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. The trailing regions, which have the opposite
polarity to the polar cap magnetic fields, give rise to
open "unipolar magnetic regions" (and coronal holes)
that are stretched as they drift poleward under the
simultaneous influence of differential rotation. When
these regions arrive at the polar caps, they erode the
polar field as a result of magnetic reconnection
eventually causing it to disappear. Flux of the opposite
REFERENCES
1. Fisk, L. A. and N. A. Schwadron, Space Sci. Rev., 97, 21
(2001).
2. Foukal, P. V., Solar Astrophysics, John Wiley and Sons,
New York, 387 (1990).
3. Lockwood, M. and R. Stamper, Geophys. Res. Lett., 26,
2461 (1999).
4. Lockwood, M., J. Geophys. Res., 106, 16,021 (2001).
5. Smith, E. J. and A. Balogh, Geophys. Res. Lett., 22, 3317
(1995).
6. Smith, E. J., A. Balogh, R. J. Forsyth and D. J.
McComas, Geophys. Res. Lett., 28, 4159 (2001).
7. Suess, S. T. and E. J. Smith, Geophys. Res. Lett., 23,
3267 (1996).
8. Stamper, R., M. Lockwood and M. N. Wild, J. Geophys.
Res., 104, 28, 325 (1999).
9. Wang, Y.-M and N.R.Sheeley, Astrophys. J., 447, L143
(1995).
10. Wang, Y.- M., J. Lean and N. R. Sheeley, Geophys. Res.
Lett., 27, 505 (2000).
70