The Onset of Neutrino-Driven Convection in Core-Collapse Supernovae

1. The Onset of Neutrino-Driven Convection in Core-Collapse Supernovae Brendan Krueger CEA Saclay 2013 October 18

2. Outline • Background • Structure of CC SNe in the stalled-shock phase • Instabilities in CC SNe • Convective instability • Standing accretion shock instability • Research at CEA • The big picture • A simple model • Small-scale perturbations • Large-scale perturbations • Were do we go from here? • Summary & Conclusions Brendan Krueger | CEA Saclay

3. Background Convection vs. SASI in CC SNe Brendan Krueger | CEA Saclay

4. Post-Bounce CC SNe Structure • Above the proto-neutron star matter is cooling from neutrino emission • Neutrino emission weakens farther out until the gain radius • Above the gain radius is the gain region, where a fraction of the neutrinos are re-absorbed • Gain region is bounded by the stalled shock • Outward of the shock matter is infalling supersonically shock gain radius PNS cooling region gain region supersonic infall Brendan Krueger | CEA Saclay

5. Post-Bounce CC SNe Structure • Neutrino heating mechanism • Reabsorb sufficient neutrinos in the gain region • Re-energize the shock • Wilson (1985), Bethe & Wilson (1985), Bethe (1990), Janka et al. (2007) • Magnetorotational • Rapid rotator • Burrows et al. (2007) shock shock gain radius gain radius PNS neutrinosphere cooling region cooling region gain region gain region supersonic infall supersonic infall Brendan Krueger | CEA Saclay

6. Instabilities in the Gain Region • Several may exist • Not mutually exclusive (e.g., Guilet et al. 2010) • Two very important: • Convective instability • Standing Accretion Shock Instability (SASI) • Generally believed that one of these two will dominate dynamics of the gain region Brendan Krueger | CEA Saclay

7. SASI I: Background • Discovered by Blondin et al. (2003) in a simplified context • Since observed in variety of simulations: e.g., Blondin & Mezzacappa (2006, 2007), Ohnishi et al. (2006), Scheck et al. (2008), Iwakami et al. (2008, 2009), Fernández & Thompson (2009), Fernández (2010), Müller et al. (2012), Hanke et al. (2013) • Studied analytically: Foglizzo et al. (2006, 2007), Blondin & Mezzacappa (2006), Yamasaki & Yamada (2007), Fernández & Thompson (2009) • Blondin & Mezzacappa (2006) suggest SASI is purely-acoustic • Sato et al. (2009) and Guilet & Foglizzo (2012) provide evidence for an advective-acoustic cycle • Demonstrated experimentally in a shallow-water analogue of CC SNe: Foglizzo et al. (2012) Brendan Krueger | CEA Saclay

8. SASI II: Advective-Acoustic Cycle • Perturbations in the shock front create perturbations of entropy and vorticity in the flow • Perturbations advect downward • Deceleration of perturbations generates acoustic wave • Acoustic wave perturbs shock front entropy-vorticity wave acoustic wave Brendan Krueger | CEA Saclay

9. SASI III: Properties • Generally dominated by low-order (l=1,2) modes • l=1, m=0: sloshing • l=1, m=±1: spiral • Increase dwell time • Increases energy gain from neutrino absorption • Push shock outward • May give neutron star a “kick” • Spiral modes may redistribute angular momentum and “spin up” neutron star Brendan Krueger | CEA Saclay

10. Convection I: Background • Seen in numerous CC SN simulations: e.g. Herant et al. (1992, 1994), Burrows et al. (1995), Janka & Müller (1995, 1996), Fryer & Heger (2000), Ott et al. (2013), Murphy et al. (2013) • Studied analytically: Foglizzo et al. (2006) • Generally higher-order modes (l~5-7) • Convection may cause low-order modes (especially in 2D): Burrows et al. (2012), Dolence et al. (2013) • Difficult to distinguish SASI from convection in nonlinear regime using naïve spherical-harmonic decomposition Brendan Krueger | CEA Saclay

11. Convection II: Stabilization • Due to negative entropy gradient, the gain region is unstable to convection • May be stabilized through advection: Foglizzo et al. (2006) • Compare the advection time across the gain region and the growth rate of convective modes • Foglizzo’sχ parameter: • χ > 3 is unstable to convection Brendan Krueger | CEA Saclay

12. SASI vs. Convection • Different behavior results from these two instabilities • SASI has the potential to generate high-velocity and/or fast-rotating neutron stars (through sloshing and spiral modes) • Convection requires much more focus on heating (i.e., neutrino transport) to capture the driving physics • SASI can lead to gravitational wave emission: Marek et al. (2009), Murphy et al. (2009) • Neutrino emission from CC SNe could depend on flow dynamics • Different shock evolution (asymmetry) • Dimensionality of simulations could be significant for different reasons • Convection: inverse cascade in 2D vs. forward cascade in 3D • SASI: tendency towards sloshing in 2D vs. spiral in 3D Brendan Krueger | CEA Saclay

13. Nonlinear Behavior of Convection Research from CEA Saclay Brendan Krueger | CEA Saclay

14. The Big Questions • What can we learn about instabilities in the gain region of a collapsing massive star? • Can we use this information to develop criteria for when a particular instability will dominate the post-shock dynamics? • We have started with convection • We also have a study of nonlinear effects relating to SASI in progress, currently being led by a graduate student, and we hope to be seeing new results soon Brendan Krueger | CEA Saclay

15. Break Down the Problem • The linear theory provides a good prediction • There is evidence that nonlinear effects could be important under some circumstances (e.g., Scheck et al. 2008) • Can we determine a criterion for the non-linear triggering of buoyancy-induced turbulence? • Step 1: When is the flow unstable? For example: When is a bubble buoyant against advection? • Step 2: When does a buoyant perturbation lead to convective instability and/or turbulence? Brendan Krueger | CEA Saclay

16. Simplified Model • Simplify to the minimal physics necessary to capture buoyant instabilities analogous to what is seen in the gain region of CC SNe • Cartesian geometry • Ideal ϒ-law equation of state (ϒ= 4/3) • Buoyant layer • Shape function s(x) • Gravitational acceleration • g = g0 s(x) • Heating function • H = H0 (ρ/ρ0) s(x) Brendan Krueger | CEA Saclay

17. Current Criteria Scheck et al. (2008) Fernández et al. (2013) Balance of gravitational, buoyant, and drag forces: No mention of heating Adiabatic bubble No reference to the background entropy gradient No mention of the size of the gain region Criterion on constant velocity, not buoyancy • Change in velocity due to buoyancy: (~0.2%) • No mention of heating • Adiabatic bubble • No reference to the background entropy gradient • No mention of drag from rising against accretion flow • Neither details the relationship between a rising bubbleand the development of turbulent convection Brendan Krueger | CEA Saclay

18. Modified Criterion • Attempt to improve the physics of previous estimates • Numerically integrate the equations of motion • Begin with an adiabatic bubble and no drag force • Approximates the inputs to the Scheck et al. (2008) criterion • Add drag force and heating of the bubble • Results • Heating had very minimal effect • Drag force caused the bubble to reach an equilibrium position instead of escaping to infinity and is thus more physical, but changed the critical value very little • Net result is a critical value within an order of magnitude of the Scheck et al. (2008) and Fernández et al. (2013) predictions for our model Brendan Krueger | CEA Saclay

19. Hydrodynamic Simulations • Same model as previously described • RAMSES fluid dynamics code • Parallel (MPI) • AMR • We are using a uniform-grid version to avoid overhead • MHD algorithm based on the MUSCL method • No magnetic field yet; that is on the to-do list Brendan Krueger | CEA Saclay

20. Bubble Test • Add a bubble to the upstream flow • Low-density/high-entropy, pressure equilibrium • Varied the density contrast to explore (in)stability limit • Result: • Density contrast must be approximately 100 times what was given by the bubble trajectory models Brendan Krueger | CEA Saclay

21. Bubble Test • Why? • Appears to be largely due to multidimensional effects that a simple trajectory will not capture • Upon entering the buoyant region, the bubble flattens, then splits into two counter-rotating bubbles (would be a ring in 3D) • This splitting lowers the density contrast • The rotating flows will dissipate, further lowering the density contrast • For an isolated bubble, multidimensional effects appear to modify the situation sufficiently that a simple trajectory is not predictive Brendan Krueger | CEA Saclay

22. Other Perturbations • Are small, localized perturbations the expectation? • Consider a SASI-like perturbation in the upstream flow • Large-scale • Oscillatory • Tilted Brendan Krueger | CEA Saclay

23. Large-Scale Perturbations • If δ < δ*: • Perturbation compresses (due to lower downstream velocity) • No significant nonlinear effects in buoyant layer • Nonlinear effects only appear downstream due to velocity difference • Unimportant to our study: this is not turbulence in the buoyant region • If δ> δ*: • Nonlinear effects dominate buoyant layer flow • As of yet, no estimate of δ* based on the initial state • Appears that δ* is approximately an order of magnitude larger than what is predicted by Scheck et al. or Fernández et al. criteria. • Why?  work in progress Brendan Krueger | CEA Saclay

24. Large-Scale Perturbations • Correlation with χ • As discussed by Fernández et al. (2013), care must be taken in computing χ for multidimensional flow • If χ(t>t0) crosses into the linearly-unstable regime (χ>χcrit), nonlinear effects appear in buoyant layer • Even in these cases χ(t) drops back into linearly-stable regime Brendan Krueger | CEA Saclay

25. Large-Scale Perturbations • How does δ* behave? • Variation with Χ • As Χ Χcrit: δ*decreases • Expected: less stable, more easily perturbed • Connection with Rayleigh-Taylor • As horizontal wavelength decreases, δ*increases • RT instability between alternating over-dense/under-dense layers • RT growth rate increases with shorter horizontal wavelength Brendan Krueger | CEA Saclay

26. Large-Scale Perturbations • Self-sustained buoyancy-driven turbulence? • Distinction between δ* and δcrit • Nonlinear effects vs. nonlinear instability threshold • A single perturbation can spike into linearly-unstable regime, but drops back to linearly-stable regime • Continuously-driven perturbation simulations not yet informative • Acoustic cavity in the upstream flow • Inherent issue with fixed-inflow boundary condition • As of yet inconclusive • Can nonlinear effects trigger instabilities even in the linearly-stable regime? Brendan Krueger | CEA Saclay

27. Next Steps • Continue refining the study of a buoyancy criterion • Determine whether a buoyant perturbation is sufficient to initiate sustained convection and turbulence • What is the missing piece in the reasoning behind the Scheck et al. and Fernández et al. criteria? • Can a single perturbation initiate buoyancy-driven turbulence? • Explore continuously-driven perturbations • Three dimensions • Convection is known to be difference in 2D and 3D due to the inverse cascade. 2D simulations can explore more effectively due to the lower computing cost, but 3D simulations will be necessary to confirm any conclusions • “Missing” physics • Are any of the physical ingredients we removed important? Brendan Krueger | CEA Saclay

28. Conclusions • We are exploring the nonlinear behavior of convection in CC SNe • A simple bubble trajectory seems to miss multidimensional effects that are important to the buoyancy of bubbles • Critical density contrasts as predicted by bubble trajectories tend to overestimate the capacity of bubbles to become nonlinearly unstable • Still a work in progress • We hope to tie in similar work on the SASI in order to develop a coherent picture of the instabilities that govern dynamics in the gain region Brendan Krueger | CEA Saclay