Some Effects of Tensor Conductivity in Magnetohydrodynamics

Abstract

A theoretical investigation concerning some Hall-current effects in inviscid, incompressible magnetohydrodynamics is made. The generalized Ohm's law and the concept of tensor conduc­ tivity are discussed. The macroscopic equations governing steady small-perturbation flow of a fully ionized gas are de­ veloped for three orientations of an applied magnetic field of in­ finite extent. These equations are then solved for flow past an insulating sinusoidal wall. The general solutions, pertaining to arbitrary values of magnetic Reynolds number and Hall param­ eter, are studied in the limits of large and small magnetic Rey­ nolds number, and characteristic Hall-current loops are detected. The pressure distribution on the wav}^ wall is investigated as a function of magnetic Reynolds number and Hall parameter. For the particular cases investigated here the influence of nonscalar conductivity is found to be much smaller than elementary considerations would indicate. For completeness, a brief study of transverse hydromagnetic waves is included. The classical plane Alfven wave is modified to have a slowly rotating plane of polarization in some analogy with the behavior of a plane-polarized light beam in an optically active medium. The modified Alfven wave is used to explain cer­ tain features of the wavy-wall solution for large magnetic Rey­ nolds number and an applied magnetic field perpendicular to the plane of the wall.