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Physics of Plasmas / Volume 19 / Issue 1 / ARTICLES / Basic Plasma Phenomena, Waves, Instabilities
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Phys. Plasmas 19, 012103 (2012); http://dx.doi.org/10.1063/1.3662115 (8 pages)

Adiabatic nonlinear waves with trapped particles. II. Wave dispersion

I. Y. Dodin and N. J. Fisch

Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey 08544, USA

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(Received 15 July 2011; accepted 27 October 2011; published online 6 January 2012)

A general nonlinear dispersion relation is derived in a nondifferential form for an adiabatic sinusoidal Langmuir wave in collisionless plasma, allowing for an arbitrary distribution of trapped electrons. The linear dielectric function is generalized, and the nonlinear kinetic frequency shift ωNL is found analytically as a function of the wave amplitude a. Smooth distributions yield ωNL∝math, as usual. However, beam-like distributions of trapped electrons result in different power laws, or even a logarithmic nonlinearity, which are derived as asymptotic limits of the same dispersion relation. Such beams are formed whenever the phase velocity changes, because the trapped distribution is in autoresonance and thus evolves differently from the passing distribution. Hence, even adiabatic ωNL(a) is generally nonlocal.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. GENERAL DISPERSION RELATION
    1. Wave Lagrangian
    2. Sinusoidal-wave approximation
    3. Weight function G
    4. Small-amplitude cold-plasma limit
  3. SMOOTH DISTRIBUTIONS
    1. Asymptotic representation
    2. Nonlinear frequency shift
    3. Comparison with existing theories
  4. BEAM-LIKE DISTRIBUTIONS
  5. DISCUSSION
  6. CONCLUSIONS

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KEYWORDS and PACS

Keywords

plasma dielectric properties, plasma kinetic theory, plasma Langmuir waves, plasma nonlinear processes

PACS

  • 52.35.Mw

    Nonlinear phenomena: waves, wave propagation, and other interactions (including parametric effects, mode coupling, ponderomotive effects, etc.)

  • 52.25.Dg

    Plasma kinetic equations

  • 52.25.Mq

    Dielectric properties

  • 52.35.Fp

    Electrostatic waves and oscillations (e.g., ion-acoustic waves)

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3662115

PUBLICATION DATA

ISSN

1070-664X (print)  
1089-7674 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

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Figures (click on thumbnails to view enlargements)

FIG.1
(Color online) Schematic of single-particle trajectories in phase space (x,  p), illustrating the definition of 2πJ (shaded area): (a) for a passing particle and (b) for a trapped particle. For J to be continuous at the separatrix (dashed line), with the passing-particle action defined as 2πJ = ∮pdx, for trapped particles, one must use the definition 2πJ = (1/2)∮pdx. This area is encircled by a particle within half of the bounce period; thus, the corresponding canonical frequency Ω equals twice the bounce frequency.

FIG.1 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.2
Weight function G(j). Dashed lines are approximate solutions given by Eq. ( 19 ), the asymptote, and j = j*.

FIG.2 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.3
(a) Function Ψ(j), Eq. ( 22 ). (b) Function Q(j), Eq. ( 26 ). The dashed lines show j = j* and also the asymptote of Q(j). For details, see the Appendix.

FIG.3 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.4
(a) Normalized action j(r), Eq. ( A1 ). (b) Weight function g(r), Eq. ( A6 ). The dashed lines denote the separatrix r = 1 and also the asymptote of g(r).

FIG.4 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.5
(a) Function ψ(r), Eq. ( A8 ). (b) Function q(r), Eq. ( A9 ). The dashed lines denote the separatrix r = 1 and also the asymptote of q(r).

FIG.5 Download High Resolution Image (.zip file) | Export Figure to PowerPoint



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