Issue 
J. Space Weather Space Clim.
Volume 10, 2020
Topical Issue  Space climate: The past and future of solar activity



Article Number  40  
Number of page(s)  10  
DOI  https://doi.org/10.1051/swsc/2020041  
Published online  27 August 2020 
Research Article
Twodipole model of the asymmetric Sun
^{1}
Center for Space Physics, Boston University, 725 Commonwealth Avenue, Boston, MA 02215, USA
^{2}
ReSoLVE Centre of Excellence, Space Climate Research Unit, University of Oulu, PO Box 3000, 90014 Oulu, Finland
^{*} Corresponding author: berci@bu.edu
Received:
10
March
2020
Accepted:
28
July
2020
The largescale photospheric magnetic field is commonly thought to be mainly dipolar during sunspot minima, when magnetic fields of opposite polarity cover the solar poles. However, recent studies show that the octupole harmonics contribute comparably to the spatial power of the photospheric field at these times. Also, the even harmonics are nonzero, indicating that the Sun is hemispherically asymmetric with systematically stronger fields in the south during solar minima. We present here an analytical model of two eccentric axial dipoles of different strength, which is physically motivated by the dipole moments produced by decaying active regions. With only four parameters, this model closely reproduces the observed largescale photospheric field and all significant coefficients of its spherical harmonics expansion, including the even harmonics responsible for the solar hemispheric asymmetry. This twodipole model of the photospheric magnetic field also explains the southward shift of the heliospheric current sheet observed during recent solar minima.
Key words: photospheric magnetic field / north–south asymmetry / spatial power spectrum / twodipole model
© B. Zieger & K. Mursula, Published by EDP Sciences 2020
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
It is known since the early 20th century that sunspot activity is often different in the two solar hemispheres. A long period of stronger sunspot activity in the northern hemisphere occurred from the late 1950s to early 1970s, covering most of solar cycles 19 and 20. During the last several solar cycles, the northern hemisphere has been more active in the ascending phase of the solar cycle, and the southern hemisphere in the descending phase. This disparate evolution between the two solar hemispheres culminated during the ongoing cycle 24, where each of the two hemispheres formed its distinct maximum, separated in time by almost one and a half years.
Not only sunspots and other active regions (all different forms of the toroidal magnetic field) depict hemispheric asymmetry. The whole largescale pattern of magnetic fields on the solar surface (photosphere) has also been found to be systematically north–south asymmetric (Zhao et al., 2005; Virtanen & Mursula, 2014; Petrie, 2015; Zieger et al., 2019). This is particularly true for the solar polar fields, which are unipolar around solar minima. Polar fields develop when the magnetic fields of decaying active regions that were generated at lower latitudes, are transported to high latitudes. Therefore, the asymmetry of polar fields naturally reflects asymmetry in poloidal magnetic field generation and/or transport. Using polar faculae as proxies of the polar fields, Goel & Choudhuri (2009) estimated the hemispheric asymmetry at solar minima during the 20th century. While the asymmetry of polar fields was negative (southern hemisphere dominance) at the beginning of solar cycles 21–24, it was predominantly positive (northern hemisphere dominance) at the beginning of the earlier solar cycles 15–20, which is more or less in agreement with the observed hemispheric asymmetry in the total sunspot area during these solar cycles.
Measuring the polar fields in the photosphere is challenging because of the small lineofsight component of the magnetic field at high solar latitudes. In addition, the solar poles are not equally visible during most of the year because of the tilt of the solar rotation axis with respect to the ecliptic, which is commonly referred to as the vantage point effect. The authors have recently developed a new method of eliminating the vantage point effect in photospheric magnetic field observations, which allows a more accurate calculation of the spatial power spectrum during solar minima (Zieger et al., 2019). It was shown that the zonal octupole component has a large contribution to the spatial power of the photospheric magnetic field during solar minimum conditions, matching the power of the dipole component. Also, the power of the lowdegree even zonal harmonics was found to be statistically significant (Zieger et al., 2019), confirming the north–south asymmetry of the largescale photospheric magnetic field.
Several studies have verified by now that the southern polar fields were stronger than the northern fields around solar minima during the recent solar cycles (21–24). This difference gives one of the most tangible evidence for solar hemispheric asymmetry that is systematically repeated from one solar minimum to the next. The detailed harmonic structure of the photospheric magnetic field, including its systematic hemispheric asymmetry, cannot yet be explained by current solar dynamo models (for a review, see Charbonneau, 2010). However, neither the global structure of the photospheric field nor its hemispheric asymmetry is only of academic or theoretical interest. They affect the distribution of solar eruptions, flares and coronal mass ejections, as well as the occurrence of highspeed solar wind streams. They have, therefore a great practical significance for space weather observed at the Earth.
In this paper we present a novel, analytical model of the largescale photospheric magnetic field, which very accurately reproduces the observed distribution of photospheric magnetic fields during solar minima, and naturally explains the observed hemispheric asymmetry. We introduce the model in Section 2 and validate it with photospheric magnetic field data in Section 3. The modeling results of the photospheric and coronal north–south asymmetry are discussed in Section 4. We present our conclusions in Section 5.
2 Twodipole model
Sunspots and other active regions are concentrations of strong magnetic fields, which appear at low to midlatitudes on the solar surface in pairs of opposite magnetic polarity. Sunspot pairs follow Hale’s law, according to which the polarity of the leading spots in one hemisphere is opposite that of the leading spots in the other hemisphere and the polarities reverse from one solar cycle to the next. Thus, two solar cycles make one 22year Hale cycle of the solar magnetic field (Hale, 1924). The leading spots tend to appear slightly closer to the equator than the trailing spots, a phenomenon known as Joy’s law (Hale et al., 1919; Howard, 1991; Wang & Sheeley, 1991; Sivaraman et al., 1999; DasiEspuig et al., 2010). During the rising phase of the solar cycle, the trailing spots have an opposite magnetic polarity compared to the polar fields in the same hemisphere during the previous minimum. Thus, as the trailing fields of decaying active regions are transported towards the pole (Wang et al., 1989; Hathaway & Rightmire, 2010), they reverse the polar field around sunspot maximum. Accordingly, solar polar fields change polarity from one minimum to another, following the 22year magnetic cycle. Joy’s law and the poleward transport of trailing fields produce a north–south (poloidal) component of the magnetic field in the declining to minimum phase of the sunspot cycle. This was the observational motivation for the flux transport dynamo model introduced by Babcock (1961) and Leighton (1964, 1969). These empirical laws regulating the spatial distribution and temporal evolution of the photospheric magnetic field form the physical basis of the twodipole model.
The trailing fields that are transported poleward and the leading fields that are transported equatorward can be represented as a set of magnetic dipoles (ideally a ring of dipoles) in each hemisphere, located at a typical latitude of active region generation, as shown in Figure 1a for an ideal, axisymmetric case. The authors would like to point out that this ring of dipoles is not equivalent mathematically with the double ring of opposite radial fluxes introduced as source terms in some early axisymmetric dynamo simulations (Durney, 1995; Nandy & Choudhuri, 2001) to mimic the Babcock–Leighton poloidal magnetic flux emergence. The authors believe that it is more physical to represent decaying bipolar active regions with their dipole moments, which does not violate the divergencefree nature of the magnetic field and does not introduce artificial discontinuities in the magnetic potential.
Fig. 1 Twodipole model. (a) Two rings of surface dipoles representing the north–south magnetic component of decaying active regions in the photosphere at the latitude of about ±30° in an axisymmetric case. (b) Two axial dipoles placed at the center of each dipole ring of panel a. The magnetic potential of the two dipole rings of panel a and of the two axial dipoles of panel b are nearly identical at the solar surface. 
The combined magnetic potential of several dipoles along the solar circumference can be closely approximated by a virtual north–south oriented (axial) dipole that is placed in the middle of the dipole ring as shown in Figure 1b. Although the individual dipoles in Figure 1a possess nonzonal harmonic terms as well, these terms vanish at the limit of axisymmetry when the number of dipoles approaches infinity. Because the polarities of the leading and trailing fields are oppositely ordered in the two hemispheres by Hale’s law (Hale, 1924), the dipole moments of the two axial dipoles in the northern and southern hemisphere in Figure 1b should point in the same direction. The superposition of the two dipole fields reduces the photospheric magnetic field at low latitudes and enhances the magnetic field at high latitudes, which leads to strong polar fields, as demonstrated in Figure 3. We note that there is an analytical solution for the magnetic potential produced by such an eccentric dipole (Hurwitz, 1960; Winch, 1968), which greatly facilitates the calculation of the combined magnetic potential produced by the many surface sources.
In an axially symmetric case of Figure 1b, only the zonal Gauss coefficients ${g}_{l}^{0}$ of the spherical harmonics expansion of the magnetic potential are nonzero. We find the following form for ${g}_{l}^{0}$ for the eccentric twodipole model (see detailed derivation in Appendix):
$${g}_{l}^{0}=l{a}_{1}{z}_{1}^{l1}+l{a}_{2}{z}_{2}^{l1},$$(1)
where l is the spherical harmonic degree, a _{1} and a _{2} are the strengths of the two dipoles, and z _{1}, z _{2} are their locations along the zaxis of symmetry (effectively the latitudes of the dipole rings). Note that for a north–south symmetric model (a _{1} = a _{2}, z _{1} = −z _{2}), equation (1) gives zero for all evendegree coefficients. Nonzero evendegree harmonics can be obtained only by breaking the hemispheric symmetry either having different strengths (a _{1} ≠ a _{2}) or distances (z _{1} ≠ z _{2}) for the two dipoles.
3 Model validation with photospheric magnetic field data
In order to estimate the success of the model of two eccentric dipoles, we compare the model prediction to the spherical harmonics expansion of the photospheric magnetic potential. We use here the synoptic maps of the photospheric radial magnetic field measured at the Mount Wilson Observatory (MWO) during twoyear periods of three solar minima in 1975–1976, 1985–1986, and 1995–1996. The largescale photospheric magnetic field during solar minima is axisymmetric and dominated by a few low harmonics (DeRosa et al., 2012; Petrie, 2013; Zieger et al., 2019). Therefore, it is sufficient to use only the lowest zonal Gauss coefficients (here up to l = 24) of the magnetic potential. The method of deriving the zonal Gauss coefficients from the synoptic maps of the radial magnetic field and estimating their error is described in detail in Zieger et al. (2019). Figure 2 presents the zonal Gauss coefficients and the zonal spatial power spectra (for definition, see Zieger et al., 2019) for the MWO photospheric magnetic field as well as for the twodipole model field during these three solar minima. The Gauss coefficients and the power spectra are normalized by the dipole term (l = 1) of the respective solar minimum so that the relative strengths of the harmonics can be compared among the three minima with different overall magnetic powers (Smith & Balogh, 2008; Zhou & Smith, 2009). Note that the (normalized) Gauss coefficients for the three minima agree with each other (at least up to l = 8) within the estimated errors. This shows that the distribution of magnetic fields on the solar surface remains closely similar from one solar minimum to the next, despite different total powers.
Fig. 2 Spherical harmonic structure of the photospheric magnetic field. (a) Normalized zonal Gauss coefficients of the photospheric magnetic potential for three consecutive solar minima. The twodipole model obtained from the first four Gauss coefficients is shown in black. (b) Normalized zonal spatial power spectra of the photospheric magnetic field for the same solar minima as in panel a, showing the relative contributions of the different harmonics (multipoles) to the total power. 
The twodipole model has four parameters (a _{1}, a _{2}, z _{1}, z _{2}), which are exact solutions of equation (1). They depend on the first four zonal Gauss coefficients as follows:
$${z}_{1}=(9{g}_{1}^{0}{g}_{4}^{0}6{g}_{2}^{0}{g}_{3}^{0}+\sqrt{3}(27\left({g}_{1}^{0}{)}^{2}\right({g}_{4}^{0}{)}^{2}108{g}_{1}^{0}{g}_{2}^{0}{g}_{3}^{0}{g}_{4}^{0}+64{g}_{1}^{0}({g}_{3}^{0}{)}^{3}+54({g}_{2}^{0}{)}^{3}{g}_{4}^{0}36\left({g}_{2}^{0}{)}^{2}\right({g}_{3}^{0}{)}^{2}{)}^{1/2})/(24{g}_{1}^{0}{g}_{3}^{0}18\left({g}_{2}^{0}{)}^{2}\right),$$(2)
$${z}_{2}=(2{g}_{3}^{0}3{g}_{2}^{0}{z}_{1})/(3{g}_{2}^{0}6{g}_{1}^{0}{z}_{1}),$$(3)
$${a}_{1}=({g}_{2}^{0}2{g}_{1}^{0}{z}_{2})/(2{z}_{1}2{z}_{2}),$$(4)
$${a}_{2}={g}_{1}^{0}{a}_{1}.$$(5)
We have used the average values of the first four Gauss coefficients of the synoptic maps during the three minima to derive the values of these four parameters. Once the four model parameters were fixed, we calculated the higherdegree coefficients (up to l = 24) of the twodipole model using equation (1). This twodipole model was then plotted in black in Figures 2a and 2b. Figure 2 shows that the twodipole model yields Gauss coefficients and power spectra that have an excellent agreement with observations. Moreover, Figure 2b shows that the octupole component (l = 3) has even a slightly higher power in the power spectrum than the dipole component (l = 1).
As seen in Figure 2b, the observed spatial power spectra during the three solar minima are similar up to l = 8 but become uncorrelated for higher degrees (l > 8). This suggests that the MWO observations are dominated by noise beyond l = 8, in agreement with earlier results (Zieger et al., 2019). Figure 2b also shows that the twodipole model closely reproduces the observed power spectrum until l = 8. Accordingly, the twodipole model yields a close representation of the axisymmetric magnetic field during solar minima, reproducing all the statistically significant zonal Gauss coefficients (l < 9) that are responsible for most of the spectral power.
4 Modeling the north–south asymmetry in the photosphere and in the corona
Most of the low evendegree Gauss coefficients (l = 2, 4, 6, 8) in Figure 2a are small but significantly different from zero during the three solar minima. This means that the photospheric magnetic field is north–south asymmetric during these solar minima. The fact that all significant evendegree coefficients are negative implies that these terms weaken the northern polar field but strengthen the southern polar field. Thus, Figure 2a verifies the earlier result that the southern solar polar magnetic fields were stronger than the northern fields during the recent solar minima (Zhao et al., 2005; Erdős & Balogh, 2010; Virtanen & Mursula, 2010; Petrie, 2015).
In order to study this hemispheric asymmetry in more detail, we fitted the four parameters of the twodipole model to the first four Gauss coefficients individually for each solar minimum. The latitudinal profiles of the photospheric radial magnetic field obtained from the first 24 Gauss coefficients of the twodipole model is plotted in Figure 3 for the three solar minima. We normalized the radial magnetic field to the dipole term (${g}_{1}^{0}$) so that the three solar minima would be directly comparable. The mean of the three radial magnetic field profiles is shown in black. The latitudinal distribution of the radial magnetic field is very similar for the three solar minima, showing a clear asymmetry with a southern polar field ~31% stronger than the northern polar field.
Fig. 3 Latitudinal distribution of the photospheric radial magnetic field obtained from the bestfitting twodipole model for each of the three solar minima (1975–1976, 1985–1986 and 1995–1996). The radial magnetic field has been normalized to the signed dipole term (${g}_{1}^{0}$) at each solar minimum to allow direct comparison. The mean latitudinal profile is plotted in black. 
The strengths of the northern and southern axial dipoles (parameters a _{1} and a _{2}) for the three minima are shown in Figure 4a and the latitudinal locations of the dipoles (parameters z _{1} and z _{2} converted to solar latitude) in Figure 4b. Figure 4b shows that the latitudes of the northern and southern dipoles are almost exactly the same, about 30°, which is a typical latitude of emerging active regions in the beginning of the solar cycle (Petrie, 2013). Accordingly, flux emergence is roughly symmetric with respect to the equator. Moreover, the fact that the dipole latitudes remain the same from one solar minimum to another suggests that the spatial location of processes producing new flux to the surface remains constant during time scales of, at least, a few solar cycles.
Fig. 4 Parameters of the northern and southern dipoles. (a) Strengths of the northern and southern dipole for the three solar minima (1975–1976, 1985–1986 and 1995–1996). The southern dipole is stronger than the northern dipole during all three solar minima. (b) Locations of the northern and southern dipoles in solar latitude (southern latitudes are reversed to positive). 
Figure 4a shows that the strength of the southern axial dipole is systematically stronger than the northern dipole during the three solar minima. Accordingly, the southern polar field is stronger than the northern polar field during solar minima, which confirms earlier results (Zhao et al., 2005; Erdős & Balogh, 2010; Virtanen & Mursula, 2010; Petrie, 2015). The southern dipole was 62% stronger than the northern dipole in 1975–1976, but only 22% stronger in 1985–1986 and 1995–1996. Based on the results depicted in Figure 4, it is clear that the observed hemispheric asymmetry of the photospheric magnetic field is mainly due to the different strengths, not due to different locations of decaying active regions in the two hemispheres. We also note that a single eccentric axial dipole can produce a north–south asymmetric magnetic field and explain the dipole (l = 1) and quadrupole (l = 2) Gauss coefficients (Wang, 1996; Pulkkinen et al., 1999; Mursula & Hiltula, 2004; Zhao et al., 2005; Virtanen & Mursula, 2016), but cannot reproduce the higherdegree Gauss coefficients, especially the observed strong octupole moment (l = 3).
An asymmetric photospheric field also affects the largescale structure of the solar coronal magnetic field. We have investigated this by computing the potentialfield sourcesurface (PFSS) model of the coronal field (Wang & Sheeley, 1992) between the solar surface and the coronal source surface at 2.5 solar radii, using the magnetic field of the twodipole model as an inner boundary condition. The coronal magnetic field corresponding to the average twodipole model for the three minima is depicted in Figure 5, where we have also included the MWO photospheric field observed during Carrington rotation 1910 in year 1996, as a typical example of photospheric magnetic field distribution during solar minimum. The dashed circle at 2.5 solar radii marks the source surface, where the coronal magnetic field becomes radial in the PFSS model.
Fig. 5 Coronal magnetic field from the twodipole model. Average coronal magnetic field during the three solar minima obtained from the twodipole model of the photospheric field using the PFSS model. The coronal source surface, where magnetic field lines (in blue) become purely radial is marked with a dashed circle. The photospheric radial magnetic field observed during Carrington rotation 1910 in year 1996 is shown on the solar surface, featuring typical unipolar regions at the poles, with yellow and blue colors indicating positive (away from the Sun) and negative (toward the Sun) magnetic polarity, respectively. The heliospheric current sheet is shifted towards the south by 4.1°. 
Figure 5 shows that there is a significant north–south asymmetry also in the largescale structure of the solar corona. Most notably, the heliospheric current sheet (HCS), the magnetic equator outside the source surface (also called the ballerina skirt), where the field lines of opposite polarity meet each other (see Fig. 5), is shifted or coned southwards. Note also that the magnetic field lines forming closed loops in the inner corona are also asymmetric with respect to the solar equator. Interestingly, some of the closed field lines at low southern latitudes are anchored to the southern hemisphere at both ends. The southward shift of the HCS is in a good agreement with earlier results using the heliospheric magnetic field observed near the Earth (Mursula & Hiltula, 2003; Koskela et al., 2018) and by the Ulysses probe (Erdős & Balogh, 2010; Virtanen & Mursula, 2010). We find that the average shift angle of the HCS for the three minima is 4.1°, and varies from 5.5° for the first minimum to 3.6° for the second and 3.0° for the last minimum. These estimates for the southward shift of the HCS agree with an earlier estimate of HCS shift in the corona (Koskela et al., 2018) and support the result that the HCS shift in the corona is larger than the shift observed further out in the heliosphere (Mursula & Hiltula, 2003; Erdős & Balogh, 2010; Virtanen & Mursula, 2010; Koskela et al., 2018). The decreasing HCS shift suggests that the asymmetry was larger during the earlier, stronger cycles, and has decreased with the reduction of the overall solar activity. Accordingly, the north–south asymmetry of the photospheric magnetic field is expected to be the smallest in the beginning of low solar activity cycles like solar cycle 24. As mentioned in Section 1, solar cycle 24 was exceptional in many ways, therefore we did not include it in this study, but it will be subject of future work.
5 Conclusions
We have shown here that the largescale structure of the photospheric magnetic field during solar minima can be closely described with an analytical model consisting of two eccentric axial dipoles that represent the combined magnetic potential of decaying active regions. We find that, while the effective location of the dipoles is the same in the two hemispheres, the southern dipole is systematically stronger than the northern dipole during recent solar minima. This difference explains the observed north–south asymmetry in the solar polar fields and in the heliospheric current sheet location. Our analytical twodipole model also offers a new method for reconstructing the poorly observed or missing values of the solar polar fields using reliably measured lowerlatitude observations, which is critical for more accurate predictions of future solar activity (Choudhuri et al., 2007; Hathaway & Upton, 2016), space weather, and space climate. This novel polar field filling method is based on an iterative procedure where the Gauss coefficients and the highlatitude solar magnetic fields are updated in each iteration step until the polar field converges to a constant value. More specific details on this iterative procedure will be given in a followup paper.
Acknowledgments
The authors acknowledge the financial support by the Academy of Finland to the ReSoLVE Centre of Excellence (Project No. 307411). Mt. Wilson data are publicly available at ftp://howard.astro.ucla.edu/pub/obs/synoptic_charts/fits/ courtesy of Roger Ulrich. The editor thanks two anonymous reviewers for their assistance in evaluating this paper.
Appendix
Potential of a shifted zonal dipole on the solar surface
For completeness, we derive the magnetic potential of an eccentric dipole on the solar surface. We use a different method of derivation than those given by Hurwitz (1960) or Winch (1968), but we verified that our method led to the same solution.
Let us derive the photospheric potential of a zonal multipole of degree l that is shifted from the center of the Sun by $\mathit{\rho}=\xi \widehat{\mathit{x}}+\eta \widehat{\mathit{y}}+\zeta \widehat{\mathit{z}}$. The potential of a zonal multipole of degree l located at the center of the Sun has the following form in spherical coordinates:
$$U\left(\mathit{r}\right)={R}_{S}{\left(\frac{{R}_{S}}{r}\right)}^{l+1}{g}_{l}^{0}{P}_{l}^{0}\left(\mathrm{cos}\theta \right),$$(A.1)
where R _{ S } is the radius of the Sun and ${P}_{l}^{0}$ is the associated Legendre function. Normalizing the radial distance by R _{ S }, the potential becomes,
$$U\left(\mathit{r}\right)={r}^{\left(l+1\right)}{g}_{l}^{0}{P}_{l}^{0}\left(\mathrm{cos}\theta \right).$$(A.2)
First, we evaluate the potential U in Cartesian coordinates on a sphere of unit radius that is shifted by $\mathit{\rho}=\xi \widehat{\mathit{x}}+\eta \widehat{\mathit{y}}+\zeta \widehat{\mathit{z}}$ from the center of the Sun. This can be done by the Taylor series expansion of U:
$$U\left(\widehat{\mathit{r}}+\mathit{\rho}\right)=U\left(\widehat{\mathit{r}}\right)+\frac{1}{1!}\nabla U\left(\widehat{\mathit{r}}\right)\mathit{\rho}+\frac{1}{2!}{\nabla}^{2}U\left(\widehat{\mathit{r}}\right){\mathit{\rho}}^{2}+\frac{1}{3!}{\nabla}^{3}U\left(\widehat{\mathit{r}}\right){\mathit{\rho}}^{3}+\dots ={g}_{l}^{0}{P}_{l}^{0}\left(\mathrm{cos}\theta \right)\left(l+1\right){g}_{l}^{0}{P}_{l}^{0}\left(\mathrm{cos}\theta \right)\mathrm{sin}\theta \mathrm{cos}\varphi \xi {g}_{l}^{0}\frac{\mathrm{\partial}{P}_{l}^{0}\left(\mathrm{cos}\theta \right)}{\mathrm{\partial}\mathrm{cos}\theta}\mathrm{sin}\theta \mathrm{cos}\theta \mathrm{cos}\varphi \xi \left(l+1\right){g}_{l}^{0}{P}_{l}^{0}\left(\mathrm{cos}\theta \right)\mathrm{sin}\theta \mathrm{sin}\varphi \eta {g}_{l}^{0}\frac{\mathrm{\partial}{P}_{l}^{0}\left(\mathrm{cos}\theta \right)}{\mathrm{\partial}\mathrm{cos}\theta}\mathrm{sin}\theta \mathrm{cos}\theta \mathrm{sin}\varphi \eta \left(l+1\right){g}_{l}^{0}{P}_{l}^{0}\left(\mathrm{cos}\theta \right)\mathrm{cos}\theta \zeta +{g}_{l}^{0}\frac{\mathrm{\partial}{P}_{l}^{0}\left(\mathrm{cos}\theta \right)}{\mathrm{\partial}\mathrm{cos}\theta}{\mathrm{sin}}^{2}\theta \zeta +\frac{1}{2!}{\nabla}^{2}U\left(\widehat{\mathit{r}}\right){\mathit{\rho}}^{2}+\dots ={g}_{l}^{0}{P}_{l}^{0}\left(\mathrm{cos}\theta \right){g}_{l}^{0}\xi \left[\left(l+1\right)\sqrt{1{\mathrm{cos}}^{2}\theta}{P}_{l}^{0}\left(\mathrm{cos}\theta \right)+\mathrm{cos}\theta \sqrt{1{\mathrm{cos}}^{2}\theta}\frac{\mathrm{\partial}{P}_{l}^{0}\left(\mathrm{cos}\theta \right)}{\mathrm{\partial}\mathrm{cos}\theta}\right]\mathrm{cos}\varphi {g}_{l}^{0}\eta \left[\left(l+1\right)\sqrt{1{\mathrm{cos}}^{2}\theta}{P}_{l}^{0}\left(\mathrm{cos}\theta \right)+\mathrm{cos}\theta \sqrt{1{\mathrm{cos}}^{2}\theta}\frac{\mathrm{\partial}{P}_{l}^{0}\left(\mathrm{cos}\theta \right)}{\mathrm{\partial}\mathrm{cos}\theta}\right]\mathrm{sin}\varphi {g}_{l}^{0}\zeta \left[\left(l+1\right)\mathrm{cos}\theta {P}_{l}^{0}\left(\mathrm{cos}\theta \right)+\left({\mathrm{cos}}^{2}\theta 1\right)\frac{\mathrm{\partial}{P}_{l}^{0}\left(\mathrm{cos}\theta \right)}{\mathrm{\partial}\mathrm{cos}\theta}\right]+\frac{1}{2!}{\nabla}^{2}U\left(\widehat{\mathit{r}}\right){\mathit{\rho}}^{2}+\dots .$$(A.3)
Using the following recurrence formulae of the associated Legendre function,
$$\sqrt{1{x}^{2}}{P}_{l}^{m}\left(x\right)=\frac{1}{2l+1}\left[{P}_{l+1}^{m+1}\left(x\right)+{P}_{l1}^{m+1}\left(x\right)\right],$$(A.4)
$$\sqrt{1{x}^{2}}\frac{\mathrm{d}}{\mathrm{d}x}{P}_{l}^{m}\left(x\right)=\frac{1}{2}\left[(l+m)(lm+1){P}_{l}^{m1}\left(x\right){P}_{l}^{m+1}\left(x\right)\right],$$(A.5)
$$({x}^{2}1)\frac{\mathrm{d}}{\mathrm{d}x}{P}_{l}^{m}\left(x\right)=(l+1)x{P}_{l}^{m}\left(x\right)+(lm+1){P}_{l+1}^{m}\left(x\right),$$(A.6)
$$(lm+1){P}_{l+1}^{m}\left(x\right)=(2l+1)x{P}_{l}^{m}\left(x\right)(l+m){P}_{l1}^{m}\left(x\right),$$(A.7)
$${P}_{l}^{m}=(1{)}^{m}\frac{(lm)!}{(l+m)!}{P}_{l}^{m},$$(A.8)
the potential U at $\widehat{\mathit{r}}$ + ρ becomes,
$$U\left(\widehat{\mathit{r}}+\mathit{\rho}\right)={g}_{l}^{0}{P}_{l}^{0}\left(\mathrm{cos}\theta \right){g}_{l}^{0}\xi \left[\frac{l+1}{2l+1}\left[{P}_{l+1}^{1}\left(\mathrm{cos}\theta \right)+{P}_{l1}^{1}\left(\mathrm{cos}\theta \right)\right]+\mathrm{cos}\theta \frac{1}{2}\left[l\left(l+1\right){P}_{l}^{1}\left(\mathrm{cos}\theta \right){P}_{l}^{1}\left(\mathrm{cos}\theta \right)\right]\right]\mathrm{cos}\varphi {g}_{l}^{0}\eta \left[\frac{l+1}{2l+1}\left[{P}_{l+1}^{1}\left(\mathrm{cos}\theta \right)+{P}_{l1}^{1}\left(\mathrm{cos}\theta \right)\right]+\mathrm{cos}\theta \frac{1}{2}\left[l\left(l+1\right){P}_{l}^{1}\left(\mathrm{cos}\theta \right){P}_{l}^{1}\left(\mathrm{cos}\theta \right)\right]\right]\mathrm{sin}\varphi {g}_{l}^{0}\zeta \left(l+1\right){P}_{l+1}^{0}\left(\mathrm{cos}\theta \right)+\frac{1}{2!}{\nabla}^{2}U\left(\widehat{\mathit{r}}\right){\mathit{\rho}}^{2}+\dots ={g}_{l}^{0}{P}_{l}^{0}\left(\mathrm{cos}\theta \right)\frac{1}{2\left(2l+1\right)}{g}_{l}^{0}\xi \left[\begin{array}{c}2\left(l+1\right){P}_{l+1}^{1}\left(\mathrm{cos}\theta \right)+2\left(l+1\right){P}_{l1}^{1}\left(\mathrm{cos}\theta \right)\\ +\left(2l+1\right)l\left(l+1\right)\mathrm{cos}\theta {P}_{l}^{1}\left(\mathrm{cos}\theta \right)\left(2l+1\right)\mathrm{cos}\theta {P}_{l}^{1}\left(\mathrm{cos}\theta \right)\end{array}\right]\mathrm{cos}\varphi \frac{1}{2\left(2l+1\right)}{g}_{l}^{0}\eta \left[\begin{array}{c}2\left(l+1\right){P}_{l+1}^{1}\left(\mathrm{cos}\theta \right)+2\left(l+1\right){P}_{l1}^{1}\left(\mathrm{cos}\theta \right)\\ +\left(2l+1\right)l\left(l+1\right)\mathrm{cos}\theta {P}_{l}^{1}\left(\mathrm{cos}\theta \right)\left(2l+1\right)\mathrm{cos}\theta {P}_{l}^{1}\left(\mathrm{cos}\theta \right)\end{array}\right]\mathrm{sin}\varphi {g}_{l}^{0}\zeta \left(l+1\right){P}_{l+1}^{0}\left(\mathrm{cos}\theta \right)+\frac{1}{2!}{\nabla}^{2}U\left(\widehat{\mathit{r}}\right){\mathit{\rho}}^{2}+\dots ={g}_{l}^{0}{P}_{l}^{0}\left(\mathrm{cos}\theta \right)\frac{1}{2\left(2l+1\right)}{g}_{l}^{0}\xi \left[\begin{array}{c}2\left(l+1\right){P}_{l+1}^{1}\left(\mathrm{cos}\theta \right)+2\left(l+1\right){P}_{l1}^{1}\left(\mathrm{cos}\theta \right)\\ \frac{\left(2l+1\right)l\left(l+1\right)\left(l1\right)!}{\left(l+1\right)!}\mathrm{cos}\theta {P}_{l}^{1}\left(\mathrm{cos}\theta \right)\left(2l+1\right)\mathrm{cos}\theta {P}_{l}^{1}\left(\mathrm{cos}\theta \right)\end{array}\right]\mathrm{cos}\varphi \frac{1}{2\left(2l+1\right)}{g}_{l}^{0}\eta \left[\begin{array}{c}2\left(l+1\right){P}_{l+1}^{1}\left(\mathrm{cos}\theta \right)+2\left(l+1\right){P}_{l1}^{1}\left(\mathrm{cos}\theta \right)\\ \frac{\left(2l+1\right)l\left(l+1\right)\left(l1\right)!}{\left(l+1\right)!}\mathrm{cos}\theta {P}_{l}^{1}\left(\mathrm{cos}\theta \right)\left(2l+1\right)\mathrm{cos}\theta {P}_{l}^{1}\left(\mathrm{cos}\theta \right)\end{array}\right]\mathrm{sin}\varphi {g}_{l}^{0}\zeta \left(l+1\right){P}_{l+1}^{0}\left(\mathrm{cos}\theta \right)+\frac{1}{2!}{\nabla}^{2}U\left(\widehat{\mathit{r}}\right){\mathit{\rho}}^{2}+\dots ={g}_{l}^{0}{P}_{l}^{0}\left(\mathrm{cos}\theta \right)\frac{1}{2l+1}{g}_{l}^{0}\xi \left[\left(l+1\right){P}_{l+1}^{1}\left(\mathrm{cos}\theta \right)l{P}_{l+1}^{1}\left(\mathrm{cos}\theta \right)\right]\mathrm{cos}\varphi \frac{1}{2l+1}{g}_{l}^{0}\eta \left[\left(l+1\right){P}_{l+1}^{1}\left(\mathrm{cos}\theta \right)l{P}_{l+1}^{1}\left(\mathrm{cos}\theta \right)\right]\mathrm{sin}\varphi {g}_{l}^{0}\zeta \left(l+1\right){P}_{l+1}^{0}\left(\mathrm{cos}\theta \right)+\frac{1}{2!}{\nabla}^{2}U\left(\widehat{\mathit{r}}\right){\mathit{\rho}}^{2}+\dots ={g}_{l}^{0}{P}_{l}^{0}\left(\mathrm{cos}\theta \right)+{g}_{l}^{0}\xi {P}_{l+1}^{1}\left(\mathrm{cos}\theta \right)\mathrm{cos}\varphi +{g}_{l}^{0}\eta {P}_{l+1}^{1}\left(\mathrm{cos}\theta \right)\mathrm{sin}\varphi {g}_{l}^{0}\zeta \left(l+1\right){P}_{l+1}^{0}\left(\mathrm{cos}\theta \right)+\frac{1}{2!}{\nabla}^{2}U\left(\widehat{\mathit{r}}\right){\mathit{\rho}}^{2}+\dots .$$(A.9)
Let us now move the center of our coordinate system to $\mathit{\rho}$. Then the potential on the sphere of unit radius is the potential of a zonal multipole that is shifted by $\mathit{\rho}$ from the center of the sphere. Thus, the potential of a shifted zonal multipole at the solar surface is,
$$U\left(\widehat{\stackrel{\u20d7}{r}}\right)={g}_{l}^{0}{P}_{l}^{0}\left(\mathrm{cos}\theta \right){g}_{l}^{0}\xi {P}_{l+1}^{1}\left(\mathrm{cos}\theta \right)\mathrm{cos}\varphi {g}_{l}^{0}\eta {P}_{l+1}^{1}\left(\mathrm{cos}\theta \right)\mathrm{sin}\varphi +{g}_{l}^{0}\zeta (l+1){P}_{l+1}^{0}\left(\mathrm{cos}\theta \right)+\frac{1}{2!}{\nabla}^{2}U\left(\widehat{\mathit{r}}\right)(\mathit{\rho}{)}^{2}+\dots $$(A.10)
where $\mathit{\rho}=\xi \widehat{\mathit{x}}+\eta \widehat{\mathit{y}}+\zeta \widehat{\mathit{z}}$ is the displacement of the zonal multipole from the center of the Sun.
Now we introduce the Schmidt seminormalized associated Legendre functions defined as,
$${\stackrel{\u0303}{P}}_{l}^{0}\left(x\right)={P}_{l}^{0}\left(x\right)={P}_{l}\left(x\right)\hspace{1em}\mathrm{for}\hspace{1em}\mathrm{m}=0,\hspace{1em}\mathrm{and}\hspace{1em}{\stackrel{\u0303}{P}}_{l}^{m}\left(x\right)=(1{)}^{m}\sqrt{\frac{2\left(lm\right)!}{\left(l+m\right)!}}{P}_{l}^{m}\left(x\right)\hspace{1em}\mathrm{for}\hspace{1em}\mathrm{m}\ne 0.$$(A.11)
The potential of a shifted zonal multipole becomes,
$$U\left(\widehat{\mathit{r}}\right)={g}_{l}^{0}{\stackrel{\u0303}{P}}_{l}^{0}\left(\mathrm{cos}\theta \right){g}_{l}^{0}\xi (1{)}^{1}\sqrt{\frac{\left(l+1+1\right)!}{2\left(l+11\right)!}}{\stackrel{\u0303}{P}}_{l+1}^{1}\left(\mathrm{cos}\theta \right)\mathrm{cos}\varphi {g}_{l}^{0}\eta (1{)}^{1}\sqrt{\frac{\left(l+1+1\right)!}{2\left(l+11\right)!}}{\stackrel{\u0303}{P}}_{l+1}^{1}\left(\mathrm{cos}\theta \right)\mathrm{sin}\varphi +{g}_{l}^{0}\zeta (l+1){\stackrel{\u0303}{P}}_{l+1}^{0}\left(\mathrm{cos}\theta \right)+\frac{1}{2!}{\nabla}^{2}U\left(\widehat{\mathit{r}}\right)(\mathit{\rho}{)}^{2}+\dots ={g}_{l}^{0}{\stackrel{\u0303}{P}}_{l}^{0}(\mathrm{cos}\theta )+{g}_{l}^{0}\xi \sqrt{\frac{\left(l+1\right)\left(l+2\right)}{2}}{\stackrel{\u0303}{P}}_{l+1}^{1}\left(\mathrm{cos}\theta \right)\mathrm{cos}\varphi +{g}_{l}^{0}\eta \sqrt{\frac{\left(l+1\right)\left(l+2\right)}{2}}{\stackrel{\u0303}{P}}_{l+1}^{1}\left(\mathrm{cos}\theta \right)\mathrm{sin}\varphi +{g}_{l}^{0}\zeta (l+1\left){\stackrel{\u0303}{P}}_{l+1}^{0}\right(\mathrm{cos}\theta )+\frac{1}{2!}{\nabla}^{2}U(\widehat{\mathit{r}}\left)\right(\mathit{\rho}{)}^{2}+\dots .$$(A.12)
From the general solution of equation (A.12), we obtain the potential of a shifted zonal dipole (l = 1) in terms of the Schmidt seminormalized associated Legendre functions:
$$U\left(\widehat{\mathit{r}}\right)={g}_{1}^{0}{\stackrel{\u0303}{P}}_{1}^{0}\left(\mathrm{cos}\theta \right)+{g}_{1}^{0}\xi \sqrt{3}{\stackrel{\u0303}{P}}_{2}^{1}\left(\mathrm{cos}\theta \right)\mathrm{cos}\varphi +{g}_{1}^{0}\eta \sqrt{3}{\stackrel{\u0303}{P}}_{2}^{1}\left(\mathrm{cos}\theta \right)\mathrm{sin}\varphi +{g}_{1}^{0}\zeta 2{\stackrel{\u0303}{P}}_{2}^{0}\left(\mathrm{cos}\theta \right)+\frac{1}{2!}{\nabla}^{2}U\left(\widehat{\mathit{r}}\right)(\mathit{\rho}{)}^{2}+\dots .$$(A.13)
For a zonal dipole that is shifted along the symmetry axis (ξ = 0, η = 0), the potential can be written as,
$${U}_{z}\left(\widehat{\mathit{r}}\right)={g}_{1}^{0}{\stackrel{\u0303}{P}}_{1}^{0}\left(\mathrm{cos}\theta \right)+2\zeta {g}_{1}^{0}{\stackrel{\u0303}{P}}_{2}^{0}\left(\mathrm{cos}\theta \right)+\frac{1}{2!}\frac{{\mathrm{\partial}}^{2}}{\mathrm{\partial}{z}^{2}}U\left(\widehat{\mathit{r}}\right)(\zeta {)}^{2}+\frac{1}{3!}\frac{{\mathrm{\partial}}^{3}}{\mathrm{\partial}{z}^{3}}U(\widehat{\mathit{r}}\left)\right(\zeta {)}^{3}+\dots .$$(A.14)
We can rewrite equation (A.14) in recursive form where each term is calculated from the preceding term,
$${U}_{z}\left(\widehat{\mathit{r}}\right)={g}_{1}^{0}{\stackrel{\u0303}{P}}_{1}^{0}\left(\mathrm{cos}\theta \right)\frac{1}{1!}\frac{\mathrm{\partial}}{\mathrm{\partial}z}U\left(\widehat{\mathit{r}}\right)\zeta \frac{1}{2!}\frac{\mathrm{\partial}}{\mathrm{\partial}z}\left(\frac{\mathrm{\partial}}{\mathrm{\partial}z}U\left(\widehat{\mathit{r}}\right)\left(\zeta \right)\right)\zeta \frac{1}{3!}\frac{\mathrm{\partial}}{\mathrm{\partial}z}\left(\frac{{\mathrm{\partial}}^{2}}{\mathrm{\partial}{z}^{2}}U\left(\widehat{\mathit{r}}\right)(\zeta {)}^{2}\right)\zeta \dots \frac{1}{\left(l1\right)!}\frac{\mathrm{\partial}}{\mathrm{\partial}z}\left(\frac{{\mathrm{\partial}}^{l2}}{\mathrm{\partial}{z}^{l2}}U\left(\widehat{\mathit{r}}\right)(\zeta {)}^{l2}\right)\zeta \dots .$$(A.15)
Now we make use of equations (A.3) and (A.9), where it was shown that,
$${\frac{\mathrm{\partial}}{\mathrm{\partial}z}\left({r}^{(l+1)}{g}_{l}^{0}{P}_{l}^{0}\left(\mathrm{cos}\theta \right)\right)}_{\widehat{r}}={g}_{l}^{0}(l+1){P}_{l+1}^{0}\left(\mathrm{cos}\theta \right).$$(A.16)
Inserting equation (A.16) into equation (A.15) recursively, the photospheric potential of a zonal dipole shifted in the $\widehat{\mathbf{z}}$ direction becomes,
$${U}_{z}\left(\widehat{\mathit{r}}\right)={g}_{1}^{0}{\stackrel{\u0303}{P}}_{1}^{0}\left(\mathrm{cos}\theta \right)+{g}_{1}^{0}\sum _{l=2}^{\mathrm{\infty}}\mathrm{}\frac{l!}{(l1)!}{\zeta}^{l1}{\stackrel{\u0303}{P}}_{l}^{0}\left(\mathrm{cos}\theta \right)={g}_{1}^{0}\sum _{l=1}^{\mathrm{\infty}}\mathrm{}l{\zeta}^{l1}{\stackrel{\u0303}{P}}_{l}^{0}\left(\mathrm{cos}\theta \right).$$(A.17)
Thus, the Gauss coefficients of a dipole shifted by ζ along the symmetry axis are the following:
$${g}_{l}^{0}={g}_{1}^{0}l{\zeta}^{l1},$$(A.18)
which is used in the twodipole model of equation (1).
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All Figures
Fig. 1 Twodipole model. (a) Two rings of surface dipoles representing the north–south magnetic component of decaying active regions in the photosphere at the latitude of about ±30° in an axisymmetric case. (b) Two axial dipoles placed at the center of each dipole ring of panel a. The magnetic potential of the two dipole rings of panel a and of the two axial dipoles of panel b are nearly identical at the solar surface. 

In the text 
Fig. 2 Spherical harmonic structure of the photospheric magnetic field. (a) Normalized zonal Gauss coefficients of the photospheric magnetic potential for three consecutive solar minima. The twodipole model obtained from the first four Gauss coefficients is shown in black. (b) Normalized zonal spatial power spectra of the photospheric magnetic field for the same solar minima as in panel a, showing the relative contributions of the different harmonics (multipoles) to the total power. 

In the text 
Fig. 3 Latitudinal distribution of the photospheric radial magnetic field obtained from the bestfitting twodipole model for each of the three solar minima (1975–1976, 1985–1986 and 1995–1996). The radial magnetic field has been normalized to the signed dipole term (${g}_{1}^{0}$) at each solar minimum to allow direct comparison. The mean latitudinal profile is plotted in black. 

In the text 
Fig. 4 Parameters of the northern and southern dipoles. (a) Strengths of the northern and southern dipole for the three solar minima (1975–1976, 1985–1986 and 1995–1996). The southern dipole is stronger than the northern dipole during all three solar minima. (b) Locations of the northern and southern dipoles in solar latitude (southern latitudes are reversed to positive). 

In the text 
Fig. 5 Coronal magnetic field from the twodipole model. Average coronal magnetic field during the three solar minima obtained from the twodipole model of the photospheric field using the PFSS model. The coronal source surface, where magnetic field lines (in blue) become purely radial is marked with a dashed circle. The photospheric radial magnetic field observed during Carrington rotation 1910 in year 1996 is shown on the solar surface, featuring typical unipolar regions at the poles, with yellow and blue colors indicating positive (away from the Sun) and negative (toward the Sun) magnetic polarity, respectively. The heliospheric current sheet is shifted towards the south by 4.1°. 

In the text 
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