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.