Construction of Hamiltonian and Nambu forms for the shallow water equations

TL;DR

This paper presents a systematic, coordinate-independent method to derive Hamiltonian and Nambu forms for the shallow water equations, highlighting different conservation laws and their physical mechanisms.

Contribution

It introduces a novel, systematic approach using exterior differential forms and self-adjoint operators to construct Hamiltonian and Nambu structures for shallow water equations.

Findings

Derivation of Hamiltonian and Nambu forms using conservation laws

Identification of mechanisms like vortical flows and gravity waves

Coordinate-independent formulation of the equations

Abstract

A systematic method to derive the Hamiltonian and Nambu form for the shallow water equations, using the conservation for energy and potential enstrophy, is presented. Different mechanisms, such as vortical flows and emission of gravity waves, emerge from different conservation laws (CLs) for total energy and potential enstrophy. The equations are constructed using exterior differential forms and self-adjoint operators and result in the sum of two Nambu brackets, one for the vortical flow and one for the wave-mean flow interaction, and a Poisson bracket representing the interaction between divergence and geostrophic imbalance. The advantage of this approach is that the Hamiltonian and Nambu forms can be here written in a coordinate independent form.