Stable and unstable roots of ion temperature gradient driven mode using curvature modified plasma dispersion functions

TL;DR

This paper reexamines the local kinetic theory of the ion temperature gradient driven mode using curvature-modified plasma dispersion functions, providing a new method to identify roots and analyze stability near the threshold.

Contribution

It introduces a bracketing technique with modified plasma dispersion functions to accurately find and analyze roots of the ITG dispersion relation, including unstable and damped solutions.

Findings

Unstable branch is asymmetric near the instability threshold.

Oscillating solutions dominate at lower wave numbers, damping at higher.

Inverse cascade process may influence nonlinear evolution of ITG.

Abstract

Basic, local kinetic theory of ion temperature gradient driven (ITG) mode, with adiabatic electrons is reconsidered. Standard unstable, purely oscillating as well as damped solutions of the local dispersion relation are obtained using a bracketing technique that uses the argument principle. This method requires computing the plasma dielectric function and its derivatives, which are implemented here using modified plasma dispersion functions with curvature and their derivatives, and allows bracketing/following the zeros of the plasma dielectric function which corresponds to different roots of the ITG dispersion relation. We provide an open source implementation of the derivatives of modified plasma dispersion functions with curvature, which are used in this formulation. Studying the local ITG dispersion, we find that near the threshold of instability the unstable branch is rather asymmetric with oscillating solutions towards lower wave numbers (i.e. drift waves), and damped solutions toward higher wave numbers. This suggests a process akin to inverse cascade by coupling to the oscillating branch towards lower wave numbers may play a role in the nonlinear evolution of the ITG, near the instability threshold. Also, using the algorithm, the linear wave diffusion is estimated for the marginally stable ITG mode.