Quasilinear theory is developed by using canonical variables for a relativistic plasma. It is self-consistent, including momentum, pitch angle, and spatial diffusions. By assuming the wave field as a superposition of known toroidal and poloidal Fourier modes, the quasilinear diffusion coefficients are written in a form which can be directly evaluated using the output of a spectral full-wave solver of Maxwell equations in toroidal plasmas. The formalism is special for tokamas and, therefore, simple and suitable for simulations of cyclotron heating, current drive, and radio-frequency wave-induced radial transport in ITER.

Interaction of radio-frequency (RF) wave with plasma in magnetic confinement devices has been a very important discipline of plasma physics. To approach more realistic description of wave-plasma interaction in a time scale longer than the kinetic time scales bounce-average is needed. The long-time evolution of the kinetic distribution can be treated with Fokker-Planck equation. The behavior of the plasma and the most interesting macroscopic effects are obtained by balancing the diffusion term with a collision term.

For the relativistic particles the action and angle variables initiated by Kaufman [

The rest of the paper is organized as follows. In next section the exact guiding center variables are derived with Hamiltonian transformation. The center variables are derived with Hamiltonian transformation. The bounce-averaged quasi-linear equation is carried out in Section

In tokamak configuration, the relativistic Hamiltonian of a charged particle can be expressed as

The magnetic field can be expressed as

We introduce a generating function [

The Jacobian in the area-conserved transformation is unity [

If the Larmor radius is smaller than the scale length of the system, a small parameter

For the gyrokinetics the Hamiltonian in (

We form a new invariant [

For trapped particles in a large aspect ratio configuration, that is,

For the circulating particles,

Lamalle [

In the extended phase space the Hamiltonian is written as follows [

According to Liouville’s theorem, the distribution function,

The linear solution of (

For one harmonic from (

For the relativistic circulating particles the resonant term in (

From (

The action and angle variables initiated by Kaufman are used [

The authors would like to give special thanks to Dr. A. Cardinali for his report and his code left here during his visit to SWIP. Z. T. Wang would like to thank Dr. R. D. Hazeltine and Dr. P. Morrision for checking the canonical transformation procedure and many helpful discussions when he worked at the Institute for Fusion Studies University of Texas at Austin. This research is supported by the National Natural Science Foundations of China under Grant nos. 10475043, 10535020, 10375019, and 10135020.