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 ) has now been detected helioseismically, down to $r/Ro . -~ 0.85$ (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 (Haber et al., 2002 ; Basu and Antia, 2003 ; Zhao and Kosovichev, 2004 ).

Accordingly, we now add a steady meridional circulation to our basic $a_O_$ 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 $rb$ . Circulation streamlines are shown in Figure 11 , together with radial cuts of the latitudinal component at mid-latitudes ( $h = p/4$ ). 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.

Figure 11:

Streamlines of meridional circulation (Panel A), together with the total magnetic diffusivity profile defined by Equation (19

) (dash-dotted line) and a mid-latitude radial cut of $uh$ (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 $a_O_$ 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 $u0$ :

Using the value $u0 = 1500 cm s-1$ from observations of the observed poleward surface meridional flow leads to $Rm -~ 200$ , again with $jT = 5× 1011 cm2 s-1$ .

4.4.1 Representative results

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 (31 ). This was nicely demonstrated by Choudhuri et al. (1995 ), in the context of a mean-field $a_O_$ model with a positive $a$ -effect concentrated near the surface, and a latitude-independent, purely radial shear at the core-envelope interface. With a solar-like differential rotation profile, however, once again the situation is far more complex.

Starting from the three $a_O_$ dynamo solutions with the $a$ -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 $Rm$ . At $Rm = 50$ , little difference is seen with the circulation-free solutions, except for the $Ca = +10$ solution with equatorially-concentrated $a$ -effect (see Panel A of Figure 12 ), where the equatorial branch is now dominant and the polar branch has shifted to mid-latitudes and has become doubly-periodic. At $Rm = 200$ , corresponding here to a solar-like circulation speed, drastic changes have materialized in all solutions. The negative $Ca$ solution has now transited to a steady dynamo mode, that in fact persists to higher $Rm$ values (panels F and I). The $Ca = +10$ solution with $a oc cosh$ is decaying at $Rm = 200$ , while the solution with equatorially-concentrated $a$ -effect starts to show a hint of equatorward propagation at mid-latitudes (Panel D). At $3 Rm = 10$ , the circulation has overwhelmed the dynamo wave, and both positive $Ca$ solutions show equatorially-propagating toroidal fields (Panels G and H). Qualitatively similar results were obtained by Küker et al. (2001 ) using different prescriptions for the $a$ -effect and solar-like differential rotation (see in particular their Figure 11; see also Rüdiger and Elstner, 2002 ; Bonanno et al., 2003 ).

Figure 12:

Time-latitude diagrams for three of the $a_O_$ solutions depicted earlier in Panels B to D of Figure 7

, except that meridional circulation is now included, with $Rm = 50$ (top row), $Rm = 200$ (middle row), and $Rm = 103$ (bottom row). For the turbulent diffusivity value adopted here, $jT = 5 × 1011 cm2 s-1$ , $Rm = 200$ corresponds to a solar-like circulation speed. Corresponding animations are available in Resource 3.

Evidently, meridional circulation can have a profound influence on the overall character of the solutions. 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. This can be seen in Figure 12

: At $Rm = 50$ the solutions in Panels A and B have markedly distinct (primary) cycle periods, while at $3 Rm = 10$ (Panels G and H) the cycle periods are nearly identical. Note however that significant effects require a large $Rm$ ( $>~~ 103$ for the circulation profile used here), which, $u0$ being fixed by surface observations, translates into a magnetic diffusivity $jT <~ 1011$ ; by most orders-of-magnitude estimates constructed in the framework of mean-field electrodynamics this is rather low.

Meridional circulation can also dominate the spatio-temporal evolution of the radial surface magnetic field, as shown in Figure 13 for a sequence of solutions with $Rm = 0$ , $50$ , and $200$ (corresponding toroidal butterfly diagram at the core-envelope interface are plotted in Panel B of Figure 7 and in Panel A and D of Figure 12 ).

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 an $a_O_$ solution with the $a$ -effect concentrated at low-latitude and at the base of the convective envelope (see Section 4.2.5 and Panel B of Figure 7

). Recall that the $Rm = 0$ solution in Panel A exhibits amplitude modulation (cf. Panel B of Figure 7

and Panel A of Figure 8

In the circulation-free solution ( $Rm = 0$ ), the equatorward drift of the surface radial field is a direct reflection of the equatorward drift of the deep-seated toroidal field (see Panel B of Figure 7

). With circulation turned on, however, the surface magnetic field is swept instead towards the pole (see Panel B of Figure 13

), becoming strongly concentrated and amplified there for solar-like circulation speeds ( $Rm = 200$ , see Panel C of Figure 13

4.4.2 Critical assessment

From the modelling point-of-view, 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 $a_O_$ 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 $Rm$ is also somewhat problematic. A more fundamental and potential serious difficulty harks back to the kinematic approximation, whereby the form and speed of $up$ 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 equipartition-strength magnetic fields should not be taken for granted.

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