A Boltzmann-kinetical description of an MHD shock with arbitrary field inclination

Argelander Institut für Astronomie der Universität Bonn, Abteilung f. Astrophysik und Extraterrestrische Forschung, Auf dem Huegel 71, 53121 Bonn, Germany e-mail: msiewert@astro.uni-bonn.de

Received: 24 October 2007
Accepted: 27 March 2008

Abstract

Aims. We revisit the general problem of the anisotropic MHD shock for arbitrary magnetic field inclinations, where the jump conditions are underdetermined. To describe the transition region of the shock, we derive a variant of a kinetic Boltzmann-Vlasov equation previously used to describe the perpendicular shock in the absence of dissipative processes.

Methods. We derive effective force terms, for the kinetic equation, that are based on the conservation of the Chew-Goldberger-Low (CGL) MHD invariants which appear in the standard model for anisotropic MHD. This approach is based on a generalisation of the well-known equivalence between the first CGL invariant and the integral over the magnetic moments of the underlying particles.

Results. Assuming an arbitrary distribution function on the upstream side, we integrate the kinetic equation across the shock. This result allows us to establish further relations between the MHD velocity moments on both sides. Using this additional information, we close the anisotropic MHD jump conditions. In addition, the now unique solution of the jump conditions allows us to present explicit cuts through a representative Maxwellian distribution function on both sides of the shock. In the kinetic equation, one only requires two parameters that need to be derived from the classical jump conditions, the classical MHD compression ratio and an equivalent ratio for the magnetic field strengths.