---
abstract: 'The success of the neutrino mechanism of core-collapse supernovae relies on the supporting action of two hydrodynamic instabilities: neutrino-driven convection and the Standing Accretion Shock Instability (SASI). Depending on the structure of the stellar progenitor, each of these instabilities can dominate the evolution of the gain region prior to the onset of explosion, with implications for the ensuing asymmetries. Here we examine the flow dynamics in the neighborhood of explosion by means of parametric two-dimensional, time-dependent hydrodynamic simulations for which the linear stability properties are well understood. We find that systems for which the convection parameter $\chi$ is sub-critical (SASI-dominated) develop explosions once large-scale, high-entropy bubbles are able to survive for several SASI oscillation cycles. These long-lived structures are seeded by the SASI during shock expansions. Finite-amplitude initial perturbations do not alter this outcome qualitatively, though they can lead to significant differences in explosion times. Supercritical systems (convection-dominated) also explode by developing large-scale bubbles, though the formation of these structures is due to buoyant activity. Non-exploding systems achieve a quasi-steady state in which the time-averaged flow adjusts itself to be convectively sub-critical. We characterize the turbulent flow using a spherical Fourier-Bessel decomposition, identifying the relevant scalings and connecting 