to an explosion that shed insight into the operation of each instability?

Our approach is experimental, employing hydrodynamic simulations that model neutrino source terms, the equation of state, and gravity in a parametric way (e.g., @FT09b). This setup has the advantage that its linear stability properties are well understood [@FT09a], allowing the development of model sequences that probe different parameter regimes. Our experimental approach to studying SASI and convection follows similar works [@foglizzo06; @ohnishi06; @scheck08; @FT09b; @burrows2012], to which we relate our findings. To connect with more realistic models, we also test the generality of our analysis results on the simulations of @mueller2012.

The structure of the paper is the following. Section \[s:methods\] describes the numerical models employed and introduces the Spherical Fourier-Bessel decomposition. Section \[s:results\] presents results, separated by exploding and quasi-steady state behavior. A summary and discussion follows in Section \[s:summary\]. Appendix \[s:L0\] addresses the stability of the $\ell=0$ mode in the parametric setup, and Appendix \[s:sfb\_appendix\] provides details about the spherical basis functions for the cases of Dirichlet and Neumann boundary conditions.

Methods {#s:methods}
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Parametric Hydrodynamic Simulations {#s:parametric}
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### Numerical Setup {#s:setup}

The parametric, two-dimensional stalled supernova shock simulations employed for the majority of the 