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Analogous formulation of electrodynamics and two-dimensional fluid dynamics

Published online by Cambridge University Press:  18 November 2014

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

A single, simply stated approximation transforms the equations for a two-dimensional perfect fluid into a form that is closely analogous to Maxwell’s equations in classical electrodynamics. All the fluid conservation laws are retained in some form. Waves in the fluid interact only with vorticity and not with themselves. The vorticity is analogous to electric charge density, and point vortices are the analogues of point charges. The dynamics is equivalent to an action principle in which a set of fields and the locations of the point vortices are varied independently. We recover classical, incompressible, point vortex dynamics as a limiting case. Our full formulation represents the generalization of point vortex dynamics to the case of compressible flow.

JFM classification

Mathematical Foundations: Mathematical Foundations Vortex Flows: Vortex dynamics
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 761 , 25 December 2014 , R2
Copyright
© 2014 Cambridge University Press 

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References

Aref, H. 2007 Point vortex dynamics: a classical mathematics playground. J. Math. Phys. 48, 065401 1–23.CrossRefGoogle Scholar
Bühler, O. 1998 A shallow-water model that prevents nonlinear steepening of gravity waves. J. Atmos. Sci. 55, 2884–2891.2.0.CO;2>CrossRefGoogle Scholar
Howe, M. S. 1999 On the scattering of sound by a rectilinear vortex. J. Sound Vib. 227 (5), 1003–1017.CrossRefGoogle Scholar
Howe, M. S. 2003 Theory of Vortex Sound. Cambridge University Press, 207 pp.Google Scholar
Kraichnan, R. H. 1959 The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech. 5 (4), 497–543.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1975 The Classical Theory of Fields. Pergamon, 397 pp.Google Scholar
Powell, A. 1964 Theory of vortex sound. J. Acoust. Soc. Am. 36 (1), 177–195.CrossRefGoogle Scholar
Salmon, R. 1988 Hamiltonian fluid mechanics. Annu. Rev. Fluid Mech. 20, 225–256.CrossRefGoogle Scholar
Salmon, R. 2009 An ocean circulation model based on operator splitting, Hamiltonian brackets, and the inclusion of sound waves. J. Phys. Oceanogr. 39, 1615–1633.CrossRefGoogle Scholar
Shepherd, T. G. 1990 Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics. Adv. Geophys. 32, 287–338.CrossRefGoogle Scholar