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Stochastic instability and turbulent transport. Characteristic scales, increments, diffusion coefficients


National Research Centre ‘Kurchatov Institute’, pl. akad. Kurchatova 1, Moscow, 123182, Russian Federation

Stochastic instability and its associated turbulent diffusion models are reviewed with particular attention to obtaining estimates and scaling characterizing correlation effects and increments. Specific models considered include the quasilinear Kazantsev approximation, stochasticity in the system of convective cells, Kadomtsev—Pogutse scaling, percolation models, Rechester—Rosenbluth balance, etc. The primary goal is to highlight the importance of determining the functional dependence of the stochastic instability increments and transport coefficients on turbulent pulsation amplitudes and other key parameters (characteristic pulsation frequencies, drift velocities, spectral energy flow, etc.) describing systems under discussion.

Fulltext pdf (736 KB)
Fulltext is also available at DOI: 10.3367/UFNe.0185.201503b.0271
Keywords: stochastic instability, turbulent transport, diffusion coefficients, dynamical chaos, stochastic magnetic field, plasma
PACS: 05.40.−a, 47.27.−i, 47.53.+n (all)
DOI: 10.3367/UFNe.0185.201503b.0271
URL: https://ufn.ru/en/articles/2015/3/b/
000356096100002
2015PhyU...58..252B
Citation: Bakunin O G "Stochastic instability and turbulent transport. Characteristic scales, increments, diffusion coefficients" Phys. Usp. 58 252–285 (2015)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Received: 17th, June 2014, revised: 17th, December 2014, 23rd, December 2014

Оригинал: Бакунин О Г «Стохастическая неустойчивость и турбулентный перенос. Характерные масштабы, инкременты, коэффициенты диффузии» УФН 185 271–306 (2015); DOI: 10.3367/UFNr.0185.201503b.0271

References (219) ↓ Cited by (9) Similar articles (20)

  1. Krylov N S Raboty Po Obosnovaniyu Statisticheskoi Fiziki (M. - L.: Izd-vo AN SSSR, 1950); Krylov N S Works On The Foundations Of Statistical Physics (Princeton, N.J.: Princeton Univ. Press, 1979)
  2. Arnold V I, Avez A Problèmes Ergodiques De La Mécanique Classique (Paris: Gauthier-Villars, 1967); Arnold V I, Avez A Ergodic Problems Of Classical Mechanics (New York: Benjamin, 1968)
  3. Lichtenberg A J, Lieberman M A Regular And Stochastic Motion (New York: Springer-Verlag, 1983); Likhtenberg A, Liberman M Regulyarnaya i Stokhasticheskaya Dinamika (M.: Mir, 1984)
  4. Zaslavskii G M, Sagdeev R Z Vvedenie v Nelineinuyu Fiziku. Ot Mayatnika do Turbulentnosti i Khaosa (M.: Nauka, 1988); Sagdeev R Z, Usikov D A, Zaslavsky G M Nonlinear Physics: From The Pendulum To Turbulence And Chaos (Chur: Harwood Acad. Publ., 1988)
  5. Zaslavskii G M i dr. Usp. Fiz. Nauk 156 193 (1988); Zaslavskii G M et al. Sov. Phys. Usp. 31 887 (1988)
  6. Reichl L E The Transition To Chaos. Conservative Classical Systems And Quantum Manifestations (New York: Springer, 2004); Raikhl L E Perekhod k Khaosu v Konservativnykh Klassicheskikh i Kvantovykh Sistemakh (M. - Izhevsk: Inst. komp’yut. issled., 2008)
  7. Bol’tsman L Izbrannye Trudy. Molekulyarno-kineticheskaya Teoriya Gazov. Termodinamika. Statisticheskaya Mekhanika. Teoriya Izlucheniya. Obshchie Voprosy Fiziki (M.: Nauka, 1984)
  8. Gibbs D V Termodinamika. Statisticheskaya Mekhanika (M.: Nauka, 1982)
  9. Bogolyubov N N Problemy Dinamicheskoi Teorii v Statisticheskoi Fizike (M. - L.: Gostekhizdat, 1946); Bogoliubov N N The Dynamical Theory In Statistical Physics (Delhi: Hindustan Publ. Corp., 1965)
  10. Leontovich M A Zh. Eksp. Teor. Fiz. 5 211 (1935)
  11. Hopf E Math. Ann. 117 590 (1940)
  12. Khinchin A Ya Usp. Fiz. Nauk 9 141 (1929)
  13. Khinchin A Ya Matematicheskie Osnovaniya Statisticheskoi Mekhaniki (M. - L.: Gostekhizdat, 1943)
  14. Krylov N S Diss. ... kand. fiz.-mat. nauk (1941)
  15. Bogolyubov N N, Krylov N M Zapiski Kafedry Matematicheskoi Fiziki Instituta Nelineinoi Mekhaniki AN USSR 4 5 (1939); pereizdano v kn., Bogolyubov N N, Krylov N M Bogolyubov N N. Sobranie Nauchnykh Trudov v 12 Tomakh T. 5 Neravnovesnaya Statisticheskaya Mekhanika, 1939 - 1980 (M.: Nauka, 2006)
  16. Borel É Mécanique Statistique Classique (Paris: Gauthier-Villars, 1925) p. 22
  17. Davydov B I Dokl. Akad. Nauk SSSR 50 131 (1945)
  18. Terletskii Ya P Zh. Eksp. Teor. Fiz. 17 837 (1947)
  19. Davydov B I Dokl. Akad. Nauk SSSR 50 135 (1945)
  20. Terletskii Ya P Dinamicheskie i Statisticheskie Zakony Fiziki (M.: MGU, 1949)
  21. Davydov B I Zh. Eksp. Teor. Fiz. 7 1069 (1937)
  22. Davydov B I, Shmushkevich I M Zh. Eksp. Teor. Fiz. 10 1043 (1940)
  23. Vlasov A A Zh. Eksp. Teor. Fiz. 18 291 (1938)
  24. Landau L D Zh. Eksp. Teor. Fiz. 16 574 (1946); Landau L D J. Phys. USSR 10 25 (1946)
  25. Davydov B I Fizika Plazmy i Problema Upravlyaemykh Termoyadernykh Reaktsii Vol. 1 (Otv. red. M A Leontovich) (M.: Izd-vo AN SSSR, 1958) p. 77; Davydov B I Plasma Physics And The Problem Of Controlled Thermonuclear Research Vol. 1 (Ed. M A Leontovich) (Oxford: Pergamon Press, 1959) p. 77
  26. Galitskii V M Diss. ... kand. fiz.-mat. nauk (1954); Galitskii V M Izbrannye Trudy. Issledovaniya Po Teoreticheskoi Fizike (M.: Nauka, 1983)
  27. Vlasov A A Teoriya Mnogikh Chastits (M.: GITTL, 1950); Vlasov A A Many-Particle Theory And Its Application To Plasma (New York: Gordon and Breach, 1961)
  28. Bohm D, Gross E P Phys. Rev. 75 1864 (1949)
  29. Ertel H Meteorol. Z. 59 277 (1942)
  30. Eckart C J. Mar. Res. 7 265 (1948)
  31. Batchelor G K Proc. Cambr. Phil. Soc. 48 345 (1952)
  32. Corrsin S Atmospheric Diffusion And Air Pollution (Advances in Geophysics) Vol. 6 (Eds F N Frenkiel, P A Sheppard) (New York: Academic Press, 1959) p. 161
  33. Monin A S Usp. Fiz. Nauk 67 119 (1959); Monin A S Sov. Phys. Usp. 2 50 (1959)
  34. Monin A S, Yaglom A M Statisticheskaya Gidromekhanika Vol. 1, 2 (M.: Nauka, 1965, 1965, 1967); Monin A S, Yaglom A M Statistical Fluid Mechanics Vol. 1, 2 (Cambridge, Mass.: MIT Press, 1971, 1971, 1975)
  35. Batchelor G K Theory Of Homogeneous Turbulence (Cambridge: Univ. Press, 1953); Betchelor Dzh Teoriya Odnorodnoi Turbulentnosti (M.: IL, 1955)
  36. Kolmogorov A N Dokl. Akad. Nauk SSSR 119 861 (1958)
  37. Kolmogorov A N Dokl. Akad. Nauk SSSR 124 754 (1959)
  38. Sinai Ya G Dokl. Akad. Nauk SSSR 124 768 (1959)
  39. Sinai Ya G Dokl. Akad. Nauk SSSR 125 1200 (1959)
  40. Sinai Ya G Dokl. Akad. Nauk SSSR 153 1261 (1963); Sinai Ya G Sov. Math. Dokl. 4 1818 (1963)
  41. Sinai Y G Selecta Vol. 1 Ergodic Theory And Dynamical Systems (New York: Springer, 2010)
  42. Sinai Y G Selecta Vol. 2 Probability Theory, Statistical Mechanics, Mathematical Physics And Mathematical Fluid Dynamics (New York: Springer, 2010)
  43. Sinai Ya G Teoriya Fazovykh Perekhodov: Strogie Rezul’taty (M.: Nauka, 1980); Sinai Ya G Theory Of Phase Transitions: Rigorous Results (Oxford: Pergamon Press, 1982)
  44. Kornfel’d I P, Sinai Ya G, Fomin S V Ergodicheskaya Teoriya (M.: Nauka, 1980); Cornfeld I P, Fomin S V, Sinai Ya G Ergodic Theory (New York: Springer-Verlag, 1982)
  45. Kornfel’d I P, Sinai Ya G Itogi Nauki i Tekhniki (Sovremennye problemy matematiki. Fundamental’nye napravleniya) Vol. 2 (M.: VINITI, 1985) p. 44
  46. Khanin K M, Sinai Ya G Nonlinear Phenomena In Plasma Physics And Hydrodynamics (Ed. R Z Sagdeev) (Moscow: Mir Publ., 1986) p. 93
  47. Sinai Ya G Sovremennye Problemy Ergodicheskoi Teorii (M.: Fizmatlit, 1995)
  48. Lazutkin V F Vypuklyi Billiard i Sobstvennye Funktsii Operatora Laplasa (L.: Izd-vo LGU, 1981)
  49. Gal’perin G A, Zemlyakov A N Matematicheskie Bil’yardy (M.: Nauka, 1990)
  50. Kozlov V V, Treshchev D V Billiardy. Geneticheskoe Vvedenie v Dinamiku Sistem s Udarami (M.: Izd-vo MGU, 1991); Kozlov V V, Treshchev D V Billiards: A Genetic Introduction To The Dynamics Of Systems With Impacts (Providence, R.I.: American Mathematical Society, 1991)
  51. Tikhomirov V M (Gl. red.) Matematicheskoe Prosveshchenie Tret’ya Seriya, Vyp. 5 (M.: MTsNMO, 2001)
  52. Tabachnikov S Geometriya i Billiardy (M. - Izhevsk: Inst. komp’yut. issled., 2011)
  53. Loskutov A Yu Usp. Fiz. Nauk 180 1305 (2010); Loskutov A Phys. Usp. 53 1257 (2010)
  54. Chernov N, Markarian R Chaotic Billiards (Providence, R.I.: American Mathematical Society, 2006); Chernov N, Markaryan R Khaoticheskie Billiardy (M. -- Izhevsk: Inst. komp’yut. issled., 2012)
  55. Dragovich V, Radnovich M Integriruemye Billiardy, Kvadriki i Mnogomernye Prizmy Ponsele (M. - Izhevsk: RKhD. Inst. komp’yut. issled., 2010)
  56. Beck C, Schlögl F Thermodynamics Of Chaotic Systems: An Introduction (Cambridge: Cambridge Univ. Press, 1993)
  57. Dorfman J R An Introduction To Chaos In Nonequilibrium Statistical Mechanics (Cambridge: Cambridge Univ. Press, 1999)
  58. Puankare A, Erenfest P, Erenfest T, fon Neiman Dzh Raboty Po Statisticheskoi Mekhanike (M. - Izhevsk: Inst. komp’yut. issled., 2011)
  59. Halmos P R Lectures On Ergodic Theory (Tokyo: The Mathematical Society of Japan, 1956); Khalmosh P R Lektsii Po Ergodicheskoi Teorii (M.: IL, 1959)
  60. Kac M Lectures in Applied Mathematics. Proc. of the Summer Seminar, Boulder, Colorado, 1957 Vol. 1 Probability and Related Topics in Physical Sciences (London: Interscience Publ., 1959); Kats M Veroyatnost’ i Smezhnye Voprosy v Fizike (M.: Mir, 1965)
  61. Reichl L E A Modern Course In Statistical Physics (New York: Wiley, 1998)
  62. Zaslavsky G M The Physics Of Chaos In Hamiltonian Systems (London: Imperial College Press, 2007); Zaslavskii G M Fizika Khaosa v Gamil’tonovykh Sredakh (M. - Izhevsk: RKhD, Inst. komp’yut. issled., 2004)
  63. Childress S, Gilbert A D Stretch, Twist, Fold: The Fast Dynamo (Berlin: Springer, 1995)
  64. Moffatt H K Rep. Prog. Phys. 46 621 (1983)
  65. Bakunin O G Turbulence And Diffusion. Scaling Versus Equations Monograph On Complexity (New York: Springer, 2008)
  66. Chirikov B V Atomnaya Energiya 6 630 (1959); Chirikov B V J. Nucl. Energy C Plasma Phys. 1 253 (1960)
  67. Vedenov A, Velikhov E P, Sagdeev R Z Plasma Phys. Control. Nucl. Fusion Res. 2 82 (1962)
  68. Lifshits E M, Pitaevskii L P Fizicheskaya Kinetika (M.: Nauka, 1979); Lifshitz E M, Pitaevskii L P Physical Kinetics (Oxford: Pergamon Press, 1981)
  69. Kadomtsev B B, Pogutse O P Plasma Physics and Controlled Nuclear Fusion Research 1978. Proc. of the 7th Intern. Conf., IAEA, Innsbruck, Austria, August 23 - 30, 1978 Vol. 1 (Vienna: Intern. Atomic Energy Agency, 1978) p. 649
  70. Osipenko M V, Pogutse O P, Chudin N V Fizika Plazmy 13 953 (1987); Osipenko M V, Pogutse O P, Chudin N V Sov. J. Plasma Phys. 13 550 (1987)
  71. Isichenko M B Rev. Mod. Phys. 64 961 (1992)
  72. Bakunin O G Rep. Prog. Phys. 67 965 (2004)
  73. Zelenyi L M i dr. Usp. Fiz. Nauk 183 365 (2013); Zelenyi L M et al. Phys. Usp. 56 347 (2013)
  74. Bakunin O G Usp. Fiz. Nauk 183 257 (2013); Bakunin O G Phys. Usp. 56 243 (2013)
  75. Galeev A A, Zeleny L M Physica D 2 90 (1981)
  76. Moffatt H K et al. (Eds) Topological Aspects Of The Dynamics Of Fluids And Plasmas (Dordrecht: Kluwer Acad. Publ., 1992)
  77. Ottino J M The Kinematics Of Mixing: Stretching, Chaos, And Transport (Cambridge: Cambridge Univ. Press, 1989)
  78. Ott E Chaos In Dynamical Systems (Cambridge: Cambridge Univ. Press, 1993)
  79. Aref H (Guest Ed.), El Naschie M S (Ed.) Chaos Applied To Fluid Mixing (Oxford: Pergamon, 1995)
  80. Priest E R, Hood A W (Eds) Advances In Solar System Magnetohydrodynamics (Cambridge: Cambridge Univ. Press, 1991); Prist E, Khud A (Red.) Kosmicheskaya Magnitnaya Gidrodinamika (M.: Mir, 1995)
  81. Koshel’ K V, Prants S V Khaoticheskaya Advektsiya v Okeane (M. - Izhevsk: RKhD, Inst. komp’yut. issled., 2008)
  82. Koshel’ K V, Prants S V Usp. Fiz. Nauk 176 1177 (2006); Koshel K V, Prants S V Phys. Usp. 49 1151 (2006)
  83. Klyatskin V I, Koshel’ K V Usp. Fiz. Nauk 170 771 (2000); Klyatskin V I, Koshel’ K V Phys. Usp. 43 717 (2000)
  84. Bakunin O G Chaotic Flows. Correlation Effects, Transport, And Structures (Springer Series in Synergetics) (Heidelberg: Springer, 2011)
  85. Frisch U Turbulence: The Legacy Of A.N. Kolmogorov (Cambridge: Cambridge Univ. Press, 1995); Frish U Turbulentnost’: Nasledie A.N. Kolmogorova (M.: Fazis, 1998)
  86. Tsinober A An Informal Introduction To Turbulence (Dordrecht: Kluwer Acad. Publ., 2001)
  87. Monin A S, Ozmidov R V Okeanskaya Turbulentnost’ (L.: Gidrometeoizdat, 1981); Monin A S, Ozmidov R V Turbulence In The Ocean (Dordrecht: Reidel, 1985)
  88. Monin A S, Zhikharev G M Usp. Fiz. Nauk 160 (5) 1 (1990); Monin A S, Zhikharev G M Sov. Phys. Usp. 33 313 (1990)
  89. Monin A S Usp. Fiz. Nauk 150 61 (1986); Monin A S Sov. Phys. Usp. 29 843 (1986)
  90. Monin A S Usp. Fiz. Nauk 125 97 (1978); Monin A S Sov. Phys. Usp. 21 429 (1978)
  91. Monin A S Usp. Fiz. Nauk 109 333 (1973); Monin A S Sov. Phys. Usp. 16 121 (1973)
  92. Bouchaud J-P et al. Phys. Rev. Lett. 64 2503 (1990)
  93. Ben-Avraham D, Havlin S Diffusion And Reactions In Fractals And Disordered Systems (Cambridge: Cambridge Univ. Press, 2000)
  94. Krommes J A Phys. Rep. 360 1 (2002)
  95. Bakunin O G Chaos Solitons Fractals 23 1703 (2005)
  96. Barenblatt G I Podobie, Avtomodel’nost’, Promezhutochnaya Asimptotika: Teoriya i Prilozheniya k Geofizicheskoi Gidrodinamike (L.: Gidrometeoizdat, 1982); Barenblatt G I Scaling, Self-similarity, And Intermediate Asymptotics (Cambridge: Cambridge Univ. Press, 1996)
  97. Barenblatt G I Scaling (Cambridge: Cambridge Univ. Press, 2003)
  98. Kadomtsev B B Usp. Fiz. Nauk 151 3 (1987); Kadomtsev B B Rep. Prog. Phys. 50 115 (1987)
  99. Barenblatt G I, Korin A Dzh, Prostokishin V M Usp. Fiz. Nauk 184 265 (2014); Barenblatt G I, Chorin A J, Prostokishin V M Phys. Usp. 57 250 (2014); Barenblatt G I, Korin A Dzh, Prostokishin V M Usp. Fiz. Nauk 184 1371 (2014), popravka k stat’e; Vigdorovich I I Usp. Fiz. Nauk 185 213 (2015); Vigdorovich I I Phys. Usp. 58 (2) (2015); Barenblatt G I, Korin A Dzh, Prostokishin V M Usp. Fiz. Nauk 185 217 (2015); Barenblatt G I, Chorin A J, Prostokishin V M Phys. Usp. 58 (2) (2015)
  100. Kadomtsev B B Dinamika i Informatsiya (M.: Redaktsiya zhurnala "Uspekhi fizicheskikh nauk", 1997)
  101. Kadomtsev B B Usp. Fiz. Nauk 164 449 (1994); Kadomtsev B B Phys. Usp. 37 425 (1994)
  102. Tabor M Chaos And Integrability In Nonlinear Dynamics: An Introduction (New York: Wiley, 1989); Tabor M Khaos i Integriruemost’ v Nelineinoi Dinamike (M.: Editorial URSS, 2001)
  103. Squires T M, Quake S R Rev. Mod. Phys. 77 977 (2005)
  104. Stroock A D, McGraw G J Philos. Trans. R. Soc. Lond. A 362 971 (2004)
  105. Welander P Tellus 7 141 (1955)
  106. Batchelor G K, Townsend A A Surveys In Mechanics (Eds G K Batchelor, R M Davies) (Cambridge: Cambridge Univ. Press, 1956) p. 352
  107. Richardson L F Proc. R. Soc. London A 110 709 (1926)
  108. Kolmogorov A N Dokl. Akad. Nauk SSSR 30 299 (1941); Kolmogorov A N Proc. R. Soc. Lond. A 434 9 (1991)
  109. Obukhov A M Dokl. Akad. Nauk SSSR 32 19 (1941); Obukhov A M C.R. (Dokl.) Acad. Sci. USSR 32 19 (1941)
  110. Brunt D Physical And Dynamical Meteorology (Cambridge: The Univ. Press, 1934); Brent D Fizicheskaya i Dinamicheskaya Meteorologiya (L. - M.: Gidrometeoizdat, 1938)
  111. Obukhov A M Izv. AN SSSR Ser. Geogr. Geofiz. 5 453 (1941)
  112. Obukhov A M Turbulentnost’ i Dinamika Atmosfery (L.: Gidrometeoizdat, 1988)
  113. Reid W H Proc. Cambr. Phil. Soc. 51 350 (1955)
  114. Cocke W J Phys. Fluids 12 2488 (1969)
  115. Cocke W J Phys. Fluids 14 1624 (1971)
  116. Kazantsev A P Zh. Eksp. Teor. Fiz. 53 1806 (1967); Kazantsev A P Sov. Phys. JETP 26 1031 (1968)
  117. Vainshtein S I, Zel’dovich Ya B, Ruzmaikin A A Turbulentnoe Dinamo v Astrofizike (M.: Nauka, 1980)
  118. Zeldovich Ya B, Ruzmaikin A A, Sokoloff D D The Almighty Chance (World Scientific Lecture Notes in Physics) Vol. 20 (Singapore: World Scientific, 1990)
  119. Zel’dovich Ya B i dr. Usp. Fiz. Nauk 152 3 (1987); Zel’dovich Ya B et al. Sov. Phys. Usp. 30 353 (1987)
  120. Taylor G I Proc. London Math. Soc. 2 20 196 (1921)
  121. Galeev A A, Sudan A V (Red.) Osnovy Fiziki Plazmy Vol. 1, 2 (M.: Energoatomizdat, 1983, 1984); Galeev A A, Sudan R N (Eds) Basic Plasma Physics Vol. 1, 2 (Amsterdam: North-Holland, 1983, 1984)
  122. Kadomtsev B B Tokamak Plasma: A Complex Physical System (Bristol: Institute of Physics Publ., 1992)
  123. Horton W, Ichikawa Y-H Chaos And Structures In Nonlinear Plasmas (Singapore: World Scientific, 1996)
  124. Balescu R Aspects Of Anomalous Transport In Plasmas (Bristol: IOP Publ., 2005)
  125. "Progress in the ITER Physics Basis" Nucl. Fusion 47 (6) (2007)
  126. Zelenyi L M, Veselovskii I S (Red.) Plazmennaya Geliogeofizika Vol. 1, 2 (M.: Fizmatlit, 2008)
  127. Bakunin O G Review Of Plasma Physics Vol. 24 (Ed. V D Shafranov) (Berlin: Springer-Verlag, 2008) p. 53
  128. Diamond P H, Itoh S-I, Itoh K Modern Plasma Physics (Cambridge: Cambridge Univ. Press, 2010)
  129. Kadomtsev B B "Turbulentnost’ plazmy" Voprosy Teorii Plazmy Vol. 4 (Pod red. M A Leontovicha) (M.: Gosatomizdat, 1964) p. 188; Kadomtsev B B Plasma Turbulence (London: Academic Press, 1965)
  130. Kadomtsev B B Kollektivnye Yavleniya v Plazme (M.: Nauka, 1988); Kadomtsev B B Reviews Of Plasma Physics Vol. 22 (Ed. V D Shafranov) (New York: Kluwer Acad./Plenum Publ., 2001) p. 1
  131. Van Kampen N G Phys. Norvegica 5 279 (1971)
  132. Novikov E A Izv. AN SSSR. Fizika Atmosfery Okeana 7 1087 (1971)
  133. Arnold V C.R. Acad. Sci. Paris 260 5668 (1965)
  134. Arnold V I, Khesin B A Topological Methods In Hydrodynamics (New York: Springer, 1999); Arnol’d V I, Khesin B A Topologicheskie Metody v Gidrodinamike (M.: Izd-vo MTsNMO, 2007)
  135. Hénon M C.R. Acad. Sci. Paris 262 312 (1966)
  136. Lorenz E N J. Atmos. Sci. 20 130 (1963)
  137. Berry M V et al. Ann. Physics 122 26 (1979)
  138. Simiu E Chaotic Transitions In Deterministic And Stochastic Dynamical Systems: Applications Of Melnikov Processes In Engineering, Physics, And Neuroscience (Princeton: Princeton Univ. Press, 2002); Simiu E Khaoticheskie Perekhody v Determinirovannykh i Stokhasticheskikh Sistemakh: Primenenie Metoda Mel’nikova v Tekhnike, Fizike i Neirofiziologii (M.: Fizmatlit, 2007)
  139. Kolmogorov A N Dokl. Akad. Nauk SSSR 98 527 (1954); Kolmogorov A N Lecture Notes Phys. 93 51 (1979)
  140. Moser J Math. Ann. 126 325 (1953)
  141. Moser J Commun. Pure Appl. Math. 11 81 (1958)
  142. Zel’dovich Ya B Zh. Eksp. Teor. Fiz. 31 154 (1956); Zel’dovich Ya B Sov. Phys. JETP 4 460 (1957)
  143. Zel’dovich Ya B, Ruzmaikin A A, Sokoloff D D Magnetic Fields In Astrophysics (New York: Gordon and Breach, 1983); Zel’dovich Ya B, Ruzmaikin A A, Sokolov D D Magnitnye Polya v Astrofizike (M. - Izhevsk: RKhD, Inst. komp’yut. issled., 2006)
  144. Sokolov D D, Stepanov R A, Frik P G Usp. Fiz. Nauk 184 313 (2014); Sokoloff D D, Stepanov R A, Frick P G Phys. Usp. 57 292 (2014)
  145. Molchanov S A, Ruzmaikin A A, Sokolov D D Usp. Fiz. Nauk 145 593 (1985); Molchanov S A, Ruzmaikin A A, Sokolov D D Sov. Phys. Usp. 28 307 (1985)
  146. Zaslavskii G M, Chirikov B V Usp. Fiz. Nauk 105 3 (1971); Zaslavskii G M, Chirikov B V Sov. Phys. Usp. 14 549 (1972)
  147. Zaslavskii G M Stokhastichnost’ Dinamicheskikh Sistem (M.: Nauka, 1984)
  148. Kleva R G, Drake J F Phys. Fluids 27 1686 (1984)
  149. Solomon T H, Gollub J P Phys. Rev. A 38 6280 (1988)
  150. Mel’nikov V K Tr. Mosk. Matem. Obshchestva 12 3 (1963)
  151. Morozov A I, Solov’ev L S Voprosy Teorii Plazmy Vol. 2 (Pod red. M A Leontovicha) (M.: Gosatomizdat, 1963) p. 3; Morozov A I, Soloviev L S Reviews Of Plasma Physics Vol. 2 (Ed. M A Leontovich) (New York: Consultants Bureau, 1966) p. 1
  152. Samelson R M, Wiggins S Lagrangian Transport In Geophysical Jets And Waves: The Dynamical Systems Approach (New York: Springer, 2006); Samel’son R M, Viggins S Lagranzhev Perenos v Geofizicheskikh Struyakh i Volnakh: Podkhod na Osnove Teorii Dinamicheskikh Sistem (M. - Izhevsk: RKhD, Inst. komp’yut. issled., 2009)
  153. Ponty Y et al. Solar and Planetary Dynamos. Proc. of a NATO Advanced Study Institute, Isaac Newton Institute, Cambridge, September 1992 (Eds M R E Proctor, P C Matthews, A M Rucklidge) (Cambridge: Cambridge Univ. Press, 1993) p. 241
  154. Kida S, Takaoka M Annu. Rev. Fluid Mech. 26 169 (1994)
  155. Flor J-B (Ed.) Fronts, Waves, And Vortices In Geophysical Flows (Heidelberg: Springer, 2010)
  156. Saleur H, Duplantier B Phys. Rev. Lett. 58 2325 (1987)
  157. Stauffer D Introduction To Percolation Theory (London: Taylor and Francis, 1985)
  158. Klyatskin V I Usp. Fiz. Nauk 178 419 (2008); Klyatskin V I Phys. Usp. 51 395 (2008)
  159. Klyatskin V I Usp. Fiz. Nauk 181 457 (2011); Klyatskin V I Phys. Usp. 54 441 (2011)
  160. Galeev A A, Kuznetsova M M, Zeleny L M Space Sci. Rev. 44 1 (1986)
  161. Isichenko M B i dr. Zh. Eksp. Teor. Fiz. 96 913 (1989); Isichenko M B et al. Sov. Phys. JETP 69 517 (1989)
  162. Bakunin O G Usp. Fiz. Nauk 173 757 (2003); Bakunin O G Phys. Usp. 46 733 (2003)
  163. Gruzinov A V, Isichenko M B, Kalda Ya L Zh. Eksp. Teor. Fiz. 97 476 (1990); Gruzinov A V, Isichenko M B, Kalda Ya L Sov. Phys. JETP 70 263 (1990)
  164. Hussain F, Duraisamy K Phys. Fluids 23 021701 (2011)
  165. Bakunin O G J. Plasma Phys. 71 435 (2005)
  166. Misguich J H et al. Physicalia Mag. 20 103 (1998)
  167. Zimbardo G, Veltri P, Pommois P Phys. Rev. E 61 1940 (2000)
  168. Reuss J-D, Misguich J H Phys. Rev. E 54 1857 (1996)
  169. Vlad M et al. J. Plasma Phys. 59 707 (1998)
  170. Isichenko M B et al. Phys. Fluids B 4 3973 (1992)
  171. Ottaviani M Europhys. Lett. 20 111 (1992)
  172. Dreizin Yu A, Dykhne A M Zh. Eksp. Teor. Fiz. 63 242 (1972); Dreizin Yu A, Dykhne A M Sov. Phys. JETP 36 127 (1973)
  173. Bakunin O G J. Plasma Phys. 72 647 (2006)
  174. Isichenko M B Plasma Phys. Control. Fusion 33 809 (1991)
  175. Isichenko M B Plasma Phys. Control. Fusion 33 795 (1991)
  176. Bakunin O G Plasma Phys. Control. Fusion 45 1909 (2003)
  177. Lifshits I M, Slutskin A A, Nabutovskii V M Zh. Eksp. Teor. Fiz. 41 939 (1961); Lifshits I M, Slutskin A A, Nabutovskii V M Sov. Phys. JETP 14 669 (1962)
  178. Timofeev A V Zh. Eksp. Teor. Fiz. 75 1303 (1978); Timofeev A V Sov. Phys. JETP 48 656 (1978)
  179. Escande D F Plasma Theory and Nonlinear and Turbulent Processes in Physics. Proc. of the International Workshop, Kiev, USSR, 13 - 25 April 1987 (Eds V G Bar’yakhtar et al.) (Singapore: World Scientific, 1988) p. 398
  180. Neishtadt A I, Chaikovskii D K, Chernikov A A Zh. Eksp. Teor. Fiz. 99 763 (1991); Neishtadt A I, Chaikovskii D K, Chernikov A A Sov. Phys. JETP 72 423 (1991)
  181. Vainshtein D L, Vasiliev A A, Neishtadt A I Chaos 6 514 (1996)
  182. Jokipii J R, Parker E N Astrophys. J. 155 777 (1969)
  183. Ptuskin V S Astrophys. Space Sci. 61 251 (1979)
  184. Jokipii J R Astrophys. J. 183 1029 (1973)
  185. Wesson J Tokamaks (Oxford: Clarendon Press, 1987)
  186. Miyamoto K Plasma Physics And Controlled Nuclear Fusion (Berlin: Springer, 2005); Miyamoto K Osnovy Fiziki Plazmy i Upravlyaemogo Sinteza (M.: Fizmatlit, 2007)
  187. Rosenbluth M N et al. Nucl. Fusion 6 297 (1966)
  188. Berezinskii V S i dr. Astrofizika Kosmicheskikh Luchei (Pod red. V L Ginzburga) 2-e izd. (M.: Nauka, 1990); Berezinskii V S et al. Astrophysics Of Cosmic Rays (Ed. V L Ginzburg) (Amsterdam: North-Holland, 1990)
  189. Golitsyn G S Issledovanie Konvektsii s Geofizicheskimi Prilozheniyami i Analogiyami (L.: Gidrometeoizdat, 1980)
  190. Rechester A B, Rosenbluth M N Phys. Rev. Lett. 40 38 (1978)
  191. Falgarone E, Passot T (Eds) Turbulence And Magnetic Fields In Astrophysics (Berlin: Springer, 2003)
  192. Chandran B D G, Cowley S C Phys. Rev. Lett. 80 3077 (1998)
  193. Dinklage A et al. (Eds) Plasma Physics. Confinement, Transport And Collective Effects (Lecture Notes in Physics) Vol. 670 (Berlin: Springer-Verlag, 2005)
  194. Bakunin O G Usp. Fiz. Nauk 173 317 (2003); Bakunin O G Phys. Usp. 46 309 (2003)
  195. Iroshnikov P S Astron. Zhurn. 40 742 (1963); Iroshnikov P S Sov. Astron. 7 566 (1964)
  196. Kraichnan R H Phys. Fluids 8 1385 (1965)
  197. Goldreich P, Sridhar S Astrophys. J. 438 763 (1995)
  198. Goldreich P, Sridhar S Astrophys. J. 485 680 (1997)
  199. Boldyrev S Phys. Rev. Lett. 96 115002 (2006)
  200. Narayan R, Medvedev M V Astrophys. J. 562 L129 (2001)
  201. Matthaeus W H, Goldstein M L J. Geophys. Res. 87 6011 (1982)
  202. Armstrong J W, Rickett B J, Spangler S R Astrophys. J. 443 209 (1995)
  203. Maron J, Goldreich P Astrophys. J. 544 1175 (2001)
  204. Cho J, Lazarian A, Vishniac E T Astrophys. J. 564 291 (2002)
  205. Tobias S M, Cattaneo F, Boldyrev S Ten Chapters In Turbulence (Eds P A Davidson, Y Kaneda, K R Sreenivasan) (New York: Cambridge Univ. Press, 2013)
  206. Schekochihin A A, Cowley S C Magnetohydrodynamics. Historical Evolution And Trends (Fluid Mechanics and Its Applications) Vol. 80 (Eds S S Molokov et al.) (Berlin: Springer-Verlag, 2007) p. 85
  207. Galtier S et al. J. Plasma Phys. 63 447 (2000)
  208. Galtier S Nonlin. Proces. Geophys. 16 83 (2009)
  209. Galtier S, Chandran B D G Phys. Plasmas 13 114505 (2006)
  210. Galtier S et al. Astrophys. J. Lett. 564 L49 (2002)
  211. Nazarenko S Wave Turbulence (Lecture Notes in Physics) Vol. 825 (Berlin: Springer-Verlag, 2011)
  212. Batchelor G K Quart. J. R. Meteor. Soc. 76 133 (1950)
  213. Kraichnan R H J. Fluid Mech. 47 525 (1971)
  214. Batchelor G K Phys. Fluids 12 II-233 (1969)
  215. Bakunin O G Physica A 345 1 (2005)
  216. Biskamp D Magnetohydrodynamic Turbulence (Cambridge: Cambridge Univ. Press, 2003)
  217. Pouquet A J. Fluid Mech. 88 1 (1978)
  218. Chernyshov A A, Karel’skii K V, Petrosyan A S Usp. Fiz. Nauk 184 457 (2014); Chernyshov A A, Karelsky K V, Petrosyan A S Phys. Usp. 57 421 (2014)
  219. Zel’dovich Ya B Pis’ma ZhETF 38 51 (1983); Zel’dovich Ya B JETP Lett. 38 57 (1983)
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