5.1 Global scales

Flows in the upper convection zone have been measured by local helioseismology on a wide variety of scales, including differential rotation and meridional circulation, local flows around complexes of magnetic activity and sunspots, and convective flows. Here we mostly discuss the longitudinal averages of flows measured by local helioseismology, and their solar-cycle variations. The focus is on temporal variations that are slow compared to the typical lifetime of active regions. By computing longitudinal averages of rotation and meridional circulation over a few solar rotation periods it is possible to filter out small-scale flows and to reach a sensitivity of the order of $-1 1 m s$ near the surface. The evolution of flows through cycle 23 reveals connections between mass motions in the solar interior and the large-scale characteristics of the magnetic cycle.

5.1.1 Rotation and torsional oscillations

Measuring the mean rotation using local techniques is a useful exercise, in particular to compare with estimates from inversions of frequency-splitting coefficients. Giles et al. (1998 ) and Giles (1999 ) measured rotation with time-distance helioseismology applied to MDI data (structure and dynamics programs; see Scherrer et al., 1995 ). Figure 30 shows the angular velocity as a function of latitude for radii $r > 0.886Ro .$ . There is a good qualitative agreement between the time-distance and the global-mode inversions, although the agreement is not as good at high latitudes. Basu et al. (1998 , 1999 ) used the ring-diagram method to obtain the rotational velocity for $r > 0.97Ro .$ in the latitude range $|c|< 50o$ . The north-south symmetric component of rotation compares favorably with the rotation determined from the frequency splittings of the global p modes (Figure 31 ). Since the inversion results of p-mode splittings were not expected to be particularly reliable near the surface, the ring diagram results complement and support the earlier conclusion that rotation increases with depth, near the surface, at low latitudes. Possible explanations for the existence of this near-surface radial shear have been discussed by Gilman (2000 ) and Robinson and Chan (2001 ). The local helioseismology results also show measurable differences in the rotation profile between the northern and southern hemispheres (see, e.g., Basu et al., 1999 ; Giles, 1999 ; González Hernández and Patrón, 2000 ). It would be premature, however, to trust the validity of the north-south asymmetry in the rotation rate, since systematic errors are not fully understood and do vary with latitude.

Figure 30:

Comparison of the time-distance and normal mode methods for determining the solar rotation. The angular velocity is plotted versus latitude for six different depths. The solid curve is the symmetric component of the time-distance results, and the dashed lines are formal errors from the inversion. The dotted curve is the result of an OLA inversion of MDI frequency splittings. From Giles (1999

Figure 31:

North-south average of rotational velocity at different latitude from ring-diagram analysis. The results obtained using RLS inversions are shown by dashed lines (with dotted lines marking the 1 $s$ error limits) and the results obtained by using OLA inversions by crosses with error bars. For comparison, the solid line shows the rotational velocity obtained from inversion of global-mode frequency splittings (after subtracting out the surface rotation velocity used in tracking each region). From Basu et al. (1999

Howard and Labonte (1980 ) discovered small ( $± 10 m s-1$ ) latitudinal variations in the surface Doppler rotation profile that propagate toward the equator with the sunspot latitudes. These variations, known as torsional oscillations, have since been confirmed by global helioseismology (Kosovichev and Schou, 1997 ; Schou, 1999 ). Inversions of global-mode frequency splittings have shown that torsional oscillations persist at a depth over a large fraction of the convection zone (Howe et al., 2000a ), especially at high latitudes where they extend down to the bottom of the convection zone (Vorontsov et al., 2002

). At low latitudes, the data support the idea that the torsional oscillation pattern originates inside the convection zone and propagates both upward and equatorward (Vorontsov et al., 2002

; Basu and Antia, 2003

). Above $45o$ latitude, bands of faster and slower rotation appear to move toward the poles (Antia and Basu, 2001 ; Schou, 2003a ) suggesting a connection with the poleward migration of high-latitude magnetic activity seen at the surface (see, e.g., Callebaut and Makarov, 1992 ).

Both time-distance (Giles et al., 1998 ; Beck et al., 2002 ; Zhao and Kosovichev, 2004 ) and ring-diagram (Basu et al., 1999 ; Basu and Antia, 2000 ; Haber et al., 2000 , 2002 ) analyses have confirmed the global-mode measurements of torsional oscillations. Figure 32 shows a plot of the zonal flows at depths of $0.9 Mm$ and $7.1 Mm$ obtained from ring-diagram analysis. At both these depths, propagating zonal fast bands are seen to migrate toward the equator as the solar cycle progresses. As will be later discussed in Section 5.2, local techniques can also provide information about the longitudinal structure of the zonal flows.

Figure 32:

Zonal flows from ring-diagram analysis at depths of $0.9 Mm$ (dashed curve) and $7.1 Mm$ (solid curve). Each panel corresponds to the average over yearly MDI Dynamics Program intervals as indicated. The zonal velocity plots for each year have been offset by subtracting a depth-dependent constant. The error bars shown are 10 times larger than the estimated formal errors in order to be visible. From Haber et al. (2002

Schüssler (1981 ) and Yoshimura (1981 ) suggested that the torsional oscillations may be driven by the Lorentz force due to a migrating dynamo wave. Recent numerical calculations by Covas et al. (2000 ) seem to be consistent with this hypothesis; their calculations also reproduce the poleward propagating torsional oscillations at high latitudes. Other explanations attribute the torsional oscillations to the feedback of the smaller-scale magnetic fields on the angular momentum transport mechanisms responsible for differential rotation, e.g., changes in the Reynolds/Maxwell stresses (Küker et al., 1996 ; Kitchatinov et al., 1999 ) or the local suppression of turbulent viscosity by active regions (Petrovay and Forgács-Dajka, 2002 ). An alternative explanation by Spruit (2003

) suggests that zonal flows may be driven by temperature perturbations near the surface due to the magnetic field: Local flows around regions of enhanced magnetic activity act through the Coriolis force to balance horizontal pressure gradients. The model of Spruit (2003

) naturally predicts a solar-cycle variation in the meridional flows. The reader is referred to the review paper of Shibahashi (2004 ) for other possible explanations.

5.1.2 Meridional flow and its variations

Surface Doppler measurements have shown the existence of a meridional flow from the equator to the poles with an amplitude of about $20 m s- 1$ (e.g. Hathaway, 1996 ). Meridional circulation is an important ingredient of solar dynamo models where magnetic flux is transported by a deep equatorward flow (flux-transport dynamo models). In such models, the solar-cycle period is largely determined by the amplitude of the meridional flow near the base of the convection zone (Dikpati and Charbonneau, 1999 ). Hathaway et al. (2003 ) claimed that the butterfly diagram is evidence for flux-transport dynamo models, and estimated a $1.2 m s-1$ flow at the bottom of the convection zone from the drift of sunspots toward the equator. However, Schüssler and Schmitt (2004 ) argue that the butterfly diagram is equally well reproduced by a conventional dynamo model with migrating dynamo waves without transport of magnetic flux by a flow. The depth of penetration of the meridional flow is another parameter in some dynamo models (see, e.g., Nandy and Choudhuri, 2002 ). Knowing meridional circulation is also important to understand its feedback on differential rotation (Rekowski and Rüdiger, 1998 ; Gilman, 2000 ).

Meridional circulation was first detected at depth by Giles et al. (1997 ) with time-distance helioseismology. The inversions, constrained by mass conservation, show that the data are consistent with a meridional circulation which is $20 m s- 1$ poleward at the solar surface and about $3 m s- 1$ equatorward at the base of the convection zone, with a turnover point near $0.8Ro .$ (Giles, 1999 , see Figure 33 ). Other estimates of the meridional circulation have been obtained in the near-surface layers with time-distance helioseismology (see, e.g., Duvall Jr and Gizon, 2000 ; Zhao and Kosovichev, 2004 ), Hankel analysis (Braun and Fan, 1998 ), acoustic imaging (e.g. Chou and Dai, 2001 ), and ring-diagram analysis (see, e.g., Basu et al., 1999 ; González Hernández et al., 1999 , 2000 ; Haber et al., 2000 ). Giles (1999 ) and Haber et al. (2002 ) found a submerged equatorward flow in the north during the years 1998-2001. This last result, however, may be contaminated in part by errors in the orientation of the MDI Dopplergrams which cause rotation to leak into the meridional flow signal: a $P$ -angle error due to the misalignement of the MDI telescope on the SOHO spacecraft (C. Toner, 2000, private communication) and an error in the Carrington elements that specify the direction of the solar rotation axis (Giles, 1999 ).

Figure 33:

Meridional circulation in 1996-98 inferred from p-mode travel times at various latitudes as a function of scaled radius $r/R o .$ . The blue (red) curves are for the northern (southern) latitudes. The turnover point is roughly at $r = 0.8Ro .$ . The assymetry between the two hemispheres is probably caused by an error in the orientation of the MDI camera. From Giles (1999 ).

Meridional circulation (averaged in longitude over several rotation periods) is observed to change from year to year through the solar cycle. To study the time-varying component of meridional circulation, we consider the residuals obtained after subtraction of a temporal average at each latitude. Meridional flow residuals have an amplitude which is less than $10 m s-1$ . Like torsional oscillations, the north-south flow residuals drift equatorward with active latitudes. Unlike torsional oscillations, the residuals of meridional circulation change sign across the uppermost $70 Mm$ . Near the surface, the time-varying residuals converge toward active latitudes (Gizon, 2003

; Basu and Antia, 2003 ; Zhao and Kosovichev, 2004

, see Figure 34

), while at greater depths meridional flow residuals diverge from the dominant latitude of activity (Chou and Dai, 2001 ; Beck et al., 2002

, see Figure 35

). The time-varying component of meridional circulation has also been measured near the surface from the advection of supergranulation (Gizon and Duvall Jr, 2004

, see Section 5.3.5).

Figure 34:

(a) Meridional flow measured by time-distance helioseismology at depths of $4 Mm$ (solid lines) and $7 Mm$ (dash-dotted lines) as a function of latitude for different Carrington rotations. (b) Northward residual flows, computed by removing the CR1911 flow at each Carrington rotation. The grey regions show the latitudes of activity. The residuals are consistent with a converging flow toward the mean latitude of activity. From Zhao and Kosovichev (2004

Figure 35:

(a) Meridional circulation residuals as a function of time and latitude, measured by time-distance helioseismology at a depth of about $50 Mm$ . The residuals are obtained by removing a time average. The green (red) shades correspond to excess poleward (equatorward) velocities, with values in the range $-1 ± 10 m s$ . The thick black line is the mean latitude of activity. The residuals are consistent with a flow diverging from the mean latitude of activity. (b) Zonal flow residuals (torsional oscillations) as a function of time and latitude. The red (blue) shades correspond to flows that are faster (slower) than average. From (Beck et al., 2002 ).

These observations, summarized in Figure 36

, lead Zhao and Kosovichev (2004

) to postulate the existence of extra meridional ciculation rolls on each side of the mean latitude of activity, superimposed on the mean global-scale poleward flows. However, it is suggested in Section 5.2 that the changes in the longitudinal averages of meridional flows can be attributed for a large part to flows that are localized both in longitude and latitude around active regions.

Figure 36:

Sketch of the time-varying components of the large-scale flows, averaged in longitude over several rotation periods. Shown is a meridional plane in the northern hemisphere; the prograde direction is coming out of the page. Zonal flows (a) introduce a $± 10 m s-1$ shear around the mean latitude of activity (AR). Residual meridional flows ( $-1 < 10 m s$ ) converge toward active latitudes near the surface (e) and diverge deeper inside the Sun (d). The whole pattern of flows drifts equatorward through the solar cycle. The dashed streamlines that connect the horizontal flows are a suggestion by Zhao and Kosovichev (2004

). From Gizon (2003

5.1.3 Vertical flows

Because of their small amplitude, vertical flows are much harder to measure than horizontal flows. The ring-diagram analysis is not, in principle, sensitive to vertical flows. However, Komm et al. (2004 ) showed how to estimate the vertical flows from ring-diagram inversions by adding a mass conservation constraint (using GONG data). This mass conservation constraint is enforced locally for each $15o× 15o$ region, under the assumption is that horizontal density fluctuations are negligible because of the large size of the region and the large integration time. Figure 37 shows the longitudinal average of the near-surface vertical velocity over one Carrington rotation (Komm et al., 2004 ). Weak downflows (less than $-1 1 m s$ ) are observed near active latitudes. This result is consistent with the observations of a residual horizontal flow converging toward the mean latitude of activity near the surface (see Figure 36 ). For depths in the range $0 -16 Mm$ , the amplitude of the downflow increases with depth. A small upflow is seen near the equator.

Figure 37:

Vertical velocity from ring-diagram inversions constrained by mass conservation, averaged over Carrington rotation CR 1988 (2002 March 30 - April 25), as a function of latitude and depth. Top panel: Surface magnetic flux as a function of latitude (solid line) and averaged over $15o$ (dotted curve). Bottom panel: Vertical velocity derived from GONG data after removing the large-scale flow components. The dashed line indicates the zero contour; the dotted lines indicate 20%, 40%, 60%, and 80% of the minimum and the maximum of the color scale. The dots indicate the depth-latitude grid. From Komm et al. (2004 ).

5.1.4 Search for variability at the tachocline

Howe et al. (2000b ) discovered temporal changes in the rotational velocity near the bottom of the convection zone with a period of approximately $1.3 yr$ and a peak amplitude at $0.72Ro .$ . The angular velocity variations at $0.72Ro .$ and $0.63Ro .$ are anticorrelated (the signal seemed much weaker near the second half of 2001 but it has come back since). These results are somewhat controversial since they have not been confirmed by others (see, e.g., Basu and Antia, 2001 ; Vorontsov et al., 2002 ). However, using wavelet analysis over cycles 21-23, Boberg et al. (2002 ) found signals with a 1-2 $yr$ period in the solar mean magnetic field and Knaack et al. (2005 ) detected transient $1.3 yr$ oscillations in the unsigned photospheric magnetic flux, which may be related to the large-scale magnetic surges towards the poles. The $1.3 yr$ periodicity is also seen in observations of the interplanetary magnetic field and geomagnetic activity (Lockwood, 2001 ). Gough (2000 ) suggested that a $500 G$ radial magnetic field across the tachocline may lead to exchange of angular momentum with the right time scale, while Covas et al. (2001 ) suggested a highly non-linear mechanism that does not require different physical time scales for torsional and tachocline oscillations.

A deeply challenging task for local helioseismology is to detect the signature of the magnetic field in the tachocline region. The results of Howe et al. (2000b ) have not been confirmed by local helioseismology yet, but there have been a few attempts to search for wave-speed perturbations near the base of the convection zone. Chou and Serebryanskiy (2002 ) selected wave packets that return to the same spatial point after traveling around the Sun with an integral number of bounces $N$ . The ray paths taking $N = 8$ bounces to go around the Sun have a lower turning point close to the bottom of the convection zone. Chou and Serebryanskiy (2002 ) observed that the change in one-bounce travel time at solar maximum relative to minimum is approximately the same for all $N$ values studied, except for $N = 8$ which shows an additional decrease in travel time. Figure 38 shows the relative decrease in the one-bounce travel time at $N = 8$ relative to the other $N$ values as a function of time (about $15 ms$ at solar maximum). The correlation with the sunspot number suggests that the additional decrease in travel time at $N = 8$ might be caused by wave-speed perturbations at the base of the convection zone. With the simplified assumption that this additional decrease is caused only by a magnetic field perturbation at the base of the convection zone, the field strength was estimated to be $0.4 MG$ at solar maximum. However, Chou and Serebryanskiy (2002 ) caution that this interpretation is oversimplified since it would also imply an additional decrease in travel time for the wave packets with $N > 7$ , which is not observed. We note here that Ruzmaikin and Lindsey (2003 ) argue that only field strengths greater than $1 MG$ should be detectable near the bottom of the convection zone with currently available data.

Figure 38:

Decrease in the one-bounce travel time at $N = 8$ relative to the other $N$ ’s as a function of time. The filled circles denote the MDI results, and the open circles the GONG results. The horizontal bar associated with each point indicates the duration of each observation (the sequence of observing runs is labelled by a series of increasing numbers). The thick horizontal line indicates the range of solar minimum period used for MDI, and the dashed line for GONG. The solid line is the sunspot number from the Greenwich sunspot data. From Chou and Serebryanskiy (2002 ).

A different approach by Birch (2002 ) and Duvall Jr (2003

) was to search for longitudinal structures near the bottom of the convection zone. Raw travel times measured between surface points separated by $45o$ were observed to be correlated from day to day, indicating a real solar signal (Duvall Jr, 2003 ). However, it is not known whether this signal is due to surface effects or deep perturbations.

It should be mentioned that global seismology may also be used to detect a toroidal magnetic field at the level of a fraction of a megagauss, if such a field were confined to a thin layer near the base of the convective envelope (e.g., Basu, 1997 ; Dziembowski and Goode, 2004 ). For example, Basu (1997 ) gave an upper limit of $0.3 MG$ near solar minimum.

It is fair to say that local helioseismological techniques have encountered difficulties in probing the tachocline. In order to reach a depth of $200 Mm$ , it is necessary to consider ray paths that connect surface points separated by almost $o 45$ . All local methods thus suffer from foreshortening near the limb. Also, the plane-wave approximation is obviously not appropriate for probing the deep convection zone. Certain methods, like ring-diagram analysis, need to be adapted to take this into account (see, e.g., Haber et al., 1995 ).