Magnetically confined plasma flows with helical symmetry

Stationary 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 the Ohm's law usually employed in ideal MHD is no more 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 a symmetry makes it 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 result to be independent. We choose the current function of the magnetic induction field and that of the plasma flux, $\psi$ and $\chi$, as independent. Then we can express the other Stokes functions of our problem as functions of $\psi$ and $\chi$. We suppose helical symmetry, as it is the most general form of spatial symmetry (Solov'ev 1967), and impose incompressibility, that is $v\grad\rho=0$. 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 the Hall's effect that tends to zero, in order to make a comparison with the results obtained for the case without Hall's effect (Palumbo and Platzeck 1998). Finally, we find the necessary conditions to obtain confined plasma columns embedded in a force-free plasma or in a vacuum.