Practical Gyrokinetics

The gyrokinetic change of variables was introduced 40 years ago (1) to deal with complicated magnetic field geometries such as those found in tokamaks and stellarators. Unlike drift kinetic variables, gyrokinetic variables retain the distinction between the guiding center and particle location. The use of nonlinear gyrokinetic codes began in the 1980s in slab geometry, and by 2000 turbulent transport was being evaluated electromagnetically across tokamak flux surfaces for specified profiles. At present, it is difficult to think of any turbulent magnetic fusion simulation that is not gyrokinetic. This tutorial will introduce the basic orderings and techniques used to derive the simplest form of the gyrokinetic Fokker-Planck equation for the distribution function when the turbulence is electrostatic. In typical magnetic fusion geometries such as tokamaks and stellarators, the presence of surfaces of constant pressure allows the lowest order background distribution functions to be Maxwellian. As a result, only its correction (delta f) need be evaluated on each flux surface to determine turbulent heat and particle fluxes. The nonlinear gyrokinetic equation solved in simulations is normally derived for axisymmetric tokamaks with the adiabatic term or Maxwell-Boltzmann response removed. This single flux surface form used for most turbulent simulations has important symmetry properties that must be honored to avoid introducing momentum transport in up-down symmetric tokamaks. More advanced gyrokinetic treatments will be briefly mentioned by indicating the procedure for obtaining gyrokinetic variables to higher order when required, for example, in the pedestal, and for turbulent momentum transport and profile evolution studies, and at cyclotron frequencies (for unperturbed trajectories).

(1) P. J. Catto, Plasma Phys. 20, 719 (1978)