Вестник КРАУНЦ. Физ.-мат. науки. 2018. № 4(24). C. 97-108. ISSN 2079-6641

Содержание

DOI: 10.18454/2079-6641-2018-24-4-97-108

MSC 86A10

THE CLOUD DROPLETS EVOLUTION IN VIEW OF THE IMPACT OF FRACTAL ENVIRONMENT: MATHEMATICAL MODELING

T. S. Kumykov

Institute of Applied Mathematics and Automation of Kabardin-Balkar Scientific Center of RAS, 360000, Nalchik, Shortanova st., 89 A, Russia

E-mail: macist20@mail.ru

In this paper, we investigate the effect of the medium with fractal structure on the growth of small cloud droplets at the initial condensation stage of cloud formation using a fractional differential equation. An electrodynamic model of coagulation of droplets under the action of an electric field is constructed in the cloud medium with a fractal structure. Numerical experiments are performed for assessment of the effect of the medium with fractal structure on the growth of cloud particles involving various combinations of microphysical parameters. A general dependence of the growth of cloud particles on different parameters of fractal structure in medium is established.

Key words: cloud droplet, fractal dimension, mathematical model, convective cloud.

УДК 517.98

РАЗВИТИЕ ОБЛАЧНЫХ КАПЕЛЬ В РЕЗУЛЬТАТЕ ВОЗДЕЙСТВИЯ ФРАКТАЛЬНОЙ СРЕДЫ: МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ

Т. С. Кумыков

Институт прикладной математики и автоматизации КБНЦ РАН, 360000, г. Нальчик, ул. Шортанова, 89 А

E-mail: macist20@mail.ru

В настоящей работе мы исследуем влияние среды с фрактальной структурой на рост малых облачных капель на начальной стадии конденсации образования облаков с использованием дробного дифференциального уравнения. В облачной среде с фрактальной структурой построена электродинамическая модель коагуляции капель под действием электрического поля. Проведены численные эксперименты для оценки влияния среды на фрактальную структуру на рост облачных частиц с использованием различных комбинаций микрофизических параметров. Установлена общая зависимость роста частиц облака от разных параметров фрактальной структуры в среде.

Ключевые слова: облачная капля, фрактальная размерность, математическая модель, конвективное облако.

 

References

  1. Mandelbrot В. В., “Fractals in physics: Squig clusters, diffusions, fractal measures, and the unicity of fractal dimensionality”, J. Stat. Phys., 34 (1983), 895–930.
  2. Feder E., Fractals, Mir, M., 1991, 214 pp.
  3. Rys F., Waldfogel A., “Analysis of the fractal dimension in clouds with powerful convective currents”, Fractals in Physics Proceedings of the VI International Symposium on Fractals in Physics, ICTP, Trieste, Italy, 1985, 644–649.
  4. Iudin D. I., Trakhtengerts V. Y., Hayakawa M., “Fractal dynamics of electric discharges in a thundercloud”, Phys. Rev. E., 68 (2003), 016601.
  5. Kumykov T. S., Parovik R. I., “Mathematical modeling of changes in the charge cloud droplets in a fractal environment”, Bulletin KRASEC. Phys. and Math. Sci., 10:1 (2015), 11–15.
  6. Kumykov T. S., “Dynamics of cloud drops charge in fractal medium”, Mathematical modeling, 2016, № 12, 56–62.
  7. Kumykov T. S., “Modeling the emergence of fractal structures «babstons»in the
    atmosphere”, Scientific Bulletins of BelGU. Series: Mathematics. Physics, 44:20 (2016), 145–153.
  8. Samko S. G., Kilbas A. A., Marichev O. I., Integrals and derivatives of fractional order and some of their applications Minsk, Science and Technology, 1987, 688 pp.
  9. Pshu A. V., Boundary value problems for partial differential equations of fractional and continual order, Publishing house KBSC RAS, Nalchik, 2005, 185 pp.
  10. Timofeev M. P., Shvets M. E., “Evaporation of small drops of water”, Meteorology and Hydrology, 1948, № 2, 14–18.
  11. Taukenova F. I., Shkhanukov-Lafishev M. Kh., “Difference methods for solving boundary value problems for fractional differential equations”, Comput. Math. Math. Phys., 46:10 (2006), 1785-–1795.
  12.  Sweilam N. H., Khader M. M., Mahdy A. M. S., “Numerical studies for solving fractional- order logistic equation”, International Journal of Pure and Applied Mathematics, 78:8. (2012), 1199–1210.
  13. Alikhanov A. A., “Apriori estimates of solutions to boundary value problems for fractional-order equations”, Differential equations, 46:5 (2010), 658–664.
  14. Shishkin N. S., Clouds, rainfall and thunderstorms, Gidrometeoizdat, L, 1964, 402 pp.

References (GOST)

  1. Mandelbrot В. В. Fractals in physics: Squig clusters, diffusions, fractal measures, and the unicity of fractal dimensionality // J. Stat. Phys. 1983. vol. 34. pp. 895-930.
  2. Feder E. Fractals: Trans. from English. M.: Mir, 1991. 214 p.
  3. Rys F., Waldfogel A. Analysis of the fractal dimension in clouds with powerful convective currents // Fractals in Physics Proceedings of the VI International Symposium on Fractals in Physics. ICTP, Trieste, Italy. 1985. pp. 644-649.
  4. Iudin D.I., Trakhtengerts V.Y., Hayakawa M. Fractal dynamics of electric discharges in a thundercloud // Phys. Rev. E. 2003. vol. 68. P. 016601.
  5. Kumykov T.S., Parovik R.I. Mathematical modeling of changes in the charge cloud droplets in a fractal environment // Bulletin KRASEC. Phys. and Math. Sci. 2015. vol. 10. no. 1. pp. 11-15.
  6. Kumykov T.S. Dynamics of cloud drops charge in fractal medium // Mathematical modeling. 2016. № 12. pp. 56-62.
  7. Kumykov T.S. Modeling the emergence of fractal structures «babstons» in the atmosphere. Scientific Bulletins of BelGU. Series: Mathematics. Physics. 2016. Issue 44, № 20, pp. 145-153.
  8. Samko S.G., Kilbas A.A., Marichev O.I. Integrals and derivatives of fractional order and some of their applications. Minsk: Science and Technology. 1987, p. 688.
  9. Pshu A. V. Boundary value problems for partial differential equations of fractional and continual order. Nalchik: Publishing house KBSC RAS. 2005. 185 p.
  10. Timofeev M. P., Shvets M. E. Evaporation of small drops of water // Meteorology and Hydrology. 1948. no. 2. pp. 14-18.
  11. Taukenova F. I., Shkhanukov-Lafishev M. Kh. Difference methods for solving boundary value problems for fractional differential equations // Comput. Math. Math. Phys. 2006. vol. 46. no. 10. pp. 1785-1795.
  12. Sweilam N. H., Khader M. M., Mahdy A. M. S. Numerical studies for solving fractional-order logistic equation // International Journal of Pure and Applied Mathematics. 2012. vol. 78. no. 8. pp. 1199-1210.
  13. Alikhanov A. A. Apriori estimates of solutions to boundary value problems for fractional-order equations // Differential equations. 2010, vol. 46. no. 5. pp. 658-664.
  14. Shishkin N.S. Clouds, rainfall and thunderstorms. L: Gidrometeoizdat, 1964. 402 p.

 

Для цитирования: Kumykov T. S. The cloud droplets evolution in view of the impact of fractal environment: mathematical modeling // Вестник КРАУНЦ. Физ.-мат. науки. 2018. № 4(24). C. 97-108. DOI: 10.18454/2079-6641-2018-24-4-97-108.
For citation: Kumykov T. S. The cloud droplets evolution in view of the impact of fractal environment: mathematical modeling, Vestnik KRAUNC. Fiz.-mat. nauki. 2018, 24: 4, 97-108. DOI: 10.18454/2079-6641-2018-24-4-97-108.

Поступила в редакцию / Original article submitted: 18.09.2018

kum   Кумыков Тембулат Сарабиевич – кандидат физико-математических наук, старший научный сотрудник отдела математического моделирования геофизических процессов Института прикладной математики и автоматизации, республика Кабардино-Балкария, г. Нальчик, Россия.
   Kumykov Tembulat Sarabievich – Ph.D. (Phys. & Math.), Senior Research of Dep. Mathematical Modeling of Geophysical Processes, Institute of Applied Mathematics and Automation, Kabardino-Balkaria, Nalchik, Russia.

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