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Restoring momentum conservation to magnetized quasilinear diffusion

I. E. Ochs

Abstract

Wave interactions with magnetized particles underly many plasma heating and current drive technologies. Typically, these interactions are modeled 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.

Restoring momentum conservation to magnetized quasilinear diffusion

Abstract

Wave interactions with magnetized particles underly many plasma heating and current drive technologies. Typically, these interactions are modeled 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.

Paper Structure

This paper contains 6 sections, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Generic form of local diffusion tensor for a resonant wave-particle interaction
  3. Restoring conservation to the Kennel-Engelmann diffusion tensor
  4. Constants-of-motion space diffusion
  5. Discussion
  6. Description of simulations

Figures (2)

  • Figure 1: Simulation of ion interaction with a second-harmonic left-hand-polarized wave over many orbits. The particle diffuses in both normalized energy $\bar{K}$ and gyrocenter position $(\bar{X},\bar{Y})$ [points, with time corresponding to color], but stays on the line determined by Eq. (\ref{['eq:DiffusionPathDeltaFormulation']}) [solid line]. The normalized variables and details of the simulation can be found in Appendix \ref{['sec:SimDescription']}.
  • Figure 2: Amplitude of normalized electric field and axial magnetic field in the simulation. The resonant interaction occurs around $z_m = 0$; the measurements of gyrocenter and energy are taken arund $z_m = L/2$.