A Ritz variational principle for local collisionless gyrokinetic instabilities

TL;DR

This paper develops a mathematically correct Ritz variational principle for local collisionless gyrokinetic instabilities, enabling more accurate modeling of turbulence in magnetic fusion devices.

Contribution

It derives a correct variational principle for local collisionless gyrokinetics and provides a weak form of the equations for numerical use.

Findings

Corrected the mathematical formulation of the variational principle.

Derived a weak form suitable for numerical implementation.

Facilitates improved turbulence modeling in fusion research.

Abstract

Turbulence driven by gyrokinetic instabilities is largely responsible for transport in magnetic fusion devices. To estimate this turbulent transport, integrated modeling codes often use mixing length estimates in conjunction with reduced models of the linearized gyrokinetic equation. One common method of formulating and solving the linearized gyrokinetic eigenvalue problem equation uses a Ritz variational principle, particularly in the local collisionless limit. However, the variational principle as typically stated in the literature is mathematically incorrect. In this work, we derive a mathematically correct form of the variational principle that applies to local linear collisionless gyrokinetics in general geometry with electromagnetic effects. We also explicitly derive a weak form of the gyrokinetic field equations suitable for numerical applications.