Pinch dynamics in a low-$β$ plasma

Abstract

The relaxation of a helical magnetic field ${\bf B}({\bf x}, t)$ in a high-conductivity plasma contained in the annulus between two perfectly conducting coaxial cylinders is considered. The plasma is of low density and its pressure is negligible compared with the magnetic pressure; the flow of the plasma is driven by the Lorentz force and and energy is dissipated primarily by the viscosity of the medium. The axial and toroidal fluxes of magnetic field are conserved in the perfect-conductivity limit, as is the mass per unit axial length. The magnetic field relaxes during a rapid initial stage to a force-free state, and then decays slowly, due to the effect of weak resistivity $η$, while constrained to remain approximately force-free. Interest centres on whether the relaxed field may attain a Taylor state; but under the assumed conditions with axial and toroidal flux conserved inside every cylindrical Lagrangian surface, this is not possible. The effect of an additional $α$-effect associated with instabilities and turbulence in the plasma is therefore investigated in exploratory manner. An assumed pseudo-scalar form of $α$ proportional to $q\,η\, ({\bf j}\cdot {\bf B})$ is adopted, where $ {\bf j}=\nabla\times {\bf B}$ and $q$ is an $\mathcal{O}(1)$ dimensionless parameter. It is shown that, when $q$ is less that a critical value $q_c$, the evolution remains smooth and similar to that for $q=0$; but that if $q>q_c$, negative-diffusivity effects act on the axial component of $\bf B$, generating high-frequency rapidly damped oscillations and an associated transitory appearance of reversed axial field. However, the scalar quantity $γ={\bf j}\cdot {\bf B}/B^2$ remains highly non-uniform, so that again the field shows no sign of relaxing to a Taylor state for which $γ$ would have to be constant.