Earth's core convection: Boussinesq approximation or incompressible approach?

Abstract

Most geodynamo models are based on the Boussinesq approximation that simplifies the mathematics of the problem by neglecting the compressibility of the liquid in the Earth's core. However, the Boussinesq approximation not only disregards the compressibility in the momentum and the continuity equations, but also neglects adiabatic cooling in the heat transport equation which is connected with the reference state temperature distribution . Whereas the former simplification introduces an acceptable inaccuracy, the latter changes the scene in a substantial way and makes the Boussinesq approximation nonadequate for the problem, especially at the onset of the convection. We prove that adiabatic cooling does not vanish in the incompressible limit used by the Boussinesq approximation and propose a new incompressible approach (IA), which neglects the direct influence of compressibility, but takes into consideration its indirect cooling effects. There is not only adiabatic cooling, but also Archimedean cooling (heating) connected with Archimedean work. This last effect allows Ohmic dissipation to be included consistently in the heat transport equation. Using the adiabatic cooling as a natural inhomogeneity in the heat equation allows the typical amplitudes of the superadiabatic temperature, flow velocity, and magnetic field in the solution to be estimated. These amplitudes are in good agreement with those given by the simulation of Glatzmaier and Roberts (An anelastic evolutionary geodynamo simulation driven by compositional and thermal convection. Physica D, 1996, 97, 81–94). Finally these amplitudes are used to obtain dimensionless equations of the IA. The coefficient with the Archimedean force, usually called the Rayleigh number, takes a moderate value of the order of R m instead of the enormous quantities which the Boussinesq approximation usually attributes to it.