How Properties that Distinguish Solids from Fluids and Constraints of Spherical Geometry Suppress Lower Mantle Convection

The large magnitude of the dimensionless Rayleigh number (Ra ∼10⁸) for Earth’s ∼3 000 km thick mantle is considered evidence of whole mantle convection. However, the current formulation assumes behavior characteristic of gases and liquids and also assumes Cartesian geometry. Issues arising from neglecting physical properties unique to solids and ignoring the spherical shapes for planets include: (1) Planet radius must be incorporated into Ra, in addition to layer thickness, to conserve mass during radial displacements. (2) The vastly different rates for heat and mass diffusion in solids, which result from their decoupled transport mechanisms, promote stability. (3) Unlike liquids, substantial stress is needed to deform solids, which independently promotes stability. (4) High interior compression stabilizes the mantle in additional minor ways. Therefore, representing conditions for convection in solid, self-gravitating spheroids, requires modifying formulae developed for bottomheated fluids near ambient conditions under an invariant gravitational field. To derive stability criteria appropriate to solid spheres, we use dimensional analysis, and consider the effects of geometry, force competition, and microscopic behavior. We show that internal heating has been improperly accounted for in the Ra. We conclude that the lower mantle is stable for two independent reasons: heat diffusion far outpaces mass diffusion (creep) and yield strength of solids at high pressure exceeds the effective deviatoric stress. We discuss the role of partial melt in lubricating plate motion, and explain why the Ra is not applicable to the multi-component upper mantle. When conduction is insufficient to transport heat in the Earth, melt production and ascent are expected, not convection of solid rock.