2. Indices of the Global Magnetic Field
[4] Long-term variations in the large-scale magnetic field at
different latitudes can be studied by using synoptic Ha charts.
The synoptic Ha charts show the neutral lines of
magnetic field polarities. As neutral line tracers, the reference
points obtained from optical observations of filaments, channels
of filaments, and protuberances are used. Thus, in contrast to
magnetographical observations whose spatial resolution constantly
increases with modernization of telescopes, the spatial detailing
of the areas occupied by the magnetic field of one or another
polarity on the Ha charts is limited. At present
Ha charts are available for the period from 1887 till
now. The length of this series of Ha charts is
commensurable with the length of the series of sunspot group
positions.
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Figure 1
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[5] For the formation of such indices we have used the
calculations based on separate synoptic charts. Later on the
obtained series were smoothed by a sliding window with the width
of 12 rotations.
2.1. Dipole-Octupole Index of the Large-Scale
Field A(t)
[6] The surface magnetic field can be presented as a function
of latitude
q and longitude
j by using the
expansion in spherical harmonics. The low modes of the expansion
can be used to analyze the global magnetic field. The
dipole-octupole index
A=m12+m32/3 was proposed by Makarov and Tlatov [1999, 2000a]. The
A(t) index demonstrates well pronounced 11-year activity cycles,
and there is a phase shift of, on the average, 5-6 years between
A(t) and the 11-year curves of Wolf numbers. As can be seen from
Figure 1b, the
A(t) index cycles are ahead of the cycles of Wolf
numbers,
W(t). It should be noted that the
A(t) parameter
includes only the dipole and octupole components of the
large-scale background magnetic field, i.e., modes
L = 1 and 3.
The even modes
L = 2, 4 and the higher-order modes have
considerably lower intensities. The low-order odd modes of
expansion with respect to Ha synoptic charts
characterize the configuration of the solar global magnetic field
at the activity minimum. The shift of
A(t) relative to
W(t)
of
about 5.5 years can be used as an index for prediction of the
maximum Wolf number
W(t) of solar activity. For example,
according to the Gnevyshev-Ol rule, the current sunspot cycle 23
has to be higher than sunspot cycle 22. However, one can see from
Figure 1a, the
A(23) index for cycle 23 was noticeably lower
than the
A(22) index for cycle 22. Therefore the index of Wolf
numbers
W(23) for sunspot cycle 23 was lower than
W(22) for
cycle 22, which does not agree with the Gnevyshev-Ol rule. At the
maximum, the
A(t) index for the current cycle 23 was equal to
11.4, and this corresponds to the Wolf number at the activity
maximum
W(23) of
130 
10. This value is consistent with the
observational data. The correlation coefficient between maximums
of parameters
W max and
A max is equal to
R=0.96. The
regression formula can be presented in the form
W max = 132 A max - 53. The forecast for May 2006 of the cycle 24
comprised
W max = 110 10.
2.2. Index of Complexity of Synoptic Charts
K(t)
[7] It is possible to introduce another characteristic of the
topological pattern of Ha synoptic charts, i.e., the
number of crossings of magnetic neutral lines with the
heliographic longitudinal net. In analysis of Ha charts,
the series giving the coordinates of crossings of meridians with
the neutral lines of the magnetic field are frequently used. The
series comprise the coordinates taken in
10o longitudinal
intervals for each Carrington rotation. The number of crossings of
meridians with the neutral lines of the magnetic field
N CRS characterizes the complexity of the topological pattern of
the background magnetic field of the Sun and shows characteristic
sizes of the unipolar regions. Let us introduce the parameter
equal to the inverse number of crossings
K = 1/N CRS.
During the activity minima, the degree of complexity of
H
a synoptic charts, and hence the background magnetic
field are lower than during the activity maxima. For this reason,
the
K = 1/N CRS index that depends on the number of
crossings of the neutral lines with the meridian net has its
maximum during the solar cycle minimum. Figure 1b shows the
behavior of the
K(t) = 1/N CRS(t) index during the period
1887-2003. This index is compared with the time dependence of
annual mean Wolf numbers (lower curve) during the same period.
[8] Figure 1b demonstrates that the topological
K(t) =1/N CRS(t) index has a maximum during the solar activity minimum.
Before the high cycles of Wolf numbers
W(t) this index is even
higher than before the cycles with lower intensity. As can be seen
from Figure 1b,
K(t) = 1/N CRS(t) is ahead of
W(t) by
approximately 5.5 years. Therefore the
K(t) = 1/N CRS(t) index can be used for the purposes of prediction, like other
indices characterizing the topology of the large-scale magnetic
field. The correlation coefficient between maximums of parameters
W max and
K max is equal to
R=0.89. The regression
formula can be presented in the form
W max = 2.47 
104 K max - 52. The forecast for the cycle 24 by this index
comprised
W max =130 16.
2.3. Area of High-Latitude Unipolar Zones
Apz(t)
[9] During the periods of sunspot activity minima the magnetic
fields of solar poles have different polarities. Near the poles
there are large unipolar zones on the background of which coronal
holes can exist. The area of the high-latitude unipolar zones can
be calculated from Ha charts. The area of the polar
zones shows the latitudinal variations of the zonal boundary of
large-scale magnetic field. Figure 1a shows the index of unipolar
zone area
Apz at the latitudes higher than
35o expressed in
units of area
1016 m2 [Tlatov and Makarov,
4]. The local maxima of the
Apz index are seen
to occur during the periods of solar activity minima. The largest
area of the unipolar zones was observed at the activity minimum
before the activity cycle 19. The correlation coefficient between
the index smoothed with a window of 1 year and the annual mean
Wolf numbers
W is
R 
0.93, the
Apz index being 5 years
ahead. The regression between the
Apz index and Wolf numbers can
be written in the form
W max = 2.72 Apz - 523.9. The
amplitude of cycle 24 can be estimated to be about
W max =115 13.
2.4. Neutral Line Length
L(t) as an Activity Index
[10] There are several morphological characteristics used to
describe the solar magnetic field topology. One of them is the
total length of magnetic neutral lines
l(t) on the Ha synoptic chart which was calculated for the period from 1887 till
2005 in units of the solar radius. It has been shown that the
value of
l(t) changes in 11-year cycles. The trend it had over 9
cycles increased it by a factor of 1.3 during the period from 1915
till 1999 [Makarov and Tlatov, 2000b].
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Figure 2
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[11] To eliminate the trend, a new value, i.e., the index of the
background magnetic field activity in the form
was introduced. It demonstrates the 11-year topology of the solar
magnetic field (see Figure 2). A comparison with the curve of the
Wolf numbers
W(t) for this period shows that the
L(t)
index
changes in antiphase to
W(t) and is ahead of the 11-year cycles
of
W(t) by, on the average, 5.5 years. One can see from
Figure 1c that the
L(t) index increases with increasing of the
solar activity during the period 1887-1960 and then sharply
decreases before the sunspot activity cycle 20. Though the
activity expressed by Wolf numbers slightly increases during
cycles 20, 21, and 22, the response of the
L(t) index to this
increase is weak. Probably it is caused by utilizing for these
cycles of the synoptic charts of P. McIntosh created in somewhat
different system. The correlation coefficient between maximums of
parameters
W max and
L max is equal to
R=0.94. The
regression formula can be presented in the form
W max = 1.07 104 L max + 53. The new
L(t) index can be used for
the purposes of prediction of the sunspot activity cycles because
the maximum
L(t) magnitude is observed before the highest
sunspot activity cycles.
2.5. Correlation Between Polarities of the Large-Scale
Fields of the Northern and Southern Hemispheres shape R(t)
[12] Synoptic H
a charts give information on the sign
of the background large-scale magnetic field. The background
magnetic field in polar areas has typically the same sign as the
polarity of the global dipole magnetic field of the Sun. At middle
latitudes, there are fields of different signs, but in the area of
equal longitudes the large-scale background fields of the northern
and southern hemispheres have the same or opposite polarities.
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Figure 3
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[13] It is possible to introduce the value of the average
magnetic field polarity
P(t) averaged in the latitude interval
40o and
10o longitudinal interval of the meridian
net. The central meridian position in the Carrington coordinates
is unequivocally related to the Sun's revolution time. Therefore
the obtained series for the northern and southern hemispheres are
the functions of time
P N(t),
P S(t). For these series, the
correlation analysis for the period from 1905 till 2003 has been
carried out. The correlation
R(t) index was calculated by using
a spectral window having a width of about one year. Then the
window was moved along the series, and the calculations were
repeated. It has been found that on the whole the correlation
index
R(t) is positive. This means that the polarities of the
large-scale magnetic fields in the middle-latitude zone
(40o) of the northern and southern hemispheres have mostly
the same signs. The highest correlation was found to take place
during the activity minimum. The correlation
R(t)
index reached
maximum values during the activity minimum prior to the
high-intensity sunspot cycles [Tlatov and Makarov,
4]. This phenomenon is also well pronounced for
the centennial cycle. The maximum in the
R(t) index is observed
before the most intense 19 cycle of solar activity
(Figure 3).
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