Solar Cycle Precursors and the Outlook for Cycle 25

Activity on the Sun varies with a periodicity of about eleven years. This variability is characterized by fluctuations in the appearance of sunspots but also includes the evolution of coronal holes, changes in the solar wind speed, and changes in the frequency of eruptive events such as solar flares and coronal mass ejections [Hathaway (\APACyear2015)]. Together, these and other solar phenomena drive changes in the interplanetary environment, i.e. space weather. As space weather events interact with the Earth, they cause geomagnetic storms that impact the geospace environment in a variety of ways. Extreme ultraviolet and x-ray emissions give rise to ionization in the ionosphere and heating and increased density in the thermosphere. This increases the drag on satellites and debris in low Earth orbits and increases the risk of satellites colliding with debris or even completely deorbiting - as occurred in early 2022 with the SpaceX Starlink satellites [Hapgood \BOthers. (\APACyear2022)]. Increases in energetic charged particles can further disrupt, damage, or cause failure of satellites or their electrical components. While satellites essential for communications and national defense are most at risk, geomagnetic storms are not a threat to just them. Radiation from these storms also pose a threat to astronauts in space and crew on airline flights over the poles. Closer to home, ground-induced currents produced by geomagnetic storms can overwhelm power grids, resulting in power outages. Predicting the size of a solar cycle is an important step in improving our ability to prepare for space weather years in advance.

Two (related) physics based precursors for predicting the amplitude of a solar cycle have risen to the top as being the most reliable: geomagnetic activity levels and the Sun’s magnetic configuration (polar fields and axial dipole moment) near the time of sunspot cycle minimum. Cycle predictions can be made well before cycle minimum by using models to simulate the evolution of the Sun’s surface magnetic field. Once the cycle is well under way, curve fitting can be used to determine the amplitude of the cycle. In this paper, we present a comprehensive picture of the outlook for Solar Cycle 25 based on each of these methods, individually and as a combined prediction.

Cycle 24/25 minimum occurred in December of 2019 and we are now over three years into Cycle 25. At this stage of the solar cycle, curve fitting becomes quite reliable. \citeA1994Hathaway_etal proposed a parametric curve for fitting to the monthly sunspot numbers, $R$, in each cycle with:

In 2015, the sunspot number was revised to version 2.0 [Clette \BOthers. (\APACyear2016), \APACcitebtitleSmoothed Sunspot Number (\APACyear2022), hereafter, V2.0], which increases the sunspot numbers by changing the multiplicative factor from 0.6 to 1.0 and by using the average of multiple observers. We find that the values for $B$ and $C$ have changed for V2.0, with the best fit for B and C now given by:

We investigate the accuracy of the fitting as the cycles progress by performing the curve fitting at six month intervals from each cycle’s minimum. We then compare the amplitude of the fit at each interval to the observed final amplitude fit. We find that using the the two parameters, the interval fit typically converges to within 10% of the final fit values after about 3 years into each cycle (near the inflection point on the rising curve), as illustrated in the bottom panel of Figure 1. Note that there is a tendency to overestimate the amplitude early in each cycle.

The curve fitting results for Cycle 25 were variable and quite uncertain until a year ago (May 2022 – 30 months after minimum). Since then the results have been virtually unchanged - the curve fitting has given $R_{max}(25)=135\pm 10$ with $t_{0}=2019.8\pm 0.3$. This determination of the amplitude of Cycle 25 is unlikely to change by more than 10% as the cycle continues to develop.

Geomagnetic precursors to the solar cycle were first suggested by the Russian researcher Ohl in 1966 [Ohl \BBA Ohl (\APACyear1979)]. He found that the minimum in geomagnetic activity, as indicated by the aa index [\APACcitebtitleGeomagnetic aa Index (\APACyear2022), plotted in the top panel of Figure 2], is strongly correlated with the amplitude of the oncoming cycle . This can be seen by comparing the heights of the minima, shown in blue, to the following peaks in sunspot number, shown in red. (Note that we include a correctional offset of +3 nT prior to 1957, when the English observing station was moved from Abinger to Hartland [Svalgaard \BBA Cliver (\APACyear2005)].) A slight drawback to this method comes from the fact that these geomagnetic minima usually occur just after the sunspot cycle minimum. Other geomagnetic precursor methods have been devised to provide earlier predictions but are either based on data that don’t cover as many sunspot cycles or require data processing with free parameters [Feynman (\APACyear1982), Thompson (\APACyear1993)].

The Sun’s polar magnetic field configuration near the time of cycle minimum is gaining popularity as a precursor for the amplitude of the following cycle. At minimum, the Sun’s magnetic field is well characterized by a simple axial dipole. This simple configuration is often the starting point for models of the Sun’s magnetic dynamo [Babcock (\APACyear1961)]. The geomagnetic precursors near minimum are thought to perform well because they are driven by high-speed solar wind streams and are thus reflections of the strength of the Sun’s polar magnetic fields [Schatten \BBA Sofia (\APACyear1987), Wang \BBA Sheeley (\APACyear2009)]. Measurements of the Sun’s polar fields over the last four solar cycles have proven to be successful predictors of the following cycle amplitude [Schatten \BOthers. (\APACyear1978), Svalgaard \BOthers. (\APACyear2005), Petrovay (\APACyear2010), Muñoz-Jaramillo \BOthers. (\APACyear2013)].

Direct, systematic (daily) measurements of the Sun’s polar fields have been made at the Wilcox Solar Observatory (WSO) since 1976 [\APACcitebtitleSolar Polar Field Strength (\APACyear2022), Hoeksema (\APACyear1995), Svalgaard \BOthers. (\APACyear1978)]. The polar magnetic field configuration can be characterized in two different ways: by calculating the axial dipole component of the Sun’s magnetic field (Figure 2, middle panel) or by averaging the flux density over each polar region (i.e., the polar field strength, shown in Figure 2, bottom panel). The latitudes and field component (e.g., radial or line-of-sight) used to calculate the flux density at each pole are somewhat arbitrary. For WSO, the polar field measurement was set by the spatial resolution of the instrument and defined using the highest latitude pixel, which measures the line-of-sight fields nominally between 55^∘ and the poles. Magnetic data from other instruments have often employed the radial component and used different latitude ranges. While the flux density over each polar region offers insight into hemispheric asymmetries, the innate ambiguity associated with this measurement may make the axial component of the Sun’s magnetic dipole a better metric for solar cycle prediction [Upton \BBA Hathaway (\APACyear2014)].

We now have observations of the Sun’s polar fields as well as measurements of geomagnetic activity during (and for three years after) the Cycle 24/25 minimum. For the geomagnetic precursor, we focus on the minimum in geomagnetic activity as measured by the aa index (Ohl’s method) because it does not require any decomposition and the measurements date back to 1868, allowing the relationship to be determined for 13 cycles. For the polar field precursor, we look at both the polar field strength and the axial dipole strength during solar cycle minimum, as measured by WSO. These measurements are all shown in Figure 2, along with the smoothed sunspot number V2.0. While the polar field measurements do indeed appear to be indicative of the strength of the next cycle, they only provide three solar cycles (Cycles 22-24) for determining the relationship between the polar fields at minimum and the amplitude of the next cycle.

We begin by relating each precursor measurement to the strength of following cycle as indicated by $R_{max}$, i.e., the maximum smoothed sunspot number V2.0 for that cycle. This is shown in Figure 3. We calculate the ratio of $R_{max}$ to the minimum of the geomagnetic aa index for each cycle and find:

Both the magnetic precursors and the curve fitting indicate that Cycle 25 sunspot number maximum will be slightly bigger than the 116 of Cycle 24 but significantly smaller than the average of 179 for all 24 previous cycles, which ranged from 81 for Cycle 6 to 285 for Cycle 19.

[Hathaway \BOthers. (\APACyear1999)] proposed a combined prediction based on a weighted mean of the precursor predictions and the curve fitting prediction with more weight given to the curve fitting as the cycle progresses (50/50 at 36 months past minimum). This combined prediction gives a maximum smoothed sunspot number of $R_{max}(25)=134\pm 8$ with maximum occurring late in the fall of 2024. Figure 4 shows a plot of the predicted curve, along with the observed monthly sunspot numbers V2.0.

Modelers have sought to extend the predictive range of the polar field precursors by using physics based dynamo models [Charbonneau (\APACyear2020), Nandy (\APACyear2021)] or Surface Flux Transport (SFT) models [Sheeley (\APACyear2005), Jiang \BOthers. (\APACyear2014)] to obtain the polar fields years ahead of minimum. \citeA2022Pawan_etal showed that the rise rate of the polar fields during the first three years after the reversal gave similar results as the rise rate of the cycle, potentially providing an avenue for modelers to further extend their predictive range. Before cycle minimum in 2019, the Solar Cycle Prediction Panel, which represents NOAA, NASA and the International Space Environmental Services (ISES), released its official sunspot number forecast [\APACcitebtitleSolar Cycle 25 Forecast Update (\APACyear2019)], predicting a sunspot number maximum of $115\pm 10$. Their consensus forecast was based largely on the model projections for the polar fields at minimum.

In 2016 and 2018, we used our Advective Flux Transport (AFT) model to create forecasts of the Sun’s polar fields in order to predict the strength of Solar Cycle 25 [Hathaway \BBA Upton (\APACyear2016), Upton \BBA Hathaway (\APACyear2018)]. Our results indicated that the polar fields at Cycle 24/25 minimum would be similar to those at the previous minimum, suggesting Solar Cycle 25 would be a weak cycle, very similar in amplitude to Solar Cycle 24. Another SFT model [Cameron \BOthers. (\APACyear2016), Jiang \BOthers. (\APACyear2018)] reported similar results, predicting that Solar Cycle 25 would be slightly larger than Cycle 24 (with a maximum of 116.4), but less than the more average sized Cycle 23 (with a maximum of 180.3, very close to the Cycle 1-24 average of 179).

The AFT model assimilates the observed magnetic field from the SOHO/MDI [Scherrer \BOthers. (\APACyear1995)] and SDO/HMI [Scherrer \BOthers. (\APACyear2012)] magnetograms in order to provide the closest contact with the observations. This mode, known as the AFT Baseline, provides the most accurate representation of the magnetic field over the entire surface of the Sun based solely on near-side observation. In it’s predictive mode, AFT starts with a map from the AFT Baseline and advances it forward to make predictions about how the Sun’s magnetic field will evolve. In 2016 and 2018, we used AFT to create predictions of the Sun’s polar field evolution in order to predict the strength of Solar Cycle 25. Now that we are well past minimum, and Cycle 25 is well under way, we revisit those predictions to assess their performance.

In [Hathaway \BBA Upton (\APACyear2016)], we predicted an axial dipole for the solar minimum of $1.36\pm 0.20$ G and in [Upton \BBA Hathaway (\APACyear2018)] we predicted an axial dipole of $1.56\pm 0.05$ G. (A comparison of those results and detailed discussion can be found in [Upton \BBA Hathaway (\APACyear2018)].) We show our 2018 polar field predictions in Figure 5 (blue lines), along with the corresponding measurements from WSO (in black), and the AFT Baseline constructed with the MDI, and HMI observations (in red). AFT was able to successfully predict the evolution of the AFT/HMI axial dipole. However, the axial dipole in the WSO observations was about 20% higher than HMI and AFT at the time of cycle minimum.

The AFT polar field strengths (bottom panel of Figure 5) are calculated from the radial magnetic field above 55^∘ (red line) and 60^∘ (pink line). The LOS WSO measurements (nominally 55^∘ and above) are shown in black and the HMI derived Mean Radial Fields 60^∘ [\APACcitebtitleHMI Polar Field (\APACyear2022), Sun \BOthers. (\APACyear2015)] are shown in light gray (smoothed values in dark gray). AFT was able to successfully predict the evolution of the AFT/HMI polar fields in the North, but that the polar fields in the South did diverge somewhat starting in late 2019. The difference between the north and south polar field strengths in the 2018 predictions ranged from 5.49 to 5.92, with an average of 5.68. The observed difference was 5.25, or about 10% smaller than the AFT 2018 ensemble of predictions. Note that the radial AFT/HMI polar fields above 55^∘ are smaller than the WSO single pixel LOS polar fields corresponding to the same latitude range. The AFT Baseline polar fields above 60^∘ at the start of 2020 agree with the WSO fields above 55^∘ and with the HMI derived fields above 60^∘.

Geomagnetic and solar magnetic precursors have been shown to be most reliable predictors of an impending solar cycle. However, there are still some uncertainties associated with these predictors. Currently, the geomagnetic precursors are more robust due to the length of the dataset, but the physical mechanism behind their success is less direct. The polar precursors have a stronger foundation in physics, but with a much shorter time line the functional relationship is poorly defined. This is confounded by the fact that the polar measurements, in and of themselves, are not well constrained. This is primarily due to innate observational limitations. This is highlighted in Figure 5 by a) the mismatch between the WSO and the MDI/HMI/AFT axial dipole measurements (top panel), b) the annual oscillation in the WSO LOS polar field measurements (bottom panel), and c) the spread in the HMI radial polar field measurements (bottom panel).

The WSO axial dipole and the MDI/HMI/AFT axial dipole appear to be offset in both time and amplitude. The offset in time is most apparent during the dipole reversals, as the WSO reversals precede the MDI/HMI/AFT reversal by about a year or two. The offset in amplitude is most apparent during last solar minimum, when the axial dipole is relatively flat. These offsets are a consequence of the limited resolution of the WSO observations convolved with the changing latitude range of the WSO pixel (due to an orbit around the inclined Sun). As previously mentioned, the highest latitude WSO pixel measures the line-of-sight fields nominally between $55^{\circ}$ and the poles. However, the Sun’s rotation axis is inclined $\sim 7.15^{\circ}$ with respect to the ecliptic plane. So, while on average the highest latitude pixel measures from $55^{\circ}$ and above, the latitudes actually being measured actually vary between $48^{\circ}$ and above to $62^{\circ}$ and above. This produces a seasonal oscillation in the measurements of the Sun’s polar fields, which is clearly visible as an annual signal in the plot of the hemispheric polar field strengths (bottom panel of Figure 5). Furthermore, the northern/southern (solid/dashed) hemisphere measurements are 6-months out of phase with one another, such that the northern/southern WSO measurements come into better agreement with the MDI/HMI/AFT measurements in the Spring/Fall, when the the Southern/Northern pole is inclined toward the Earth. It should be noted that these annual signals are present in the MDI and HMI data as well, though to a lesser extent. This is because the higher resolution of MDI and and HMI provide more detailed latitudinal coverage, but the inclined orbit causes the line-of-sight angle of the magnetic field (with respect to the radial component) to also change by $\pm 7^{\circ}$ over the course of the yearly orbit.

The deviations in the polar field measurements are most notable when the polar fields are rapidly evolving, during the polar field reversals. At this time, new polarity flux is being transported to the poles to cancel with the old polarity polar flux, causing a large latitudinal gradient in the high latitude flux. As the Earth orbits the Sun and the latitudes measured by the most poleward WSO pixel change, the pixel samples lower/higher latitudes and more of the new/old polarity flux is present in that pixel. Consequently, the average polar field strength measured by that pixel changes substantially over the course of the orbit resulting in the observed annual oscillation.

These deviations in the hemispheric polar field strengths feed into the measurements of the axial dipole moment (top panel of Figure 5). However, rather than appearing as an annual oscillation, they instead present as the offsets in amplitude and time noted above. One might expect that the deviations would cancel out, since the northern and southern hemispheres are out of phase with one another and one of the poles is always favorable. However, the unfavorable pole always appears “ahead” in the polar reversal process (because the lower latitudes that are being include have more new polarity flux). This means that at nearly all times, one of the two poles will seem to be “ahead” and thus the evolution of the axial dipole as measured by WSO will precede the true polar reversal in time, producing the apparent temporal offset seen in the top panel of Figure 5. This is evidenced by the fact that the WSO polar field measurements preferentially precede the MDI/HMI polar field measurements.

If not properly accounted for, the temporal deviations can also produce an offset in amplitude. The WSO axial dipole moment during minima is the “yardstick” used to determine the strength of the next cycle for prediction purposes. Since, the axial dipole moment is most crucial during solar minima, one might naturally scale the the WSO and MDI/HMI measurements such that they are in agreement during the Solar Cycle 22/23 and Solar Cycle 23/24 minima (1999 and 2009), as done in Figure 5. However, doing so results in the WSO dipole being about 20 percent higher than the HMI dipole for the Solar Cycle 24/25 minimum in December 2019. This faulty scaling resulted in the AFT solar cycle prediction underestimating the strength of Solar Cycle 25, not because the flux transport did not accurately predict the evolution of polar fields, but rather because the “yardstick” used to make the prediction was not calibrated properly.

While outside the scope of this paper, a detailed accounting of these deviations and a proper calibration of the WSO observations with respect to the modern space observations is needed in order to reduce the uncertainty in the polar field-SSN relationship. Uncertainty in the polar measurements and the polar field-SSN relationship would also greatly benefit from polar mission to measure the magnetic fields from directly overheard thus removing uncertainty associated with LOS observations of the magnetic field.

Uncertainty in the polar measurements not withstanding, the predictions presented in this paper firmly point to Solar Cycle 25 being slight larger than Solar Cycle 24, but no where close to the amplitude of Solar Cycle 23 or the average solar cycle strength ($\sim 180$). This will be the first increase to the cycle amplitude that we’ve seen since solar Cycle 21 (e.g., about 50 years). This may mean that we have reached the inflection point in the current Gleissberg cycle and might start to see bigger cycles again.

The results presented in this paper rely on geomagnetic indices [\APACcitebtitleGeomagnetic aa Index (\APACyear2022)] calculated and made available by ISGI Collaborating Institutes from data collected at magnetic observatories. We thank the involved national institutes, the INTERMAGNET network and ISGI (isgi.unistra.fr). Wilcox Solar Observatory data [\APACcitebtitleSolar Polar Field Strength (\APACyear2022)] used in this study was obtained via the web site http://wso.stanford.edu/Polar.html courtesy of J.T. Hoeksema. The Wilcox Solar Observatory is currently supported by NASA. HMI Polar Field data [\APACcitebtitleHMI Polar Field (\APACyear2022)] used in this study was obtained via the web site http://jsoc.stanford.edu/ajax/lookdata.html?ds=hmi.meanpf_720s. HMI is currently supported by NASA. SILSO sunspot number data [\APACcitebtitleSmoothed Sunspot Number (\APACyear2022)] was obtained via the web site https://www.sidc.be/silso/ courtesy of the Royal Observatory of Belgium, Brussels.