Vestnik КRAUNC. Fiz.-Mat. nauki. 2023. vol. 45. no. 4. P. 52-66. ISSN 2079-6641

Research Article
Full text in Russian
MSC 34D08, 76W05

Contents of this issue

Read Russian Version

Chaotic Modes in the Low-Mode Model \alpha\Omega-Dynamo with Hereditary \alpha-Quenching by the Field Energy

O. V. Sheremetyeva^\ast

Institute of Cosmophysical Research and RadioWave Propagation, Far Eastern Branch of the Russian Academy of Sciences, 684034 Kamchatka region, Elizovskiy district, Paratunka, Mirnaya str., 7, Russia

Abstract. This article considers the conditions under which it is possible to simulate the chaotic regime of the magnetic field in a large-scale model \alpha\Omega-dynamo in a low-mode approximation. The intensity of the \alpha– and \Omega-generators is regulated by the Lorentz force. The quenching of the \alpha-effect is determined by the action of the Lorentz force through a process with hereditarity properties (finite «memory»). The nature of the impact of the process is determined by an alternating kernel with variable damping frequency and damping coefficient. The effect of large-scale and turbulent generators on the magnetohydrodynamic system is embedded in the control parameters — the Reynolds number and the measure of the \alpha-effect, respectively. Within the framework of this work, the solutions of the magnetohydrodynamic system are investigated for Lyapunov stability in the vicinity of the rest point, depending on the set values of the input parameters. Based on the results of the numerical experiment, the limitations of the stability characteristic and parameters of the system are determined, under which it is possible to simulate the chaotic regime of the magnetic field.

Key words: \alpha\Omega-dynamo, hereditarity, \alpha-quenching, low-mode dynamo model, magnetic field, chaotic regime, reversals.

Received: 26.11.2023; Revised: 12.12.2023; Accepted: 13.12.2023; First online: 14.12.2023

For citation. Sheremetyeva O. V. Chaotic modes in the low-mode model \alpha\Omega-dynamo with hereditary \alpha\alpha-quenching by the field energy. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 45: 4, 52-66. EDN: WJOTLU.

Funding. The work was carried out within the framework of realization of the State task AAAA-A21-121011290003-0.

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 submitting the final version of the article to the press.

^\astCorrespondence: E-mail:

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

© Sheremetyeva O. V., 2023

© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)


  1. Vodinchar G. M. Using Modes of Free Oscillation of a Rotating Viscous Fluid in the Large-Scale Dinamo, Vestnik KRAUNC. Fiz.-Mat. Nauki, 2020, vol. 7, no. 2, pp. 33–42. DOI: 10.18454/2079-6641-2013-7-2-33-42 (In Russian)
  2. Vodinchar G. M., Feschenko L. K. 6-jet kinematic model of geodinamo, Nauchnye Vedomosti BelGU Matematika Fizika, 2014, no. 5, pp. 94–102. (In Russian)
  3. Vodinchar G. M., Feshenko L. K. Reversals in the 6-cells convection driven, Bull. KRASEC. Phys.& Math. Sci., 2015, vol. 11, no. 2, pp. 41–50. DOI: 10.18454/2313-0156-2015-11-2-41-50
  4. Feschenko L. K., Vodinchar G. M. Reversals in the large-scale \alpha\Omega-dynamo with memory, Nonlinear Processes in Geophysics, 2015, vol. 22, no. 4, pp. 361–369. DOI: 10.5194/npg-22-361-2015
  5. Vodinchar G. M., Feshchenko L. K. Model of geodynamo dryven by six-jet convection in the Earth’s core, Magnetohydrodynamics, 2016, vol. 52, no. 1, pp. 287–300.
  6. Vodinchar G. M., Godomskaya A. N., Sheremetyeva O. V. Reversal of magnetic field in the dynamic system with stochastic \alpha\Omega-generators, Vestnik KRAUNC. Fiz.-Mat. Nauki, 2017, vol. 20, no. 4, pp. 76–82. DOI: 10.18454/2079-6641-2017-20-4-76-82 (In Russian)
  7. Vodinchar G. M., Parovik R. I., Perezhogin A. S., Sheremetyeva O. V. Simulation of physical processes and systems in institute of cosmophysical research and radio wave propagation FEB RAS [Raboty po modelirovaniyu fizicheskih processov i sistem v institute kosmofizicheskih issledovanij i rasprostraneniya radiovoln DVO RAN], History of Science and Engineering [Istoriya nauki i tehniki], 2017, no. 8, pp. 100–112. (In Russian)
  8. Godomskaya A. N., Sheremetyeva O. V. Reversals in the low-mode model dynamo with \alpha\Omega-generators, E3S Web of Conferences, 2018, vol. 62, no. 02016. DOI: 10.1051/ e3sconf/20186202016
  9. Sheremetyeva O. V., Godomskaya A. N. Modelling the magnetic field generation modes in the low-mode model of the \alpha\Omega-dynamo with varying intensity of the \alpha\alpha-effect, Bulletin of the South Ural State University, Series «Mathematical Modelling, Programming & Computer Software», 2021, vol. 14, no. 2, pp. 27–38. DOI: 10.14529/mmp210203 (In Russian)
  10. Godomskaya A. N., Sheremetyeva O. V. The modes of magnetic field generation in a lowmode model of \alpha\Omega-dynamo with \alpha\alpha-generator varying intensity regulated by a function with an alternating kernel, EPJ Web of Conferences, 2021, vol. 254, no. 02015. DOI: 10.1051/epjconf/202125402015
  11. Sheremetyeva O. V. Modes of magnetic field generation in the low-mode \alpha\Omega-dynamo model with dynamic regulation of the \alpha-effect by the field energy, Vestnik KRAUNC. Fiz.-Mat. Nauki, 2021, vol. 37, no. 4, pp. 92–103. DOI: 10.26117/2079-6641-2021-37-4-92-103 (In Russian)
  12. Sheremetyeva O. V. Dynamics of generation modes changes in magnetic field depending on the oscillation frequency of the \alpha\alpha-effect suppression process by field energy in the \alpha\Omega-dynamo model, Vestnik KRAUNC. Fiz.-mat. nauki, 2022, vol. 41, no. 4, pp. 107–119. DOI: 10.26117/2079-6641-2022-41-4-107-119 (In Russian)
  13. Sheremetyeva O. Magnetic Field Dynamical Regimes in a Large-Scale Low-Mode αΩ-Dynamo Model with Hereditary \alpha\alpha-Quenching by Field Energy, Mathematics, 2023, vol. 11, no. 10, pp. 2297. DOI: 2023.10.3390/math11102297.
  14. Vodinchar G. M., Feshchenko L. K. Computational Technology for the Basis and Coefficients of Geodynamo Spectral Models in the Maple System, Mathematics, 2023, vol. 11, no. 13, pp. 3000. DOI: 10.3390/math11133000.
  15. Kolesnichenko A. V., Marov M.Ya. Turbulence and self-organization. Problems of modeling space and natural environments [Turbulentnost’ i samoorganizatsiya. Problemy modelirovaniya kosmicheskikh i prirodnykh sred]. Voscow: BINOM, 2009, 632 pp. (In Russian)
  16. Merril R. T., McElhinny M. W., McFadden P. L. The Magnetic Field of the Earth: Paleomagnetism, the Core, and the Deep Mantle. Academic Press: London, 1996, 531 pp.
  17. Zheligovsky V. A., Chertovskih R. A. On the kinematic generation of magnetic modes of the bloch type, Izvestiya Phys. Solid Earth, 2020, vol. 56, no. 1, pp. 118–132. DOI: 10.31857/S0002333720010159 (In Russian)
  18. Rozenknop L. M., Reznikov E. L. On the free oscillations of a rotating viscous in the outer Earth core, Vychislitelnaya Seismologiya: Pryamye Zadachi Matematicheskoi Fiziki, 1998, no. 30, pp. 121–132. (In Russian)
  19. Vodinchar G. M., Feshenko L. K. Library of Programs for the Research of «Low-Mode Geodynamo Model»: «LowModedGeodinamoModel», Certificate of State Registration No. 50201100092, 2011.
  20. Vodinchar G. M. Database «Parameters of Eigenmodes of Free Oscillations of MHD Fields in the Earth’s Core», Certificate of State Registration No. 2019620054, 10.01.2019.
  21. Vodinchar G. M. Using symbolic calculations to calculate the eigenmodes of the free damping of a geomagnetic field, E3S Web of Conferences, 2018, vol. 62, no. 02018. DOI: 10.1051/e3sconf/20186202018
  22. Sokoloff D. D., Nefedov S. N. A small-mode approximation in the stellar dynamo problem, Num. Meth. Prog., 2007, vol. 8, no. 2, pp. 195–204. (In Russian)
  23. Gledzer E. B., Dolzhanskiy F. V., Obukhov A. M. Sistemy gidrodinamicheskogo tipa i ikh primenenie [Hydrodynamic Type Systems and Their Application]. Nauka: Moscow, 1981, 368 pp. (In Russian)
  24. Elsholts L. E. Differential Equations and Calculus of Variations. Nauka: Moscow, Russia, 1965, 424 pp. (In Russian)
  25. Kurosh A.G. Course of Higher Algebra. Nauka: Moscow, Russia, 1968, 431 pp. (In Russian)
  26. Benettin G., Galgani L., Giorgilli A., Strelcyn J.-M. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them. Part I: Theory, Meccanica, 1980, vol. 15, no. 1, pp. 9–20.
  27. Kuznetsov S.P. Dynamic Chaos and Hyperbolic Attractors: From Mathematics to Physics. Institute of Computer Science Izhevsk: Izhevsk, Russia, 2013, 488 pp. (In Russian)

Information about author

Sheremetyeva Olga Vladimirovna – Cand.Sci.(Tech.), Research Scientist, Laboratory of Physical Process Modeling, Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Paratunka, Kamchatka, Russia, ORCID 0000-0001-9417-9731.