Flux-driven convective layer cooling ($H\dot{\theta }=F-C{\theta }^{4/3}$)

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 ${\theta }_{\ast }=(F/C{)}^{3/4}$. The near-equilibrium relaxation time is $\tau =3H{\theta }_{\ast }/(4F)=3d{\delta }_{\ast }/(4\kappa )$, with ${\delta }_{\ast }=k{\theta }_{\ast }/F$. A full large-amplitude cooling history is nonlinear rather than a single exponential.