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We derive the mechanical energy budget for shallow, ice-covered lakes energized by penetrative solar radiation. Radiation increases the available and background components of the potential energy at different rates. Available potential energy drives under-ice motion, including diurnally active turbulence in a near-surface convective mixing layer. Heat loss at the ice-water interface depletes background potential energy at a rate that depends on the available potential energy dynamics. Expressions for relative energy transfer rates show that the pathway for solar energy is sensitive to the convective mixing layer temperature through the nonlinear equation of state. Finally, we show that measurements of light penetration, temperature profiles resolving the diffusive boundary layer, and an estimate of the kinetic energy dissipation rate can be combined to estimate the forcing rate, the rate of heat loss to the ice, and efficiencies of the energy pathways for radiatively driven flows.

Plain Language Summary Global observations reveal a pervasive decline in the annual ice cover duration of inland waters. This has stimulated urgent new research into cold and polar aquatic environments. Predicting thermal changes in ice-covered waters requires the extension of current fluid-dynamical theories to incorporate the physics that governs cold water near its temperature of maximum density. In this work, we present new mathematical expressions for the transformation of solar energy that penetrates the ice and show that feasible under-ice measurements can be used to estimate the fraction of this energy that is transferred to the ice as heat, contributing to its rate of melting.