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A length-scale formula for confined quasi-two-dimensional plasmas

Published online by Cambridge University Press:  01 August 2009

TIMOTHY D. ANDERSEN  and
CHJAN C. LIM
TIMOTHY D. ANDERSEN
Affiliation:
Mathematical Sciences, RPI, 110 8th Street, Troy, NY 12180, USA (andert@alum.rpi.edu)
CHJAN C. LIM
Affiliation:
Mathematical Sciences, RPI, 110 8th Street, Troy, NY 12180, USA (andert@alum.rpi.edu)
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Abstract

Typically a magnetohydrodynamical model for neutral plasmas must take into account both the ionic and the electron fluids and their interaction. However, at short time scales, the ionic fluid appears to be stationary compared to the electron fluid. On these scales, we need consider only the electron motion and associated field dynamics, and a single fluid model called the electron magnetohydrodynamical model which treats the ionic fluid as a uniform neutralizing background applies. Using Maxwell's equations, the vorticity of the electron fluid and the magnetic field can be combined to give a generalized vorticity field, and one can show that Euler's equations govern its behavior. When the vorticity is concentrated into slender, periodic, and nearly parallel (but slightly three-dimensional) filaments, one can also show that Euler's equations simplify into a Hamiltonian system and treat the system in statistical equilibrium, where the filaments act as interacting particles. In this paper, we show that, under a mean-field approximation, as the number of filaments becomes infinite (with appropriate scaling to keep the vorticity constant) and given that angular momentum is conserved, the statistical length scale, R, of this system in the Gibbs canonical ensemble follows an explicit formula, which we derive. This formula shows how the most critical statistic of an electron plasma of this type, its size, varies with angular momentum, kinetic energy, and filament elasticity (a measure of the interior structure of each filament) and in particular it shows how three-dimensional effects cause significant increases in the system size from a perfectly parallel, two-dimensional, one-component Coulomb gas.

Type
Papers
Information
Journal of Plasma Physics , Volume 75 , Issue 4 , August 2009 , pp. 437 - 454
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Onsager, L. 1949 Statistical hydrodynamics. Nuovo Cimento Suppl. 6, 279–287.CrossRefGoogle Scholar
[2]Joyce, G. R. and Montgomery, D. 1973 Negative temperature states for a two-dimensional guiding center plasma. J. Plasma Phys. 10, 107.CrossRefGoogle Scholar
[3]Edwards, S. F. and Taylor, J. B. 1974 Negative temperature states for two-dimensional plasmas and vortex fluids. Proc. R. Soc. Lond. A. 336, 257–271.Google Scholar
[4]Caglioti, E., Lions, P. L., Marchioro, C. and Pulvirenti, M. 1992 A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Commun. Math. Phys. 143, 501–525.CrossRefGoogle Scholar
[5]Kiessling, M. 1993 Statistical mechanics of classical particles with log interactions. Commun. Pure Appl. Math. 46, 27–56.CrossRefGoogle Scholar
[6]Kiessling, M. and Spohn, H. 2001 A note on the eigenvalue density of random matrices. Commun. Math. Phys. 199, 683–695.CrossRefGoogle Scholar
[7]Lim, C. C. 2005 Recent advances in 2d and 2.5d vortex statistics and dynamics. In: Proc. 35th American Institute of Aeronautics and Astronautics Fluid Dynamics Conf., Toronto, Canada (paper no. 2005–5158).CrossRefGoogle Scholar
[8]Lim, C. C. and Nebus, J. 2006 Vorticity Statistical Mechanics and Monte Carlo Simulations. New York: Springer.Google Scholar
[9]Hasimoto, H. 1972 A soliton on a vortex filament. J. Fluid Mech. 51, 472.CrossRefGoogle Scholar
[10]Callegari, A. J. and Ting, L. 1978 Motion of a curved vortex filament with decaying vortical core and axial velocity. SIAM J. Appl. Math. 35 (1), 148–175.CrossRefGoogle Scholar
[11]Klein, R. and Majda, A. J. 1991 Self-stretching of a perturbed vortex filament I: the asymptotic equation for deviations from a straight line. Physica D 49, 323.Google Scholar
[12]Ting, L. and Klein, R. 1991 Viscous Vortical Flows (Lecture Notes in Physics, 374). Berlin: Springer.Google Scholar
[13]Uby, L., Isichenko, M. B. and Yankov, V. V. 1995 Vortex filament dynamics in plasmas and superconductors. Phys. Rev. E 52, 932–939.Google Scholar
[14]Kinney, R., Tajima, T. and Petviashvili, N. 1993 Discrete vortex representation of magnetohydrodynamics. Phys. Rev. Lett. 71, 1712–1715.CrossRefGoogle ScholarPubMed
[15]Klein, R., Majda, A. and Damodaran, K. 1995 Simplified equation for the interaction of nearly parallel vortex filaments. J. Fluid Mech. 288, 201–248.CrossRefGoogle Scholar
[16]Weinan, E. 1994 Dynamics of vortex liquids in Ginsburg–Landau theories with applications to superconductivity. Phys. Rev. B 50 (2), 1126–1135.Google Scholar
[17]Gordeev, A. V., Kingsep, A. S. and Rudakov, L. I. 1994 Electron magnetohydrodynamics. Phys. Rep. 243, 215.Google Scholar
[18]Molenaar, D., Clercx, H. and van Heijst, G. 2005 Transition to chaos in a confined two-dimensional fluid flow. Phys. Rev. Lett. 95, 104503.CrossRefGoogle Scholar
[19]Chorin, A. J. 1994 Vorticity and Turbulence. New York: Springer.Google Scholar
[20]Haken, H. 1975 Cooperative phenomena in systems far from thermal equilibrium and in nonphysical systems. Rev. Mod. Phys. 47 (1), 67–121.Google Scholar
[21]Andersen, T. D. and Lim, C. C. 2007 Negative specific heat in generalized quasi-2d vorticity. Phys. Rev. Lett. 99, 165001.CrossRefGoogle ScholarPubMed
[22]Lim, C. C. and Assad, S. M. 2005 Self-containment radius for rotating planar flows, single-signed vortex gas and electron plasma. Regul. Chaot. Dyn. 10, 240–254.CrossRefGoogle Scholar
[23]Lions, P.-L. and Majda, A. J. 2000 Equilibrium statistical theory for nearly parallel vortex filaments. Commun. Pure Appl. Math. 53, 76–142.3.0.CO;2-L>CrossRefGoogle Scholar
[24]Abrikosov, A. A. 1957 On the magnetic properties of superconductors of the second group. Sov. Phys. JETP 5 (6), 1442–1452.Google Scholar
[25]Ivonin, I. A. 1992 Stability of two-dimensional vortices against three-dimensional perturbations in a fluid and in electron hydrodynamics. Sov. J. Plasma Phys. 18, 302.Google Scholar
[26]Assad, S. M. and Lim, C. C. 2005 Statistical equilibrium of the coulomb/vortex gas in the unbounded two-dimensional plane. Discrete Contin. Dyn. Syst. B 5 (1), 1–14.Google Scholar
[27]Berlin, T. H. and Kac, M. 1952 The spherical model of a ferromagnet. Phys. Rev. 86 (6), 821.CrossRefGoogle Scholar
[28]Zee, A. 2003 Quantum Field Theory in a Nutshell. Princeton: Princeton University Press.Google Scholar
[29]Feynman, R. P. and Wheeler, J. W. 1948 Space–time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20, 367.Google Scholar
[30]Ceperley, D. M. 1995 Path integrals in the theory of condensed helium. Rev. Mod. Phys. 67, 279.CrossRefGoogle Scholar
[31]Kiessling, M. K.-H. and Neukirch, T. 2003 Negative specific heat of a magnetically self-confined plasma torus. Proc. Natl. Acad. Sci. USA 100, 1510–1514.CrossRefGoogle ScholarPubMed