Rotational Components of the Sun’s Mean Field

This paper uses wavelet transforms to look for the rotational frequencies of the Sun’s mean line-of-sight magnetic field. For a sufficiently high wavelet frequency, the spectra of the dipole, quadrupole, and hexapole field components each show a time-dependent fine structure with periods in the range of 26.5–30 days and their harmonics. These maps confirm that a large enhancement of power at 30 days occurred in the dipole field during 1989–1990, as recorded previously using Fourier techniques. In addition, during some years the maps show power at 26.5 days (or its harmonics), which is clearly distinguishable from the rotation period of 26.9–27.0 days at the Sun’s equator. In at least one case, the 26.5-day period was a wave phenomenon caused by the systematic eruption of active regions at progressively more western locations in the Carrington coordinate system, as if the flux were emerging from a fixed longitude in a faster-rotating subsurface layer. Based on previous studies of the mean field, I conclude that the enhanced wavelet patterns in this paper are regions where magnetic flux is emerging in configurations that strengthen the Sun’s horizontal dipole, quadrupole, and hexapole fields, and (in the case of the more slowly rotating patterns) where this flux is being transported to midlatitudes whose rotation periods are in the range of 28–30 days.


INTRODUCTION
In a previous paper, I used Fourier transforms of Wilcox Solar Observatory (WSO) measurements from 16 May 1975 to 16 November 2021 to study the rotational components of the Sun's mean line-of-sight magnetic field (Sheeley 2022).With this 46.5-yrsequence of daily measurements, the transforms resolved long-lived 2-sector recurrence patterns of 27, 28.5, and 30 days, and showed the existence of 4-sector and 6-sector patterns.For suitable choices of the limits of integration, it was possible to determine the temporal origin of some of the 2-sector recurrence patterns.Now, in this paper, I present the results of applying wavelet transforms to the WSO mean-field measurements updated though 5 July 2022.These transforms are designed to provide spectral power as a function of time and wavelet scale (equivalent to spectral frequency, ω, or period, T = 2π/ω).The analysis differs from an earlier wavelet analysis of the WSO data by Boberg et al. (2002).They considered the general properties of oscillating structures over a broad range of periods from 90 minutes to 11 years, whereas in this study, I am focussing on the detailed spectral properties for periods comparable to the 27-day rotational period of the Sun and its second and third harmonics at 13.5 days and 9 days.
By selecting the wavelet frequency sufficiently high, I have been able to resolve the components with periods of ∼27 days from those of periods 28.5 days and 30 days.In addition, the resulting two-dimensional maps show resolved spectral power, not only for the horizontal dipole fields with azimuthal number m = 1 found in the earlier paper (Sheeley 2022), but also for the horizontal quadrupole (m = 2) and hexapole (m = 3) fields.As we will see, this approach verifies my previous results using Fourier transforms.Also, it reveals a long-lived quadrupole pattern in 1990-1991, having a period that is shorter than the equatorial rotation period at the Sun's surface (∼26.5 days, compared to ∼26.9-27.0days of synodic rotation).

Sheeley
After describing the wavelet technique, I will show the resulting maps of spectral power (the so-called wavelet scaleograms), and present a possible interpretation of this 2-dimensional power.Appendices A and B contain derivations of mathematical relations used in the text, and Appendices C and D contain derivations of the mean field and open flux of an idealized magnetic doublet representing the flux in a nominal bipolar magnetic region.

Range of Frequencies
The WSO mean-field used in this paper consists of a single measurement every day for N = 17, 218 days, corresponding to the 47-yr interval from 16 May 1975 to 5 July 2022.The corresponding range of frequencies can be interpreted as follows: The maximum frequency, ω max (the Nyquist frequency), is determined by one point every half wave.Consequently, ω max = π rad point −1 , which is π rad day −1 at the observing rate of 1 point day −1 .The minimum frequency, ω min , is determined by putting N/2 points in the half wave, so that ω min = π/(N/2) = 2π/N rad day −1 .
Thus, we can express the range, R, in rad day −1 as The smallest frequency, ω min = 2π/N , is also the spectral resolution, ∆ω, of the data.In terms of this resolution, the full set of frequencies, Ω, becomes Of course, it was not possible to obtain mean-field measurements on every day due to a variety of conditions that presumably included adverse weather, and occasional hardware and software problems.Consequently, there were data gaps on 3069 of the 17218 days, corresponding to 17.8% of the data.On these days, I set the mean field equal to zero before computing the wavelet transforms.To get a feeling for how that may have affected the resulting maps, I also replaced those values by random numbers in the range (-0.32, +0.32) G, where 0.32 G was the root-mean-square value of the other measurements.This had essentially no effect on the distribution of wavelet power, and even when the threshold was set higher than 0.32 G, only the faint background distribution was affected.The enhanced regions of power were unchanged.I did not try interpolating between adjacent non-gap values because the gaps were not always isolated days, but came in a variety of combinations ranging from one to several days and sometimes to a few weeks or more.

The Wavelet and the Wavelet Transform
When using Fourier techniques, one combines the time series, f(t), with an oscillating factor, e iωt , that spans the entire time series.This gives the power as a function of frequency, ω, but does not indicate when the spectral peaks of this distribution originated.With wavelets, one introduces a damping factor of the form, e −(1/2){(t−τ )/s} 2 , to limit the range of oscillation to a temporal scale, s, around the time t = τ , and then combines the time series, f(t), with this damped and phase-shifted oscillation to obtain power as a function of the temporal shift, τ , and the temporal scale, s.In particular, the damped, but unshifted wavelet, ψ, analogous to the Fourier factor, e iωt , is where γ is a dimensionless constant, indicating the number of radians of oscillation in a scale length, s.Equivalently, γ/2π is the number of oscillations in a decay time scale, s.Now, ω 0 is a true frequency in rad day −1 , given by from these wavelet expressions, but I will include it when the actual calculations are performed.The idea is to shift this wavelet by an amount, τ , to form and then find the value of where B m (τ ) is the mean-field time series, and the factor of π −1/4 has now been included in the wavelet.Also, I have interchanged the roles of t and τ so that t survives the integration and now refers to the time shift of the wavelet from its peak value.Because F (t, s) is usually a complex number, we plot the real number, F (t, s)F * (t, s) = |F (t, s)| 2 , as a function of t (on the horizontal axis) and as a logarithmic function of spectral period, T = 2π/ω, (on the vertical axis).In this case, the vertical axis indicates log 2 (2T ) increasing downward from 1 at the top of the map.
As we shall see in the next section, it will be necessary to choose γ >> 6 to resolve the individual solar rotation periods and see the effects of differential rotation.As shown in Appendix A, for such very large values of γ, there is essentially no difference between ω 0 , the frequency defined by Eq(4), and ω, the frequency of the oscillating mean field, obtained from the wavelet analysis.This means that there is no need to retain the 0 on ω 0 or on T 0 = 2π/ω 0 .Finally, as a point of terminology, ψ is a Gabor wavelet that depends on γ, and for the special case of γ = π 2/ln2 ≈ 5.336, the Gabor wavelet reduces to what is usually called a Morlet wavelet.

RESULTS
Figure 1 compares the wavelet power in the WSO mean field (top panel) with the monthly averaged sunspot number from the Royal Observatory of Belgium (SILSO) shown in the lower panel.Here, γ = π 2/ln2 ≈ 5.33, corresponding to a Morlet wavelet transform of the WSO mean-field measurements.This is essentially what Boberg et al. (2002) obtained for the WSO data through 2001 using γ = 6.(See their Figure 1).
The oscillation period, T , is indicated in powers of 2 (like octaves of a musical scale), running logarithmically downward along the vertical axis as log 2 (2T ).For convenience, I use the approximation that log 2 (2T ) ≈ log 2 (T /27) + 5.75 (equivalent to log 2 27 ≈ 4.75), which shows directly that the 27-day equatorial rotation period occurs at 5.75 on the vertical axis.Also, I used 32 steps per octave so that with n = 32 and j running from 1 to 416 (13×32).In this case, j = 416 gives log 2 (2T ) = 14.Note that log 2 N = log 2 17218 ≈ 14.07, so that the ordinate values from 1 to 14 almost span the full set of 17218 measurements.
The map shows a variety of spectral and temporal features.Toward the bottom of the map where ordinate values are in the range 11-13 (corresponding to times of years), coarse structures are aligned horizontally.Near the top of the map where ordinate values are in the range 4-7 (8-64 days), fine structures are aligned vertically.In between, the transition seems to consist of a multitude of round features scattered through the ordinate range of log 2 2T = 7 − 9, corresponding to periods in the range 64-256 days.
Looking more closely, we can see that the vertical fine structures are distributed in rows running horizontally at locations of 5.75 (27 days), 4.75 (27/2 days), and more faintly at 4.16 (27/3 days), corresponding to the equatorial rotation period and its second and third harmonics.These features indicate rotational contributions from the dipole, quadrupole, and hexapole moments of the Sun's field.They fall into four time intervals corresponding to sunspot cycles 21-24.Their intensities are roughly correlated with the strengths of the sunspot cycles and are relatively weak during cycle 24.However, these features are widely distributed over each sunspot cycle, and often occur early in the declining phase of the cycle, as Sheeley & Wang (2015) found in their analysis of the Sun's large-scale field.
If we represent the vertical dimension of this map by y = log 2 (2T ), then we can relate small changes in y to changes in the period, T. We do this by converting to natural logarithms and then differentiating to obtain Thus, small changes of y are proportional to the fractional change ∆T /T .In Figure 1, the rotational fine structures have a vertical extent of about ∆y = 0.5, corresponding to ∆T /T = 0.5 ln2 = 0.346, which is ∆T = 9.4 days at T = 27 days and 4.7 days at T = 27/2 days.So γ = 5.33 gives a relatively coarse picture of these rotational features.In other words, the Morlet transform shows the dipole, quadrupole and hexapole components with good temporal resolution (≲ 1 solar rotation period), but it does not show the rotational fine structures of these components.On the other hand, with 32 pixels per octave (corresponding to dy = 1/32 and dT /T = 0.02), the display is capable of resolving features whose periods differ by dT = 0.54 days, which includes the 27-day, 28.5-day, and 30-day periods.However, to achieve this spectral resolution, it is necessary to increase γ, or γ/2π, which is the number of waves of period, T, in a decay time, s, as indicated by Eq(4) rewritten in the form To understand this, recall from section 2.1 that a Fourier transform of the entire set of N = 17218 points has a frequency resolution of 1/N cycles day −1 .In other words, the frequency resolution is inversely related to the number of data points in the sample.Accordingly, we would expect wavelets of scale s to give a frequency resolution ∼1/s cycles day −1 .As shown in Eq(B18b) of Appendix B, the root-mean-square angular resolution is (∆ω) rms = 1/(s √ 2).Dividing by the angular frequency, ω, given by Eq(4), we obtain ∆ω/ω = 1/(γ √ 2).Also, because ∆ω/ω = −∆T /T , it follows that the spectral resolution is For a given period, T = 27 days, we can obtain a large value of s (corresponding to a high spectral resolution) by choosing a large value of γ.As a specific case, let's take γ/2π = 20, corresponding to |∆T /T | = 0.0056 and ∆T = 0.15 days.Then, using Eq(9) and Eqs(B18a), we obtain Thus, for this example of T = 27 days and γ/2π = 20, the spectral resolution is 0.15 days, which is sufficient to distinguish the 27-day, 28.5-day, and 30-day spectral features, but is obtained at the cost of degrading the temporal resolution to about 1 yr.
Figure 2 illustrates this effect by changing γ/2π in powers of 2 ranging from 1 in the top panel to 16 in the bottom panel.In the top panel, the narrow slivers of power have widths on the order of a 27-day solar rotation or less.But their vertical extensions are about 0.5 unit, corresponding to several days of frequency resolution, as mentioned above.In the bottom panel where γ/2π = 16, the spectral features are aligned in horizontal strips of widths ≲ 0.03 unit (corresponding to about 0.5 day) and separations of about 0.1 unit (corresponding to ≈ 2 days, so that the rotation periods of 27, 28.5, and 30 days are clearly resolved.These three components are especially noticeable in 1990 when the dipole field has significant power at all three frequencies. By comparison, in the second panel from the top where γ/2π = 2, the distribution of wavelet power looks very much like the temporal plots of power in the m = 1, m = 2, and m = 3 modes, as shown in Figure 7 of Sheeley (2022).This is as one would expect because those plots were obtained by inverting the Fourier transform of the entire data set for the three frequency bands corresponding to m = 1, m = 2, and m = 3.
I used maps like those in Figure 2 to make a movie of the wavelet power, but with a wider and finer range of γ/2π.When viewed in a single-step mode, the movie allows one to investigate the transition from high temporal resolution to high spectral resolution in detail.The idea is to identify individual features and track them backward and forward as a function of γ/2π.In this way, one might be able to identify the separate eruptions of flux that contribute to the long-term patterns of the mean field.Figure 2 is sufficient to make these identifications for some simple cases.For example, the 30-day, 2-sector pattern in 1989-1990 seems to have originated in two bursts, one around April 1989 and the other in January 1990.Likewise, the 27-day, 2-sector pattern in 2003-2004 seems to have originated in several eruptions during that time.Further discussion of this transition between high temporal resolution and high spectral resolution is contained in Appendix B.
Figure 3 shows the wavelet power in the dipole (m=1), quadrupole (m=2), and hexapole (m=3) regions of the spectrum near 5.75, 4.75, and 4.16 on the vertical axis.In the middle and upper panels, the values of γ/2π have been increased to 20 and 40 cycles per decay time, respectively, and in both panels, the display resolution has been increased to 64 pixels per octave, corresponding to dT /T = 0.022 and dT = 0.27 days.For reference, faint dotted lines are drawn at values of T = 27 days, 28.5 days, and 30-days in the middle panel, and at their second harmonics in the top panel.Only the line corresponding to T = 9 days is shown for the third harmonic in the top panel.As before, the bottom panel shows the monthly averaged sunspot number from the Royal Observatory of Belgium plotted for cycles 21-24 and the rising phase of cycle 25.
In Figure 3, the rotational fine structure is clearly visible in the dipole and quadrupole modes, and to a lesser extent in the hexapole mode.As found previously (Sheeley 2022), the splitting is mainly one-sided, ranging from the equatorial rotation period of about 27 days to 30 days, as one might expect for magnetic patterns that are subject to differential rotation at latitudes between the equator and 45 • (Newton & Nunn 1951;Snodgrass 1983;Sheeley & DeVore 1986).In addition, meridional flow and the poleward component of supergranular diffusion ought to affect the rotation by making its latitudinal profile more rigid (Sheeley et al. 1987;DeVore 1987;Wang 1998).The strong 30-day period is visible for the dipole field in 1989-1990, and there is some weak 27-day power in 2022 at the start of the rising phase of sunspot cycle 25.As noted previously (Sheeley 2022), the fine structures in one sectoral mode are not reproduced in the other sectoral modes.These fine structures refer to different multipole patterns of field, and are not the same structures rotating at different rates.
A puzzling feature of this map is the presence of power at 4.73 on the vertical scale in the years 1990-1991.This corresponds to a magnetic quadrupole rotating with a period of 13.24 days.Interpreted as a second harmonic,  this would imply that the fundamental is recurring with a period of ∼26.48 days, which is noticeably smaller than the equatorial period of 26.9-27 days.There is a similar feature near 5.73 in the middle panel during 2002-2004, corresponding to a magnetic dipole field rotating with period of 26.5 days, which is again noticeably smaller than the equatorial rotation period of the Sun's surface.How can these magnetic features drift in longitude faster than the equatorial surface rotates?By means of waves, as I will explain in the next section of this paper.

SUMMARY AND DISCUSSION
The purpose of this paper was to see if a wavelet analysis would reveal the fine structure within each of the harmonic components of the mean field and show how that fine structure varied with time during past sunspot cycles.These objectives were achieved, but, to resolve the rotational fine structures, it was necessary to increase γ/2π, the number of waves per decay time of the wavelet, well beyond the value of approximately 1 that is customarily used in the conventional Morlet transform.This was illustrated in Figures 1-3, which showed the distribution of wavelet power when γ/2π ranged from 0.85 to 40.

Sheeley
The middle and upper panels of Figure 3 showed the rotational fine structure in the 2-sector, 4-sector, and 6-sector fields, expressed as a function of time during sunspot cycles 21-24 and early in the rising phase of cycle 25.In addition to power with the 27-day equatorial rotation period during each sunspot cycle, a substantial amount of 2-sector power was visible with a 30-day rotation period during 1989-1990, reproducing the result obtained previously using Fourier transforms (Sheeley 2022).
Figure 3 showed puzzling examples of apparent solar rotation with periods of approximately 26.5 days, noticeably lower than the 26.9-27 day equatorial rotation period of the Sun's surface.In an attempt to understand this behavior, I examined a time-lapse movie constructed from MWO Carrington maps of Ca II K-line emission during the years 1915-1979 and from NSO Carrington maps of the photospheric magnetic field during 1980-2002.This movie was made previously as part of the work for a paper on the long-term variation of K-line emission (Sheeley et al. 2011).Because these Carrington maps were displayed with the 27.27-day Carrington cadence, magnetic patterns poleward of about 15 • (where the synodic rotation period is about 27.27 days) moved rapidly eastward, and features equatorward of about 15 • drifted westward.The visual appearance was dramatic; high-latitude regions of each new sunspot cycle seemed to 'fly' off to the left, and the flux diffusing away from the last equatorial active regions of each cycle drifted very slowly to the right.
In addition to these obvious indications of differential rotation, there were some flux concentrations that seemed to move more rapidly to the west.Their speeds were about twice the drift speed of long-lived active regions at the Sun's equator, which is consistent with a rotation period ∼26.5 days.(A period of 26.5 days corresponds to a rotation rate of 13.58 • day −1 , or 0.38 • day −1 faster than the Carrington rate of 13.20 • day −1 .By comparison, the 26.9-day equatorial rate is 13.38 • day −1 , or 0.18 • day −1 faster than the Carrington rate.Thus, in the Carrington maps, the 26.5-day feature moves westward at a rate that is 0.38/0.18= 2.1 times faster than flux drifting at the equatorial rate.) However, a close examination of the movie showed that these fast westward motions were not due to the differential rotation of long-lived magnetic regions.Rather, they were waves -apparent motions caused by the emergence of new active regions at progressively more western longitudes in the Carrington frame.Similar wavelike motions have been observed previously in Carrington stackplots of the NSO magnetic measurements by Gaizauskas et al. (1983).In fact, their oxymoron, 'simple complex IIIa', is a very clear example of this fast wavelike motion in the N10-40 • latitude band during Carrington rotations 1656-1659(14 June -03 September 1977).My measurements of their Figure 2 give an effective rotation period of 26.3 days, compared to their value of 26.5 days.But the essential point is that the period was significantly smaller than the 26.9-27.0-dayequatorial rotation period of the Sun.Consequently, the motion cannot be caused by longitudinal transport of flux by solar rotation, at least at the Sun's surface.
It is interesting that the 26.5-day period obtained from our wavelet measurements is nearly the same as the ∼26.4day equatorial rotation period found at a depth of 0.06R ⊙ (∼42 Mm) using GONG helioseismology measurements (Howe 2009).Thus, the fast wavelike motion in Figure 3 occurs as if the flux were emerging from a fixed longitude at that depth.In terms of the popular word, 'nesting' which Castenmiller et al. (1986) introduced for repeated eruptions of sunspot activity at the same location (see also the reviews by van Driel-Gesztelyi & Green (2015) and by Hathaway (2015)), we could call this fast wavelike motion 'subsurface nesting'.
This raises the question of whether all of the enhanced wavelet patterns of the mean field are due to nesting.We know that the long-lived recurrent patterns of 28-30-day periods owe their existence to the continued emergence of bipolar regions with their polarities in phase.Otherwise, differential rotation winds up their flux and leaves a weaker field rotating with the 27-day period of the unwound flux at the equator.This is consistent with our more recent demonstration that the mean field is dominated by the horizontal dipole and quadrupole fields (and to a lesser extent the hexapole field) (Sheeley 2022).So, the active regions must emerge in a way that reinforces the lower-order multipoles of the non-axisymmetric field.It does not matter whether this systematic emergence occurs in active longitudes as described by Gaizauskas et al. (1983), or whether it occurs by chance as in the randomized doublets of Sheeley & DeVore (1986).More recently, Hudson et al. (2014) have discussed this effect as a way of reinforcing long-lived polarity patterns at Hale sector boundaries.
It is interesting that when Sheeley & DeVore (1986) removed supergranular diffusion and meridional flow from their simulations during the declining phase of the sunspot cycle (when all of the sources emerged below 20 • latitude), the 28-29-day recurrence patterns disappeared.In other words, a poleward component of transport was essential for obtaining the 28-29-day periods, and with this transport, the slowly rotating patterns could be obtained from the flux in low-latitude active regions (providing that they continued to emerge in phase.Also, this is consistent with the results of Sheeley et al. (1987), who found that the large-scale field rotates differentially when the source rate is high, but begins to weaken and rotate rigidly with the 27-day equatorial period when the flux stops emerging and the surviving non-equatorial flux migrates poleward and is eliminated by differential rotation and supergranular diffusion.Thus, it seems likely that spectrally resolved wavelet patterns correspond to active-region nests at different latitudes and with strong horizontal dipole and/or quadrupole moments.Wavelet maps like those in Figure 3 indicate the sunspot-cycle distribution of those nests, and γ-sequences, like those in Figure 2, provide a way of finding the individual sources that maintain the strengths of these patterns.
I am grateful to Kristopher Klein (LPL/UA) for suggesting wavelet transforms as a way of tracking solar features of different rotational periods.Also, I am grateful to Chen Shi (UCLA) for providing details of how he made wavelet maps of the radial component of the interplanetary magnetic field observed at the Parker Solar Probe.Wilcox Solar Observatory data used in this study were obtained via the web site http://wso.stanford.educourtesy of J.T. Hoeksema.NSO data were acquired by SOLIS instruments operated by NISP/NSO/AURA/NSF.Sunspot numbers were obtained from WDC-SILSO, Royal Observatory of Belgium, Brussels.
If we regard Eq(A14) as the first-order solution, x 1 , then we can write x = x 1 − ϵ and substitute it into Eq(A13) to obtain ϵ = [2x 1 /(2x 1 − 1)] e −2γ 2 x1 as the second-order correction.In this case, the second-order solution for x becomes where x 1 is the first-order solution given by Eq(A14).Thus, for γ ≳ 6, this second-order correction is negligible and the first-order solution, given by Eq(A14), is accurate to about 1%.There is no difference between ω and ω 0 for the larger values of γ that resolve the 27-day, 28.5-day, and 30-day components of differential rotation.For smaller values of γ/2π, one can use Eqs(A14) and (A15).

B. THE TRANSITION BETWEEN HIGH SPECTRAL AND SPATIAL RESOLUTION
Regardless of the values of s and ω 0 (or γ), the wavelet and its Fourier transform satisfy the Heisenberg uncertainty principle in the form ∆ω∆t = 1/2, provided that ∆ω and ∆t are both defined as the root-mean-square differences from their average values.As mentioned below, this provides a basis for understanding the transition between the maps of high spectral resolution and high spatial resolution like those shown in Figure 2.
Let's begin with ψ(t) and its Fourier transform, ψ(ω): In this case, the mean squared averages are Consequently, the root-mean-square values are and their product is (∆ω) rms (∆t) rms = 1 2 , (B19) independent of s.Also, if the energy, E, of the wave packet were E = ℏω, then which is the conventional form of the Heisenberg uncertainty principle.
As mentioned in the text, the spectral resolution varies as 1/s, so that Eq(B18b) gives the spectral resolution in terms of the scale, s.Likewise, Eq(B18a) gives the temporal resolution in terms of s.When combined with Eq(9) of Section 3 (s = (γ/2π)T ), these equations provide a way of calculating these resolutions as a function of γ/2π, and therefore of tracking the evolution of wavelet power from high temporal resolution to high spectral resolution as was done in Figure 2.
Let's pursue this matter further by recalling from Figure 2 that well-defined frequencies in the high spectral resolution maps (with γ/2π ∼ 8 − 16) typically correspond to bursts of short-lived features in the high temporal resolution maps (with γ/2π ∼ 1).In particular, the quadrupole feature at 4.73 on the vertical axis during 1990-1991 (also marked by an arrow in Figure 3) corresponds to a 0.5-yr burst of 2-3 short-lived features in the high temporal resolution maps at the top of Figure 2. Other bursts are visible with lifetimes ranging from 0.4 yr to about 1 yr.The closely spaced If this idealized magnetic doublet has a pole strength Φ 0 = 10×10 21 Mx and a longitudinal pole separation ∆ϕ ∼ 10 • (∼10 5 km), corresponding to a mid-sized active region, then its contribution to the Sun's mean field would be only about 0.08 G.This is an order of magnitude smaller than the 1 G peaks that typically occur when the Sun's equatorial dipole and quadrupole moments are strong.To achieve these strong fields, it would take a nest of several large bipolar magnetic regions arranged with their polarities in phase, as we have found in spatially resolved observations around sunspot maximum and during the initial declining phase of the sunspot cycle (Sheeley & Wang 2015).

D. THE OPEN FLUX OF A PHOTOSPHERIC DOUBLET
Here, we extend the previous calculation to determine how much open flux is contributed by the idealized magnetic doublet given by Eq(C21).To obtain this open flux, we assume a potential field whose angular components, B θ and B ϕ vanish at a spherical source surface located at a radial distance R ss , and whose radial component, B r matches the radial component of the doublet field given by Eq(C21) .Eventually, we will select R ss = 2.5R, where R is the solar radius, but, for the moment, let's consider a general value of R ss .Our objective is to find the source surface field and then integrate its positive value over the source surface.
Rather than repeating the solution of this well-known boundary-value problem, I will take the solution from Eq(4a) of a previous paper by Nash et al. (1988).Setting r = R ss in their equation, the source-surface field becomes: where c lm are the spherical harmonic components of the doublet, given by When B ph is given by the doublet field of Eq(C21) with θ L = θ F = θ and ϕ L − ϕ F = ∆ϕ << π/2, the solution to Eq(D29) is Next, we suppose that the resulting field will be dominated by the contributions of the horizontal dipole and quadrupole, and limit the sum in Eq(D28) to terms of the form Y ±1 where the approximation was obtained by neglecting terms of order (R/R ss ) 3 = (0.4) 3 = 0.064 in the expression for a and (0.4) 5 = 0.010 in b.Without this approximation, the combined open flux drops slightly to 0.54 of the doublet moment, mainly due to a drop of the dipole contribution from 0.45 to 0.44.The essential point is that for a current-free corona, the open flux of an idealized doublet located near the equator is about half of its doublet moment.This is comparable to 0.71(Φ 0 ∆ϕ), the flux obtained from Eq(C27) by multiplying the rms mean field by the area of the visible solar disk.Like the mean field, several suitably arranged doublets of modest size will be required to contribute appreciably to the amount of open flux.

Figure 2 .
figure call

Figure 3 .
Figure 3. Wavelet power in the region of the solar rotation period (middle panel) and its second and third harmonics (top panel), displayed using γ/2π = 20 and γ/2π = 40, respectively.Faint dotted lines indicate periods of T = 27 days, 28.5 days, and 30 days near 5.8 on the vertical scale, and for their second harmonics near 4.8.A single dotted line indicates the third harmonic of T = 27 days near 4.2.The arrows indicate features at 4.73 in 1990-1991 and 5.73 during 2002-2004, corresponding to rotation periods ∼26.5 days.(bottom) Monthly averaged sunspot number from the Royal Observatory of Belgium (SILSO).
! , and P m l (0) is an Associated Legendre polynomial function of x evaluated at x = 0.