Open Magnetic Flux: Variation with Latitude and Solar Cycle Edward J. Smith1 and Andre Balogh2 1 2 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA The Blackett Laboratory, Imperial College of Science, Technology and Medicine, London, UK. Abstract. Recent Ulysses observations of the heliospheric magnetic field (HMF) reveal that the sun's open magnetic flux in the solar wind, as measured by r2 Br, is independent of heliographic latitude at solar maximum and at solar minimum. It follows that 4π r2 Br at any latitude provides an accurate estimate of the total open flux from the sun. An additional Ulysses result is that long term averages of r2 Br are very nearly the same at the recent solar maximum as at the preceding minimum. The model of the HMF developed by L. Fisk and his colleagues, which includes several features absent from the well- known Parker model, leads to the prediction that the open flux is relatively constant or perhaps invariant. Motivated by these considerations, Br has been analyzed over the longer interval of four sunspot cycles (20-23) using the archived OMNI field measurements obtained in the ecliptic at 1 AU. Averages of Br, after separation into inward and outward sectors as with Ulysses data, agree within a few percent with previous results based on |Br|, the modulus of the radial component. Averages of Br increase by a factor of about 2 from solar minimum to maximum in cycles 21 and 22 but change only slightly in cycles 20 and 23. The variations in open flux are better correlated with total unsigned flux in the photosphere than with sunspot number, both reaching peak values after sunspot maximum. The open flux decreases at solar minima and secondary decreases occur when the polar cap fields vanish. It appears that the near equality of the open flux at minimum and maximum observed by Ulysses was, in fact, caused by a time variation with the measurements near maximum being obtained while a secondary decrease was occurring. Thus, the open flux is generally variable and only constant during shorter intervals when the solar field is stable. If the latitude invariance is not restricted to the present solar cycle but is a general result, the nearly continuous measurements of Br made in the ecliptic at 1 AU by various spacecraft determines the open flux over previous sunspot cycles. Several independent estimates of open flux have been compared with F(|Br|). Lockwood and Stamper (1999) used a correlation between Br and the aa index of geomagnetic activity to determine F(aa) and compared it with F(|Br|) during solar cycles 20-22. Wang et al. (2000) estimated open flux from a potential field source surface model, F(PFSS) between 1971 and 1999. Both approaches revealed a close correspondence between the three estimates of open flux over the last several solar cycles and are consistent with the latitude invariance in Br being a general condition. INTRODUCTION Ulysses measurements at both solar minimum and maximum show that r2 Br, i.e., the radial component of the heliospheric magnetic field multiplied by the square of the radial distance at which the observations were made, r, is independent of heliographic latitude (Smith and Balogh, 1995; Smith et al., 2001). This independence (Figure 1) is explained by excess magnetic pressure in the solar wind source regions causing a spreading out of the open magnetic fields (and non-radial solar wind flow) until a pressure balance is achieved (Smith and Balogh, 1995; Suess and Smith, 1996). Values of r2 Br multiplied by 2π are a measure of the signed open flux in the solar wind, the usual units, nT (AU)2, are equivalent to webers (maxwells) and the total unsigned open flux from the sun is simply 4πr2 |Br| = F(|Br|). CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference, edited by M. Velli, R. Bruno, and F. Malara © 2003 American Institute of Physics 0-7354-0148-9/03/$20.00 67 6 -70o -80.2o -70o -40o 0o +40o +70o +80.2o +70o OUTWARD Mean 3.27 St Dev 0.91 4 2 0 SOLAR MAXIMUM -2 INWARD Mean -3.48 St Dev 0.75 -4 2 R r B Sol Rot Avgs, nT-AU 2 The availability of the aa index between 1868 and the present enables an extrapolation of the open flux backward in time throughout the previous century. The open flux estimates enabled studies of their correlation with other solar parameters such as total radiance (Lockwood and Stamper (1999) and other variables such as cosmic rays (Lockwood, 2001). Wang et al. (2000) found that the open flux in the photosphere evolved systematically from high to low latitudes during the solar cycle. The average value of Br at Ulysses during solar maximum extrapolated to 1 AU is ≅3 nT approximately the same as at minimum. This result is surprising considering the gross changes in the solar magnetic field between the two phases. However, a model of the HMF developed recently by Fisk and Schwadron (2001) that includes magnetic reconnection at the solar surface implies that the amount of open flux on the sun tends to be conserved. The argument is made that open flux at the surface reconnects predominantly with closed flux causing a migration or diffusion of field lines across the surface without changing the net open flux. The availability of Br measurements in the ecliptic over the past 4 sunspot cycles allows this invariance to be tested more thoroughly. The previous studies used the modulus of the radial component, |Br|, in order to avoid the compensating effect of the signs in opposing magnetic sectors. However, use of the modulus is equivalent to rectifying the radial component without regard to the sector structure. Rectification is a non-linear process that can add the power in the ever-present HMF fluctuations to the mean and affect the value obtained for the open flux. On the other hand, the Ulysses results have been based on Br in the two sectors separately, i.e., Br(+) and Br(-). In the following, we compare the signed Br at 1 AU in the ecliptic (OMNI data) with |Br| and use it to reexamine its variation in this and previous sunspot cycles. Details of the variation in Br, the sunspot number and total magnetic flux on the sun are examined to try to understand their relationship and to ascertain whether or not a time variation may have contributed to the apparent invariance of the Ulysses results. -6 6 -70o -80.2o -70o 2001.5 -40o 0o +40o 2002.0 +70o +80.2o 4 0 2 +70o OUTWARD Mean 3.03 St Dev 0.27 2 SOLAR MINIMUM -2 INWARD Mean -3.30 St Dev 0.32 -4 R r B Sol Rot Avgs, nT-AU 2 2001.0 -6 1994.5 1995.0 1995.5 FIGURE 1. Ulysses observations of open flux at sunspot minimum and maximum. Averages and errors of r2 Br over successive solar rotations are shown as well as overall values for inward and outward sectors and for sunspot minimum and maximum. The two intervals cover the fast latitude scans (or perihelion passages or pole-to-pole transits). covered are the two fast latitude (or perihelion) scans in 1994-1995 (sunspot minimum, lower panel) and 2000-2002 (maximum, upper panel). Both Br(+) and Br(-) are included with only single polarities present above +/- 20˚ during minimum and a single negative polarity above 70˚ at maximum (indicative of the polar cap polarity reversal that occurs near solar maximum). The averages and standard deviations are more variable at maximum because the variable low speed wind extends to essentially all latitudes and the number of coronal mass ejections increases. Nevertheless, no significant latitude variation is evident at maximum or at minimum. Although the means are slightly higher in the upper panel, the standard deviations imply that all four are consistent with same value for r2 Br and the open flux in both phases of the sunspot cycle. Before investigating the behavior of the in-ecliptic Br during past cycles, we address the issue of how mean values are affected by using the modulus rather than separating the measurements into the two sectors before averaging (symbolized by brackets, < >). The result of this comparison, Figure 2, is a plot of <|Br|> vs. the combined average, < Br(+,-)> = <Br(+)> – <Br(-)>. The least squares fit indicates that |Br| is about 2.5% larger on average than Br(+,-) as expected. This difference is probably insignificant in previous or future studies provided that longer term averages (such as shown here) are based on averages over shorter intervals (hourly averages here). The power in the field fluctuations increases with increasing period (decreasing frequency) and the differences due rectification will grow larger if the base data set consists of longer period averages. The succeeding ANALYSIS The independence of r2 Br on heliolatitude is demonstrated in Figure 1, solar rotation averages of Ulysses measurements plotted against time with latitude shown along the upper scales. The intervals 68 7 27Day Avgs of OMNI_M Hour Avgs 1967-2001 27 Day Avgs of OMNI_M Hour Data Br(+,-) SSN 6 Br(+,-) smoothed, nT 6 y = 0.02 + 1.02x R= 0.982 5 4 5 160 120 4 80 3 2 40 1 Smoothed Sunspot Number Weighted Avg of Br(+) and |Br(-)|, nT 7 3 0 2 2 3 4 |Br|, nT 5 6 0 FIGURE 3. Smoothed averages of Br(+,-) and sunspot number during cycles 20-23. The solar rotation averages of Br(+,-) are smoothed further by 7 point running averages. Sunspot numbers are dotted. Horizontal bars indicate when the Ulysses measurements were made. 1 1 1970.0 1975.0 1980.0 1985.0 1990.0 1995.0 2000.0 7 both the limited increase in Br during the present cycle and the timing of the Ulysses pole-to-pole scan evidently kept r2 Br nearly the same at maximum as at minimum. The solar magnetic flux is compared more directly with Br in Figure 4. The total unsigned flux, from which the open flux is derived, is shown between 1975 and the present with sunspot maxima and minima indicated (courtesy of K. Harvey). The salient features are the large variation over the sunspot cycle by a factor of about 5 and the appearance of maxima in total flux following the sunspot maxima by about 2 years. The leveling-off of the flux near sunspot maximum is suggestive of the decrease in the polar cap fields at that time. The smoothed means of <Br(+,–)> are shown and the values are converted to open flux on the right hand scale. Solid horizontal lines again show the two Ulysses intervals. The correlation between Br and total flux is better than with R and provides supporting evidence that significant changes in the axial field occurred during the Ulysses measurements making them more nearly equal at minimum and maximum. FIGURE 2. The modulus of Br compared with the weighted average of Br(+) and Br(-). Parameters derived from a least squares fit are shown. analyses are based on <Br(+,-)> but are generally consistent with the previous publications. In-ecliptic values of <Br(+,-)> obtained from the OMNI data set are plotted in Figure 3 along with the sunspot number, R, over the four cycles, 20-23. The 27 day means are filtered by applying 7 point smoothing to emphasize only the longer period variations that may correlate with R. In cycles 21, 22, <Br(+,-)> varies systematically by about a factor of two (2.8 5.4 nT). The variation in Br is correlated with R but is more complex. Minima occur near sunspot minimum but the maxima occur later in Br than in R. Furthermore, Br has secondary minima near sunspot maximum suspiciously near the times at which the polar cap magnetic fields vanish and then reappear with the reversed polarity. The increases in Br following sunspot maximum also coincide with the times at which the sun's axial dipole reaches maximum values. The open flux is more closely related to the solar dipole and its changes rather than to the sunspot number. Cycles 20 and 23 exhibit only modest variations in Br along with a lesser variability in R. Cycle 20 shows a small maximum prior to the minimum in R consistent with cycles 21,22. A slight decrease in 1970 may be correlated with the disappearance of the polar cap fields but is of questionable significance. Of greater relevance to the present study are the recent variations in Br, R and r2 Br. The intervals during which the Ulysses measurements were made are indicated by the horizontal solid lines. The variation in Br in the ecliptic in cycle 23 is modest (2.6 -3.5 nT) and exhibits another of the decreases, this time in 2001, accompanying the polar cap reversals. Thus, DISCUSSION The Ulysses observations of nearly constant open magnetic flux at solar minimum and maximum during the recent cycle were influenced by the low values of the sunspot numbers and the related total unsigned flux on the sun. Time variations on the sun associated with the disappearance and reversal of the axial dipole and polar cap fields coincided with the Ulysses measurements and also contributed to low values of the Ulysses observations during maximum. However, the open magnetic flux was shown by Ulysses to be 69 sign then builds up to produce the strong dipole field seen after solar maximum. Since the solar fields, unipolar regions and open flux are ultimately derived from sunspots, a correlation of the open flux with total flux and of both with sunspot number is not surprising. Furthermore, the reconnection of open fields on the sun, indicated by the dip in open flux when the unipolar regions are reversing the polar caps, violates the conditions assumed in the Fisk and Schwadron model where the oppositely-directed fields are alternately open and closed not both open field regions. 24 18 12 Open Magnetic Flux (10 22 30 Mx) 36 6 FIGURE 4. Open magnetic flux compared with total flux in the photospheric. The values of <Br(+,-)> are converted to flux units on the right scale. Horizontal bars indicate when the Ulysses observations in Figure 1 were made. ACKNOWLEDGMENTS Joyce Wolf provided invaluable assistance in the analysis, preparation of the figures and formatting of the text. The research performed at the Jet Propulsion Laboratory was carried out under a contract between the California Institute of Technology and the Aeronautics and Space Administration. Ulysses research at Imperial College is supported by the UK Particle Physics and Astronomy Research Council. independent of solar latitude during both maximum and minimum. In-ecliptic observations of Br could then be used to extend the investigation of the constancy of the open flux to the previous three cycles. Significant variations in Br and, presumably, the open flux occurred during cycles 21 and 22 but were again at a low level in cycle 20. It seems evident that the open flux is not typically conserved over the sunspot cycle contrary to models which envision a simple rotation of the heliospheric current sheet during the solar cycle with the flux above and below the current sheet being conserved. It may be that when solar conditions are stable the open flux remains unchanged for shorter time intervals. The interesting correlations between open flux, sunspot number and total photospheric flux are consistent with current understanding of solar cycle variations in the sun's magnetic field represented, for example, by the Babcock- Leighton phenomenological model (Foukal, 1990). As solar activity increases, the sun's field becomes increasingly structured as the strong bipolar fields emerge in association with sunspots. 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