3.4 Meridional circulation

The differential rotation is the axisymmetric component of the mean longitudinal flow, $< vf >$ . The axisymmetric flow in the meridional plane, $< vh >$ and $< vr >$ , is generally known as the meridional circulation.

The meridional circulation in the solar envelope is much weaker than the differential rotation, making it relatively difficult to measure (e.g., Hathaway, 1996a ). Furthermore, although it can in principle be probed using global helioseismology (Woodard, 2000 ), the effect of meridional circulation on global acoustic oscillations is small and may be difficult to distinguish from rotational and magnetic effects (Giles et al., 1997 ). Thus, we must currently rely on surface measurements and local helioseismology.

Early attempts to measure the mean meridional circulation in the solar photosphere by both Doppler and tracer techniques (reviewed by Hathaway, 1996a ; Snodgrass and Dailey, 1996 ; Latushko, 1996 ) varied dramatically. Many suggested a poleward flow of $~ 10- 20 m s-1$ , but others found amplitudes ranging from $-1 1- 100 m s$ and complex latitudinal structure with both poleward and equatorward flows, multiple cells, and large asymmetries about the equator.

Figure 2:

Spatial and temporal variation of the meridional circulation in the surface layers of the Sun. (a) The colatitudinal velocity $<vh>$ in the solar photosphere obtained from Doppler measurements, averaged over longitude and time. Positive values represent southward flow and different curves correspond to adjacent 6-month averaging intervals between 1992 and 1995 (from Hathaway, 1996b

). (b) $<vh>$ as a function of latitude and depth inferred from ring-diagram analysis. Each inversion is averaged over a 3-month interval and results are shown for 1997, 1999, and 2001. Grey and white regions represent southward and northward flow, respectively. A contour plot of the velocity amplitude underlies the arrow plots, with contours labeled in $-1 m s$ . Flow near the surface and in the southern hemisphere is generally poleward but beginning in 1998, equatorward circulation is found in the northern hemisphere at depths below $~ 3 Mm$ (from Haber et al., 2002

More recent Doppler measurements of the photospheric meridional circulation by Hathaway (1996b

) and Hathaway (1996a

) yield a poleward flow of about $20 m s- 1$ on average, confirming many of the earlier results. This mean poleward flow is nearly symmetric about the equator and peaks at latitudes of about 40 $o$ . However, Hathaway (1996a ) found substantial monthly and yearly variations in the flow amplitude and profile, reaching speeds of up to $- 1 50 m s$ (panel a in Figure 2

). Doppler measurements by Ulrich et al. (1988 ) showed even larger fluctuations, sometimes reversing sign and becoming equatorward.

Recent estimates of the meridional circulation obtained from the cross-correlation of magnetic features yield an average latitudinal flow which is poleward at low latitudes and weakly equatorward at high latitudes, with a peak amplitude of about $-1 15 m s$ (Komm et al., 1993 ; Snodgrass and Dailey, 1996 ; Latushko, 1996 ). However, these methods too exhibit large temporal variations. In the 26-year interval studied by Snodgrass and Dailey (1996 ), the meridional flow achieves amplitudes as large as $50 m s-1$ and often becomes equatorward at latitudes below $20o$ and above $o 40$ .

Local helioseismology provides an alternative to surface measurements and gives us the capability of probing the meridional flow below the photosphere. Near the surface the results are generally consistent with Doppler and tracer measurements, showing poleward flow of about $20 m s-1$ with substantial time variation and significant asymmetry about the equator (Giles et al., 1997 ; Chou and Dai, 2001 ; Haber et al., 2002 ; Basu and Antia, 2003 ; Zhao and Kosovichev, 2004 ).

Below the surface, Haber et al. (2002 ) have reported a flow reversal in the northern hemisphere where the circulation becomes equatorward at depths greater than about 3 Mm below the photosphere ( $r ~ 0.99Ro .$ ), down to the limit of their sampling domain which lies at a depth of $15 Mm$ (panel b in Figure 2 ). Their ring-diagram analysis spans six years, from $1996 -2001$ , with the flow reversal occurring in the latter four, from $1998 - 2001$ . Such a flow reversal is not evident in the time-distance results of Zhao and Kosovichev (2004 ) who present meridional flows averaged over depths of $3- 4.5 Mm$ and $6 -9 Mm$ . Several local helioseismic studies have attempted to probe deeper still. Giles et al. (1997 ) presented time-distance results for the upper $4%$ of the solar interior and concluded that the meridional flow throughout this region was poleward. Braun and Fan (1998 ) similarly find no evidence for a return equatorward flow down to $0.85Ro .$ . Inferring the circulation at depth below about $0.98Ro .$ is a difficult task and it is still too early to know what to make of these efforts.

There is evidence from both surface measurements and local helioseismology that the amplitude of the meridional circulation may be anticorrelated with magnetic activity, decreasing during solar maximum and increasing during solar minimum (Komm et al., 1993 ; Chou and Dai, 2001 ; Basu and Antia, 2003 ). Furthermore, a weak meridional circulation component of a few $-1 m s$ has been found which diverges out of magnetic activity belts and propagates with them toward the equator as the activity cycle progresses (Snodgrass and Dailey, 1996 ; Beck et al., 2002 ). However, Zhao and Kosovichev (2004 ) report the opposite: weak meridional flows which converge toward activity belts. They argue that the convergence occurs in the outermost layers, less than $~ 12 Mm$ below the photosphere whereas the divergence occurs deeper down.

Although much progress has been made in recent years, improving our understanding of the meridional circulation throughout the convective envelope remains an important challenge for local helioseismology in particular and will be a major research focus in the near future.