Duvall Jr and Harvey (1984) and Duvall Jr et al. (1984) observed from the South Pole, using only sectoral modes; their instrument used intensity images in a Calcium absorption line, scanning rather than imaging the whole Sun at once. Their main conclusion was that: “Most of the Sun’s volume rotates at a rate close to that of the surface”.

Brown (1985) had a different instrument, the Fourier Tachometer, which produced 100 100 pixel velocity images. Brown’s initial crude analysis of five days of data used cross-correlation, and expanded the multiplet frequencies using low-order polynomial fits; the results showed little sign of any depth variation in the differential rotation.

Duvall Jr et al. (1986), again using data from South Pole observations but now covering the full range of azimuthal orders, found values of the coefficient (the first-order measure of differential rotation) consistent with the surface rotation and rather larger than was consistent with the results of Brown (1985).

Brown and Morrow (1987), with 15 days (not all consecutive) of Fourier Tachometer data, could not distinguish between rotation on cylinders and latitude-dependence, but found that there was definitely less differential rotation in the radiative interior below the convection zone; their values were now closer to those of Duvall Jr et al. (1986), and they declared the previous ones erroneous. Brown et al. (1989) carried out a much more detailed analysis of the Brown and Morrow (1987) data, strengthening the evidence for mostly depth-independent rotation in the convection zone, as shown in Figure 16.

Both the South Pole observations and those of Brown and collaborators were relatively noisy and of poor resolution; although they strongly hinted at a picture with little radial differential rotation in the convection zone and little differential rotation at all below it, other interpretations were possible.

Libbrecht (1989) published splittings from 100 days of BBSO observations in summer 1986, broadly confirming the results of Brown et al. (1989) with substantially smaller uncertainties. Dziembowski et al. (1989) inverted these data, and inferred a sharp radial gradient at the base of the convection zone and roughly constant rotation at each latitude above that. They also found a bump in the rotation rate in the middle of the convection zone, to which we will return below. Other inversions of the same data set were presented by Christensen-Dalsgaard and Schou (1988) and Libbrecht (1988b), with similar results, though not all the early inversions (cf. Korzennik et al. 1988; Sekii 1991) produce such recognizable results; this may be an example of the difficulty of using OLA-type techniques for data with insufficient higher-degree modes. Another (2dOLA) inversion of these data, shown in Figure 17, was carried out by Schou et al. (1992), who illustrated their averaging kernels; these were rather broad, but adequate to rule out a rotation-on-cylinders model. This paper was also the first to make the important point that the so-called “polar” rotation rate inferred from inversions is actually localized somewhat away from the pole.

Gough et al. (1993) continued to challenge the observers to completely exclude rotation on cylinders, pointing out that it was possible to construct a cylindrical model that satisfied the constraint of the BBSO data, but Schou and Brown (1994b) showed that such a model could not be made consistent with both the Fourier Tachometer data and the gravitational stability of the rotating Sun.

Bachmann et al. (1993) analyzed Fourier Tachometer observations from 1989 and pointed out a “wiggle” in the splitting coefficients at , (corresponding to a turning-point radius of about ); attributed to daily modulation of the observations, this now well-known effect accounts for the “feature” seen in the middle of the convection zone in many inversions of single-site data.

Better data, with long time series free from daily modulation, were obviously needed before much more progress could be made, and with the advent of the GONG network in 1995 and the MDI instrument aboard SOHO in 1996 such data became available. Preliminary rotation profiles were presented by Thompson et al. (1996) for GONG and by Kosovichev et al. (1997) for MDI, both showing the now familiar pattern of almost-constant rotation in the convection zone, with shear layers both at the base of the convection zone and below the surface.

Schou et al. (1998) carried out a comprehensive analysis of the rotation profile based on the first 144 days of observations from MDI, using and comparing several different rotation inversion techniques with an input data set consisting of coefficients up to for modes up to and modes up to . They were able to obtain consistent and robust results from the surface to about at low latitudes; at higher latitudes the domain of reliability was shallower. Roughly speaking, the inversions could not be well localized within about of the rotation axis. The results (Figure 18) showed that the rotation in the bulk of the convection zone, below , had a slow increase with radius at most latitudes, but was definitely incompatible with rotation on cylinders.

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