Solution to a collisionless shallow-angle magnetic presheath with kinetic ions

Using a kinetic model for the ions and adiabatic electrons, we solve a steady state, electronrepelling magnetic presheath in which a uniform magnetic field makes a small angle α ≫ 1 (in radians) with the wall. The presheath characteristic thickness is the typical ion gyroradius p_i. The Debye length lD and the collisional mean free path of an ion λ_mfp satisfy the ordering λ_D = p_i = α λ_mfp, so a quasineutral and collisionless model is used. We assume that the electrostatic potential is a function only of distance from the wall, and it varies over the scale p_i. Using the expansion in α = 1, we derive an analytical expression for the ion density that only depends on the ion distribution function at the entrance of the magnetic presheath and the electrostatic potential profile. Importantly, we have added the crucial contribution of the orbits in the region near the wall. By imposing the quasineutrality equation, we derive a condition that the ion distribution function must satisfy at the magnetic presheath entrance-the kinetic equivalent of the Chodura condition. Using an ion distribution function at the entrance of the magnetic presheath that satisfies the kinetic Chodura condition, we find numerical solutions for the selfconsistent electrostatic potential, ion density and flow across the magnetic presheath for several values of α. Our numerical results also include the distribution of ion velocities at the Debye sheath entrance. We find that at small values of α there are substantially fewer ions traveling with a large normal component of the velocity into the wall.