Turbulent Chemical Diffusion in Convectively Bounded Carbon Flames

It has been proposed that mixing induced by convective overshoot can disrupt the inward propagation of carbon deflagrations in super-asymptotic giant branch stars. To test this theory, we study an idealized model of convectively bounded carbon flames with 3D hydrodynamic simulations of the Boussinesq equations using the pseudospectral code Dedalus. Because the flame propagation timescale is much longer than the convection timescale, we approximate the flame as fixed in space, and only consider its effects on the buoyancy of the fluid. By evolving a passive scalar field, we derive a {\it turbulent} chemical diffusivity produced by the convection as a function of height, $D_{\rm t}(z)$. Convection can stall a flame if the chemical mixing timescale, set by the turbulent chemical diffusivity, $D_{\rm t}$, is shorter than the flame propagation timescale, set by the thermal diffusivity, $\kappa$, i.e., when $D_{\rm t}>\kappa$. However, we find $D_{\rm t}<\kappa$ for most of the flame because convective plumes are not dense enough to penetrate into the flame. Extrapolating to realistic stellar conditions, this implies that convective mixing cannot stall a carbon flame and that "hybrid carbon-oxygen-neon" white dwarfs are not a typical product of stellar evolution.