Relaxed states of an ideal MHD plasma with, external magnetic field

摘要

We study an ideal MHD plasma with the non-vanishing invariants energy, cross-helicity and magnetic helicity, confined in a cylinder with infinitely conducting wallsand an externally applied magnetic field Bo- The magnetic and velocity fields are expanded in base vector fields, satisfying V xB2 = B Boundary conditions are imposed, to make the curl a self-adjoint operator. The three invariants, depend onthe time-dependent coefficients of the base vector fields, and are; used to construct the partition function to gather statistical information about the equlibrjum ther-raodynarnic staje 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 obtmned'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 andfinite external magnetic field, the average induced;rnagnetic field exactly, cancels the external field. This indicates that, on a statistical average, the plasma is diamagnetic or superconduQting, Superimposed on the average statistical state are fluctuations that may become-large if the magnetic helicity is large.