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

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.

Polarization and magnetization in collisional and turbulent transport processes

“…( 1). In the Eulerian picture (or the Euler-Poincaré formulation), [17][18][19][36][37][38][39] we use the expression in the last line of Eq. ( 67) and consider the variations of f a and Ż as functions of (Z,t) to derive the gyrokinetic Vlasov equation from δ I = 0.…”

“…Effects of the collision term, if included, on the local energy and momentum balance equations can be clarified following the same procedure as shown in Refs. 17,18 .…”

“…[40][41][42][43][44] Based on the presented Lagrangian, local energy and momentum balance equations for the gyrokinetic system with electromagnetic turbulence and collisions can be derived following the same formulation as given by our previous work in the case of electrostatic turbulence. 18 Details of the derivation will be reported in a future work. and…”

“…10,11 Conservation properties possessed by such gyrokinetic equations are suitable for global and long-time transport simulations and they have been extensively investigated based on Lagrangian and Hamiltonian formulations. [11][12][13][14][15][16][17][18][19][20][21][22] It is well-known that the finite gyroradius representing the distance between particle and gyrocenter positions generates so-called polarization and magnetization, 11,12 in terms of which the relations of the density and mean velocity of particles to those of gyrocenters are expressed. These relations are important for using gyrokinetic simulation results to correctly evaluate particle transport, as well as to accurately calculate the charge density and the electric current in Poisson and Ampère equations, which are required to self-consistently determine electromagnetic fields in the simulation.…”

Metriplectic foundations of gyrokinetic Vlasov-Maxwell-Landau theory

“…Despite several attempts at constructing collision operators for gyrokinetic applications [30][31][32][33][34][35] , no fullf gyrokinetic Vlasov-Maxwell-Landau field theory has yet been presented that would conserve energy, produce entropy monotonically, and conserve the toroidal angular momentum functional in an axially symmetric background magnetic field B 0 . Only the full-f electrostatic theory has been properly covered 36,37 , and the succesa) Electronic mail: eero.hirvijoki@aalto.fi full δf -formulations [38][39][40][41] rely on various model linearized collision operators. Exploiting the general framework of metriplectic dynamics [42][43][44][45][46][47][48][49] , this letter proposes a theory for the electromagnetic full-f case, securing the as-ofyet-missing closed form expression for the gyrokinetic Landau collision operator.…”