  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
\chi of the advection time through the gain region divided by the local
timescale of buoyancy. The gain region is linearly stable if \chi< 3. The
classical convective instability is recovered in the limit \chi\gg 3. For
\chi>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}\propto 1/{\chi} and Hk_{max}\propto \chi. 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.
