6.4 Meridional circulation

In numerical simulations, as in the Sun, the meridional circulation is weak relative to the differential rotation. The kinetic energy is typically smaller by about two orders of magnitude. Sample profiles are illustrated in Figure 17

for case M3 and case P, which is a continuation of Case TUR of Miesch et al. (2000

), with increased resolution and lower dissipation⁸ .

Figure 17:

Streamlines are shown for the mean meridional mass flux in Case M3 (a) and Case P (b), as defined by the streamfunction $Y$ in Equation (13

). Red/orange tones and black contours denote clockwise circulation whereas blue tones and green contours denote counter-clockwise circulations. The right frames show the corresponding latitudinal velocity (positive southward) near the top (c) and bottom (d) of the convection zone for each simulation (represented by blue and red lines, respectively). All results are averaged over longitude and time (60 days for Case M3 and 72 days for case P).

The first thing to note about the profiles shown in Figure 17

is that they are much more spatially complex than is assumed in many kinematic dynamo models and other applications. Multiple cells are present in both latitude and radius, and flow patterns are generally not symmetric about the equator. The temporal dependence is equally complex, exhibiting large fluctuations on timescales of weeks and months, as shown in Figure 18

. This spatial and temporal complexity can be attributed to the turbulent nature of the convection and to the sensitivity of the meridional circulation to small variations in the differential rotation and Reynolds stress, as will be discussed further later in this section.

Figure 18:

Movie showing streamlines for the longitudinally-averaged mass flux in Case M3 are shown evolving over the course of 60 days. Contours are indicated as in panel a of Figure 17

, which represents a temporal average of this sequence of images. The inset illustrates the mean latitudinal velocity $< vh >$ near the top of the domain ( $r = 0.96Ro .$ ) as in the temporal average of panel c in Figure 17

Although the spatial and temporal fluctuations are generally chaotic, systematic patterns emerge when the circulation profiles are averaged over several months. In the equatorial plane, the circulation in the upper convection zone is typically outward, giving rise to poleward flow at low latitudes near the surface. This can be seen for case M3 in panel a of Figure 17

and panel c of Figure 17

⁹ . This outward flow arises primarily as a result of the centrifugal force acting on the prograde differential rotation at low latitudes. Another systematic trend which is robust in simulations of penetrative convection is a persistent equatorward circulation in the overshoot region of a few $m s-1$ (Figure 17

, panel d). This can be attributed to the turbulent alignment of downflow plumes as illustrated in panel b of Figure 15

and as discussed in Section 4.3. In turbulent parameter regimes, convective overshoot is dominated by helical downflow plumes which are tilted toward the rotation axis with respect to the vertical. When these plumes reach the overshoot region, negative buoyancy removes their vertical momentum but an equatorward latitudinal momentum remains. This equatorward circulation does not occur in more laminar simulations which do not exhibit turbulent plumes (Miesch et al., 2000

Near the poles, simulations generally exhibit several circulation cells which span about $o o 10 -15$ in latitude and extend from the top of the convection zone to the bottom. The sense of the circulation can vary with time and may or may not be the same in the northern and southern hemispheres. Without exception, simulations which exhibit solar-like differential rotation profiles have such localized circulation cells near the poles. Since axisymmetric circulations tend to conserve angular momentum, a single, global cell extending from low to high latitudes would tend to spin up the poles, driving a polar vortex which is inconsistent with helioseismic inversions (see Section 6.3).

How do these simulation results compare with what we know about the meridional circulation in the Sun? Our knowledge of the solar circulation is currently limited to the uppermost regions of the convection zone (see Section 3.4). There the circulation is generally poleward, although it does fluctuate substantially and is not in general symmetric about the equator. Some of these fluctuations appear to be associated with magnetic activity and exhibit a systematic equatorward propagation over the course of the solar activity cycle in conjunction with torsional oscillations (Snodgrass and Dailey, 1996 ; Beck et al., 2002 ; Zhao and Kosovichev, 2004 ). Fluctuations of comparable amplitude occur in simulations both with and without magnetic fields, but they do not exhibit such systematic latitudinal propagation.

The poleward circulation in the Sun is about the same amplitude as in simulations, $~ 20 m s-1$ , but it extends to higher latitudes. Doppler measurements and local helioseismic inversions indicate poleward flow in the solar surface layers up to latitudes of at least $o 60$ . By comparison, the poleward flow near the outer boundary in simulations generally only extends to latitudes of about $30o -50o$ (Figure 17 , panel c). Little is currently known about circulation patterns in the polar regions of the Sun but surface tracer measurements do show some hints of flow reversals at latitudes above $o 60$ (Komm et al., 1993 ; Snodgrass and Dailey, 1996 ; Latushko, 1996 ). Multiple-cell structure in the polar regions such as that seen in simulations has not yet been unambiguously found in surface measurements or helioseismic inversions but it cannot be ruled out.

Further insight into the maintenance of meridional circulation in global convection simulations can be obtained by considering the balance Equation (15 ). If we apply a Legendre transform to this equation, we obtain an evolution equation for the mass flux vorticity in spectral space: $~p$ . We may then multiply by $p~$ and integrate over radius to obtain

where $2 W(l) = ~p /2$ is the mass flux enstrophy spectrum associated with the circulation and the terms on the right-hand-side reflect contributions from the Reynolds stress, axisymmetric advection, the buoyancy force, and viscous diffusion (see Appendix A.5). The spectrum $W(l)$ is shown in Figure 19

, frames (a) and (b), along with the corresponding spectrum for the streamfunction, $Y$ , defined in Equation (13

Figure 19:

(a) Power spectra are shown for the mass flux vorticity $p$ (red) and the streamfunction $Y$ (blue) for case P, averaged over radius and time [see Equations (12

) and (13

)]. The former curve (red) is equivalent to $W$ in Equation (24

). Spectra are normalized such that they sum to unity. Exponential fits to each curve are also shown for comparison. Frame (b) exhibits the same curves as in frame (a) but with a linear vertical axis and a logarithmic horizontal axis. Frame (c) shows the relative contributions of the maintenance terms in Equation (24

), using the same normalization as for $W$ in frames (a) and (b). In frame (d), the Reynold stress contribution, represented by the blue curve in (c), is decomposed into contributions from radial advection, radial tipping, and latitudinal transport as described in the text. The plots in (b)-(d) extend only to $l = 100$ as contributions beyond this point are negligible.

The density-weighted enstrophy spectrum, $W(l)$ , decays roughly exponentially with the spherical harmonic degree, $l$ , with an e-folding scale of $lp ~ 31$ . The streamfunction spectrum is steeper, with an e-folding scale of $lY ~ 22$ over the range shown in panel a of Figure 19

. However, it is not as well approximated by an exponential distribution, being somewhat more intermittent. As is most evident in panel b of Figure 19

, most of the power in both $p$ and $Y$ is concentrated at large scales, $l < 20$ , and in odd values of $l$ . Odd $l$ values correspond to $p$ , $Y$ , and $< vh >$ profiles which are antisymmetric about the equator and $< vr >$ profiles which are symmetric. For example, a single large-scale circulation cell per hemisphere with upflow at the equator and downflow at the poles is generally dominated by the $l = 1$ and $l = 3$ components of $p$ and $Y$ (depending on the latitude at which it turns over).

The maintenance terms on the right-hand-side of Equation (24 ) are shown in panel c of Figure 19 . The sum of all contributions is nearly zero, indicating a statistically steady state. Although the $GAD$ and $GBF$ terms dominate the total flux represented in Equation (16 ), they are largely offset by pressure gradients. Large-scale circulations ( $l = 1- 3$ ) are driven by the Reynolds stress (RS) and the residual buoyancy force (BF), which are balanced by axisymmetric advection (AD) and viscous diffusion (VD). On intermediate scales, $4 < l < 10$ , axisymmetric advection is the primary driving mechanism and the Reynolds stress plays an inhibiting role.

It is instructive to further decompose the Reynolds stress contribution in order to clarify which processes are most relevant. According to Equation (76 ) in Appendix A.4, the radial component of the Reynolds stress includes contributions from vorticity advection $< > oc v'rw'f$ and vortex tipping, $< > oc v'fw'r$ . These contributions are plotted separately in panel d of Figure 19 along with that due to the latitudinal component of the Reynolds stress. This figure indicates that the radial tipping term is most important, followed closely by the radial advection term. The latitudinal Reynolds stress is less significant.

Given the important role of the turbulent Reynolds stress, it is perhaps no surprise that the circulation patterns are complex. If the solar meridional circulation is as spatially and temporally variable as the simulations suggest, then this has important implications for kinematic dynamo models. It may pose problems for flux-transport dynamo models in particular which rely on a steady large-scale circulation component to set the period and other aspects of the magnetic activity cycle (Choudhuri et al., 1995 ; Durney, 1995 ; Dikpati and Charbonneau, 1999 ; Dikpati and Gilman, 2001a ; Charbonneau, 2005 ). On the other hand, the success of flux-transport dynamo models in reproducing many features of the solar cycle may point to some shortcomings of global convection simulations. The maintenance of differential rotation in the solar convection zone is subtle, involving small imbalances among relatively large forces. Simulations may be sensitive to dynamics which are not sufficiently resolved or otherwise missing from the model. Still, in light of this delicate balance, it would be surprising if the solar meridional circulation did not fluctuate substantially in space and time.

One feature that global convection simulations and flux-transport dynamo models have in common is an equatorward circulation in the overshoot region. Hathaway et al. (2003 ) argue that the observed drift speeds of sunspots as a function of latitude support the presence of such a flow. Some flux-transport models require that this equatorial circulation extend even below the overshoot region (Nandy and Choudhuri, 2002 ). However, any circulation which is driven in the convection zone is unlikely to penetrate deeper than $r ~ 0.7Ro .$ due to the strongly limiting influence of buoyancy and rotation (Gilman and Miesch, 2004 ). Secondary circulations may be driven by waves and turbulence in the radiative interior but these are likely to be much weaker than those in the convection zone (see Section 8).