Generalised Lagrangian Solutions for Atmospheric and Oceanic Flows

Abstract

Atmospheric or oceanic flows strongly constrained by rotation and stratification can be described by a set of Lagrangian partial differential equations called the semigeostrophic equations. In these equations the trajectories must be determined implicitly. Generalised solutions of these equations are defined as a sequence of rearrangements of the fluid, which need not be smooth. These solutions are closely related to generalised solutions of the Monge-Ampere equation. Existence and uniqueness of such solutions is proved. The evolution is shown to be a sequence of minimum energy states of the fluid, giving strong physical plausibility to the solutions.