Tightening energetic bounds on linear gyrokinetic instabilities

TL;DR

This paper develops tighter energetic bounds on linear gyrokinetic instability growth, especially for the slab ion-temperature-gradient mode, improving predictions of the most unstable eigenmode behavior.

Contribution

It introduces a new approach to derive energetic upper bounds specifically for linear instability growth, including constrained optimal modes that better capture real frequency effects.

Findings

Tightest bounds given by energy of eigenmode projection coefficients

Constrained optimal modes closely match linear growth rates

Bounds now incorporate effects of real frequencies in instabilities

Abstract

Bounding energetic growth of gyrokinetic instabilities is a complementary approach to linear instability analyses involving normal eigenmodes. Previous work has focused on upper bounds which are valid linearly and nonlinearly. However, if an upper bound on linear instability growth is desired, these nonlinearly valid bounds may be a poor predictor of the growth of the most unstable eigenmode. This is most evident for the simplest of instabilities: the ion-temperature-gradient (ITG) mode in slab geometry. In this work, we derive energetic upper bounds specifically for linear instability growth, focusing on the slab ITG. We show that there is no fundamental limitation on how tightly linear growth can be bounded by an energetic norm, with the tightest possible bound being given by a special energy comprised of projection coefficients of the linear eigenmode basis. Additionally, we consider `constrained optimal modes' that maximise energy growth subject to constraints that are also obeyed by the linear eigenmodes. This yields computationally efficient upper bounds that closely resemble the linear growth rate, capturing effects connected to the real frequency of instabilities, which have been absent in the energetic bounds considered thus far.