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

TL;DR

This paper develops an analytical kinetic model for the magnetic presheath with a small angle to the wall, deriving conditions and solutions for ion density, potential, and velocity distributions in a collisionless, quasineutral regime.

Contribution

It introduces a kinetic model incorporating ion orbits near the wall and derives an analytical expression for ion density, extending the understanding of magnetic presheath physics at small angles.

Findings

Derived an analytical ion density expression depending on entrance distribution and potential.

Established a kinetic Chodura condition for ion distribution at the presheath entrance.

Numerical solutions show fewer ions with large normal velocity at small angles.

Abstract

Using a kinetic model for the ions and adiabatic electrons, we solve a steady state, electron-repelling magnetic presheath in which a uniform magnetic field makes a small angle $\alpha \ll 1$ (in radians) with the wall. The presheath characteristic thickness is the typical ion gyroradius $\rho_{\text{i}}$. The Debye length $\lambda_{\text{D}}$ and the collisional mean free path of an ion $\lambda_{\text{mfp}}$ satisfy the ordering $\lambda_{\text{D}} \ll \rho_{\text{i}} \ll \alpha \lambda_{\text{mfp}}$, 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 $\rho_{\text{i}}$. Using the expansion in $\alpha \ll 1$, 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 self-consistent electrostatic potential, ion density and flow across the magnetic presheath for several values of $\alpha$. Our numerical results also include the distribution of ion velocities at the Debye sheath entrance. We find that at small values of $\alpha$ there are substantially fewer ions travelling with a large normal component of the velocity into the wall.