Long-wavelength limit of gyrokinetics in a turbulent tokamak and its intrinsic ambipolarity

TL;DR

This paper derives the long-wavelength limit of gyrokinetic equations in tokamaks, revealing that the turbulent system inherently maintains ambipolarity, meaning the radial electric field remains undetermined.

Contribution

It provides a detailed derivation of the long-wavelength gyrokinetic equations and proves the intrinsic ambipolarity in turbulent tokamak plasmas at second order.

Findings

Long-wavelength components include neoclassical and turbulent effects.

The system leaves the radial electric field undetermined.

Turbulent tokamaks are intrinsically ambipolar.

Abstract

Recently, the electrostatic gyrokinetic Hamiltonian and change of coordinates have been computed to order $\epsilon^2$ in general magnetic geometry. Here $\epsilon$ is the gyrokinetic expansion parameter, the gyroradius over the macroscopic scale length. Starting from these results, the long-wavelength limit of the gyrokinetic Fokker-Planck and quasineutrality equations is taken for tokamak geometry. Employing the set of equations derived in the present article, it is possible to calculate the long-wavelength components of the distribution functions and of the poloidal electric field to order $\epsilon^2$. These higher-order pieces contain both neoclassical and turbulent contributions, and constitute one of the necessary ingredients (the other is given by the short-wavelength components up to second order) that will eventually enter a complete model for the radial transport of toroidal angular momentum in a tokamak in the low flow ordering. Finally, we provide an explicit and detailed proof that the system consisting of second-order gyrokinetic Fokker-Planck and quasineutrality equations leaves the long-wavelength radial electric field undetermined; that is, the turbulent tokamak is intrinsically ambipolar.