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RECONSTRUCTION OF THE SOLAR LARGE-SCALE MAGNETIC FIELD MAIN CHARACTERISTICS IN THE 20-TH CENTURY
E.V. Miletsky and V.G. Ivanov Central astronomical observatory RAS; solar1@gao.spb.ru

Abstract Set of models are constructed, which connect yearly averaged values of main energetic characteristics of the large-scale solar magnetic field by Stanford observation in 1976-1989 with analogous characteristics obtained from synoptic H-alpha charts and with indices of solar and geomagnetic activity. The constructed models have sufficiently high accuracy and small number of input variables. Basing on the obtained models, reconstruction of time series of three main energetic components of large-scale solar magnetic field in 1915-1975 years is made. The analysis of long-term (more than 11 years) variations of the reconstructed series provides base for statement that since 1915 and until the end of the fifties the components of large-scale magnetic field, corresponding both to closed and to open configurations, as well as indices of sunspot and geomagnetic activity, have clearly pronounced tendency to grow. It allows making conclusion about general increasing of solar magnetic fields intensities in the time interval under consideration, which is related, probably, with growth phase of secular cycle of solar activity. Keywords: solar activity, solar cycle, large-scale solar magnetic fields Searching for regularities in variations of the solar magnetic field characteristics on decennial and centennial time scales is a task of great importance for determination of long-term changes in the internal structure of the Sun and clarification of nature of solar activity. Authors of several papers [1,2], which were published recently, came to conclusion that the average magnitude of heliospheric magnetic field, which is determined by open magnetic field configurations, doubled in the 20th century. However, this conclusion was criticized [3], with referring to data of the large-scale magnetic field direct observations, which is available since 1967. On the other hand, the authors of the paper [4] explain the centennial increasing of the solar magnetic flux by growth of polar zone area that is occupied by a magnetic field of a certain polarity during sunspot activity minima. It is known that processes of evolution and interaction of magnetic fields on the Sun is related with various manifestations of solar activity, which, in its turn, is described by corresponding indices (characteristics). So, in particular, in series of investigations [5-9] it was found that certain characteristics of the global (large-scale) magnetic field (GMF), obtained from data of line-of-sight magnetic field component observation in Stanford, proves to be related with set of solar activity indices. Such links were found for series of solar rotation averages during 21st and 22nd cycles of solar activity. In the present paper we find relations between main parameters of the GMF and another solar characteristics during solar cycles 21 and 22, reconstruct, on this base, the annual averages of these parameters in 1915-1975 and estimate, by means of these reconstructions, their centennial trends. As input data for description of the GMF we use the spherical harmonic coefficients Ylm [10], which are published on the site of Wilcox observatory [11]. These coefficients were obtained from observations of the line-of-sight component of the solar photospheric magnetic field. For each of 241 solar rotations in 1976-1994 the decomposition coefficients

g

lm

and

harmonics is equal to 101. However, they are not independent. In order to separate groups of the harmonics

hlm , where 0

with qualitatively different behavior, we made cluster analysis of these harmonics. As a result, three sets of harmonics were separated -- "quasidipole" E13 (l=1,3, m=0), "intermediate" E5 (l=5, m=0) and "quasiquadrupole" ELM (all other harmonics). We took as set of input variables the sum of powers of the corresponding harmonics. The behaviour of these modes is presented in Fig.1. As one should expect, ELM evolves in phase with the sunspot cycle, E13 -- in antiphase, E5 has intermediate behavior. Having information about variations of each of these modes, one can describe the main features of the solar magnetic field evolution. It is known [12], that the energy of individual GMF components can be expressed by an expression

P0l =
for zonal modes with m=0 and by

l +1 2 gl0 , 4l + 2
2 lm

Pl =



l

m =1

l +1 (g 4l + 2

2 + hlm ) ,

for non-zonal modes with m>0, where l=1,...,9. Therefore, the energy of the quasidipole mode 13 = P01 + P03, where P01 is energy of dipole (l=1, m=0) (Fig.1a), and P03 is one of octopole (l=1, m=0) (Fig.1b), the energy of the intermediate mode is 5 = P03 (l=1, m=0) and the energy of the quasiquadrupole mode is

ELM = P04 +


l =6

9

P0l +


l =1

9

Pl .

Since 1976 the coefficients glm hlm were calculated from direct observations of solar magnetic field. Recently these coefficients were also reconstructed by data of synoptic Halpha charts for wider period [13,14]. We used the data presented in [15] for calculation of these three modes E13, E5 and ELM during 1915-1979 (we shall denote them below as iE13, i0E5 and i0ELM correspondingly). Besides, we employ following data for period 1976-1989: a) average absolute values of the line-of-sight magnetic field component in polar (with latitude >55o) zones of the Sun (PMF) from Wilcox observatory (Stanford) [11]; b) averaged over solar disk absolute values of the line-of-sight magnetic field strength (MX) from NSO (Kitt Peak) [16]; c) absolute values of the interplanetary magnetic field strength (IMF) from OMNIWEB [17]. The solar activity for this investigation is presented by the following series of annual average indices for period 1915-1989: the total sunspot areas (SA), Wolf numbers (W) and the number of polar faculae (PF). AA-index is selected as a characteristic of geomagnetic activity for this period. All series are normalized to zero mean and unit dispersion. The Stanford series of GMF observations started as late as in 1976. Hence, for determination of long-term evolution of the GMF we reconstructed the three aforementioned energy modes E13, E5 and ELM, as well as the series PMF, MX and IMF, for the time interval 19151975. To find these indices, we used Group Method of Data Handling (GMDH) [18-21], which was discussed in detail in papers [8, 9, 21]. On the first stage we found by this method polynomial models, which optimally (in terms of a certain criterion) link each of the reconstructed characteristics of the GMF (as output parameter of the model) with one or several indices of solar and geomagnetic activity i0E13, i0E5, i0ELM, SA, W and AA (as input parameters of the model). The obtained optimal models (for normalized variables) are schematically presented in Fig.2. The names of output "model" variables (with letter "m") are outlined by ovals, and the attached arrows denote input variables selected by GMDH method. The regression coefficient are placed near the arrows, and in brackets correlation coefficients between the model and the real series are put down, which show quality of the corresponding model. First of all, we must


note, that all models proved to be linear and include only 1-2 input variables. Nevertheless, the accuracy of the models is rather high, that gives promise good quality of the subsequent reconstruction. Then, with use of the obtained models and the known for years 1915-1975 values of the indices, which play role of input parameters of the models, the reconstruction of GMF characteristics was made. Fig.3 shows the plots of the time series of the normalized reconstructed characteristics. Among them there are pairs that have some similar features, e.g. the quasidipole mode (E13) and the polar magnetic field (PMF), the quasiquadrupole mode (ELM) and the average strength (total flux) of magnetic field (MX), the interplanetary magnetic field (IMF) and the geomagnetic activity (AA) (observed values), the intermediate GMF mode (E5) and the same mode obtained by H-alpha charts (i0E5) (observed values). The good agreement between the observed and the reconstructed series is an evidence of sufficiently good quality of the reconstruction. For additional check of the applied method we made "reconstruction" of Wolf numbers (W) for years 1915-1975 and then estimated the difference between the observed series and the reconstructed one. For this purpose the model expressed by equation W = 0.85·ELMm (for normalized variables) was obtained for time range 1976-1989. This model, having accuracy r=0.97, was used for reconstruction of W series during 1915-1975. The similarity of the "reconstructed" and the observed series turned to be rather high (r=0.95), being additional argument for reliability of the reconstruction. On the next stage we select long-term (trend) components in all observed and reconstructed series during the entire interval 1915-1989. To obtain the trends we used 7-point smoothing with harmonic weights, which provides result similar to ones for usual 11-points adjacent averaging, but the resulting curves describe trend components in a better way. The graphs of the obtained series are plotted in Fig.4. On all panels of the figure values, corresponding to factors of increasing from minimal value to point of inflection, are placed near names of the series and in brackets square roots of these factors are shown. The presented data allow us to make quite definite conclusions. It can be seen that, in spite of similarity of 11-year cycles in indices PMF and E13, their long-term components are essentially different. While the quasidipole mode E13 exhibits explicit growth before beginning of the fifties, the polar magnetic field strength does not significantly increase during interval under consideration. This fact agrees well with the conclusion of the paper [4] about absence of significant increasing of the polar magnetic field absolute value in the 20th century. The conclusion of the authors of this paper about increasing of areas of the solar polar zones, occupied by magnetic field of a certain sign, is confirmed by increasing of the intermediate mode power E5, presented on the lower panel of Fig.4. If to interpret this fact geometrically, it means decreasing of the three-zone structure in each of hemispheres and increasing of the dipole and octopole structures, which must lead to growth of the corresponding areas. The quasiquadrupole mode ELM (second panel of Fig.4), which puts the largest contribution into the total energy of the GMF, has period of increasing since beginning of the twenties until the middle of the fifties (the maximum of cycle 19) and since the middle of the sixties until the end of the eighties (the maximum of cycle 21). The index MX has similar behavior, but the increasing factors for these two characteristics are different: 2.7 for ELM and 1.4 for MX. However, taking into account the fact that the energy mode ELM is a quadratic value and calculating the square root of the increasing factor, we obtain value 1.64, which is fairly close to the increasing factor of MX. The reconstructed interplanetary magnetic field IMF also has similar behavior and similar increasing factor 1.4. Fig.5 shows plots of the long-term components of the observed solar activity indices, separated by the same method. Comparing character of long-term variations of indices SA


and W (presented on the upper panel of Fig.5) with the reconstructed mode ELM, one can see high similarity between them. The increasing factors 2.1 (1.45) for SA, 2.6 (1.61) for W and 2.7 (1.64) for ELM are also close to each other. Incidentally, this similarity shows that the indices SA and W, as well as the mode ELM, are approximately quadratically depend upon magnetic field strength. On the other hand, comparison of evolution and the increasing factors of the series AA and IMF (second panel of Fig.5) indicates their similarity and linearity with respect to the magnetic field strength. Variations of the mode i0E13 are also similar, but have an advance of about half of cycle with respect to the rest of the curves presented on Fig.4. Therefore, one can conclude that the energy modes E13 and ELM, which determine the total energy of the GMF, exhibited long-term growth in the first half of the 20th century. The characteristics of magnetic field MX and IMF had similar increasing. It is not the case for the polar magnetic field strength PMF and for the intermediate mode E5, but the latter makes very small contribution to the total energy of the GMF. This conclusion is in agreement with the statements of papers [1,2]. But in the second part of the 20th century decreasing, followed by some increasing, took place (see curves in Fig.4 and 5). If one averages the indices only in the time interval from the sixties to the eighties (where data of direct observations are available), these two tendencies compensate each other. From this viewpoint the statement of paper [3] about absence of significant solar magnetic field growth in this epoch can be understood. We should also remark that, since the variations of long-term component of traditional indices of solar activity SA and W in the 20th century fit well with analogous variations of the GMF total flux (see Fig.4 and 5), information about behavior of these indices on such time scales allows to estimate corresponding variations of the GMF. Wolf numbers (W) and aa-index (AA) are available for a long time interval, and we can determine long-range components of these indices since 1900 (see two lower panels of Fig.5). The dotted lines on these panels are trends obtained by common 11-point averaging, and the continuous lines are trends, obtained by 7-point harmonic smoothing. First of all, we can see that both obtained curves are very similar. However, on the 11-point curve one can see residual short-range oscillations, which slightly distort the shape of the long-term variations. The increasing factors for AA and W between minimal and maximal values of their trend components, which are marked in Fig.5 by vertical dashes, were calculated. For W the increasing factors, determined from 11-year cycles maxima and from trend, are equal to 3.00 and 2.96 respectively. For aa-index the factors, determined by local maxima, minima and trends are equal correspondingly to 1.95, 3.17 and 2.14. Taking into account that on the interval of reconstruction the magnitude of the total GMF flux increased 1.4 times, and the magnitude of aa-index became 1.5 times as large (i.e. their growth is approximately the same), and knowing that on the whole interval (from the beginning of the century) AA increased 2.14 times, one can conclude that increasing factor for MX is about 2. It agrees well with the conclusion of papers [1, 2] about "doubling of magnetic flux" in the first half of the 20th century. In our opinion, there is good reason to suppose that such an increasing is caused by the growth phase of the solar secular cycle. The work is supported by grant INTAS 01-550.
References [1] Lockwood M., Stamper R., Wild M. N. / Nature. 1999. V.399. P.437 [2] Solanki S. K., Schussler, M., Fligge M./ Astr. Astrophys.. 2002. V.383. P.706-712. [3] Arge C. N., Hildner E., Pizzo V. J. / Astr. Astrophys. 2002. V.383. P.706-712.. [4] Makarov V.I., Obridko V.N., Tlatov A.G. / Asronomicheskij zhurnal. 2001. V.78. P. 859-564. [5] Obridko V.N., Shelting B.D. / Solar Phys. 1992. V.137. P.167-177. [6] Ikhsanov R.N., Miletsky E.V. / Proceedings of conference "Large-scale structure of solar activity". St.Petersburg, 1999.


[7] Ikhsanov R.N., Miletsky E.V. / Izvestija GAO. 2000. V.215. P.69. [8] Miletsky E.V., Ivanov V.G. / Proceedings of conference "The Sun in epoch of magnetic field reversal". St.Petersburg. 2001. [9] Ivanov V.G., Miletsky E.V. / Proceedings of conference "Solar activity and cosmic rays in epoch of magnetic field reversal". St.Petersburg. 2002. P. 195. [10] Hoeksema J.T., Scherrer P.H. / Solar magnetic Field: 1976 ­1985, WDCA, Boulder, 1986. [11] http://quake.stanford.edu/~wso/ [12] Mihajlutsa V.P. / Issledovanija po geomagnetizmu, aeronomii i fizike Solntsa.1989. Iss.87. P.199206. [13] Makarov V.I., Tlatov A.G./ in A.Wilson (ed.), Magnetic Fields and Solar Processes. Proc. 9th Europ. Meeting on Solar Physics. 1999. P.125. [14] Obridko V.N., Shelting B.D. / Solar Phys. 1999. V.184. P.187. [15] http://www.izmiran.ru [16] http:/www.nso.edu/ [17] http://www.omniweb.edu/ [18] Farlow, S. J. (ed.), Self-organizing Method in Modelling: GMDH Type Algorithms. Statistics: Textbooks and Monographs, 54, 1984. [19] Madala,H.R., Ivakhnenko,A.G. Inductive Learning Algorithms for Complex Systems Modeling. CRC Press Inc., Boca Raton, 1994. [20] Ivahnenko A.G., Yurachkovskij Yu.P. Modelling of complex systems by experimental data. Moscow. Radio i svjaz. 1987. 115 P. [21] Miletsky E.V., Ivanov V.G., Nagovitsyn Yu.A. / Solnechno-zemnaja fizika. 2002. Iss.2. P.137139.


E13

E5

ELM

1980

1985

1990

1995

Fig. 1. The behaviour of the quasidipole (E13), intermediate (E5) and quasiquadrupole (ELM) modes of the GMF.

Fig. 2. The optimal polynomial models of links between indices of solar and geomagnetic activity and characteristics of the GMF. See the text for comments.


3 2 1 0 -1 -2

E13m PMFm

1910 1920 1930 1940 1950 1960 1970 1980 1990

2 1 0 -1

ELMm MXm

1910 1920 1930 1940 1950 1960 1970 1980 1990

2 1 0 -1 -2

IMFm AA

4 3 2 1 0 -1 -2

1910 1920 1930 1940 1950 1960 1970 1980 1990

E5m i0E5

1910 1920 1930 1940 1950 1960 1970 1980 1990

Fig. 3. The normalized reconstructed characteristics of the GMF


Polar magnetic field(PMF)
0.5 0.0 -0.5 1.0

2.5(1.58) E13

1910 1920 1930 1940 1950 1960 1970 1980 1990

-0.5 1.0 0.5 0.0 -0.5 -1.0 2.0 1.5 1.0 0.5 0.0 -0.5

Total Magnetic Flux(MX) 1.4 0.5 ELM(L,M=1-9)(2.7) 0.0 ELM (L,M=1-9) 2.7(1.64) ELM
1910 1920 1930 1940 1950 1960 1970 1980 1990

Interpl. mag. field(IMF) 1.4

1910 1920 1930 1940 1950 1960 1970 1980 1990

E5

E5-H-alpha

1910 1920 1930 1940 1950 1960 1970 1980 1990
Fig. 4. The long-term components of all observed and reconstructed series of the GMF on the entire time interval 1915-1989.


1.0 0.5 0.0 -0.5 1900 1.5 1.0 0.5 0.0 -0.5 -1.0 1900 32 28 24 20 16 12 8

SA 2.1(1.45)

W 2.6(1.61)

2.7(1.64) ELM
1920 1940 1960

i0E13 2.6(1.61)

AA 1.5

1980

2000

IMF 1.4
1920 1940 1960 1980 2000

aa-index

(1.95)

(3.17)
1920 1940 1960

(2.14)
1980 2000

1900 200 150 100 50 0 1900

W olf number 3.00 2.96(1.72)

1920

1940

1960

1980

2000

Fig. 5. The same as in Figure 4 for the solar and geomagnetic indices.