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Review of the geomagnetic secular variations on the historical time scale
Physics of the Earth and Planetary Interiors, 20 (1979) 83-95
83
© Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
REVIEW OF THE GEOMAGNETIC SECULAR VARIATIONS ON THE HISTORICAL TIME SCALE TAKESI YUKUTAKE
Earthquake Research Institute, University of Tokyo, Bunkyo-ku, Tokyo (Japan)
(Accepted for publication in revised form April 24, 1979)
Yukutake, T., 1979. Review of the geomagnetic secular variations on the historical time scale. Phys. Earth Planet. Inter., 20." 83-95. In view of the classification of the geomagnetic field into its axisymmetric and non-axisymmetrie parts, studies of geomagnetic secular variations on the historical time-scale ate reviewed. The westward drift of the geomagnetic field, which is one of the most conspicuous features of its secular variation, is examined first. The nOn-axisymmetric field during the past several hundred years can be well approximated by the superposition of two constant-magnitude fields, a standing and a drifting field, whose lifetimes are supposed to be longer than 1000 years. It is pointed out that the seetorial term of the non-dipole standing field is small compared with the drifting one. The lick of the n = m = 2 term of the standing field is particularly remarkable. On the other hand, the equatorial dipole field is likely to consist of two components which are both drifting. One drifts westwards with a normal velocity and the other eastwards with a small velocity. Besides the pronounced westward drift in an east-west direction, the poleward movements of particular loci of the secular variation are noted. This may, however, be related to the rapid growth of the axisymmetrie quadrupole field. The time variation of the dipole field is briefly examined. As far as the data on the historical time scale are concerned, an antiparallel relationship seems to exist between the variations in the dipole and the quadrupole field. As the dipole moment decreases, the magnitude of the quadrupole moment increases. Finally, characteristic oscillation periods of the dipole field are examined. Although the data are few, a 60-70-year period, a 400-600-year and a 8000-year period emerge as the dominant periods.
1. Introduction Geomagnetic secular variations originate mostly in the Earth's liquid core. Assuming that the magnetic field is frozen in the conducting core, geomagnetic secular variations are sometimes regarded as manifestations o f fluid motion near the surface o f the core (Kahle et al., 1967a,b; Rikitake, 1967; Booker, 1969). On the other hand, some features o f the geomagnetic field, such as represented b y the equatorial dipole component, are considered to be related to the intrinsic process o f the geodynamo that creates the dipole field (Braginskiy, 1964; Pekeris et al., 1973; Krause, 1977). Geomagnetic secular variations provide, therefore, some o f the basic data for investigating the magnetohydrodynamic processes taking place in the Earth's core.
It is well known that the rotation o f the Earth exerts a predominant effect on core motions. Such highly axisymmetric motions as a shear motion around the rotation axis are supposed to be generated b y the Earth's rotation. Besides these axisymmetric motions, non.axisymmetric motions are required to maintain the geomagnetic field (see Roberts, 1971). The non-axisymmetric motions are expected to induce the non-axisymmetric magnetic field when they interact with'an axisymmetric field such as that o f the dipole. Therefore, examination o f the non-axisymmetric field should provide information on the nonaxisymmetric parts o f the fluid motion. The secular variation o f the non-axisymmetric field will be reviewed first. One o f the outstanding features o f the secular variation that has been known
84 for centuries is the westward-drift phenomenon. Besides a westward-drifting component of the field, a standing component or one that is drifting eastwards with a small velocity has been noticed. The predominance of movement of the geomagnetic field in the east-west direction may be contrasted with the equatorward migration of the sunspot field. Therefore, it is of interest to examine whether any movement in the north-south direction can be discernible in the geomagnetic secular variation. It is pointed out that a poleward movement of particular foci of the geomagnetic secular variation is likely to exist. But this might be related to rapid growth of the quadrupole field rather than to real migration of the particular patterns of the field. Finally, the time-variation of the axisymmetric field associated with the dipole and the quadrupole fields is briefly mentioned.
2. Spherical harmonic analysis of the geomagnetic field Spherical harmonic series have been commonly used to express the geomagnetic field since Gauss exploited the technique and made an analysis of the field for 1835. Fritsche (1899) applied this technique to earlier data back to 1600. Since around the end of the 15th century, many data on the magnetic declination and inclination were acquired with the development of navigation, whereas no instrumental observations of magnetic intensity were made before the 19th century. Fritsche's analyses are based on three components, the north, the east and the vertical downward components, which were computed from declination and inclination on an implicit assumption that the intensity before the 19th century had been the same as that for the 19th century. Without knowledge of the magnetic intensity, one can only determine the relative magnitudes of the Gauss coefficients of the spherical harmonic series from the data of declination and inclination. Braginskiy (1972a) and Barraclough (1974) avoided this difficulty by extrapolating the variation in the dipole moment backwards with a specific algebraic expression, such as a quadratic form, and determined sets of, Gauss coefficients back to 1550. Without assuming the dipole term, however, it is possible to determine a complete set of Gauss coefficients with absolute magnitudes, provided that the intensity data are
available at a single point (see Appendix). Selecting places where both archeomagnetic and recent data on intensity are available, Yukutake (1971) and Benkova et al. (1974) estimated the intensity values for the 17th and I8th centuries, a time when instrumental observations of the intensity had not been initiated. The intensity data thus interpolated were incorporated with the declination and the inclination data to determine the Gauss coefficients up to the degree and the order of 4. In this process, the absolute magnitudes of the Gauss coefficients are linearly dependent on the • interpolated intensity. Since the archeomagnetic intensity data used for estimating the intensity value for the specific epochs contain many more uncertainties than those of declination and inclination, the magnitudes of the coefficients thus determined are less accurate than the relative magnitudes between the coefficients. In spite of these difficulties, however, the analyses conducted by various authors for the earlier epochs are still sufficiently consistent with each other to analyse the long-term variations over centuries, as can be seen in Fig. 4, where trajectories of the equatorial dipole component computed from different analyses are plotted in the coordinates of the cosine (gl) and the sine (h l)coefficients. The spherical harmonic coefficients thus obtained will be used to analyse the secular variations of the axisymmetric and the nonaxisymmetric field separately.
3. The westward drift of the geomagnetic field The largest component of the non-axisymmetric field is the equatorial component which causes the best-fitting geocentric axial dipole to deviate from the geographic axis. Figure 1 shows the movement of the geomagnetic north pole since 1550. The locations of the pole were plotted from the results of Benkova et al. (1974) and Yukutake (1971) together with those of analyses after 1829. Although only two sets of the analyses are plotted here for the earlier epochs, the other analyses give almost the same polar paths. It is evident from this figure that the geomagnetic pole has rotated clockwise. Since the clockwise rotation of the inclined dipole is equivalent to the westward drift of the equatorial dipole, the data of Fig. 1 imply that the equatorial dipole has been drifting
85
/2e't Fig. 1. Locations of the geomagneticnorth pole for various epochs since 1550. Clockwiserotation of the pole position is seen. Open circlesare taken from the analysis by Benkova et al. (1974); solid circles from Yukutake's analysis for 1600 to 1770, and from various authors for 1829 to 1975. westwards since the 17th century, although the speed has been apparently slower since the beginning of the 19th century. It is well known that the non-dipole field, the residual field after the dipole component is removed from the observed geomagnetic field, drifts westwards (see Bullard et al., 1950; Yukutake, 1962). Figure 2 shows the vertical component of the non-dipole field for 1770 and 1965 synthesized from the Gauss coefficients (Yukutake and Tachinaka, 1968a; Yukutake, 1971). By comparing the similar charts for different epochs, 1907 and 1945, Bullard et al. (1950) estimated a mean drift rate of the whole pattern of the non-dipole field to be 0.18 ° year -1. Figure 2 shows, however, that there are some non-dipole anomalies that have not been noticeably displaced during the period concerned. Therefore it is inconceivable that the whole pattern of the non-dipole field drifts westwards with a uniform rate. Comparison of the two charts in Fig. 2 suggests rather that there exist different types of non-dipole anomaly. One is the anomaly that displaces its location with time. The African negative anomaly belongs to this type. It was located in the western Indian Ocean in 1770 and reached the west coast of Africa in 1965. The mean rate of westward drift of this anomaly was estimated to be
0.28 ° year -1 by Yukutake and Tachinaka (1968a). The other type of anomaly is that standing almost at the same place, such as the anomaly that covers North America. A positive anomaly with its center in Mongolia has not moved either, but grown up rapidly to become the largest non-dipole anomaly during the past 400 years, with a mean growth rate of 53 nT year -1 (Yukutake and Tachinaka, 1968a). It is possible to consider that the individual anomalies change independently as described above. However, it is also possible to interpret the complicated features of the secular variations in a completely different way based on a steady field model (Yukutake and Tachinaka, 1969). Consider two steady fields as shown in Fig. 3. One is the field standing at the same place, and the other is the field of which the whole pattern drifts westwards with a constant velocity of 0.3 ° year -1. It is demonstrated by Yukutake and Tachinaka (1969) that simple superposition of the two kinds of field represents well the various aspects of the observed geomagnetic secular variations. Not only has the steady movement of the African anomaly been successfully approximated by this steady drifting-standing field model, but so has the rapid growth of the Monogolian anomaly.
4. Eastward drift of the equatorial component Separation of the drifting from the standing field was originally made for the time variation of the individual spherical harmonic components. Figure 4 shows to what extent the standing-drifting field model can approximate the observed variation in the equatorial dipole component, i.e. g] and h~. In this figure, the time variations in the equatorial dipole component are expressed by successive points in the coordinates ofg~ and h[. The results of five analyses by Fritsche (1899), Yukutake (1971), Braginskiy (1972a), Barraelough (1974) and Benkova et al. (1974) are all plotted, together with recent analyses, covering the period from 1550 to the present. As in Fig. 1, westward drift, or clockwise rotation of the equatorial component, is seen. In Fig. 4 the standing field is represented by a vector from the origin of the coordinates to the point X, while the drifting field is represented by a vector rotating clockwise around the point X. Therefore, the circle in Fig. 4 represents the
86
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88
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the drifting field changes its amplitude during its rotation, the trajectory will expand or shrink across the circle expected from the steady model. Accordingly it would be possible to adjust amplitude change so that the model may fit the observations. However, the adjustment seems too arbitrary and would bring unnecessary complications into the model. The simplest way of adjusting the discrepancies is to introduce a component that drifts eastwards steadily. Replacing the standing field b y a field drifting eastwards, Braginskiy (1972b) and Yukutake (1976) approximated the variation in the equatorial dipole component by superposition of two components drifting steadily in an opposite direction, eastwards and westwards, with constant amplitudes. The ampli-
-FR o BR "BL Fig. 4. The equatorial dipole component,g[ and hi, for various epochs obtained by different analyses, starting with either 1550 or 1600 A.D. For convenience sake, each epoch is labelled near an approximate position of the mean of the different analyses for the specific epoch. The two vectors on the diagram represent the standing and drifting field components. The vector from the origin to the point X is the standing component, and the vector from the point X to the circle the drifting component which rotates clockwise around the point X with a velocity of 0.293 = year-1 . Accordingly the circle is the trajectory of the equatorial dipole component expected from the standing-drifting field model. Solid circles connected by solid lines are from Yukutake (1971); open circles from Benkova et al. (1974); triangles from Fritsche (1899); open squares from Braginskiy (1972a) and solid squares from Barraelough (1974). Solid circles for the epochs from 1829 to 1965 are taken from the analyses based on the recent instrumental observations.
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trajectory around which the equatorial dipole component is expected to move for the standing-drifting field model. Although the general trend is well approximated by the standing-drifting field model, if one scrutinizes the detailed features, it may be noticed that there exist significant deviations o f the model from the observations. The recent trend of the variation is obviously different from the circular trajectory expected from the model. One possible way o f reducing the deviations is to assume time-variation in the amplitude o f either the drifting or the standing field. If
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Fig. 5. Trajectories of the equatorial dipole component obtained by three different models. One is the standing-drifting field model whose trajectory is shown by a circle. The other two, Braginskiy's and Yukutake's model, consist of two components drifting in the opposite direction with the velocities and the amplitudes listed in Table I. Open circles connected with dotted lines are after Braginskiy's modal, and open circles with solid lines after Yukutake's model. Rerults of analyses are also ~own for comparison with the models. Solid squares with solid tines are the rerults of Braglnskiy's analysis, and solid circles with solid lines arc those of Yukutake
89 TABLE I Velocities and amplitudes of the components drifting westwards and eastwards for the equatorial dipole field. Vw and A w represent the velocity and the amplitude of the westward drifting component, while Ve and A e are those of the eastward drifting one
nT
• Standing field
Braginsky Yukutake
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3880 2850
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2780 3616
tudes and the velocities were obtained as listed on Table I, where A e and Ve represent the amplitude and velocity of the eastward component, respectively, and A w and Vw those of the westward component. The eastward velocity is taken as positive. Figure 5 shows the trajectories of the equatorial dipole component obtained by these models. Open circles connected by solid lines are from Yukutake's model, and open circles connected by broken lines are after Braginskiy's model. The circular trajectory with small open circles is from the standing-drifting field model. Solid circles and solid squares indicate the observations. An improvement of the fit is clearly seen in the revised models. The models that include an eastward-drifting component are applicable not only to the equatorial dipole component but also to other higher harmonic components. However, the approximation is not improved as much as that for the equatorial dipole component. Therefore, the standing-drifting field model for the higher harmonics will be retained.
5. Lack of sectorial terms in the standing field Regarding the structure of the standing and drifting field, one interesting feature is the lack of sectorial terms in the standing field. In Fig. 6 the amplitude of the standing field as well as that of the drifting field is plotted against the degree and the order of the spherical harmonics. There are minima of amplitudes at n = m = 2 and 3 for the standing field. The amplitude is also small with n = m = 4. Except for the equatorial component, n = m = 1, the sectorial terms are smaller for the standing component than for the drifting one. This feature is particularly conspicuous
o Drifting field
= m=l
1
2 2
3 1
3 2
3 3
4 1
4 2
4 3
4 4
Fig. 6. Amplitudes of standing and drifting components of the individual spherical harmonics. Open circles represent the amplitudes of the drifting field, and the solid circles those of the standing field except for the equatorial dipole component, n = m = 1. For the equatorial dipole component the amplitude of the component drifting eastwards is plotted. Small amplitudes are noted for the sectorial terms of the standing field, n = m = 2, 3, 4.
with the term n = m = 2. So far no separation of the standing field from the drifting field has been made for the components higher than n = m = 4. It is important to examine whether the lack of the sectorial terms can be seen with the standing field of the higher components. It is also interesting to note that the amplitude of the drifting field for n --- 3 and m = 2 is small.
6. Lifetime of the drifting field If proper values are taken as the parameters of the core, such as 3 X 10 -6 emu for the electrical conductivity, the free decay time of the non-dipole field, e.g. n = 2, can be as long as 7000 years (Yukutake, 1968). Therefore, unless vigorous fluid motions destroy the field, the drifting and the standing field may have a lifetime that covers the whole historic age.
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91
Bullard et al. (1950) considered that the non-dipole field was generated by turbulent fluid motions near the surface of the core and suggested that the lifetime would be shorter than a hundred years. If the nonaxisymmetric field is merely a manifestation of the turbulent phenomena taking place near the surface of the core, its lifetime will be relatively short as suggested by Bullard et al. (1950). However, if it is a phenomenon related to the intrinsic dynamo process in some way, it may possibly have a longer lifetime. Comparison of the non-dipole charts in Fig. 2 indicates that the main features of the non-dipole field have persisted over the period of comparison, i.e. 195 years, which is longer than the lifetime suggested by Bullard et al. It is, therefore, highly important to examine how long the non-axisymmetric field has existed. Yukutake (1967) examined the secular variations of declination and inclination at various places with periods of several hundred years, which were collected from a variety of sources: observatory data; results of repeat surveys and paleomaguetic data, He noticed westward drift o f the maxima and the minima of the variations with an average velocity of 0.36 ° year -l. It was recognized that some features, which were traced back to about 1000 A.D., were still propagating, and that the propagation of the maximum deviations of declination is consistent with that of the eastward component obtained from the recent analyses (Yukutake and Tachinaka, 1968b). These data suggest that the drifting field has persisted longer than i000 years.
7. Poleward motion of the magnetic foei and the growth of the quadrupole field Although the movement of the magnetic pattems in the east-west direction is a predominant feature of the geomagnetic secular variation, it will be interesting to examine whether a meridional component exists in the movement of the magnetic field. The northward shift of the eccentric dipole is another pronounced aspect of the geomagnetic secular variation that is frequently cited with the westward-drift phenomenon (Vestine, 1953; also see Rikitake, 1966). Since the distance of the eccentric dipole from the center of the Earth is closely related to the magnitude
of the quadrupole term, gO, the northward shift of the eccentric dipole was considered to be caused by an oscillation of the quadrupole field (Nagata and Rikitake, 1963). Poleward movement of particular foci of the secular variation was suggested by Yukutake and Tachinaka (1968b). Figure 7 shows the rate of change in the vertical component (Z) for different epochs, for 1806 and for 1965. A negative zone dominates the equatorial region. Outside this negative zone, pairs of positive anomalies exist, one in the area of India and the other near the American continent. These positive anomalies seem to have moved polewards with the elapse of time. Figure 8 shows how the centers of the positive foci changed their latitudes with time. Poleward movement is dearly seen. The rate of the apparent poleward migration was estimated to be 0.14 ° year -z for the Indian feel and 0.26 ° year -1 for the American foci. The apparent poleward migration seems to be caused by the expansion of the equatorial negative zone. This zone is well expressed by a change in the quadrupole field. Consequently the expansion of this zone may be understood as a rapid growth of the
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92
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Fig. 9. Time-variationin the quadrupole term, gO. quadrupole field. Figure 9 shows the time-variation in the quadrupole term gO; the results of the spherical harmonic analyses for earlier epochs (Yukutake, 1971) are shown together with those of the recent ones. Since the beginning of the 18th century, gO has been decreasing. Because of the negative sign o f g °, this means that the absolute value o f g ° is increasing. This may perhaps be another way of representing the expansion of the equatorial negative zone.
8. Variations in the dipole field
The most predominant axisymmetric field in the geomagnetic field is the dipole field. The time variation in the dipole coefficient gO since the 17th century is shown in Fig. 10 (Yukutake, 1971). The negative value is taken upwards in Fig. 10 for the sake of convenience to show the change in the absolute value of gO (which is linearly proportional to the axial dipole moment). The general trend is approximated by a parabolic curve with the peak sometime around 1800 A.D., as shown on the diagram. By comparison with Fig. 9, similarity of the general trend between the dipole and quadrupole variations is noted. This feature can also be seen in the results of the analysis of Benkova et al. (1974). In spite of the difficulty already mentioned in determining the absolute values of the Gauss coefficients before the 19th century, Benkova et al. (1974) obtained time-variations in the dipole and the quadrupole coefficients that are similar to those shown in Figs, 9 and 10, although the
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Fig. 10. Time variation in the axial dipole component, gO. In order to see the change in the magnitude of the dipole intensity, negative value is taken upwards. epochs of the peaks are somewhat earlier than in Figs. 9 and 10. The signs o f g ° andg ° are both negative, except for the limited period from the early 18th century through the middle of the 19th century, wheng ° is likely to have been positive, and the magnitude o f g ° increases as the magnitude o f g ° decreases. Therefore, as far as these historical data are concerned, it may be said that there exists an antiparaUel relationship between the variations of the dipole and the quadrupole m o m e n t * On the geologic time scale, however, this antiparallel nature does not seem to exist. As was mentioned in a previous section, the ratio o f g ° and gO is related to the distance of the eccentric dipole from the center of the Earth. The eccentricity is sometimes modeled as an "offset of the dipole" in paleomagnetic studies. The present offset is about 150 km north of the equator. If the change in the quadrupole field is independent of the variation in the dipole field, the ratio of the quadrupole to the dipole term should have equal probability of having either positive or negative sign when there is a change in the dipole polarity. * Verosub and Cox (1971) examined time variations in the magnetic energy contained in the dipole and the non-dipole field separately by making use of the results of spherical harmonic analyses since 1845, and concluded that exchange of energy was taking place between the dipole and the non-dipole field. However, in their analyses, the non-dipole field includes not only the axisymmetric quadrupole field but also the nonaxisymmetrie components. In this text, energy transfer between the axial dipole and the axial quadmpole field is discussed.
93
However, investigation of the dipole offset in paleomagnetism strongly suggests that the quadrupole field has also changed its sign when the dipole field reversed (Cox, 1975), and the ratios of the quadrupole to the dipole term are nearly the same between the normal and the reversed periods (Merrill and McElhinny, 1978). This suggests a positive correlation between the changes in the dipole and the quadrupole field, contrary to the historical time scale variations. A tentative conclusion derivable from these results is that the quadrupole field changes in parallel with the dipole field on a time scale as long as a million years, but that there are shorter period fluctuations that are superposed on this long-term change and behave in the opposite way.
9. Spectrum of the dipole variations The spectrum of the dipole variations will be mentioned very briefly in this section. In order to determine the geomagnetic field intensity in prehistoric times, paleomagnetic intensity techniques have been applied to samples collected from old kilns, potteries and dated lavas. Results of this application indicate that the geomagnetzc field has greatly changed its intensity during the past 9000 years. The equivalent dipole moment was computed from the intensity data on an assumption that the geomagnetic field was dipolar when the sample was magnetized. These dipole moments were averaged for small time blocks giving the curve shown in Fig. 11 (Cox, 1968). A1-
xlO2~.,ouss.cm3
Q
-m~,~-'~
~
I
•
u year(AD)
Fig. 11. Variation in the dipole moment obtained from paleomagnetic data (Cox, 1968).
- 30500
~o
\
- 30000 1900
I
I
I
I
I
I
1950 A.D.
I
Fig. 12. Decreasein the magnitude of the axial dipole component since 1900. Negativevalue is taken upwards, gO obtained from the annual values of the observatory data is plotted. The rate of decrease in the dipole intensity is not uniform.
though the data are still very few and are not uniformly distributed in space, Fig. 11 seems to suggest a large fluctuation of the dipole moment with a period of about 8000 years. The amplitude amounts to about half of the present dipole moment. According to this result, the dipole moment has been decreasing for the last 1500 years. On the other hand, historical data suggest the existence of a maximum in the dipole moment in a recent epoch around 1800 A.D., as seen in Fig. 10. In order to reconcile the historical data with the archeomagnetic results, one must assume a minimum of the dipole moment sometime around 1600 A.D. From this consideration the dipole moment is also presumed to have fluctuations of several hundred years with intensity variations around 3%. It is well known that the dipole moment has been decreasing rather monotonously since the beginning of the 19th century when Gauss made the first spherical harmonic analysis of the geomagnetic field. In order to examine detailed features, spherical harmonic analyses were made of the annual mean values of the selected observatory data for the period from 1900 to 1965 (Fig. 12). It is seen that the rate of decrease in the magnitude o f g ° is not constant with time. If the linear trend is removed, a minimum is observable around 1930 (see Yukutake, 1973b). A spectrum analysis technique, the maximum entropy method, was applied to this time series, and the largest energy content was obtained for the periods of 4 0 - 8 0 years
94 If
Appendix - Spherical harmonic analyses for the
OLIn!
epochs prior to the 19th century
PERIOD
i'g~ y k r
Fig. 13. Amplitude spectrum of the dipole variation. Width of the column represents an approximate range of uncertainty of the characteristic period. The spectrum is tentative at present. CYukutake, 1973a). The amplitude amounts to about 0.4% of the present dipole moment. Applying the same technique to the results of their own analysis, Jin and Thomas (1977) obtained the characteristic period of 66.7 years for the dipole variation. The characteristic period thus obtained agrees well with those widely noticed from the analyses of the time variations in the geomagnetic components observed at various observatories, i.e. 60-70-year periods. Although it is premature to draw conclusions about the spectrum, the above data are summarized very tentatively in Fig. 13. The height of each column represents the amplitude, and the width covers an approximate range of uncertainty of the characteristic period. For the period range from 50 years to 10 000 years, there are at least three distinct peaks, i.e. at the period around 6 0 - 7 0 years, 400-600 years and 8000 years. The existence of these distinct frequencies in the variations of the dipole field could be of fundamental importance in investigations of the magnetohydrodynamic process in the Earth's core.
Acknowledgements The author would like to thank Drs. R.T. Merrill and J.C. Cain for reading the manuscript and giving valuable comments.
It is evident that, from data of declination and inclination alone, a unique solution of the Gauss coefficients is unobtainable. Suppose all the coefficients are multiplied by a factor of k, the intensity and the field components synthesized from them become k-times larger, but the declination and the inclination remain constant, since they are derived from the ratios of two field components. Therefore, the intensity data are indispensable for obtaining the absolute values of the Gauss coefficients. However, once a term, gO for example, is known, a definite set of the Gauss coefficients can be determined. In Yukutake (1971) and Benkova et al. (1974), the m relative values of the tesseral terms, (Gn, H~m ), where G~n = gr~n/gOl, Hrffn =hn/h m oI (m = 0), are determined from declination data, and the zonal terms G° (=g~/gO) from those of inclination. Then gO is estimated so that the synthesized values fit best to the intensity data. In Yukutake (1971), grid point values of the declination at an interval of 10° in latitude and longitude were used. The number of grid point values ranges from 260 to 330 for different epochs; and 11 to 12 values of inclination were used in Yukutake (1971). Since gO is the only quantity to be determined from the intensity data, a reliable intensity value at a single point would be sufficient theoretically to determine a definite set of coefficients. The number of intensity values used'forg ° is 5 in Yukutake (1971) and 9 in Benkova et al. (1974). This implies that the gO values are heavily dependent on the quality of the archeomagnetic data which were used to estimate the intensity for specific epochs. Unfortunately, however, the quality of the archeomagnetic data is not so high as that of the declination and the inclination data that are available since the 16th century. Yukutake (1971) estimated the probable error to be about 500 nT for gO and about 1000 nT forg °. Therefore much room is left for improvement regarding the absolute magnitudes of the coefficients compared with the relative relation between the coefficients.
References Barraclough, D.R., 1974. Sphericalharmonic analyses of the geomagnetic field for eight epochs between 1600 and 1910. Geophys. J. R. Astron. Soc., 36: 497-513.
95 Benkova, N.P., Kolomiytseva, G.I. and Cherevko, T.N., 1974. Analytical model of the geomagnetic field and its secular variations over a period of 400 years (1550-1950). Geomagn. Aeron. (USSR), 14:751-755 (English). Booker, J.R., 1969. Geomagnetic data and core motions. Proc. R. Soc. London, Ser. A, 309: 27-40. Braginskiy, S.I., 1964. Kinematic models of the Earth's hydromagnetic dynamo. Geomagn. Aeron. (USSR), 4: 572-583 (English). Braginskiy, S.I., 1972a. Spherical analyses of the main geomagnetic field in 1550-1800. Geomagn. Aeronom. (USSR), 12:464-468 (English). Braginskiy, S.I., 1972b. Analytical description of the geomagnetic field of past epochs and determination of the spectrum of magnetic waves in the core of the Earth. Geomagn. Aeron. (USSR), 12: 947-957. Bullard, E.C., Freedman, C., Gellman, H. and Nixon, J., 1950. The westward drift of the Earth's magnetic field. Philos. Trans. R. Soc. London, Ser. A, 243: 67-92. Cox, A., 1968. Lengths of geomagnetic polarity intervals. J. Geophys. Res., 73: 3247-3260. Cox, A., 1975. The frequency of geomagnetic reversals and the symmetry of the non-dipole field. Rev. Geophys. Space Phys., 13: 35-51. Fritsche, H., 1899. Die Elemente des Erdmagnetismus fttr die Epochen 1600, 1650, 1700, 1780, 1842 und 1885, und ihre saecularen Aenderungen. St. Petersburg. Jin, R.S. and Thomas, D.M., 1977. Spectral line similarity in the geomagnetic dipole field variations and length of day fluctuations. J. Geophys. Res., 82: 828-834. Kahle, A.B., Ball, R.H. and Vestine, E.H., 1967. Comparison of estimates of surface fluid motions of the Earth's core for various epochs. J. Geophys. Res., 72: 4917-4925. Kahle, A.B., Vestine, E.H. and Ball, R.H., 1967. Estimated surface motions of the Earth's core. J. Geophys. Res., 72: 1095-1108. Krause, F., 1977. Mean-field electrodynamics and dynamo theory of the Earth's magnetic field. J. Ge0PhyS., 43: 421-440. Merrill, R.T. and McElhinny, M.W., 1978. An outline of some of the properties of the geomagnetic field. Abstr. Conf. Origins Planet. Magn., 67-68. Nagata, T. and Rikitake, T., 1963. The northward shifting of the geomagnetic dipole and stability of the axial magnetic quadrupole of the Earth. J. Geomagn. Geoeleetr., 14: 213-220.
Pekeris, C.L., Accad, Y. and Shkoller, B., 1973. Kinematic dynamos and the Earth's magnetic field. Philos. Trans. R. Soc. London, Ser. A, 275: 425-461. Rikitake, T., 1966. Electromagnetism and the Earth's interior. Elsevier, Amsterdam, pp. 1-308. Rikitake, T., 1967. Non-dipole field and fluid motion in the Earth's core. J. Geomagn. Geoelectr., 19: 129-142. Roberts, P.H., 1971. Dynamo theory. In: W.H. Reid (Editor), Mathematical Problems in the Geophysical Sciences. Lectures in Applied Mathematics, American Mathematical Society, Providence, R.I., 14: 129-206. Verosub, K.L. and Cox, A., 1971. Changes in the total magnetic energy external to the Earth's core. J. Geomagn. Geoelectr., 23: 235-242. Vestine, H., 1953. On variations of the geomagnetic field, fluid motions, and the rate of the Earth's rotation. J. Geophys. Res., 58: 127-145. Yukutake, T., 1962. The westward drift of magnetic field of the Earth. Bull. Earth. Res. Inst., 4 0 : 1 - 6 5 . Yukutake, T., 1967. The westward drift of the Earth's magnetic field in historic times. J. Geomagn. Geoelectr., 19: 103-116. Yukutake, T., 1968. Free decay of non-dipole components of the geomagnetic field. Phys. Earth Planet. Inter., 1: 93-96. Yukutake, T., 1971. Spherical harmonic analysis of the Earth's magnetic field for the 17th and the 18th centuries. J. Geomagn. Geoelectr., 23: 11-23. Yukutake, T., 1973a, Power spectrum analysis of the geomagnetic secular variation (Abstract). Int. Assoc. Geomagn. Aeron. Bull., 34: 86. Yukutake, T., 1973b. Fluctuations in the Earth's rate of rotation related to changes in the geomagnetic dipole field, J. Geomagn. Geoelectr., 25: 195-212. Yukutake, T., 1976. A model of the geomagnetic secular variation for the past 2000 years. Abstr. Fail Meeting Soc. Terr. Magn. Electr. Jpn. (in Japanese), p. 154. Yukutake, T. and Tachinaka, H., 1968a. The non-dipole part of the Earth's magnetic field. Bull. Earthquake Res. Inst. (Tokyo Univ.), 46: 1027-1074. Yukutake, T. and Tachinaka, H., 1968b. The westward drift of the geomagnetic secular variation. Bull. Earthquake Res. Inst. (Tokyo Univ.), 46: 1075-1102. Yukutake, T. and Tachinaka, H., 1969. Separation of the Earth's magnetic field into the drifting and the standing parts. Bull. Earthquake Res. Inst. (Tokyo Univ.) 47: 65 -97.
83
© Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
REVIEW OF THE GEOMAGNETIC SECULAR VARIATIONS ON THE HISTORICAL TIME SCALE TAKESI YUKUTAKE
Earthquake Research Institute, University of Tokyo, Bunkyo-ku, Tokyo (Japan)
(Accepted for publication in revised form April 24, 1979)
Yukutake, T., 1979. Review of the geomagnetic secular variations on the historical time scale. Phys. Earth Planet. Inter., 20." 83-95. In view of the classification of the geomagnetic field into its axisymmetric and non-axisymmetrie parts, studies of geomagnetic secular variations on the historical time-scale ate reviewed. The westward drift of the geomagnetic field, which is one of the most conspicuous features of its secular variation, is examined first. The nOn-axisymmetric field during the past several hundred years can be well approximated by the superposition of two constant-magnitude fields, a standing and a drifting field, whose lifetimes are supposed to be longer than 1000 years. It is pointed out that the seetorial term of the non-dipole standing field is small compared with the drifting one. The lick of the n = m = 2 term of the standing field is particularly remarkable. On the other hand, the equatorial dipole field is likely to consist of two components which are both drifting. One drifts westwards with a normal velocity and the other eastwards with a small velocity. Besides the pronounced westward drift in an east-west direction, the poleward movements of particular loci of the secular variation are noted. This may, however, be related to the rapid growth of the axisymmetrie quadrupole field. The time variation of the dipole field is briefly examined. As far as the data on the historical time scale are concerned, an antiparallel relationship seems to exist between the variations in the dipole and the quadrupole field. As the dipole moment decreases, the magnitude of the quadrupole moment increases. Finally, characteristic oscillation periods of the dipole field are examined. Although the data are few, a 60-70-year period, a 400-600-year and a 8000-year period emerge as the dominant periods.
1. Introduction Geomagnetic secular variations originate mostly in the Earth's liquid core. Assuming that the magnetic field is frozen in the conducting core, geomagnetic secular variations are sometimes regarded as manifestations o f fluid motion near the surface o f the core (Kahle et al., 1967a,b; Rikitake, 1967; Booker, 1969). On the other hand, some features o f the geomagnetic field, such as represented b y the equatorial dipole component, are considered to be related to the intrinsic process o f the geodynamo that creates the dipole field (Braginskiy, 1964; Pekeris et al., 1973; Krause, 1977). Geomagnetic secular variations provide, therefore, some o f the basic data for investigating the magnetohydrodynamic processes taking place in the Earth's core.
It is well known that the rotation o f the Earth exerts a predominant effect on core motions. Such highly axisymmetric motions as a shear motion around the rotation axis are supposed to be generated b y the Earth's rotation. Besides these axisymmetric motions, non.axisymmetric motions are required to maintain the geomagnetic field (see Roberts, 1971). The non-axisymmetric motions are expected to induce the non-axisymmetric magnetic field when they interact with'an axisymmetric field such as that o f the dipole. Therefore, examination o f the non-axisymmetric field should provide information on the nonaxisymmetric parts o f the fluid motion. The secular variation o f the non-axisymmetric field will be reviewed first. One o f the outstanding features o f the secular variation that has been known
84 for centuries is the westward-drift phenomenon. Besides a westward-drifting component of the field, a standing component or one that is drifting eastwards with a small velocity has been noticed. The predominance of movement of the geomagnetic field in the east-west direction may be contrasted with the equatorward migration of the sunspot field. Therefore, it is of interest to examine whether any movement in the north-south direction can be discernible in the geomagnetic secular variation. It is pointed out that a poleward movement of particular foci of the geomagnetic secular variation is likely to exist. But this might be related to rapid growth of the quadrupole field rather than to real migration of the particular patterns of the field. Finally, the time-variation of the axisymmetric field associated with the dipole and the quadrupole fields is briefly mentioned.
2. Spherical harmonic analysis of the geomagnetic field Spherical harmonic series have been commonly used to express the geomagnetic field since Gauss exploited the technique and made an analysis of the field for 1835. Fritsche (1899) applied this technique to earlier data back to 1600. Since around the end of the 15th century, many data on the magnetic declination and inclination were acquired with the development of navigation, whereas no instrumental observations of magnetic intensity were made before the 19th century. Fritsche's analyses are based on three components, the north, the east and the vertical downward components, which were computed from declination and inclination on an implicit assumption that the intensity before the 19th century had been the same as that for the 19th century. Without knowledge of the magnetic intensity, one can only determine the relative magnitudes of the Gauss coefficients of the spherical harmonic series from the data of declination and inclination. Braginskiy (1972a) and Barraclough (1974) avoided this difficulty by extrapolating the variation in the dipole moment backwards with a specific algebraic expression, such as a quadratic form, and determined sets of, Gauss coefficients back to 1550. Without assuming the dipole term, however, it is possible to determine a complete set of Gauss coefficients with absolute magnitudes, provided that the intensity data are
available at a single point (see Appendix). Selecting places where both archeomagnetic and recent data on intensity are available, Yukutake (1971) and Benkova et al. (1974) estimated the intensity values for the 17th and I8th centuries, a time when instrumental observations of the intensity had not been initiated. The intensity data thus interpolated were incorporated with the declination and the inclination data to determine the Gauss coefficients up to the degree and the order of 4. In this process, the absolute magnitudes of the Gauss coefficients are linearly dependent on the • interpolated intensity. Since the archeomagnetic intensity data used for estimating the intensity value for the specific epochs contain many more uncertainties than those of declination and inclination, the magnitudes of the coefficients thus determined are less accurate than the relative magnitudes between the coefficients. In spite of these difficulties, however, the analyses conducted by various authors for the earlier epochs are still sufficiently consistent with each other to analyse the long-term variations over centuries, as can be seen in Fig. 4, where trajectories of the equatorial dipole component computed from different analyses are plotted in the coordinates of the cosine (gl) and the sine (h l)coefficients. The spherical harmonic coefficients thus obtained will be used to analyse the secular variations of the axisymmetric and the nonaxisymmetric field separately.
3. The westward drift of the geomagnetic field The largest component of the non-axisymmetric field is the equatorial component which causes the best-fitting geocentric axial dipole to deviate from the geographic axis. Figure 1 shows the movement of the geomagnetic north pole since 1550. The locations of the pole were plotted from the results of Benkova et al. (1974) and Yukutake (1971) together with those of analyses after 1829. Although only two sets of the analyses are plotted here for the earlier epochs, the other analyses give almost the same polar paths. It is evident from this figure that the geomagnetic pole has rotated clockwise. Since the clockwise rotation of the inclined dipole is equivalent to the westward drift of the equatorial dipole, the data of Fig. 1 imply that the equatorial dipole has been drifting
85
/2e't Fig. 1. Locations of the geomagneticnorth pole for various epochs since 1550. Clockwiserotation of the pole position is seen. Open circlesare taken from the analysis by Benkova et al. (1974); solid circles from Yukutake's analysis for 1600 to 1770, and from various authors for 1829 to 1975. westwards since the 17th century, although the speed has been apparently slower since the beginning of the 19th century. It is well known that the non-dipole field, the residual field after the dipole component is removed from the observed geomagnetic field, drifts westwards (see Bullard et al., 1950; Yukutake, 1962). Figure 2 shows the vertical component of the non-dipole field for 1770 and 1965 synthesized from the Gauss coefficients (Yukutake and Tachinaka, 1968a; Yukutake, 1971). By comparing the similar charts for different epochs, 1907 and 1945, Bullard et al. (1950) estimated a mean drift rate of the whole pattern of the non-dipole field to be 0.18 ° year -1. Figure 2 shows, however, that there are some non-dipole anomalies that have not been noticeably displaced during the period concerned. Therefore it is inconceivable that the whole pattern of the non-dipole field drifts westwards with a uniform rate. Comparison of the two charts in Fig. 2 suggests rather that there exist different types of non-dipole anomaly. One is the anomaly that displaces its location with time. The African negative anomaly belongs to this type. It was located in the western Indian Ocean in 1770 and reached the west coast of Africa in 1965. The mean rate of westward drift of this anomaly was estimated to be
0.28 ° year -1 by Yukutake and Tachinaka (1968a). The other type of anomaly is that standing almost at the same place, such as the anomaly that covers North America. A positive anomaly with its center in Mongolia has not moved either, but grown up rapidly to become the largest non-dipole anomaly during the past 400 years, with a mean growth rate of 53 nT year -1 (Yukutake and Tachinaka, 1968a). It is possible to consider that the individual anomalies change independently as described above. However, it is also possible to interpret the complicated features of the secular variations in a completely different way based on a steady field model (Yukutake and Tachinaka, 1969). Consider two steady fields as shown in Fig. 3. One is the field standing at the same place, and the other is the field of which the whole pattern drifts westwards with a constant velocity of 0.3 ° year -1. It is demonstrated by Yukutake and Tachinaka (1969) that simple superposition of the two kinds of field represents well the various aspects of the observed geomagnetic secular variations. Not only has the steady movement of the African anomaly been successfully approximated by this steady drifting-standing field model, but so has the rapid growth of the Monogolian anomaly.
4. Eastward drift of the equatorial component Separation of the drifting from the standing field was originally made for the time variation of the individual spherical harmonic components. Figure 4 shows to what extent the standing-drifting field model can approximate the observed variation in the equatorial dipole component, i.e. g] and h~. In this figure, the time variations in the equatorial dipole component are expressed by successive points in the coordinates ofg~ and h[. The results of five analyses by Fritsche (1899), Yukutake (1971), Braginskiy (1972a), Barraelough (1974) and Benkova et al. (1974) are all plotted, together with recent analyses, covering the period from 1550 to the present. As in Fig. 1, westward drift, or clockwise rotation of the equatorial component, is seen. In Fig. 4 the standing field is represented by a vector from the origin of the coordinates to the point X, while the drifting field is represented by a vector rotating clockwise around the point X. Therefore, the circle in Fig. 4 represents the
86
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Fig. 3. The upper diagram shows the vertical component of the drifting part of the non-dipole field and the lower diagram that of the standing part. The non-dipole field due to the zonal terms is included in the standing field. Contour intervals are 2000 nT.
88
cl
-heonr '
the drifting field changes its amplitude during its rotation, the trajectory will expand or shrink across the circle expected from the steady model. Accordingly it would be possible to adjust amplitude change so that the model may fit the observations. However, the adjustment seems too arbitrary and would bring unnecessary complications into the model. The simplest way of adjusting the discrepancies is to introduce a component that drifts eastwards steadily. Replacing the standing field b y a field drifting eastwards, Braginskiy (1972b) and Yukutake (1976) approximated the variation in the equatorial dipole component by superposition of two components drifting steadily in an opposite direction, eastwards and westwards, with constant amplitudes. The ampli-
-FR o BR "BL Fig. 4. The equatorial dipole component,g[ and hi, for various epochs obtained by different analyses, starting with either 1550 or 1600 A.D. For convenience sake, each epoch is labelled near an approximate position of the mean of the different analyses for the specific epoch. The two vectors on the diagram represent the standing and drifting field components. The vector from the origin to the point X is the standing component, and the vector from the point X to the circle the drifting component which rotates clockwise around the point X with a velocity of 0.293 = year-1 . Accordingly the circle is the trajectory of the equatorial dipole component expected from the standing-drifting field model. Solid circles connected by solid lines are from Yukutake (1971); open circles from Benkova et al. (1974); triangles from Fritsche (1899); open squares from Braginskiy (1972a) and solid squares from Barraelough (1974). Solid circles for the epochs from 1829 to 1965 are taken from the analyses based on the recent instrumental observations.
.I 10 IN0
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trajectory around which the equatorial dipole component is expected to move for the standing-drifting field model. Although the general trend is well approximated by the standing-drifting field model, if one scrutinizes the detailed features, it may be noticed that there exist significant deviations o f the model from the observations. The recent trend of the variation is obviously different from the circular trajectory expected from the model. One possible way o f reducing the deviations is to assume time-variation in the amplitude o f either the drifting or the standing field. If
,
,
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/
,
-~d
•--BR o - - Yukutake mode o - - - Broginsk y model
Fig. 5. Trajectories of the equatorial dipole component obtained by three different models. One is the standing-drifting field model whose trajectory is shown by a circle. The other two, Braginskiy's and Yukutake's model, consist of two components drifting in the opposite direction with the velocities and the amplitudes listed in Table I. Open circles connected with dotted lines are after Braginskiy's modal, and open circles with solid lines after Yukutake's model. Rerults of analyses are also ~own for comparison with the models. Solid squares with solid tines are the rerults of Braglnskiy's analysis, and solid circles with solid lines arc those of Yukutake
89 TABLE I Velocities and amplitudes of the components drifting westwards and eastwards for the equatorial dipole field. Vw and A w represent the velocity and the amplitude of the westward drifting component, while Ve and A e are those of the eastward drifting one
nT
• Standing field
Braginsky Yukutake
Vw ~ y - l )
Aw(nT)
Ve( o y - l )
Ae(nT)
-0.290 -0.36
3880 2850
0.215 0.08
2780 3616
tudes and the velocities were obtained as listed on Table I, where A e and Ve represent the amplitude and velocity of the eastward component, respectively, and A w and Vw those of the westward component. The eastward velocity is taken as positive. Figure 5 shows the trajectories of the equatorial dipole component obtained by these models. Open circles connected by solid lines are from Yukutake's model, and open circles connected by broken lines are after Braginskiy's model. The circular trajectory with small open circles is from the standing-drifting field model. Solid circles and solid squares indicate the observations. An improvement of the fit is clearly seen in the revised models. The models that include an eastward-drifting component are applicable not only to the equatorial dipole component but also to other higher harmonic components. However, the approximation is not improved as much as that for the equatorial dipole component. Therefore, the standing-drifting field model for the higher harmonics will be retained.
5. Lack of sectorial terms in the standing field Regarding the structure of the standing and drifting field, one interesting feature is the lack of sectorial terms in the standing field. In Fig. 6 the amplitude of the standing field as well as that of the drifting field is plotted against the degree and the order of the spherical harmonics. There are minima of amplitudes at n = m = 2 and 3 for the standing field. The amplitude is also small with n = m = 4. Except for the equatorial component, n = m = 1, the sectorial terms are smaller for the standing component than for the drifting one. This feature is particularly conspicuous
o Drifting field
= m=l
1
2 2
3 1
3 2
3 3
4 1
4 2
4 3
4 4
Fig. 6. Amplitudes of standing and drifting components of the individual spherical harmonics. Open circles represent the amplitudes of the drifting field, and the solid circles those of the standing field except for the equatorial dipole component, n = m = 1. For the equatorial dipole component the amplitude of the component drifting eastwards is plotted. Small amplitudes are noted for the sectorial terms of the standing field, n = m = 2, 3, 4.
with the term n = m = 2. So far no separation of the standing field from the drifting field has been made for the components higher than n = m = 4. It is important to examine whether the lack of the sectorial terms can be seen with the standing field of the higher components. It is also interesting to note that the amplitude of the drifting field for n --- 3 and m = 2 is small.
6. Lifetime of the drifting field If proper values are taken as the parameters of the core, such as 3 X 10 -6 emu for the electrical conductivity, the free decay time of the non-dipole field, e.g. n = 2, can be as long as 7000 years (Yukutake, 1968). Therefore, unless vigorous fluid motions destroy the field, the drifting and the standing field may have a lifetime that covers the whole historic age.
90
isos2 0°
40"
80 °
160" E
160 ° W
120 °
80"
40 °
I
N 75'
120 °
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~.
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,
,.
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70 °
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li
~
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l
,
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: . _--~' ~ _~_~_~.':77[..~_'~:_-Z_.._D-~__.~ : r i..1--_,:-'4 ~. . . . "~~_!.., -: : : - - : ~ ~ ..-:~...,~. :,,:,. ),,'.I _- --D---. , ~ - ~ ..... "'-...[""7,, :... ,~ "~-/-__
600 ---.-.
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~/,.~L.~i;;t~_ rIXi~l)
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40 °
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i
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j
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t
/
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i
~.
/ ~ 7--/z,,z..,mt~:-4-
C[~"--.~D.
120 °
il,..,L/'~__-..~
801
40 I
Fig. 7. Isopofic charts of the vertical component of the non-dipole field for the epochs 1806 and 1965. The vector indicates the horizontal component of the secular variation in the non-dipole component. A negative zone runs along the equator. Outside the zone pairs of positive foci are seen. These foci seem to have been pushed away from the low latitudes towards the poles as they drift westwards. Units: nT y - l .
91
Bullard et al. (1950) considered that the non-dipole field was generated by turbulent fluid motions near the surface of the core and suggested that the lifetime would be shorter than a hundred years. If the nonaxisymmetric field is merely a manifestation of the turbulent phenomena taking place near the surface of the core, its lifetime will be relatively short as suggested by Bullard et al. (1950). However, if it is a phenomenon related to the intrinsic dynamo process in some way, it may possibly have a longer lifetime. Comparison of the non-dipole charts in Fig. 2 indicates that the main features of the non-dipole field have persisted over the period of comparison, i.e. 195 years, which is longer than the lifetime suggested by Bullard et al. It is, therefore, highly important to examine how long the non-axisymmetric field has existed. Yukutake (1967) examined the secular variations of declination and inclination at various places with periods of several hundred years, which were collected from a variety of sources: observatory data; results of repeat surveys and paleomaguetic data, He noticed westward drift o f the maxima and the minima of the variations with an average velocity of 0.36 ° year -l. It was recognized that some features, which were traced back to about 1000 A.D., were still propagating, and that the propagation of the maximum deviations of declination is consistent with that of the eastward component obtained from the recent analyses (Yukutake and Tachinaka, 1968b). These data suggest that the drifting field has persisted longer than i000 years.
7. Poleward motion of the magnetic foei and the growth of the quadrupole field Although the movement of the magnetic pattems in the east-west direction is a predominant feature of the geomagnetic secular variation, it will be interesting to examine whether a meridional component exists in the movement of the magnetic field. The northward shift of the eccentric dipole is another pronounced aspect of the geomagnetic secular variation that is frequently cited with the westward-drift phenomenon (Vestine, 1953; also see Rikitake, 1966). Since the distance of the eccentric dipole from the center of the Earth is closely related to the magnitude
of the quadrupole term, gO, the northward shift of the eccentric dipole was considered to be caused by an oscillation of the quadrupole field (Nagata and Rikitake, 1963). Poleward movement of particular foci of the secular variation was suggested by Yukutake and Tachinaka (1968b). Figure 7 shows the rate of change in the vertical component (Z) for different epochs, for 1806 and for 1965. A negative zone dominates the equatorial region. Outside this negative zone, pairs of positive anomalies exist, one in the area of India and the other near the American continent. These positive anomalies seem to have moved polewards with the elapse of time. Figure 8 shows how the centers of the positive foci changed their latitudes with time. Poleward movement is dearly seen. The rate of the apparent poleward migration was estimated to be 0.14 ° year -z for the Indian feel and 0.26 ° year -1 for the American foci. The apparent poleward migration seems to be caused by the expansion of the equatorial negative zone. This zone is well expressed by a change in the quadrupole field. Consequently the expansion of this zone may be understood as a rapid growth of the
I
]'
0
T
o
•
N
6~
!
,
3~
o" -30 •
o
-60'
.e
o
I
1800
I
1900
'
2 0 0 0 A.D.
Fig. 8. Latitudes of the centers of positive isoporic foci. Solid circles represent the latitudes of the Indian foci, and open circles those of the American loci. Apparent poleward velocity is estimated to be about 0.14 ° year -1 for the Indian foci and 0.26 ° year -1 for the American foei.
92
Y
g:
-3200C
0
".
.
-31000 °X)O0
-30000 -200C
~oo
~oo
'
,~oo
'
,~oo
'
,~oo
'
~oo
AD
Fig. 9. Time-variationin the quadrupole term, gO. quadrupole field. Figure 9 shows the time-variation in the quadrupole term gO; the results of the spherical harmonic analyses for earlier epochs (Yukutake, 1971) are shown together with those of the recent ones. Since the beginning of the 18th century, gO has been decreasing. Because of the negative sign o f g °, this means that the absolute value o f g ° is increasing. This may perhaps be another way of representing the expansion of the equatorial negative zone.
8. Variations in the dipole field
The most predominant axisymmetric field in the geomagnetic field is the dipole field. The time variation in the dipole coefficient gO since the 17th century is shown in Fig. 10 (Yukutake, 1971). The negative value is taken upwards in Fig. 10 for the sake of convenience to show the change in the absolute value of gO (which is linearly proportional to the axial dipole moment). The general trend is approximated by a parabolic curve with the peak sometime around 1800 A.D., as shown on the diagram. By comparison with Fig. 9, similarity of the general trend between the dipole and quadrupole variations is noted. This feature can also be seen in the results of the analysis of Benkova et al. (1974). In spite of the difficulty already mentioned in determining the absolute values of the Gauss coefficients before the 19th century, Benkova et al. (1974) obtained time-variations in the dipole and the quadrupole coefficients that are similar to those shown in Figs, 9 and 10, although the
'
~oo
'
~oo
'
,~oo
'
2booAo
Fig. 10. Time variation in the axial dipole component, gO. In order to see the change in the magnitude of the dipole intensity, negative value is taken upwards. epochs of the peaks are somewhat earlier than in Figs. 9 and 10. The signs o f g ° andg ° are both negative, except for the limited period from the early 18th century through the middle of the 19th century, wheng ° is likely to have been positive, and the magnitude o f g ° increases as the magnitude o f g ° decreases. Therefore, as far as these historical data are concerned, it may be said that there exists an antiparaUel relationship between the variations of the dipole and the quadrupole m o m e n t * On the geologic time scale, however, this antiparallel nature does not seem to exist. As was mentioned in a previous section, the ratio o f g ° and gO is related to the distance of the eccentric dipole from the center of the Earth. The eccentricity is sometimes modeled as an "offset of the dipole" in paleomagnetic studies. The present offset is about 150 km north of the equator. If the change in the quadrupole field is independent of the variation in the dipole field, the ratio of the quadrupole to the dipole term should have equal probability of having either positive or negative sign when there is a change in the dipole polarity. * Verosub and Cox (1971) examined time variations in the magnetic energy contained in the dipole and the non-dipole field separately by making use of the results of spherical harmonic analyses since 1845, and concluded that exchange of energy was taking place between the dipole and the non-dipole field. However, in their analyses, the non-dipole field includes not only the axisymmetric quadrupole field but also the nonaxisymmetrie components. In this text, energy transfer between the axial dipole and the axial quadmpole field is discussed.
93
However, investigation of the dipole offset in paleomagnetism strongly suggests that the quadrupole field has also changed its sign when the dipole field reversed (Cox, 1975), and the ratios of the quadrupole to the dipole term are nearly the same between the normal and the reversed periods (Merrill and McElhinny, 1978). This suggests a positive correlation between the changes in the dipole and the quadrupole field, contrary to the historical time scale variations. A tentative conclusion derivable from these results is that the quadrupole field changes in parallel with the dipole field on a time scale as long as a million years, but that there are shorter period fluctuations that are superposed on this long-term change and behave in the opposite way.
9. Spectrum of the dipole variations The spectrum of the dipole variations will be mentioned very briefly in this section. In order to determine the geomagnetic field intensity in prehistoric times, paleomagnetic intensity techniques have been applied to samples collected from old kilns, potteries and dated lavas. Results of this application indicate that the geomagnetzc field has greatly changed its intensity during the past 9000 years. The equivalent dipole moment was computed from the intensity data on an assumption that the geomagnetic field was dipolar when the sample was magnetized. These dipole moments were averaged for small time blocks giving the curve shown in Fig. 11 (Cox, 1968). A1-
xlO2~.,ouss.cm3
Q
-m~,~-'~
~
I
•
u year(AD)
Fig. 11. Variation in the dipole moment obtained from paleomagnetic data (Cox, 1968).
- 30500
~o
\
- 30000 1900
I
I
I
I
I
I
1950 A.D.
I
Fig. 12. Decreasein the magnitude of the axial dipole component since 1900. Negativevalue is taken upwards, gO obtained from the annual values of the observatory data is plotted. The rate of decrease in the dipole intensity is not uniform.
though the data are still very few and are not uniformly distributed in space, Fig. 11 seems to suggest a large fluctuation of the dipole moment with a period of about 8000 years. The amplitude amounts to about half of the present dipole moment. According to this result, the dipole moment has been decreasing for the last 1500 years. On the other hand, historical data suggest the existence of a maximum in the dipole moment in a recent epoch around 1800 A.D., as seen in Fig. 10. In order to reconcile the historical data with the archeomagnetic results, one must assume a minimum of the dipole moment sometime around 1600 A.D. From this consideration the dipole moment is also presumed to have fluctuations of several hundred years with intensity variations around 3%. It is well known that the dipole moment has been decreasing rather monotonously since the beginning of the 19th century when Gauss made the first spherical harmonic analysis of the geomagnetic field. In order to examine detailed features, spherical harmonic analyses were made of the annual mean values of the selected observatory data for the period from 1900 to 1965 (Fig. 12). It is seen that the rate of decrease in the magnitude o f g ° is not constant with time. If the linear trend is removed, a minimum is observable around 1930 (see Yukutake, 1973b). A spectrum analysis technique, the maximum entropy method, was applied to this time series, and the largest energy content was obtained for the periods of 4 0 - 8 0 years
94 If
Appendix - Spherical harmonic analyses for the
OLIn!
epochs prior to the 19th century
PERIOD
i'g~ y k r
Fig. 13. Amplitude spectrum of the dipole variation. Width of the column represents an approximate range of uncertainty of the characteristic period. The spectrum is tentative at present. CYukutake, 1973a). The amplitude amounts to about 0.4% of the present dipole moment. Applying the same technique to the results of their own analysis, Jin and Thomas (1977) obtained the characteristic period of 66.7 years for the dipole variation. The characteristic period thus obtained agrees well with those widely noticed from the analyses of the time variations in the geomagnetic components observed at various observatories, i.e. 60-70-year periods. Although it is premature to draw conclusions about the spectrum, the above data are summarized very tentatively in Fig. 13. The height of each column represents the amplitude, and the width covers an approximate range of uncertainty of the characteristic period. For the period range from 50 years to 10 000 years, there are at least three distinct peaks, i.e. at the period around 6 0 - 7 0 years, 400-600 years and 8000 years. The existence of these distinct frequencies in the variations of the dipole field could be of fundamental importance in investigations of the magnetohydrodynamic process in the Earth's core.
Acknowledgements The author would like to thank Drs. R.T. Merrill and J.C. Cain for reading the manuscript and giving valuable comments.
It is evident that, from data of declination and inclination alone, a unique solution of the Gauss coefficients is unobtainable. Suppose all the coefficients are multiplied by a factor of k, the intensity and the field components synthesized from them become k-times larger, but the declination and the inclination remain constant, since they are derived from the ratios of two field components. Therefore, the intensity data are indispensable for obtaining the absolute values of the Gauss coefficients. However, once a term, gO for example, is known, a definite set of the Gauss coefficients can be determined. In Yukutake (1971) and Benkova et al. (1974), the m relative values of the tesseral terms, (Gn, H~m ), where G~n = gr~n/gOl, Hrffn =hn/h m oI (m = 0), are determined from declination data, and the zonal terms G° (=g~/gO) from those of inclination. Then gO is estimated so that the synthesized values fit best to the intensity data. In Yukutake (1971), grid point values of the declination at an interval of 10° in latitude and longitude were used. The number of grid point values ranges from 260 to 330 for different epochs; and 11 to 12 values of inclination were used in Yukutake (1971). Since gO is the only quantity to be determined from the intensity data, a reliable intensity value at a single point would be sufficient theoretically to determine a definite set of coefficients. The number of intensity values used'forg ° is 5 in Yukutake (1971) and 9 in Benkova et al. (1974). This implies that the gO values are heavily dependent on the quality of the archeomagnetic data which were used to estimate the intensity for specific epochs. Unfortunately, however, the quality of the archeomagnetic data is not so high as that of the declination and the inclination data that are available since the 16th century. Yukutake (1971) estimated the probable error to be about 500 nT for gO and about 1000 nT forg °. Therefore much room is left for improvement regarding the absolute magnitudes of the coefficients compared with the relative relation between the coefficients.
References Barraclough, D.R., 1974. Sphericalharmonic analyses of the geomagnetic field for eight epochs between 1600 and 1910. Geophys. J. R. Astron. Soc., 36: 497-513.
95 Benkova, N.P., Kolomiytseva, G.I. and Cherevko, T.N., 1974. Analytical model of the geomagnetic field and its secular variations over a period of 400 years (1550-1950). Geomagn. Aeron. (USSR), 14:751-755 (English). Booker, J.R., 1969. Geomagnetic data and core motions. Proc. R. Soc. London, Ser. A, 309: 27-40. Braginskiy, S.I., 1964. Kinematic models of the Earth's hydromagnetic dynamo. Geomagn. Aeron. (USSR), 4: 572-583 (English). Braginskiy, S.I., 1972a. Spherical analyses of the main geomagnetic field in 1550-1800. Geomagn. Aeronom. (USSR), 12:464-468 (English). Braginskiy, S.I., 1972b. Analytical description of the geomagnetic field of past epochs and determination of the spectrum of magnetic waves in the core of the Earth. Geomagn. Aeron. (USSR), 12: 947-957. Bullard, E.C., Freedman, C., Gellman, H. and Nixon, J., 1950. The westward drift of the Earth's magnetic field. Philos. Trans. R. Soc. London, Ser. A, 243: 67-92. Cox, A., 1968. Lengths of geomagnetic polarity intervals. J. Geophys. Res., 73: 3247-3260. Cox, A., 1975. The frequency of geomagnetic reversals and the symmetry of the non-dipole field. Rev. Geophys. Space Phys., 13: 35-51. Fritsche, H., 1899. Die Elemente des Erdmagnetismus fttr die Epochen 1600, 1650, 1700, 1780, 1842 und 1885, und ihre saecularen Aenderungen. St. Petersburg. Jin, R.S. and Thomas, D.M., 1977. Spectral line similarity in the geomagnetic dipole field variations and length of day fluctuations. J. Geophys. Res., 82: 828-834. Kahle, A.B., Ball, R.H. and Vestine, E.H., 1967. Comparison of estimates of surface fluid motions of the Earth's core for various epochs. J. Geophys. Res., 72: 4917-4925. Kahle, A.B., Vestine, E.H. and Ball, R.H., 1967. Estimated surface motions of the Earth's core. J. Geophys. Res., 72: 1095-1108. Krause, F., 1977. Mean-field electrodynamics and dynamo theory of the Earth's magnetic field. J. Ge0PhyS., 43: 421-440. Merrill, R.T. and McElhinny, M.W., 1978. An outline of some of the properties of the geomagnetic field. Abstr. Conf. Origins Planet. Magn., 67-68. Nagata, T. and Rikitake, T., 1963. The northward shifting of the geomagnetic dipole and stability of the axial magnetic quadrupole of the Earth. J. Geomagn. Geoeleetr., 14: 213-220.
Pekeris, C.L., Accad, Y. and Shkoller, B., 1973. Kinematic dynamos and the Earth's magnetic field. Philos. Trans. R. Soc. London, Ser. A, 275: 425-461. Rikitake, T., 1966. Electromagnetism and the Earth's interior. Elsevier, Amsterdam, pp. 1-308. Rikitake, T., 1967. Non-dipole field and fluid motion in the Earth's core. J. Geomagn. Geoelectr., 19: 129-142. Roberts, P.H., 1971. Dynamo theory. In: W.H. Reid (Editor), Mathematical Problems in the Geophysical Sciences. Lectures in Applied Mathematics, American Mathematical Society, Providence, R.I., 14: 129-206. Verosub, K.L. and Cox, A., 1971. Changes in the total magnetic energy external to the Earth's core. J. Geomagn. Geoelectr., 23: 235-242. Vestine, H., 1953. On variations of the geomagnetic field, fluid motions, and the rate of the Earth's rotation. J. Geophys. Res., 58: 127-145. Yukutake, T., 1962. The westward drift of magnetic field of the Earth. Bull. Earth. Res. Inst., 4 0 : 1 - 6 5 . Yukutake, T., 1967. The westward drift of the Earth's magnetic field in historic times. J. Geomagn. Geoelectr., 19: 103-116. Yukutake, T., 1968. Free decay of non-dipole components of the geomagnetic field. Phys. Earth Planet. Inter., 1: 93-96. Yukutake, T., 1971. Spherical harmonic analysis of the Earth's magnetic field for the 17th and the 18th centuries. J. Geomagn. Geoelectr., 23: 11-23. Yukutake, T., 1973a, Power spectrum analysis of the geomagnetic secular variation (Abstract). Int. Assoc. Geomagn. Aeron. Bull., 34: 86. Yukutake, T., 1973b. Fluctuations in the Earth's rate of rotation related to changes in the geomagnetic dipole field, J. Geomagn. Geoelectr., 25: 195-212. Yukutake, T., 1976. A model of the geomagnetic secular variation for the past 2000 years. Abstr. Fail Meeting Soc. Terr. Magn. Electr. Jpn. (in Japanese), p. 154. Yukutake, T. and Tachinaka, H., 1968a. The non-dipole part of the Earth's magnetic field. Bull. Earthquake Res. Inst. (Tokyo Univ.), 46: 1027-1074. Yukutake, T. and Tachinaka, H., 1968b. The westward drift of the geomagnetic secular variation. Bull. Earthquake Res. Inst. (Tokyo Univ.), 46: 1075-1102. Yukutake, T. and Tachinaka, H., 1969. Separation of the Earth's magnetic field into the drifting and the standing parts. Bull. Earthquake Res. Inst. (Tokyo Univ.) 47: 65 -97.