On Nusselt numbers and the relative resolution of plumes and boundary layers in mantle convection

Abstract

SUMMARY The fact that vertical plumes and horizontal boundary layers have different cross-sectional dimensions in idealized models of mantle convection is quantified and then exploited to provide a criterion for the selection of an optimal ratio of horizontal and vertical spatial increments in finite difference solutions to the equations governing mantle convection. The effects of varying the ratio r = &/Az on the computed value of the Nusselt number, Nu,, is assessed from a suite of 21 model solutions of BCnard convection at the same Rayleigh number but with varying grid dimensions. It is shown that: (i) for any constant value of r, Nu, varies linearly with (Ax)’ and may be extrapolated to the limit Ax = 0; (ii) the extrapolated value, Nuo, is independent of the value of r employed; (iii) for r = 1 the discretization error (Nu,-Nu,) introduced when Ax>O may be parametrized in terms of the Rayleigh number and Ax; (iv) it is possible to choose r such that Nu, equals Nuo, the value at Ax = 0, and is independent of Ax; (v) such solutions may be obtained on surprisingly coarse grids when r > 1; and (vi) in general Nu, is much less sensitive to the loss of horizontal resolution than vertical. Implications of these results for future developments in the modelling of convection in the Earth’s mantle are discussed.