Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-25T02:00:25.906Z Has data issue: false hasContentIssue false
  • English
  • Français

A multiscale dynamo model driven by quasi-geostrophic convection

Published online by Cambridge University Press:  02 September 2015

Michael A. Calkins ,
Keith Julien ,
Steven M. Tobias  and
Jonathan M. Aurnou
Michael A. Calkins*
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA
Keith Julien
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA
Steven M. Tobias
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
Jonathan M. Aurnou
Affiliation:
Department of Earth, Planetary and Space Sciences, University of California, Los Angeles, CA 90095, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A convection-driven multiscale dynamo model is developed in the limit of low Rossby number for the plane layer geometry in which the gravity and rotation vectors are aligned. The small-scale fluctuating dynamics are described by a magnetically modified quasi-geostrophic equation set, and the large-scale mean dynamics are governed by a diagnostic thermal wind balance. The model utilizes three time scales that respectively characterize the convective time scale, the large-scale magnetic evolution time scale and the large-scale thermal evolution time scale. Distinct equations are derived for the cases of order one and low magnetic Prandtl number. It is shown that the low magnetic Prandtl number model is characterized by a magnetic to kinetic energy ratio that is asymptotically large, with ohmic dissipation dominating viscous dissipation on the large scale. For the order one magnetic Prandtl number model, the magnetic and kinetic energies are equipartitioned and both ohmic and viscous dissipation are weak on the large scales; large-scale ohmic dissipation occurs in thin magnetic boundary layers adjacent to the horizontal boundaries. For both magnetic Prandtl number cases the Elsasser number is small since the Lorentz force does not enter the leading order force balance. The new models can be considered fully nonlinear, generalized versions of the dynamo model originally developed by Childress & Soward (Phys. Rev. Lett., vol. 29, 1972, pp. 837–839), and provide a new theoretical framework for understanding the dynamics of convection-driven dynamos in regimes that are only just becoming accessible to direct numerical simulations.

JFM classification

MHD and Electrohydrodynamics: Dynamo theory Geophysical and Geological Flows: Geodynamo Geophysical and Geological Flows: Geophysical and Geological Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 780 , 10 October 2015 , pp. 143 - 166
Check for updates
Copyright
© 2015 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bender, C. M. & Orszag, S. A. 2010 Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory. Springer.Google Scholar
Busse, F. H. 1975 A model of the geodynamo. Geophys. J. R. Astron. Soc. 42, 437–459.Google Scholar
Busse, F. H. 1986 Asymptotic theory of convection in a rotating, cylindrical annulus. J. Fluid Mech. 173, 545–556.Google Scholar
Calkins, M. A., Julien, K. & Marti, P. 2013 Three-dimensional quasi-geostrophic convection in the rotating cylindrical annulus with steeply sloping endwalls. J. Fluid Mech. 732, 214–244.Google Scholar
Calkins, M. A., Julien, K. & Marti, P. 2014 Onset of rotating and non-rotating convection in compressible and anelastic ideal gases. Geophys. Astrophys. Fluid Dyn. 109 (4), 422–449.Google Scholar
Calkins, M. A., Julien, K. & Marti, P. 2015 The breakdown of the anelastic approximation in rotating compressible convection: implications for astrophysical systems. Proc. R. Soc. Lond. A 471, 20140689.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Charney, J. G. 1948 On the scale of atmospheric motions. Geophys. Publ. 17, 3–17.Google Scholar
Childress, S. & Soward, A. M. 1972 Convection-driven hydromagnetic dynamo. Phys. Rev. Lett. 29 (13), 837–839.Google Scholar
Chini, G., Julien, K. & Knobloch, E. 2009 An asymptotically reduced model of turbulent Langmuir circulation. Geophys. Astrophys. Fluid Dyn. 103 (2–3), 179–197.Google Scholar
Christensen, U. & Aubert, J. 2006 Scaling properties of convection-driven dynamos in rotating shells and applications to planetary magnetic fields. Geophys. J. Intl 166, 97–114.Google Scholar
Davidson, P. A. 2001 An Introduction to Magnetohydrodynamics. Cambridge University Press.Google Scholar
Dolaptchiev, S. I. & Klein, R. 2009 Planetary geostrophic equations for the atmosphere with evolution of barotropic flow. Dyn. Atmos. Oceans 46, 46–61.Google Scholar
Fautrelle, Y. & Childress, S. 1982 Convective dynamos with intermediate and strong fields. Geophys. Astrophys. Fluid Dyn. 22 (3), 235–279.Google Scholar
Favier, B. & Proctor, M. R. E. 2013 Kinematic dynamo action in square and hexagonal patterns. Phys. Rev. E 88, 053011.Google Scholar
Finlay, C. C. & Amit, H. 2011 On flow magnitude and field-flow alignment at Earth’s core surface. Geophys. J. Intl 186, 175–192.Google Scholar
Gillet, N., Brito, D., Jault, D. & Nataf, H.-C. 2007 Experimental and numerical studies of magnetoconvection in a rapidly rotating spherical shell. J. Fluid Mech. 580, 123–143.Google Scholar
Gillet, N., Jault, D., Canet, E. & Fournier, A. 2010 Fast torsional waves and strong magnetic field within the Earth’s core. Nature 465, 74–77.Google Scholar
Gillet, N., Schaeffer, N. & Jault, D.2012 Rationale and geophysical evidence for quasi-geostrophic rapid dynamics within the Earth’s outer core. Phys. Earth Planet. Inter. 202–203, 78–88.Google Scholar
Grooms, I., Julien, K. & Fox-Kemper, B. 2011 On the interactions between planetary geostrophy and mesoscale eddies. Dyn. Atmos. Oceans 51, 109–136.Google Scholar
Guervilly, C., Hughes, D. W. & Jones, C. A. 2015 Generation of magnetic fields by large-scale vortices in rotating convection. Phys. Rev. E 91, 041001.Google Scholar
Haut, T. & Wingate, B. 2014 An asymptotic parallel-in-time method for highly oscillatory PDES. SIAM J. Sci. Comput. 36 (2), A693–A713.Google Scholar
Jackson, A., Bloxham, J. & Gubbins, D. 1993 Time-dependent flow at the core surface and conservation of angular momentum in the coupled core-mantle system. Dyn. Earth’s Deep Interior Earth Rotation 72, 97–107.Google Scholar
Jault, D., Gire, C. & Mouël, J. L. L. 1988 Westward drift, core motions and exchanges of angular momentum between core and mantle. Nature 333, 353–356.Google Scholar
Jones, C. A. 2011 Planetary magnetic fields and fluid dynamos. Annu. Rev. Fluid Mech. 43, 583–614.CrossRefGoogle Scholar
Jones, C. A., Mussa, A. I. & Worland, S. J. 2003 Magnetoconvection in a rapidly rotating sphere: the weak-field case. Proc. R. Soc. Lond. A 459, 773–797.Google Scholar
Julien, K. & Knobloch, E. 2007 Reduced models for fluid flows with strong constraints. J. Math. Phys. 48, 065405.Google Scholar
Julien, K., Knobloch, E., Milliff, R. & Werne, J. 2006 Generalized quasi-geostrophy for spatially anistropic rotationally constrained flows. J. Fluid Mech. 555, 233–274.Google Scholar
Julien, K., Knobloch, E., Rubio, A. M. & Vasil, G. M. 2012a Heat transport in Low-Rossby-number Rayleigh–Bénard Convection. Phys. Rev. Lett. 109, 254503.Google Scholar
Julien, K., Knobloch, E. & Werne, J. 1998 A new class of equations for rotationally constrained flows. Theor. Comput. Fluid Dyn. 11, 251–261.CrossRefGoogle Scholar
Julien, K., Rubio, A. M., Grooms, I. & Knobloch, E. 2012b Statistical and physical balances in low Rossby number Rayleigh–Bénard convection. Geophys. Astrophys. Fluid Dyn. 106 (4-5), 392–428.Google Scholar
King, E. M. & Buffett, B. A. 2013 Flow speeds and length scales in geodynamo models: the role of viscosity. Earth Planet. Sci. Lett. 371, 156–162.Google Scholar
Klein, R. 2010 Scale-dependent models for atmospheric flows. Annu. Rev. Fluid Mech. 42, 249–274.Google Scholar
Malecha, Z., Chini, G. & Julien, K. 2014 A multiscale algorithm for simulating spatially-extended Langmuir circulation dynamics. J. Comput. Phys. 271, 131–150.Google Scholar
Miesch, M. S. 2005 Large-scale dynamics of the convection zone and tachocline. Living Rev. Solar Phys. 2 (1), available online at http://solarphysics.livingreviews.org/Articles/lrsp-2005-1/.Google Scholar
Mizerski, K. A. & Tobias, S. M. 2013 Large-scale convective dynamos in a stratified rotating plane layer. Geophys. Astrophys. Fluid Dyn. 107 (1-2), 218–243.Google Scholar
Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.Google Scholar
Moffatt, H. K. 2008 Magnetostrophic turbulence and the geodynamo. In IUTAM Symposium on Computational Physics and New Perspectives in Turbulence, pp. 339–346. Springer.Google Scholar
Nataf, H.-C. & Schaeffer, N. 2015 Turbulence in the core. In Treatise on Geophysics (ed. Olson, P. & Schubert, G.), vol. 8. Elsevier.Google Scholar
Parker, E. N. 1955 Hydromagnetic dynamo models. Astrophys. J. 122, 293–314.Google Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics. Springer.Google Scholar
Phillips, N. A. 1963 Geostrophic motion. Rev. Geophys. 1 (2), 123–176.Google Scholar
Pozzo, M., Davies, C. J., Gubbins, D. & Alfé, D. 2013 Transport properties for liquid silicon–oxygen–iron mixtures at Earth’s core conditions. Phys. Rev. B 87, 014110.Google Scholar
Roberts, P. H. 1988 Future of geodynamo theory. Geophys. Astrophys. Fluid Dyn. 44, 3–31.Google Scholar
Robinson, A. & Stommel, H. 1959 The oceanic thermocline and the associated thermohaline circulation. Tellus 11, 295–308.Google Scholar
Rotvig, J. & Jones, C. A. 2002 Rotating convection-driven dynamos at low Ekman number. Phys. Rev. E 66, 056308.Google Scholar
Rubio, A. M., Julien, K., Knobloch, E. & Weiss, J. B. 2014 Upscale energy transfer in three-dimensional rapidly rotating turbulent convection. Phys. Rev. Lett. 112, 144501.Google Scholar
Schaeffer, N. & Pais, M. A. 2011 On symmetry and anisotropy of Earth-core flows. Geophys. Res. Lett. 38, L10309.CrossRefGoogle Scholar
Schubert, G. & Soderlund, K. 2011 Planetary magnetic fields: observations and models. Phys. Earth Planet. Inter. 187, 92–108.Google Scholar
Soderlund, K. M., King, E. M. & Aurnou, J. M.2012 The influence of magnetic fields in planetary dynamo models. Earth Planet. Sci. Lett. 333–334, 9–20.Google Scholar
Soward, A. M. 1974 A convection-drive dynamo. Part I: the weak field case. Phil. Trans. R. Soc. Lond. A 275, 611–646.Google Scholar
Sprague, M., Julien, K., Knobloch, E. & Werne, J. 2006 Numerical simulation of an asymptotically reduced system for rotationally constrained convection. J. Fluid Mech. 551, 141–174.CrossRefGoogle Scholar
St Pierre, M. G. 1993 The strong field branch of the Childress–Soward dynamo. In Theory of Solar and Planetary Dynamos (ed. Proctor, M. R. E., Matthews, P. C. & Rucklidge, A. M.), pp. 295–302. Cambridge University Press.Google Scholar
Stanley, S. & Glatzmaier, G. A. 2010 Dynamo models for planets other than Earth. Space Sci. Rev. 152, 617–649.Google Scholar
Steenbeck, M. & Krause, F. 1966 The generation of stellar and planetary magnetic fields by turbulent dynamo action. Z. Naturforsch. a 21, 1285–1296.Google Scholar
Steenbeck, M., Krause, F. & Rädler, K.-H. 1966 A calculation of the mean electromotive force in an electrically conducting fluid in turbulent motion, under the influence of coriolis forces. Z. Naturforsch. a 21, 369–376.Google Scholar
Stellmach, S. & Hansen, U. 2004 Cartesian convection driven dynamos at low Ekman number. Phys. Rev. E 70, 056312.Google Scholar
Stellmach, S., Lischper, M., Julien, K., Vasil, G., Cheng, J. S., Ribeiro, A., King, E. M. & Aurnou, J. M. 2014 Approaching the asymptotic regime of rapidly rotating convection: boundary layers versus interior dynamics. Phys. Rev. Lett. 113, 254501.Google Scholar
Stewartson, K. & Cheng, H. K. 1979 On the structure of inertial waves produced by an obstacle in a deep, rotating container. J. Fluid Mech. 91, 415–432.Google Scholar
Weinan, E., Engquist, B., Li, X., Ren, W. & Vanden-Eijnden, E. 2007 Heterogeneous multiscale methods: a review. Commun. Comput. Phys. 2 (3), 367–450.Google Scholar
Welander, P. 1959 An advective model for the ocean thermocline. Tellus 11, 309–318.Google Scholar