Neutrino-driven Convection versus Advection in 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.