Viscous heating, adiabatic heating and energetic consistency in compressible mantle convection

Although it has been suggested that the total viscous heating, Q(v), should be exactly balanced by the total adiabatic heating, Q(a), for compressible mantle convection, previous numerical studies show a significant imbalance of up to several percent between Q(v) and Q(a) for simple isoviscous compressible convection. The cause of this imbalance and its potential effects on more complicated convective systems remain largely unknown. In this study, we present an analysis to show that total viscous heating and adiabatic heating for compressible mantle convection with anelastic liquid approximation (ALA) and the Adams-Williamson equation of state are balanced out at any instant in time, and that the previously reported imbalance between Q(v) and Q(a) for numerical models with a truncated anelastic liquid approximation (TALA) is caused by neglecting the effect of the pressure on the buoyancy force. Although we only consider the Adams-Williamson equation of state in our analysis, our method can be used to check the energetic consistency for other forms of equation of state. We formulate numerical models of compressible mantle convection under both TALA and ALA formulations by modifying the Uzawa algorithm in Citcom code. Our numerical results confirm our analysis on the balance between total viscous heating and total adiabatic heating.