Linear response of the two-dimensional pure electron plasma: Quasimodes for some model profiles

Abstract

After examining the initial value problem for the linear, diocotron response of a long cylinder of pure-electron plasma, the "quasimodes" associated with convex, power-law density profiles are studied. For these profiles, exact, analytic results are available. The "quasimodes," which are damped by phase mixing, may be characterized by their angular variation, hamess, and the magnitude of the gap separating the plasma from the containing wall. 0 I995 American institute of Physics. I. lNTRODUCTlONThis paper is concerned with diocotron waves in a nonneutral plasma. In a recent experiment,' Pillai and Gould excited the m = 2, diocotron mode of a column of pureelectron plasma, and observed its subsequent, damped "ringing." When the amplitude of the short, exciting burst is relatively small, one expects a linear analysis of the response to be fruitful. (deGrassie and Malmberg, in an earlier study of diocotron waves,' emphasized the nonlinear regime.) Linear "ringing" is usually associated with a normal mode of oscillation of a system, or with a discrete eigenvalue. or a point in the point spectrum of some operator describing the evolution of the system. But it has been known for some time3 that such modes do not exist in the collisionless dynamical model most commonly used to describe the pure-electron plasma, under the conditions believed to prevail in these experiments. There is, however, a continuous spectrum associated with the system, and it brings what appears to be a paradox: the evolution of a system governed by a continuous spectrum is generally nonexponential, but the experimenter sees exponential behavior. Two responses are available. One can simply remark that the integrand in the integration over the continuous range of frequencies is peaked sharply at the observed frequency. Or, one can ask why?-is the peak the "shadow" of an interesting singularity sitting nearby?-a "quasimode?" And might the quasimode be viewed as a signature, or used for diagnostics?Of course, the situation is not new. It resembles that encountered in the study of Landau damping. To understand it fully one must-in the language of the theory-examine a multivalued Green's function which has been continued analytically onto a sheet adjacent to the "physical sheet." This problem was discussed generally and thoroughly, two decades ago, by Briggs, Daugherty, and Levy3 (henceforth BDL). This paper supplements BDL in showing that such an analysis may be carried out analytically for all angular modes-not merely m = 2, or the peculiar m = 1 -in plasmas characterized by convex, power-law profiles. Then, the quantities of interest may be expressed in terms of hypergeometric functions and the continuation is straightforward. One finds the "quasimode," in an appropriate place. One obtains formulas for the dependence of frequency and damping of the mode upon profile, mode number, and gap size. To make our presentation coherent, we shall have to develop material that is presented either explicitly or implicitly in BDL. We hope that the repetition wil...

Linear and nonlinear decay of cat's eyes in two-dimensional vortices, and the link to Landau poles

“…The quasi-modes of a circular vortex have been studied by taking the Laplace transform of the initial value problem for linear, inviscid disturbances to the vortex profile (Briggs et al 1970;Corngold 1995;Schecter et al 2000). Using this method, the frequency −mα qm and the decay rate γ qm appear as the real and imaginary parts of the complex growth rate −ip qm from a Landau pole at p qm in the notation below.…”

“…Solving (4.4) by means of the appropriate Green's function (Briggs et al 1970;Corngold 1995), the Laplace transform of Q m can be written…”

Neutral-plasma oscillations at zero temperature

“…Because this is a driven system, we first look for normal modes. Notice, however, that with E 0 = 0 and γ = 0 (undamped free response) this mode equation has a continuous spectrum, similar to the diocotron mode equation in non-neutral plasmas [14,15,16,17,18,19]. And, as with the diocotron mode, this equation also has a damped quasimode.…”

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“…The quasimode is a solution of the linearized Euler equations that is not separable in time, and decays on a much longer time-scale than the turn-over time-scale of the underlying vortex. Quasimodes of an axisymmetric circular vortex have been studied analytically via Laplace transforms of the linearized Euler equation; 9,10,13 an attraction of this method is that the angular velocity and the decay rate of the disturbance stream function can be deduced from the real and imaginary parts of a simple pole of the governing system. The location of this pole, known as a Landau pole, can be calculated by analytic continuation techniques 9 and it follows that the decay rate and angular velocity of the quasimode depend only on the form of the axisymmetric base profile of the vortex.…”