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A quasi-local inhomogeneous dielectric tensor for arbitrary distribution functions

S. J. Frank, J. C. Wright, P. T. Bonoli

TL;DR

The paper addresses how parallel inhomogeneity in plasmas modifies wave propagation and damping by deriving a quasilocal inhomogeneous correction to the hot-plasma dielectric tensor for arbitrary distribution functions. The authors generalize previous Maxwellian-limited formulations by expanding the phase-correlation integral and implementing a novel contour integration in velocity space, enabling accurate evaluation for arbitrary distributions and resonances $n$, with the plasma dispersion function $\mathrm{Z}$ and weights $w_n$. They apply the correction to lower-hybrid current drive (LHCD) in Alcator C-Mod, showing that inhomogeneous damping does not alter the linear damping condition for Maxwellian electrons, while non-Maxwellian Landau plateau distributions introduce small, high-$\zeta$ effects near the plateau edge. The work provides a practical tool to estimate inhomogeneity effects in tokamaks and open-field devices, while acknowledging that a full energy-conserving, global treatment would require bounce-averaged approaches that are computationally intensive.

Abstract

Treatments of plasma waves usually assume homogeneity, but the parallel gradients ubiquitous in plasmas can modify wave propagation and absorption. We derive a quasilocal inhomogeneous correction to the plasma dielectric for arbitrary distributions by expanding the phase correlation integral and develop a novel integration technique that allows our correction to be applied in many situations and has greater accuracy than other inhomogeneous dielectric formulas found in the literature. We apply this dielectric tensor to the lower-hybrid current drive problem and demonstrate that inhomogeneous wave damping does not affect the lower-hybrid wave's linear damping condition, and in the non-Maxwellian problem damping and propagation should remain unchanged except in the case of waves with very large phase velocities.

A quasi-local inhomogeneous dielectric tensor for arbitrary distribution functions

TL;DR

The paper addresses how parallel inhomogeneity in plasmas modifies wave propagation and damping by deriving a quasilocal inhomogeneous correction to the hot-plasma dielectric tensor for arbitrary distribution functions. The authors generalize previous Maxwellian-limited formulations by expanding the phase-correlation integral and implementing a novel contour integration in velocity space, enabling accurate evaluation for arbitrary distributions and resonances , with the plasma dispersion function and weights . They apply the correction to lower-hybrid current drive (LHCD) in Alcator C-Mod, showing that inhomogeneous damping does not alter the linear damping condition for Maxwellian electrons, while non-Maxwellian Landau plateau distributions introduce small, high- effects near the plateau edge. The work provides a practical tool to estimate inhomogeneity effects in tokamaks and open-field devices, while acknowledging that a full energy-conserving, global treatment would require bounce-averaged approaches that are computationally intensive.

Abstract

Treatments of plasma waves usually assume homogeneity, but the parallel gradients ubiquitous in plasmas can modify wave propagation and absorption. We derive a quasilocal inhomogeneous correction to the plasma dielectric for arbitrary distributions by expanding the phase correlation integral and develop a novel integration technique that allows our correction to be applied in many situations and has greater accuracy than other inhomogeneous dielectric formulas found in the literature. We apply this dielectric tensor to the lower-hybrid current drive problem and demonstrate that inhomogeneous wave damping does not affect the lower-hybrid wave's linear damping condition, and in the non-Maxwellian problem damping and propagation should remain unchanged except in the case of waves with very large phase velocities.
Paper Structure (4 sections, 7 equations, 3 figures)

This paper contains 4 sections, 7 equations, 3 figures.

Table of Contents

  1. Introduction & Background
  2. Generalizing The Quasilocal Formulation
  3. Application To LHCD
  4. Conclusions

Figures (3)

  • Figure 1: Example $\textrm{w}_0$ and $\textrm{w}_1$ integration paths, plotted in blue and red respectively, over contours of the $Z$-Function and its phase in the complex plane. The initial paths are plotted with dots and the shifted paths with crosses. Initially both $\textrm{w}_0$ and $\textrm{w}_1$ pass into the oscillatory lower midplane quadrant making numerical integration by most techniques impossible, but after being shifted to the modified $p_{\parallel}$ contour (\ref{['eq:vcont']}) neither enter the region removing oscillation in the integrand.
  • Figure 2: (a) Plots of $\chi_{zz}$ versus $\zeta = \omega/k_\parallel v_{the}$, (b) the normalized damping parameter $\Gamma_e$ versus $\zeta$ comparing the homogeneous solution with the inhomogeneous solution for $dk_\parallel/dl > 0$ and $dk_\parallel/dl < 0$, and (c) ratio of the Berry dielectric Berry2016 and our dielectric for $dk_\parallel/dl > 0$ and $dk_\parallel/dl < 0$.
  • Figure 3: Plots comparing the non-Maxwellian homogeneous and inhomogeneous $\chi_{zz}$ versus $\zeta$ with a Landau plateau distribution for $dk_\parallel/dl > 0$ and $dk_\parallel/dl < 0$. (a) The real part of $\chi_{zz}$ with a Landau plateau distribution versus $\zeta$, and (b) the imaginary part of $\chi_{zz}$ for a Landau plateau distribution versus $\zeta$.