Classification of sideband instability regimes for whistler waves with trapped electrons

The instability of a finite amplitude whistler mode wave propagating in a collisionless plasma along an external magnetic field is analyzed within the framework of a simple model wherein all trapped electrons are assumed to be concentrated at the bottom of the potential energy troughs. The dispersion equation describing the instability is solved in combination with a nonlinear dispersion relation of the primary equilibrium wave. Such a self-consistent approach allows one to obtain analytical expressions for the growth rates in all limiting cases corresponding to different regimes of the instability. It is shown that scaling of the growth rates relative to the amplitude of the equilibrium wave and trapped electron number density is similar to analogous scaling for electrostatic plasma waves. However, the whistler sideband instability exhibits new specific features due to the difference between the dispersion properties of the whistler modes and Langmuir waves. It is established that the phase velocity of the exponentially growing large scale modulation of the primary wave in the wave frame of reference vanishes on the condition that the frequency of the equilibrium wave equals one half of the electron gyrofrequency. As a result, the instability takes on the character of an aperiodic growth of the modulation. The influence of the trapped electron oscillations in the field of the equilibrium whistler wave on the development of electrostatic instabilities is also discussed.