We study hydrodynamic instabilities during the first seconds of core-collapse supernovae by means of 2D simulations with approximative neutrino transport and boundary conditions that parameterize the effects of the contracting neutron star and allow us to obtain sufficiently strong neutrino heating and, hence, neutrino-driven explosions. Confirming more idealised studies, as well as supernova simulations with spectral transport, we find that random seed perturbations can grow by hydrodynamic instabilities to a globally asymmetric mass distribution in the region between the nascent neutron star and the accretion shock, leading to a dominance of dipole (l = 1) and quadrupole (l = 2) modes in the explosion ejecta, provided the onset of the supernova explosion is sufficiently slower than the growth time scale of the low-mode instability. By gravitational and hydrodynamic forces, the anisotropic mass distribution causes an acceleration of the nascent neutron star, which lasts for several seconds and can propel the neutron star to velocities of more than 1000 km s −1 . Because the explosion anisotropies develop chaotically and change by small differences in the fluid flow, the magnitude of the kick varies stochastically. No systematic dependence of the average neutron star velocity on the explosion energy or the properties of the considered progenitors is found. Instead, the anisotropy of the mass ejection, and hence of the kick, seems to increase when the nascent neutron star contracts more quickly, and thus low-mode instabilities can grow more rapidly. Our more than 70 models separate into two groups, one with high and the other with low neutron star velocities and accelerations after one second of post-bounce evolution, depending on whether the l = 1 mode is dominant in the ejecta or not. This leads to a bimodality of the distribution when the neutron star velocities are extrapolated to their terminal values. Establishing a link to the measured distribution of pulsar velocities, however, requires a much larger set of calculations and ultimately 3D modelling.

We investigate the behavior and consequences of the reverse shock that terminates the supersonic expansion of the baryonic wind which is driven by neutrino heating off the surface of (non-magnetized) new-born neutron stars in supernova cores. To this end we perform long-time hydrodynamic simulations in spherical symmetry. In agreement with previous relativistic wind studies, we find that the neutrino-driven outflow accelerates to supersonic velocities and in case of a compact, ∼1.4 M (gravitational mass) neutron star with a radius of about 10 km, the wind reaches entropies of about 100 k B per nucleon. The wind, however, is strongly influenced by the environment of the supernova core. It is decelerated and shock-heated abruptly by a termination shock that forms when the supersonic outflow collides with the slower preceding supernova ejecta. The radial position of this reverse shock varies with time and depends on the strength of the neutrino wind and the explosion conditions in progenitor stars with different masses and structure. Its basic properties and behavior can be understood by simple analytic considerations. We demonstrate that the entropy of the matter going through the reverse shock can increase to a multiple of the asymptotic wind value. Seconds after the onset of the explosion it therefore can exceed 400 k B per nucleon in low-mass progenitors around 10 M , where the supernova shock and the reverse shock propagate outward quickly. The temperature of the shocked wind has typically dropped to about or less than 10 9 K, and density and temperature in the shock-decelerated matter continue to decrease only very slowly. For more massive progenitors with bigger and denser metal cores, the explosion expands more slowly so that the termination shock stays at smaller radii and affects the wind at higher temperatures and densities. In this case the termination shock might play a non-negligible, strongly time-and progenitordependent role in discussing supernova nucleosynthesis.

Performing two-dimensional hydrodynamic simulations including a detailed treatment of the equation of state of the stellar plasma and for the neutrino transport and interactions, we investigate here the interplay between different kinds of non-radial hydrodynamic instabilities that can play a role during the postbounce accretion phase of collapsing stellar cores. The convective mode of instability, which is driven by the negative entropy gradients caused by neutrino heating or by variations in the shock strength in transient phases of shock expansion and contraction, can be identified clearly by the development of typical Rayleigh-Taylor mushrooms. However, in those cases where the gas in the postshock region is rapidly advected towards the gain radius, the growth of such a buoyancy instability can be suppressed. In this situation the shock and postshock flow can nevertheless develop non-radial asymmetry with an oscillatory growth in the amplitude. This phenomenon has been termed "standing (or spherical) accretion shock instability" (SASI). It is shown here that the SASI oscillations can trigger convective instability, and like the latter, they lead to an increase in the average shock radius and in the mass of the gain layer. Both hydrodynamic instabilities in combination stretch the advection time of matter accreted through the neutrino-heating layer and thus enhance the neutrino energy deposition in support of the neutrino-driven explosion mechanism. A rapidly contracting and more compact nascent neutron star turns out to be favorable for explosions, because the accretion luminosity and neutrino heating are greater and the growth rate of the SASI is higher. Moreover, we show that the oscillation period of the SASI observed in our simulations agrees with the one estimated for the advective-acoustic cycle (AAC), in which perturbations are carried by the accretion flow from the shock to the neutron star and pressure waves close an amplifying global feedback loop. A variety of other features in our models, as well as differences in their behavior, can also be understood on the basis of the AAC hypothesis. The interpretation of the SASI in our simulations as a purely acoustic phenomenon, however, appears difficult.

We analyze the linear stability of a stalled accretion shock in a perfect gas with a parameterized cooling function L / À P . The instability is dominated by the l ¼ 1 mode if the shock radius exceeds 2Y3 times the accretor radius, depending on the parameters of the cooling function. The growth rate and oscillation period are comparable to those observed in the numerical simulations of Blondin & Mezzacappa. The instability mechanism is analyzed by separately measuring the efficiencies of the purely acoustic cycle and the advective-acoustic cycle. These efficiencies are estimated directly from the eigenspectrum and also through a WKB analysis in the high-frequency limit. Both methods prove that the advective-acoustic cycle is unstable and that the purely acoustic cycle is stable. Extrapolating these results to low frequency leads us to interpret the dominant mode as an advective-acoustic instability, different from the purely acoustic interpretation of Blondin & Mezzacappa. A simplified characterization of the instability is proposed, based on an advective-acoustic cycle between the shock and the radius r 9 where the velocity gradients of the stationary flow are strongest. The importance of the coupling region in this mechanism calls for a better understanding of the conditions for an efficient advective-acoustic coupling in a decelerated, nonadiabatic flow, in order to extend these results to core-collapse supernovae.

A toy model is analyzed in order to evaluate the linear stability of the gain region immediately behind a stalled accretion shock, after core bounce. This model demonstrates that a negative entropy gradient is not sufficient to warrant linear instability. The stability criterion is governed by the ratio χ of the advection time through the gain region divided by the local timescale of buoyancy. The gain region is linearly stable if χ < 3. The classical convective instability is recovered in the limit χ ≫ 3. For χ > 3, perturbations are unstable in a limited range of horizontal wavelengths centered around twice the vertical size H of the gain region. The threshold horizontal wavenumbers k min and k max follow simple scaling laws such that Hk min ∝ 1/χ and Hk max ∝ χ. The convective stability of the l = 1 mode in spherical accretion is discussed, in relation with the asymmetric explosion of core collapse supernovae. The advective stabilization of long wavelength perturbations weakens the possible influence of convection alone on a global l = 1 mode.