The theory of quasi-linear diffusion of runaway electrons in the field of electromagnetic waves (HF field) excited by external sources is given. The frequency of the waves is somewhat less than the electron cyclotron frequency. The time-averaged distribution function of resonant electrons is derived. The presence of the toroidal electric field changes the process of quasi-linear relaxation considerably. It also leads to a continuous absorption of energy and momentum from the HF field. On the other hand, the HF field increases the number of high-energy electrons. As a consequence, beam instabilities can arise. A comparison with experimental results is given.

A statistical description of the Vlasov equation is made possible by truncation of phase space in the velocity co-ordinates and writing the equation in terms of Fourier components in configuration and velocity space. Invariants of the resulting nonlinear turbulence equations are discussed. Expectation values and in particular an electric field spectrum of the form (β+ ακ2)-1 are derived. α and β are constants; α is always positive; β may be negative, depending on the initial conditions of the plasma. The spectrum is in reasonable agreement with available experiments and simulations.