Abstract
In this thesis we shall be looking at rigorously proving the local in time existence of C2 classical solutions to the semigeostrophic equations. We present the semigeostrophic model as a formal approximation to the primitive equations. We explicitly state the scaling parameter representing the Rossby number. We then use Hoskins' coordinate frame to transform these equations into a coupled Monge- Ampere-transport system of hyperbolic-elliptic nonlinear partial differential equations. This system is singularly perturbed by the large Coriolis force in the form of the small Rossby number appearing in the denominator. We prove a new a priori estimate for classical solutions of the fully nonlinear Monge-Ampere equations with right hand side close to a positive constant, and use this to prove that there exist classical solutions to the Semigeostrophic equations in dual coordinates for a time interval that increases as the Rossby number decreases, which confirms the stabilising effect of the Coriolis force. As an immediate consequence, classical solutions in the physical coordinates also exist for at least the same time interval.