= Flux-driven convective layer cooling
{title2=$H\dot\theta=F-C\theta^{4/3}$}
For <heat capacity> per area $H=kd/\kappa$, imposed basal flux $F$, fixed top <temperature>, and <marginal-stability thermal boundary-layer scaling>, the well-mixed interior has $H\dot\theta=F-C\theta^{4/3}$. Its steady excess <temperature> is $\theta_*=(F/C)^{3/4}$. The near-equilibrium <relaxation time> is $\tau=3H\theta_*/(4F)=3d\delta_*/(4\kappa)$, with $\delta_*=k\theta_*/F$. A full large-amplitude cooling history is nonlinear rather than a single exponential.
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