6 Simulations Over Many 11-Year Sunspot Cycles

The next phase was stimulated by interest in the long-term evolution of the Sun’s open magnetic flux. From an analysis of geomagnetic observations, Lockwood et al. (1999 ) had concluded that the Sun’s open flux doubled during the twentieth century. The variations of open flux and heliospheric magnetic field strength also affect the occurrence of the cosmogenic isotopes of $10Be$ and $14C$ , and are thus of interest to researchers working on cosmic rays and solar irradiance variations.

This was the reason that our NRL colleague Judith Lean encouraged us to deduce the long-term variation of the Sun’s open flux from potential-field source-surface extrapolations of the observed photospheric field. We found that the magnitude of the open flux followed the evolution of the Sun’s total dipole during 1971-1999. There was relatively little variation during the sunspot cycle as the dipole evolved from an axisymmetric configuration in the years around sunspot minimum to an equatorial configuration around maximum (Wang et al., 2000a ,b ) without changing its strength very much. The open flux was approximately constant, except for intermittent fluctuations of a factor of two in the years around or just after sunspot maximum. These fluctuations came from the equatorial dipole, which strengthened when active regions happened to erupt in longitudinal phase with the dipole, and weakened as meridional flow carried the flux to midlatitudes where it was eroded by differential rotation and supergranular diffusion on a 1-2 year time scale (Wang and Sheeley Jr, 2003 ).

Meanwhile, in Scotland, Duncan Mackay was combining the flux-transport program with the potential-field source-surface model to simulate the evolution of open flux during the sunspot cycle (Mackay et al., 2002a ,b ). For nominal values of meridional flow speed, the open flux peaked at sunspot minimum, rather than 1-2 years after maximum when the observed open flux (as determined from in situ spacecraft magnetometer measurements) obtained its peak. Mackay et al. concluded that something important must be missing from the model and suggested that the problem might lie with the source-surface component. However, the source-surface model seems to be an unlikely culprit because it correctly reproduces the observed variation of open flux when it is applied to the observed photospheric field (as opposed to the field simulated from the doublet sources) (Wang et al., 2000a ; Wang and Sheeley Jr, 2002 ).

Nevertheless, we also found that the open flux peaked at sunspot minimum when we repeated the Mackay et al. simulation using our estimated doublet sources and nominal flux-transport parameters. Rather than supposing that something was missing from the model, we tried other values of the input parameters to see if we could find a combination that would match the observations. In principle, stronger source fluxes ought to increase the open flux around sunspot maximum. In addition, a faster flow speed with a slightly smaller diffusion rate would weaken the polar field strength (and reduce the open flux) around sunspot minimum. Keep in mind that the ultimate strength of the polar field is determined by the amount of unbalanced flux that is available for meridional flow to transport to the pole, and not on the rate at which the flux is carried there. Thus, a stronger flow speed (relative to diffusion) leaves less unbalanced flux in each hemisphere and produces a weaker, rather than a stronger, polar field. This idea is the key to understanding the polar field reversal when a relatively inactive sunspot cycle follows an active one, as will be discussed below.

To match the simulated and observed open flux, we needed to increase the doublet strengths by a factor of 3, reduce the diffusion rate slightly from $600 km2 s- 1$ to $500 km2 s-1$ , and increase the meridional flow rate from $~ 10 m s-1$ to $~ 25 m s- 1$ (Wang et al., 2002b ). The factor-of-2 increase of flow speed was appreciable, but not unreasonable because the speed still lay in the range of uncertainty obtained from Doppler measurements. By contrast, the factor-of-3 increase of the source strengths was surprising, and made me wonder if active regions really contained 3 times as much flux as I had always supposed. One could rationalize that the process of estimating fluxes from photographic prints of Kitt Peak magnetograms (by measuring areas and using empirical correlations) is not the same as deriving magnetic fluxes from the digital data, and that errors of this magnitude might easily be involved. However, the systematic nature of the correction suggests that it must involve something more fundamental than the use of photographic prints. This means that prior measurements of fluxes in active regions, even though they led to average field strengths of several hundred Gauss in network features, must have been systematically underestimated, perhaps due to weakenings of the $Ca Ic6103$ line used in those measurements (Sheeley Jr, 1966 ; Chapman and Sheeley Jr, 1968 ). This possibility is supported by the fact that, during sunspot cycle 21, the total flux on the Sun (as derived from Kitt Peak magnetograms) was 2-3 times larger than that derived from the fields simulated using the doublet sources (cf. Figure 3 of Wang et al. (2002b )). We need to resolve this issue by making improved measurements of the fluxes in newly emerging bipolar magnetic regions, paying careful attention to the fluxes both inside and outside sunspots.

We did not run into this problem until we compared interplanetary field measurements with the open flux simulated using our doublet sources. Evidently, the spacecraft magnetometer measurements provide a constraint on the flux in active regions, and therefore on our doublet-strength calibration. Prior studies, which were not concerned with the absolute amount of flux on the Sun, would not be affected by the doublet-strength calibration (except indirectly through the use of a lower meridional flow speed), nor would studies that used the observed photospheric field as input, instead of the doublet sources. As mentioned above, the open flux, determined from the source-surface extrapolation of the observed photospheric field, agreed well with the open flux derived from interplanetary measurements, assuming that the interplanetary flux is distributed isotropically (Wang and Sheeley Jr, 1995 ). It is interesting that the Ulrich correction, used to convert the saturated $Fe Ic5250$ synoptic measurements of the photospheric field to unsaturated (and therefore more reliable) $Fe I c5233$ measurements, increased the low-latitude flux by a factor $~$ 4 (Ulrich, 1992 ), and thus had roughly the same effect on the open flux as enhancing the doublet strengths by a factor of 3.

In 2002, flux-transport simulations over many sunspot cycles were providing examples in which the polar fields would not reverse. Schrijver et al. (2002 ) had recently used source fluxes derived from past sunspot records to simulate the polar field from the mid-17th century to the present, and had found several cases for which the simulated polar fields did not reverse. They noted that this problem would not have occurred if flux emerged on the Sun with a finite life expectancy of about 5 years. At the same time, Judith Lean had used our list of doublets from the years 1976-1986 to create a 110-year series of hypothetical sunspot cycles with intervals in which each cycle was stronger than the next. Judith was motivated by the long-term variation of solar irradiance, and was looking for systematic variations of field strength from one sunspot minimum to the next. As part of that study, she found that the simulated polar fields ultimately became so strong that they would not reverse if the sequence of progressively stronger sunspot cycles were interrupted by a weaker cycle (Lean et al., 2002 ).

As mentioned above, a variable meridional flow speed (relative to the diffusion rate) provides a possible way of avoiding these puzzling non-reversals of the simulated polar fields. Scaling our doublet sources by past levels of sunspot activity, Yi-Ming simulated the polar field evolution during the past 100 years (Wang et al., 2002a ). He varied the meridional flow speed from one cycle to the next so that it was slightly faster in an active cycle and slightly slower during an inactive cycle. For only modest $±6 m s-1$ variations of the flow speed, polar field reversals were obtained in every sunspot cycle, with strengths that were consistent with the polar faculae measurements (Sheeley Jr, 1991 ). A reassuring aspect of this result is that the positive correlation between flow speed and cycle amplitude is consistent with a negative correlation between flow speed and cycle period. In particular, the flux-transport dynamo calculation of Dikpati and Charbonneau (1999 ) gives short sunspot cycles when the flow is fast. Because short (or at least rapidly rising) cycles tend to be relatively active according to the sunspot records (Schatten and Hedin, 1984 ), this implies that fast flow occurs during active cycles.

The flux-transport model has evolved since its birth more than forty years ago. In 1963, it contained differential rotation and supergranular diffusion. Twenty years later, the model included poleward meridional flow and was telling us about coronal holes and sector structure. Now, after twenty more years, the flow speed has temporal variations that are telling us about the long-term behavior of the sunspot cycle, and vice versa. What will be next?