We use time-dependent, axisymmetric, hydrodynamic simulations to study the linear stability of the stalled, spherical accretion shock that arises in the postbounce phase of core-collapse supernovae. We show that this accretion shock is stable to radial modes, with decay rates and oscillation frequencies in close agreement with the linear stability analysis of Houck and Chevalier. For nonspherical perturbations we find that the l = 1 mode is always unstable for parameters appropriate to core-collapse supernovae. We also find that the l = 2 mode is unstable, but typically has a growth rate smaller than that for l = 1. Furthermore, the l = 1 mode is the only mode found to transition into a nonlinear stage in our simulations. This result provides a possible explanation for the dominance of an l = 1 "sloshing" mode seen in many two-dimensional simulations of core-collapse supernovae.
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