# Cartesian Hydrodynamics

Poisson equation

5230CvMagnetohydrodynamics (including electron magnetohydrodynamics)

5272+vLaboratory studies of space- and astrophysical-plasma processes

5225FiTransport properties

I.

INTRODUCTION

There has been significant recent work on Vlasov-Maxwell (VM) equilibria that are consistent with nonlinear force-free1–8 and “nearly force-free”9 magnetic fields in Cartesian geometry. Therein, force-free refers to a magnetic field for which the associated current density is exactly parallel, which is the definition we shall also use

∇×B=μ0j,

j×B=0.These works consider one-dimensional (1D) collisionless current sheets, with Refs. 1–8 specifically calculating

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This clearly demonstrates that “sourcing” an exactly force-free macroscopic equilibrium with an equilibrium DF in a 1D cylindrical geometry is inherently a more difficult task than in the Cartesian case. The presence of “extra” positive definite inertial forces and, almost inevitably, forces associated with charge separation raises the question of whether exactly force-free equilibria are possible at all in this

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In the case of zero scalar potential, the result of the calculation is to give a distribution function that is not a solution of the Vlasov equation as it is not a function of the constants of motion only. In essence, an additional exp(−r2) factor is required in the DF to counter exp(r2) terms that manifest by completing the square in the integration. The physical cause here would appear to be the inertial forces associated with the rotational bulk flow.

If one assumes a non-zero scalar potential, then it seems impossible to satisfy the Ampère 's law. The physical cause seems to be that, in the case of force-free fields, one would require a “different” electrostatic potential to balance the inertial forces for the ions and electrons, which is of course nonsensical. Thus, our investigation seems to suggest that it is not possible to calculate a DF of the form of Equation (6) for the exact GH field.

C.

The magnetic field: A Gold-Hoyle flux tube plus a background field

To make progress, we introduce a background field in the negative z direction. The mathematical motivation for this change is to balance the