Wave turbulence in magnetized plasmas

Nonlinear Processes in Geophysics

Volumn 16, Issue 1, 2009, Pages 83-98

  Galtier, S ^a,b  

^a UNIVERSITÉ PARIS SUD   (France)

^b INSTITUT UNIVERSITAIRE DE FRANCE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

PLASMA; SOLAR WIND; THEORETICAL STUDY; TURBULENCE;

EID: 69849094879     PISSN: 10235809     EISSN: 16077946     Source Type: Journal    
DOI: 10.5194/npg-16-83-2009     Document Type: Article

References (127)

1. Abdurakhimov L. V. Brazhnikov M. Y. Levchenko A. A.: Statistic of capillary waves on the surface of liquid hydrogen in a turbulent regime J. Low Temp. Phys. 150, 431-434, 2008.
2. Alexakis, A., Bigot, B., Politano, H., and Galtier, S.: Anisotropic fluxes and nonlocal interactions in magne-tohydrodynamic turbulence, Phys. Rev. E., 76, 056313, doi:10.1103/PhysRevE.76.056313, 2007. (Pubitemid 350127471)
3. Alexakis, A., Mininni, P. D., and Pouquet, A.: Shell-to-shell energy transfer in magnetohydrodynamics. I. Steady state turbulence, Phys. Rev. E., 72, 046301, 2005.
4. Alexandrova, O., Carbone, V., Veltri, P., and Sorriso-Valvo, L.: Small-scale energy cascade of the solar wind turbulence, Astro-phys. J., 674, 1153-1157, 2008.
5. Armstrong, J. W., Coles, W. A., Rickett, B. J., and Kojima, M.: Observations of field-aligned density fluctuations in the inner solar wind, Astrophys. J., 358, 685-692, 1990.
6. Aschwanden, M.J., Poland, A.I., and Rabin, D.M.: The new solar corona, Annu. Rev. Astron. Astrophys., 39, 175-210, 2001.
7. Bale, S. D., Kellogg, P. J., Mozer, F. S., Horbury, T. S., and Reme, H.: Measurement of the electric field fluctuation spectrum of MHD turbulence, Phys. Rev. Lett., 94, 215002, doi:10.1103/PhysRevLett.94.215002, 2005.
8. Barnes, A.: Interplanetary Alfvénic fluctuations: a stochastic model, J. Geophys. Res., 86, 7498-7506, 1981.
9. Batchelor, G. K.: The theory of homogeneous turbulence, Cambridge University Press, 1953.
10. Belcher, J. W. and Davis, L.: Large-Amplitude Alfvén waves in the interplanetary medium, J. Geophys. Res., 76, 3534-3563, 1971.
11. Belmont, G. and Rezeau, L.: Magnetopause reconnection induced by magnetosheath Hall-MHD fluctuations, J. Geophys. Res., 106, 10751-10760, 2001.
12. Benney, J., and Saffman, P. G.: Nonlinear interactions of random waves in a dispersive medium, Proc. Roy. Soc. A., 289, 301-320, 1966.
13. Benney, J. and Newell, A. C.: Sequential time closures for interacting random wavers, J. Maths. Phys., 46, 363-393, 1967.
14. Benney, J. and Newell, A. C.: Random wave closures, Studies in Applied Math., 48, 29-53, 1966.
15. Bhattacharjee, A.: Impulsive magnetic reconnection in the earth's magnetotail and the solar corona, Ann. Rev. Astron. Astrophys., 42, 365-384, 2004.
16. Bieber, J. W., Wanner, W., and Matthaeus, W. H.: Dominant two-dimensional solar wind turbulence with implications for cosmic ray transport, J. Geophys. Res., 101, 2511-2522, 1996.
17. Bigot, B., Galtier, S., and Politano, H.: Energy decay laws in strongly anisotropic MHD turbulence, Phys. Rev. Lett., 100, 074502, doi:10.1103/ PhysRevLett.100.074502, 2008a.
18. Bigot, B., Galtier, S., and Politano, H.: Development of Anisotropy in Incompressible Magnetohydrodynamic Turbulence, Phys. Rev. E., 78, 066301, doi:10.1103/PhysRevE.78.066301, 2008b.
19. Bigot, B., Galtier, S., and Politano, H.: An anisotropic turbulent model for solar coronal heating, Astron. Astrophys., 490, 325-337, 2008c.
20. Biskamp, D.: Collision and collisionless magnetic reconnection, Phys. Plasmas, 4, 1964-1968, 1997.
21. Biskamp, D., Schwarz, E., and Drake, J.F.: 2-D electron mag-netohydrodynamic turbulence, Phys. Rev. Lett., 76, 1264-1267, 1996.
22. Biskamp, D., Schwarz, E., Zeiler, A., Celani, A., and Drake, J. F.: Electron magnetohydrodynamic turbulence, Phys. Plasmas, 6, 751-758, 1999.
23. Boldyrev, S.: Spectrum of magnetohydrodynamic turbulence, Phys. Rev. Lett., 96, 115002, doi:10.1103/PhysRevLett.96.115002, 2006.
24. Brekke, P., Kjeldseth-Moe, O., and Harrison, R. A.: High-velocity flows in an active region loop system observed with the coronal diagnostic spectrometer on SOHO, Sol. Phys., 175, 511-521, 1997.
25. Bruno, R. and Carbone, V.: The solar wind as a turbulence laboratory, Living Rev. Solar Phys., 2, 1-186, 2005.
26. Buchlin, É. and Velli, M.: Shell models of RMHD turbulence and the heating of solar coronal loops, Astrophys. J., 662, 701-714, 2007.
27. Chae, J., Schühle, U., and Lemaire, P.: Sumer measurements of nonthermal motions: constraints on coronal heating mechanism, Astrophys. J., 505, 957-973, 1998.
28. Chandran, B. D. G.: Weak compressible magnetohydrodynamic turbulence in the solar corona, Phys. Rev. Lett., 95, 265004, doi:10.1103/PhysRevLett.95. 265004, 2005. (Pubitemid 43061558)
29. Chandran, B. D. G.: Strong anisotropic MHD turbulence with cross helicity, Astrophys. J., 685, 646-658, 2008.
30. Chandrasekhar, S.: The invariant theory of isotropic turbulence in magneto-hydrodynamics, Proc. Roy. Soc. A., 204, 435-449, 1951.
31. Cho, J. and Vishniac, E. T.: The anisotropy of magnetohydrody-namic Alfvénic turbulence, Astrophys. J., 539, 273-282, 2000.
32. Cho, J. and Vishniac, E. T.: The anisotropy of electron magnetohy-drodynamic turbulence, Astrophys. J., 615, L41-L44, 2004.
33. Coleman, P. J.: Turbulence, viscosity, and dissipation in the solar-wind plasma, Astrophys. J., 153, 371-388, 1968.
34. Connaughton, C., Newell, A. C., and Pomeau, Y.: Non-stationary spectra of local wave turbulence, Physica D., 184, 64-85, 2003.
35. Coroniti, F. V., Kennel, C. F., and Scarf, F. L.: Whistler mode turbulence in the disturbed solar wind, J. Geophys. Res., 87, 6029-6044, 1981.
36. Cranmer, S. R., van Ballegooijen, A. A., and Edgar, R. J.: Self-consistent coronal heating and solar wind acceleration from anisotropic magnetohydrodynamic turbulence, Astrophys. J., 171, 520-551, 2007.
37. Dastgeer, S., Das, A., Kaw, P., and Diamond, P. H.: Whistlerisa-tion and anisotropy in 2-D electron magnetohydrodynamic turbulence, Phys. Plasmas, 7, 571-579, 2000.
38. David, C., Gabriel, A. H., Bely-Dubau, F., Fludra, A., Lemaire, P., and Wilhelm, K.: Measurement of the electron temperature gradient in a solar coronal hole, Astron. Astrophys., 336, L90-L94, 1998.
39. -5 Hz to 470Hz from HELIOS-observations during solar minimum conditions., J. Geophys., 54, 60-67, 1983.
40. Doschek, G. A., Mariska, J. T., Warren, H. P., Brown, C. M., Cul-hane, J. L., Hara, H., Waranabe, T., Toung, P. R., and Mason, H. E.: Nonthermal velocities in solar active regions observed with the extreme-ultraviolet imaging spectrometer on Hinode, Astro-phys. J., 667, L109-L112, 2007.
41. Dyachenko, S., Newell, A. C., Pushkarev, A. N., and Zakharov, V. E.: Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation, Physica D., 57, 96-160, 1992.
42. Elmegreen, B. G. and Scalo, J.: Interstellar turbulence I: observations and processes, Annu. Rev. Astron. Astrophys., 42, 211-273, 2004.
43. Falcon, É., Laroche, C., and Fauve, S.: Observation of gravitycapillary wave turbulence, Phys. Rev. Lett., 98, 094503, doi:10.1103/PhysRevLett.98.094503, 2007. (Pubitemid 46363382)
44. Falkovich, G. E. and Shafarenko, A. V.: Nonstationary wave turbulence, J. Nonlinear Sci., 1, 457-480, 1991.
45. Forsyth, R. J., Balogh, A., Horbury, T. S., Erdös, G., Smith, E. J., and Burton, M. E.: The heliospheric magnetic field at solar minimum: Ulysses observations from pole to pole, Astron. As-trophys., 316, 287-295, 1996.
46. Frisch, U.: Turbulence, Cambridge University Press, 1995.
47. Galtier, S., Nazarenko, S. V., Newell, A. C., and Pouquet, A.: A weak turbulence theory for incompressible MHD, J. Plasma Phys., 63, 447-488, 2000.
48. Galtier, S., Nazarenko, S. V., and Newell, A. C.: On wave turbulence in MHD, Nonlin. Processes Geophys., 8, 141-150, 2001, http://www.nonlin-processes- geophys.net/8/141/2001/.
49. Galtier, S., Nazarenko, S. V., Newell, A. C., and Pouquet, A.: Anisotropic turbulence of shear-Alfvén waves, Astrophys. J., 564, L49-L52, 2002.
50. Galtier, S.: A weak inertial wave turbulence theory, Phys. Rev. E., 68, 015301, doi:10.1103/PhysRevE.68.015301, 2003.
51. Galtier, S. and Bhattacharjee, A.: Anisotropic weak whistler wave turbulence in electron magnetohydrodynamics, Phys. Plasmas, 10, 3065-3076, 2003.
52. Galtier, S. and Bhattacharjee, A.: Anisotropic wave turbulence in electron MHD, Plasma Phys. Control. Fusion, 47, B691-B701, 2005.
53. Galtier, S., Pouquet, A., and Mangeney, A.: On the spectral scaling laws for incompressible anisotropic MHD turbulence, Phys. Plasmas, 12, 092310, doi:10.1063/1.2052507, 2005.
54. Galtier, S.: Multi-scale turbulence in the solar wind, J. Low Temp. Phys., 145, 59-74, 2006a.
55. Galtier, S.: Wave turbulence in incompressible magnetohydrody-namics, J. Plasma Phys., 72, 721-769, 2006b.
56. Galtier, S. and Chandran, B. D. G.: Extended spectral scaling laws for shear-Alfvén wave turbulence, Phys. Plasmas, 13, 114505, doi:10.1063/1.2399466, 2006.
57. Galtier, S. and Buchlin, É.: Multiscale Hall magnetohydrodynamic turbulence in the solar wind, Astrophys. J., 656, 560-566, 2007.
58. Galtier, S.: von Karman-Howarth equations for Hall Magnetohy-drodynamic flows, Phys. Rev. E., 77, 015302(R)-1-4, 2008a.
59. Galtier, S.: Exact scaling laws for 3D electron MHD turbulence, J. Geophys. Res., 113, A01102, doi:10.1029/2007JA012821, 2008b.
60. Galtier, S. and Nazarenko, S. V.: Large-scale magnetic field sus-tainment by forced MHD wave turbulence, J. Turbulence, 9(40), 1-10, 2008.
61. Ghosh, S., Siregar, E., Roberts, D. A., and Goldstein, M. L.: Simulation of high-frequency solar wind power spectra using Hall magnetohydrodynamic, J. Geophys. Res., 101, 2493-2504, 1996.
62. Goldreich, P. and Reisenegger, A.: Magnetic field decay in isolated neutron stars, Astrophys. J., 395, 250-258, 1992.
63. Goldreich, P. and Sridhar, S.: Toward a theory of interstellar turbulence. II. Strong alfvénic turbulence, Astrophys. J., 438, 763-775, 1995.
64. Goldstein, M. L., Roberts, D. A., and Fitch, C. A.: Properties of the fluctuating magnetic helicity in the inertial and dissipative ranges of solar wind turbulence, J. Geophys. Res., 99, 11519-11538, 1994.
65. Goldstein, M. L. and Roberts, D. A.: Magnetohydrodynamics turbulence in the solar wind, Phys. Plasmas, 6, 4154-4160, 1999.
66. Hendrix, D. L. and Van Hoven, G.: Magnetohydrodynamic turbulence and implications for solar coronal heating, Astrophys. J., 467, 887-893, 1996.
67. Higdon, J. C.: Density fluctuations in the interstellar medium: evidence for anisotropic magnetogasdynamic turbulence. I Model and astrophysical sites, Astrophys. J., 285, 109-123, 1984.
68. Hollweg, J.V., and Isenberg, P.A.: Generation of the fast solar wind: a review with emphasis on the resonant cyclotron interaction, J. Geophys. Res., 107, 1147-1-37, 2002.
69. Howes, G. G., Cowley, S. C., Dordland, W., Hammett, G. W., Quataert, E., and Schekochihin, A. A.: A model of turbulence in magnetized plasmas: implications for the dissipation range in the solar wind, J. Geophys. Res., 113, A05103, doi:10.1029/2007JA012665, 2008.
70. Iroshnikov, P. S.: Turbulence of a conducting fluid in a strong magnetic field, Soviet Astron., 7, 566-571, 1964.
71. Kingsep, A. S., Chukbar, K. V., and Yankov, V. V.: Electron magne-tohydrodynamics, in: Reviews of Plasma Physics, 16, 243-291, Consultant Bureau, New York, 1990.
72. Klein, L., Bruno, R, Bavassano, B., and Rosenbauer, H: Anisotropy and minimum variance of magnetohydrodynamic fluctuations in the inner heliosphere, J. Geophys. Res., 98, 17461-17466, 1993.
73. Kolmakov, G. V. and Pokrovsky, V. L.: Stability of weak turbulence spectra in superfluid helium, Physica D., 86, 456-469, 1995.
74. Kolmakov, G. V., Levchenko, A. A., Brahnikov, M. Y., Mezhov-Deglin, L. P., Silchenko, A. N., and McClintock, P. V. E.: Quasiadiabatic decay of capillary turbulence on the charged surface of liquid hydrogen, Phys. Rev. Lett., 93, 074501, doi:10.1103/PhysRevLett.93.074501, 2004.
75. Kolmogorov, A. N.: The local structure of turbulence in incompressible viscous fluid for very high Reynolds numbers, Dokl. Akad. Nauk. SSS, 30, 301-305, 1941.
76. Kraichnan, R. H.: Inertial range spectrum in hydromagnetic turbulence, Phys. Fluids, 8, 1385-1387, 1965.
77. Kraichnan, R. H.: Inertial ranges in two-dimensional turbulence, Phys. Fluids, 10, 1417-1423, 1967.
78. Krishan, V. and Mahajan, S. M.: Magnetic fluctuations and Hall MHD turbulence in the solar wind, J. Geophys. Res., 109, A111051, doi:10.1029/2004JA010496, 2004.
79. Krommes, J. A.: Fundamental statistical descriptions of plasma turbulence in magnetic fields, Phys. Reports, 360, 1-352, 2002.
80. Kuznetsov, E. A.: Turbulence of ion sound in a plasma located in a magnetic field, Sov. Phys. J. Exp. Theor. Phys., 35, 310-314, 1972.
81. Kuznetsov, E. A.: Weak magnetohydrodynamic turbulence of a magnetized plasma, Sov. Phys. J. Exp. Theor. Phys., 93, 1052-1064, 2001.
82. Lacaze, R., Lallemand, P., Pomeau, Y., and Rica, S.: Dynamical formation of a Bose-Einstein condensate, Physica D., 152-153, 779-786, 2001.
83. Laveder, D., Passot, T., and Sulem, P.-L.: Transverse dynamics of dispersive Alfvén waves: I. direct numerical evidence of filamen-tation, Phys. Plasmas, 9, 293-304, 2002.
84. Leamon, R. J., Smith, C. W., Ness, N. F., and Matthaeus, W. H.: Observational constraints on the dynamics of the interplanetary magnetic field dissipation range, J. Geophys. Res., 103, 4775-4787, 1998.
85. Lithwick, Y. and Goldreich, P.: Imbalanced weak magnetohydrody-namic turbulence, Astrophys. J., 582, 1220-1240, 2003.
86. Lvov, Y., Nazarenko, S.V, and West, R.: Wave turbulence in Bose-Einstein condensates, Physica D., 184, 333-351, 2003.
87. MacBride, B. T., Smith, C. W., and Forman, M.: The turbulent cascade at 1 AU: energy transfer and the third-order scaling for MHD, Astrophys. J., 679, 1644-1660, 2008.
88. Markovskii, S. A., Vasquez, B. J., and Smith, C. W.: Statistical analysis of the high-frequency spectral break of the solar wind turbulence at 1 AU, Astrophys. J., 675, 1576-1583, 2008.
89. Maron, J. and Goldreich, P.: Simulations of incompressible mag-netohydrodynamic turbulence, Astrophys. J., 554, 1175-1196, 2001.
90. Matthaeus, W. H., Ghosh, S., Oughton, S., and Roberts, D. A.: Anisotropic three-dimensional MHD turbulence, J. Geophys. Res., 101, 7619-7629, 1996.
91. Mininni, P. D., Gómez, D. O., and Mahajan, S. M.: Dynamo action in magnetohydrodynamics and Hall-magnetohydrodynamics, Astrophys. J., 587, 472-481, 2003
92. Mininni, P. D., Alexakis, A., and Pouquet, A.: Energy transfer in Hall-MHD turbulence: cascades, backscatter, and dynamo action, J. Plasma Physics, 73, 377-401, 2007
93. Montgomery, D. and Turner, L.: Anisotropic magnetohydrody-namic turbulence in a strong external magnetic field, Phys. Fluids, 24, 825-831, 1981.
94. Montgomery, D. C. and Matthaeus, W. H.: Anisotropic modal energy transfer in interstellar turbulence, Astrophys. J., 447, 706-707, 1995.
95. Morize, C., Moisy, F., and Rabaud, M.: Decaying grid-generated turbulence in a rotating frame, Phys. Fluids, 17, 095105, 2005.
96. Nazarenko, S. V., Newell, A. C., and Galtier, S.: Non-local MHD turbulence, Physica D., 152-153, 646-652, 2001.
97. Nazarenko, S. V.: 2D enslaving of MHD turbulence, New J. Phys., 9, 1-14, 2007.
98. Newell, A. C., Nazarenko, S. V., and Biven, L.: Wave turbulence and intermittency, Physica D., 152-153, 520-550, 2001.
99. Ng, C. S. and Bhattacharjee, A.: Interaction of shear-Alfvén wave packets: implication for weak magnetohydrodynamic turbulence in astrophysical plasmas, Astrophys. J., 465, 845-854, 1996.
100. Ng, C. S., Bhattacharjee, A., Germaschewski, K., and Galtier, S.: Anisotropic fluid turbulence in the interstellar medium and solar wind, Phys. Plasmas, 10, 1954-1962, 2003.
101. Nishizuka, N., Shimizu, M., Nakamura, T., Otsuji, K., Okamoto, T. J., Katsukawa, Y., and Shibata, K.: Giant chromospheric anemone jet observed with Hinode and comparison with MHD simulations: evidence of propagating Alfvén waves and magnetic reconnection, Astrophys. J., 683, L83-L86, 2008.
102. Perez, J.-C. and Boldyrev, S.: On weak and strong magnetohydro-dynamic turbulence, Astrophys. J., 672, L61-L64, 2008.
103. Politano, H. and Pouquet, A.: von-Kárman-Howarth equation for MHD and its consequences on third-order longitudinal structure and correlation functions, Phys. Rev. E., 57, R21-R24, 1998.
104. Pouquet, A.: Magnetohydrodynamic turbulence, in: Astrophysical fluid dynamics, edited by: Zahn, J.-P. and Zinn-Justin, J., Elsevier science publishers, 139-227, 1993.
105. Priest, E. R.: Solar magnetohydrodynamics, D. Reidel Pub. Comp., 1982.
106. Rappazzo, A.F., Velli, M., Einaudi, G., and Dahlburg, R.B.: Coronal heating, weak MHD turbulence, and scaling laws, Astrophys. J., 657, L47-L51, 2007.
107. Rappazzo, A. F., Velli, M., Einaudi, G., and Dahlburg, R. B.: Nonlinear dynamics of the Parker scenario for coronal heating, As-trophys. J., 677, 1348-1366, 2008.
108. Regnier, S., Parnell, C. E., and Haynes, A. L.: A new view of quiet-Sun topology from Hinode/SOT, Astron. Astrophys., L47-L50, 2008.
109. Roberts, D. A., Goldstein, M. L., Klein, L. W., and Matthaeus, W. H.: Origin and evolution of fluctuations in the solar wind: Helios observations and Helios-Voyager comparison, J. Geoph. Res., 92, 12023-12035, 1987.
110. Sagaut, P. and Cambon, C.: Homogeneous Turbulence Dynamics, Cambridge University Press, 2008.
111. Sahraoui, F., Belmont, G., and Rezeau, L.: Hamiltonian canonical formulation of Hall MHD: toward an application to weak turbulence, Phys. Plasmas, 10, 1325-1337, 2003
112. Sahraoui, F., Belmont, G., Rezeau, L., Cornilleau-Wehrlin, N., Pinçon, J. L., and Balogh, A.: Anisotropic turbulent spectra in the terrestrial magnetosheath as seen by the Cluster spacecraft, Phys. Rev. Lett., 96, 075002, doi:10.1103/PhysRevLett.96.075002, 2006. (Pubitemid 43290794)
113. Sahraoui, F., Galtier, S., and Belmont, G.: Incompressible Hall MHD waves, J. Plasma Phys., 73, 723-730, 2007.
114. Scalo, J. and Elmegreen, B. G.: Interstellar turbulence II: implications and effects, Annu. Rev. Astron. Astrophys., 42, 275-316, 2004.
115. Schekochihin, A. A., Cowley, S. C., Dorland, W., Hammett, G. W., Howes, G. G., Quataert, E., and Tatsuno, T.: Astrophysical gy-rokinetics: kinetic and fluid turbulence cascades in magnetized weakly collisional plasmas, Astrophys. J. Sup. Series, submitted, 2008.
116. Shaikh, D. and Zank, G.: Anisotropic cascades in interstellar medium turbulence, Astrophys. J., 656, L17-L20, 2007.
117. Shebalin, J. V., Matthaeus, W. H., and Montgomery, D.: Anisotropy in MHD turbulence due to a mean magnetic field, J. Plasma Phys., 29, 525-547, 1983.
118. Smith, C. W., Hamilton, K., Vasquez, B. J., and Leamon, R. J.: Dependence of the dissipation range spectrum of interplanetary magnetic fluctuations on the rate of energy cascade, Astrophys. J., 645, L85-L88, 2006.
119. Sorriso-Valvo, L., Marino, R., Carbone, V., Noullez, N., Lepreti, F., Veltri, P., Bruno, R., Bavassano, B., and Pietropaolo, E.: Observation of inertial energy cascade in interplanetary space plasma, Phys. Rev. Lett., 99, 115001, doi:10.1103/PhysRevLett.99.115001, 2007. (Pubitemid 47429445)
120. Sridhar, S. and Goldreich, P.: Toward a theory of interstellar turbulence 1. Weak alfvénic turbulence, Astrophys. J., 432, 612-621, 1994.
121. Stawicki, O., Gary, P. S., and Li, H.: Solar wind magnetic fluctuations spectra: dispersion versus damping, J. Geoph. Res., 106, 8273-8281, 2001.
122. Verma, M. K.: Statistical theory of magnetohydrodynamic turbulence: recent results, Phys. Rep., 401, 229-380, 2004.
123. Warren, H. P.: Multithread hydrodynamic modeling of a solar flare Astrophys. J., 637, 522-530, 2006.
124. Yamada, M.: Progress in understanding magnetic reconnection in laboratory and space astrophysical plasmas, Phys. Plasmas, 14, 1-16, doi:10.1063/1. 2740595, 2007. (Pubitemid 46872574)
125. Zakharov, V. E., and Filonenko, N. N.: The energy spectrum for stochastic oscillations of a fluid surface, Doclady Akad. Nauk. SSSR, 170, 1292-1295, 1966 [Sov. Phys. Docl., 11, 881-884, 1967].
126. Zakharov, V. E.: On the spectrum of turbulence in plasma without magnetic field, J. Exp. Theor. Phys., 24, 455-459, 1967.
127. Zakharov, V. E., L'vov, V., and Falkovich, G. E.: Kolmogorov Spectra of Turbulence I: Wave Turbulence, Springer-Verlag, Berlin, Germany, 1992.