Theory of one-dimensional Vlasov-Maxwell equilibria: with applications to collisionless current sheets and flux tubes

Abstract

Vlasov-Maxwell equilibria are characterised by the self-consistent descriptions of the steady-states of collisionless plasmas in particle phase-space, and balanced macroscopic forces. We study the theory of Vlasov-Maxwell equilibria in one spatial dimension, as well as its application to current sheet and flux tube models.The 'inverse problem' is that of determining a Vlasov-Maxwell equilibrium distribution function self-consistent with a given magnetic field. We develop the theory of inversion using expansions in Hermite polynomial functions of the canonical momenta. Sufficient conditions for the convergence of a Hermite expansion are found, given a pressure tensor. For large classes of DFs, we prove that nonnegativity of the distribution function is contingent on the magnetisation of the plasma, and make conjectures for all classes.The inverse problem is considered for nonlinear 'force-free Harris sheets'. By applying the Hermite method, we construct new models that can describe sub-unity values of the plasma beta (β pl ) for the first time. Whilst analytical convergence is proven for all β pl , numerical convergence is attained for β pl = 0.85, and then β pl = 0.05 after a 're-gauging' process.We consider the properties that a pressure tensor must satisfy to be consistent with 'asymmetric Harris sheets', and construct new examples. It is possible to analytically solve the inverse problem in some cases, but others must be tackled numerically. We present new exact Vlasov-Maxwell equilibria for asymmetric current sheets, which can be written as a sum of shifted Maxwellian distributions. This is ideal for implementations in particle-in-cell simulations.We study the correspondence between the microscopic and macroscopic descriptions of equilibrium in cylindrical geometry, and then attempt to find Vlasov-Maxwell equilibria for the nonlinear force-free 'Gold-Hoyle' model. However, it is necessary to include a background field, which can be arbitrarily weak if desired. The equilibrium can be electrically non-neutral, depending on the bulk flows.

Collisionless distribution functions for force-free current sheets: using a pressure transformation to lower the plasma beta

“…In the cases considered by Channell (1976) and Attico & Pegoraro (1999), the plasma beta is constrained to be at least one, due to the summative nature of the pressure function (of the form P zz = P 1 (A x ) + P 2 (A y )). This is straightforward to show and, as discussed by Allanson (2017b), it can be shown that, for certain types of force-free magnetic fields, the plasma beta is constrained to be at least one whenever a summative pressure function is assumed (see Appendix B for further details). In the cases considered by Sestero (1967); Bobrova et al (2001Bobrova et al ( , 2003, the plasma beta can be smaller than one, and so the exponential transformation has allowed for a reduction in the plasma beta, in a similar way to the nonlinear force-free Harris sheet case considered by Allanson et al (2015Allanson et al ( , 2016a.…”

“…then the plasma beta is constrained to be at least one. This has been considered in a more general sense by Allanson (2017b); in this appendix, we will summarise the discussion given therein. Firstly, for summative pressure functions of the form in Equation (B 1), the force balance equation for force-free fields, given by…”