Eulerian Variational Principles for Stratified Hydrostatic Equations
Abstract The aim of this paper is to advocate the use of variational and Hamiltonian formulations of the hydrostatic equations of motion in finding new conservative numerical techniques for forecast models. For that reason, the fundamental conservative structure of the hydrostatic equations of motion is presented. Variational principles and Hamiltonian formulations of various hydrostatic equations, stratified continuously or in layers, are derived systematically from an Eulerian perspective in the horizontal and a Lagrangian or material perspective in the vertical direction. Variational formulations are derived for the hydrostatic incompressible Boussinesq system and the hydrostatic equations in multiple isentropic and isopycnic layers. The various Hamiltonian formulations presented share similar Poisson brackets; that is, the (contribution to the) Poisson bracket in each layer or in the continuously stratified case is the one for the shallow-water equations with the depth replaced by the appropriate pseudodensity, while the potential (and internal) energy in the Hamiltonian differs in each case. For hydrostatic equations in material coordinates, either stratified continuously or in layers, the coupling between layers happens solely through the Hamiltonian, an observation that may aid in searching for conservative numerical discretizations.