Analysis of the semi-geostrophic shallow water equations

The semi-geostrophic shallow water equations are an accurate approximation to the rotating shallow water equations on scales larger than the Rossby radius of deformation. The global existence of weak solutions of the Lagrangian form of the equations is known. However, uniqueness is not known except for short times for smooth initial data. Similarly, a rigorous proof of existence of solutions has not been achieved for variable rotation rate, which is important for atmospheric applications. The obstacle is the lack of regularity of solutions to the Monge–Ampere equation, which underlies the solution procedure. In this note, it is shown formally that the modified form of the Monge–Ampere equation that appears in the solution procedure for the semi-geostrophic shallow water equations is better controlled on large scales than that of the standard equation that is used in the incompressible case. This suggests that more progress should be possible with the rigorous theory, and in particular that it should be possible to prove uniqueness.