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The relaxation of a helical magnetic field ${\bf B}({\bf x}, t)$ in ahigh-conductivity plasma contained in the annulus between two perfectlyconducting coaxial cylinders is considered. The plasma is of low density andits pressure is negligible compared with the magnetic pressure; the flow of theplasma is driven by the Lorentz force and and energy is dissipated primarily bythe viscosity of the medium. The axial and toroidal fluxes of magnetic fieldare conserved in the perfect-conductivity limit, as is the mass per unit axiallength. The magnetic field relaxes during a rapid initial stage to a force-freestate, and then decays slowly, due to the effect of weak resistivity $\eta$,while constrained to remain approximately force-free. Interest centres onwhether the relaxed field may attain a Taylor state; but under the assumedconditions with axial and toroidal flux conserved inside every cylindricalLagrangian surface, this is not possible. The effect of an additional$\alpha$-effect associated with instabilities and turbulence in the plasma istherefore investigated in exploratory manner. An assumed pseudo-scalar form of$\alpha$ proportional to $q\,\eta\, ({\bf j}\cdot {\bf B})$ is adopted, where ${\bf j}=\nabla\times {\bf B}$ and $q$ is an $\mathcal{O}(1)$ dimensionlessparameter. It is shown that, when $q$ is less that a critical value $q_c$, theevolution remains smooth and similar to that for $q=0$; but that if $q>q_c$,negative-diffusivity effects act on the axial component of $\bf B$, generatinghigh-frequency rapidly damped oscillations and an associated transitoryappearance of reversed axial field. However, the scalar quantity $\gamma={\bfj}\cdot {\bf B}/B^2$ remains highly non-uniform, so that again the field showsno sign of relaxing to a Taylor state for which $\gamma$ would have to beconstant.

Pinch dynamics in a low-$\beta$ plasma