Scaling relationships and physics for mixed heating convection in planetary interiors: Isoviscous spherical shells

Matthew B. Weller, Adrian Lenardic, and William B. Moore

Abstract We use a suite of 3-D numerical experiments to test and expand 2-D planar isoviscous scaling
relationships of Moore (2008) for mixed heating convection in spherical geometry mantles over a range of
Rayleigh numbers (Ra). The internal temperature scaling of Moore (2008), when modified to account for
spherical geometry, matches our experimental results to a high degree of fit. The heat flux through the
boundary layers scale as a linear combination of internal (Q) and basal heating, and the modified theory
predictions match our experimental results. Our results indicate that boundary layer thickness and surface
heat flux are not controlled by a local boundary layer stability condition (in agreement with the results of Moore
(2008)) and are instead strongly influenced by boundary layer interactions. Subadiabatic mantle temperature
gradients, in spherical 3-D, are well described by a vertical velocity scaling based on discrete drips as opposed
to a scaling based on coherent sinking sheets, which was found to describe 2-D planar results. Root-meansquare
(RMS) velocities are asymptotic for both low Q and high Q, with a region of rapid adjustment between
asymptotes for moderate Q. RMS velocities are highest in the low Q asymptote and decrease as internal
heating is applied. The scaling laws derived by Moore (2008), and extended here, are robust and highlight the
importance of differing boundary layer processes acting over variable Q and moderate Ra.

Weller, M. B., A. Lenardic, and W. B. Moore (2016), Scaling relationships and physics for mixed heating convection in
planetary interiors: Isoviscous spherical shells, J. Geophys. Res. Solid Earth, 121, doi:10.1002/2016JB013247.

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