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 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.

Figure 17:

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.

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 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 $a$ -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 $a$ -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 $a$ -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.

4.8.2 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 ).

Figure 18:

Time-latitude diagrams of the surface 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 $Dj = 1/300$ , the envelope diffusivity $11 2 -1 jT = 2.5 × 10 cm s$ , 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 18

peaks at much higher latitude ( $o ~ 45$ ) than the sunspot butterfly diagram ( $~ 15o- 20o$ , 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

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 $j T$ and on the magnitude $s 0$ 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.

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 $a$ -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.