  We investigate the stability of a neutrino-dominated accretion flow (NDAF), which is expected to be formed in the gravitational collapse of a massive star or the merger of a neutron star binary, based on the variable-$\alpha$ prescription. Recent magnetohydrodynamic (MHD) simulations shows that the viscosity parameter $\alpha$ is proportional to the power of the magnetic Prandtl number ${\rm Pm}=\nu/\eta$, where $\nu$ and $\eta$ are the kinematic viscosity and electric resistivity of the fluid, respectively. In the inner region of a hyperaccretion flow, the viscosity and resistivity are carried by mildly, relativistically degenerated electrons. We fit the dependence of the magnetic Prandtl number on density and temperature by a simple analytic form, and derive the condition for an NDAF to be dynamically unstable. As demonstrations we perform simple one-dimensional simulations of NDAFs with the variable-$\alpha$ and show that the mass accretion becomes highly time-variable in the unstable branch. This mechanism may account for the rapid variability observed in the prompt emission of gamma-ray bursts (GRBs). The mass ejection from a hyperaccretion flow due to viscous heating, which makes a kilonova/macronova emission in the merger of a neutron star binary, is also briefly discussed. 
Neutrino-Dominated Accretion Flows with Magnetic Prandtl   Number-Dependent MRI-driven Turbulence