Fully electromagnetic nonlinear gyrokinetic equations for tokamak edge turbulence

Abstract

An energy conserving set of the fully electromagnetic nonlinear gyrokinetic Vlasov equation and Maxwell's equations, which is applicable to both L-mode turbulence with large amplitude and H-mode turbulence in the presence of high E × B shear has been derived.The phase-space action variational Lie perturbation method ensures the preservation of the conservation laws of the underlying Vlasov-Maxwell system. Our generalized ordering takesis the thermal ion Larmor radius and ρ θi = B B θ ρ i ), as typically observed in the tokamak H-mode edge, with L E and L p being the radial electric field and pressure gradient lengths. We take k ⊥ ρ i ∼ 1 for generality, and keep the relative fluctuation amplitudes eδφ/T i ∼ δB/B up to the second order. Extending the electrostatic theory in the presence of high E × B shear [Hahm, Phys. Plasmas 3, 4658 (1996)], contributions of electromagnetic fluctuations to the particle charge density and current are explicitly evaluated via pull-back transformation from the gyrocenter distribution function in the gyrokinetic Maxwell's equation.

Generalized expression for polarization density

“…Therefore, we neglect the other components of the generator. This is very similar to the case in gyrokinetics 14 .…”

“…where the effective gyrocenter perturbed potential is The classical polarization 10,14 originates from the pull-back transformation from gy to gc, and then to particle, phase space.…”

“…15 Some detailed theoretical discussions of the classical polarization have been also presented. 10,14 Furthermore, if the characteristic frequency ω is even much smaller than the bounce (transit) frequency ω b(t) , there exists another adiabatic invariant J b(t) , corresponding to the bounce motion of trapped particles (denoted by index b) or transit motion for passing particles (denoted by index t) in a nonuniform magnetic field. Based on the existence of this second adiabatic invariant, bounce-kinetics was derived by using a conventional multiplescale-ordering expansion.…”

“…1 After that, modern nonlinear gyrokinetics based on a Lie-transform perturbation method, previously used by Littlejohn 2-4 to derive guiding-center (gc) equa-tions, was developed and has been improved in recent years. [5][6][7][8][9][10][11][12][13][14] The classical polarization density plays an important role in gyrokinetic simulations. 15 Some detailed theoretical discussions of the classical polarization have been also presented.…”

Fluid Moments in the Reduced Model for Plasmas with Large Flow Velocity

“…Hence, extension of the standard model is needed to treat the formation of transport barriers. Several gyrokinetic Lagrangians have been presented to treat this situation [4][5][6][7][8][9][10]. Here we give an account of the underlying representations of the particle density in each of these models.…”