Phase-space Lagrangian derivation of electrostatic gyrokinetics in general geometry

TL;DR

This paper rigorously derives the electrostatic gyrokinetic phase-space Lagrangian in general magnetic geometry to second order in the small parameter, providing explicit formulas for improved simulation accuracy.

Contribution

It presents a new, rigorous derivation of the second-order gyrokinetic Hamiltonian and Lagrangian in general geometry, linking magnetic geometry and turbulence.

Findings

Derived explicit second-order gyrokinetic Hamiltonian.

Provided a phase-space Lagrangian suitable for simulations.

Showed the interdependence of magnetic geometry and turbulence.

Abstract

Gyrokinetic theory is based on an asymptotic expansion in the small parameter $\epsilon$, defined as the ratio of the gyroradius and the characteristic length of variation of the magnetic field. In this article, this ordering is strictly implemented to compute the electrostatic gyrokinetic phase-space Lagrangian in general magnetic geometry to order $\epsilon^2$. In particular, a new expression for the complete second-order gyrokinetic Hamiltonian is provided, showing that in a rigorous treatment of gyrokinetic theory magnetic geometry and turbulence cannot be dealt with independently. The new phase-space gyrokinetic Lagrangian gives a Vlasov equation accurate to order $\epsilon^2$ and a Poisson equation accurate to order $\epsilon$. The final expressions are explicit and can be implemented into any simulation without further computations.