Symmetries, Conservation Laws, and Hamiltonian Structure in Geophysical Fluid Dynamics

Abstract

This chapter discusses symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics. Some basic definitions and results for finite-dimensional Hamiltonian dynamical systems, leading up to the introduction of the symplectic notation that proves crucial in the generalization to noncanonical, infinite-dimensional systems, such as the Eulerian representation of fluid flow are presented. It is found that in addition to the invariants, such as energy and momentum that are associated with explicit symmetries, noncanonical Hamiltonian systems generally possess what are sometimes known as Casimir invariants or distinguished functions. The equations of fluid dynamics are continuous in space and thus represent infinite-dimensional dynamical systems. The existence of nonlinearly stable solutions whose stability relies on the full infinity of Casimir invariants strongly suggests a failure of ergodicity, because trajectories originating sufficiently close to those states cannot fill out the entire energy–enstrophy hypersurface. It is found that the physical approximations made to derive the shallow-water equations from 3D incompressible flow consist of taking the fluid to be homogeneous and constrained to move in vertical columns.