Fluctuations in geomagnetic secular variation rate according to data from the global network of magnetic observatories

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ScienceDirect Russian Geology and Geophysics 57 (2016) 483–497 www.elsevier.com/locate/rgg

Fluctuations in geomagnetic secular variation rate according to data from the global network of magnetic observatories A.V. Ladynin * Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia Received 16 May 2014; received in revised form 7 July 2015; accepted 28 August 2015

Abstract The secular variation rate (SVR) of geomagnetic-field components (horizontal intensity H, vertical intensity Z, and inclination I) shows two types of fluctuations: with a short period (3 ± 0.15 years) and a long period (10–70 years). The amplitudes of the short-period fluctuations (SPFs) were estimated. The SPFs are uniform throughout the Earth, the Z and I SVR fluctuations are synchronous and have the same phases, and H SVR fluctuations are opposite in phase. Modeling of an eccentric-dipole field with a variable axial-pole latitude has shown that the SPFs in SVR are caused by the nutation of the dipole axis (and by the outer-core current systems responsible for the dipole field). LPFs in SVR manifest themselves differently in different regions, and their nature is dominated by the effect of currents in the liquid core near the mantle. © 2016, V.S. Sobolev IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved. Keywords: geomagnetic field; secular variation rate; fluctuation; fluctuation period; fluctuation amplitude; nutation of the dipole axis; magnetic dipole moment variation

Introduction The study of geomagnetic secular variations is of great theoretical importance for improving models for the generation of the main geomagnetic field (GMF) (Ben’kova and Pushkov, 1980; Braginskii, 1978; Kalinin, 1984; Langel, 1987; Parkinson, 1983; Wardinski, 2005; Yanovskii, 1978) and of practical importance for providing adequate models of the normal field for magnetic surveys over large areas (Demina and Petrova 2010). In recent years, geomagnetic secular variations have been widely studied using data of satellite measurements; the same data are used to construct models for the main geomagnetic field, the best-known of which are the International Geomagnetic Reference Field (IGRF) and the Definitive Geomagnetic Reference Field (DGR) models (Ben’kova and Pushkov, 1980; Langel, 1987). However, the use of these models in regions with a low-density network of magnetic observatories (MOs) creates great difficulties in interpreting magnetic anomaly maps widely used to solve various geological problems (Demina and Petrova, 2010). On the other hand, the data on secular variations of the GMF, in particular, the secular * Corresponding author. E-mail address: [email protected] (A.V. Ladynin)

variation rate of the GMF, obtained using these models also have significant drawbacks (Barraclough, 1985; Ladynin et al., 2006a). Much more reliable results in this respect are obtained by observations at magnetic observatories (MOs) and secular variation stations (SVSs) (Ladynin et al., 2006b). It is necessary to give some explanations in connection with the ambiguous use of the terms variation, trend, and variation rate. Trend is the general direction of something; variation is a change or deviation (Small Academic Dictionary, http://encdic.com/academic/Mal-19104.html) (Evgen’ev, 1999). In geomagnetism, variation (trend) is the time dependence of a variable, for example, H(t), and the variation rate is the time derivative of the variation (the first finite difference of discrete values of the variable). Some authors refer to the variation rate as the trend, with which we cannot agree. The SVR is defined as the difference between successive annual means of the GMF (for each component). Below, it is denoted by D (the first finite difference) in series of initial data; in this study, these are the force components H and Z and the inclination I, for which the SVR is denoted by DH, DZ, and DI, respectively. Quasi-periodic fluctuations in the secular variation rate of the geomagnetic field (components H and Z) with a period of ~3 years were found in MO data in comparing satellite and observatory measurements (Ladynin et al., 2006a). They are

1068-7971/$ - see front matter D 201 6, V.S. So bolev IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.rgg.201 + 6.03.010

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also noted in (Ladynin et al., 2006c). In previous studies, they were considered as noise (Ben’kova and Pushkov, 1980; Kalinin, 1984; Wardinski, 2005). In the dissertation of Simonyan (2005) devoted to the nature of high-frequency secular variations of the main geomagnetic field, the spectrum of these variations is limited to a 10-year period. Quasi-periodic fluctuations in geomagnetic secular variation rate have been studied in greater (Ladynin and Popova, 2008) using data from 110 magnetic observatories worldwide for the 1985–2005 period. It has been found that the short-period fluctuations in DH and DZ are synchronous and opposite in phase and are identical in different regions of the world. It has been shown that these fluctuations are independent of the external field. It has been suggested that they are related to the nutation of the eccentric-dipole axis. This paper presents the results of more comprehensive studies on this topic. The initial data were the annual means of the geomagnetic-field components from the global network of MOs available from the website www.geomag.bgs.ac. uk/cgi-bin/means. Here we use the complete data of the global network of MOs (264 MOs) over the entire period of their work up to 2012. MO data with observation periods less than 6 years and with long breaks are not included. Three GMF components are analyzed: the horizontal intensity H, the vertical intensity Z, and the inclination I. Data on the inclination I are important for clarifying the nature of SVR fluctuations. With a change in the magnitude of the magnetic induction T (other symbols F and B), the changes in both intensity components have the same sign, and with a change in the angle of the magnetic induction (inclination I), the signs of the changes in H and Z are opposite (Z = T sin I; H = T cos I). The secular variation rates of DH, DZ, and DI are calculated from the annual means of the GMF components (H, Z, and I) at each MO. We study the spaciotemporal structure (regime) of the fluctuations in the secular variation rate of the GMF, i.e., the periods and amplitudes of SVR fluctuations as functions of the MO location on the Earth’s surface and time. For the study of the spatial structure of SVR fluctuations, all MOs are distributed, according to their coordinates, into 12 regions of the Earth’s surface. Changes in the periods and amplitudes in time are studied over 30-year observation intervals at each MO and on average over the regions.

Separation of variations with different periods (3 years and more than 10 years) was performed by moving-averaging of DH, DZ, and DI values on an interval of 10 years. Secular fluctuations in geomagnetic SVR with periods of more than 10 years (LPFs) were identified at all MOs and over 12 regions of the Earth’s surface. In contrast to fluctuations with periods of ~3 years (SPFs), their averaging over wider areas is meaningless since they are individual at each MO (given that their network is sparse everywhere except in Western Europe), and the more so in each region, and are likely to be related to the near-surface layer of the liquid core. The assumption about the relationship between short-period fluctuations in SVR and nutations of the eccentric-dipole (ED) axis is verified using a model for the magnetic field of this dipole (Ladynin and Popov, 2009).

Factual data It has been shown (Ladynin and Popova, 2008) that the short-period fluctuations in geomagnetic SVR during 1985– 2005 vary little in period and amplitude in averaging MO data over regional groups, then over large regions, continents, hemispheres, and over the entire Earth. On this basis, the objective of the first step of this study was to verify this pattern over longer time intervals and for a larger factual base (246 MOs) and determine its stability in time. Data over regions of the Earth’s surface were analyzed using a procedure different from that described in (Ladynin and Popova, 2008). The Earth’s surface was divided into 12 regions: two regions in each of the polar areas at latitudes above 60° and –60° (PN_E and PN_W in the Northern Hemisphere, and PS_E and PS_W in the Southern Hemisphere; E and W denote the Eastern and Western Hemispheres) and four regions in each of the Northern and Southern Hemispheres at latitudes of 0°–60° and 0°...–60° and in longitudinal intervals of 0°–90°, 90°–180°, 180°–270°, and 270°–360°. The number of MOs in these 12 regions and the mean length of the observation series in years (Nav) are given in Table 1. The MOs worked over different times: some began working in the mid-19 century and finished in the 20 century, and others, the majority, have been working since the 1960s to the present. When averaging the MO data for each region in

Table 1. Distribution of MOs over regions NP region

Number of MOs

Nav

SP region

Number of MOs

Nav

PN_E

20

51

PS_E

11

40

PN_W

16

36

PS_W

6

22

N_0–90

77

37

S_0–90

20

44

N_90–180

39

26

S_90–180

13

29

N_180–270

12

45

S_180–270

2

62

N_270–360

40

36

S_270–360

8

44

Total in NP

204

–

Total in SP

60

–

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485

Fig. 1. Short-period fluctuations in SVR from the data of MOs in three regions. 1, North America; 2, Southern Hemisphere; 3, Europe.

different years of observations, we used two to several tens of time series of MO data. For this reason, and most importantly, due to the lower accuracy of measurements (especially of Z) before the mid-20th century, the SVR means in these years vary in accuracy. It is evident from Table 1 that the series of annual means of the GMF components at most MOs were rather short, N = 20–50 years, although among 264 MOs there are 16 MOs at which the observation series exceeded 70 years (maximum N = 145 years at the COI MO, Coimbra, Spain). Hereinafter, MOs are denoted by three-letter codes in accordance with the IAGA international standard.

Methods for identifying periodicity in SPFs of SVR The following characteristics of observation series and SVR fluctuations are important for the choice of a suitable method for estimating SVR periods. First, short series at most MOs do not allow a reliable estimation of LPF periods. This is the main limitation for the application of mathematical methods to identify periodicities. The second limitation is due to the fact that short-period fluctuations in SVR have periods close, on average, to 3 years (Ladynin and Popova, 2008), i.e., at a data spacing of one year, only 3–4 points fall into the period. The third limitation is the instability of the period of short-period fluctuations in SVR; as an illustration, Fig. 1 shows SPFs in SVR—the DZ values averaged over a number of MOs in three regions: Europe (20 MOs), North America (14 MOs), and the Southern Hemisphere (12 MOs). The plot (Fig. 1) was constructed for an interval of 30 years (1975–2005). Short-period fluctuations in DZ at each MO

were determined as the differences of the initial DZ values and their values averaged in a moving window 10 years long. Therefore, the series of the initial DZ values were 40 years. MOs with series of observations of more than 40 years are few in number. The exception is the region of Europe. Here only the data of the compact group of MOs of Western and Central Europe were used. In the other two regions, all MOs were used. Averaging DZ values over all MOs within each region allowed a more reliable identification of regular DZ fluctuations due to a reduction in random effects. The main patterns of short-period DZ fluctuations are similar in all regions (periods of 2–5 years are present), although they are superimposed with random fluctuations. The periods of SVR fluctuations were evaluated from these data using the Periodogram procedure of the software for analysis of time series in the AtteStat system (Gaidyshev, 2001, 2012). The results are discussed below. The periodogram of the time series yn, n = 1, 2,…, N consists of r = N/2 intensity values calculated by the formula I(fi) = N(ai2 + bi2)/2,

(1)

where ai and bi are the coefficients of the Fourier series, i = 1, 2, …, r, and fi is the ith harmonic. Evaluation of the capabilities of the periodogram method as applied to the problem of identifying short-period fluctuations in DH and DZ required testing on synthetic models and some real data. Model 1 is the sum of three sinusoids with markedly different periods (3, 10, and 30 years) and defined on different time intervals—from 8 to 200 years. Without giving relevant plots, we give the main conclusions. 1. The periods of fluctuations are reliably evaluated for a series length 2.5 times greater than the defined period. Periods

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Table 2. Periods and intensities of Model 2 of three sinusoids; the periodogram method Task interval, yr 31 T, yr

24 2

I, (nT/yr)

T, yr

18 2

I, (nT/yr)

T, yr

12 2

I, (nT/yr)

T, yr

8 2

I, (nT/yr)

T, yr

I, (nT/yr)2

3.9

12.2

4.0

11.9

3.0

12.2

4.0

5.0

2.7

6.1

3.1

12.1

3.0

10.7

2.7

6.6

2.4

4.5

4.0

4.2

2.4

7.7

2.4

7.4

3.6

3.9

3.0

3.8

–

–

2.6

4.0

–

–

4.5

3.8

–

–

–

–

Table 3. Periods and intensities of DZ fluctuations from the NVS MO data Task interval, yr 35 T, yr

24 2

I, (nT/yr)

T, yr

18 2

I, (nT/yr)

T, yr

12 2

I, (nT/yr)

T, yr

8 2

I, (nT/yr)

T, yr

I, (nT/yr)2

2.7

259

2.7

270

3.6

406

3.0

258

4.0

123

3.9

189

2.4

226

2.6

356

4.0

184

2.7

78

2.3

179

4.0

206

–

–

–

–

–

–

2.5

149

3.4

169

–

–

–

–

–

–

3.2

147

–

–

–

–

–

–

–

–

of ~3 years are clearly identified for a series length of more than 8 years. 2. In the periodogram method, the set of periods depends on the length of the series N (1); the greatest period is equal to N; subsequent periods are obtained by dividing N by 2, 3, 4, ..., N/2 (integer). In series of different length, a particular period of fluctuations (e.g., ~3 years) corresponds to numbers close to 3 (e.g., 2.87, 3.12). Model 2 is the sum of three sinusoids with the same amplitude of 1 nT/year and close periods (2.5, 3, and 4 years);

the definition intervals are 8 to 31 years. Table 2 shows the values of the periods T in decreasing order of the intensity I. All periods for large intervals of data definition—31 and 24 years—were identified, although the shortest period on an interval of 24 years has much lower intensity, and on an interval of 31 years, it is divided into two periods—2.4 and 2.6 years. On shorter intervals, the pattern is varied: on a 16-year interval, a period of 3 years dominates, and on an 8-year interval, periods of 2.5 and 3 years are replaced by one

Fig. 2. Periodogram of short-period DZ fluctuations from the data of the COI MO.

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period of 2.7 years. Satisfactory identification of periods, as we see, takes place only up to 12 years. The third test is based on DZ periodograms from data of the NVS MO (Klyuchi, Novosibirsk) for intervals of 8 to 35 years (Table 3); the required periods are not known beforehand. Notation is the same as in Table 2. Here a period of 2.7 years is clearly identified on intervals of 35 and 24 years. The same period was determined by the method of calculating extrema (Ladynin and Popov, 2008). On intervals of 18 years or less, this period is poorly identified. This may be due to significant noise. The fourth test was aimed at estimating the effect of noise on the spectrum of periods. For this, the periodicity in the DZ series was estimated from data of the COI MO (145 years) using the same program. The periodogram of the initial series is not given—because of the high intensity of long periods, it is difficult to estimate details of short-period SVR fluctuations. The periodogram of the latter is shown in Fig. 2; here the series is 10 years shorter because of the averaging in a window of length 10 years to identify LPFs of DZ. Here a period of 3.1 years is also clearly identified and periods of 3.9 and 2.6 years are present, but a period of 5.2 years has the greatest intensity; a period of 9.0 years is seen. The plot shows a high noise level, which is manifested in numerous intensity extrema. The plot is constructed from the data of one MO, unlike the periodogram in Fig. 3. It is expected that for short series based on the data of one MO, it is difficult to reliably identify periodicity using this method. Figure 3 shows a periodogram of the short-period fluctuations in SVR (DZ) shown in Fig. 1. We can clearly distinguish a period of 3.1 years, common to the three regions, and a less intense period of 2.4 years. A longer period of about 4–5 years (the most intense according to the COI MO data, Fig. 2) is less definitely established, and the longest periods are different for the three regions (from 8 to 15 years). These differences may be real, but may also be a consequence of the relatively short length of the initial series. As we can see, random factors, including MO measurement errors, lead to an increase in the number of local maxima in the periodogram (Fig. 3), which does not allow identifying the main periods. Previously, we (Ladynin and Popov, 2008; Ladynin et al., 2006c) have used the method of counting extrema (peaks of different sign) to estimate the average minimum period of SVR fluctuations over the observation interval. This period for DZ of the COI MO was 3.1 years (Fig. 2). Due to the instability of the period of SPFs in SVR, this method was found to be preferred to others for almost any observation interval. Further, to estimate periods, we used the method of counting the number of extrema of SVR curves for each field component (DH, DZ, and DI). This method was tested on the sinusoidal model of LPFs in SVR with definition of values on a sparse grid of points (Fig. 4). The sinusoid S = A sin (2π/T) and a similar cosine have a period T = 3.03 years (which is not a multiple of the interval

487

Fig. 3. Periodogram of short-period SVR fluctuations. Notation is the same as in Fig. 1.

of definition of A(t) equal to 1 year) and amplitude A = 1. The curve DS has 9.5 peaks—maxima and minima—for 29 years, which corresponds to a period of 3.05; the curve DC has the same number of peaks. For a period of 3.03, the number of peaks of the same sign for 29 years would be equal to 9.57. Thus, the error of counting peaks in this model is 0.7% (the period estimates have the same error). To identify short-period fluctuations in SVR, it is possible to use the second finite difference (DD), which is used to identify jerks (Courtillot and Mouël, 1984; Golovkov and Simonyan, 1989). In the model (Fig. 4), it is shown by the thin line DDS. As we can see, it is not an analog of the derivative of the sinusoid due to the sparse grid of points of definition. It has a larger root-mean-square amplitude (RMSA), 1.24 against 0.71, although the period determined from the number of peaks, is the same. Note that the sinusoid RMSA of 0.71 is less than the defined value of 1 and its apparent value of 0.87 for a sparse grid of points (when the 3.03-year period of the sinusoid is not a multiple of the 1-year interval). However, the RMSA is the only of these values that can be estimated from observation data to compare the SPF amplitudes in different MOs and regions. This parameter can be used to characterize the range of values of even nonperiodic functions; as such, it is used to estimate the apparent amplitude of long-period fluctuations in SVR. In real data, to reduce the period estimate error in each plot (for a MO or a region), we counted SVR peaks of different sign for three components (DH, DZ, and DI) and derived the average from them with an accuracy of 0.1. The main source of error in estimating the number of peaks (and period) of SPFs is a superposition of these fluctuations on LPFs with periods of 10–70 years. The largest error occurs for a large slope of the curve of these fluctuations (large absolute values of DDH, DDZ, and DDI). As an example, Fig. 5 presents data from the ABG MO (Alibaug, India). In this and subsequent plots, DI values are in angular minutes per year; for brevity, we write min/year as misunderstanding is impossible.

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Fig. 4. Sinusoidal model of short-period SVR fluctuations.

Fig. 5. Secular variation rates of DH, DZ, and DI at the ABG MO.

SVR fluctuations with periods of more than 10 years make it difficult to count peaks and estimate SPF periods on a large time interval, as in Fig. 2; at the ABG MO, this is 105 years. Periods on shorter intervals are more precisely estimated. For this purpose, an interval of 30 years was used (at the beginning of a series, if its length is not a multiple of 30 years, it may be larger, as for the ABG MO data, or smaller). From the material such as that presented in Fig. 5, where SVR fluctuations with different periods are superimposed, it

is difficult to estimate the amplitude of fluctuations of different periods, in particular, a period of ~3 years, especially important in our case. This was the reason for dividing our study into two steps: – in the first step, periods of ~3 are estimated from the initial SVR plots, as in Fig. 5, over the entire interval of observations at the MO; – in the second step, after separation of the SVR fluctuations into two parts, ~3 years and 10–70 years, the periods

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489

Fig. 6. Long-period fluctuations in DH, DZ, and DI at the ABG MO. Notation is the same as in Fig. 5.

Fig. 7. Short-period fluctuations in DH and DZ at the ABG MO. Notation is the same as in Fig. 5.

are verified and the amplitudes of short-period fluctuations in SVR and the periods of long-period fluctuations are estimated. This separation is performed by averaging the initial SVR values over moving 10-year intervals. The average values are SPFs in SVR, and the differences between the initial and average values are LPFs. Naturally, the length of the series is thus decreased by 10 years. It is not possible to estimate the periods of SVR fluctuations of more than 10 years, as seen in Fig. 6. Here DZ is fairly

well correlated with DI; for other MOs, the pattern is different. These fluctuations vary among different MOs and regions. Short-period fluctuations in DH and DZ at the ABG MO are shown in Fig. 7. To make the plot more readable, DI values are not shown; they are largely correlated with DZ. In the DZ periodogram derived from the ABG MO data, a period of 3.1 (3.13) years is identified, and the intensity of periods of 2.7 and 2.3 years is lower, and a period of about 5 years is absent, unlike at the COI MO.

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Fig. 8. Short-period SVR fluctuations at the ABG MO from the second finite differences of the initial GMF values.

The identification of short-period fluctuations in SVR based on the second finite differences (DDs) of the initial field values is presented in Fig. 8 for the ABG MO as an example. The geometry of the curves in Figs. 7 and 8 is somewhat different, but the number of peaks is the same. These figures do not show DDI values, which are fairly well correlated with DDZ values.

Periods of short-period fluctuations in SVR The results of the first step of the estimation of the periods of sub-three-year fluctuations in SVR are summarized in Table 4. This table gives period estimates over the entire time of observation at the MOs whose number is indicated in Table 1. The period averaged over all regions is 2.97 ± 0.07 years, and the average period for all 264 MOs without division into regions is 2.95 ± 0.15 years.

Similar estimates obtained in the second step after the identification of short SVR fluctuations are as follows: the period averaged over all regions is 2.97 ± 0.07 years, and the average period for 264 MOs is 2.96 ± 0.15 years. The estimates of the period of SPFs in SVR in the first and second steps coincide. Table 5 shows data on the sub-three-year fluctuations in SVR over time at 30-year intervals. Here the data on the polar regions (PN and PS) are not separated for the Eastern and Western Hemispheres because of the small number of MOs in each of them. There are no significant differences in the periods of short fluctuations in SVR from one 30-year interval to another. Thus: – The method of counting extrema (peaks of different sign) to estimate the average minimum period of SVR fluctuations over the observation interval provides reliable values of LPF periods ~3 years in the series of data of any length; the

Table 4. Periods of short-period SVR fluctuations over regions NP region

Period, yr

SP region

Period, yr

PN_E

2.95

PS_E

2.91

PN_W

2.98

PS_W

2.87

N_0–90

2.99

S_0–90

3.12

N_90–180

2.94

S_90–180

2.99

N_180–270

2.99

S_180–270

2.94

N_270–360

2.88

S_270–360

3.02

Average over NP

2.96 ± 0.04

Average over SP

2.98 ± 0.09

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491

Table 5. Periods (yr) of short-period SVR fluctuations over 30-year intervals Region

Interval, years 1977–2007

1977–2007

1977–2007

PN

2.88

2.88

2.88

N_0–90

2.73

2.73

2.73

N_90–180

2.86

2.86

2.86

N_180–270

2.73

2.73

2.73

N_270–360

3.00

3.00

3.00

S_0–90

3.00

3.00

3.00

S_90–180

2.91

2.91

2.91

S_180–270

3.09

3.09

3.09

S_270–360

2.91

2.91

2.91

PS

3.16

3.16

3.16

Average

2.93 ± 0.14

2.93 ± 0.14

2.93 ± 0.14

periodogram method gives the same period values; if the length of the series is ≥18 years, the first method is easier to implement; – fluctuations in the secular variation rate of the geomagnetic field with periods of ~3 years are stable in space and time. However, one observes regular instability (especially pronounced for relatively short data series) of the periods— they vary from 2 to 4–5 years on different time intervals.

Amplitudes of short-period SVR fluctuations The attempt to estimate the amplitudes of fluctuations in geomagnetic SVR without separating them into short-period fluctuations (SPFs) and long-period (LPFs) fluctuations, as one would expect, was not successful. The reason for this is the large changes in the SVR level in the long-period part. As an example, Fig. 9 shows the LPFs in SVR for three regions: N_270–360, S_0–90, and PS (union of the regions PS_E and PS_W). These figures show the complex relationships between the signs of long-period changes: for DH and DZ, they are sometimes the same and sometimes opposite; DZ is sometimes poorly correlated with DI. This pattern of change cannot be attributed to the dipole field. In the region N_270–360, the root-mean-square amplitude of LPFs in Z SVR is 65 nT/year at an average level of –37 nT/year (Fig. 9a); in the region S_0–90, these quantities are 58 and –41 nT/year, respectively (Fig. 9b); in the united region PS, the RMSA H = 40 nT/year, and the average level is –28 nT/year (Fig. 9c). These figures show variations in the ratio of the DH, DZ, and DI curves. Figure 10 compares the short-period fluctuations in SVR at the BFE MO (Brofelde, Denmark) (a) and the fluctuations at the same MO from the second differences DDH and DDZ (b). In these figures, we can see instability of the fluctuations, in particular, the presence of periods of 2, 3, and 4 years.

The amplitudes of SVR fluctuations are roughly estimated from the standard deviation in the series of DH, DZ, and DI, which gives the root-mean-square amplitudes of fluctuations of these quantities (RMSA of DH, DZ, and DI). Table 6 gives estimates of the RMSA of short-period fluctuations in DH, DZ, and DI in 11 regions (except the region S_270–360, for which there are little data) and DH fluctuations for one MO with the longest series of observations in each region. The RMS amplitude of SPFs is almost the same in individual MOs and in zones after averaging the data from all MO in each region. This, along with the similarity of the period values, indicates the stability of SPFs on the Earth’s surface. The amplitude over the regions may be greater than the amplitude at particular MOs as, in the averaging, we used data from all MOs in the region, and, among them, there were MOs with significant measurement errors. The H and Z values on the Earth’s surface depend on latitude: due to the predominance of the dipole part in the main field, the H values are maximal in the equatorial region, and the |Z| values are maximal in the polar regions; latitudinal dependence can also be expected for short-period SVR fluctuations, which associated with the dipole field (Ladynin and Popova, 2008). It was found that the ratio of the RMSA of DH to the RMSA of DZ in the polar regions (for 29 MOs at latitudes above ± 67°) is equal to 0.56 and the same ratio in the equator region (for 26 MOs at latitudes below ± 20°) is equal to 0.81. Latitudinal dependence exists, but it is much weaker than that for the H and Z field components.

Amplitudes of long-period SVR fluctuations Long-period SVR fluctuations, as shown in Figs. 6–8, are various. Naturally, the amplitude of these fluctuations depends as a rule on the length of the series.

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Fig. 9. Long-period SVR fluctuations in the regions N_270–360 (a) and S_0–90 (b) and in the united region PS (c).

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Fig. 10. Comparison of the short-period SVR fluctuations at the BFE MO calculated: after their separation from the LPFs in SVR (a) and from the second differences DDH and DDZ (b).

Table 7 gives RMSA estimates for six MOs with long observation series. The longitude ϕ and latitude λ of the MOs are indicated. The upper part of Table 7 shows the root-mean-square amplitudes (RMSA) for three MOs with the longest observation periods; here, three 30-year intervals are identified. The lower part of the table shows the same data for three MOs with observation periods over 60 years; here two 30-year intervals are identified. As we see, the RMSA values of long-period SVR fluctuations vary quite widely not only at different MOs, but also at particular MOs over time. This is evident in the above plots (Figs. 6 and 9); this is inferred from the data of many MOs not included in Table 7. Further evidence for this is estimates of the ratio of the amplitudes (RMSA) of long-period fluctuations for the same factual basis that was used above to estimate the ratio of the RMSA of short-period fluctuations in DH and DZ: in the North Pole region, this ratio is 0.73, in the equator

region, 0.47, and in the South Pole region, 0.27. Here, as we see, there is no regularity, which is obviously related to the independence of the long-period fluctuations in SVR on the dipole field. Although root-mean square amplitude values give only an approximate (and underestimated, as noted above) estimate of the amplitudes, they can be used to compare different MOs at different times and, more interestingly, different regions in this parameter. They show, in particular, that the RMSA of DH, DZ, and DI, as a rule, have significantly lower values in a region than in most of the MOs in the region. Of course, this is a consequence of the averaging of fluctuations at MOs with different ratios of amplitudes and phases. It can be argued that the regionalization of the period of LPFs in SVR in space and time is difficult. In the averaging of long-period SVR fluctuations over regions, the individual features of the RMSA at the MOs are lost and significant information for particular regions is not acquired. This effect

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Table 6. RMSA of short-period fluctuations over the regions and single MO Region

DH, nT/yr

DZ, nT/yr

DI, min/yr

MO

DH, nT/yr

PN_E

7

13

0.9

TRO

7

PN_W

7

11

0.5

LER

6

N_0–90

10

13

0.8

NGK

6

N_90–180

10

11

0.8

MMB

8

N_180–270

7

10

0.5

MEA

8

N_270–360

8

12

0.9

VAL

6

PS_E

11

8

1.2

MIR

12

PS_W

7

7

0.7

AIA

9

S_0–90

9

11

1.1

HER

8

S_90–180

8

8

0.6

GNA

7

S_180–270

11

10

1.0

API

11

S_270–360

9

8

1.0

PIL

2

Table 7. RMSA of long-perion fluctuations of DH, DZ, and DI in MO Data period, years

Code

ϕ, deg

λ, deg

Code

ϕ, deg

λ, deg

Code

ϕ, deg

λ, deg

ABG

18.63

72.87

VAL

51.93

349.75

API

–13.8

188.22

DH, nT/yr

DZ, nT/yr

DI, min/yr

DH, nT/yr

DZ, nT/yr

DI, min/yr

DH, nT/yr

DZ, nT/yr

DI, min/yr

Total

30.6

39.6

3.4

14.3

22.0

0.6

10.4

16.3

1.5

1917–1947

16.2

38.9

3.5

7.4

20.6

0.2

8.4

11.8

1.2

1947–1977

30.1

12.5

1.5

5.0

4.4

0.4

8.4

5.4

0.7

1977–2007

14.9

23.3

1.4

4.3

5.0

0.3

6.8

11.8

1.2

Data period, years

Code

ϕ, deg

λ, deg

Code

ϕ, deg

λ, deg

Code

ϕ, deg

λ, deg

HER

–34.42

19.23

MEA

54.62

246.65

NGK

52.07

12.68

DH, nT/yr

DZ, nT/yr

DI, min/yr

DH, nT/yr

DZ, nT/yr

DI, min/yr

DH, nT/yr

DZ, nT/yr

DI, min/yr

Total

26.3

11.8

2.4

13.5

27.2

0.8

8.4

6.0

0.7

1947–1977

4.7

4.3

0.5

5.5

9.3

0.2

6.0

2.6

0.4

1977–2007

19.5

14.3

1.7

8.8

23.0

0.8

5.0

3.7

0.3

is even more pronounced in averaging over all 12 regions or over all MOs. Table 8 presents the periods and amplitudes (RMSAs) of short-period SVR fluctuations and the RMSAs of long-period SVR fluctuations in averaging over 12 regions with division into 30-year intervals. The SPF periods are almost identical throughout the observation time. The SPF amplitudes vary little in the last three intervals, starting from 1915, and their high values in 1850–1915 are likely due to the low accuracy of observations (especially Z) at some magnetic observatories. As we see, the amplitudes of LPFs in SVR averaged over 12 regions are 1–2 orders of magnitude smaller than those in individual regions. This again suggests that these regions are significantly individual in terms of LPFs in SVR and that these fluctuations are not related to the dipole field. They are probably related to the more or less localized current systems at the boundary between the outer core and the mantle.

Discussion of the results Fluctuations in the geomagnetic secular variation rate with periods of ~3 years are stable in space and time; the period is 3.0 (±5%). These fluctuations are presumably caused by nutations of the dipole axis in the plane containing the axis of the eccentric dipole and the center of the Earth. This assumption was tested on models of the geomagnetic eccentric dipole field (Ladynin and Popov, 2009) with a variable latitude of the dipole axial pole and a variable magnetic moment modulus. Figure 11 shows a plot of the model SPFs in SVR for the ED geomagnetic field of the 1955 epoch with fixed parameters: the relative distance (as a fraction of the Earth’s radius) from the dipole to the center of the Earth rd = 0.0815; the colatitude of the center of the dipole θd = 83.78°; the longitude of the center of the dipole λd = 148.84°; the

A.V. Ladynin / Russian Geology and Geophysics 57 (2016) 483–497

495

Table 8. Periods and amplitudes (RMSA) of the short-period and RMSA of the long-period SVR fluctuations Parameter

Interval, years 1850–1885

1885–1915

1915–1945

1945–1975

1975–2005

Period, yr

3.09

2.95

2.95

3.05

2.90

RMSA SPF DH, nT/yr

20

8

5

6

5

RMSA SPF DZ, nT/yr

81

11

5

3

3

RMSA SPF DI, min/yr

0.6

0.6

0.3

0.4

0.3

RMSA LPF DH, nT/yr

16

14

3

1

3

RMSA LPF DZ, nT/yr

14

12

14

8

3

RMSA LPF DI, min/yr

0.9

0.8

0.4

0.5

0.4

longitude of the North axial pole (NAP) λP = –80.42°; the dipole magnetic moment m in 1955 was 31002 nT; the colatitude of the NAP θP is equal to 8.17°. Short-period fluctuations in SVR were calculated at the point of the Earth’s surface with coordinates ϕ = 60° and λ = 60°; the choice of the location of the point is not critical since these fluctuations are almost independent of the coordinates. The fitted parameters are the changes in m and θP. The best fit to the actual data on SPF periods and the ratio of the amplitudes of DH and DZ fluctuations is obtained for the model with the following parameters: m = 31002 – 5 cos (2πt/30); θP = 8.17 – 0.02 cos (2πt/3), where t is time in years from 1955. The following features of the model fit real data (Fig. 11): – the opposition of the phases of DZ and DH SPFs; – the amplitudes of DZ and DH LPFs and their ratio. The number of peaks in the plots of DZ and DN is equal to 9.5, which corresponds to a period of 3.05 years and is in satisfactory agreement with the defined period of 3 years.

The only difference between the real SPFs in SVR and this model is a marked change in the peak amplitudes in the real plots, apparently due to the instability of the process generating SPFs in geomagnetic SVR. The fit of this model to real data leads to the conclusion that the SPFs in geomagnetic SVR are due to variations in the colatitude of the NAP of the eccentric dipole, i.e., nutations of the dipole axis in the plane containing the center of the Earth and the dipole axis with an amplitude of about 0.02° (22 km on the Earth’s surface). It can be suggested that these nutations of the dipole axis are related to changes in the geometry of the current system generating the dipole field. This high-frequency vibration of the generally unstable hydromagnetic dynamo mechanism (if we take this mechanism as the cause of the GMF) can promote the occurrence of GMF reversals (Braginskii, 1978; Langel, 1987; Parkinson, 1983). The nature of LPFs in geomagnetic SVR is more complex, as evidenced by the great variety of types of curves in different regions and even at MOs within the same regions. In regions

Fig. 11. Short-period SVR fluctuations based on the ED model for the 1955 epoch. Notation is the same as in Fig. 5.

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with a large number of MOs, particularly in Western Europe, it is possible to identify some predominant form of plots (Golovkov and Kolomiitseva, 1971), but in the other regions, the LPF curves are quite individual for each MO. Modeling of LPFs in SVR based on the ED field of the 1955 epoch showed that if magnetic moment variations are dominant, the curves for N and Z are in phase. Through this modeling, it could be possible to pick up time distributions of dipole axis nutations and magnetic moment variations that would give an effect similar to the SVR curves for individual MOs. The main obstacle to this is that the proposed mechanisms are global in nature, whereas the SVR curves at individual MOs for the same time vary greatly. It is suggested that the reason for these differences are electrical currents in the conductive outer core at the coremantle boundary, which vary among different regions (Golovkov and Kolomiitseva, 1983; Wardinski, 2005; Yaremenko et al., 2004). The existence of current systems in the lower mantle is possible as a result of the high conductivity of wustite, an important mantle component. At depths of 1000– 2200 km, wustite is converted to a low-spin state of iron with increased density and electrical conductivity (Ladynin, 2014; Lin et al., 2007; Plotkin et al., 2014).

Conclusions The secular variation rate (SVR) of the geomagnetic field components (horizontal intensity H, vertical intensity Z, and inclination I) shows two types of fluctuations: short-period fluctuations (SPFs) (3 ± 0.15 years) and long-period fluctuations (LPFs) (10–70 years). The amplitudes of the short-period and long-period fluctuations in SVR were estimated. The root-mean-square amplitudes of the short-period SVR fluctuations averaged over all 264 MOs are as follows: H = 8 ± 6 nT/year, Z = 13 ± 10 nT/year, and I = 0.6 ± 0.6 m/year. The rather large variation is due to the low accuracy of data from some MOs, especially those referring to the 19th and early 20th century. If they are excluded, the amplitudes of short fluctuations will be: H = 7 ± 4 nT/year, Z = 10 ± 7 nT/year, and I = 0.7 ± 0.5 m/year. The ratio of the DH and DZ amplitudes is equal to 0.6; in the polar regions, it is 0.55, and in the equatorial region, 0.8; thus, there is a small latitudinal dependence. The root-mean-square amplitudes of long fluctuations in SVR vary quite widely not only at different MOs, but also at individual MOs over time. Their values are 5 to 31 nT/year for H, 4 to 40 nT/year for Z, and 0.2 to 3.5 m/year for I; however, large amplitudes of SVR refer to the data of the 19th and the early 20th century. Short-period SVR fluctuations are uniform throughout the Earth. The DZ and DI fluctuations are synchronous and have the same phases, and the DH fluctuations are opposite in phase; this indicates that changes in the inclination angle of the dipole magnetic moment vector, rather than its magnitude, play a major role in the short-period SVR fluctuations.

Modeling of an eccentric-dipole field with variable latitude of the axial pole has shown that the SPFs in SVR are caused by fluctuations of the dipole axis (and the outer-core current systems responsible for the dipole field). Long-period SVR fluctuations manifest themselves differently in different regions, and their nature is dominated by the influence of currents in the liquid core at the core-mantle boundary and possibly in the lower mantle. The author expresses his deep gratitude to the staff of the global network of magnetic observatories for the opportunity of using the materials at the website www.geomag.bgs.ac.uk/ cgi–bin/means. The author is grateful to the reviewers of the paper P.G. Dyad’kov and S.Yu. Khomutov for the careful reading of the manuscript and valuable comments, which significantly improved the presentation of our results.

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