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Modeling Thermal Convection Using Citcom 2 - Governing Equations |
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The governing equations for thermal convection are the conservation equations of the mass, momentum, and energy. Over large time scales (>10,000 years), the mantle may be approximated as incompressible fluid, and then the mass conservation becomes continuity equation. The large mantle viscosity or Prandtl number (viscosity divided by thermal diffusivity) enables us to ignore the inertial terms in the momentum equation. To the first order, we can also ignore viscous heating and adiabatic heating in the energy equation. The governing equations can be written as 
In fluid dynamics, we often like to deal with dimensionless numbers. For example, we normalize depth of mantle by the thickness of mantle, which yields dimensionless thickness of 1. The real advantage of this practice is to identify controlling parameters that result from nondimensionalize the governing equations. For example, for thermal convection problems, two important nondimensional parameters can be immediately identified, Rayleigh number, Ra, and internal heating parameter, H. Then for thermal convection problems, whether they are for the upper mantle convection, or whole mantle convection, or convection for Mars, as long as Ra and H are chosen to be the same, the dynamics as dictated by the nondimensional equations is the same. 
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