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Astron. Astrophys. 323, 231-234 (1997)

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3. Meaning of the [FORMULA] 's

There is no doubt that the [FORMULA] 's are a measure of the Sun's magnetic activity. Kuhn (1988) first noted that the antisymmetric part of the frequency splittings in the fine structure change through the solar cycle. He found a correlation between the phase of the activity cycle and the size of the perturbation with the largest effect corresponding to activity maximum and with the perturbation nearly vanishing at activity minimum. He made this argument using splitting data from 1986 and earlier. His conclusion was strenghtened by Libbrecht and Woodard (1990) who employed observational data covering the subsequent period of high solar activity beyond the 1986 minimum. Near the last solar activity maximum, Woodard et al. (1991) found a strong correlation between oscillation frequency changes and solar magnetic variations from monthly averages of their data.

There has been much work devoted to the interaction of p-modes with magnetic fields. Nevertheless, we don't have a satisfactory theory allowing the calculation of the [FORMULA] 's for a specified, realistic field structure. We don't know whether they arise primarily from direct effect of the Lorentz force or through an induced sound speed perturbation. Futhermore, inertial effects from recently discovered (Duvall et al. 1996) rapid flows related to active regions may also contribute. Therefore, using the [FORMULA] 's as a probe of the Sun's near surface field is for the future.

Of course, a precise localization of the perturbation would be possible only after such a theory is in hand. However, the weak dependence on [FORMULA] seen in Fig. 1 suggests that the perturbing agent must reside very close to the photosphere where the radial eigenfunctions were normalized. We have calculated the frequency shift induced by a localized perturbation of the sound speed. When the perturbation resided high above the photosphere, we observed a rapid increase of the magnitude of the shift with increasing mode frequency. If, instead, we located it far beneath the base of photosphere, we observed, at low frequencies, a rapid decrease of the shift, and at higher frequencies, an oscillatory behavior reflecting the nodal structure of the modes. The behavior, such as seen in the [FORMULA] 's shown in Fig. 1, was found only for a perturbation localized within about a megameter of the base of the photosphere. A similar localization was suggested by Dziembowski and Goode (1991). An equivalent conclusion, based on centroid shifts, was reached by Goldreich, et al.(1991).

We should stress that such a localization of the aspherical perturbation in the Sun is at variance with the recent result of Kosovichev (1996), who inverted the time-distance seismic data of Duvall, et al. (1996). In these data, the same asphericity as manifested in the [FORMULA] 's is seen. Kosovichev finds that it may be interpreted as a pertubation in the sound speed extending down some 30 Mm. The works of Duvall, et al. (1996) and Kosovichev (1996) clearly demonstrate the utility and potential of time-distance seismology, but advances in the theory are required for this field to reach its potential. We view the time-distance approach as being complementary to traditional helioseismology. Its greatest strength lies in probing local structure and, especially, velocity fields.

In spite of the fact that we don't have detailed theoretical knowledge of the origin of the [FORMULA] 's, measurements of them are important and interesting. The angular structure of the non-axisymmetric perturbation, as given through the [FORMULA] coefficients, reflects the underlying global structure of the magnetic field. In this context, the observation that in the years of minimum solar activity, the fact that [FORMULA] is the only significant component seems particularly interesting. We take this as evidence that the Sun's asphericity is dominated by its [FORMULA] component at activity minimum. Here we assume that asphericity of high polynomial order, of which we have no direct information yet, does not contribute significantly to low order [FORMULA] 's. If such components are present, then the [FORMULA] condition may not be satisfied for bulk of the modes in the data set used to determine the [FORMULA] 's. This is the condition for a one-to-one correspondence between the [FORMULA] and the [FORMULA] -asphericity. The successful fit of the data to Eq. (2) suggests that high-k are not significant because they should affect the l -dependence of the [FORMULA] coefficents.

The [FORMULA] geometry corresponds to the quadrupole toroidal magnetic field generated by an [FORMULA] -dynamo action operating at the base of the convection zone where there exists a large radial gradient in rotation. The perturbation that we see in the [FORMULA] 's is localized in the outer layers, and we can only speculate as to how a buried toroidal field may affect these layers. Following changes in the [FORMULA] 's through the whole activity cycle may yield essential clues to understanding the physics of solar activity.

The BBSO data used in Fig. 1 are too spread out in time to allow one to follow the transition of the [FORMULA] from low activity years to high activity years. However, with the LOWL, GONG and SOHO instruments, we will have the chance to follow that transition. In Fig. 2, we may observe the behavior of [FORMULA] and [FORMULA] for the four quarters between Feb '94 and Feb '95. This was a period of low activity. The [FORMULA] 's are the weighted averages over the frequency range. We see no significant change in [FORMULA] over the four quarters. The determination of [FORMULA] in the first two quarters appears significant, and its sign is the same as in years of high activity, but it is about an order of magnitude smaller.

[FIGURE] Fig. 2. [FORMULA] and [FORMULA], averaged over frequency, are shown for each of the four quarters of the LOWL data .

We remark that the use of frequency-averaged [FORMULA] to follow cycle-dependent changes is justified if one compares averages from sets of modes covering the same frequency ranges. Otherwise, the calculated changes may reflect difference in frequency range rather than a genuine temporal evolution of [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998