Generalized curvature modified plasma dispersion functions and Dupree renormalization of toroidal ITG

TL;DR

This paper introduces a generalized class of plasma dispersion functions to express Dupree renormalized dispersion relations, enabling improved modeling of toroidal ITG modes in plasma physics.

Contribution

It develops a new mathematical framework using generalized curvature modified plasma dispersion functions for Dupree renormalization in gyrokinetic ITG mode analysis.

Findings

The new functions allow expressing Dupree's dispersion relations entirely.

An iterative scheme for calculating the Dupree diffusion coefficient converges rapidly.

The method's practical usefulness depends on application-specific wave-number spectrum assumptions.

Abstract

A new generalization of curvature modified plasma dispersion functions is introduced in order to express Dupree renormalized dispersion relations used in quasi-linear theory. For instance the Dupree renormalized dispersion relation for gyrokinetic, toroidal ion temperature gradient driven (ITG) modes, where the Dupree's diffusion coefficient is assumed to be a low order polynomial of the velocity, can be written entirely using generalized curvature modified plasma dispersion functions: Knm's. Using those, Dupree's formulation of renormalized quasi-linear theory is revisited for the toroidal ITG mode. The Dupree diffusion coefficient has been obtained as a function of velocity using an iteration scheme, first by assuming that the diffusion coefficient is constant at each v (i.e. applicable for slow dependence), and then substituting the resulting v dependence in the form of complex polynomial coefficients into the Knm's for verification. The algorithm generally converges rapidly after only a few iterations. Since the quasi-linear calculation relies on an assumed form for the wave-number spectrum, especially around its peak, practical usefullness of the method is to be determined in actual applications.