Each and every one of the \(T\rightarrow P\) mechanisms described in Sect. 3.2 relies on fundamentally non-axisymmetric physical effects, yet these must be “forced” into axisymmetric dynamo equations for the mean magnetic field. There are a great many different ways of doing so, which explains the wide variety of dynamo models of the solar cycle to be found in the recent literature. The aim of this and the following section is to provide representative examples of various classes of models, to highlight their similarities and differences, and illustrate their successes and failings. In all cases, the model equations are to be understood as describing the evolution of the mean field \(\left\langle {\varvec{B}}\right\rangle \), namely the large-scale, slowly varying, axisymmetric component of the total solar magnetic field. For those wishing to code up their own versions of these (relatively) simple models, Jouve et al. (2008) have set up a suite of benchmark calculations against which numerical dynamo solutions can be validated.
Common model ingredients
All kinematic solar dynamo models have some basic “ingredients” in common, most importantly (i) a solar structural model, (ii) a differential rotation profile, and (iii) a magnetic diffusivity profile (possibly depth-dependent).
Helioseismology has pinned down with great accuracy the internal solar structure, including the exact location of the core–envelope interface (Basu 2016), as well as the internal differential rotation (Howe 2009). Unless noted otherwise, all illustrative models discussed in this section are computed using the following analytic formulae for the angular velocity \(\varOmega (r,\theta )\) and magnetic diffusivity \(\eta (r)\):
$$\begin{aligned} {\varOmega (r,\theta )\over \varOmega _{\rm E}}=\varOmega _{\rm C}+ {\varOmega _{\rm S}(\theta )-\varOmega _{\rm C}\over 2} \left[ 1+{{\,\mathrm{erf}\,}}\left( {r-r_{\rm c}\over w}\right) \right] , \end{aligned}$$
(20)
with
$$\begin{aligned} \varOmega _{\rm S}(\theta )=1-a_2\cos ^2\theta -a_4\cos ^4\theta , \end{aligned}$$
(21)
and
$$\begin{aligned} {\eta (r)\over \eta _{\rm T}}= \varDelta \eta +{1-\varDelta \eta \over 2} \left[ 1+{{\,\mathrm{erf}\,}}\left( {r-r_{\rm c}\over w}\right) \right] . \end{aligned}$$
(22)
With appropriately chosen parameter values, Eq. (20) describes a solar-like differential rotation profile, namely a purely latitudinal differential rotation in the convective envelope, with equatorial acceleration and smoothly matching a core rotating rigidly at the angular speed of the surface mid-latitudes.Footnote 4 This rotational transition takes place across a spherical shear layer of half-thickness w coinciding with the core–envelope interface at \(r_{\rm c}/R_\odot =0.7\) (see Fig. 4b, with parameter values listed in caption). As per Eq. (22), a similar transition takes place with the net diffusivity, falling from some large, “turbulent” value \(\eta _{\rm T}\) in the envelope to a much smaller diffusivity \(\eta _{\rm c}\) in the convection-free radiative core, the diffusivity contrast being given by \(\varDelta \eta =\eta _{\rm c}/\eta _{\rm T}\). Given helioseismic constraints, these represent minimal yet reasonably realistic choices.Footnote 5
Such a solar-like differential rotation profile is quite complex, in that it is characterized by three partially overlapping shear regions: a strong positive radial shear in the equatorial regions of the tachocline, an even stronger negative radial shear in its the polar regions, and a significant latitudinal shear throughout the convective envelope and extending partway into the tachocline. For a tachocline of half-thickness \(w/R_\odot =0.05\), the mid-latitude latitudinal shear at \(r/R_\odot =0.7\) is comparable in magnitude to the equatorial radial shear; its potential contribution to dynamo action should not be casually dismissed.
\(\alpha \varOmega \) mean-field models
Calculating the \(\alpha \)-effect and turbulent diffusivity
Mean-field electrodynamics is a subject well worth its own full-length review, so the foregoing discussion will be limited to the bare essentials. Detailed discussion of the topic can be found in Krause and Rädler (1980), Moffatt (1978), Rüdiger and Hollerbach (2004), chapter 3 in Schrijver and Siscoe (2009), and in the recent review articles by Ossendrijver (2003) and Hoyng (2003).
The task at hand is to calculate the components of the \({\varvec{\alpha }}\) and \({\varvec{\beta }}\) tensor in terms of the statistical properties of the underlying turbulence. A particularly simple case is that of homogeneous, weakly anisotropic turbulence, which reduces the \({\varvec{\alpha }}\) and \({\varvec{\beta }}\) tensor to simple scalars, so that the mean electromotive force becomes
$$\begin{aligned} \varvec{\mathcal {E}}=\alpha \left\langle {\varvec{B}}\right\rangle -\beta \nabla \times \left\langle {\varvec{B}}\right\rangle . \end{aligned}$$
(23)
This is the form commonly used in solar dynamo modelling, even though turbulence in the solar interior is most likely inhomogeneous and anisotropic. There are three (kinematic) regimes in which simple closed form expressions for \(\alpha \) and \(\beta \) can be obtained in terms of the small-scale flow \({\varvec{u}}^\prime \), all ultimately amounting to the large-scale field \(\left\langle {\varvec{B}}\right\rangle \) suffering little deformation by the turbulent flow \({\varvec{u}}^\prime \):
-
1.
weak turbulent magnetic fields, in the sense \(|{\varvec{B}}^\prime | \ll |\left\langle {\varvec{B}}\right\rangle |\),
-
2.
low (\(<1\)) magnetic Reynolds number \(\mathrm {Rm}=v\ell /\eta \),
-
3.
short coherence time turbulence, in the sense that the lifetime of turbulent eddies \(\tau _{\rm c}\) is smaller than their turnover time \(\ell /v\), i.e., the Strouhal Number \(\mathrm {St}=\tau _c v/\ell <1\).
With mixing length theory of convection suggesting \(v\sim 10^4 \,{\text{cm s}}^{-1}\) and \(\ell \sim 10^9 \,\mathrm {cm}\) as characteristic velocities and length scales for the dominant turbulent eddies, and \(\eta \sim 10^4 \,\mathrm {cm}^2\,\mathrm {s}^{-1}\), one finds \(\mathrm {Rm}=v\ell /\eta \sim 10^9\); mixing length convection also implicitly assumes \(\mathrm {St}\simeq 1\), and high-\(\mathrm {Rm}\) MHD turbulence simulations suggest that \(|{\varvec{B}}^\prime | \gg |\left\langle {\varvec{B}}\right\rangle |\) if a mean-field is present at all. Equation (23) should be dubious already. Nonetheless, if either of the three conditions above is satisfied, it can be shown that in the kinematic regime (i.e., \(\alpha \) and \(\beta \) are not affected by either \(\left\langle {\varvec{B}}\right\rangle \) or \({\varvec{B}}^\prime \)):
$$ \alpha\sim - {\tau _{\rm c}\over 3} \left\langle {\varvec{u}}^\prime \cdot \nabla \times {\varvec{u}}^\prime \right\rangle , $$
(24)
$$ \beta\sim {\tau _{\rm c}\over 3} \left\langle ({\varvec{u}}^\prime )^2\right\rangle . $$
(25)
Order-of-magnitude estimates of the scalar coefficients yield \(\alpha \sim \varOmega \ell \) and \(\beta \sim v\ell \), where \(\varOmega \) is the solar angular velocity. At the base of the solar convection zone, one then finds \(\alpha \sim 10^3 \,{\text{cm s}}^{-1}\) and \(\beta \sim 10^{12} \,\mathrm {cm}^2\,\mathrm {s}^{-1}\), these being understood as very rough estimates. Because the kinetic helicity may well change sign along the longitudinal (averaging) direction, thus leading to cancellation, the resulting value of \(\alpha \) may be much smaller than its r.m.s. deviation about the longitudinal mean. In contrast the quantity being averaged on the right hand side of Eq. (25) is positive definite, so one would expect a more “stable” mean value (see Hoyng 1993; Ossendrijver et al. 2001, for further discussion). Equations (24)–(25) certainly indicate that one cannot have an \(\alpha \)-effect without turbulent diffusivity being also present, but that the converse is possible, e.g. for non-helical flows. At any rate, difficulties in computing \(\alpha \) and \(\beta \) from first principle (whether as scalars or tensors) have led to these quantities often being treated as free parameters of mean-field dynamo models, to be adjusted (within reasonable bounds) to yield the best possible fit to observed solar cycle characteristics, most importantly the cycle period. One finds in the literature numerical values in the approximate ranges \(10{-}10^3 \,{\text{cm s}}^{-1}\) for \(\alpha \) and \(10^{10}{-}10^{13} \,\mathrm {cm}^2\,\mathrm {s}^{-1}\) for \(\beta \).
The cyclonic character of the \(\alpha \)-effect also indicates that it is equatorially antisymmetric and positive in the Northern solar hemisphere, except perhaps at the base of the convective envelope, where the horizontal divergence of downflows can lead to a sign change. These expectations have been confirmed in a general sense by theory and numerical simulations (see, e.g., Rüdiger and Kitchatinov 1993; Brandenburg et al. 1990; Ossendrijver et al. 2001; Käpylä et al. 2006a, also Sect. 6 herein).
In cases where the turbulence is more strongly inhomogeneous, an additional effect comes into play: turbulent pumping. Mathematically it is associated with the antisymmetric part to the \(\alpha \)-tensor in Eq. (19), whose three independent components can be recast as a velocity-like vector field \({\varvec{\gamma }}\) that acts as an additional (and non-solenoidal) contribution to the mean flow:
$$\begin{aligned} {\varvec{\mathcal{E}}}= \alpha \left\langle {\varvec{B}}\right\rangle +{\varvec{\gamma }}\times \left\langle {\varvec{B}}\right\rangle +\beta \nabla \times \left\langle {\varvec{B}}\right\rangle . \end{aligned}$$
(26)
with
$$\begin{aligned} {\varvec{\gamma }}\sim -{1\over 6}\tau _c\nabla \left\langle ({\varvec{u}}^\prime )^2\right\rangle , \end{aligned}$$
(27)
in the same kinematic physical regimes in which Eqs. (24)–(25) hold.
Algebraic \(\alpha \)-quenching
Assuming the dynamo-generated magnetic field grows in time, magnetic tension will increasingly resist deformation by the small-scale turbulent fluid motions. Something is bound to happen when the growing dynamo-generated mean magnetic field reaches a magnitude such that its energy per unit volume is comparable to the kinetic energy of the underlying turbulent fluid motions:
$$\begin{aligned} {\left\langle {\varvec{B}}\right\rangle ^2\over 8\pi }={1\over 2}\rho ({\varvec{u}}^\prime )^2 . \end{aligned}$$
(28)
Denoting the corresponding equipartition field strength by \(B_{\rm eq}\), one often introduces an ad hoc nonlinear dependency of \(\alpha \) directly on the mean-field \(\left\langle {\varvec{B}}\right\rangle \) by writing:
$$\begin{aligned} \alpha \rightarrow \alpha (\left\langle {\varvec{B}}\right\rangle )={\alpha _0\over 1+(\left\langle {\varvec{B}}\right\rangle /B_{\rm eq})^2} . \end{aligned}$$
(29)
This expression “does the right thing”, in that \(\alpha \rightarrow 0\) as \(\left\langle {\varvec{B}}\right\rangle \) starts to exceed \(B_{\rm eq}\). It remains an extreme oversimplification of the complex interaction between flow and field that characterizes MHD turbulence,Footnote 6 but its wide usage in solar dynamo modeling makes it a nonlinearity of choice for the illustrative purpose of this section.
Dynamical \(\alpha \)-quenching
The nonlinear feedback of the small-scale magnetic field \({\varvec{B}}^\prime \) on small-scale cyclonic turbulence can be also understood in terms of magnetic helicity conservation. Magnetic helicity (\(\mathcal{H}\)) is a topological measure of linkage between magnetic flux systems linking a volume of fluid (Berger 1999). It is mathematically defined as
$$\begin{aligned} \mathcal{H}_B= \int _V {\varvec{A}}\cdot {\varvec{B}}\mathrm{d}V, \end{aligned}$$
(30)
where \({\varvec{B}}=\nabla \times {\varvec{A}}\). In a closed system, i.e. without helicity flux through its boundaries, magnetic helicity can be shown to evolve according to:
$$\begin{aligned} {\mathrm{d}\over \mathrm{d}{t}}\int {\varvec{A}}\cdot {\varvec{ B}}\mathrm{d}V = -{8\pi \eta \over c}\int {\varvec{J}}\cdot {\varvec{B}}\mathrm{d}V . \end{aligned}$$
(31)
In the ideal limit \(\eta \rightarrow 0\), which is the relevant limit for dynamo action in the interior of the sun and stars, the RHS vanishes and Eq. (31) then indicates that total helicity must be conserved, or at best vary on the (long) diffusive timescale. Conservation of magnetic helicity thus puts a strong constraint on the high-\(\mathrm {Rm}\) amplification of any magnetic field that carries a net helicity, which is certainly the case with the large-scale solar magnetic field.
Following the scale separation logic introduced in Sect. 3.2.1, and because both the current density \({\varvec{J}}\) and vector potential \({\varvec{A}}\) are linearly related to \({\varvec{B}}\), the total vector potential and electric current density can be written as \({\varvec{A}}=\left\langle {\varvec{A}}\right\rangle +{\varvec{A}}^\prime \) and \({\varvec{J}}=\left\langle {\varvec{J}}\right\rangle +{\varvec{J}}^\prime \), with again \(\left\langle {\varvec{A}}^\prime \right\rangle =0\) and \(\left\langle {\varvec{J}}^\prime \right\rangle =0\). Substituting into Eq. (31) and averaging leads to an evolution equation for the mean helicity of the large-scale field:
$$\begin{aligned} {\mathrm{d}\over \mathrm{d}{t}}\int \left\langle {\varvec{A}}\right\rangle \cdot \left\langle {\varvec{B}}\right\rangle \mathrm{d}V = +2\int \varvec{\mathcal {E}}\cdot \left\langle {\varvec{B}}\right\rangle \mathrm{d}V -{8\pi \eta \over c}\int \left\langle {\varvec{J}}\right\rangle \cdot \left\langle {\varvec{B}}\right\rangle \mathrm{d}V , \end{aligned}$$
(32)
where \(\varvec{\mathcal {E}}=\left\langle {\varvec{u}}^\prime \times {\varvec{B}}^\prime \right\rangle \) is the usual turbulent emf (see, e.g., Sect. 3.4.7 in Schrijver and Siscoe 2009). Subtracting Eq. (32) from the unaveraged form of (31) yields a companion equation for the evolution of small-scale magnetic helicity:
$$\begin{aligned} {\mathrm{d}\over \mathrm{d}{t}}\int \left\langle {\varvec{A}}^\prime \cdot {\varvec{B}}^\prime \right\rangle \mathrm{d}V = -2\int \varvec{\mathcal {E}}\cdot \left\langle {\varvec{B}}\right\rangle \mathrm{d}V -{8\pi \eta \over c}\int \left\langle {\varvec{J}}^\prime \cdot {\varvec{B}}^\prime \right\rangle \mathrm{d}V . \end{aligned}$$
(33)
Because the first terms on the RHS of Eqs. (32) and (33) are identical but for their sign, the total helicity given by the sum of Eqs. (32) and (33) is still conserved in the ideal limit \(\eta \rightarrow 0\). But these expressions also indicate that the turbulent emf leads to the buildup of helicity of opposite signs at large and small spatial scales. This corresponds to a dual helicity cascade away from the scale at which the emf is operating (Brandenburg 2001). Buildup of a helical large-scale magnetic field is only possible in the \(\mathrm {Rm}\rightarrow \infty \) regime because an equal amount of oppositely-signed magnetic helicity is cascading down to dissipative scales. In this way \(\left\langle {\varvec{B}}\right\rangle \) can be amplified by the turbulent electromotive force \(\varvec{\mathcal {E}}\), with its growth rate ultimately determined by the rate at which helicity can be transported and dissipated at small scales, or evacuated from the region where dynamo action is taking place (Pipin et al. 2013; Blackman 2015).
Following Pouquet et al. (1976), the total (isotropic) \(\alpha \)-effect is often written as the sum of a two contributions, proportional respectively to the kinetic and magnetic (current) helicities:
$$\begin{aligned} \alpha =\alpha _K+\alpha _M = - {\tau _{\rm c}\over 3}\left( \left\langle {\varvec{u}}^\prime \cdot \nabla \times {\varvec{u}}^\prime \right\rangle - {1\over \rho } \left\langle {\varvec{B}}^\prime \cdot \nabla \times {\varvec{B}}^\prime \right\rangle \right) . \end{aligned}$$
(34)
A key finding of Pouquet et al. (1976) is that these two contributions have opposite signs, i.e, the magnetic helicity contribution to the total \(\alpha \)-effect opposes that of kinetic helicity. This forms the basis of the various dynamical \(\alpha \)-quenching formulations that have been proposed in the literature (e.g., Kleeorin et al. 1995; Blackman and Brandenburg 2002, and references therein). For example, Brandenburg et al. (2009) take \(\alpha _K\) to be temporally steady and given by Eq. (24), and the evolution of the magnetic contribution to be described by:
$$\begin{aligned} {\partial {\alpha _M}\over \partial {t}}=-2\eta k_f^2 \left( {\varvec{\mathcal {E}}\cdot \left\langle {\varvec{B}}\right\rangle \over B_{\rm eq}^2}+{\alpha _M\over \mathrm {Rm}}\right) , \end{aligned}$$
(35)
in the absence of helicity fluxes in or out of the dynamo region. The quantity \(k_f\) is a scale factor relating current to magnetic helicity. Stable cycles amplitudes can be obtained by quenching the \(\alpha \)-effect in this manner (see also Schmalz and Stix 1991; Chatterjee et al. 2011; Pipin et al. 2012). Indeed, the quenching can even become “catastrophic”, in the sense that it sets in long before the mean-field reaches significant strength (see Brandenburg and Subramanian 2005).
An interesting situation can arise if the growth of \(\alpha _M\) is such that \(|\alpha _M|>|\alpha _K|\) over a substantial fraction of the magnetic cycle. The resulting sign change in the total \(\alpha \)-effect can then lead to a reversal in the direction of dynamo wave propagation (viz. Sect. 4.2.9 below). The effect has been observed in the mean-field model of Chatterjee et al. (2011), and may also be at play in some of the MHD simulations discussed in Sect. 6 further below.
Diffusivity quenching
The same small-scale magnetic field that quenches the \(\alpha \)-effect can in principle also reduce the turbulent diffusivity \(\beta \) (Sect. 4.2.1). This effect has been included in some mean-field and mean-field-like solar cycle models, sometimes via a simple algebraic parametrization similar to Eq. (29) (e.g., Tobias 1996; Guerrero et al. 2009), sometimes in a more elaborate manner through specific turbulence models (e.g., Rüdiger et al. 1994; Rüdiger and Arlt 1996), and sometimes through a dynamical equation for \(\beta \) in the spirit of dynamical \(\alpha \)-quenching (e.g., Muñoz-Jaramillo et al. 2011). The nature and magnitude of the consequent impact on cyclic amplitude and period are highly model-dependent. A noteworthy effect of magnetic diffusivity quenching is the possibility to produce super-equipartition magnetic fields in the tachocline (Tobias 1996; Gilman and Rempel 2005). On the other hand, the stability analyses of Arlt et al. (2007a, b) suggests that there exist a lower limit to the magnetic diffusivity, below which equipartition-strength toroidal magnetic field beneath the core–envelope interface become unstable.
Backreaction on large-scale flows
The backreaction of the growing magnetic field on the large-scale flows contributing to induction and transport can also quench the growth of the dynamo. In the context of solar cycle models, one could expect the Lorentz force to reduce the amplitude of differential rotation, gradually decreasing its inductive effect until the magnetic field amplitude stabilizes, as it does under \(\alpha \)-quenching. In the mean-field literature it has become costumary to distinguish two classes of (related) amplitude-limiting mechanisms:
-
The Malkus–Proctor effect (after the groudbreaking numerical investigations of Malkus and Proctor 1975): this is the Lorentz force associated with the mean magnetic field directly affecting the large-scale flow \(\left\langle {\varvec{u}}\right\rangle \).
-
\(\varLambda \)-quenching (e.g., Kitchatinov and Rüdiger 1993; Kitchatinov et al. 1994): this is the Lorentz force impacting small-scale turbulence and the associated Reynolds stresses powering large-scale flows.
An efficient approach to model the Malkus–Proctor effect consists in simply dividing the large-scale flow into two components, the first (\({\varvec{U}}\)) corresponding to some prescribed, steady profile, and the second (\({\varvec{U}}^\prime \)) to a time-dependent flow field driven by the Lorentz force (see, e.g., Tobias 1997; Beer et al. 1998; Moss and Brooke 2000; Thelen 2000b; Covas et al. 2001; Brooke et al. 2002; Bushby 2006; Simard and Charbonneau 2020):
$$\begin{aligned} {\varvec{u}}={\varvec{U}}({\varvec{x}})+{\varvec{U}}^\prime ({\varvec{x}},t,\left\langle {\varvec{B}}\right\rangle ), \end{aligned}$$
(36)
with the (non-dimensional) governing equation for \({\varvec{U}}^\prime \) including only the Lorentz force and a viscous dissipation term on its right hand side:
$$\begin{aligned} {\partial {{\varvec{U}}^\prime }\over \partial {t}}= {\varLambda \over 4\pi \rho }(\nabla \times \left\langle {\varvec{B}}\right\rangle )\times \left\langle {\varvec{B}}\right\rangle + \mathrm {Pm}\nabla ^2{\varvec{U}}, \end{aligned}$$
(37)
where time has been scaled according to the magnetic diffusion time \(\tau =R_\odot ^2/\eta _{\rm T}\). Two dimensionless parameters appear in Eq. (37). The first (\(\varLambda \)) is a numerical parameter setting the absolute scale of the magnetic field, and can be set to unity without loss of generality (cf. Tobias 1997; Phillips et al. 2002). The second, \(\mathrm {Pm}=\nu /\eta \), is the magnetic Prandtl number. It measures the relative importance of viscous and Ohmic dissipation. An additional, long timescale is thus introduced in the system, associated with the evolution of the magnetically-driven flow; the smaller \(\mathrm {Pm}\), the longer that timescale.
Incorporating \(\varLambda \)-quenching in mean-field or mean-field-like dynamo models requires a turbulence model allowing to calculate Reynolds stresses and their quenching by the magnetic field. Various such prescriptions have been developed (see Kitchatinov et al. 1994), and, upon being inserted in dynamo models, can lead to stable magnetic cycles (Küker et al. 1996; Rempel 2006a).
Nonlinear magnetic backreaction, whether through \(\varLambda \) quenching or the Malkus–Proctor effect, can lead to strong modulation of the cycle amplitude and large-scale flow unfolding on timescales much longer than the primary cycle if the Prandlt number is significantly smaller than unity (see Brooke et al. 1998; Küker et al. 1999; Pipin 1999; Rempel 2006a); more on this in Sect. 7.2.3 further below.
Flux loss through magnetic buoyancy
Another amplitude-limiting mechanism is the loss of magnetic flux through magnetic buoyancy. Magnetic fields concentrations are buoyantly unstable in the convective envelope, and so should rise to the surface on time scales much shorter than the cycle period (see, e.g., Parker 1975; Schüssler 1977; Moreno-Insertis 1983, 1986). This is often incorporated on the right-hand-side of the dynamo equations by the introduction of an ad hoc loss term of the general form \(-f(\left\langle {\varvec{B}}\right\rangle )\left\langle {\varvec{B}}\right\rangle \); the function f measures the rate of flux loss, and is often chosen proportional to the magnetic pressure \(\left\langle {\varvec{B}}\right\rangle ^2\), thus yielding a cubic damping nonlinearity in the mean-field.
The degree to which flux emergence actually depletes the internal toroidal flux is not trivial to estimate quantitatively, as it hinges critically on the longitudinal extend of the buoyantly destabilized loop and on the manner in which the emerging flux disconnects from the underlying axisymmetric toroidal magnetic flux system; see Sect. 2.3 in Miesch and Teweldebirhan (2016) for an insightful discussion of this issue. In addition to regulating cycle amplitude in dynamo models, (see, e.g., Schmitt and Schüssler 1989; Moss et al. 1990), magnetic flux loss can also have a large impact on the cycle period (Kitchatinov et al. 2000).
The \(\alpha \varOmega \) dynamo equations
Adding the mean-electromotive force given by Eq. (23) to the MHD induction equation leads to the following form for the axisymmetric mean-field dynamo equations:
$$ {\partial {\left\langle A\right\rangle }\over \partial {t}}= \underbrace{(\eta +\beta ) \left( \nabla ^2-{1\over \varpi ^2}\right) \left\langle A\right\rangle }_{{\rm turbulent}\,{\rm diffusion}}- {{\varvec{u}}_{\rm p}\over \varpi }\cdot \nabla (\varpi \left\langle A\right\rangle )+ {\underbrace{\alpha \left\langle B\right\rangle }_{{\rm MFE}\,{\rm source}}}, $$
(38)
$$\begin{aligned} {\partial {\left\langle B\right\rangle }\over \partial {t}}& {} = \underbrace{(\eta +\beta ) \left( \nabla ^2-{1\over \varpi ^2}\right) \left\langle B\right\rangle + {1\over \varpi }{\partial {\varpi \left\langle B\right\rangle }\over \partial {r}} {\partial {(\eta +\beta )}\over \partial {r}}}_{\mathrm {turbulent}\,{\rm diffusion}}- \varpi {\varvec{u}}_{\rm p}\cdot \nabla \left( {\left\langle B\right\rangle \over \varpi }\right) - \left\langle B\right\rangle \nabla \cdot {\varvec{u}}_{\rm p} \nonumber \\&\quad +\underbrace{\varpi (\nabla \times (\left\langle A\right\rangle \hat{{\varvec{e}}}_{\phi }))\cdot \nabla \varOmega }_{\rm shearing}+ \underbrace{\nabla \times [\alpha \nabla \times (\left\langle A\right\rangle \hat{{\varvec{e}}}_{\phi })]}_{{\rm MFE}\,{\rm source}}, \end{aligned}$$
(39)
[compare to Eqs. (8)–(9)]. There are now source terms on both right hand sides, so that dynamo action becomes possible at least in principle. For solar-like convective turbulence one expects \(\beta \gg \eta \), and in what follows the total magnetic diffusivity is denoted \(\eta _{\rm T}=\eta +\beta \) (\(\simeq \beta \) in the turbulent fluid layers). The relative importance of the \(\alpha \)-effect and shearing terms in Eq. (39) is measured by the ratio of the two dimensionless dynamo numbers
$$\begin{aligned} C_\alpha ={\alpha _0 R_\odot \over \eta _0}, \quad C_\varOmega ={(\varDelta \varOmega )_0R_\odot ^2\over \eta _0}, \end{aligned}$$
(40)
where in the spirit of dimensional analysis, \(\alpha _0\), \(\eta _0\), and \((\varDelta \varOmega )_0\) are “typical” values for the \(\alpha \)-effect, turbulent diffusivity, and angular velocity contrast. These quantities arise naturally in the non-dimensional formulation of the mean-field dynamo equations, when time is expressed in units of the magnetic diffusion time \(\tau \) based on the envelope (turbulent) diffusivity:
$$\begin{aligned} \tau ={R_\odot ^2\over \eta _0}. \end{aligned}$$
(41)
In the solar case, it is usually estimated that \(C_\alpha \ll C_\varOmega \), so that the \(\alpha \)-term is neglected in Eq. (39); this results in the class of dynamo models known as \(\alpha \varOmega \) dynamos, which will be the only ones discussed in the remainder of this section. Models retaining both \(\alpha \)-terms are dubbed \(\alpha ^2\varOmega \) dynamos, and may be relevant to the solar case even in the \(C_\alpha \ll C_\varOmega \) regime, in particular if the latter operates in a very thin layer, e.g. the tachocline (see, e.g., DeLuca and Gilman 1988; Gilman et al. 1989; Choudhuri 1990).Footnote 7
Eigenvalue problems and initial value problems
With the large-scale flows, turbulent diffusivity and \(\alpha \)-effect considered given, Eqs. (38, 39) become truly linear in A and B. It becomes possible to seek eigensolutions in the form
$$\begin{aligned} \left\langle A\right\rangle (r,\theta ,t)= a(r,\theta )\exp (s t), \quad \left\langle B\right\rangle (r,\theta ,t)= b(r,\theta )\exp (s t), \end{aligned}$$
(42)
with \(s=\sigma +i\omega \). Substitution of these expressions into Eqs. (38, 39) yields an eigenvalue problem for s and associated eigenfunction \(\{a,b\}\). The real part \(\sigma \) of the eigenvalue is then a growth rate, and the imaginary part \(\omega \) an oscillation frequency. One typically finds that \(\sigma <0\) until the total dynano number
$$\begin{aligned} D=C_\alpha \times C_\varOmega , \end{aligned}$$
(43)
exceeds a critical value \(D_{\rm crit}\) beyond which \(\sigma >0\), corresponding to a growing solutions. Such solutions are said to be supercritical, while the solution with \(\sigma =0\) is critical. A dynamo solution is considered weakly supercritical if its dynamo number only slightly exceeds \(D_{\rm crit}\); cyclic solution exhibiting polarity reversals require \(\omega \not =0\). In the weakly supercritical regime such cyclic solutions typically have \(\sigma \ll \omega \), while \(\sigma \gg \omega \) in the strongly supercritical regime.
With any amplitude-limiting nonlinearity included, the dynamo equations are usually solved as an initial-value problem, with some arbitrary low-amplitude seed field used as initial condition. Equations (38, 39) are then integrated forward in time using some appropriate time-stepping scheme. A useful quantity to monitor in order to ascertain saturation is the magnetic energy within the computational domain:
$$\begin{aligned} \mathcal{E}_B={1\over 8\pi }\int _V \left\langle {\varvec{B}}\right\rangle ^2 \, \mathrm {d}V. \end{aligned}$$
(44)
Figure 5 shows time series of this quantity in a sequence of \(\alpha \)-quenched kinematic \(\alpha \varOmega \) mean-field dynamo solutions. The four solutions have increasing values for the dynamo number D, and all start from the same initial condition of very weak magnetic field.
The linear phase of exponential growth (gray lines), at rates increasing with D, is followed by saturation at an energy level also increasing with D; these are behaviors typical of \(\alpha \)-quenched mean-field and mean-field-like dynamo models operating not too far in the supercritical regime. Here \(\alpha \)-quenching has the desired effect, namely stabilizing the cycle amplitude at field strengths corresponding to a significant fraction of the equipartition value \(B_{\rm eq}\) introduced in the quenching parametrization (29). Dynamo models achieving amplitude saturation through backreaction on large-scale flows (viz. Sect. 4.2.5) behave similarly, provided the magnetic Prandtl number is not much smaller than unity.
Dynamo waves and cycle period
One of the most remarkable property of the (linear) \(\alpha \varOmega \) dynamo equations is that they support travelling wave solutions. This was first demonstrated in Cartesian geometry by Parker (1955), who proposed that a latitudinally-travelling “dynamo wave” was at the origin of the observed equatorward drift of sunspot emergences in the course of the cycle. This finding was subsequently shown to hold in spherical geometry, as well as for non-linear models (Yoshimura 1975; Stix 1976). Dynamo wavesFootnote 8 travel in a direction \({\varvec{s}}\) given by
$$\begin{aligned} {\varvec{s}}=\alpha \nabla \varOmega \times \hat{{\varvec{e}}}_{\phi }, \end{aligned}$$
(45)
a result now known as the “Parker–Yoshimura sign rule”. Dynamo waves also materialize in \(\alpha ^2\varOmega \) mean-field dynamos (Choudhuri 1990), as long as the ratio \(C_\alpha /C_\varOmega \) is not too high (see, e.g., Charbonneau and MacGregor 2001).
Recalling the rather complex form of the helioseismically inferred solar internal differential rotation (cf. Fig. 4b), even an \(\alpha \)-effect of uniform sign in each hemisphere can produce complex migratory patterns, as will be apparent in the illustrative \(\alpha \varOmega \) dynamo solutions to be discussed presently. If the seat of the dynamo is to be identified with the low-latitude portion of the tachocline, and if the (positive) radial shear therein dominates over the latitudinal shear, then equatorward migration of dynamo waves will require a negative \(\alpha \)-effect in the low latitudes of the Northern solar hemisphere.
In linear \(\alpha \varOmega \) mean-field models without a significant meridional flow, the cycle frequency increases with the total dynamo number D (viz. Eq. 43). In nonlinearly saturated models, the cycle frequency shows reduced sensitivity to D and becomes equal to some approximately fixed fraction of the magnetic diffusion time (41). The primary determinant of the (dimensional) period then becomes the adopted value for the turbulent diffusivity. Although model dependent to some extent, decadal periods typically require a few \(10^{11}\) to \(10^{12}\,\hbox {cm}^2\,\hbox {s}^{-1}\), roughly consistent with estimates from mixing length models of convective energy transport; values lower by a factors of \(\sim 10\) are required for dynamos contained in radially thin layers, because the smaller radial length scale enhances dissipation. Similarly low values are also possible (and in fact expected) in the upper tachocline, where residual turbulent diffusivity presumably results from convective overshoot. The ratio of poloidal-to-toroidal field strength, in turn, is found to scale as some power (usually close to 1/2) of the ratio \(C_\alpha /C_\varOmega \), at a fixed value of the product \(C_\alpha \times C_\varOmega \).
Representative results
We first consider \(\alpha \varOmega \) models without meridional circulation [\({\varvec{u}}_{\rm p}=0\) in Eqs. (38, 39)], with the \(\alpha \)-term omitted in Eq. (39), and using the magnetic diffusivity and angular velocity profiles of Fig. 4. We investigate the behavior of \(\alpha \varOmega \) models, with the \(\alpha \)-effect concentrated just above the core–envelope interface (green line on Fig. 4a). We also consider two latitudinal dependencies, namely \(\alpha \propto \cos \theta \), which is the “minimal” possible latitudinal dependency compatible with the required equatorial antisymmetry of the Coriolis force, and an \(\alpha \)-effect concentrated towards the equatorFootnote 9 via an assumed latitudinal dependency \(\alpha \propto \sin ^2\theta \cos \theta \). Unless otherwise noted all models have \(C_\varOmega =25{,}000\), \(|C_\alpha |=10\), \(\eta _{\rm T}/\eta _{\rm c}=10\), and \(\eta _{\rm T}=5\times 10^{11} \,\mathrm {cm}^2\,\mathrm {s}^{-1}\), which leads to \(\tau \simeq 300\,{\text {years}}\). To facilitate comparison between solutions, here antisymmetric parity is imposed via the boundary condition at the equator (via Eq. 11). Algebraic \(\alpha \)-quenching, in the form of Eq. (29), is chosen as the amplitude-limiting nonlinearity.
Figures 6 and 7 show a selection of such dynamo solutions, in the form of animations in meridional planes and time–latitude diagrams of the toroidal field extracted at the core–envelope interface, here \(r_{\rm c}/R_\odot =0.7\). If sunspot-producing toroidal flux ropes form in regions of peak toroidal field strength, and if those ropes rise radially to the surface, then such diagrams are directly comparable to the sunspot butterfly diagram of Fig. 2.
Examination of these animations reveals that the dynamo is concentrated in the vicinity of the core–envelope interface, where the adopted radial profile for the \(\alpha \)-effect is maximal (cf. Fig. 4a). In conjunction with a fairly thin tachocline, the radial shear therein then dominates the induction of the toroidal magnetic component. With an eye on Fig. 4b, notice also how the dynamo waves propagates along isocontours of angular velocity, in agreement with the Parker–Yoshimura sign rule (cf. Sect. 4.2.9). Note that even for an equatorially-concentrated \(\alpha \)-effect (Panels b and c), a strong polar branch is nonetheless apparent in the butterfly diagrams, a direct consequence of the stronger radial shear present at high latitudes in the tachocline (see also corresponding animations). Models using an \(\alpha \)-effect operating throughout the whole convective envelope, on the other hand, would feed primarily on the latitudinal shear therein, so that for positive \(C_\alpha \) the dynamo mode would propagate radially upward in the envelope (see Lerche and Parker 1972).
It is noteworthy that co-existing dynamo branches, as in Panel b of Fig. 7, can have distinct dynamo periods (on this see also Belvedere et al. 2000), which in nonlinearly saturated solutions leads to long-term amplitude modulation. This is typically not expected in dynamo models where the only nonlinearity present is a simple algebraic quenching formula such as Eq. (29). This does not occur for the \(C_\alpha <0\) solution, where both branches propagate away from each other, but share a common latitude of origin and so are phased-locked at the onset (cf. Panel c of Fig. 7).
The models discussed above are based on rather minimalistics and partly ad hoc assumptions on the form of the \(\alpha \)-effect. More elaborate models have been proposed, relying on calculations of the full \(\alpha \)-tensor based on an underlying turbulence model (see, e.g., Kitchatinov and Rüdiger 1993). While this approach usually displaces the ad hoc assumptions into the turbulence model, it has the definite merit of offering an internally consistent approach to the calculation of turbulent diffusivities and large-scale flows. Rüdiger and Brandenburg (1995) and Rempel (2006b) remain a good example of the current state-of-the-art in this area; see also Rüdiger and Arlt (2003), Inceoglu et al. (2017), and references therein.
Critical assessment
From a practical point of view, the outstanding success of the mean-field \(\alpha \varOmega \) model remains its robust explanation of the observed equatorward drift of toroidal field-tracing sunspots in the course of the cycle in terms of a dynamo wave. On the theoretical front, the model is also buttressed by mean-field electrodynamics which, in principle, offers a physically sound theory from which to compute the (critical) \(\alpha \)-effect and magnetic diffusivity. The models’ primary uncertainties turn out to lie at that level, in that the application of the theory to the Sun in a tractable manner requires additional assumptions that are most likely not met under solar interior conditions. Those uncertainties are exponentiated when taking the theory into the nonlinear regime, to calculate the dependence of the \(\alpha \)-effect and diffusivity on the magnetic field strength. This latter problem remains very much open at this writing.
Interface dynamos
Strong \(\alpha \)-quenching and the saturation problem
The \(\alpha \)-quenching expression (29) used in the preceding section amounts to saying that dynamo action saturates once the mean, dynamo-generated field reaches an energy density comparable to that of the driving turbulent fluid motions [viz. Eq. (28)]. At the base of the solar convective envelope, one finds \(B_{\rm eq}\simeq 8 \,\mathrm {kG}\), for \(v\simeq 5\times 10^3 \,{\text{cm s}}^{-1}\), according to mixing length theory of convection. However, various calculations and numerical simulations have indicated that long before the mean field \(\left\langle {\varvec{B}}\right\rangle \) reaches this strength, the helical turbulence reaches equipartition with the small-scale, turbulent component of the magnetic field (e.g., Cattaneo and Hughes 1996, and references therein), ultimately as a consequence of the constraint posed by magnetic helicity conservation (viz. Sect. 4.2.3 herein; see also Brandenburg and Subramanian 2005). Such calculations also indicate that the ratio between the small-scale and mean magnetic components should itself scale as \(\mathrm {Rm}^{1/2}\), where \(\mathrm {Rm}=v\ell /\eta \) is a magnetic Reynolds number based on the microscopic magnetic diffusivity. This then leads to the alternate algebraic quenching expression
$$\begin{aligned} \alpha \rightarrow \alpha (\left\langle {\varvec{B}}\right\rangle )={\alpha _0\over 1+\mathrm {Rm}(\left\langle {\varvec{B}}\right\rangle /B_{\rm eq})^2}, \end{aligned}$$
(46)
known in the literature as strong \(\alpha \)-quenching or catastrophic quenching. Since \(\mathrm {Rm}\sim 10^{9}\) in the solar convection zone, this leads to quenching of the \(\alpha \)-effect for very low amplitudes for the mean magnetic field, of order \(10^{-1}\) G. Even though significant field amplification is likely in the formation of a toroidal flux rope from the dynamo-generated magnetic field, we are now a very long way from the 10–100 kG demanded by simulations of buoyantly rising magnetic flux ropes (see Fan 2009).
A beautifully simple way out of this difficulty was proposed by Parker (1993), in the form of interface dynamos. In a situation where a radial shear and \(\alpha \)-effect are segregated on either side of a discontinuity in magnetic diffusivity (taken to coincide with the core–envelope interface), the \(\alpha \varOmega \) dynamo equations support solutions in the form of travelling surface waves localized on the discontinuity in diffusivity. The key aspect of Parker’s solution is that for supercritical dynamo waves, the ratio of peak toroidal field strength on either side of the discontinuity surface is found to scale with the diffusivity ratio as
$$\begin{aligned} {\max (B_2)\over \max (B_1)} \sim \left( {\eta _2\over \eta _1} \right) ^{-1/2}, \end{aligned}$$
(47)
where the subscript “1” refers to the low-\(\eta \) region below the core–envelope interface, and “2” to the high-\(\eta \) region above. If one assumes that the envelope diffusivity \(\eta _2\) is of turbulent origin then \(\eta _2\sim \ell v\), so that the toroidal field strength ratio then scales as \(\sim (v\ell /\eta _1)^{1/2}\equiv \mathrm {Rm}^{1/2}\). This is precisely the factor needed to bypass strong \(\alpha \)-quenching (Charbonneau and MacGregor 1996). Somewhat more realistic variations on Parker’s basic model were later elaborated (MacGregor and Charbonneau 1997; Zhang et al. 2004), and, while differing in important details, nonetheless confirmed Parker’s overall picture. Tobias (1996) discusses in detail a related Cartesian model bounded in both horizontal and vertical direction, but with constant magnetic diffusivity \(\eta \) throughout the domain. Like Parker’s original interface configuration, his model includes an \(\alpha \)-effect residing in the upper half of the domain, with a purely radial shear in the bottom half. The introduction of diffusivity quenching then reduces the diffusivity in the shear region, “naturally” turning the model into a bona fide interface dynamo, supporting once again oscillatory solutions in the form of dynamo waves travelling in the “latitudinal” x-direction. This basic model was later generalized by various authors (Tobias 1997; Phillips et al. 2002) to include the nonlinear backreaction of the dynamo-generated magnetic field on the differential rotation (as described in Sect. 4.2.5).
Representative results
The next obvious step is to construct an interface dynamo in spherical geometry, using a solar-like differential rotation profile. Such numerical models can be constructed as a variation on the \(\alpha \varOmega \) models considered earlier, introducing a continuous but rapidly varying diffusivity profile at the core–envelope interface, an \(\alpha \)-effect concentrated at the base of the envelope, and the radial shear immediately below, but without significant overlap between these two source regions (see Panel b of Fig. 8).
In spherical geometry, and especially in conjunction with a solar-like differential rotation profile, making a working interface dynamo model is markedly trickier than if only a radial shear is operating, as in the Cartesian models discussed earlier (see Charbonneau and MacGregor 1997; Markiel and Thomas 1999; Zhang et al. 2003a). Panel a of Fig. 8 shows a butterfly diagram for a numerical interface solution with \(C_\varOmega =2.5\times 10^5\), \(C_\alpha =+10\), and a core-to-envelope diffusivity contrast \(\varDelta \eta =10^{-2}\). The poleward propagating equatorial branch is what one would expect from the combination of positive radial shear and positive \(\alpha \)-effect according to the Parker–Yoshimura sign rule.Footnote 10 Here the \(\alpha \)-effect is (artificially) concentrated towards the equator, by imposing a latitudinal dependency \(\alpha \sim \sin (4\theta )\) for \(\pi /4\le \theta \le 3\pi /4\), and zero otherwise.
The model does achieve the kind of toroidal field amplification one would like to see in interface dynamos. This can be seen in Panel b of Fig. 8, which shows radial cuts of the toroidal field taken at latitude \(\pi /8\), and spanning half a cycle. Notice how the toroidal field peaks below the core–envelope interface (vertical dotted line), well below the \(\alpha \)-effect region and near the peak in radial shear. Panel c of Fig. 8 shows how the ratio of peak toroidal field below and above \(r_{\rm c}\) varies with the imposed diffusivity contrast \(\varDelta \eta \). The dashed line is the dependency expected from Eq. (47). For relatively low diffusivity contrast, \(-1.5\le \log (\varDelta \eta ) \lesssim 0 \), both the toroidal field ratio and dynamo period increase as \(\sim (\varDelta \eta )^{-1/2}\). Below \(\log (\varDelta \eta )\sim -1.5\), the \(\max (B)\)-ratio increases more slowly, and the cycle period falls, contrary to expectations for interface dynamos (see, e.g., MacGregor and Charbonneau 1997). This is basically an electromagnetic skin-depth effect; the cycle period is such that the poloidal field cannot diffuse as deep as the peak in radial shear in the course of a half cycle. The dynamo then runs on a weaker shear, thus yielding a smaller field strength ratio and weaker overall cycle.
Critical assessment
The great success of interface dynamos remains their ability to evade \(\alpha \)-quenching even in its “strong” formulation, and so produce equipartition or perhaps even super-equipartition mean toroidal magnetic fields immediately beneath the core–envelope interface. They represent the only variety of dynamo models formally based on mean-field electrodynamics that can achieve this without additional physical effects introduced into the model. All of the uncertainties regarding the calculations of the \(\alpha \)-effect and magnetic diffusivity carry over from \(\alpha \varOmega \) to interface models, with diffusivity quenching becoming a particularly sensitive issue in the latter class of models (see, e.g., Tobias 1996).
Interface dynamos suffer acutely from “structural fragility”. A given model’s dynamo behavior often end up depending sensitively on what one would normally hope to be minor details of the model’s formulation. For example, the interface solutions of Fig. 8 are found to behave very differently if the \(\alpha \)-effect region is displaced slightly upwards, or assumes other latitudinal dependencies. Moreover, as exemplified by the calculations of Mason et al. (2008), this sensitivity carries over to models in which the coupling between the two source regions is achieved by transport mechanisms other than diffusion. This sensitivity is exacerbated when a latitudinal shear is present in the differential rotation profile; compare, e.g., the behavior of the \(C_\alpha >0\) solutions discussed here to those discussed in Markiel and Thomas (1999). Often in such cases, a mid-latitude \(\alpha \varOmega \) dynamo mode, powered by the latitudinal shear within the tachocline and envelope, interferes with and/or overpowers the interface mode [see also Dikpati et al. (2005)]. Because of this structural fragility, interface dynamo solutions also end up being annoyingly sensitive to choice of time-step size, spatial resolution, and other purely numerical details. From a modelling point of view, interface dynamos lack robustness.
Including meridional circulation: flux transport dynamos
Meridional circulation is as unavoidable as differential rotation in turbulent, compressible rotating convective shells (see Featherstone and Miesch 2015, and references therein). Long considered unimportant from the dynamo point of view, meridional circulation has gained popularity in recent years, initially in the Babcock–Leighton context but now also in other classes of models.
Accordingly, we now add a steady meridional circulation to our basic \(\alpha \varOmega \) models of Sect. 4.2. The convenient parametric form developed by van Ballegooijen and Choudhuri (1988) is used here and in all later illustrative models including meridional circulation (Sects. 4.5 and 5). This “minimal” parameterization defines a steady quadrupolar circulation pattern, with a single flow cell per quadrant extending from the surface down to a depth \(r_{\rm b}\). Circulation streamlines are shown in Fig. 4c; the flow is poleward in the outer convection zone, with an equatorial return flow peaking slightly above the core–envelope interface, and rapidly vanishing below.
The inclusion of meridional circulation in the non-dimensionalized \(\alpha \varOmega \) dynamo equations leads to the appearance of a new dimensionless quantity, again a magnetic Reynolds number, but now based on an appropriate measure of the meridional circulation speed \(u_0\) and turbulent diffusivity \(\eta _{\rm T}\):
$$\begin{aligned} \mathrm {Rm}={u_0R_\odot \over \eta _{\rm T}}. \end{aligned}$$
(48)
Using the value \(u_0=1500 \,{\text{cm s}}^{-1}\) from observations of the poleward surface meridional flow leads to \(\mathrm {Rm}\simeq 200\), again with \(\eta _{\rm T}=5\times 10^{11} \,\mathrm {cm}^2 \,\mathrm {s}^{-1}\). In the solar cycle context, using higher values of Rm thus implies proportionally lower turbulent diffusivities.
Representative results
Meridional circulation can bodily transport the dynamo-generated magnetic field [terms labeled “transport” in Eqs. (8, 9)], and therefore, for a (presumably) solar-like equatorward return flow that is vigorous enough—in the sense of Rm being large enough—overpower the Parker–Yoshimura propagation rule (see, e.g. Choudhuri et al. 1995; Küker et al. 2001; Pipin and Kosovichev 2011a). The behavioral turnover from dynamo wave-like solutions to circulation-dominated magnetic field transport sets in when the circulation speed becomes comparable to the propagation speed of the dynamo wave. In the circulation-dominated regime, the cycle period loses sensitivity to the assumed turbulent diffusivity value, and becomes determined primarily by the circulation’s turnover time. Models achieving equatorward propagation of the deep toroidal magnetic component in this manner are now often called flux-transport dynamos (see Dikpati and Gilman 2009; Karak et al. 2014, and references therein).
With a solar-like differential rotation profile, however, once again the situation is far more complex. Starting from the most basic \(\alpha \varOmega \) dynamo solution with \(\alpha \sim \cos \theta \) (Fig. 7a), new solutions are now recomputed, this time including meridional circulation. An animation of a typical solution is shown in Fig. 9, and a sequence of time–latitude diagrams for four increasing values of the circulation flow speed, as measured by Rm, are plotted in Fig. 10.
At \(\mathrm {Rm}=50\), little difference is seen with the circulation-free solutions (cf. Fig. 7a), except for an increase in the cycle frequency, due to the Doppler shift experienced by the equatorwardly propagating dynamo wave (Roberts and Stix 1972). At \(\mathrm {Rm}=100\) (part B), the cycle frequency has further increased and the poloidal component produced in the high-latitude region of the tachocline is now advected to the equatorial regions on a timescale becoming comparable to the cycle period, so that a cyclic activity, albeit with a longer period, becomes apparent at low latitudes. At \(\mathrm {Rm}=10^{3}\) (panel c and animation in Fig. 9) the dynamo mode now peaks at mid-latitude, a consequence of the inductive action of the latitudinal shear, favored by the significant stretching experienced by the poloidal fieldlines as they get advected equatorward. At \(\mathrm {Rm}=2000\) the original high latitude dynamo mode has all but vanished, and the mid-latitude mode is dominant. The cycle period is now set primarily by the turnover time of the meridional flow; this is the telltale signature of flux-transport dynamos.
All this may look straightforward, but it must be emphasized that not all dynamo models with solar-like differential rotation behave in this (relatively) simple manner. For example, the \(C_\alpha =-10\) solution with \(\alpha \sim \sin ^2\theta \cos \theta \) (Fig. 7c) transits to a steady mode as Rm increases above \(\sim 10^2\). Moreover, the sequence of \(\alpha \sim \cos \theta \) shown in Fig. 10 actually presents a narrow window around \(\mathrm {Rm}\sim 200\) where the dynamo is decaying, due to a form of destructive interference between the high-latitude \(\alpha \varOmega \) mode and the mid-latitude advection-dominated dynamo mode that emerges at higher values of Rm. Qualitatively similar results were obtained by Küker et al. (2001) using different prescriptions for the \(\alpha \)-effect and solar-like differential rotation (see in particular their Fig. 11; also Rüdiger and Elstner 2002; Bonanno et al. 2003).
When transport by turbulent pumping is included (see Käpylä et al. 2006b), \(\alpha \varOmega \) models including meridional circulation can provide time–latitude “butterfly” diagrams that are closer to solar-like, even without an equatorward return flow in the deep convection zone (Pipin and Kosovichev 2013).
Even if the meridional flow is too slow—or the turbulent magnetic diffusivity too high—to force the dynamo model in the advection-dominated regime, being much faster at the surface the poleward flow can dominate the spatio-temporal evolution of the radial surface magnetic field. For the dynamo solutions of Fig. 10, at low circulation speeds (\({\mathrm {Rm}}\lesssim 50\)) the spatiotemporal evolution of the surface radial field is simply a diffused imprint of the equatorward drift of the deep-seated toroidal field. At higher circulation speeds, however, the surface magnetic field is swept instead towards the pole becoming strongly concentrated and amplified there for Rm exceeding a few hundreds.
Critical assessment
From the modelling point-of-view, in the kinematic regime at least the inclusion of meridional circulation yields a much better fit to observed surface magnetic field evolution, as well as a robust setting of the cycle period. Whether it can provide an equally robust equatorward propagation of the deep toroidal field is less clear. The results presented here in the context of mean-field \(\alpha \varOmega \) models suggest a rather complex overall picture, and in interface dynamos the cartesian solutions obtained by Petrovay and Kerekes (2004) even suggest that dynamo action can be severely hindered. Yet, in other classes of models discussed below (Sects. 4.5 and 5), circulation does have this desired effect.
On the other hand, dynamo models including meridional circulation tend to produce surface polar field strength largely in excess of observed values, unless magnetic diffusion is significantly enhanced in the surface layers, and/or field submergence takes place very efficiently. This is a direct consequence of magnetic flux conservation in the converging poleward flow. This situation carries over to the other types of models to be discussed in Sects. 4.5 and 5, unless additional modelling assumptions are introduced (e.g., enhanced surface magnetic diffusivity, see Dikpati et al. 2004), or if a counterrotating meridional flow cell is introduced in the high latitude regions (Dikpati et al. 2004; Jiang et al. 2009), a feature that has actually been detected in surface Doppler measurements as well as helioseismically during cycle 22 (Haber et al. 2002; Ulrich and Boyden 2005).
A more fundamental and potential serious difficulty harks back to the kinematic approximation, whereby the form and speed of \({\varvec{u}}_{\rm p}\) is specified a priori. Meridional circulation is a relatively weak flow in the bottom half of the solar convective envelope (see Miesch 2005), and the stochastic fluctuations of the Reynolds stresses powering it are expected to lead to strong spatiotemporal variations, an expectation verified by both analytical models (Rempel 2005) and numerical simulations (Miesch 2005; Passos et al. 2017). The ability of the meridional flow to merrily advect equipartition-strength magnetic fields should not be taken for granted (but do see Rempel 2006a, b).
Before leaving the realm of mean-field dynamo models it is worth noting that many of the conceptual difficulties associated with calculations of the \(\alpha \)-effect and turbulent diffusivity are not unique to the mean-field approach, and in fact carry over to all models discussed in the following sections. In particular, to operate properly all of the upcoming solar dynamo models require the presence of a strongly enhanced magnetic diffusivity, presumably of turbulent origin, at least in the convective envelope. In this respect, the rather low value of the turbulent magnetic diffusivity needed to achieve high enough \(\mathrm {Rm}\) in flux transport dynamos is also somewhat problematic, since the corresponding turbulent diffusivity ends up at least one order of magnitude smaller than the (uncertain) mean-field estimates. However, the model calculations of Muñoz-Jaramillo et al. (2011) indicate that magnetic diffusivity quenching may offer a viable solution to this latter quandary.
Models based on HD and MHD instabilities
The various rotationally-influenced hydrodynamical and magnetohydrodynamical instabilities described in Sect. 3.2.3 have been invoked as \(T\rightarrow P\) inductive mechanisms that can, usually acting in conjunction with rotational shear, form the basis of viable solar cycle models. These models are all mean-field-like, in the sense that the axisymmetric mean-field dynamo equations (38)–(39) are solved, usually in their \(\alpha \varOmega \) form and sometimes including a meridional flow, with mean-field turbulent diffusivity also implicitly invoked. The inductive action of the chosen instability is parametrized by a source term replacing the \(\alpha \)-effect (see, e.g., Ferriz-Mas et al. 1994; Dikpati and Gilman 2001; Ossendrijver 2000a).
Hydrodynamical shear instabilities
Perhaps the most thoroughly studied class of instability-based models is that relying on the shear instability of the latitudinal shear within the tachocline (Dikpati and Gilman 2001; Dikpati et al. 2004). The resulting “tachocline \(\alpha \)-effect” ends up proportional to the longitudinally-averaged kinetic helicity of the hydrodynamical instability planform, the latter computed in the framework of shallow-water theory. The Dikpati and Gilman (2001) dynamo model is of the flux transport variety, with the advective action of the deep meridional flow setting equatorward propagation of the deep toroidal field; it uses a solar-like differential rotation, depth-dependent magnetic diffusivity and meridional circulation pattern much similar to those shown in Fig. 4 herein. The usual ad hoc \(\alpha \)-quenching formula [cf. Eq. (29)] is introduced as the sole amplitude-limiting nonlinearity.
The model can be adjusted to yield equatorward propagating dominant activity belts, solar-like cycle periods, and correct phasing between the surface polar field and the tachocline toroidal field. These features can be traced primarily to the advective action of the meridional flow. It also yields the correct solution parity, and is self-excited. Its primary weakness, in its present form, is the reliance on a linear stability analysis that altogether ignores the known destabilizing effect of magnetic fields (see, e.g., Gilman and Fox 1997; Zhang et al. 2003b). Progress has been made in studying non-linear development of both the hydrodynamical and MHD versions of the shear instability (see Cally 2001; Cally et al. 2003; Dikpati et al. 2009), so that the needed improvements on the dynamo front are potentially forthcoming.
Instability of sheared magnetic layers
Dynamo models relying on the buoyant instability of sheared magnetized layers have been presented in Thelen (2000b), the layer being identified with the tachocline. Here also the resulting azimuthal electromotive force is parameterized as a mean-field-like \(\alpha \)-effect, introduced into the standard \(\alpha \varOmega \) dynamo equations. The model is nonkinematic, in that it includes the magnetic backreaction on the large-scale, purely radial velocity shear within the layer. The analysis of Thelen (2000a) indicates that the \(\alpha \)-effect is negative in the upper part of the shear layer. Cyclic solutions are found in substantial regions of parameter space, and the solutions exhibit migratory wave patterns compatible with the Parker–Yoshimura sign rule. These models are not yet at the stage where they can be meaningfully compared with the solar cycle. They do have a number of attractive features, including their ability to operate in the strong field regime (see also Chatterjee et al. 2011).
Buoyant instability of magnetic flux tubes
Dynamo models relying on the non-axisymmetric buoyant instability of toroidal magnetic fields were first proposed by Schmitt (1987), and further developed by Ferriz-Mas et al. (1994); Schmitt et al. (1996) and Ossendrijver (2000a, b) for the case of toroidal flux tubes. Working in the framework of the thin-flux tube approximation (Spruit 1981), it is possible to construct “stability diagrams” taking the form of growth rate contours in a parameter space comprised of flux tube strength, latitudinal location, depth in the overshoot layer, etc. One such diagram, taken from Ferriz-Mas et al. (1994), is reproduced in Fig. 11. Dynamo action is possible when the instability is weak (growth rates \(\gtrsim 1 \,\mathrm {year}\)). In the case shown in Fig. 11, these regions are restricted to flux tube strengths in the approximate range 60–150 kG. The correlation between the flow and field perturbations is such as to yield a mean azimuthal electromotive force operationally equivalent to a positive \(\alpha \)-effect in the N-hemisphere (Ferriz-Mas et al. 1994; Brandenburg and Schmitt 1998).
This dynamo mechanism operates without difficulty in the strong field regime (in fact it requires strong fields to operate). Difficulties include the need of a relatively finely tuned magnetic diffusivity to achieve a solar-like dynamo period, and a finely tuned level of subadiabaticity in the overshoot layer for the instability to turn on at the appropriate toroidal field strengths (compare Figs. 1 and 2 in Ferriz-Mas et al. 1994). Because the instability model predicts a positive \(\alpha \)-effect-like poloidal source term in the Northern hemisphere, equatorward propagation of the low latitude deep toroidal field would require the addition of a meridional flow, as it does in true mean-field models with positive \(\alpha \)-effect (cf. Sect. 4.4).