that for which $2\ell_{\rm crit}$ eddies of size $(r_s-r_g)$ fit into a transverse wavelength [@foglizzo06] $$\label{eq:lmax_def}
\lambda_{\perp,\rm crit} \equiv \frac{\pi (r_s+r_g)}{\sqrt{\ell_{\rm crit}(\ell_{\rm crit}+1)}} \sim 2(r_s - r_g).$$ Modes with larger or smaller $\ell$ bifurcate at higher heating rate.

Figure \[f:growth\_timescales\] also shows two important timescale ratios in the stationary solution as a function of heating rates. The first one is the ratio of advection times in the gain and cooling regions, $t_{\rm adv-g}$ and $t_{\rm adv-c}$, respectively. On the basis of numerical simulations with a realistic EOS, @F12 found that equality between these two timescales at $t=0$ corresponds approximately to the onset of oscillatory instability. Figure \[f:growth\_timescales\] shows that this relation is valid for the $\varepsilon=0$ sequence, losing accuracy when nuclear dissociation is included.

The *instantaneous* value of the ratio of advection to heating timescales in the gain region has for long been used to quantify proximity to an explosion in numerical simulations [@janka98; @thompson00; @thompson05]. @F12 found that equality between these two timescales in the *initial condition* – or equivalently, at the time of shock stalling – marks approximately the subsequent onset of non-oscillatory $\ell=0$ instability in numerical simulations. However, Figure \[f:growth\_timescales\] shows that the point where the linear 