  We study the transition to runaway expansion of an initially stalled
core-collapse supernova shock. The neutrino luminosity, mass accretion rate,
and neutrinospheric radius are all treated as free parameters. In spherical
symmetry, this transition is mediated by a global non-adiabatic instability
that develops on the advection time and reaches non-linear amplitude. Here we
perform high-resolution, time-dependent hydrodynamic simulations of stalled
supernova shocks with realistic microphysics to analyze this transition. We
find that radial instability is a sufficient condition for runaway expansion if
the neutrinospheric parameters do not vary with time and if heating by the
accretion luminosity is neglected. For a given unstable mode, transition to
runaway occurs when fluid in the gain region reaches positive specific energy.
We find approximate instability criteria that accurately describe the behavior
of the system over a wide region of parameter space. The threshold neutrino
luminosities are in general different than the limiting value for a
steady-state solution. We hypothesize that multidimensional explosions arise
from the excitation of unstable large-scale modes of the turbulent background
flow, at threshold luminosities that are lower than in the laminar case.
