Marsden, Jerrold E. and Weinstein, Alan J.
(1983)
*Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids.*
Physica D, 7
(1-3).
pp. 305-323.
ISSN 0167-2789.
http://resolver.caltech.edu/CaltechAUTHORS:20100910-121621517

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## Abstract

This paper is a study of incompressible fluids, especially their Clebsch variables and vortices, using symplectic geometry and the Lie-Poisson structure on the dual of a Lie algebra. Following ideas of Arnold and others it is shown that Euler's equations are Lie-Poisson equations associated to the group of volume-preserving diffeomorphisms. The dual of the Lie algebra is seen to be the space of vortices, and Kelvin's circulation theorem is interpreted as preservation of coadjoint orbits. In this context, Clebsch variables can be understood as momentum maps. The motion of N point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit, and the standard canonical variables for them are a special kind of Clebsch variables. Point vortices with cores, vortex patches, and vortex filaments can be understood in a similar way. This leads to an explanation of the geometry behind the Hald-Beale-Majda convergence theorems for vorticity algorithms. Symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the KdV and the Schrödinger equations respectively.

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Additional Information: | © 1983 North-Holland. Available online 19 August 2002. Research partially supported by NSF grants MCS 81-07086 and MCS 80-23356, DOE Contract DE-AT03-82ERI2097, and the Miller Institute. The work described here is an outgrowth of our work on plasmas, which was inspired by Phil Morrison and Allan Kaufman. Conversations with Darryl Holm were important in our understanding of Clebsch variables. The hospitality of the Aspen Center for Physics made possible some useful discussions with Jim Meiss and Phil Morrison on vorticity equations. Finally, we thank Alex Chorin and Andy Majda for their helpful comments on vorticity algorithms. | ||||||||||

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Record Number: | CaltechAUTHORS:20100910-121621517 | ||||||||||

Persistent URL: | http://resolver.caltech.edu/CaltechAUTHORS:20100910-121621517 | ||||||||||

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ID Code: | 19871 | ||||||||||

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Deposited By: | Ruth Sustaita | ||||||||||

Deposited On: | 10 Sep 2010 20:37 | ||||||||||

Last Modified: | 27 Jan 2015 18:27 |

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