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

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 = Δx/Δz on the computed value of the Nusselt number, Nuc, is assessed from a suite of 21 model solutions of Bénard convection at the same Rayleigh number but with varying grid dimensions. It is shown that: (i) for any constant value of r, Nuc varies linearly with (Δx)2 and may be extrapolated to the limit Δx = 0; (ii) the extrapolated value, Nu0, is independent of the value of r employed; (iii) for r = 1 the discretization error (Nu0—Nuc) introduced when Δx > 0 may be parametrized in terms of the Rayleigh number and Δx; (iv) it is possible to choose r such that Nuc equals Nu0, the value at Δx = 0, and is independent of Δx; (v) such solutions may be obtained on surprisingly coarse grids when r > 1; and (vi) in general Nuc 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.