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Setso $0<x<\Delta$. Apart from material constants, the heat flux in part ii isThe derivative of its logarithm vanishes whenwhich gives the maximizing interface temperatureSubstitution into the flux law giveswhereThus the proportionality coefficient depends on the thermal conductivity, nonlinear thermal-expansion coefficient $\alpha$, kinematic viscosity, thermal diffusivity, gravity, and critical Rayleigh number. This is the maximum-flux law for penetrative convection in an ice-covered lake.

The stagnant layer transports heat by thermal conduction, so its upward heat flux isAt the maximum-flux interface from part iii,Equating $F_s$ to $F_{\max }$ givesThe stagnant layer has positive thickness only ifFor an ice-covered freshwater lake, take $T_s\simeq 0{}^{\circ }\mathrm{C}$ and $T_m\simeq 4{}^{\circ }\mathrm{C}$. The largest interior temperature compatible with this penetrative convection in an ice-covered lake model is therefore