Each and every one of the T → P mechanisms described in Section 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 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 〈B〉, namely the large-scale, axisymmetric component of the total solar magnetic field.
Model ingredients
All 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). Meridional circulation in the convective envelope, long considered unimportant from the dynamo point of view, has gained popularity in recent years, initially in the Babcock-Leighton context but now also in other classes of models.
Helioseismology has pinned down with great accuracy the internal solar structure, including the internal differential rotation, and the exact location of the core-envelope interface. Unless noted otherwise, all illustrative models discussed in this section were computed using the following analytic formulae for the angular velocity Ω(r, θ) and magnetic diffusivity η(r):
$${{\Omega (r,\theta)} \over {{\Omega _{\rm{E}}}}} = {\Omega _{\rm{C}}} + {{{\Omega _{\rm{S}}}(\theta) - {\Omega _{\rm{C}}}} \over 2}\left[ {1 + {\rm{erf}}\left({{{r - {r_{\rm{c}}}} \over w}} \right)} \right],$$
(17)
with
$${\Omega _{\rm{S}}}(\theta) = 1 - {a_2}{\cos ^2}\theta - {a_4}{\cos ^4}\theta,$$
(18)
and
$${{\eta (r)} \over {{\eta _{\rm{T}}}}} = \Delta \eta + {{1 - \Delta \eta } \over 2}\left[ {1 + {\rm{erf}}\left({{{r - {r_{\rm{c}}}} \over w}} \right)} \right].$$
(19)
With appropriately chosen parameter values, Equation (17) 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-latitudesFootnote 4. This rotational transition takes place across a spherical shear layer of half-thickness w coinciding with the core-envelope interface at rc/R⊙ = 0.7 (see Figure 5, with parameter values listed in caption). As per Equation (19), a similar transition takes place with the net diffusivity, falling from some large, “turbulent” value ηT in the envelope to a much smaller diffusivity ηC in the convection-free radiative core, the diffusivity contrast being given by Δη = ηC/ηt. Given helioseismic constraints, these represent minimalistic yet reasonably realistic choices.
It should be noted already that such a solar-like differential rotation profile is quite complex from the point of view of dynamo modelling, 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. As shown on Panel B of Figure 5, for a tachocline of half-thickness w/R⊙ = 0.05, the mid-latitude latitudinal shear at r/R⊙ = 0.7 is comparable in magnitude to the equatorial radial shear; its potential contribution to dynamo action should not be casually dismissed.
Ultimately, the magnetic diffusivities and differential rotation in the convective envelope owe their existence to the turbulence therein, more specifically to the associated Reynolds stresses. While it has been customary in solar dynamo modelling to simply assume plausible functional forms for these quantities (such as Equations (17, 18, 19) above), one recent trend has been to calculate these quantities in an internally consistent manner using an actual model for the turbulence itself (see, e.g., Kitchatinov and Rüdiger, 1993). While this approach introduces additional — and often important — uncertainties at the level of the turbulence model, it represents in principle a tractable avenue out of the kinematic regime.
αΩ mean-field models
Calculating the α-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), and in the recent review article by Hoyng (2003).
The task at hand is to calculate the components of the α and β tensor in terms of the statistical properties of the underlying turbulence. A particularly simple case is that of homogeneous, weakly isotropic turbulence, which reduces the α and β tensor to simple scalars, so that the mean electromotive force becomes
$${\cal E} = \alpha \langle {\bf{B}} \rangle - {\eta _{\rm{T}}}\nabla \times \langle {\bf{B}} \rangle.$$
(20)
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 υℓ/η is much smaller than unity, where υ ∼ 103 cm s−1 and ℓ ∼ 109 cm are characteristic velocities and length scales for the dominant turbulent eddies, as estimated, e.g., from mixing length theory. With η ∼ 104 cm2 s−1, one finds υℓ/η ∼ 108, so that on that basis alone Equation (20) should be dubious already. In the kinematic regime, α and β are independent of the magnetic field fluctuations, and end up simply proportional to the averaged kinetic helicity and square fluctuation amplitude:
$$\alpha \sim - {{{\tau _{\rm{c}}}} \over 3}\langle {{{\bf{u}}^{\prime}} \cdot \nabla \times {{\bf{u}}^{\prime}}} \rangle,$$
(21)
$${\eta _{\rm{T}}}\sim{{{\tau _{\rm{c}}}} \over 3}\langle {{{\bf{u}}^{\prime}} \cdot {{\bf{u}}^{\prime}}} \rangle,$$
(22)
where τc is the correlation time of the turbulent motions. Order-of-magnitude estimates of the scalar coefficients yield α ∼ Ωℓ and ηT ∼ υℓ, where Ω is the solar angular velocity. At the base of the solar convection zone, one then finds α ∼ 103 cm s−1 and ηT ∼ 1012 cm2 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 α 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 (22) 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 α and ηT 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 10–103 cm s−1 for α and 1010–1013 cm2 s−1 for ηT.
The cyclonic character of the α-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 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).
Leaving the kinematic regime, it is expected that both α and β 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 (21) should be replaced by something like
$$\alpha \sim - {{{\tau _{\rm{c}}}} \over 3}[\langle {{{\bf{u}}^{\prime}} \cdot \nabla \times {{\bf{u}}^{\prime}}} \rangle - \langle {{{\bf{a}}^{\prime}} \cdot \nabla \times {{\bf{a}}^{\prime}}} \rangle],$$
(23)
where \({\bf a}^{\prime}={\bf b}^{\prime}\sqrt{4\pi\rho}\) 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 Beq, one often introduces an ad hoc nonlinear dependency of α (and sometimes ηT as well) directly on the mean-field 〈B〉 by writing:
$$\alpha \rightarrow \alpha (\langle {\bf{B}} \rangle) = {{{\alpha _0}} \over {1 + {{(\langle {\bf{B}} \rangle/{B_{{\rm{eq}}}})}^2}}}.$$
(24)
This expression “does the right thing”, in that α → 0 as 〈B〉 starts to exceed Beq. 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.
The αΩ dynamo equations
Adding this contribution to the MHD induction equation leads to the following form for the axisymmetric mean-field dynamo equations:
$${{\partial \langle A \rangle} \over {\partial t}} = \underbrace {(\eta + {\eta _{\rm{T}}})\left({{\nabla ^2} - {1 \over {{\varpi ^2}}}} \right)\langle A \rangle }_{{\rm{turbulent}}\,{\rm{diffusion}}} - {{{{\bf{u}}_{\rm{p}}}} \over \varpi } \cdot \nabla (\varpi \langle A \rangle) + \underbrace {\alpha \langle B \rangle }_{{\rm{MFE}}\,{\rm{source}}},$$
(25)
$$\matrix{{{{\partial \langle B \rangle } \over {\partial t}} = \underbrace {(\eta + {\eta _{\rm{T}}})\left({{\nabla ^2} - {1 \over {{\varpi ^2}}}} \right)\langle B \rangle }_{{\rm{turbulent}}\;{\rm{diffusion}}} + \underbrace {{1 \over \varpi }{{\partial \varpi \langle B \rangle } \over {\partial r}}{{\partial (\eta + {\eta _{\rm{T}}})} \over {\partial r}}}_{{\rm{turbulent}}\;{\rm{diamagnetic}}\;{\rm{transport}}} - \varpi {{\bf{u}}_{\rm{p}}} \cdot \nabla \left({{{\langle B \rangle } \over \varpi }} \right) - \langle B \rangle \nabla \cdot {{\bf{u}}_{\rm{p}}}} \hfill \cr {\quad \quad \quad + \underbrace {\varpi (\nabla \times (\langle A \rangle {{{\bf{\hat e}}}_\phi })) \cdot \nabla \Omega }_{{\rm{shearing}}} + \underbrace {\nabla \times [\alpha \nabla \times (\langle A \rangle {\bf{\hat e}}\;\phi)]}_{{\rm{MFE}}\;{\rm{source}}},} \hfill \cr}$$
(26)
where, in general, ηT ≫ η There are source terms on both right hand sides, so that dynamo action is now possible in principle. The relative importance of the α-effect and shearing terms in Equation (26) is measured by the ratio of the two dimensionless dynamo numbers
$${C_\alpha } = {{{\alpha _0}{R_ \odot }} \over {{\eta _0}}},\quad \quad {C_\Omega } = {{{{(\Delta \Omega)}_0}R_ \odot ^2} \over {{\eta _0}}},$$
(27)
where, in the spirit of dimensional analysis, α0, η0, and (ΔΩ)0 are “typical” values for the α-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 t based on the envelope (turbulent) diffusivity:
$$\tau = {{R_ \odot ^2} \over {{\eta _{\rm{T}}}}}.$$
(28)
In the solar case, it is usually estimated that Cα ≪ CΩ, so that the α-term is neglected in Equation (26); this results in the class of dynamo models known as αΩ dynamo, which will be the only ones discussed hereFootnote 5.
Eigenvalue problems and initial value problems
With the large-scale flows, turbulent diffusivity and α-effect considered given, Equations (25, 26) become truly linear in A and B. It becomes possible to seek eigensolutions in the form
$$\langle A \rangle (r, \theta, t) = a(r,\theta)\exp (st),\quad \quad \langle B \rangle (r, \theta, t) = b(r,\theta)\exp (st).$$
(29)
Substitution of these expressions into Equations (25, 26) yields an eigenvalue problem for s and associated eigenfunction {a, b}. The real part σ ≡ Re s is then a growth rate, and the imaginary part w ≡ Im s an oscillation frequency. One typically finds that σ < 0 until the product Cα × CΩ exceeds a certain critical value Dcrit beyond which σ > 0, corresponding to a growing solutions. Such solutions are said to be supercritical, while the solution with α = 0 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 α-quenching. If α-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 (25, 26) 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:
$${{\cal E}_B} = {1 \over {8\pi }}\int_V {{{\langle {\bf{B}} \rangle }^2}} dV.$$
(30)
Dynamo waves
One of the most remarkable property of the (linear) αΩ 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 6 travel in a direction s given by
$${\bf{s}} = \alpha \nabla \Omega \times {{\bf{\hat e}}_\phi},$$
(31)
a result now known as the “Parker-Yoshimura sign rule”. Recalling the rather complex form of the helioseismically inferred solar internal differential rotation (cf. Figure 5), even an α-effect of uniform sign in each hemisphere can produce complex migratory patterns, as will be apparent in the illustrative αΩ dynamo solutions to be discussed shortly. Note already at this juncture that if the seat of the dynamo is to be identified with the low-latitude portion of the tachocline, and if the latter is thin enough for the (positive) radial shear therein to dominate over the latitudinal shear, then equatorward migration of dynamo waves will require a negative α-effect in the low latitudes of the Northern solar hemisphere.
Representative results
We first consider αΩ models without meridional circulation (up = 0 in Equations (25, 26)), with the α-term omitted in Equation (26), and using the diffusivity profile and angular velocity profile of Figure 5. We will investigate the behavior of αΩ models with the α-effect operating throughout the bulk of the convective envelope (red line in Figure 6), as well as with an α-effect concentrated just above the core-envelope interface (green line in Figure 6). We also consider two latitudinal dependencies, namely α ∝ cos θ, which is the “minimal” possible latitudinal dependency compatible with the required equatorial antisymmetry of the Coriolis force, and an α-effect concentrated towards the equatorFootnote 7 via an assumed latitudinal dependency α ∝ sin2 θ cos θ.
Figure 7 shows a selection of such dynamo solutions, in the form of time-latitude diagrams of the toroidal field extracted at the core-envelope interface, here rc/R⊙ = 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 Figure 3. All models have CΩ = 25000, |Cα| = 10, ηT/ηc = 10, and ηT = 5 × 1011 cm2 s−1, which leads to τ ≃ 300 yr. To facilitate comparison between solutions, here antisymmetric parity was imposed via the boundary condition at the equator.
Models using the radially broad, full convection zone α-effect (Panel A of Figure 7) feed mostly on the latitudinal shear in the envelope, so that the dynamo mode propagates radially upward in the envelope, with some latitudinal propagation in the tachocline only at the onset of the cycle (see animation). Models with positive Cα nonetheless yield oscillatory solutions, but those with Cα < 0 produce steady modes over a wide range of parameter values. Models using an α-effect concentrated at the base of the envelope (Panels B through D), on the other hand, are powered by the radial shear therein, and show the expected tilt in the butterfly diagrams, as expected from the Parker-Yoshimura sign rule (cf. Section 4.2.4). Note that even for an equatorially-concentrated α-effect (Panels B and D), 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).
It is noteworthy that co-existing dynamo branches, as in Panel B of Figure 7, 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 (24). A portion of the magnetic energy time-series for that solution is shown in Panel A of Figure 8 to illustrate the effect. Note that this does not occur for the Cα < 0 solution (Panel B of Figure 8), 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 D of Figure 7).
A common property of all oscillatory αΩ solutions discussed so far is that their period, for given values of the dynamo numbers Cα, CΩ, is inversely proportional to the numerical value adopted for the (turbulent) magnetic diffusivity ηT. 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 Cα/CΩ, at a fixed value of the product Cα × CΩ.
Vector magnetograms of sunspots active regions make it possible to estimate the current helicity j · B which is closely related to the usual magnetic helicity A · B, and the amount of twist in the sunspot-forming toroidal flux ropes (see, e.g., Hagyard and Pevtsov, 1999, and references therein). Upon assuming that this current helicity reflects that of the diffuse, dynamo-generated magnetic field from which the flux ropes formed, one obtains another useful constraint on dynamo models. In the context of classical αΩ mean-field models, predominantly negative current helicity in the N-hemisphere, in agreement with observations, is usually obtained for models with negative α-effect relying primarily on positive radial shear at the equator (see Gilman and Charbonneau, 1999, and discussion therein).
The models discussed above are based on rather minimalistics and partly ad hoc assumptions on the form of the α-effect. More elaborate models have been proposed, relying on calculations of the full α-tensor based on some underlying turbulence models. While this approach usually displaces the ad hoc assumptions away from the α-effect and into the turbulence model, it has the definite advantage 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.
Critical assessment
From a practical point of view, the outstanding success of the mean-field αΩ 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) α-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 α-effect and diffusivity on the magnetic field strength. This latter problem remains very much open at this writing.
Interface dynamos
Strong α-quenching and the saturation problem
The α-quenching expression (24) 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, i.e., \(B_{\rm eq}\sim \sqrt{4\pi\rho} \ v\), where υ is the turbulent velocity amplitude. This appears eminently sensible, since from that point on a toroidal fieldline would have sufficient tension to resist deformation by cyclonic turbulence, and so could no longer feed the α-effect. At the base of the solar convective envelope, one finds Beq ∼ 1 kG, for υ ∼ 103 cm s−1, according to standard mixing length theory of convection. However, various calculations and numerical simulations have indicated that long before the mean field 〈B〉 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). Such calculations also indicate that the ratio between the small-scale and mean magnetic components should itself scale as Rm1/2, where Rm = υℓ/η is a magnetic Reynolds number based on the microscopic magnetic diffusivity. This then leads to the alternate quenching expression
$$\alpha \rightarrow \alpha (\langle {\bf{B}} \rangle) = {{{\alpha _0}} \over {1 + {\rm{Rm(}}\langle {\bf{B}} \rangle/{B_{{\rm{eq}}}}{{\rm{)}}^{\rm{2}}}}},$$
(32)
known in the literature as strong α-quenching or catastrophic quenching. Since Rm ∼ 108 in the solar convection zone, this leads to quenching of the α-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 flux ropes (see Fan, 2004).
A way out of this difficulty was proposed by Parker (1993), in the form of interface dynamos. The idea is beautifully simple: If the toroidal field quenches the α-effect, amplify and store the toroidal field away from where the α-effect is operating! Parker showed that in a situation where a radial shear and α-effect are segregated on either side of a discontinuity in magnetic diffusivity (taken to coincide with the core-envelope interface, see Panel A of Figure 9), the αΩ 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
$${{\max ({B_2})} \over {\max ({B_1})}}\sim\left({{{{\eta _2}} \over {{\eta _1}}}} \right){,^{ - 1/2}}$$
(33)
where the subscript “1” refers to the low-η region below the core-envelope interface, and “2” to the high-η region above. If one assumes that the envelope diffusivity η2 is of turbulent origin then η2 ∼ ℓυ, so that the toroidal field strength ratio then scales as ∼ (υℓ/η1)1/2 = Rm1/2. This is precisely the factor needed to bypass strong α-quenching (Charbonneau and MacGregor, 1996). Somewhat more realistic variations on Parker’s basic models were later elaborated (MacGregor and Charbonneau (1997); Zhang et al. (2004), see also Panels B and C of Figure 9), and, while differing in important details, nonetheless confirmed Parker’s overall picture.
Tobias (1996a) discusses in detail a related Cartesian model bounded in both horizontal and vertical direction, but with constant magnetic diffusivity η throughout the domain. Like Parker’s original interface configuration, his model includes an α-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; further discussion of such nonlinear models is deferred to Section 5.3.1.
Representative results
The next obvious step is to construct an interface dynamo in spherical geometry, using a solar-like differential rotation profile. This was undertaken by Charbonneau and MacGregor (1997). Unfortunately, the numerical technique used to handle the discontinuous variation in η at the core-envelope interface turned out to be physically erroneous for the vector potential A describing the poloidal fieldFootnote 8 (see Markiel and Thomas, 1999, for a discussion of this point), which led to spurious dynamo action in some parameter regimes. The matching problem is best avoided by using a continuous but rapidly varying diffusivity profile at the core-envelope interface, with the α-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 Figure 10). Such numerical models can be constructed as a variation on the αΩ models considered earlier, and stand somewhere between Parker’s original picture (see Panel A of Figure 9) and the models with spatially localized α-effect and shear (see Panels B and C of Figure 9).
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). Panel A of Figure 10 shows a butterfly diagram for a numerical interface solution with CΩ = 2.5 × 105, Cα = +10, and a core-to-envelope diffusivity contrast Δη = 10−2. A magnetic energy time series for this solution is plotted in Panel D of Figure 8, together with a solution with a smaller diffusivity contrast Δη = 0.1 (see Panel C of Figure 8). The poleward propagating equatorial branch is precisely what one would expect from the combination of positive radial shear and positive α-effect according to the Parker-Yoshimura sign ruleFootnote 9. Here the α-effect is (artificially) concentrated towards the equator, by imposing a latitudinal dependency α ∼ sin (49) for π/4 ≤ θ ≤ 3π/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 Figure 10, which shows radial cuts of the toroidal field taken at latitude π/8, and spanning half a cycle. Notice how the toroidal field peaks below the core-envelope interface (vertical dotted line), well below the α-effect region and near the peak in radial shear. Panel C of Figure 10 shows how the ratio of peak toroidal field below and above rc varies with the imposed diffusivity contrast Δη. The dashed line is the dependency expected from Equation (33). For relatively low diffusivity contrast, −1.5 ≤ log(Δη) ≲ 0, both the toroidal field ratio and dynamo period increase as ∼ (Δη)−1/2. Below log(Δη) ∼ −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 (cf. Panels C and D of Figure 8; on the energetics of interface dynamos; see also Ossendrijver and Hoyng, 1997)
Zhang et al. (2003a) have presented results for a fully three-dimensional α2Ω interface dynamo model where, however, dynamo solutions remain largely axisymmetric when a strong shear is present in the tachocline. They use an α-effect spanning the whole convective envelope radially, but concentrated latitudinally near the equator, a core-to-envelope magnetic diffusivity contrast Δη = 10−1, and the usual algebraic α-quenching formula. Unfortunately, their differential rotation profile is non-solar. However, they do find that the dynamo solutions they obtain are robust with respect to small changes in the model parameters. The next obvious step here is to repeat the calculations with a solar-like differential rotation profile.
Critical assessment
So far the great success of interface dynamos remains their ability to evade α-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 α-effect and magnetic diffusivity carry over from αΩ to interface models, with diffusivity quenching becoming a particularly sensitive issue in the latter class of models (see, e.g., Tobias, 1996a).
Interface dynamos suffer acutely from something that is sometimes termed “structural fragility”. Many gross aspects of the 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 Figure 10 are found to behave very differently if either
-
the α-effect region is displaced upwards by a mere 0.05 R⊙, or
-
the α-effect is less concentrated towards the equator, for example via the ∼ sin2 θ cos θ form used in Section 4.2, or
-
the tachocline thickness is increased by 50%, leading to somewhat greater overlap between the α-effect and shear source regions.
Compare also the behavior of the Cα > 0 solutions discussed here to those discussed in Markiel and Thomas (1999). Once again the culprit is the latitudinal shear. Each of these minor variations on the same basic model has the effect that a parallel mid-latitude dynamo mode, powered by the latitudinal shear within the tachocline and envelope, interferes with and/or overpowers the interface mode. This interpretation is not inconsistent with the robustness claimed by Zhang et al. (2003a), since these authors have chosen to omit the latitudinal shear throughout the convective envelope in their model. Because of this structural sensitivity, 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.
Mean-field models including meridional circulation
Meridional circulation is unavoidable in turbulent, compressible rotating convective shells. It basically results from an imbalance between Reynolds stresses and buoyancy forces. The ∼ 15 m s−1 poleward flow observed at the surface (see, e.g., Hathaway, 1996) has now been detected helioseismically, down to r/R⊙ ≃ 0.85 (Schou and Bogart, 1998; Braun and Fan, 1998), without significant departure from the poleward direction except locally and very close to the surface, in the vicinity of active region belts (Haber et al., 2002; Basu and Antia, 2003; Zhao and Kosovichev, 2004).
Accordingly, we now add a steady meridional circulation to our basic αΩ models of Section 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 (Sections 4.5 and 4.8). This parameterization defines a steady quadrupolar circulation pattern, with a single flow cell per quadrant extending from the surface down to a depth rb. Circulation streamlines are shown in Figure 11, together with radial cuts of the latitudinal component at mid-latitudes (θ = π/4). 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 αΩ 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 circulation speed u0:
$${\rm{Rm}}={{{u_0}{R_ \odot }} \over {{\eta _{\rm{T}}}}}.$$
(34)
Using the value u0 = 1500 cm s−1 from observations of the observed poleward surface meridional flow leads to Rm ≃ 200, again with ηT = 5 × 1011 cm2 s−1.
Representative results
Meridional circulation can bodily transport the dynamo-generated magnetic field (terms labeled “advective transport” in Equations (11, 12)), and therefore, for a (presumably) solar-like equator-ward return flow that is vigorous enough — in the sense of Rm being large enough — overpower the Parker-Yoshimura propagation rule embodied in Equation (31). This was nicely demonstrated by Choudhuri et al. (1995), in the context of a mean-field αΩ model with a positive α-effect concentrated near the surface, and a latitude-independent, purely radial shear at the core-envelope interface. With a solar-like differential rotation profile, however, once again the situation is far more complex.
Starting from the three αΩ dynamo solutions with the α-effect concentrated at the base of the convective envelope, (see Figure 7, Panels B through D), new solutions are now recomputed, this time including meridional circulation. Results are shown in Figure 12, for three increasing values of the circulation flow speed, as measured by Rm. At Rm = 50, little difference is seen with the circulation-free solutions, except for the Cα = +10 solution with equatorially-concentrated α-effect (see Panel A of Figure 12), where the equatorial branch is now dominant and the polar branch has shifted to mid-latitudes and has become doubly-periodic. At Rm = 200, corresponding here to a solar-like circulation speed, drastic changes have materialized in all solutions. The negative Cα solution has now transited to a steady dynamo mode, that in fact persists to higher Rm values (panels F and I). The Cα = +10 solution with α ∝ cos θ is decaying at Rm = 200, while the solution with equatorially-concentrated α-effect starts to show a hint of equatorward propagation at mid-latitudes (Panel D). At Rm = 103, the circulation has overwhelmed the dynamo wave, and both positive Cα solutions show equatorially-propagating toroidal fields (Panels G and H). Qualitatively similar results were obtained by Küker et al. (2001) using different prescriptions for the α-effect and solar-like differential rotation (see in particular their Figure 11; see also Rüdiger and Elstner, 2002; Bonanno et al., 2003).
Evidently, meridional circulation can have a profound influence on the overall character of the solutions. 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. This can be seen in Figure 12: At Rm = 50 the solutions in Panels A and B have markedly distinct (primary) cycle periods, while at Rm = 103 (Panels G and H) the cycle periods are nearly identical. Note however that significant effects require a large Rm (≳ 103 for the circulation profile used here), which, u0 being fixed by surface observations, translates into a magnetic diffusivity ηT ≲ 1011; by most orders-of-magnitude estimates constructed in the framework of mean-field electrodynamics this is rather low.
Meridional circulation can also dominate the spatio-temporal evolution of the radial surface magnetic field, as shown in Figure 13 for a sequence of solutions with Rm = 0, 50, and 200 (corresponding toroidal butterfly diagram at the core-envelope interface are plotted in Panel B of Figure 7 and in Panel A and D of Figure 12).
In the circulation-free solution (Rm = 0), the equatorward drift of the surface radial field is a direct reflection of the equatorward drift of the deep-seated toroidal field (see Panel B of Figure 7). With circulation turned on, however, the surface magnetic field is swept instead towards the pole (see Panel B of Figure 13), becoming strongly concentrated and amplified there for solar-like circulation speeds (Rm = 200, see Panel C of Figure 13).
Critical assessment
From the modelling point-of-view, 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 αΩ models suggest a rather complex overall picture, yet in other classes of models discussed below (Sections 4.5 and 4.8), circulation does have this desired effect. The effects of envelope meridional circulation on interface dynamos (Section 4.3), however, remains unexplored.
On the other hand, dynamo models including meridional circulation invariably produce surface polar field strength largely in excess of observed values. This is 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 Sections 4.5 and 4.8, unless additional modelling assumptions are introduced (e.g., enhanced surface magnetic diffusivity, see Dikpati et al., 2004). The rather low value of the turbulent magnetic diffusivity needed to achieve high enough Rm is also somewhat problematic. A more fundamental and potential serious difficulty harks back to the kinematic approximation, whereby the form and speed of up is specified a priori. Meridional circulation is a relatively weak flow in the bottom half of the solar convective envelope (see Miesch, 2005), so its ability to merrily advect equipartition-strength magnetic fields should not be taken for granted.
Before leaving the realm of mean-field dynamo models it is worth noting that many of the conceptual difficulties associated with calculations of the α-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.
Models based on shear instabilities
We now turn to a recently proposed class of dynamo models relying on the latitudinal shear instability of the angular velocity profiles in the upper radiative portion of the solar tachocline (Dikpati and Gilman, 2001). Although the study of dynamo action in this context has barely begun, results published so far (see Dikpati and Gilman, 2001; Dikpati et al., 2004) make this class of models worthy of further consideration.
From instability to α-effect
Dikpati and Gilman (2001) work with what are effectively the mean field αΩ dynamo equations including meridional circulation. They design their “tachocline α-effect” in the form of a latitudinal parameterization of the longitudinally-averaged kinetic helicity associated with the planforms they obtain from a linear hydrodynamical stability analysis of the latitudinal differential rotation in the part of the tachocline coinciding with the overshoot region (see Dikpati and Gilman, 2001). Figure 14 shows some typical latitudinal profiles of kinetic helicity for various model parameter settings and azimuthal wavenumbers, all computed in the framework of shallow-water theory. In analogy with mean-field theory, the resulting ±-effect is assumed to be proportional to kinetic helicity but of opposite sign (see Equation (21)), and so is here predominantly positive at mid-latitudes in the Northern solar hemisphere. In their dynamo model, Dikpati and Gilman (2001) use a solar-like differential rotation, depth-dependent magnetic diffusivity and meridional circulation pattern much similar to those shown on Figures 5, 6, and 11 herein, and the usual ad hoc ±-quenching formula (cf. Equation (24)) is introduced as the sole amplitude-limiting nonlinearity.
Representative solutions
Many representative solutions for this class of dynamo models can be examined in Dikpati and Gilman (2001) and Dikpati et al. (2004), where their properties are discussed at some length. Figure 15 shows time-latitude diagrams of the toroidal field at the core-envelope interface, and surface radial field. This is a solar-like solution with a mid-latitude surface meridional (poleward) flow speed of 17 m s−1, envelope diffusivity ηT = 5 × 1011 cm2 s−1, and a core-to-envelope magnetic diffusivity contrast Δη = 10−3.
Note the equatorward migration of the deep toroidal field, set here by the meridional flow in the deep envelope, and the poleward migration and intensification of the surface poloidal field, again a direct consequence of advection by meridional circulation, as in the mean-field dynamo models discussed in Section 4.4 in the advection-dominated, high Rm regime. The three-lobe structure of each spatio-temporal cycle in the butterfly diagram reflects the presence of three peaks in the latitudinal profile of kinetic helicity (see Figure 14).
Critical assessment
While these models are only a recent addition to the current “zoo” of solar dynamo models, they have been found to compare favorably to a number of observed solar cycle features. In many cases they 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. They also yield the correct solution parity, and are self-excited. Like conventional αΩ models relying on meridional circulation to set the propagation direction of dynamo waves (see Section 4.4.2), the meridional flow must remain unaffected by the dynamo-generated magnetic field at least up to equipartition strength, a potentially serious difficulty also shared by the Babcock-Leighton models discussed in Section 4.8 below.
The applicability of shallow-water theory to the solar tachocline notwithstanding, the primary weakness of these models, in their present form, is their reliance on a linear stability analysis that altogether ignores the destabilizing effect of magnetic fields. Gilman and Fox (1997) have demonstrated that the presence of even a weak toroidal field in the tachocline can very efficiently destabilize a latitudinal shear profile that is otherwise hydrodynamically stable (see also Zhang et al., 2003b). Relying on a purely hydrodynamical stability analysis is then hard to reconcile with a dynamo process producing strong toroidal field bands of alternating polarities migrating towards the equator in the course of the cycle, especially since latitudinally concentrated toroidal fields have been found to be unstable over a very wide range of toroidal field strengths (see Dikpati and Gilman, 1999). In the MHD version of the shear instability studied by P. Gilman and collaborators, the structure of the instability planforms is highly dependent on the assumed underlying toroidal field profile, so that the kinetic helicity can be expected to (i) have a time-dependent latitudinal distribution, and (ii) be intricately dependent on 〈B〉, in a manner that is unlikely to be reproduced by a simple amplitude-limiting quenching formula such as Equation (24). Linear calculations carried out to date in the framework of shallow-water MHD indicate that the purely hydrodynamical α-effect considered here is indeed strongly affected by the presence of an unstable toroidal field (see Dikpati et al., 2003). However, progress has been made in studying non-linear development of both the hydrodynamical and MHD versions of the shear instability (see, e.g., Cally, 2001; Cally et al., 2003), so that the needed improvements on the dynamo front are hopefully forthcoming.
Models based on buoyant instabilities of sheared magnetic layers
Dynamo models relying on the buoyant instability of 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 α-effect, introduced into the standard αΩ dynamo equations. The model is nonlinear, 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 α-effect is negative in the upper part of the shear layer. Cyclic solutions are found in substantial regions of parameter space, and, not surprisingly, the solutions exhibit migratory wave patterns compatible with the Parker-Yoshimura sign rule.
Representative solutions for this class of dynamo models can be examined in Thelen (2000b). 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.
Models based on flux tube instabilities 4.7.1 From instability to α-effect
To date, stability studies of toroidal flux ropes stored in the overshoot layer have been carried out 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 Figure 16. The key is now to identify regions in such stability diagrams where weak instability arises (growth rates ≳ 1 yr). In the case shown in Figure 16, 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 equivalent to a positive α-effect in the N-hemisphere (Ferriz-Mas et al., 1994; Brandenburg and Schmitt, 1998).
Representative solutions
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) for the case of toroidal flux tubes. These dynamo models are all mean-field-like, in that the mean azimuthal electromotive force arising from instability of the flux tubes is parametrized as an α-effect, and the dynamo equations solved are then the same as those of the conventional αΩ mean-field model (see Section 4.2.2), including various forms of algebraic α-quenching as the sole amplitude-limiting nonlinearity. As with mean-field models, the dynamo period presumably depends sensitively on the assumed value of (turbulent) magnetic diffusivity, and equatorward propagation of the dynamo wave requires a negative α-effect at low latitudes.
Critical assessment
Although it has not yet been comprehensively studied, this dynamo mechanism has a number of very attractive properties. It operates without difficulty in the strong field regime (in fact it requires strong fields to operate). It also naturally yields dynamo action concentrated at low latitudes, so that a solar-like butterfly diagram can be readily produced from a negative α-effect even with a solar-like differential rotation profile, at least judging from the solutions presented in Schmitt et al. (1996) and Ossendrijver (2000a,b).
Difficulties include the need of a relatively finely tuned magnetic diffusivity to achieve a solar-like dynamo period, and a relatively finely-tuned level of subadiabaticity in the overshoot layer for the instability to kick on and off at the appropriate toroidal field strengths (compare Figures 1 and 2 in Ferriz-Mas et al., 1994). The non-linear saturation of the instability is probably less of an issue here than with the α-effect based on purely hydrodynamical shear instability (see Section 4.5 above), since, as the instability grows, the flux ropes leave the site of dynamo action by entering the convection zone and buoyantly rising to the surface.
The effects of meridional circulation in this class of dynamo models has yet to be investigated; this should be particularly interesting, since both analytic calculations and numerical simulations suggest a positive α-effect in the Northern hemisphere, which should then produce poleward propagation of the dynamo wave at low latitude. Meridional circulation could then perhaps produce equatorward propagation of the dynamo magnetic field even with a positive α-effect, as it does in true mean-field models (cf. Section 4.4). At any rate, further studies of dynamo models relying on this poloidal field regeneration mechanism should be vigorously pursued.
Babcock-Leighton models
Solar cycle models based on what is now called the Babcock-Leighton mechanism were first proposed by Babcock (1961) and further elaborated by Leighton (1964, 1969), yet they were all but eclipsed by the rise of mean-field electrodynamics in the mid- to late 1960’s. Their revival was motivated not only by the mounting difficulties with mean-field models alluded to earlier, but also by the fact that synoptic magnetographic monitoring over solar cycles 21 and 22 has offered strong evidence that the surface polar field reversals are indeed triggered by the decay of active regions (see Wang et al., 1989; Wang and Sheeley Jr, 1991, and references therein). The crucial question is whether this is a mere side-effect of dynamo action taking place independently somewhere in the solar interior, or a dominant contribution to the dynamo process itself.
The mode of operation of a generic solar cycle model based on the Babcock-Leighton mechanism is illustrated in cartoon form in Figure 17. Let Pn represent the amplitude of the high-latitude, surface (“A”) poloidal magnetic field in the late phases of cycle n, i.e., after the polar field has reversed. The poloidal field Pn is advected downward by meridional circulation (A→B), where it then starts to be sheared by the differential rotation while being also advected equatorward (B→C). This leads to the growth of a new low-latitude (C) toroidal flux system Tn+1, which becomes buoyantly unstable (C→D) and starts producing sunspots (D) which subsequently decay and release the poloidal flux Pn+1 associated with the new cycle n +1. Poleward advection and accumulation of this new flux at high latitudes (D→A) then obliterates the old poloidal flux Pn, and the above sequence of steps begins anew. Meridional circulation clearly plays a key role in this “conveyor belt” model of the solar cycle, by providing the needed link between the two spatially segregated source regions.
Formulation of a poloidal source term
As with all other dynamo models discussed thus far, the troublesome ingredient in dynamo models relying on the Babcock-Leighton mechanism is the specification of an appropriate poloidal source term, to be incorporated into the mean-field axisymmetric dynamo equations. In essence, all implementations discussed here are inspired by the results of numerical simulations of the buoyant rise of thin flux tubes, which, in principle allow to calculate the emergence latitude and tilts of BMRs, which is at the very heart of the Babcock-Leighton mechanism.
The first post-helioseismic dynamo model based on the Babcock-Leighton mechanism is due to Wang et al. (1991); these authors developed a coupled two-layer model (2 × 1D), where a poloidal source term is introduced in the upper (surface) layer, and made linearly proportional to the toroidal field strength at the corresponding latitude in the bottom layer. A similar non-local approach was later used by Dikpati and Charbonneau (1999) and Charbonneau et al. (2005) in their fully 2D axisymmetric model implementation, using a solar-like differential rotation and meridional flow profiles similar to Figures 5 and 11 herein. The otherwise much similar implementation of Nandy and Choudhuri (2001, 2002), on the other hand, uses a mean-field-like local α-effect, concentrated in the upper layers of the convective envelope and operating in conjunction with a “buoyancy algorithm” whereby toroidal fields located at the core-envelope interface are locally removed and deposited in the surface layers when their strength exceed some preset threshold. The implementation developed by Durney (1995) is probably closest to the essence of the Babcock-Leighton mechanism (see also Durney et al., 1993; Durney, 1996, 1997); whenever the deep-seated toroidal field exceeds some preset threshold, an axisymmetric “double ring” of vector potential is deposited in the surface layer, and left to spread latitudinally under the influence of magnetic diffusion.
In all cases the poloidal source term is concentrated in the outer convective envelope, and, in the language of mean-field electrodynamics, amounts to a positive α-effect, in that a positive dipole moment is being produced from a positive deep-seated mean toroidal field. The Dikpati and Charbonneau (1999) and Nandy and Choudhuri (2001) source terms both have an α-quenching-like upper operating threshold on the toroidal field strength. This is motivated by simulations of rising thin flux tubes, indicating that tubes with strength in excess of about 100 kG emerge without the E-W tilt required for the Babcock-Leighton mechanism to operate. The Durney (1995), Nandy and Choudhuri (2001), and Charbonneau et al. (2005) implementations also have a lower operating threshold, as suggested by thin flux tubes simulations.
Representative results
Figure 18 shows N-hemisphere time-latitude diagrams for the toroidal magnetic field at the core-envelope interface (Panel A), and the surface radial field (Panel B), for a representative Babcock-Leighton dynamo solution computed following the model implementation of Dikpati and Charbonneau (1999). The equatorward advection of the toroidal field by meridional circulation is here clearly apparent, as well as the concentration of the surface radial field near the pole. Note how the polar radial field changes from negative (blue) to positive (red) at just about the time of peak positive toroidal field at the core-envelope interface; this is the phase relationship inferred from synoptic magnetograms (see, e.g., Figure 4 herein) as well as observations of polar faculae (see Sheeley Jr, 1991).
Although it exhibits the desired equatorward propagation, the toroidal field butterfly diagram in Panel A of Figure 18 peaks at much higher latitude (∼ 45°) than the sunspot butterfly diagram (∼ 15° −20°, cf. Figure 3). This occurs because this is a solution with high magnetic diffusivity contrast, where meridional circulation closes at the core-envelope interface, so that the latitudinal component of differential rotation dominates the production of the toroidal field. This difficulty can be alleviated by letting the meridional circulation penetrate below the core-envelope interface. Solutions with such flows are presented, e.g., in Dikpati and Charbonneau (1999) and Nandy and Choudhuri (2001, 2002). These latter authors have argued that this is in fact essential for a solar-like butterfly diagram to materialize, but this conclusion appears to be model-dependent at least to some degree (Guerrero and Muñoz, 2004), and others have failed to reproduce some of their numerical results (see Dikpati et al., 2005), leaving the issue somewhat muddled at this juncture. At any rate, Babcock-Leighton dynamo solutions often do tend to produce strong polar branches, a consequence of both the strong radial shear present in the high-latitude portion of the tachocline, and of the concentration of the poloidal field taking place in the high latitude-surface layer prior to this field being advected down into the tachocline by meridional circulation (viz. Figure 17)
A noteworthy property of this class of model is the dependency of the cycle period on model parameters; over a wide portion of parameter space, the meridional flow speed is found to be the primary determinant of the cycle period P. For example, in the Dikpati and Charbonneau (1999) model, this quantity is found to scale as
$$P = 56.8 \, u_0^{- 0.89}s_0^{- 0.13}\eta _{\rm{T}}^{0.22}\;\;[{\rm{yr}}].$$
(35)
This behavior arises because, in these models, the two source regions are spatially segregated, and the time required for circulation to carry the poloidal field generated at the surface down to the tachocline is what effectively sets the cycle period. The corresponding time delay introduced in the dynamo process has rich dynamical consequences, to be discussed in Section 5.4 below.
Note finally that the weak dependency of P on ηT and on the magnitude s0 of the poloidal source term is very much unlike the behavior typically found in mean-field models, where both these parameters play a dominant role in setting the cycle period.
Critical assessment
As with most models including meridional circulation published to date, Babcock-Leighton dynamo models usually produce excessively strong polar surface magnetic fields. While this difficulty can be fixed by increasing the magnetic diffusivity in the outermost layers, in the context of the Babcock-Leighton models this then leads to a much weaker poloidal field being transported down to the tachocline, which can be problematic from the dynamo point-of-view. On this see Dikpati et al. (2004) for illustrative calculations, and Mason et al. (2002) on the closely related issue of competition between surface and deep-seated α-effect.
Because of the strong amplification of the surface poloidal field in the poleward-converging meridional flow, Babcock-Leighton models tend to produce a significant — and often dominant — polar branch in the toroidal field butterfly diagram. Many of the models explored to date tend to produce symmetric-parity solutions when computed pole-to-pole over a full meridional plane (see, e.g., Dikpati and Gilman, 2001), but it is not clear how serious a problem this is, as relatively minor changes to the model input ingredients may flip the dominant parity (see, e.g., Chatterjee et al., 2004, for a specific, if physically curious, example).
Because the Babcock-Leighton mechanism is characterized by a lower operating threshold, the resulting dynamo models are not self-excited. On the other hand, the Babcock-Leighton mechanism is expected to operate even for toroidal fields exceeding equipartition, the main uncertainties remaining the level of amplification taking place when sunspot-forming toroidal flux ropes form from the dynamo-generated mean magnetic field.
The nonlinear behavior of this class of models, at the level of magnetic backreaction on the differential rotation and meridional circulation, remains unexplored.
Numerical simulations of solar dynamo action
Ultimately, the solar dynamo problem should be tackled as a (numerical) solution of the complete set of MHD partial differential equations in a rotating, spherical domain undergoing thermally-driven turbulent convection in its outer 30% in radius. It is a peculiar fact that the last full-fledged attempts to do so go back some some twenty years, to the simulations of Gilman and Miller (1981); Gilman (1983); Glatzmaier (1985a,b), despite the remarkable advances in computational capabilities having taken place in the intervening years.
These epoch-making simulations did produce cyclic dynamo action and latitudinal migratory patterns suggestive of the dynamo waves of mean-field theory. However, the associated differential rotation profile turned out non-solar, as did the magnetic field’s spatio-temporal evolution. In retrospect this is perhaps not surprising, as limitations in computing resources forced the simulations to be carried out in a parameter regime far removed from solar interior conditions. Since then and until recently, efforts on the full-sphere simulation front have gone mostly into the purely hydrodynamical problem of reproducing the large-scale flows in the solar convective envelope, as inferred by helioseismology (see, e.g., Miesch, 2005, and references therein). However, the recent numerical simulations of Brun et al. (2004) have reached a strongly turbulent regime, and have managed to produce a reasonably strong mean magnetic field, but without equatorward migration or polarity reversals. These authors suggest that these failings can be traced to the absence of a tachocline-like stable region of strong shear at the bottom of their simulation domain. If this is the case, then direct numerical simulation of the solar cycle will turn out to be even more demanding computationally than hitherto believed.
A number of attempts have also been made to reproduce some salient features of the solar cycle by carrying out high-resolution, local simulations in parameters regimes closer to solar-like (see, e.g., Nordlund et al., 1992; Tobias et al., 2001; Ossendrijver et al., 2002). One issue that has received much attention in the past decade is the interaction of a turbulent, convecting fluid (“convection zone”) with a shear flow concentrated in an underlying stably stratified “tachocline” region (see, e.g., Brummell et al., 2002). Such simulations have shown that magnetic fields can be pumped into the stable regions by convective downdrafts, and produce elongated structures reminiscent of magnetic flux tubes aligned with the flow. The recent simulations of Cline et al. (2003) are particularly interesting in this respect, as they have managed to produce sustained, reasonably regular cyclic activity with occasional polarity reversals for extended periods of simulation time. At first sight this may look superficially like an αΩ dynamo, but the turbulent flow has no net helicity since rotation is not included in the simulations; the poloidal field is not being regenerated by a mean-field-like turbulent α-effect, but rather by the interaction between buoyancy and the Kelvin-Helmholtz instability in the shear layer. Such fascinating numerical experiments must clearly be pursued.