MHD stationary symmetric flows with Hall effect

Abstract

Stationary magnetohydrodynamic (MHD) flows with a symmetry are analyzed for very-low-density plasmas in strong magnetic fields, in the infinite conductivity limit. In this particular case the simplified form of Ohm's law usually employed in the ideal MHD case is no longer valid, because Hall's and electron pressure terms in the generalized Ohm's law must be retained. Thus we include them in our equations. The assumption of symmetry makes possible the definition of an ignorable coordinate and of current functions for the divergence free magnitudes. In the case without Hall's effect and electron pressure terms in Ohm's law, there is only one independent current function; in the present case, instead, two Stokes functions are independent. We choose the current function of the magnetic induction field and that of the plasma flux, $\psi$ and $\chi$, to be independent. Then we can express the other Stokes functions of our problem as functions of $\psi$ and $\chi$. We assume helical symmetry, as it is the most general form of spatial symmetry and impose incompressibility, that is ${\bf v\cdot\nabla\rho =}\,0$. We obtain a system of two differential equations for $\psi$ and $\chi$, and solve them for the particular case of constant density. Then we calculate the limit of our solution for Hall's effect that tends to zero, in order to make a comparison with the results obtained for the case without Hall's effect. Finally, we find the necessary conditions to obtain confined plasma columns embedded in a force-free plasma or in a vacuum.