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 10, together with radial cuts of the latitudinal component at mid-latitudes (). 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 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 :
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 equatorward return flow that is vigorous enough – in the sense of Rm being large enough – overpower the Parker–Yoshimura propagation rule embodied in Equation (30). This was nicely demonstrated by Choudhuri et al. (1995), in the context of a mean-field model with a positive -effect concentrated near the surface, and a latitude-independent, purely radial shear at the core-envelope interface. 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. Models achieving equatorward propagation of the deep toroidal magnetic component in this manner are now often called flux-transport dynamos.
With a solar-like differential rotation profile, however, once again the situation is far more complex. Starting from the most basic dynamo solution with (Figure 8A), new solutions are now recomputed, this time including meridional circulation. An animation of a typical solution is shown in Figure 11, and a sequence of time-latitude diagrams for four increasing values of the circulation flow speed, as measured by Rm, are plotted in Figure 12.
At Rm = 50, little difference is seen with the circulation-free solutions (cf. Figure 8A), except for an increase in the cycle frequency, due to the Doppler shift experienced by the equatorwardly propagating dynamo wave (see Roberts and Stix, 1972). At Rm = 100 (part B), the cycle frequency has further increased and the poloidal component produced in the high-latitude region of the tachocline is now advected to the equatorial regions on a timescale becoming comparable to the cycle period, so that a cyclic activity, albeit with a longer period, becomes apparent at low latitudes. At Rm = 103 (panel C and animation in Figure 11) the dynamo mode now peaks at mid-latitude, a consequence of the inductive action of the latitudinal shear, favored by the significant stretching experienced by the poloidal fieldlines as they get advected equatorward. At Rm = 2000 the original high latitude dynamo mode has all but vanished, and the mid-latitude mode is dominant. The cycle period is now set primarily by the turnover time of the meridional flow; this is the telltale signature of flux-transport dynamos.
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All this may look straightforward, but it must be emphasized that not all dynamo models with solar-like differential rotation behave in this (relatively) simple manner. For example, the solution with (Figure 8C) transits to a steady mode as Rm increases above . Moreover, the sequence of shown in Figure 12 actually presents a narrow window around where the dynamo is decaying, due to a form of destructive interference between the high-latitude mode and the mid-latitude advection-dominated dynamo mode that dominates at higher values of Rm. Qualitatively similar results were obtained by Küker et al. (2001) using different prescriptions for the -effect and solar-like differential rotation (see in particular their Figure 11; see also Rüdiger and Elstner, 2002; Bonanno et al., 2003). When field transport by turbulent pumping are included (see Käpylä et al., 2006b), models including meridional circulation can provide time-latitude “butterfly” diagrams that are reasonably solar-like.
Even if the meridional flow is too slow – or the turbulent magnetic diffusivity too high – to force the dynamo model in the advection-dominated regime, being much faster at the surface the poleward flow can dominate the spatio-temporal evolution of the radial surface magnetic field, as shown in Figure 13, for the same sequence of solutions with as in Figure 12, at Rm = 0, 50, 100, and 500 (panels A – D). For low circulation speeds (), the equatorward drift of the surface radial field is simply a diffused imprint of the equatorward drift of the deep-seated toroidal field (cf. Figure 8A and 12A). At higher circulation speeds, however, the surface magnetic field is swept instead towards the pole (see Figure 13C), becoming strongly concentrated and amplified there for Rm exceeding a few hundreds (Figures 11 and 13D).
From the modelling point-of-view, in the kinematic regime at least 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 models suggest a rather complex overall picture, and in interface dynamos the cartesian solutions obtained by Petrovay and Kerekes (2004) even suggest that dynamo action can be severely hindered. Yet, in other classes of models discussed below (Sections 4.5 and 4.8), circulation does have this desired effect (see also Seehafer and Pipin, 2009, for an intriguing mean-field model calculation not relying on the -effect).
On the other hand, dynamo models including meridional circulation tend to produce surface polar field strength largely in excess of observed values, unless magnetic diffusion is significantly enhanced in the surface layers, and/or field submergence takes place very efficiently. This is a 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), or if a counterrotating meridional flow cell is introduced in the high latitude regions (Dikpati et al., 2004; Jiang et al., 2009), a feature that has actually been detected in surface Doppler measurements as well as helioseismically during cycle 22 (see Haber et al., 2002; Ulrich and Boyden, 2005).
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), and the stochastic fluctuations of the Reynolds stresses powering it are expected to lead to strong spatiotemporal variations, and expectation verified by both analytical models (Rempel, 2005) and numerical simulations (Miesch, 2005). The ability of thus meridional flow to merrily advect equipartition-strength magnetic fields should not be taken for granted (but do see Rempel, 2006a,b).
Before leaving the realm of mean-field 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 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. In this respect, the rather low value of the turbulent magnetic diffusivity needed to achieve high enough Rm in flux transport dynamos is also somewhat problematic, since the corresponding turbulent diffusivity ends up some two orders of magnitude below the (uncertain) mean-field estimates. However, the model calculations of Muñoz-Jaramillo et al. (2010a) indicate that magnetic diffusivity quenching may offer a viable solution to this latter quandary.
Living Rev. Solar Phys. 7, (2010), 3
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