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Force balance in convectively driven dynamos with no inertia

Published online by Cambridge University Press:  01 October 2019

David W. Hughes Open the ORCID record for David W. Hughes [Opens in a new window]  and
Fausto Cattaneo
David W. Hughes*
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
Fausto Cattaneo
Affiliation:
Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA
*
†Email address for correspondence: [email protected]
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Abstract

We study dynamo action in rotating, plane layer Boussinesq convection in the absence of inertia. This allows a decomposition of the velocity into a thermal part driven by buoyancy, and a magnetic part driven by the Lorentz force. We have identified three families of solutions, defined in terms of what is the dominant contribution to the velocity. In weak field dynamos the dominant contribution is the thermal component, in super strong field dynamos the dominant contribution is magnetic and in strong field dynamos the two components are comparable. For each of these solutions we investigate the force balance in the momentum equation to determine the relative importance of the viscous, buoyancy, Coriolis and magnetic forces. We do this by extracting the solenoidal part of the individual terms in the momentum equation, thereby removing their pressure contributions. This is numerically preferable to the more common practice of taking the curl of the momentum equation, which introduces an extra derivative. We find that, irrespective of the type of dynamo solution, the dynamics is controlled by the horizontal forces (in projection). Furthermore, in the progression from weak to strong to super strong dynamos, we find that the viscous forces in the thermal equation become negligible, thereby leading to a balance between buoyancy and Coriolis forces. On the other hand, no corresponding trend is observed in the magnetic part of the momentum equation: the viscous stresses always remain significant. This can be attributed to the different degrees of smoothness of the Coriolis and Lorentz forces, the latter having contributions from strong, filamentary structures. We discuss how our findings relate to dynamo solutions in which viscosity plays no role whatsoever – so-called Taylor states.

JFM classification

MHD and Electrohydrodynamics: Dynamo theory Turbulent Flows: Rotating turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 879 , 25 November 2019 , pp. 793 - 807
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Copyright
© 2019 Cambridge University Press 

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References

Aubert, J., Gastine, T. & Fournier, A. 2017 Spherical convective dynamos in the rapidly rotating asymptotic regime. J. Fluid Mech. 813, 558–593.10.1017/jfm.2016.789Google Scholar
Cattaneo, F. 1999 On the origin of magnetic fields in the quiet photosphere. Astrophys. J. 515, L39–L42.10.1086/311962Google Scholar
Cattaneo, F., Emonet, T. & Weiss, N. 2003 On the interaction between convection and magnetic fields. Astrophys. J. 588, 1183–1198.Google Scholar
Cattaneo, F. & Hughes, D. W. 2006 Dynamo action in a rotating convective layer. J. Fluid Mech. 553, 401–418.10.1017/S0022112006009165Google Scholar
Cattaneo, F. & Hughes, D. W. 2017 Dynamo action in rapidly rotating Rayleigh–Bénard convection at infinite Prandtl number. J. Fluid Mech. 825, 385–411.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon Press.Google Scholar
Dormy, E. 2016 Strong-field spherical dynamos. J. Fluid Mech. 789, 500–513.10.1017/jfm.2015.747Google Scholar
Eltayeb, I. A. 1972 Hydromagnetic convection in a rapidly rotating fluid layer. Proc. R. Soc. Lond. A 326, 229–254.Google Scholar
Eltayeb, I. A. & Roberts, P. H. 1970 Note: on the hydromagnetics of rotating fluids. Astrophys. J. 162, 699.10.1086/150701Google Scholar
Hughes, D. W. & Cattaneo, F. 2016 Strong-field dynamo action in rapidly rotating convection with no inertia. Phys. Rev. E 93, 061101.Google Scholar
Jones, C. A. & Roberts, P. H. 2000 Convection-driven dynamos in a rotating plane layer. J. Fluid Mech. 404, 311–343.10.1017/S0022112099007363Google Scholar
Roberts, P. H. 1978 Magneto-convection in a rapidly rotating fluid. In Rotating Fluids in Geophysics (ed. Roberts, P. H. & Soward, A. M.). Academic.Google Scholar
Roberts, P. H. & Soward, A. M. 1992 Dynamo theory. Annu. Rev. Fluid Mech. 24, 459–512.Google Scholar
Rotvig, J. & Jones, C. A. 2002 Rotating convection-driven dynamos at low Ekman number. Phys. Rev. E 66, 056308.Google Scholar
Schaeffer, N., Jault, D., Nataf, H.-C. & Fournier, A. 2017 Turbulent geodynamo simulations: a leap towards Earth’s core. Geophys. J. Intl 211, 1–29.Google Scholar
Taylor, J. B. 1963 The magneto-hydrodynamics of a rotating fluid and the Earth’s dynamo problem. Proc. R. Soc. Lond. A 274, 274–283.Google Scholar
Yadav, R. K., Gastine, T., Christensen, U. R., Wolk, S. J. & Poppenhaeger, K. 2016 Approaching a realistic force balance in geodynamo simulations. Proc. Natl Acad. Sci. 113, 12065–12070.Google Scholar