Vestnik КRAUNC. Fiz.-Mat. Nauki. 2025. vol. 53. no. 4. P. 29 – 44. ISSN 2079-6641

MATHEMATICAL MODELING
https://doi.org/10.26117/2079-6641-2025-53-4-29-44
Research Article
Full text in Russian
MSC 86-10, 76W05, 86A25, 35Q86

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Multiscale Geodynamo Model: Coupling Large-Scale and Small-Scale Submodels

G. M. Vodinchar^{\ast}, L. K. Feshchenko

Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Mirnaya Str., 7, Paratunka, Kamchatka region, 683003, Russia

Abstract. The formation of the main geomagnetic field is ensured by the hydromagnetic dynamo mechanism — the generation of a magnetic field by the movement of a conducting medium. The Earth’s dynamo (geodynamo) is driven by the convection of liquid metal in the Earth’s outer core. This mechanism is fundamentally nonlinear and three-dimensional, so solving the geodynamo equations is only possible numerically. Convection in the core is characterized by developed turbulence, and a direct numerical solution with resolution of all turbulent scales is impossible even on high-performance computing systems. One solution is to separate scales into large and small. At large scales, the geodynamo equations are solved using a suitable numerical method, but using turbulent values of the diffusion coefficients. These values are properties of the flow, not the medium, and therefore must be calculated based on small-scale movements. Small-scale dynamics can be described by a shell model of magneto convection. Combining two such models forms a multiscale geodynamo model. This paper derives equations for a complex shell model of magneto convection that satisfy the conservation laws of energy, cross-helicity, magnetic helicity, and temperature fluctuation energy in the dissipation-free limit. The model coefficients are consistent with the probabilities of interaction between scale shells. A scheme is described for coupling a large-scale spectral model of the geodynamo and a shell model of turbulent convection, which exchange information with each other. The spectral model determines the values of the phase variables of the largest scales in the shell model. Turbulent values of the diffusion coefficients for the spectral model are calculated from the phase variables of the shell model. Explicit expressions are obtained for calculating the parameters of one model from the phase variables of the other.

Key words: geodynamo, geomagnetic field, low-mode models, spectral models, shell models of turbulence, multiscale models.

Received: 12.11.2025; Revised: 26.11.2025; Accepted: 28.11.2025; First online: 29.11.2025

For citation. Vodinchar G. M., Feshchenko L. K. Multiscale geodynamo model: coupling large-scale and small-scale submodels. Vestnik KRAUNC. Fiz.-mat. nauki. 2025, 53: 4, 29-44. EDN: UKRQBI. https://doi.org/10.26117/2079-6641-2025-53-4-29-44.

Funding. The study was carried out under the State Subject of IKIR FEB RAS (reg. № 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: gvodinchar@ikir.ru

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

© Vodinchar G. M., Feshchenko L. K., 2025

© Institute of Cosmophysical Research and Radio Wave Propagation, 2025 (original layout, design, compilation)

References

  1. 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. 532 pp.
  2. Jones C. A. Convection-driven geodynamo models // Phil. Trans. R. Soc. Lond. A, 2000. vol. 358, pp. 873–897. DOI: 10.1098/rsta.2000.0565
  3. Aurnou J., King E. The cross-over to magnetostrophic convection in planetary dynamo systems // Proc. R. Soc. A Math. Phys. Eng. Sci., 2017. vol. 473, 20160731. DOI: 10.1098/rspa.2016.0731
  4. Zeldovich Ya. B., Rusmaikin A. A., Sokoloff D.D. Magnitny‘e polya v astrofizike [Magnetic fields in astrophysics]. Moscow-Izhevsk, NIC RHD, 2006. 384 p. (In Russian).
  5. Frick P., Sokoloff D., Stepanov R. Large-small scales interactions and quenching in α2-dynamo, Physical Rev. E., 2006, vol. 74, pp. 4155–4164.
  6. Frick P., Reshetnyak M., Sokoloff D., Combined grid-shell approach for convection in a rotating spherical layer, EuroPhys. Letters, 2002, vol. 59, no. 2, pp. 212–217.
  7. Vodinchar G., Feshchenko L. Computational Technology for the Basis and Coefficients of Geodynamo Spectral Models in the Maple System, Mathematics, 2023, vol. 11, no. 13, 3000. DOI:10.3390/math11133000
  8. Frick P.G. Turbulence: approaches and models [Turbulence: approaches and models]. Moscow, Izhevsk, 2010, 332 p. (In Russian).
  9. Ditlevsen P. D. Turbulence and shell models, New York: Cambridge Univ. Press, 2011, 332 p.
  10. Plunian F., Stepanov R., Frick P. Shell models of magnetohydrodynamic turbulence, Physics Reports, 2013, vol. 523, no. 1, pp. 1–60. DOI: 10.1016/j.physrep.2012.09.001
  11. Vodinchar G.M., Feshchenko L.K. Avtomatizirovannaya generatsiya kaskadnykh modeley turbulentnosti metodami komp’yuternoy algebry [Automated Generation of Cascade Models of Turbulence Using Computer Algebra Methods].Computation Technology, 2021, vol. 26, no 5, pp. 65–80. DOI: 10.25743/ICT.2021.26.5.006 (In Russian).
  12. Vodinchar G.M., Feshchenko L.K. Vy‘chislitel‘naya texnologiya postroeniya kaskadny‘x modelej magnitogidrodinamicheskoj turbulentnosti [Computational Technology for Shell Models of Magnetohydrodynamic Turbulunce]. Informatics and Automation, 2024, vol. 23, no 6, pp. 1665–1697. DOI: 10.15622/ia.23.6.4 (In Russian).
  13. Smagorinsky J. General circulation experiments with the primitive equations: 1. The basic experiment, Monthly Weather Review, 1963, vol. 91, pp 99–164.
  14. Yakhot V., Orszag S., Yakhot A. Heat transfer in turbulent fluid. 1. Pipe flows, Inter. J. Heat Mass Transfer, 1987, vol. 30, no. 1, pp. 15–22.
  15. Kirillov P.L. Teploobmen v turbulentnom potoke. CH.1. Turbulentnoye chislo Prandtlya [Heat Transfer in Turbulent Flow. Part 1. Turbulent Prandtl Number], Atomnaya energiya, 2017, vol. 122, no. 3, pp. 133–145. (In Russian).

Information about the authors

Vodinchar Gleb Mikhailovich – PhD (Phys. & Math.), Leading Researcher, Acting Head of the Laboratory for Simulation of Physical Processes, IKIR FEB RAS, Russia, ORCID 0000-0002-5516-1931.

Feshchenko Liubov Konstantinovna – PhD (Phys. & Math.), Researcher, Laboratory for Modeling Physical Processes, IKIR FEB RAS, Russia, ORCID 0000-0001-5970-7316.

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