Interaction of a low frequency coherent magneto-acoustic wave with a turbulent spectrum of extra-ordinary waves

Starting from nonlinear evolution equations of a three-dimensional extra-ordinary wave-packet in a magnetized plasma and introducing a correlation function, equations are derived that describe interaction of a coherent low-frequency magneto-acoustic wave with a turbulent spectrum of high frequency extra-ordinary waves. These equations are used to derive the dispersion relation for a low frequency magneto-acoustic wave in the presence of a spectrum of turbulent high frequency extra-ordinary waves. In the narrow spectrum case, it is found that a small but finite spread in the spectrum of turbulence has a stabilizing influence on the instability which exists for vanishing spectral width. In the wide spectrum case, the effect of a weak turbulent spectrum of extra-ordinary waves is to produce a slight shift in frequency of the coherent low frequency magneto-acoustic wave and to damp or excite the wave.