Penetrative convection in an ice-covered lake

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 $(T_w-T_m{)}^{5/3}$.