Natural convection in shallow water

Starting from the 3D Boussinesq model and taking the limit as the domain thickness tends to zero, we derive rigorously the 2D model for natural convection in shallow water. The model reduces to a degenerated elliptic equation for the pressure, an explicit formula for horizontal components of the velocity and a vertical diffusion for the vertical component. The macroscopic flow is driven by temperature variations as well as the bottom topography.