In our study, the determination of the integrated value of over r and leads to at the surface of the Sun (that is to say equal to (Godier and Rozelot, 1999)). These values are sightly weaker in comparaison to the values quoted in the introduction of this paper and also in comparaison to the values obtained by Pijpers (1998) deduced from the ponderate GONG and MDI data, that is to say (. The main reason comes from the assumptions made in the computations, which are slightly reductive. In a first approach, and to improve these assumptions, one could change the boundary conditions. In a second study, one should take into account the variation of temperature at the surface of the Sun (Kuhn et al., 1998), if any. This variation would question the assumption of the hydrostatic equilibrium and of the equipotential surfaces. On the other hand, the chosen assumptions are sufficient, for the time being, to study and interpret the profiles of oblateness. Indeed at least two observations have been made to attempt to measure the successive moments in order to be free from the atmospheric turbulences. The first one has been made by Lydon and Sofia (1996) which found for J2 and for J4. The second one has been performed by Kuhn et al. (1998) which report along a Legendre expansion of the measurements made on board the SOHO spacecraft on March 1997, a quadrupolar distortion (l=2; the oblateness) of mas and a hexadecapole shape term (l=4) of mas. The l=4 value are of very high amount and in complete contradiction with Eq. 9. To be perfectly self consistent with the formalism, it must be noted that J2() which is in fact a differential form of J2, must not be confused with the l=4 coefficient. A paper under preparation will bring more detailed comments on that multipole moments applied to the Solar case.
4.1. The properties of the tachocline
The profile of is one of the direct information through which the latitudinal variation of the solar quantities can be reached, such as the position of the center or the width of the transition layers. The first perturbed layer, located between and , where the rotation rate changes from differential rotation in the convection zone to an almost latitudinally independent rotation rate in the radiative interior, is called the tachocline (Spiegel and Zahn, 1992). The helioseismic data show that this region of rapid change has its center located only slightly below the convective zone (Charbonneau et al., 1999) at , and that the thickness of this layer is . Current studies of this layer show that the tachocline is a shear layer and could accomodate the dynamo-effect and a magnetic field. In particular, the helioseismic data show that the endpoint of the adiabatically stratified part of the convection zone is located at (Kosovichev, 1996a) and (Christensen-Dalsgaard et al., 1991). This would mean, that the tachocline ends where the convection zone begins. Nevertheless, if the tachocline becomes slightly thicker and shifts at higher latitudes, a part of the tachocline might be extended into the convection zone, at least at these latitudes (Basu and Antia, 1998) and (Antia et al., 1998); in this case, it is possible that the position of the base of the convection zone might also depend on the latitude. At (determinated at the equator), the value of the oblateness varies with the latitude. So, according to the chosen latitude, the value of the position of the base of the convection zone can be larger or smaller than its position at the equator. Therefore, if the position of the base of the convection zone changes with the latitude, the position of the top also changes, not necessarily in phase with the base. This is clearly visible in Figs. 1 and 3, which show that when r is increasing, the departure between each profile is not constant. Thus, the properties of the tachocline, described by the position of its center and its width, vary with the latitude. As the Sun is a bit more elliptic at the equator, each layer is thinner at the poles than at other latitudes: the tachocline would be certainly thinner near the poles than near the equator.
Charbonneau et al. (1999) suggest that the tachocline is prolate and may have a central radius larger at 60 than at the equator. This suggestion is directly deduced from the most recent helioseismic study of the differential rotation, and may not be contradictory to our result. This would simply mean that if a bump does exist around 60 of latitude, at the surface, and as previously seen, this geometry with a bump, is preserved inside the Sun.
4.2. The tachocline and the overshoot layer
Canuto (1998) suggests that the tachocline can originate in the part of the overshooting region, where the convective flux is still positive. Another scenario assumes that the solar dynamo has a measurable effect on the stratification of the overshoot layer where the magnetic field can be stored (Monteiro et al., 1998), if the convective flux inside this layer is negative. These two arguments would mean that the overshooting layer might be divided in two regions. The first one would be located at the base of the convection zone, where the convective flux is negative, that is, where the magnetic field is stored. The second region, where the convective flux would become positive, would correspond to a part of the tachocline. According to the values of the position of the center of the tachocline and the position of the base of the convection zone, the top of the tachocline is located at around while the base of the convection zone is located at around . The difference in thickness, which accounts for approximately 700 km only, may be associated to the first overshoot layer, with a weak thickness, if this layer begins at the base of the convection zone. If this layer begins within the convection zone, this first part of the overshoot layer should be more extended. A priori, our model of oblateness, for a Sun splitted in shells, shows a variation in the curves near , more or less visible according to the latitude, and even more marked at the latitudes of 60, 75 and 90, which is typically associated to the borderline of the convection zone. Then, we observe on these same curves a weaker gradient up to for latitudes 60, 75 and 90. This means that the oblateness does not vary along the radius for these considered latitudes until a certain depth, that defines a new region. As the length of the plateau is not the same according to the considered latitude, the thickness of this new region also varies with the latitude. This shows that this region could be associated with the extend of the first overshoot layer, whose thickness would be around : in this case, our model shows, if we can trust the helioseismic data, that the extension of this region would be larger, accounting for approximately km. In our opinion, this layer, which would accomodate the magnetic field, would be a transfer layer of the field. The dynamo-effect, located in the tachocline (not in the convection zone, to eliminate a radial propagation), would transform the poloidal field into a toroidal field, both stored in the transfer layer. Then, this toroidal field would climb throughout the convection zone, up to the surface to produce the sunspots. Boruta (1996) estimated the limit strength of the magnetic field in a transition layer around 0.5G, but this value depends on the thickness of this layer. Recently, Tobias et al.(1998) proposed a scenario where the required transport of magnetic field, from the convection zone to the overshoot region, can be achieved on a convective timescale by a pumping mechanism in turbulent penetrative compressible convection. This scenario may explain the passage of the magnetic field from the transfer layer to the tachocline to sustain the dynamo-effect, but we do not have yet a scenario to explain the increase of the transformed magnetic field.
4.3. The structure of the subsurface
Apart from the tachocline, another shear layer near the solar surface has been put into evidence at about , where the rotation rate increases with depth (Antia et al., 1998). This shear layer has already appeared in our study of the quadrupole moment of the Sun (Godier and Rozelot, 1999). It is certainly directly linked to the passage from the convective zone to a new thin radiative layer (Richard et al., 1996) and (Morel et al., 1997), the top of which being the surface of the Sun. This layer is associated to several events which appear just below the surface, such as the meridional and zonal flows, or the seismic events and the jets. The meridional flows, large-scale mass motions from the equator to the poles, are located between and (Hernandez et al., 1999) and the velocities are predominantly poleward. The zonal flow bands have been detected with a velocity variation of 5 m.s-1 at a depth located between and beneath the surface from helioseismic MDI data (Kosovichev, 1997). These zonal bands, characterised by faster and slower rotation, and consistent with surface observations of the torsional oscillations, migrate towards the equator. The seismic events (or sunquakes) have been identified in the photosphere and seem generated by the collapse of the intergranular lane (Goode et al., 1998). The total duration of the expansive phase of the events is of about 5 minutes. These events are confirmed by Nigam and Kosovichev (1999) who found that the solar acoustic modes are excited in a thin superadiabatic layer of turbulent convection of about km (i.e. between and ) below the Sun surface. If we carefully examine the curves of our model in the region defined in the range [; ], where the meridional and zonal flows are found, it is difficult to associate observed variations with a given flow. In any case, the maximum observed around (Fig. 1 and 2) can not be considered, because it depends on the Kosovichev rotation law which presents the same maximum at the same value of the radius. Apart from these maxima, we notice a first change of curvature around for each latitude, and a second one around , at each latitude (less marked at 75 and 90 degrees). These two changes can be associated with the zonal and meridional flows. They both verify the definition range of these flows and one of them might be the signature of the zonal flows. These variations exist at each latitude and confirm the faster and slower rotation of the bands. The other one might be the signature of the meridional flows, which circulate from the equator to the poles. These variations mean that the set of shells belonging to this range of values presents a larger oblateness than the previous shells and than the following ones. Around , our curves present another change of curvature at latitudes 0, 15, 30, 45, and 60 degrees, which might be associated with a borderline of the layer. Indeed, from the surface up to this depth, the acoustic cutoff frequency is much larger (until 5000 µHz) than in the solar interior ( µHz). The acoustic cutoff frequency is linked to the gradient of density which rapidly increases above (Corbard, 1998a). Here, we can remark that the curve, corresponding to a rigid rotation of the Sun (, plotted in Fig. 1), presents the same variations around , and . This indicates that these variations are not brought by the rotation model but present a physical reality. Above , we observe new variations for certain latitudes. At 0, 15 and 30, we have two changes of curvature and an increasing ended curve. At 45 and 60, we notice two changes of curvature and a decreasing ended curve. At 75 and 90, the curves do not clearly present variations, but we observe a slight shift to the linearity of the curves around . The changes of curvature are located around and . These values belong to the range [; ] where the seismic events seem to take place. But, these seismic events would occur at precise latitudes, since we do not observe the same variation at every latitude. Our model does not present events near the poles. Concerning the jets, our model gives no variations around .
The comparison between the solar events and our model of oblateness, is very important to improve the understanding of the stucture of the subsurface. The shear layer beneath the surface is not without calling on the shear layer located around and described in (Sect. 4.2). In these two cases, this transition represents the passage between a convection zone and a radiation zone, and thus we can assume that similar physical characteristics might be found again. The helioseismic data permit us to deduce an increasing of the rotation below the surface (with a maximum around ) up to nHz (Corbard, 1998b). This value corresponds to the rotation of the small magnetic structures observed at the surface (Komm et al., 1993). This suggests that these structures might be stored around (Corbard et al., 1997).
This set of considerations strongly suggests that, just below the surface, may exist a double layer which could be constituted in the same way as that of the transition region located at . The first layer would be a shear layer like the tachocline, located between (which seems to correspond to a borderline) and , which would be the top of the convection zone, given by the observation of our integrated profile (Fig. 6). The second layer would be an overshoot layer, extended from up to . In this layer, the magnetic field at small-scale would be stored and the zonal and meridional flows would circulate. This scenario of a double layer beneath the surface is also proposed by Basu et al. (1999) who gave a depth of 4 Mm () for the outer part. Thus, the seismic events would happen at the base of the shear layer. In this approach, the shear layer would have a thickness of and the overshoot layer of .
4.4. The variation of the solar radius with latitude
The profiles of the oblateness allow us to estimate the shape of the Sun with respect to the latitude. Our results show that the radius decreases slightly from the equator (latitude at which the radius is the largest), then at 15, 30, 45, 60, 75 up to 90 (latitude at which the radius is the smallest). At latitudes 0, 15, and 30, the radius is slightly larger than the mean radius. It is more difficult to give an estimation of the radius around 45, but the values found are sufficient to evaluate the shape of the Sun, which seems to present through our model, a lenghtening at the equator and a bump around 60. This estimation is globally in agreement with the observations of the semi diameter (Fig. 4) made both by F. Laclare (1996) at CERGA in France and E. Noïl (1999) in Santiago, Chili, by means of a solar astrolabe. The consistency of our model with the variability found through observations of the semi diameter shows that the data obtained from ground-based experiments seem to have a physical reality. Nevertheless, it seems that the observed difference between the two extreme values, is very large: 80 mas in the data given by F. Laclare and 170 mas in the data given by E. Noïl. The oblateness deduced from Laclare's data would be larger than 10.10-5, whereas we find an oblateness of .
© European Southern Observatory (ESO) 2000
Online publication: March 17, 2000