Asymmetric kinetic equilibria: Generalization of the BAS model for rotating magnetic profile and non-zero electric field

Published in Physics of Plasmas DOI : http://dx.doi.org/10.1063/1.4930210

Nicolas Dorville, Gérard Belmont, Nicolas Aunai, Jérémy Dargent and Laurence Rezeau

Finding kinetic equilibria for non-collisional/collisionless tangential current layers is a key issue as well for their theoretical modeling as for our understanding of the processes that disturb them, such as tearing or Kelvin Helmholtz instabilities. The famous Harris equilibrium [E. Harris, Il Nuovo Cimento Ser. 10 23, 115–121 (1962)] assumes drifting Maxwellian distributions for ions and electrons, with constant temperatures and flow velocities; these assumptions lead to symmetric layers surrounded by vacuum. This strongly particular kind of layer is not suited for the general case: asymmetric boundaries between two media with different plasmas and different magnetic fields. The standard method for constructing more general kinetic equilibria consists in using Jeans theorem, which says that any function depending only on the Hamiltonian constants of motion is a solution to the steady Vlasov equation [P. J. Channell, Phys. Fluids (1958–1988) 19, 1541 (1976); M. Roth et al., Space Sci. Rev. 76, 251–317 (1996); and F. Mottez, Phys. Plasmas 10, 1541–1545 (2003)]. The inverse implication is however not true: when using the motion invariants as variables instead of the velocity components, the general stationary particle distributions keep on depending explicitly of the position, in addition to the implicit dependence introduced by these invariants. The standard approach therefore strongly restricts the class of solutions to the problem and probably does not select the most physically reasonable. The BAS (Belmont-Aunai-Smets) model [G. Belmont et al., Phys. Plasmas 19, 022108 (2012)] used for the first time the concept of particle accessibility to find new solutions: considering the case of a coplanar-antiparallel magnetic field configuration without electric field, asymmetric solutions could be found while the standard method can only lead to symmetric ones. These solutions were validated in a hybrid simulation [N. Aunai et al., Phys. Plasmas (1994-present) 20, 110702 (2013)], and more recently in a fully kinetic simulation as well [J. Dargent and N. Aunai, Phys. Plasmas (submitted)]. Nevertheless, in most asymmetric layers like the terrestrial magnetopause, one would indeed expect a magnetic field rotation from one direction to another without going through zero [J. Berchem and C. T. Russell, J. Geophys. Res. 87, 8139–8148 (1982)], and a non-zero normal electric field. In this paper, we propose the corresponding generalization: in the model presented, the profiles can be freely imposed for the magnetic field rotation (although restricted to a 180 rotation hitherto) and for the normal electric field. As it was done previously, the equilibrium is tested with a hybrid simulation.

Kinetic equilibrium for an asymmetric tangential layer

(ADS Link, DOI)

Belmont, G., Aunai, N., and Smets, R.

Finding kinetic (Vlasov) equilibria for tangential current layers is a long standing problem, especially in the context of reconnection studies, when the magnetic field reverses. Its solution is of pivotal interest for both theoretical and technical reasons when such layers must be used for initializing kinetic simulations. The famous Harris equilibrium is known to be limited to symmetric layers surrounded by vacuum, with constant ion and electron flow velocities, and with current variation purely dependent on density variation. It is clearly not suited for the “magnetopause-like” layers, which separate two plasmas of different densities and temperatures, and for which the localization of the current density j=nδv is due to the localization of the electron-to-ion velocity difference δv and not of the density n. We present here a practical method for constructing a Vlasov stationary solution in the asymmetric case, extending the standard theoretical methods based on the particle motion invariants. We show that, in the case investigated of a coplanar reversal of the magnetic field without electrostatic field, the distribution function must necessarily be a multi-valued function of the invariants to get asymmetric profiles for the plasma parameters together with a symmetric current profile. We show also how the concept of “accessibility” makes these multi-valued functions possible, due to the particle excursion inside the layer being limited by the Larmor radius. In the presented method, the current profile across the layer is chosen as an input, while the ion density and temperature profiles in between the two asymptotic imposed values are a result of the calculation. It is shown that, assuming the distribution is continuous along the layer normal, these profiles have always a more complex profile than the profile of the current density and extends on a larger thickness. The different components of the pressure tensor are also outputs of the calculation and some conclusions concerning the symmetries of this tensor are pointed out.