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), and Rüdiger and Hollerbach (2004), and in the recent review articles by Ossendrijver (2003) and Hoyng (2003).

The task at hand is to calculate the components of the and 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 and tensor to simple scalars, so that the mean electromotive force becomes

This is the form commonly used in solar dynamo modelling, even though turbulence in the solar interior is most likely inhomogeneous and anisotropic. Moreover, hiding in the above expressions is the assumption that the small-scale magnetic Reynolds number is much smaller than unity, where and are characteristic velocities and length scales for the dominant turbulent eddies, as estimated, e.g., from mixing length theory. With , one finds , so that on that basis alone Equation (18) should be dubious already. In the kinematic regime, and are independent of the magnetic field fluctuations, and end up simply proportional to the averaged kinetic helicity and square fluctuation amplitude: where is the correlation time of the turbulent motions. Order-of-magnitude estimates of the scalar coefficients yield and , where is the solar angular velocity. At the base of the solar convection zone, one then finds and , 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 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 Equation (20) is positive definite, so one would expect a more “stable” mean value (see Hoyng, 1993; Ossendrijver et al., 2001, for further discussion). At any rate, difficulties in computing and from first principle (whether as scalars or tensors) have led to these quantities often being treated as adjustable 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 for and for .The cyclonic character of the -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 rapid variation of the turbulent velocity with depth 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).

In cases where the turbulence is more strongly inhomogeneous, an additional effect comes into play: turbulent pumping. Mathematically it arises through an antisymmetric contribution to the -tensor in Equation (14), whose three independent components can be recast as a velocity-like vector field that acts as an additional (and non-solenoidal) contribution to the mean flow:

The tensor now contains only the symmetric part of the original tensor. Measurements of the -tensor in MHD numerical simulations of turbulence in a box (see Ossendrijver et al., 2002; Käpylä et al., 2006a, and references therein) indicate that pumping is directed mostly downwards throughout the solar convection, as a result of stratification, and that a significant equatorward latitudinal pumping also arises once rotation becomes important, in the sense that the Coriolis number exceeds unity. Turbulent pumping speeds of a few m s

Leaving the kinematic regime, it is expected that both and should depend on the strength of the magnetic field, since magnetic tension will resist deformation by the small-scale turbulent fluid motions. The groundbreaking numerical MHD simulations of Pouquet et al. (1976) suggested that Equation (19) should be replaced by something like

where is the Alfvén speed based on the small-scale magnetic component (see also Durney et al., 1993; Blackman and Brandenburg, 2002). This is rarely used in solar cycle modelling, since the whole point of the mean-field approach is to avoid dealing explicitly with the small-scale, fluctuating components. On the other hand, 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. Denoting this equipartition field strength by , one often introduces an ad hoc nonlinear dependency of (and sometimes as well) directly on the mean-field by writing: This expression “does the right thing”, in that as starts to exceed . It remains an extreme oversimplification of the complex interaction between flow and field that characterizes MHD turbulence, but its wide usage in solar dynamo modeling makes it a nonlinearity of choice for the illustrative purpose of this section.Diffusivity-quenching is an even more uncertain proposition than -quenching, with various quenching models more complex than Equation (23) having been proposed (e.g., Rüdiger et al., 1994). Measurements of the components of the and tensors in the convective turbulence simulations of Brandenburg et al. (2008) do suggest a much stronger magnetic quenching of the -effect than of the turbulent diffusivity, but many aspects of this problem remain open. One appealing aspect of diffusivity quenching is its potential ability to produce localized concentrations of strong magnetic fields, exceeding equipartition strength under some conditions (Gilman and Rempel, 2005). On the other hand, the stability analyses of Arlt et al. (2007b,a) 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.

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 ; the function measures the rate of flux loss, and is often chosen proportional to the magnetic pressure , thus yielding a cubic damping nonlinearity in the mean-field.

Adding the mean-electromotive force given by Equation (18) to the MHD induction equation leads to the following form for the axisymmetric mean-field dynamo equations:

where, in general, . There are now source terms on both right hand sides, so that dynamo action becomes possible at least in principle. The relative importance of the -effect and shearing terms in Equation (25) is measured by the ratio of the two dimensionless dynamo numbers where, in the spirit of dimensional analysis, , , and are “typical” values for the -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 based on the envelope (turbulent) diffusivity: In the solar case, it is usually estimated that , so that the -term is neglected in Equation (25); this results in the class of dynamo models known as dynamos, which will be the only ones discussed here

With the large-scale flows, turbulent diffusivity and -effect considered given, Equations (24, 25) become truly linear in and . It becomes possible to seek eigensolutions in the form

Substitution of these expressions into Equations (24, 25) yields an eigenvalue problem for and associated eigenfunction . The real part is then a growth rate, and the imaginary part an oscillation frequency. One typically finds that until the product exceeds a certain critical value beyond which , corresponding to a growing solutions. Such solutions are said to be supercritical, while the solution with is critical.Clearly exponential growth of the dynamo-generated magnetic field must cease at some point, once the field starts to backreact on the flow through the Lorentz force. This is the general idea embodied in -quenching. If -quenching – or some other nonlinearity – is included, then the dynamo equations are usually solved as an initial-value problem, with some arbitrary low-amplitude seed field used as initial condition. Equations (24, 25) 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:

One of the most remarkable property of the (linear) 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
waves^{5}
travel in a direction given by

We first consider models without meridional circulation ( in Equations (24, 25)), with the
-term omitted in Equation (25), and using the diffusivity and angular velocity profiles of Figure 5. We
will investigate the behavior of models with the -effect concentrated just above the
core-envelope interface (green line in Figure 6). We also consider two latitudinal dependencies,
namely , which is the “minimal” possible latitudinal dependency compatible with the
required equatorial antisymmetry of the Coriolis force, and an -effect concentrated towards the
equator^{6}
via an assumed latitudinal dependency .

Figures 7 and 8 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 . 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 Figure 3. All models have , , , and , which leads to . To facilitate comparison between solutions, here antisymmetric parity was imposed via the boundary condition at the equator.

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 -effect is maximal (cf. Figure 6). In conjunction with a fairly thin tachocline, the radial shear therein then dominates the induction of the toroidal magnetic component. With an eye on Figure 5, notice also how the dynamo waves propagates along isocontours of angular velocity, in agreement with the Parker–Yoshimura sign rule (cf. Section 4.2.5). In the butterfly diagram, this translates a systematic tilt of the isocontours of toroidal magnetic field. Note that even for an equatorially-concentrated -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 -effect operating throughout the whole convective envelope, on the other hand, would feed primarily on the latitudinal shear therein, so that for positive 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 Figure 8, can have distinct dynamo periods, 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 Equation (23). Note that this does not occur for the 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 Figure 8).

A common property of all oscillatory solutions discussed so far is that their period, for given values of the dynamo numbers , , is inversely proportional to the numerical value adopted for the (turbulent) magnetic diffusivity . 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 , at a fixed value of the product .

The models discussed above are based on rather minimalistics and partly ad hoc assumptions on the form of the -effect. More elaborate models have been proposed, relying on calculations of the full -tensor based on some underlying turbulence models (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) remain a good example of the current state-of-the-art in this area; see also Rüdiger and Arlt (2003), and references therein.

From a practical point of view, the outstanding success of the mean-field 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) -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 certainly not met under solar interior conditions. Those uncertainties are exponentiated when taking the theory into the nonlinear regime, to calculate the dependence of the -effect and diffusivity on the magnetic field strength. This latter problem remains very much open at this writing.

Living Rev. Solar Phys. 7, (2010), 3
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