Exact quadratic longitudinal response of a Maxwellian plasma and its application to the rate of electrostatic decay

Brett Layden
University of Sydney

John Percival
Defence Science and Technology Organisation

Iver Cairns
University of Sydney

Peter Robinson
University of Sydney

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     Last modified: July 25, 2011

Abstract
Langmuir waves driven by electron beams are commonly observed in both space and laboratory plasmas. Once generated, these Langmuir waves may undergo a variety of mode conversion processes, in which wave energy is transferred between wave modes. One such process is electrostatic decay, in which a Langmuir wave decays into a product Langmuir wave and an ion-sound wave; this is an important mechanism in the generation of electromagnetic radiation at multiples of the plasma frequency, known as plasma emission. Until now the rate of electrostatic decay has been calculated using an approximate expression for the quadratic longitudinal response, which describes the coupling between three electrostatic waves. We derive an exact expression for the quadratic longitudinal response in terms of generalized plasma dispersion functions. This expression reduces to the approximate result in the appropriate limit. We develop numerical expressions for the generalized plasma dispersion functions, correcting various aspects of previous calculations and positioning us to calculate the rate of electrostatic decay.