ISSN 1063-7737, Astronomy Letters, 2009, Vol. 35, No. 4, pp. 247–252. c ? Pleiades Publishing, Inc., 2009.
Original Russian Text c ? V.N. Obridko, B.D. Shelting, 2009, published in Pis’ma v Astronomicheski˘ ı Zhurnal, 2009, Vol. 35, No. 4, pp. 279–285.
Anomalies in the Evolution of Global and Large-Scale Solar Magnetic
Fields as the Precursors of Several Upcoming Low Solar Cycles
V. N. Obridko*and B. D. Shelting
Pushkov Institute of Terrestrial Magnetism, Ionosphere, and Radiowave Propagation, Russian Academy of
Sciences, Troitsk, Moscow oblast’ 142190, Russia
Received August 18, 2008
Abstract—Anomalies in the solar magnetic fields of various scales are studied. The polar magnetic field
strength is shown to have decreased steadily during the last three solar cycles.This is because the increase
in the dipole magnetic moment observed from 1915 to 1976 has changed into a decrease in the last three
cycles. At the same time, the intermediate-scale magnetic fields (like those of isolated coronal holes) have
beenunusuallystronginthe lastcycle.Asaresult,the tiltoftheheliosphericcurrentsheetisstillabout30◦.
Thelargeeffectivecontributionfromthe intermediate-scalefieldstothe total energyofthe large-scalefields
is also confirmed by our calculations of the effective multipolarity index. The aa-index at the cycle minima
is correlated with the height of the succeeding maxima. The set of data considered may be indicative of the
possible approach of a sequence of low solar cycles.
PACS numbers : 96.60.Hv
Key words: solar cyclicity, large-scale magnetic field.
The present solar cycle (23 according to the
in several respects, breaking many of the previously
established typical characteristics of solar cycles.
This primarily applies to the data on local magnetic
fields. Below, we list the best-known characteristics.
Violation of the Gnevyshev–Ohl Rule
As is well known, according to this rule, an odd
the adopted cycle numbering. This rule was formu-
lated in 1948 by Gnevyshev and Ohl for the sums of
monthly Wolf numbers and it had only one exception
over 26 cycles in the pair (4, 5) until the present
pair of cycles (22, 23). This rule was updated by
Kopeck ´ y (1950), who generalized it for the maximum
(in the cycle) Wolf numbers. Unfortunately, this rule
(which, for definiteness, should have been called the
Gnevyshev–Ohl–Kopeck ´ y rule) already has three vi-
olations in the pairs (−2, −1), (4, 5), and (8, 9). The
current pair of cycles (22, 23) violatesthis rulein both
A Very Long Cycle
By August 2008, there had been no reliable data
on the termination of cycle 23, but now it is clear that
its duration approaches 12 years and it is one of the
longest cycles or just the longest one in the recording
history of solar activity since 1848.
A more detailed list of anomalies in solar activ-
ity related to the behavior of sunspots and nonsta-
tionary processes can be found in Ishkov (2005). In
this paper, we would like to draw attention to some
peculiar features in the behavior of global and large-
scale magnetic fields in the present cycle, which, in
our opinion, suggest a transition to the period of low
First of all, let us define what we mean by the
large-scale and global magnetic fields. Looking at
modern high-resolution magnetograms, for example,
SOHO/MDI magnetograms, it is immediately ap-
parent that the magnetic fields have a small-scale
patchy pattern. However, we see at once that these
small-scale elements are distributed over the surface
not randomly but form extended regions in which
one of the polarities dominates, i.e., they form quasi-
unipolar regions of various scales. Undoubtedly, this
is due to the existence of weak large-scale (or global)
248 OBRIDKO, SHELTING
In what follows, we will call the fields with a char-
acteristic spatial scale comparable to the solar ra-
dius global ones and the fields with a slightly smaller
scale (0.3–0.7 of the solar radius) large-scale ones.
Naturally, separating these relatively weak fields re-
quires a complex filtering procedure. Such filtering
is usually performed by decomposing the observed
magnetograms into spherical harmonics followed by
the summation in the selected range of spatial fre-
Obviously, the largest-scale, global magnetic field
is related to the first harmonic of the decomposition
and is called a dipole field. The usually discussed
polarity reversal of the global field primarily means
the polarity reversal of the global dipole. Direct ob-
servations of the polar magnetic field give an idea of
this polarity reversal. However, it should be kept in
mind thatthesetwo times,thepolarityreversals ofthe
global and polar fields, may not coincide. Moreover,
times when the fields had the same sign at the two
poles of the Sun were repeatedly observed in the past,
which, naturally, cannot be for a dipole.
CHARACTERISTICS OF THE GLOBAL
AND LARGE-SCALE FIELDS ESTABLISHED
FROM THE PAST CYCLES
First, recall the universally accepted characteris-
tics of the behavior of the global fields in a solar cycle.
— The global (in particular, polar) magnetic fields
develop in antiphase with the local fields, which are
commonly characterized by Wolf numbers.
— The global fields change their sign at the max-
imum of the Wolf numbers and reach their maximum
at the minimum of the Wolf numbers.
— If the most global fields are related to a dipole,
then it is reversed at the cycle maximum.
— Occasionally, the total magnetic moment of the
dipole decreases greatly but it never becomes zero.
During one or two years at the decay phase of the
solar cycle, the vertical (i.e., coaxial with the Sun’s
rotation axis) and horizontal (whose axis lies in the
solar equatorial plane) dipoles are comparable. This
situation is known in astrophysics as an oblique rota-
tor (Livshits and Obridko 2006).
—Thepolar(and, ingeneral, large-scale) fieldcan
serve as a prognostic index of the upcoming cycle
(Makarov et al. 2001a).
— Observations of the sector structure (Wilcox
and Ness 1965) revealed the existence of a helio-
spheric current sheet(HCS)as thesurfacethatsepa-
rates the heliosphere into two magnetic hemispheres
with opposite magnetic polarities (Schulz 1973;
Hundhausen 1977). This surface is also commonly
called a heliospheric equator. The HCS structure
is determined by the field structure on the source
— The corrugated surface formed by the HCS is
often compared with the flaring skirt of a ballerina.
At the cycle minimum, the HCS lies in the solar
equatorial plane; at the cycle maximum, it is tilted
and its tilt becomes very large, up to 90◦. Mursula
and Hiltula (2003) analyzed the direct observations
of the interplanetary magnetic field polarity for 1965–
2001 and showed that the frequency of the polarity
corresponding to the northern solar hemisphere dur-
ing solar minima is higher than that of the polarity
minima. They called this effect “bashful ballerina”,
comparing the Sun at the decay of activity to a bal-
lerina pushing her high flaring skirt downward.
in the course of time. Depending on the cycle phase,
the main component of the global magnetic field, the
global dipole, is tilted and passes from the position
coaxial with the Sun’s rotation axis to the position
in the equatorial plane and then passes into the other
hemisphere (Livshits and Obridko 2006). The HCS
becomes corrugated and the size of this corruga-
tion or the tilt is denoted by T. Mathematically, this
quantity can be found by calculating the maximum
latitude of the neutral line of the large-scale field
on the source surface. Subsequently, the southern
(negative) latitude TSis subtracted from the northern
latitudeTNandthemean valueisfound. Basically,the
heliospheric equator is found in this way. The defini-
tion of the tilt T includes only the odd antisymmetric
harmonics of the global field multipoles. (The term
“tilt” is not quite appropriate. In fact, not the HCS
but the axis of the corresponding equivalent dipole is
The even harmonics (the quadrupole and higher),
which do not change their sign when the equator is
crossed, lead to an HCS asymmetry. This means an
HCS shift ∆T relative to the solar equator. Mathe-
matically, this shift is defined as the half-sum of TN
and TS. The southward shift described by Mursula
and Hiltula (2003) must give TSlarger in magnitude
than TNand, hence, ∆T < 0. Previously (Obridko
and Shelting 2008), we calculated the yearly mean
TN, TS, and ∆T for 1915–2000. The center of grav-
ity of the HCS undergoes quasi-periodic oscillations
with a period close to 11 yr. The minima of the curve
usually lie in the negative half-plane and are often
close to the solar minima. This confirms the con-
clusions by Mursula and Hiltula (2003). However,
in addition to this fact, another effect is observed:
near the cycle maxima, positive values of ∆T are
encountered more frequently, which is indicative of a
northward HCS shift. Occasionally, the curve near
ASTRONOMY LETTERSVol. 35 No. 42009
ANOMALIES IN THE EVOLUTION 249
19181928193819481958 19681978 198819982008
Fig. 1. Magnetic moment of the solar dipole versus time.
the maximum does not pass through zero but the
value of ∆T at maximum is always higher than that
Thus, the intensity and structure of the global and
large-scale magnetic fields can be characterized by
the following indices:
—thefieldstrengths atthesolarpolesBNand BS;
— the magnetic moment of the effective solar
— the HCS tilt T;
— the HCS shift ∆T, the negative sign of ∆T
points to a southward shift.
A more sophisticated analysis of the variation in
the ratio of the fields of various scales is provided
by calculations of the mean square of the magnetic
field strength on a selected spherical surface. These
indices were introduced by Obridko and Ermakov
(1989), Shelting et al. (1989), and Obridko and
Shelting (1992). Two indices are used most com-
monly: the mean square of the magnetic field strength
on the photospheric surface IBr(ph) and the mean
square of the magnetic field strength on the source
surface at a height of 2.5 solar radii from the center
In addition, the effective multipolarity index intro-
duced by Ivanov et al. (1997) is commonly used. Its
definition isrelatedto thefactthatthemultipolemag-
netic field strength decreases with height as 2l + 1,
where l is themultipolenumber. Thus, comparing the
mean squares of the fields at two levels, we can find
the effective index of multipole n using the formula
n = lnIBr(ph)
An increase in this parameter at the decay phase
points to a decrease in the effective field scale. During
the minima when mainly the dipole component re-
mains on the Sun, the parameter n approaches three;
at the cycle maxima, it reaches five, six, or higher.
PECULIARITIES IN THE THE BEHAVIOR
OF THE GLOBAL AND LARGE-SCALE
FIELDS IN CYCLE 23
According to the data of the John Wilcox Solar
Observatory (WSO) in Stanford, a decrease in the
Here, we will not present the full plot of the polar
field against time; it can be seen at the WSO site
http://wso.stanford.edu/gifs/Polar.gif. Although we
cannot confidently talk about the polar field strength
in 1976 (at that time, the observations had just be-
gun), it is higher than 2 G in absolute value. In 1985,
it was higher than 2.5 G in absolute value. The field
strength did not exceed 2 G in 1995 and was about
1 G in 2007. This means that the polar field strength
decreased by a factor of 2–2.5 over three cycles.
This strongly suggests that the global magnetic
fields gradually decrease. This conclusion is con-
firmed by an analysis of the magnetic moment of the
effective global solar dipole. Makarov et al. (2001b,
2002) showed that the magnetic moment increased
until 1984–1985 and then decreased sharply (but the
data at that time were sufficient only until 1991).
These calculations are indicated in Fig. 1 by the thick
curve; a linear fit is drawn. At present, we have calcu-
lated the magnetic moment for another three cycles.
These calculations are indicated by the thin dotted
curve. It turned out that after 1980, the magnetic
moment showed a tendency to gradually decrease
and, at present, it has already reached values lower
than those at the beginning of the 20th century.
This points to a tendency for the largest-scale
fields to decrease. However, this cannot be said about
the intermediate-scale fields, as confirmed by ourcal-
culations of the effective multipolarity index n. The
variation of this parameter for 1976–2008 is shown
in Fig. 2.
Note that this parameter behaved in a standard
way in cycles 21 and 22. It reached values of the order
ASTRONOMY LETTERSVol. 35 No. 4 2009
19861991 19962001 2006
Fig. 2. Effective multipolarity index versus time.
19951990 19952000 20052010
Fig. 3. Indices of the large-scale magnetic field versus time: (a) the index on the photosphere and (b) the index on the source
surface. The thin curve indicates the indices for each half Carrington rotation; the thick curve indicates the smoothed values.
The dotted line indicates the relative sunspot numbers.
ASTRONOMY LETTERSVol. 35No. 4 2009
ANOMALIES IN THE EVOLUTION 251
Fig. 4. HCS tilt versus time.
of 5 at the cycle maxima, which was determined by
the contribution from the local fields, and fell to ∼3
at the minima, which is completely determined by the
behavior of the global quasi-dipole field. However, its
behavior has changed sharply in the current cycle.
After the short-term decrease in 2003–2004, this
index has again increased to values comparable to
those at the cycle maximum. The decrease in 2003
per se agrees well with the behavior of the large-scale
field shown in Fig. 3.
It stems from the fact that in 2003–2004, the
indices shown in Fig. 3, the mean square of the
magnetic field strength on the photospheric surface
IBr(ph) and the mean square of the magnetic field
strength on the source surface Ibr(ss), reach their
maxima. However, the further rise in n is not quite
clear. The increase in this parameter at the decay
phase points to a decrease in the effective scale of
the fields compared to the global field. However, these
cannot be the sunspot fields, because only the open
fields remain on the source surface by definition.
Moreover, the number of sunspots decreased greatly
by 2006–2008. Theeffective sizeof the elements with
n ∼ 4.5 can reach 40 heliographic degrees. These
may be the large-scale unipolar regions with which
equatorial coronal holes are often associated. Note
that equatorial coronal holes have been continu-
ously observed on the Sun since 2003. It is these
coronal holes that probably cause the HCS tilt to
be above 30◦. Indeed, whereas part of the flux on
the photospheric surface can occasionally be closed
back through low-lying loops, a coronal hole on the
neutral line cannot pass through a coronal hole. In
this case, it is shifted away from the equator, causing
the HCS tilt to increase.
Turning again to Fig. 3, which shows the Wolf
numbers together with IBr(ph) on the upper panel
and with IBr(ss) on the lower panel, note that the
maximum of the large-scale field is increasingly re-
ceded from the maximum of the Wolf numbers. The
meaning of this effect is not yet clear.
the minimum, the HCS is smoothed out and sinks to
the equator. As we see from Fig. 4, the typical values
of the tilt at minimum are less than 10◦. In this cycle,
it exceeded 25◦by June 2008. This phenomenon is
not completely clear either and may be related to
the unusually wide distribution of intermediate-scale
ON THE HEIGHT OF THE UPCOMING
SOLAR CYCLE 24
Since the behavior of the magnetic fields in the
present cycle is anomalous, the prediction of the
height of the upcoming cycle is additionally com-
plicated. Previously (Obridko and Shelting 2008),
we analyzed a large number of predictions of cycle
24. Today, forecasters cannot reach any firm con-
clusion. The debate conducted in June 2008 at the
NOAA Space Weather Prediction Center (SWPC)
that two alternative viewpoints exist: the cycle will
be either high (140 units) or below the average one
(90 units). The choice between these two possibil-
ities cannot yet be made. Previously (Obridko and
Shelting 2008), we also suggested several possible
scenarios for the development of cycle 24. The data
on the polar field are more likely indicative of a low
cycle (the maximum value is of the order 80). This is
confirmed by Fig. 1. We see from this figure that the
magnetic moment of the global solar dipole in 2008
decreased to values typical of those at the beginning
of the 20th century, when three low cycles were
observed (the maximum in 1907.0 was 64.2 in cycle
ASTRONOMY LETTERS Vol. 35No. 4 2009
80 100 120 140 160 180 200
Fig. 5. Correlation between the geomagnetic aa-index in
the years of minimum solar activity (vertical axis) and the
relative sunspotnumber at the upcomingsolar maximum
(horizontal axis). The solid curve represents a quadratic
the maximum in 1928.4 was 78.1 in cycle 16). On the
other hand, in this paper, using the characteristics
of the large-scale field and geomagnetic activity, we
obtained moderate values for the maximum of cycle
24, 128 and 113 units, respectively. Using the data
on the geomagnetic field seems most reliable. Some
increase in the intermediate-scale fields associated
with the fields of extended unipolar regions must lead
to an increase in the geomagnetic disturbances.
The height of the upcoming maximum can be
correlated with the geomagnetic disturbance level at
the minimum. In Fig. 5, the yearly mean monthly
geomagnetic aa-indices for the year of minimum, one
year before, and one year after it are shown along
the vertical axis. This index can be determined from
thedataofseveral geomagnetic observatoriesandhas
been tabulated from 1968 until the present time. The
height of the upcoming maximum is shown along
the horizontal axis. A satisfactory quadratic depen-
dence with a correlation coefficient of 0.89 ± 0.03 is
date of the current minimum, not to mention the aa-
index in the next year. If we use the data for 2007 and
2008, then we will obtain 14–15 (these values are in-
dicated in the figure by the shaded rectangle) and this
corresponds to aheightofthenextmaximumof about
110. Analyzing the shift between the maximum of
the Wolf numbers and the maximum of the aa-index,
Georgieva (2008) obtained the same value. The shift
obtained in this work may be somehow related to the
the maximum of the sunspot number shown in Fig. 3.
Thus, in general, one may expect the beginning of
the 21st century to be characterized by one or two
cycles with a fairly low or just low intensity. A more
serious, Maunder-type decline of activity or at least
the decline that was observed at the beginning of the
20th century cannot be ruled out either.
This work was supported by the Russian Founda-
tion for Basic Research (project no. 08-02-00070).
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Translated by G. Rudnitskii
ASTRONOMY LETTERS Vol. 35 No. 42009