Theory of coherent parametric instabilities excited by two or more pump waves

A method based on energy balance is used to obtain the thresholds of parametric instabilities of the ‘purely growing’ and ‘arithmetic mean’ varieties which can exist in a plasma irradiated by two or more electromagnetic pump waves. These instabilities produce growing standing electron plasma waves at frequencies satisfying either ω = ω i , where ω i is the frequency of the i th pump wave, or ω = (ω i +ω j )/2, where i and j correspond to two different pump waves. Additionally growing ion standing waves of zero frequency or of frequency (ω i −ω j )/2 are produced. These coherent instabilities are found to be coupled in a sense with the usual ‘decay’ instability and, in the case of three or more pump waves, with each other. The results of this investigation are applied to a recent ionospheric heating experiment described in a companion paper.