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Let ${\delta }_T$ be the thickness of the thermal boundary layer immediately below $z=h$, let $\nu$ be the kinematic viscosity, let $\kappa$ be the thermal diffusivity, and let ${\mathrm{Ra}}_c$ be the critical local Rayleigh number. The density contrast driving the boundary layer follows from the density anomaly of water:A Local Rayleigh-number closure setsand thereforeBy Fourier's law, the upward heat flux through this layer iswhere $k$ is the water's thermal conductivity. This expression applies while the quantity inside braces is positive.

The density anomaly of water makes mass density increase with temperature up to $T_m$ and decrease above $T_m$. Since temperature increases with depth in the stated layer, the density profile first increases to its maximum ${\rho }_m$ at the $T=T_m$ isotherm and then decreases toward the warm bottom.

The upper region has denser water below lighter water and hence stable density stratification. Below the density maximum, density decreases downward, so lighter water lies beneath denser water and gives unstable density stratification. The lower region therefore overturns by thermal convection, while the cold upper region remains a stagnant cap. This coexistence is penetrative convection.