tl;dr: For snow on top of ice, is the temperature at the base of the snowpack always (practically) the same as the temperature at the top of the ice?
I was trying to understand how the thermodynamics in a sea ice model (LIM3) were set up and how one could set up a finite element method (FEM) for the heat equation in the sea ice system. My focus is on the skin temperature and its importance for the coupled atmosphere-sea ice dynamics. The whole point of skin temperature is that properties at the interface between two substances can be different from the bulk properties of either substance.
My first attempt at setting up an FEM following the same logic as LIM3's finite difference formulation resulted in an underdetermined system by 1 equation. I then realized that the model allows density, heat capacity, and thermal conductivity to be discontinuous between snow and ice (obviously) but the temperature was continuous. I get that the conductive heat flux has to be continuous, but is the temperature? If not, where might an additional constraint for heat diffusion come from? I can describe my FEM in more detail if that helps, but that might be better suited to a different SE.