Stability of Longitudinal Electrostatic Oscillations in Plasmas of Finite Dimensions

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Abstract
This paper discusses the stability of longitudinal electrostatic oscillations in plasmas of finite dimensions. It is shown that in general the effect of simple boundary conditions is to increase the domain of stability. Thus, a cylindrical plasma of radius a, consisting of electron and ion streams of equal temperatures and surrounded by a coaxial conducting boundary of radius b, where b >> a, is stable for all stream velocities when λD > 0.53 a(In b/a)1/2, where λD is the Debye length. In the case of a plane-shaped plasma of semi-width a, between parallel plane-conducting boundaries at distance b, the equivalent condition is λD > [2(ab-a)2]1/2. More generally, for multi-stream plasmas, in which the component streams have arbitrary temperatures and densities, it is possible to give a sufficient condition of stability for all stream velocities. The stability of three-stream plasmas is briefly discussed, and it is shown that the experimental results of Looney and Brown, and their failure to observe any excitation of plasma oscillations, are in accord with the stability criteria for plasmas of finite dimensions. Attention is drawn to the similarity between plane shaped plasmas and auroral arcs, and also the similarity between cylindrical shaped plasmas and auroral rays. It is suggested that the difference of the stability conditions for plane and cylindrical shaped plasmas may account for the tendency of auroral arcs to disrupt into rayed structures.

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