analysis follow the setup of @FT09b [@FT09a]. These models have been calibrated to the global linear stability analysis of @F07. The linear analysis has been extended to include the effects of parameterized nuclear dissociation and lightbulb neutrino heating [@FT09b].

In our time-dependent models, the equations of mass, momentum, and energy conservation are solved in spherical polar coordinates ($r,\theta$), subject to the gravity from a point mass $M$ at the origin and parameterized neutrino heating and cooling: $$\begin{aligned}
\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{v}) & = & 0\\
\frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v}\cdot \nabla)\mathbf{v}
                     &  = & -\frac{1}{\rho}\nabla p - \frac{GM}{r^2}\hat r\\
\frac{\totd e_{\rm int}}{\totd t} -\frac{1}{\rho}\frac{\totd p}{\totd t} & = & Q_\nu.\end{aligned}$$ Here $\rho$, $\mathbf{v}$, $p$, and $e_{\rm int}$ are the fluid density, velocity, pressure, and specific internal energy, respectively. The equation of state is that of an ideal gas with adiabatic index $\gamma$, i.e., $p = (\gamma-1)\rho e_{\rm int}$. To connect with previous studies, the net neutrino source term is set to $$\label{eq:neutrino_source_term}
Q_\nu = \left[ \frac{B}{r^2} - A p^{3/2}\right]
                   \,e^{-(s/s_{\rm min})^2}\,\Theta(\mathcal{M}_0-\mathcal{M}),$$ where $s$ is the fluid entropy, $\mathcal{M}$ the Mach number, and $\Theta$ the step function. This functional form models heating as a lightbulb, with $B$ a normalization constant 