POP Nameplate
  • Volume/Page
  • Keyword
  • DOI
  • Citation
  • Advanced
   
 
 
 

Flickr Twitter UniPHY Group iResearch App Facebook

Physics of Plasmas / Volume 19 / Issue 1 / ARTICLES / Basic Plasma Phenomena, Waves, Instabilities
Author Select

FULL-TEXT OPTIONS:

Phys. Plasmas 19, 012104 (2012); http://dx.doi.org/10.1063/1.3673065 (9 pages)

Adiabatic nonlinear waves with trapped particles. III. Wave dynamics

I. Y. Dodin and N. J. Fisch

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

View MapView Map

(Received 15 July 2011; accepted 8 December 2011; published online 6 January 2012)

The evolution of adiabatic waves with autoresonant trapped particles is described within the Lagrangian model developed in Paper I, under the assumption that the action distribution of these particles is conserved, and, in particular, that their number within each wavelength is a fixed independent parameter of the problem. One-dimensional nonlinear Langmuir waves with deeply trapped electrons are addressed as a paradigmatic example. For a stationary wave, tunneling into overcritical plasma is explained from the standpoint of the action conservation theorem. For a nonstationary wave, qualitatively different regimes are realized depending on the initial parameter S, which is the ratio of the energy flux carried by trapped particles to that carried by passing particles. At S < 1/2, a wave is stable and exhibits group velocity splitting. At S > 1/2, the trapped-particle modulational instability (TPMI) develops, in contrast with the existing theories of the TPMI yet in agreement with the general sideband instability theory. Remarkably, these effects are not captured by the nonlinear Schrödinger equation, which is traditionally considered as a universal model of wave self-action but misses the trapped-particle oscillation-center inertia.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. BASIC MODEL
  3. WAVE EQUATIONS
  4. WAVE TRANSFORMATIONS IN PLASMA WITH VARYING PARAMETERS
  5. PULSE PROPAGATION
    1. Group velocity splitting
    2. TPMI
  6. DISCUSSION
    1. Stability criterion
    2. Comparison with the existing theories
  7. CONCLUSIONS

EDITORIALLY RELATED

    Related Articles

  1. Adiabatic nonlinear waves with trapped particles. I. General formalism
    I. Y. Dodin et al.
    Phys. Plasmas 19, 012102 (2012)PHPAEN000019000001012102000001
  2. Adiabatic nonlinear waves with trapped particles. II. Wave dispersion
    I. Y. Dodin et al.
    Phys. Plasmas 19, 012103 (2012)PHPAEN000019000001012103000001

RELATED DATABASES

Web of Science

View this record in Web of Science

View Web of Science related articles

KEYWORDS and PACS

Keywords

modulational instability, plasma Langmuir waves, plasma nonlinear waves, plasma oscillations, plasma transport processes, Schrodinger equation

PACS

  • 52.35.Mw

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

  • 52.35.Fp

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

  • 52.35.Py

    Macroinstabilities (hydromagnetic, e.g., kink, fire-hose, mirror, ballooning, tearing, trapped-particle, flute, Rayleigh-Taylor, etc.)

  • 52.25.Fi

    Transport properties

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3673065

PUBLICATION DATA

ISSN

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

Publisher

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

  1. G. B. Whitham, J. Fluid Mech. 22, 273 (1965). [ISI]
  2. G. B. Whitham , Linear and Nonlinear Waves (Wiley, New York, 1974), Chaps. 14 and 15.
  3. I. Y. Dodin and N. J. Fisch, Phys. Plasmas 19, 012102 (2012).
  4. I. Y. Dodin and N. J. Fisch, Phys. Plasmas 19, 012103 (2012).
  5. I. Y. Dodin and N. J. Fisch, Phys. Rev. Lett. 107, 035005 (2011).
  6. V. I. Karpman , Non-Linear Waves in Dispersive Media (Pergamon, New York, 1974), Chap. V.
  7. The indexes a, k, omega, A, and [fraktur I] (except in nu<sub>[fraktur I]</sub>) denote differentiation with respect to the corresponding variables.
  8. From the standpoint of time-averaged dynamics, particle detrapping is an irreversible process, which thus cannot be described by a Lagrangian in principle. Of course, the plasma true Lagrangian still exists; however, its wave part cannot be represented in the Whitham's form, L(a,omega,k) .
  9. Of course, an alternative is the kinetic approach, which yet leads to solutions that are not easily tractable and are generally specific to particular settings. See Ref. 23 and V. L. Krasovskii , Zh. Eksp. Teor. Fiz. 95, 1951 (1989) [Sov. Phys. JETP 68, 1129 (1989)]. Also, see references therein and more recent Ref. 27 and A. I. Matveev , Fiz. Plazmy 34, 1001 (2008) [Plasma Phys. Rep. 34, 924 (2008)]; A. I. Matveev , Russ. Phys. J. 52, 885 (2009); A. I. Matveev , Fiz. Plazmy 35, 351 (2009) [Plasma Phys. Rep. 35, 315 (2009)]; A. I. Matveev , Russ. Phys. J. 53, 369 (2010); A. I. Matveev , Russ. Phys. J. 53, 657 (2010).
  10. V. L. Krasovsky, Phys. Lett. A 163, 199 (1992).
  11. V. L. Krasovsky, Fiz. Plazmy 18, 739 (1992), [Sov. J. Plasma Phys. 18, 382 (1992)].
  12. V. L. Krasovsky, J. Plasma Phys. 47, 235 (1992).
  13. V. L. Krasovsky, Phys. Scr. 49, 489 (1994). [Inspec]
  14. R. L. Dewar, W. L. Kruer, and W. M. Manheimer, Phys. Rev. Lett. 28, 215 (1972). [ISI]
  15. H. Ikezi, K. Schwarzenegger, and A. L. Simons, Phys. Fluids 21, 239 (1978)PFLDAS000021000002000239000001.
  16. H. A. Rose, Phys. Plasmas 12, 012318 (2005)PHPAEN000012000001012318000001.
  17. H. A. Rose and L. Yin, Phys. Plasmas 15, 042311 (2008)PHPAEN000015000004042311000001.
  18. V. I. Karpman and E. M. Krushkal, Zh. Eksp. Teor. Fiz. 55, 530 (1968) [Inspec]
    [Sov. Phys. JETP 28, 277 (1969)]. [ISI]
  19. W. L. Kruer, J. M. Dawson, and R. N. Sudan, Phys. Rev. Lett. 23, 838 (1969). [ISI]
  20. M. V. Goldman, Phys. Fluids 13, 1281 (1970)PFLDAS000013000005001281000001.
  21. M. V. Goldman and H. L. Berk, Phys. Fluids 14, 801 (1971)PFLDAS000014000004000801000001.
  22. V. L. Krasovsky, Plasma Phys. Controlled Fusion 51, 115011 (2009).
  23. V. L. Krasovskii, Zh. Eksp. Teor. Fiz. 107, 741 (1995), [JETP 80, 420 (1995)].
  24. H. A. Rose and D. A. Russell, Phys. Plasmas 8, 4784 (2001)PHPAEN000008000011004784000001.
  25. B. J. Winjum, J. Fahlen, and W. B. Mori, Phys. Plasmas 14, 102104 (2007)PHPAEN000014000010102104000001.
  26. I. Y. Dodin, V. I. Geyko, and N. J. Fisch, Phys. Plasmas 16, 112101 (2009)PHPAEN000016000011112101000001.
  27. A. I. Matveev, Fiz. Plazmy 34, 114 (2008), [Plasma Phys. Rep. 34, 95 (2008)].
  28. Notice, in particular, that Eq. (18) can be written as L=[integral]<sub>0</sub><sup>a</sup>La da+n(t)mu2/2. Yet, this form holds only for a delta-shaped trapped-particle distribution; for other distributions, there is a minimum a below which L is undefined.
  29. P. F. Schmit, I. Y. Dodin, and N. J. Fisch, Phys. Rev. Lett. 105, 175003 (2010).
  30. I. Y. Dodin and N. J. Fisch, Phys. Rev. D 82, 044044 (2010).
  31. T. H. Stix , Waves in Plasmas (AIP, New York, 1992), Sec. 4.2.
  32. From S ~ 1, one gets another estimate, [cursive phi] ~ (nu/kappa3)1/2. Thus, kappa*~[cursive phi]<sub>0</sub><sup>-1</sup><>(nu/kappa0)1/2 and [cursive phi]*~[cursive phi]<sub>0</sub><sup>3/2</sup>kappa<sub>0</sub><sup>3/4</sup>nu-1/4 .
  33. M. J. Lighthill, J. Inst. Math. Appl. 1, 1 (1965).
  34. C. D. Decker and W. B. Mori, Phys. Rev. Lett. 72, 490 (1994). [MEDLINE]
  35. C. D. Decker and W. B. Mori, Phys. Rev. E 51, 1364 (1995). [MEDLINE]
  36. C. B. Schroeder, C. Benedetti, E. Esarey, and W. P. Leemans, Phys. Rev. Lett. 106, 135002 (2011).
  37. Considered here is nondissipative dynamics only. For dissipative effects, see D. Bénisti, O. Morice, L. Gremillet, E. Siminos, and D. J. Strozzi, Phys. Plasmas 17, 082301 (2010)PHPAEN000017000008082301000001;, D. Bénisti, D. J. Strozzi, L. Gremillet, and O. Morice, Phys. Rev. Lett. 103, 155002 (2009); [MEDLINE]
    D. Bénisti, and L. Gremillet, Phys. Plasmas 14, 042304 (2007)PHPAEN000014000004042304000001. See also our Ref. 3 for comparison.
  38. L. D. Landau and E. M. Lifshitz , Fluid Dynamics (Pergamon, New York, 1987), Secs. 103 and 104.
  39. S. Brunner and E. J. Valeo, Phys. Rev. Lett. 93, 145003 (2004). [ISI] [MEDLINE]
  40. To our knowledge, the results of Ref. 13 have never been associated with the TPMI before.
  41. P. E. Masson-Laborde, W. Rozmus, Z. Peng, D. Pesme, S. Hüller, M. Casanova,V. Yu. Bychenkov, T. Chapman, and P. Loiseau, Phys. Plasmas 17, 092704 (2010)PHPAEN000017000009092704000001.
  42. G. B. Whitham, Proc. R. Soc., London Ser. A 283, 238 (1965).
  43. M. J. Lighthill, J. Inst. Math. Appl. 1, 269 (1965).
  44. M. J. Lighthill, Proc. R. Soc., London Ser. A 299, 28 (1967).
  45. W. D. Hayes, Proc. R. Soc., London Ser. A. 332, 199 (1973).
  46. V. E. Zakharov and L. A. Ostrovsky, Physica D 238, 540 (2009). [Inspec]
  47. E. E. Kunhardt and B. R. -S. R.-S. Cheo, Phys. Rev. Lett. 37, 1688 (1976).

Figures (click on thumbnails to view enlargements)

FIG.1
1D stationary wave with trapped particles in inhomogeneous plasma. Shown is the normalized amplitude of the wave potential, φ/φc, vs. the plasma normalized density n/nc = Ωp2 for different ν. Here φc = 2Θ, with Θ = 0.1 taken as an example; nc is the critical density. Dashed is the linear solution (ν = 0) and the location of the linear cutoff (n = nc).

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

FIG.2
(Color online) Schematic of the parameter domain assumed for Sec. 5 in space (κ, ϑ, ΩE) and (b) in space (κ, ϑ, S). Combined here are the following assumptions: the plasma is cold [Eq. ( 10 )], the wave is sinusoidal [Eq. ( 14 )] and weak enough [Eq. ( A7 )], the quasistatic field due to trapped particles is negligible [Eq. ( A11 )]; also, the bulk motion and the nonlinear effects are weak, i.e., V0≪Δνg≪νg0 [Eq. ( 30 ); see also Eqs. ( 8 , 28 )]. The inequalities Eqs. ( 9 , A12 ) reduce to V0≪νg0 and thus are satisfied automatically.

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

FIG.3
(a) Nonlinear group velocities, νg+ and νg- [Eq. ( 31 )] vs. κ, for sample ΩE and ϑ; the dashed line shows the linear group velocity νg0. At S > 1/2, corresponding to κ<κS ≡ ΩE-1math, no real solutions exist for νg, rendering the wave unstable. (b) Close-up at κ ≈ κS.

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

FIG.4
(Color online) Evolution of the perturbation Δ(a02) = -0.03a02<>exp(-x4/ℓ4) to a homogeneous wave with the initial amplitude a0. The solution is obtained by numerical integration of Eqs. ( B1 , B3 ), with math taken from Eq. ( 18 ), for the same parameters as in Fig. 3 and ℓ = 20λD. Shown is Δ(a2) (arbitrary color scaling), vs. t and x in the frame moving with the linear group velocity νg0; the units are ωp-1 and λD, correspondingly. (a) κ = 0.2, so S = 1/4; the wave is stable, resulting in signal splitting. (b) κ = 0.1, so S = 1; the wave is TPMI-unstable.

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



Close

Google Calendar

ADVERTISEMENT

Your Recent History Recent History Help

Recently Viewed

  1. Adiabatic nonlinear waves with trapped particles. III. Wave dynamics
  2. Adiabatic nonlinear waves with trapped particles. II. Wave dispersion
  3. Adiabatic nonlinear waves with trapped particles. I. General formalism
  4. Poynting vector, energy densities, and pressure of collective transverse electromagnetic fluctuations in unmagnetized plasmas
  5. A Maxwell formulation for the equations of a plasma
Go to AIP Home
Copyright © American Institute of Physics
close