  We study the three-dimensional (3D) hydrodynamics of the post-core-bounce
phase of the collapse of a 27-solar-mass star and pay special attention to the
development of the standing accretion shock instability (SASI) and
neutrino-driven convection. To this end, we perform 3D general-relativistic
simulations with a 3-species neutrino leakage scheme. The leakage scheme
captures the essential aspects of neutrino cooling, heating, and lepton number
exchange as predicted by radiation-hydrodynamics simulations. The 27-solar-mass
progenitor was studied in 2D by B. Mueller et al. (ApJ 761:72, 2012), who
observed strong growth of the SASI while neutrino-driven convection was
suppressed. In our 3D simulations, neutrino-driven convection grows from
numerical perturbations imposed by our Cartesian grid. It becomes the dominant
instability and leads to large-scale non-oscillatory deformations of the shock
front. These will result in strongly aspherical explosions without the need for
large-scale SASI shock oscillations. Low-l-mode SASI oscillations are present
in our models, but saturate at small amplitudes that decrease with increasing
neutrino heating and vigor of convection. Our results, in agreement with
simpler 3D Newtonian simulations, suggest that once neutrino-driven convection
is started, it is likely to become the dominant instability in 3D. Whether it
is the primary instability after bounce will ultimately depend on the physical
seed perturbations present in the cores of massive stars. The gravitational
wave signal, which we extract and analyze for the first time from 3D
general-relativistic models, will serve as an observational probe of the
postbounce dynamics and, in combination with neutrinos, may allow us to
determine the primary hydrodynamic instability.
