The Radial Interplanetary Field Strength at Sunspot Minimum as Polar Field Proxy and Solar Cycle Predictor

The minimum value of the geomagnetic aa index has served as a remarkably successful predictor of solar cycle amplitude. This value is reached near or just after sunspot minimum, when both the near-Earth solar wind speed and interplanetary magnetic field (IMF) strength fall to their lowest values. At this time, the heliospheric current sheet is flattened toward the heliographic equator and the dominant source of the IMF is the Sun’s axial dipole moment, which, in turn, has its source in the polar fields. As recognized previously, the success of aamin as solar cycle precursor provides support for dynamo models in which the sunspots of a given cycle are produced by winding up the poloidal field built up during the previous cycle. Because they are highly concentrated toward the poles by the surface meridional flow, the polar fields are difficult to measure reliably. Here we point out that the observed value of the radial IMF strength at solar minimum can be used to constrain the polar field measurements, and that this parameter, which is directly proportional to the Sun’s axial dipole strength, may be an even better solar cycle predictor than geomagnetic activity.


Introduction
In many models of the solar cycle, such as those of Babcock (1961) and Leighton (1969), the sunspots of a given cycle are generated by winding up the poloidal field built up during the previous cycle.This has led to the idea of using the strength of the polar field near solar minimum to predict the amplitude of the following cycle (see, e.g., Schatten & Sofia 1987;Layden et al. 1991;Schatten & Pesnell 1993;Schatten 2005;Svalgaard et al. 2005;Upton & Hathaway 2023).Another widely used and successful predictor has been geomagnetic activity late in the solar cycle (Ohl 1966(Ohl , 1971;;Feynman 1982;Thompson 1993;Lantos & Richard 1998;Hathaway et al. 1999;Hathaway 2015;Upton & Hathaway 2023).Here, it should be emphasized that it is the minimum level of geomagnetic activity that reflects the polar field strength and is the physically relevant parameter.
The Sun's polar fields are notoriously difficult to measure, while geomagnetic activity is also sensitive to parameters such as the local wind speed.In Wang & Sheeley (2009), we suggested that the radial interplanetary magnetic field (IMF) strength near solar minimum could be used as a solar cycle predictor.In this Letter, we provide further support for this idea by showing that the Sun's dipole component provides the main contribution to the radial IMF through most of the solar cycle, with the axial dipole component being heavily dominant near sunspot minimum.We then compare predictions for the peak sunspot number of cycle 25 using the aa index, the measured axial dipole strength, and the radial IMF strength during the last solar minimum.Finally, we show how the observed IMF strength can be used to constrain the axial dipole measurements.

Dependence of the Radial IMF Strength on the Sun's Dipole Component
Magnetometer measurements by Ulysses showed that the radial IMF strength is approximately independent of heliographic latitude (Balogh et al. 1995;Smith & Balogh 2008).When averaged over a 27.The total open flux may be estimated by applying a potential-field source-surface (PFSS) extrapolation (Altschuler & Newkirk 1969;Schatten et al. 1969) to line-of-sight measurements of the photospheric field.In this model, the coronal field is assumed to satisfy ∇ × B = 0 out to a spherical "source surface" at heliocentric distance r = R ss , where the transverse field components are required to vanish.At the inner boundary, B r is matched to the observed photospheric field, deprojected on the assumption that it is radial at the height where it is measured (Wang & Sheeley 1992).The total open flux is given by where L is heliographic latitude, f Carrington longitude, and Ω solid angle.For the source surface radius, the only free parameter in the PFSS model, we take R ss = 2.5 R e , which has been shown to approximately reproduce the IMF sector structure during 1976-1982(Hoeksema 1984) ) and the configuration of He I 1083.0 nm coronal holes during 1976-1995 (Figure 2 in Wang et al. 1996).
We employ photospheric field maps from the Mount Wilson Observatory (MWO), available from 1966 December to 2013 Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
January (CR 1516(CR -2132)), and from the Wilcox Solar Observatory (WSO), available from 1976 May to the present (CR 1642(CR -2271)).Both observatories use the narrow, magnetically sensitive (Landé g = 3.0) Fe I 525.0 nm absorption line to measure the shift between the left-and right-circularly polarized intensities.To correct for magnetograph saturation, where the Zeeman shift brings the nonlinear core of the line profile into the exit slit, comparison measurements are customarily made in the wide, nonsaturating Fe I 523.3 nm line (g = 1.3), and the ratio δ −1 of the 523.3 and 525.0 nm flux densities is used as the correction factor (see Ulrich et al. 2002, and references therein).Here, it is important that the position of the exit slit along the 523.3 nm line wing be such that the signal comes from the same height as the 525.0 nm signal (Wang et al. 2022).The saturation correction is then 4.5 at the disk center, falling to 2.0 at the limb, as found originally by Ulrich (1992).Accordingly, the MWO and WSO photospheric field measurements will be scaled upward by Because the multipole components l of the photospheric field fall off as r −( l+2) , the main contribution to the coronal field at r = 2.5 R e , and thus to Φ open , comes from the dipole (l = 1) component; the only exception is a brief interval around the time of polar field reversal, when the (l = 2, |m| = 2) quadrupole component may be of comparable strength or even dominate (splitting the heliospheric current sheet (HCS) into two conical surfaces located 180°apart; see Wang et al. 2014).The Sun's total dipole strength, D tot , may be expressed as with the axial (l = 1, m = 0) and equatorial (l = 1, |m| = 1) components related to the photospheric flux distribution by , , , sin , 4 , , , cos cos , 6 , , , cos sin .7  From Figures 1(a) and 2(a), we see that the variation of the MWO and WSO total open fluxes roughly reproduces that of the observed radial IMF strength, although the predicted values tend to be too low (by ∼30%) during the rising and maximum phases of the solar cycle.As expected, the open fluxes derived from the l = 1 component alone are similar to those calculated by summing over all multipoles (up to l = 31), with the largest deviations occurring when the axial dipole strength approaches zero near 1980, 1989, 1999, and 2012.Comparing Figure 1(a) with (b) and 2(a) with (b), we see that the large peaks in the IMF strength observed in 1982IMF strength observed in , 1991IMF strength observed in , 2002IMF strength observed in -2003IMF strength observed in , and 2014IMF strength observed in -2015 correspond to peaks in D eq .At these times, |D ax | is also rapidly increasing toward its solar minimum level.Near sunspot minimum (marked by the vertical dashed lines), |D ax | ?D eq , implying that the Sun's axial dipole component is the (overwhelmingly) dominant source of the radial IMF strength during this period.
The discrepancy between the derived open flux and the measured radial IMF strength near sunspot maximum can at least partially be attributed to the presence of interplanetary coronal mass ejections (ICMEs).Following the procedure in Wang & Sheeley (2015), we now add to our MWO and WSO total open fluxes the contribution of the 556 ICMEs listed in the catalog of Richardson & Cane (2010),2 assigning to each ICME a radial field strength extracted from the OMNIWeb database.It should be evident that high spatial resolution is not required when extrapolating photospheric field measurements into the heliosphere, since the high-order multipoles that dominate at the solar surface fall off rapidly with height and do not contribute to Φ open .The MWO and WSO synoptic maps have very low resolution but provide continuous coverage over several decades, making them especially suitable for long-term studies of the IMF.More recent synoptic databases, such as those of the Global Oscillation Network Group and the Helioseismic and Magnetic Imager, provide much higher spatial resolution but relatively noisy measurements of the dipole component, with little agreement among different instruments (see Figures 5 and 6 in Wang et al. 2022).

The Minimum Level of Geomagnetic Activity as Solar Cycle Predictor
On timescales greater than a month, the geomagnetic aa index is highly correlated with the V B n sw tot IMF , where V sw is the solar wind speed, B tot IMF the total magnitude of the IMF at Earth, and n ∼ 2 (see, e.g., Murayama & Hakamada 1975;Crooker et al. 1977;Rouillard et al. 2007;Svalgaard & Cliver 2007;Lockwood et al. 2022).Near solar minimum, when sunspot activity is negligible and |D ax | ?D eq , the HCS is flattened toward the ecliptic and V sw falls to its lowest values.Although D ax is strong, the near-absence of the equatorial dipole and quadrupole components and of ICMEs means that both | | B r IMF and B tot IMF reach their minima.As a result, the aa index also falls to a minimum.Figure 4(a) shows 13-CR running averages of the aa index, total and radial IMF strength, and wind speed at Earth during 1976-2023.Clearly defined minima are seen in these four quantities near (but often several months after) each sunspot minimum (marked by vertical dashed lines).The overall correlation between the aa index and | | B r IMF is as high as 0.93, while that between aa and V sw is 0.78.The correlation between aa and B tot IMF (0.86) is lower than that between aa and IMF because of the effect of the Parker spiral: increases in the wind speed act to reduce IMF .Figure 4(b) shows the variation of the MWO and WSO total and axial dipole strengths during this same period, with successive sunspot minima again marked by vertical dashed lines.It may be noted that peaks in the aa index, IMF strength, and wind speed tend to coincide with peaks in D tot and D eq .
The recurrent geomagnetic activity characteristic of the declining phase of the cycle is closely related to increases in D eq due to low-latitude sunspot activity.However, the equatorial dipole decays on the timescale of the surface meridional flow (∼1-2 yr) and does not act as a source of sunspots for the next cycle.When sunspot activity falls to a minimum, the Sun's axial dipole component becomes the dominant source of the IMF, the HCS lies very close to the ecliptic plane, and the wind speeds at Earth are low; the aa index then reaches a minimum value that depends on |D ax |.This is the physical basis of the original finding of Ohl (1966) that it is the lowest value of aa, and not just the level of geomagnetic activity during the declining phase of the cycle (as assumed in some subsequent studies), which is the precursor of the following cycle.
An interesting question raised by Figure 4(a) is why the aa index, IMF strength, and solar wind speed tend to reach their lowest values several months after sunspot minimum itself.In the case of V sw , remnant coronal holes may still be present at low latitudes even in the absence of sunspot activity; such holes and their wind streams finally decay after their nonaxisymmetric magnetic flux has been transported to midlatitudes by meridional flow and annihilated by the combined action of for the amplitude of cycle 25, which is lower than that predicted using the classical aa index.
A striking outlier in Figures 5(a) and (b), located well above the regression lines, is cycle 19 (1954)(1955)(1956)(1957)(1958)(1959)(1960)(1961)(1962)(1963)(1964), which has by far the highest amplitude ( = R 269 max ) of the 13 cycles.Although the associated values of aa min and aaH min are smaller than for cycle 21, the polar faculae counts of Sheeley (2008) indicate that the polar fields were roughly twice as strong in 1954 than in 1976.A possible explanation is that the HCS was so flattened toward the equator during the cycle 19 sunspot minimum that the Earth remained inside the relatively weak field near the polarity reversal; this is supported by the pronounced semiannual modulation in the geomagnetic activity recorded at that time, when the inclination of the HCS was determined by the Sun's 7°tilt (see Rosenberg & Coleman 1969).

Measurements of the Sun's Axial Dipole Strength as Solar Cycle Predictor
The Sun's axial dipole strength during the 1976, 1986, 1996, and 2008 sunspot minima may be derived from both MWO and WSO photospheric field maps, and during the 2019 minimum from the WSO synoptic data.Figure 6 We remark that the axial dipole strength near solar minimum is relatively insensitive to the detailed structure of the polar fields, which depends on the poorly known latitudinal profile of the surface meridional flow and may vary from cycle to cycle (see, e.g., Ulrich 2010;Hathaway et al. 2022).However, variations in the polar field distribution may affect the process of converting the internal poloidal field into toroidal flux and sunspots, contributing to the scatter in the solar cycle predictions.
Although the physical connection between | | D ax ssmin and R max is more direct than that between aa min or aaH min and R max , obvious disadvantages of this predictor are the small number of cycles for which measurements of | | D ax ssmin are available and the uncertainty in these measurements, as suggested by comparing the MWO and WSO values for a given sunspot minimum.While in situ measurements of the IMF strength exist only for a similarly small number of cycles, they are likely to be more reliable than observations of the weak line-of-sight fields near the Sun's poles.

The Radial IMF Strength at Sunspot Minimum as Solar Cycle Predictor
In Figure 7(a), we plot the minimum value of the radial IMF strength against the amplitude of the next cycle, for cycles 21-24.Here, rotation by rotation, we have calculated 13-CR running means of the (absolute value of) the daily averaged B x from the OMNIWeb, and selected the lowest value occurring within a year of solar minimum.This gives  Since D ax provides the dominant contribution to the radial IMF strength near sunspot minimum, we may attempt to correct the axial dipole measurements by reversing the process of extrapolating the photospheric field into the heliosphere.The source surface field at solar minimum may be written as Integrating |B r (R ss , L)| over the source surface to obtain the total open flux, we find, using Equation (1), Setting R ss = 2.5 R e and r E = 215 R e , we obtain the following relation: . 10 IMF , we take an unsigned 13-CR average, centered on the solar minimum date, of the daily values of B x from the OMNIWeb.

Summary and Discussion
We now summarize our results.).The difference is partly a result of their assuming a constant, cycle-independent amplitude-toprecursor ratio (so that the associated straight line passes through the origin).Also, they omitted the high-amplitude cycle 21 when calculating the ratio ssmin .Finally, we note that the close relationship between the aa or aaH index, the radial IMF strength/total open flux, and the Sun's dipole moment may be used to constrain the long-term evolution of the Sun's large-scale field and irradiance.This may be especially helpful in multicycle flux transport simulations that otherwise rely almost entirely on the historical sunspot record.
3 day Carrington rotation (CR), the strength of the radial IMF at Earth, denotes the Sun's total open flux, t is time, and r E = 1 au.

Figure 1
Figure 1(a) compares the observed radial IMF strength during 1967-2012 with the values predicted from the MWO total open fluxes; also plotted are the values of | | B r IMF obtained when only the dipole contribution to Φ open is included.Here, daily averages of the near-Earth B x were downloaded from the OMNIWeb site 1 and averaged without the sign over each CR. Figure 1(b) shows the variation of the MWO axial and equatorial dipole strengths during the same time interval.All curves (which also include the sunspot numbers) represent 3-CR running averages.Figure 2 is analogous to Figure 1, but with the MWO data replaced by WSO data from 1976-2023.From Figures1(a) and 2(a), we see that the variation of the MWO and WSO total open fluxes roughly reproduces that of the observed radial IMF strength, although the predicted values tend to be too low (by ∼30%) during the rising and maximum phases of the solar cycle.As expected, the open fluxes derived from the l = 1 component alone are similar to those calculated by summing over all multipoles (up to l = 31), with the largest deviations occurring when the axial dipole strength approaches zero near1980, 1989, 1999, and 2012.
Figure 1(a) compares the observed radial IMF strength during 1967-2012 with the values predicted from the MWO total open fluxes; also plotted are the values of | | B r IMF obtained when only the dipole contribution to Φ open is included.Here, daily averages of the near-Earth B x were downloaded from the OMNIWeb site 1 and averaged without the sign over each CR. Figure 1(b) shows the variation of the MWO axial and equatorial dipole strengths during the same time interval.All curves (which also include the sunspot numbers) represent 3-CR running averages.Figure 2 is analogous to Figure 1, but with the MWO data replaced by WSO data from 1976-2023.From Figures1(a) and 2(a), we see that the variation of the MWO and WSO total open fluxes roughly reproduces that of the observed radial IMF strength, although the predicted values tend to be too low (by ∼30%) during the rising and maximum phases of the solar cycle.As expected, the open fluxes derived from the l = 1 component alone are similar to those calculated by summing over all multipoles (up to l = 31), with the largest deviations occurring when the axial dipole strength approaches zero near1980, 1989, 1999, and 2012.
Figure 3(a) shows the effect of adding the ICME flux to the MWO open flux; Figure 3(b) shows the result for the WSO open flux.The ICMEs contributed ∼20% of the near-Earth radial IMF strength during 1998-2005 and ∼18% during 2011-2015.Their inclusion improves the correlation between the predicted and observed variation of the radial IMF strength from 0.77 to 0.82 in the case of MWO during 1996-2012, and from 0.82 to 0.84 in the case of WSO during 1996-2023.

Figure 1 .
Figure 1.(a) The observed variation of the radial IMF strength during 1967-2012 (red curve) is compared with the total open flux derived from the MWO photospheric field (blue curve) and with the MWO open flux including only the dipole (l = 1) contribution (black curve); also plotted are the (SIDC version 2.0) sunspot numbers (dotted curve).All curves represent 3-CR running averages; vertical dashed lines mark the successive sunspot minima.A PFSS extrapolation with R ss = 2.5 R e was applied to the MWO photospheric field maps for CR 1516-2132, which were corrected for the saturation of the Fe I 525.0 nm measurements by multiplying by -d = -L 4.5 2.5 sin 1 2 .The total open flux is dominated by the dipole component, except during polar field reversal, when the quadrupole (l = 2) component becomes significant.(b) Variation of the MWO axial dipole (l = 1, m = 0) (red curve) and equatorial dipole (l = 1, |m| = 0) (black curve) strengths.The peaks in the IMF strength observed just after sunspot maximum correspond to peaks in D eq ; the axial dipole component is heavily dominant near sunspot minimum.

Figure 2 .
Figure 2. As in Figure 1, except that the open fluxes and dipole components are derived from WSO photospheric field maps (CR 1642-2271) instead of MWO data, with the time interval now extending from 1976 to 2023.
(a) shows | | D ax ssmin plotted against R max of the following cycle based on the MWO measurements, and Figure 6(b) shows the same based on WSO.Here, the axial dipole strengths represent 13-CR averages centered on the month when the sunspot number reaches its minimum, except for the WSO value for 1976, which is centered on November rather than March (the WSO synoptic data started in 1976 May).A linear fit to the four points in the MWO scatterplot (96.A least-squares fit to the WSO scatterplot (Figure 6(b)) gives (measured value during the 2019 sunspot minimum, we obtain =  R 133 40 max for the amplitude of cycle 25.

Figure 3 .
Figure 3.Effect of adding the contribution of ICMEs to the (a) MWO and (b) WSO total open fluxes, corrected using the ( -) d = -

Figure 4 .
Figure4.(a) Variation of the geomagnetic aa index (red curve), (near-Earth) total IMF strength (thin black curve), radial IMF strength (thick black curve), and solar wind speed (dashed-dotted blue curve) during 1976-2023; also plotted is the monthly sunspot number (dotted curve).All curves represent 13-CR running means; vertical dashed lines mark the successive sunspot minima.The aa index shows well-defined minima that occur a few months to a year after each sunspot minimum and that coincide with dips inB tot IMF , | | B rIMF , and V sw .(b) Variation of the MWO axial dipole (thick blue curve) and total dipole (thin blue curve) strengths, and of the WSO axial dipole (thick black curve) and total dipole (thin black curve) strengths.Again, all curves represent 13-CR running means, and successive sunspot minima are indicated by the vertical dashed lines.Note that the 1977Note that the  , 1986Note that the  , 1997Note that the  , 2009Note that the  , and 2020   minima in aa occur when D tot ; |D ax | ?D eq .

Figure 5 .
Figure 5. (a) Scatterplot relating the minimum value of the classical aa index to the maximum annual sunspot number of the following cycle, for cycles 12-24.Here, we have calculated 13-CR running averages of aa centered on successive CRs and selected the lowest value within ∼1 yr of each sunspot minimum.A least-squares fit yields ( ) = +  R aa 51 9.5 nT 23 max min , with correlation coefficient cc = 0.85.Setting = aa 10.8 min nT, as observed in 2020 March, we find =  R 153 23 max for the amplitude of cycle 25.(b) Here the minimum value of the "homogeneous" aa index, aaH min , is used as the precursor.In this case, ( ) = +  R aaH 37 11.6 nT 22 max min , with cc = 0.88.Setting aaH 9.3 min  nT, based on the average difference (1.4 nT) between aa min and aaH min preceding cycles 23-24, we obtain  R 145 22 max  for the amplitude of cycle 25.

Figure 6 .
Figure 6.(a) A 13-CR average of the MWO axial dipole strength centered at sunspot minimum is plotted against the amplitude of the following cycle for cycles 21-24.A least-squares fit gives (| | ) = +  R D 8 49.4 G 37 max ax ssmin , with correlation cc = 0.96.(b) Here the WSO axial dipole strength at sunspot minimum is plotted against the amplitude of the next cycle.In this case, a linear fit yields (| | ) = +  R D 63 32.1 G 40 max ax ssmin , with cc = 0.91.With | | = D 2.2 ax ssmin

((
1.The Sun's total dipole (l = 1) component provides by far the dominant contribution to the radial IMF strength through the solar cycle, except during polar field reversal, when the nonaxisymmetric quadrupole (l = 2, |m| = 2) component becomes significant or even dominant for a period of a year or less.During the rising and maximum phases of the cycle, ICMEs contribute on the order of 20% to the near-Earth radial IMF strength.2. Near sunspot minimum, the IMF strength falls to relatively low values that are determined by the Sun's axial dipole (l = 1, m = 0) component.Because the HCS is highly flattened toward the equator, the solar wind speed at Earth is also very low.As a result, the geomagnetic aa index reaches a minimum value that depends on |D ax |.This minimum value often occurs several months after the actual solar minimum date because |D ax | begins to decrease as the first new-cycle active regions emerge with axial dipole moments of the opposite sign, and because the wind speed continues to decrease until the last remnants of low-latitude coronal holes have decayed away.3.In Babcock-Leighton or flux-transport dynamo models, the sunspots of a given cycle are produced by winding up the poloidal field built up during the previous cycle, whose strength may be represented by the value of |D ax | near sunspot minimum, of which aa min may be considered a proxy.Employing aa min as a solar cycle precursor, the predicted amplitude of cycle 25 is found to be = Figure 5(a)).However, if we replace the classical aa index with the "homogenized" version aaH of Lockwood et al. (2018a, 2018b), which corrects for secular changes in the geomagnetic field, we obtain a lower value of  the Sun's large-scale field is highly concentrated toward the poles at solar minimum, measurements of | | D ax ssmin are subject to considerable uncertainties.In contrast, the radial IMF strength at Earth, | | B rIMF , is well observed and may be taken as a proxy for the Sun's axial dipole strength near solar minimum.Depending on whether the lowest value or the sunspot-minimumcentered value is used as the precursor, we obtain = Figure 7(b)) as the predicted amplitude of cycle 25.6.Since the source surface field at solar minimum is dominated by the axial dipole component, we may reverse the process of extrapolating the photospheric field into the heliosphere and use the observed radial IMF strength to determine | | D ax ssmin .We deduce that the measured WSO axial dipole strength was 35% too high during 1986 and 24% too high during 1996.If these two values are replaced by the IMF-derived ones, the correlation between | | D ax ssmin and R max increases from 0.907 to 0.999, but the predicted value of R max for cycle 25 changes only from 133 to 134.By curve-fitting the monthly sunspot numbers through early 2023, Upton & Hathaway (2023) found that cycle 25 will peak at =  R 135 10 max in late 2024.The predicted amplitude is smaller than those that we derived using the minimum value of the classical aa index (

Figure 7 .
Figure 7. (a) The lowest (13-CR averaged) value of the radial IMF strength (generally occurring just after sunspot minimum and coinciding with the minimum in aa) is plotted against the amplitude of the following cycle, for cycles 21-24.A least-squares fit gives (| | ) = +  R B 14 90.9 nT 21 r max IMF min , with cc = 0.985.Setting | | = B 1.45 r IMF min nT, the (13-CR averaged) value recorded in early 2020, we obtain a predicted amplitude of =  R 146 21 max for cycle 25.(b) The (13-CR averaged) radial IMF strength centered on sunspot minimum is plotted against the amplitude of the following cycle.The linear fit to cycles 21-24 has the form (| | ) = +  R B 1 88.4 nT 10 r max IMF ssmin , with cc = 0.997.With | | = B 1.53 r IMF ssmin nT near the end of 2019, the predicted amplitude of cycle 25 is =  R 136 10 max

Table 1
Axial Dipole Strengths at Solar Minimum a Solar Minimum Date Radial IMF Strength IMF-Inferred |D ax | | D ax | (MWO) |D ax | (WSO) All quantities represent 13-CR averages centered on the solar minimum date, which itself is determined from a 13-CR average of the monthly mean sunspot numbers.b Since WSO photospheric field maps are available only from 1976 May (CR 1642) onward, the 13-CR averaged value of |D ax | for WSO given here and the corresponding | | B r IMF and IMF-inferred |D ax | are centered on 1976 November. a