Vestnik КRAUNC. Fiz.-Mat. nauki. 2024. vol. 48. no. 3. P. 70 – 82. ISSN 2079-6641

MATHEMATICAL MODELING
https://doi.org/10.26117/2079-6641-2024-48-3-70-82
Research Article
Full text in Russian
MSC 47G20

Contents of this issue

Read Russian Version

Stochastic Two-Mode Hereditary Model of a Cosmic Dynamo

E. A. Kazakov^{\ast}, G.M Vodinchar

Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, 684034, Paratunka, Mirnaya str. 7, Russia

Abstract. The paper is devoted to a class of stochastic two-mode hereditary models of the cosmic dynamo. The models include two magnetic field generators — large-scale and turbulent (α-effect). The influence of the magnetic field on the motion of the medium is presented through the suppression of the α-effect by a functional of the field components, which introduces memory (hereditary) into the model. The model describes the dynamics of only large-scale components, but takes into account the possible impact of smallscale modes using a stochastic term. This term models the influence of possible spontaneous synchronization of small-scale modes. The paper also presents a numerical scheme for solving the integro-differential equations of the model. The numerical scheme consists of two parts, for the differential part the Adams «predictor-corrector» method of the fourth order is used, and for the integral part the Simpson method.The main result of the work is a generalized model of a dynamo system, with an additive addition of a random correction to the α-generator. Taking into account such a correction significantly diversifies the dynamic modes in the model.

Key words: hydromagnetic dynamo, memory, heredity, integro-differential equations, stochastic model, α-effect, coherent structures.

Received: 11.11.2024; Revised: 13.11.2024; Accepted: 15.11.2024; First online: 20.11.2024

For citation. Kazakov E. A., Vodinchar G. M. Stochastic two-mode hereditary model of a cosmic dynamo. Vestnik KRAUNC. Fiz.-mat. nauki. 2024, 48: 3, 70-82. EDN: YFWFOR. https://doi.org/10.26117/2079-6641-2024-48-3-70-82.

Funding. The work was supported by IKIR FEB RAS State Task (Reg. No. NIOKTR 124012300245-2).

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

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

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

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

© Kazakov E. A., Vodinchar G. M., 2024

© Institute of Cosmophysical Research and Radio Wave Propagation, 2024 (original layout, design, compilation)

References

  1. Zeldovich Ya. B., Ruzmaikin A. A., Sokolov D. D. Magnitie polay v astrofizike [Magnetic fields in astrophysics]Moscow-Izhevsk: SIC «RHD», 2006. (In Russian).
  2. Krause F., R¨adler K.-H. Mean-field magnetohydrodynamics and dynamo theory. – New York: PergamonPress, 1980.
  3. Merril R.T., McElhinny M.W., McFadden P.L. The Magnetic Field of the Earth: Paleomagnetism, the Core, and the Deep Mantle. London: Academic Press, 1996.
  4. Brandenburg A. Memory effects in turbulent transport, Astrophys. J, 2009. vol. 706. P. 712–726.
  5. Hori K., Yoshida S. Non-local memory effects of the electromotive force by fluid motion with helicity and two-dimensional periodicity. Geophys. Astro. Fluid. 2008. vol. 102. P. 601–632.
  6. Vodinchar G.M. Hereditary Oscillator Associated with the Model of a Large-Scale αω-Dynamo, Mathematics, 2020, vol. 8, no. 11, p. 2065. DOI: 10.3390/math8112065.
  7. Kazakov E. A. Hereditary low-mode dynamo model. Vestnik KRAUNC. Fiz.-mat. nauki. 2021. vol. 35. no. 2. P. 40-47. DOI: 10.26117/2079-6641-2021-35-2-40-47 (In Russian).
  8. Kazakov E. A. Two-mode model of a hydromagnetic dynamo with memory. Computational Technologies. 2022. vol. 27. no. 6. P.19–32. DOI:10.25743/ICT.2022.27.6.003. (In Russian).
  9. Vodinchar G. M., Kazakov E. A. Elimination of the integral term in the equations of one hereditary system related to the hydromagnetic dynamo. Vest. KRAUNC. Fiz.-mat. nauki. 2023. vol. 42. no. 1. P. 180-190. DOI: 10.26117/2079-6641-2023-42-1-180-190. (In Russian).
  10. Vodinchar G., Kazakov E. The Lorenz system and its generalizations as dynamo models with memory. E3S Web of Conf. 2018, vol. 62. 02011. DOI: 10.1051/e3sconf/20186202011.
  11. Kolisnichenko A. V., Marov M. Ya. Turbulence and self-organization. Problems of modeling space and natural environments. Moscow. Binom, 2009. (In Russian).
  12. Vodinchar G., Feshchenko L. Fractal Properties of the Magnetic Polarity Scale in the Stochastic Hereditary αω-Dynamo Model, Fractal Fract, 2022. vol. 6, no. 6, p. 328. DOI: 10.3390/math8112065.
  13. Korn G., Korn T. Spravochnik po matimatike dlay nauchnih rabotikov i ingenerov [Handbook of mathematics for scientists and engineers]. Moscow. Nauka. 1968. (In Russian).
  14. Kalitkin N. N. Numerical methods Moscow: Science. 1978. 512. (In Russian).
  15. Tutueva A. V., et al. G. Semi-explicit multistep Adams-Bashforth-Moulton methods for solving stiff systems of ordinary differential equations. Jour. of Instrum. Enginer. 2021. vol. 64, no. 8. P. 599-607 (in Russian).
  16. Moheuddin М. М., Titu M.A.S., Hossain S. A New Analysis of Approximate Solutions for Numerical Integration Problems with Quadrature-based Methods. Pure and Applied Mathem. Jour. 2020. vol. 9. no. 3. P. 46-54. DOI: 10.11648/j.pamj.20200903.11.

Information about the authors

Kazakov Evgeny Anatolevich – Junior Researcher, Lab. of electromagnetic propogation, Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Paratunka, Russia, ORCID 0000-0001-7235-4148.

Vodinchar Gleb Mikhailovich – Ph. D. (Phys. & Math.), Leading Researcher, Lab. for Simulation of Physical Processes, Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Paratunka, Russia, ORCID 0000-0002-5516-1931.

Read articleDownload article