Russian Meteorology and Hydrology

, Volume 43, Issue 3, pp 135–142 | Cite as

Laws of Random Walks Derived by A.N. Kolmogorov in 1934

  • G. S. Golitsyn


In the first half of the 1930s A.N. Kolmogorov was developing analytical methods for the probability theory and presented the solution of the Fokker–Planck type equation. This solution contains scales for the distribution function moments of the mean squares for velocities and relative displacements of the analyzed objects and for the mixed moments of velocities and coordinates. The exclusion of time from these moments leads to the 2/3 law for the velocity structure function and to the Richardson–Obukhov law for the eddy diffusion. The analysis of the fetch laws for wind waves demonstrates that the Kolmogorov laws are manifested in the growth of wave amplitudes and in the form of elevation spectra. These laws also work in the statistics of the planetary surface relief, in the size distribution of the lithospheric plates, in the energy spectra of cosmic rays, and in other processes. In the equation deduced in 1934, probability distribution functions are derived only under the condition of homogeneity of these functions and thereby allow describing a broad range of phenomena and processes.


Random walks in the coordinate and velocity space turbulence laws wind wave fetch laws elevation spectra relief statistics 


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  1. 1.
    Astrophysics of Cosmic Rays, Ed. by V. L. Ginzburg (Amsterdam, North Holland, 1990).Google Scholar
  2. 2.
    E. B. Gledzer and G. S. Golitsyn, “Scaling and Finite Ensembles of Particles in Motion with the Energy Influx,” Dokl. Akad. Nauk, No. 4, 433 (2010) [Dokl. Phys., No. 8, 55 (2010)].Google Scholar
  3. 3.
    G. S. Golitsyn, “Coefficient of Horizontal Eddy Diffusion of a Tracer on the Water Surface as a Function of the Wave Age,” Izv. Akad. Nauk, Fiz. Atmos. Okeana, No. 3, 47 (2011) [Izv., Atmos. Oceanic Phys., No. 3, 47 (2011)].Google Scholar
  4. 4.
    G. S. Golitsyn, “Size Distribution of the Number of Lithospheric Plates,” Fizika Zemli, No. 6 (2008) [Izv. Phys. Solid Earth, No. 3, 44 (2008)].Google Scholar
  5. 5.
    G. S. Golitsyn, “On the Nature of Spiral Eddies on the Surface of Seas and Oceans,” Izv. Akad. Nauk, Fiz. Atmos. Okeana, No. 3, 48 (2012) [Izv., Atmos. Oceanic Phys., No. 3, 48 (2012)].Google Scholar
  6. 6.
    G. S. Golitsyn, Statistics and Dynamics of Natural Processes and Events (KRASAND, Moscow, 2012) [in Russian].Google Scholar
  7. 7.
    G. S. Golitsyn, “Statistical Laws of Macroprocesses: Random Walks in the Momentum Space,” Dokl. Akad. Nauk, No. 2, 398 (2004) [Dokl. Phys., No. 9, 49 (2004)].Google Scholar
  8. 8.
    G. S. Golitsyn, “Statistical Description of the Topography of a Planet and Its Evotution,” Fizika Zemli, No. 7 (2003) [Izv. Phys. Solid Earth, No. 7, 39 (2003)].Google Scholar
  9. 9.
    G. S. Golitsyn, “Similarity and Dimensional Theory for Galaxies: Explanation of Long-known Results of Observations,” Dokl. Akad. Nauk, No. 4, 475 (2017) [Dokl. Phys., No. 8, 62 (2017)].Google Scholar
  10. 10.
    G. S. Golitsyn, “Polar Lows and Tropical Hurricanes: Their Energy and Sizes and a Quantitative Criterion for Their Generation,” Izv. Akad. Nauk, Fiz. Atmos. Okeana, No. 5, 44 (2008) [Izv., Atmos. Oceanic Phys., No. 5, 44 (2008)].Google Scholar
  11. 11.
    G. S. Golitsyn, “Phenomenological Explanation of the Form of the Spectrum of Cosmic Rays with the Energy E > 10 GeV,” Pis'ma v Astronomicheskii Zhurnal, No. 7, 31 (2005) [in Russian].Google Scholar
  12. 12.
    G. S. Golitsyn, “The Energy Cycle of Wind Waves on the Sea Surface,” Izv. Akad. Nauk, Fiz. Atmos. Okeana, No. 1, 46 (2010) [Izv., Atmos. Oceanic Phys., No. 1, 46 (2010)].Google Scholar
  13. 13.
    I. N. Davidan, L. I. Lopatukhin, and V. V. Rozhkov, Wind Waves in the World Ocean (Gidrometeoizdat, Leningrad, 1985).Google Scholar
  14. 14.
    S. A. Kitaigorodskii, “Application of SimHarity Theory to the Analysis of Wind-induced Waves as a Random Process,” Izv. Akad. Nauk, Ser. Geofizicheskaya, No. 1 (1962) [in Russian].Google Scholar
  15. 15.
    A. V. Karelin, O. Adriani, G. Barbarino, et al., “New Measurements of the Energy Spectra of High-energy Cosmic Ray Pro tons and Helium Nuclei with the Calorimeter in the PAMELA Experiment,” Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki, No. 3, 146 (2014) [J. Experimental and Technical Phys., No. 3, 119 (2014)].Google Scholar
  16. 16.
    A. N. Kolmogorov, Random Motions. Transactions by A. N. Kolmogorov (Fizmatlit, Moscow, 1993) [in Russian].Google Scholar
  17. 17.
    A. S. Monin and A. M. Yaglom, Statistical Hydromechanics, Vol. 2 (MIT Press, 1975).Google Scholar
  18. 18.
    A. M. Obukhov, “Description of Turbulence in Terms of Lagrangian Variables,” Adv. Geophys., 6 (1959).Google Scholar
  19. 19.
    A. M. Obukhov, “On the Energy Distribution in the Spectrum of Turbulent Flow,” Dokl. Akad. Nauk, No. 1, 32 (1941) [in Russian].Google Scholar
  20. 20.
    A. M. Yaglom, “Correlation Theory with Random Stationary nth Increments,” Matematicheskii Sbornik, No. 1, 37 (1955) [in Russian].Google Scholar
  21. 21.
    P. Bird, “An Updated Digital Model of Plate Boundaries,” Geochemistry, Geophysics, Geosystems, No. 3, 4 (2003).Google Scholar
  22. 22.
    K. S. Gage, “Evidence of a k-5/3 Law Inertial Range in Mesoscale Two-dimensional Turbulence,” J. Atmos. Sci., 36 (1979).Google Scholar
  23. 23.
    E. Gagnaire-Renou, M. Benoit, and S. I. Badulin, “On Weakly Turbulent Scaling of Wind Sea in Simulations of Fetch-limited Growth,” J. Fluid Mech., 669 (2011).Google Scholar
  24. 24.
    A. B. Kazansky, “Studying the Motion of Energy Dissipation Process in Highly Fractured Glaciers Using Remote Sensing Technique,” Arch. Glaciology, 19 (1987).Google Scholar
  25. 25.
    A. N. Kolmogorov, “Zufallige Bewegungen,” Ann. Math., 35 (1934).Google Scholar
  26. 26.
    G. J. Komen, L. Cavaleri, M. Donelan, et al., Dynamics and Modelling ofOcean Waves (Cambridge Univ. Press, Cambridge, 1994).CrossRefGoogle Scholar
  27. 27.
    E. Lindborg, “Can the Atmospheric Kinetic Energy Spectrum be Explained by Two-dimensional Turbulence?”, J. Fluid Mech., 388 (1999).Google Scholar
  28. 28.
    B. Mandelbrot, The Fractal Geometry of Nature (W.H. Freeman, N.Y., 1982)Google Scholar
  29. 29.
    A. Okubo, “Oceanic Diffusion Diagrams,” Deep-Sea Res., No. 5, 18 (1971).Google Scholar
  30. 30.
    O. M. Phillips, The Dynamics of the Upper Ocean (Cambridge Univ. Press, Cambridge, 1977).Google Scholar
  31. 31.
    L. F. Richardson, “Atmospheric Diffusion on a Distance-neighbour Graph,” Proc. Roy. Soc. London A, No. 686, 97 (1926).Google Scholar
  32. 32.
    L. F. Richardson, “A Search for the Law of Atmospheric Diffusion,” Beitrage Phys. fur Atmos., 15 (1929).Google Scholar
  33. 33.
    Y.-h. Taguchi, “Numerical Study of Granular Turbulence and an Appearance of the k- 5 3 Energy Spectrum in the Absence of Mean Flow,” Physica D, No. 1, 80 (1995).Google Scholar
  34. 34.
    Y. J. Toba, “Stochastic Form of the Growth of Wind Waves in a Single-parameter Representation with Physical Implications,” J. Phys. Oceanogr., 8 (1978).Google Scholar
  35. 35.
    D. L. Turcotte, Fractals and Chaos in Geology and Geophystcs, 2nd ed. (Cambridge Univ. Press, Cambridge, 1997).CrossRefGoogle Scholar

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© Allerton Press, Inc. 2018

Authors and Affiliations

  1. 1.Obukhov Institute of Atmospheric PhysicsRussian Academy of SciencesMoscowRussia

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