Earth
Mantle convection
Heat a layer of slow fluid from below: at a small Rayleigh number a disturbance dies away and the conducting profile stays, above the critical value hot plumes rise and cold sheets sink and the Nusselt number climbs above one. Make the lid rigid and watch the surface stop moving and the heat transport fall; raise the Rayleigh number and the cells speed up and thin their boundary layers.
What happens
Two-dimensional Boussinesq convection at infinite Prandtl number in a 2:1 box heated from below, in stream-function and vorticity form: ∇²ω = −Ra·∂T/∂x and ∇²ψ = −ω are relaxed together by Gauss–Seidel sweeps (a free lid has ω = 0 at the top, a rigid lid ω = −2ψ_1/Δz²), the velocity u = ∂ψ/∂z, w = −∂ψ/∂x is discretely divergence-free, and the temperature is advected upwind and diffused explicitly with the step held under both the diffusive and the Courant limits. The Nusselt numbers at the top and the bottom, the largest speed and the Poisson residual are read; the 32 × 16 grid is coarse, so the numbers are those of this grid.
Try this
Lower the Rayleigh number to a few hundred and watch the disturbance die; raise it and watch the plumes form and the Nusselt number climb; switch to a rigid lid and compare.
The model and its limits
A coarse 32 × 16 grid, first-order upwind advection, a fixed number of relaxation sweeps (the residual is shown), constant viscosity, no internal heating or phase changes. Dimensionless; not the mantle’s numbers.
Equations used in the module
Equations ∇²ω = −Ra·∂T/∂x, ∇²ψ = −ω u = ∂ψ/∂z, w = −∂ψ/∂x ∂T/∂t + u·∂T/∂x + w·∂T/∂z = ∇²T Nu = −(1/W)·∫ ∂T/∂z dx
