Abstract
In 1934, A.N. Kolmogorov considered the random walks of the 6D vector X, U of velocities generated by the Markov process and corresponding coordinates. The Fokker–Planck type equation using the scales of mean squares of velocity and coordinates as solutions was proposed to describe the time evolution of probability density of random process p(X, U, t). In 1959, A.M. Obukhov excluded time from these scales and obtained formulas in the form of asymptotes for the small-scale turbulence formulated in 1941. However, these results allow deeper insight into a wider range of phenomena, such as the size distribution of lithospheric plates (Bird [6]); the fetch laws describing wind wave growth (Toba [15]); and other phenomena (see specifically (Golitsyn [25]) considering the scales manifested in galaxies). Random accelerations and their integration set the velocities, which, being integrated, set the coordinates of particle ensembles. All these factors promote the energy input into the system increasing with time, whose growth rate ε is the doubled diffusion coefficient in the space of velocities, according to the Kolmogorov equation. It was found that the mean square velocity obtained upon solving the Fokker–Planck equation grows with time—theoretical physicists have long been aware of this result.
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References
A. N. Kolmogorov, “The local structure of turbulence in an incompressible fluid at very large Reynolds numbers,” Dokl. Akad. Nauk SSSR 30 (4), 299–303 (1941).
A. M. Obukhov, “On energy distribution in the turbulent flow spectrum,” Izv. Akad. Nauk 32 (1), 22–24 (1941).
A. S. Monin and A. M. Yaglom, Statistical Hydromechanics, Vol. 1 (MIT Press, 1971; Fizmatlit, Moscow, 1965); Vol. 2 (MIT Press, 1975; Fizmatlit, Moscow, 1967).
Y. Y. Kagan, “Observational evidence for earthquakes as a non-linear dynamic process,” Phys. D (Amsterdam) 77 (2), 160–192 (1994).
G. S. Golitsyn, “The place of the Gutenberg–Richter law among other statistical laws of nature,” Comput. Seismol., No. 32, 138–161 (2003).
P. Bird, “An updated digital model of plate boundaries,” Geochem. Geophys. Geosyst. 4 (3), 1027 (2003). doi 10.1029/2001GC000252
A. B. Kazansky, “Studying the motion of energy dissipation process in highly fractured glaciers using remote sensing technique,” Arch. Glaciol. 19 (3), 239–248 (1987).
Y. Taguchi, “Numerical study of granular turbulence and an appearance of the energy spectrum in the absence of mean flow,” Phys. D 80 (1), 61–71 (1995).
G. S. Golitsyn, Statistics and Dynamics of Natural Processes and Phenomena (Krasand, Moscow, 2012) [in Russian].
E. A. Novikov, “Functionals and the random-force method in the turbulence theory,” Zh. Eksp. Teor. Fiz. 47 (5), 1919–1926 (1964).
A. N. Kolmogorov, “Zufallige Bewegungen,” Ann. Math 35, 116–117 (1934).
A. M. Obukhov, “Description of turbulence in terms of Lagrangion variables,” Adv. Geophys. 6, 113–116 (1959).
E. B. Gledzer and G. S. Golitsyn, “Scaling and finite ensembles of particles in motion with the energy influx,” Dokl. Phys. 55 (8), 369–373 (2010).
G. J. Komen, L. Cavaleri, M. Donelan, K. Hasselmann, S. Hasselmann and P. A. E. M. Janssen, Dynamics and Modelling of Ocean Waves (Cambridge University Press, Cambridge, 1994).
Y. Toba, “Stochastic form of the growth of wind waves in a single-parameter representation with physical implications,” J. Phys. Oceanogr. 8 (5), 494–507 (1978).
G. I. Barenblatt, Scaling (Cambridge University Press, Cambridge, 2002).
G. Schubert, D. Turcotte, and P. Olson, Mantle Convection in the Earth and Planets (Cambridge University Press, Cambridge, 2001).
G. S. Golitsyn, “Energy cycle of geodynamics and the seismic process,” Izv., Phys. Solid Earth 43 (6), 443–446 (2007).
P. W. Bridgman, Dimensional Analysis (Yale Univ. Press, New Haven, 1922; RKhD, Izhevsk, 2001).
L. D. Landau and E. M. Lifshits, Mechanics of Continuous Media (GITTL, Moscow, 1944) [in Russian].
B. B. Kadomtsev, Collective Phenomena in Plasma (Fizmatgiz, Moscow, 1976) [in Russian].
G. S. Golitsyn, “Phenomenological explanation for the shape of the spectrum of cosmic rays with energies E >10 GeV,” Astron. Lett. 31 (7), 446–451 (2005).
A. V. Karelin, O. Adriani, G. C. Barbarino, G. A. Bazilevskaya, et al., “New measurements of the energy spectra of high-energy cosmic-ray protons and helium nuclei with the calorimeter in the PAMELA experiment,” J. Exp. Theor. Phys. 119 (3), 448–452 (2014).
G. S. Golitsyn, “Earthquakes from the standpoint of scaling theory,” Dokl. Earth Sci. 346 (1), 166–169 (1996).
G. S. Golitsyn, “Similarity and dimensional theory for galaxies: Explanation of long-known results of observations,” Dokl. Phys. 62 (8), 403–406 (2017).
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Original Russian Text © G.S. Golitsyn, 2018, published in Izvestiya Rossiiskoi Akademii Nauk, Fizika Atmosfery i Okeana, 2018, Vol. 54, No. 3.
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Golitsyn, G.S. Random Walk Laws by A.N. Kolmogorov as the Basics for Understanding Most Phenomena of the Nature. Izv. Atmos. Ocean. Phys. 54, 223–228 (2018). https://doi.org/10.1134/S0001433818030064
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DOI: https://doi.org/10.1134/S0001433818030064