1.1 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 $〈B〉$,

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

Note that $〈u\times b〉$ is analguous to the Reynolds stress used in hydrodynamical turbulence studies.

The aim of mean-field theory is to find an expression for $〈u\times b〉$. 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 $〈u\times b〉$ can be written as a functional of $B,\frac{\partial B}{\partial x_i},\frac{{\partial }^{2}B}{\partial x_i\partial x_j}$, . . . where $x_i$ are the coordinates, and $\frac{\partial B}{\partial t},\frac{{\partial }^{2}B}{\partial t^2}$, . . . 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 in$〈B〉$, and assuming that the first few terms suffice to capture the spatial dependence, gives

where ${〈B〉}_j$ is the $j^{th}$ component of $〈B〉$, and $\alpha$ and $\beta$ are pseudo-tensors, which under the assumptions that have been made, will depend upon $〈U〉$, $\eta$ and the statistical properties of $u$, all of which can vary in space.

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

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 $U_{\text{pump}}$ 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 $\alpha$ and $\beta$ 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 $\alpha$ term is dominated by the structure of the turbulent flows. An important consequence of $\alpha$ being a pseudo-tensor is that its symmetric part vanishes if all the statistical properties of$u$ are unchanged under the transformation $r’=-r’$ (i.e., if the turbulence has reflexional symmetry). The simplest breaking of this symmetry occurs if the $〈u\cdot \left(\nabla \times u\right)〉\ne 0$. The integral of the scalar product of the velocity and vorticity over some volume is called kinetic helicity,

and a flow with helicity gives rise to a non-zero symmetric component of $\alpha$. In the special case of homogeneous isotropic turbulence, the $\alpha$ tensor has components ${\alpha }_{ij}={\alpha }_0{\delta }_{ij}$, where ${\delta }_{ij}=1$ if $i=j$ and 0 otherwise, and ${\alpha }_0$ is a pseudo-scalar with a value which can be approximated (after further assumptions) as a weighted integral of the spectral transform of $u\cdot \left(\nabla \times u\right)$.

The$\beta$ tensor has even more terms, and under a number of restrictive assumptions unlikely to apply in the Sun, one can obtain the approximation ${\beta }_{ijk}={\eta }_t{\text{ε}}_{ijk}$ where ${\eta }_t$ is a scalar and ${\epsilon }_{ijk}=1$ 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

where again 〈. . .〉 indicates an averaging over azimuth. Note that, in the special case ${\alpha }_0=0$, Cowling’s theorem rules out dynamo action. Cases with ${\alpha }_0\ne 0$, however, fall outside the scope of Cowling’s theorem, and depending on ${\eta }_t$ (which is usually much larger than $\eta$) 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, $〈U_{\text{pump}}〉$, which adds to $〈U〉$ appearing in Equation (7).

Particularly relevant for the Sun is the case where the toroidal field $〈B_{\varphi }〉$ is generated by the differential rotation acting on the poloidal field $(〈B_{\gamma }〉,〈B_{\theta }〉)$, and the poloidal field is generated by the ${\alpha }_0$ term from the toroidal field. This is the case studied by Parker (1955), where the $\alpha$ 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 $0.95R_{\odot }$, 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 $\alpha$ effect of Parker, indicates that, while differential rotation and the $\alpha$ 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?

1.2 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

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 $\alpha$ 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).

2.1 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 $\alpha$ 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 $\alpha$ effect. Theory and numerical simulations (Yoshimura, 1975a), however, suggest that the $\alpha$ 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 $\alpha$ 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 $\alpha$ effect, or a Babcock–Leighton source term implemented as an $\alpha$ 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 $U_{\text{pump}}$ 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.

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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 $\alpha$ 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 $\alpha$ 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.

2.2 Solar-Stellar Observations

2.2.3 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).

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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.

2.2.4 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).

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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.