  By 2D hydrodynamic simulations including a detailed equation of state and
neutrino transport, we investigate the interplay between different non-radial
hydrodynamic instabilities that play a role during the postbounce accretion
phase of collapsing stellar cores. The convective mode of instability, which is
driven by negative entropy gradients caused by neutrino heating or by time
variations of the shock strength, can be identified clearly by the development
of typical Rayleigh-Taylor mushrooms. However, in 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 such a situation the shocked
flow nevertheless can develop non-radial asymmetry with an oscillatory growth
of the amplitude. This phenomenon has been termed ``standing 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 of the
average shock radius and of the mass in the gain layer. Both hydrodynamic
instabilities in combination stretch the advection time of matter 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 NS turns out to be favorable for explosions, because the
accretion luminosity and neutrino heating are larger and the growth rate of the
SASI is higher. Moreover, we show that the oscillation period of the SASI and a
variety of other features in our simulations agree with estimates 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. (abridged)
