Accordingly, we now add a steady meridional circulation to our basic 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 . Circulation streamlines are shown in Figure 11, together with radial cuts of the latitudinal component at midlatitudes (). The flow is poleward in the outer convection zone, with an equatorial return flow peaking slightly above the coreenvelope interface, and rapidly vanishing below.

Meridional circulation can bodily transport the dynamogenerated magnetic field (terms labeled “advective transport” in Equations (11, 12)), and therefore, for a (presumably) solarlike equatorward return flow that is vigorous enough  in the sense of being large enough  overpower the ParkerYoshimura propagation rule embodied in Equation (31). This was nicely demonstrated by Choudhuri et al. (1995), in the context of a meanfield model with a positive effect concentrated near the surface, and a latitudeindependent, purely radial shear at the coreenvelope interface. With a solarlike differential rotation profile, however, once again the situation is far more complex.
Starting from the three dynamo solutions with the 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 . At , little difference is seen with the circulationfree solutions, except for the solution with equatoriallyconcentrated effect (see Panel A of Figure 12), where the equatorial branch is now dominant and the polar branch has shifted to midlatitudes and has become doublyperiodic. At , corresponding here to a solarlike circulation speed, drastic changes have materialized in all solutions. The negative solution has now transited to a steady dynamo mode, that in fact persists to higher values (panels F and I). The solution with is decaying at , while the solution with equatoriallyconcentrated effect starts to show a hint of equatorward propagation at midlatitudes (Panel D). At , the circulation has overwhelmed the dynamo wave, and both positive solutions show equatoriallypropagating toroidal fields (Panels G and H). Qualitatively similar results were obtained by Küker et al. (2001) using different prescriptions for the effect and solarlike differential rotation (see in particular their Figure 11; see also Rüdiger and Elstner, 2002; Bonanno et al., 2003).

Meridional circulation can also dominate the spatiotemporal evolution of the radial surface magnetic field, as shown in Figure 13 for a sequence of solutions with , , and (corresponding toroidal butterfly diagram at the coreenvelope interface are plotted in Panel B of Figure 7 and in Panel A and D of Figure 12).

From the modelling pointofview, 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 meanfield 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 is also somewhat problematic. A more fundamental and potential serious difficulty harks back to the kinematic approximation, whereby the form and speed of 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 equipartitionstrength magnetic fields should not be taken for granted.
Before leaving the realm of meanfield dynamo models it is worth noting that many of the conceptual difficulties associated with calculations of the effect and turbulent diffusivity are not unique to the meanfield 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.
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