  We investigate the structure of the stalled supernova shock in both 2D and 3D
and explore the differences in the effects of neutrino heating and the standing
accretion shock instability (SASI). We find that early on the amplitude of the
dipolar mode of the shock is factors of 2 to 3 smaller in 3D than in 2D.
However, later in both 3D and 2D the monopole and dipole modes start to grow
until explosion. Whereas in 2D the (l,m) = (1,0) mode changes sign
quasi-periodically, producing the "up-and-down" motion always seen in modern 2D
simulations, in 3D this almost never happens. Rather, in 3D when the dipolar
mode starts to grow, it grows in magnitude and wanders stochastically in
direction until settling before explosion to a particular patch of solid angle.
In 2D we find that the amplitude of the dipolar shock deformation separates
into two classes. For the first, identified with the SASI and for a wide range
of "low" neutrino luminosities, this amplitude remains small and roughly
constant. For the other, identified with higher luminosities and
neutrino-driven convection, the dipolar amplitude grows sharply. Importantly,
it is only for this higher luminosity class that we see neutrino-driven
explosions within ~1 second of bounce. Moreover, for the "low" luminosity runs,
the power spectra of these dipolar oscillations peak in the 30-50 Hz range
associated with advection timescales, while for the high-luminosity runs the
power spectra at lower frequencies are significantly more prominent. We
associate this enhanced power at lower frequencies with slower convective
effects and the secular growth of the dipolar shock amplitude. On the basis of
our study, we hypothesize that neutrino-driven buoyant convection should almost
always dominate the SASI when the supernova explosion is neutrino-driven.
