Sheath constraints on turbulent magnetised plasmas

Abstract A solid target in contact with a plasma charges (negatively) to reflect the more mobile species (electrons) and thus keep the bulk plasma quasineutral. To shield the bulk plasma from the charged target, there is an oppositely (positively) charged sheath with a sharp electrostatic potential variation on the Debye length scale <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">D</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> . In magnetised plasmas where the magnetic field is inclined at an oblique angle α with the target, some of the sheath potential variation occurs also on the ion sound gyroradius length scale <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">S</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mi>cos</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:math> , caused by finite ion gyro-orbit distortion and losses. We consider a collisionless and steady-state magnetised plasma sheath whose thickness <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>l</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>ms</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo>∼</mml:mo> <mml:mo movablelimits="true">max</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">D</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">S</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mi>cos</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> is smaller than the characteristic length scale L of spatial fluctuations in the bulk plasma, such that the limit <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>l</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>ms</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>L</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> is appropriate. Spatial structures are assumed to be magnetic field-aligned. In the case of small magnetic field angle <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∼</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>≡</mml:mo> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">S</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>L</mml:mi> <mml:mo>≪</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , electric fields tangential to the target transport ions towards the target via E × B drifts at a rate comparable to the one from parallel streaming. A generalised form of the kinetic Bohm–Chodura criterion at the sheath entrance is derived by requiring that the sheath electric field have a monotonic spatial decay far from the target. The criterion depends on tangential gradients of potential and ion distribution function, with additional nontrivial conditions.