= Solution
\b[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, and
$$
J=C_i(\theta_l-\theta_u)^{4/3},\qquad E=C_s\theta_u^{4/3}.
$$
With no internal heating the budgets are $H_l\dot\theta_l=F-J$ and $H_u\dot\theta_u=J-E$. At <steady state>,
$$
\theta_u=(F/C_s)^{3/4},\qquad
\theta_l=\theta_u+(F/C_i)^{3/4}.
$$
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.
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