Restoring momentum conservation to magnetised quasilinear diffusion

Wave interactions with magnetised particles underlie many plasma heating and current drive technologies. Typically, these interactions are modelled by bounce averaging the quasilinear Kennel–Engelmann diffusion tensor over the particle orbit. However, as an object derived in a two-dimensional space, the Kennel–Engelmann tensor does not fully respect the conservation of four-momentum required by the action conservation theorem, since it neglects the absorption of perpendicular momentum. This defect leads to incorrect predictions for the wave-induced cross-field particle transport. Here, we show how this defect can easily be fixed, by extending the tensor from two to four dimensions and matching the form required by four-momentum conservation. The resulting extended tensor, when bounce averaged, recovers the form of the diffusion paths required by action-angle Hamiltonian theory. Importantly, the extended tensor should be easily implementable in Fokker–Planck codes through a mild modification of the existing Kennel–Engelmann tensor.