Vestnik KRAUNC. Fiz.-Mat. Nauki. 2022. vol. 39. no. 2. pp. 80–90. ISSN 2079-6641

Contents of this issue

Read Russian Version US Flag


MSC 86A10

Research Article

Modeling the growth of flat snow crystals in clouds with fractal structure

T. S. Kumykov

Institute of Applied Mathematics and Automation KBSC RAS, 360004, Nalchik, Shortanova st., 89 a, Russia


In this paper, a universal model is proposed to describe the growth process of flat round-shaped snow crystals in mixed-type clouds with a fractal structure. Snow crystals were chosen as the object of research, as they can have a significant impact on the weather conditions and climate of the Earth. In an analytical form, the solution of the equation of the model is found, in which the fractal property of the cloud environment is taken into account through a phenomenological parameter that determines the intensity of the growth of snow crystals using the fractional integro-differentiation apparatus. It is shown that the growth of snow crystals under sublimation and coagulation growth mechanisms mainly depends not only on temperature and water content, but also on the fractal parameter of the cloud environment. The snow crystal growth curves are presented depending on the experimental parameters of the fractality of the cloud medium in the general case and with rapid diffusion. It is noted that the fractality index is responsible for the intensity of the process, the greater the fractality, the more intense the process of snow crystal growth. The considered model can be used to calculate the growth of snow crystals taking into account the fractal parameters of the cloud environment.

Key words: snow crystal, dynamic model, fractal medium, cloud water content.

DOI: 10.26117/2079-6641-2022-39-2-80-90

Original article submitted: 04.08.2022

Revision submitted: 14.09.2022

For citation. Kumykov T. S. Modeling the growth of flat snow crystals in clouds with fractal structure. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 39: 2, 80-90. DOI: 10.26117/2079-6641-2022-39-2-80-90

Competing interests. The authors declare that there are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

The content is published under the terms of the Creative Commons Attribution 4.0 International License (

© Kumykov T. S., 2022


  1. Shishkin N. S. O roste snezhinok v oblakakh. Voprosy fiziki oblakov i aktivnykh vozdejstvii [About the growth of snowflakes in the clouds. Questions of cloud physics and active influences]. Leningrad, 1965, pp. 144 (In Russian).
  2. Mandelbrot В. В. The Fractal Geometry of Nature. N. Y., Freeman, 1982, pp. 468.
  3. Kumykov T. S., Parovik R. I. Mathematical modeling of changes in the charge cloud droplets in a fractal environment, Vestnik KRAUNC. Fiz.-Mat. Nauki, 2015, vol. 10, no. 1, pp. 12–17. DOI: 10.18454/2079-6641-2015-10-1-12-17 (In Russian).
  4. Kumykov T. S. Charge accumulation in thunderstorm clouds: fractal dynamic model, E3S Web of Conferences. 2019. vol. 127, 01001. DOI: 10.1051/e3sconf/201912701001.
  5. Kumykov T. S. Mathematical modeling of the evolution of cloud drops with the influence of the fractality of the cloud environment, Journal of Mathematical Sciences. 2021. vol. 253. no. 4. pp. 520–529. DOI: 10.1007/s10958-021-05249-x
  6. Haughton N. G. А preliminary quantitative analysis of precipitation mechanisms, J. Meteorology. 1950. vol. 7, no. 6.
  7. Shogenov V. KH., Akhkubekov A. A., Akhkubekov R. A. Metod drobnogo differencirovaniia v teorii brounovskogo dvizheniia [Fractional differentiation method in the theory of Brownian motion], Izvestiia vysshikh uchebnykh zavedenii. Severo-Kavkazskii region. Seriia: Estestvennye nauki, 2004, no. 1, pp. 46–50. (In Russian).
  8. Potapov A. A. Fraktaly v radiofizike i radiolokacii: Topologiya vyborki [Fractals in radiophysics and radar: Sampling topology.] , M., Universitetskaya kniga, 2005, 848 pp. (In Russian).
  9. Rekhviashvili S. Sh. Formalizm Lagranzha s drobnoj proizvodnoj v zadachah mekhaniki [Lagrange formalism with fractional derivative in problems of mechanics], Pis’ma ZHTF, 2004. vol.30. no. 2, pp. 33–37. (In Russian).
  10. Pskhu A.V. Uravneniia v chastnykh proizvodnykh drobnogo poriadka [Partial differential equations of fractional order]. Moscow, Nauka, 2005, p. 199 (In Russian).
  11. Rys F., Waldfogel A. Fractals in Physics Proceedings of the VI International Symposium on Fractals in Physics, ICTP, Trieste, Italy. 1985. pp. 644–649.

Kumykov Tembulat Sarabievich – Ph. D. (Phys. & Math.), Head of the Department of Mathematical modelling of geophysical processes Institute of Applied Mathematics and Automation, Kabardino-Balkaria, Nalchik, Russia, ORCID 0000-0002-9259-4509.