Neutrino-Driven Turbulent Convection in Stalled Supernova Cores - PowerPoint PPT Presentation

Neutrino-Driven Turbulent Convection in Stalled Supernova Cores David Radice Collaborators: E. Abdikamalov, S. Couch, R. Haas, C. Ott, E. Schnetter

Contents 1.Turbulence in core-collapse supernovae 2.Numerical simulations 3.Conclusions

Contents 1.Turbulence in core-collapse supernovae � 2.Numerical simulations 3.Conclusions

The Supernova Problem Core-Collapse Supernovae: • End of massive stars • Birthplace of heavy elements, neutron stars, black holes … • Regulate star formation • … Cassiopeia-A Problem: how do they explode?

Core-Collapse Supernovae ������������������������������������������� ���������������� �������������������������� ��������������� � � ��������������� ���������������� ��������� �� �� � ��������� �� � �� �� ���� � � �� �� ������ �� ������ ����� ����� �� ������������������ ������������������ ��������������������������� From Janka et al. 2012

Shock Revival by Neutrinos From Janka 2001

The Roles of Turbulence Regulates accretion Turbulent pressure Transports heat Increase dwelling time Difficult to simulate!

Turbulent Pressure Rankine-Hogoniot jump condition: ρ d v 2 d + p d = ρ u v 2 u + p u EOS: γ th ' 4 p d = ( � th − 1) ⇢ d ✏ d 3 Effect of downstream turbulence (Murphy et al. 2013): γ turb > γ th ! ρ d v 2 v 2 d + ρ d ( δ v ) 2 d + p d → ρ d ¯ d + p d Turbulence can be modeled with an effective EOS ⇢ d ( � v ) 2 γ turb ' 2 d ↔ ( � turb − 1) ⇢ d ✏ turb Jump conditions for a shock with downstream turbulence: v 2 d + ( � turb − 1) ⇢ d ✏ turb + ( � th − 1) ⇢ d ✏ d = ⇢ u v 2 ⇢ d ¯ u + p u

Contents 1.Turbulence in core-collapse supernovae 2.Numerical simulations � 3.Conclusions

Resolution Dependance 200 s 27ULR f heat 1.05 s 27ULR f heat 1.05 s 27LR f heat 1.05 s 27LR f heat 1.05 R shock,avg [km] 180 s 27MR f heat 1.05 s 27MR f heat 1.05 ULR 3.78 km s 27IR f heat 1.05 s 27IR f heat 1.05 160 s 27HR f heat 1.05 s 27HR f heat 1.05 LR 1.89 km 140 120 MR 1.42 km IR 1.24 km 0.06 HR 1.06 km 0.04 σ 0.02 Resolutions 0 20 40 60 80 100 120 140 t − t bounce [ms] Explosion more difficult at higher resolution!

Why? • Lower resolution favors the formation of larger, longer lived structures • Secondary instabilities (Kelvin-Helmholtz) is suppressed by numerical viscosity • When is the resolution good enough?

Turbulent Cascade I Energy flux Energy injection @ t E + @ k Π = − 2 ⌫ k 2 E + ✏ Specific kinetic energy Energy dissipation

Turbulent Cascade II ✏ ∂ k Π 2 ν k 2 E Adapted from Frisch 1996 E ⇠ k − 5 / 3 Kolmogorov 1941: Π ' const = )

The Water-Spill Analogy 222 f.P. Boris et al. / Large eddy simulations ✏ Navier-Stokes ( a) I :_ Navier-Stokes ∂ k Π .. I "inertial range" 2 ν k 2 E Bottleneck: water piles up (b) Finite Volumes MILES with Built-in Subgrid Model Numerical viscosity (c) Adapted from Boris 1992 Conventional LES Fig. 8. LES water-spill analogy. faster so the mass flow past any radius is constant. Increasing radius away from the center of the table is analogous to increasing wavenumber of the eddies in a turbulent cascade. The decreasing depth of the water is analogous to the decreasing energy content in each wavelength scale of the turbulence. The incompressibility of the water gives effectively the same influence at a distance that the nonlocal interaction of disparate scales does in considering turbulence. The inertial range of the turbulent cascade is represented by the region between the vertical dashed lines where the profile is smoothly decreasing in fig. 8a. The radius of the table and how the water eventually falls off the table is analogous to the viscous dissipation of turbulent energy at the Kolmogorov scale in very high Re flows. This dropoff clearly does not significantly affect the depth of the water near the center of the table. In this hydrodynamic analogy, different possible contours at the edge of the table correspond to the different properties of various high Re Navier-Stokes models, conventional filtered LES models, and MILES models. In MILES models based on monotone convection algorithms, the nonlinear flux limiter acts as a built-in subgrid model coupled intrinsically to the short wavelength errors in the solution. Turbulent energy reaching the grid scale is extracted from the calculation and converted to the correct conserved quantities. This has the effect of curving the table edge sharply downward, as illustrated in fig. 8b, so that the water can flow smoothly off at a finite radius without significant perturbations reaching the center of the table. The dissipation in MILES algorithms is physically matched to the grid-scale errors to minimize effects on long wavelengths which are accurately resolved. With conventional, high-order CFD algorithms which are not monotone, dissipation is added through the physical viscosity. Thus a blocking or damming up phenomenon [13] occurs

The Bottleneck Effect 10 − 1 E ( k ) k 5 / 3 Bottleneck 10 − 2 PPM HLLC N64 PPM HLLC N128 PPM HLLC N256 PPM HLLC N512 10 − 3 10 0 10 1 10 2 k

Energy Cascade: PPM 0 . 06 0 . 05 0 . 04 0 . 03 Π ( k ) Need very high resolution!!! 0 . 02 0 . 01 PPM HLLC N64 PPM HLLC N128 0 . 00 PPM HLLC N256 PPM HLLC N512 10 0 10 1 10 2 k

Turbulent Energy Spectrum 10 26 E ( ` ) [erg cm − 3 ] 10 25 ` − 1 ` − 1 s 27ULR f heat 1.05 s 27ULR f heat 1.05 s 27LR f heat 1.05 s 27LR f heat 1.05 s 27MR f heat 1.05 s 27MR f heat 1.05 10 24 s 27IR f heat 1.05 s 27IR f heat 1.05 t − t bounce = 90 ms s 27HR f heat 1.05 s 27HR f heat 1.05 1 10 100 `

Semi-Global Convection Study ULR ∆ r ' 1 . 9 km r s ' 190 km 60 km XLR LR ∆ r ' 960 m ∆ r ' 3 . 8 km MR ∆ r ' 640 m

300 4.0 300 4.0 XLR ULR 250 3.5 250 3.5 200 3.0 200 3.0 z [ km ] z [ km ] 150 2.5 150 2.5 100 2.0 100 2.0 50 1.5 50 1.5 0 1.0 0 1.0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 x [ km ] x [ km ] 300 4.0 300 4.0 LR MR 250 3.5 250 3.5 200 3.0 200 3.0 z [ km ] z [ km ] 150 2.5 150 2.5 100 2.0 100 2.0 50 1.5 50 1.5 0 1.0 0 1.0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 x [ km ] x [ km ]

Convective Instability 0.10 0.05 Ω [ rad/ms ] 0.00 XLR ULR − 0.05 LR MR − 0.10 0.4 0.6 0.8 1.0 r / r s

Radial Reynolds Stresses XLR 6 r / c 2 [ × 10 40 g cm/s 2 ] ULR LR 4 MR 2 R r 0 0.2 0.4 0.6 0.8 1.0 1.2 r / r s

Not Quite There Yet 10 26 10 25 10 24 E ( k ) [ erg/cm 3 ] ∼ k − 1 10 23 10 22 XLR ULR 10 21 LR 10 20 MR 10 19 10 0 10 1 10 2 k / ∆ θ

A New Ingredient: Intermittency I 6 4 XLR θ / c 2 [ × 10 40 g cm/s 2 ] E turb [ × 10 26 erg/cm 3 ] ULR 4 LR 2 MR XLR 2 ULR LR R θ 0 MR 0 0.2 0.4 0.6 0.8 1.0 1.2 0.2 0.4 0.6 0.8 1.0 1.2 r / r s r / r s Turbulent energy density Tangential Reynolds stress

A New Ingredient: Intermittency II 400 300 r s [ km ] XLR ULR LR 200 MR 0 200 400 600 t [ ms ] Shock radius evolution

Contents 1.Turbulence in core-collapse supernovae 2.Numerical simulations 3.Conclusions

Conclusions • Turbulence: crucial role for supernova explosions • Local simulations: very high resolution is needed • Idealized global simulations: rich dynamics of turbulent convection

The Standing Shock Flow From Janka 2001

Initial Data 10 11 0.05 5 0.3 s υ r 0.2 10 10 0.00 0 0.1 Ω [ rad/ms ] ρ [ g/cm 3 ] 10 9 − 0.05 − 5 υ / c 0.0 ρ s 10 8 − 0.10 − 10 − 0.1 10 7 − 0.15 − 15 − 0.2 Ω BV 10 6 − 0.20 − 20 − 0.3 50 100 150 200 50 100 150 200 r [ km ] r [ km ] Stationary initial data