and the unit of time is the free-fall time at this position ($\sim 3$ ms for a central mass of $1.3M_\sun$ and $r_{\rm s0}\sim 150$ km, §\[s:setup\]).[]{data-label="f:shock_entropy_pert"}](f5.eps){width="\columnwidth"}

### Effect of Initial Perturbations

The effect of different initial perturbations on the exploding model of the e0 sequence is illustrated in Figure \[f:shock\_entropy\_pert\]. Models with large amplitude random cell-to-cell density perturbations (p0B10R1 and pB10R3) follow the same path as the model where $\ell=1$ is directly perturbed (p0B10L1, Fig. \[f:shock\_entropy\_sasi\]). The model with an $\ell=2$ perturbation (p0B10L2) undergoes a weak convective phase over a number of advection times, during which $\ell=0$ grows and $\ell=2$ saturates at a small amplitude. After a delay of $\sim 100t_{\rm ff0}$, however, $\ell=1$ oscillations of the shock emerge, and the model joins the usual SASI-dominated explosion path. The models with large amplitude density perturbations in the gain region with a fixed $\ell=4$ and $5$ dependence (p0B10G4 and p0B10G5) trigger less regular sloshings of the shock, which however still result in the formation of large-scale bubbles. As with the $\ell=2$ perturbation, an even-$\ell$ convective perturbation takes longer to couple to an $\ell=1$ SASI mode.

The time to explosion appears to be a non-trivial function of the perturbation form and 