Calculation of the gyroviscosity using a gyrokinetic formalism

Gyrokinetic variables for parallel velocity, magnetic moment, gyroangle and gyrocenter are obtained for arbitrary perpendicular wavelength electrostatic fluctuations to second order in the expansion for small Larmor radius of the background magnetic field by demanding the gyrokinetic equation contain no gyrophase dependent terms order by order. The resulting nonlinear gyrokinetic equation is valid for arbitrary collisionalities and perpendicular wavelengths for electrostatic turbulence. The distribution function in these gyrokinetic variables when Taylor expanded for large perpendicular wavelengths recovers the correct gyroviscosity and Reynold's stress tensor found by drift kinetic [1] and moment [2] approaches. The full gyrokinetic generalization of the gyroviscosity and Reynold's stress tensor is expected to be required to evaluate the coupling of the axisymmetric portion of the radial electric field associated with zonal flow and neoclassical toroidal angular momentum transport. \newline \newline [1] A. N. Simakov and P. J. Catto, Phys. Plasmas 12, 012105 (2005). \newline [2] J. J. Ramos, Phys. Plasmas 12, 052102 (2005).