Plasma losses through an adiabatic cusp

Previous calculations of β = 1 cusp losses are extended to the case where there exists a third ‘adiabatic’ invariant of the particle motion, related to the magnetic moment, and additional to the particle energy and angular momentum. Simple analytic results are obtained for the particle and energy loss rate from a Maxwellian plasma in both plane and axisymmetric geometry, electric fields being assumed zero everywhere. The results are valid for a plasma radius large relative to the ion gyro-radius, in which case the adiabaticity in the sense used reduces the loss rate by a factor ⅔λ⅔ to give an overall scaling as λ-⅔ with external mirror ratio λ. The absolute value is about a factor two less than the previously accepted result (for λ = 1 only) for an axisymmetric cusp. cusp. The enhanced scaling with A arises from the absence of a certain class of particles in the sheath, which must be everywhere adiabatic; its applicability to a simple theta pinch is queried, however. The end loss rate of angular momentum is also obtained, and is non-zero at a point cusp. This is shown to lead to rapid rotational instability in a collisionless theta pinch.