Vestnik КRAUNC. Fiz.-Mat. Nauki. 2026. vol. 54. no. 2. P. 57 – 67. ISSN 2079-6641

MATHEMATICAL MODELING
https://doi.org/10.26117/2079-6641-2026-54-2-57-67
Research Article
Full text in English
MSC 26A33, 86A10

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Dedicated to the 85th anniversary of Professor P.P. Firstov

Modeling of Particle Electrification in a Fractal Cloud Environment Taking into Account Charge Leakage Due to Conductivity

T. S. Kumykov¹²^{\ast}

¹Institute of Applied Mathematics and Automation KBSC RAS, 360004 Nalchik, Shortanova st., 89 a, Russia
²Kabardino-Balkarian State University named after H.M. Berbekov, Chernyshevskogo St, 173, Nalchik, 360004, Russia

Abstract. This study develops a mathematical model of cloud particle electrification that integrates fractal cloud geometry and charge leakage through atmospheric conductivity. Using fractional calculus with the Gerasimov–Caputo operator of variable order α, related to the fractal dimension by \alpha= 6−2D_f, the model captures multiscale memory effects missing in classical approaches. Analytical solutions for hailstones and shed droplets are obtained via Mittag-Leffler functions. Numerical simulations over a 50-second interval for \alpha = 0.1, 0.3, 0.5, 0.7, 0.9 reveal strong sensitivity of charge dynamics to the fractality parameter: for \alpha < 0.3 charging is smooth and monotonic, whereas for \alpha > 0.7 it rapidly approaches saturation; the most probable scenario with \alpha \approx 0.5 yields accumulation times of 20–40 s, consistent with field observations. By determining fractal dimension parameters and leakage coefficients, the model predicts thunderstorm probability and intensity, improves lightning forecasting, supports critical infrastructure protection, and facilitates the inclusion of atmospheric electricity in climate models under increasing extreme weather conditions.

Key words: mathematical model, fractional equation, Gerasimov-Caputo derivative, thunderstorm cloud, fractal medium, charge generation.

Received: 20.07.2026; Revised: 28.07.2026; Accepted: 30.07.2026; First online: 14.09.2026

For citation. Kumykov T. S. Modeling of particle electrification in a fractal cloud environment taking into account charge leakage due to conductivity. Vestnik KRAUNC. Fiz.-mat. nauki. 2026, 54: 2, 57-67. EDN: GYIWOF. https://doi.org/10.26117/2079-6641-2026-54-2-57-67.

Funding.The study was conducted without the support of foundations.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. The author participated in the writing of the article and is fully responsible for the submission of the final version of the article for publication.

^{\ast}Correspondence: E-mail: macist20@mail.ru

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

© Kumykov T. S., 2026

© Institute of Cosmophysical Research and Radio Wave Propagation, 2026 (original layout, design, compilation)

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