Nonlocal dc electrical conductivity of a Lorentz plasma in a stochastic magnetic field

Abstract

The stochastic wandering of magnetic field lines allows momentum of even perfectly magnetized electrons to be transported across the mean $\overrightarrow{\mathrm{B}}$. We include this effect, along with the usual acceleration and scattering terms, in a spatially one-dimensional Boltzmann equation for the electron distribution function. For an electric field $E_{\parallel }$ (along local $\overrightarrow{\mathrm{B}}$) which varies versus position normal to $\overrightarrow{\mathrm{B}}$, the momentum transport leads to a nonlocal electrical conductivity. We apply the formalism to sheared, force-free magnetoplasmas, in which the $E_{\parallel }$ gradient is caused by variable twisting of $\overrightarrow{\mathrm{B}}$ with respect to an externally applied uniform $\overrightarrow{\mathrm{E}}$. We examine in particular the experimentally documented phenomenon of field-aligned current density $j_{\parallel }>0\mathrm{}$ in regions of the sheared magnetic field where $E_{\parallel }\approx 0$ or even $E_{\parallel }<0\mathrm{}$. This phenomenon is in apparent violation of Ohm's law. Under suitable conditions of stochasticity and collisionality, we find that the spatial structure and temporal persistence of these force-free configurations can be directly caused by electron-momentum transport. This result is derived solely on the basis of electron dynamics. In contrast to fluid-turbulent models, our kinetic derivation requires no hypothetical "plasma dynamo" and no conjecture on the decay rates of magnetic helicity versus magnetic energy.