= Radiogenically heated lava-lake crust model
For a well-mixed <lava lake> of depth $H$ and a thin conductive crust of thickness $a$, put $\Delta T=T_m-T_s$, and use the convective flux $K(\overline T-T_m)$ with constant <heat transfer coefficient> $K>0$. The <Stefan condition> and liquid heat balance give, to leading order in $a/H$,
$$
\rho L_f\dot a=\frac{k\Delta T}{a}-K(\overline T-T_m),\qquad \rho c_pH\dot{\overline T}=\rho QH-K(\overline T-T_m).
$$
Here <latent heat> $L_f$ is per unit mass and $k=\rho c_p\kappa$. Crust heating and changes of liquid depth are neglected. The linear conductive profile uses a <quasi-steady approximation>; thinness alone does not justify neglecting crust thermal storage. During initial growth, a small <Stefan number> supplies a sufficient rapid-adjustment regime. Define $h=a/H$, $\theta=(\overline T-T_m)/\Delta T$, and $\tau=\kappa t/H^2$. Scaling by the <thermal diffusion time> $H^2/\kappa$ and $\Delta T$ gives $h'=S(1/h-B\theta)$ and $\theta'=R-B\theta$, where $B=KH/k$, $R=QH^2/(c_p\kappa\Delta T)$, and $S=c_p\Delta T/L_f$ is the <Stefan number>. Thus $\theta=R/B+(\theta_0-R/B)e^{-B\tau}$ exactly. The reduction applies while $a/H\ll1$, the conductive profile is approximately linear, and the <thermal convection> closure remains valid; it does not model complete freezing of a deep lake.
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