Langmuir wave self-focusing versus decay instability

Physics of Plasmas

Volumn 12, Issue 1, 2005, Pages 1-14

  Rose, Harvey A ^a  

^a LOS ALAMOS NATIONAL LABORATORY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ELECTRON BOUNCE FREQUENCY; ELECTRON PLASMA FREQUENCY; STIMULATED RAMAN SCATTER (SRS); TRAPPED PARTICLE MODULATIONAL INSTABILITY (TPMI);

APPROXIMATION THEORY; DAMPING; DISPERSIONS; ELECTRON ENERGY LEVELS; FUNCTIONS; PLASMAS; RAMAN SCATTERING; VECTORS;

ELECTRON TRAPS;

EID: 18844457979     PISSN: 1070664X     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1829066     Document Type: Article

References (66)

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13. J. L. Kline, D. S. Montgomery, B. Bezzerides, J. A. Cobble, D. F. DuBois, R. P. Johnson, H. A. Rose, and H. X. Vu, "Observation of a transition from kinetic to fluid nonlinearities for Langmuir waves driven by stimulated Raman backscatter," Phys. Rev. Lett. (submitted).
14. H. A. Rose and D. A. Russell, Phys. Plasmas 8, 4784 (2001).
15. Although there only appear to be two traveling wave solutions that bifurcate from thermal equilibrium [M. Buchanan and J. Doming, Phys. Rev. E 52, 3015 (1995)]
16. Afeyan [B. Afeyan et al., "Kinetic electrostatic electron nonlinear (KEEN) waves and their interactions driven by the ponderomotive force of crossing laser beams," IFSA Proceedings, 2003, edited by B. A. Hammel, D. D. Meyerhofer. J. Meyer-ter-Vehn, and H. Azechi (American Nuclear Society, Inc., La Grange Park, IL, 2004), Chap. V, pp. 213-217;
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18. 0 is implicitly assumed for the rest of this paper.
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23. See Sec. III B 6 of Ref. 11.
24. 2.
25. See Fig. 1 of Ref. 11. The particular class of related BGK modes is also discussed there.
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28. e)>0, as is the case for the Langmuir and electron-acoustic modes.
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31. See Sec. III B 4 of Ref. 11.
32. Because of ion inertia, the actual ponderomotive induced shift tends to be smaller.
33. b. The analysis presented here is for the case of a coherent wave.
34. The expression for Re Ξ, however, as given by Eq. (4), may be valid for φ larger than its loss of resonance value for a particular value of k, because the definition and evaluation of Π does not require a plasma resonance.
35. The apex of this region is not quantitatively accurate since Eq. (9), and hence Eq. (12), breaks down near loss of resonance.
36. The anharmonic components of the exact solution can be quite small. See Eq. (60) in Ref. 11.
37. The external potential and relaxation to the background distribution function are omitted here.
38. References 7 provides the estimate that the temporal variation is "slow" if its associated frequency is small compared to the bounce frequency.
39. B. I. Cohen and A. N. Kaufman, Phys. Fluids 21, 404 (1978). There is an enormous literature on the derivation of nonlinear wave equations, going back at least to the seminal work of
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43. 0(k,ω)=0.
44. Terms which are proportional to products of ∂/∂t and space derivatives are ignored.
45. So long as there is positive dispersion.
46. 2, in thermal units.
47. Ref. 38, p. 182, Eq. (7.347).
48. Any direction for the LDI daughter wave, in principle, can support the instability, though absent strong damping anisotropy, backscatter is favored over sidescatter, both because the intrinsic growth rate is a maximum in the former and because of the highly elongated geometry of the laser beam used in experiment (Ref. 10). While this needs to be reexamined, in view of anisotropic trapped particle effects, since experimental data (Ref. 10) is in quantitative agreement with LDI backscatter, this is the point of view adopted here (Ref. 41).
49. These considerations strictly apply near threshold. If the LDI daughter wave attains finite amplitude, its damping may decrease.
50. G allows for change of frequency. However this leads to an artificial instability. Instead, one may assume that at a given spatial location, the LW is close to local equilibrium, whose damping is then determined by the local values of k and φ. This is the nonlinear analog of the customary inclusion of linear Landau damping in Zakharov's model, as discussed by N. R. Pereira, R. N. Sudan, and J. Denavit, Phys. Fluids 20, 936 (1977).
51. For example, see Fig. 9 of Ref. 11. In a strongly damped regime, where the resonant response is proportional to the source and inversely proportional to linear damping, multiple solutions are still possible due to the trapped particle frequency shift. For example, this is discussed in the context of stimulated Brillouin scatter by E. A. Williams, B. I. Cohen, L. Divol, M. R. Dorr, J. A. Hittinger, D. E. Hinkel, A. B. Langdon, R. K. Kirkwood, D. H. Froula, and S. H. Glenzer, Phys. Plasmas 11, 231 (2004) (Fig. 1).
52. D,as in Fig. 14 of Ref. 45.
53. As given, e.g., by Eq. (15) in H. A. Rose, Phys. Plasmas 10, 1468 (2003).
54. If the average laser intensity 〈I〉 is above its critical value for SRS, then the linear power gain over a speckle length is at least as large as the ratio of a particular speckle's intensity to 〈I〉 [see H. A. Rose and D. F. DuBois, Phys. Rev. Lett. 72, 2883 (1994)]. Since the most probable speckle intensity (absent self-focusing) is about 3〈I〉
55. [see J. Garnier, Phys. Plasmas 6, 1601 (1999)], this condition is then easily satisfied. For the experimental conditions in Ref. 10, there is no SRS seed, and this assumption is valid.
56. J. Kline (private communication).
57. For experimental parameters (Ref. 10), Epperlein's nonlocal transport theory, [M. Epperlein, Phys. Rev. Lett. 65, 2145 (1990)] predicts temperature fluctuations which are too large to trust linear theory. In lieu of the nonlinear generalization of such a model proposed by
58. O. V. Batishchev, V. Yu. Bychenkov, F. Detering, W. Rozmus, R. Sydora, C. E. Capjack, and V. N. Novikov, Phys. Plasmas 9, 2302 (2002), in which spatial variations of the collisional mean free path are allowed, a simpler model is used to obtain a qualitative estimate for a localized heat source: the largest electron thermal fluctuation is self-consistently added to the background electron temperature. This variant of Epperleins's model yields a lower bound to the actual temperature fluctuation.
59. See Sec. V of D. A. Russell, D. F. DuBois, and H. A. Rose, Phys. Plasmas 6, 1294(1999).
60. The spectrum of unstable modes cuts off at a wave number only √2 larger.
61. e, instead of being related, as in Eq. (25) of Ref. 11.
62. H. A. Rose, Bull. Am. Phys. Soc. 46 (8), 284 (2001).
63. D≈0.3
64. D, due to an increase in optic f/#, from that due to the change in v.
65. P. M. Lushnikov and H. A. Rose, Phys. Rev. Lett. 92, 255003 (2004).
66. In the strongly damped regime, with LW Landau damping comparable to, or greater than γSRS, actual SRS time scales may be large compared with 1/γSRS, enabling the TPMI regime.