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Astron. Astrophys. 324, 298-310 (1997)

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5. Results and discussions

The 2D solar rotation rate obtained by inverting the a-coefficients of LOWL data and the corresponding averaging kernels are given respectively in Figs. 5.1 and 9. In order to see what could hypothetically be achieved with individual splittings, we made the (unjustified) assumption that the higher a-coefficients are all identically zero and built the corresponding individual splittings. The inversion of these splittings and the corresponding averaging kernels are given respectively in Figs. 5.1 and 10. The averaging kernels are presented in Appendix C and can be used to assess the quality of the solution and the resolution that we can obtain at different target locations.

5.1. About the [FORMULA] value

Let us discuss first, the [FORMULA] values obtained at the corner of L-curves. This value is not enough to quantify the quality of the solution but can reveal some problems in the analysis or in the data themselves. The inversion of a-coefficients with [FORMULA] leads to a value around [FORMULA]. The value of 2.0 for this parameter is highly improbable for a system with many degrees of freedom and reveals that we can not produce a rotation profile by our RLS inversion that agrees strictly with the LOWL data. We note however that this value was higher (around 2.5 at the corner of the L-curves) with a data set covering only the first year of observations. With the two years dataset, the inversion of only the modes for which [FORMULA] leads to the same value of [FORMULA], so that the more superficial p-modes do not appear to be particularly subject to systematic errors which was a concern in an analysis of the first 3 months of LOWL data (Tomczyk et al. 1995b).

We remark that the [FORMULA] value obtained by inverting individual splittings with weights (or errors) specified as explained in Appendix B (where N is the difference between the number of splittings and the number of searched coefficients), is around [FORMULA]. Nevertheless, this value is not significant because although the hypothesis of independence of individual splittings is useful to compute their weights in the minimization process (see Appendix B, Eq. (15) implies that individual splittings are dependent, so that the real number of degrees of freedom is still given from the number of a-coefficients (assumed to be independent) even when individual splittings are inverted. Since, the total number of splittings is on average about 15 times higher than the number of odd indexed a-coefficients, the resulting [FORMULA] value is still around [FORMULA]. We note however that this discussion is valid only on average because the actual ratio between the number of individual splittings and a-coefficients is obviously l -dependent leading to a radial gradient in the apparent improvement on the errors.

The input errors of the a-coefficients are derived from the formal errors when fitting the power spectra and are known to underestimate the true errors. This is the most likely cause for the large values of the [FORMULA]. Additionally, systematic errors in the data could contribute to the value of the [FORMULA]. Also, our results are obtained under the assumption that the hypothesis made in Sect. 2, and the integral expression Eq. (2) for the splittings, are valid for all observed modes. Overly constraining the hypothesis in the forward analysis, as well as some unknown bugs in the inversion process, may also increase the value of [FORMULA].

[FIGURE] Fig. 7. Inferred rotation rate obtained by inverting a-coefficients, plotted against the solar radius for ten latitudes from the equator up to the pole. Bold curves correspond to colatitudes [FORMULA] from the top to the bottom. Dotted curves are the corresponding 1 [FORMULA] errors.

[FIGURE] Fig. 8. The same as Fig. 5.1 but using individual splittings

5.2. Inversion of a-coefficients

Fig. 5.1 shows the variations of the inferred rotation rates against the solar radius from the inversion of a-coefficients. The different inversion parameters have been chosen as discussed in the previous sections with, in particular, [FORMULA] for the plasma surface constraints. The solution shown in this figure is in good agreement with the result obtained at [FORMULA] between 0.2 and 0.85 solar radius by Tomczyk et al. (1995b) who have inverted a-coefficients from the first three months of LOWL data. The rotation rate presents no variation with latitude at 0.2 [FORMULA] with a value around 410 nHz and a transition, between 0.65 and 0.75 [FORMULA], to a latitudinal dependent rotation that leads to rotation rates around 370 nHz at [FORMULA] of colatitude, 410 nHz at mid-latitude and 460 nHz at the equator for depths between 0.75 and 0.85 [FORMULA]. By inverting the a-coefficients, the inferred rotation has no significant latitudinal variation between 0.75 and 0.95 [FORMULA] in zones covering [FORMULA] around the pole and [FORMULA] around the equator.

In the radiative interior, between 0.45 and 0.65 [FORMULA] we find no significant variation of the rotation rate with radius and latitude. Tomczyk et al. (1995b) found a local maximum at 0.4 [FORMULA] occurring for all latitudes. This latitudinal independence was due to the fact that at these radii the kernels for all latitudes were all centered at the equator and very similar. In our results with two years of data, the rotation rate for colatitudes from the equator to [FORMULA] is found constant for radii between 0.3 to 0.7 [FORMULA] at a value of 430 nHz. Around [FORMULA] [FORMULA] we find a small latitudinal dependence with a maximum at the pole. This difference with previous work may be due to a better latitudinal localization of the averaging kernels obtained at this depth with the two year dataset (kernels at [FORMULA], [FORMULA] and [FORMULA], [FORMULA] are clearly distinguishable (Fig. 9)). However, this small latitudinal dependence remains marginally significant if we take into account the errors found on the solution at this depth ([FORMULA] over 5 nHz at the equator and at the pole).

[FIGURE] Fig. 9. Averaging kernels corresponding to the inversion of a-coefficients (Fig. 5.1). For each panel, the contour spacing is defined by the value of the averaging kernel at [FORMULA] divided by eight. Positive contours are shown solid, and negative contours dotted. For clarity the zero contour have been omitted. [FORMULA] is the geometrical distance between the position [FORMULA] of the maximum value of the peak and the point [FORMULA] shown by a star. [FORMULA] and [FORMULA] denote the radial and latitudinal resolution as defined in Appendix C. [FORMULA] is the 1 [FORMULA] error (in nHz) calculated at [FORMULA] and shown by dotted lines on Fig. 5.1.

5.3. Inversion of individual splittings

Fig. 5.1 shows the inferred rotation rate deduced from the inversion of individual splittings. Doing so is equivalent to assuming that the higher (unmeasured) a-coefficients are zero. The rotation rate is very close to that of Fig. 5.1 except in the convection zone near the equator and the pole. These differences may be analyzed by looking at the averaging kernels.

The kernels obtained by inverting individual splittings (Fig. 10) indicate that we can obtain a better latitudinal resolution than by inverting a-coefficients (Fig. 9) for targets located at radii larger than [FORMULA]. In fact, by using a-coefficients directly, the number of these coefficients (i.e. 3) seems to set a limit to the latitudinal resolution around [FORMULA] ([FORMULA] is the best latitudinal resolution reached in Fig. 9). This result is not surprising since, in a first approximation, the [FORMULA] coefficients correspond to rotation constant on spheres and higher order coefficients specify the deviation from this solid rotation, then their number is strongly related to the latitudinal resolution that we can expect. Obviously, the better latitudinal resolution reached in individual splittings inversion induces higher 1 [FORMULA] errors on the solutions (e.g. [FORMULA] nHz at [FORMULA] at the equator in Fig. 10 against [FORMULA] nHz for the same location in Fig. 9) so that the rotation obtained near the equator remains compatible, at the 1 [FORMULA] level, with that given by the inversion of a-coefficients. Nevertheless, this is not the case near the pole where the difference in the rotation rates is over [FORMULA]. In this zone we must look not only at the resolution but also at the localization of averaging kernels. By inverting a-coefficients, averaging kernels calculated at the pole remain localized at best at [FORMULA] of colatitude for all depths (see [FORMULA] for the four lower panels in Fig. 9) although, by inverting individual splittings, we can obtain a peak with a maximum value separated only by [FORMULA] from the pole at [FORMULA]. These remarks could be enough to explain the differences in rotation rates obtained near the pole in the convection zone and seems to argue in favor of the use of the inversion of individual splittings to probe these zones since the inversion of a-coefficients does not allow us to constrain latitudes higher than [FORMULA]. Nevertheless, we must keep in mind that the rotation obtained near the pole is related to the real rotation rate only under the assumption that the [FORMULA] coefficients are null for [FORMULA]. These coefficients are certainly small but not null, therefore this result will change when more accurate data will become available.

[FIGURE] Fig. 10. Averaging kernels corresponding to the inversion of individual splittings (Fig. 5.1) (See caption Fig. 9).
[FIGURE] Fig. 11. Inferred rotation rate obtained by inverting individual splittings with [FORMULA] in the regularization term [FORMULA] Eq. (13), [FORMULA], [FORMULA].

Thus, when using a-coefficients inversion, the regularization forces flatness in latitude when there are no data and the resolution is poor. On the other hand, when using individual splittings, we are forcing the behavior of the rotation near the pole leading to an apparent, but not real, increase in the resolution. This result is however interesting from the point of view of exploring what one might get in terms of averaging kernel and latitudinal resolution if one had more a-coefficients or even individual splittings.

5.4. The rotation of surface layers

The rotation of layers just beneath the solar surface, and in the convection zone is of great importance for our understanding of the solar dynamo and its observed consequences. Some radial gradient of the solar rotation has been suspected in order to explain the different rotation rates, deduced from the observations of various surface indicators, as a consequence of the different depths where these tracers are anchored (e.g. Snodgrass & Ulrich 1990). Therefore it is of interest to look at the rotation rate calculated in this zone by inverting helioseismic data. In the two year dataset, modes with [FORMULA] are no longer thought to be subject to systematic errors and according to Figs. 1 and 4.2.1 we believe that the number of superficial p-modes are now enough to try to describe the rotation between 0.85 and 0.95 solar radii.

The introduction of the surface constraints does not modify the solution below [FORMULA]. Above this depth, the contribution of surface constraints increases and represents more than 70 percent in the calculation of the inferred rotation rate at the surface (Fig. 4.2.3).

In Figs. 5.1 and 5.1 the solution reaches a maximum at [FORMULA] between the pole and [FORMULA] of latitude. The radial and latitudinal resolution obtained at [FORMULA] ([FORMULA], [FORMULA] at [FORMULA] in Fig. 10) indicates that the positive gradient between 0.85 and [FORMULA] may be real in zones between [FORMULA] and [FORMULA] of latitudes.

The discussion in Sect. 4.2.3 has shown that the LOWL data are more compatible with the plasma observations than with the small magnetic feature observations. Fig. 5.1 shows that the inferred rotation rate at [FORMULA] is close to the small magnetic features rate [FORMULA] ([FORMULA] at the equator). Therefore our inversion should argue in favor of this depth for the location where small magnetic features are anchored. Nevertheless, this result is different from the one obtained by Thompson et al. (1996) with Global Oscillation Network Group (GONG) data in which the inferred rotation rate at the surface (without setting surface constraints) is close to the rate deduced from the observation of small magnetic features and reaches a maximum near [FORMULA] with a value [FORMULA] at the equator which can correspond to the value observed by Snodgrass & Ulrich (1990) for the rotation of supergranular network. From the inversion of LOWL data, this value is never reached but the observation of modes with higher degrees is certainly necessary for making a more reliable inference about the rotation of these layers.

Finally, we note that Antia et al. (1996), who have investigated the Sun's rotation rate in the equatorial plane by inverting BBSO datasets for the years 1986, 1988, 1989 and 1990, have found a locally enhanced rotation rate near [FORMULA]. They have pointed out that this behavior shows variation with time. Our solution covering years 1994 to 1996 does not show a bump with significant amplitude near [FORMULA] in the equatorial plane.

5.5. The solar tachocline

At the base of the convection zone, from 0.75 down to 0.65 solar radii, the rotation rate makes a transition to a latitudinally independent behavior which persists in the whole radiative interior. This transition layer is sometimes called the solar tachocline and the evaluation of its thickness which can be related to the horizontal behavior of the turbulent viscosity is of primary importance for our understanding of the eddy diffusivity (Spiegel & Zahn 1992). If we assume that this transition occurs at all latitudes with roughly the same thickness, we can use in this zone the results obtained by inverting a-coefficients that provide worse latitudinal resolution but better radial resolution than the inversion of individual splittings. Unfortunately, the radial resolution reached in the transition zone ([FORMULA] at the equator down to [FORMULA] at the pole in Fig. 9) does not allow us to specify how sharp this transition is. It is not more than 0.1 solar radius but it could be less. Thus in our analysis, the solar tachocline remains unresolved, even with a two year dataset. The radial resolution reached at [FORMULA] with the one year dataset was slightly poorer (namely [FORMULA] compared to [FORMULA] at the equator). This small increase in the radial resolution could be due to the lower errors of the 2 year dataset but we think that we are approaching the fundamental limit of resolution at least at the base of the convection zone with this modeset. Further improvement will be very difficult and we may need to resort to non-linear inversion methods. For this work, continuing the ground-based observations in addition to the space missions would be very important if the width and position of the solar tachocline does not vary too much during the solar cycle.

5.6. The rotation of the core

Below [FORMULA] our solution is compatible with a core that rotates slower than the radiative interior and gives [FORMULA] nHz for the value in the center. As already discussed in Tomczyk et al. (1996), this low value of [FORMULA] is partly due to the low frequency splittings measured for the modes [FORMULA] and that we use in our inversion. Nevertheless, at these depths the averaging kernels are large, not well localized and consist of several peaks, so that the result and the corresponding errors are difficult to interpret. In particular, the latitudinal independence found at these depths results from the choice [FORMULA] in the regularization term [FORMULA] Eq. (13).

Fig. 5.6 shows an instructive example of a solution obtained by setting [FORMULA] and taking the trade-off parameters given by the corner of the L-curves that correspond to this choice (see Sect. 4.2.2). Above [FORMULA] the solution is roughly identical to the solution of Fig. 5.1. The fact that we insure the regularity of the solution at the center avoids finding several values at [FORMULA] and gives in that case [FORMULA] nHz, but the solution shows a significant latitudinal variation below [FORMULA] contrary to the case with [FORMULA]. Therefore the latitudinal dependence is very sensitive to the order of the derivative used in the regularization term and reveals that a reliable description of the latitudinal dependence in this region requires data with lower errors for the low-degree p-modes. Thus we think that the choice [FORMULA] in our code provides an initial way to sound the very deep interior from such global inversions, without searching for a description of a latitudinal dependence in the core that requires very low errors in the data.

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998