An analytical form of the dispersion function for local linear gyrokinetics in a curved magnetic field

TL;DR

This paper derives an analytical, closed-form solution for the velocity-space integrals in local linear gyrokinetics with curved magnetic fields, simplifying analysis and computation of plasma dispersion relations.

Contribution

It presents the first known analytical solution for these integrals in curved magnetic fields, connecting them to the plasma dispersion function for easier analysis.

Findings

Results converge to known solutions in special limits

Good agreement with existing numerical solvers

Exact dispersion relation for ion-temperature-gradient instability

Abstract

Starting from the equations of collisionless linear gyrokinetics for magnetised plasmas with an imposed inhomogeneous magnetic field, we present the first known analytical, closed-form solution for the resulting velocity-space integrals in the presence of resonances due to both parallel streaming and constant magnetic drifts. These integrals are written in terms of the well-known plasma dispersion function (Faddeeva & Terentev 1954; Fried & Conte 1961), rendering the subsequent expressions simpler to treat analytically and more efficient to compute numerically. We demonstrate that our results converge to the well-known ones in the straight-magnetic-field and two-dimensional limits, and show good agreement with the numerical solver by G\"urcan (2014). By way of example, we calculate the exact dispersion relation for a simple electrostatic, ion-temperature-gradient-driven instability, and compare it with approximate kinetic and fluid models.