Space Sci Rev (2010) 152: 591–616 DOI 10.1007/s11214-009-9579-5 The Solar Dynamo Chris A. Jones · Michael J. Thompson · Steven M. Tobias Received: 24 June 2009 / Accepted: 12 October 2009 / Published online: 19 December 2009 © Springer Science+Business Media B.V. 2009 Abstract Observations relevant to current models of the solar dynamo are presented, with emphasis on the history of solar magnetic activity and on the location and nature of the solar tachocline. The problems encountered when direct numerical simulation is used to analyse the solar cycle are discussed, and recent progress is reviewed. Mean field dynamo theory is still the basis of most theories of the solar dynamo, so a discussion of its fundamental prin- ciples and its underlying assumptions is given. The role of magnetic helicity is discussed. Some of the most popular models based on mean field theory are reviewed briefly. Dynamo models based on severe truncations of the full MHD equations are discussed. Keywords Solar magnetism · Solar dynamo · Sunspots · Solar cycles · Solar interior · Helioseismology 1Observations 1.1 Sunspots and Solar Magnetic Activity Sunspots are perhaps the most obvious manifestation of solar magnetism (Thomas and Weiss 2008). Sunspots were observed already by the ancients, and were shown by Galileo to be features on the Sun itself; but the observational evidence that sunspots possess magnetic fields was established only a hundred years ago by Hale (1908). The number of sunspots, or the area occupied by them, follows an irregular 11-year activity cycle (lower panel of C.A. Jones () · S.M. Tobias Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK e-mail: [email protected] S.M. Tobias e-mail: [email protected] M.J. Thompson School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK e-mail: michael.thompson@sheffield.ac.uk 592 C.A. Jones et al. Fig. 1 Solar-cycle activity on the Sun since 1874. The lower panel shows the daily sunspot area, as a percent- age of the visible hemisphere that is covered by sunspots and averaged over individual solar rotations. The butterfly diagram in the upper panel shows the corresponding incidence of sunspots as a function of latitude ◦ and time. At the beginning of a new cycle, spots appear around latitudes of ±30 . The activity zones spread until they extend to the equator, and then gradually die away, disappearing at the equator as the first spots of the next cycle appear at higher latitudes (courtesy of D.H. Hathaway) Fig. 1). Sunspots exist for anything from a few days to more than two solar rotation periods (i.e. more than two months). Early in the solar cycle, new sunspots occur preferentially at mid-latitudes on the Sun, and occur successively closer to the solar equator as the cycle progresses. Plotting the locations of sunspots as a function of latitude and time thus gives rise to the so-called butterfly diagram (upper panel of Fig. 1). Sunspots are cooler and darker than the rest of the solar photosphere (the visible surface of the Sun), with typical sunspot temperatures being 4000 K compared with the photo- sphere’s 6000 K. The outer 30 per cent by radius of the Sun constitutes the convection zone, where convective motions of the plasma transport heat from the deeper interior to the solar surface. But the strong magnetic fields in sunspots inhibits the convective motions (a typical strength at the centre of a spot is 2–3000 G and the plasma beta is of order unity), causing the spots to be cooler and hence darker. Actually the structure of a sunspot is rather com- plex and beautiful, with a dark inner region known as the umbra and a surrounding region called the penumbra which exhibits a filamentary structure and is quite dynamic, as revealed for example by the exquisite observations from the Hinode satellite (e.g. Jurcak˘ and Bellot Rubio 2008). Further work by Hale and his collaborators (Hale et al. 1919) concerning the polarity of sunspot magnetic fields and their distribution and orientation in pairs established the basis for what are now known as Hale’s Laws and Joy’s Law. Sunspots occur in bipolar pairs consisting of a leading spot and a trailing spot, aligned approximately parallel to the solar equator. Hale’s Laws are that the magnetic polarities of the leading and trailing spot in a pair are opposite, with the polarities being reversed in the two hemispheres; and that the polarities reverse in successive solar cycles. Joy’s Law is that the line joining leading and trailing spots in a pair tends to be inclined at an angle of about 4◦ to the equator. Taking the change in polarity into account, it is seen therefore that the quasi-period of the Sun’s magnetic cycle as revealed by sunspots is about 22 years. The Solar Dynamo 593 As a theoretical aside, we may note the generally accepted theoretical picture is that bipolar sunspot pairs are caused by toroidal field in the form of magnetic flux tubes ris- ing through the convection zone and forming so-called Omega loops. These loops emerge through the surface, the two regions of intersection of the loops with the photosphere being identified with the bipolar sunspot pair (e.g. Fan 2004). Sunspots typically occur within larger magnetic complexes called active regions. There is a reported tendency for active regions to recur at the same longitudinal position on the Sun during a solar cycle and even from one cycle to another: these locations are called active longitudes. Above active regions one observes magnetic loops or arcades of loops in the overlying chromosphere and in the corona which is the outer atmosphere of the Sun. These loops are made manifest by emission from the plasma confined within them. Occasional explosive reconfigurations of these magnetic structures, caused by magnetic reconnection, give rise to flares and coronal mass ejections (CMEs). As well as the sunspot fields, there is magnetic field on a wide range of smaller scales. Smaller bipolar structures called ephemeral regions appear at all latitudes and show little cycle variation. A discovery of recent years is that the Sun has a lot of small scale magnetic flux that has been dubbed the solar ‘magnetic carpet’. The flux that comprises the magnetic carpet is constantly emerging and disappearing (by subduction or reconnection), such that the magnetic carpet is renewed over a timescale of order one day. Whether this magnetic flux is the product of local small-scale dynamo action or whether it is recycled old magnetic flux is a matter of debate. The Sun also possesses a weak (of order 1–2 G) global magnetic field that is predomi- nantly dipolar. In the polar regions, the global field rises to around 10 G. Like the sunspot field, the polar field reverses its polarity roughly every 11 years. The polar field reversal occurs at about the time that the azimuthal sunspot field reaches its peak strength, at about 15–20◦ latitude (Parker 2007). It may be noted that other observables that are related to the solar magnetic field also vary over the solar cycle: these include the 10.7 cm radio flux and the solar irradiance (particularly at short UV wavelengths). Solar-cycle studies can be extended further back in time using proxy data. The magnetic field from the Sun deflects galactic cosmic rays. In the Earth’s atmosphere these rays produce radioactive isotopes such as 14C, which is preserved in trees, and 10Be, which is preserved in polar ice-caps. The abundances of these isotopes therefore vary in antiphase with the solar magnetic field. Studies of both the ice-core record and tree-ring data allow the 11-yr cycle to be traced back over hundreds of years, while its envelope modulation can be traced back over millennia (see for example the review by Weiss and Thompson 2009). From about 1645 to 1715, soon after Galileo had made the first telescopic studies of sunspots, there were very few sunspots seen on the solar disc and those that were seen were almost all in the Sun’s southern hemisphere. This period is known as the Maunder Minimum (Eddy 1976; Ribes and Nesme-Ribes 1993). Thus in this period the sunspot cycle seems to be broken. Yet ice-core studies show that a cyclical modulation continued through the Maunder Minimum, indicating that the Sun’s large-scale magnetic field continued even though there were few sunspots (Beer et al. 1998). The north-south symmetry of the sunspot distribution was soon re-established after the end of the Maunder Minimum. Studies of the proxy datasets indicate that the Maunder Minimum is only the latest of the grand minima that have occurred in solar activity over time. Conversely, the present era of rather large-amplitude solar cycles (cf. Fig. 1) is a grand maximum, though this is perhaps now coming to an end (Abreu et al. 2008). 594 C.A. Jones et al. 1.2 Solar Rotation and Helioseismology It is generally accepted that the strong toroidal field that gives rise to sunspots is produced by a large-scale dynamo. Differential rotation of the solar plasma is likely an important ingredient of that dynamo. The Sun’s rotation at the surface can be observed directly by spectroscopic Doppler measurements and by observing the motion of sunspots and other tracers. These observations show that the surface rotation period at the solar equator is about 25 days, increasing with latitude to more than a month at high latitudes. There is some variation in the periods obtained from different methods of measurement (see Beck 1999 for a comprehensive review).