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Toward scale-separated weak-field spherical dynamos

R. J. Teed*

E. Dormy†

  • School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8SQ, United Kingdom

  • Département de Mathématiques et Applications, UMR-8553, École Normale Supérieure, CNRS, PSL University, 75005 Paris, France

  • *Contact author: Robert.Teed@glasgow.ac.uk
  • †Contact author: Emmanuel.Dormy@ens.fr
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Phys. Rev. Fluids 10, 103702 – Published 15 October, 2025

DOI: https://doi.org/10.1103/9r2f-t4wk

Abstract

Recent numerical experiments of dynamo action relevant to the generation of the geomagnetic field have produced different regime branches identified within bifurcation diagrams. Notable are separate branches in which the resultant magnetic field is either weak or strong. Weak-field solutions can be identified by the prominent role of viscosity on the motion whereas the magnetic field has a leading order effect on the flow in strong-field solutions. For a given Ekman number, E (measuring the ratio of viscosity to rotational effects), the existence of these branches and bistability between them is reliant on a small enough magnetic Ekman number, Em (measuring the ratio of magnetic diffusion to rotational effects, so E/Em=Pm, the magnetic Prandtl number). Both branches are known to produce large scale dipolar magnetic fields but do not exhibit an expected scale separation between the flow and magnetic field. In this work, by reducing Em, we identify a variety of dynamo states on the weak-field branch beyond the known dipolar solutions. Specifically, hemispherical and nondipolar dynamos were found, in addition to the usual dipolar solutions. Some solutions exhibit clear scale separation between small-scale flow and large-scale magnetic field, despite the large ratio of viscosity to magnetic diffusion. Numerical solutions in this regime have not been observed before and they offer a first connection with earlier theoretical work based on mean-field theory.

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References (50)

  1. E. Dormy, Rapidly rotating magnetohydrodynamics and the geodynamo, Annu. Rev. Fluid Mech. 57, 335 (2025).
  2. C. Guervilly, P. Cardin, and N. Schaeffer, Turbulent convective length scale in planetary cores, Nature (London) 570, 368 (2019).
  3. M. Steenbeck, F. Krause, and K.-H. Rädler, Berechnung der mittleren Lorentz-Feldstärke für ein elektrisch leitendes Medium in turbulenter, durch Coriolis-Kräfte beeinflußter Bewegung, Z. Naturforsch. 21, 369 (1966) [English Translation: Roberts & Stix (1971), pp. 29–47].
  4. F. Krause and K.-H. Rädler, Mean-Field Magnetohydrodynamics and Dynamo Theory (Pergamon Press, Oxford, UK, 1980).
  5. H. K. Moffatt and E. Dormy, Self-Exciting Fluid Dynamos, Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, UK, 2019).
  6. K. Mizerski, Non-Equilibrium Hydromagnetic Dynamos (IOP Publishing, Bristol, UK, 2023).
  7. U. Christensen, P. Olson, and G. Glatzmaier, Numerical modelling of the geodynamo: A systematic parameter study, Geophys. J. Int. 138, 393 (1999).
  8. C. Kutzner and U. Christensen, From stable dipolar towards reversing numerical dynamos, Phys. Earth Planet. Int. 131, 29 (2002).
  9. V. Morin and E. Dormy, The dynamo bifurcation in rotating spherical shells, Int. J. Mod. Phys. B 23, 5467 (2009).
  10. E. Dormy, Strong-field spherical dynamos, J. Fluid Mech. 789, 500 (2016).
  11. N. Schaeffer, D. Jault, H.-C. Nataf, and A. Fournier, Turbulent geodynamo simulations: A leap towards Earth's core, Geophys. J. Int. 211, 1 (2017).
  12. J. Aubert, State and evolution of the geodynamo from numerical models reaching the physical conditions of Earth's core, Geophys. J. Int. 235, 468 (2023).
  13. C. Jones, A dynamo model of jupiter's magnetic field, Icarus 241, 148 (2014).
  14. M. Schrinner, K.-H. Rädler, D. Schmitt, M. Rheinhardt, and U. R. Christensen, Mean-field concept and direct numerical simulations of rotating magnetoconvection and the geodynamo, Geophys. Astrophys. Fluid Dyn. 101, 81 (2007).
  15. M. Schrinner, L. Petitdemange, and E. Dormy, Oscillatory dynamos and their induction mechanisms, Astron. Astrophys. 530, A140 (2011).
  16. K. Soderlund, A. Sheyko, E. King, and J. Aurnou, The competition between Lorentz and Coriolis forces in planetary dynamos, Prog. Earth Planet. Sci. 2, 24 (2015).
  17. T. Schwaiger, T. Gastine, and J. Aubert, Force balance in numerical geodynamo simulations: A systematic study, Geophys. J. Int. 219, S101 (2019).
  18. T. Schwaiger, T. Gastine, and J. Aubert, Relating force balances and flow length scales in geodynamo simulations, Geophys. J. Int. 224, 1890 (2020).
  19. R. J. Teed and E. Dormy, Solenoidal force balances in numerical dynamos, J. Fluid Mech. 964, A26 (2023).
  20. A. Soward, A convection-driven dynamo I. The weak field case, Philos. Trans. R. Soc. London A 275, 611 (1974).
  21. A. Soward, Convection driven dynamos, Phys. Earth Planet. Inter. 20, 134 (1979).
  22. D. R. Fearn, Thermal and magnetic instabilities in a rapidly rotating fluid sphere, Geophys. Astrophys. Fluid Dyn. 14, 103 (1979).
  23. S. Horn and J. M. Aurnou, The Elbert range of magnetostrohic convection. I. Linear theory, Proc. R. Soc. A 478, 20220313 (2022).
  24. S. Horn and J. M. Aurnou, The Elbert range of magnetostrophic convection. II. Comparing linear theory to nonlinear low-Rm simulations, Proc. R. Soc. A 481, 20240016 (2025).
  25. P. Roberts, Pure dynamo theory and geomagnetic dynamo theory, in Proceedings of the 1st International Workshop on Dynamo Theory and the Generation of the Earth's Magnetic Field, edited by I. Cupal (Czech Academy of Sciences, Prague, 1979), pp. 7–12.
  26. E. Dormy, L. Oruba, and L. Petitdemange, Three branches of dynamo action, Fluid Dyn. Res. 50, 011415 (2018).
  27. R. D. Simitev and F. H. Busse, Bistability and hysteresis of dipolar dynamos generated by turbulent convection in rotating spherical shells, Europhys. Lett. 85, 19001 (2009).
  28. L. Petitdemange, Systematic parameter study of dynamo bifurcations in geodynamo simulations, Phys. Earth Planet. Inter. 277, 113 (2018).
  29. E. Dormy, A. Soward, C. Jones, D. Jault, and P. Cardin, The onset of thermal convection in rotating spherical shells, J. Fluid Mech. 501, 43 (2004).
  30. C. A. Jones, A. I. Mussa, and S. J. Worland, Magnetoconvection in a rapidly rotating sphere: The weak–field case, Proc. R. Soc. London A 459, 773 (2003).
  31. A. Willis, B. Sreenivasan, and D. Gubbins, Thermal core–mantle interaction: Exploring regimes for “locked” dynamo action, Phys. Earth Planet. Inter. 165, 83 (2007).
  32. “Leeds spherical dynamo” code, https://github.com/Leeds-Spherical-Dynamo.
  33. R. J. Teed, Data for: Toward scale separated weak-field spherical dynamos (2025), https://doi.org/10.5525/gla.researchdata.1935.
  34. P. Mauersberger, Das mittel der energiedichte des geomagnetischen hauptfeldes an der Erdoberfläche und seine säkulare anderung, Gerlands Beitr. Geophys. 65, 207 (1956).
  35. F. Lowes, Spatial power spectrum of the main geomagnetic field, and extrapolation to the core, Geophys. J. Int. 36, 717 (1974).
  36. P. Roberts, On the thermal instability of a rotating-fluid sphere containing heat sources, Philos. Trans. R. Soc. London A 263, 93 (1968).
  37. F. H. Busse, Thermal instabilities in rapidly rotating systems, J. Fluid Mech. 44, 441 (1970).
  38. E. Dormy, Modélisation numérique de la dynamo terrestre, Ph.D. thesis, Institut de Physique du Globe de Paris (1997).
  39. P. Roberts and A. Soward, Dynamo theory, Annu. Rev. Fluid Mech. 24, 459 (1992).
  40. A. Guseva, L. Petitdemange, and S. M. Tobias, Run-away transition to turbulent strong-field dynamo, JGR Planets 130, e2024JE008496 (2025).
  41. W. Dietrich and J. Wicht, A hemispherical dynamo model: Implications for the martian crustal magnetization, Phys. Earth Planet. Inter. 217, 10 (2013).
  42. H. Fuchs, K.-H. Rädler, and M. Rheinhard, On self–killing and self–creating dynamos, Astron. Nachr. 320, 129 (1999).
  43. F. H. Busse, Generation of magnetic fields by convection, J. Fluid Mech. 57, 529 (1973).
  44. F. H. Busse, A model of the geodynamo, Geophys. J. R. Astron. Soc. 42, 437 (1975).
  45. C. Guervilly and E. Dormy, The cross-over from viscous to inertial lengthscales in rapidly-rotating convection, Geophys. Res. Lett. 52, e2024GL111593 (2025).
  46. Indeed, if Pm is chosen to be too small (for a given E), then dynamo action ceases altogether!
  47. S. Stanley, L. Elkins-Tanton, M. T. Zuber, and E. M. Parmentier, Mars’ paleomagnetic field as the result of a single-hemisphere dynamo, Science 321, 1822 (2008).
  48. Y. Wu, W. Dietrich, and X. Tao, Parameter regimes of hemispherical dynamo waves in a spherical shell from 3D MHD simulations, JGR Planets 129, e2023JE007976 (2024).
  49. A. Sheyko, C. Finlay, J. Favre, and A. Jackson, Scale separated low viscosity dynamos and dissipation within the Earth's core, Sci. Rep. 8, 12566 (2018).
  50. www.dirac.ac.uk

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Phys. Rev. Fluids 10, 103702– Published 15 October, 2025
  • Received 13 June 2025
  • Accepted 4 September 2025
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