At this point, we realized that the doublet source list and flux-transport code could be used to study a variety of solar magnetic field problems, and we successively considered the decay of the mean field (Sheeley Jr and DeVore, 1986a), the origin of the 28 – 29 day recurrent patterns (Sheeley Jr and DeVore, 1986b), the total flux on the Sun (Sheeley Jr et al., 1986), and the Sun’s polar magnetic fields (DeVore and Sheeley Jr, 1987). When Yi-Ming Wang and Ana Nash joined the project, we studied the effects of solar rotation on the field, and found that the meridional component of flux transport was responsible for the quasi-rigid rotation of large-scale photospheric magnetic field patterns (Sheeley Jr et al., 1987). This was the result that Schatten et al. (1972) had found, but did not recognize, fifteen years earlier. In his talk at the Tenth Workshop on Cool Stars, Stellar Systems, and the Sun in Cambridge, Massachusetts, Yi-Ming used the ‘swimming duck’ analogy of Figure 2 to illustrate this effect (Wang, 1998).
With a few exceptions (Sheeley Jr, 1981a; Wang et al., 1988), we tended to solve the flux-transport equation for the evolving magnetic field, rather than for the amplitudes and phases of its spherical harmonic components. However, in an elegant analytical treatment, Rick DeVore found an approximate solution for the rigidly rotating eigenmodes of the field when both diffusion and meridional flow were included (DeVore, 1987). Fourteen years later, Mike Schulz (2001) would perform a similar study, calculating the eigenmodes numerically for the case of diffusion alone.
When Jay, Rick, and I began the flux-transport simulations, we did not know if meridional flow was present on the Sun, and we couched our language in phrases like, ‘meridional flow (if present) and diffusion will do ...’. However, as time passed, it became clear that a better fit between simulations and observations would be obtained if diffusion were assisted by a 10 – 20 m s–1 poleward meridional flow. For each problem, diffusion alone worked fairly well, but it always worked better if it were accompanied by meridional flow.
The most persuasive arguments concerned the polar fields and the poleward surges of flux from the sunspot belts. First, the topknot structure inferred by Svalgaard et al. (1978) could be explained as a quasi-equlibrium between dispersal via supergranular diffusion and concentration by a poleward flow (DeVore et al., 1984; Sheeley Jr et al., 1989). Second, ‘butterfly diagrams’ of the simulated field, such as those shown in Figure 3, did not reproduce the observed poleward surges of flux unless meridional flow was included in the model (Wang et al., 1989a; Sheeley Jr, 1992). In particular, Wang et al. (1989a) found that enhanced eruption rates and accelerated flows were required to match the episodic poleward surges of flux during sunspot cycle 21. Thus, we became confident that meridional flow was present even though the Doppler measurements were near the limit of credibility, and we dropped the ‘if present’ from our papers.
Seven years earlier, Topka et al. (1982) had reached the same conclusion from their analysis of the poleward migration of H filaments. In fact, their description was similar to that of Babcock (1961) who referred to large regions of trailing polarity drifting to the poles. But, except for Leighton (1964) and later Giovanelli (1984), most researchers did not appreciate that supergranular diffusion was essential for neutralizing the leading polarities and leaving those large regions of trailing polarity in each hemisphere.
Having been convinced that there really is a poleward meridional flow on the Sun’s surface, we wondered if the implied subsurface return flow might be responsible for the equatorward migration of sunspot eruptions (the butterfly diagram) (Wang and Sheeley Jr, 1991). In quantitative simulations of the sunspot cycle without meridional flow, Leighton (1969) obtained stable oscillations with the flux eruptions migrating toward the equator. However, his solution required a negative radial gradient of the Sun’s angular rotation rate, which was contrary to the new helioseismology observations (Duvall Jr et al., 1986). Consequently, we performed quantitative modeling in the spirit of Leighton (1969), and found that a subsurface return flow 1 m s–1 and a subsurface turbulent diffusion rate 10 km2 s–1 gave cyclic solutions with the flux eruptions migrating toward the equator (Wang et al., 1991). Two-dimensional MHD versions of this flux transport dynamo have since been developed by Choudhuri et al. (1995), Dikpati and Charbonneau (1999), and Charbonneau and Dikpati (2000).
van Ballegooijen et al. (1998) modified the flux-transport model to include the effect of diffusion and flow on the horizontal component of the magnetic field. Their objective was to study the formation of filament channels and to look for systematic chirality relations in the northern and southern hemispheres of the Sun. Using a diffusion rate of 450 km2 s–1, a meridional flow speed of 10 m s–1, and the Snodgrass (1983) differential rotation formula, they obtained good agreement with the longitudinally averaged radial magnetic field observed at the Kitt Peak Observatory, and they reproduced the locations of the observed filament channels. The model seemed to be less successful in predicting the chirality relations, but after modifying the model further and increasing the diffusion rate to 600 km2 s–1, Mackay et al. (2000) were able to recover the observed hemispheric dependence. As described in Section 6, Mackay et al. (2002a,b) would eventually use the model to simulate the evolution of the Sun’s open flux.
Yi-Ming coupled the flux-transport code to the potential-field source-surface extrapolation, and we began to study the rigid rotation of the corona (Wang et al., 1988) and of coronal holes (Nash et al., 1988). The coronal field always rotated rigidly because it originated in a few low-order harmonic components. However, the rigid rate varied during the sunspot cycle as the sources of non-axisymmetric flux changed their latitudes. The same was true of coronal holes. They sheared rapidly when the large-scale non-axisymmetric flux was spread over a wide range of latitudes. But when the flux was concentrated at a specific latitude, a hole would rotate rigidly at the rate of that latitude. Toward sunspot minimum, active regions erupted at low latitudes, causing the meridional extensions of the polar coronal holes to rotate almost rigidly at the equatorial rate, as illustrated in Figure 4.
Finally, with the link between flux-tube expansion in the corona and solar wind speed far from the Sun (Levine et al., 1977; Wang and Sheeley Jr, 1990a), it became possible to relate solar wind speed variations to the eruption and evolution of photospheric flux (Wang and Sheeley Jr, 1990b; Sheeley Jr and Wang, 1991). For example, it was shown that the small open-field regions that form in active region remnants at sunspot maximum are characterized by rapid flux-tube expansion and therefore produce relatively slow wind. On the other hand, the interaction between two holes of the same polarity results in flux tubes with very low expansion factors. Thus the wind originating from the equatorward extensions of the polar coronal holes is expected to be even faster than that from the poles, a prediction confirmed by Ulysses spacecraft measurements (Phillips et al., 1994).
Although the model had become a powerful tool for analyzing problems involving the large-scale magnetic fields, it was still not universally accepted. Jan Stenflo (1989a,b, 1992) argued that our explanation for the quasi-rigid rotation could not be correct because, for short time lags, the cross-correlations of the simulated field gave the quasi-rigid rotation rate. The model did not reproduce the differential rate that Snodgrass (1983) obtained from cross-correlations of Mount Wilson daily magnetograms. We argued that the Snodgrass differential rate was a consequence of small scale field components, which, of course, are not contained in a diffusing large-scale field. But this argument had little effect until we replaced the diffusion by a discrete random walk and obtained the Snodgrass rate for short time lags (Wang and Sheeley Jr, 1994; Sheeley Jr and Wang, 1994). As we expected, the small-scale features gave the Snodgrass differential rate and the large-scale features gave the quasi-rigid rate. The quasi-rigid rotation could be understood in terms of surface motions alone, and no appeal to the unknown subsurface field was needed.
Although the higher resolution of the random-walk approach provided an appealing match to the resolution of the Mount Wilson magnetograms, we have not used it in our subsequent studies. The diffusion approximation has been adequate for studying the evolution of the large-scale field. However, as mentioned below, Schrijver (2001) has developed a random-walk model whose resolution mimics that of the Kitt Peak and Solar and Heliospheric Observatory (SOHO) magnetographs, and which also includes a procedure for simulating the initial breakup of flux concentrations in active regions.
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