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

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.