Parametric instabilities of Alfvén waves in a dusty plasma

The excitation of low frequency modes of oscillations in a magnetized dusty plasma with parametric pumping of the magnetic field is analyzed. The dust is taken to be stationary, but acts as a sink for negative charge: The absence of a proportion of the electrons in the overall neutral plasma modifies the dispersion properties of the pump and excited waves. The pump is a magnetoacoustic wave modified by the dust, and the excited waves are modifications of Alfvén waves due to the dust and due to ion cyclotron effects. The equation of motion governing the perturbed plasma is derived and pump wave fields are found. The stability of the pump to modes propagating parallel to the magnetic field is investigated as a function of the plasma parameters. It is found that pairs of slow and fast parallel propagating modes are parametrically excited. The growth rates of the various interactions are calculated, and it is shown that the growth rates of slow–fast and fast–fast mode interactions can be maximized by varying the proportion of negative charge on the dust.