Following our last two weeks of mantle convection talks I will be concluding our trilogy by discussing ways to emulate spherical shell thermal properties in plane-layer convection models. These plane-layer geometry convection models remain a useful tool for investigating planetary mantle dynamics but yield significantly warmer geotherms than spherical shell systems.

In particular, I will discuss the implementation of heat sinks in isoviscous plane-layer models to mimic the temperature found in spherical shell systems. We derived a single equation relating mean temperature, Rayleigh number, internal heating (or cooling) rate, and f, the inner to outer radius ratio. For a given Rayleigh number, the derived expression can be used to calculate an appropriate heating or cooling rate for a plane-layer convection model in order to obtain the temperature of a spherical system described by f. With increasing Rayleigh number it is necessary to implement more cooling in the plane-layer models to match the spherical shell temperatures. This effect is amplified when considering systems with a depth-dependent viscosity. We further derived a set of equations valid for systems featuring a uniform upper mantle viscosity and a lower mantle viscosity increased by a factor of 30 or 100.  These equations can again be used to determine the heating rate necessary in a plane-layer model to emulate the temperature in a spherical shell. I will show examples of mean temperatures and geotherms from plane-layer models which mimic the results from spherical shell convection models.