## Convection

Convection is much more efficient than conduction at removing internal heat from a planet and dominates in the deeper parts of a planet' s interior. Convection occurs when the thermal gradient exceeds some critical value, causing hot material to rise and cooler material to sink. This occurs when material is hotter and therefore less dense than its surroundings, causing the hotter material to be buoyant. As the mass of hotter material rises, it expands and begins to cool. Under adiabatic conditions, it will continue to rise, expand, and cool until its temperature matches that of the surroundings. Convective motions are governed by the conservation equations combined with fluid dynamical equations (e.g., the Navier-Stokes equation). The full description of convection involves finite element modeling and is beyond the scope of this discussion. Interested readers are referred to Turcotte and Schubert (2002).

The dimensionless Rayleigh number (1a) is used to determine if convection will occur. Assume we have a region of the mantle of thickness d which is warmer than its surroundings by a factor of AT. If the mantle has a density pm, thermal expansion coefficient a, thermal diffusivity j, and viscosity g, 1a can be computed from

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