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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.

Vigorous convection keeps the interior nearly uniform, so the upper thermal boundary layer carries a temperature drop of order $\theta$. A boundary layer becomes unstable when its local Rayleigh number reaches an order-fixed critical value $R_c$:This is marginal-stability thermal boundary-layer scaling using the Local Rayleigh-number closure: boundary-layer buoyancy overcomes viscous and thermal-diffusive stabilization. Equivalently, a buoyancy-driven velocity scale $u\sim {\rho }_{0}g\alpha \theta {\delta }^2/\eta$ makes $u\delta /\kappa$ order $R_c$. ThusThere is no $d$ in this closure, as required for a thin boundary layer controlled by its own stability. The exponent is fixed by scaling; its absolute coefficient is not. Here $R_c$ represents the geometry and boundary conditions, together with any order-one boundary-layer coefficient. The question supplies no numerical value for it. The assumption requires $\delta \ll d$, positive $\theta$, and properties sufficiently constant over the temperature contrast.

In an ice-covered freshwater lake, a stagnant layer colder than the density-maximum temperature can cap a warmer turbulent region. Maximizing the boundary-layer heat flux obtained from a Local Rayleigh-number closure gives an interface temperature $T_h$ satisfyingand a maximum upward heat flux proportional to $(T_w-T_m{)}^{5/3}$.