The development of the nonlinear theory of gyroresonant quasi-monochromatic wave–particle interactions in an inhomogeneous plasma was stimulated by experiments on the injection of quasi-monochromatic VLF radio waves into the magnetosphere, which led to the discovery of a broad class of triggered VLF emissions. We shall discuss this phenomenon in Chapter 11. The aim of this chapter is to give the foundations of the theory, putting off the applications to later chapters.

The case of cyclotron resonant wave–particle interactions in an inhomogeneous plasma is much more complicated than the case for a homogeneous plasma considered in the previous chapter. It is connected with the mismatch of the cyclotron resonance condition, which occurs due to the change of the electron gyrofrequency along the inhomogeneous magnetic field, and to the change of the magnitude of the wave vector. Helliwell (1967) made a fundamental contribution to the mechanism generating whistler-mode waves in an inhomogeneous magnetic field when he suggested the idea of second-order cyclotron resonance (CR). Then the CR mismatch is compensated by the change with time of the frequency of a generated wave packet. The first papers on a quantitative consideration of the nonlinear CR interaction of electrons with a quasi-monochromatic whistler wave in an inhomogeneous magnetic field were published between 1971 and 1974 (Dysthe, 1971, Nunn, 1971, 1973, 1974, Sudan and Ott, 1971, Helliwell and Crystal, 1973, and Brinca 1973).