Kinetic equilibrium for an asymmetric tangential layer
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.