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Relaxed states of an ideal MHD plasma with external magnetic field

Published online by Cambridge University Press:  13 March 2009

G. Knorr ,
M. Mond  and
C. Grabbe
G. Knorr
Affiliation:
The Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
M. Mond
Affiliation:
The Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
C. Grabbe
Affiliation:
Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242, U.S.A.
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Abstract

We study an ideal MHD plasma with the non-vanishing invariants energy, crosshelicity and magnetic helicity, confined in a cylinder with infinitely conducting walls and an externally applied magnetic field B0. The magnetic and velocity fields are expanded in base vector fields, satisfying Δ × Bλ = Bλ. Boundary conditions are imposed to make the curl a self-adjoint operator. The three invariants depend on the time-dependent coefficients of the base vector fields, and are used to construct the partition function to gather statistical information about the equlibrium thermodynamic state to which the plasma relaxes after a turbulent transition. For zero external magnetic field but large magnetic helicity, the energy resides preferentially in magnetic field fluctuations. A sizeable fraction of the kinetic energy initially present is transformed into magnetic energy. The energy condenses via an inverse cascade predominantly to the lowest energy eigenstate, in agreement with results obtained by Taylor. However, since we consider the whole spectrum of eigenstates, the energy does not exclusively occupy the lowest eigenstate. If the eigenvalues are densely spaced (as in a thin torus), the higher eigenmodes also contain appreciable amounts of energy, resulting in a finite pressure of the plasma. For constant and finite external magnetic field, the average induced magnetic field exactly cancels the external field. This indicates that, on a statistical average, the plasma is diamagnetic or superconducting. Superimposed on the average statistical state are fluctuations that may become large if the magnetic helicity is large.

Type
Research Article
Information
Journal of Plasma Physics , Volume 53 , Issue 3 , June 1995 , pp. 373 - 385
Copyright
Copyright © Cambridge University Press 1995

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