Neutrino‐driven Convection versus Advection in Core‐Collapse Supernovae

A toy model is analyzed in order to evaluate the linear stability of the gainregion immediately behind a stalled accretion shock, after core bounce. Thismodel demonstrates that a negative entropy gradient is not sufficient towarrant linear instability. The stability criterion is governed by the ratio\chi of the advection time through the gain region divided by the localtimescale of buoyancy. The gain region is linearly stable if \chi< 3. Theclassical convective instability is recovered in the limit \chi\gg 3. For\chi>3, perturbations are unstable in a limited range of horizontal wavelengthscentered around twice the vertical size H of the gain region. The thresholdhorizontal wavenumbers k_{min} and k_{max} follow simple scaling laws such thatHk_{min}\propto 1/{\chi} and Hk_{max}\propto \chi. The convective stability ofthe l=1 mode in spherical accretion is discussed, in relation with theasymmetric explosion of core collapse supernovae. The advective stabilizationof long wavelength perturbations weakens the possible influence of convectionalone on a global l=1 mode.