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An alternative view of generalized Lagrangian mean theory

Published online by Cambridge University Press:  19 February 2013

Rick Salmon*
Affiliation:
Scripps Institution of Oceanography, La Jolla, CA 92093-0213, USA
*
Email address for correspondence: rsalmon@ucsd.edu

Abstract

If the variables describing wave–mean flow interactions are chosen to include a set of fluid–particle labels corresponding to the mean flow, then the generalized Lagrangian mean (GLM) theory takes the form of an ordinary classical field theory. Its only truly distinctive features then arise from the distinctive feature of fluid dynamics as a field theory, namely, the particle-relabelling symmetry property, which corresponds by Noether’s theorem to the many vorticity conservation laws of fluid mechanics. The key feature of the formulation is that all the dependent variables depend on a common set of space–time coordinates. This feature permits an easy and transparent derivation of the GLM equations by use of the energy–momentum tensor formalism. The particle-relabelling symmetry property leads to the GLM potential vorticity law in which pseudo-momentum is the only wave activity term present. Thus the particle-relabelling symmetry explains the prominent importance of pseudo-momentum in GLM theory.

Type
Papers
Copyright
©2013 Cambridge University Press

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References

Andrews, D. G. & McIntyre, M. E. 1978a An exact theory of nonlinear waves on a Lagrangian mean flow. J. Fluid Mech. 89, 609646.Google Scholar
Andrews, D. G. & McIntyre, M. E. 1978b On wave action and its relatives. J. Fluid Mech. 89, 647664.Google Scholar
Bretherton, F. P. 1971 The generalized theory of wave propagation. In Mathematical Problems in the Geophysical Sciences, part 1, Geophysical Fluid Dynamics , Lectures in Applied Mathematics , vol. 13, pp. 61102. American Mathematical Society.Google Scholar
Bretherton, F. P. 1976 Conservation of wave action and angular momentum in a spherical atmosphere. (Unpublished manuscript).Google Scholar
Bühler, O. 2009 Waves and Mean Flows. Cambridge University Press.CrossRefGoogle Scholar
Bühler, O. & McIntyre, M. E. 1998 On non-dissipative wave–mean interactions in the atmosphere or oceans. J. Fluid Mech. 354, 301343.CrossRefGoogle Scholar
Gjaja, I. & Holm, D. D. 1996 Self-consistent Hamiltonian dynamics of wave mean-flow interaction for a rotating stratified incompressible fluid. Physica D 98, 343378.Google Scholar
Grimshaw, R. 1984 Wave action and wave–mean flow interaction, with applications to stratified shear flows. Annu. Rev. Fluid Mech. 16, 1144.CrossRefGoogle Scholar
Salmon, R. 1988 Hamiltonian fluid mechanics. Annu. Rev. Fluid Mech. 20, 225256.CrossRefGoogle Scholar
Salmon, R. 1998 Lectures on Geophysical Fluid Dynamics. Oxford University Press.Google Scholar
Whitham, G. B. 1965 A general approach to linear and nonlinear waves using a Lagrangian. J. Fluid Mech. 22, 273283.CrossRefGoogle Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar