influence the hydrodynamic development of the system. The laser-driven, quasi-steady state system never reaches thesupersonicregime...... 107

D.4 Magnetic field and plasma β over time for the self-generating and pre-existing field cases. (top panel) Magnitude of the magnetic field. The thick curves indicate the arithmetic mean, while the thinner lines above and below show the 75th and 25th percentiles, respectively. For the pre-existing case, the initial mean magnetic field is amplified by a factor of 2 to 3 during the driving stage. The field then decreases when driving is halted. Note that the self-generating case produces only kilogauss fields. (bottom panel) Plasma β values (β = p/(8πB2)). The top and bottom lines for each case represent the 75th and 25th percentiles, respectively. The mean is not shown, as localized magnetic field voids produce small regions of extremely high β, skewing the results. The preexisting case generates β values on the order of 1 to 100. The self-generating case generates very high β values, resulting in a thermally dominated flowfield...... 108

D.5 Pseudocolor plot of the bivariate probability distribution of magnetic field strength and plasma density for the two magnetized cases. (a) The field strength for the pre- existing field correlates linearly with density, indicative of compression of the magnetic

xi field lines as the gas compresses. (b) The self-generated field strength indicates that more complicated physics is involved than pure compression effects. There is no visible correlation between field strength and density for this case, indicating that amplification of field strength due to material compression is, at least, strongly suppressed. . . 109

D.6 Pseudocolor plots of density (g cm−3) for the self-generated magnetic ﬁeld case in the turbulent core. Mesh resolution in the region is 4 µm . (a) t = 23 ns (b) t = 130 ns (c) t = 230 ns ...... 110

D.7 Isocontours of the bivariate probability distribution for scaled velocity vector components in the turbulent core. The individual components of velocity for each grid cell are scaled by the maximum velocity magnitude at each snapshot to generate ux/y. The insets show a polar histogram indicating directionality of the velocity field. The solid circle shows how a uniform (isotropic) distribution of velocity would appear.e (a) 6 −1 t = 30 ns, u max = 9.3 10 cm s . The initial imprint from the first laser triple is | | × 6 −1 clearly visible. (b) t = 130 ns, u max = 9.4 10 cm s . During the driving phase, | | × 6 the velocity field achieves an isotropic distribution. (c) t = 230 ns, u max = 5.5 10 − | | × cm s 1. The absence of confining ram pressure allows the compressed plasma to expand when laser driving ceases, which manifests as the of the velocity space. . . . 111

D.8 Streamlines of the velocity ﬁeld and its decomposition at t = 130 ns. This square region is centered in the TC and is the area which the spectra is computed on. (a) Total velocity. Note the combination of point sources, link sinks, and vortical motions. (b) Compressive velocity component. There are multiple point sources visible, indicating expanding regions. The coalescing streamlines in the domain are line sinks, and correspond to compression of the material. (c) Solenoidal velocity component. The underlying vortical ﬂow structure is showcased, with a nested structure to the vortices. Note that for two dimensional turbulence, the smaller vortices are likely inducing the formation of the larger vortical cells...... 113

D.9 Radially averaged distributions of density (left column) and pressure (right column) inside the turbulent core at t = 130 ns, t = 200 ns, and t = 230 ns. Thick solid lines denote the mean value while thin solid lines show the minimum and maximum values. 115

D.10 Probability distribution function for the density in the turbulent core at t = 130 ns, t = 200 ns, and t = 230 ns. The distribution at t = 230 ns is scaled by 1/2 for visualization purposes. During the driving phase the density follows a lognormal distribution. However, once laser driving ceases the distribution becomes bimodal. . 116

D.11 Kinetic energy spectra (total, compressive, and solenoidal) at t = 130 ns, t = 200 ns, and t = 230 ns for the self-generated magnetic field case. (a,b) During the driving phase the power spectra in the inertial range (5 ≲ k ≲ 100) scales as P k−2.3. This − ∝ deviates from the traditional two-dimensional k 3 behavior due to laser-driven stirring and not magnetic field effects. (c) After driving ceases, the solenoidal component becomes the dominant source of kinetic energy and scales as P k−2. In the absence ∝

xii of inertial forces, the dominant process is the interaction between vortical cells, leading to a k−2 dependence...... 117

D.12 Evolution of the velocity power spectra exponent for 10 k 50. The turbulent ≤ ≤ ﬂow reaches a quasi-steady state near t = 75 ns with α 2.3 for all components. ≈ − This deviation from α = 3 for two-dimensional turbulence is a consequence of the − laser-drivenstirring...... 117

xiii LIST OF TABLES

2.1 Estimated explosion energetics of SN 1987A...... 19

3.1 Parameters and main properties of the explosion models...... 21

3.2 Characteristics of the M187A model series with varying angular resolution...... 39

B.1 Area integrals for planar curves in Cartesian geometry...... 75

B.2 Area integrals for planar curves in cylindrical geometry...... 76

B.3 Area integrals for planar curves in spherical geometry...... 77

B.4 Summary of area integrals for the diﬀerent coordinate systems...... 78

B.5 Final area integrals for Cartesian coordinates...... 78

B.6 Final area integrals for cylindrical coordinates...... 79

B.7 Final area integrals for spherical coordinates...... 81

B.8 Monomial area integrals for Cartesian coordinates...... 82

B.9 Monomial area integrals for cylindrical coordinates...... 82

B.10 Monomial area integrals for spherical coordinates...... 83

D.1 Target geometry and initial conditions...... 102

xiv LIST OF ALGORITHMS

1 FloodFill recursively ﬁnds all pixels that can be reached from an initial pixel . . . 87

2 FindClusters partitions an image into “clusters” of disjoint pixels ...... 89

xv ABSTRACT

We study the evolution of the collapsing core of a 15 M⊙ blue supergiant supernova progenitor from the moment shortly after core bounce until 1.5 seconds later. We present a sample of two- and three- dimensional hydrodynamic models parameterized to match the explosion energetics of supernova

SN 1987A. We focus on the characteristics of the ﬂow inside the gain region and the interplay between hydrodynamics, self-gravity, and neutrino heating, taking into account uncertainty in the nuclear equation of state.

We characterize the evolution and structure of the ﬂow behind the shock in terms the accretion

ﬂow dynamics, shock perturbations, energy transport and neutrino heating eﬀects, and convective and turbulent motions. We also analyze information provided by particle tracers embedded in the

ﬂow.

Our models are computed with a high-resolution ﬁnite volume shock capturing hydrodynamic code. The code includes source terms due to neutrino-matter interactions from a light-bulb neutrino scheme that is used to prescribe the luminosities and energies of the neutrinos emerging from the core of the proto-neutron star. The proto-neutron star is excised from the computational domain, and its contraction is modeled by a time-dependent inner boundary condition.

We find the spatial dimensionality of the models to be an important contributing factor in the explosion process. Compared to two-dimensional simulations, our three-dimensional models require lower neutrino luminosities to produce equally energetic explosions. We estimate that the convective engine in our models is 4% more efficient in three dimensions than in two dimensions. We propose that this is due to the difference of morphology of convection between two- and three-dimensional models. Specifically, the greater efficiency of the convective engine found in three-dimensional simulations might be due to the larger surface-to-volume ratio of convective plumes, which aids in distributing energy deposited by neutrinos.

We do not ﬁnd evidence of the standing accretion shock instability in our models. Instead we identify a relatively long phase of quasi-steady convection below the shock, driven by neutrino

xvi heating. During this phase, the analysis of the energy transport in the post-shock region reveals characteristics closely resembling that of penetrative convection. We ﬁnd that the ﬂow structure grows from small scales and organizes into large, convective plumes on the size of the gain region.

We use tracer particles to study the flow properties, and find substantial differences in residency times of fluid elements in the gain region between two-dimensional and three-dimensional models.

These appear to originate at the base of the gain region and are due to differences in the structure of convection. We also identify differences in the evolution of energy of the fluid elements, how they are heated by neutrinos, and how they become gravitationally unbound. In particular, at the time when the explosion commences, we find that the unbound material has relatively long residency times in two-dimensional models, while in three dimensions a significant fraction of the explosion energy is carried by particles with relatively short residency times.

We conduct a series of numerical experiments in which we methodically decrease the angular resolution in our three-dimensional models. We observe that the explosion energy decreases dramatically once the resolution is inadequate to capture the morphology of convection on large scales.

Thus, we demonstrated that it is possible to connect successful, energetic, three-dimensional models with unsuccessful three-dimensional models just by decreasing numerical resolution, and thus the amount of resolved physics. This example shows that the role of dimensionality is secondary to correctly accounting for the basic physics of the explosion.

The relatively low spatial resolution of current three-dimensional models allows for only rudimentary insights into the role of turbulence in driving the explosion. However, and contrary to some recent reports, we do not ﬁnd evidence for turbulence being a key factor in reviving the stalled supernova shock.

xvii CHAPTER 1

INTRODUCTION

Observations of core-collapse supernova (ccSNe) remnants show that asymmetric mass motion is a fundamental characteristic of these explosive events. The most studied of these objects, Cas A and SN 1987 A, provide direct evidence of this intrinsic asymmetry. The jet-like emission and heavy element distribution of Cas A (Hwang et al., 2000) has made it famous in the astrophysics community.

Analysis of these objects indicates that nucleosynthetic products become mixed on large scales by hydrodynamic instabilities within minutes to hours after the onset of explosion (Hanuschik and

Dachs, 1987; Hanuschik, 1991; Phillips and Heathcote, 1989; Utrobin, 2005). Additionally, high quality speckle observations show the presence of a localized, high velocity feature (Meikle et al.,

1987; Nisenson et al., 1987; Nisenson and Papaliolios, 1999). This emission characteristic may indicate a bidirectional mass ejection, possibly a weak jet (Wang et al., 2002).

The initial mixing and asymmetries in core-collapse supernovae are entrenched in the explosion mechanism itself. Revival of the supernova shock is believed to be driven by neutrino heating of material in the immediate vicinity of the core; a process potentially aided by neutrino driven convection, the standing accretion shock instability (SASI), and turbulence (Herant, 1995; Scheck et al.,

2004; Burrows et al., 2006). As the supernova shock passes through the stratified stellar atmosphere conditions become favorable for the development of Rayleigh-Taylor, Richtmyer-Meshkov, and Kelvin-Helmholtz instabilities. Additional effects from rotation and magnetic fields may launch jets or highly collimated outflows, affecting the morphology at late times (Akiyama et al., 2003;

Khokhlov et al., 1999; Dessart et al., 2008).

The rooting of asymmetries at the SNR stage ( 100 days) in the early phases of explosion ∼ implies that the late time evolution can not be studied in isolation from the onset of explosion.

1 Therefore, understanding the morphology and evolution of young core-collapse supernova remnants

(SNRs) cannot be achieved without capturing at least the most relevant aspects of the very early explosion phase ( 100 ms, 700 km). At present, this represents a signiﬁcant computational ∼ ∼ challenge. Indeed, the ﬁeld is divided along this computational line; work is either focused on the very early time dynamics with no connection to the SNR phase, or on the SNR phase with questionable physics required for driving the explosion.

Models attempting to capture the early time explosion dynamics from ﬁrst principles using sophisticated multi-group and multi-angle radiation-hydrodynamic simulations with Boltzmann neutrino transport are currently restricted to two dimensions due to the enormous computational cost. At present, this brand of models leads to at best marginal explosions, especially for high-mass progenitors. Additionally, the results produced using this method are in disagreement.

As a result, approximate and less expensive prescriptions of neutrino transport, neutrino heating, and neutrino cooling are the only feasible way to treat long-term evolution, especially in three dimensions. These types of schemes allow for parameterizations or semi-parameterizations of the neutrino fluxes emitted by the collapsed stellar core, or proto-neutron star (PNS). This parameterization allows for comparatively cheap parameter studies to probe supernova models with varying characteristics (e.g. explosion energies), subsequently enabling the study of large samples of computed SNRs. These methods are expected to yield accurate descriptions for multidimensional flows, provided that the ratio of neutrino heating timescale to the growth timescale for hydrodynamic instabilities is not significantly impacted (Scheck et al., 2006). The use of this parameterized models for late-time SNR is preferable to current approaches which neglect the early explosion physics entirely.

1.1 Signiﬁcance of the research

The results of our study will allow for obtaining supernova explosion model observables and test predictions of stellar evolution theory. A large amount of observational data has been collected over the last several years, and this trend is ﬁrmly expected to continue and strengthen. Furthermore,

2 recent observations revealed close connections between massive stars and gamma ray bursts (GRBs), with the long GRBs seemingly directly associated with core-collapse SNe. These extreme events are expected to be strongly asymmetric, producing highly collimated relativistic outﬂows, a property expected to be intimately linked to the central engine driving the explosion, in particular its rotation rate.

In contrast to thermonuclear supernova modeling, no more than perhaps a dozen of 3-dimensional core-collapse models exist. Among those,

No 3D model has been studied from the moment of core bounce to homologous expansion. •

No such studies have been conducted for zero metallicity progenitors. •

No long-term evolution results are available for rotating progenitors. •

Very little is known about the morphology of young supernova remnants. •

Our long term aim is to address the above deﬁciencies and provide realistic computational models to enable the testing of stellar evolution theory in new regimes. Our current work focuses on enabling an intuition about the early explosion phase, developing new, novel methods for analysis (particularly Section 3.2), and delevoping results for comparison when moving to our new computational code.

1.2 Stellar evolution and core-collapse supernovae

1.2.1 Pre-collapse

In a star’s formative years, it subsists by fusing hydrogen to helium in its core. As time proceeds, and the hydrogen is used up, the core contracts. This shrinkage increases the density and temperatures deep in the core, making conditions suitable to begin burning the new helium atoms into carbon. From here, the process repeats by fusing carbon into oxygen and heavier elements.

At this point, the remainder of the star’s life can follow one of two possible paths. First, if it is of small to modest size, the process ends here. A star in this category is referred to as a white

3 Figure 1.1 The onion-like distribution of elements in a star toward the end of its life. The shaded areas represent the interface between two layers where the conditions are right for fusion. The light areas are stable distributions of the elements in the star. The pressure, temperature, and density in these regions are outside of the range required for nuclear burning. Not to scale.

dwarf. These carbon-oxygen stars can continue to accrete matter, and eventually result in a nova or a Type Ia supernova. A nova occurs when conditions are ripe on the surface to cause a runaway nuclear explosion. This does not tend to destroy the star, but produces intense radiation emissions and a small amount of heavy elements.

In a Type Ia supernova, the star accretes mass until it approaches the Chandrasekhar mass from below. Before the star reaches this critical mass, conditions become favorable in the core for nuclear burning to begin. The fusion of carbon starts, and a nuclear burning ﬂame expands through the star. This process produces enough energy to tear apart the star, ejecting material at several percent of the speed of light.

If the star has a mass greater than roughly 8 9 M⊙, further evolution occurs. More burning − stages occur, ﬁrst burning oxygen to silicon, and then silicon to iron. Other elements up to iron are produced (neon, aluminum, magnesium, etc.) in the process, but the primary products generated are the elements mentioned.

Eventually, the stellar interior has an onion-like structure with respect to atomic composition

(see Fig. 1.1). There exists an iron core, surrounded by a silicon shell, which is itself surrounded by oxygen, carbon, helium, and hydrogen shells. At the interfaces of these shells conditions are right for nuclear burning. This only occurs near the transitions, as conditions inside the shell are not

4 extreme enough to cause the entire shell to be burning at once. As atoms are advected from their shell toward the core, the temperature and densities increase to values making nuclear combustion feasbile.

All great things must come to an end. In the case of these massive stars, the same nuclear chemistry that has fueled their lives for thousands of millenia betrays them in the end. Fusion of elements higher than iron is an endothermic process, i.e. it requires a supply of energy to occur rather than adding energy to the surroundings. The result of this is that no new elements heavier than iron form, and the iron core continues to grow as silicon is burnt. Simultaneously, the extreme conditions in the core produce highly energetic photons. These photons can rapidly undo the results of the star’s life by tearing the iron group nuclei apart. Photo-disintegration is an endothermic process that ends up removing sustaining energy (and subsequently gravity-countering pressure) from the core. The ﬁnal process occuring is named electron capturing (EC) and results when free electrons and free protons interact. The result of the electron capture is a neutron and an electron neutrino. This results in a decrease in the lepton number due to the neutrinos easily streaming from the core, which decreases the mass required for the critical Chandasekhar mass. Thus, the iron core is growing while simultaneously losing size-preserving pressure and destabilizing. The relative importance of these last two aﬀects depend on the mass of the original star, with photo-disintegration dominating when the progenitor mass ≳ 20M⊙ and EC dominating otherwise.

1.2.2 Collapse & shock bounce

Ultimately, the iron core reaches the Chandasekhar mass ( 1.4 M⊙), gravity overpowers the ≈ pressure in the core, and quantum mechanical eﬀects come into play. Electrons are pushed inside protons resulting in neutrons, in a process aptly named neutronization, and the core rapidly contracts as it can no longer sustain its size. This shrinking happens homologously, meaning the velocity of the infalling material is proportional to the radius (farther out material falling faster).

At a certain point inside the star the the ﬂow becomes supersonic (rsonic), and shortly thereafter peaks at a maximum infalling velocity (rvmax ). From this point outward, the magnitude of the

5 velocity decreases nonlinearly and the shells higher than this radius appear to be suspended in

“mid-air,” resulting in a “rain” of material.

The homologously infalling shells begin decelerating as they reach lower heights. This highly inertial material overshoots its equilibrium position, piling up in a dense fog and releasing strong pressure waves that collect at the sonic point mentioned earlier. Much like a spring, the material begins bouncing back, attempting to reach equilibrium. During this process the central core begins exceeding the nuclear matter density, resulting in a stiﬀening of the equation of state, and essentially becoming a rigid wall as the new protoneutron star (PNS). The result is an expanding density wave which travels behind the shockwave until they separate at rvmax . The shockwave continues propagating outward, shocking the infalling material.

1.2.3 Explosion

As the outward traveling shock passes through the rain of outer shells, photo-disintegration reenters the picture. Energetic photons present in the post-shock region begin to strip apart heavy elements, extracting energy from the ﬂow, and decreasing the pressure behind the shock.

Additionally, the shock is moving outward, sweeping through the infalling matter. This expansion requires an increasing ammount of energy to sustain its progress. These two factors conspire to bring the shock to a halt. Typically, the shock will overshoot its equilibrium position and reverse direction, falling back toward the star. It will then enter a period of oscillation about some nominal radius.

If the shock was to stay in one location indeﬁnitely then this would not be much of an explosion.

Therefore, shock motion must eventually revive, so that it can push through the outer layers of our onion-star, expelling them into space.

The revival of the shock depends on the introduction of new energy sources to the post-shock region. These sources must exist prior to the shock stalling and/or emerge from external perturbations after the shock has stalled. One possible source of energy is neutrino heating due to electron capture near the core. Other processes participating in the explosion may include magnetic ﬁelds and the rotation the progenitor star.

6 1.2.4 Post-explosion phase

Assuming the supernova shock successfully revives, it will begin traveling through the layers of the star. As it passes through the interfaces of the onion-like structure, hydrodynamic instabilities will begin to develop. These include Rayleigh-Taylor, Richtmyer-Meshkov, and Kelvin-Helmholtz.

All of these instabilities are known for their ability to mix the ﬂow they’re embedded in. As this occurs, the heavy elements from deep in the star will be partially brought into the outer layers of the star. In addition, it is not overly irrational to think that these instabilities will eﬀect the shock structure, and therefore the remnant morphology.

When the shock reaches the “edge” of the star, it undergoes shock breakout. Prior to this the shock had expended energy heating the envelope of the star, and essentially trapping light in a bottle. This event is characterized by a sudden burst of X-ray/UV light which occurs when energetic photons are abruptly able to stream through the shock and make it through the stellar atmosphere; essentially taking the lid oﬀ the bottle. The radiation released interacts with the surrounding medium, shedding light (no pun intended) on the properties of the progenitor that produced it.

1.3 Previous work

1.3.1 Early studies

Beginning in the 1970’s, the idea of neutrino emission from the proto-neutron star as a component of supernovae explosions began its acceptance among the astrophyics community (Wilson,

1971, 1974; Wilson et al., 1975). The work of Arnett (1977) laid the stage for Bethe et al. (1980) to make initial estimates on the parameters and eﬀects of neutrino absorption.

By the end of 1982, Bowers and Wilson (1982a,b) were generating one-dimensional, non- exploding models. These models managed to produce stalled shocks, but were unable to revive them. However, it seems that later in 1982 Wilson was able to begin seeing revival due to neutrino heating (c.f. Bethe and Wilson (1985) wherein they references unavailable conference proceedings from Wilson).

7 The work of Bethe and Wilson (1985) was able to successfully revive a shock produced from a

25M⊙ star. However, this victory proposed an interesting requirement that a “quasi-vacuum” is formed between the shock and the star caused by a bifurcation in the radial velocity (and subsequent evacuation of material). At the time this was believed to be a fundamental requirement due to estimates of binding energy and other reasonable arguments.

Models including convection (via mixing length theory) were produced by Mayle and Wilson

(1988). Comparing their explosion models with and without convection, they arrived at the conclusion that “convection will roughly double the explosion energy”. This put to rest an argument that neutrino-driven shock revival produced weak explosions, and paved the way for multidimensional considerations.

1.3.2 Neutrino-driven convection paradigm

Nearly two decades after neutrinos arose as a fundamental component of core-collapse supernova explosions, Burrows and Goshy (1993) proposed the concept of a critical neutrino luminosity. As- suming a standing shock and a steady state, they modeled the supernova as a system of eigenvalue

ODEs subjected to simpliﬁed physics. Their analysis resulted in a smooth, one-to-one relation between the minimum neutrino luminosity required for a successful explosion and the accretion rate of material through the stalled shock. It must be noted that this work has stood the test of time, in some sense. It has driven the ﬁeld in a particular direction (the hunt for a true critical luminosity relation), which persists to this day, 20 years later.

During the mid-1990’s, simulations in two dimensions began to illustrate the effects of convection on the explosion. Müller and Janka (1994) produced models indicating that convection inside the proto-neutron star resulted in more efficient neutrino heating, as energy could now be transported due to advection and no longer relied soley on neutrino diffusion. Additionally, they reasoned that convection occuring between the proto-neutron star and the shock front aids the explosion by increasing the residency time of material in the gain region.

Burrows et al. (1995) came to similar conclusions as M¨uller and Janka (1994), noting that

8 “Our two-dimensional calculations imply that the breaking of spherical symmetry may

be central to the supernova phenomenon.”

In the view of the critical neutrino luminosity concept, they postulated that convection would allow the minimum luminosity to be lowered, making explosions more feasible. Their simulations found that penetration below the gain radius due to convection caused events of enhanced neutrino heating, increasing the neutrino luminosity by 10%-20%. In an interesting case of foreshadowing, they state in no uncertain terms that there is no evidence of energy accumulation prior to explosion.

In fact, the total energy behind the shock decreases monotonically with time. The robustness of this result would be challenged by Blondin et al. (2003) with the standing accretion shock instability.

1.3.3 The role of multidimensional instabilities

Beginning in 2000, Foglizzo and collaborators put forth the concept of “vortical-acoustic feedback,” now known as the advective-acoustic cycle (AAC) (Foglizzo and Tagger, 2000; Foglizzo,

2002; Foglizzo and Galetti, 2003). This process is dependent upon having a system in which vorticity can be generated and turned into acoustics. In the supernova setting, a misalignment of the shock normal with the accreting ﬂow will generate vorticity. As these vorticity perturbations are advected downstream and compressed together. The interactions between perturbations generates acoustic (pressure) waves, which travel back toward the shock. The interaction of these pressure waves with the shock results in the deformation of the shock, creating (or at least sustaining), the misalignment of the shock normal and the accreting ﬂow. The process then repeats, resulting in a positive feedback loop.

The work of Blondin et al. (2003) applied this concept directly to core-collapse supernovae.

They produced two-dimensional models starting from convectively stable, steady-state supernova conditions (including the standing shock). These models used no neutrino heating and treated the inner boundary like a hydrostatic atmosphere, letting perturbations advect out of the domain.

They then sent perturbations from the upper boundary down through the shock.

The results of this study showed the AAC in full operation. The manifestation of the AAC in the supernova setting was characterized by a side-to-side sloshing motion, with large angular

9 velocities. This feature is now known as the standing accretion shock instability (SASI). As accretion progressed, they observed a buildup of turbulent energy in the post-shock region, resulting in a net-positive energy increase and an expansion of the supernova shock. At late times, the shock position took on a low mode (l = 1) conﬁguration, which was predicted to exist from linear perturbation studies performed by Houck and Chevalier (1991) over a decade earlier. Blondin et al.

(2003) themselves note that the simulations produced by Burrows et al. (1995) are done in one quadrant and therefore disallowed any low order modes from appearing.

With the work of Blondin et al. (2003), the ﬁeld now had two possible drivers for core-collapse supernova explosions. This discovery brought about a surge of SASI studies Foglizzo et al. (2006);

Laming (2007); Scheck et al. (2008); Foglizzo (2009) to determine whether the dominant process driving the explosion was neutrino driven convection or the more exotic SASI.

1.3.4 Late-term ccSNe explosions

Studies investigating the late-time evolution of ccSNe toward the remnant stage are few in number. The ﬁrst of such models were produced in two dimensions by Kifonidis et al. (2000) utilizing a two step process. From bounce to 1s was evolved using the Herakles code, including the eﬀects of neutrino physics and one-dimensional general relativity (GR). For later times, the

Herakles data was mapped to the Amra code and evolved for 300s, excluding neutrino and GR effects. These initial models indicated that the mixing of elements observed in SN 1987 A could be produced by hydrodynamic instabilities. However, their results failed to reproduce the mass ejecta velocities recorded by astronomers. Subsequent work by Kifonidis et al. (2006) expanded on their initial work in 2000 by evolving the models to 5h after bounce. These were the first multidimensional models produced in which the shock expanded past the photosphere. These results improved on the previous work, showing both qualitative remnant morphology and quantitiative Ni9 velocity similarity to SN 1987 A. The work of Gawryszczak et al. (2010) evolved two dimensional models for the first week after bounce, starting at 1s. The initial conditions were provided by Kifonidis et al. (2006), and subsequent evolution did not include neutrino transport or GR.

10 Hammer et al. (2010) produced the ﬁrst three dimensional explosions. Their work used the three dimensional, neutrino-driven simulations from Scheck (2007) as their initial conditions. The use of Scheck’s models as initial conditions was questionable, as these models are only evolved until

0.5 s after bounce, and consequently the explosion energy are not yet saturated. To correct for this,

Hammer et al. (2010) artificially inflated the energy distribution to reproduce typical supernova explosion energies. The bias caused from this intrusion is difficult to quantify. Additionally, their results were evolved to only 3h after bounce, making it difficult to infer the morphology of the resulting supernova remnant.

1.3.5 Current state of the ﬁeld

There is substantial controversy over the importance of physics processes involved in core- collapse supernova (ccSN) explosions (see, e.g., Janka et al., 2012; Burrows, 2013, and references therein). One problem is a multitude of relevant processes which include hydrodynamics, gravity, nuclear equation of state, and neutrino-matter interactions. Despite much eﬀort in the last four decades, no model accounting for these physics eﬀects exists that consistently produces energetic ccSNe explosions for suitable progenitor models. Early theories of ccSNe explosions such as neutrino-driven convection (see, e.g., Burrows and Goshy, 1993; Janka and Mueller, 1996; Mezza- cappa et al., 1998) and the more recent standing accretion shock instability (SASI) (Blondin et al.,

2003) remain the centerpiece of computational core-collapse supernova studies.

The SASI mechanism has been discovered in the course of numerical simulations in the work by Blondin et al. (2003). Subsequent SASI studies included theoretical analysis (Laming, 2007;

Foglizzo, 2009) and numerical investigations (Ohnishi et al., 2006; Blondin and Mezzacappa, 2007;

Scheck et al., 2008). This shock instability is characterized by global, low order oscillations of the supernova shock which generate substantial non-radial ﬂow in the post-shock region. These

ﬂuid motions result in increased residence times of the accreted material and possible increase the heating eﬀect by neutrinos emitted by the nascent proto-neutron star.

Since SASI is only one of the participating processes, it is often times diﬃcult to unambiguously quantify its contribution to the overall explosion process. Only recently has careful analysis been

11 attempted to quantify the role of SASI and compare its contribution to the other major participating process of neutrino-driven convection. In particular, SASI may only be present in progenitors with higher masses when neutrino-driven convection might be suppressed by increased accretion rates

(M¨uller et al., 2012, see also Bruenn et al. (2013)). Furthermore, the situation is additionally complicated by eﬀects due to model dimensionality observed in some simulations (Murphy and

Burrows, 2008; Nordhaus et al., 2010; Hanke et al., 2012; Dolence et al., 2013; Couch, 2013; Takiwaki et al., 2013). Those findings are, however, not confirmed by all studies (Janka et al., 2012). (In passing, we would like to note that is conceivable that the high frequency of SASI observations in two-dimensional simulations can be due to the assumed symmetry of such models. In either case, disentangling this kind of effects has not been attempted in a systematic manner.) There is also little known about the difference in the efficiency of convective heat transport between two and three dimensions, especially in the context of core-collapse supernovae. Although three-dimensional ccSNe explosion models have been available for some time (see, for example, Fryer and Warren,

2002), no systematic study aimed at comparing the convection eﬃciency between two and three dimensions exists.

1.4 Current work

In this work, we investigate the energetic explosions of a 15 M⊙ post-collapse WPE15 model of Bruenn (1993) in one, two, and three spatial dimensions. This model has been used extensively in the past by Kifonidis and his collaborators (Kifonidis et al., 2003, 2006; Gawryszczak et al.,

2010, see also Wongwathanarat et al. (2013)), mainly in application to post-explosion mixing of nucleosynthetic products. Here our focus is solely on the explosion mechanism. To this end, we tune the parameterized neutrino luminosity such that the explosion energy near saturation matches the observationally constrained energetics of SN 1987A. In the process, we have constructed a database containing more than 250 individual model realizations, diﬀering in model dimensionality, parameterized neutrino luminosity, and the small amplitude noise added to the initial velocity.

Obtaining such a large database of models was required given the extreme sensitivity of models to

12 perturbations, and one cannot draw conclusions about the nature of the explosion process except in a statistical manner. Furthermore, the observed diﬀerences between individual model realizations reﬂect the combined model sensitivity and may provide information about the role of, and coupling between, participating physics processes.

In Section 2 we describe the physical and numerical models adopted in our study. In Section

3.1 we present overall properties of the explosion models. In particular, we focus on dimensionality effects, provide estimates of neutron star kick velocities, and analyze the flow dynamics in the context of neutrino-driven convection and SASI. Next, in Section 3.2, we examine the role of convection in more detail. We analyze the energy flow in gain region using energy flux decomposition, and introduce and apply new methods for characterizing the morphology of the post-shock flow.

Section 3.3 discusses the potential role of turbulence in the explosion process, while in Section 3.3.3 we use the Lagrangian view of the ﬂow and track the energy histories of a large sample of ﬂuid parcels in the gain region in order to characterize conditions leading up to shock revival. Finally,

Section 3.4 contains our discussion of results and conclusions of this work.

13 CHAPTER 2

NUMERICAL MODEL

2.1 Hydrodynamics

We model the conservation of mass, momentum, energy, and electron fraction as

∂ρ + ∇· (ρu) = 0, (2.1) ∂t

∂ρu + ∇· (ρu ⊗ u)+ ∇P = ρ∇Φ+ Q , (2.2) ∂t − M ∂ρE + ∇· [u (ρE + P )] = ρu ·∇Φ+ Q + u · Q , (2.3) ∂t − E M ∂ρY e + ∇· (ρY u)= Q . (2.4) ∂t e N where ρ is the mass density, u the fluid velocity, P the thermal pressure, Φ the gravitational potential, E = u2/2+ e the specific total energy (with e being the specific internal energy), and

Ye is the electron fraction. The source terms QM , QE, and QN account for the eﬀects of neutrino- matter interaction (see, e.g., Janka and Mueller (1996)).

Equations (2.1)–(2.4) are evolved using the neutrino-hydrodynamics code of (Janka and Mueller,

1996; Kifonidis et al., 2003; Janka et al., 2003). The hydrodynamics solver of this code is built upon the Piecewise Parabolic Method (PPM) of Colella and Woodward (1984), and applies Strang splitting (Strang, 1968) to evolve multidimensional problems.

The numerical ﬂuxes inside grid-aligned shocks are calculated using the HLLE Riemann solver of Einfeldt (1988) to avoid the problem of even-odd decoupling (Kifonidis et al., 2006); otherwise, the hydrodynamic ﬂuxes are calculated using the Riemann solver of Colella and Glaz (1985) is used.

14 2.2 Equation of state

We adopt the equation of state (EOS) outlined by Janka and Mueller (1996). This EOS consists of contributions from free nucleons, α-particles, and a heavy nucleus in nuclear statistical equilibrium. Nuclei and nucleons are treated as ideal, nonrelativistic Boltzmann gases, with electrons and positrons contributing as arbitrarily degenerate and arbitrarily relativistic, ideal Fermi gases.

Thermal eﬀects between photons and massive particles are taken into account. Nuclear statistical equilibrium was assumed for T ≳ 0.5 MeV ( 5.8 109 K). Good agreement has been reported ∼ × between this EOS and the Lattimer and Swesty (1991) EOS for ρ ≲ 5 1013 g cm−3 (Janka and × · Mueller, 1996).

2.3 Gravity

The Newtonian gravitational potential ﬁeld is treated as a composite of a point mass ﬁeld

(for the excised PNS), and a distributed ﬁeld (for the mass on the mesh). The composite ﬁeld is computed using a multipole expansion of the Poisson equation for gravity,

2 ∇ ΦNewton = 4πGρ, (2.5) where G is Newton’s gravitational constant. Spherically symmetric eﬀects due to general relativity are accounted for using the eﬀective relativistic potential of Rampp and Janka (2002) such that,

Φ =Φ Φ1D +Φ1D , (2.6) final Newton − Newton TOV

1D 1D where ΦNewton and ΦTOV are the spherically symmetric Newtonian and Tolman-Oppenheimer- Volkoﬀ potentials, respectively. See Kifonidis et al. (2003); Marek et al. (2006); Murphy and Bur- rows (2008) for the details of this method. Thus, only in two-dimensions we account for deviations from spherical symmetry in the Newtonian part of the potential. Otherwise, both the Newtonian and general relativistic contributions are spherically symmetric.

15 2.4 Numerical grid

All simulations were performed in spherical geometry. In the radial direction we utilize 450 log- arithmically spaced zones. The inner boundary is located at a time-dependent location to mimic contraction of the collapsing proto-neutron star core (see Section 2.9 for details of this parameterization). The outer boundary is located at 4 109 cm. The choice of outer boundary radius ensures × that the total mass on the mesh remains essentially constant throughout the simulation. In multidimensional models we excise a 6cone around the symmetry axis in order to minimize potential numerical artifacts associated with the symmetry axis (see Scheck (2007) for justiﬁcation of this approach).

Multidimensional models have angular resolution of 2and 3in 2D and 3D, respectively. Addi- tionally, a representative 3D model was selected to conduct a series of coarse resolution simulations at 6 12 and 24(cf. Section 3.4).

2.5 Boundary conditions

We impose reﬂecting boundary conditions at the symmetry axis. In three dimensions, we use periodic boundaries in the azimuthal direction. The outer radial boundary is transmitting (zero gradient), while the inner boundary is an impenetrable wall with the radial pressure gradient matching the hydrostatic equilibrium condition.

In order to account for continued contraction of the neutron star, we parameterize the inner boundary position, rib, according to

i rib rib (t)= , (2.7) 1 + [1 exp( t/t )] ri /rf 1 − − ib ib ib − ( ) as described in Janka and Mueller (1996). The contraction of the proto-neutron star is parameterized by a characteristic timescale, tib. We make this parameterization under the assumption that even in fully consistent models, uncertain physics such as the neutrino-matter interaction rates and the (stiﬀness of) the nuclear equation of state will inevitably lead to uncertainties in the cooling

16 of the neutron star, and therefore its contraction timescale. Larger values of contraction timescale correspond to stiﬀer nuclear equations of state (slow contraction), while smaller timescales lead to softer nuclear equations of state (fast contraction).

In the present work, we consider families of both fast and slow contracting PNS core models, denoted as SC and FC models, respectively, to reﬂect the above uncertainty. The ﬁrst family are f the slowly contracting proto-neutron star models with rib = 15 km and tib = 1 s. The second family f are fast contracting proto-neutron star models with rib = 10.5 km and tib = 0.25 s. In both cases i we use rib = 54.47 km.

2.6 Neutrino physics eﬀects

In our implementation of neutrino energy deposition we use a light-bulb approximation (Janka and Mueller, 1996) with modiﬁcations introduced later by Scheck et al. (2006) (see below). Accord- tot,0 ingly, we impose an isotropic, parameterized neutrino luminosity, Lν . The initial total neutrino luminosity is chosen such that it asymptotically represents the gravitational binding energy released by the neutron star core (Janka and Mueller, 1996; Scheck et al., 2006; Scheck, 2007),

∞ ∞ tot,0 tot,0 ∆Ecore = Lν h(t)dt = 3Lν tib, (2.8) ∫0

The neutrino luminosity varies in time as

tot,0 Lνx (rib,t)= Lν Kνx h (t) , (2.9) where ν ν , ν ,ν , ν ,ν , ν . The K terms represent the constant, fractional contributions x ≡ e e µ µ τ τ νx from each neutrino type to the total neutrino luminosity. The h(t) function describes the temporal evolution of the neutrino luminosity and is given by Scheck et al. (2006)

1.0, if t tib, h (t)=  ≤ (2.10)  3/2 (tib/t) if t>tib.   17 2.7 Tracer particles

We utilize passive tracer particles to provide additional analysis of the ﬂow structure. Particles are initially distributed uniformly in Lagrangian mass coordinate. Each particle represents a ﬂuid

−6 −7 parcel with a mass of 1 10 M⊙ in two dimensions and 1 10 M⊙ in three dimensions × × (thus, the total number of particles is 7.6 105 and 7.6 106 in two and three-dimensional models, × × respectively). We note that our choice of distributing particles uniformly in Lagrangian mass results in large numbers of particles near the inner boundary due to high densities there. Consequently, not all particles sample the most dynamically active parts of the ﬂow. Therefore, we limit our analysis of particle-based data to those inside the gain region.

2.8 Initial conditions

For the initial conditions we use the post-collapse model WPE15 ls(180) of Bruenn (1993) based on the 15 M⊙ blue supergiant progenitor model by Woosley et al. (1988). We map the spherically symmetric collapse model to the grid and in multidimensions we add small velocity perturbations with relative amplitude 1 10−3 of the radial velocity. ×

2.9 Tuning explosion energies

In the current work we are interested in explosion models with energetics (with energy per unit mass) matching that of SN 1987A. Estimates of the SN 1987A explosion energetics are summarized

50 −1 in Table 2.1. Based on this summary, we adopt E/M = 0.75 10 erg M⊙ , which for the × adopted progenitor model corresponds to an explosion energy of approximately 1.05 1051 erg. [b] ×

We deﬁne the total model positive binding energy as

1 E = ρ u2 + ε +Φ (2.11) exp 2 ∑ [ ]

18 Table 2.1. Estimated explosion energetics of SN 1987A.

E/M Mej Eexp 50 51 (10 erg/M⊙) (M⊙) (10 erg)

Arnett (1987) 1–2 ··· ··· Chugai (1988) 1.3 ··· ··· Woosley et al. (1988) 0.8 0.8–1.5 ··· Shigeyama et al. (1990) 1.10 0.3 1.0 0.4 ± ··· ± Imshennik and Popov (1992) 0.75 0.05 1.05–1.2 ± ··· Utrobin (1993) 0.85 15–19 ··· Blinnikov (1999) 0.75 0.17 14.7 1.05 0.25 ± ± Utrobin (2004) 0.67 18 1.2 Utrobin (2005) 0.83 18 1.5 1.5 0.12 ± ± Utrobin and Chugai (2005) 0.83 18 1.5 Pumo and Zampieri (2011) 0.59 16–18 1.0

where the summation is taken over grid cells for which the binding energy is positive, 1/2u2+ε+Φ >

0. Here u is the velocity magnitude, ε is the internal energy, and Φ is the gravitational potential.

We deﬁne the explosion time as the moment when the positive binding energy exceeds 1 1048 × erg (Janka and Mueller, 1996). From that point in time on the total positive binding energy is identiﬁed as the explosion energy.

In order to match the model energetics with that of SN 1987A, we computed a large number of trial explosion models with various neutrino luminosities recording their explosion energies at the

ﬁnal time, t = 1.5 s. Once we identiﬁed the range of viable neutrino luminosities, we computed additional explosion models varying the seed velocity perturbations.

The above procedure follows from the realization that due to strong model sensitivity to small perturbations a unique mapping between the neutrino luminosity and the ﬁnal energy does not exist. Consequently, one can only study the model energetics in a statistical sense. Then, for a given neutrino luminosity one can consider an explosion model characterized by average properties with possibly large dispersion. Our numerical experiments indicate that one can produce equally energetic explosions for neutrino luminosities varying by as much as 3%.

19 CHAPTER 3

RESULTS

3.1 General properties of explosion models

We obtained a large database of supernova explosion models in one-, two-, and three-dimensions.

The database contained 26 spherically symmetric models, 169 2D models, and 61 3D models.

The individual model realizations diﬀered in mesh resolution, parameterized neutrino luminosity, and random perturbation pattern. Table 3.1 provides a summary of a subset of the database of explosion models that most closely match the energetics of SN 1987A. The subset contains 10 models in two-dimensions for both slow and fast contracting families, and 5 slow contracting models in three-dimensions.

The model explosion times vary from about 200 ms in the case of slow contracting models to slightly above 100 ms in the case of fast contracting models. Every family of models shows intrinsic dispersion in both the explosion times and the explosion energies. For example, the explosion times for slow contracting models vary by about 30% in 2D, and about 5% in 3D. The variations in explosion times are comparatively modest in the case of fast contracting models in 2D, with observed variations not exceeding 10%. The corresponding variations in the energetics are about

20% in 2D for slow and fast contracting models. The observed variations of our 3D models does not exceed 4%.

We discuss the accretion rates and shock radii in Sections 3.1.1 and 3.1.4, respectively.

3.1.1 Eﬀects of neutrino luminosity on explosion energy

Figure 3.1 shows the explosion energy in models with both slow and fast contracting PNS cores.

We show the mean explosion energy and the explosion energy envelope for 2D FC models (dash-dot),

20 Table 3.1. Parameters and main properties of the explosion models.

a b c d ˙ e exp Model Lνe texp Eexp Mexp rs 52 51 (10 erg/s) (ms) (10 erg) (M⊙/s) (km)

Slow contracting models (SC) - 2D

M194A 1.943 161 0.973 0.282 657 M194B 1.943 184 1.038 0.265 765 M194C 1.943 202 0.933 0.254 820 M194D 1.943 186 1.025 0.260 847 M194E 1.943 167 1.145 0.274 744 M194F 1.943 200 0.925 0.258 755 M194G 1.943 166 1.080 0.276 759 M194H 1.943 207 0.937 0.248 867 M194I 1.943 194 1.105 0.260 732 M194J 1.943 148 1.054 0.288 721

Slow contracting models (SC) - 3D

M187A 1.871 189 1.066 0.259 814 M187B 1.871 193 1.034 0.256 822 M187C 1.871 184 1.054 0.265 793 M187D 1.871 187 1.036 0.260 818 M187E 1.871 182 1.058 0.264 805

Fast contracting models (FC) - 2D

M352A 3.528 109 1.074 0.315 697 M352B 3.528 114 1.037 0.311 711 M352C 3.528 115 1.089 0.310 724 M352D 3.528 112 0.995 0.312 707 M352E 3.528 110 0.990 0.311 702 M352F 3.528 114 1.139 0.313 707 M352G 3.528 110 0.968 0.316 667 M352H 3.528 107 1.221 0.317 666 M352I 3.528 117 0.962 0.305 772 M352J 3.528 113 1.019 0.309 729

a Parameterized electron neutrino luminosity. The model anti-electron neutrino luminosity is 7.5% larger. b Explosion time determined when the total positive binding energy is greater than 1 × 1048 erg. c Total positive binding energy (i.e. explosion energy) evaluated at t =1.5 s post-bounce. d Mass ﬂow rate evaluated immediately upstream from the shock at

t = texp. e Average shock radius evaluated at t = texp.

1.2

] 1.0 g r e 0.8 51 10

[ 0.6 p x e

E 0.4

0.2 2D FC 2D SC 3D SC

0.0 0.2 0.5 0.8 1.0 1.2 1.5 t [s]

Figure 3.1 Explosion energy for models tuned to the energetics of SN 1987A (saturation at ≈ 1.05 1051 erg). Thick curves denote the average over all models in the group; thin curves denote × the respective minimum and maximum values. Note that the fast contracting proto-neutron star models begin unbinding material in roughly half the explosion time of the standard contracting models.

2D SC models (dash), and 3D SC models (solid). In all cases, the explosion energy rises rapidly at the onset of explosion and approximately saturates by the final time. In the case of SC models, the explosion energy threshold is reached between 140 ms to 200 ms post-bounce depending on model realization. The two dimensional models tend to explode faster and reach their final energies later than the three dimensional models. The FC models explode significantly earlier, at approximately

110 ms on average. The short explosion times found in FC models, as compared to SC models, are not unexpected. This is because signiﬁcantly greater neutrino luminosities, approximately by a factor of two, are required to energize the explosion in the presence of the deeper gravitational potential well in the FC models. Regardless of the contraction times of the PNS core, by the

ﬁnal time, tf = 1.5 s, and given the required explosion energetics and envelope mass, the average explosion energy in our models reaches about 1.05 1051 erg. × It is instructive to discuss our models in terms of the critical luminosity curve (Burrows and

Goshy, 1993). We shall note that ﬁnding the curve was not the goal of our study, and therefore, we are not in a position to provide the exact relation. Our results only provide an upper limit on the

3.5 1D Nordhaus Couch 3.5 1D Nordhaus Couch 2D Hanke Handy 2D Hanke Handy 3D 3D ] ]

s 3.0 3.0 s / / g g r r e e

52 2.5 52 2.5 10 10 [ [ e e ν ν L 2.0 L 2.0

1.5 1.5

0.1 0.2 0.3 0.4 0.5 0.6 100 300 500 700 900 1100 M˙ [M⊙/s] texp [ms]

Figure 3.2 Comparison of our explosion models with the results of critical neutrino luminosity studies conducted by other groups. The results of individual studies are shown with diﬀerent symbols, while model dimensionality is represented by line style (1D: solid; 2D: dashed; 3D: dash- dotted). (left panel) The dependence of the critical neutrino luminosity on the accretion rate at the time of explosion. We observe a systematic decrease in the required neutrino luminosity as the model dimensionality increases. (right panel) Parameterized electron neutrino luminosity as a function of explosion time. For a given dimensionality, our models tend to explode sooner and require lower neutrino luminosities. critical neutrino luminosity for the (relatively small) range of accretion rates present in our models

(see Table 3.1).

Figure 3.2 provides a comparison of our explosion models with the results of critical neutrino luminosity studies conducted by other groups. In the ﬁgure, the results of Nordhaus et al. (2010),

Hanke et al. (2012), and Couch (2013) are shown with open circles, triangles, and squares, respectively, while our data are shown with solid circles; the model dimensionality is indicated with line style with solid, dashed, and dash-dotted lines connecting models obtained in 1D, 2D, and 3D, respectively. In the left panel, the neutrino luminosity is shown as a function of the accretion rate, which is measured by integrating the mass ﬂux through the surface located just upstream of the shock at the onset of explosion.

This ﬁgure indicates that our choice to constrain the ﬁnal explosion energy by observations required a relatively narrow range of neutrino luminosities, compared to other studies. We also

ﬁnd that the required neutrino luminosity systematically decreases as the dimensionality of the model increases. Neutrino luminosities found in our models only provide an upper limit for the critical luminosity for the mass accretion rates present in our simulations. Furthermore, explosion

23 times found in our models are shorter than those of other groups for similar neutrino luminosities.

The simplest explanation for this finding is that the neutrino heating is more efficient in our models. However, we cannot exclude the possibility that this observed, systematic difference is due to differences in the progenitors. Detailed investigation of possible differences between our findings and those of other groups is beyond the scope of the current work.

3.1.2 Neutron star recoil

Since we excise the neutron star from the grid, a special method has to be used to estimate its recoil velocity. To this end, we use the approach of Scheck et al. (2006), and exploit momentum conservation. In this method, the momentum of the neutron star balances the momentum of the surrounding envelope (total momentum on the mesh in our simulations). An additional correction can be applied to the neutron star recoil velocities by considering that the neutron star will continue accreting material beyond the ﬁnal time in our simulations. We estimate the corrected velocity as

∞ 1.5 3−1.5 vH = vH + vH , (3.1)

1.5 3−1.5 where vH is the recoil velocity obtained in our simulations at tf = 1.5 s, and vH is the approximate correction due to late time accretion onto the neutron star. To calculate the velocity correction, we assumed the rate of the recoil velocity change obtained by Scheck et al. (2006) in his

B18 model series between the times t = 1.5 s and t = 3.0 s (see Figure 19 in Scheck et al., 2006),

3−1.5 vS 1.5−1 0.862. (3.2) ⟨vS ⟩ ≈

Then the recoil velocity correction has been computed as

3−1.5 3−1.5 vS 1.5−1 vH = 1.5−1 vH , (3.3) ⟨vS ⟩

In the above formulae superscripts denote post-bounce explosion times, and the subscripts S and

H denote the values from Scheck et al. (2006) and the current work, respectively.

24 500 2D SC 3D SC 400 ] s 300 / m [k

s 200 n v

100

0 0 500 1000 1500 t [ms]

Figure 3.3 Proto-neutron star kick velocity for our SC models, shown with dashed and solid lines, respectively. The outset shows the estimated saturation velocities based on estimates utilizing the results of Scheck et al. (2006).

Figure 3.3 shows the evolution of recoil velocities in our models until the ﬁnal time (left panel), and the estimated saturation velocities based on Equation 3.1 (right panel). Our results indicate relatively modest recoil velocities compared to Scheck et al. (2006). The approximate ﬁnal recoil velocities are in the range between 100 km/s and 300 km/s, with the lowest and highest recoil velocities of 30 km/s and 475 km/s. These results are in qualitative agreement with the results reported by Scheck et al. (2006, Figure 20) and Wongwathanarat et al. (2013, B-series models in

Table 2).

3.1.3 Gain region characteristics

In this section we report the results following the analysis of a quasi-steady state period in the pre-explosion epoch. If no such evolutionary stage is reached then the entire pre-explosion phase could be considered a transient phenomenon with little importance for the energetic core-collapse supernova explosions considered here. If the opposite is true however, we can adopt methods appropriate for analysis of quasi-steady state ﬂows and apply them to analyze the dynamics of the gain region.

25 0.18 1.00 2D FC 2D FC 2D SC 2D SC 3D SC 3D SC 0.75

] 0.12 ] ⊙ s / M ⊙ [ 0.50 n M i [ ga ˙ M

M 0.06 0.25

0.00 0.00 0 50 100 150 200 250 300 0 50 100 150 200 250 300 t [ms] t [ms]

Figure 3.4 (left panel) Mass in the gain region. The evolution of mass in the gain region is shown with solid lines for SC models (thin solid lines in 2D and thick solid lines in 3D) and dashed lines for FC models. Note that the mass in the gain region is, on average, greater in FC models than in SC models. Also, after the initial transient oscillations, the mass in the gain region stabilizes, indicating that evolution in the gain region has reached a quasi-steady state period. (right panel) Shock accretion rate. The line types are associated with model family and model dimensionality as in the left panel.Note that after the initial period of fast accretion the accretion rates progressively decrease in both families of models. The accretion rates are, on average, greater in SC models than in FC models after the initial transient. See Section 3.1.3 for details.

We identify the quasi-steady state period in the post-shock ﬂow evolution by calculating the mass contained in the gain region, Mgain. Its time-dependence is shown in the left panel of Figure 3.4 for our set of multidimensional models. As one can see, the mass in the gain region for the case of SC models (shown with solid lines) changes only slightly between t = 100 ms and t = 150 ms.

A similar period of modest increase in the mass of the gain region can be found in the case of FC models (shown with dashed lines) for times between t = 30 ms and t = 80 ms. We note that in both cases, the moment when the mass in the gain region increases precedes the explosion times, which is consistent with a scenario in which the shock revival process occurs over an extended period of time in a quasi-steady state fashion. Also note the presence of strong oscillations in the gain region mass at early times in both families of models. Since, as we discuss below (Section

3.1.4), the shock radius is increasing steadily at these early times, the observed oscillations in the gain region mass are either due to density changes of the material entering the gain region or the changes in the position of the gain radius during these times. Those oscillations, however, have a

26 transient character and do not play any role in the subsequent evolution of the system, and cease as soon as the ﬂow in the gain region becomes multidimensional.

The mass accretion rate (shown in the right panel of Figure 3.4) as measured at the position of the shock front shows a rapid decline during the first 50 ms of the simulation time after which it steadily decreases at a progressively slower rate. The evolution of accretion rate does not differ significantly between SC and FC models. This is expected, as we consider only a single progenitor model. At later times when the quasi-steady state is reached, the accretion rates are between

−1 −1 0.29 0.35 M⊙ s for SC models and between 0.40 0.67 M⊙ s for FC models. The FC models − − are characterized by consistently lower accretion rates than the SC models for times later than 50 ms.

To better understand the dynamics in the gain region in the context of neutrino heating, we consider the advection timescale, τadv,

Mgain τadv = , (3.4) M˙ where Mgain is the mass in the gain region, and M˙ is the accretion rate. The advection time scale is the characteristic time a fluid parcel spends in the gain region. This quantity neglects multidimensional effects, which are known to be important in the process of shock revival. In the same context, we consider the heating efficiency, η,

τ η = adv , (3.5) ρQdV˙ ∫ where Q˙ is the net neutrino energy deposition rate, and the integral in the above equation is taken

1 2 over the gravitationally bound ( 2 u + ε +Φ < 0) material inside the gain region. The quantity in the denominator of the above equation is the heating timescale, i.e. the time it takes to heat a gravitationally bound ﬂuid parcel so that it becomes unbound. The heating eﬃciency is a measure of the competition between the advection and heating processes. In particular, η > 1 implies that the material in the gain region gains energy due to heating faster than it is removed by the advection process.

27 500

400 ]

s 300 m [ v d

a 200 τ

100 2D FC 2D SC 3D SC 0 0 50 100 150 200 250 t [ms]

Figure 3.5 Advective timescale for the gain region. The oscillatory behavior of the advective timescale at early times for SC models likely has the same origin as the transient oscillations observed in the gain region mass. Note that the advective times are similar between model realizations in the same family. The diﬀerences in advective times for diﬀerent model realizations are particularly large for 2D SC models that develop a few large-scale nonuniformities in the gain region. On average, the advective times are longer for FC models than for SC models. See Section 3.1.3 for details.

The time evolution of advective time for our sample of models is shown in Figure 3.5. Since the advective time depends on the mass in the gain region, the oscillations in the advective time visible in Figure 3.5 prior to 50 ms are simply a reflection of the oscillations in the gain region mass (c.f. left panel in Figure 3.4). At later times, both SC and FC families (shown with solid and dashed lines in Figure 3.5) show qualitatively similar behavior. Again, this is expected due to a common progenitor model used in these calculations. Also, and as we discussed above, the variations between model realizations are due to differences that develop in multidimensional flow structure during the shock revival process.

We can compare the advective time to the length of the quasi-steady state phase of the evolution.

For the SC models, we estimate the steady state lasts about 50 ms, while the advective time in this case is about 160 ms. In the case of FC models, we estimate the steady state lasts about 50 ms, but the advective time is somewhat shorter (about 100 ms). We conclude that in our models, on average, ﬂuid parcels enter the gain region and reside in it during the entire process of shock

2D FC 9 2D SC 3D SC

η 5

0 50 100 150 200 250 t [ms]

Figure 3.6 Heating efficiency for the gain region. The evolution of heating efficiency is shown for SC and FC models with solid and dashed lines, respectively. The results from the 2D SC models are shown with thin solid lines. Note that the heating efficiency in the FC models is always greater than one. This indicates that a strong, immediate neutrino heating is required in order to produce energetic explosions in this case. On the other hand, the heating becomes efficient in the SC models only once the quasi-steady state is established. Again, as in the case of the mass in the gain region and advective times, the evolution of heating efficiency is qualitatively similar between SC and FC families. revival. This conclusion is consistent with the history of individual fluid parcels (see Section 3.3.3 below).

The advective times found in our simulations are 3 to 10 times larger than those reported by

Müller et al. (2012). We initially suspected that this significant difference in advective times might be due to general relativistic effects in the potential carefully accounted for by Müller et al. in their model. However, we believe the most probable explanation is the difference in the model energetics obtained in the two studies. This is because in the models of Müller et al. the shock resides during the shock revival phase for long times at much lower radii ( 100 km) than in our ≈ models ( 400–500 km). Therefore, provided the accretion rates are similar, the gain region has ≈ much lower mass in their models, resulting in correspondingly shorter advective times.

The evolution of heating eﬃciency is shown in Figure 3.6. It is interesting to note a qualitative similarity between the SC and FC model families (shown with solid and dashed lines in the ﬁgure).

29 For example, we note a period of modest increase in heating efficiency between t = 20 ms and t = 50 ms in FC models, and t = 50 ms and t = 90 ms in SC models. This increase in the heating efficiency is associated with the onset of convection prior to the quasi-steady state phase (see Section 3.2). At later times, after the quasi-steady state is established, the heating efficiency continues to steadily increase in the SC models while the increase in the heating efficiency is significantly greater in the FC models. Nevertheless, the heating efficiency continues to evolve in a qualitatively similar manner in both families of models. As we noted before, the similarities between SC and FC models can be easily explained by the fact that we consider only a single progenitor model in this work.

On the quantitative level, the heating eﬃciency in the FC models is greater than one from the beginning of the evolution. On the other hand, the heating becomes eﬃcient in the SC models only after the quasi-steady state is established. We conclude that a strong, immediate neutrino heating is required in order to produce energetic explosions.

3.1.4 Shock evolution

The shock revival process can also be studied by analyzing the time evolution of the shock radius. In particular, prolonged periods of shock stagnation may indicate the onset of the standing accretion shock instability (SASI). Conversely, a steady increase in the shock radius may indicate that other processes (e.g. neutrino-driven convection) operate eﬃciently. Additionally, the unbinding of shocked matter is directly dependent on radius (through the gravitational potential), and large shock radii may aid in reaching the explosion threshold. Finally, hydrodynamic perturbations present at early times can be imprinted on the shock and aﬀect the morphology of the supernova ejecta. To this end, in the following discussion we use the shock aspect ratio to quantify the explosion asymmetry.

The evolution of the average shock radius, Rs, in our models is shown in Figure 3.7 with solid and dashed lines for the SC and FC models, respectively. The shock radius increases steadily in both families of models, although the shock expansion is about 25% faster in the FC models. At the time when the early convection sets in, the shock expands to about 270 km in the FC models and about 200 km in the SC models. By the time the shock is revived, its average radius for diﬀerent

30 3000 2D FC 2D SC 2500 3D SC

2000 ]

m 1500 [k s r 1000

500

0 0 50 100 150 200 250 t [ms]

Figure 3.7 Evolution of the average shock radius. The average shock radius is shown with solid and dashed lines for SC and FC models, respectively, with 2D SC models shown with thin solid lines. Shaded areas denote regions where (texp, rs) pairs for SC and FC model families are located. Note that the FC and SC models explode at similar shock radii but at significantly different times. The shock expansion rate for FC models is, on average, twice that of the SC models. model realizations varies between 645 km and 855 km in the 2D SC models, and between 790 km and 810 km in the 3D SC models. We believe the greater variation in the average shock radius in the 2D models compared to 3D may hint at differences in the dynamics of the gain region between

2D and 3D. Moreover, 2D FC models display a similar amount of variation in the average shock position as 2D SC models. Also, in that case, the average shock position at the time of explosion varies between 640 km and 730 km. Overall, the average shock radius at the time of explosion is similar in SC and FC families. One should keep in mind, however, that the FC models explode, on average, twice faster than the SC models (c.f. Table 3.1).

max min The shock aspect ratio, rs /rs , found in our models (see Figure 3.8) appears relatively modest compared to the shock asymmetry reported in other studies (Hanke et al., 2012; M¨uller et al., 2012). Although an extreme shock aspect ratio of 2 was found in 2D SC simulations, the average aspect ratios are only 1.25 in 2D and 1.4 in 3D. This should be compared to the shock aspect ratios between 3 and 4 we estimated based on the data presented by M¨uller et al. (2012).

31 2.0 2D FC 2D SC 3D SC 1.8 , n i 1.6 m s r / x a m s 1.4 r +

1.2

1.0 0.0 0.5 1.0 1.5 t [ms]

Figure 3.8 Shock aspect ratio for our models. Thick lines denote the average over all models in the group. Thin lines denote the minimum and maximum values for the particular families. Our results show relatively mild asymmetries (rmax/rmin 1.3). Visual inspection of SASI-producing s s ≈ models from M¨uller et al. (2012) show aspect ratios upwards of 3 to 4.

The most striking feature of the evolution of the shock aspect ratios in our models are strong asymmetries observed in the 2D SC models at early times. These asymmetries seem to rapidly decrease for times > 400 ms. In contrast, the shock aspect ratios in the 2D FC models appear to evolve more smoothly and the most deformed FC models seem to retain their asymmetric character as time progresses. Qualitatively, however, the degree to which the shock is deformed in 2D models is modest and on average is 1.3. In 3D models, the average degree of shock asymmetry at late ≈ times is somewhat greater ( 1.4), and the variations between the individual 3D model realizations ≈ are less than 20%.

We speculate that the smooth evolution and persistent character of deformations found in the 2D

FC models might be due to weaker convection which operates over a shorter period. Conversely, in

2D SC models convection has more time to organize the ﬂow inside the gain region. Occasionally, the dominant l = 1 mode develops in those models (cf. Figure 3.10(a)) and results in a strong asymmetry. In other 2D SC models, the higher order modes make the shock more spherical (cf.

32 Figure 3.10(b)). The situation is similar in the case of 3D SC models, in which the post-shock region shows richer convective structure and evolves on a similar timescale as in 2D.

The evolution of shock radius also provides information that is helpful in the context of the standing accretion shock instability (SASI). Speciﬁcally, SASI manifests itself as low-order (l = 1, 2) oscillations of the shock radius (Foglizzo et al., 2006; Laming, 2007; Scheck et al., 2008; Foglizzo,

2009). Therefore, the ﬁrst step in the SASI analysis of the shock revival is to decompose the shock radius in terms of spherical harmonics. The expansion coeﬃcients are given by

∗ alm = Ylm (θ,ϕ) f (θ,ϕ) dΩ, (3.6) ∫Ω

∗ 2l+1 (l−m)! where the spherical harmonic Ylm is normalized by a factor 4π (l+m)! , and f (θ,ϕ)= rs (θ,ϕ). In order to enable direct comparison between the 2D (m√= 0) and 3D (m = l...l) cases, we − consider a suitably normalized contribution of the m-mode coeﬃcients (Ott et al., 2013),

l 1 2 αl = alm. (3.7) a00 v um=−l u ∑ t

The evolution of α1 and α2 coeﬃcients is shown in Figure 3.9. The larger values of the coeﬃcients indicate that the 2D SC models, on average, exhibit stronger variability than their three-dimensional counterparts. However, we also observe a certain dichotomy among 2D SC models with one sub- group showing variability distinctly larger than the remaining set of models. 2D SC models also show relatively weak l = 1 mode contributions. perturbations, while others rise to a moderate fraction of the average shock position. The 2D FC models (data not shown in Figure 3.9) show a similar degree of variability as the 2D SC models at all times.

Prior to explosion, the 2D models exhibit at most one or two weak oscillations. This is a quantitatively different behavior than found in the simulations of Müller et al. (2012), who reported strong, multiple cycles leading up to shock revival. In addition, the shock perturbations in our models are weak compared to those considered as evidence for SASI. Furthermore, we do not find

33 0.20 0.20 2D SC 2D SC 3D SC 3D SC

0.15 0.15

1 0.10 2 0.10 α α

0.05 0.05

0.00 0.00 0 50 100 150 200 250 0 50 100 150 200 250 t [ms] t [ms]

Figure 3.9 Evolution of the leading coefficients in the spherical harmonic decomposition of the shock radius. Only the results of the decomposition for SC models are shown. The runs of the α1 (left panel) and α2 (right panel) coefficients are presented with solid and dashed lines for 3D and 2D models, respectively. Prior to the onset of convection (t 50 ms), the shock front remains ≈ essentially spherically symmetric. The emergence of low order perturbations around t 80 ms is ≈ due to buoyant convective plumes deforming select sections of the shock front rather than SASI. Note that the degree of shock perturbation is relatively modest compared to some recent results (see, e.g., Müller et al., 2012). qualitative differences between the 2D and 3D models, as we mentioned above. Note that most evidence for SASI presented in the literature is essentially restricted to 2D, non-exploding models.

In our SC models, the low order modes emerge at about t = 75 ms, shortly before the quasi- steady state is established in the gain region. Visual inspection of the ﬂow morphology during the quasi-steady state provides no evidence for large-scale “sloshing” motions in the gain region, considered a deﬁning signature of SASI. We conclude it is unlikely the SASI plays any important role in the evolution of our models.

3.1.5 Morphology of the gain region

As soon as the neutrino-driven convection sets in and the related perturbations reach the shock, one faces a difficult problem of disentangling various physics processes participating in the shock revival, including fluid instabilities and neutrino-matter interactions. Analysis of the overall flow morphology is the first step in the process of understanding the explosion mechanism. It provides initial evidence for the physics of fluid flow participating in re-energizing the stalled shock (Herant et al., 1994). It also may provide evidence for the possible role of model parameters, such as assumed

34 symmetries and discretization errors, e.g. near the symmetry axis (Scheck et al., 2006; Gawryszczak et al., 2010), on the ﬂow dynamics. In this section we present the morphological evolution of our

SC models in two and three dimensions. In particular, we are interested in identifying when the

ﬂuid ﬂow instabilities imprint their structure on the just-formed inner core of the supernova ejecta.

Entropy pseudocolor maps of the gain region for two two-dimensional and two three-dimensional

SC models at their respective explosion times are shown in Figure 3.10. Prior to reaching the explosion threshold, the 2D SC models evolve to have either dipolar (l = 1, Figure 3.10(a)) or quadrupolar (l = 2, Figure 3.10(b)) m = 0 ejecta morphology. These flow features emerge when strong downflows in the gain region are formed shortly after the onset of convection. Furthermore, our models indicate that these flow structures do not evolve once they set in and continue to persist after the explosion is launched, evolving in a self-similar fashion at later times (see below).

Additionally, these downﬂows carry accreted material deep into the gain region close to the gain radius. This behavior does not seem to be as extreme in 3D SC models, because a comparable amount of accreted material is transported through many more downﬂows (see Figure 3.10(c) and

(d)). The presence of many downflows in three-dimensions reflects the fact that the flow structure is inherently different in 2D and 3D. We quantify this behavior later in Section 3.2.

The entropy distribution in the 3D SC model M187A is shown in Figure 3.11 for select times. At early times, the flow evolves away from initially spherically symmetric accretion, with the first signs of convective instability emerging around t = 50 ms (Figure 3.11(a)). At this time, the shock is not affected by convection. At intermediate times (t = 100 ms, Figure 3.11(b)), the initial convective plumes gradually merge into larger structures and begin deforming large segments of the shock front. As time progresses, the process of bubble merging continues (Figure 3.11(c)) and the shock eventually is launched around t = 189 ms (Figure 3.11(d)).

The large-scale morphology of the post-shock region appears essentially frozen after the explosion commences. For example, three ejecta plumes seen in the southern hemisphere (Figure

3.11(d)), can still be easily identiﬁed more than 300 ms later (Figure 3.11(e)). However, during the same period the convective bubbles in the northern hemisphere show signiﬁcant evolution and

35 (a) (b) ] ] km km [ [ r r

M194J M194C texp = 148 ms texp = 202 ms

z [km] z [km]

M187A M187B texp = 189 ms texp = 193 ms

x [km] x [km]

Figure 3.10 Entropy distribution in the post-shock region at the explosion time for select two- dimensional and three-dimensional models. Entropy distribution is shown with pseudocolor maps for models M194J (panel a), M194C (panel b), M187A (panel c), and M187B (panel d). The entropy distribution maps for two-dimensional models (panels (a), (b)) are reﬂected across the symmetry axis. For three-dimensional models, the entropy is shown for a slice through the computational domain at the equatorial plane. In two dimensions models tend to have either dipolar (panel a) or quadrupolar (panel b) ejecta structures which result in an oblate shape of the shock. In three dimensions the ejecta appear to have more regular structure, and the shock is less deformed. See Section 3.1.5 for discussion. appear to merge into a single structure. This morphology persists until the end of the simulation and the central region becomes ﬁlled with the neutrino-driven wind (Figure 3.11(f)).

36 (a) (b) (c) ] ] ] cm cm cm 5 5 5 10 10 10 [ [ [ y y y

t=50ms t = 100 ms t = 150 ms x 105 cm x 105 cm x 105 cm [ ] [ ] [ ] (d) (e) (f) ] ] ] cm cm cm 6 6 7 10 10 10 [ [ [ y y y

t = 189 ms t = 500 ms t = 1500 ms x 106 cm x 106 cm x 107 cm [ ] [ ] [ ] Figure 3.11 Distribution of entropy in the 3D SC model M187A. The entropy is shown with pseudocolor maps for a slice through the computational domain at the equatorial plane. The entropy distribution is shown at t = 50, 100, 150, 189, 500, and 1500 ms in panels (a)-(f). The development of convection can be seen as early as t = 50 ms (panel a). The ﬂow structure becomes progressively more complex at intermediate times (panels (b) and (c)) and the explosion is launched at t = 189 ms (panel (d)). Relatively little change in the structure of the gain region takes place at later times (panels (e) and (f)).

3.2 Convection

Our analysis of the explosion process presented above revealed that the explosion is preceded by a relatively short (compared to other studies) quasi-steady state inside the gain region (Section

3.1.3) during which the shock only slowly expands and SASI does not seem to a major role (Section

3.1.4). We also found that the heating becomes eﬃcient shortly before the quasi-steady state is established (Section 3.1.3).

There exists a large body of evidence from numerical simulations of core-collapse supernova explosions indicating that multidimensionality, and speciﬁcally the presence of convection, decreases

37 the requirements for the neutrino luminosity produced by the contracting core of the proto-neutron star (Janka and Mueller, 1996; Nordhaus et al., 2010; Hanke et al., 2012; Couch, 2013). In this section we focus on characterization and quantiﬁcation of the eﬀects of neutrino-driven convection during the quasi-steady state of the gain region. We introduce and apply novel methods dedicated to investigating this aspect of the supernova explosion process.

3.2.1 Minimum resolution requirements

We perform a series of three-dimensional simulations in order to assess the dependence of model physics operating inside the gain region on numerical resolution. In particular, we are interested in

ﬁnding conditions required to suppress neutrino-driven convection on large scales. If indeed convection is a critical component driving the explosion, one could expect that by suppressing convection

(by whatever means) the explosion would not occur, even in models with neutrino luminosities that in other situations are suﬃciently large enough to revive the shock. This expectation is strongly supported by numerous calculations which show that, for example, much higher luminosities are required to produce an explosion in one-dimension than in multidimensions (see, e.g., Janka and

Mueller (1996), Section 3.1.3, and Figure 3.2 for discussion and compilation of relevant recent results).

In our series of 3D simulations, we keep the radial resolution ﬁxed while the angular resolution is gradually decreasing by a factor of 2 between the models, from 3◦ (our base resolution) down to

24◦. In addition, we compute a 2◦ resolution model. For each model, we recorded the explosion times and explosion energies at t = 250 ms. The basic characteristics of models obtained in this series is given in Table 3.2. Inspection of data shown in the table reveals that the explosion times and the explosion energies change little as long as the angular resolution is not worse than 6◦. For coarser angular meshes, we observe a signiﬁcant increase in the explosion times and decrease in the explosion energies. No explosions were found at resolutions coarser than 24◦.

It is conceivable that the observed changes in global model characteristics should be correlated with changes in the ﬂow structure inside the gain region. Figure 3.12 shows the entropy maps in a subset of the models used in this resolution study. One can easily note profound structural

38 Table 3.2. Characteristics of the M187A model series with varying angular resolution.

Angular texp Eexp (t = 250 ms) Resolution (ms) (1051 erg)

2◦ 183 0.114 3◦ 183 0.107 6◦ 175 0.106 12◦ 198 0.077 24◦ 216 0.028

changes in the morphology of the gain region as the angular model resolution decreases. Large- and small-scale convective bubbles can clearly be seen in the 3◦ resolution model (panel (a)). There are significantly fewer small, convective bubbles present in the 6◦ model (panel (b)). In those two models the overall shock radius appears comparable. In contrast, no well-defined, small-scale bubbles can be identified in the 12◦ or 24◦ models (shown in panels (c) and (d), respectively). In addition, the shock radius is visibly smaller in the 24◦ model.

We interpret the observed dependence of energetics and explosion timing on angular resolution described above as follows. Consider that the hydrodynamic solver used in our simulations is the PPM scheme (Colella and Woodward, 1984). PPM is nominally third order (and practically second order) accurate in space, and uses piece-wise parabolic interpolation to describe proﬁles of the hydrodynamic state. Therefore, the minimal number of mesh resolution elements required to resolve a convective bubble is about 3 mesh cells. In our 12◦ and 24◦ resolution models there are only 13 and 7 cells in angle, respectively. This implies our hydrodynamic solver can only represent between 2 to 4 large-scale structures in those models. The number of those structures does not represent the actual number of bubbles, as some mesh cells are also required to describe the ﬂow structure between the bubbles. This explains the lack of bubbles on smaller scales and the overall degraded appearance of the neutrino-driven convection in these cases.

We demonstrated through this resolution study that the neutrino-driven convection is a critical component of the explosion mechanism in our models. Speciﬁcally, we have shown that one can turn

39 (a) (b) ] ] km km [ [ y y

3◦ 6◦

x [km] x [km]

12◦ 24◦

x [km] x [km]

Figure 3.12 The distribution of entropy in the M187A series of models with varying angular resolution. The entropy is shown for a slice taken at the equatorial plane at t = 100 ms in models with angular resolutions of (a) 3◦, (b) 6◦, (c) 12◦, and (d) 24◦. Note the overall appearance of convection is degraded as the angular resolution decreases, indicating a drop in the efficiency of the neutrino-driven convective engine. The shock radius is significantly smaller in the coarsest model. See Section 3.2.1 for discussion. an energetic, multidimensional model into a failed multidimensional model simply by degrading its angular resolution. Therefore, to correctly capture the efficiency of the convective engine one needs to resolve its basic components. This picture is also consistent with the analysis of Herant et al.

(1994), who argued that, independent of dimensionality, large scales will play the dominant role in convection. Furthermore, it is important to resolve not only structures on the largest scales, but also the structures on 1◦ are important. We conclude with the somewhat obvious statement ∼

40 1.00

0.75

up 0.50 ˆ m

0.25 2D FC 2D SC 3D SC 0.00 0 100 200 300 400 t [ms]

Figure 3.13 The temporal evolution of the upflowing mass in the gain region. The fraction of the total mass inside the gain region with (ur > 0) is shown with dash-dotted, dashed, and solid lines for 2D FC, 2D SC, and 3D SC models, respectively. See Section 3.2.2 for discussion. that capturing the relevant physics of the problem requires adequate numerical resolution. It is conceivable that the next generation of ccSNe models with much higher resolution will begin uncovering new physics effects that cannot be observed in the current generation of models due to their insufficient quality.

3.2.2 Dynamics

One of the quantities of interest in the context of dynamics of convection is the amount of mass contained inside rising, convective bubbles. To characterize that quantity, we introduce the upﬂow mass fraction,

ρ ur>0dV gain | mˆ up (t)= ∫∫∫ , (3.8) Mgain where the integration is performed over the gain region, and takes into account only the fluid elements (grid zones) with positive radial velocity. Thus, the upflow mass fraction characterizes the mass inside the gain that is moved away from the proto-neutron star toward the shock. The evolution of the upflow mass fraction in our models is shown in Figure 3.13. As one can see, the initial behavior of the upflowing mass is characterized by a transient period until t 50 ms. Soon ≈

41 after that, the amount of upflowing mass begins to steadily increase. It levels off during the quasi- steady state phase lasting approximately between 85–140 ms for SC models (shown with solid lines in Figure 3.13) and 65–105 ms for FC models (shown with dashed lines in Figure 3.13), and rises again after the shock is launched. Interestingly, the net mass flux inside the gain region is close to zero in the SC models (the horizontal dashed line atm ˆ up = 0.5), while upflows dominate in the FC models. This can be qualitatively understood considering that energetic explosions in the case of fast contracting proto-neutron star cores require much higher neutrino luminosities in order to unbind a sufficiently large amount of material inside the gain region. This indicates that perhaps convection plays a smaller role in FC models compared to SC models. Also note that the amount of upflowing mass begins to differ between various model realizations shortly before the quasi-steady state is established. Those variations are greater in 2D than in 3D (for SC models). Ultimately, after the explosion is in progress, the amount of upflowing mass steadily increases and eventually levels off, signifying the fact of the overall expansion of the material inside the gain region.

It is interesting to note that the observed behavior in the upflowing mass fraction, in particular its steady increase after the transient period, is closely correlated with the evolution of the heating efficiency (cf. Figure 3.6). For example, the heating efficiency becomes greater than 1 in the SC models before the amount of upflowing material becomes comparable to the amount of accreted material. In addition, the heating efficiency is always greater than 1 in the case of FC models, where upflows dominate at all times.

Figure 3.14 shows the time evolution of the fluctuations in the upflowing mass in the gain region in the 3D SC models as a function of time around the quasi-steady phase. It is interesting to note that the increase of the outflowing mass in these models is non-monotonic. One can identify several short (∆t 10 ms) episodes during which the amount of mass in upflows fluctuates. These ≈ fluctuations have an amplitude of several percent. Given that the typical number of convective bubbles, and therefore also the number of downflows, in those models during the quasi-steady state phase is about 10 to 25 (see Section 3.2.3), it is conceivable that perhaps individual downflows are responsible for the observed fluctuations.

42 0.10

0.05 up

ˆ 0.00 m ∆

−0.05

−0.10

0.08 0.10 0.12 0.14 0.16 0.18 0.20 t [ms]

Figure 3.14 The evolution of relative mass fluctuations inside the gain region shortly before and during the quasi-steady state phase in the 3D SC explosion models. Fluctuations are calculated relative to a linear fit of the upflowing mass in the gain region from the beginning of the quasi- steady state phase until the explosion time, where individual curves terminate. (thin solid) M157A; (medium solid) M157B; (thick solid) M157C; (dashed) M157D; (dotted) M157E. See Section 3.2.2 for discussion.

3.2.3 Characterization of convective structures

Having in mind the continuing debate regarding the role of dimensionality in numerical models of ccSNe explosions, we seek information about the dependence of large-scale structure of convection inside the gain region on model dimension. As the ﬁrst step in our analysis, we wish to measure the net radial momentum over a fraction of the gain region,

rmid(θ,φ)+αw 1, ρurdr > 0 − Imom (θ,ϕ)=  rmid(θ,φ) αw (3.9)  ∫ 0, otherwise   where rmid = (rs + rg) /2 is the angle-dependent midpoint between the shock and gain radii (rs and r , respectively), w = r r is the angle-dependent gain region width, and α is a fractional g s − g oﬀset between 0 and 0.5 which controls the extent of the gain region included in the integration.

We use the above equation to project average properties of the radial flow and distinguish between regions dominated by either upflows (Imom = 1) or downflows (Imom = 0). Recall that our models

43 1.00

0.75 i Ω 0.50 X π 1 4

0.25

2D SC 3D SC 0.00 0 100 200 300 400 500 t [ms]

Figure 3.15 Estimate of the total solid angle spanned by buoyant, rising material. The onset of convection at t 50 ms results in a rapid increase in the spanned solid angle as the ﬁrst convective ≈ plumes begin to form. During the quasi-steady convective phase the plumes continue to grow. By the time the explosion is launched they cover roughly 2/3 of the total solid angle. See Section 3.2.3 for discussion. are essentially free of possible SASI contributions, and therefore this method should be applicable to any convective ﬂows.

Apart from the situation when the shock and gain radius are constant, the radial extent of the integration bounds in the above equation will vary with angle. In this way, we avoid possible ambiguities that may arise near those two regions, and make the results of this method insensitive to ﬂow asymmetries naturally developing especially along the shock front. In the above equation we heuristically set α = 0.1 so that our results are based on the data over the middle 20% of the gain region (at a given angle).

We use the maps produced with the help of the above equation to create connected clusters of upflow-like and downflow-like sections. We identify the upflow-dominated clusters with the rising convective bubbles. The solid angle spanned by each bubble is simply computed by integrating the area of the upflow-dominated cluster.

The time evolution of the total solid angle occupied by bubbles for the SC models is shown in

Figure 3.15 (with data for 2D and 3D models shown by dashed and solid lines, respectively). Prior

44 to the onset of convection (t < 50 ms), the whole gain region oscillates radially. As convective structures begin to emerge, the solid angle occupied by bubbles starts to increase. This process starts to slow down at the beginning of the quasi-steady state phase (t 80 ms). However, and ≈ unlike in the case of other diagnostics we have discussed above, the evolution of the solid angle occupied by bubbles does not provide any clear signature of the explosion time. Rather, the growth continues at a similar pace until about 150 ms after the explosion commences. It starts to level off soon after that time, and by 400 ms after bounce the whole gain region is filled-in with upflowing material.

Apart from large-amplitude oscillations, the evolution of the solid angle occupied by bubbles in

2D SC models is qualitatively similar to that seen in 3D, with the average value similar between the models. However, in a few 2D cases persistent downﬂows develop (see also Section 3.1.5). For these models the solid angle occupied by bubbles does not reach 1 by the end of the period we analyzed here. Note that our diagnostics can be use to identify situations when strong accretion onto the proto-neutron star continues at late times.

Figure 3.16 shows the time evolution of the number of bubbles inside the gain region averaged separately for 2D and 3D SC models (shown with dashed and solid lines, respectively). Initially, our method identifies only a single bubble or finds no bubble during the initial transient (t < 50 ms), in agreement with the radial oscillations of the gain region during that period as we discussed earlier. Soon after convection sets in, the number of distinct upflows sharply increases, reaching

12 in 2D and 200 in 3D. (Note, for the purpose of presentation we have limited the scale in ≈ ≈ Figure 3.16 to 25.) Given that at those early times the solid angle occupied by bubbles is small, we conclude that the bubbles are initially small and grow in angular extent over time. By the time of the explosion, the bubbles are roughly 4 times smaller in 3D than in 2D.

Recall that in 2D the bubbles are truly tori rather than quasi-spherical plumes. We recognize this may have certain implications for the bubble dynamics (see, e.g., Kifonidis et al., 2003; Hammer et al., 2010; Couch, 2013). Furthermore, the observed evolution of bubble sizes from small to large

(see also Herant et al., 1994) oﬀers an interesting parallel between the evolution of the structure of neutrino-driven convection (and perhaps convection in general), and that of bubble merger observed

45 25 2D SC 3D SC 20

15 i N h 10

0 0 100 200 300 400 500 t [ms]

Figure 3.16 Evolution of the number of rising bubbles inside the gain region. The number of bubbles is averaged separately for 2D and 3D SC model realizations (shown with dashed and solid lines, respectively). For the clarity of presentation, the vertical scale is limited to N = 25. The ⟨ ⟩ maximum average number of bubbles found in 3D is 200. See Section 3.2.3 for discussion. ≈ in a multi-mode Rayleigh-Taylor instability (see, e.g., Alon et al., 1994; Miles, 2004). The study of the process of merging convective bubbles is beyond the scope of the present work.

The growth of convective bubbles after t = 50 ms coincides with time at which the heating eﬃciency starts to increase (cf. Figure 3.6) and accumulation of mass in the gain region (cf. Figure

3.4). We believe this indicates that early, small-scale convection begins to interact with the incom- ing accretion ﬂow, resulting in rather rapid increase in the gain region mass. Subsequently, the advection time scale also increases. We believe this provides evidence for the direct dependence of the heating eﬃciency on the intensity of convection.

3.2.4 Convective energy transport

In our discussion of convective energy transport, we adopt the approach originally introduced for the analysis of stellar convection by Hurlburt et al. (1986). In this approach, the individual components of the energy transport equation, Equation 2.3, are averaged over lateral directions.

Then, one computes radial distributions of deviations away from the lateral averages, f ′ f f¯. ≡ −

46 1.5 1.5 $ 1

1.0 − 1.0 $ 1 m − c s 1 g − r

0.5 s 0.5 e g r 51 e 10 44

# 0.0 0.0 10 i # K i F h −0.5 P −0.5 , P

h − i FC − 2D PA 2D C − , − i F FC 3D PA 3D h −1.0 A −1.0 − FK − 2D PP 2D P

h − FK − 3D PP 3D

−1.5 −1.5 100 200 300 400 500 600 100 200 300 400 500 600 r [km] r [km]

Figure 3.17 Decomposition of the total energy flux and the distribution of the related work source terms in the SC models. Quantities are averaged over time over the period between t = 110 ms and t = 120 ms. The individual curves represent averages taken over all model realizations for a particular model dimension (shown with dashed and solid lines for 2D and 3D models, respectively). The vertical lines denote the approximate locations of the gain radius and the minimum shock radii for the two families. (left panel) Convective flux, FC , and kinetic flux, FK . (right panel) Buoyant (PA) and expansion (PP ) work. See Section 3.2.4 for discussion.

In the discussion here, we use the notation of Mocáket al. (2009). We define the convective flux,

FC , kinetic ﬂux, FK , buoyant work, PA, and expansion work, PP , as follows:

′ P F = u ρ ε + r2dΩ, (3.10) C r ρ I ( )

′ 1 F = u ρ u · u r2dΩ, (3.11) K r 2 I ( ) ∂Φ P = u ρ′ r2dΩ, (3.12) A − r ∂r I ′ 2 PP = (∇· u) P r dΩ, (3.13) I where the integrals are taken over the suitable solid angle (in our models with the excised 12◦ cone around the symmetry axis this amounts to 99.5% of the full solid angle). These terms describe ≈ the total energy transported per unit time through a surface of a sphere of radius r.

Figure 3.17 shows the time averages of convective and kinetic energy ﬂuxes (left panel) and the buoyant and expansion work terms (right panel) for the SC models. The main characteristics of the energy transport follow from the direction of the energy transport and the relative contributions

47 of individual terms. We begin our analysis at the gain radius located at 115 km in both 2D and ≈ 3D models (marked with the solid, vertical line). There, the convective flux is initially negative but rapidly increases and reaches the maximum at 15% of the radial extent of the gain region. ≈ Further out, the convective flux gradually decreases and ultimately ceases well upstream of the shock. On the other hand, the kinetic energy flux is negative below the gain radius and remains negative through most of the gain region. The observed run of the convective flux provides clear evidence for convective transport in which material heated near the gain radius buoyantly rises toward the shock.

The relation between buoyancy and convection also manifests itself through the buoyant work,

PA (shown with gray lines in the right panel of Figure 3.17). The work done by buoyancy becomes positive starting at the gain radius, indicative of buoyancy driving the convection upwards. It

ﬁrst rapidly increases through the bottom layers of the gain region, and reaches its maximum at

10% of the gain region extent. Further out, the work done by buoyancy gradually decreases ≈ and eventually vanishes as expected at the shock, where the ﬂow becomes, on average, spherically symmetric.

As we noted, the buoyant work is positive throughout the gain region. This can be caused by either overdense material sinking or underdense material rising. Since the work due to buoyancy is negative below the gain region, therefore the opposite scenario with either overdense, cold material is rising or underdense, hot material is sinking. Since we do not observe sinking of the underdense, hot material in that region, we conclude that the negative work due to buoyancy indicates the presence of rising overdense and cold matter. In our models, this rising, overdense material originates from the downﬂows that are turned around by the pressure gradient. This process is known as buoyant braking (Brummell et al., 2002).

The most conspicuous diﬀerence between 2D and 3D lies in the distribution and value of the expansion work source term, PP (shown with solid lines in the right panel of Figure 3.17). When integrated over time and space, this term represents the PdV work of ﬂuid elements. Below the gain radius this term is negative, indicative of the work done in the process of compressing the

ﬂuid. It changes sign across the gain radius, and remains positive through the lower one-third of

48 the gain region. This indicates that the fluid is expanding in that region. We quantify the amount of work due to expansion by integrating Equation 3.13 over the portion of domain between r 60 ≈ km and r 250 km, where the expansion work source term is significant and differs most between ≈ 2D and 3D. Assuming a characteristic timescale of 100 ms, we estimate the (kinetic) energy change due to PdV work to be 1 1050 erg and 3 1049 erg in 2D and 3D, respectively, or about a ≈ × ≈− × few percent of the explosion energy.

We believe the difference in the PdV work, as we described above, reflects the fact that the structural integrity of upflows and downflows in 2D is much greater than in 3D. The structural integrity of flow features depends on the surface-to-volume ratio. For a given volume, 2D structures have less surface area than their 3D counterparts. Thus, flow structures in 2D are less susceptible to

(surface) perturbations. We alluded to this property in our discussion of morphological differences between 2D and 3D simulations in Section 3.1.5. Consequently, in 3D, the downflows do not penetrate as deep, which can be clearly seen in the left panel of Figure 3.17, which shows that both the kinetic fluxes are negative and of comparable magnitude in both 2D and 3D around r 250 km. ≈ However, as one moves into deeper layers the kinetic flux much more rapidly decreases in 2D than in 3D, and vanishes in the layers closer to the proto-neutron star surface. The convective fluxes in 2D and 3D behave in a qualitatively similar way, although they are positive and equal closer to the gain radius, at r 150 km. From there inward, the convective flux decreases at a significantly ≈ greater rate across the gain radius and ceases closer to the proto-neutron star surface in 2D than in

3D. Furthermore, the greater structural integrity of upﬂows and their coherent nature in 2D makes the expansion process dramatically more eﬃcient than in 3D. This is evidenced by much greater work done by expansion above the gain radius, as we discussed earlier.

As one can see in the right panel of Figure 3.17, the buoyant and expansion terms are both negative below the gain region. This is because below the gain region the system is no longer energy conserving due to intense neutrino heating. This situation is diﬀerent, however, across the shock front. There, both terms are of diﬀerent sign, which is consistent with the fact that the neutrino heating is weak and the energy is approximately conserved in that region.

49 Finally, we would like to note an interesting possibility that the difference in efficiency of the convective engine between 2D and 3D might be another consequence of the difference in surface-to- volume ratio between those models. This is because in 3D the surface area of the interface between hot and cold material where the mixing, and thus also heat exchange, takes place is relatively larger. Quantifying this possibly important effect is, however, beyond the scope of this paper.

3.3 Turbulence

Motivated by the recent discussion of the possible role of turbulence in the process of shock revival (Murphy and Meakin, 2011; Hanke et al., 2012), we present spectra of the kinetic energy in a lateral direction and turbulent Reynolds stresses characteristic of the gain region for our SC models. In particular, we will determine if the kinetic energy spectra are consistent with model dimensionality, and discuss the qualitative diﬀerences in the Reynolds stresses found between 2D and 3D.

3.3.1 Spectra

We investigate the spectral properties of turbulence by considering the spherical harmonic decomposition of Equation 3.6 with f (θ,ϕ) = √ρuθ, where uθ is the polar component of the velocity vector. Our choice of this particular velocity component hinges on the assumption that the ﬂow is isotropic in the lateral directions, which is supported by our analysis of the Reynolds stresses presented below in Section 3.3.2.

To ensure that the energy contained in the spherical harmonic modes adequately represents the lateral kinetic energy, we include the factor √ρ (Endeve et al., 2012). Following Hanke et al. (2012), we deﬁne the lateral kinetic energy spectrum, E (l), as

l 2 ∗ E (l)= Ylm (θ,ϕ) √ρuθdΩ , (3.14) − Ω m∑= l [∫ ] where the integral is taken over the solid angle included in our simulations. In order to account for spatial and temporal variations of the lateral kinetic energy spectrum, we average the spectrum

50 26 10 2D SC 3D SC −0.21 1024 l ] −1.86

3 l m

c − / l 3.74

g 22

r 10 e [ )

l −0.37

( l 20 E 10 l −2.59

−4.21 18 l 10 100 101 102 l

Figure 3.18 Lateral kinetic energy spectra for SC models. The spectra are shown with dashed and solid lines for 2D and 3D models, respectively. The data are averaged spatially over a spherical shell 25 km thick centered at 290 km, and over the time period between t = 120 ms and t = 130 ms. Piecewise powerlaw functions are ﬁt separately to spectra for 2D and 3D model families, and are shown with dashed and solid lines for 2D and 3D, respectively. The ﬁt process allows for optimal locations of the break points in the piecewise powerlaw functions. See Section D.5.6 for discussion.

over a shell of thickness 25 km centered at the midpoint of the gain region (r 290 km). In mid ≈ addition, in our analysis the spectra are averaged over snapshots from the time interval between t = 120 ms and t = 130 ms (soon after the quasi-steady state phase). In passing we note that there exists other choices of physical quantities for the analysis of turbulence, for example density

fluctuations (Borriello et al., 2013), which are more suited for the analysis of neutrino- matter interactions rather than fluid flow dynamics.

Figure 3.18 shows the lateral kinetic energy spectra for our SC models with 2D and 3D models shown with dashed and solid lines, respectively, averaged over time and space as described above.

We ﬁnd that one can identify three distinct spectral regions in the lateral kinetic energy spectrum.

The ﬁrst region occupies large scales (l ≲ a few), the intermediate region extends up to l ≲ several

10s, and the third region extends toward still smaller scales (l ≳ 60). This resembles spectra routinely found in turbulence studies, with the highest end of our spectrum closely resembling the

51 region found in numerical simulations. It is customary to characterize those regions by a piecewise- linear combination of powerlaws, E (l) lα. In particular, it is expected that in the intermediate ∝ regime, in which the energy is transported from large to small scale (and possibly also in the opposite direction in 2D situations), a cascade develops with the power law exponent, α, chieﬂy dependent on the particular characteristics of the system.

To quantify the shape of the power spectra found in our simulations, we ﬁt power law functions inside each region. We found that the power law exponent that best characterizes the shape of the spectrum at large scales is α 0.2 (in 3D) to α 0.4 (in 2D). On the other extreme, ≈ − ≈ − the spectrum steepens around l 35 with α 4. At still smaller scales (l ≳ 60), the spectrum ≈ ≈ − abruptly changes its shape presumably due to numerical dissipation.

Of the greatest interest from the point of view of turbulence are energy-transporting intermediate scales (a few ≲ l ≲ 35). We ﬁnd that in this region the power spectra in 2D simulations are signiﬁcantly steeper (α = 2.59) than in 3D (α = 1.86). This is qualitatively consistent with − − the results of turbulence studies in 2D. We also note that the power law exponent found in 3D is not too dissimilar from that of the classic Kolmogorov spectrum (α = 5/3; Monin and I’Aglom − (1971)).

On a ﬁnal note, the Reynolds number estimated for our models is quite small (Re 10s ≃ ... 100s, see Appendix A), characteristic of perturbed laminar ﬂow rather than turbulence. We caution one should be careful with drawing any conclusions regarding the turbulent character of such underresolved numerical models (Murphy and Burrows, 2008; Murphy and Meakin, 2011;

Hanke et al., 2012). Furthermore, our models do not account for additional physics eﬀects that may play an important role at the base of the convective region, such as neutrino viscosity (see discussion in Section 3.1 of Fryer and Young (2007)). These eﬀects may actually reduce the physical

Reynolds number down to 100, which is comparable to the numerical Reynolds number found in ≈ our simulations. Studying such eﬀects, however, is beyond the scope of the present work.

3.3.2 Turbulent Reynolds stresses

Figure 3.19 shows the time- and model realization-averaged turbulent Reynolds stress compo-

52 4

2D Rrr Rφφ 3D Rθθ 3 Rφφ + Rθθ ] Pa 16 2 10 [ 2D 3D i ii R h 1

0 0 100 200 300 400 500 600 700 r [km]

Figure 3.19 Distributions of Reynolds stresses in the gain region for the SC models. The data shown in the figure is obtained by first averaging stress distributions for individual model realizations over the period between t = 110 ms and t = 130 ms, and then by calculating mean values based on the time-averages separately for 2D and 3D. The results for 2D and 3D families are shown with thin and thick lines, respectively. The dashed vertical lines at r 110 km, 400 km, and 570 km mark ≈ the approximate positions of the gain radius, minimum shock radius, and maximum shock radius, respectively. Note that the stresses for 2D models rise much more rapidly across the gain radius region than in 3D. See Section 3.3.2 for discussion. nents for the SC models. The data shown in the figure were generated as follows. First, the tensor components were computed for individual model realizations using

′ ρuiuj Rij = ⟨ ⟩, (3.15) ρ0 where denotes the averaging operation over solid angle, u′ = u u is the perturbation away ⟨·⟩ j j −⟨ j⟩ from the background of the j-th component of velocity, and ρ = ρ is the background density. 0 ⟨ ⟩ After averaging over the solid angle, the stress components were then averaged over the period between t = 110 ms and t = 130 ms. In the ﬁnal step, we used these averages to compute the model-averaged values of turbulent Reynolds stresses.

In 3D, the radial (shown with the thick solid line in Figure 3.19) and lateral stress components

(shown with thick dash-dotted and dotted lines; their sum is shown with the thick dashed line),

53 increase from the inner boundary and across the gain radius (marked with a vertical, dashed line at r 115 km). The radial component reaches the maximum around r 300 km, roughly in the ≈ ≈ middle of the gain region, while the overall contribution of lateral stresses peaks slightly deeper at r 255 km. The radial stress is about a factor of 2 greater than the sum of lateral stresses. Farther ≈ out, the stresses decrease and reach approximately similar magnitude just below the shock. In the

ﬁgure, the three dashed, vertical lines located between 400 km and 600 km mark the innermost, average, and outermost shock radii. The stresses show a mild increase across the region occupied by the shock and eventually vanish upstream of the shock inside the accretion ﬂow. The sum of lateral stresses exceeds the radial stress by 30% in a relatively narrow region behind the shock ≈ (at r 430 km). Before we begin discussing the qualitative and quantitative between the average ≈ turbulent Reynolds stresses in 2D and 3D, we note that the radial dependence of stresses in 2D

(shown with thin solid and dash-dotted lines for the radial and lateral stresses, respectively, in

Figure 3.19) is quite similar to that in 3D. Moreover, ﬂow appears roughly isotropic below the gain radius, with radial and lateral stresses in equipartition.

We note that the stresses in 2D are much larger in the region near the gain radius than in

3D. This can be understood by recalling that the convection is most vigorous in that region in our two-dimensional models (see Section 3.2.4). Farther out, the dominant, buoyantly-generated radial stresses undergo redistribution in lateral directions (see, e.g., Murphy and Meakin (2011)). This process of redistribution of the radial stress is expected to be isotropic in the lateral directions, as observed in our 3D simulations (thick dash-dotted and dotted lines in Figure 3.19). By volume- integrating the radial averages of the stress components inside the gain region, we ﬁnd that in both

2D and 3D the integrated radial stress is larger by about 60% than the lateral stresses. Furthermore, the integrated turbulent stresses are greater by about 30% in 3D than in 2D. The latter result may appear at ﬁrst surprising, judging by the run of turbulent stresses shown in Figure 3.19. However it is important to recognize that the curves shown in the ﬁgure represent angular averages rather than volume-integrated contributions. Therefore, even though the stresses are greater in 2D than in 3D near the gain radius, the larger stresses in 3D in the upper layers of the gain region result in their overall greater volume-integrated value.

54 The fact that the volume-integrated turbulent stresses are greater in 3D than in 2D is consistent with the surface-to-volume diﬀerence between 2D and 3D simulations (the argument we made in

Section 3.2.4). In this scenario, 3D structures are more susceptible to perturbations and the flow becomes disorganized over larger regions (in our case, a greater fraction of the radial extent of the gain region). Specifically, full-width at half-maximum measure of the radial stress distribution is 25% greater in 3D than in 2D. (The corresponding volume factor is still larger due to the 3D distribution being centered at a higher radius.) Finally, we note that the process of redistribution of the radial stress is appears more gradual in 3D than in 2D (compare the two curves connected with the double arrow in Figure 3.19). In fact, this may only be a misimpression. Note that the contribution of lateral stresses in 2D differs qualitatively from that in 3D only in the lower half of the gain region. We find preliminary evidence that this may be due to the difference in the work done by expansion in that region, where the expansion work source term is large in 2D, while it is nearly zero in 3D (see Figure 3.17). We speculate that the expansion increases the velocity

ﬂuctuations and thus contributes to the increase of the turbulent stresses.

3.3.3 Lagrangian analysis of the explosion mechanism

We believe that potentially important insights into the core-collapse supernova explosion mechanism can be gained through analyzing the history of fluid parcels as they enter the gain region and participate in the revival of the shock. Our motivation partially follows from the realization that the amount of evidence provided by the Eulerian analysis appears insufficient to disentangle and clearly identify the role various physics plays in the explosion process. To this end, we seeded our simulations with a large number of tracer particles from the beginning of the simulations. Recall from Section 2 than in our models, each tracer particle represents a fluid element with a mass of

−6 −7 1 10 M⊙ in two-dimensions and 1 10 M⊙ in three-dimensions. × × Residence times in the gain region. In our analysis we assume a particle resides inside the gain region if its position is between the shock radius and the gain radius (both of which are angle- and time-dependent). We deﬁne the residency time, tres, as the total time a tracer particle spends in the gain region. Thus, the maximum residency time of any particle never exceeds the

55 2000 2D SC 3D SC

1500 10 / , D 3 p

N 1000 + , , D 2 p

N 500 +

0 0 50 100 150 200 tres [ms]

Figure 3.20 Distributions of tracer particles in the gain region as a function of particle residency time for SC models. This distributions are shown with dashed and solid lines for 2D and 3D families, respectively, at t = 50 ms, 100 ms, and 200 ms. The data are averaged over all model realizations for a particular model dimension. For legibility, we smooth the curves with a boxcar method using a time window of 5 ms. Note that the scale diﬀers for 3D models. See Section 3.3.3 for discussion. model time elapsed since the beginning of the simulation. We construct a distribution of particle residency times by binning the residency times with a resolution of 1 ms.

Figure 3.20 shows the model realization-averaged distributions of particle numbers for our SC models at t = 50 ms, 100 ms, and 200 ms, as a function of particle residency time. In the ﬁgure, the data for 2D and 3D models are shown with dashed and solid lines, respectively. The total number of particles in the gain region are approximately 5.2 104, 6.3 104, and 7.3 104 in 2D × × × and 5.1 105, 5.9 105, and 7.3 105 in 3D, for progressively later evolutionary times. We note × × × that particles occupying bins near tres = 0 should be interpreted as material that has been just accreted (entered the gain region), while the farther to the right the particle bin is the earlier it entered the gain region.

There is little diﬀerence in distributions of particle numbers between the 2D and 3D models

(shown with thick dashed and solid lines, respectively, in Figure 3.20) for particles that enter the gain region prior to the onset of convection. This is expected, given that at those early times the

56 shock is nearly spherically symmetric and the post-shock ﬂow is essentially radial. These initial distributions are expected to evolve due to both accretion of new material (particles passing through the shock), and loss of particles already residing in the gain region through the gain radius and their settling onto the proto-neutron star. Therefore, one may expect certain diﬀerences in the distributions of particle numbers at later times. First, notice that the number of particles with maximum residency times is always smaller in 3D than in 2D, except at the early time (t = 50 ms).

This means that a portion of particles with the longest residency times that are accreted through the gain radius is larger in 3D than in 2D. Second, this trend reverts at t 75, and at t = 200 ≈ ms there are more particles with short residency times present in 3D than in 2D. One possible explanation for this behavior might be a systematic diﬀerences in accretion rates and shock radii between 2D and 3D, but we found no evidence to support this possibility (cf. Section 3.1.3).

Additional information about the origin and evolution of particle number distributions can be obtained by examining the spatial dependence of particle residency times. These distributions reveal that, in 3D, there exists a population of particles with low residency times located near the gain radius. The systematic differences between particle number distributions in 2D and 3D can therefore be explained by particles leaving and reentering the gain region across the gain radius in 3D. This leads to the apparent deficit of particles with long residency times and causes excess of particles with low residency times in this case. We quantified the magnitude of this effect by examining the contours of cumulative particle numbers at a given residency time. We found that between 10% and 20% of particles located near the gain radius participate in this process. This estimate is consistent with the difference between the particle number distributions seen in Figure

3.20. The above effect signifies a difference in the flow dynamics near the gain radius between 2D and 3D. In particular, we believe this behavior is due to a larger amount of mass (by 10–20%) involved in convection at the bottom of the gain region, where the convection is driven, and may be the reason for the greater efficiency of convection in 3D (see Section 3.3.3 for discussion).

Thermodynamic evolution of the shocked material. One quantity of particular interest in the context of the ccSNe explosion mechanism is the boost to the internal energy of the material residing in the gain region provided by neutrino heating. This process can be analyzed by consid-

2D SC 3D SC 6 $ ⊙ M / g

r 4 e 51 10 #

i 2 e ∆ h

0 0 50 100 150 200 tres [ms]

Figure 3.21 Distribution of the speciﬁc internal energy change for tracer particles in the gain region as a function of particle residency time for SC models. The distributions are shown at t = 50, 100, and 200 ms with thick, medium, and thin lines, respectively. The data are averaged over all model realizations for a particular dimension, and are shown separately with dashed and solid lines for 2D and 3D. For legibility, we smoothed the curves with a boxcar method using a time window of 5 ms. See Section 3.3.3 for discussion. ering the increase in the internal energy of shocked ﬂuid parcels (represented in our study by tracer particles) as a function of the particle residency time.

Figure 3.21 shows the model realization-averaged internal energy gain per unit mass, ∆e , as ⟨ ⟩ a function of particle residency time for the 2D and 3D SC models (shown with dashed and solid lines, respectively). We define the internal energy gain of a tracer particle as the difference between the particle internal energy at the evolutionary time shown and its internal energy when it enters the gain region. We find that at early times (t ≲ 50 ms) there is essentially no difference in the way particles gain energy in 2D and 3D (thick dashed and solid lines). At later times, we begin to observe modest at first (t = 100 ms) changes with the newly accreted material progressively gaining energy faster in 3D than in 2D (the solid, medium thick line representing 3D data lies always to the left of the dashed, medium thick line representing the 2D data).

58 In addition to the apparent asymmetry in the 3D distribution at late times, we also find qualitatively new unique features, in particular at t = 200 ms. First, we identify a region with almost constant increase in the energy gain for particles that entered the gain region between t = 50 ms and t = 100 ms. During this time, and as we discussed earlier in Section 3.3.3, the gain radius shrinks faster in 3D than in 2D. This allows the particles residing close to the gain radius to gain comparatively more energy in 3D. This process is responsible for the observed shape of the internal energy gain distribution for residency times between t 100 ms and t 150 ms. Second, res ≈ res ≈ there is a pronounced hump visible in Figure 3.21 which broadly occupies a 35 ms time interval for particles that entered the gain region shortly after the vigorous convection developed. This is marked by a rapid increase in the 3D energy gain curve taken at t = 200 ms (thin solid line) at t 100 ms. We believe this signifies the increase in the thermodynamic efficiency of the res ≈ convective engine in 3D. Third, there is a smaller hump for a population of particles that entered the gain region shortly before (tres < 50 ms) the shock was launched. Finally, we would like to point out that the similarity in the shapes of the 3D distributions at late times is quite striking. The arrows in the figure connect specific features in the energy gain distributions that are well preserved in 3D between t = 100 ms and t = 200 ms. This indicates that the particles which entered the post-shock region during the first 100 ms of the simulation gained internal energy at a similar rate during the process of shock revival.

Unbinding of the shocked material. The explosion process necessarily requires gravitationally unbinding a substantial fraction of the collapsing core. Alas, very little information about this process can be found in the literature, perhaps with the exception of discussions of global quantities such as the explosion energy. However, armed with the tracer particles we are now in a position to investigate this process in more detail. In particular, the results discussed in the previous section indicate that profound differences in the heating efficiency exist between 2D and 3D models. Although we are able to identify populations of particles responsible for those differences, our focus was more on the interplay between the advection and neutrino heating of the shocked material. In the following discussion of the process of gravitational unbinding of the shocked, stellar core we also take into account the flow dynamics inside the gain region.

59 12 2D SC 3D SC $ ⊙

M 8 / g r e 48 10 # 4 i b E h

0 0 50 100 150 200 tres [ms]

Figure 3.22 Distribution of the positive speciﬁc gravitational binding energy for tracer particles in the gain region as a function of particle residency time for SC models. The data are taken at t = 200 ms, and are averaged over all model realizations for a particular model dimension. The results are shown with dashed and solid lines for 2D and 3D, respectively. For legibility, we smooth the curves using a boxcar method with a time window of 5 ms. Note that in 2D, unbound material is composed primarily of matter which entered the gain region prior to the quasi-steady phase. However, in 3D, there are substantial contributions from material which entered the gain region during the period of vigorous convection. See Section 3.3.3 for discussion.

Figure 3.22 shows the model realization-average of the positive part of the specific gravitational binding energy, Eb, as a function of residency time for 2D and 3D families of models at t = 200 ms (marked by dashed and solid lines, respectively). The data indicates dramatic differences in the energetics of fluid elements in the gain region between 2D and 3D. In 2D, the unbound portion of the gain region is almost exclusively made of material that entered the gain region during the first

100 ms of the evolution. However, in 3D one can identify two diﬀerent populations of particles. The

ﬁrst 3D population consists of particles that entered the gain during the ﬁrst 100 ms. Although this is the same material that is responsible for the explosion in 2D, in 3D this material is substantially less energetic. A careful examination of the data shown in Figure 3.22 reveals that, indeed, around the explosion time the explosion energies in 2D are greater than in 3D. This is consistent with the

2D models exploding, on average, faster than 3D models. Also, as discussed in the previous section,

60 the particles that enter the gain region early on gain energy more eﬃciently in 2D than in 3D. It is only at later times when the particles in 3D show a dramatic increase in their internal energies.

In the scenario presented above, the explosion is launched in 2D early on by a relatively stronger central neutrino source. In 3D, however, the more efficient convective engine is able to launch the shock at neutrino luminosities lower by 4%. (Recall that we found evidence for up to 20% more ≈ mass participating in convection across the gain radius in 3D, as discussed in Section 3.3.3). By integrating the particle energies, we found that in 3D the same amount of positive binding energy is carried by particles that entered the gain region during the first 100 ms of evolution as by those which entered the gain region during the following 50 ms. Thus, the process of unbinding material in the gain region appears more gradual in three-dimensions, with significant energy gain during the quasi-steady phase.

The cause of the observed differences between energetics in 2D and 3D ccSNe simulations, and so the question how the efficiency of the convective engine depends on dimensionality, remains unknown at this time. However, we believe the Lagrangian approach to analyzing the evolution of the gain region offers significant advantages over Eulerian statistics for investigating the efficiency of convection. This will problem will be the subject of a forthcoming publication.

3.4 Summary and future work

We have presented an analysis of two- and three-dimensional models of early core-collapse supernova development in the collapsing core of a 15 M⊙ progenitor. In our study, the simulations start at shock bounce, continue through the onset of neutrino-driven convection, quasi-steady state of the gain region, and early post-revival shock expansion until the energy of the explosion had approximately saturated at 1.5 seconds. Our models were tuned to match the estimated explosion energetics of SN 1987A by careful selection of the parameterized neutrino luminosity. We considered the cases of slow and fast contraction rates of the proto-neutron star to reﬂect on uncertainty in the nuclear equation of state. We introduced and applied new diagnostic methods for the explosion process, morphology and structure of convection, and energetics of material inside the gain region.

61 Our conclusions are formulated based on a large database of explosion models. This computa- tionally expensive approach is necessary since the core-collapse supernova problem involves highly nonlinear, strongly coupled physics which results in the extreme sensitivity of realistic computational models to perturbations. Therefore, conclusions of core-collapse supernova studies can only be understood in a statistical sense. Moreover, such ensemble-based conclusions are potentially aﬀected by resolution eﬀects, especially in three-dimensions, due to the high computational cost of individual model realizations.

The main ﬁndings of our work can be summarized as follows:

We found that the same explosion energy can be obtained in three-dimensions with less energy • than in two-dimensions. This trend also holds between two-dimensions and one-dimension.

This result is in agreement with some previous studies.

The stochastic character of the explosion process results in non-uniqueness of the relation • between neutrino luminosity and explosion energy. In particular, in the process of tuning the

neutrino luminosities, we were able to obtain equally energetic 3D models with the neutrino

luminosities diﬀering by as much as 3%.

Dimensionality also contributes to the non-uniqueness of the above relation. In particular, • equally energetic explosions can be obtained in 3D using neutrino luminosities smaller by 4%

than in 2D. This implies the eﬃciency of the convective engine is greater by that amount in

3D.

We proposed that the observed dependence of the eﬃciency of the convective engine on the • model dimensionality might be a consequence of the diﬀerence in the surface-to-volume ratio

of ﬂow structures between 2D and 3D. In this scenario, the greater eﬃciency of the convective

engine in 3D is simply due to a larger surface-to-volume ratio in that case, which helps the

process of heat exchange between hot and cold ﬂow structures. Quantifying this possibly

important eﬀect is, however, beyond the scope of this paper.

62 We note that the relations between neutrino luminosities, accretion rates, and explosion times • show, in our models, less variation with the model dimensionality than in other studies (see

Figure 3.2). We believe this is purely due to the fact that the objective of our study was

to obtain models with similar energetics, rather than to determine the threshold neutrino

luminosity required for the explosion.

We found no evidence for the standing accretion shock instability (SASI) in our models. Our • results showed no persistent, oscillatory l = 1 or l = 2 mode behavior, considered to be a

trademark of SASI. The absence of SASI in our models is consistent with short advective times

through the gain region. Instead, the explosions are boosted in multidimensions by strong,

neutrino- driven convection. At the same time, we would like to stress that SASI cannot be

excluded as an important, or even decisive, physics mechanism for diﬀerent parameters of

collapsing core (e.g., progenitor structure, equation of state, etc.).

We found that the shock front shows a rather modest degree of asymmetry in our models, • rmax/rmin 1.3. We found tentative evidence between the large-scale structure of convection s s ≈ and the shock asymmetry. In particular, the shock is less deformed in the case when convection

has a richer structure (higher-order modes are present).

We found the neutron star recoil velocities reaching up to 475 km/s, with typical values • ≈ between 100 km/s and 300 km/s. These results are in good agreement with the previous

studies (e.g. Scheck et al., 2006).

We introduced and applied new metrics to diagnose the dynamics and structure of the gain • region. In particular, we studied the characteristics of the upﬂowing material (which we

identiﬁed as the buoyant, convective bubbles) and the solid angle it occupies. Using these

measures, we found that the structure of the gain region on large scales and its evolution did

not signiﬁcantly diﬀer between 2D and 3D. However, for the fast contracting models, we found

that the mass inside upﬂows always exceeded 50% of the total gain region, hinting at prompt-

like explosions. This situation is qualitatively diﬀerent in the case of the slow contracting

models, where a prolonged period of quasi-steady state in the gain region can be clearly

63 recognized. It appears that during that phase the mass in the upﬂows ﬂuctuates around the

slowly increasing mean value with amplitude of several percent, and on a timescale of 10 ms.

We speculate that individual downﬂows are responsible for the observed ﬂuctuations.

Our method of identifying the convective bubbles allowed us to study the statistical properties • of the bubble distributions. We found that the solid angle occupied by bubbles is initially

small and progressively increases with time (see Figure 3.15). This indicates that the bubbles

are small at ﬁrst and steadily grow as the evolution progresses. The process appears to

bear some similarity to the bubble merging process observed in multi-mode Rayleigh-Taylor

instabilities. Furthermore, we ﬁnd a good correlation between the solid angle occupied by

rising material and the upﬂowing mass. This highlights the convection takes place inside the

whole gain region, from the gain radius all the way up to the shock.

The development and evolution of convection inside the gain region can also be studied by • considering the temporal evolution of the number of bubbles. We found that their number

rapidly increased at t 50 ms (see Figure 3.16), indicating the onset of convection. After ≈ the convection set in, the number of bubbles progressively decreased with time. We used the

information about the number of bubbles combined with the solid angle they occupy, and

estimated that the convective bubbles are, on average, about 4 times smaller in 3D than in

2D at the time of the explosion.

We used the decomposition of the total energy ﬂux to understand the energy transport in the • gain region. We found that the work done by expansion and compression in that region diﬀers

between 2D and 3D, and we proposed that this is due to the greater structural integrity of

ﬂow structures in 2D.

The analysis of the components of the total energy ﬂux also provided evidence that the • region near the gain radius was not in thermal equilibrium at the beginning of the quasi-

steady phase. We found evidence for strong cooling below and strong heating above the gain

radius. Furthermore, we found that the buoyant work was positive throughout the gain region

in our models, in accordance with the neutrino-driven convection scenario.

64 We demonstrated that, for the same neutrino luminosities that produce energetic explosions • in well-resolved models, the explosion energies dramatically decrease once the angular mesh

resolution decreases below 6◦. This implies there exists a minimal mesh resolution required

for the central engine to eﬃciently operate. This is expected. This also implies that the

neutrino-driven convection is the key ingredient of the explosion mechanism in the energetic

models considered here. We cannot exclude the possibility that more physics (e.g. turbulence)

may emerge at still higher resolutions (i.e. less than about 0.05 r) to power the explosion × more eﬃciently than convection.

We found that the the structure of turbulent energy spectra are steeper in our 2D models • (E l−2.68) than in our 3D models (E l−1.86). These results should be taken with caution ∼ ∼ given the low resolution of our models and an estimated Reynolds number on the order of

100.

Analysis of the turbulent Reynolds stresses revealed that the post-shock ﬂow is anisotropic • in the radial direction due to buoyant driving. At the same time, we found the ﬂow in the

gain region is isotropic in the lateral directions.

Using a Lagrangian representation of the gain region, we found substantial differences in the • energetics of the shocked fluid elements between 2D and 3D. We identified material near the

base of the gain region that experiences stronger heating during the period when the gain

radius shrinks. We also found that material accreted during the phase when the convection

is fully developed undergoes especially strong heating in 3D.

We found significant differences in the evolution of residency times and binding energies of • fluid elements inside the gain region between 2D and 3D. In 3D, more mass is involved in the

process of driving convection in the region around the gain radius. At the same time, the

same amount of positive binding energy is carried by material that entered the gain region

during the ﬁrst 100 ms of evolution as by those which entered the gain region during the

following 50 ms. However, in 2D, a smaller portion of the gain region mass participates in

65 driving convection, and practically all of the positive binding energy is carried by material

that entered the gain region during the ﬁrst 100 ms.

Several possible directions for future research emerge from the current work. It remains unclear why the convective engine is more efficient in three-dimensions. Although we were able to quantify the effect, we did not provide a clear explanation for its origin. We obtained only rudimentary information about the properties of turbulence in our largely underresolved and dominated by numerical viscosity models. It is clear that to gain meaningful insight into the role of turbulence in the process of core-collapse supernova explosion a new generation of far better-resolved models exploiting more efficient computational approaches is required. Also, one would like to evolve those models until the neutron star recoil velocities saturate. Finally, our current model does not include nuclear burning, and therefore we are unable to discuss the nucleosynthetic yields and the compositional structure of our models. The above issues will be the subject of a forthcoming publication.

66 APPENDIX A

ESTIMATING NUMERICAL VISCOSITY AND

REYNOLDS NUMBER

In order to estimate the numerical viscosity and the numerical Reynolds number, we use the information provided by the Lagrangian tracer particles. Consequently, in our approach the velocity

ﬁeld is represented at discrete points by tracer particles. We deﬁne the longitudinal Eulerian velocity structure function of order-p as

SL (r )= [(u (x) u (x + r rˆ)) · rˆ]p , (A.1) p sep ⟨ − sep ⟩

where x is the particle position, u is the particle velocity, rsep is the separation between particles, and rˆ is the particle displacement unit vector. The averaging operator, , is applied over a given ⟨·⟩ rsep at a speciﬁc model time. Kolmogorov’s turbulence theory (Monin and I’Aglom, 1971; Frisch, 1995), assuming negligible

L intermittence, states that for p = 2, the structure function, S2 , is related to the average speciﬁc turbulent dissipation, ϵ , via ⟨ ⟩ SL (r )= C ϵ 2/3r2/3, (A.2) 2 sep 2⟨ ⟩ sep where C2 is a constant on the order of 1. This result allows one to estimate the turbulent dissipation (physical + numerical) present in a numerical simulation. Within the same framework, the local, speciﬁc turbulent dissipation is related to the average strain rate as

ϵ = 2ν S:S , (A.3) ⟨ ⟩ where ν is the kinematic viscosity and S is the strain rate tensor.

67 5 6 10 2D(M194F) 2D(M194F) 3D(M187D) 3D(M187D) 5 104 $ s

/ 4 2 103 m c 3 e R 14 102 10

# 2 ν 1 1 10

0 0 10 0 200 400 600 800 1000 0 200 400 600 800 1000 rsep [km] rsep [km]

Figure A.1 Analysis results for models M194J and M187D. (left panel) Estimated numerical kinematic viscosity. (right panel) Estimated Reynolds number as a function of particle separation distance. We may expect our models to reﬂect ﬂows with Reynolds numbers on the order of 10s or 100s.

In order to estimate the numerical Reynolds number in our models, we assume that the local turbulent dissipation is representative of the average dissipation rate, ϵ ϵ . This allows us to ≈ ⟨ ⟩ compute the kinematic viscosity as

3/2 SL ν = 2 . (A.4) 2C3/2 S:S r 2 ⟨ ⟩ sep

Using rsep as our characteristic length scale and approximating the characteristic velocity in the

L ﬁeld by S2 , we estimate the Reynolds number as a function of separation distance via √

L rsep S2 Re = . (A.5) √ν

Figure A.1 shows the numerical kinematic viscosity and numerical Reynolds number estimates for the 2D SC model M194J and the 3D SC model M187D at t = 150 ms as a function of particle separation distance. We ﬁnd numerical viscosities on the order of 1 1014 cm2s−1 for the 2D model × and 1 1013 cm2s−1 for the 3D model. At this time in the model evolution, the average shock radius ×

68 is about 500 km, providing an upper limit on structure size. On that scale, the numerical Reynolds number is 200 in 2D and 1000 in 3D. However, as one can see, the numerical viscosity, and so ≈ ≈ the numerical Reynolds number, vary with scale. Therefore, our models cannot be represented by a single numerical Reynolds number. Consequently, and given relatively low values of the numerical

Reynolds number present in our simulations, we tend to characterize the ﬂuid ﬂow inside the gain region of our models as perturbed laminar, rather than turbulent.

Finally, we would like to remark that given the anisotropy of the flow field obtained in our simulations (chiefly due to large-scale convection), the assumption that the average dissipation rate can be represented by the local dissipation rate may not hold true in all regions in our models.

However, we believe that the obtained estimates still provide valuable insights into the nature of our simulations. In particular, given the low estimated numerical Reynolds numbers, conclusions regarding the role of turbulence in the core-collapse supernova explosions should be considered, at most, tentative. Since turbulence was not a particular focus of this study, we are not in a position to estimate the resolution required to capture these eﬀects.

69 APPENDIX B

LAVAFLOW: LIBRARY FOR ADVANCED

VARIABLE ANALYSIS

The data produced by modern computer simulations results in interesting challenges. For example, high resolution simulations of multiphysics problems can generate datasets on the order of tens of terabytes. As a result, analyzing the data often requires the same systems as the initial simulation.

Furthermore, the current generation of large-scale multiphysics simulations requires the development of novel analytics methods in order to identify and disentangle the roles of participating physics. This development typically requires multiple iterations, and it is ideal that the analytics framework allow for ﬂexibility.

In order to address these problems we have developed the analytics tool LAVAflow (Library for Advanced Variable Analysis). LAVAflow is a parallelized collection of fundamental mathematical modules which can be combined to produce specific metrics for simulation data. Analytics capabilities exist for both grid-based and particle-based data.

LAVAflow’s library modules are developed in C++03 (with potential redevelopment for C++11 features). Initially, LAVAflow was developed for serial analytics of Visualization Toolkit-based grid and particle data (Schroeder et al., 2006). While this approach allowed for analysis of the bulk of our simulation data at the time, we have since expanded the scope of the LAVAflow project to encompass our future research goals. This includes extending mesh support to block-based adaptive grids (Berger and Oliger, 1984) and the enabling of parallel capabilities.

Appendices B.1-B.3 present a subset of the modules I have added to LAVAﬂow. Appendix B.1 describes how LAVAﬂow performs integration and averaging over non-trivial regions on grids (e.g. inside a sphere embedded in a Cartesian grid). Appendix B.2 provides the algorithms implemented

70 for image segmentation which were used in Section 3.2.3 to quantify convective upﬂows. Finally,

Appendix B.3 covers the mathematics behind the vector ﬁeld decomposition used in Appendix D.

B.1 LAVAﬂow feature highlight: Non-grid-aligned variable

averaging for orthogonal coordinate systems

Consider an n-dimensional mesh with positions given by an orthogonal coordinate system

(OSC). Any orthogonal coordinate system can be viewed as a uniform Cartesian coordinate system

(or logically Cartesian coordinate system, LCCS) with an appropriate scaling ﬁeld, or metric, which decribes how geometric properties (length, area, volume) change in this Cartesian space.

Returning to our mesh, every cell is an orthogonal, axis aligned brick in the LCCS. If we arbitrarily truncate or clip this cell, we will obviously have a change in the geometry when viewing the result in its native coordinate system.

Example. Consider the LCCS r θ z for cylindrical coordinates. Fixing r and z produces − − circles as θ varies. If we consider the domain r r r , 0 θ 2π, and z = 0, we produce a in ≤ ≤ out ≤ ≤ ﬁlled ring. The area of this ring is given by

A = π r2 r2 . (B.1) ring out − in ( ) If we naively attempt to calculate the area of the ring as

ALCCS =(r r )(θ θ ) (B.2) ring out − in max − min

we see that

A r + r ring = π out in = 1. (B.3) ALCCS θ θ ̸ ring max − min This result stresses the importance of including the metric when computing quantities on an

LCCS.

71 B.1.1 Computing Area

The divergence theorem in a plane is given by

F dA = F nˆds, (B.4) ∇· · ∫S I∂S where F is a vector field, nˆ is the normal to the curve, and ds is the differential arc length. By choosing a vector field F such that F = 1, it is possible to compute the area directly by reducing ∇· Equation B.4 to

A = dA = F nˆds (B.5) S · ∫S I∂S Area of a generic surface in 3-space. In general, the boundary of the surface (∂S) may be a curve in 3-space, which complicates computation of the normal component. In the case where the curve lies in one of the coordinate planes, the normal is easy to directly calculate. Let α,β, and γ deﬁne coordinates of a LCCS. In the following we consider cases where the curve is either in the αβ plane, the αγ plane, the βγ plane, or along an embedded surface in αβγ. For all cases, N denotes the normal to the surface the curve lies on and may be written as

ˆ N = Nα(α,β,γ)αˆ + Nβ(α,β,γ)β + Nγ(α,β,γ)γˆ. (B.6)

Additionally, the general line element can be written as

ˆ ds = hα(α,β,γ,dα)αˆ + hβ(α,β,γ,dβ)β + hγ(α,β,γ,dγ)γˆ. (B.7)

From these two statements, the unit normal to the curve is, in general, given by

ds N nˆ = × . (B.8) ds N | × |

αβ plane. For a curve in the αβ plane, the surface normal is

N = 0αˆ + 0βˆ + 1γˆ. (B.9)

72 Application of Equation B.8 gives

h αˆ h βˆ + 0γˆ nˆ = β − α . (B.10) 2 2 hα + hβ √ The diﬀerential arc length becomes (with h 0) γ ≡

2 2 ds = hα + hβ, (B.11) √ and Equation B.5 reduces to

A = F [h , h , 0] . (B.12) S · β − α I∂S αγ plane. For a curve in the αγ plane, the surface normal is

N = 0αˆ + 1βˆ + 0γˆ. (B.13)

Application of Equation B.8 gives

hαˆ + 0βˆ + h γˆ nˆ = − α . (B.14) 2 2 hα + hγ √ The diﬀerential arc length becomes (with h 0) β ≡

2 2 ds = hα + hγ, (B.15) √ and Equation B.5 reduces to

A = F [ h , 0,h ] . (B.16) S · − γ α I∂S βγ plane. For a curve in the βγ plane, the surface normal is

N = 1αˆ + 0βˆ + 0γˆ. (B.17)

73 Application of Equation B.8 gives

0αˆ + h βˆ h γˆ nˆ = γ − β (B.18) 2 2 hβ + hγ √ The diﬀerential arc length becomes (with h 0) α ≡

2 2 ds = hβ + hγ, (B.19) √ and Equation B.5 reduces to

A = F [0,h , h ] . (B.20) S · γ − β I∂S Arbitrary surface in 3-space. For a curve bounding an arbitrary surface embedded in

3-space, the surface normal is given by Equation B.6. Application of Equation B.8 gives

(h N h N ) αˆ +(h N h N ) βˆ +(h N h N ) αˆ nˆ = β γ − γ β γ α − α γ α β − β α . (B.21) (h N h N )2 +(h N h N )2 +(h N h N )2 β γ − γ β γ α − α γ α β − β α √ This quickly becomes analytically intractible as the denominator does not cancel with the diﬀeren- tial arc length (ds) and one is left with

(F [(h N h N ) , (h N h N ) , (h N h N )]) h2 + h2 + h2 · β γ − γ β γ α − α γ α β − β α α β γ AS = . (B.22) ∂S 2 2 √ 2 I (hβNγ hγNβ) +(hγNα hαNγ) +(hαNβ hβNα) − − − √ Area of a generic surface in Cartesian geometry. The coordinate notation change for the LCCS is given by

(α,β,γ) (x, y, z) . (B.23) ←

74 Table B.1. Area integrals for planar curves in Cartesian geometry.

plane area integrals

αβ A = F dβ F dα αβ ∂S α − β αγ A = F dα F dγ αγ H∂S γ − α βγ A = F dγ F dβ βγ H∂S β − γ H

In Cartesian geometry, the metric is given by

hα = dα,

hβ = dβ, (B.24)

hγ = dγ, and the diﬀerential line element is

ds = dααˆ + dββˆ + dγγˆ, (B.25) which implies a diﬀerential arc length of

ds = √ds ds = dα2 + dβ2 + dγ2. (B.26) · √ For planar curves, the area integrals are summarized in Table B.1.

Area of a generic surface in cylindrical geometry. The coordinate notation change for the LCCS is given by

(α,β,γ) (r, θ, z) . (B.27) ←

75 Table B.2. Area integrals for planar curves in cylindrical geometry.

plane area integrals

αβ A = αF dβ F dα αβ ∂S α − β αγ A = F dα F dγ αγ H∂S γ − α βγ A = F dγ αF dβ βγ H∂S β − γ H

The metric is given by

hα = dα,

hβ = αdβ, (B.28)

hγ = dγ, and the diﬀerential line element is

ds = dααˆ + αdββˆ + dγγˆ, (B.29) which implies a diﬀerential arc length of

ds = √ds ds = dα2 + α2dβ2 + dγ2. (B.30) · √ For planar curves, the area integrals are summarized in Table B.2.

Area of a generic surface in spherical geometry. The coordinate notation change for the LCCS is given by

(α,β,γ) (r,θ,ϕ) . (B.31) ←

76 Table B.3. Area integrals for planar curves in spherical geometry.

plane area integrals

αβ A = α sin(γ)F dβ F dα αβ ∂S α − β αγ A = F dα αF dγ αγ H∂S γ − α βγ A = αF dγ α sin(γ)F dβ βγ H∂S β − γ H

The metric is given by

hα = dα,

hβ = α sin(γ)dβ, (B.32)

hγ = αdγ, and the diﬀerential line element is

ds = dααˆ + αdγγˆ + α sin(γ)dββˆ, (B.33) which implies a diﬀerential arc length of

ds = √ds ds = dα2 + α2dγ2 + α2 sin2 (γ)dβ2. (B.34) · √

For planar curves, the area integrals are summarized in Table B.3.

Summary of general area integrals. The area integrals for Cartesian, cylindrical, and spherical geometry are provided in Table B.4.

Choosing F when F = 1 . Given the general contour integrals as a function of F we ∇· must now choose the components of the vector ﬁeld. Recall that ﬁnding the area of a surface implies that

F = 1. (B.35) ∇·

77 Table B.4. Summary of area integrals for the diﬀerent coordinate systems.

plane Cartesian cylindrical spherical

αβ F dβ F dα αF dβ F dα α sin(γ)F dβ F dα α − β α − β α − β αγ F dα F dγ F dα F dγ F dα αF dγ H γ − α H γ − α H γ − α βγ F dγ F dβ F dγ αF dβ αF dγ α sin(γ)F dβ H β − γ H β − γ H β − γ H H H

Table B.5. Final area integrals for Cartesian coordinates.

plane Fα = α (F1) Fβ = β (F2) Fγ = γ (F3)

αβ αdβ βdα − ··· αγ αdγ γdα −H H ··· βγ βdγ γdβ H ··· − H H H

In order to reduce the number of integrals that need to be calculated, it is best to choose F such that there is only 1 non-zero component. As we have three coordinate planes, each geometry will need three F vectors.

Cartesian. The divergence in Cartesian coordinates is given by

∂F ∂F ∂F F = α + β + γ . (B.36) ∇· ∂α ∂β ∂γ

This leads to choosing our three F vectors as

F1 =[α, 0, 0] , (B.37)

F2 = [0,β, 0] , (B.38)

F3 = [0, 0,γ] . (B.39)

The area integrals for Cartesian coordinates reduce to the expressions in Table B.5.

78 Table B.6. Final area integrals for cylindrical coordinates.

1 1 2 3 plane Fα = 2 α (F ) Fβ = αβ (F ) Fγ = γ (F ) αβ 1 α2dβ αβdα 2 − ··· αγ 1 αdγ γdα −H 2 H ··· βγ αβdγ αγdβ H ··· − H H H

Cylindrical. The divergence in cylindrical coordinates is given by

1 ∂ 1 ∂ ∂ F = (αF )+ F + F . (B.40) ∇· α ∂α α α ∂β β ∂γ γ

This leads to choosing our three F vectors as

1 F1 = α, 0, 0 (B.41) 2 [ ] F2 = [0,αβ, 0] (B.42)

F3 = [0, 0,γ] (B.43)

The area integrals for Cylindrical coordinates reduce and are shown in Table B.6. At ﬁrst glance, the integrals in the βγ plane may seem confusing. If we are in the βγ plane, how can there be any area if there is no α component? The answer is that we are in a βγ plane (α =const), not strictly the βγ plane corresponding to α = 0. Intuitively, the βγ plane is a tube, and we know that the radius (α) scales the area of a tube (Atube = 2πr∆z).

Note on Divergence. Until now, we have been using the standard 3d divergence in the various coordinate systems to evaluate F , despite evaluating in planes in the systems. This has worked so far because the metrics do not change when viewing only a plane in the systems. One must be careful, however, as this is not always true and violation of this will give incorrect results.

79 The general divergence equations for an orthogonal coordinate system, given the scale factors, is 1 ∂ ∂ ∂ F = (h h F )+ (h h F )+ (h h F ) . (B.44) ∇· h h h ∂α β γ α ∂β α γ β ∂γ α β γ α β γ [ ] We will see in the next subsection on spherical coordinates that this will come into play and we will have to be more careful about choosing F .

Spherical. The divergence in spherical coordinates is given by

1 ∂ 1 ∂ 1 ∂ F = α2F + F + (sin(γ)F ) (B.45) ∇· α2 ∂α α α sin(γ) ∂β β α sin(γ) ∂γ γ ( ) This is, in general, not compatible with our work so far when on planes in αβγ. For the αβ plane:

1 ∂ 1 ∂ 1 ∂ F = (αF )+ F + (sin(γ)F ) (B.46) ∇αβ · α ∂α α α sin(γ) ∂β β sin(γ) ∂γ γ

For the αγ plane: 1 ∂ ∂ 1 ∂ F = (αF )+ F + F (B.47) ∇αγ · α ∂α α ∂β β α ∂γ γ

For the βγ plane:

1 ∂ 1 ∂ 1 ∂ F = α2F + F + (sin(γ)F ) (B.48) ∇βγ · α2 ∂α α α sin(γ) ∂β β α sin(γ) ∂γ γ ( ) This leads to choosing our three F vectors as

1 F1 = α, 0, 0 (B.49) 2 [ ] F2 = [0,αβ sin(γ), 0] (B.50)

F3 = [0, 0, α cot (γ)] (B.51) −

80 Table B.7. Final area integrals for spherical coordinates.

1 plane F = α (F1) F = αβ sin(γ)(F2) F = α cot (γ)(F3) α 2 β γ − αβ 1 α2 sin(γ)dβ αβ sin(γ)dα 2 − ··· αγ 1 α2dγ α cot (γ)dα H − 2 H ··· − βγ α2β sin(γ)dγ α2 cos (γ)dβ H ··· H H H

The choice of Fα is initially unintuitive if one just attempts to plug into the divergence. In the rθ and rϕ planes, the ﬁrst term in the divergence reduces to

1 ∂ 1 ∂ α2F (αF ) (B.52) α2 ∂α α → α ∂α α ( ) as though we were in cylindrical coordinates, and we lose the multidimensional aspect of radial divergence.

The area integrals for Spherical coordinates reduce to the expressions in Table B.7.

B.1.2 The case when F is a monomial ∇· Consider now the case when the divergence of F is a monomial in α, β, and γ such that

F = Kαaβbγc. This arises when calculating the volume of a set of faces, or when interpolating ∇· quantities between grids.

Cartesian. For the Cartesian case, our three vectors become

K a+1 b c F1 = α β γ , 0, 0 (B.53) a + 1 [ ] K a b+1 c F2 = 0, α β γ , 0 (B.54) b + 1 [ ] K a b c+1 F3 = 0, 0, α β γ (B.55) c + 1 [ ]

The ﬁnal contour integrals for Cartesian coordinates are given in Table B.8.

81 Table B.8. Monomial area integrals for Cartesian coordinates.

K K K a+1 b c 1 a b+1 c 2 a b c+1 3 plane Fα = a+1 α β γ (F ) Fβ = b+1 α β γ (F ) Fγ = c+1 α β γ (F ) αβ K αa+1βbγcdβ K αaβb+1γcdα a+1 − b+1 ··· αγ K αa+1βbγcdγ K αaβbγc+1dα −H a+1 H ··· c+1 βγ K αaβb+1γcdγ K αaβbγc+1dβ H ··· b+1 − H c+1 H H

Table B.9. Monomial area integrals for cylindrical coordinates.

K K K a+1 b c 1 a+1 b+1 c 2 a b c+1 3 plane Fα = a+2 α β γ (F ) Fβ = b+1 α β γ (F ) Fγ = c+1 α β γ (F ) αβ K αa+2βbγcdβ K αa+1βb+1γcdα a+2 − b+1 ··· αγ K αa+1βbγcdγ K αaβbγc+1dα −H a+2 H ··· c+1 βγ K αa+1βb+1γcdγ K αa+1βbγc+1dβ H ··· b+1 − cH+1 H H

Cylindrical. For the Cylindrical case, our three vectors become

K a+1 b c F1 = α β γ , 0, 0 (B.56) a + 2 [ ] K a+1 b+1 c F2 = 0, α β γ , 0 (B.57) b + 1 [ ] K a b c+1 F3 = 0, 0, α β γ (B.58) c + 1 [ ]

The ﬁnal contour integrals for Cylindrical coordinates are provided in Table B.9

82 Table B.10. Monomial area integrals for spherical coordinates.

plane plane plane plane Fα Fβ Fγ αβ K αa+2βbγc sin(γ)dβ K αa+1βb+1γc sin(γ)dα a+2 − b+1 ··· αγ K αa+2βbγcdγ K αa+1βbγc+1dα H − a+2 H ··· c+1 βγ K αa+2βb+1γc sin(γ)dγ H ··· b+1 H ··· H

Spherical. For the Spherical case, our 6 vectors become

K F αβ = αa+1βbγc, 0, 0 (B.59) α a + 2 [ ] K F αβ = 0, αa+1βb+1γc, 0 (B.60) β b + 1 [ ] K F αγ = αa+1βbγc, 0, 0 (B.61) α a + 2 [ ] K F αγ = 0, 0, αa+1βbγc+1 (B.62) γ c + 1 [ ] K F βγ = 0, αa+1βb+1γc sin(γ), 0 (B.63) β b + 1 [ ] F βγ = (B.64) γ ···

The ﬁnal contour integrals for Spherical coordinates are listed in Table B.10.

B.1.3 Application to Volumetric Quantities

We will now attempt to build upon the area formulations in order to calculate volumetric quantities. Consider a function g(α,β,γ) deﬁned over some volume V . It is possible to write the integral of the quantity as

g(α,β,γ)dV = GdV. (B.65) ∇· ∫∫∫V ∫∫∫V

83 Furthermore, Equation B.65 can be rewritten as an area integral via Stoke’s theorem:

GdV = G Nˆ dA. (B.66) ∇· · ∫∫∫V ∫∫S

Additionally, Green’s theorem applied in a plane states that

F dA = F nˆds, (B.67) ∇· · ∫∫S I∂S and consequently

F = G Nˆ . (B.68) ∇· ·

If we consider that G has only one nonzero component such that G =[Gα, 0, 0] and apply the general deﬁnition of the divergence (Equation B.4), we arrive at

h h h g(α,β,γ)dα F = N G = N α β γ . (B.69) α α α h h ∇· ∫ β γ

The case when g(α,β,γ) is a monomial. Now let us consider the case when g(α,β,γ) =

Cαµβηγξ. Then, for each coordinate system, we arrive at

N C F = α αµ+1βηγξ, (B.70) ∇· d + 1 where d = 1, 2, 3 for Cartesian, cylindrical, and spherical geometries, respectively. This is a fairly remarkable result given the complexity of the formulas prior to this.

B.1.4 Computing area and volume over polyhedra

Integration of polyhedra. Given a closed surface in αβγ space composed of planar polygons, we can condense the results of the previous section as the sum of the area integrals over faces

Nfaces f µ η ξ Nα C µ+1 η ξ Cα β γ dV = α β γ dSf . (B.71) V d + 1 Sf ∫∫∫ f∑=1 ∫∫

84 The surface integrals in the above equation can be reduced to line integrals as per Section B.1.2.

Note on the evaluation of non-planar polygon surface integrals. The previous work is only applicable if the surface lies in a plane (not necessarily containing the origin) in αβγ. We can evaluate the surface integral of non-planar polygons by projecting them onto a plane, applying the integrals we have previously derived, and then unprojecting (thereby scaling the resulting integral).

Let Nˆf be the constant normal to the polygon in αβγ space. In order to decide which plane to project to, we determine the maximum absolute component Nˆc, where c = α,β, or γ. We then determine the minimum value of the cth component over all vertices, and denote that value as minComp. We can project onto the plane with normal only in the cth direction by setting the cth component equal to minComp.

Choosing the maximum absolute component of the normal gives us the largest projection, which helps with numerical errors. One can see this by projecting a planar polygon into one of its orthogonal planes, which gives a line (and results in incorrect answers).

From here, it is possible to evaluate the surface integrals as described in this work.

We can easily rescale the resulting answer as

Nˆf I = | |I (B.72) true N proj | c|

A consequence of the underlying coordinate system given by αβγ being a LCCS is that the

Jacobian does not need to be included in this projection.

Integration Over Straight Lines in αβγ. Given a planar polygon, we can now evaluate the contour integral to determine the surface integral. Let a line in the αβγ plane connect two points, with the starting vertex index by v and the end vertex indexed by v + 1. Each coordinate

85 is then given by

α(λ) = (1 λ)α + λα (B.73) − e e+1 β(λ) = (1 λ)β + λβ (B.74) − e e+1 γ(λ) = (1 λ)γ + λγ (B.75) − e e+1

Consider the integral of two quantities µ(λ) and η(λ) deﬁned analogously to the above. Then the integral of the product of these quantities raised to the powers p and q, respectively, is

p q r p q r 1 1 α(λ)pβ(λ)qγ(λ)rdλ = i j k αi αp−iβj βq−jγk γr−k (B.76) p + q + r + 1 p+q+r v+1 v v+1 v v+1 v 0 ( i)(+j+)(k ) ∫ ∑i=0 ∑j=0 ∑k=0 ( ) For the spherical case, a sin/cos term is included in the kernel of the integral, resulting in

αpβqγr sin γ or αpβqγr cos γ

1 p q r p q r 1 p q r i j k i p−i j q−j k r−k p+q+r α(λ) β(λ) γ(λ) sin γdλ = p+q+r αv+1αv βv+1βv γv+1γv Bi+j+k (λ)sin γdλ 0 ( i)(+j+)(k ) 0 ∫ ∑i=0 ∑j=0 ∑k=0 ∫ ( ) (B.77)

p+q+r where Bi+j+k is the i + j + k term of the p + q + r order Bernstein basis. This also appears in the simpler integral above, but reduces to the factor 1/(1 + p + q + r) that appears infront of the summations. The integrals in Equation B.77 can be evaluated as

1 n Bn (λ)sin ξdλ =π2−n−3Γ(m + 1) Γ( m + n + 1) (B.78) m m − ∫0 ( ) m + 1 m + 2 1 n + 2 n + 3 1 2 4sin(ξv)2F˜3 , ; , , ; (ξv ξv+1) 2 2 2 2 2 −4 − − ( ( ) m + 2 m + 3 3 n + 3 n + 4 1 2 (m + 1)(ξv ξv+1) cos(ξv)2F˜3 , ; , , ; (ξv ξv+1) − 2 2 2 2 2 −4 − ( )) where 2F˜3 (a; b; z) is the regularized, generalized hypergeometric function.

86 B.2 LAVAﬂow feature highlight: Image segmentation

In Section 3.2.3 we presented an analysis the flow structure emerging at the onset of neutrino- driven convection. To perform this analysis, we projected the radial momentum for a slice of the post-shock region onto the surface of a sphere (or circle in two dimensions; see Equation 3.9) and then algorithmically identified the number of disjoint upflowing (positive radial momentum) regions and estimated the solid angle they spanned. In this section, we present the algorithms implemented in LAVAflow used to perform this analysis.

B.2.1 The ﬂood ﬁll algorithm

The flood fill algorithm (also known as seed fill; Lieberman, 1978; Heckbert, 1990) traditionally used in computer graphics programs to replace all pixels of a certain color with another, provided that a starting pixel is given. While the core algorithm has many variations (4-way versus 8-way

ﬁll, stack versus queue), the fundamental approach has remained the same. For LAVAﬂow, we implement the simple 4-way, stack-based algorithm as outlined in Algorithm 1.

Algorithm 1: FloodFill recursively ﬁnds all pixels that can be reached from an initial pixel Data: Image I comprised of values unused, unfilled, filled Input: Pixel indices i, j Output: List of pixels, list, that can be reached by the starting pixel if I(i,j) = unfilled then return̸ else Add pixel (i,j) to list I(i,j) filled ← /* Recursively fill neighboring pixels */ FloodFill(i-1,j ) /* West */ FloodFill(i+1,j ) /* East */ FloodFill(i,j+1 ) /* North */ FloodFill(i,j-1 ) /* South */ return

Prior to performing the ﬂood ﬁll algorithm, one must initialize an image (i.e. array) with three possible pixel values: filled; unfilled; unused. The pixel values filled and unfilled denote pixels that have or have not been replaced by the chosen color, respectively. The value unused

87 represents pixels whose value is not allowed to change. A toy analogy would be water filling a cave system, where filled is flowing water, unfilled is dry floor, and unused is an impermeable rock wall.

The algorithm begins by selecting an initial pixel from the image. If the pixel is either filled or unused, the method terminates. However, if the pixel’s value is unfilled, we record the pixel location, change its value, and continue the algorithm. For the 4-way fill, the remainder of the method involves recursively applying the flood fill method to the initial pixels four cardinal neighbors (west, east, north, south). This process visits every pixel in the unfilled region until all pixels have been replaced. The 8-way variation of this method is straightforward by incorporating the diagonal neighbors (northwest, northeast, southwest, southeast) in the search.

This method is also easily generalized to one- and three-dimensional images. In one dimension, it is only necessary to check the east and west neighbors. For three-dimensions, one must check the neighbors above and below (top and bottom), as well as the original two-dimensional neighbors.

(Note that for a three-dimensional “8-way” ﬁll, this involves 26 neighbor checks.)

B.2.2 Identifying disjoint pixel clusters

In the analysis performed in Section 3.2.3, the interest was in identifying the quantity and size of upﬂows in the domain. This problem is equivalent to identifying the number of disjoint clusters of pixels in an image with initial values of unfilled. This is accomplished by the algorithm outlined in Algorithm 2.

The initialization of the image is obtained via Equation 3.9. The method begins by ﬁnding the

first unfilled pixel in the image. We then apply the flood fill operation to obtain a list of all pixels reachable by the starting pixel. This list is stored for later use and we increment the number of clusters by one. This process is repeated until there are no unfilled pixels remaining in the image, with the final result being a set of disjoint pixels.

This process is illustrated in Figure B.1, with white and black pixels representing unfilled and unused, respectively. The left panel shows the initial image with no filled pixels. To the human eye, it is quite easy to count the number of unconnected white regions (16 in total). The middle

88 Algorithm 2: FindClusters partitions an image into “clusters” of disjoint pixels Data: Image I comprised of values unused, unfilled, filled Output: Collection of pixel lists, with each list comprised of all pixels belonging to a “cluster” numClusters 0 ← clusters ← ∅ while the number of unfilled pixels in I > 0 do Find index (i,j) of ﬁrst unfilled pixel list ← ∅ /* Find all connected pixels */ list FloodFill(i,j ) ← /* Add the new cluster given by list to the set of all clusters */ clusters clusters + list ← numClusters numClusters +1 return ←

Figure B.1 Demonstration of the process of identifying clusters of disjoint pixels in an image as outlined in Algorithm 2. The colors black and white represent unused pixels (where no flooding should occur) and unfilled pixels (where flooding has not yet reached), respectively. Gray denotes filled pixels which have been found by the flood fill algorithm. (left panel) Initial unclustered image. (middle panel) Intermediate step during which Algorithm 1 is finding all pixels in a new cluster. (right panel) Fully segmented image after all unfilled pixels have been visited. panel shows an intermediate step where some clusters have been identified (colored regions), one is in the process of being flooded, and the others remain unvisited. The end result of Algorithm 2 is shown in the right panel, and all clusters have been found. The pixel lists generated during the cluster finding procedure allow further analysis, such as the calculation of spanned solid angle (see

Figure 3.15).

89 B.2.3 Comments on the 4-way, stack-based ﬂood ﬁll

The choice to use a 4-way fill instead of an 8-way fill for the analysis performed in Section 3.2.3 was guided by physical interpretation. The 8-way fill is allowed to “jump” a diagonal aliased line of unused values if the line is only one pixel thick. When projecting the upflows as given in Equation

3.9, it is possible that a particularly narrow downflow (equivalent to unused) could occur. If this was to happen, two human-identifiable upflow features separated by a narrow downflow could become merged into a single feature when using the 8-way fill.

The decision to use the stack-based fill over the queue-based fill was for simplicity. While the potential to overflow the stack for large images exists, this was not an issue in practical application of the method. This is due to a combination of relatively small images (restricted to the angular resolution of the mesh which is no greater than 360x180), and inhomogeneous projections (no ≈ images consisting of entirely unfilled pixels). In the event future projects consume the entire stack, the queue-based flood fill should be used instead.

B.3 LAVAﬂow feature highlight: Vector ﬁeld decomposition

To enable studies of turbulence in thermodynamically compressible systems (i.e. with the potential for shocks), it is desirable to eliminate the divergent component of the vector field prior to analysis (see Appendix D). Failure to remove the divergent component can result in high- frequency polution of the kinetic energy spectrum due to discontinuities in the velocity field which occur across shock waves. In order to clean the velocity field, we decompose the total velocity field into the sum of compressive and solenoidal fields. The following sections address the mathematical background behind this process and an overview of the implementation in LAVAflow.

90 B.3.1 Helmholtz Decomposition

The Fundmental Theorem of Vector Calculus states that a continuous vector ﬁeld F can be decomposed into the sum of a gradient term and a curl term such that

F = ϕ + A, (B.79) ∇ ∇× where ϕ is the scalar potential and A is the vector potential. The terms ϕ and A represent the ∇ ∇× longitudinal (irrotational) and transverse (solenoidal) components of the vector ﬁeld, respectively.

We note that there can exist an additional harmonic term that serves as a correction due to non- periodicity of the domain and/or a non-compactness of the original vector ﬁeld F. In what remains, we will neglect the contribution from the harmonic term (ψ 0), as its implications for physically ≡ motivated vector ﬁeld decomposition are minimal.

Additional constraints due to the nature of scalar and vector potentials are described by the relations ϕ = 0, ∇×∇ (B.80) A = 0, ∇·∇× and are known vector calculus relations.

Let us rewrite Equation B.79 in the form

F (x)= Ft (x)+ Fl (x) , (B.81)

where Ft and Fl are the transverse and longitudinal components, respectively, as deﬁned in Equa- tion B.79. The Fourier transform of this equation results in

Fˆ (k)= Fˆ t (k)+ Fˆ l (k) . (B.82)

91 Fourier transformation of the relations given in Equation B.80 produces

k Fˆ t = 0, · (B.83) k Fˆ = 0, × l where k is the wave vector. Equation B.83 indicates that in k space the longitudinal field − is parallel to the wave vector k and the transverse field is orthogonal to k. These geometric arguments provide a direct method of computing the spectral longitudinal and transverse fields from the original original spectral field using

Fˆ l = kˆ Fˆ kˆ, · (B.84) [ ] Fˆ = kˆ Fˆ kˆ, t × × [ ] where kˆ = k/ k . The physical velocity ﬁelds can be obtained by taking the inverse Fourier | | transform of Equation B.84.

92 APPENDIX C

CASANOVA: COMPUTATIONAL ASTROPHYSICS

SIMULATION ARCHIVE

CASAnova (Computational Astrophysics Simulation Archive) is a data repository whose goal is to store and provide access to simulation results produced during the course of a research group’s work. The development of CASAnova accomplishes two tasks: 1) Internally compare and contrast simulation results to guide future research directions; 2) Disseminate our data to external users for comparison, veriﬁcation, and further analysis. This second point is particularly important in the

field of computational astrophysics, where the results of different research groups can be conflicting and irreproducible due to differing simulation codes, initial conditions, runtime parameters, and other elements of simulations.

In order to facilitate the previously stated goals, CASAnova was envisioned as a web-based system in which user interaction happens on a traditional website. This website would then com- municates with the CASAnova backend which consists of a collection of databases and server-side analysis tools. Initial development was undertaken in the summer of 2012 by myself and another graduate student.

The initial implementation of the CASAnova project consisted of three parts: 1) The frontend website; 2) The backend databases; 3) The framework that provides the interaction between the frontend and the backend, as well as the capabilities for parsing simulation ﬁles to populate the databases. To this end, we relied heavily upon Python to dynamically construct the website. Addi- tionally, a collection of library functions was developed for querying and populating the databases

(also in Python). This included reading metadata from raw simulation data, parsing integral ﬁles

(generally time-varying scalar quantities that summarize the simulation data, e.g. total mass in the system), and rudimentary plot generation.

93 For the database backend we opted to use MySQL to hold the metadata extracted from simulation results. The drawback of using MySQL or any other relational database for this task is the lack of ﬂexibility. Relational databases rely on an underlying structure (called tables) that organize the data into a reusable component for all entries in the database. For example, a table named

Student may have ﬁelds for Name and Entrance Year. Then, at the university level, all students in the database would have these characteristics. Attempting to ﬁt all computer simulations into the same type of table was determined to be unsustainable for long-term development of the project.

Therefore, development of CASAnova has continued, with the primary focus being to redesign the backend system. The primary change is the switch from the relational MySQL database to the document-oriented MongoDB (Plugge et al., 2010). MongoDB is part of the NoSQL family of databases and uses a graph-based, key- value system instead of the traditional table structure of MySQL. The tangible beneﬁt from this change is enabling the integration of non-homogeneous simulation results that may have very little in common. This allows CASAnova to incorporate data produced by drastically diﬀerent numerical models/codes with minimal adjustments to the database backend.

We have also redesigned the frontend website as well. In particular, we have transitioned from the previous Python-based website to a JavaScript-enabled (Flanagan, 2002) version. All communication with the initial framework now occurs through JSON objects (Sriparasa, 2013), enabling simpler data communication between the front and backend systems and easier implementation of new front end features. This transition was done primarily for extending CASAnova’s longevity and ease the work of future developers and users; since the majority of students that will work on

CASAnova in the future will have minimal web development experience, it is crucial that development and maintenance are made as simple as possible.

C.1 Current capabilities

This section summarizes the current capabilities of CASAnova.

94 Sort, select, and view all or a subset of models archived by CASAnova. Query results include • list of simulation data files, parameter files, notes, and integral quantity files. Models are

classiﬁed by a set of high-level identiﬁers (e.g. Astrophysics, High Energy Density Physics,

etc.).

Generation of in-browser plots of integral quantities for one or more models matching search • criteria. This capability is provided through the Flot1 package.

In-browser connection to tools in the LAVAﬂow library (see Appendix B). This allows for a • predetermined analysis method to be run from the user’s browser (if the analysis data does

not already exist). These results can then be visualized using the plotting capabilities of Flot.

Job distribution and queuing for in-browser LAVAﬂow tools. For analytics that can not be • performed in real-time, the Celery2 package is used to queue, run, and inform the user when

the analysis is ﬁnished.

User management. To avoid abuse by anonymous users, as well as facilitate user-speciﬁc • features, the user management system cakePHP3 is incorporated into the website.

1http://www.ﬂotcharts.org/ 2http://www.celeryproject.org/ 3http://cakephp.org/

95 APPENDIX D

APPLICATION OF LAVAFLOW TO STUDIES OF

TURBULENT, SELF-MAGNETIZED PLASMA

D.1 Introduction

The main physics processes describing the evolution of the interstellar medium (ISM) are hydrodynamics, magnetization, and radiation processes such as ionization. In addition, the stars are localized sources of mass and energy, some of them, such as supernovae, are very powerful and capable of shaping the evolution and structure of the ISM on global scales. Those interactions eventually result in small-scale structures, including supersonic turbulence.

Absorption and emission at infrared and radio wavelengths are the primary messengers for turbulence in the interstellar medium (ISM). Observations have shown that velocity dispersion is correlated to region size via a power law dependence from sub-parsec to kilo-parsec scales (Larson,

1979, 1981; Ossenkopf and Mac Low, 2002). Observations of velocity and density are consistent with supersonic turbulence driven on large scales (at or above the size of molecular clouds), and exhibit velocity structures indicative of a shock-dominated medium (Ossenkopf and Mac Low, 2002).

There is further evidence that small-scale driving from star formation is negligible (Brunt and Heyer,

2002a,b; Brunt et al., 2009), and the observational velocity scaling is inertial. Radio scintillation measurements provide direct evidence for turbulence on small-scales ( 1012 cm) (Rickett, 1990; ∼ Boldyrev and Gwinn, 2003).

The existence of supersonic (compressible) turbulence plays an important role in star formation

(Mac Low and Klessen, 2004; Krumholz and McKee, 2005; Padoan and Nordlund, 2011), the stellar initial mass function (Padoan and Nordlund, 2002; Hennebelle and Chabrier, 2008, 2009), and more fundamentally, the density and velocity statistics of the ISM (Elmegreen and Scalo, 2004; Scalo

96 and Elmegreen, 2004; McKee and Ostriker, 2007). The primary metric in these analyzes is the dependence on the density variance with the rms Mach number; for log-normal distributions of the density contrast (s ln(ρ/ρ )), the density variance is given by σ = ln 1+ b2M 2 , where b is ≡ 0 s related to the energy injection mechanism (Kritsuk et al., 2007; Lemaster and( Stone, 2008;) Padoan and Nordlund, 2011; Molina et al., 2012). Despite analytical (Padoan and Nordlund, 2011; Molina et al., 2012) and numerical (Ostriker et al., 2001; Lemaster and Stone, 2008) investigations, the eﬀect of magnetic ﬁelds on the density variance remains unclear.

Information about the magnetic field in the ISM can be inferred via Faraday rotation and polar- ization of synchrotron radiation (Troland and Crutcher, 2008; Crutcher et al., 2009). Observations based on Zeeman measurements indicate that magnetic effects may play an important role in hydrodynamic evolution (Crutcher, 1999). However, the sustenance of the interstellar magnetic field may be be coupled to turbulence via generation due to folding and stretching of the field, resulting in small-scale dynamo effects (Schekochihin et al., 2004).

Earth-bound methods for investigating turbulence in the ISM have come only from numerical modeling. Numerical simulations indicate that the webbed structure of the ISM may result from the nonlinear advection operator (Scalo et al., 1998). Additionally, pure hydrodynamic simulations reproduce similar behavior (Kritsuk et al., 2007). Marginal quantitative differences are found between hydrodynamic and magnetohydrodynamic simulations when subjected to a background magnetic field (de Avillez and Breitschwerdt, 2005); however, filaments tend to orient along magnetic field lines. There is numerical evidence that energy transfer between spatial scales may be regulated by shocks, as opposed to a turbulent cascade (Lemaster and Stone, 2009).

While turbulence has been studied extensively in ﬂuids (Frisch, U., 1995; Kellay and Goldburg,

2002; Boﬀetta and Ecke, 2012), it has not received the same treatment for plasmas. However, advancements in high-energy-density physics (HEDP) experiments may provide a way to investigate properties of the ISM in the laboratory. Laser systems at the Omega Laser Facility and the National

Ignition Facility provide the ability to deposit kilojoule to megajoule energies on the surface of millimeter scale targets over a timescale of picoseconds to nanoseconds. Experimental approaches to potential turbulence inducing eﬀects from Rayleigh-Taylor, Kelvin-Helmholtz, and Richtmyer-

97 Meshkov instabilities have been summarized in Drake et al. (2008a). Replication of turbulence in the ISM will require long driving times, implying the need to drive large amounts of material (large volumes) presumably using multiple blast wave-like impulses created by an array of laser beams.

An initial experimental design to study compressible turbulence in the laboratory was proposed by Drake et al. (2008b). In this design, a gas-ﬁlled target box is embedded in a medium. As this surrounding medium is exposed to laser energy deposition, small holes in the box allow driven material to enter the cavity. The interactions of these focused blast waves are then expected to produce turbulent behavior in the interior of the container.

In this work we present a preliminary high-energy-density experimental scenario to investigate shock generated turbulence. Our initial, two-dimensional scenario focuses on stirring a target composed of concentric, circular layers with blast waves generated from laser irradiation. We include the effects of magnetic fields in this work by considering three cases; pure hydrodynamics, a pre-existing magnetic field, and a magnetic field generated from the Biermann battery source term. Section D.2 outlines our experimental design and the corresponding computational model, with supplementary material found in D.3.3. Section D.4 provides a comparison between the three magnetization cases. Section D.5 offers an in-depth analysis of the self-generated magnetic field case. Discussion and conclusions are offered in Section D.6

D.2 Experimental scenario

Our preliminary experimental design is based on two concentric spheres as illustrated in Figure

D.1. The inner sphere (core) is composed of low density material and provides a medium for the driven mixing process. The outer sphere (shell) is of higher density. The target is embedded in a very low density ambient medium (essentially vacuum). All materials are initially in pressure equilibrium. The aim of this work is to produce an initial investigation of the two-dimensional problem.

In order to reproduce a multiply shocked section of interstellar medium, the shell layer is irradiated on its surface by a set of laser drives. The drive- shell interactions result in pressure and

98 Figure D.1 Proposed experimental setup. (left panel) Two-material target configuration. The dashed circle shows the region we consider for our analysis, the “turbulent core”. (right panel) Schematic for one round of laser driving. Three lasers (a triple) are positioned 120◦apart. Each laser is a precomputed hydrodynamic profile from the CRASH code, with the leading edge of the shock touching the thick solid line. material waves propagating toward the center of the target. As the shocks pass through the core at different times complex hydrodynamic conditions are created. The primary purpose of the high density shell is to absorb the laser drive and convert thermal energy into kinetic energy. As a result, the perturbations reaching the core region should be kinetically dominated. Our hope is repeated exposure of the shell material to laser pulses should produce a proxy for the effects of supernovae in the ISM. Ideally, the laser system would be arranged in a spherically symmetric configuration to help confinement of the target material and allow for longer evolution of the shocked system.

The eﬀects of laser driving on the shell material is a transient problem that is not of particular interest in this work. We are primarily interested in the eﬀects of mixing in the light core, and therefore consider a “turbulent core” (TC) encompassing the target core and part of the shell.

Consequently, we view the area exterior to the TC as generating the boundary conditions for the interior of the TC. The results of this work focus only on data interior to the TC, unless otherwise stated.

99 D.3 Computational model

We assume that the laboratory setting can be modeled via the extended magnetohydrodynamic equations (Braginskii, 1965; Nishiguchi, 2002):

∂ρ + (ρu) = 0, (D.1) ∂t ∇·

∂ρu B 2 + (ρu u)= p + | | , (D.2) ∂t ∇· ⊗ −∇ ( 8π ) ∂ρε + (ρεu)= (pu) , (D.3) ∂t ∇· −∇ · ∂B c p ( B) B = (u B)+ ∇ e ∇× × , (D.4) ∂t ∇× × e ∇× n −∇× 4πn [ e e ] where ρ, p, ε, u, B are the fluid density, thermal pressure, specific total energy, material velocity, and magnetic field, respectively. The electron pressure and electron density are denoted by pe and ne, respectively. The leading constants on the bracketed term of Eqn. D.4 are the speed of light (c) and electron charge (e).

Equation D.4 describes the evolution of the magnetic ﬁeld using the generalized Ohm’s law, where the bracketed term indicates the components that result in self-generation of the magnetic

field. The first term inside the brackets is the Biermann battery term, and causes field generation when the temperature and density gradients are misaligned. The second term is the Hall term, and indicates that pre-existing fields can self amplify. When no initial magnetic fields exist, the

Biermann battery term may begin creating a ﬁeld, which the Hall term will then act upon. We do not include the electron-ion friction term in our models.

We utilize an ideal gas equation of state with γ = 1.6. The plasma composition is represented with a single species with atomic mass, A, and atomic charge, Z. Accordingly, the ion number density is ni = ρNA/A, where NA is the Avogadro constant. To obtain the electron number density, ne, we use the Thomas-Fermi equation of state (Salzmann, D., 1998). The required electron number density is calculated as ne = Zn¯ i.

100 D.3.1 Proteus

The set of Equations D.1-D.4 are solved numerically using the ﬁnite volume Proteus code, which is our developmental fork of the FLASH code (Fryxell et al., 2000). In this work we use the unsplit staggered mesh solver of Lee, D. and Deane, A.E. (2009). This magneto-hydrodynamics (MHD) solver is a variant of the constrained transport method (Evans and Hawley, 1988) and is used for all cases considered. The MHD solver is formally second-order accurate in space and time.

The method used for driving the turbulence in these models (see Section D.3.3) produces numerical diﬃculties for many Riemann solvers. In order to maintain a robust simulation environment, we use the Harten-Lax-van Leer-Einfeldt (HLLE) Riemann solver (Einfeldt, 1988).

Computations are performed on a square, Cartesian domain with sides of length 9000 µm . This allows for the entire target to be placed inside, with additional room to develop in the ambient medium. We treat the boundary conditions as open outﬂows. While we limit our analysis to the smaller region of interest surrounding the core, simulation of the complete target allows for the interaction of laser-driven material. As the area exterior to the TC can be viewed as boundary conditions for the TC, it is possible that these interactions could aﬀect our results and are accounted for as much as possible.

We utilize statically reﬁned Cartesian meshes for our domain decomposition. We reﬁne the grid inside of the region of interest to a uniform spacing. We perform each case on three separate TC grid sizes: 16 µm for the coarsest run; 8 µm for the medium resolution run; and 4 µm for the

ﬁne resolution run. Exterior to the TC we allow the mesh to coarsen radially (subjected to proper nesting). The coarsest cell resolution for all computed models is 64 µm .

D.3.2 Initial conditions

We consider three distinct magnetization cases in this work: pure hydrodynamics; a pre-existing, out of plane magnetic field; and self-generated magnetic fields (no pre-exiting field). The pure hydrodynamics case provides a reference from which to gauge the impact of magnetic fields on the

ﬂow structure. With these three cases, we aim to begin quantifying the eﬀects of turbulence in high

101 Table D.1. Target geometry and initial conditions.

quantity value units

−4 −3 ρambient 1 10 g cm × −1 −3 ρshell 1 10 g cm × − − ρ 2 10 2 g cm 3 core × p 1 103 bar target × rshell 3250 µm rcore 500 µm rTC 1500 µm Tcore 11, 600 K

energy density conditions, attempting to include increasing levels of physical complexity. Table D.1 shows the shared conditions for all of our models.

The pre-existing magnetic field case consists of an initial magnetic field whose only nonzero component is out of the plane. This field provides an additional pressure component (in the form of magnetic pressure), without forcing the advection of material to be along field lines. Such a configuration is difficult to realize in the laboratory, and should be thought of as a toy model.

However, it allows us to gauge any magnetic effects in a controlled manner, where we have an a priori estimate of the plasma β. (For the entirety of this work we define β as β = p/(8πB2).) In order to initialize the magnetic field we use a characteristic reference pressure of p = 1 106 bar ref × and choose βref = 20 to solve for the magnitude of the initial magnetic field. Our chosen reference pressure is characteristic of conditions in the turbulent core during the laser driving period, and should put our pre-existing field simulations near βref .

D.3.3 Shock generation

The requirement of producing high-energy density turbulent plasma in a state close to steady state and as isotropic as possible imposes certain restrictions on the laser drive. In a viable design it would be highly desirable that average linear and angular momenta of the system are close to zero. One possible way to achieve this is to compensate for the linear momentum injected by any

102 laser pulse using a some combination of remaining laser pulses. For example, one could fire two laser beams from opposite directions. This configuration, however, would not allow for lasting plasma confinement. Therefore, a more complex setting is needed, such as a triple (with laser beams originating at the tips of an equilateral triangle in 2D) or a quadruple (with laser beams originating at the tips of a tetrahedron in 3D) laser drive configuration. Furthermore, one would wish to add a certain degree of randomness to the drive in order to promote the development of turbulence. (Although such a drive configuration cannot be realized at existing HEDP laser facilities, our primary goal is to assess the feasibility of possible future designs for turbulent plasma experiments.)

In our two-dimensional study, the laser drive configuration is defined by a set of three lasers arranged 120◦ apart (a triple) in order to improve confinement. Over the course of driving we sample 100 triples every 2 nanoseconds with each laser in the triple fired at the same time. Each triple is offset by a random angle, θ, sampled from a uniform distribution over 0◦ <θ< 120◦. The initial triplet has an angular offset of √2 180◦/π 81◦ in order to avoid initial grid symmetries. × ≈ While we use Proteus to evolve the target, we do not use it to simulate the laser-target interaction. Instead, we precompute a two-dimensional “laser drive profile” (LDP) and then map it onto the Proteus mesh when a triple is activated. The LDP is a fixed time, two dimensional, cylindrical set of primitive variables, and is further described in Section D.3.3. When mapping the LDP to the Proteus mesh, we place the inward moving tip along a circle of radius rmap = 2000 µm , with the angle of the LDP axis given by the sampled angle θ.

We note that the process of mapping the LDP onto the Proteus mesh is not conservative. We feel this approach is justified, as the drive is mapped away from the TC, which is an open system in its own right. Thus, the area outside is of ancillary importance in terms of conservation. This method also enables dramatically faster turnaround on model generation, as we do not have to compute laser-target interactions in the complex media surrounding the TC. More realistic studies aimed at evaluating specific laboratory experiments may have to abandon the LDP concept and compute the laser energy deposition along with the interior calculations at the significant increase

103 of computational time. Again, our aim in this work is to provide initial insights into the behavior of such an experiment.

Calculating the laser drive profiles. We compute the laser drive profile (LDP) using the CRASH code (van der Holst et al., 2011). In this configuration, a 2.5D cylinder of carbon foam is irradiated along the positive z-axis. The physical domain for the LDP simulations is

120 µm z 2000 µm , and 0 r 2000 µm . The initial density is set to ρ for − ≤ ≤ ≤ ≤ ambient z < 0, and ρ for z 0. Initially, both layers are in pressure equilibrium at 1 108 Pa. These shell ≥ × conditions match those of the target in the full Proteus simulation. The LDP simulations do not contain any magnetic ﬁeld eﬀects.

The incident irradiation is deﬁned by by a super-Gaussian laser beam of order 4.2, with a standard deviation of 250 µm in the radial direction, and whose center is coincident to the axis of symmetry. The pulse is a constant 3.3 1013 W for t = 1.8 ns with a linear rise and decay time of × 0.1 ns. Laser energy deposition is accomplished using CRASH’s geometric ray-tracing functionality with 400 individual rays representing the beam.

We performed two runs with uniform mesh resolutions of 8 µm and 4 µm . These resolutions are at least a factor of two smaller than the Proteus mesh the LDP is mapped to. We determined there were minimal diﬀerences between the two resolutions.

We choose the evolutionary snapshot to be used as the LDP by ﬁnding the moment when the rarefaction fan catches up to the shock front. The laser driven ﬂow is thermally developed at this point, and the bulk of the energy transferred to the shell will be in the form of kinetic energy.

This corresponds to the snapshot at t = 2.1 ns in both the 8 µm and 4 µm simulations. This evolutionary time is after the CRASH laser drive turns oﬀ and we are not truncating any laser physics by choosing the t = 2.1 ns snapshot.

Depicted in Figure D.2 are the pseudocolor plots for density, thermal pressure, and velocity magnitude. The laser drive produces a parabolic structure moving predominately in the radial direction. By the time the ﬂow becomes thermally developed the extent the blast wave reaches nearly 1500 µm . The ablation near the beam leaves a low density, high pressure region behind the

104 (a) (b) (c) Figure D.2 Pseudocolor plots of quantities for the prototypical laser drive calculated with CRASH. All plots are colored in log scale. The solid line which forms an envelope over the on all plots 9 −3 indicates our cutoﬀ region for mapping (pcut = 3 10 Pa). (a) Density [g cm ]. (b) Pressure − × [Pa]. (c) Velocity magnitude [cm s 1]. shock, implying that the bulk of the kinetic energy that will reach the core is compressed near the shock structure. In addition, the model shows a substantial preheat region ahead of the shock.

The only qualitative diﬀerence between the 8 µm and 4 µm models is near the symmetry axis; the 4 µm model shows a slightly more bulbous structure. We do not think this will aﬀect the generation of turbulence. Therefore, in the interest of memory constraints and mapping time, we choose to use the 8 µm model as the prototypical laser drive.

D.3.4 Mapping the laser drive proﬁles

We interpret the conﬁgurations described in Section D.3.2 as a set of cylinders (rectangles in two-dimensions) embedded into the shell of the target. The geometry of these cylinders are computed at code startup on each processor. Additionally, LDP hydrodynamic variables (obtained in Section D.3.3) are loaded onto each core.

When it is decided that a laser should be ﬁred in the Proteus simulation (described in Section

D.3.2), the cells owned by a processor are searched to determine if any reside inside, or are clipped by, the laser cylinder. If any do, the location of the cell in the cylinder’s local coordinate system are determined. From this information, we determine where the cell lies in the two-dimensional

CRASH data set. In the case of two dimensions, this is intuitive as we are rotating and shifting a rectangle onto another rectangle. In three dimensions we only consider the axial and radial

105 coordinates in the cylindrical coordinate system, and discard the angular component. In our study this is reasonable, as we are mapping an axisymmetric data set. However, we note that mapping a three-dimensional data set into the cylinder (or any other geometric primitive) requires a physically meaningful deﬁnition of the rotation about the axis, complicating the modeling of the system.

After determining our current cell’s position in the data set frame, we interpolate the data set quantities using bi-linear interpolation. In order to enhance the mapping obtained, we perform the interpolation on a uniform grid of 10 points per dimension, weighting by sub-cell volume.

The quantities interpolated from LDP are forcibly written onto the mesh in the pre-computed location. This entails overwriting the values of density, pressure, and velocity components (trans- formed into the coordinates of the cylinder axis). We lessen obtrusiveness of mapping on the surroundings by only using the LDP data inside of a pressure cutoﬀ of p = 3 108 Pa. This cut- cut × oﬀ is shown as the gray or black solid line moving through the domain in Figure D.2. Additionally, strong discontinuities near the edge of the mapped data can lead to failures of the Riemann solver.

To circumvent this problem we smooth the ﬁelds with an arithmetic averaging ﬁlter of 5 5 cells × near the edge.

D.4 Eﬀect of magnetic ﬁelds on hydrodynamics

D.4.1 Compressibility eﬀects

In order to judge the effects of compressibility, we consider the evolution of the rms-Mach number, Mrms, shown in Figure D.3 for the hydrodynamic, pre-existing field, and self-generated cases. The first laser-driven shocks penetrate the low density core at t = 30 ns, causing a sharp rise in Mrms. By t = 50 ns the core has been completely overrun and the rms-Mach number reaches a quasi-steady value of approximately 0.2.

For the remaining t = 150 ns of evolution, continual driving via laser triplets stirs the region of interest. Over this period the rms-Mach number marginally decays due to the combined effects of suppressed material accelerations (resulting from the confinement via the laser arrangement) and sound speed increase via compression. During this driving phase there is no discernible effect

106 1.0 self-generated

0.8 pre-existing hydrodynamic

0.6 rms

M 0.4

0.2

0.0 0 100 200 300 400 time [ns]

Figure D.3 RMS-Mach number for the hydrodynamic, pre-existing magnetic field, and self- generating magnetic field cases. The Mach number behaves similarly for all magnetization cases, indicating that the magnetic field is likely too low to influence the hydrodynamic development of the system. The laser-driven, quasi-steady state system never reaches the supersonic regime. from either a priori or in situ magnetic fields. The low rms-Mach numbers obtained during the driving phase indicates that we are not reaching the supersonic regime. When laser driving ceases at t = 200 ns there is a marked increase in the rms- Mach number. This trend peaks at t 230 ns, ≈ after which Mrms decays to a nominal value of 0.25. After the laser drive turns off, there is no confining ram pressure to balance the thermal pressure in the core. This results in outward expansion which simultaneously increases material velocity and decreases sound speed. The Mach number is able to stabilize at later times as conditions in the region of interest homogenize. We note that the deviation between Mrms and the density-weighted turbulence Mach number is approximately 3% during the driving phase.

D.4.2 Magnetic ﬁeld evolution

While magnetic ﬁelds do not appear to play a role in the hydrodynamic development of the driven system, the role of generation and ampliﬁcation in this driven turbulence scenario is interesting in its own right.

107 7 7 10 10 self-generated 106 106 pre-existing 105

5 10 104

[G] self-generated | β 3

B 4 10 | 10 pre-existing 102 103 101

2 0 10 10 0 100 200 300 400 0 100 200 300 400 time [ns] time [ns]

Figure D.4 Magnetic field and plasma β over time for the self-generating and pre-existing field cases. (top panel) Magnitude of the magnetic field. The thick curves indicate the arithmetic mean, while the thinner lines above and below show the 75th and 25th percentiles, respectively. For the pre-existing case, the initial mean magnetic field is amplified by a factor of 2 to 3 during the driving stage. The field then decreases when driving is halted. Note that the self-generating case produces only kilogauss fields. (bottom panel) Plasma β values (β = p/(8πB2)). The top and bottom lines for each case represent the 75th and 25th percentiles, respectively. The mean is not shown, as localized magnetic field voids produce small regions of extremely high β, skewing the results. The preexisting case generates β values on the order of 1 to 100. The self-generating case generates very high β values, resulting in a thermally dominated flow field.

The evolution of the magnetic field strength and plasma β for the two magnetized cases is shown in Figure D.4. For the case of a weak pre-existing magnetic field with βref = 20, the initial magnitude of the magnetic field 5 105 G. During the driving phase the field is amplified by a × factor of 2, with a mean field strength of 1 106 G. The distribution of field strength is roughly × Gaussian during this period with a negative skew. The cessation of laser driving results in the distribution narrowing and the field strength decaying as the magnetic field is advected from the turbulent core.

The self-generating case quickly reaches its peak shortly after the TC is overrun by the first shocks and produces field strengths on the order of 1 104 G. The resulting field distribution is also × a negative skew non-Gaussian. Unlike the pre-existing case, the shape of the distribution remains static post-driving. However, this distribution undergoes a shift as the magnetic field decays at the same rate regardless of field strength.

The plasma β distribution for the pre-existing case predominantly covers the moderate-to-weak

ﬁeld range of 10 β 100, which agrees with our initial magnetic ﬁeld estimate of β = 20. This ≤ ≤ ref

108 distribution contains ﬂuctuations in the larger-β region and remains roughly constant for smaller-β.

In the post-driving phase β decreases to a uniform value on the order of 10 throughout the domain.

In contrast, the distribution for the self-generating case maintains the same structure throughout the driving phase, much like the distribution of B. The bulk of the distribution is in the range of

104 β 105, indicating very weak effects on hydrodynamics due to the magnetic field. After ≤ ≤ driving β increases drastically in contrast to the pre-existing field case.

(a) pre-existing ﬁeld (b) self-generated ﬁeld

Figure D.5 Pseudocolor plot of the bivariate probability distribution of magnetic field strength and plasma density for the two magnetized cases. (a) The field strength for the pre-existing field correlates linearly with density, indicative of compression of the magnetic field lines as the gas compresses. (b) The self-generated field strength indicates that more complicated physics is involved than pure compression effects. There is no visible correlation between field strength and density for this case, indicating that amplification of field strength due to material compression is, at least, strongly suppressed.

Figure D.5 shows the probability distributions of the magnetic field magnitude with respect to the density and gradient of density. For the pre-existing field case, there is a linear correlation between the magnetic field strength and the density. For compressible flows, this is indicative that the compression of the fluid is also compressing the field lines. In contrast, no correlation is seen between the density and field strength for the self-generating case. This does not indicate an absence of field line compression, but rather the complicated physics involved with generation hides such a correlation. Preliminary investigation into dependence for the self-generated case shows that short term, strong field events occur (on the order of megagauss), but it is difficult to

109 quantify such behavior with the available data. We recommend further work in this area, with a focus on Lagrangian particle analysis.

D.5 Hydrodynamic evolution of the self-generated case

D.5.1 Morphology

(a) t = 23 ns (b) t = 130 ns (c) t = 230 ns Figure D.6 Pseudocolor plots of density (g cm−3) for the self-generated magnetic ﬁeld case in the turbulent core. Mesh resolution in the region is 4 µm . (a) t = 23 ns (b) t = 130 ns (c) t = 230 ns

Pseudocolor plots of the density in the region of interest at multiple times is depicted in Figure

D.6. At t = 23 ns the initial laser-driven shocks have penetrated the light core material. The small triangular region at the center of the TC is still unshocked and the preheated region in front of the hydrodynamic shocks can be seen. There are no discernible ﬂuid instabilities at this time, although at previous times small Rayleigh-Taylor and Kelvin-Helmholtz instabilities could be seen before being overrun by later shocks. The interaction of initial shocks has compressed the triple point areas by a factor of 13 from the shell density.

In the middle of the driving phase (t = 130 ns) conﬁnement due to the laser arrangement results in a dense core embedded in a lower density medium. The ﬂow structure appears very chaotic, with the multitude of shocks passing through the core clearly visible. This image is representative of the remaining 70 ns of driving.

At the peak of the post-driving phase (230 ns), the density ﬁeld exhibits interesting ﬂow features.

In particular, there appears to be low density bubbles embedded in the more dense remnant of the core. We note that the maximum density has decreased by a factor of nearly 6 from the snapshot at

110 t = 130 ns. The bulk distribution of density appears (visually) to be quite anisotropic, with a low density imprint at 120◦, 220◦, and 320◦. Note that the last laser triplet was ﬁred nearly t = 30 ns prior, and the resulting shocks have left the domain already.

D.5.2 Isotropy

1 1 1

y 0 y 0 y 0 e u e u e u

−1 (a) t = 30 ns −1 (b) t = 130 ns −1 (c) t = 230 ns −1 0 1 −1 0 1 −1 0 1 uex uex uex

Figure D.7 Isocontours of the bivariate probability distribution for scaled velocity vector components in the turbulent core. The individual components of velocity for each grid cell are scaled by the maximum velocity magnitude at each snapshot to generate ux/y. The insets show a polar histogram indicating directionality of the velocity field. The solid circle shows how a uniform 6 −1 (isotropic) distribution of velocity would appear. (a) t = 30 ns, u maxe = 9.3 10 cm s . The | | × − initial imprint from the first laser triple is clearly visible. (b) t = 130 ns, u = 9.4 106 cm s 1. | |max × During the driving phase, the velocity field achieves an isotropic distribution. (c) t = 230 ns, u = 5.5 106 cm s−1. The absence of confining ram pressure allows the compressed plasma | |max × to expand when laser driving ceases, which manifests as the of the velocity space.

The amount of physical material available in the experimental setting is limited due to the balance between target size and available driving energy. Consequently, confinement of the shocked plasma in the region of interest is of great benefit to the experiment. Therefore, we investigate the isotropy of the velocity field over time, with the most beneficial outcome being that the velocity evolves to a fully isotropic state where the bulk momentum is zero. Additionally, isotropy of the

ﬂow ﬁeld has additional implications for turbulence and the application of Kolmogorov theory.

Probability density contours of the velocity components in the region of interest are shown in

Figure D.7. The axes are scaled by the maximum instantaneous velocity magnitude to standardize the ﬁgures. The inset shows the angular probability distribution mapped to polar coordinates with the solid line. A fully isotropic distribution is shown with the dashed circle.

111 As flow features (such as shocks) penetrate the TC and evolve, large velocity gradients are formed. This behavior is reflected in velocity space by tightly packed isocontours. Additionally, the velocity within fluid structures has relatively mild gradients, leading to high probability regions.

Therefore, the distribution of isocontour lines is a direct result of the ﬂuid structures inside of the region and one can infer information about the aggregate behavior of ﬂow features in the domain.

At t = 30 ns three “ejecta” can be seen extending from the origin. In physical space this corresponds to the three dense, multiply shocked regions in Figure D.6. This distribution is highly anisotropic, which is illustrated in the inset. Note that the inset probability plot is rotated by

180◦ from the apparent position of the shocks because it is a representation of the direction of motion, not position.

In the middle of the driving phase (t = 130 ns), the core has been overrun repeatedly with shocks and the chaotic motion induced by the laser driving results in a much more isotropic distribution of velocities shown in the inset. The circular banding of isocontours around the origin implies that

ﬂow structures are causing a roughly isotropic distribution of velocity gradients. This indicates that the shocks (in particular, their normals) are isotropically distributed throughout the domain, as the primary source of acceleration in this system comes from the kinetic, laser-driven material

(since magnetic ﬁeld eﬀects play a minor role).

The final snapshot at t = 230 ns illustrates that the velocity components have begun to fill out the velocity space. There are strong, localized flow features that result in directionally biased velocities; however, the bulk of the distribution remains isotropic. The localized features are the result of the final shocks passing through the domain. The bulk outward motion is due to the lack of laser-driven confinement which allows for pressure gradients to drive material away from the origin.

D.5.3 Flow ﬁeld structure during driving

Additional information can be obtained about the ﬂow ﬁeld during the driving phase by de- composing the velocity into constituent components. Using the Helmholtz theorem (Arfken and

Weber, 2005), we can reformulate the velocity ﬁeld as u = uc + us + uh, where uc, us, and uh

112 (a) total velocity (b) compressive velocity (c) solenoidal velocity

Figure D.8 Streamlines of the velocity ﬁeld and its decomposition at t = 130 ns. This square region is centered in the TC and is the area which the spectra is computed on. (a) Total velocity. Note the combination of point sources, link sinks, and vortical motions. (b) Compressive velocity component. There are multiple point sources visible, indicating expanding regions. The coalescing streamlines in the domain are line sinks, and correspond to compression of the material. (c) Solenoidal velocity component. The underlying vortical ﬂow structure is showcased, with a nested structure to the vortices. Note that for two dimensional turbulence, the smaller vortices are likely inducing the formation of the larger vortical cells. are the compressive, solenoidal, and harmonic components, respectively. We do not consider the harmonic component in our analysis as it exists only as a correction for non-periodic domains and plays no dynamical role in the system. In order to compute these terms, we rewrite them in terms of their Fourier transforms,

ˆuc (k)=[k ˆu (k)] k, (D.5) ·

ˆus (k)=[k ˆu (k)] k. (D.6) × ×

Equations D.5 and D.6 are easily computed and inverted to produce the corresponding velocity

ﬁelds in physical space. Additional details of our computations involving the Fourier transform are found in Section D.5.6.

Figure D.8 shows streamlines of the velocity ﬁeld and its components at t = 130 ns for a square subsection of the region of interest. The left most image shows the streamlines for the total velocity

ﬁeld (u).

113 The middle image illustrates the compressive component of the velocity ﬁeld (uc). The sources and sinks in the ﬁeld are now clearly visible. The point-like areas where streamlines fan out are indicative of the expansion of material ( u > 0). ∇· The domain is also populated by “line sinks” where streamlines abruptly end along a curve.

These features are regions of fluid compression ( u < 0), induced by either strong perturbations ∇· or weak shocks. Strong shocks in the domain are rare, and the flow field is predominantly populated by strong perturbations.

The final set of streamlines are for the solenoidal component of the velocity field (us). One can easily identify the underlying vortical structure of the flow field now. In particular, the domain consists of nested vortices typical of turbulence. As we are dealing with two dimensional turbulence, it is conceivable that the small vortices induce the larger structures due to the inverse enstrophy cascade. For the subsection pictured, the largest cell is approximately 1600 µm in diameter, while the smallest cell contained within has a diameter of roughly 100 µm .

D.5.4 Radial distribution

Figure D.9 quantiﬁes the descriptions provided in Sect. D.5.1, showing density and pressure as a function of distance from the center of the target. These results are obtained by constructing probability distribution functions for grid cells contained in circular shells of 50 µm width. Thick solid lines denote the mean value while thin solid lines show the minimum and maximum values.

During the middle of the driving phase (t = 130 ns), the density peaks at 600 µm away from the core, and decays as the radius increases. The pressure is relatively low at the center of the core, and increases until 600 µm when it peaks and decays slightly.

By the time the laser drives cease firing at t = 200 ns, the density profile has flattened near the core, but continues to decay with radius. The pressure near the center of the core has increased, bringing the region of interest into approximate thermal equilibrium.

At t = 230 ns, at the peak of the rms-Mach number, the average density proﬁle in the core has homogenized, staying roughly constant out to 800 µm . Note that the maximum mean density has

114 0.8 2.5 t = 130 ns 2.0 t = 130 ns 0.6 ] 3

− 1.5 0.4 1.0 [g cm ρ

0.2 p [10 Mbar] 0.5

0.0 0.0 0 375 750 1125 1500 0 375 750 1125 1500 radius [µm] radius [µm] 0.8 2.5 t = 200 ns 2.0 t = 200 ns 0.6 ] 3

− 1.5 0.4 1.0 [g cm ρ

0.2 p [10 Mbar] 0.5

0.0 0.0 0 375 750 1125 1500 0 375 750 1125 1500 radius [µm] radius [µm] 0.2 1.8 t = 230 ns t = 230 ns 1.6 ] 3 − 0.1 1.4 [g cm p [Mbar] ρ 1.2

0.0 1.0 0 375 750 1125 1500 0 375 750 1125 1500 radius [µm] radius [µm]

Figure D.9 Radially averaged distributions of density (left column) and pressure (right column) inside the turbulent core at t = 130 ns, t = 200 ns, and t = 230 ns. Thick solid lines denote the mean value while thin solid lines show the minimum and maximum values. decreased by approximately a factor of 4. While the pressure proﬁle indicates that the region is still roughly in thermal equilibrium, the mean value has decreased by an order of magnitude.

D.5.5 Density probability distribution functions

In addition to investigating the spatial dependence on density, we can also look at the probability distributions in the region of interest. Figure D.10 shows these distributions during and after the driving phase for the self-generating case. Comparing the distributions at early and late times during driving, we note that the proﬁles are generally similar and follow log-normal distributions with parameters µ = 1.6435, σ = 0.4571 for the t = 130 ns distribution, and µ = 1.8020, 130 − 130 200 −

115

130 ns 200 ns 230 ns ) ρ PDF (

0.0 0.1 0.2 0.3 0.4 0.5 − ρ [g cm 3]

Figure D.10 Probability distribution function for the density in the turbulent core at t = 130 ns, t = 200 ns, and t = 230 ns. The distribution at t = 230 ns is scaled by 1/2 for visualization purposes. During the driving phase the density follows a lognormal distribution. However, once laser driving ceases the distribution becomes bimodal.

σ200 = 0.4619 for t = 200 ns. These long tails are the byproduct of shock-driven turbulence. The late-time distribution also indicates that mass has been added to the region from the driving, but this was to be expected due to the nature of the open system.

The post-driving distribution of density is quite diﬀerent from that of the driving phase. It has become strongly bimodal and no longer exhibits the extended tail that characterized the driving phase distributions. In addition, the densities have shifted to lower values with the low and high density peaks at approximately 4.5 10−2 g cm−3 and 9.4 10−2 g cm−3, respectively. This result × × quantiﬁes the structures at late times shown in Figure D.6.

D.5.6 Kinetic energy power spectra

In order to compute the velocity power spectra on the mesh we consider a square area of

1000 µm by 1000 µm inside the region of interest. Our ﬁnest mesh (4 µm resolution) provides a uniform 5122 grid. Likewise, the 8 µm and 16 µm meshes produce 2562 and 1282 grids, respectively.

From these grids we compute the numeric Fourier transform of u (x), ˆu (k), using the fast Fourier

116 24 24 24 10 10 10 Pt (k) Pt (k) Pt (k)

Ps (k) Ps (k) Ps (k)

Pc (k) Pc (k) Pc (k) − − − 1021 P (k) ∝ k 5/3 1021 P (k) ∝ k 5/3 1021 P (k) ∝ k 5/3 − − − P (k) ∝ k 3 P (k) ∝ k 3 P (k) ∝ k 3 P (k) P (k) P (k) 1018 1018 1018

t = 130 ns t = 200 ns t = 230 ns 15 15 15 10 10 10 100 101 102 103 100 101 102 103 100 101 102 103 k k k

Figure D.11 Kinetic energy spectra (total, compressive, and solenoidal) at t = 130 ns, t = 200 ns, and t = 230 ns for the self-generated magnetic field case. (a,b) During the driving phase the power spectra in the inertial range (5 ≲ k ≲ 100) scales as P k−2.3. This deviates from the traditional − ∝ two-dimensional k 3 behavior due to laser-driven stirring and not magnetic field effects. (c) After driving ceases, the solenoidal component becomes the dominant source of kinetic energy and scales as P k−2. In the absence of inertial forces, the dominant process is the interaction between ∝ − vortical cells, leading to a k 2 dependence.

−1.5

−2.0 α

−2.5

αt αs αc −3.0 0 50 100 150 200 250 time [ns]

Figure D.12 Evolution of the velocity power spectra exponent for 10 k 50. The turbulent flow ≤ ≤ reaches a quasi-steady state near t = 75 ns with α 2.3 for all components. This deviation from ≈− α = 3 for two-dimensional turbulence is a consequence of the laser-driven stirring. − transform. We define the multi-frequency, total velocity power spectra as P (k) = û (k) 2 /2. In t | | order to reduce this to a function of a single, mean wavenumber (k) we average Pt (k) over circular shells of unit thickness in k-space. In addition to the spectrum for the total velocity field, we also

117 examine the power spectra for the compressive and solenoidal components of the velocity ﬁeld,

Pc (k) and Ps (k). Figure D.11 shows estimated velocity power spectra at t = 130 ns, t = 200 ns, and t = 230 ns for the 4 µm resolution, self-generating case. Figure D.12 shows the evolution of the power law exponent, α, for the total, solenoidal, and compressive components. A least-squares fit over the range 10 k 50 is used to estimate α. The profiles are smoothed over a t = 5 ns window to ≤ ≤ remove high frequency oscillations and allowing differentiation of the curves.

In all snapshots the inverse energy cascade (P k−5/3) is visible up to the driving mode at ∝ k = 5.

During the driving phase, the power spectra in the inertial range (5 ≲ k ≲ 100) scales as

P k−2.3. This behavior is similar to the classical theory of magnetohydrodynamic turbulence, ∝ where two dimensional behavior obeys P k−7/3 (Kraichnan, 1965; Kraichnan and Montgomery, ∝ 1980). This relation is likely coincidental as the magnetic ﬁeld in these two dimensional simulation is generated out-of-plane and acts only as an additional pressure term. The self-generated magnetic

ﬁelds produce negligible pressures that are unable to drive material motion over the timescales considered. Indeed, these results agree with the same analysis run on the pure hydrodynamic case.

As a result, the deviation from the expected P k−3 behavior is most likely due to the laser-driven ∝ stirring.

In the post-driving phase the behavior of the compressive and solenoidal power spectra diverge.

The equilibration of pressure in the region of interest results in a substantially weaker contribution from compression effects. In the absence of inertial forces, the dominant process is the interaction between vortical cells. This makes the system appear diffusion-dominated rather than advection- dominated. This has certain consequences on the kinetic energy spectra, causing it to assume the form P k−2 (Figure D.12, t > 225 ns). This can be explained by transforming the diffusion ∝ operator into k-space, which gives a time-independent spectra with k−2 dependence.

We note that it is typical to see the inertial region smoothly transition into numerical dissipation at high wave numbers. However, the inertial region ﬂattens before abruptly dropping into the

118 dissipation range (not shown is a smoothly decaying knee beginning after the sudden drop). We believe there are two possible causes for this bottleneck eﬀect.

A potential physical explanation is that the stirring mechanism limits the cascade of energy to smaller scales. In particular, shocks moving through the turbulent core will overrun and destroy small-scale features. As such, the driving process may impose a lower-limit on feature size resulting in a buildup of power near this lower bound.

An alternative explanation for the bottleneck is that numerical eﬀects halt the transfer of energy to smaller scales (Sytine et al., 2000). As we were unable to successfully compute models using

Riemann solvers other than HLLE, this hypothesis is diﬃcult to either conﬁrm or refute.

D.6 Discussion and conclusions

We have presented the results of a computational study of a high-energy density physics laser- driven experiment aimed at producing supersonic turbulence in plasma. The design included a target irradiated by sets of laser beams to provide plasma conﬁnement and induce turbulence. To this end we computed a generic laser drive proﬁle (LDP). During the evolution, the LDP has been mapped at select positions and times in such a way as to create turbulent conditions in the central region of the target.

We found that:

The turbulence Mach number reaches nominal, quasi-steady value of 0.2 throughout the • driving phase for all cases. There is a minor downward trend as the material in the turbulent

core is increasingly thermalized. Shortly after the driving phase, the turbulence Mach number

rises due to the conversion of suppressed thermal energy into kinetic energy as the conﬁning

ram pressure is removed. In the post-driving phase the turbulence Mach number reaches

steady values of about 0.25.

The magnetic fields produced for the out-of-plane a priori field are on the order of megagauss. • These fields correspond to β 10 100. Amplification of the magnetic field due to driving ≈ −

119 results in a factor of 2 increase during the driving phase. In the post-driving phase the spatial

distribution becomes uniform with a nominal value on the order of 100 kG.

The self-generated magnetic ﬁelds obtain kilogauss strengths during the driving phase. These

fields correspond to β 104 105, indicating that the effects due the magnetic field on the ≈ − hydrodynamic development of the system are minimal.

The distribution of material velocity obtains an isotropic distribution during the driving phase • for the self-generated case.

The velocity power spectra show the expected inverse energy cascade and forward enstrophy • cascade during the driving phase. The forward enstrophy cascade obeys P k−2.3 in the in- ∝ ertial range (5 ≲ k ≲ 100). The deviation from the two-dimensional, hydrodynamic behavior

of P k−3 is due to the laser-driven mechanism and not magnetic ﬁeld eﬀects. ∝

The solenoidal and compressive kinetic energies are roughly in equipartition during active • driving. In the post-driving phase, the solenoidal component contains the bulk of the kinetic

energy (with a ratio on the order of 100:1).

We conclude that, in principle, one can produce a weakly compressible, quasi-steady state, turbulent plasma in laser driven experiments for as long as driving is provided. We note, with some disappointment, that the particular choice of parameters did not produce supersonic turbulence.

We found this primarily due to the turbulent central region being ﬁlled with the ablating material.

It would be interesting to consider a scenario in which the entire target is composed of a single material. In the case that the target is made out of the low density material used in this study, one could expect the turbulent Mach number to increase by a factor of √5 ( 2.2). ≈ Furthermore, it is conceivable that by adjusting the ﬁring frequency of the laser, and the energy and pulse length of individual beams one can change the thermodynamic conditions in the turbulent core. This provides a way to control the sound speed, and therefore the turbulent Mach number opening a possibility of reaching the supersonic regime.

120 One aspect of the proposed design we did not discuss in this work in detail is experimental diagnostics. We defer the discussion of this crucial component of the experiment until more realistic computations are performed in three-dimensions.

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132 BIOGRAPHICAL SKETCH

The author was born in northern California and lived across the western United States until arriving in Tallahassee. After ﬁnishing high school in Florida at age 16, the author completed a Bachelor of

Science degree in Engineering Physics at the University of Central Oklahoma. Much to the author’s surprise, he then returned to Tallahassee to pursue his graduate education at The Florida State

University.

133