Past exam of the physics of the Earth as a Planet course of the University of Cambridge / 2012 / Paper 1 / 1

Write $T_s$ for the fixed surface temperature and $\theta =\overline{T}-T_s$. The heat capacity per unit volume is ${\rho }_0{c}_p=k/\kappa$. Per unit horizontal area the layer stores sensible heat $(k/\kappa )d\theta$. Its increase equals the input less the output:The same equation holds for $\theta$ because $T_s$ is constant. It assumes the supplied constant-property, fixed-thickness layer, without internal heat production, latent heat, or mechanical contributions to its heat budget. The infinite horizontal area is handled by taking the budget per unit area.

Vigorous convection keeps the interior nearly uniform, so the upper thermal boundary layer carries a temperature drop of order $\theta$. A boundary layer becomes unstable when its local Rayleigh number reaches an order-fixed critical value $R_c$:This is marginal-stability thermal boundary-layer scaling using the Local Rayleigh-number closure: boundary-layer buoyancy overcomes viscous and thermal-diffusive stabilization. Equivalently, a buoyancy-driven velocity scale $u\sim {\rho }_{0}g\alpha \theta {\delta }^2/\eta$ makes $u\delta /\kappa$ order $R_c$. ThusThere is no $d$ in this closure, as required for a thin boundary layer controlled by its own stability. The exponent is fixed by scaling; its absolute coefficient is not. Here $R_c$ represents the geometry and boundary conditions, together with any order-one boundary-layer coefficient. The question supplies no numerical value for it. The assumption requires $\delta \ll d$, positive $\theta$, and properties sufficiently constant over the temperature contrast.

For flux-driven convective layer cooling, use $E=C{\theta }^{4/3}$ with the same unspecified coefficient as in (b). The steady heat balance givesThe corresponding boundary-layer thickness isSet $\theta ={\theta }_{\ast }+ϵ$ and expand the heat loss to first order. Since $E^{\prime }({\theta }_{\ast })=(4/3)C{\theta }_{\ast }^{1/3}=4F/(3{\theta }_{\ast })$,This is the near-equilibrium relaxation time. The full equation is nonlinear, so an arbitrary large initial temperature departure need not decay as one exponential. Positive heat loss derivative makes the steady state stable. In particular, $\tau$ scales with $d{\delta }_{\ast }$, rather than $d^2$, because convection shortens the effective heat-transport distance to the boundary layer.

A persistent barrier to whole-mantle circulation adds an internal thermal bottleneck and permits the deep mantle to retain a larger temperature excess. In a single well-mixed layer there is one interior temperature and one surface heat-loss law. The coupled thermal histories of layered mantle convection arise because separately mixed layers can have different temperatures, exchanging heat through an interfacial boundary layer as well as losing heat at the surface. Their thermal histories then have coupled relaxation modes rather than the single mode in (c).

For example, let ${\theta }_l,{\theta }_u$ be lower- and upper-layer temperature excesses, $H_l,H_u$ their heat capacities per area, andWith no internal heating the budgets are $H_l{\dot{\theta }}_l=F-J$ and $H_u{\dot{\theta }}_u=J-E$. At steady state,Thus a fixed basal flux still leaves the surface eventually with $E=F$, but the lower layer is hotter than it would be in the single-temperature model. For a given deep temperature instead, the extra interfacial drop reduces outward heat transfer. This favours delayed loss of deep primordial heat and distinct upper- and lower-mantle cooling histories. A numerical change in cooling time cannot be obtained from layer count alone: interface transport, layer thicknesses, temperature-dependent viscosity, internal heating, and whether material can cross the interface matter. Merely replacing $d$ by $d/2$ in (c) misses the added interface and is not a model of two-layer cooling.