Two-dimensional Vlasov solution for a collisionless plasma jet across transverse magnetic field lines with a sheared bulk velocity

We consider a two-dimensional (2D) stationary stream of a collisionless plasma injected across an external stationary magnetic field and a background stagnant plasma. The solution is found by solving the Vlasov equation for each species (electrons and protons), the Maxwell-Ampere equation for the magnetic vector potential, and the equation of plasma quasineutrality for the electrostatic potential. The solution of the stationary Vlasov equation is given in terms of two constants of motion and one adiabatic invariant. The partial charge and current densities are given by analytical expressions of the moments of the velocity distribution functions for each particle species. The 2D distribution of the plasma bulk velocity, V-x(y,z), is roughly uniform inside the jet. There is no plasma bulk flow in the direction of the magnetic field. Inside the boundary layer interfacing the jet and the stagnant plasma, the bulk velocity has gradients (i.e., shears) in the direction parallel as well perpendicular to the magnetic field. The parallel component of this gradient, del V-parallel to(perpendicular to), produces a nonzero electric field component parallel to the magnetic field lines, E center dot B not equal 0. The parallel electric field within the transition layer is a basic element allowing plasma elements to be transported across magnetic field lines in astrophysical systems as well as in laboratory experiments where plasmoids are injected across magnetic fields.