A stalled spherical accretion shock, such as that arising in core-collapse supernovae, is unstable to non-spherical perturbations. In three dimensions, this Standing Accretion Shock Instability (SASI) can develop spiral modes that spin-up the protoneutron star. Here we study these non-axisymmetric modes by combining linear stability analysis and three-dimensional, time-dependent hydrodynamic simulations with Zeus-MP, focusing on characterizing their spatial structure and angular momentum

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... . We do not impose any rotation on the background accretion flow, and use simplified microphysics with no neutrino heating or nuclear dissociation. Spiral modes are examined in isolation by choosing flow parameters such that only the fundamental mode is unstable for a given polar index, leading to good agreement with linear theory. We find that any superposition of sloshing modes with non-zero relative phases survives in the nonlinear regime and leads to angular momentum redistribution. It follows that the range of perturbations required to obtain spin-up is broader than that needed to obtain the limiting case of a phase shift of pi/2. The bulk of the angular momentum redistribution occurs during a phase of exponential growth, and arises from internal torques that are second order in the perturbation amplitude. This redistribution gives rise to at least two counter rotating regions, with the maximum angular momentum of a given sign approaching a significant fraction of the mass accretion rate times the shock radius squared (Mdot * r_shock^2 ~ 1E+47 g/cm^2/s, spin period ~60 ms). Nonlinear mode coupling at saturation causes the angular momentum to fluctuate in all directions with much smaller amplitudes.