Inversion of Polar Motion Data: Chandler Wobble, Phase Jumps, and Geomagnetic Jerks Dominique Gibert, Jean-Louis Le Mouël

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Inversion of polar motion data: Chandler wobble, phase jumps, and geomagnetic jerks Dominique Gibert, Jean-Louis Le Mouël

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Dominique Gibert, Jean-Louis Le Mouël. Inversion of polar motion data: Chandler wobble, phase jumps, and geomagnetic jerks. Journal of Geophysical Research : Solid Earth, American Geophysical Union, 2008, 113 (B10), pp.B10405. 10.1029/2008JB005700. insu-00348004

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Inversion of polar motion data: Chandler wobble, phase jumps, and geomagnetic jerks Dominique Gibert1 and Jean-Louis Le Moue¨l1 Received 18 March 2008; revised 11 July 2008; accepted 24 July 2008; published 23 October 2008.

[1] We reconsider the analysis of the polar motion with a method totally different from the wavelet analysis used in previous papers. The total polar motion is represented as the sum of oscillating annual and Chandler terms whose amplitude and phase perturbations are inverted with a nonlinear simulated annealing method. The phase variations found in previous papers are confirmed with the huge phase change (3p/2) occurring in the 1926–1942 period and another less important one (p/3) in the 1970–1980 epoch. The best Chandler period, Tc, is found to be 434 ± 0.5 mean solar days. The epochs of significant phase changes found in the Chandler wobble are also those where important geomagnetic jerks occur in the secular variation of the geomagnetic field. Turbulent viscous friction produced by small-scale topography at the core-mantle boundary might be responsible for the observed phase jumps. Citation: Gibert, D., and J.-L. Le Moue¨l (2008), Inversion of polar motion data: Chandler wobble, phase jumps, and geomagnetic jerks, J. Geophys. Res., 113, B10405, doi:10.1029/2008JB005700.

1. Introduction and constitutes a new clue to the role of core-mantle coupling onto the Earth’s polar motion. [2] The movement of the Earth’s pole has been observed [4] In the present paper, we reconsider the analysis of the for more than 150 years, and, at the end of the 19th century, Earth’s polar motion with a new method totally different Chandler [1891a, 1891b] showed that this motion is the from the wavelet analysis used in our earlier papers [Gibert sum of two periodic oscillations, the forced annual oscilla- et al., 1998; Bellanger et al., 2002]. Also, the inverse tion and the Chandler wobble with a period close to 434 approach followed in the present study is more direct in mean solar days [e.g., Guinot, 1972; Lambeck, 1980; Wahr, the sense that the detrended total polar motion is analyzed 1988; Cazenave and Feigl, 1994]. Both the amplitude and without the complicated preprocessing stages of the wavelet the phase of the Chandler wobble vary with time [e.g., analysis. Moreover, we hope this new and simpler method Lambeck, 1980], and Danjon and Guinot [1954] first will be more easily used by independent teams in order to reported a phase change of about 150° which occurred perform analysis similar to the one presented here. around 1926 and was achieved in less than 10 years [5] In the remaining of the present paper, we first present [Guinot, 1972]. This phase jump remained poorly docu- the polar motion series and describe the model used to mented until other jumps were reported by Gibert et al. compute realistic synthetic series to be compared with the [1998] and suggested to be in temporal coincidence with data. Then, we detail the parameterization of the inverse geomagnetic jerks [Gibert et al., 1998; Bellanger et al., problem where the amplitudes of the annual and Chandler 2002]. Similar phase jumps have been detected in the Free components of the polar motion are to be retrieved together Core Nutation time series and also found in coincidence with the phase curve of the Chandler term. Next, the with geomagnetic jerks [Shirai et al., 2005]. nonlinear inversion is performed with the simulated anneal- [3] Different hypotheses have been proposed as the ing method and a Bayesian assessment of the uncertainty excitation mechanism of the Chandler wobble (see Dehant attached to the estimated parameters is done with the and de Viron [2002] for a short review), the main ones being Metropolis algorithm. the atmosphere [e.g., Aoyama and Naito, 2001], the pressure at the bottom of the oceans [Gross, 2000; Brzezin˜ski and Nastula, 2002; Gross et al., 2003], the earthquakes 2. Modeling of the Polar Motion [Mansinha and Smylie, 1967; Gross, 1986], or core motions 2.1. Polar Motion Data [Wahr,1988;Dickman, 1993]). The proposed temporal [6] The total polar motion, m(t), provided by the Interna- coincidence between geomagnetic jerks and phase jumps tional Earth Rotation Service (IERS) (data series eopc01) is in the Chandler wobble constitutes an exciting challenge shown in Figure 1a. Let us recall that the coordinate system is left-handed with the X axis coinciding with the Green- wich meridian and the Y axis corresponding to the W90

1 meridian. This signal is the superimposition of a smooth Institut de Physique du Globe, CNRS/INSU UMR 7154, Paris, France. secular drift d(t) and of an oscillating term o(t) resulting Copyright 2008 by the American Geophysical Union. from interferences between the annual prograde and retro- 0148-0227/08/2008JB005700$09.00 grade components, ap(t) ar(t), and the prograde Chandler

B10405 1of8 B10405 GIBERT AND LE MOUE¨ L: INVERSION OF POLAR MOTION DATA B10405 wobble c(t)[Lambeck, 1980]. In previous studies [Gibert et [10] As can be seen in equation (2), the amplitudes are al., 1998; Bellanger et al., 2002], we showed that the annual linear parameters, while the phase variation is a nonlinear prograde component is dominant with respect to the retro- one. In what follows, the inversion will not deal directly grade one which, however, remains significant (see the inset with these unknown functions, but instead will be per- in Figure 2c for an illustration of the importance of the formed for primary parameters from which the Apa(t), Ara(t), retrograde annual component). In the same studies, it was Ac(t) and Df(t) functions are deduced. More explicitly, the also shown that the Chandler wobble is affected by a small amplitude functions are obtained by linearly interpolating number of phase perturbations Df(t) whose temporal shape coarsely sampled amplitude functions. In practice, this reads may be modeled as sigmoid phase jumps with a short duration of the order of one year. Accounting for these ApaðÞ¼LIt Apa;1; t1 ; ÁÁÁ; Apa;k ; tk ; ÁÁÁ; Apa;N ; tN ; ð4Þ observations the total polar motion may be written as, mtðÞ¼otðÞþdtðÞþntðÞ AraðÞ¼LIt Ara;1; t1 ; ÁÁÁ; Ara;k ; tk ; ÁÁÁ; Ara;N ; tN ; ð5Þ ¼ apðÞþt arðÞþt ctðÞþdtðÞþntðÞ; ð1Þ where n(t) represents the noise presents in the data AcðÞ¼LIt Ac;1; t1 ; ÁÁÁ; Ac;k ; tk ; ÁÁÁ; Ac;N ; tN ; ð6Þ (Figure 1c). [7] In the present study we remove the secular drift before where LI represents the linear interpolation operator, and inverting the data. Owing to the smooth nature of d(t), we where the dates tk are fixed and regularly distributed with a represent it by the least squares fitted polynomials of degree sampling interval of 3 months. Consequently, for ampli- 5 shown in Figure 1a. No significant differences were tudes, the parameters to be actually determined are the observed when using either fourth and sixth degree poly- elements of the three sets {Apa,k, k =1,N}, {Ara,k, k =1,N}, nomials. The noisy oscillating part, o(t)+n(t), is obtained and {Ac,k, k =1,N}. by subtracting the fitted polynomials from the total polar [11] An identical parameterization is adopted for the motion and is shown in Figure 1b. phase variation function Df(t), [8] We remark that IERS data at face value, and the polar motion series shown in Figure 1a results from the merging 0 0 0 0 DfðÞ¼LIt Df1; t1 ; Df2; t2 ; ÁÁÁ; Dfk ; tk ; ÁÁÁ; DfM ; tM of series with data of different nature (for more details, see 7 IERS Web site http://www.iers.org). For instance, it cannot ð Þ be eliminated that some inconsistencies affect the data in the 0 1970–1980 period due to the merging of classical astrome- with the important difference that the dates t k are not fixed tric and space geodetic measurements [Gross and Vondra´k, but, instead, are to be determined together with the phase 1999]. variations Dfk. This parameterization makes the inversion highly non linear but, as will be seen later, it permits us to 2.2. Parameterization of the Polar Motion Model eventually represent the phase function with a very small [9] The signal to be inverted is the o(t) part of the total number M of parameters. polar motion and, in the present study, the model of signal [12] To summarize, the model to be determined is the set used in the forward problem reads, of parameters, M¼ A ; ÁÁÁ; A ; ÁÁÁ; A [pa;1 pa;k pa;N m 2ipt o ðÞ¼t ApaðÞt exp þ iDfpa Ara;1; ÁÁÁ; Ara;k ; ÁÁÁ; Ara;N Ta [ A ; ÁÁÁ; A ; ÁÁÁ; A 2ipt [ c;1 c;k c;N þ AraðÞt exp À À iDfra Ta fgDf1; ÁÁÁ; Dfk ; ÁÁÁ; DfM [ 2ipt 0 0 0 þ AcðÞt exp þ iDfðÞt ; ð2Þ t1; ÁÁÁ; tk ; ÁÁÁ; tM Tc [ Dfpa; Dfra ; ð8Þ where the real amplitudes Apa(t), Ara(t), Ac(t), the phase from which the functions Apa(t), Ara(t), Ac(t) and Df(t)are variation Df(t), and the constant phase shifts Dfpa, Dfra of the annual components are to be determined. The period deduced. We determine these parameters through a two-step T = 365.25 mean solar days (msd) and the Chandler period procedure by, first, searching for a good model fitting with a the data, and, second, exploring the nonuniqueness of the Tc is assumed known and constant during a given inversion. The Chandler period is not considered as a parameter to be solution. The first step is accomplished by using a global inverted since the phase variation Df(t) may incorporate a search method, namely simulated annealing, and the second period change dT through the linear correcting term, step uses a heat bath method based on the Metropolis c algorithm as explained in the sections 3 and 4. 2pt dT DfðÞt ¼À Â c : ð3Þ Tc Tc þ dTc 3. Inversion by Simulated Annealing 3.1. Method Equation (3) shows that a change in the choice of the [13] The search for a good model, Mbest, is done with the Chandler period Tc will result in a change in the linear drift simulated annealing algorithm which allows us to easily of the phase Df (see discussion below).

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Figure 1. Polar motion data. (a) Total polar motion series eopc01 provided by the International Earth Rotation Service, the X and Y components correspond to the thick and thin lines, respectively. Also shown are the order 5 polynomials representing the smooth secular drift, d(t), of the pole. (b) The oscillating polar motion, o(t), obtained by subtracting the order 5 polynomials from the original series, m(t), shown in Figure 1a. (c) The IERS’s estimate of the noise level for the original series. A logarithmic scale must be used for the ordinate axis in order to account for the huge reduction of the noise during the 20th century. For comparison with the signals shown above, average values of the error are 0.025 arc sec before 1960 and 0.012 arc sec in 1960–1970 period. The dramatic decrease of the noise level which initiates in 1976 is due to the incorporation of space geodetic measurements into the polar motion time series [Gross and Vondra´k, 1999]. deal with nonlinear models and sophisticated likelihood Mosegaard and Tarantola, 1995; Pessel and Gibert, functions. In the following, we shall only give the technical 2003]. The transformed likelihood is given by, details specifically adjusted for the present study, and the 1=T reader is referred to Metropolis et al. [1953], Kirkpatrick et LT ðÞ¼M L ðÞM : ð10Þ al. [1983], Bhanot [1988], and Sambridge and Mosegaard [2002] for general considerations concerning the Metropolis The simulated annealing begins with T !1for which L1 and the simulated annealing algorithms in the framework of equals a constant and stops at either T = 1 when the nonlinear inversion. likelihood is the one given by equation (9) or at T 0 with [14] In what follows, the likelihood function used to L0 d (Mbest ÀM) where Mbest is the searched best assess for the quality of a model M is taken Gaussian-like, fitting model (i.e., L(Mbest)>M). For each temperature T, hia Metropolis loop produces a sequence of models m 2 LðÞ¼M exp ÀkkotðÞþntðÞÀo ðÞt ; ð9Þ distributed according to LT. In practice, this consists in generating a sequence of models where the next model, j+1 2 2 M , to be added in the sequence is obtained from the where jjÁjj represent the L norm of a vector. For the preceding one, j, according to the random choice, moment, this likelihood function does not need to be either M scaled or normalized as it will serve only to quantify the "# L try relative goodness of fit of the models. jþ1 try T M prob M ¼M ¼ min 1; j ; ð11Þ [15] Simulated annealing consists in a sequence of LT M Metropolis loops while simultaneously applying a topolog- ical transformation to the likelihood L by decreasing a where Mtry is a candidate model to be eventually included control parameter T traditionally called the temperature in the sequence of models. Equation (11) indicates that a [Kirkpatrick et al., 1983; Gibert and Virieux, 1991;

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0.20 0.15 a) 0.10 0.05 0.00 -0.05

arc seconds -0.10 X residuals (T = 434 m.s.d.) -0.15 Y residuals (T = 434 m.s.d.) -0.20 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 years 40 -4 10 35 b) c) -5 10 30 -6 10 25 -7 10 20 -4 10 -8 10 15 -5 10

Spectral energy -9 10 10

normalized abundance -6 10 -10 5 10 -7 10 0.80 0.90 1.00 1.10 -11 0 10 -0.10 -0.05 0.00 0.05 0.10 0.1 1 10 arc seconds Period (years)

Figure 2. (a) Residuals (data minus synthetics) for the X and Y components obtained for the model fitted with Tc = 434 msd. (b) Histogram of the residuals: the wide histograms correspond to the residuals before 1960 (SD = 0.04435), and the narrow histograms are for the residuals after 1965 (SD = 0.01426). Thick and thin curves are for the X and Y components respectively. (c) Energy spectrum of the residuals for the X (thick curve) and Y (thin curve) components; inset is magnification of the spectra of the X component for residuals of models with (thick line) and without (thin line) the retrograde annual component. The same result is observed for the Y component. more likely model is always accepted and that a less likely that the random generator returns a zero value. The phase model is sometimes accepted. When the random assignment variation is perturbed by randomly drawing a parameter j given by equation (11) fails, i.e., when the less likely Dfk in M and perturbing it with the rule, candidate model has been rejected, the replicating transition Mj+1 = Mj is applied. As the Metropolis loop proceeds, a try j Dfk ¼ Dfk þ rand0:5½À30 ; þ30 : ð13Þ sequence of models is generated such that more models fall in the most likely regions of the model space. When the 0 The corresponding date, t k, is also modified according to, temperature decreases, these regions have their likelihood dramatically (i.e., exponentially) augmented, and if the ;try ;j t0 ¼ t0 þ rand ½À0:75 year; þ0:75 year : ð14Þ temperature is slowly lowered, the chain of models is gently k k 0:5 guided toward the regions of the model space where the likelihood is maximum [Metropolis et al., 1953; Bhanot, [18] Observe that the random generator is biased in both 1988]. equations (13) and (14), so that a drawn parameter is try [16] The candidate model, M , is obtained by perturbing actually perturbed only half of the time. We found this rule the current model Mj in order to give some memory to the necessary because of the very different roles played by the Metropolis chain. This point is particularly important since various parameters. We indeed observed that the phase it makes the algorithm not a simple Monte Carlo search. In parameters are more rapidly determined than the amplitude the present study, we observed that the perturbation strategy ones. A way to deal with this situation is to reduce the is not a critical matter, and the method we retained consists perturbation of the phase parameters which become deter- in randomly choosing the parameters to be perturbed in the mined first. The biased random generator provides a solu- j 0 parameter set M . tion in the sense that no perturbation of both Dfk and t k occurs 25% of the time (equations (13) and (14)). Further- [17] The perturbation applied to the amplitudes Apa,k, more, the phase variations Df have quite a complicated Ara,k, Ac,k is randomly assigned with the following rule: k contribution to the misfit budget, i.e., in the likelihood L, try j since those with dates falling at the beginning of the 20th A ¼ A þ rand ½À0:02 arc sec; þ0:02 arc sec ; ð12Þ :;k :;k 1 century concern longer portions of signal than those place near the end of the century. When running the simulated where randa[x1, x2] represents the biased uniform random annealing algorithm, we observed a hierarchical determination generator in the [x1, x2] interval with the probability 1 À a

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0.30 a) Chandler envelope so that a global sampling of the model space is performed [Kirkpatrick et al., 1983; Mosegaard and Tarantola, 1995] 0.20 In the present study, we use Tstart = 1 for which almost all models Mtry (starting, for example, from a first model with 0.10 arc seconds a constant amplitude and a null phase) are accepted in the Metropolis sequence, and 20 trial models Mtry were tested 0.00 1900 1920 1940 1960 1980 2000 (and a fraction of them retained, see equation (11)) for each 0.30 Metropolis sequence. The final temperature was taken as b) Annual envelopes T =10À5. A total of 107 models were tested during the whole 0.20 process, and several inversions were done with Tc = 431, 432, 433, 434, and 435 mean solar days. For these inver- 0.10 sions, we set M = 23 and N = 425 (see equation (8)). A large arc seconds value was chosen for N to ensure a good modeling of the 0.00 envelopes of the signals and eliminate the possibility that a 1900 1920 1940 1960 1980 2000 bad envelope modeling could produce phase perturbations. date (years) On the opposite, a rather small value was chosen for M in Figure 3. (a) Envelopes of the Chandler component, and order to comply with the hypothesis that only a small (b) the prograde (top curves) and the retrograde (bottom number of phase impulses are present in the Chandler curves) annual components obtained by inverting the component. This strong hypothesis was relaxed inverting the most recent part of the polar motion series with a large detrended polar motion (Figure 1) Tc = 434 mean solar days. The thick curves are the envelopes of the average M = 71 (see below). model of the models produced by Metropolis sequences, [21] Figure 2a shows the residuals corresponding to the and the thin curves represent the extremes envelopes of all best fitting model obtained for Tc = 434 msd. Similar results models accepted in these sequences. The horizontal dashed were obtained for the other Chandler periods. The most conspicuous feature visible in Figure 2a is the dramatic line represents the zero level in order to show that the 5 retrograde annual amplitude remains significantly different reduction of the residuals level for t 1962. Histograms of from zero in the recent period where the data are more the residuals (Figure 2b) which give standard deviations of accurate. 0.04435 arc sec and 0.01426 arc sec before 1960 and after 1965, respectively. Note that the decrease of the amplitude of the residuals in 1962 coincides with the decrease of the noise starting in 1962 as estimated by IERS (Figure 1c). of the parameters, with those placed at the beginning of the That means that the big residuals before 1962 (Figure 2b) century being locked more quickly than those placed later. was essentially noise for the largest part. Afterward, the [19] The temperature schedule, i.e., the manner by which IERS estimation of noise keeps decreasing strongly, a the temperature T is lowered, must be carefully designed to characteristics which is not reflected in our residuals. We obtain a successful convergence toward models with a high have to conclude that signal remains in our residuals. likelihood. In the present study, the main difficulty we [22] Figure 2c shows the energy spectrum of the residuals encountered was related to the hierarchical importance of Figure 2a. This spectrum is reasonably white in its high- among the parameters to be determined. In practice we frequency domain and becomes more similar to a pink noise observed that too rapid a cooling made the Metropolis for periods greater than about 3 years. The inset in Figure 2c sequences of models get trapped in local minimums located shows a comparison of the energy spectra of the residuals far from the best models. A detailed analysis of this obtained for models with and without the retrograde annual difficulty revealed that the trapping was due to parameters component. It can be seen that the spectrum for the model associated with dates early in the century. Indeed, changing without the retrograde annual term has a significant peak at the value or the date of such parameters produced huge the period of 1 year. This peak disappears in the spectrum perturbations in the more recent part of the synthetic signals, corresponding to the model with the retrograde annual resulting in a high misfit and, consequently, in a low component. This result favors the use of a retrograde annual probability. Since the lower the temperature, the lower the term in the fitted model (equation (2)). probability to accept a poorer model in the Metropolis sequence, the perturbations applied to the parameters placed early in the 20th century were always rejected, and the 4. Posterior Probability Metropolis sequence was unable to escape from the local 4.1. Method minimum. This problem was solved by controlling the [23] The residuals obtained for the best fitting models cooling with the constant thermodynamical speed algorithm found with the simulated annealing can now be used to alreadyusedbyGibert and Virieux [1991]. The main perform a statistical Bayesian inversion. Indeed, the likeli- advantage of this algorithm is to automatically reduce the hood function given by equation (9) can be completed to perturbations of the parameters when the convergence obtain a normalized probability density function, accelerates. "# kkotðÞþntðÞÀomðÞt 2 3.2. Results LðÞ¼M b exp À ; ð15Þ 2s2 [20] The starting temperature, Tstart, of the simulated annealing must be large enough to ensure that LTstart 1

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differ by the global drift of their linear segments where no significant phase jumps occur (e.g., the two segments corresponding to the horizontal arrows in Figure 4). Owing to the fact that a global linear drift may be due to a wrong choice for the value of Tc as shown by equation (3), the right period is the one for which these drifts are minimized. Accordingly, one can observe that the curve corresponding to Tc = 434 msd appears free of linear drift while those for Tc = 433 and 435 msd display a small but significant negative and positive drift, respectively. We conclude that the Chandler base period may be taken as Tc = 434 ± 0.5 mean Figure 4. Phase variation, Df(t), inverted for Tc = 434 mean solar days. The thick curve is the phase curve of the solar days in full agreement with our previous determina- average model of all models produced by a Metropolis tion based on parsimony arguments [Gibert et al., 1998]. sequence, and the thin curves represent the extreme [27] Another inversion restricted to t 1970 was performed with a model counting a large number (M = 71) of envelopes of the models accepted during the sequence. 0 The dashed curve represents the phase variation obtained by parameters Dfk and t k. The resulting phase curve is shown integrating the ridge function (i.e., the instantaneous in Figure 6; interestingly, we observe that this curve is very frequency curve) of the Chandler prograde component similar to the one obtained with a small number of param- shown in Figure 1 (top) of Bellanger et al. [2002]. The eters, confirming that the angular behavior of the phase vertical bands represent the dates of occurrence of the variations is not due to modeling artifacts. geomagnetic jerks. The width of each band is tuned according to the dating uncertainties reported by Alexandrescu 5. Discussion et al. [1995, 1996]. The multiple bands associated with the [28] Let us write the Chandler wobble time series in the 1969, 1980, and 1991 jerks represent the bimodal nature of form their occurrence date histograms [Alexandrescu et al., 1996; De Michelis and Tozzi, 2005]. ctðÞ¼A ðÞt exp½iðÞw t þ fðÞt c Zc t ¼ AcðÞt exp i ðÞwc þ dfðÞ t dt ; ð16Þ where b is a suitable normalizing constant and s equals 0 either 0.04435 arc sec or 0.01426 arc sec for t smaller or greater than 1962, respectively. where wc =2p/Tc. The origin of time is taken at the [24] As explained before, the Metropolis algorithm beginning of the time series. The two aboveR expressions are t generates a sequence of models distributed according to obviously strictly equivalent, with f(t)= 0df(t)dt + f0, 0 the likelihood functions. So, by running a Metropolis loop and df(t)=f (t). It is also clear that changing wc or Tc for T = 1 and L given by equation (15), we obtain a set of comes down to adding a linear trend to f(t): wc ! wc + dwc, acceptable models from which uncertainties may be com- f(t) ! f(t) À dwct. For the reasons given above, we chose puted. In practice, the computer code is the one used for the Tc = 434 (Figures 4, 5, and 6). simulated annealing except that the starting model is the last [29] Let us compare the phase variations shown in one obtained in the simulated annealing inversion and that Figures 4 and 6 with our previous results [Gibert et al., the temperature is maintained equal to 1. Let us also remark 1998; Bellanger et al., 2002], i.e., the phase variation curve that the normalizing constant b in equation (15) has not to obtained by integrating the ridge function of the wavelet be computed in practice since it cancels in the likelihood analysis of polar motion data [Bellanger et al.,2002]. ratio used in the Metropolis test (11). Figure 4 shows that the latter curve is in full general agreement with the phase variations obtained in the present 4.2. Results study with a totally different method; however, the view of [25] Figure 3 shows the amplitudes Apa(t), Ara(t), and Figures 4, 5, and 6 reveals that the phase variations Ac(t) interpolated from the model with Tc = 434 msd. The thick lines represent the average amplitudes for a Metropolis sequence of 106 models. The thin lines are the envelopes of the whole set of models accepted during the sequence. As expected, the envelopes are closer after 1962, when the noise level is smaller. Interestingly, at that date, the envelope of the retrograde annual component becomes smaller but with its lower envelope slightly above zero. At the same time, a clear decrease is observed in the prograde annual component amplitude which recovers a lower level similar to the one of the period 1900–1906. [26] Figure 4 shows the phase variation, Df(t), interpolated from the model with Tc = 434 msd. Again, a reduction of the uncertainty domain is observed for t 1962. Figure 5 Figure 5. Phase variation Df(t) evolution of the Mbest models obtained for five reference periods Tc = 431, 432, shows the phase curves for Tc = 431, 432, 433, 434, and 435 mean solar days. All curves possess similar features but 433, 434, and 435 mean solar days.

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Figure 6. The thin solid line represents the phase variation, Df(t), inverted for a reference period Tc = 434 mean solar days and for a large number M = 71 of pilot points. The thick solid line represents the phase curve inverted for a small number (M = 9 in the represented time interval) of pilot points (Figure 5). The dashed curve represents the phase variation obtained by integrating the ridge function (i.e., the instantaneous frequency curve) of the Chandler prograde component shown in Figure 1 (top) of Bellanger et al. [2002]. The vertical bands represent the dates of the recognized geomagnetic jerks (see caption of Figure 4 for details). recovered in the present paper appear sharper than the secular variation [Gibert et al., 1998; Le Huy et al., 2000; smooth curve obtained by wavelet analysis, which does not Bellanger et al., 2001]. Other interpretations have been display by construction [see Gibert et al., 1998, Figure 7], the proposed, for example, that the jerks can be explained by segmented nature of the phase variations. Note that this a core flow consisting of steady motion and torsional segmented nature of the curve of the present study oscillations [Bloxham et al., 2002]. (Figure 5) is not an effect of the algorithm characteristics [32] But in this paper we focus on the spectacular phase since the main pattern is preserved in the high-resolution change of 3p/2 which corresponds to a decrease of the inversion of Figure 6; it mean that the o(t) data presents apparent Chandler period of 23 days (equation (3)) from important phase variations at discrete times. The last 1926 to 1942 [Danjon and Guinot, 1954], and which increasing part of the integrated curve starts in 1994, while follows the 1926 magnetic jerk. As far as we know, this only in 1997 for the curves obtained in the present study. large variation has never been convincingly explained by an This discrepancy is probably due to a small edge effect in external forcing, including the atmosphere and/or the ocean, the wavelet analysis. nor by an unusual series of big earthquakes. If external or [30] Let us come back to the analysis of Gibert et al. superficial causes fail to provide the right mechanism, it is [1998] and Bellanger et al. [2002]. In this paper we located reasonable to call for deep sources, i.e., interactions with the phase jumps near the dates of geomagnetic jerks determined core, which is a big reservoir of angular momentum. That is from geomagnetic observatory series, and we adjusted the why we are motivated to look for the exchanges of angular amplitudes and durations of the phase jumps, supposed to momentum between the core and the mantle as a possible have a sigmoid shape, in such a way that the phase curves cause for these large phase changes, to some extent inde- determined from this modeling be identical to the integrated pendently of the suggested coincidence of magnetic jerks ridge function obtained from the data by wavelet analysis. with wobble phase changes. The close similarity between the phase curve obtained in the [33] The difficulty is that we need big enough core- present paper and the one computed earlier buttresses of our mantle torques with short time constants are needed. More former analysis (i.e., locating phase jumps at the jerks time). precisely, torques of the order of 1021 N.m with a duration [31] Geomagnetic jerks are short-duration events which shorter than 430 days, frequent enough, and which do not punctuate the evolution of the fluid core motion at the core- cancel each other, are requested. In some previous papers mantle boundary (CMB); such events have been detected in [Hinderer et al., 1987, 1990; Bellanger et al., 2001] we 1901, 1926, 1932, 1949, 1969, 1980 [Alexandrescu et al., called for the geostrophic torque resulting from the pressure 1995, 1996], and 1991 [Macmillan, 1996; De Michelis and p associated with the geostrophic flow ~u at the top of the Tozzi, 2005]. The time of occurrence of a jerk is not core, acting on the large-scale bumps of the CMB [Jault and determined with a high accuracy for two main reasons: Le Moue¨l, 1989; Hide, 1969] to explain slow irregularities (1) the (weak) conductivity of the mantle smooths out the in the drift of the Earth’s mean pole with time constants of a shortest periods of the event [Mandea-Alexandrescu et al., few tens of years. But the difficulty with the excitation of 1999] and (2) the time of the event can be somewhat the Chandler wobble is, as said above, the need for a dense different in different observatories; for example, in 1969 sequence of short torques which, however, do not lead to and 1978, the Southern Hemisphere lags the northern one unacceptable velocity field ~u. This requirement practically by some 2 years [Alexandrescu et al., 1996]. We have rules out the topographic torque [Jault and Le Moue¨l, reported the times of the 20th century jerks, with their 1993], taken in the sense given above. Recently, however, uncertainties, on Figure 4. It seems to us again that it is not it was proposed [Narteau et al., 2001] that the CMB, in so far-fetched to propose a correlation between the sequence addition to the large-scale topography dynamically sus- of these events and the sequence of accidents in the phase of tained by the mantle convection, might present a small- the Chandler wobble. We interpret the jerks as marking scale roughness, with typical length scales ranging from a changes in the flow at the top of the core responsible for the meter to hundreds of meters, due to physical-chemical

7of8 B10405 GIBERT AND LE MOUE¨ L: INVERSION OF POLAR MOTION DATA B10405 interactions between the core and the mantle. A torque Gross, R. S. (1986), The influence of earthquakes on the Chandler wobble during 1977–1983, Geophys. J. R. Astron. Soc., 85, 161–177. between the core and the mantle then results from the Gross, R. S. (2000), The Excitation of the Chandler wobble, Geophys. Res. mantle topography displacing volumes of core fluid, some Lett., 27, 2329–2332. kind of turbulent viscosity coupling, when there is a relative Gross, R. S., and J. Vondra´k (1999), Astrometric and Space-Geodetic motion ~u between the two envelopes. This torque happens Observations of Polar Wander, Geophys. Res. Lett., 26, 2085–2088. 2 Gross, R. S., I. Fukumori, and D. Menemenlis (2003), Atmospheric and to be proportional to jj~ujj [Le Moue¨l et al., 2006]. Without oceanic excitation of the Earth’s wobbles during 1980–2000, J. Geophys. changing significantly the parameters adopted in the latter Res., 108(B8), 2370, doi:10.1029/2002JB002143. paper, it is possible, although difficult, to meet the require- Guinot, B. (1972), The Chandlerian wobble from 1900 to 1970, Astron. Astrophys., 84, 207–214. ments stated at the beginning of this paragraph for the Hide, R. (1969), Interaction between the Earth’s liquid core and solid observed 1926 phase change; this phase change does exist mantle, Nature, 222, 1055–1056. and has not received, to our knowledge, any other convinc- Hinderer, J., C. Gire, H. Legros, and J.-L. Le Moue¨l (1987), Geomagnetic secular variation, core motions and implications for the Earth’s wobble, ing explanation. In this view, geomagnetic jerks mark Phys. Earth Planet. Inter., 49, 121–132. changes in the top core flow characteristics and, as a Hinderer, J., D. Jault, H. Legros, and J.-L. Le Moue¨l (1990), Core-mantle consequence, in the torque exerted by the core on the topographic torque: A spherical harmonic approach and implications for the excitation of the Earth’s rotation by core motion, Phys. Earth Planet. mantle and conversely. We are currently developing this Inter., 59, 329–341. analysis, its fair to say that the subject is not free of Jault, D., and J.-L. Le Moue¨l (1989), The topographic torque associated uncertainties. with a tangential geostrophic motion at the core surface and inferences on the flow inside the core, Geophys. Astrophys. Fluid Dyn., 48, 273–296. Jault, D., and J.-L. Le Moue¨l (1993), Circulation in the liquid core and [34] Acknowledgments. The data analyzed in this study were pro- coupling with the mantle, Adv. Space. Res., 13, 221–233, doi:10.1016/ vided by the International Earth Reference Service (http://www.iers.org). 0273-1177(93)90225-Z. Associate Editor Giorgio Spada and referees Mathieu Dumberry, Richard Kirkpatrick, S., C. D. 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