of 3. Here $r_g$ is the gain radius, $\omega_{\rm BV}$ is the buoyancy frequency, $$\label{eq:brunt}
\omega^2_{\rm BV} = \frac{GM}{r^2}\left[\frac{1}{\gamma}\frac{\partial \ln p}{\partial r} 
          -\frac{\partial\ln\rho}{\partial r}\right]$$ and $v_r$ is the radial velocity. This critical value of $\chi$ is the number of e-foldings by which an infinitesimal buoyant perturbation needs to grow to counter advection out of the gain region. Larger heating rates and longer advection times are favorable for the growth of convection, as they increase $\chi$.

A finite-amplitude density perturbation can also overcome the stabilizing effect of advection when $\chi < 3$. The minimum amplitude required for a perturbation to rise buoyantly against the accretion flow is [@thompson00; @scheck08; @FT09b; @dolence2013; @couch2012] $$\label{eq:delta_crit}
\left(\frac{\Delta \rho}{\rho}\right)_{\rm c} \simeq \frac{C_D\, v_2^2}{2l_v\, g_s}$$ where $v_2$ and $g_s$ are the postshock velocity and gravitational acceleration at the shock, respectively, $C_D$ is the drag coefficient of the perturbation ($\simeq 0.5$ for a sphere), and $l_v$ is the ratio of the volume to the cross-sectional area of the perturbation in the direction of gravity ($4/3$ times the radius, for a sphere).

[f1a.eps]{} (18,63)[$\ell$]{} (61,78.5)[[$\ell$]{}]{} (76,80)[[$\ell$]{}]{} (82,88.5)[[$\ell$]{}]{}

[f1b.eps]{} (20,64.5)[[$\ell$]{}]{} (40,78.7)[[$\ell$]{}]{} (70,87.5)[[$\ell$]{}]{} (78,63.5)[$\ell$]{}

We evolve two different sequences of models for which the heating rate $B$ is varied from 