Collisional Transport in Magnetoplasmas in the Presence of Differential Rotation

Abstract

The aim of this paper is to propose a generalization of the treatment of collisional transport theory in a strongly toroidally rotating magnetoplasma recently developed by Catto et al. ( Ion transport in toroidally rotating Tokomak discharges, Physics of Fluids, Vol.30, 1987, pp.2784), in order to allow the theoretical description of the decay of relative rotation in an axisymmetric plasma. By investigating the macroscopic transport equations describing the angular momentum and the parallel current density balance, the problem of determination of the divergenceless inductive electric self-field ( E ) is addressed. It is found that, although E in a lowbeta plasma does not influence transport directly, being of higher order in the Larmor radius, its determination is actually relevant for closing the transport equations. In particular it is pointed out that E can always be chosen in such a way to produce a stationary current density vector. However, since such a constraint is not automatically fulfilled by the kinetic distribution function, it is found that a generalization of the strong rotation driftkinetic equation becomes necessary in order to describe firstorder velocity perturbations of the equilibrium distribution function. Copyright © 1989 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved, *Associate Professor in Mathematical Physics, Dipartimento di Scienze Matematiche, tResearch Plasma Physicist, Member of Professional Staff, 89 90 M. TESSAROTTO AND P. J. CATTO Introduction It is well known that the current kinetic transport theories are unable to account for electron transport in weakly collisional magnetically confined plasmas, such as in Tokamaks, which results in unexpectedly large fluxes even in apparently quiescent plasmas,in contrast to theoretical predictions based on kinetic theory-. In reference to this problem, we want to point out the possible influence on enhanced electron transport produced by purely collisional effects on a quiescent magnetoplasma, related to collisional relaxation and diffusion of angular momentum in weakly collisional systems. Ve recall that, in customary transport theory' it is assumed that collisions produce a rapid relaxation in the plasma differential ( toroidal) rotation and consequently that rotation speed is small. Recent theoretical work of Hinton and Vong,and Catto et al., have systematically extended the theory to strong toroidal rotation speeds ( i.e. comparable to the thermal speed of some particle species), showing that the inclusion of such effects in the model may yield more acccurate predictions as far as the ion transport is concerned. In the present paper, instead, by an extension of the model developed in Ref.5, we address the problem of relaxation of plasma relative toroidal rotation ( i.e. of the relative rotation speeds of the various particle species present ). Ve find that a correct description of this relaxation process requires a generalization of the drift-kinetic equation in order to include first order velocity perturbations with respect to a Larmor radius expansion. Inspection of appropriate moment equations shows that in a transport regime first-order velocity perturbations are uniquely determined in terms of the relevant thermodynamics forces and that they decay on a time scale comparable to that of customary collisional transport. Moment Equations and Constraints In order to discuss transport it is convenient to investigate, first of all, some general consequences stemming from appropriate moment equations of the Fokker-Planck kinetic equation in the presence of spatial symmetry ( i.e., an axisymmetric torus ) and taking into account a transport ordering appropriate to describe a magnetically confined plasma in the presence of strong drifts, i.e. in the presence of toroidal rotation speeds comparable to the ion thermal velocity ( see e.g.,Refs.4 and 5 ). COLLISIONAL TRANSPORT IN MAGNETOPLASMAS 91 It suff ic ies for this purpose to take into account the -4 -> moment equations corresponding to X =1,M Rv-0 where R is the s s distance from the principal axis of the torus and ~6 the too roidal unit vector. By denoting, therefore, N =/d vf the s s number density, and P =M ft R*N = R/dvM v-#£ the toroidal k> O O U S S angular momentum, with f =f (r,v,t) the kinetic distribus s tion function, we obtain the moment equations of continuity and toroidal angular momentum balance, i.e. respectively: