The Cooling Box Problem: What does the water surface freeze?

In environmental systems, such as lakes, the boundary condition at the air-water interface depends upon the atmospheric forcing and the induced convection. These processes are particularly important for seasonally ice-covered lakes, as they determine, in part, when the lake will freeze. To study this, we performed three-dimensional direct numerical simulations of surface-driven convection near the temperature of maximum density $\tilde T_{md}$. In this system with a nonlinear (quadratic) equation of state, we identified three convective regimes: (1) free convection when the water temperature is above $\tilde T_{md}$, (2) penetrative convection when the surface water temperature is below $\tilde T_{md}$ and the system is vigorously convecting, and (3) decaying convection when the system transitions to a laminar state. We discuss predictions for the regime transitions. Our hope is that this will lead to better estimates for the timing of ice-formation in lakes and other natural systems.