Abstract
The calculation of the velocity moment in the gyrokinetic theory and its application in the study of zonal flows in a tokamak plasma are investigated. Based on the scalar invariance property, the expression for the particle velocity in the gyrocenter coordinates is obtained, and a method to systematically calculate the velocity moment in the gyrokinetic theory is proposed, especially for the calculation of the first-order perpendicular velocity moment. The kinetic equation which describes the evolution of the perturbed distribution on the meso-scale of the zonal flow is derived. The effects of the turbulent particle flux, the turbulent energy flux, the turbulent toroidal Reynolds stress and the turbulent poloidal Reynolds stress (PRS) are explicitly included in the phase-space particle flux. In the cylindrical geometry, the zonal radial electric field is driven by the turbulent PRS and the turbulent energy flux, and the result agrees with the radial force balance equation. In the toroidal geometry, the zonal radial electric field is driven by the turbulent PRS, the turbulent energy flux and the turbulent toroidal Reynolds stress; in contrast to the case in the cylindrical geometry, the effect of the turbulent PRS is shielded by the toroidal effect. By combining the radial force balance equation, the poloidal momentum equation in the toroidal geometry is obtained.
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