Show Summary Details

Page of

PRINTED FROM the OXFORD RESEARCH ENCYCLOPEDIA, PHYSICS (oxfordre.com/physics). (c) Oxford University Press USA, 2020. All Rights Reserved. Personal use only; commercial use is strictly prohibited (for details see Privacy Policy and Legal Notice).

date: 28 September 2020

Solar Dynamo

Summary and Keywords

The solar dynamo is the action of flows inside the Sun to maintain its magnetic field against Ohmic decay. On small scales the magnetic field is seen at the solar surface as a ubiquitous “salt-and-pepper” disorganized field that may be generated directly by the turbulent convection. On large scales, the magnetic field is remarkably organized, with an 11-year activity cycle. During each cycle the field emerging in each hemisphere has a specific East–West alignment (known as Hale’s law) that alternates from cycle to cycle, and a statistical tendency for a North-South alignment (Joy’s law). The polar fields reverse sign during the period of maximum activity of each cycle.

The relevant flows for the large-scale dynamo are those of convection, the bulk rotation of the Sun, and motions driven by magnetic fields, as well as flows produced by the interaction of these. Particularly important are the Sun’s large-scale differential rotation (for example, the equator rotates faster than the poles), and small-scale helical motions resulting from the Coriolis force acting on convective motions or on the motions associated with buoyantly rising magnetic flux. These two types of motions result in a magnetic cycle. In one phase of the cycle, differential rotation winds up a poloidal magnetic field to produce a toroidal field. Subsequently, helical motions are thought to bend the toroidal field to create new poloidal magnetic flux that reverses and replaces the poloidal field that was present at the start of the cycle.

It is now clear that both small- and large-scale dynamo action are in principle possible, and the challenge is to understand which combination of flows and driving mechanisms are responsible for the time-dependent magnetic fields seen on the Sun.

Keywords: Sun, dynamo, magnetic fields, magnetohydrodynamics, solar cycle

Historical Background

The discovery of non-terrestrial magnetic fields took place when Hale (1908) showed that sunspots have magnetic fields. This discovery led to the question of why magnetic fields should exist on the surface of the Sun (a similar question was already being posed in terms of the Earth’s magnetic field). “Why” questions are hard, and at least initially admit a number of possible answers, which can only be excluded one by one. The most fundamental of the proposed answers was that the magnetic field reflects a general relationship between rotating bodies and their magnetic fields (Blackett, 1947). This proposal is now seen as being very unlikely. The explanation which is now generally accepted is that from Larmor (1919), which is that the Sun acts as a self-excited dynamo.

Astrophysical dynamo theory usually employs the assumptions of magnetohydrodynamics. For the conditions of the solar interior and photosphere (see Chitta et al., Forthcoming), these assumptions are reasonable, and the evolution of the magnetic field is given by the induction equation

Bt=×(U×B η×B)
(1)

Where B(r,t) is the magnetic field, U(r,t) is the velocity field and η(r,t) is the magnetic diffusivity, which can be a function of the position vector r and time t. For the solar dynamo problem, the field is assumed not to be driven from the outside which implies a boundary condition which, in the absence of a flow (U=0), allows Ohmic diffusion to cause the magnetic field to decay to zero as t. Assuming that the magnetic field is potential above the solar surface (the photosphere) and continuous across it, is a sufficient boundary condition.

The solar dynamo explanation posits that flows inside the Sun may interact with the magnetic field inside the Sun to sustain the field against Ohmic decay. For purposes of the solar dynamo, the kinematic dynamo problem considers the long-term evolution of a magnetic field for prescribed magnetic diffusivity η and flow U, which is zero above the solar surface. If there exists any magnetic field that does not asymptotically decay to zero as t then the flow and diffusivity are said to be capable of dynamo action.

The search for dynamo solutions to the induction equation proved to be difficult, and the first results were anti-dynamo theorems, the most famous of which is due to Cowling (1933). In essence, Cowling’s theorem states that (for flows in a bounded volume which are finite everywhere, and for appropriate boundary conditions on the magnetic field) a purely axisymmetric magnetic field cannot be sustained against Ohmic decay. In particular, while an axisymmetric toroidal (azimuthal) magnetic field can be produced by the action of differential rotation on an axisymmetric poloidal field (the radial and latitudinal components), a purely axisymmetric poloidal magnetic field cannot be maintained by an axisymmetric toroidal field. This result, and similar theorems, ruled out the possibility of dynamos with very simple spatial structures. More complicated magnetic fields were finally analytically shown to be able to be produced dynamo action (e.g., Herzenberg, 1958). These were important in showing that there was no universal anti-dynamo theorem.

The question at this point became whether or not the magnetic fields observed on the Sun are the result of a dynamo, and for this, observations need to be discussed. The most prominent of the observational conclusions, which were already available prior to the mid 1930s were

  1. 1. Sunspots are magnetic (Hale, 1908).

  2. 2. The number of sunspots varies over the course of an 11-year activity cycle (Schwabe, 1849).

  3. 3. Sunspots typically emerge in bipolar pairs, with the east-west orientation of the two polarities switching sign each eleven years (Hale, Ellerman, Nicholson, & Joy, 1919). This property of the emergence process is known as Hale’s law. Hale’s law implies that the Sun has a large-scale toroidal magnetic field. The north–south orientation also flips sign each cycle, with the leading polarity (the component to the west) of each bipolar pair being closer to the equator in latitude than the following (eastward) polarity (Hale et al., 1919). This property is called Joy’s law.

  4. 4. At the beginning of a cycle, sunspots emerge at mid-latitudes (approximately 35°) and move closer to the equator as the cycle progresses (Spoerer, 1879). This behavior is captured in the butterfly diagram (Maunder, 1904), a recent version of which is shown in Figure 1.

  5. 5. The number of sunspots varies from cycle to cycle (Spoerer, 1889).

  6. 6. The Sun rotates non-uniformly (Carrington, 1863).

  7. 7. The Sun has an outer convection zone (Schwarzschild, 1906; Unsöld, 1930).

Solar Dynamo

Figure 1. The observed sunspot areas as a function of time and latitude (based on RGO/SOON observations).

A few years before the formal proof that dynamo action in astrophysical plasmas was possible, the physical framework for the astrophysical dynamo was described in a fundamental paper by Parker (1955). It identifies the action of cyclonic motions inside the Sun as the mechanism for generating poloidal field from toroidal field. Together with the winding up of poloidal magnetic field by differential rotation as the source of toroidal field, such cyclonic flows are sufficient to maintain a magnetic field. The field thus maintained is oscillatory in time, and has the form of a wave. The cyclonic motions themselves are expected in astrophysical bodies with convection, as the convective motions will be subject to the Coriolis force (similar to cyclones on the Earth), and the Sun is but one system with the necessary ingredients (rotation and convection of a conducting fluid).

Mean-Field MHD Models

From the early and mid-1960s to the early 1970s, two major schools of thought formed. One school, led by Steenbeck, Krause, and Rädler (1966), developed the theory of the interaction between magnetic fields, turbulence and rotation into mean-field magnetohydrodynamics. While Parker (1955) concentrated on the effect of one cyclone, then the combined effect of many cyclones, the East German group led by Krause, Rädler, and Steenbeck considered directly the action of the Coriolis force on turbulent motions. They showed that systematic correlations would be created between the different components of the flow, and how these would then affect the magnetic field. The effects include Parker’s mechanism as well as others.

The mean-field theory plays a central role in dynamo theory, and so the following will sketch the theory as detailed in Krause and Rädler (1980) and Moffatt (1978). The main idea is to expand the velocity and magnetic fields U and B into (appropriately chosen) mean components U and B and fluctuating components u and b, with U=U+u and B=B+b

. For the case of the solar dynamo, the obvious choice is to consider azimuthal averaging to define the mean quantities. The induction equation then gives for the mean component B,

Bt=×(U×B+u×bη×B)
(2)

and subtracting this from the induction equation gives the evolution equation for the fluctuating component,

bt=×(u×bu×b+U×b+u×Bη×b).
(3)

Note that u×b is analguous to the Reynolds stress used in hydrodynamical turbulence studies.

The aim of mean-field theory is to find an expression for u×b. In the kinematic case, the statistical properties of u are assumed to be known. Equation (3), however, remains highly non-trivial, and admits exponentially growing solutions of b, which will be returned to in “Small-Scale Dynamo.” For now, it is best to overlook this issue and assume, as is often done, that the turbulent nature of u leads to purely local correlations between u and b, so that u×b can be written as a functional of B,Bxi,2Bxixj, . . . where xi are the coordinates, and Bt,2Bt2, . . . Note that this implies that the fluctuations are localized to much shorter scales in space and time than the large-scale field. Further noting that Equation (3) is linear inB, and assuming that the first few terms suffice to capture the spatial dependence, gives

u×bi=αijBj +βijkBj xk
(4)

where Bj is the jth component of B, and α and β are pseudo-tensors, which under the assumptions that have been made, will depend upon U, η and the statistical properties of u, all of which can vary in space.

To gain some physical insight into the α term, one can begin by breaking it into the sum of its symmetric and antisymmetric components

αij=αij+αji2+αijαji2
(5)

The antisymmetric part has the same effect in the induction equation as a mean velocity, and is called turbulent pumping. It can be treated as an additional mean velocity Upump which is added to U in Equation (2). The symmetric part is much more interesting because it cannot be treated as a simple flow or diffusivity and therefore the structure of the mean-field induction equation is different from that of the induction equation.

The problem of determining α and β is not trivial and requires extra information or assumptions about the system. To give a sense of the relevant physics, consider a case where the α term is dominated by the structure of the turbulent flows. An important consequence of α being a pseudo-tensor is that its symmetric part vanishes if all the statistical properties ofu are unchanged under the transformation r=r (i.e., if the turbulence has reflexional symmetry). The simplest breaking of this symmetry occurs if the u(×u)0. The integral of the scalar product of the velocity and vorticity over some volume is called kinetic helicity,

H=vU(×U)dV
(6)

and a flow with helicity gives rise to a non-zero symmetric component of α. In the special case of homogeneous isotropic turbulence, the α tensor has components αij=α0δij, where δij=1 if i=j and 0 otherwise, and α0 is a pseudo-scalar with a value which can be approximated (after further assumptions) as a weighted integral of the spectral transform of u(×u).

Theβ tensor has even more terms, and under a number of restrictive assumptions unlikely to apply in the Sun, one can obtain the approximation βijk=ηtεijk where ηt is a scalar and εijk=1 if i,j,k is an even permutation of 1, 2, 3, -1 if it is an odd permutation and 0 otherwise (e.g., Krause & Rädler, 1980; Moffatt, 1978).

With all these assumptions, the mean-field induction equation becomes

Bt=×[U×B+α0B(ηt+η)×B]
(7)

where again 〈. . .〉 indicates an averaging over azimuth. Note that, in the special case α0=0, Cowling’s theorem rules out dynamo action. Cases with α00, however, fall outside the scope of Cowling’s theorem, and depending on ηt (which is usually much larger than η) and U, dynamo action has been demonstrated in a large number of cases. Note that additional terms appear when the turbulence becomes anisotropic (for example, when the background is gravitationally stratified as is the case for the Sun). One of the additional terms which appears is magnetic pumping which can be included as an effective velocity, Upump, which adds to U appearing in Equation (7).

Particularly relevant for the Sun is the case where the toroidal field Bϕ is generated by the differential rotation acting on the poloidal field (Bγ,Bθ), and the poloidal field is generated by the α0 term from the toroidal field. This is the case studied by Parker (1955), where the α term here is a formalization of the action of Parker’s cyclonic motions.

At this point in history, there was a description of the some of the necessary physical ingredients by Parker (1955) and a rich formalism for dynamo action by turbulent motions in a system with helicity, such as expected in rotating stars with convection. The specific description by Parker falls within the general framework, and in terms of Equation (7) corresponds to U consisting only of differential rotation.

The model of Parker leads to an explanation of why the Sun has a cycle (why the dynamo mode is oscillatory). Together with the observed equatorial propagation of the locations where sunspots emerge as a cyclic progresses (Schwabe, 1849), this model lead to the prediction that the rotation rate of the Sun about its axis increases inward (Yoshimura, 1975b). When in the 1980s this prediction was put to the test, it was found that outside of a near-surface layer covering above 0.95R, the rotation rate decreases inward (Schou et al., 1998) until beneath the convection zone, where the rotation is almost uniform. This failure of the models, including only the simplest form of the α effect of Parker, indicates that, while differential rotation and the α term from the expected cyclonic motions are enough to explain dynamo action, other terms must be important in the case of the Sun to explain the equatorial drift of the emergence locations of sunspots. The mean-field framework offers many such terms. So by this point in the story it was clear that the interaction of (sufficiently strong) turbulence, differential rotation, and magnetic fields in a conducting fluid are sufficient to produce large-scale dynamo action, explaining why astrophysical objects with these properties have large-scale magnetic fields. The question in the solar context then becomes: what combination of bulk and turbulent flows determine how the solar dynamo works in detail?

Babcock–Leighton Models

Around the same time as the mean-field framework was being developed, a more empirically focused effort was taking place in California. This effort was undertaken by Babcock and Leighton (Babcock, 1961, 1963; Leighton, 1964, 1969). The essential idea behind their model is that the observed behavior of flux emergence and the observed subsequent evolution of the magnetic field on the surface are critical for the Sun’s dynamo. The key assumption is that the poloidal field relevant to the global dynamo threads through the solar surface. In its current form, the most important steps in the Babcock–Leighton model are

  1. 1. During activity minima, the poloidal field threading through the solar surface is concentrated at the poles (observed).

  2. 2. This field threads through the solar interior to re-emerge at the opposite pole (B=0).

  3. 3. Because the polar regions are rotating slower than the equator at all depths, the poloidal field threading from pole to pole gets wound up by latitudinal differential rotation.

  4. 4. Some bulk flow or turbulent process acts to transport the field equatorward.

  5. 5. The toroidal field thus produced is subject to some instability or turbulent motions so that it emerges at the solar surface.

  6. 6. The opposite polarities in each hemisphere emerge with a systematic latitudinal separation (Joy’s law, observed). The detailed cause of the latitudinal separation is currently unknown. The two major possible causes are (a) the turbulent cyclonic motions act on the magnetic field prior to, during, or immediately after emergence and/or (b) flows produced as a consequence of the buoyant rise of the magnetic field are acted on by the Coriolis force (Choudhuri & Gilman, 1987). This question is currently open.

  7. 7. The magnetic flux passing through the surface after emergence is carried passively by small- and large-scale flows (the empirically based surface flux transport model of DeVore, Boris, & Sheeley, 1984, and Wang, Nash, & Sheeley, 1989a, observed). This transport eventually reverses the polar fields and the next cycle begins not long thereafter (observed).

It should be noted that the Babcock–Leighton model does not specify the physical cause of Joy’s law, nor the cause of the equatorial propagation (even if the original papers of Babcock and Leighton tried to include these). In this sense, the model is physically incomplete and instead describes a class of dynamos, with a dynamo being of the Babcock–Leighton type if it is consistent with the given steps.

The critical ingredients of the model are latitudinal differential rotation (which is observed) and the physical processes that transport the toroidal field equatorward, cause the eruptions, and produce Joy’s law. These processes may be due to correlations between the magnetic field and turbulent convective flows in the rotating Sun, or to mean flows, or due to flows driven, for example, by instabilities associated mainly with the magnetic field and the flows induced thereafter in the rotating Sun. Distinguishing between these possibilities will require further observations and numerical simulations, as will be discussed.

The Babcock–Leighton model can be formulated in the mathematical mean-field framework (Stix, 1974), because this framework is rather general. Mathematically, Joy’s law can be expressed as an α effect, but the physical interpretation may be very different from that of Parker’s cyclonic motions or the kinetic helicity more often associated with mean-field theory. In particular there is the school of thought (Caligari, Moreno-Insertis, & Schüssler, 1995; Choudhuri & Gilman, 1987; D’Silva & Choudhuri, 1993) which argues that Joy’s law is the consequence of the Coriolis force acting not on the turbulent convective motions, but rather acting on flows set up as magnetic flux tubes rise from the base of the convection zone to emerge through the solar surface.

When the Babcock–Leighton model was formulated, its central distinguishing feature, that the relevant poloidal field for the dynamo passes through the solar surface, was just an assumption. Babcock and Leighton showed that dynamo action is possible given Joy’s law and this assumption, but that does not show that the assumption is valid. Observations show a strong correlation between the polar field strength at the end of a cycle (or proxies of the polar fields at the end of a cycle) and the strength of the next cycle (Muñoz-Jaramillo, Dasi-Espuig, Balmaceda, & DeLuca, 2013; Schatten, Scherrer, Svalgaard, & Wilcox 1978; Svalgaar, Cliver, & Kamide, 2005; Wang & Sheeley 2009).

Current State

The state of the field in 1970 was described by Parker (1970, p. 168) as follows: “The basic ideas are sound, I think, but, when we come to the details, there are so many theoretical alternatives that need yet to be explored and so much that is not yet made clear by observations.” This section will discuss how synoptic observations collected from the mid-1970s onward have clarified the description of how dynamo action occurs in the Sun, and how numerical simulations have allowed an exploration of the basic processes of dynamo action driven by turbulent convection.

Solar Observations

For a reasonably long period from the early 1970s until 2006, new observations were kept at the periphery of dynamo theory. During this time, DeVore et al. (1984) and Wang et al. (1989b) developed the part of the Babcock–Leighton model dealing with the evolution of the surface radial field into the surface flux transport model. This model is empirical in nature and distills the observed behavior of the magnetic flux after emergence. According to this model, the radial magnetic field at the surface behaves like passive tracer particles, being transported by the surface flows. The model is quantitatively in agreement with the observed evolution of the radial field. The simplicity of the evolution of the surface radial flux after emergence, and in particular the lack of a complicated interaction with the subsurface magnetic field dynamics is remarkable. A possible reason why the model works in the mean-field framework was provided by Cameron, Schmitt, Jiang, and Isık (2012) in terms of near-surface radial pumping. The linkage of surface models to the evolution of the coronal magnetic field by Mackay and van Ballegooijen (2005) and Yeates, Mackay, and van Ballegooijen(2008) has been remarkably successful in predicting the location of prominences as twisted flux tubes and of their eruption.

The development of global helioseismology (for a review, see Basu, 2016) led to the result that, throughout the bulk of the convection zone, the rotation rate decreases slightly with depth. The two exceptions are in the near-surface shear layer where the rotation rate increases with depth and near the base of the convection zone where, at low latitudes, the rotation rate decreases rapidly with depth (Thompson et al., 1996). This presents a problem for both the Parker and Babcock–Leighton models. For the Babcock–Leighton model, the sign of the α effect follows directly from Joy’s law and is observed to be positive in the northern hemisphere and negative in the southern hemisphere. Equatorward propagation of the subsurface toroidal field then requires a rotation rate that increases with depth (Yoshimura, 1975b), which is only found near the solar surface.1 There are arguments against the field being so shallow (Dikpati, Corbard, Thompson, & Gilman,2002). The same problem exists in the Parker model, where the physics envisioned leads to the same sign for the α effect. Theory and numerical simulations (Yoshimura, 1975a), however, suggest that the α effect can have the opposite sign (negative in the northern hemisphere) near the bottom of the convection zone due to diverging flows.

This problem can be remedied for both models by including other terms from mean-field theory, or if α has the opposite sign (the sign is, however, fixed in the Babcock–Leighton model by observations). One way of obtaining the equatorward propagation involves using a U with a meridional component, leading to the flux transport dynamo model (Choudhuri, Schussler, & Dikpati,1995; Wang & Sheeley, 1991). This class of models can employ either a mean-field α effect, or a Babcock–Leighton source term implemented as an α coefficient in Equation (7), and a mean velocity field U including the observed differential rotation as well as a non-zero meridional flow. The latter is based upon observations of a poleward flow at the surface (Duvall, 1979) and an assumed return flow at depth. The model is sensitive to the assumed form and strength of the return flow as well as to the strength of the unobserved subsurface turbulent magnetic diffusivity. It should be noted that Upump or other turbulent transport coefficients also could play a role.

A conceptual return of dynamo theory to observations began with Dikpati, de Toma, and Gilman. (2006) who used a dynamo model together with observations in order to predict the strength of cycle 23. While the prediction eventually proved to be substantially too high, the idea that models should be tested against reality and include observational constraints when appropriate, was reestablished. The observations themselves have vastly improved since the time the Babcock–Leighton model was developed. For instance, very important observations are the poloidal and toroidal fields at the solar surface. These are available thanks to synoptic observing programs starting from the mid-1970s at the Wilcox Solar Observatory, the Mount Wilson Observatory, and the Kitt Peak/National Solar Observatory. These programs have been supplemented by other observatories including the spaceborn SOHO and SDO platforms. Figure 2 shows the toroidal field inferred from Wilcox Solar Observatory observations (reduced as per Duvall, Scherrer, Svalgaard, & Wilcox, 1979) and the radial field based on Kitt Peak/National Solar Observatory and SOLIS observations. In both cases the data have been averaged over one year.

Solar Dynamo

Figure 2. Upper panel: Time latitude diagram of the yearly averaged radial component of the surface magnetic field from KPNSO/SOLIS Observations. Lower panel: Yearly averaged toroidal component of the surface magnetic field from WSO Observations, oversaturated to bring out the high latitude fields.

Striking from Figure 2 is the simplicity of the large-scale surface field. In particular, the toroidal field during times of activity maxima (1979, 1989, 2001, 2014) is almost unidirectional in each hemisphere, and reverses from one cycle to the next. As the surface toroidal field reflects flux emergence (Cameron, Duvall, Schüssler, & Schunker, 2018), these surface observations reflect the evolution of the subsurface field. The change in the sign of the toroidal field over the whole hemisphere raises the question of how the net toroidal flux in each hemisphere can change. This question was addressed by Cameron & Schüssler (2015), who applied Stokes theorem to the induction equation to calculate the rate of change of the toroidal flux in the convection zone in a hemisphere. They showed that the rate is dominated by the contribution from the surface. The rate was calculated from the surface observations, and the amount of flux thus generated was shown to be comparable with the amount of flux that emerges.

The argument presented by Cameron & Schüssler (2015), together with the observed toroidal and poloidal magnetic field at the solar surface, provides strong support for the Babcock–Leighton assumption that the magnetic field which threads through the surface is the poloidal flux which gets wound up to produce the toroidal field. In addition, there is strong observational support for the empirical surface flux transport model. This essentially restricts dynamo models to those of the Babcock–Leighton type. The cause of the α effect, which then involves Joy’s law, could be a turbulent effect or could be due to flows associated with buoyantly rising magnetic flux. Equatorial transport of the locations where sunspots emerge could be due to bulk flows or turbulent effects.

Given that the solar dynamo appears to be of the Babcock–Leighton type, the scatter in the tilt angle of sunspot groups about Joy’s law introduces randomness into the amount of activity from one cycle to the next. This was studied quantitatively, based on the observed distribution of tilt angles by Jiang, Cameron, and Schüssler (2014). The observed drop in the amplitude of cycle 24 with respect to cycles 21 to 23 was shown by Jiang, Cameron, and Schüssler (2015) to be caused by the actual distribution of the tilt in cycle 23 as measured from spatially and temporally resolved magnetograms by Li and Ulrich (2012). The conclusion that the weakness of cycle 24 was due to the statistics of the tilt angles of active regions during cycle 23 is supported by Whitbread, Yeates, and Muñoz-Jaramillo (2018). Even single active regions can substantially affect the strength of the subsequent cycle (Nagy, Lemerle, Labonville, Petrovay, & Charbonneau, 2017).

In addition to the results from helioseismology and the four cycles of highly detailed observations of the Sun’s surface magnetic fields, some of the observational advances relevant to the global solar dynamo include: local helioseismology, which has started to measure the three- dimensional structure of convection, including the correlations between the different flow components (e.g., Langfellner, Gizon, & Birch, 2015), attention given to the proper interpretation of the observational records from the time of the invention of the telescope onwards (for example Clette, Svalgaard, Vaquero, & Cliver 2014; Willamo, Usoskin, & Kovaltsov 2018), and the reconstruction of the level of solar activity over 9,000 years using cosmogenic isotopes (Solanki, Usoskin,Kromer, Schüssler, & Beer, 2004; Steinhilber et al., 2012; Usoskin, Gallet, Lopes, Kovaltsov, & Hulot, 2016).

The statistics of the cycle-to-cycle and long-term variations of solar activity, found in the records of telescopic observations and inferred from the cosmogenic isotopes, are consistent with the Babcock–Leighton model (Cameron & Schüssler, 2017b). The local-helioseismic results are important for understanding the α effect. In the Babcock–Leighton framework, this can be traced back to Joy’s law—the systematic latitudinal separation of the two polarities of emerging bipolar regions. The cause of this is related to the Coriolis force. The Coriolis force is acting on the convective motions (as per Parker, 1955) and on motions associated with buoyantly rising magnetic fields (Caligari et al., 1995; Choudhuri & Gilman, 1987). Both effects are of the correct order of magnitude (the Coriolis force acting on flows on the order of 100 m/s over a day or two would be sufficient) and it seems likely that both effects will contribute. The relative importance of the two effects requires further study.

Solar-Stellar Observations

Before leaving the discussion of the observations, a natural question at this point is why is the solar dynamo is of the Babcock–Leighton type? To obtain a sketch of a possible answer, it is necessary to turn to the solar-stellar connection. This is a burgeoning field which considers the Sun in the context of other stars. It is not the purpose here to summarize, review, or describe what has or is being learned about stellar dynamos. Instead the focus is on two issues.

Why Is the Observed Large-Scale Field so Simple?

The first result is that of van Saders et al. (2016), who argue that the Sun’s rotation has decreased (due to magnetic breaking) to the point where it is near to the critical rotation rate below which the Sun ceases to be a large-scale dynamo. The argument results from the observation that the rotational deceleration of main sequence stars apparently comes to an end approximately half way through their main-sequence lifetimes. From the theoretical side, it is easy to see that magnetic breaking will weaken as the star approaches the rotation rate at which large-scale dynamo action switches off, and it is not difficult to imagine that in the later half of their main-sequence evolution, isolated stars will asymptotically approach the critical rotation, with magnetic breaking becoming weaker as the dynamo action slowly switches off (Kitchatinov & Nepomnyashchikh, 2017a).

Near the critical rotation rate for large-scale dynamo action, the competition between the inductive effects of the velocity field and diffusion is finely balanced. The diffusive effects are stronger for (mean-field) magnetic fields with shorter wavelengths, and weaker for large-wavelength magnetic fields. This suggests a possible reason why the (observed) toroidal field is unipolar in each hemisphere at maximum—this is the largest wavelength available for the dynamo.

Where Is the Toroidal Flux Stored?

A second set of results from stellar observations show that fully convective red dwarfs have been shown to have activity cycles (Route, 2016). They also have activity levels which fit the same dependence on rotation rate as non-fully convective stars (Wright & Drake, 2016). This indicates that both a bottom to the convection zone and the strong radial shear in the rotation rate near the bottom of the convection zone known as the tachocline may not be crucial elements of the solar dynamo (for additional arguments against the role of the tachocline, see Spruit, 2011, and for arguments showing that dynamo action in the bulk of the convection zone is possible, see “Numerical Experiments”). The role of the overshoot region at the base of the convection zone for the solar dynamo remains an open question: stellar observations show that this region is not necessary for activity levels and dynamo periods similar to those of the Sun in fully convective stars but this leaves open the possibility that it still plays a role in the solar case.

Numerical Experiments

So far, the focus has been on the kinematic induction equation in order to understand how the flows on the Sun can generate magnetic fields. The more ambitious question is to ask how convection, rotation, and magnetic fields interact to produce both the magnetic field and the large-scale flows. One approach is to use mean-field theory. The generation of the large-scale flows has been investigated using mean-field hydrodynamics (see Ruediger, 1989) and the approach can be coupled with mean-field magnetohydrodynamics to extend the Babcock–Leighton model to stellar dynamos (Kitchatinov & Nepomnyashchikh, 2017b). A second, self-consistent, approach is to solve numerically for the turbulent and three-dimensional structure of the flows, thermodynamics, and magnetic field using the full system of magnetohydodynamic equations. The advantage of this approach is in obtaining fully self-consistent solutions with no free parameters or assumptions. The simulations can be performed in a spherical convective shell, or in a Cartesian box intended to model part of the solar convection zone (see Figure 3).

Solar Dynamo

Figure 3. A Cartesian box direct numerical simulation (DNS) covering a fraction of the Sun (upper-right corner; see Käpylä, Brandenburg, Kleeorin, Käpylä, & Rogachevskii, 2016a for details) and a spherical shell DNS (left; see Käpylä et al., 2016b for details) of turbulent convection producing magnetic-field concentrations in solar-like dynamo simulations. The greyscale shows the radial magnetic-field strength and polarity in the topmost layers in of the simulations with maxima of +/- 3 kG in the Cartesian domain and +/-10 kG in the spherical shell model. The butterfly diagram from the global simulation is shown in the lower right panel (image courtesy of M. Käpylä and P. Käpylä).

Such numerical simulations have been remarkably successful. They have demonstrated that the convectively driven turbulence in a rotating spherical shell, such as the solar convection zone, can act as a large-scale dynamo (Gilman & Miller, 1981). The point of these calculations is that dynamo action was shown for a fully three-dimensional convectively driven plasma. This was done using the full, self-consistent, magnetohydrodynamic equations without the need for further approximations. Increasing the resolution has produced dynamos with cycles (Gilman, 1983), has allowed the differential rotation to become less rotationally constrained and more solar like (Miesch et al., 2000), and has produced oscillatory solutions with equatorial propagation (Käpylä, Mantere, & Brandenburg, 2012; Strugarek, Beaudoin, Charbonneau, Brun, & do Nascimento, 2017). However, it is unclear if the mechanisms for the equatorial propagation in the simulations are those responsible for the observed equatorial propagation. Subsurface concentrations of buoyant magnetic fields appear in simulations (Fan & Fang, 2014; Nelson, Brown, Brun, Miesch, & Toomre, 2011); these may be identified as the precursors of features which would later emerge at the surface. The simulations, however, have not reached the stage where they self-consistently model the emergence in sufficient detail to describe sunspots—for this purpose they need to be coupled to simulations capable of treating the photosphere (Chen, Rempel, & Fan, 2017).

Another important advance due to the simulations is the confirmation of the relevance of the mean-field model. This was achieved by deriving the mean-field coefficients from a dynamo simulation and using these to drive a mean-field model for comparison (e.g., Schrinner, Rädler, Schmitt, Rheinhardt, & Christensen, 2007). More generally, the mean-field coefficients derived from the numerical simulations are useful in understanding the different mean-field effects and the behavior of the simulations themselves (e.g., for the importance of the Parker–Yoshimurra rule in determining the direction of propagation of dynamo waves in the simulations, see Warnecke et al., 2018).

The main limitation of the simulations in trying to exactly understand the solar dynamo is that simulations do not operate in the solar parameter regime, and will not for the foreseeable future—the diffusivities are too high by many orders of magnitude. Hotta, Rempel, and Yokoyama (2016) and Käpylä, Käpylä, Olspert,Warnecke, and Brandenburg (2017) showed that the small-scale magnetic fields that can be generated by the turbulent motions even in non-rotating systems (Kazantsev, 1968) have a substantial effect on the large-scale flows and magnetic fields. This suggests that any hope of being in an asymptotic regime, where further decreasing the diffusivities has little effect, requires at least reaching the regime where the small-scale dynamo operates robustly.

One reason for concern is that the solar-stellar observations indicate that the Sun may be near a bifurcation point where dynamo action switches off. Near the bifurcation point, the solution is qualitatively sensitive to the parameters. More worrying than this is that the simulations suggest that this bifurcation is near in parameter space (where the parameter is the rotation rate) to another qualitative change where the latitudinal differential rotation changes from being solar-like to anti-solar like. The idea here is that as the rotation rate of the Sun decreases, it reduces the latitudinal rotation to the point where the large-scale (presumably Babcock–Leighton) dynamo switches off, and if the rotation rate decreases further then the latitudinal differential rotation switches sign. This implies that the system is sensitive to the parameters. Given this sensitivity of the qualitative nature of the solutions to the parameters, the numerical solutions have been remarkably successful in reproducing some aspects of the solar dynamo.

An important aspect of the discrepancy between the simulations and the Sun is that those simulations that self-consistently describe the transport of the energy associated with the solar luminosity through the convection zone have too much energy in the convective motions at large spatial scales (scales beyond those of supergranulation). This discrepancy in the power of the convection at large wavelengths is seen by comparison to helioseismic observations (this is debated; see Greer, Hindman, & Toomre 2016; Hanasoge, Duvall, & Sreenivasan, 2012), surface observations (Gizon & Birch, 2012), and most importantly the simulations themselves: most of the simulations self-consistently transporting the energy by convection and with the solar luminosity produce anti-solar differential rotation unless a rotation rate about three times faster than the Sun is assumed. This problem is known as the convective conundrum (O’Mara, Miesch, Featherstone, & Augustson, 2016) and is a major open question in the dynamics of the solar interior.

Given that the diffusivities used in the simulations are non-solar by many orders of magnitude, and that the velocity amplitudes are only too large at large-spatial scales, getting within a factor of three should be seen as a remarkable success of the modeling. The success is shared by different models with different assumptions. The simulations have more trouble reproducing the observed surface meridional flow, which, although much weaker than the differential rotation, is a critical ingredient of our understanding of the observed evolution of the surface magnetic fields.

Small-Scale Dynamo

At the outset of our discussion of the mean-field approach, an assumption was made that the small-scale field was local in time and space and therefore tightly linked to the mean field. Even at the time that mean field theory was being developed, however, it was recognized that the small-scale field could be self-sustained by the action of the small-scale turbulent-flow field (Kazantsev, 1968). The turbulence on the Sun is, however, stratified, and has characteristic time-scales that vary with depth. To show that small-scale dynamo action is possible in these circumstances, numerical simulations of the Sun’s photospheric layer and the upper part of the convection zone were necessary. Simulations by Vögler and Schüssler (2007) showed that even when the stratification is present, and the motions are convectively driven and the effect of radiation and partial ionization are included, dynamo action can take place. The only cautionary note here is that the diffusivity was much higher than in the Sun, and more importantly the ratio of the viscous and magnetic diffusivities was also far from that on the Sun (Figure 4).

Solar Dynamo

Figure 4. A Cartesian box direct numerical simulation of small-scale dynamo action. Shown is the vertical magnetic field (blue positive, red negative) at a constant geometrical height near the solar surface, saturated at +/-50G. The simulation that this figure is based on is run 3 of Graham Cameron, & Schüssler (2010). The magnetic field is concentrated in turbulent downflow lanes between smooth upflowing plasma.

This type of simulation has been extended to larger domains, and comparison with observations indicates that the magnetic field of the small-scale dynamo is in equipartition with the convective flows (Rempel, 2014). It should therefore be understood that the small-scale dynamo has a substantial effect on the flows inside the Sun, and thus the operation of the small-scale dynamo is likely to be crucial for understanding the flows inside the Sun and, from there, the operation of the large-scale dynamo (Hotta et al., 2016). This implicitly links the small- and large-scale solar dynamos.

Conclusion

At this point it is appropriate to return to the statement of Parker (1970), “The basic ideas are sound, I think, but, when we come to the details, there are so many theoretical alternatives that need yet to be explored and so much that is not yet made clear by observations.” Since 1970, the synoptic observations have provided some limited clarity, and strongly support the conclusion that the solar-dynamo is of the Babcock–Leighton type. Simulations have probed a number of dynamo mechanisms with remarkable success. Outstanding questions concerning the solar dynamo, the answers to which will require more observations and simulations and increased interaction between the two, include:

  1. 1. What is the physical cause of Joy’s law? The Coriolis force acting on some combination of the cyclonic motions of Parker (1955) and motions driven by the buoyant rise of the magnetic fields (Choudhuri & Gilman, 1987) seems to be very likely, but can the relative importance of these two effects be quantified?

  2. 2. Where in depth is the toroidal flux stored? The stellar observations showing that fully convective stars can have dynamos with properties similar to that of the Sun are extremely important in this regard, showing that a base to the convection zone is not critical for these properties.

  3. 3. What is the transport mechanism for the subsurface toroidal flux? A return flow of the poleward meridional flow observed at the surface and/or latitudinal turbulent pumping are obvious possibilities but there are possibly other mean-field terms which may be important.

  4. 4. Are other dynamo mechanisms, besides that of the Babcock–Leighton model, operating? Such mechanisms may not be dominant but would certainly be interesting.

  5. 5. In terms of the flow fields, what is the solution to the convective conundrum? Does solving it produce a meridional flow compatible with observations? From the observational side, what is the meridional flow beneath the surface?

The number of open questions increases dramatically when stars are considered. The most likely way progress will be made here is through a continued development of observational techniques and long-time series combined with a continued exploration of the theoretical possibilities using ever more detailed numerical simulations.

Acknowledgments

The author gratefully acknowledges comments on a draft from Prof. M. Schüssler, Dr. M. Käpylä, and Dr. J. Warnecke.

References

Babcock, H. W. (1961). The topology of the sun’s magnetic field and the 22-YEAR cycle. Astrophysical Journal, 133, 572.Find this resource:

Babcock, H. W. (1963). The sun’s magnetic field. Annual Review of Astronomy and Astrophysics, 1, 41–58.Find this resource:

Basu, S. (2016). Global seismology of the sun. Living Reviews in Solar Physics, 13(2).Find this resource:

Blackett, P. M. S. (1947). The magnetic field of massive rotating bodies. Nature, 159, 658–666.Find this resource:

Caligari, P., Moreno-Insertis, F., & Schüssler, M. (1995). Emerging flux tubes in the solar convection zone. 1: Asymmetry, tilt, and emergence latitude. Astrophysical Journal, 441, 886–902.Find this resource:

Cameron, R. H., Duvall, T. L., Schüssler, M., & Schunker, H. (2018). Observing and modeling the poloidal and toroidal fields of the solar dynamo. Astronomy & Astrophysics, 609, A56.Find this resource:

Cameron, R. H., Schmitt, D., Jiang, J., & Isık, E. (2012). Surface flux evolution constraints for flux transport dynamos. Astronomy & Astrophysics, 542, A127.Find this resource:

Cameron, R. H., & Schüssler, M. (2015). The crucial role of surface magnetic fields for the solar dynamo. Science, 347(6228), 1333–1335.Find this resource:

Cameron, R. H., & Schüssler, M. (2017a). An update of Leighton’s solar dynamo model. Astronomy & Astrophysics, 599, A52.Find this resource:

Cameron, R. H., & Schüssler, M. (2017b). Understanding solar cycle variability. Astrophysical Journal, 843(111).Find this resource:

Carrington, R. C. (1863). Spots on the sun, from November 9, 1853 to March 24, 1861, made at Redhill. London, U.K.: Williams and Norgate.Find this resource:

Chen, F., Rempel, M., & Fan, Y. (2017). Emergence of magnetic flux generated in a solar convective dynamo. I. The formation of sunspots and active regions, and the origin of their asymmetries. Astrophysical Journal, 846(149).Find this resource:

Chitta, L. P., Smitha, H. N., & Solanki, S. K. (Forthcoming). Solar Photosphere.Find this resource:

Choudhuri, A. R., & Gilman, P. A. (1987). The influence of the Coriolis force on flux tubes rising through the solar convection zone. Astrophysical Journal, 316, 788–800.Find this resource:

Choudhuri, A. R., Schussler, M., & Dikpati, M. (1995). The solar dynamo with meridional circulation. Astronomy & Astrophysics, 303, L29.Find this resource:

Clette, F., Svalgaard, L., Vaquero, J. M., & Cliver, E. W. (2014). Revisiting the sunspot number. A 400-year perspective on the solar cycle. Space Science Reviews, 186(1–4), 35–103.Find this resource:

Cowling, T. G. (1933). The magnetic field of sunspots. Monthly Notices of the Royal Astronomical Society, 94(1), 39–48.Find this resource:

DeVore, C. R., Boris, J. P., & Sheeley, N. R., Jr. (1984). The concentration of the large-scale solar magnetic field by a meridional surface flow. Solar Physics, 92(1–2), 1–14.Find this resource:

Dikpati, M., Corbard, T., Thompson, M. J., & Gilman, P. A. (2002). Flux transport solar dynamos with near-surface radial shear. Astrophysical Journal Letters, 575(1), L41–L45.Find this resource:

Dikpati, M., de Toma, G., & Gilman, P. A. (2006). Predicting the strength of solar cycle 24 using a flux-transport dynamo-based tool. Geophysical Research Letters, 33, L05102.Find this resource:

D’Silva, S., & Choudhuri, A. R. (1993). A theoretical model for tilts of bipolar magnetic regions. Astronomy & Astrophysics, 272, 621.Find this resource:

Duvall, T. L., Jr. (1979). Large-scale solar velocity fields. Solar Physics, 63(1), 3–15.Find this resource:

Duvall, T. L., Jr., Scherrer, P. H., Svalgaard, L., & Wilcox, J. M. (1979). Average photospheric poloidal and toroidal magnetic field components near solar minimum. Solar Physics, 61(2), 233–245.Find this resource:

Fan, Y., & Fang, F. (2014). A simulation of convective dynamo in the solar convective envelope: Maintenance of the solar-like differential rotation and emerging flux. Astrophysical Journal, 789(1), 35.Find this resource:

Gilman, P. A. (1983). Dynamically consistent nonlinear dynamos driven by convection in a rotating spherical shell. II—Dynamos with cycles and strong feedbacks. Astrophysical Journal Supplement, 53, 243–268.Find this resource:

Gilman, P. A., & Miller, J. (1981). Dynamically consistent nonlinear dynamos driven by convection in a rotating spherical shell. Astrophysical Journal Supplement, 46, 211–238.Find this resource:

Gizon, L., & Birch, A. C. (2012). Helioseismology challenges models of solar convection. Proceedings of the National Academy of Science, 109(30), 11896–11897.Find this resource:

Greer, B. J., Hindman, B. W., & Toomre, J. (2016). Helioseismic measurements of the Rossby number in the sun’s near-surface shear layer. Astrophysical Journal, 824(1), 4.Find this resource:

Hale, G. E. (1908). On the probable existence of a magnetic field in sun-spots. Astrophysical Journal, 28, 315.Find this resource:

Hale, G. E., Ellerman, F., Nicholson, S. B., & Joy, A. H. (1919). The magnetic polarity of sun-spots. Astrophysical Journal, 49, 153.Find this resource:

Hanasoge, S. M., Duvall, T. L., & Sreenivasan, K. R. (2012). Anomalously weak solar convection. Proceedings of the National Academy of Science, 109(30), 11928–11932.Find this resource:

Herzenberg, A. (1958). Geomagnetic dynamos. Philosophical Transactions of the Royal Society of London Series A, 250(986), 543–583.Find this resource:

Hotta, H., Rempel, M., & Yokoyama, T. (2016). Large-scale magnetic fields at high Reynolds numbers in magnetohydrodynamic simulations. Science, 351(6280), 1427–1430.Find this resource:

Jiang, J., Cameron, R. H., & Schüssler, M. (2014). Effects of the scatter in sunspot group tilt angles on the large-scale magnetic field at the solar surface. Astrophysical Journal, 791(1), 5.Find this resource:

Jiang, J., Cameron, R. H., & Schüssler, M. (2015). The cause of the weak solar cycle 24. Astrophysical Journal Letters, 808(1), L28.Find this resource:

Käpylä, P. J., Brandenburg, A., Kleeorin, N., Käpylä, M. J., Rogachevskii, I. (2016a). Magnetic flux contributions from turbulent stratified convection. Astronomy & Astrophysics, 588, A150.Find this resource:

Käpylä, M. J., Käpylä, P. J., Olspert, N., Brandenburg, A., Warnecke, J., Karak, B. B., Pelt, J., (2016b). Multiple dynamo modes as a mechanism for long-term solar activity variations. Astronomy & Astrophysics, 589, A56.Find this resource:

Käpylä, P. J., Käpylä, M. J., Olspert, N., Warnecke, J., & Brandenburg, A. (2017). Convection-driven spherical shell dynamos at varying Prandtl numbers. Astronomy & Astrophysics, 599, A4.Find this resource:

Käpylä, P. J., Mantere, M. J., & Brandenburg, A. (2012). Cyclic Magnetic activity due to turbulent convection in spherical wedge geometry. Astrophysical Journal Letters, 755(1), L22.Find this resource:

Kazantsev, A. P. (1968). Enhancement of a magnetic field by a conducting fluid. Soviet Journal of Experimental and Theoretical Physics, 26, 1031.Find this resource:

Kitchatinov, L. L., & Nepomnyashchikh, A. A. (2017a). How supercritical are stellar dynamos, or why do old main-sequence dwarfs not obey gyrochronology? Monthly Notices of the Royal Astronomical Society, 470(3), 3124–3130.Find this resource:

Kitchatinov, L. L., & Nepomnyashchikh, A. A. (2017b). A joined model for solar dynamo and differential rotation. Astronomy Letters, 43(5), 332–343.Find this resource:

Krause, F., & Rädler, K.-H. (1980). Mean-field magnetohydrodynamics and dynamo theory. Berlin, Germany: Akademie Verlag.Find this resource:

Langfellner, J., Gizon, L., & Birch, A. C. (2015). Spatially resolved vertical vorticity in solar supergranulation using helioseismology and local correlation tracking. Astronomy & Astrophysics, 581, A67.Find this resource:

Larmor, J. (1919). How could a rotating body such as the Sun become a magnet? Report of the British Association for the Advancement of Science, 87th Meeting, 159–160.Find this resource:

Leighton, R. B. (1964). Transport of magnetic fields on the sun. Astrophysical Journal, 140, 1547.Find this resource:

Leighton, R. B. (1969). A magneto-kinematic model of the solar cycle. Astrophysical Journal, 156, 1.Find this resource:

Li, J., & Ulrich, R. K. (2012). Long-term measurements of sunspot magnetic tilt angles. Astrophysical Journal, 758(2), 115.Find this resource:

Mackay, D. H., & van Ballegooijen, A. A. (2005). New results in modeling the hemispheric pattern of solar filaments. Astrophysical Journal Letters, 621(1), L77–L80.Find this resource:

Maunder, E. W. (1904). Note on the distribution of sun-spots in heliographic latitude, 1874–1902. Monthly Notices of the Royal Astronomical Society, 64, 747–761.Find this resource:

Miesch, M. S., Elliott, J. R., Toomre, J., Clune, T. L., Glatzmaier, G. A., & Gilman, P. A. (2000). Three-dimensional spherical simulations of solar convection. I. Differential rotation and pattern evolution achieved with laminar and turbulent states. Astrophysical Journal, 532(1), 593–615.Find this resource:

Moffatt, H. K. (1978). Magnetic field generation in electrically conducting fluids. Cambridge, U.K.: Cambridge University Press.Find this resource:

Muñoz-Jaramillo, A., Dasi-Espuig, M., Balmaceda, L. A., & DeLuca, E. E. (2013). Solar cycle propagation, memory, and prediction: Insights from a century of magnetic proxies. Astrophysical Journal Letters, 767(2), L25.Find this resource:

Nagy, M., Lemerle, A., Labonville, F., Petrovay, K., & Charbonneau, P. (2017). The effect of “rogue” active regions on the solar cycle. Solar Physics, 292, 167.Find this resource:

Nelson, N. J., Brown, B. P., Brun, A. S., Miesch, M. S., & Toomre, J. (2011). Buoyant magnetic loops in a global dynamo simulation of a young sun. Astrophysical Journal Letters, 739(2), L38.Find this resource:

O’Mara, B., Miesch, M. S., Featherstone, N. A., & Augustson, K. C. (2016). Velocity amplitudes in global convection simulations: The role of the Prandtl number and near-surface driving. Advances in Space Research, 58(8), 1475–1489.Find this resource:

Parker, E. N. (1955). Hydromagnetic dynamo models. Astrophysical Journal, 122, 293.Find this resource:

Parker, E. N. (1970). The origin and dynamical effects of the magnetic fields and cosmic rays in the disk of the galaxy. In H. J. Habing (Ed.), Interstellar gas dynamics (Vol. 39, pp. 168–183). Dordrecht, the Netherlands: D Reidel.Find this resource:

Pietarila G. J., Cameron, R., & Schüssler, M. (2010). Turbulent small-scale dynamo action in solar surface simulations. Astrophysical Journal, 714(2), 1606–1616.Find this resource:

Rempel, M. (2014). Numerical simulations of quiet Sun magnetism: On the contribution from a small-scale dynamo. Astrophysical Journal, 789, 132Find this resource:

Route, M. (2016). The discovery of solar-like activity cycles beyond the end of the main sequence?Astrophysical Journal Letters, 830(2), L27.Find this resource:

Ruediger, G. (1989). Differential rotation and stellar convection. Sun and the solar stars. Berlin, Germany: Akademie Verlag.Find this resource:

Schatten, K. H., Scherrer, P. H., Svalgaard, L., & Wilcox, J. M. (1978). Using dynamo theory to predict the sunspot number during solar cycle 21. Geophysical Research Letters, 5, 411–414.Find this resource:

Schou, J., Antia, H. M., Basu, S., Bogart, R. S., Bush, R. I., Chitre, S. M., . . ., Toomre, J. (1998). Helioseismic studies of differential rotation in the solar envelope by the solar oscillations investigation using the Michelson doppler imager. Astrophysical Journal, 505(1), 390–417.Find this resource:

Schrinner, M., Rädler, K.-H., Schmitt, D., Rheinhardt, M., & Christensen, U. R. (2007). Mean-field concept and direct numerical simulations of rotating magnetoconvection and the geodynamo. Geophysical and Astrophysical Fluid Dynamics, 101(2), 81–116.Find this resource:

Schwabe, M. (1849). Sonnen-und Saturn-Beobachtungen im Jahre 1848, von Herrn Hofrath Schwabe in Dessau. Astronomische Nachrichten, 28, 302.Find this resource:

Schwarzschild, K. (1906). On the equilibrium of the sun’s atmosphere. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse, 195, 41–53.Find this resource:

Solanki, S. K., Usoskin, I. G., Kromer, B., Schüssler, M., & Beer, J. (2004). Unusual activity of the sun during recent decades compared to the previous 11,000 years. Nature, 431, 1084–1087.Find this resource:

Spoerer, G. (1889). Ueber die periodicitat der Sonnenflecken seit dem Jahr 1618. Nova Acta der Ksl. Leop.- Carol. Deutschen Akademie der Naturforscher, 53(2), 281–324.Find this resource:

Spoerer, G. (1879). Beobachtung der Sonnenflecken etc.Astronomische Nachrichten, 96, 23.Find this resource:

Spruit, H. C. (2011). Theories of the solar cycle: A Critical view. In M. P. Miralles & J. Sanchez Almeida (Eds.), The sun, the solar wind, and the heliosphere (pp. 39–50). Dordrecht, the Netherlands: Springer.Find this resource:

Steenbeck, M., Krause, F., & Rädler, K.-H. (1966). Berechnung der mittleren Lorentz-Feldstärke v × B für ein elektrisch leitendes Medium in turbulenter, durch Coriolis-Kräfte beeinflußter Bewegung. Zeitschrift Naturforschung Teil A, 21, 369.Find this resource:

Steinhilber, F., Abreu, J. A., Beer, J., Brunner, I., Christl, M., Fischer, H., . . .Wilhelms, F. (2012). 9,400 years of cosmic radiation and solar activity from ice cores and tree rings. Proceedings of the National Academy of Science, 109(16), 5967–5971.Find this resource:

Stix, M. (1974). Comments on the solar dynamo. Astronomy & Astrophysics, 37, 121–133.Find this resource:

Strugarek, A., Beaudoin, P., Charbonneau, P., Brun, A. S., & do Nascimento, J.-D. (2017). Reconciling solar and stellar magnetic cycles with nonlinear dynamo simulations. Science, 357(6347), 185–187.Find this resource:

Svalgaard, L., Cliver, E. W., & Kamide, Y. (2005). Sunspot cycle 24: Smallest cycle in 100 years?Geophysical Research Letters, 32(1), L01104.Find this resource:

Thompson, M. J., Toomre, J., Anderson, E. R., Antia, H. M., Berthomieu, G., Burtonclay, D., . . . Wilson, P. R. (1996). Differential rotation and dynamics of the solar interior. Science, 272(5266), 1300–1305.Find this resource:

Unsöld, A. (1930). Konvektion in der Sonnenatmosphäre (nebst einer Bemerkung zur Deutung der Novae). (Eingegangen am 28. Juni 1930). Zeitschrift für Astrophysik, 1, 138.Find this resource:

Usoskin, I. G., Gallet, Y., Lopes, F., Kovaltsov, G. A., & Hulot, G. (2016). Solar activity during the Holocene: The Hallstatt cycle and its consequence for grand minima and maxima. Astronomy & Astrophysics, 587, A150.Find this resource:

van Saders, J. L., Ceillier, T., Metcalfe, T. S., Silva Aguirre, V., Pinsonneault, M. H., García, R. A., . . . Davies, G. R. (2016). Weakened magnetic braking as the origin of anomalously rapid rotation in old field stars. Nature, 529, 181–184.Find this resource:

Vögler, A., & Schüssler, M. (2007). A solar surface dynamo. Astronomy & Astrophysics, 465(3), L43–L46.Find this resource:

Wang, Y.-M., Nash, A. G., & Sheeley, N. R., Jr. (1989a). Evolution of the sun’s polar fields during sunspot cycle 21—Poleward surges and long-term behavior. Astrophysical Journal, 347, 529–539Find this resource:

Wang, Y.-M., Nash, A. G., & Sheeley, N. R., Jr. (1989b). Magnetic flux transport on the sun. Science, 245, 712–718.Find this resource:

Wang, Y.-M., & Sheeley, N. R., Jr. (1991). Magnetic flux transport and the sun’s dipole moment—New twists to the Babcock-Leighton model. Astrophysical Journal, 375, 761–770.Find this resource:

Wang, Y.-M., & Sheeley, N. R., Jr. (2009). Understanding the geomagnetic precursor of the solar cycle. Astrophysical Journal Letters, 694(1), L11–L15.Find this resource:

Warnecke, J., Rheinhardt, M., Tuomisto, S., Käpylä, P. J., Käpylä, M. J., & Brandenburg, A. (2018). Turbulent transport coefficients in spherical wedge dynamo simulations of solar-like stars. Astronomy & Astrophysics, 609, A51.Find this resource:

Whitbread, T., Yeates, A. R., & Muñoz-Jaramillo, A. (2018). How many active regions are necessary to predict the solar dipole moment?Astrophysical Journal, 863(2), 116.Find this resource:

Willamo, T., Usoskin, I. G., & Kovaltsov, G. A. (2018). A test of the active-day fraction method of sunspot group number calibration: Dependence on the level of solar activity. Solar Physics, 293, 69.Find this resource:

Wright, N. J., & Drake, J. J. (2016). Solar-type dynamo behaviour in fully convective stars without a tachocline. Nature, 535, 526–528.Find this resource:

Yeates, A. R., Mackay, D. H., & van Ballegooijen, A. A. (2008). Modelling the global solar corona II: Coronal evolution and filament chirality comparison. Solar Physics, 247(1), 103–121.Find this resource:

Yoshimura, H. (1975a). *A model of the solar cycle driven by the dynamo action of the global convection in the solar convection zone. Astrophysical Journal Supplement, 29, 467–494.Find this resource:

Yoshimura, H. (1975b). Solar-cycle dynamo wave propagation. Astrophysical Journal Supplement, 201, 740–748.Find this resource:

Notes:

(1.) Leighton (1969) claims his model gives equatorward propagation in the absence of radial shear but this is a consequence of a mistake identified in Cameron and Schüssler (2017a).