Abstract
A brief review of the emergence and development of the nonlocal approach to the problem of turbulent diffusion with a discussion of the physical reasons of the nonlocality is given. The main attention is paid to fractional differential operators. In concluding the paper, the author’s original results on applications to the diffusion of cosmic rays in the interstellar galactic medium are presented.
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References
G. K. Batchelor, “Diffusion in a field of homogeneous turbulence. II. The relative motion of particles,” Phil. Soc., 40, No. 2, 345–362 (1952).
G. K. Batchelor and A. A. Townsend, “Turbulent diffusion,” in: Surveys in Mechanics (G. K. Batchelor and R. M. Davis, eds.), Cambridge Univ. Press, New York (1956), pp. 352–399.
R. Bourret, “An hypothesis concerning turbulent diffusion,” Can. J. Phys., 38, 665–676 (1959).
Chan-Mou Tchen, “Transport processes as foundation of the Heisenberg and Obukhov theories of turbulence,” Phys. Rev., 93, 4–14 (1954).
P.-H. Chavanis, “Statistical mechanics of two-dimensional vortices and stellar systems,” in: Dynamics and Thermodynamics of Systems with Long-Range Interactions, Lect. Notes Phys., 602(T. Dauxois, S. Ruffo, E. Arimondo, and M. Wilkens, eds.), Springer-Verlag, Berlin–Heidelberg (2002), pp. 208–289.
W. Heisenberg, “Zur statistischen Theorie der Turbulenz,” Z. Phys., 124, 628–657 (1948).
A. S. Monin, “Turbulent diffusion equation,” Dokl. Akad. Nauk SSSR, 105, 256–259 (1955).
L. Onsager, “Statistical hydrodynamics,” Nuovo Cim., 6, 279–287 (1949).
S. J. Pigolotti, M. H. Jensen, and A. Vulpiani, “Absorbing processes in Richardson diffusion: Analytical results,” Phys. Fluids, 18, 048104 (2006).
L. F. Richardson, “Atmospheric diffusion shown on a distance-neighbor graph,” Proc. Roy. Soc. Ser. A., 110, No. 756, 709–720 (1926).
N. Romanov, J. Meteorology, 8, No. 1-2, 37–45 (2006).
A. I. Saichev and G. M. Zaslavsky, “Fractional kinetic equation: Solutions and applications,” Chaos, 7, 753–764 (1997).
J. C. Schönfeld, “Integral diffusivity,” J. Geophys. Res., 67, No. 8, 3187–3199 (1962).
M. F. Shlesinger, B. J. West, and J. Klafter, “Lévy dynamics of enhanced diffusion: Application to turbulence,” Phys. Rev. Lett., 58, No. 11, 1100–1103 (1987).
V. V. Uchaikin, “Anomalous diffusion and fractional stable distributions,” Zh. Eksp. Teor. Fiz., 124, 903–920 (2003).
V. V. Uchaikin, “On the fractional differential model of the transfer of cosmic rays in the Galaxy,” Pisma Zh. Eksp. Teor. Fiz., 91, No. 3, 115–120 (2010).
V. V. Uchaikin, “Fractional phenomenology of anomalous diffusion of cosmic rays,” Usp. Fiz. Nauk, 183, No. 11, 1175–1223 (2013).
V. V. Uchaikin, “Nonlocal models of cosmic ray transport in the Galaxy,” J. Appl. Math. Phys., 3, 187–200 (2015).
V. V. Uchaikin and V. M. Zolotarev, Chance and Stability: Stable Distributions and Their Applications, VSP, Utrecht (1999).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 154, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, 2018.
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Uchaikin, V.V. Nonlocal Turbulent Diffusion Models. J Math Sci 253, 573–582 (2021). https://doi.org/10.1007/s10958-021-05255-z
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DOI: https://doi.org/10.1007/s10958-021-05255-z