# Variational Principles in Geophysical Fluid Dynamics and Approximated Equations

Chapter
Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM)

## Abstract

In this chapter, the variational principle of Hamilton is applied to different examples from Geophysical Fluid Dynamics. Hamilton’s principle is extended to uniformly rotating flows and to incompressible flows. After an example in finite dimensions consisting of the motion of point vortices, a set of approximated equations is considered, that is, rotating shallow water equations, rotating Green–Naghdi equations and semi-geostrophic equations. Equations of the first and second kind conserve potential vorticity as a consequence of the invariance of the related action functional under relabelling symmetry. Equation of the third kind takes into account also an ageostrophic part of the flow and conserves the so-called transformed potential vorticity which is based on a special Legendre transformation on the coordinates. The case of continuously stratified fluids is then analysed. Finally, the variational approach is applied to wave dynamics, where it can be used to both derive the equations of motion and to obtain the dispersion relation for nonlinear problems as well as the conservation of the wave activity of the system.

## Keywords

Fluid dynamics Geophysical fluid dynamics Ideal fluid Variational principle Conservation laws Rotating flows Stratified flows Potential vorticity Ertel’s theorem Circulation Shallow water equations Quasi-geostrophic equations Lagrangian labels Relabelling symmetry Point vortices Approximated equations Semi-geostrophy Green–Naghdi equations Wave dynamics Surface waves Luke’s variational principle Whitham’s averaged variational principle Wave Activity Klein–Gordon equation Korteweg–deVries (KdV) equation

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## Authors and Affiliations

1. 1.Universität HamburgHamburgGermany
2. 2.University of TriesteTriesteItaly