The mode of operation of a generic solar cycle model based on the Babcock–Leighton mechanism is illustrated in cartoon form in Figure 16. Let represent the amplitude of the high-latitude, surface (“A”) poloidal magnetic field in the late phases of cycle , i.e., after the polar field has reversed. The poloidal field is advected downward by meridional circulation (AB), where it then starts to be sheared by the differential rotation while being also advected equatorward (BC). This leads to the growth of a new low-latitude (C) toroidal flux system , which becomes buoyantly unstable (CD) and starts producing sunspots (D) which subsequently decay and release the poloidal flux associated with the new cycle . Poleward advection and accumulation of this new flux at high latitudes (DA) then obliterates the old poloidal flux , 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. 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., 2004; Jiang et al., 2009).
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 (1999), Charbonneau et al. (2005) and Guerrero and de Gouveia Dal Pino (2008) in their 2D axisymmetric model implementation, using a solar-like differential rotation and meridional flow profiles similar to Figures 5 and 10 herein. The otherwise much similar implementation of Nandy and Choudhuri (2001, 2002) and Chatterjee et al. (2004), on the other hand, uses a mean-field-like local -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. 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 -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 -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 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.
Figure 17 is a meridional plane animation of a representative Babcock–Leighton dynamo solution computed following the model implementation of Charbonneau et al. (2005). 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 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 Babcock–Leighton dynamo solution now computed following the closely similar model implementation of Dikpati and Charbonneau (1999). 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, a situation that persists in models using more realistic differential profiles taken from helioseismic inversions (see Muñoz-Jaramillo et al., 2009). 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; Guerrero and de Gouveia Dal Pino, 2007; Muñoz-Jaramillo et al., 2009). 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 (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 . For example, in the Dikpati and Charbonneau (1999) model, this quantity is found to scale aset 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 (2008) is shown in Figure 19. In this specific solution the circulation penetrates only down to , and the radial and latitudinal peak pumping speed are and , respectively.
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 (2008) are found to obey a scaling law of the form
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 -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., 2004; Charbonneau, 2007a, 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, 2005, 2007a; Chatterjee and Choudhuri, 2006).
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.
Living Rev. Solar Phys. 7, (2010), 3
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