Quantifying the Solar Cycle Modulation of Extreme Space Weather

By obtaining the analytic signal of daily sunspot numbers since 1818 we construct a new solar cycle phase clock that maps each of the last 18 solar cycles onto a single normalized 11 year epoch. This clock orders solar coronal activity and extremes of the aa index, which tracks geomagnetic storms at the Earth's surface over the last 14 solar cycles. We identify geomagnetically quiet intervals that are 40% of the normalized cycle, ±2π/5 in phase or ±2.2 years around solar minimum. Since 1868 only two severe (aa>300 nT) and one extreme (aa>500 nT) geomagnetic storms occurred in quiet intervals; 1–3% of severe (aa>300 nT) geomagnetic storms and 4–6% of C‐, M‐, and X‐class solar flares occurred in quiet intervals. This provides quantitative support to planning resilience against space weather impacts since only a few percent of all severe storms occur in quiet intervals and their start and end times are quantifiable.


Introduction
Extreme space weather events can disrupt power distribution, aviation, communication, and satellites. They are driven by large-scale plasma structures emitted from the solar corona, but the geoeffectiveness of an event depends on many factors, including how the event propagates from Sun to Earth and how it interacts with Earth's magnetosphere (Baker & Lanzerotti, 2016;Hathaway, 2015;Yermolaev et al., 2013). Events that lead to geomagnetically induced currents that affect power grids are statistically more likely close to a solar maximum and in the descending phase of the solar cycle, but importantly they also occur at all other times in the solar activity cycle (Hayakawa et al., 2020;Thomson et al., 2010). As the largest events can result in significant societal impact and financial loss (Hapgood, 2019;Oughton et al., 2017), quantifying the chance of occurrence of extreme space weather events is essential to planning the resilience of vulnerable systems to catastrophic failure.
When more frequent, moderate space weather storms are aggregated across different solar cycles, there is a well-established correlation between occurrence rate and solar cycle modulated activity (Tsubouchi & Omura, 2007;Tsurutani et al., 2006). However, due to their rarity, the likelihood of more extreme geomagnetic storms is challenging to quantify and thus most estimates (Thomson et al., 2011) average over

10.1029/2020GL087795
Key Points: • New Sun clock maps solar cycles onto a regular time base quantifying the occurrence times of quiet intervals of each cycle • 1-3% of severe (aa > 300 nT) geomagnetic storms and 4-6% of C-, M-, and X-class solar flares occurred in solar cycle quiet intervals • Solar cycle modulation of the chance of severe (aa > 300 nT) geomagnetic storms is much stronger than that of moderate (aa > 100 nT) storms multiple solar cycles. Estimates based on extrapolating a power law event distribution (Riley, 2012) suggest a 12% probability of a "Carrington-class" (Tsurutani et al., 2003) extreme event in the next solar cycle, but are highly uncertain, and an underlying solar cycle modulation contributes to this uncertainty (Riley & Love, 2016). Some estimates based on extreme value statistics (Thomson et al., 2011) suggest much lower probabilities (Elvidge & Angling, 2018;Silbergleit, 1996Silbergleit, , 1999Siscoe, 1976;Tsubouchi & Omura, 2007).
Crucially, each solar cycle is unique in amplitude and duration (see, e.g., Hathaway, 2015;Russell et al., 2019) and geomagnetic activity tracks the different levels of activity of each solar maximum and declining phase Lockwood et al., 2018). Quantifying how solar coronal activity, and the chance of an extreme space weather event, varies within and between solar cycles is central to space weather resilience planning. A uniform normalized time base for the solar cycle is needed in order to collate data across solar cycles of different duration in order to quantify correlation between the frequency of occurrence of severe geomagnetic storms and solar cycle activity phase. In this letter we propose a new solar cycle "clock" that stretches (or shrinks) the observed sunspot cycle of activity onto a single (normalized 11 year) time base. Once we have shown that this clock can be constructed, we find a sharp demarcation between active and quiet intervals of both the level of solar coronal activity and severe geomagnetic activity. We then obtain quantitative estimates of the solar cycle variation in the chance of occurrence of severe and moderate space weather events.

Constructing the Solar Cycle Clock
The daily sunspot number record provides an almost uninterrupted measure of solar coronal activity since 1818 and is plotted in Figure 1a. Both the amplitude and duration of each solar cycle varies from one cycle to the next. We express this time series S(t) in terms of a time-varying amplitude A(t) and phase (t) by obtaining its analytic signal (Boashash, 1992;Gabor, 1946) such that the real part of this signal is S(t) and the imaginary part is obtained such that . This is a standard approach that is used to test for synchronization (e.g., Chapman, Lang, et al. 2018) and for amplitude-frequency relationships (Palȗs & Notovná, 1999). Here it is used to provide a mapping between time and signal phase that converts the (variable) duration of each solar cycle into a corresponding uniform phase interval, from 0 to 2 . We use a standard method to obtain the discrete analytic signal (Marple, 1999) which satisfies both invertability and orthogonality. While defined for an arbitrary time series, the analytic signal will only give a physically meaningful decomposition of the original time series if the instantaneous frequency (t)=d (t)/dt remains positive (Boashash, 1992). We therefore need to remove fast fluctuations and, for a positive definite signal such as the daily sunspot number, a background trend (see, e.g., Boashash, 1992;Chapman, Lang et al. 2018). Before performing the Hilbert transform we performed a 180 day moving average and obtained a slowly varying trend by performing a robust local linear regression which down weights outliers ("rlowess") using a T B =40 year window. Figure 1 charts how we construct the analytic signal for the daily sunspot record. We first subtract the long-timescale trend (blue line in Figure 1a) to give a sunspot time series that is unambiguously zero crossing (Figure 1b). We then obtain the Hilbert transform H(t) for this (T s = 180 day moving average) smoothed and detrended signal which then gives the analytic signal. Figures 1c and 1d plot A(t) and (t), respectively. Each cycle maximum (red circles) and minimum (green circles) is also overplotted on the analytic phase. The analytic phase (t) plotted in Figure 1d will be used to read across between phase and occurrence time t. This read-across is robust across a reasonable range of T s and T B , for example, the zero crossing times (t)=0 vary by less than ±3 months when calculated over the range T s = [150-360] days and T B = [20-80] years. We used the most recent sunspot record, Sunspot Index and Long-term Solar Observations (SILSO) version 2.0 released in 2015 to determine (t) for this study. There are differences in the way that sunspot records from different eras are combined (see, e.g., Lockwood, Owens, Barnard, Scott, Lockwood et al., 2016;, and references therein). We have verified that our results are insensitive to this by repeating our analysis using the archived SILSO version 1.0; we find that the zero crossing times (t) = 0 differ at most by less than 2 months.
The times of cycle terminators (McIntosh et al., 2019) are also plotted in Figure 1d   which indicate the end of one cycle of activity and the beginning of the next. The termination of a solar cycle, or terminator, has a three-component global signature (McIntosh et al., 2019). It is initially observed as a very abrupt reduction in the density of extreme ultraviolet bright point density around the solar equator, marking the final cancelation of the old cycle's (magnetic) activity bands at the equator . The equatorial reduction occurs in close conjunction with a very rapid growth of bright point density in the (magnetic) activity bands at midlatitudes. This switch in magnetic flux emergence patterns occurs at the same time as the rapid increase in the number of midlatitude sunspots which belong to the new solar cycle. At higher solar latitudes, the terminator is signaled as the start of the polar magnetic reversal process (McIntosh et al., 2019) or the "rush to the poles" phenomenon (Babcock, 1961;Sheeley et al., 1989). We may therefore expect terminators to feature in how solar cycle activity is ordered.
We can use the mapping between time and phase shown in Figure 1d to construct a Sun clock which for each solar cycle has a different duration in time, but which maps onto a regular 0-2 interval in phase. We will see how this orders observations that are available over multiple solar cycles. The F10.7 index (solar radio flux at 10.7 cm Tapping, 2013) is available since 1947 giving six solar cycles of observations. As well as providing an index of the state of the solar corona, it is used by many operational space weather models as their prime solar input. It is correlated with the density of the upper atmosphere which in turn has consequences for the design and operation of satellites in low Earth orbit (e.g., Vedder & Tabor, 1992). The intensity and occurrence times of solar flares seen in X-ray have been continuously observed by GOES, and these are cataloged since 1975, that is, over the last four solar cycles. The intensities of space weather events are routinely characterized by geomagnetic indices derived from ground-based magnetometer observations. The aa index Black lines indicate the average analytic phase for the maxima, minima, and terminators. The pre-terminator (dashed black line) is at the same phase difference (clock angle) in advance of the minimum as that phase difference by which the terminator lags the minimum. These phase differences are close to ±2 /5 either side of the average minimum phase; these are indicated by blue lines. Blue dots overplot daily F10.7 and overplotted red, blue, and green histograms show counts in nonoverlapping (normalized) 3-month-long bins for X-class, M-class, and C-class flare occurrence (scaled relative to each other in ratio 75:500:2,000). is constructed (Mayaud, 1972;1980) from the 3-hourly K indices determined at two antipodal observatories (invariant magnetic latitude 50 • ) and is available over the last 14 solar cycles, from 1868 to the present. An important consideration is that the aa index (units, nT) is discretized in amplitude (Bubenik & Fraser-Smith, 1977;Chapman et al., 2020) since the underlying K index (Bartels et al., 1939) is a quasi-logarithmic 0-9 integer scale that characterizes the maximum positive and negative magnetic deviations that occur during each 3 hr period at a given observatory.

Sun Clocks for Solar and Geomagnetic Activity
We map the last 18 solar cycles onto a regular [0 2 ] interval in phase to produce a clock as shown in Figures  2 and 3. On the inner ring of both figures we plot the minima and maxima of the last 18 cycles and the terminators of the last 12 cycles. Lines indicate the average of each of these; this forms the basis of the solar cycle clock which we can read off as having a (normalized) 11 year period corresponding to 2 in phase. Increasing time (phase) is read clockwise as plotted. We now add to this clock multi-solar cycle observations of solar and geomagnetic activity. Solar flares (cataloged from GOES X-ray flux observations) occurrence is plotted as (scaled) counts in nonoverlapping (normalized) 3 month binned histograms in Figure 2, X-class, M-class, and C-class flare counts indicated by red, blue, and green histogram bars, respectively. In Figure  3, rings of red, blue, and green dots indicate (nonoverlapping) days in which X-class, M-class, and C-class flares, respectively, occurred. As we would expect, the occurrence of flares is modulated by the solar cycle. Daily F10.7 observations are overplotted for the full six-solar cycle record in Figure 2 (blue dots) and again are clearly ordered by the Sun clock.
To see to what extent the solar cycle clock orders geomagnetic activity, in Figure 3 we consider all 14 solar cycles of the 3 hr aa index (Mayaud, 1972). We aim to characterize extreme space weather events, but since aa is a coarse-grained measure (Bubenik & Fraser-Smith, 1977), its maximum excursions are not well resolved . Rather than plot poorly resolved aa maximum values, we flag (nonoverlapping) calendar days in which any of the 3 hr aa index records in a given day exceeds a given threshold. The outer rings of Figure 2 (black dots) plot these flagged days with successively increasing radius for increasing threshold, aa > 100, 200, 300, 400, 500, and 600 nT. Radial "spokes" on this plot then indicate severe space weather events where multiple thresholds are simultaneously exceeded.
As expected, severe events are clustered more toward solar maximum. However the clock provides more quantitative detail on how the solar cycle orders solar coronal activity and severe space weather. We see that the average terminator time (phase) identifies a clear "switch on," that is, an increase in solar flare and severe space weather occurrence as we move from minimum to maximum in each cycle. Terminators have previously been identified solely from observations of solar coronal activity as the start time of each solar cycle; here we see the corresponding response in geomagnetic activity. Furthermore, as we move from maximum to minimum, there is a decrease or "switch off" in solar flare rates and severe space weather activity for 10.1029/2020GL087795 which we will introduce the terminology "pre-terminator." The "switch on" at the average terminator location occurs at a phase difference following average minimum of =1.2769 radians (2.23 normalized years). We locate the pre-terminator at approximately the same phase difference preceding the average solar minimum on the solar cycle clock. We then see that between the pre-terminator and the terminator, there is a significantly lowered occurrence rate for severe storms; only one aa > 500 nT and a further two aa > 300 nT events occurred in the entire 151 year aa record. This identifies a specific "quiet interval" of the solar cycle which begins approximately ≃2 /5 (or 2.2 normalized years) before and ends approximately ≃2 /5 (or 2.2 normalized years) after the 18 cycle average phase of solar minimum as indicated by the blue lines on the clock; these can be seen to closely coincide with the terminator and pre-terminator. The 1903 storm that is associated with a quiet Sun outburst (Hayakawa et al., 2020) can be seen in Figure 3 just after the average terminator location outside this quiet interval. The terminator time, estimated from solar observations, then is potentially a tool to support operational decision making as it flags an imminent increase in the likelihood of more severe space weather activity. Since we have obtained the sunspot record analytic phase as a function of time, we can translate the terminator (activity "switch on") and pre-terminator (activity "switch off") phases into occurrence times. In Figure 1d, zero phase is that of the terminator average and we indicate in gray the range of phases [−4 /5 0]. These gray shaded regions then indicate the times of the quiet interval of each cycle. The sunspot number analytic phase can in principle be extrapolated forward in time, albeit with some uncertainty, to forecast when a specific phase will occur. This has been done for the terminator at the start of the next solar cycle (Leamon et al., 2020), and a similar procedure could also forecast when the "switch off" of activity will occur.

Quantifying Solar Cycle Modulation of the Level of Activity
To quantify solar coronal activity and occurrence rates of space weather of different severity, in Figure 4 we plot the information shown on the Sun clocks as histograms. The abscissa plots the range [0 2 ] of phase on the Sun clock as a normalized 11 year cycle, with Year zero at the average terminator occurrence time (phase). We form histograms of occurrences as counts in 3 month nonoverlapping bins within this 11 year cycle from the entire observational record for each quantity. Panels (a) and (b) plot the counts per 3 month bin of the number of days, during the full aa record since 1868, in which aa exceeds the above thresholds. Panels (c) and (d) count the number of C, M, and X flares per 3 month bin that were observed over the GOES catalog since 1975. Panel (e) overplots all individual records of F10.7; the index is available since 1947 giving six overplotted solar cycles.
In Figure 4 a gray shaded region indicates the quiet interval of the cycle, centered on the average location of solar minimum and demarcated by the pre-terminator (at −4.4 years) and terminator (at Year zero) as obtained from the Sun clock (Figures 2 and 3). The quiet interval clearly coincides with reduced occurrence rates for flares and severe space weather and low values of F10.7 solar radio flux. Indeed, only 12 of the 453 X-flares from the GOES flare catalog occurred when F10.7 was <90 sfu. Including weaker flares, only ∼4-6% of all X, M, or C-flares occurred in the quiet interval; the relative chance of a flare occurring in the quiet interval is roughly the same for all flare classes. Over the 14 solar cycles of the aa index record there were 19 occurrences of the most intense, aa > 600 nT events, and none of these occurred in the quiet interval. There were three events with aa > 300 nT, one of which reached aa > 500 nT in the quiet interval; ∼1-3% of all aa ∼ 300-500 nT days occurred in the quiet interval. This significantly modulates solar cycle averaged estimates of the occurrence rates of severe geomagnetic storms. If the occurrence rates were uniform across the solar cycle, a quiet interval of 4.4 years within an 11 year cycle would translate to 40% of all events occurring in the quiet interval. An overall number of C events in N years implies an average return period N∕C, so that the 130 aa > 300 nT days seen in the entire aa record implies a return time of ∼ 1.2 years. Given that approximately 4.4 out of 11 years are quiet, finding 3 aa > 300 nT events in quiet intervals translates to a quiet interval return period of ∼20 years for these events. Similarly, the active interval return period is ∼ 0.7 years. Counts and return periods are listed in supporting information Table S1.
From 14 cycles of aa index data we find that more moderate storm days are less strongly modulated by the solar cycle, with ∼21% of aa > 100 nT days occurring in the quiet interval. This is consistent with previous estimates based on the last five solar cycles. More moderate storms are more frequent, and hence an estimate of the solar cycle modulation of their occurrence rates can be attempted using observations over fewer solar cycles for which there are geomagnetic indices that are well resolved in amplitude so that individual storm peak disturbance values can be identified. Based on the five solar cycles of available D ST observations, the occurrence rate of more moderate storms has been estimated to vary by a factor of 2-3 between solar maximum and minimum (Tsubouchi & Omura, 2007). Extrapolating distributions sorted by solar maximum and minimum to the most extreme events (Riley & Love, 2016) gives an occurrence likelihood that is more strongly solar cycle modulated, with 1.4% during solar minimum conditions and 28% in solar maximum conditions. This is again consistent with our findings; however, the analysis presented here does not require the assumption of any specific distribution or its extrapolation.

Conclusions
In summary, we constructed a new solar cycle clock by using a Hilbert transform of the daily sunspot number record to map the variable duration solar cycle onto a uniform [0 2 ] interval of analytic phase. We found this clearly identifies a ∼4.4 year (normalized) quiet interval centered on solar minimum in a (normalized) 11 year cycle. The start and end of this quiet interval occur at specific analytic phases of the daily sunspot number which in principle are forecastable in real time by forward extrapolation of the relationship between time and analytic phase of the daily sunspot number record. Since F10.7 solar radio flux is also ordered by analytic phase, it could provide an additional signal with which to make this forecast.
Knowing when the next quiet interval will start and end has considerable implications for planning resilience of systems to the impacts of severe space weather events. Approximately 1-3% of all aa > 300 nT days in the aa record occurred in solar cycle quiet intervals giving a return period of ∼ 20 years in quiet intervals and ∼ 0.7 years in active intervals. Averaged over the solar cycle, the return period is ∼ 1.2 years. The active interval return period differs from the solar cycle average by about a factor of 2. It is, however, significantly reduced during quiet intervals.
Across the aa record, we find that the occurrence rate of severe events is significantly more strongly solar cycle modulated than that of more moderate ones. Estimates of the likely occurrence rate based on more frequently occurring, moderate events may therefore underestimate the solar cycle modulation of more severe events. This pattern is not seen as strongly in the solar cycle modulation of solar flares where we find that the proportion that occurs in the quiet interval is roughly the same for C-, M-, and X-class flares. This may reflect the fact that more severe geomagnetic storms tend to be more directly correlated with flare activity, whereas more moderate storms can result from other drivers in the solar wind such as high-speed streams.

Data Availability Statement
The aa index data set is available from the International Service of Geomagnetic Indices (at http://isgi. unistra.fr/). The daily sunspot number data set is available from the SILSO, World Data Center-Sunspot Number and Long-term Solar Observations, Royal Observatory of Belgium, online Sunspot Number catalog (http://www.sidc.be/silso/datafiles). The dates of solar cycle maxima and minima are as determined from the smoothed sunspot number record by SILSO (http://www.sidc.be/silso/cyclesmm). The solar radio flux at 10.7 cm (the F10.7 index) is available since 1947 (at https://www.spaceweather.gc.ca/solarflux/sx-en.php). The GOES X-ray Flare data set is available through the NOAA National Geophysical Data Center (NGDC) (https://www.ngdc.noaa.gov/stp/space-weather/solar-data/solar-features/solar-flares/x-rays/).