Vol. 34, issue 09, article # 9

Nosov V. V., Lukin V. P., Kovadlo P. G., Nosov E. V., Torgaev A. V. Intermittency of Kolmogorov and coherent turbulence in the mountain boundary layer (overview). // Optika Atmosfery i Okeana. 2021. V. 34. No. 09. P. 726–749. DOI: 10.15372/AOO20210909 [in Russian].
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This overview is devoted to the intermittency of atmospheric turbulence of different types (of Kolmogorov and coherent) in the mountain boundary layer. A brief overview of the world scientific literature on the types of turbulence intermittency is made for a better understanding of the place of intermittency of different types among them. Due to available in literature various interpretations of the coherent turbulence concept, our earlier overviews of the world scientific literature on coherent turbulence and coherent structures are supplemented; these overviews describe the mechanisms of the origination (formation) of coherent turbulence and the key properties of coherent turbulence, as well as indicate the differences and relationships between the Kolmogorov and coherent turbulence. Discussion of the contemporary scientific understanding of turbulence structure can be attributed to the overviews results. Thus, the authors previously (2008–2019) independently showed that atmospheric turbulence can be considered as an incoherent mixture of various coherent structures. At the same time, there is a conjecture by E. Hopf (1948) about the finite dimensionality of attractors in the phase space of solutions of the Navier–Stokes equations. Physical interpretation of this conjecture, as pointed out by A.S. Monin and A.M. Yaglom (1991, 1992), is the representation of turbulence structure as a spatio-temporal chaos of a finite number of interacting coherent structures. From comparison of these representations, it can be seen that the authors’ results are in fact a proof of E. Hopf's conjecture in interpretation, formulated by A.S. Monin and A.M. Yaglom, and that the “chaos" of a turbulence is to a large extent determinate.
The overview results of the study of the turbulence intermittency of different types are the consequent of many years experimental studies of turbulence by acoustic and optical methods performed by the authors at mountain astronomical observatories. Lifetimes of the turbulence different types: Kolmogorov and coherent, recorded in optical and meteorological measurements are given. It is shown that the effect of the intermittency of turbulence types characterizes the turbulence local structure over the observatory location region and allows to develop practical recommendations for the most favorable observation regime at astronomical observatories.


intermittency of turbulence, coherent turbulence, coherent structure, coherent turbulence spectrum, coherent structure spectrum, Kolmogorov and coherent turbulence relationship


  1. Monin A.S., Yaglom A.M. Statisticheskaya gidromekhanika. Mekhanika turbulentnosti: v 2-h pt. M: Nauka, 1965. Pt. 1. 640 p.
  2. Monin A.S., Yaglom A.M. Statisticheskaya gidromekhanika. Mekhanika turbulentnosti: v 2-h pt. M: Nauka, 1967. Pt. 2. 720 p.
  3. Monin A.S., Yaglom A.M. Statisticheskaya gidromekhanika. V. 1. SPb.: Gidrometeoizdat, 1992. 696 p.
  4. Monin A.S., Yaglom A.M. Statisticheskaya gidromekhanika. V. 2. SPb.: Gidrometeoizdat, 1996. 742 p.
  5. Rejnol'ds O. Dinamicheskaya teoriya dvizheniya neszhimaemoj vyazkoj zhidkosti i opredelenie kriteriya // Problemy turbulentnosti. M.: ONTI, 1936. P. 185–227. Perevod izd.: Reynolds O. On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion. 1894.
  6. Fridman A.A., Keller L.V. Differentsial'nye uravneniya turbulentnogo dvizheniya szhimaemoj zhidkosti // Izbrannye trudy. M.: Nauka, 1966. P. 45–57. Perevod izd.: Friedman A., Keller L. Differentialgleichungen für die turbulente Bewegung einer kompressiblen Flüssigkeit. Delft, 1925.
  7. Nosov V.V., Lukin V.P., Nosov E.V., Torgaev A.V. The emergence and evolution of the concept “coherent turbulence” // J. Phys.: Conf. Ser. 2020. V. 1499, N 012005. DOI: 10.1088/1742-6596/1499/1/012005.
  8. Nosov V.V., Lukin V.P., Nosov E.V., Torgaev A.V. Formirovanie turbulentnosti v astronomicheskih observatoriyah yuga Sibiri i Severnogo Kavkaza // Optika atmosf. i okeana. 2019. V. 32, N 3. P. 228–246; Nosov V.V., Lukin V.P., Nosov E.V., Torgaev A.V. Formation of turbulence at astronomical observatories in Southern Siberia and North Caucasus // Atmos. Ocean. Opt. 2019. V. 32, N 4. P. 464–482. DOI: 10.15372/ AOO20190309.
  9. Tokovinin A. Where is the surface-layer turbulence? // Proc. SPIE. 2010. V. 7733. P. 77331N. DOI: 10.1117/ 12.856409.
  10. Gur'yanov A.E. O pul'satsiyah temperatury v vozdushnoj srede vblizi teleskopa noch'yu // Astroklimat i effektivnost' teleskopov. L.: Nauka, 1984. P. 164–168.
  11. Sрlihting G. Vozniknovenie turbulentnosti. M.: IIL, 1962. 204 p. Perevod izd.: Schlichting H. Entstehung der turbulenz. Heidelberg, 1959.
  12. Drajden H.L. Perekhod laminarnogo techeniya v turbulentnoe // Turbulentnye techeniya i teploperedacha. M.: IL, 1963. P. 9–82. Perevod izd.: Dryden H.L. Transition from Laminar to Turbulent Flow, 1959.
  13. Stuart J.T. Hydrodynamic stability // Laminar Boundary Layers. Oxford, UK: Clarendon Press. 1963. P. 492–579.
  14. Lin' Ts.Ts. Teoriya gidrodinamicheskoj ustojchivosti. M.: IL, 1958. 194 p. Perevod izd.: Lin C.C. The Theory of Hydrodynamic Stability. Cambridge, 1955.
  15. Chandrasekhar S. Hydrodynamic and Hydromagnetic Stability. England: Oxford University Press, 1961. 652 p.
  16. Reynolds O. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels // Proc. Roy. Soc. 1883. V. 35. P. 84–99. DOI: 10.1098/rspl.1883.0018.
  17. Schiller L. Neue quantitative versuche zur turbulenzentstehung // ZAMM J. Appl. Math. Mech. / Zeitschrift für Angewandte Mathematik und Mechanik. 1934. V. 14, N 1. P. 36–42. DOI: 10.1002/zamm.19340140105.
  18. Sрlihting G. Teoriya pogranichnogo sloya. M.: IIL, 1956. 528 p. Perevod izd.: Schlichting H. Grenzschicht-Theorie. Karlsruhe, 1951.
  19. Letellier C. Intermittency as a transition to turbulence in pipes: A long tradition from Reynolds to the 21st century // Comptes Rendus Mécanique. 2017. V. 345, iss. 9. P. 642–659. DOI: 10.1016/j.crme.2017.06.004.
  20. Frisch U. Turbulence: The Legacy of A.N. Kolmogorov. Cambridge: University Press, 1995. 313 p.
  21. Fage A., Townend H.C.H. An examination of turbulent flow with an ultramicroscope // Proc. Roy. Soc. A. 1932. V. 135, N 828. P. 656–677. DOI: 10.1098/rspa. 1932.0059.
  22. Sandborn V.A. Measurements of intermittency of turbulent motion in a boundary layer // J. Fluid Mech. 1959. V. 6, N 2. P. 221–240. DOI: 10.1017/ S0022112059000581.
  23. Novikov E.A. Izmenchivost' dissipatsii energii v turbulentnom potoke i raspredelenie energii po spektru // Prikladnaya matematika i mekhanika. 1963. V. 27, N 5. P. 944–946.
  24. Novikov E.A., Styuart R.V. Peremezhaemost' turbulentnosti i spektr flyuktuatsij dissipatsii energii // Izv. AN SSSR. Ser. geofiz. 1964. V. 3. P. 408–413.
  25. Mahrt L. Intermittency of atmospheric turbulence // J. Atmos. Sci. 1989. V. 46, N 1. P. 79–95.
  26. She Z.S., Jackson E., Orszag S.A. Intermittent vortex structures in homogeneous isotropic turbulence // Nature. 1990. V. 344, N 6263. P. 226–228. DOI: 10. 1038/344226a0.
  27. Hagelberg C.R., Gamage N.K.K. Structure-preserving wavelet decompositions of intermittent turbulence // Bound.-Lay. Meteorol. 1994. V. 70, N 3. P. 217–246. DOI: 10.1007/BF00709120.
  28. Young G.S., Kristovich D.A., Hjelmfelt M.R., Foster R.C. Rolls, streets, waves, and more: A review of quasi-two-dimensional structures in the atmospheric boundary layer // Bull. Am. Meteorol. Soc. 2002. V. 83, N 7. P. 997–1002.
  29. Acevedo O.C., Moraes O.L., Degrazia G.A., Medeiros L.E. Intermittency and the exchange of scalars in the nocturnal surface layer // Bound.-Lay. Meteorol. 2006. V. 119, N 1. P. 41–55. DOI: 10.1007/s10546-005-9019-3.
  30. Chowdhuri S., Prabhakaran T., Banerjee T. Persistence behavior of heat and momentum fluxes in convective surface layer turbulence // Phys. Fluids. 2020. V. 32, N 11. P. 115107. DOI: 10.1063/5.0027168.
  31. Barthlott C., Drobinski P., Fesquet C., Dubos T., Pietras C. Long-term study of coherent structures in the atmospheric surface layer // Bound.-Lay. Meteorol. 2007. V. 125. N 1. P. 1–24. DOI: 10.1007/s10546-007-9190-9.
  32. Velikanov M.A. Dinamika ruslovyh potokov. V. 1: Struktura potoka. M.: Gostekhizdat, 1954. 324 p.
  33. Zommerfel'd A. Mekhanika deformiruemyh sred. M.: IIL, 1954. 491 p. Perevod izd.: Sommerfeld A. Mechanik der deformierbaren Medien. Wiesbaden, 1949.
  34. Birkgof G. Gidrodinamika. Metody. Fakty. Podobie. M.: IIL, 1963. 244 p. Perevod izd.: Birkhoff G. Hydrodynamics. A study in logic, fact and similitude. Princeton, 1960.
  35. Kochin N.E., Kibel' I.A., Roze N.V. Teoreticheskaya gidromekhanika. Pt. 2. M.: Fizmatgiz, 1963. 728 p.
  36. Betchov R., Kriminale V. Voprosy gidrodinamicheskoj ustojchivosti. M.: Mir, 1971. 350 p. Perevod izd.: Betchov R., Criminale W.O. Stability of Parallel Flows. New York, 1967.
  37. Kolchinskij I.G. Opticheskaya nestabil'nost' zemnoj atmosfery po nablyudeniyam zvezd. Kiev: Naukova dumka, 1967. 184 p.
  38. Tatarskij V.I. Rasprostranenie voln v turbulentnoj atmosfere. M.: Nauka, 1967. 548 p.
  39. Nosov V.V., Emaleev O.N., Lukin V.P., Nosov E.V. Poluempiricheskie gipotezy teorii turbulentnosti v anizotropnom pogranichnom sloe // Optika atmosf. i okeana. 2005. V. 18, N 10. P. 845–863.
  40. Nosov V.V. Atmospheric Turbulence in the Anisotropic Boundary Layer. In Optical waves and laser beams in the irregular atmosphere / N. Blaunshtein, N. Kopeika (eds.) London, New York, USA: Boca Raton, Taylor & Francis Group, CRC Press, 2018. Chap. 3, P. 67–180.
  41. Lukin V.P., Nosov V.V., Nosov E.V., Torgaev A.V. Causes of non-Kolmogorov turbulence in the atmosphere // Appl. Opt. 2016. V. 55, N 12. P. B163–B168. DOI: 10.1364/AO.55.00B163.
  42. Nosov V.V., Lukin V.P., Kovadlo P.G., Nosov E.V., Torgaev A.V. Opticheskie svojstva turbulentnosti v gornom pogranichnom sloe atmosfery. Novosibirsk: Izd-vo SO RAN, 2016. 153 p.
  43. Lukin V.P., Bol'basova L.A., Nosov V.V. Comparison of Kolmogorov’s and coherent turbulence // Appl. Opt. 2014. V. 53, N 10. P. B231–B236. DOI: 10.1364/AO.53.00B231.
  44. Nosov V.V., Kovadlo P.G., Lukin V.P., Torgaev A.V. Atmosfernaya kogerentnaya turbulentnost' // Optika atmosf. i okeana. 2012. V. 25, N 9. P. 753–759; Nosov V.V., Kovadlo P.G., Lukin V.P., Torgaev A.V. Atmospheric coherent turbulence // Atmos. Ocean. Opt. 2013. V. 26, N 3. P. 201–206.
  45. Nosov V.V., Grigoriev V.M., Kovadlo P.G., Lukin V.P., Nosov E.V., Torgaev A.V. Intermittency of the astronomical images jitter in the high-mountain observations // Proc. SPIE. 2014. V. 9292. P. 92920V1–4. DOI: 10.1117/12.2074614.
  46. Nosov V.V., Lukin V.P., Nosov E.V., Torgaev A.V. The effect of intermittency of astronomical images in the high-altitude observations // Imag. Appl. Opt. 2019. OSA. 2019. DOI: 10.1364/COSI.2019.JW2A.36.
  47. Nosov V.V., Lukin V.P., Nosov E.V., Torgaev A.V., Grigoriev V.M., Kovadlo P.G. Coherent structures in the turbulent atmosphere  / A.B. Nadycto et al. (eds.) // Math. Models Non-lin. Phenomen., Process. Systems: From Molecular Scale to Planetary Atmosphere. New York: Nova Science Publishers, 2013. Chap. 20. P. 297–330.
  48. Nosov V.V., Lukin V.P., Nosov E.V., Torgaev A.V. The Evolution of Turbulence Structure over Inhomogeneously Heated Surfaces. Nonlinearity: Problems, Solutions and Applications. V. 1 / L.A. Uvarova, A.B. Nadicto, A.V. Latyshev (eds.). New York: Nova Science Publishers, 2017. Chap. 17. P. 335–411.
  49. Nosov V.V., Lukin V.P., Nosov E.V., Torgaev A.V., Grigoriev V.M., Kovadlo P.G. Coherent structures in turbulent atmosphere // Proc. SPIE. 2009. V. 7296-09. P. 53–70. DOI: 10.1117/12.823804.
  50. Nosov V.V., Lukin V.P., Nosov E.V., Torgaev A.V. Struktura turbulentnosti nad nagretymi poverhnostyami. CHislennye resheniya // Optika atmosf. i okeana. 2016. V. 29, N 1. P. 23–30; Nosov V.V., Lukin V.P., Nosov E.V., Torgaev A.V. Turbulence structure over heated surfaces: Numerical solutions // Atmos. Ocean. Opt. 2016, V. 29, N 3. P. 234–243. DOI: 10.15372/AOO20160103.
  51. Nosov V.V., Lukin V.P., Nosov E.V., Torgaev A.V. Kolmogorov's and coherent turbulence in the atmosphere // Imag. Appl. Opt. 2019. DOI: 10.1364/PCAOP.2019. PM3C.3.
  52. Nosov V.V., Lukin V.P., Nosov E.V., Torgaev A.V., Afanas'ev V.L., Balega Yu.Yu., Vlasyuk V.V., Panchuk V.E., Yakopov G.V. Issledovaniya astroklimata v Spetsial'noj astrofizicheskoj observatorii RAN // Optika atmosf. i okeana. 2018. V. 31, N 8. P. 616–627; Nosov V.V., Lukin V.P., Nosov E.V., Torgaev A.V., Afanas’ev V.L., Balega Yu.U., Vlasyuk V.V., Panchuk V.E., Yakopov G.V. Astroclimate studies in the special astrophysical observatory of the Russian Academy of Sciences // Atmos. Ocean. Opt. 2019, V. 32, N 1. P. 8–18. DOI: 10.15372/ AOO20180804.
  53. Nosov V.V., Lukin V.P., Nosov E.V., Torgaev A.V. Representation of the synoptic spectra of atmospheric turbulence by sums of spectra of coherent structures // IOP Conf. Ser.: Earth and Environ. Sci. 2019. V. 231, N 012040. P. 1–7. DOI: 10.1088/1755-1315/231/1/ 012040.
  54. Nosov V.V., Grigor'ev V.M., Kovadlo P.G., Lukin V.P., Nosov E.V., Torgaev A.V. Kogerentnye struktury – elementarnye sostavlyayushchie atmosfernoj turbulentnosti // Izv. vuzov. Fizika. 2012. V. 55, N 9/2. P. 236–238.
  55. Kolmogorov A.N. Lokal'naya struktura turbulentnosti v neszhimaemoj vyazkoj zhidkosti pri ochen' bol'shih chislah Rejnol'dsa // Dokl. AN SSSR. 1941. V. 30, N 4. P. 299–303.
  56. Kolmogorov A.N. K vyrozhdeniyu izotropnoj turbulentnosti v neszhimaemoj vyazkoj zhidkosti // Dokl. AN SSSR. 1941. V. 31, N 6. P. 538–541.
  57. Kolmogorov A.N. Rasseyanie energii pri lokal'no izotropnoj turbulentnosti // Dokl. AN SSSR. 1941. V. 32, N 1. P. 19–21.
  58. Kolmogorov A.N. Uravneniya turbulentnogo dvizheniya neszhimaemoj zhidkosti // Izv. AN SSSR. Ser. fiz. 1942. V. 6, N 1–2. P. 56–58.
  59. Kolmogorov A.N. Utochnenie predstavlenij o lokal'noj strukture turbulentnosti v neszhimaemoj vyazkoj zhidkosti pri bol'shih chislah Rejnol'dsa // Mecanique de la turbulence: Colloq. Intern. CNRS, Marseille, aout-sept. 1961 / Na rus. i fr. yaz. Paris, 1962. P. 447–458.
  60. Prandtl L. Bericht über Untersuchungen zur ausgebildeten Turbulenz // ZAMM – J. Appl. Math. Mech. / Zeitschrift für Angewandte Mathematik und Mechanik. 1925. Bd. 5. N 2. P. 136–139. DOI: 10.1002/zamm. 19250050212.
  61. Reynolds O. An Experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels // Proc. Roy. Soc. 1883. V. 35. P. 84–99. DOI: 10.1098/rspl.1883.0018.
  62. Reynolds O. On the dynamical theory of incompressible viscous fluids and the determination of the criterion // Proc. Roy. Soc. 1894. V. 56. P. 40–45. DOI: 10.1098/rspl.1894.0075.
  63. Liepmann H.W. Aspects of the turbulence problem. Part 1 // J. Appl. Math. Phys. (ZAMP). 1952. V. 3, iss. 5. P. 321–342. DOI: 10.1007/BF02008148.
  64. Batchelor G.K. The Theory of Homogeneous Turbulence. New York: Cambridge University Press, 1953. 197 p.
  65. Townsend A.A. The structure of Turbulent Shear Flow. Cambridge: Cambridge University Press, 1956. 315 p.
  66. Bradshaw P. The turbulence structure of equilibrium boundary layers // J. Fluid Mech. 1967. V. 29. P. 625–645. DOI: 10.1017/S0022112067001089.
  67. Kaplan R.E., Laufer J. The intermittently turbulent region of the boundary layer // Appl. Mech. Proc. 12th Internat. Congr. of Appl. Mech., 1968. Berlin, Heidelberg: Springer, 1969. P. 236–245. DOI: 10. 1007/978-3-642-85640-2_17.
  68. Liepmann H.W. Experimental fluid mechanics: The impact of modern instrumentation // Proc. 13 Internat. Congr. Theor. Appl. Mech. Moscow Univers. Aug. 21–26, 1972. Berlin, Heidelberg: Springer, 1973. P. 200–212. DOI: 10.1007/978-3-642-65590-6_13.
  69. Brown G., Roshko A. On density effects and large structure in turbulent mixing layers // J. Fluid Mech. 1974. V. 64, N 4. P. 775–816. DOI: 10.1017/ S002211207400190X.
  70. Davies P., Yule A. Coherent structures in turbulence // J. Fluid Mech. 1975. V. 69, iss. 3. P. 513–537. DOI: 10.1017/S0022112075001541.
  71. Frost W., Moulden T.H. (eds.) Handbook of Turbulence. Vol. 1: Fundamentals and Applications. New York: Plenum Press. 1977. 498 p. DOI: 10.1007/978-1-4684-2322-8.
  72. Kantuell B.Dzh. Organizovannye dvizheniya v turbulentnyh potokah // Vihri i volny. Sb. statej. M.: Mir, 1984. P. 9–79. DOI: 10.1146/annurev.fl.13. 010181.002325.
  73. Feigenbaum M.J. Quantitative universality for a class of nonlinear transformations // J. Stat. Phys. 1978. V. 19. P. 25–52. DOI: 10.1007/BF01020332.
  74. Michalke A., Fuchs H. On turbulence and noise of an axisymmetric shear flow // J. Fluid Mech. 1975. V. 70. P. 179–205. DOI: 10.1017/S0022112075001966.
  75. Adrian R.J. On the role of conditional averages in turbulence theory // Conf. Proc. 4th Biennial Symp. on Turbulence in Liquids. University of Missouri-Rolla, 22-24 Sept. 1975. Princeton: Science Press, 1977. P. 323–332.
  76. Hussain A.K.M.F. Coherent structures and studies of perturbed and unperturbed jets // The Role of Coherent Structures in Modelling Turbulence and Mixing. Lect. Not. in Phys. Berlin, Heidelberg: Springer, 1981. V. 136. P. 252–291. DOI: 10.1007/3-540-10289-2_30.
  77. Lojtsyanskij L.G. Mekhanika zhidkosti i gaza. 6-e izd. M.: Nauka, 1987. 840 p.
  78. Sedunov Yu.S., Avdyushin S.I., Borisenkov E.P., Volkovitskij O.A., Petrov N.N., Rejtenbah R.G., Smirnov V.I., Chernikov A.A. (red.). Atmosfera. Spravochnik (spravochnye dannye, modeli). L.: Gidrometeoizdat, 1991. 510 p.
  79. Vernet A., Kopp G.A., Ferré J.A., Giralt F. Three-dimensional structure and momentum transfer in a turbulent cylinder wake // J. Fluid Mech. 1999. V. 394. P. 303–337. DOI: 10.1017/S0022112099005807.
  80. Ryu J., Cheong C., Kim S., Lee S. Computation of internal aerodynamic noise from a quick-opening throttle valve using frequency-domain acoustic analogy // Appl. Acoustics. 2005. V. 66, N 11. DOI: 10.1016/j.apacoust. 2005.04.002.
  81. Millet C., Robinet J.C., Roblin C. On using computational aeroacoustics for long range propagation of infrasounds in realistic atmospheres // Geophys. Res. Lett. 2007. V. 34, N 14. DOI: 10.1029/2007GL029449.
  82. Borovik A.V., Konyaev P.A. Astroklimat Bajkal'skoj astrofizicheskoj observatorii ISZF SO RAN // Izv. Irkutskogo gos. uni-ta. Ser. Nauki o Zemle. 2014. V. 8. P. 25–34.
  83. Volkov M.V., Garanin S.G., Kozlova T.I., Konoval'tsov M.I., Kopalkin A.V., Lebedev R.S., Starikov F.A., Techko O.L., Tyutin S.V., Hohlov S.V., Tsykin V.S. Fazirovka izlucheniya 7-kanal'nogo optovolokonnogo lazera s dinamicheskimi turbulentnymi iskazheniyami fazy s ispol'zovaniem stohasticheskogo parallel'nogo gradientnogo algoritma pri shirine polosy 450 kGz // Kvant. elektron. 2020. V. 50, N 7. P. 694–699. DOI: 10.1070/QEL17193.
  84. Nosov V.V., Grigor'ev V.M., Kovadlo P.G., Lukin V.P., Nosov E.V., Torgaev A.V. Kogerentnye struktury v turbulentnoj atmosfere. Eksperiment i teoriya // Solnechno-zemnaya fizika. 2009. Iss. 14. P. 97–113.
  85. Mironov V.L., Nosov V.V. O vliyanii vneshnego masshtaba atmosfernoj turbulentnosti na prostranstvennuyu korrelyatsiyu sluchajnyh smeshchenij svetovyh puchkov // Izv. vuzov. Radiofizika. 1974. V. 17, N 2. P. 274–281. DOI: 10.1007/BF01037408.
  86. Nosov V.V., Lukin V.P., Nosov E.V., Torgaev A.V. Modelirovanie kogerentnyh struktur (topologicheskih solitonov) v zakrytyh pomeshcheniyah putem chislennogo resheniya uravnenij gidrodinamiki // Optika atmosf. i okeana. 2015. V. 28, N 2. P. 120–133.
  87. Nosov V.V., Lukin V.P., Nosov E.V., Torgaev A.V. Approximations of the synoptic spectra of atmospheric turbulence by sums of spectra of coherent structures // Proс. SPIE. 2016. V. 9910. P. 99101Y1–6. DOI: 10. 1117/12.2231941.
  88. Korn G.A., Korn T.M. Spravochnik po matematike dlya uchenyh i inzhenerov. M.: Nauka, 1973. 834 p.
  89. Monin A.S. Ob opredelenii kogerentnyh struktur // Dokl. AN SSSR. 1991. V. 318, N 4. P. 853–856.
  90. Monin A.S.. O kogerentnyh strukturah v turbulentnyh techeniyah // Etyudy o turbulentnosti. M.: Nauka, 1994. P. 7–17.
  91. Hopf E. A mathematical example displaying features of turbulence // Commun. Pure Appl. Math. 1948. V. 1, N 4. P. 303–322. DOI: 10.1002/CPA.3160010401.
  92. Zhukovskij N.E. O snezhnyh zanosah i zailenii rek. 1919. P. 451–474 // Polnoe sobranie sochinenij. V. 3. Gidrodinamika. M., L.: ONTI, 1936. 487 p.
  93. Prandtl' L., Tit'ens O. Gidro- i aeromekhanika. V. 1. M.: GITTL, 1933. 224 p.
  94. Monin A.S. Struktura atmosfernoj turbulentnosti // Teoriya veroyatnostej i ee primeneniya. 1958. V. 3, iss. 3. P. 285–317.
  95. Nosov V.V., Grigor'ev V.M., Kovadlo P.G., Lukin V.P., Nosov E.V., Torgaev A.V. Astroklimat spetsializirovannyh pomeshchenij Bol'shogo solnechnogo vakuumnogo teleskopa. Pt. 1 // Optika atmosf. i okeana. 2007. V. 20, N 11. P. 1013–1021.
  96. Nosov V.V., Grigor'ev V.M., Kovadlo P.G., Lukin V.P., Nosov E.V., Torgaev A.V. Astroklimat spetsializirovannyh pomeshchenij Bol'shogo solnechnogo vakuumnogo teleskopa. Pt. 2 // Optika atmosf. i okeana. 2008. V. 21, N 3. P. 207–217.
  97. Nosov V.V., Grigor'ev V.M., Kovadlo P.G., Lukin V.P., Nosov E.V., Torgaev A.V. Kogerentnaya turbulentnost' na territorii Bajkal'skoj astrofizicheskoj observatorii // Izv. vuzov. Fizika. 2012. V. 55, N 9/2. P. 204–205.
  98. Brandt P.N., Mauter H.A., Smartt R. Day-time seeing statistics at Sacramento Peak Observatory // Astron. Astrophys. 1987. V. 188, N 1. P. 163–168.
  99. Kovadlo P.G., Ivanov V.I., Darchiya Sh.P. Fotoelektricheskij registrator drozhaniya izobrazheniya Solntsa // Issledovaniya po geomagnetizmu, aeronomii i fizike Solntsa. 1975. Iss. 37. P. 196–202.
  100. Brandt P.N. Frequency spectra of solar image motion // Solar Phys. 1969. V. 7. P. 187–203. DOI: 10.1007/BF00224897.
  101. Nosov V.V., Lukin V.P., Nosov E.V., Torgaev A.V., Bogushevich A.Ya. Measurement of atmospheric turbulence characteristics by the ultrasonic anemometers and the calibration processes // Atmosphere. 2019. V. 10, N 8. P. 1–15. DOI: 10.3390/atmos10080460.
  102. Azbukin A.A., Bogushevich A.Ya., Lukin V.P., Nosov V.V., Nosov E.V., Torgaev A.V. Apparatno-programmnyj kompleks dlya issledovanij struktury polej turbulentnyh fluktuatsij temperatury i vetra // Optika atmosf. i okeana. 2018. V. 31, N 5. P. 378–384; Azbukin A.A., Bogushevich A.Ya., Lukin V.P., Nosov V.V., Nosov E.V., Torgaev A.V. Hardware-software complex for studying the structure of the fields of temperature and turbulent wind fluctuations // Atmos. Ocean. Opt. 2018. V. 31, N 5. P. 479–485. DOI: 10.15372/AOO20180507.
  103. Azbukin A.A., Bogushevich A.Ya., Il'ichevskij V.S., Korol'kov V.A., Tihomirov A.A., Shelevoj V.D. Avtomatizirovannyj ul'trazvukovoj meteorologicheskij kompleks AMK-03 // Meteorol. i gidrol. 2006. N 11. P. 89–97.
  104. Liepmann H.W. Aspects of the turbulence problem. Part 2 // J. Appl. Math. Phys. (ZAMP) V. 3, iss. 6. P. 407–426. DOI: 10.1007/BF02025569.
  105. Townsend A.A. The Structure of Turbulent Shear Flow. 2nd ed. Cambridge: Cambridge University Press, 1976. 429 p. DOI: 10.1002/zamm.19760560921.