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Dependence on ion temperature of shallow-angle magnetic presheaths with adiabatic electrons

The magnetic presheath is a boundary layer occurring when magnetized plasma is in contact with a wall and the angle $α$ between the wall and the magnetic field $\vec{B}$ is oblique. Here, we consider the fusion-relevant case of a shallow-angle, $α\ll 1$, electron-repelling sheath, with the electron density given by a Boltzmann distribution, valid for $α/ \sqrt{τ+1} \gg \sqrt{m_{\text{e}}/m_{\text{i}}}$, where $m_{\text{e}}$ is the electron mass, $m_{\text{i}}$ is the ion mass, $τ= T_{\text{i}}/ZT_{\text{e}}$, $T_{\text{e}}$ is the electron temperature, $T_{\text{i}}$ is the ion temperature, and $Z$ is the ionic charge state. The thickness of the magnetic presheath is of the order of a few ion sound Larmor radii $ρ_{\text{s}} = \sqrt{m_{\text{i}} \left(ZT_{\text{e}} + T_{\text{i}} \right) } / ZeB$, where $e$ is the proton charge and $B = |\vec{B}|$ is the magnitude of the magnetic field. We study the dependence on $τ$ of the electrostatic potential and ion distribution function in the magnetic presheath by using a set of prescribed ion distribution functions at the magnetic presheath entrance, parameterized by $τ$. The kinetic model is shown to be asymptotically equivalent to Chodura's fluid model at small ion temperature, $τ\ll 1$, for $|\ln α| > 3|\ln τ| \gg 1$. In this limit, despite the fact that fluid equations give a reasonable approximation to the potential, ion gyro-orbits acquire a spatial extent that occupies a large portion of the magnetic presheath. At large ion temperature, $τ\gg 1$, relevant because $T_{\text{i}}$ is measured to be a few times larger than $T_{\text{e}}$ near divertor targets of fusion devices, ions reach the Debye sheath entrance (and subsequently the wall) at a shallow angle whose size is given by $\sqrtα$ or $1/\sqrtτ$, depending on which is largest.