  The neutrino-heated "gain layer" immediately behind the stalled shock in a
core-collapse supernova is unstable to high-Reynolds-number turbulent
convection. We carry out and analyze a new set of 19 high-resolution
three-dimensional (3D) simulations with a three-species neutrino
leakage/heating scheme and compare with spherically-symmetric (1D) and
axisymmetric (2D) simulations carried out with the same methods. We study the
postbounce supernova evolution in a $15$-$M_\odot$ progenitor star and vary the
local neutrino heating rate, the magnitude and spatial dependence of
asphericity from convective burning in the Si/O shell, and spatial resolution.
Our simulations suggest that there is a direct correlation between the strength
of turbulence in the gain layer and the susceptability to explosion. 2D and 3D
simulations explode at much lower neutrino heating rates than 1D simulations.
This is commonly explained by the fact that nonradial dynamics allows accreting
material to stay longer in the gain layer. We show that this explanation is
incomplete. Our results indicate that the effective turbulent ram pressure
exerted on the shock plays a crucial role by allowing multi-D models to explode
at a lower postshock thermal pressure and thus with less neutrino heating than
1D models. We connect the turbulent ram pressure with turbulent energy at large
scales and in this way explain why 2D simulations are erroneously exploding
more easily than 3D simulations.
