Conservation of canonical circulation and its relation to finite Hall term magnetohydrodynamics

Abstract

The axisymmetric, compressible visco-resistive two-fluid plasma equations are examined under the constraint that the current is purely poloidal and the pressure is a function of density only ("barotropic"). For ideal plasmas (zero resistivity and zero viscosity) the Kelvin circulation theorem of fluid mechanics and the concept of frozen-in field lines turn out to be limiting cases of a more general concept, namely, that the canonical circulation Ssigma=[contour-integral] (msigmausigma+qsigmaA) ·dl of a toroidal fluid element, is exactly conserved as the toroidal element convects and/or is compressed. Appropriate linear combinations of the electron and ion fluid equations give a magnetohydrodynamic vorticity transport equation and an induction equation with a nonlinear Hall term. The finite Hall term is identical to the source term in the vorticity transport equation [P. M. Bellan, Phys. Rev. Lett. 69, 3515 (1992)], except for a constant factor related to the ion collisionless skin depth.