Thermal stratification in lakes: Analytical and laboratory studies

Abstract

Theories are developed for the time dependent vertical temperature distribution in a deep lake during the yearly cycle of solar heating and cooling. A portion of the incoming solar radiation is assumed to be absorbed at the water surface, whereas the remainder is absorbed exponentially beneath the surface. Heat is also conducted downward by molecular diffusion. A heat flux balance at the water surface, which accounts for back radiation and evaporative heat loss, is formulated as a boundary condition. The linear, second‐order heat transfer equation is solved by superposition of solutions for the surface absorbed radiation and the internally absorbed radiation. Analytical solutions are given for three different assumptions regarding the time dependence of the incoming radiation and the surface heat losses. At certain times, the resulting temperature and density distribution in the epilimnion are unstable. Under these conditions convective mixing and turbulent diffusion are accounted for by generating a surface mixed layer of uniform temperature. The depth of the surface mixed layer is determined by a thermal energy balance. The theory shows good agreement with field observations of temperature distributions in Lake Tahoe. Experiments are performed using artificial insolation (mercury vapor and infrared lamps) on a laboratory tank. We conclude that it is possible to simulate the development of thermal stratification under laboratory conditions.