Numerical Analysis on Standing Accretion Shock Instability with Neutrino Heating in the Supernova Cores

We have numerically studied the instability of the spherically symmetric standing accretion shock wave against non-spherical perturbations. We have in mind the application to the collapse-driven supernovae in the post bounce phase, where the prompt shock wave generated by core bounce is commonly stalled. We take an experimental stand point in this paper. Using spherically symmetric, completely steady, shocked accretion flows as unperturbed states, we have clearly observed both the linear growth and the subsequent nonlinear saturation of the instability. In so doing, we have employed a realistic equation of state together with heating and cooling via neutrino reactions with nucleons. We have done a mode analysis based on the spherical harmonics decomposition and found that the modes with l=1, 2 are dominant not only in the linear regime, but also after the nonlinear couplings generate various modes and the saturation occurs. Varying the neutrino luminosity, we have constructed the unperturbed states both with and without a negative entropy-gradient. We have found that in both cases the growth of the instability is similar, suggesting the convection does not play a dominant role, which also appears to be supported by the recent linear analysis of the convection in accretion flows by Foglizzo et al. The real part of the eigen frequency seems to be mainly determined by the advection time rather than by the sound-crossing time. Whatever the cause may be, the instability is favorable for the shock revival.