4.4 Mean-field models including meridional circulation

Meridional circulation is unavoidable in turbulent, compressible rotating convective shells. It basically results from an imbalance between Reynolds stresses and buoyancy forces. The ∼ 15 m s–1 poleward flow observed at the surface (see, e.g., Hathaway, 1996; Ulrich and Boyden, 2005Jump To The Next Citation Point) has now been detected helioseismically, down to r∕R ⊙ ≃ 0.85 (Schou and Bogart, 1998; Braun and Fan, 1998), without significant departure from the poleward direction except locally and very close to the surface, in the vicinity of active region belts (see Gizon, 2004; Gizon and Rempel, 2008Jump To The Next Citation Point, and references therein), and in polar latitudes at some phases of the solar cycle (Haber et al., 2002Jump To The Next Citation Point). Long considered unimportant from the dynamo point of view, meridional circulation has gained popularity in recent years, initially in the Babcock–Leighton context but now also in other classes of models.

Accordingly, we now add a steady meridional circulation to our basic αΩ models of Section 4.2. The convenient parametric form developed by van Ballegooijen and Choudhuri (1988Jump To The Next Citation Point) 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 rb. Circulation streamlines are shown in Figure 10View Image, together with radial cuts of the latitudinal component at mid-latitudes (𝜃 = π∕4). 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.

View Image

Figure 10: Streamlines of meridional circulation (Panel A), together with the total magnetic diffusivity profile defined by Equation (17View Equation) (dash-dotted line) and a mid-latitude radial cut of u 𝜃 (bottom panel). The dotted line is the core-envelope interface. This is the analytic flow of van Ballegooijen and Choudhuri (1988), with parameter values m = 0.5, p = 0.25, q = 0, and rb = 0.675.

The inclusion of meridional circulation in the non-dimensionalized αΩ 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 u0:

u R Rm = --0-⊙-. (33 ) ηT
Using the value u0 = 1500 cm s−1 from observations of the observed poleward surface meridional flow leads to Rm ≃ 200, again with η = 5 × 1011 cm2 s− 1 T. In the solar cycle context, using higher values of Rm thus implies proportionally lower turbulent diffusivities.

4.4.1 Representative results

Meridional circulation can bodily transport the dynamo-generated magnetic field (terms labeled “advective transport” in Equations (11View Equation, 12View Equation)), 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 (30View Equation). This was nicely demonstrated by Choudhuri et al. (1995), in the context of a mean-field αΩ model with a positive α-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 αΩ dynamo solution with α ∼ cos𝜃 (Figure 8View ImageA), new solutions are now recomputed, this time including meridional circulation. An animation of a typical solution is shown in Figure 11Watch/download Movie, and a sequence of time-latitude diagrams for four increasing values of the circulation flow speed, as measured by Rm, are plotted in Figure 12View Image.

At Rm = 50, little difference is seen with the circulation-free solutions (cf. Figure 8View ImageA), 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 11Watch/download Movie) 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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Figure 11: mpg-Movie (2347 KB) Meridional plane animations for an α Ω dynamo solutions including meridional circulation. With Rm = 103, this solution is operating in the advection-dominated regime as a flux-transport dynamo. The corresponding time-latitude “butterfly” diagram is plotted in Figure 12View ImageC below. Color-coding of the toroidal magnetic field and poloidal fieldlines as in Figure 7Watch/download Movie.
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Figure 12: Time-latitude “butterfly” diagrams for the α-quenched αΩ solutions depicted earlier in Panel A of Figure 8View Image, except that meridional circulation is now included, with (A) Rm = 50, (B) Rm = 100, (C) Rm = 1000, and (D) Rm = 2000 For the turbulent diffusivity value adopted here, 11 2 −1 ηT = 5 × 10 cm s, Rm = 200 would corresponds to a solar-like circulation speed.

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 Cα = − 10 solution with α ∼ sin2 𝜃cos 𝜃 (Figure 8View ImageC) transits to a steady mode as Rm increases above ∼ 102. Moreover, the sequence of α ∼ cos𝜃 shown in Figure 12View Image actually presents a narrow window around Rm ∼ 200 where the dynamo is decaying, due to a form of destructive interference between the high-latitude αΩ 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 α-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), αΩ 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 13View Image, for the same sequence of α Ω solutions with α ∼ cos 𝜃 as in Figure 12View Image, at Rm = 0, 50, 100, and 500 (panels A – D). For low circulation speeds (Rm ≲ 50), 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 8View ImageA and 12View ImageA). At higher circulation speeds, however, the surface magnetic field is swept instead towards the pole (see Figure 13View ImageC), becoming strongly concentrated and amplified there for Rm exceeding a few hundreds (Figures 11Watch/download Movie and 13View ImageD).

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Figure 13: Time-latitude diagrams of the surface radial magnetic field, for increasing values of the circulation speed, as measured by the Reynolds number Rm. This is for the same reference αΩ with α ∼ cos𝜃 as in Figures 8View ImageA and 12View Image. Note the marked increased of the peak surface field strength as Rm exceeds ∼ 100.

4.4.2 Critical assessment

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 αΩ 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 α-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., 2004Jump To The Next Citation Point), or if a counterrotating meridional flow cell is introduced in the high latitude regions (Dikpati et al., 2004Jump To The Next Citation Point; Jiang et al., 2009Jump To The Next Citation Point), 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 u p is specified a priori. Meridional circulation is a relatively weak flow in the bottom half of the solar convective envelope (see Miesch, 2005Jump To The Next Citation Point), 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, 2006aJump To The Next Citation Point,bJump To The Next Citation Point).

Before leaving the realm of mean-field dynamo models it is worth noting that many of the conceptual difficulties associated with calculations of the α-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.

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