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Journal of Geodynamics 72 (2013) 5358

Contents lists available at ScienceDirect

Journal of Geodynamics
j ourna l h om epage: http://www.elsevier.com/locate/jog

Free core nutation and geomagnetic jerks
Z. Malkin
a b

a,b,

Pulkovo Observatory, St. Petersburg 196140, Russia St. Petersburg University, St. Petersburg 198504, Russia

article

info

abstract
Variations in free core nutation (FCN) are associated with different processes in the Earth's fluid core and coremantle coupling. The same processes are generally caused the variations in the geomagnetic field (GMF) particularly the geomagnetic jerks (GMJs), which are rapid changes in GMF secular variations. Therefore, the joint investigation of variations in FCN and GMF can elucidate the Earth's interior and dynamics. In this paper, we investigated the FCN amplitude and phase variations derived from VLBI observations. Comparison of the epochs of the changes in the FCN amplitude and phase with the epochs of the GMJs indicated that the observed extremes in the FCN amplitude and phase variations were closely related to the GMJ epochs. In particular, the FCN amplitude begins to grow one to three years after the GMJs. Thus, processes that cause GMJs are assumed as sources of FCN excitation. © 2013 Elsevier Ltd. All rights reserved.

Article history: Received 27 March 2013 Received in revised form 6 June 2013 Accepted 7 June 2013 Available online 26 June 2013 Keywords: Earth's rotation Free core nutation Geomagnetic field Geomagnetic jerks VLBI

1. Introduction Free core nutation (FCN) is one of the free rotational modes of the Earth with a rotating liquid ellipsoidal core and an ellipsoid mantle. FCN causes variations in the position of the celestial pole, with an average amplitude of about 0.2 mas and a period of approximately 430 days. Although FCN has been investigated for several decades, its excitation mechanism remains not fully understood. The period, amplitude, and phase of the FCN depend on factors such as structure, dynamic flattening and moments of inertia of the core, differential rotation of the core and mantle, and coremantle couґ pling (Dehant and Mathews, 2003; Brzezinski, 2005). Knowledge of processes that can predict the FCN behavior with microarcsecond accuracy is currently insufficient. Thus, only observations derived primarily from very long baseline interferometry (VLBI) provide the variations in FCN parameters. These observational data can be used to build FCN models, thereby elucidating the Earth's interior and dynamics. Previous FCN studies have indicated that the FCN amplitude and phase significantly vary with time (Souchay et al., 1995; Shirai and Fukushima, 2001; Herring et al., 2002; Malkin and Terentev, 2003a,b; Malkin, 2004b; VondrАk et al., 2005; Gubanov, 2010; KrАsnА et al., 2013). Given that FCN parameters depend on the processes in the fluid core and at the coremantle boundary and that the same processes are responsible for geomagnetic field (GMF)

Correspondence address: Pulkovo Observatory, St. Petersburg 196140, Russia. Tel.: +7 812 363 7442; fax: +7 812 704 2427. E-mail addresses: malkin@gao.spb.ru, zmalkin@zmalkin.com 0264-3707/$ see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jog.2013.06.001

variations, the latter may be identified as the cause of FCN amplitude and phase variations. Among the remarkable types of GMF variations are geomagnetic jerks (GMJs), which are observed as rapid changes in the GMF secular variations. The cause of these GMJs is most likely associated with the motions in the fluid core and coupling between the core and the mantle (Courtillot and Le Mouel, 1984; Gibert et al., 1998; De Michelis et al., 2005). The GMJs occur on a time scale of about one year, once or twice per decade, and are registered at geomagnetic observatories (as well as satellite observations) from the measurements of the vertical and the horizontal components of the GMF and magnetic declination, among others. They can be observed worldwide (global jerks) or within certain geographic regions (regional jerks). GMJs are not always registered simultaneously at all observatories. The difference between the epochs of the same jerk registered at different observatories can reach two years, and for some jerks a bimodal epoch distribution occurs (Alexandrescu et al., 1996; De Michelis et al., 2000; Gibert and Le MouКL, 2008). The difference in the registration epoch among observatories can be attributed to the inhomogeneity and anisotropy of mantle conductivity. Many studies published in recent decades have been devoted to the study of the connection between GMFs and the Earth's rotation variations. The correlation between the GMF variations and the length of day (i.e., the Earth's rotation velocity) (Le MouКl et al., 1981, 1992; Mandea et al., 2000; Holme and de Viron, 2005; Olsen and Mandea, 2007; Silva et al., 2012; Gorshkov et al., 2012) or the amplitude and phase of the Chandler wobble (CW) (Gibert et al., 1998; Bellanger et al., 2001, 2002; Gibert and Le MouКL, 2008; Malkin and Miller, 2010; Gorshkov, 2010) are usually investigated.

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Z. Malkin / Journal of Geodynamics 72 (2013) 5358

1

3

dX/dY error, mas
1985 1990 1995 2000 Year 2005 2010

dX, mas

2

0

1

-1

0 1985 1990 1995 2000 Year 2005 2010

1

Fig. 2. Errors in the IVS CPO series, mean values for dX and dY. Each point corresponds to one VLBI observing session. Unit: mas.

dY, mas

0

-1 1985 1990 1995 2000 Year
Fig. 1. IVS CPO series. Each point corresponds to one VLBI observing session. Unit: mas.

2005

2010

A possible association between the GMJs and FCN was considered for the first time in the study by Shirai et al. (2005), which revealed a close coincidence of two FCN phase jumps with two GMJs that occurred in 1992 and 1999. In the present study, we revisit the results of Shirai et al. (2005) by using a longer series of VLBI observations and new GMF data. We also consider three recent FCN models (two of which are proposed by the author) that are currently publicly available and continuously maintained by the authors. The remainder of this paper is organized as follows. In Section 2, we describe the FCN models used for this study. In Section 3, the amplitude and phase variations derived from these models are compared with GMJs. Section 4 concludes the paper. 2. FCN models All FCN models are constructed based on the analysis of the celestial pole offset (CPO) series obtained from VLBI observations. The CPOs are the differences dX, dY between the observed celestial pole position and the International Astronomical Union (IAU) official celestial intermediate pole, currently modeled by the IAU 2000/2006 precession-nutation theory (Petit and Luzum, 2010) Fig. 1 shows the combined CPO series provided by the International VLBI Service for Geodesy and Astrometry (IVS), (BЖckmann et al., 2010; Schuh and Behrend, 2012). The CPO data comprise two principal components, namely, the (quasi) periodic FCN term with a period of approximately 430 days and an average amplitude of about 0.2 mas, as well as low-frequency changes, including trend and long-period harmonics of similar amplitude caused mainly by the inaccuracy of the precession-nutation model. Regarding data quality, the results obtained before the 1990s are very noisy and involve large uncertainties (see Fig. 2). Malkin (2004a) analyzed in detail the evolution of the VLBI results and found similar error behavior for other geodynamical and astronomical parameters derived from the VLBI observations. Thus, CPO data collected prior to 1990 should be used carefully. The accuracy of the FCN models directly depends on the CPO data. Significant improvements in the accuracy of the VLBI results

are attributed to both the development in the VLBI technology (see, e.g., Ryan and Ma, 1998) and the increasing number of stations. The latter provides a large VLBI network that directly affects the precision and accuracy of the results, which is associated with the volume of network by exponential law (Malkin, 2009). The accuracy of the CPO results also largely depends on the accuracy of the radio source positions used during data processing (Sokolova and Malkin, 2007; Malkin, 2008). Therefore, the new version of the International Celestial Reference Frame ICRF2 (Ma et al., 2009) further improves CPO series. Three FCN models are currently available for users. These models are regularly updated so that they always contain the most recent data. However they differ in their underlying CPO data and method of analysis. The first model, ZM1, was proposed by Malkin (2004b) in 2003. This model, as well as the ZM3 model described below, is based on the ZM2 CPO series obtained by smoothing the IVS CPO series (Malkin, 2007). Smoothing has a twofold purpose. First, the VLBIderived CPO series include a significant noise component mainly caused by observational errors, and this noise is transferred to the FCN model. The original VLBI CPO series is then provided for the middle of the 24-h VLBI observing sessions, that is, for unevenly spaced epochs. Similarly with noise, it can lead to significant interpolation errors (Malkin, 2003). To prevent these problems, the ZM2 CPO series is computed by simultaneous Gaussian smoothing of the original IVS CPO series and its interpolation at the midnight epochs. In the ZM1 model, the FCN contribution to nutation is computed by dX = A(t ) sin dY = A(t ) cos A(t ) =
2

(t ), (t ),
2

(1)

dX + dY ,

where A(t) and (t) represent the FCN amplitude and phase, respectively. The FCN phase (t) is computed as follows. First, the band-pass filter is applied to the CPO series to extract the signal in the FCN frequency band. Subsequently, the apparent variations in time of the observed FCN frequency (t) are computed using the wavelet technique applied to the filtered CPO series. The FCN phase is finally given by
t

(t ) =
t0

dt + 0 ,

(2)

where 0 is the phase at the initial epoch J2000.0, which is an adjustable parameter of the model. Let us notice that the wavelet results can be distorted at the beginning and the end of the interval because of the edge effect. For FCN, this effect was investigated in Malkin and Terentev (2003a) and Shirai et al. (2005). Hence, although the ZM1 model is computed within the whole interval of the data, the resulting FCN series is published after removing

Z. Malkin / Journal of Geodynamics 72 (2013) 5358

55

FCN-dX, mas

the first and the last 18 months. Unfortunately, the method used to evaluate the ZM1 model does not allow the determination of the errors of the FCN values. It should be noted that from the mathematical point of view, the phase variation obtained from the CPO series analysis cannot be distinguished from the variations in the FCN period, as was previously discussed in Malkin (2004b, 2007). Strictly speaking, we have the variations in both the period and the phase that can be separated only by invoking other types of observations such as resonance of the forced nutations and gravity measurements, as well as geophysical analysis. Studies (Hinderer et al., 2000; Lambert and Dehant, 2007; VondrАk and Ron, 2009) indicated that the FCN period remains constant within ±2 days. Therefore, the dependence found is most likely caused by phase variations, which can be correlated with other geophysical observations. In 2004, S. Lambert proposed the FCN model1 (hereafter referred to as SL) constructed similarly with the MHB model (Herring et al., 2002), which was the first ever FCN model. The SL model is based on the C04 CPO series2 produced by the International Earth Rotation and Reference Systems Service (IERS) Earth Orientation Product Center at the Paris Observatory (Bizouard and Gambis, 2009). The parameters of the SL model are computed by the running two-year intervals with one-year shift. At each interval, the four parameters are adjusted according to the relations below: dX = Ac cos - As sin + X0 , dY = Ac sin + As cos + Y0 , (3)

5 4 ZM3 error, µas 3 2 1 0 1990 1995 2000 Year 2005 2010

Fig. 3. Errors in the ZM3 FCN series, the same values for dX and dY. Unit: mas.

0.4

ZM1 ZM3 SL

0.2

0

-0.2

FCN-dY, mas

where = 2 /PFCN (t - t0 ), PF CN is the FCN period equal to -430.21 solar days. Each pair of the Eq. (3) corresponds to one C04 epoch given with one-day step. The model parameters Ac , As , X0 , and Y0 are computed at the middle epoch of the two-year interval. Thus the resulting table of the FCN parameters is given with one-year step. To compute the FCN contribution, the same Eq. (3) is used without the shift terms X0 and Y0 . The Ac and As parameters should be interpolated at the given date using linear interpolation as recommended by the author of the SL model. This model forms part of the IERS Conventions (2010) (Petit and Luzum, 2010). The third FCN model, ZM3, developed by the author of this study, is computed by using a method similar to that used by Lambert for the SL model with three differences: -- the IVS CPO series is used as the input; -- the running reference interval of 431 days (one nominal FCN period) is used; -- the shift between sequential reference intervals is equal to one day. The first difference is a main source of the systematic differences between ZM3 and SL models. Using a shorter running interval only slightly influences the difference between models. Using a shorter running interval has small influence on the difference between models. The result is a slightly less smoother ZM3 model compared with the SL model. However, for this study we computed the ZM3 series with a two-year running interval to obtain a series exhibiting the same smoothness as that of the SL model. The use of a one-day shift between running intervals instead of the one-year shift used in the SL model was proven to facilitate interpolation and smooth variations in the FCN amplitude and phase (see Fig. 5). Fig. 3 shows the random errors in the ZM3 time series.

-0.4 1990 1995 2000 Year ZM1 ZM3 SL 2005 2010

0.4

0.2

0

-0.2

-0.4 1990 1995 2000 Year 2005 2010

Fig. 4. Three FCN series used in this study. Unit: mas.

These data also suggests that the FCN model prior to 1990 is not reliable. The ZM1 and ZM3 models are maintained at the Pulkovo Observatory and available at its Web site.3 All the three FCN series computed with the models described above are depicted in Fig. 4. These series are generally highly similar except at certain intervals such as near the minimum of the FCN amplitude in the 1990s.

1 2

http://syrte.obspm.fr/lambert/fcn/. http://hpiers.obspm.fr/eop-pc/

3

http://www.gao.spb.ru/english/as/persac/.

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Z. Malkin / Journal of Geodynamics 72 (2013) 5358

0.3

ZM1 ZM3 SL

0.2

0.1

0 1990 1995 2000 Year 200 ZM1 ZM3 SL 2005 2010

100 FCN phase, deg

0

-100

-200 1990 0.2 1995 2000 Year 2005 2010

ZM1 ZM3 SL

0.1

0

-0.1

-0.2 1990 1995 2000 Year
Fig. 5. Variations in the FCN amplitude (top), phase with the linear trend removed (middle), and FCN phase derivative (bottom) for the three FCN models. GMJs are indicated by vertical lines.

2005

2010

frequency was removed from the FCN phase series. We also computed the series of the first derivative of the FCN phase shown in Fig. 5 for enhanced detection of the epochs of changes in the phase variations. We evaluated the second derivative of the phase (not shown here); however, this result contained no useful information. Fig. 5 also shows the GMJs registered after 1990, namely, observed in 1991, 1999, 2003/2004, 2007/2008 (Mandea et al., 2010; Silva and Hulot, 2012; Chulliat et al., 2010; KotzИ et al., 2011). The latest jerk in 2007 is not thoroughly examined yet, and its epoch is not well established. We accepted epoch 2007.5 for this jerk as the average of the results of Chulliat et al. (2010) and KotzИ et al. (2011). We also added a possible jerk in 1994, which was observed in the geomagnetic data obtained at several observatories (Nagao et al., 2002; Olsen and Mandea, 2007; Mandea et al., 2010; PavСnCarrasco et al., 2013). This event could manifest of the 1991 jerk though. Comparison of the FCN amplitude and phase variation observed for the three FCN models reveals their similarity, especially after 2000. The three FCN series show the same epochs of the amplitude and phase derivative extremes. Despite the smoother curve of the ZM1 phase variation than those of the other two, the epochs of the characteristic points for this model are also well distinguished. The ZM3 and the SL models exhibit similar amplitude and phase behaviors; however, the latter shows inflection points resulting from the method used in constructing the SL model, as described in the previous section. Differences in the extreme epochs between the FCN models do not exceed one year. As mentioned previously, the uncertainty of the GMJ epochs is also approximately one year. Hence, the epochs of the FCN variations and the epochs of the GMJ coincide when the difference between the two epochs is not greater than two years with a possible general lag. The comparison (Fig. 5) indicates that all minima of the FCN amplitude follow corresponding GMJs with a delay of one year to three years. The maximum three-year lag is observed for the jerk in 2007; however, we can expect that the epoch of this jerk will further rectified after more detailed studies. This result is consistent with the conclusion drawn (Gibert et al., 1998; Bellanger et al., 2002), stating that the rapid changes in the CW phase follow the GMJs with a delay of one year to three years. Such coincidence shows that the same processes, presumably in the fluid core and/or at the coremantle boundary induce GMJs as well as variations in the FCN and CW amplitudes and phases. Fig. 5 shows a similar association between the GMJ and the FCN phase variations. However, a greater spread between the GMJ epochs and the epochs of the FCN phase changes is observed. In addition, an ambiguous situation occurs around the jerks in 1991 and 1994 (or one lasting jerk in the 19911994 period). The similarity of the top and the bottom plots in Fig. 5 suggests a significant correlation between the FCN amplitude and the phase (or period, as mentioned earlier) variations. A specific study investigating the presence of a physical connection between them or merely a coincidence arising from common geophysical processes should be conducted to clarify this point.

FCN phase, 1st derivative, deg/day

FCN amplituide, mas

4. Concluding remarks In this paper, we compared the variations in FCN with the GMJ epochs. Three different FCN models based on different CPO data and developed using different analysis techniques were adopted to derive a reliable conclusion. All three models yielded similar results. It was found that the observed extremes in the FCN amplitude and phase variations were closely related to the GMJ epochs. Moreover, GMJs immediately preceded the FCN amplitude minima, and the FCN amplitude began to grow immediately after the GMJs. This observation suggests that the FCN can be excited by

More different CPO and FCN series were discussed and compared in Malkin (2007, 2011). 3. FCN variations and GMJs We computed the variations of the FCN amplitude and phase for the three models described in the previous section. The results are shown in Fig. 5. The linear trend corresponding to the nominal FCN

Z. Malkin / Journal of Geodynamics 72 (2013) 5358

57

the same processes that cause GMJs. This assumption seems close to reality because the GMF is mostly generated by the flows in the core, and the same flows lead to variations in the moments of inertia of the core (as well as the whole Earth, to a lesser extent), hence, the variations in FCN (Dehant and Mathews, 2003). Such a relation between FCN and the GMJ can potentially improve our understanding of FCN excitation mechanisms. It is widely accepted that the FCN is mainly excited by the atmosphere with ocean contribution (Sasao and Wahr, 1981; Dehant ґ et al., 2003; Brzezinski, 2005; Lambert, 2006). However, this mechanism does not explain all the details of the FCN variations. Thus, the effect of GMF variations or other internal processes that cause variations in both GMF and FCN can help to eliminate the difference between the geophysical and the observed excitation. Meanwhile, new evidence on the correlation between the FCN variations and GMJs found in this study can provide additional insights on the constraints of the mechanism of GMJ generation as it occurs in CW (Bellanger et al., 2001) and more generally, on the physics of the flows in the fluid core and coremantle coupling. The correlation between the FCN variations and GMJs is verified for all five GMJs considered in the present study, including the supposed jerk in 1994, which is not officially recognized in the literature. In this respect, the following is worth considering. Current investigations of the association between the Earth rotation and GMF variations were "oneway" confrontations. Correlations were sought between the changes in the Earth rotation parameters and the given set of GMJs. However, the GMJs emanated from the upper core and coremantle boundary regions and registered at or above the Earth's surface after passing through the mantle, causing a variable delay, damping, or even screening of the signal. Consequently, some jerks originating from the core could not be registered reliably at the geomagnetic observatories or satellites. However, the same processes causing GMJs most likely influence the Earth's rotation. Thus, the observed changes in the Earth's rotation can indicate the preceding GMJs, including those not detected by modern geophysical methods. In our case, the presence of two consequent GMJs in 1991 and 1994 (see Fig. 5) can be concluded from the FCN amplitude analysis. Let us finally notice that the unpredictability of the FCN variations could be attributed to the relation between GMJs and FCN, considering that GMJs thus far remain unpredictable. Acknowledgement The author is grateful to two anonymous reviewers for their valuable comments and suggestions which helped to improve the paper. References
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Z. Malkin / Journal of Geodynamics 72 (2013) 5358 Shirai, T., Fukushima, T., 2001. Construction of a new forced nutation theory of the nonrigid earth. Astronom. J. 121 (June), 32703283. Silva, L., Hulot, G., 2012. Investigating the 2003 geomagnetic jerk by simultaneous inversion of the secular var