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Convective dynamo action in a spherical shell: symmetries and modulation

Published online by Cambridge University Press:  28 June 2016

Raphaël Raynaud  and
Steven M. Tobias
Raphaël Raynaud*
Affiliation:
School of Astronomy, Institute for Research in Fundamental Sciences (IPM), 19395-5531, Tehran, Iran LERMA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, École normale supérieure, F-75005, Paris, France
Steven M. Tobias
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
*
†Email address for correspondence: raphael.raynaud@ipm.ir
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Abstract

We consider dynamo action driven by three-dimensional rotating anelastic convection in a spherical shell. Motivated by the behaviour of the solar dynamo, we examine the interaction of hydromagnetic modes with different symmetries and demonstrate how complicated interactions between convection, differential rotation and magnetic fields may lead to modulation of the basic cycle. For some parameters, type 1 modulation occurs by the transfer of energy between modes of different symmetries with little change in the overall amplitude; for other parameters, the modulation is of type 2, where the amplitude is significantly affected (leading to grand minima in activity) without significant changes in symmetry. Most importantly, we identify the presence of ‘supermodulation’ in the solutions, where the activity switches chaotically between type 1 and type 2 modulation; this is believed to be an important process in solar activity.

JFM classification

Convection: Convection MHD and Electrohydrodynamics: MHD and Electrohydrodynamics

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Rapids
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Journal of Fluid Mechanics , Volume 799 , 25 July 2016 , R6
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© 2016 Cambridge University Press 

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References

Arlt, R. 2009 The butterfly diagram in the eighteenth century. Solar Phys. 255, 143–153.Google Scholar
Arlt, R. & Weiss, N. 2014 Solar activity in the past and the chaotic behaviour of the dynamo. Space Sci. Rev. 186, 525–533.Google Scholar
Baliunas, S. L., Donahue, R. A., Soon, W. & Henry, G. W. 1998 Activity cycles in lower main sequence and post main sequence stars: the HK Project. Astron. Soc. Pacific Conf. Ser. 154, 153–172.Google Scholar
Beer, J., Tobias, S. & Weiss, N. 1998 An active Sun throughout the Maunder minimum. Solar Phys. 181, 237–249.Google Scholar
Braginsky, S. I. & Roberts, P. H. 1995 Equations governing convection in Earth’s core and the geodynamo. Geophys. Astrophys. Fluid Dyn. 79, 1–97.Google Scholar
Bushby, P. & Mason, J. 2004 Solar dynamo: understanding the solar dynamo. Astron. Geophys. 45 (4), 4.07–4.13.Google Scholar
Busse, F. H. & Simitev, R. D. 2006 Parameter dependences of convection-driven dynamos in rotating spherical fluid shells. Geophys. Astrophys. Fluid Dyn. 100, 341–361.Google Scholar
Charbonneau, P. 2014 Solar dynamo theory. Annu. Rev. Astron. Astrophys. 52, 251–290.Google Scholar
Choudhuri, A. R. & Karak, B. B. 2012 Origin of grand minima in sunspot cycles. Phys. Rev. Lett. 109 (17), 171103.Google Scholar
Christensen, U. R. 2002 Zonal flow driven by strongly supercritical convection in rotating spherical shells. J. Fluid Mech. 470, 115–133.Google Scholar
Dietrich, W., Schmitt, D. & Wicht, J. 2013 Hemispherical Parker waves driven by thermal shear in planetary dynamos. Europhys. Lett. 104, 49001.CrossRefGoogle Scholar
Dormy, E., Cardin, P. & Jault, D. 1998 MHD flow in a slightly differentially rotating spherical shell, with conducting inner core, in a dipolar magnetic field. Earth Planet. Sci. Lett. 160, 15–30.Google Scholar
Dubé, C. & Charbonneau, P. 2013 Stellar dynamos and cycles from numerical simulations of convection. Astrophys. J. 775, 69.Google Scholar
Eddy, J. A. 1976 The Maunder minimum. Science 192, 1189–1202.Google Scholar
Gallet, B. & Pétrélis, F. 2009 From reversing to hemispherical dynamos. Phys. Rev. E 80 (3), 035302.Google Scholar
Gastine, T., Duarte, L. & Wicht, J. 2012 Dipolar versus multipolar dynamos: the influence of the background density stratification. Astron. Astrophys. 546, A19.Google Scholar
Grote, E. & Busse, F. H. 2000 Hemispherical dynamos generated by convection in rotating spherical shells. Phys. Rev. E 62, 4457.Google Scholar
Grote, E., Busse, F. H. & Tilgner, A. 2000 Regular and chaotic spherical dynamos. Phys. Earth Planet. Inter. 117, 259–272.Google Scholar
Hackman, T., Lehtinen, J., Rosén, L., Kochukhov, O. & Käpylä, M. J. 2016 Zeeman–Doppler imaging of active young solar-type stars. Astron. Astrophys. 587, A28.Google Scholar
Hazra, S., Passos, D. & Nandy, D. 2014 A stochastically forced time delay solar dynamo model: self-consistent recovery from a Maunder-like grand minimum necessitates a mean-field alpha effect. Astrophys. J. 789, 5.Google Scholar
Jones, C. A., Boronski, P., Brun, A. S., Glatzmaier, G. A., Gastine, T., Miesch, M. S. & Wicht, J. 2011 Anelastic convection-driven dynamo benchmarks. Icarus 216, 120–135.Google Scholar
Jones, C. A., Thompson, M. J. & Tobias, S. M. 2010 The solar dynamo. Space Sci. Rev. 152, 591–616.Google Scholar
Knobloch, E. 1994 Bifurcations in rotating systems. In Lectures on Solar and Planetary Dynamos (ed. Proctor, M. R. E. & Gilbert, A. D.), p. 331. Cambridge University Press.Google Scholar
Knobloch, E. & Landsberg, A. S. 1996 A new model of the solar cycle. Mon. Not. R. Astron. Soc. 278, 294–302.Google Scholar
Knobloch, E., Tobias, S. M. & Weiss, N. O. 1998 Modulation and symmetry changes in stellar dynamos. Mon. Not. R. Astron. Soc. 297, 1123–1138.Google Scholar
Krause, F. & Rädler, K. H. 1980 Mean-Field Magnetohydrodynamics and Dynamo Theory. Pergamon.Google Scholar
Lantz, S. R. & Fan, Y. 1999 Anelastic magnetohydrodynamic equations for modeling solar and stellar convection zones. Astrophys. J. Suppl. 121, 247–264.Google Scholar
McCracken, K. G., Beer, J., Steinhilber, F. & Abreu, J. 2013 A phenomenological study of the cosmic ray variations over the past 9400 years, and their implications regarding solar activity and the solar dynamo. Solar Phys. 286, 609–627.Google Scholar
Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.Google Scholar
Oláh, K., Kolláth, Z., Granzer, T., Strassmeier, K. G., Lanza, A. F., Järvinen, S., Korhonen, H., Baliunas, S. L., Soon, W., Messina, S. et al. 2009 Multiple and changing cycles of active stars. II. Results. Astron. Astrophys. 501, 703–713.Google Scholar
Parker, E. N. 1955 Hydromagnetic dynamo models. Astrophys. J. 122, 293.Google Scholar
Pétrélis, F., Fauve, S., Dormy, E. & Valet, J.-P. 2009 Simple mechanism for reversals of Earth’s magnetic field. Phys. Rev. Lett. 102 (14), 144503.Google Scholar
Pipin, V. V. 1999 The Gleissberg cycle by a nonlinear 𝛼Λ dynamo. Astron. Astrophys. 346, 295–302.Google Scholar
Raynaud, R., Petitdemange, L. & Dormy, E. 2014 Influence of the mass distribution on the magnetic field topology. Astron. Astrophys. 567, A107.Google Scholar
Raynaud, R., Petitdemange, L. & Dormy, E. 2015 Dipolar dynamos in stratified systems. Mon. Not. R. Astron. Soc. 448, 2055–2065.Google Scholar
Schmitt, D., Schuessler, M. & Ferriz-Mas, A. 1996 Intermittent solar activity by an on–off dynamo. Astron. Astrophys. 311, L1–L4.Google Scholar
Schrinner, M., Petitdemange, L. & Dormy, E. 2011 Oscillatory dynamos and their induction mechanisms. Astron. Astrophys. 530, A140.Google Scholar
Schrinner, M., Petitdemange, L. & Dormy, E. 2012 Dipole collapse and dynamo waves in global direct numerical simulations. Astrophys. J. 752, 121.Google Scholar
Schrinner, M., Petitdemange, L., Raynaud, R. & Dormy, E. 2014 Topology and field strength in spherical, anelastic dynamo simulations. Astron. Astrophys. 564, A78.Google Scholar
Sokoloff, D. & Nesme-Ribes, E. 1994 The Maunder minimum: a mixed-parity dynamo mode? Astron. Astrophys. 288, 293–298.Google Scholar
Tobias, S. M. 1996 Grand minima in nonlinear dynamos. Astron. Astrophys. 307, L21.Google Scholar
Tobias, S. M. 1998 Relating stellar cycle periods to dynamo calculations. Mon. Not. R. Astron. Soc. 296, 653–661.Google Scholar
Tobias, S. M. 2002 Modulation of solar and stellar dynamos. Astron. Nachr. 323 (3–4), 417–423.Google Scholar
Usoskin, I. G. 2013 A history of solar activity over millennia. Living Rev. Solar Phys. 10.Google Scholar
Usoskin, I. G., Arlt, R., Asvestari, E., Hawkins, E., Käpylä, M., Kovaltsov, G. A., Krivova, N., Lockwood, M., Mursula, K., O’Reilly, J. et al. 2015 The Maunder minimum (1645–1715) was indeed a grand minimum: a reassessment of multiple datasets. Astron. Astrophys. 581, A95.Google Scholar
Weiss, N. O. 2011 Chaotic behaviour in low-order models of planetary and stellar dynamos. Geophys. Astrophys. Fluid Dyn. 105, 256–272.Google Scholar
Weiss, N. O. & Tobias, S. M. 2016 Supermodulation of the Sun’s magnetic activity: the effects of symmetry changes. Mon. Not. R. Astron. Soc. 456, 2654–2661.Google Scholar
Yadav, R. K., Gastine, T., Christensen, U. R., Duarte, L. D. V. & Reiners, A. 2016 Effect of shear and magnetic field on the heat-transfer efficiency of convection in rotating spherical shells. Geophys. J. Intl 204, 1120–1133.Google Scholar