The Eulerian variational formulation of the gyrokinetic system in general spatial coordinates

H. Sugama,S. Satake,M. Nunami,S. Matsuoka

doi:10.1063/5.0027905

H. Sugama, S. Satake + Show 2 more

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https://doi.org/10.1063/5.0027905

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Abstract

The Eulerian variational formulation of the gyrokinetic system with electrostatic turbulence is presented in general spatial coordinates by extending our previous work [H. Sugama et al., Phys. Plasmas 25, 102506 (2018)]. The invariance of the Lagrangian of the system under an arbitrary spatial coordinate transformation is used to derive the local momentum balance equation satisfied by the gyrocenter distribution functions and the turbulent potential, which are given as solutions of the governing equations. In the symmetric background magnetic field, the derived local momentum balance equation gives rise to the local momentum conservation law in the direction of symmetry. This derivation is in contrast to the conventional method using the spatial translation in which the asymmetric canonical pressure tensor generally enters the momentum balance equation. In the present study, the variation of the Lagrangian density with respect to the metric tensor is taken to directly obtain the symmetric pressure tensor, which includes the effect of turbulence on the momentum transport. In addition, it is shown in this work how the momentum balance is modified when the collision and/or external source terms are added to the gyrokinetic equation. The results obtained here are considered useful for global gyrokinetic simulations investigating both neoclassical and turbulent transport processes even in general non-axisymmetric toroidal systems.

Highlights

The governing equations of the gyrokinetic system with electrostatic turbulence are derived in the general spatial coordinates based on the Eulerian variational principle
It is shown that, when the background magnetic field satisfies the consistency condition that its rotation is given by the solenoidal part of the current density as in the Darwin model, the momentum and energy balance equations for the whole system are rewritten in the complete conservative forms where contributions of the turbulent electric field and the background magnetic field are clearly given in the expressions of the momentum and energy densities, the pressure tensor, and the energy flux
The effects of the collision and/or external source terms added into the gyrokinetic equation on the momentum and energy balance equations are clarified as well

Summary

INTRODUCTION

Gyrokinetics[1,2,3,4,5,6,7] has been used for several decades as a basic theoretical framework to study microinstabilities and turbulent processes in magnetized plasmas.[8]. ) ∼ ρ/L with L representing the equilibrium gradient scale length is assumed This second-order term is retained because it is necessary for deriving the gyrokinetic Poisson’s equation correctly including the polarization effect as shown in Sec. III.C. other second-order terms shown in Ref.[50] are neglected in the gyrocenter Hamiltonian given by Eqs.

EULERIAN VARIATIONAL PRINCIPLE FOR

The Lagrangian density represented in general spatial coordinates

Derivation of the momentum balance for a single particle species

Derivation of the momentum balance for the whole system

CONCLUSIONS