4.8 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 1960s. 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, 1991Jump To The Next Citation Point, 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 16View Image. 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.

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Figure 16: Operation of a solar cycle model based on the Babcock–Leighton mechanism. The diagram is drawn in a meridional quadrant of the Sun, with streamlines of meridional circulation plotted in blue. Poloidal field having accumulated in the surface polar regions (“A”) at cycle n must first be advected down to the core-envelope interface (dotted line) before production of the toroidal field for cycle n + 1 can take place (B→C). Buoyant rise of flux rope to the surface (C→D) is a process taking place on a much shorter timescale.

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. Not surprisingly, topologically more complex multi-cells circulation patterns can lead to markedly different dynamo behavior (see, e.g., Bonanno et al., 2006; Jouve and Brun, 2007), and can also have a profound impact on the evolution of the surface magnetic field (Dikpati et al., 2004Jump To The Next Citation Point; Jiang et al., 2009).

4.8.1 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 latitudes 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 (1999Jump To The Next Citation Point), Charbonneau et al. (2005Jump To The Next Citation Point) and Guerrero and de Gouveia Dal Pino (2008Jump To The Next Citation Point) in their 2D axisymmetric model implementation, using a solar-like differential rotation and meridional flow profiles similar to Figures 5View Image and 10View Image herein. The otherwise much similar implementation of Nandy and Choudhuri (2001Jump To The Next Citation Point, 2002Jump To The Next Citation Point) and Chatterjee et al. (2004Jump To The Next Citation Point), 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 (1995Jump To The Next Citation Point) is probably closest to the essence of the Babcock–Leighton mechanism (see also Durney et al., 1993Jump To The Next Citation Point; 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. As shown by Muñoz-Jaramillo et al. (2010b), this formulation, used in conjunction with the axisymmetric models discussed in what follows, also leads to a good reproduction of the observed synoptic evolution of surface magnetic flux.

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 (1999Jump To The Next Citation Point) and Nandy and Choudhuri (2001Jump To The Next Citation Point) 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 strengths 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 (2001Jump To The Next Citation Point), and Charbonneau et al. (2005Jump To The Next Citation Point) implementations also have a lower operating threshold, as suggested by thin flux tubes simulations.

4.8.2 Representative results

Figure 17Watch/download Movie is a meridional plane animation of a representative Babcock–Leighton dynamo solution computed following the model implementation of Charbonneau et al. (2005Jump To The Next Citation Point). The equatorward advection of the deep toroidal field by meridional circulation is here clearly apparent. Note also how the surface poloidal field first builds up at low latitudes, and is subsequently advected poleward and concentrated near the pole.

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Figure 17: mpg-Movie (6019 KB) Meridional plane animation of a representative Babcock–Leighton dynamo solution from Charbonneau et al. (2005Jump To The Next Citation Point). Color coding of the toroidal field and poloidal fieldlines as in Figure 7Watch/download Movie. This solution uses the same differential rotation, magnetic diffusivity, and meridional circulation profile as for the advection-dominated αΩ solution of Section 4.4, but now with the non-local surface source term, as formulated in Charbonneau et al. (2005Jump To The Next Citation Point), and parameter values C α = 5, C Ω = 5 × 104, Δη = 0.003, Rm = 840. Note again the strong amplification of the surface polar fields, the latitudinal stretching of poloidal fieldlines by the meridional flow at the core-envelope interface.

Figure 18View Image 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 Babcock–Leighton dynamo solution now computed following the closely similar model implementation of Dikpati and Charbonneau (1999Jump To The Next Citation Point). 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 4View Image herein) as well as observations of polar faculae (see Sheeley Jr, 1991Jump To The Next Citation Point).

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Figure 18: Time-latitude diagrams of the toroidal field at the core-envelope interface (Panel A), and radial component of the surface magnetic field (Panel B) in a Babcock–Leighton model of the solar cycle. This solution is computed for solar-like differential rotation and meridional circulation, the latter here closing at the core-envelope interface. The core-to-envelope contrast in magnetic diffusivity is Δ η = 1∕300, the envelope diffusivity 11 2 −1 ηT = 2.5 × 10 cm s, and the (poleward) mid-latitude surface meridional flow speed is u0 = 16 m s− 1.

Although it exhibits the desired equatorward propagation, the toroidal field butterfly diagram in Panel A of Figure 18View Image peaks at much higher latitude (∼ 45°) than the sunspot butterfly diagram (∼ 15° – 20°, cf. Figure 3View Image). 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, a situation that persists in models using more realistic differential profiles taken from helioseismic inversions (see Muñoz-Jaramillo et al., 2009Jump To The Next Citation Point). 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 (1999Jump To The Next Citation Point) 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; Guerrero and de Gouveia Dal Pino, 2007; Muñoz-Jaramillo et al., 2009Jump To The Next Citation Point). From the hydrodynamical standpoint, the boundary layer analysis of Gilman and Miesch (2004) (see also Rüdiger et al., 2005) indicates no significant penetration below the base of the convective envelope, although this conclusion has not gone unchallenged (see Garaud and Brummell, 2008), leaving the whole issue somewhat muddled at this juncture. The present-day observed solar abundances of Lithium and Beryllium restrict the penetration depth to r∕R ≃ 0.62 (Charbonneau, 2007b), which is unfortunately too deep to pose very useful constraints on dynamo models, so that the final word will likely come from helioseismology, hopefully in the not too distant future.

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 (1999Jump To The Next Citation Point) model, this quantity is found to scale as

P = 56.8u −0.89s−0.13η0.22[yr]. (34 ) 0 0 T
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. 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 usually play a dominant role in setting the cycle period. The analysis of Hathaway et al. (2003) supports the idea that the solar cycle period is indeed set by the meridional flow speed (but do see Schmitt and Schüssler, 2004, for an opposing viewpoint). As demonstrated by Jouve et al. (2010), interesting constraints can also be obtained from the observed dependence of stellar cycle periods on rotation rates.

An interesting variation on the above model follows from the inclusion of turbulent pumping. With the expected downward pumping throughout the bulk of the convective envelope, and with a significant equatorward latitudinal component at low latitudes, the Babcock–Leighton mechanism can lead to dynamo action even if the meridional flow is constrained to the upper portion of the convective envelope. Downward turbulent pumping then links the two sources regions, and latitudinal pumping provides the needed equatorward concentration of the deep-seated toroidal component. An example taken from Guerrero and de Gouveia Dal Pino (2008Jump To The Next Citation Point) is shown in Figure 19View Image. In this specific solution the circulation penetrates only down to r∕R = 0.8, and the radial and latitudinal peak pumping speed are γ = 0.3 m s−1 r0 and γ = 0.9 m s−1 𝜃0, respectively.

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Figure 19: Time-latitude diagrams of the toroidal field at the core-envelope interface (Panel A), and radial component of the surface magnetic field (Panel B) in a Babcock–Leighton model of the solar cycle with a meridional flow restricted to the upper half of the convective envelope, and including (parametrized) radial and latitudinal turbulent pumping. This is a solution from Guerrero and de Gouveia Dal Pino (2008Jump To The Next Citation Point) (see their Section 3.3 and Figure 5), but the overall modelling framework is almost identical to that described earlier, and used to generate Figure 18View Image. The core-to-envelope contrast in magnetic diffusivity is Δ η = 1∕100, the envelope diffusivity ηT = 1011 cm2 s− 1, and the (poleward) mid-latitude surface meridional flow speed is u = 13 m s−1 0 (figure produced from numerical data kindly provided by G. Guerrero).

With downward turbulent pumping now the primary mechanism linking the surface and tachocline, the dynamo period loses sensitivity to the meridional flow speeds, and becomes set primary by the radial pumping speed. Indeed the dynamo solutions presented Guerrero and de Gouveia Dal Pino (2008Jump To The Next Citation Point) are found to obey a scaling law of the form

− 0.12 −0.51 − 0.05 P = 181.2u0 γr0 γ𝜃0 [yr], (35 )
over a fairly wide range of parameter values. The radial pumping speed γ r0 emerges here as the primary determinant of the cycle period. Finally, one can note in Figure 19View Image that the surface magnetic field no longer shows the strong concentration in the polar region that usually characterizes Babcock–Leighton dynamo solutions operating in the advection-dominate regime. This can be traced primarily to the efficient downward turbulent pumping that subducts the poloidal field as it is carried poleward by the meridional flow.

4.8.3 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. The model calculations of Guerrero and de Gouveia Dal Pino (2008) suggest that downward turbulent pumping may be a better option to reduce the strength of the polar field without impeding dynamo action.

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 Chatterjee et al., 2004Jump To The Next Citation Point; Charbonneau, 2007aJump To The Next Citation Point, for specific examples). Nonetheless, in the advection-dominated regime there is definitely a tendency for the quadrupolar symmetry of the meridional flow to imprint itself on the dynamo solutions. A related difficulty, in models operating in the advection-dominated regime, is the tendency for the dynamo to operate independently in each solar hemisphere, so that cross-hemispheric synchrony is lost (Charbonneau, 2005Jump To The Next Citation Point, 2007aJump To The Next Citation Point; Chatterjee and Choudhuri, 2006Jump To The Next Citation Point).

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 largely unexplored.


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