Plate bending, energetics of subduction and modeling of mantle convection: A boundary element approach

Abstract

Several authors have suggested that mantle convection is primarily resisted by strong subduction zones, which if true implies small or even negative values of the exponent β in the Nusselt number/Rayleigh number relation Nu∼Raβ. To evaluate this hypothesis, we use the boundary element method (BEM) to study the energetics of subduction in a two-dimensional system comprising two purely viscous plates, a subducting plate (SP) and an overriding plate (OP), immersed in an infinitely deep ambient fluid beneath a free-slip surface. The negative buoyancy of the slab is the only driving force. The principal quantity of interest is the fraction R of the total viscous dissipation that occurs in the upper convective boundary layer comprising the SP, the OP and the subduction interface (SI) between them. Scaling analysis and BEM solutions of the instantaneous flow driven by an isolated SP yield R∼St/[St+F(θ)], where St is the flexural stiffness of the SP and F(θ) is a function of the dip θ of the plate's leading edge. More realistic time-dependent solutions for the SP+OP case show that R(t)≤0.4 for reasonable viscosity contrasts ηSP/ηambient∈[250,2500], indicating that the dissipation is dominated by the ambient mantle contribution. Finally, we formulate a parameterized model of mantle convection to evaluate the influence of subduction-zone dissipation on the effective value of β, motivated by the possibility that the use of the classical value β=1/3 in global parameterized convection models may be the cause of their failure to predict reasonable thermal histories. Using the correct length scale to describe bending (the ‘bending length’; Ribe (2010)), we find β∈[0.25,0.34], which is not much different than the classical result. We conclude that subduction zone dissipation is not large enough to change substantially the classical Nusselt number/Rayleigh number scaling law. It is therefore probably necessary to look elsewhere to reconcile geodynamical and geochemical arguments regarding the thermal history of the Earth.