  We perform three-dimensional direct numerical simulations of surface-driven
convection near the temperature of maximum density $\tilde T_{md}$. A dynamic
surface boundary condition couples heat flux through the surface to the induced
convection, creating a dynamic equilibrium between the surface water
temperature and the convection below. In this system, we identified three
convective regimes: (1) free convection when the surface water temperature is
above $\tilde T_{md}$, (2) penetrative convection when the surface water
temperature is below $\tilde T_{md}$ and the convection is actively mixing the
fluid layer, and (3) decaying convection when the convection weakens. We then
predict the transitions between these regimes. Understanding these transitions
is essential for the predicting timing of ice formation in natural systems.
