## Abstract

The origins of solar magnetism lie below the visible surface of the Sun, in the highly turbulent convection zone. Turbulent convection operates in conjunction with rotational shear, global circulations and intricate boundary layers to produce the rich diversity of magnetic activity we observe. Here, we review recent insights into the operation of the solar dynamo obtained from solar and stellar observations and numerical models.

## 1. Introduction

The year 2008 marked the 100th anniversary of George Ellery Hale's monumental detection of magnetic fields in sunspots by means of the Zeeman effect [1]. It was the first demonstration of magnetism in an astronomical object and the discovery transformed not only solar physics but all of astrophysics. The 11 year sunspot cycle, discovered 65 years earlier by Heinreich Schwabe [2], was soon seen in a new light (although not without controversy), as a remarkable example of hydromagnetic self-organization on a colossal scale, whereby bulk motions of a turbulent plasma produce organized, global patterns of magnetic activity. Hale himself appreciated the implications, drawing parallels between sunspots and laboratory experiments of magnetic fields generated by rotating electrically charged discs. Astrophysical dynamo theory was born (or at least *reborn*, because prior links between geomagnetic storms and solar activity had already provided inklings of solar magnetism).

Modern observations reveal the handiwork of the solar dynamo with marked clarity. Filamentary magnetic structures such as coronal loops and streamers permeate the solar atmosphere, and the release of magnetic energy accelerates the solar wind and powers eruptive events such as flares and coronal mass ejections. These in turn shape the heliosphere and regulate the space environment of planets. Solar and stellar variability and rotational evolution are intimately linked with magnetism. Yet, the sunspot cycle still stands as one of the most fundamental and enigmatic challenges of solar physics.

In this paper, we provide a brief overview of solar dynamo theory as it now stands, focusing on fundamental building blocks and active frontiers. In order to set the stage for the subsequent discussion, we give a brief overview of solar magnetism in §2. We then address the nature of convective dynamos in §3, highlighting the distinction between small-scale and large-scale dynamo action and its implications for stars and the Sun in particular. In §4, we focus on the solar activity cycle, highlighting current theoretical paradigms and open questions. Section 5 is a summary and a look to the future.

## 2. Solar magnetism

### (a) The solar cycle

The cyclic nature of the global solar magnetic field is most clearly demonstrated in the structure and evolution of the mean field threading through the solar surface. Figure 1*a* shows the radial component of the magnetic field in the solar photosphere inferred from Zeeman spectroscopy as a function of latitude and time. A longitudinal average is approximated by combining measurements near the centre of the disc spanning one mean rotation period, known as a Carrington rotation.

The low-latitude behaviour seen in figure 1*a* reflects the well-known solar butterfly diagram, prominent also in sunspot observations [4–6]. At the beginning of a cycle, bipolar active regions appear at latitudes of roughly ±30^{°}, signifying the emergence of coherent subsurface flux structures. As the cycle proceeds, these active regions disperse and new flux emerges at progressively lower latitudes, reaching the equator in approximately 11 years.^{1} As the active bands approach the equator, the emergence of new active regions largely ceases, as signified for example by a minimum in the fraction of the surface covered by spots (figure 1*b*). This is known as solar minimum, when the large-scale structure of the magnetic field in the photosphere and overlying corona is relatively simple,^{2} dominated by polar caps of opposite polarity (e.g. 1996–1997 in figure 1*a*). Much of the magnetic energy is in the dipole component, but higher order multi-poles also contribute significantly [7]. This was particularly evident during the recent extended minimum in 2008–2009 when the dipole contribution dropped unusually low relative to higher order contributions (J. Luhmann 2010, unpublished data). The appearance of active regions at mid-latitudes signifies the beginning of a new 11 year cycle. The polarity of the polar fields during solar minimum reverses each 11 year sunspot cycle; so the net period of the magnetic activity cycle is approximately 22 years.

Regions of positive and negative polarity within an active region are oriented approximately along an east–west axis, although the trailing spots (westward edge, following the sense of rotation) are systematically displaced poleward relative to the leading (eastward edge) polarity. The tilt of the axis increases with latitude, known as Joy's Law, and is thought to arise through the influence of the Coriolis force on buoyantly rising toroidal flux structures [8]. Furthermore, the sense of the leading and trailing polarities in active regions reverses sign across the equator, known as Hale's polarity law, implying antisymmetry in the subsurface toroidal fields that give rise to them.

The emergent flux structures that form photospheric active regions apparently decouple from deeper seated dynamics within a few days [9], fragmenting and dispersing over the course of several weeks. The dispersal and subsequent transport of the residual flux from active regions is well captured by quasi-two-dimensional models that incorporate turbulent diffusion from convective motions as well as advection by a time-varying, axisymmetric, poleward flow of order 10–30 m s^{−1} [10,11]. Such a flow has been detected in the solar surface layers by multiple independent techniques, including Doppler measurements, local helioseismic inversions and correlation tracking [12].

The combined effect of Joy's Law, active region fragmentation and surface transport by convection and by meridional flow induces a net global, mean poloidal field by means of a phenomenon known as the Babcock–Leighton (BL) mechanism [13,14]. Trailing residual flux from active regions is transported poleward, where it interacts with pre-existing polar fields of the opposite polarity (figure 1*a*). This process apparently accounts for the cyclic polarity reversals in the polar caps, although other mechanisms may also contribute to global poloidal field generation (§4).

Although sunspots and active regions are the most familiar manifestations of cyclic solar activity, corresponding variability is observed in many other diagnostics, such as solar irradiance, the frequency of eruptive events (coronal mass ejections and flares) and geomagnetic activity induced by solar forcing [15,16]. An example is shown in figure 1*c*, which traces what is known as the Mg II index, quantifying the relative emission in the core and wing of the Mg II spectral line. This Mg II emission originates in the chromosphere, and the core-to-wing ratio is observed to peak above active regions and supergranular downflow lanes where the photospheric magnetic flux is predominantly vertical. The Mg II index is commonly taken as a proxy for solar and stellar magnetic activity [17].

Although continuous multi-wavelength monitoring of solar irradiance variability and related magnetic activity is limited to the past three to five decades, available data suggest that similar variations in solar magnetic activity have been operating unabated for at least 10 000 years, with extended periods of relatively low and high activity known as grand minima and grand maxima [5,18]. Continuous sunspot records date back to the early seventeenth century, and earlier variations in solar activity can be inferred from cosmogenic chemical isotopes found in tree rings and ice cores. The data are consistent with the notion that the 22 year activity cycle has been operating continually for at least several millennia, although the most ancient isotope data often have insufficient temporal resolution to demonstrate this conclusively. Longer term variations such as grand minima and maxima appear relatively random. There is some possible evidence for quasi-periodic behaviour on time scales of 80–2000 years, but none as prominent or regular as the primary 11 year sunspot cycle [18].

### (b) Order amid chaos

Although cyclic activity has long been regarded as the most important and formidable challenge of solar dynamo theory, a closer look at solar magnetism reveals that this striking regularity exists amid a turbulent maelstrom of small-scale magnetic fluctuations. High-resolution Zeeman measurements of the solar photosphere reveal rapidly varying magnetic structure on the smallest observed scales of order 100 km, while more sophisticated measurements based on full Stokes polarimetry and the Hanle effect suggest that there is yet more magnetic energy on even smaller scales that are beyond current resolution limits [19–22]. This small-scale flux component replenishes itself daily and has been referred to as the Sun's magnetic carpet [23,24].

Photospheric magnetograms (vertical field maps inferred from Zeeman measurements) reveal patches of magnetic flux spanning a vast range of scales (figure 2*a*,*b*). By applying clumping algorithms to cross-calibrated data from the SOlar and Heliospheric Observatory (SOHO)/Michelson Doppler Imager (MDI) and Hinode/Solar Optical Telescope (SOT), Parnell *et al.* [25] have demonstrated that the flux distribution is self-similar, following a power law extending from the smallest resolved features (area ∼10^{14} cm^{2}, flux ∼10^{16} Mx) all the way to sunspots and active regions (area ∼2×10^{19} cm^{2}, flux ∼10^{22}–10^{23} Mx). The distribution persists at solar minimum and solar maximum, the main difference being the presence of active regions in the latter, which extends the distribution to higher flux values (figure 2*c*). The best-fit power-law exponent for the data in figure 2*c* is −1.85. Careful scrutiny of time series suggests that the distribution of emerging flux is even steeper, following a power law with a best-fit exponent of −2.74 [26]. Presumably, it is this emergent flux spectrum that characterizes the dynamo, while the shallower spectrum shown in figure 2*c* may reflect the coalescence of flux elements through passive advection by the granular flow or through dynamical feedback mechanisms such as convective collapse [27]. Indeed, reprocessing of surface flux through coalescence and/or fragmentation can readily yield power laws similar to observational inferences [28]. Still, the power-law behaviour of emergent bipoles and the persistence of power-law behaviour throughout the solar cycle suggests that the scale similarity may be inherent in the field generation and not just due to surface processing of emergent flux.

The distribution function *f*(*Φ*) shown in figure 2*c* is defined such that the total flux per unit area from flux concentrations with values between *Φ*_{1} and *Φ*_{2} is given by
2.1The emergent flux spectrum obtained by Thornton & Parnell [26] is defined similarly, but the integrated flux *F*(*Φ*_{1},*Φ*_{2}) now has units of Mx cm^{−2} d^{−1}. Thus, a power-law distribution *f*(*Φ*)∝*Φ*^{−α} with α>2 indicates that the unsigned flux emerging on small scales dominates over that emerging on large scales. In particular, the rate of flux emergence in structures smaller than 10^{20} Mx exceeds that in active regions by nearly three orders of magnitude [26]. This estimate must be regarded as a lower limit, since much of the small-scale flux is likely to be unresolved, as suggested by Hanle measurements [19]. Furthermore, if the mean-field strength in small and large flux concentrations is comparable (justified as a first approximation by the common saturation level of figure 2*a*,*b*), this implies that most of the magnetic energy produced by the dynamo is likewise in small-scale, turbulent fluctuations.

Thus, the solar dynamo does not just produce the solar cycle. Rather, it produces a continuous spectrum of dynamically significant fields extending from the global dipole moment to sunspots to turbulent fluctuations on the smallest sub-granule scales we can currently resolve. The common power law across this vast range of scales suggests that the sunspot cycle and the magnetic carpet are dynamically linked. Although this does not necessarily rule out meaningful models targeted at specific aspects of the solar dynamo, it is wise to keep in mind that the multi-faceted manifestations of solar magnetism have a common origin.

### (c) Convection, rotation and magnetic activity

It has been realized since the formative years of solar dynamo theory a half century ago that the key ingredient in producing organized patterns of magnetic activity from disorganized convective motions is rotation [29–31]. Rotation induces anisotropic momentum and energy transport, which in turn establishes differential rotation and global meridional circulations (MCs) that amplify and transport magnetic flux [32]. Rotation also gives rise to helical flows and fields that can promote hydromagnetic self-organization by coupling large and small scales. There are many subtleties in how such processes proceed in practice (reviewed in §3*b*) and there are still many uncertainties as to what physical mechanisms are responsible for establishing cyclic solar activity (reviewed in §4). Yet, it is still fair to say that, to the best of our current understanding, if the Sun were not rotating, it would have a magnetic carpet but not a magnetic cycle.

In the past few decades, helioseismology has revolutionized solar dynamo theory by mapping out the solar internal rotation profile throughout most of the convection zone (figure 3*a*). The solar envelope exhibits prominent rotational shear, with the angular velocity Ω decreasing monotonically by about 30 per cent from the equator to latitudes of 60–70^{°} (higher latitude inversions are unreliable owing to the insensitivity of global acoustic modes to rotation near the rotation axis). This latitudinal shear persists throughout the convection zone, with radial shear largely confined to upper and lower boundary layers. The lower boundary layer is known as the tachocline and represents a transition from latitudinal shear in the convection zone to nearly uniform rotation in the radiative zone. At the top of the convection zone is the near-surface shear layer (NSSL), where the rotation rate increases inward by about 2–3% from the photosphere to *r*∼0.95*R*. In this paper, we focus on the dynamo implications of this differential rotation profile, which will be addressed in §§3*b* and 4. For a discussion of how mean flows are established and maintained, see Miesch & Toomre [32].

It is often assumed that the solar dynamo operates in a kinematic regime such that fields are passively amplified and transported by the flow. Because this assumption helps us to justify the concept of the α-effect (§3*b*), it underlies virtually all dynamo models of the solar cycle. Yet, it is also widely realized that the kinematic assumption cannot be strictly valid for the Sun. In the kinematic limit, the magnetohydrodynamic (MHD) induction equation is linear and dynamo action can be regarded as a linear instability with an exponential growth rate. Because the amplitude of the solar cycle does not appear to be growing exponentially, this implies that there must be some nonlinear feedback between fields and flows that limits it.

Dynamo saturation, such as magnetic self-organization, is a subtle issue that we will address briefly in §3*b*. Here, we merely highlight the torsional oscillations, which represent the most compelling evidence we have that systematic feedbacks between large-scale fields and flows do occur. These are alternating bands of faster and slower rotation that appear and propagate in latitude in conjunction with the solar activity cycle (figure 3*b*). In particular, there is both a low-latitude branch that propagates equatorwards along with the bands of magnetic activity as well as a high-latitude branch that propagates poleward on a comparable time scale.

The torsional oscillations are clearly associated with magnetism but their significance, if any, with regard to the operation or saturation of the dynamo is unclear. With an amplitude of less than 0.5 per cent, they do not appear to be large enough to reflect dynamo saturation through the suppression of rotational shear (although MHD convection simulations and non-kinematic mean-field models do exhibit a reduction in rotational shear relative to non-magnetic models, providing some evidence for dynamo saturation via the so-called Ω-quenching; [35–37]). Current models generally regard them as a passive by-product of cyclic activity, although it is worth noting that they appear each cycle before the emergence of sunspots on the surface (figure 3*c*). The poleward branch is typically attributed to the mean Lorentz force or turbulent diffusion responding to cyclic lower latitude forcing, but the meridional structure of the low-latitude branch suggests that it may be thermally driven, induced by enhanced radiative losses in active bands^{3} [38].

It is clear from stellar observations that convection and rotation play a central role in magnetic activity. Late-type stars with convective envelopes such as the Sun generally exhibit similar signatures of magnetic activity, including chromospheric and coronal emission and photometric variations indicative of spots [17]. Although unresolved, the most active stars also exhibit detectable Zeeman splittings [39]. Inferred field strengths scale roughly as the square of the rotation rate, , although precise estimates for the power-law index range widely from 1.2 to 2.8 [40–42]. The increase in is likely to be due in part to an increase in rotational shear ΔΩ, which also increases with Ω, although precisely how is still being debated. Various studies have suggested power-law relationships such that ΔΩ∝Ω^{n} but estimates for the index *n* range from 0.1 to 0.7 [43–45]. Such scaling uncertainties for and ΔΩ are most probably owing to heterogeneous sampling but they may also reflect intrinsic variability owing to sensitivities in the underlying nonlinear dynamics [42]. Scaling relationships for both and ΔΩ are made somewhat tighter when Ω is replaced by the inverse Rossby number *τ*_{c}Ω (particularly for heterogeneous samples), where *τ*_{c} is a convective turnover time obtained from theoretical models or empirical fitting.

Scaling relationships for both and ΔΩ saturate beyond a threshold value that depends weakly on stellar type [41,42]. For solar-like stars, the threshold is about 10Ω_{⊙}, beyond which becomes insensitive to Ω and ΔΩ appears to be suppressed, presumably by the Lorentz force (thus providing evidence for Ω-quenching). In this saturated regime, the dynamo appears to operate at peak efficiency, with the mean-field strength determined by the stellar luminosity [46].

Many late-type stars certainly exhibit magnetic activity cycles but, despite much literature on the subject, there are few well-established results. Data are scarce, largely because observations spanning multiple years or even decades are needed to verify cycle periods and amplitudes. There is some evidence that cycle periods for solar-type stars with moderate rotation rates tend to decrease somewhat with Ω but there is much scatter and there may be multiple branches signifying distinct dynamo regimes. We refer the interested reader to the reviews by Rempel [47], Saar [42] and Lanza [48]. Still, it is clear that the interplay between convection and rotation lies at the root of solar and stellar magnetic activity, so we turn now to discuss the nature of convective dynamos.

## 3. Convective dynamos

The starting point for any solar dynamo model is the magnetic induction equation under the MHD approximation
3.1where **v** and **B** are the fluid velocity and magnetic field and *η* is the magnetic diffusivity of the plasma. The (global-scale) ratio of the advection term to the diffusive term on the right-hand side (RHS) is the magnetic Reynolds number *R*_{m}=*UL*/*η*, where *U* and *L* are characteristic velocity and length scales, respectively.

### (a) Small-scale dynamos

The best understood class of turbulent dynamos are isotropic, homogeneous, incompressible flows with large magnetic Prandtl numbers *P*_{m}=*ν*/*η*≫1, where *ν* is the kinematic viscosity. Here, the kinetic energy spectrum scales with the spatial wavenumber *k* as *E*(*k*)∝*k*^{−p} within an inertial range that extends approximately from the energy injection scale *k*_{0} to the viscous dissipation scale . Here, *R*_{e}=*UL*/*ν* is the Reynolds number and *q*=2/(*p*+1). For Kolmogorov turbulence, *p*=5/3 and *q*=3/4.

For such a flow, the magnetic field is most efficiently generated on small scales, below the viscous dissipation scale *k*>*k*_{ν}, where the velocity field is spatially smooth but temporally random [49,50]. Quasi-laminar (yet chaotic) stretching generates folded field topologies such that variations in the direction of **B** occur on a length scale comparable to the size of eddies ℓ_{e}, whereas variations perpendicular to **B** occur on a resistive scale [49]. The resulting magnetic energy peaks near the ohmic dissipation scale . From equation (3.1), we anticipate that the kinematic dynamo growth rate should scale with the strength of the shear *ku*_{k}∼*k*^{(3−p)/2}, where . For a general velocity field with *p*<3, this implies that the smallest eddies are most efficient at amplifying field because they have the shortest turnover times. Thus, ℓ_{e}∼ℓ_{ν} and where ℓ_{ν}=2*π*/*k*_{ν}.

This picture breaks down in stars where *P*_{m}<1. Here, the ohmic dissipation scale lies within the inertial range such that ℓ_{r}>ℓ_{ν}. There are no scales that appear spatially smooth from the perspective of the magnetic field. Small eddies still have the fastest turnover times but ohmic diffusion will suppress field amplification on scales smaller than ℓ_{r}. Thus, we might still expect magnetic energy generation to be most efficient near the ohmic dissipation scale ℓ_{r} but now quasi-laminar stretching by larger eddies ℓ>ℓ_{r} must compete against the disruptive effects of turbulent mixing from smaller eddies ℓ<ℓ_{r}. Dynamo action is more difficult to sustain and depends sensitively on the slope of the spectrum *p* near ℓ_{r}; steeper slopes (larger *p*) facilitate field generation [50]. This is most readily demonstrated in numerical simulations where *R*_{m} can be held fixed while *P*_{m} is varied by changing the value of *ν*. As *ν* is decreased below *η* such that *P*_{m} passes through unity, there is a sharp rise in the critical value of *R*_{m} needed to sustain dynamo action^{4} [51,52].

Despite the subtleties at low *P*_{m}, it is clear that isotropic, homogeneous, turbulent dynamos tend to generate intermittent magnetic fields on scales much smaller than the typical scale of the velocity field *L*_{B}≪*L*_{V}. In short, turbulent flows beget turbulent fields. Such dynamos have been referred to as *small-scale dynamos* in order to distinguish them from *large-scale dynamos*, which may generate magnetic fields on scales much larger than the velocity field *L*_{B}≫*L*_{V}.

Small-scale dynamo action has been proposed to account for the intricate flux distribution of the magnetic carpet [21,53,54]. This would then imply that the magnetic energy in the photosphere may well peak on scales that are currently unresolved, possibly extending down to ohmic dissipation scales of order 100 m.

### (b) Large-scale dynamos: from fluctuations to flux

The principal agents of self-organization in turbulent dynamos are helicity and shear. Although the significance of these agents was originally appreciated within the context of the kinematic mean-field theory, they continue to take centre stage as three-dimensional MHD simulations become increasingly important tools at the leading edge of dynamo research. Chaotic stretching is still required to generate magnetic energy but helicity and shear—coupled with the geometry, boundary conditions and other sources of anisotropy and instability such as density stratification and buoyancy—play a dominant role in shaping the large-scale magnetic topology.

The pioneers of mean-field dynamo theory emphasized the significance of the kinetic helicity, defined as *H*_{k}=〈*ω***⋅****v**〉_{V}, where ** ω**=

**∇**×

**v**is the fluid vorticity and 〈 〉

_{V}denotes a volume integral [29–31]. The combined influence of rotation and density stratification induces a helicity density

*ω***⋅**

**v**, which reflects the presence of spiralling flows that break anti-dynamo constraints associated with two dimensionality and mirror symmetry. More recently, the focus has shifted to the magnetic helicity

*H*

_{m}=〈

**A**

**⋅**

**B**〉

_{V}, where

**A**is the vector magnetic potential, defined such that

**B**=

**∇**×

**A**.

Magnetic helicity is produced by means of kinetic helicity through magnetic induction but has its own particular significance, largely because *H*_{m}, unlike *H*_{k}, is an ideal invariant of the MHD equations. However, because **A** is undefined within the addition of an arbitrary scalar gradient **∇***Λ* (the gauge), *H*_{m} is not, in general, uniquely defined for a finite spatial domain. This ambiguity can be resolved by defining a gauge-invariant relative helicity with respect to a reference field **B**_{p} as follows [55]:
3.2where *V* is the volume of the domain in question and **B**_{p}=**∇**×**A**_{p}. The reference field is generally taken to be potential (**∇**×**B**_{p}=0) with a Coulomb gauge (**∇****⋅****A**_{p}=0). Furthermore, it is defined such that the component normal to the surface **S** that encloses *V* is the same as the actual field: on **S**, which specifies **B**_{p} and **A**_{p} uniquely. If no field lines leave the domain, i.e. on **S**, then **A**_{p}=**B**_{p}=0 and *H*_{R}=*H*_{m}.

If we take *S* to be the spherical surface of the Sun, or a single hemisphere, then the evolution equation for *H*_{R}, obtained from equations (3.2) and (3.1), is given by [55]
3.3where *c* is the speed of light, **J**=*c*(4*π*)^{−1}**∇**×**B** is the current density and we have used cgs units throughout. Equation (3.3) is gauge invariant (with respect to **A**). If *η* is constant in *V* then the first term on the RHS is equal to −8*πηc*^{−1}*H*_{c}, where *H*_{c}=〈**J****⋅****B**〉_{V} is the current helicity. Thus, the rate of change of *H*_{R} is determined by ohmic dissipation, ∝*H*_{c}, and the helicity flux through the surface **S**, given by the integrand of the second term on the RHS of (3.3). If the field is entirely contained within *V* ( on **S**) and if the fluid is perfectly conducting (*η*=0), then the magnetic helicity is conserved (*H*_{R}=*H*_{m}, *dH*_{m}/*dt*=0).

To demonstrate the significance of helicity and shear to the generation of large-scale fields, we consider the longitudinally averaged induction equation (3.1)
3.4where is the cylindrical radius and . Throughout this paper, angular brackets 〈 〉 denote averages over longitude (not to be confused with 〈 〉_{V}, which indicates a volume *integral*) and primes indicate fluctuations about the mean, e.g. **v**^{′}=**v**−〈**v**〉. The subscript *m* denotes components in the meridional plane, e.g. .

With regard to the solar activity cycle, the primary challenge of solar dynamo theory is to determine the nature of the turbulent electromotive force (emf) . Correlations between fluctuating fields and flows are generally attributed to convective motions but they may also arise from other phenomena, such as magnetic buoyancy (MB) or magneto-shear instabilities [4,6]. Explicit expressions for can only be obtained for idealized scenarios, the best known being kinematic mean-field dynamo theory in which the field is passively advected by the flow (the Lorentz force is neglected) and it is assumed that characteristic length and time scales of the fluctuations are small relative to the mean. Under these conditions, the turbulent emf is linear in 〈**B**〉 and its components can be expressed (in Cartesian coordinates) as a converging infinite series [4,6]
3.5Under the additional assumptions of quasi-isotropy (antisymmetric under mirror reflections but otherwise isotropic), and homogeneity, the α and *β* tensors reduce to scalar coefficients proportional to the kinetic helicity and the kinetic energy of the fluctuating motions, respectively.

The α-effect is inherently helical. The generation of magnetic energy is associated with a transfer of magnetic helicity from fluctuating to mean fields [56]. In strictly kinematic models, this transfer is implicit (fluctuating fields never appear explicitly), but many mean-field models incorporate Lorentz-force feedbacks through slight adjustments to the basic mean-field paradigm [57]. To lowest order, Lorentz force feedbacks modify the α-effect by contributing an extra term such that , where *H*^{′}_{k} and *H*^{′}_{c} are the kinetic and current helicity associated with the fluctuating motions. A mean-field analogue of equation (3.3), accounting for the conservation of magnetic helicity in the absence of ohmic dissipation, can be used to derive an expression for α_{m}, the result being [32,57,58]
3.6where *B*^{2}_{eq}=4*πρu*^{2} and *R*_{m}=*u*^{2}(3*τη*)^{−1} are, respectively, the equipartition field strength and the magnetic Reynolds number associated with the turbulent flow. In short, Lorentz force feedbacks lead to a marked suppression of the turbulent α-effect for large *R*_{m} that can be attributed to the near conservation of magnetic helicity. The generation of helical mean flows gives rise to a build-up of small-scale helicity of the opposite sense that can efficiently quench the dynamo, unless it is either dissipated or removed from the domain by the helicity flux in equation (3.3).

This conclusion is not restricted to the mean-field theory. In fact, the first derivation of α_{m} was based on an eddy-damped, quasi-normal Markovian (EDQNM) turbulence model that did not adopt the kinematic and scale-separation assumptions required by traditional mean-field theory [59]. Results exhibited a self-similar inverse cascade of magnetic helicity from small to large scales,^{5} which was later demonstrated in idealized numerical simulations of MHD turbulence [60,61]. However, MHD convection simulations have demonstrated that rotation alone is not sufficient to promote the generation of large-scale fields by means of an inverse cascade of magnetic helicity; rather, results are sensitive to boundary conditions, geometry and the strength and orientation of rotational shear (reviewed by Tobias *et al.* [62] and Brandenburg [63]).

Whatever the nature of the initial saturation on a dynamical time scale, the helicity will continue to evolve on a diffusive time scale according to equation (3.3). In a closed system, a steady state can only be reached if the current helicity vanishes **J****⋅****B**=0. A helical mean field can still persist, provided the helicity of the opposite sign prevails at small scales such that 〈**J**〉**⋅**〈**B**〉=−〈**J**^{′}〉**⋅**〈**B**^{′}〉. If **J**∼*k***B**, where *k* is a characteristic wavenumber (*k*_{0} for the mean and *k*^{′} for the fluctuations), then saturation occurs when 〈*B*〉^{2}∼*k*^{′}/*k*_{0}〈(*B*^{′})^{2}〉. In other words, energy in the mean fields will grow slowly until it exceeds small-scale fields by the ratio of length scales. This was first derived and demonstrated in numerical simulations of MHD turbulence by Brandenburg [60].

The first term in equation (3.4) is the well-known Ω-effect whereby mean toroidal field 〈*B*_{ϕ}〉 is generated from mean poloidal field 〈**B**_{m}〉 and amplified by means of rotational shear. The prominence of sunspots and the large amount of flux contained in toroidal relative to poloidal fields suggests that the Ω-effect must play an essential role in cyclic solar activity (§4). Rotational shear can also influence the topology of large-scale fields by inducing a flux of small-scale magnetic helicity along isorotation surfaces that can potentially alleviate dynamical α-quenching [57,64].

Global numerical simulations of convective dynamos verify that rotation promotes and regulates dynamo action by means of helicity and shear. Figure 4 illustrates a simulation of a young solar-type star rotating at five times the solar rate [65]. Enhanced convective Reynolds stresses produce prominent differential rotation, which in turn generate strong toroidal fields (approx. 10 kG) in confined bands with opposite polarity in the Northern and Southern Hemispheres, as inferred from solar observations (figure 1). These toroidal flux structures account for over half of the total magnetic energy of the dynamo and persist amid the intense turbulence of the convection zone. This is in contrast to the simulation at the solar rotation rate where the kinetic energy of the convection exceeds that in the differential rotation and turbulent pumping effectively confines such structures to the base of the convection zone and tachocline [66–68]. Simulations such as this often exhibit quasi-cyclic polarity reversals, as demonstrated in figure 4*d* for a fast rotator. By using sophisticated numerical methods featuring implicit time stepping and minimal numerical dissipation, Ghizaru *et al.* [68] (see [37]) have recently achieved the first global MHD simulation of solar convection to exhibit cyclic polarity reversals on a decadal time scale (figure 5). Understanding the origin of such behaviour requires a closer look at the fundamental physical mechanisms that give rise to magnetic activity cycles in simulations and stars, which we now explore (see §4).

## 4. Origins of cyclic solar activity

Dynamo models of the solar cycle are generally based on the longitudinally averaged induction equation (3.4), with some theoretically or empirically motivated parametrization for the turbulent emf . Nearly all recent models invoke an emf of the form
4.1The first term on the RHS is the source term for the mean poloidal field. The simplest representation is a quasi-isotropic, homogeneous, kinematic α-effect,^{6} as described in §3*b*: . However, models often employ a nonlinear quenching formulation such as in equation (3.6) in order to limit the dynamo amplitude. Furthermore, many current models employ a non-local source term such that the value of near the solar surface is related to the integrated toroidal flux near the base of the convection zone [35,69,70]. This is intended to implicitly capture the mean magnetic induction arising from the buoyant destabilization, rise, emergence and dispersal of the toroidal flux structures that give rise to active regions; namely, the BL mechanism discussed in §2*a*. Although the BL mechanism is phenomenologically distinct from the turbulent α-effect, it is functionally similar in that is linearly proportional to the mean toroidal field 〈*B*_{ϕ}〉, apart from quenching and non-locality.

The second and third terms in equation (4.1) are transport terms and may be related directly to the kinematic expansion in equation (3.5). The ** γ** term arises from the antisymmetric off-diagonal components of the α tensor and reflects magnetic pumping by convective motions. Again, in terms of magnetic induction, this is phenomenologically distinct but functionally equivalent to a mean MC

**v**

_{m}; cf. equation (3.4). There is also a zonal component to

**which can transport poloidal magnetic flux in longitude [37,71]. The turbulent diffusion**

*γ**η*

_{t}is usually assumed to be isotropic, although this is not strictly consistent with a non-zero

**. As with the α-effect, mean-field models are increasingly incorporating implicit Lorentz-force feedbacks by means of**

*γ**η*-quenching [72,73].

In most solar dynamo models, the mean toroidal field is generated by the Ω-effect (see §3*b*) near the base of the convection zone—region III in figure 6*a* (for a notable exception focusing on the NSSL, see [74]). Although one may expect that the strong radial shear in the tachocline is responsible, many recent mean-field models find that the latitudinal shear in the lower convection zone is more efficient at generating the global, equatorially antisymmetric toroidal bands that give rise to sunspots [35,75,76]. By contrast, the principal mechanisms and even the location of the poloidal source term are currently a matter of debate.

Photospheric measurements suggest that the flux emerging in active regions is more than sufficient to account for the magnetic polarity reversal in the polar caps, as suggested by figure 1*a*. The flux in a single active region is of order 10^{22}–10^{23} Mx, with about 10^{24}–10^{25} Mx emerging over the course of an 11 year sunspot cycle [77]. If the average magnetic field strength poleward of ±70^{°} latitude is roughly 10 G, then a polarity reversal would require a flux of order 4×10^{22} Mx. Thus, even if much of the flux emerging in active regions is lost through magnetic reconnection or vertical transport of magnetic loops, the BL mechanism can account for reversals of the surface dipole moment—threading region I in figure 6*a*.

Yet, can the redistribution of active region flux in the photospheric boundary layer really determine the global dipole moment of the entire solar envelope? The flux in photospheric active regions (region I) and the associated magnetic energy is likely to be a small fraction of the total unsigned flux and energy in the remainder of the convective zone (regions II and III), where most of the mass and kinetic energy of the solar envelope reside. Photospheric observations suggest that active regions make up only a small fraction of the magnetic flux and energy in the photosphere, let alone the entire convection zone (§2*b*). Furthermore, cosmogenic isotope measurements suggest that the 22 year magnetic activity cycle persisted throughout the Maunder minimum, a 70 year interval in the seventeenth century when there were very few sunspots [78]. Convection simulations exhibit cyclic magnetic activity without capturing the dynamics of flux emergence and dispersal that underlie the BL mechanism (figures 4 and 5; although aspects of these simulations do not match solar observations). Amplitude and parity issues have also motivated several authors to advocate for additional deep-seated sources of poloidal field in BL dynamo models, including convection and tachocline instabilities [79]. Still, the BL mechanism is undeniably operating and plays a prominent role in most recent solar dynamo models.

Cyclic behaviour in the Sun is intimately linked with propagation; activity bands migrate equatorwards, and residual vertical flux migrates poleward during the course of each cycle, as evident in figure 1*a*. Within the MHD framework, there are essentially three physical mechanisms that can give rise to the propagation of mean fields and can thus establish cyclic variability: (A) a spatial offset between poloidal and toroidal sources (i.e. a dynamo wave), (B) advection by the mean MC, and (C) transport by convection or other non-axisymmetric motions (magnetic pumping, turbulent diffusion, etc.). These are not mutually exclusive; multiple mechanisms can operate simultaneously and some phenomena, such as MB instabilities, can contribute to all three.

The earliest solar dynamo models were of type (A). Cyclic behaviour is attributed to a phase shift between the α-effect and the Ω-effect that admits travelling wave solutions to the linear, kinematic induction equation (3.4) [6,30]. The propagation speed and the growth rate of the eigenmodes are proportional to (α|**∇**Ω|)^{1/2}, with waves travelling along isorotation surfaces. The natural location for such a dynamo wave is the low-latitude tachocline, where the nearly radial orientation of **∇**Ω and the sign of α (∝−*H*_{k}) are favourable for equatorwards propagation. This, together with the need to avoid dynamical α-quenching, motivated interface dynamo models [80,81] whereby the α-effect occurs in the convection zone (region II) while the Ω-effect occurs in the tachocline (region III). Cyclic behaviour still hinges on a dynamo wave but now the cycle period depends also on the coupling mechanism between regions II and III, usually taken to be turbulent diffusion.

Most recent solar dynamo models advocate mechanism (B), whereby an equatorwards circulation of 1–3 m s^{−1} near the base of the convection zone (region III) accounts for the equatorwards migration of active bands as manifested in the solar butterfly diagram and largely determines the period of the solar cycle [35,69,73,75,76,79]. These are known as flux-transport (FT) dynamo models, most of which attribute poloidal field generation to the BL mechanism. Furthermore, many BL–FT dynamo models operate in a so-called advection-dominated regime whereby the coupling between regions I and III is achieved by the MC. Such models generally employ single-celled circulation profiles as illustrated in figure 6*a*, but recent results by Dikpati *et al.* [82] based on modelling and photospheric observations suggest that the presence of a high-latitude counter-cell may account for why cycles are typically shorter (approx. 11 years) than the most recent one (approx. 13 years). An equatorwards circulation of several m s^{−1} at the base of the convection zone is consistent with global convection simulations, but estimates of the efficiency of convective transport (CT) are about an order of magnitude larger than typical circulation speeds, calling into question the advection-dominated regime [67,83,84].

Magnetic pumping or anisotropic magnetic diffusivity, mechanism (C), can in principle induce cyclic activity in mean-field models but in practice it generally operates in conjunction with other processes. For example, BL models can be constructed in which turbulent diffusion or magnetic pumping account for the coupling between regions I and III [85–87]. The challenge here is to account for the relative inefficiency of CT; if coupling were to occur on a turnover time scale (approx. 1 month), cycles would be too short.

The picture emerging from global convection simulations is only now beginning to take shape. For the simulation shown in figure 4, cyclic activity appears to be primarily owing to a nonlinear dynamo wave (A), even though is not well represented by a simple α-effect. Poleward circulations associated with polar slip instabilities, and torsional oscillations are also present and may play a role in polarity reversals. The simulation shown in figure 5 appears to be more amenable to a mean-field treatment and shows evidence for all three mechanisms for cyclic variability: (A) spatial offsets between toroidal and poloidal sources, (B) an equatorwards circulation of approximately 0.2 m s^{−1} near the base of the convection zone, and (C) equatorwards turbulent pumping of a similar amplitude, approximately 0.2 m s^{−1} [37].

## 5. Summary and outlook

Kinematic, axisymmetric mean-field models are still the workhorse of solar dynamo theory, but verifying their underlying assumptions and accounting for the full richness of solar magnetic activity will require more sophisticated three-dimensional MHD simulations. A major challenge for the future will be to link the cyclic global dynamo and the magnetic carpet into a unified theoretical paradigm. Three-dimensional dynamo simulations must also be linked to simulations and observations of flux emergence. Current dynamo models take the strength of the mean toroidal field near the base of the convection zone as a proxy for the sunspot number (figure 6*b*). However, there is no guarantee that this relationship is linear and the details of how it unfolds may have important implications for the operation of the dynamo as well as solar cycle predictability and the space weather implications of emergent flux.

The solar cycle itself still harbours many mysteries. What processes dominate the generation of large-scale poloidal fields as encapsulated in the essential but enigmatic turbulent emf? Can the BL mechanism overwhelm the turbulent α-effect? Might MHD instabilities contribute to mean-field generation? What regulates the cycle period and amplitude? Can a weak but persistent meridional flow prevail over turbulent transport? Does the spectral and spatial flux of magnetic helicity play a crucial role in determining the large-scale magnetic topology and dynamo saturation? Is flux emergence merely a by-product of dynamo action or is it essential for the establishment of cyclic activity? How does the Sun fit within the context of other stars? Inevitably, answers can only come from diligent observation and innovative modelling. Thanks to continuing advances in instrumentation and high-performance computing, the future is bright.

## Acknowledgements

We are grateful to Paul Charbonneau, Guiliana De Toma, David Hathaway, Rachael Howe, Clare Parnell, Étienne Racine, Matthias Rempel and Marty Snow for help in preparing and interpreting figures. We also thank Paul Charbonneau, Matthias Rempel, Janet Luhmann and the two anonymous referees for discussions (referees excluded) and comments on the manuscript. Research support is provided through NASA grants NNH09AK14I and NNX08AI57G and computing resources were provided by NASA HEC (Pleiades) and NSF PACI (NCSA, TACC, PSC and SDSC). The National Center for Atmospheric Research is sponsored by the National Science Foundation.

## Footnotes

One contribution of 11 to a Theme Issue ‘Astrophysical processes on the Sun’.

↵1 Individual active regions do not exhibit latitudinal propagation until after they fragment and disperse.

↵2 The photospheric magnetic field at solar minimum exhibits many small-scale, short-lived features that cover most of the surface and dominate the magnetic energy. Thus, the magnetic topology may actually be

*more*complex at solar minimum relative to solar maximum in the sense of more photospheric null points, separators and separatrix surfaces. Still, the large-scale magnetic structure is relatively simple in terms of the relative prominence of low-order, nearly axisymmetric multi-pole moments.↵3 Sunspots are famously dark relative to their surroundings but their appearance on the solar disc is accompanied by bright, more weakly magnetized regions known as faculae. Facular brightenings generally overwhelm sunspot darkenings; so the Sun is roughly 0.1 per cent brighter at solar maximum [15].

↵4 The value of

*R*_{m}in stars is sufficiently large that supercriticality is ensured, despite the small value of*P*_{m}.↵5 As opposed to the

*forward*cascade of energy from large to small scales.↵6 Strictly speaking, the α

_{k}appearing here is the symmetric part of the full α tensor appearing in equation (3.5). The antisymmetric part reflects transport and is expressed here separately as the magnetic pumping vector, which may be regarded as a pseudo-flow.*γ*

- This journal is © 2012 The Royal Society