Dynamo Models of the Solar Cycle

Part of the Saas-Fee Advanced Courses book series (SAASFEE, volume 39)


This chapter details a series of dynamo models applicable to the sun and solar-type stars. After introducing the theoretical framework known as mean-field electrodynamics, a series of increasingly complex dynamo models are constructed, with the primary aim of reproducing the various basic observed characteristics of the solar magnetic activity cycle. Global and local magnetohydrodynamcial simulations of solar convection, and dynamo action therein, are also considered, and the resulting magnetic cycles compared and contrasted to those obtained in the simpler dynamo models. The focus throughout the chapter is on the sun, simply because the amount of available observational material on the solar magnetic field and its cycle dwarfs anything else in the astrophysical realm, in terms of spatial and temporal resolution, sensitivity, and time span.


Flux Rope Meridional Flow Dynamo Model Toroidal Field Poloidal Field 
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  1. The observational literature on the solar magnetic field is immense, and many good review papers are available. One I particularly like is Solanki, S. K., Inhester, B., & Schüssler, M.: 2006, The solar magnetic field, Rep. Prog. Phys., 69, 563–668Google Scholar
  2. Hale’s original papers on sunspots are still well worth reading. The two key papers are: Hale, G. E.: 1908, On the probable existence of a magnetic field in sun-spots, Astrophys. J., 28, 315–343Google Scholar
  3. Hale, G. E, Ellerman, F., Nicholson, S.B., & Joy, A.H.: 1919, The magnetic polarity of sun-spots, Astrophys. J., 49, 153–178Google Scholar
  4. The study of rising toroidal flux ropes, a proxy for the emergence of the solar internal toroidal field in the form of sunspot pairs, is a topic that has generated a voluminous literature. Among the many noteworthy contributions in this field, the following are recommended as starting points: Moreno-Insertis, F.: 1986, Nonlinear time-evolution of kink-unstable magnetic flux tubes in the convective zone of the sun. Astron. & Astrophys. 166, 291–305Google Scholar
  5. Choudhuri, A. R., & Gilman, P. A.: 1987, The influence of the Coriolis force on flux tubes rising through the solar convection zone, Astrophys. J., 316, 788–800Google Scholar
  6. Fan, Y., Fisher, G. H., & Deluca, E. E.: 1993, The origin of morphological asymmetries in bipolar active regions, Astrophys. J., 405, 390–401Google Scholar
  7. D’Silva, S., & Choudhuri, A. R.: 1993, A theoretical model for tilts of bipolar magnetic regions, Astron. & Astrophys., 272, 621–633Google Scholar
  8. Caligari, P., Moreno-Insertis, F., & Schüssler, M.: 1995, Emerging flux tubes in the solar convection zone. I: Asymmetry, tilt, and emergence latitude. Astrophys. J., 441, 886–902Google Scholar
  9. The thin flux-tube approximation used in most of these calculations is due to Spruit, H. C.: 1981, Motion of magnetic flux tubes in the solar convection zone and chromosphere, Astron. & Astrophys., 98, 155–160Google Scholar
  10. On the storage and stability of toroidal flux ropes below the solar convective envelope, see Ferriz-Mas, A., & Schüssler, M.: 1994, Waves and instabilities of a toroidal magnetic flux tube in a rotating star, Astrophys. J., 433, 852–866Google Scholar
  11. Ferriz-Mas, A.: 1996, On the storage of magnetic flux tubes at the base of the solar convection zone, Astrophys. J. 458, 802–816Google Scholar
  12. Considerable effort is currently being put into doing away with the thin flux tube approximation, in order to see which of the above results remains robust, once the flux tube is no longer treated as a one-dimensional object. This is a rapidly moving field, so for the latest see the following recent on-line review: Fan, Y.: 2009, Magnetic fields in the solar convection zone, Liv. Rev. Solar Phys., 6, 4, http://solarphysics.livingreviews.org/Articles/lrsp-2009-4/
  13. The following three recent review papers jointly offer a good overview of dynamo models of the solar cycle: Charbonneau, P.: 2010, Dynamo models of the solar cycle, Liv. Rev. Solar Phys., 7, 3, http://solarphysics.livingreviews.org/Articles/lrsp-2010-3/
  14. Ossendrijver, M.: 2003, The solar dynamo, Astron. & Astrophys. Rev., 11, 287–367Google Scholar
  15. Hoyng, P.: 2003, The field, the mean, and the meaning, in Advances in Non-Linear Dynamos, Ferriz Mas, A., & Jiménez, M. M., eds., The Fluid Mechanics of Astrophysics and Geophysics, 9, Taylor & Francis, 1–36Google Scholar
  16. Mean-field electrodynamics grew out of the original pioneering efforts of Parker, E. N.: 1955, Hydromagnetic dynamo models. Astrophys. J., 122, 293–314Google Scholar
  17. Braginskii, S. I.: 1964, Self-excitation of a magnetic field during motion of a highly conducting fluid, Sov. Phys. JETP, 20, 726–735Google Scholar
  18. Steenbeck, M., & Krause, F.: 1969, Zur Dynamotheorie stellarer und planetarer Magnetfelder. I. Berechnung sonnenähnlicher Wechselfeldgeneratoren, Astron. Nachr., 291, 49–84, in GermanGoogle Scholar
  19. but the following three monographs are a better starting point for those wishing to dig deeper into the subject: Moffatt, H. K.: 1978, Magnetic Field Generation in Electrically Conducting Fluids, Cambridge University PressGoogle Scholar
  20. Parker, E. N.: 1979, Cosmical Magnetic Fields: Their Origin and their Activity, Clarendon Press, chap. 18Google Scholar
  21. Krause, F., & Rädler, K.-H.: 1980, Mean-Field Magnetohydrodynamics and Dynamo Theory, Pergamon PressGoogle Scholar
  22. The sketch shown on Fig.3.5 is from Parker, E. N.: 1970, The generation of magnetic fields in astrophysical bodies. I. The dynamo equations, Astrophys. J., 162, 665–673Google Scholar
  23. On empirical estimates of the \(\alpha \)-effect from numerical simulations of MHD turbulence, start with: Pouquet, A., Frisch, U., & Léorat, J.: 1976, Strong MHD helical turbulence and the nonlinear dynamo effect, J. Fluid Mech., 77, 321–354Google Scholar
  24. Ossendrijver, M., Stix, M., & Brandenburg, A.: 2001, Magnetoconvection and dynamo coefficients: Dependence of the \(\alpha \) effect on rotation and magnetic field, Astron. & Astrophys., 376, 713–726Google Scholar
  25. Käpylä, P. J., Korpi, M. J., Ossendrijver, M., & Stix, M.: 2006, Magnetoconvection and dynamo coefficients. III. \(\alpha \) -Effect and magnetic pumping in the rapid rotation regime, Astron. & Astrophys., 455, 401–412Google Scholar
  26. Hubbard, A., Del Sordo, F., Käpylä, P. J., & Brandenburg, A.: 2009, The \(\alpha \) effect with imposed and dynamo-generated magnetic fields, Mon. Not. Roy. Astron. Soc., 398, 1891–1899Google Scholar
  27. The technical literature on dynamo models of the solar cycle is truly immense. There are many hundreds of noteworthy papers out there! Those included below are just meant to be good entry points for those wishing to pursue in greater depth topics covered in this chapter. For a good overview of mean-field solar cycle models and their evolution in time, see Lerche, I., & Parker, E. N.: 1972, The generation of magnetic fields in astrophysical bodies. IX. A solar dynamo based on horizontal shear, Astrophys. J., 176, 213–223Google Scholar
  28. Yoshimura, H.: 1975, Solar-cycle dynamo wave propagation, Astrophys. J., 201, 740–748Google Scholar
  29. Ivanova, T. S., & Ruzmaikin, A. A.: 1976, A magnetohydrodynamic dynamo model of the solar cycle, Sov. Astron., 20, 227–233Google Scholar
  30. Stix, M.: 1976, Differential rotation and the solar dynamo, Astron. & Astrophys., 47, 243–254Google Scholar
  31. Rüdiger, G., & Brandenburg, A.: 1995, A solar dynamo in the overshoot layer: cycle period and butterfly diagram, Astron. & Astrophys., 296, 557–566Google Scholar
  32. Moss, D., & Brooke, J.: 2000, Towards a model for the solar dynamo, Mon. Not. Roy. Astron. Soc., 315, 521–533Google Scholar
  33. On the impact of meridional circulation on dynamo waves, see Bullard, E. C.: 1955, The magnetic fields of sunspots, Vistas in Astronomy 1, 685–691Google Scholar
  34. Choudhuri, A. R., Schüssler, M., & Dikpati, M.: 1995, The solar dynamo with meridional circulation, Astron. & Astrophys., 303, L29–L32Google Scholar
  35. Küker, M., Rüdiger, G., & Schultz, M.: 2001, Circulation-dominated solar shell dynamo models with positive alpha-effect, Astron. & Astrophys., 374, 301–308Google Scholar
  36. Roberts, P. H., & Stix, M.: 1972, \(\alpha \) -Effect dynamos, by the Bullard-Gellman formalism, Astron. & Astrophys., 18, 453–466Google Scholar
  37. The meridional circulation profile described in Sect. 3.2.1 is the creation of van Ballegooijen, A. A., & Choudhuri, A. R.: 1988, The possible role of meridional flows in suppressing magnetic buoyancy, Astrophys. J., 333, 965–977Google Scholar
  38. On \(\alpha \)-quenching, standard versus catastrophic and related dynamical issues: Blackman, E. G., & Field, G. B.: 2000, Constraints on the magnitude of \(\alpha \) in dynamo theory, Astrophys. J., 534, 984–988Google Scholar
  39. Cattaneo, F., & Hughes, D. W.: 1996, Nonlinear saturation of the turbulent \(\alpha \) effect, Phys. Rev. E, 54, R4532–R4535Google Scholar
  40. Durney, B. R., De Young, D. S., & Roxburgh, I. W.: 1993, On the generation of the large-scale and turbulent magnetic fields in solar-type stars, Solar Phys. 145, 207–225Google Scholar
  41. Rüdiger, G., & Kichatinov, L. L.: 1993, Alpha-effect and alpha-quenching, Astron. & Astrophys., 269, 581–588Google Scholar
  42. Cattaneo, F., & Hughes, D. W.: 2009, Problems with kinematic mean field electrodynamics at high magnetic Reynolds numbers, Mon. Not. Roy. Astron. Soc., 395, L48–L51Google Scholar
  43. On interface dynamos, see Charbonneau, P., & MacGregor, K. B.: 1996, On the generation of equipartition-strength magnetic fields by turbulent hydromagnetic dynamos, Astrophys. J. Lett., 473, L59–L62Google Scholar
  44. MacGregor, K. B., & Charbonneau, P.: 1997, Solar interface dynamos. I. Linear, kinematic models in Cartesian geometry, Astrophys. J., 486, 484–501Google Scholar
  45. Parker, E. N.: 1993, A solar dynamo surface wave at the interface between convection and nonuniform rotation, Astrophys. J., 408, 707–719Google Scholar
  46. Petrovay, K., & Kerekes, A.: 2004, The effect of a meridional flow on Parker’s interface dynamo, Mon. Not. Roy. Astron. Soc., 351, L59–L62Google Scholar
  47. Tobias, S. M.: 1996, Diffusivity quenching as a mechanism for Parker’s surface dynamo, Astrophys. J., 467, 870–880Google Scholar
  48. on the energetics of thin layer dynamos: Steiner, O., & Ferriz-Mas, A.: 2005, Connecting solar radiance variability to the solar dynamo with the virial theorem, Astron. Nachr., 326, 190–193Google Scholar
  49. What is now referred to as Babcock-Leighton solar-cycle models goes back to the following three seminal papers by H. W. Babcock and R. B. Leighton: Babcock, H. W.: 1961, The topology of the Sun’s magnetic field and the 22-year cycle, Astrophys. J., 133, 572–587Google Scholar
  50. Leighton, R. B., 1964, Transport of magnetic fields on the Sun, Astrophys. J., 140, 1547–1562Google Scholar
  51. Leighton, R. B.: 1969, A magneto-kinematic model of the solar cycle, Astrophys. J., 156, 1–26Google Scholar
  52. Although some details of the model are different, the 2D surface simulations described in Sect. 3.3.1 basically follow Wang, Y.-M., Nash, A. G., & Sheeley, Jr., N. R.: 1989, Magnetic flux transport on the sun, Science, 245, 712–718Google Scholar
  53. Wang, Y.-M., & Sheeley, Jr., N. R.: 1991, Magnetic flux transport and the sun’s dipole moment - New twists to the Babcock-Leighton model, Astrophys. J., 375, 761–770Google Scholar
  54. but on this general topic of surface magnetic flux evolution, see also: Schrijver, C. J., Title, A. M., van Ballegooijen, A. A., Hagenaar, H. J., & Shine, R. A.: 1997, Sustaining the quiet photospheric network: the balance of flux emergence, fragmentation, merging, and cancellation, Astrophys. J., 487, 424–436Google Scholar
  55. Schrijver, C. J.: 2001, Simulations of the photospheric magnetic activity and outer atmospheric radiative losses of cool stars based on characteristics of the solar magnetic field, Astrophys. J., 547, 475–490Google Scholar
  56. Schrijver, C. J., & Title, A. M.: 2001, On the formation of polar spots in Sun-like stars, Astrophys. J., 551, 1099–1106Google Scholar
  57. Schrijver, C. J., De Rosa, M. L., & Title, A. M.: 2002, What is missing from our understanding of long-term solar and heliospheric activity?, Astrophys. J., 577, 1006–1012Google Scholar
  58. Baumann, I., Schmitt, D., Schüssler, M., & Solanki, S. K.: 2004, Evolution of the large-scale magnetic field on the solar surface: a parameter study, Astron. & Astrophys., 426, 1075–1091Google Scholar
  59. The formulation of the Babcock-Leighton solar cycle model of Sect. 3.3 is identical to Charbonneau, P., St-Jean, C., & Zacharias, P.: 2005, Fluctuations in Babcock-Leighton dynamos. I. Period doubling and transition to chaos, Astrophys. J., 619, 613–622Google Scholar
  60. For different modelling approaches, see Wang, Y.-M., Sheeley, Jr., N. R., & Nash, A. G.: 1991, A new solar cycle model including meridional circulation, Astrophys. J., 383, 431–442Google Scholar
  61. Durney, B. R.: 1995, On a Babcock-Leighton dynamo model with a deep-seated generating layer for the toroidal magnetic field, Solar Phys., 160, 213–235Google Scholar
  62. Dikpati, M., & Charbonneau, P.: 1999, A Babcock-Leighton flux transport dynamo with solar-like differential rotation, Astrophys. J., 518, 508–520Google Scholar
  63. Nandy, D., & Choudhuri, A. R.: 2001, Toward a mean field formulation of the Babcock-Leighton type solar dynamo. I. \(\alpha \) -coefficient versus Durney’s double-ring approach, Astrophys. J., 551, 576–585Google Scholar
  64. Guerrero, G., & de Gouveia Dal Pino, E. M.: 2008, Turbulent magnetic pumping in a Babcock-Leighton solar dynamo model, Astron. & Astrophys., 485, 267–273Google Scholar
  65. Muñoz-Jaramillo, A., Nandy, D., Martens, P. C. H., & Yeates, A. R.: 2010, A double-ring algorithm for modeling solar active regions: unifying kinematic dynamo models and surface flux-transport simulations, Astrophys. J. Lett., 720, L20–L25Google Scholar
  66. On the “tachocline \(\alpha \)-effect” dynamo model described in Sect. 3.4.1, and associated stability analyses, begin with: Dikpati, M., & Gilman, P. A.: 2001, Flux-transport dynamos with \(\alpha \) -effect from global instability of tachocline differential rotation: a solution for magnetic parity selection in the Sun, Astrophys. J., 559, 428–442Google Scholar
  67. Dikpati, M., Gilman, P. A., & Rempel, M.: 2003, Stability analysis of tachocline latitudinal differential rotation and coexisting toroidal band using a shallow-water model, Astrophys. J., 596, 680–697Google Scholar
  68. Gilman, P. A., & Fox, P. A.: 1997, Joint instability of latitudinal differential rotation and toroidal magnetic fields below the solar convection zone, Astrophys. J., 484, 439–454Google Scholar
  69. and for the “flux tube \(\alpha \)-effect” dynamo model of Sect. 3.4.2, and associated stability analyses, try first: Ferriz-Mas, A., Schmitt, D.,& Schüssler, M.: 1994. A dynamo effect due to instability of magnetic flux tubes. Astron. & Astrophys., 289, 949–956Google Scholar
  70. Ossendrijver, M. A. J. H.: 2000, The dynamo effect of magnetic flux tubes, Astron. & Astrophys., 359, 1205–1210Google Scholar
  71. On the numerical simulations of global 3D MHD convection in thick, rotating stratified spherical shells, begin with Brun, A. S., Miesch, M. S., & Toomre, J.: 2004, Global-scale turbulent convection and magnetic dynamo action in the solar envelope, Astrophys. J., 614, 1073–1098Google Scholar
  72. Browning, M. K., Miesch, M. S., Brun, A. S., & Toomre, J.: 2006, Dynamo action in the solar convection zone and tachocline: pumping and organization of toroidal fields, Astrophys. J. Lett., 648, L157–L160Google Scholar
  73. Brown, B. P., Browning, M. K., Brun, A. S., Miesch, M. S., & Toomre, J.: 2010, Persistent magnetic wreaths in a rapidly rotating Sun, Astrophys. J., 711, 424–438Google Scholar
  74. Brown, B. P., Miesch, M. S., Browning, M. K., Brun, A. S., & Toomre, J.: 2011, Magnetic cycles in a convective dynamo simulation of a young solar-type star, Astrophys. J., 731, id. 69Google Scholar
  75. as well as the following two recent review articles: Miesch, M. S.: 2005, Large-scale dynamics of the convection zone and tachocline, Living Reviews Solar Phys., 2, 1, http://solarphysics.livingreviews.org/Articles/lrsp-2005-1/
  76. Miesch, M. S., & Toomre, J.: 2009, Turbulence, Magnetism, and Shear in Stellar Interiors, Ann. Rev. Fluid Mech., 41, 317–345Google Scholar
  77. See also the fascinating results presented in Cline, K. S., Brummell, N. H., & Cattaneo, F.: 2003, Dynamo action driven by shear and magnetic buoyancy, Astrophys. J., 599, 1449–1468Google Scholar
  78. Käpylä, P. J., Korpi, M. J., Brandenburg, A., Mitra, D., & Tavakol, R.: 2010, Convective dynamos in spherical wedge geometry, Astron. Nachr., 331, 73–81Google Scholar
  79. The production of solar-like magnetic cycles in such simulations is a recent breakthrough. The simulation results presented in Sect. 3.5 are taken from Ghizaru, M., Charbonneau, P., & Smolarkiewicz, P. K.: 2010, Magnetic cycles in global large-eddy simulations of solar convection, Astrophys. J. Lett., 715, L133–L137Google Scholar
  80. Racine, É., Charbonneau, P., Ghizaru, M., Bouchat, A., & Smolarkiewicz, P. K.: 2011, On the mode of dynamo action in a global large-eddy simulation of solar convection, Astrophys. J., 735, id. 46Google Scholar
  81. These simulations were computed with the MHD version, developed at the Université de Montréal, of the general purpose hydrodynamical simulation code EULAG; on the latter, Prusa, J. M., Smolarkiewicz, P. K., & Wyszogorodzki, A. A.: 2008, EULAG, a computational model for multi-scale flows, Comp. Fluids, 37, 1193–1207Google Scholar
  82. as well as the EULAG web-page: http://www.mmm.ucar.edu/eulag/
  83. The numerical simulation results displayed on Fig. 3.25 is publicly available at: http://steinr.pa.msu.edu/~bob/data.html
  84. Explanatory notes describing the simulation framework are also provided there, and discussed in greated detail in Stein, R. F., Lagerfjärd, A., Nordlund, Å., & Georgobiani, D.: 2011, Solar flux emergence simulations, Solar Phys., 268, 271–282Google Scholar
  85. In a similar vein, do not miss: Cheung, M. C. M., Rempel, M., Title, A. M., & Schüssler, M.: 2010, Simulation of the formation of a solar active region, Astrophys. J. 720, 233–244Google Scholar
  86. On the observational measurements and characterization of small-scale solar surface magnetic structures, and the potential implications for dynamo processes, see Parnell, C. E., DeForest, C. E., Hagenaar, H. J., Johnston, B. A., Lamb, D. A., & Welsch, B. T.: 2009, A power-law distribution of solar magnetic fields over more than five decades in flux, Astrophys. J. 698, 75–82Google Scholar
  87. and references therein. A simple diffusion-limited aggregation model, producing power-law distributions of magnetic structures with logarithmic slope comparable to observational inferences, is presented in Crouch, A. D., Charbonneau, P., & Thibault, K.: 2007, Supergranulation as an emergent length scale, Astrophys. J., 662, 715–729Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Département de physiqueUniversité de MontréalMontrealCanada

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