Computational aspects of calculating vertical velocity in a wind circulation model

Authors

UDC

519.63

DOI:

https://doi.org/10.31429/vestnik-21-2-14-22

Abstract

Qualitative calculation of the vertical component of the velocity does not lose its relevance even today. Basically, the results obtained are compared by calculations using other models and with a~priori information about the dynamics of waters in the studied area. More accurate testing of calculation methods can be carried out if there is an accurate analytical solution to the problem. For these purposes, a~three-dimensional analytical solution to the problem of wind circulation in a rectangular reservoir with a flat bottom is used for a given wind effect. Analytical expressions for the barotropic and additional three-dimensional components of the velocity field are obtained. The expression for the vertical component of the velocity field in this paper is used for comparison with the values calculated from the continuity equation. The vertical velocity was calculated in various ways: by standard integration of the vertical continuity equation and by the run-through method. In addition, in this work, when integrating the equation for the current function, a family of difference discretizations obtained for solving a similar class of problems is used. As the results of numerical experiments presented in the tables have shown, even with a sufficiently accurate solution of the problem for the current function, the calculation of the horizontal components of the total flow can lead to significant errors if the specifics of the problem are not taken into account. The algorithms used give a fairly good accuracy of reproducing the solution of the equation for the current function. On a fine grid, the error is hundredths of a percent of the norm used. The technique used in this work allows, within the framework of a single approach, to solve the problem for the current function and calculate the derivatives of this solution, which guarantees the accuracy of determining the horizontal components of the total flow. The paper shows that the use of the run-through method in calculating vertical velocity significantly improves the calculation results. The results of the work can be used in modeling dynamic processes in the sea.

Keywords:

wind circulation model, run-through method, vertical velocity, numerical discretization

Acknowledgement

The work was carried out within the framework of the state assignment on the topic FNNN-2021-0005 "Comprehensive interdisciplinary studies of oceanological processes that determine the functioning and evolution of ecosystems in the coastal zones of the Black and Azov Seas" (code "Coastal research").

Author Infos

Vladimir S. Kochergin

младший научный сотрудник отдела теории волн Федерального исследовательского центра «Морской гидрофизический институт РАН»

e-mail: vskocher@gmail.com

Sergei V. Kochergin

старший научный сотрудник отдела вычислительных технологий и математического моделирования Федерального исследовательского центра «Морской гидрофизический институт РАН»

e-mail: ko4ep@mail.ru

References

  1. Марчук, Г.И., Саркисян, А.С., Математическое моделирование циркуляции океана. Москва, Наука, 1988. [Marchuk, G.I., Sarkisyan, A.S., Matematicheskoe modelirovanie tsirkulyatsii okeana = Mathematical modeling of ocean circulation. Moscow, Nauka, 1988. (in Russian)]
  2. Еремеев, В.Н., Кочергин, В.П., Кочергин, С.В., Скляр, С.Н., Математическое моделирование гидродинамики глубоководных бассейнов. Севастополь, "Экоси-гидрофизика", 2002. [Eremeev, V.N., Kochergin, V.P., Kochergin, S.V., Sklyar, S.N., Matematicheskoe modelirovanie gidrodinamiki glubokovodnykh basseynov = Mathematical modeling of hydrodynamics of deep-water basins. Sevastopol, "Ekosi-gidrofizika", 2002. (in Russian)]
  3. Кочергин, В.С., Кочергин, С. В., Скляр, С.Н., Аналитическая тестовая задача ветровых течений. Процессы в геосредах, № 2(20), 2019, с. 198–203. [Kochergin, V.S., Kochergin, S.V., Sklyar, S.N., Analytical test problem of wind currents. Protsessy v geosredakh = Processes in geoenvironments, No. 2(20), 2019, pp. 198–203. (in Russian)]
  4. Stommel, H., The Gulf Stream. A Physical and Dynamical Description. University of California Press, 1965.
  5. Стоммел, Г., Гольфстрим. Москва, ИЛ, 1965. [Stommel, G., Gol'fstrim = Gulf Stream. Moscow, IL, 1965. (in Russian)]
  6. Годунов, С.К., Рябенький, В.С., Разностные схемы. Введение в теорию. Москва, Наука, 1977. [Godunov, S.K., Ryabenkiy, V.S., Raznostnye skhemy. Vvedenie v teoriyu = Difference Schemes. Introduction to Theory. Moscow, Nauka, 1977. (in Russian)]
  7. Кочергин, В.С., Кочергин, С.В., Скляр, С.Н., Вычисление компонент полного потока в моделях ветрового движения жидкости. Морской гидрофизический журнал, 2023, т. 39, № 3, c. 299–313. [Kochergin, V.S., Kochergin, S.V., Sklyar, S.N., Calculation of the components of the total flow in models of wind-induced fluid motion. Morskoy gidrofizicheskiy zhurnal = Marine Hydrophysical Journal, 2023, vol. 39, no. 3, pp. 299–313.] EDN: RUBXPZ. DOI: 10.29039/0233-7584-2023-3-299-313
  8. Ekman, V.W., On the influence of the Earth rotation on ocean currents. Arkiv Mat., Astron., or Fysik, 1905, bd. 2, no. 11, pp. 1–52.
  9. Stommel, H., The westward intensification of wind-driven ocean currents. Trans. Amer. Geoph. Un., 1948, vol. 29, pp. 202–206.
  10. Sklyar, S.N., A projective version of the integral-interpolation method and it's application for the discretization of the singular perturbation problems. Proc. of the Int. Conf. "Advanced Mathematics: Computations and Applications" (AMCA-95). NCC Pablisher, Novosibirsk, 1995, pp. 380–385.
  11. Роуч, П., Вычислительная гидродинамика. Москва, Мир, 1980. [Roach, P., Vychislitel'naya gidrodinamika = Computational Fluid Dynamics. Moscow, Mir, 1980. (in Russian)]
  12. Самарский, А.А., Теория разностных схем. Москва, Наука, 1983. [Samarskii, A.A., Theory of difference schemes. Moscow, Nauka, 1983. (in Russian)]
  13. Булеев, Н.И., Тимухин, Г.И., О составлении разностных уравнений гидродинамики вязкой неоднородной среды. Численные методы механики сплошной среды, 1972, т. 3, № 4, с. 19–26. [Buleev, N.I., Timukhin, G.I., On the formulation of difference equations for the hydrodynamics of a viscous inhomogeneous medium. Chislennye metody mekhaniki sploshnoy sredy = Numerical Methods of Continuum Mechanics, 1972, vol. 3, no. 4, pp. 19–26. (in Russian)]
  14. Булеев, Н.И., Пространственная модель турбулентного обмена. Москва, Наука, 1983. [Buleev, N.I., Prostranstvennaya model' turbulentnogo obmena = Spatial model of turbulent exchange. Moscow, Nauka, 1983. (in Russian)]
  15. Ильин, А.М., Разностная схема для дифференциального уравнения с малым параметром при старшей производной. Математические заметки, 1969, т. 6, вып. 2, с. 237–248. [Ilyin, A.M., Difference scheme for a differential equation with a small parameter at the highest derivative. Matematicheskie zametki = Mathematical notes, 1969, vol. 6, iss. 2, pp. 237–248. (in Russian)]
  16. Дулан, Э., Миллер, Дж., Шилдерс, У., Равномерные численные методы решения задач с пограничным слоем. Москва, Мир, 1983. [Doolan, E., Miller, J., Shielders, W., Ravnomernye chislennye metody resheniya zadach s pogranichnym sloem = Uniform Numerical Methods for Solving Boundary Layer Problems. Moscow, Mir, 1983. (in Russian)]
  17. Коллатц, Л., Функциональный анализ и вычислительная математика. Москва, Мир, 1969. [Collatz, L., Funktsional'nyy analiz i vychislitel'naya matematika = Functional Analysis and Computational Mathematics. Moscow, Mir, 1969. (in Russian)]

Downloads

Issue

2024. Vol. 21. No. 2

Section

Mathematics

Pages

14-22

Submitted

2024-05-27

Published

2024-06-28

How to Cite

Kochergin V.S., Kochergin S.V. Computational aspects of calculating vertical velocity in a wind circulation model. Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2024, vol. 21, no. 2, pp. 14-22. DOI: https://doi.org/10.31429/vestnik-21-2-14-22 (In Russian)

Copyright (c) 2024 Kochergin V.S., Kochergin S.V.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.