Reflections on dissipation associated with thermal convection

Abstract

AbstractBuoyancy-driven convection is modelled using the Navier–Stokes and entropy equations. It is first shown that the coefficient of heat capacity at constant pressure, ${c}_{p} $, must in general depend explicitly on pressure (i.e. is not a function of temperature alone) in order to resolve a dissipation inconsistency. It is shown that energy dissipation in a statistically steady state is the time-averaged volume integral of $- \mathrm{D} P/ \mathrm{D} t$ and not that of $- \alpha T(\mathrm{D} P/ \mathrm{D} t)$. Secondly, in the framework of the anelastic equations derived with respect to the adiabatic reference state, we obtain a condition when the anelastic liquid approximation can be made, $\gamma - 1\ll 1$, independent of the dissipation number.