We have used a numerical scheme based on higher-order finite differences to investigate effects of adiabatic heating and viscous dissipation on 3-D rapidly rotating thermal convection in a cartesian box with an aspect-ratio of 2x2x1. Although we omitted coupling with the magnetic field, which can play a key role in the dynamics of the Earth's core, the understanding of non-linear rotating convection including realistic thermodynamic effects is a necessary prerequisite for understanding full complexity of the Earth's core dynamics. The system of coupled partial differential equations has been solved in terms of the principal variables vorticity $\bh \omega$, vector potential $\bh A$ and temperature $T$. The use of the vector potential $\bh A$ allows the velocity field to be calculated with one spatial differentiation in contrast to the spheroidal and toroidal function approach. The temporal evolution is governed by a coupled time-dependent system consisting of $\bh \omega$ and $T$. The equations are discretized in all directions by using an eighth-order, variable spaced scheme. Rayleigh number $Ra$ of 10$^6$, Taylor number $Ta$ of 10$^8$ and a Prandtl number $Pr$ of 1 have been employed. The dissipation number of the outer core was taken to be 0.2. A stretched grid has been employed in the vertical direction for resolving the thin shear boundary layers at the top and bottom. This vertical resolution corresponds to around 240 regularly spaced points with an eighth-order accuracy. For the regime appropriate for the Earth's outer core, the dimensionless surface temperature $T_0$ takes on a large value, around 4. This large value in the adiabatic heating/cooling term is found to cause stabilization of both the temperature and velocity fields.
Keyword(s):High Rayleigh number, 3-D convection, higher-order finite differences, rotating fluid, finite Prandtl number.
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