An analysis of heliospheric magnetic field flux based on sunspot number from 1749 to today and prediction for the coming solar minimum

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1 JOURNAL OF GEOPHYSICAL RESEARCH: SPACE PHYSICS, VOL. 118, , doi: /2013ja019404, 2013 An analysis of heliospheric magnetic field flux based on sunspot number from 1749 to today and prediction for the coming solar minimum Molly L. Goelzer, 1,2 Charles W. Smith, 1 Nathan A. Schwadron, 1 and K. G. McCracken 3 Received 4 September 2013; revised 25 October 2013; accepted 24 November 2013; published 20 December [1] It is now well established that many bulk properties of the solar wind rise and fall with the solar cycle, and the heliospheric magnetic field (HMF) intensity is no exception. The HMF intensity is seen to be maximum around the time of solar maximum, lowest during solar minimum, and lower still during the recent protracted solar minimum One explanation of this behavior can be found in the theory of Schwadron et al. (2010) that argues magnetic flux is injected into interplanetary space by coronal mass ejection eruptions and removed by reconnection in the low solar atmosphere. This produces an HMF intensity that is correlated with sunspot number, and the rapid injection of flux followed by the slow removal by reconnection results in a hysteresis effect that is readily evident in the observations. Here for the first time we apply this theory to the sunspot record going back to 1749 and compare favorably our predictions to the results derived from 10 Be observations. We also make a prediction for the coming solar minimum based on results from the Dalton Minimum. Citation: Goelzer, M. L., C. W. Smith, and N. A. Schwadron (2013), An analysis of heliospheric magnetic field flux based on sunspot number from 1749 to today and prediction for the coming solar minimum, J. Geophys. Res. Space Physics, 118, , doi: /2013ja Introduction [2] Numerous observational studies have reported that the heliospheric magnetic field (HMF) intensity rises and falls with the solar cycle. These analyses include in situ observations [Smith and Balogh, 2008; Lockwood et al., 2009; Connick et al., 2009, 2011; Smith et al., 2013], groundbased measurements [Lockwood, 2003], and examinations of cosmogenic data such as 10 Be, 14 C, and 36 Cl samples obtained from tree rings and ice cores [McCracken, 2007; McCracken et al., 2013] to name just a few. Smith and Balogh [2008] and Connick et al. [2009, 2011] have shown that the HMF intensity during the recent protracted solar minimum ( ) falls to an all-time low for space age in situ measurements. McComas et al. [2008, 2013] have shown that this decline in HMF intensity coincides with a global reduction in the average wind speed, density, flux, and temperature of the solar wind. McComas et al. [2013] showed that the mini solar maximum of cycle 24 resulted 1 Physics Department, Space Science Center, University of New Hampshire, Durham, New Hampshire, USA. 2 Also in Department of Chemical Engineering, University of New Hampshire, Durham, New Hampshire, USA. 3 Institute of Physical Science and Technology, University of Maryland, College Park, Maryland, USA. Corresponding author: C. W. Smith, Physics Department and Space Science Center, University of New Hampshire, Room 207 Morse Hall, 39 College Rd., Durham, NH 03824, USA. (Charles.Smith@unh.edu) American Geophysical Union. All Rights Reserved /13/ /2013JA in only a small recovery in particle and magnetic fluxes. Therefore, the cycle 24 mini solar maximum is more like the protracted solar minimum than the previous maxima observed throughout the space age. We will show here that this failure to restore flux levels to traditional solar maximum values will result in even lower flux levels during the coming solar minimum. Explanations for the observed solar cycle behavior include flux injection at small scales via emerging loops as described by potential field theory [Wang, 1994; Sheeley et al., 2007] and flux injection at large scales via coronal mass ejections (CMEs) [Owens and Crooker, 2006, 2007; Schwadron et al., 2010]. [3] This work builds on the theory of Schwadron et al. [2010] which attempts to account for the HMF intensity through a balance of flux injection via CME eruption and flux removal via magnetic reconnection [McComas, 1995; Fisk and Schwadron, 2001; Crooker et al., 2002; Owens and Crooker, 2006; Owens and Lockwood, 2012]. This theory has been applied by Smith et al. [2013] in a comparison between predicted HMF intensity and the observed HMF as recorded on the Omni2 data set. That work was limited to the years 1963 onward when in situ HMF measurements are available. The analysis uses monthly sunspot numbers as a proxy for CME ejection rates and validates that substitution by comparing values when statistics for both are available. Using the same analysis technique and parameters, we go back to the sunspot record starting in 1749 and compute the predicted HMF intensity over 24 sunspot cycles. We compare the predicted HMF intensity with results derived from 10 Be [McCracken, 2007] with favorable conclusions.

2 2. Theory [4] We apply the theory of Schwadron et al. [2010] to observations of sunspot number from 1749 to the present. The theory contains two components of the HMF: the CMEassociated magnetic flux, CME, associated with ejecta and the open magnetic flux, o, associated with the steady solar wind. Schwadron et al. write the time derivative of CMEassociated flux as dˆej dt = f (1 D) CME ˆej 1 ic + 1 d + 1 o where ˆej is the ejecta-associated flux, f is the rate of CME injection as a function of time, D is the fraction of CME flux reconnected immediately after release, CME is the flux content of a typical CME, ic represents the interchange timescale between ejecta-associated flux and open flux, d represents the timescale over which ejecta-associated and open magnetic flux are destroyed through disconnection, and o represents the timescale over which ejecta-associated flux is converted into open flux. Owens [2008] sets CME = Wb. See Schwadron et al. [2010, Figure 1] for further description of these three timescales. [5] CME field lines are converted into steady open magnetic field lines via interchange reconnection on a timescale o. Reconnection at the foot points of open field lines can shed open field line flux. The evolution of open field line flux in the Schwadron et al. theory is given by dˆo dt = ˆo ˆflr d (1) + ˆej o. (2) The above allows for ˆflr, a minimum floor of open flux [Wang et al., 2000; Fisk and Schwadron, 2001; Svalgaard and Cliver, 2007; Owens et al., 2008; Zhao et al., 2009; Lockwood et al., 2009; Crooker and Owens, 2010], which we set to zero. We do so, in part, because the observed HMF intensities of the recent protracted solar minimum fell below previous expectations for ˆflr,andSmith et al. [2013] were able to reproduce the observed behavior without invoking a finite value for ˆflr. We will return to this idea in section 5. [6] Summing equations (1) and (2), we obtain the time derivative for the total flux of the HMF dˆtot dt = ˆo d + f (1 D) CME ˆej d. (3) It is important to understand that the Schwadron et al. [2010] theory assumes that all field lines follow the Parker [1958, 1963] spiral direction. From ˆtot one can estimate an average magnetic field which we will call B P that is consistent with the flux. It is given by Z ˆtot = B P OndS (4) where the surface integral is hemispherical to allow for the changing HMF direction across the heliospheric current sheet. Although it ignores latitudinal dependence, we will evaluate equation (4) as ˆtot = 4R 2 B P for the sake of obtaining an estimate for B P where R =1AU is heliocentric distance. We will see below that B P is not the same as the average HMF intensity which contains a significant component perpendicular to the spiral direction Figure 1. (black) Plot of monthly average sunspot number from 1749 to the present. (red) Corresponding predicted Parker component of the HMF intensity at 1 AU using theory and parameters in paper. (green circles) Yearly average value of HMF intensity B derived from Be 10 data. (blue curve) Measured yearly averages of HMF intensity h B i as determined by Smith et al. [2013] from the Omni2 data. 3. Data Analysis [7] We have obtained the monthly average sunspot number data from NOAA Geophysical Data Center and used it as a normalized input for our theory. As with Smith et al. [2013] who compare sunspot number to observed CME ejection rates, we take f = (8/200) sunspot number (SSN), where SSN is the monthly sunspot number. We also take D =1/2, ic =20d, d =6.0year, and o =2.5years as did Smith et al. [8] Figure 1 shows the monthly sunspot number (black line) from 1749 onward. Using these values as input to obtain f, we compute the predicted value for B P (red line) using the equations above where the absolute value,..., is meant to remove the ambiguity associated with solar latitude and the solar dipole. Smith et al. [2013] has compared these same results to observations contained within the Omni2 data set for solar wind observations at 1 AU and found very good agreement from 1975 onward. This analysis extends those predictions back to the earliest reliable monthly sunspot record. While there have been periods of greater B P, most notably , the bulk of the predicted field intensities are less than recent measurements during solar maximum. Likewise, while the low value of field intensity during the recent protracted solar minimum ( ) has attracted attention [Smith and Balogh, 2008; Connick et al., 2009, 2011; Smith et al., 2013], we see from Figure 1 that no fewer than seven solar minima of the past have had field intensities lower than what was seen during the recent minimum. This includes most years of the Dalton Minimum (especially ). [9] An important feature of the theory is the hysteresis seen when comparing sunspot number to the computed field intensity: The field intensity rises sharply with sunspot number but decays more slowly. This behavior is clearly evident

3 Figure 2. Hysteresis plots for given years and cycles plotting smoothed monthly sunspot numbers versus smoothed computed HMF flux as derived from the above theory. Because the flux rises quickly with sunspots, which we use as a proxy for CME activity, and falls more slowly as sunspot activity decreases, there is a noted hysteresis effect. in Figure 1 when comparing rising and falling phases of the cycle. It can be readily quantified in hysteresis plots such as Figure 2 where we smooth both the monthly sunspot number and monthly computed field intensity by a 7 month running boxcar average and subset over an appropriate set of years. The smoothing is necessary because fluctuations about the trend obscure the center of the hysteresis curve that is generally open. The subset is necessary because changing peak heights also tend to obscure the center of the plots. It is readily evident from the figures that there exists a hysteresis in the field intensity made evident by the rapid rise and slow decay of field intensity driven by sunspot number. 4. Comparison With 10 Be Data [10] The paleocosmic radiation (PCR) data (primarily 10 Be and 14 C) enable us to determine the time dependence of the HMF and solar activity for more than 10,000 years into the past. On entering the heliosphere, galactic cosmic rays (GCR) are scattered by magnetic irregularities embedded in the outward moving solar wind. As shown by Parker [1965] and Jokipii [1991], this reduces the flux of GCR that reaches Earth in an approximately reciprocal relationship to the intensity of the HMF. Upon entering the atmosphere, each cosmic ray initiates a nucleonic cascade and subsequent 7527 spallation reactions with atmospheric atoms. The resulting radionuclides ( 10 Be, 14 C, and others) make their way to the surface of Earth and are sequestered, year by year, in ice, tree rings, and sediments [Beer et al., 2011]. [11] In the case of 10 Be, the radionuclides are deposited into polar ice within 1 to 2 years. At times of low solar activity (lower HMF intensity; higher GCR flux), more 10 Be is produced each year, and the 10 Be concentration is higher. High HMF intensities result in low concentrations of 10 Be. Ice cores sampled from areas with low yearly melt provide well-defined annual layers that can be used to determine the paleocosmic ray intensity (and thence, the HMF intensity and solar activity) with a resolution as high as 1 year. [12] Caballero-Lopez et al. [2004] pioneered the mathematical inversion of the 10 Be data to yield the intensity of the HMF near Earth. McCracken [2007] adapted their process to estimate the annual intensity of the HMF from 1428 to the present. It is those results that are used herein (see Figure 1). [13] As outlined above, the PCR data provide a second, independent set of HMF intensities that can be used to examine the validity of our model when driven by sunspot numbers. Figure 1 (circles) presents the HMF intensities B from McCracken [2007]. The estimates for were derived from 10 Be data, after being passed through a (1, 4, 6, 4, 1) time series (binomial) filter to reduce statistical

4 fluctuations and minimize residual atmospheric effects (The HMF estimates for the years from McCracken [2007] have been replaced here by more reliable estimate based on the 10 Be measurements.). The data after 1950 were derived from the cosmic ray intensity recorded by the worldwide neutron monitor network. Notice that all these estimates ( ) systematically overestimate B P as computed in the previous section. We will explain this below. [14] If one compares the pattern of sunspot numbers and HMF intensity derived from the 10 Be measurements, five anomalies appear to be present. Four are roughly , , , and when the sunspot numbers are falling, but the HMF estimates rise. The fifth occurs over the years when the HMF intensity derived from the 10 Be measurements falls while the sunspot number climbs. Note that all five anomalies occurred during the two periods of low solar activity in (the Dalton Minimum) and [15] McCracken et al. [2002] have shown that the PCR data exhibit a 5 year periodicity during intervals of low solar activity. During such periods, maxima in the PCR intensity occurred during sunspot minimum, and in addition coincident with sunspot maximum. They proposed that this might be a consequence of the reduction of the strength of the solar axial dipole relative the equatorial dipole, as proposed by Wang et al. [2002] and as discussed by Wang and Sheeley [2013]. That is, the high PCR intensity at sunspot maximum occurs at the time of the reversal of the solar magnetic field in epochs of low solar activity. All five of the anomalies identified above are consistent with this model, and we conclude that the five anomalies are the consequence of a secondary feature of the 11 year periodicity in the solar dynamo itself. [16] The hysteresis discussed in section 3 seems to be a common feature of both the HMF intensity computed from the theory and that deduced from 10 Be measurements. In combination with the direct comparisons made by Smith et al. [2013] using measured HMF data, this strongly suggests that hysteresis is a real and reproducible part of the heliospheric magnetic solar cycle. We can uncover hysteresis behavior in the 10 Be data if we limit our analysis to the years 1930 to present when some of the above anomolous behavior is not present. Figure 3 reveals the underlying hysteresis behavior in the theory and the data. Figure 3 (top) again shows the predicted magnetic flux derived from the theory plotted versus the monthly average sunspot number. Note the clear evidence associated with the rapid rise in the field with CME activity followed by the slow decline that results in a hysteresis effect. Figure 3 (middle) plots the last 75 years of yearly HMF intensity estimates derived from the cosmic ray measurements ( 10 Be and neutron monitor) versus yearly average sunspot number. Although partially obscured, a hysteresis effect is again evident. The scales of the two curves (Figure 3, top and middle) are different, but the behavior is the same. The years prior to 1930 seem to exhibit a similar behavior, but this is often obscured by the presence of secondary peaks between solar maxima. Figure 3 (bottom) plots the yearly average of the HMF intensity derived from the Omni2 data versus yearly averages of the sunspot number. The use of yearly averages offers a clearer comparison with the 10 Be data which behave in a similar manner. Figure 3. (top) Hysteresis plot showing the HMF flux (red) as computed by equation (3) versus sunspot number for the years 1930 to The numbers are smoothed as in Figure 2. (middle) The yearly average HMF intensity B as computed from 10 Be (green) plotted versus yearly averages of sunspot number. While the curve is offset relative to Figure 3 (top), Figure 3 (top and middle) shows similar hysteresis. (bottom) The yearly average measured HMF intensity (blue) plotted versus yearly averages of sunspot number. 5. Comparing Flux and Total Field Intensity [17] Since there remains a significant difference between B P as predicted above using equations (3) and (4) and the average total field intensity B derived by McCracken et al. [2013] using 10 Be data, we wish to account for that difference. One possible solution is a nonzero value to ˆflr. This would raise the predicted minimum flux level derived from this theory and, in turn, it would raise the maximum flux level simply by reducing depletion between times of maximum injection. However, this is not the answer as it disregards a basic difference between the flux predictions of Schwadron et al. [2010] and the HMF intensities derived from 10 Be. Schwadron et al. [2010] predicts only the magnetic flux which means the associated magnetic field intensity is that of the Parker field [Parker, 1958, 1963]. The values for B contain the azimuthal fields often associated with magnetic clouds and the turbulent magnetic fluctuations, both of which are absent from the definition of B P. Therefore, we fully expect that B P < B which we clearly see in Figure 1 when comparing our prediction for B P with the computed HMF intensity derived from 10 Be 7528

5 measurements. Furthermore, field line injection via CMEs is quantified by Owens [2008] who describes magnetic flux injection and not the injection of a volume-integrated magnetic field intensity. [18] The above explains that there should be a difference between the two field intensity estimates. We now show that the difference can be found in the data and quantified in general agreement with the above conclusions. We can access the Omni2 data for spacecraft measurements of the solar wind at 1 AU starting in This is the basis for the analysis done by Connick et al. [2009, 2011]. Smith et al. [2013, Figure 1c] show the systematic difference between the average field intensity and the average intensity of the Parker component. Figure 4 (top) reproduces those results with squares showing the average field intensity h B i and triangles showing the average intensity of the Parker component h B P i where h...i represents the computed average over the Omni2 data set. Figure 4 (bottom) shows the yearly difference between the two intensities. The difference averages about 2.4 nt with the difference slightly greater during high sunspot numbers (solar maximum) and lower during low sunspot numbers (solar minimum). The years provide a slight anomaly to this and other behaviors as noted in the papers above. [19] We can now see that the difference between B P as computed above using the theory of Schwadron et al. [2010] and B as computed from cosmogenic 10 Be measurements by McCracken [2007] is generally consistent with the measured difference between these two quantities as computed from the Omni2 data set. We can now understand this difference as the azimuthal component that remains unresolved in the Schwadron et al. [2010] theory. 6. The Next Solar Minimum [20] Based on the ability of this theory to reconcile in situ measurements of the past 40 years and the general agreement between this theory and the conclusions drawn from paleogenic 10 Be data, we can reasonably predict the HMF intensity of the next solar minimum. We do so by noting that early in 2013 marks the peak in the solar cycle and that the sunspot number is comparable to what was seen during the Figure 5. Analysis of recent solar cycle. From year 2013 onward the sunspot number is obtained from the historical record 1805 onward. The resulting B P for 2020 shown in red is 1 nt lower than in the last protracted solar minimum. Prediction for B shown in green is based on the observation that B B P averages 2.4 nt, although it is less during solar minimum. Dalton Minimum. The years following 1805 thereby serve as a prediction for the coming 10 years of solar activity. We apply the theory described above and the same parameters as have been used to date [Owens et al., 2012]. Figure 5 shows first a reproduction of the results shown in Figure 1 for the years leading up to 2013 and then the resulting prediction for the coming solar minimum computed using the Dalton Minimum years 1805 onward. The vertical dashed line marks the transition between measured sunspot numbers for this cycle and those from the Dalton Minimum. [21] This theory builds flux rapidly with the onset of CME activity. It dictates that flux decay on a slower rate in accordance with the timescale of disconnection (6 years). Because solar activity during the recent maximum (2013) is lower than in past years of in situ measurements, the HMF intensity never reaches the level seen during the last solar maximum. Consequently, the HMF intensity falls from a lower value as we now approach solar minimum. So long as the steady state flux level associated with the lower level of CME activity is not reached, this dictates that the flux will fall to lower levels than were seen in the last protracted solar minimum. The Parker field is expected to reach 1 nt. If the mean field intensity remains 2.4 nt higher as it is on average, then B is expected to drop as low as 3.4 nt. However, Figure 4 shows that this difference is smaller during solar minimum and was just 1.5 nt during the last minimum. This means that B will be in the range 2.5 to 3.4 nt at the next sunspot minimum if the solar cycle proceeds as expected. Figure 4. Analysis of Omni2 data showing difference between h B i and h B P i. (top) h B P i is shown as black triangles while h B i is shown as blue squares. (bottom) Yearly difference between the two Discussion [22] We have shown a good correlation between the HMF intensity as derived from 10 Be measurements and those predicted by this theory. There are six highly unusual intervals of disagreement. One is the first solar maximum of the space age ( ) for which we have no explanation except to note that it breaks the pattern of correlation with sunspots in all of our recent papers [Connick et al., 2009, 2011; Schwadron et al., 2010; Smith et al., 2013]. The other five intervals are times of nominal solar minimum when the 10 Be data show elevated HMF intensities. McCracken et al. [2002] have offered an explanation for these observations based on the possibility there might be a reduction of the strength of the solar axial dipole relative the equatorial dipole. We find no a priori reason to discount these results or the suggested explanation, and note that the only

6 reason that they disagree with this theory is that they lie outside the expected sunspot pattern. The fact that they reside within the two periods of low solar activity ( and ) may suggest that similar behavior may soon be observed if the pattern of low maxima and prolonged minima continues. 8. Summary [23] We have employed a theory by Schwadron et al. [2010] that attempts to explain the changing HMF flux and its dependence on solar cycle. Smith et al. [2013] have shown that this theory can accurately reproduce the measuredhmffluxwhencomparedagainsttheinsiturecord from 1975 onward. That effort showed that sunspot number can be used as an effective proxy for CME activity at least during those years of observations. We have returned to the historical record for sunspots using data from 1749 onward, and we have attempted to account for the HMF flux during those years and with it the intensity of the HMF. In an effort to validate those results, we have compared them to analyses of the HMF intensity based on 10 Be concentrations contained in paleogenic data [McCracken, 2007]. We have found good agreement between the two results, but we must note that the 10 Be predictions show five secondary peaks in the HMF intensity seen during times of low sunspot number that are not contained within this theory. These secondary peaks have not been seen within the space age in situ record from 1965 onward. [24] As is a part of this theory, we see a strong hysteresis effect wherein HMF flux and intensity rises quickly with sunspot activity and decays away slowly as reconnection in the low solar atmosphere sheds field lines from interplanetary space. The same appears true in the interplanetary magnetic field and the 10 Be data if one allows for the presence of those secondary peaks that are not well understood. The hysteresis effect is the direct result of the large lag time needed to disconnect magnetic flux previously open to the heliosphere. In contrast to the long disconnection time (6 years in the current study), CMEs introduction of new magnetic flux to the heliosphere rises quite rapidly in the phase of rising solar activity. The hysteresis therefore represents the memory of magnetic flux that was introduced by CMEs and decays in the declining phase of solar activity. The hysteresis observed in both interplanetary magnetic field data since 1975 and in 10 Be data provides direct observational support for long-term decay of magnetic flux through disconnection. [25] We have also used the results from the Dalton Minimum to predict the likely HMF flux and intensity from late 2013 to 2022 including the coming solar minimum. Because the HMF has not fully recovered to previous solar minimum values that are typical of the space age years, reconnection and the associated flux shedding will drive the HMF to lower values than were seen in the recent protracted solar minimum of The total field intensity at 1 AU is likely to be in the range 2.5 to 3.4 nt while the Parker component (the part of the field that follows the Parker spiral direction) will probably get as low as 1 nt. [26] Acknowledgments. The authors thank the National Space Science Data Center for providing data used in this study. We thank the NGDC for sunspot data used in this study. C.W.S. is funded by Caltech subcontract 44A to the University of New Hampshire in support of the ACE/MAG instrument. N.A.S. is funded by EMMREM (grant NNX07AC14G), C-SWEPA (NASA grant NNX07AC14G), Sun-2-Ice (NSF grant AGS ), and the NASA LRO/CRaTER (NNG11PA03C) projects. 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