Effects of compressibility in the mantle convection Equations

Abstract

The Boussinesq approximation of thermal convection equations results from neglecting the number of the terms which are actually not small in the conditions of the Earth’s mantle. However, the error of calculating the structure of the convective flows is lower than the discarded terms. In this work we analyze the causes of this fact by successively passing from the general equations for a heated viscous compressible fluid to the simpler thermal convection equations by rejecting small quantities with the parameters of the presentday Earth. We consider the anelastic liquid approximation (ALA), truncated anelastic liquid approximation (TALA), extended Boussinesq approximation (EBA), and the simplest classical Boussinesq approximation (BA) which fully disregards the compressibility of a fluid. With the parameters of the mantle, BA is only accurate when describing the flow velocities, while the temperature is predicted with an error of up to a few dozen percent. Therefore, it appears reasonable to consider an intermediate approximation between EBA and BA, in which the effects of compressibility are only taken into account for temperature. This approximation can be referred to as the superadiabatic Boussinesq approximation (SBA) for temperature Tsa. The corresponding equations are structurally similar to the standard Boussinesq approximation but with a superadiabatic temperature Tsa instead of total temperature T. In this simple approximation, the calculated structure of the convective flows and the distribution of total temperature (obtained by adding the known adiabatic Ta to the calculated Tsa) are more accurate than in the classical Boussinesq approximation.