Hamiltonian dynamics of internal waves

Abstract

The isopycnal elevation ζ of a stratified incompressible fluid constitutes a generalized co-ordinate in terms of which the dynamics can be described in Hamiltonian form. The description is complete, and no additional variables are necessary, when the isopycnal circulation vanishes and when there are no remote sources of flow. The Hamiltonian, constructed in a co-ordinate system that coincides with the isopycnal surfaces, generates nonlinear equations of motion for ζ and its conjugate variable π which are formally exact for arbitrary displacements of the fluid and which are equivalent at lowest order to the usual linearized equations of internal waves. The appropriate scaling parameters for the nonlinearity are the isopycnal slope [dtri ]ζ and vertical strain ∂_zζ, which emerge as the expansion parameters of an explicit powerseries representation of the Hamiltonian.