Vestnik KRAUNC. Fiz.-Mat. Nauki. 2022. vol. 41. no. 4. pp. 9–31. ISSN 2079-6641

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MSC 68W30, 76F20

Research Article

Construction of complex shell models of turbulent systems by computer algebra methods

G. M. Vodinchar¹², L. K. Feshchenko¹, N. V. Podlesnyi³

¹Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Mirnaya Str., 7, Paratunka, Kamchatka region, 683003, Russia
²Kamchatka State Technical University, Klyuchevskaya Str., 35, Petropavlovsk-Kamchatsky, 683003, Russia
³Vitus Bering Kamchatka State University, Pogranichnaya Str., 4, Petropavlovsk-Kamchatsky, 683032, Russia


One popular class of small-scale turbulence models is the class of shell models. In these models, the fields of a turbulent system are represented by time-dependent collective variables (real or complex), which are understood as the field intensity meassure in a given range of spatial scales. The model itself is a certain system of quadratically nonlinear ordinary differential equations for collective variables. Construction a new shell model requires rather complex analytical transformations. This is due to the fact that the system of model equations in the absence of dissipation must have some quadratic invariants and phase-space volume is unchanged. In addition, there are limitations associated with the impossibility of non-linear interaction of some scales ranges. All this imposes limitations on the coefficients of the nonlinear terms of the model. Constraints form a system of equations with parameters. The complexity of this system increases sharply for non-local models, when the interaction is described not only close ranges of scales and when complex collective variables are used. The paper proposes a computational technology that allows automating the process of building shell models. It makes it easy to combine different invariants and the meassure of nonlocality. The technology is based on computer algebra methods. The process of constructing equations for unknown coefficients and their solution has been automated. As a result, parametric classes of cascade models are obtained that have the required analytical properties.

Key words: turbulence, shell models, computer algebra, automation of model development.

Funding. The work was carried out within the framework of the state assignment on the topic «Physical processes in the system of near space and geospheres under solar and lithospheric influences» (No. AAAA-A21-121011290003-0).

DOI: 10.26117/2079-6641-2022-41-4-9-31

Original article submitted: 29.11.2022

Revision submitted: 06.12.2022

For citation. Vodinchar G. M., Feshchenko L. K., Podlesnyi N. V. Construction of complex shell models of turbulent systems by computer algebra methods. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 41: 4, 9-31. DOI: 10.26117/2079-6641-2022-41-4-9-31

Competing interests. The author declare that there is no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. The author is solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by the author.

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

© Vodinchar G. M., Feshchenko L. K., Podlesnyi N. V., 2022


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Vodinchar Gleb Mikhailovich – Candidate of Physical and Mathematical Sciences, Leading Researcher, Acting Head of the Laboratory for Simulation of Physical Processes, IKIR FEB RAS, Associate Professor, Department of Control Systems, Kamchatka State Technical University, Petropavlovsk-Kamchatsky, Russia, ORCID 0000-0002-5516-1931.

Feshchenko Liubov Konstantinovna – Candidate of Physical and Mathematical Sciences, Researcher, Laboratory for Modeling Physical Processes, IKIR FEB RAS, Russia, ORCID 0000-0001-5970-7316.

Podlesny Nikita Viktorovich – Master student of the Department of Mathematics and Physics of the Vitus Bering Kamchatka State University, Russia, ORCID 0000-0002-3213-5706.