Practical gyrokinetics

Abstract

Thousands of gyrokinetic papers have been published since the introduction of the gyrokinetic change of variables. The intent here is not to review the field, but rather the goal is to present a tutorial providing insight into why gyrokinetics is an appropriate description of turbulent transport. The focus is on turbulent transport in axisymmetric tokamaks, but many of the ideas and techniques are applicable to stellarators and other magnetic fusion devices with nested surfaces of constant pressure. Besides the origins of gyrokinetics and recent insights, gyrokinetic orderings and gyrokinetic variables are summarized. Then a compact, but careful, derivation of the simplest electrostatic gyrokinetic equation for tokamaks is presented, along with a brief mention of gyrokinetics for stellarators. The advantages of assuming scale separation between the finer spatial scales and faster time variation of the turbulence and the global behaviour and slow temporal evolution of the background are stressed. Moreover, the procedure for removing the adiabatic or Maxwell–Boltzmann response is emphasized. Scale separation allows the near Maxwellian behaviour of the background and the rapid variation of the turbulence to be described by separate, local gyrokinetic equations. The turbulent fluctuations are found by solving the nonlinear gyrokinetic equation for the non-adiabatic portion of the fluctuating distribution function. Also, generalizations of the gyrokinetic equation so that it retains electromagnetic and flow modifications are considered in detail along with some symmetry properties. In addition, ambipolarity, particle transport and heat transport are addressed, along with a discussion of the complications associated with describing momentum transport and profile evolution.