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Semigeostrophic theory as a Dirac-bracket projection

Published online by Cambridge University Press:  21 April 2006

Rick Salmon
Rick Salmon
Affiliation:
Scripps Institution of Oceanography A025, La Jolla, CA 92093, USA
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Abstract

This paper presents a general method for deriving approximate dynamical equations that satisfy a prescribed constraint typically chosen to filter out unwanted high frequency motions. The approximate equations take a simple general form in arbitrary phase-space coordinates. The family of semigeostrophic equations for rotating flow derived by Salmon (1983, 1985) fits this general form when the chosen constraint is geostrophic balance. More precisely, the semigeostrophic equations are equivalent to a Dirac-bracket projection of the exact Hamiltonian ønto the phase-space manifold corresponding to geostrophically balanced states. The more widely used quasi-geostrophic equations do not fit the general form and are instead equivalent to a metric projection of the exact dynamics on to the same geostrophic manifold. The metric, which corresponds to the Hamiltonian of the linearized dynamics, is an artificial component of the theory, and its presence explains why the quasi-geostrophic equations are valid only near a state isopycnals.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 196 , November 1988 , pp. 345 - 358
Copyright
© 1988 Cambridge University Press

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