The document discusses numerical relativity and simulations of core-collapse supernovae. It provides background on extreme astrophysical phenomena involving strong gravity and relativistic dynamics that require general relativity. It describes the 3+1 formalism used to evolve Einstein's equations numerically and challenges such as instabilities and gauge choices. Core-collapse supernovae involve gravitational collapse, core bounce, and reviving the stalled shock, tapping into the gravitational potential energy released. Fully modeling these explosions requires solving Einstein's equations coupled to hydrodynamics and neutrino transport.

1. Numerical Relativity &
Simulations of Core-Collapse Supernovae
Christian D. Ott
TAPIR, Caltech
cott@tapir.caltech.edu
TAPIR

2. Extreme Astrophysics, Extreme Gravity
2
• Phenomena involving:
• extreme mass-energy density,
• strongly curved dynamical spacetime,
• mass-energy dynamics with v ~ c.
Newtonian gravity &
dynamics fail
- quantitatively
- qualitatively
Octant symmetry
t tb = 67.8ms
C. D. Ott @ Tarusa 2016

3. General Relativity (GR)
C. D. Ott @ Tarusa 2016 3
Einstein, 1915
Curvature
of Spacetime:
Einstein tensor
Source of Curvature:
Stress-Energy Tensor
“Matter tells space how to curve and space tells matter how to move”
- John Archibald Wheeler
(symmetric; 10 indep. components)

6. C. D. Ott @ Tarusa 2016 6
The Dark Side of the Universe
NASA, M. WeissVela
Only observable: Gravitational Waves
(K. Thorne)
Pure curvature!
Non-linear regime of General Relativity.

7. C. D. Ott @ Tarusa 2016 7
GW150914 – Coalescence of a BH-BH Binary
LIGO Scientific Collaboration &
Virgo Collaboration, PRL 2016

8. Numerical Relativity vs. Newtonian Simulations
C. D. Ott @ Tarusa 2016 8
r2
= 4⇡G⇢
Newtonian Poisson equation
• Elliptic partial differential equation.
• Instantaneous, “action at a distance”.
• Various solution methods (direct, relaxation, integral).
General Relativity
Latin : i, j, k, ... ! {1, 2, 3}
Greek : ↵, , , ⌫, µ, ... ! {0, 1, 2, 3}

9. Numerical Relativity vs. Newtonian Simulations
C. D. Ott @ Tarusa 2016 9
Einstein Equations
Now:
G = c = M = 1
Tµ⌫
= ⇢huµ
u⌫
+ Pgµ⌫
Stress-Energy Tensor (for ideal fluid)
metric tensorpressure4-velocityrest-mass density relativistic
spec. enthalpy
Key takeaway: no derivatives!

10. Numerical Relativity vs. Newtonian Simulations
C. D. Ott @ Tarusa 2016 10
Gµ⌫
= Rµ⌫ 1
2
Rgµ⌫
Einstein Tensor
Ricci Tensor Ricci Scalar R = gµ⌫Rµ⌫
µ⌫ =
1
2
g ⇢
(g⌫⇢,µ + g⇢µ,⌫ gµ⌫,⇢)
Rµ⌫ = ↵
µ⌫,↵
↵
µ↵,⌫ + ↵
µ⌫ ↵
↵
µ ⌫↵
, ⌘
@
@xµ Rµ⌫ = gµ↵g⌫ R↵
(connection coefficients; Christoffel symbols)
-> Einstein equations: 2nd derivatives of the metric in space and time
-> similar to (inhomogeneous) wave equation:
@2
@t2
U c2 @2
@x2
U = T
-> Gravitational waves!
-> Einstein equations can be written in
hyperbolic form! (time-evolution equations)

11. Numerical Relativity
C. D. Ott @ Tarusa 2016 11
Proceedings of the GR1 Conference on the role of gravitation in physics
University of North Carolina, Chapel Hill [January 18-23, 1957]
Recommended texts:
Baumgarte & Shapiro, Numerical Relativity
Alcubierre, Introduction to 3+1 Numerical Relativity

12. Basic Idea of Numerical Relativity
C. D. Ott @ Tarusa 2016 12
Figure:
C. Reisswig
Arnowitt-Deser-Misner, Lichnerowitz
3+1 split of spacetime
Foliation of spacetime
3-hypersurface
• 12 first-order hyperbolic evolution equations.
• 4 elliptic constraint equations
• 4 coordinate gauge degrees of freedom: α, βi.

13. 3+1 Split of General Relativity
C. D. Ott @ Tarusa 2016 13
3+1 split – key objects:
g00 = ↵2
+ i
i gij = ij
Extrinsic curvature: ≈ time derivative of 3-metric
@t ij = 2↵Kij + j;i + i;j
g0i = ij
j
rµV µ
= @µV µ
+ ⌫
µ Vcovariant derivative:

14. ADM Equations
C. D. Ott @ Tarusa 2016 14
(Historic: Arnowitt-Deser-Misner 1962; York 79)
Sij = iµ j⌫Tµ⌫
⇢ADM = nµn⌫Tµ⌫
S, K – traces of Sij, Kij
R + K2
KijKij
16⇡⇢ADM = 0
Si
= iµ
n⌫
Tµ⌫
Constraint Equations:
Hamiltonian
Momentum
@tKij = ↵;ij + ↵

RijKKij 2KimKm
j
8⇡(Sij
1
2
ijS) 4⇡⇢ADM ij
+ m
Kij;m + Kim
m
;j + Kmj
m
;i
Kij
;j
ij
K;j 8⇡Si
= 0
Evolution System@t ij = 2↵Kij + j;i + i;j
+ gauge choice

15. Cauchy Evolution
C. D. Ott @ Tarusa 2016 15
Specify
constraint-satisfying
initial data & boundary values.
-> solve constraint equations.
Initial boundary
value problem.
Evolve
forward
in time &
monitor
constraints.

16. Practical Numerical Relativity
C. D. Ott @ Tarusa 2016 16
Have not yet specified gauge conditions: Anything goes?
• GR dynamics will twist, squeeze, stretch
coordinates.
• GR can develop coordinate singularities
and physical singularities.
• For numerically stable evolution, must
avoid singularities and control
coordinate distortion.
NASA
• ADM form of the Einstein equations is unstable in 2D/3D!
-> well-posednessproblems (-> see literature; e.g., Kidder+2001).
-> small errors in constraints get amplified exponentially over time!
• Spherical symmetry (1D):
-> no radiative degrees of freedom, fully constrained evolution.
-> ADM with simple gauge choices: no problem.

17. C. D. Ott @ Tarusa 2016 17
Practical Numerical RelativityKey issues:
• Initial conditions must satisfy Einstein equations.
• No unique way to formulate evolution equations.
• Gauge freedom – how choose gauge conditions?
• Need combination of evolution equations + gauges that yield
to numerically stable simulations.
BSSN Formulation
Generalized Harmonic Formulation
Nakamura+87, Shibata & Nakamura 95, Baumgarte & Shapiro 99
Friedrich 85, Pretorius 05, Lindblom+ 06
• Conformal-traceless reformulation of Arnowitt-Deser-Misner 59, York 79.
• Additional evolution equations, conditionally strongly hyperbolic.
• Sensitive to gauge choice; good gauges known.
• Most widely used evolution system today.
• Choice of coordinates so that evolution equations
wave-equation like. Symmetric hyperbolic.
• Sensitive to gauge choices, horizon boundary conditions.
• Used primarily by Caltech/Cornell SXS code SpEC.

18. Numerical Implementation
C. D. Ott @ Tarusa 2016 18
• Most common:
high-order finite difference approximation
(typically 4th-order in space & time).
• Powerful alternatives:
Spectral methods, Discontinuous Galerkin Finite Elements.
d
dt
L(q) = RHS
• Common approach: Method of Lines
Treat problem as semi-discrete; discretize in space, then treat as
ODE, integrate in time via Runge-Kutta(or similar).
Provides for high-order coupling with additional physics
(hydrodynamics/MHD, radiation).

19. The Einstein Toolkit
C. D. Ott @ Tarusa 2016 19
Mösta+14
Löffler+12
• Collection of open-source software components for the
simulation and analysis of general-relativistic
astrophysical systems.
• Supported by US National Science Foundation.
~110 users, 60 groups; ~10 active maintainers.
http://einsteintoolkit.org

20. The Einstein Toolkit
C. D. Ott @ Tarusa 2016 20
Mösta+14
Löffler+12
• Collection of open-source software components for the
simulation and analysis of general-relativistic
astrophysical systems.
• Supported by US National Science Foundation.
~110 users, 60 groups; ~10 active maintainers.
http://einsteintoolkit.org
• Cactus (framework), Carpet (adaptive mesh refinement)
• GRHydro – GRMHD solver
• McLachlan – BSSN/Z4c spacetime solvers.
(code auto-generated based on Mathematica script, GPU-enabled)
• Initial data solvers / readers.
• Analysis tools (wave extraction, horizon finders, etc.)
• Visualization via VisIt (http://visit.llnl.gov)
Available Components:

21. The Einstein Toolkit
C. D. Ott @ Tarusa 2016 21
Mösta+14
Löffler+12
• Regular releases of stable code versions.
Most recent: “Brahe” release, June 2016
• Support via mailing list and weekly open conference calls.
• Working examples for BH mergers, NS mergers, isolated
NSs, rotating, magnetized core collapse (see also arXiv:1305.5299).
Simulate a binary black hole
merger on your laptop!

22. C. D. Ott @ Tarusa 2016 22
© Anglo-Australian Observatory
Core-Collapse Supernovae:
Supernova 1987A
Large MagellanicCloud
Progenitor:
BSG Sanduleak-69° 220a, ≈18 MSUN
Explosions of Massive Stars

23. MotivationMotivation
but
ding
SN 1987A: Neutrino Detection
23C. D. Ott @ Tarusa 2016
-> First detection of extragalactic neutrinos!
Hirata+87
Bionta+87
Alekseev+87

24. The Basic Theory of Core Collapse
C. D. Ott @ Tarusa 2016 24
MCh ⇡ 1.44
✓
Ye
0.5
◆2
"
1 +
✓
⇡
⇢c ⇡ 1010
g cm 3
Tc ⇡ 0.5 MeV
Ye,c ⇡ 0.43
[not drawn to scale]
8M . M . 130M
M

25. Collapse and Bounce
C. D. Ott @ Tarusa 2016 25
Stiff Nuclear Equation
of State (EOS):
“Core Bounce”

26. Collapse and Core “Bounce”
C. D. Ott @ Tarusa 2016 26
Stiff Nuclear Equation
of State (EOS):
“Core Bounce”
Bounce:
t=0 for SN theorists.
Central rest-mass density in the collapsing core:

27. 1010 1011 1012 1013 1014
1.0
1.5
2.0
2.5
3.0
Density (g/cm3)
AdiabaticIndexG
s = 1.2 kB/baryon
Ye = 0.3
P ⇠ K⇢
“Stiffening” of the Nuclear EOS
27
“Core Bounce”
C. D. Ott @ Tarusa 2016
Schematic
nuclear force
potential
=
d ln P
d ln ⇢
“repulsive core”

28. Situation after Core Bounce
C. D. Ott @ Tarusa 2016 28

29. The Core-Collapse Supernova Problem
• The shock always stalls:
Dissociation of Fe-group nuclei @ ∼8.8 MeV/baryon.
Neutrino losses initially @ >100 B/s (1 [B]ethe = 1051 ergs).
C. D. Ott @ Tarusa 2016 29
Radius (km)
Animation
by Evan O’Connor
Caltech GR1D code
(open source!)
Hans Bethe
1906-2005

30. “Postbounce” Evolution
C. D. Ott @ Tarusa 2016 30
⌧ ⇡ 1 few s

31. “Postbounce” Evolution
C. D. Ott @ Tarusa 2016 31
What is the mechanism that revives the shock?
⌧ ⇡ 1 few s

32. Core-Collapse Supernova Energetics
C. D. Ott @ Tarusa 2016 32
• Collapse to a neutron star: ∼3 x 1053 erg = 300 [B]ethe
gravitational energy (≈0.15 MSunc2).
-> Any explosion mechanism must tap this reservoir.
• ∼1051 erg = 1 B kinetic and internal energy of the ejecta.
(Extreme cases: 10B; “hypernova”)
• ∼ 90 - 99% of the energy is radiated in neutrinos on O(10)s
-> Strong evidence from SN 1987A neutrino observations.
• If spinning with few ms spin period, proto-NS has
∼1052 erg = 10 B in spin energy.

33. C. D. Ott @ Tarusa 2016 33
Magneto-Hydrodynamics
Nuclear and Neutrino Physics
General Relativity
Boltzmann Transport (Kinetic Theory)
Dynamics of the stellar fluid.
Nuclear EOS, nuclear
reactions & ν interactions.
Gravity
Neutrino transport.
Fully coupled!
• Additional Complication: Core-Collapse Supernovae are 3D
– Rotation, fluid instabilities, magnetic fields, multi-D stellar structure
from convective burning, etc.
• Route of Attack: Computational simulation.
– Full problem is 3 (space) + 3 (momentum space) + 1 (time) dimensional
– Approach: employ reduced dimensionality in space and momentum space
(in some sensible way).
Detailed Models: Ingredients

34. Core-Collapse Supernova Simulations
34C. D. Ott @ Tarusa 2016
1D (spherical symmetry)
-First simulations: 1960s-70s by Colgate & White,
Sato, Wilson, Arnett, Nadyozhin,
Bisnovatyi-Kogan
-Colgate & White ‘66: “Neutrino Mechanism” (direct)
Bethe & Wilson ‘85: “Delayed Neutrino Mechanism”
-Bisnovatyi-Kogan ‘70: “MagnetorotationalMechanism”
Cray-I

35. Neutrino Mechanism: Heating
C. D. Ott @ Tarusa 2016
Ott+ ’08
35
¯⌫e + p ! n + e+
⌫e + n ! p + e
Cooling:
Heating via
charged-current
absorption:
Bethe & Wilson ’85; also see: Janka ‘01, Janka+ ’07
30 km 60 km 120 km 240 km
Q+
⌫ /
⌧
1
F⌫
L⌫r 2
h✏2
⌫i
Neutrino radiation field:
, T9

36. 1D Neutrino-Driven Explosions
C. D. Ott @ Tarusa 2016
36
Kitaura+ ‘06, Hüdepohl+ ’10, Fischer+ ’10, ‘12
Kitaura+ 2006
8.8 MSUN
progenitor
star
O-Ne-Mg
core
Problem:
1D neutrino mechanism fails
for more massive stars
(which explode in nature).

37. 2D and 3D Neutrino-Driven CCSNe
C. D. Ott @ Tarusa 2016
37
• Progress driven by advances in compute power!
• First 2D (axisymmetric) simulations in the 1990s:
Herant+94, Burrows+95, Janka & E. Müller 96.
Dessart+ ‘05
Bruenn+13
• 2D simulations now self-consistent & from first principles.
E.g.: Bruenn+13,16 (ORNL), Dolence+14 (Princeton),
B. Müller+12ab (MPA Garching), Nagakura+16 (Kyoto), Takiwaki+14 (NAOJ/Fukoka)

38. Standing Accretion Shock Instability (SASI)
C. D. Ott @ Tarusa 2016 38
Blondin+’03
Foglizzo+’06
Scheck+ ’08
and many
others
Movie by
Burrows,
Livne,
Dessart,
Ott, Murphy‘06

39. The 3D Frontier – Petascale Computing!
C. D. Ott @ Tarusa 2016
39
• Some early work: Fryer & Warren 02, 04
• Much work since ~2010:
Fernandez 10, Nordhaus+10, Takiwaki+11,13,
Burrows+12, Murphy+13, Dolence+13,
Hanke+12,13, Kuroda+12, Ott+13, Couch 13,
Takiwaki+13, Couch & Ott 13, 15,
Abdikamalov+15, Couch & O’Connor 14,
Lentz+15, Melson+15ab, Summa+15, Roberts+16
• Approximations currently made:
(1) Gravity (2) Neutrinos (3) Resolution

40. 40
Ott+13
Caltech,
full GR,
parameterized
neutrino heating

41. Multi-Dimensional Simulations: Effects
C. D. Ott @ Tarusa 2016 41
(1) Lateral/azimuthal flow:
“Dwell time” in gain
region increases.
(2) New: Anisotropy of convection
-> Turbulent ram pressure
(Radice+15ab,Couch&Ott 15,
Murphy+13)
(e.g., Hanke+13, Couch&Ott 15, Murphy+08, Murphy+13, Ott+13, Dolence+13)
Rij = vi vj
vi = vi vi
Rrr ⇠ 2{R✓✓, R }
Pturb = ⇢Rrr
effective
turbulent
pressure

42. 2D & 3D Explosions!
C. D. Ott @ Tarusa 2016 42
(e.g., Lentz+15, Melson+15)

43. C. D. Ott @ Tarusa 2016
43
2D vs. 3D
(e.g., Couch 13, Couch & O’Connor 14)

44. C. D. Ott @ Tarusa 2016
44
3D: Sensitivity to Resolution
Abdikamalov+15
low resolution-> less efficient turbulent cascade
-> kinetic energy stuck at large scales

45. Resolution Comparison
C. D. Ott @ Tarusa 2016 45
(Radice+16)
dθ,dφ= 1.8°
dr = 3.8 km
dθ,dφ= 0.9°
dr = 1.9 km
dθ,dφ= 0.45°
dr = 0.9 km
dθ,dφ= 0.3°
dr = 0.64 km
• semi-global simulations
of neutrino-driven
turbulence.
(typical resolution of
3D rad-hydro sims)

46. C. D. Ott @ Tarusa 2016 46

47. Summary of 2D & 3D Neutrino-Driven CCSNe
C. D. Ott @ Tarusa 2016 47els s27fheat1.00 (left column), s27fheat1.05 (center column), and s27fheat1
• More efficient neutrino heating,
turbulent ram pressure.
• 2D simulations explode but can’t be
trusted (2D turbulence is wrong).
Explosions too weak?
But see Bruenn+16.
• 3D simulations:
Much must be improved:
(1) Resolution
(2) Treatment of neutrino transport
(3) Treatment of gravity in
many codes
Ott+13

48. Magnetorotational Explosions
48C. D. Ott @ Tarusa 2016
he iron-core
Connor & Ott
Chieffi 2006;
how an anti-
2007). The
s for rate and
s in massive
hi et al. 2005
ally symmet-
s code GR1D
d through a
equation —
rotation rel-
account for
etry nor any
e equation of
racterized by
er referred to
1 10 100 1000 10000
Radius [105
cm]
0.001
0.01
0.1
1
10
100
1000
10000
Ω(r)[rads-1
]
12TJ
16SN
16OG
16TI
35OC
preSN
bounce
Figure 1. Angular velocity ⌦(r) versus radius r at both the pre-SN stage
(dashed lines) and at core bounce (solid lines) for selected models of Woosley
& Heger (2006). The inner homologously collapsing core maintains its initial
uniform rotation throughout collapse.
• Core: x 1000 spin-up
• Differential rotation -> reservoir of free energy.
• Spin energy tapped by magnetorotational instability (MRI)?
Dessart, O’Connor, Ott ‘12

49. Magnetorotational Mechanism
49C. D. Ott @ Tarusa 2016
[LeBlanc & Wilson ‘70, Bisnovatyi-Kogan ’70 & ‘74, Meier+76,
Ardeljan+’05, Moiseenko+’06, Burrows+‘07, Bisnovatyi-Kogan+’08,
Takiwaki & Kotake ‘11, Winteler+ 12, Mösta+14,15]
Rapid Rotation + B-field amplification to > 1015 G
(need magnetorotationalinstability [MRI])
2D: Energetic “bipolar” explosions.
Results in ms-period “proto-magnetar.”
-> connection to GRBs, SuperluminousSNe?
Burrows+’07
Problem: Need high core spin;
only in very few progenitor stars?
MHD stresses lead to outflows.

50. A Note on Magnetic Field Amplification
C. D. Ott @ Tarusa 2016 50
• Precollapse magnetic field in the core?
Best observational information:
White Dwarf B-fields, max ~108 – 109 G
• Amplification processes:
(1) flux compression
(2) linear winding (poloidal->toroidal)
(3) magnetorotational instability + dynamo
Useful estimates in
Shibata+06 &
Burrows+07
Example calculation for flux compression:
⇠ BR2
= const. ! B /
1
R2
BPNS = BIC
✓
R2
IC
R2
PNS
◆
RIC ⇠ 1500 km RPNS ⇠ 30 km
BPNS ⇠ 2500BIC ⇠ 2.5 ⇥ 1011
1012
G
-> Flux compression alone cannot produce 1015 G magnetic field!
Winding gives another x 10. Need the MRI.

51. 51C. D. Ott @ Tarusa 2016
Burrows+’07
(1011 G
seed field)

52. 3D Dynamics of Magnetorotational Explosions
C. D. Ott @ TC Meeting, Berkeley, 2015/02/28 52
Octant Symmetry (no odd modes) Full 3D
ß 2000 km àß 2000 km à
New, full 3D GRMHD simulations. Mösta+ 2014, ApJL.
Initial configuration as in Takiwaki+11, 1012 G seed field.

53. C. D. Ott @ Tarusa 2016 53
Mösta+ 2014
ApJL

54. What is happening here?
C. D. Ott @ Tarusa 2016 54
Mösta+14, ApJL
• B-field near proto-NS: Btor >> Bz
• Unstable to MHD screw-pinch kink instability.
• Similar to situation in Tokamak fusion reactors!
Braithwaite+ ’06
Sherwood
Richers
Philipp Mösta
Credit: Moser & Bellan, CaltechSarff+13

55. Explosion?
C. D. Ott @ Tarusa 2016 55
Mösta+16, in prep.
160 180 200 220 240 260 280
8⇥102
9⇥102
103
1.1⇥103
1.2⇥103
1.3⇥103
1.4⇥103
t tbounce [ms]
r[km]
Maximum shock radius
(low resolution
– work in progress)

56. Summary
C. D. Ott @ Tarusa 2016 56
• Core-Collapse Supernovae are fundamentally 3D:
turbulence, magnetic fields
• Neutrino & Magnetorotational Mechanism:
Possible solutions to the Supernova Problem.
− Neutrino mechanism may be too weak
(missing neutrino physics?).
− Magnetorotationalmechanism needs fast core
rotation, but stellar evolution predicts slow rotation.
• 3D simulations have made great progress,
but no final answers yet.
Much work ahead!
wikipedia.org/wiki/Magnetar

57. Supplemental Slides
C. D. Ott @ Tarusa 2016 57

58. Can this work at all?
C. D. Ott @ Tarusa 2016 58
• Simulations of the magnetorotationalmechanism assume:
MRI works + large-scale field created by dynamo.
• So far impossible to resolve
fastest-growing MRI mode in
global 3D simulations.
• Unstable regions (roughly):
• In this simulation:
fastest growing mode
λ ~ 1 km.
dark blue: most MRI unstable
Mösta+15, Nature
d ln ⌦
dr
< 0

59. 59
C. D. Ott @ Tarusa 2016
Global Field Structure
Mösta+15, Nature
dx = 500 m dx = 200 m dx = 100 m dx = 50 m

60. 0 5 10
1014
1015
1016
t tmap [ms]
Bf[G]
Maximum in
equatorial layer
500 m
200 m
Bf = 4.0 · 1014 · e(t tmap)/t
, t = 0.5 ms
100 m
50 m
100 m
50 m
60
Local Magnetic Field Saturation
• Initial exponential
growth resolved with
100m/50m
simulations.
• Saturated turbulent
state within 5 ms.
C. D. Ott @ Tarusa 2016
Mösta+15, Nature

61. 61
Energy Spectra
1 10 100
1028
1029
1030
1031
1032
1033
1034
1035
1036
k
E(k)[erg]
Emag 500 m
Emag 200 m
Emag 100 m
Emag 50 m
Emag 50 m (t = 0 ms)
Ekin 50 m
5 · 1036 erg · k 5/3
Ekin 50 m
5 · 1036 erg · k 5/3
Magnetic energy spectrum very resolution dependent.
C. D. Ott @ Tarusa 2016
Mösta+15, Nature
Inverse
Cascade:
Dynamo!

62. 62
Energy Spectra
1 10 100
1028
1029
1030
1031
1032
1033
1034
1035
1036
k
E(k)[erg]
Emag(k) t = 0 ms
t = 1 ms
t = 2 ms
t = 4 ms
t = 6 ms
t = 8 ms
t = 10 ms
5 · 1036 erg · k 5/3
Ekin(k) t = 7 ms
5 · 1036 erg · k 5/3
Ekin(k) t = 7 ms
• Turbulent saturated state after ~3 ms.
• Inverse cascade (dynamo) afterwards.
C. D. Ott @ Tarusa 2016
Mösta+15, Nature

63. Schematic Numerical Relativity Simulation
C. D. Ott @ Tarusa 2016 63
Initial Data
(satisfy constraints)
Evolve one
Timestep
Evaluate Right-Hand Side
Apply Update
High-Order
Runge-Kutta
Integrator
(typically 4th order)
Analysis/Output
spacetime
curvature
gauge
Other physics:
MHD
Radiation
Complication: Adaptive Mesh Refinement
Xn+1
= Xn
+ L(Xn
) t
(first order)
t < x/c

64. • Rapidly spinning, magnetized proto-NS.
• Global simulation in quadrant symmetry:
70 km x 70 km x 140 km box
• Resolutions:500 m/200 m/100 m/50 m
• hot nuclear eq. of state, neutrinos, fixed gravity, GRMHD.
• Simulations on 130,000 CPU cores on NSF Blue Waters,
simulate for 10-20 ms.
64
Simulation Setup
• Does the MRI efficiently build up dynamically relevant field?
• Saturation field strength? Global field structure?
Key questions:
C. D. Ott @ Tarusa 2016
Mösta+15, Nature

65. 65
B-Field Growth at Large Scales
• k=4; corresponding
roughly to width of
shear layer
• Field will grow to
saturation at
large scales within
~60 ms.
0 5 10
1
2
3
4
5
6
7
t tmap [ms]
Ek,mag(t)[1033erg]
( 2.05 + 0.75 ms 1 · (t tmap)) · 1033
5 · 1032 e(t tmap)/t
, t = 3.5 ms
k = 4
k = 6
k = 8
k = 10
k = 20
k = 50
k = 100
( 2.05 + 0.75 ms 1 · (t tmap)) · 1033
5 · 1032 e(t tmap)/t
, t = 3.5 ms
k = 4
k = 6
k = 8
k = 10
k = 20
k = 50
k = 100
C. D. Ott @ Tarusa 2016
Mösta+15, Nature

66. C. D. Ott @ Tarusa 2016
66
Some Facts about Supernova Turbulence
(e.g., Abdikamalov, Ott+ 15, Radice+15ab)
• Neutrino-driven convection is turbulent.
• Kolmogorov turbulence: Kolmogorov 1941
isotropic, incompressible, stationary.
• Supernova turbulence:
anisotropic (buoyancy), mildly compressible, quasi-stationary.
• Reynolds stresses (relevant for explosion!) dominated by
dynamics at largest scales.
Re =
lu
⌫
⇡ 1017
Rij = vi vj
E(k) / k 5/3

67. C. D. Ott @ Tarusa 2016
67
Kolmogorov Turbulence
log E(k)
/ k 5/3
inertial
range
dissipation
range
(large spatial scale) (small spatial scale)
(Fourier-space wave number)
log k
large eddies -----------------------> small eddies
Rij = vi vj

68. C. D. Ott @ Tarusa 2016
68
Turbulent Cascade: 2D vs. 3D
and large, high-entropy bubbles emerge that push the shock outward. The explosi
convection in our simulations is very similar to that of Ott et al. (2013).
100
101
102
`
1023
1024
1025
1026
E`
r = 125 km, tpb = 150 ms
` 1
` 5/3
` 3
s15 0.95 2D
s15 1.00 2D
s15 1.00 3D
s15 1.05 3D
1
Couch & O’Connor 14
see also: Dolence+13, Hanke+12,13, Abdikamalov+’15, Radice+15ab
2D
3D
• 2D: wrong; turbulent cascade unphysical.
• 3D: physical; more power at small scales, less
on large scales -> harder to explode!

69. C. D. Ott @ Tarusa 2016
69
Kolmogorov Turbulence
log E(k)
/ k 5/3
inertial
range
dissipation
range
(large spatial scale) (small spatial scale)
(Fourier-space wave number)
log k
large eddies -----------------------> small eddies
Rij = vi vj
Sensitivity to
kinetic energy flux!
-> sensitivity to resolution

70. C. D. Ott @ Tarusa 2016 70
Turbulent Kinetic Energy Spectrum
(Radice+16)
“compensated” spectrum
Core-collapse supernova turbulence obeys Kolmogorov scaling!
But: Global simulations at necessary resolutioncurrently impossible!
Way forward? -> Subgrid modeling of neutrino-driven turbulence?