Solutions and singularities of the semigeostrophic equations via the geometry of Lagrangian submanifolds

Using Monge-Amp\`ere geometry, we study the singular structure of a class nonlinear equations in three dimensions, arising geophysical fluid dynamics. We extend seminal earlier work on geometry by examining role an induced metric Lagrangian submanifolds cotangent bundle. In particular, show that signature serves as classification equation, while singularities and elliptic-hyperbolic transitions are revealed degeneracies metric. The theory is illustrated application to example solution semigeostrophic equations.