the entire computational domain with an amplitude of $10\%$ and $30\%$, overdense shells in the upstream flow that trigger specific SASI modes (see @FT09a for details), and density perturbations in the gain region which are radially-constant from $r_g$ to $r_s$, but with an angular dependence set by a Legendre polynomial. The latter are aimed at exciting convection by large-scale perturbations. The amplitude is chosen to be $50\%$.

Note that we focus on exploding models that are marginally above the heating rate for explosion, where non-radial instabilities are expected to have the maximum effect. Further increase of the heating rate yields explosions that develop earlier, eventually approaching the spherically-symmetric runaway condition (e.g., Appendix \[s:L0\]).

### Linear Stability Properties {#s:linear_stability}

The linear stability properties of the two sets of background flow configurations (e0 and e3) are shown in Figure \[f:growth\_timescales\]. The growth rates of the fundamental $\ell=1$, and $\ell=2$ modes as a function of heating rate are monotonically decreasing as long as $\chi < 3$. Above $\chi > 3$, modes transition into a non-oscillatory (convective) state with two branches, in line with the results of @yamasaki07. The mode $\ell_{\rm crit}$ that bifurcates at the lowest heating rate ($\chi \simeq 3$) is approximately 