Monge-Ampère geometry and semigeostrophic equations

Semigeostrophic equations are a central model in geophysical fluid dynamics designed to represent large-scale atmospheric flows. Their remarkable duality structure allows for a geometric approach through Lychagin's theory of Monge-Ampère equations. We extend seminal earlier work on the subject by studying the properties of an induced metric on solutions, understood as Lagrangian submanifolds of the phase space. We show the interplay between singularities, elliptic-hyperbolic transitions, and the metric signature through a few visual examples.