Amplitude Dispersion and Nonlinear Instability of Whistlers

The propagation of small but finite amplitude plane “whistlers” in a cold plasma imbedded in a uniform magnetic field is studied. The usual frequency dispersion relation of linear theory is extended to include the effect of amplitude dispersion. The present analysis is based on a perturbation procedure in which the frequency, wavenumber, and amplitude relation is represented by a power series in the energy density or the square of the amplitude of the wave, i.e., $${\mathrm{\omega \hspace{0.17em}=\hspace{0.17em}\omega }}_0\left(k\right){\mathrm{\hspace{0.17em}+\hspace{0.17em}\omega }}_2\left(k\right){\mathrm{\hspace{0.17em}a}}^2{\mathrm{\hspace{0.17em}+\hspace{0.17em}\omega }}_4\left(k\right){\mathrm{\hspace{0.17em}a}}^4\mathrm{\hspace{0.17em}+\hspace{0.17em}\dots }$$ . It is found that the effect of amplitude dispersion is to cause the “whistlers” to undergo nonlinear instability if the wave vector ${k}$ is not parallel to the direction of the undisturbed magnetic field. For plane whistlers which propagate along the mean magnetic field lines it is noted that the linear solution is also an exact solution of the two component cold plasma equations.