1887
banner image
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
oa
Adiabatic nonlinear waves with trapped particles. III. Wave dynamics
Rent:
Rent this article for
Access full text Article
/content/aip/journal/pop/19/1/10.1063/1.3673065
1.
1. G. B. Whitham, J. Fluid Mech. 22, 273 (1965).
http://dx.doi.org/10.1017/S0022112065000745
2.
2. G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974), Chaps. 14 and 15.
3.
3. I. Y. Dodin and N. J. Fisch, Phys. Plasmas 19, 012102 (2012).
4.
4. I. Y. Dodin and N. J. Fisch, Phys. Plasmas 19, 012103 (2012).
5.
5. I. Y. Dodin and N. J. Fisch, Phys. Rev. Lett. 107, 035005 (2011).
http://dx.doi.org/10.1103/PhysRevLett.107.035005
6.
6. V. I. Karpman, Non-Linear Waves in Dispersive Media (Pergamon, New York, 1974), Chap. V.
7.
7. The indexes a, k, ω, A, and (except in ) denote differentiation with respect to the corresponding variables.
8.
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, .
9.
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.
10. V. L. Krasovsky, Phys. Lett. A 163, 199 (1992).
http://dx.doi.org/10.1016/0375-9601(92)90408-E
11.
11. V. L. Krasovsky, Fiz. Plazmy 18, 739 (1992)
11. V. L. Krasovsky, [Sov. J. Plasma Phys. 18, 382 (1992)].
12.
12. V. L. Krasovsky, J. Plasma Phys. 47, 235 (1992).
http://dx.doi.org/10.1017/S0022377800024193
13.
13. V. L. Krasovsky, Phys. Scr. 49, 489 (1994).
http://dx.doi.org/10.1088/0031-8949/49/4/016
14.
14. R. L. Dewar, W. L. Kruer, and W. M. Manheimer, Phys. Rev. Lett. 28, 215 (1972).
http://dx.doi.org/10.1103/PhysRevLett.28.215
15.
15. H. Ikezi, K. Schwarzenegger, and A. L. Simons, Phys. Fluids 21, 239 (1978).
http://dx.doi.org/10.1063/1.862198
16.
16. H. A. Rose, Phys. Plasmas 12, 012318 (2005).
http://dx.doi.org/10.1063/1.1829066
17.
17. H. A. Rose and L. Yin, Phys. Plasmas 15, 042311 (2008).
http://dx.doi.org/10.1063/1.2901197
18.
18. V. I. Karpman and E. M. Krushkal, Zh. Eksp. Teor. Fiz. 55, 530 (1968)
18. V. I. Karpman and E. M. Krushkal, [Sov. Phys. JETP 28, 277 (1969)].
19.
19. W. L. Kruer, J. M. Dawson, and R. N. Sudan, Phys. Rev. Lett. 23, 838 (1969).
http://dx.doi.org/10.1103/PhysRevLett.23.838
20.
20. M. V. Goldman, Phys. Fluids 13, 1281 (1970).
http://dx.doi.org/10.1063/1.1693061
21.
21. M. V. Goldman and H. L. Berk, Phys. Fluids 14, 801 (1971).
http://dx.doi.org/10.1063/1.1693512
22.
22. V. L. Krasovsky, Plasma Phys. Controlled Fusion 51, 115011 (2009).
http://dx.doi.org/10.1088/0741-3335/51/11/115011
23.
23. V. L. Krasovskii, Zh. Eksp. Teor. Fiz. 107, 741 (1995)
23. V. L. Krasovskii, [JETP 80, 420 (1995)].
24.
24. H. A. Rose and D. A. Russell, Phys. Plasmas 8, 4784 (2001).
http://dx.doi.org/10.1063/1.1410111
25.
25. B. J. Winjum, J. Fahlen, and W. B. Mori, Phys. Plasmas 14, 102104 (2007).
http://dx.doi.org/10.1063/1.2790385
26.
26. I. Y. Dodin, V. I. Geyko, and N. J. Fisch, Phys. Plasmas 16, 112101 (2009).
http://dx.doi.org/10.1063/1.3250983
27.
27. A. I. Matveev, Fiz. Plazmy 34, 114 (2008)
27. A. I. Matveev, [Plasma Phys. Rep. 34, 95 (2008)].
28.
28. Notice, in particular, that Eq. (18) can be written as . Yet, this form holds only for a δ-shaped trapped-particle distribution; for other distributions, there is a minimum a below which is undefined.
29.
29. P. F. Schmit, I. Y. Dodin, and N. J. Fisch, Phys. Rev. Lett. 105, 175003 (2010).
http://dx.doi.org/10.1103/PhysRevLett.105.175003
30.
30. I. Y. Dodin and N. J. Fisch, Phys. Rev. D 82, 044044 (2010).
http://dx.doi.org/10.1103/PhysRevD.82.044044
31.
31. T. H. Stix, Waves in Plasmas (AIP, New York, 1992), Sec. 4.2.
32.
32. From S ∼ 1, one gets another estimate, φ ∼ (ν/κ3)1/2. Thus, and .
33.
33. M. J. Lighthill, J. Inst. Math. Appl. 1, 1 (1965).
http://dx.doi.org/10.1093/imamat/1.1.1
34.
34. C. D. Decker and W. B. Mori, Phys. Rev. Lett. 72, 490 (1994).
http://dx.doi.org/10.1103/PhysRevLett.72.490
35.
35. C. D. Decker and W. B. Mori, Phys. Rev. E 51, 1364 (1995).
http://dx.doi.org/10.1103/PhysRevE.51.1364
36.
36. C. B. Schroeder, C. Benedetti, E. Esarey, and W. P. Leemans, Phys. Rev. Lett. 106, 135002 (2011).
http://dx.doi.org/10.1103/PhysRevLett.106.135002
37.
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);
http://dx.doi.org/10.1063/1.3464467
37. D. Bénisti, D. J. Strozzi, L. Gremillet, and O. Morice, Phys. Rev. Lett. 103, 155002 (2009);
http://dx.doi.org/10.1103/PhysRevLett.103.155002
37. D. Bénisti, and L. Gremillet, Phys. Plasmas 14, 042304 (2007). See also our Ref. 3 for comparison.
http://dx.doi.org/10.1063/1.2711819
38.
38. L. D. Landau and E. M. Lifshitz, Fluid Dynamics (Pergamon, New York, 1987), Secs. 103 and 104.
39.
39. S. Brunner and E. J. Valeo, Phys. Rev. Lett. 93, 145003 (2004).
http://dx.doi.org/10.1103/PhysRevLett.93.145003
40.
40. To our knowledge, the results of Ref. 13 have never been associated with the TPMI before.
41.
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).
http://dx.doi.org/10.1063/1.3474619
42.
42. G. B. Whitham, Proc. R. Soc., London Ser. A 283, 238 (1965).
http://dx.doi.org/10.1098/rspa.1965.0019
43.
43. M. J. Lighthill, J. Inst. Math. Appl. 1, 269 (1965).
http://dx.doi.org/10.1093/imamat/1.3.269
44.
44. M. J. Lighthill, Proc. R. Soc., London Ser. A 299, 28 (1967).
http://dx.doi.org/10.1098/rspa.1967.0121
45.
45. W. D. Hayes, Proc. R. Soc., London Ser. A. 332, 199 (1973).
http://dx.doi.org/10.1098/rspa.1973.0021
46.
46. V. E. Zakharov and L. A. Ostrovsky, Physica D 238, 540 (2009).
http://dx.doi.org/10.1016/j.physd.2008.12.002
47.
47. E. E. Kunhardt and B. R. -S.R.-S. Cheo, Phys. Rev. Lett. 37, 1688 (1976).
http://dx.doi.org/10.1103/PhysRevLett.37.1688
48.
journal-id:
http://aip.metastore.ingenta.com/content/aip/journal/pop/19/1/10.1063/1.3673065
Loading
View: Figures

Figures

Image of FIG. 1.

Click to view

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 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 ).

Image of FIG. 2.

Click to view

FIG. 2.

(Color online) Schematic of the parameter domain assumed for Sec. V in space (κ, ϑ, Ω E ) 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., [Eq. (30); see also Eqs. (8) and (28)]. The inequalities Eqs. (9) and (A12) reduce to and thus are satisfied automatically.

Image of FIG. 3.

Click to view

FIG. 3.

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

Image of FIG. 4.

Click to view

FIG. 4.

(Color online) Evolution of the perturbation to a homogeneous wave with the initial amplitude a 0. The solution is obtained by numerical integration of Eqs. (B1) and (B3), with taken from Eq. (18), for the same parameters as in Fig. 3 and  = 20λD . Shown is Δ(a 2) (arbitrary color scaling), vs. t and x in the frame moving with the linear group velocity ; the units are 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.

Loading

Article metrics loading...

/content/aip/journal/pop/19/1/10.1063/1.3673065
2012-01-06
2014-04-21

Abstract

The evolution of adiabatic waves with autoresonant trapped particles is described within the Lagrangianmodel 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.

Loading

Full text loading...

/deliver/fulltext/aip/journal/pop/19/1/1.3673065.html;jsessionid=4qh8bsha2hks6.x-aip-live-02?itemId=/content/aip/journal/pop/19/1/10.1063/1.3673065&mimeType=html&fmt=ahah&containerItemId=content/aip/journal/pop
true
true
This is a required field
Please enter a valid email address
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Adiabatic nonlinear waves with trapped particles. III. Wave dynamics
http://aip.metastore.ingenta.com/content/aip/journal/pop/19/1/10.1063/1.3673065
10.1063/1.3673065
SEARCH_EXPAND_ITEM