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Statistical mechanics, Euler’s equation, and Jupiter’s Red Spot

Miller, Jonathan and Weichman, Peter B. and Cross, M. C. (1992) Statistical mechanics, Euler’s equation, and Jupiter’s Red Spot. Physical Review A, 45 (4). pp. 2328-2359. ISSN 1050-2947. http://resolver.caltech.edu/CaltechAUTHORS:MILpra92

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Abstract

We construct a statistical-mechanical treatment of equilibrium flows in the two-dimensional Euler fluid which respects all conservation laws. The vorticity field is fundamental, and its long-range Coulomb interactions lead to an exact set of nonlinear mean-field equations for the equilibrium state. The equations depend on an infinite set of parameters, in one-to-one correspondence with the infinite set of conserved variables. We illustrate the equations by solving them numerically in simple cases. In more complicated cases we use Monte Carlo techniques, with the eventual aim of detailed comparison with the Red Spot dynamical simulations of Marcus: Preliminary efforts show good agreement. We review previous literature on two-dimensional (2D) Euler flow in light of our theory. In particular, we derive the Kraichnan energy-enstrophy theory and the Lundgren and Pointin point-vortex mean-field theory as special cases of our own. Our techniques may be generalized to a number of other Coulomb-like Hamiltonian systems with an infinite number of conservation laws, including some in higher dimensions. For example, we rederive Lynden-Bell's theory of stellar-cluster formation, as well as the Debye-Huckel theory of electrolytes. Our results may also be applicable to cylindrically bound guiding-center plasmas, which under idealized conditions provide another realization of 2D Euler flow.


Item Type:Article
Additional Information:©1992 The American Physical Society. Received 18 March 1991. We would like to thank Philip Marcus and Changhoon Lee for providing the numerical data used for comparisons in Figs. 3 and 4. We thank Ali Rouhi and Daniel Meiron for conversations leading to Appendix A. The financial support of the NSF, through Grant No. DMR-9013984, and the Shell Foundation is gratefully acknowledged.
Subject Keywords:GUIDING-CENTER PLASMA; 2-DIMENSIONAL TURBULENCE; DIFFEOMORPHISM-GROUPS; PHASE-TRANSITION; VORTEX DYNAMICS; 2 DIMENSIONS; SU INFINITY; VORTICES; FLUIDS; FLOW
Record Number:CaltechAUTHORS:MILpra92
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:MILpra92
Alternative URL:http://dx.doi.org/10.1103/PhysRevA.45.2328
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:2320
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:27 Mar 2006
Last Modified:26 Dec 2012 08:48

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