The Earth's free core nutation: Formulation of dynamics and estimation of eigenperiod from the very-long-baseline interferometry data

JID:EPSL

AID:13515 /SCO

[m5G; v1.161; Prn:20/10/2015; 13:51] P.1 (1-10)

Earth and Planetary Science Letters ••• (••••) •••–•••

Contents lists available at ScienceDirect

Earth and Planetary Science Letters www.elsevier.com/locate/epsl

The Earth’s free core nutation: Formulation of dynamics and estimation of eigenperiod from the very-long-baseline interferometry data B.F. Chao ∗ , Y. Hsieh Institute of Earth Sciences, Academia Sinica, Taipei, Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 26 July 2015 Received in revised form 24 September 2015 Accepted 2 October 2015 Available online xxxx Editor: B. Buffett Keywords: free-core nutation convolution/deconvolution resonance natural period VLBI

a b s t r a c t The free-core nutation (FCN) is a rotational normal mode of the Earth’s outer core. We derive the equations of motion for FCN w.r.t. both the inertia space F 0 and the uniformly rotating frame F  , and show that the two sets of equations are invariant in form under the reference frame transformation, as required by physics. The frequency-domain formulation describes the FCN resonance (to nearby tidal signals), which has been exploited to estimate the complex eigenfrequency of FCN, or its eigenperiod P and quality factor Q . On the other hand, our time-domain formulation in terms of temporal convolution describes the response of the free FCN under a (continual) excitation. The convolution well explains the dynamic behaviors of FCN in the observed very-long-baseline interferometry (VLBI) data (in F 0 ), including the undulation of the FCN amplitude and the apparent fluctuations in the period and phase over time, as well as the temporal concurrence of a large phase jump with the near-zero amplitude during ∼1998–2000, in complete analogy to the observed behavior of the Chandler wobble (in F  ). The reverse, deconvolution process is further exploited to yield optimal estimates for FCN’s eigenfrequency using the VLBI data, following the approach of Furuya and Chao (1996) of locating minimum excitation power. While this method is found to be insensitive to Q owing to the short timespan of the data, we obtain the estimate of P = 441 ± 4.5 sidereal days (sd) where the 1-sigma uncertainty is assessed via extensive Monte Carlo simulations. This value is closer to the theoretical value of ∼460 sd predicted by Earth models assuming hydrostatic equilibrium than do the prior estimates (425–435 sd) by the resonance method. The deconvolution process also yields the excitation function as a by-product, the physical sources of which await further studies. © 2015 Elsevier B.V. All rights reserved.

1. Introduction The classical astronomical precession–nutation of the Earth’s rotational axis is driven by the luni-solar tidal torques exerted on the oblate, quasi-rigid Earth (e.g., Melchoir, 1983; Wahr, 1981a). In parallel, the Earth has a rotational normal mode known as the free-core nutation (FCN), a retrograde motion (clockwise as viewed from north) of the misalignment of the rotation axis of the spheroidal fluid outer core w.r.t. the figure axis of the spheroidal solid mantle (Toomre, 1974; Smith, 1977; Wahr, 1981b). Both the astronomical nutations and FCN have periods much longer than one day w.r.t. the inertial space, or near-diurnal periods w.r.t. the rotating Earth.

*

Corresponding author. E-mail address: [email protected] (B.F. Chao).

http://dx.doi.org/10.1016/j.epsl.2015.10.010 0012-821X/© 2015 Elsevier B.V. All rights reserved.

The FCN is just one of Earth’s rotational modes which also include the better-known Chandler wobble (CW), along with those supposedly belonging to the solid inner core (the so-called prograde free-core nutation and the inner core wobble) (e.g., Mathews et al., 2002; Dehant and Mathews, 2007). In this sense the nutations can be regarded as the rotational response of the FCNresonance system, just as the polar motion is regarded as the rotational response of the CW-resonance system, to various astronomical and geophysical forcings. The nutation terms are driven by the discrete-frequency luni-solar tidal torques but modified in amplitude and phase by the FCN resonance; only the few of these tidal components at periods in close proximity to that of FCN are modified (amplified) to an appreciable extent. The FCN itself may or may not appear in actual observations, depending on whether and how strongly it is actually excited by pertinent geophysical excitation mechanisms whatever they may be. The very-long-baseline interferometry (VLBI) technique has been measuring the Earth’s nutations since the early 1980s.

JID:EPSL

AID:13515 /SCO

2

[m5G; v1.161; Prn:20/10/2015; 13:51] P.2 (1-10)

B.F. Chao, Y. Hsieh / Earth and Planetary Science Letters ••• (••••) •••–•••

By the 1990s, as the VLBI data accumulated and continually improved in precision, a significant FCN signals began to emerge with varying amplitudes as large as 0.1–0.5 milliarcsecond (mas) peakto-peak (Herring and Dong, 1994; Shirai and Fukushima, 2001; Herring et al., 2002; Lambert and Dehant, 2007). In this paper we shall study the FCN signal in the VLBI nutation data in two fronts: First, we shall formulate the physics of FCN, that of a forced 2-D simple harmonic motion in a uniformly rotating reference frame. It takes the form of a temporal convolution of the resonance with some excitation function. We do not inspect the identity and behavior of the excitation process itself. Rather we grant the existence of the excitation for the FCN, and show that the physics of the convolution well explains the general behavior of the FCN observed in the VLBI data. Secondly, exploiting the convolution formulation we shall estimate from the VLBI data the FCN’s complex eigenfrequency, i.e. its natural period (eigenperiod) and the decay rate or quality factor. These are gross Earth data that contain important information about the property of the core and core–mantle interactions (e.g., Mathews et al., 2002; Dehant and Mathews, 2007). Prior estimates of the eigenperiod (w.r.t. to inertia frame) by means of the resonance method clustered around 425–435 sidereal days (sd) (see e.g. Rosat et al., 2009, for a review). That is significantly shorter than the 460 sd predicted by idealized Earth models under the assumption of rotation-gravitational hydrostatic equilibrium (Sasao et al., 1980; Wahr, 1981b). That suggests a core configuration that is significantly more oblate than the hydrostatic equilibrium (Gwinn et al., 1986), presumably under the influence of certain core–mantle torquing mechanism yet unattained in numerical or physical modeling (Buffett et al., 2002). Meanwhile, the estimates of FCN’s quality factor range upwards from a few thousand but remain poorly constrained. Here we shall revisit this subject and, alternative to the resonance method, we estimate the FCN’s eigenfrequency by the deconvolution approach of Furuya and Chao (1996); see also Gross (2007). We reach optimal estimates of the FCN eigenperiod which lie between the prior estimates and the theoretical value. 2. VLBI data Referencing to distant celestial quasars, the VLBI technique measures, among other things, the 3-D Earth rotation parameters starting in the early 1980s, nowadays achieving accuracies better than ∼0.1 mas (IERS Conventions, 2010). As part of the Earth rotation parameters, the 2-D nutational motion in the Earth’s rotation axis orientation in space is customarily given in terms of the celestial pole offsets dψ and dε , i.e. the deviations of the longitude ψ and the obliquity ε of the equator in the ecliptic coordinates, referenced to the model values. The VLBI nutation dψ and dε data used presently are the combined EOP 08 C04 data series from the International Earth Rotation and Reference Systems Service (IERS). The data are referenced to the IAU 2006/2000A precession–nutation model (e.g. Wallace and Capitaine, 2006) consistent with ITRF2008 reference system (Bizouard and Gambis, 2009) and adopted by IERS Conventions 2010 based on and updated from Mathews et al. (2002). The IAU2000A reference model accounts for all the nutation terms considered to be the Earth’s response to the luni-solar tidal forcings (where FCN resonances are considered), whereas the physical parameters that are poorly known yet relevant to the rotation are estimated to best match the VLBI data. The FCN signal itself is left intact. The VLBI data as provided have been homogenized and slightly smoothed to nominal intervals of 1 solar day. Our analysis is solely based on post-1992 data for their better quality, spanning 23 years of 1992–2014, long enough to resolve spectrally the FCN from

nearby tidally driven terms. We shall refer to this VLBI data series as the “full dataset”, whereas any segment thereof as n-year segment dataset. Only for comparison purposes will we present the rather noisy pre-1992 data and results derived from them. To ensure the “cleanness” of the FCN signal we remove any residual tidal terms as well as the seasonal terms of non-tidal meteorological origin (after editing out obvious out-lier points). We do so by the linear least-squares regression (on the full dataset) and subtraction of the following periodic terms: the major tidal terms of Mf (13.6608 days), Msf (14.7653 days), Mm (27.5546 days), 9.31 year and 18.6 year, two seasonal (annual and semiannual) terms, and a long-term linear trend accounting for any unmodeled precession. The removed terms are actually quite small, making no appreciable differences. Fig. 1 gives the “cleaned” FCN time series of dψ and dε , along with the time-frequency wavelet spectrum of the complex quantity m(t ) = sin ε0 dψ(t ) + idε (t ) (see Equation (5) below) calculated adopting the Morlet wavelet, a normalized Gaussian-enveloped cosine function (e.g., Chao et al., 2014). The noisiness of the pre-1992 data is evident. The dominant (retrograde) FCN signal across the wavelet spectrum at the (negative) period somewhat longer than a year is well captured. Note the fluctuations in the apparent period of this FCN signal, and the considerable time-undulations in amplitude which all but disappeared temporarily during the late 1990s. We shall return to these observations later. 3. Kinematics of FCN It is crucial to consider two distinct fundamental reference frames. The reference frame set in the inertia space is referred to as F 0 , which is equivalent to the celestial reference frame to the best of its realization. The other reference frame, referred to as F  , undergoes a uniformly rotation at a constant angular velocity  w.r.t. the inertia space,  = ˆz, where zˆ is the unit vector pointing to the mean Earth rotation axis for the last half century, and the magnitude  = 2π radians per sd = 7.292115 × 10−5 rad s−1 , equivalent to 1/86 164 s−1 or 1 cycle per sd. F  is idealized in the sense that it is not observationally realized on the Earth but adequately approximates the diurnally rotating terrestrial reference frame to describe the physics below (e.g., Smith, 1977; Chao, 1983). The transformation between F 0 and F  is purely kinematic and rather simple: For a function m(t ) = mx (t ) + im y (t ) describing a 2-D motion in the equatorial plain (xˆ , yˆ ) with the Cartesian coordinates x (the real part) and y (the imaginary part), we can write, following Brzezinski and Capitaine (1993):

m (t )[w.r.t. F  ] = m(t )[w.r.t. F 0 ] · exp(−i t ).

(1)

Here, contrary to the literature, we use the primed symbol to denote quantities w.r.t. F  . Upon taking the Fourier transform, Equation (1) amounts to a simple shift in the angular frequency:

ω [w.r.t. F  ] = ω[w.r.t. F 0 ] − ,

(2)

as depicted in Fig. 2. The positive frequency indicates prograde motion (as of CW), and negative frequency retrograde motion (as of FCN). In particular, the (retrograde) “nearly diurnal free wobble” (an old terminology for FCN) in F  transforms to the (retrograde) FCN with a near-zero frequency in F 0 (thereof Equation (2) takes on a form −1.003 = −0.003 − 1 or thereabout when expressed in the frequency unit of cycle/sd). The periodicity seen in Fig. 1 (retrograde with negative period of somewhat longer than a year) corresponds to FCN’s ω w.r.t. F 0 . For convenience we shall adopt the solar day as the time unit to conform to the VLBI observation unless specified otherwise, and convert the period to sd only in the end.

JID:EPSL

AID:13515 /SCO

[m5G; v1.161; Prn:20/10/2015; 13:51] P.3 (1-10)

B.F. Chao, Y. Hsieh / Earth and Planetary Science Letters ••• (••••) •••–•••

3

Fig. 1. The VLBI-observed celestial pole offsets dψ and dε time series for FCN (courtesy of IERS), and the time-frequency (Morlet) wavelet spectrum of the complex quantity m(t ) = sin ε0 dψ(t ) + idε (t ). The dominant FCN signal (retrograde with negative period of somewhat longer than a year, in dark red) is well captured. The dashed vertical line marks the year 1992, the beginning of dataset used in this study. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 2. Frequency spectrum of the 2-D motions of the Earth’s rotation axis: ω ’ w.r.t. F  (convenient for the polar motion, upper axis) and ω w.r.t. F 0 (convenient for the nutation, lower axis) in units of cycle/sd, where ω = ω −  and  = 2π rad/sd corresponding to 1 cycle/sd. The positive frequency means prograde motion and negative frequency means retrograde motion. Earth’s two rotational modes of resonance, namely CW and FCN, are indicated.

It is instructional to draw a counterpart analogy between CW and FCN. The CW of the solid mantle has a prograde period of about 434 days and a quality factor of about 100 in F  (e.g., Gross, 2007). The FCN of the fluid outer core has a retrograde period of ∼430 days (but see below) in F 0 and a fairly high (but poorly determined) quality factor. The main phenomenological difference between the two lies in the different frequency bands they reside, and hence in the reference frames usually chosen to describe them. The apparent nearness of the two aforementioned values of period is not fortuitous – they are largely dictated by the dynamic oblateness of the Earth under diurnal spin (although respectively modified in somewhat different manners). 4. Dynamics of FCN The analogy between the two resonance systems of CW and FCN is not only kinematic but also dynamical. For one thing, FCN presumably shares the same excitation sources that excite CW –

the variations of the atmospheric and oceanic angular momentum (e.g. Brzezinski et al., 2014), although in rather different frequency bands. The excitation amplifies the forced oscillations that have periods close enough to feel the resonance effects: in the case of CW it is the prograde annual wobble in F  , in the case of FCN they include the retrograde annual nutation in F 0 or the corresponding retrograde diurnal tides in F  . We now formulate the basic physics of the excitation of a rotational resonance system, to which FCN (and CW for that matter) belong. Using the method of spectral decomposition, Chao (1982) has developed the dynamics in a generic and rigorous manner for a conservative eigen-system in F 0 that can be completely described by a positive semi-definite, self-adjoint linear operator (as of any non-rotating musical instrument), for example the free oscillation of a non-rotating elastic Earth. The infinitesimal displacement field, in F 0 , driven by a generalized simple-harmonic forcing or source distribution f(r) exp(i ωt ) at frequency ω can be decomposed into the superposition of discrete normal modes:

S(r, t ) =

 n



1

ω −ω 2 n

2



un∗ , f un (r) exp(i ωt ).

(3)

Here un (r) and ωn are respectively the eigenfunction and eigenfrequency of the n’th normal mode satisfying the orthonormality ∗ , u ) = δ . The symbol ∗ denotes complex conjugate, relation (um n mn and (,) denotes the inner product or the (dot) product of two (vector) functions integrated over the mass of the body, so that (un∗ , f) is the “projection” of f onto the normalized eigenfunction un . Physically Equation (3) signifies the harmonic response of a resonance system dictated by the set of transfer functions 1/(ωn2 − ω2 ) thereof the amplitude grows unbounded as ω approaches one ωn . This is a well-treated problem in classical mechanics (e.g., Morse and Feshbach, 1953). Extending the formulation to the rotating reference frame F  , thereby the additional Coriolis force constitutes an anti-self-adjoint operator, Chao (1982) proceeds to derive the corresponding ex-

JID:EPSL

AID:13515 /SCO

[m5G; v1.161; Prn:20/10/2015; 13:51] P.4 (1-10)

B.F. Chao, Y. Hsieh / Earth and Planetary Science Letters ••• (••••) •••–•••

4

pression that describes the displacement field in F  driven by the simple-harmonic forcing f(r) exp(i ωt ) co-rotating in F  :

S(r, t ) =



1

n

2ωn (ωn − ω)





un∗ , f un (r) exp(i ωt ).

(4)

Here the pertinent set of eigen-elements {un (r), ωn } satisfy the ∗ , u ) − 2i /(ω + ω )(u∗ , × u ) = quasi-orthonormality relation: (um n m n n m ∗ δmn in F  (which reduces to (um , un ) = δmn as above when  = 0). Equivalent formulas were derived by Dahlen and Smith (1975) and Wahr (1981a) via approaches parallel to Chao (1982). Insofar as the Earth’s rotational motion is concerned, Equations (4) sums over the two (known) eigen-modes of CW and FCN, at the two quite different resonance frequencies as noted above. Focusing on the single FCN in the nutation frequency band (where the CW effect can be comfortably neglected), we form the complex expression

m(t ) = sin ε0 dψ(t ) + idε (t ),

m (t ) =

2(σ0 − ω )





m (t ) = −



E  exp i ω t ,

(6)

for a harmonic excitation source acting at frequency ω . Equivalent formulas were derived by Wahr (1981a) and Sasao and Wahr (1981), and further treated by Brzezinski et al. (2002) in forming their broad-band equation of motion for polar motion. Now we introduce a (slight) damping into the physical system, whereupon the eigenfrequency becomes (slightly) complex, whose (small) imaginary part is the damping factor or exponential decay rate in the modal amplitude, a crucial assumption made prevailingly. Such a damping is a physical consequence of mechanisms involving Newtonian viscosity (e.g., Feynman et al., 1964) and/or magnetic coupling at the core–mantle boundary (Buffett et al., 2002) that give rise to a resistive force proportional to and against the motion velocity. In Equation (6) the FCN eigenfrequency in F  becomes σ0 = ω0 + i β , where β is the decay rate. The nearly-diurnal eigenperiod, letting it be a positive quantity, is P  = −2π /ω0 (note ω0 is negative), and the quality factor, also positive, is defined as Q  = −ω0 /2β . Correspondingly, we have P = −2π /ω0 and Q = −ω0 /2β w.r.t. F 0 (see further discussions in Section 7.2 about the Q value). The generic function E (t ) that appears in Equation (6) denotes what is formally referred to as the excitation function. It is likewise expressed in the complex form E = E x + i E y in the equatorial plane (xˆ , yˆ ), here taken to be harmonic in time. Going from Equation (4) to (6) we have supposed that the rotational eigenfunction un (r) in F  in the present framework to be formally equal to α (ˆx + i yˆ ) × r (Chao, 1983); see also Dahlen and Smith (1975), Wahr (1981a), where α is the appropriate normalization factor. Then, following the derivation in Chao (1983), it can be shown that E  (t ) in Equation (6) is proportional to the projection of L (t ) onto xˆ + i yˆ , where L is the generalized torque distribution (derived from the generalized forcing) in F  upon integration over the Earth, and the proportionality involves certain effective moment of inertia accounting for the dimensionality. Note that the factor ωn in the denominator of Equation (4) leads to the factor  in the denominator of Equation (6) upon insertion of the normalization factor α , see Chao (1983). The above are generically true for a body possessing eigenmodes of axial-symmetric rigid-body rotation as we shall assume

t

i 2

(5)

where ε0 = 23◦ .439 is the present-day obliquity angle of the Earth of epoch J2000 according to the IERS convention. As expressed, m represents the celestial ephemeris pole undergoing nutational motion in the right-handed Cartesian coordinates xˆ (the real part) and yˆ (the imaginary part) in F 0 . On account of Equation (4) we can now write, for FCN w.r.t. F  ,

1

for the Earth, whether it is FCN or CW or any other rotational modes. We further assert the following: Equation (6) is equivalent to Equation (6.7.8) of Munk and MacDonald (1960) for the harmonic excitation of CW (and hence our Equation (8) below is equivalent to Munk and MacDonald’s Equation (6.1.7); see also Sasao and Wahr, 1981), where Munk and MacDonald’s dimensionless excitation function for CW (defined in their Equation (6.1.5)) is the projection of L (t )/(2σ0 ) onto xˆ + i yˆ in the present framework. An integration of Equation (6) over ω , along a Bromwich contour in the complex plane dictated by physical causality (Morse and Feshbach, 1953; see also Chao, 1982), results in the general solution for m (t ) that can be expressed in a temporal convolution:





exp i σ0 (t − τ ) E  (τ )dτ

(7)

−∞

for an excitation E  (t ) having an arbitrary time history. Equation (7) can be shown to be equivalent to the differential equation

i dm (t )

σ0

dt

+ m (t ) =

1 2σ0

E  (t ),

(8)

in such a way that Equation (7) constitutes the particular solution to Equation (8), all expressed in F  (cf. Munk and MacDonald, 1960, Chapter 6). Now we take a crucial step in the formulation: A straightforward substitution and some moderate algebra would show that Equations (6), (7), (8) are invariant in form under the reference frame transformation Equation (1) and frequency shift Equation (2), a fact that is in fact anticipated on the grounds of physics. As such, the dynamical Equations (6), (7), (8) are equally valid expressions when described w.r.t. F 0 , where they respectively become (now in terms of the un-primed symbols):

m(t ) =

1 2(σ0 − ω)

m(t ) = − i dm(t )

σ0 dt

i 2

t

E exp(i ωt ),



(9)



exp i σ0 (t − τ ) E (τ )dτ ,

(10)

−∞

+ m(t ) =

1 2σ0

E (t ).

(11)

Equations (9), (10), (11), respectively, constitute the basis for the ensuing discussions of Sections 5, 6 and 7 below. 5. FCN as a resonance system (Equation (9)) Equation (6) gives explicitly the harmonic response of the FCN resonance system in F  . The left-side m (t ) is the output from a harmonic input E  exp(i ω t ) of the right side through the harmonic transfer function 1/[2(σ0 − ω )]. Describing the resonance effect as ω approaches the natural frequency σ0 , this transfer function is formally distinct from the transfer function 1/(σ0 2 − ω 2 ) (in Equation (3)) that describes the resonance of a non-rotating damped simple harmonic oscillator in classical mechanics (e.g., Feynman et al., 1964). Yet we note that the two transfer functions are asymptotically identical to first-order approximation when ω ≈ σ0 and both are close to , or, as in the case of FCN in the retrograde nearly-diurnal nutation band in F  , when ω ≈ σ0 (both are negative) and both are close to −. This has been noted by Neuberg et al. (1987) and Florsch and Hinderer (2000). Equation (9) then states that all the resonance physics discussed above w.r.t. F  stays unaltered w.r.t. F 0 in corresponding terms. In

JID:EPSL

AID:13515 /SCO

[m5G; v1.161; Prn:20/10/2015; 13:51] P.5 (1-10)

B.F. Chao, Y. Hsieh / Earth and Planetary Science Letters ••• (••••) •••–•••

5

Fig. 3. Fourier amplitude spectrum of the VLBI FCN complex series m(t ) = sin ε0 dψ(t ) + idε (t ) for the 11-year segment datasets each advanced by ∼2.8 yr with 75% overlap consecutively (the respective timespan given on the right). The bottom thick curve is that for the 23-year full dataset. The range of the apparent peak periods is indicated to top.

particular, the harmonic transfer function 1/[2(σ0 − ω)] for the resonance in F 0 is, on account of Equation (2) and dictated by physics, the same as 1/[2(σ0 − ω )] in F  as above. The resonance effect has been exploited to estimate P and Q of FCN, a point we shall return to in Section 7 below. 6. Observed FCN as a temporal convolution (Equation (10)) The integral in Equation (10) is the temporal convolution of the free oscillation exp(i σ0 t ) of the rotational mode with the excitation function E (t ) when described in F 0 . In what follows we draw from Chao and Chung (2012), who conducted extensive numerical Monte Carlo experiments that fully simulated the following observed behaviors of CW: (i) The oscillation, upon excited continually by broad-band random (white) “noise”, would in general appear fairly smooth, and in fact quasi-periodic. (ii) The apparent period is never far from the natural resonance period but would fluctuate somewhat with time because of (iii) below. (iii) The apparent amplitude would undulate considerably with time by the action of the continual (random) excitation, far from just due to the exponential damping. (iv) The x and y components would undergo essentially the same undulation of amplitude over time, maintaining the (near-)circular 2-D motion; for CW (a prograde motion) the x component would lead the y component by a quarter cycle (while for the retrograde FCN the opposite is true). (v) The apparent phase would also fluctuate with time, but kept more or less steady except during (vi) below. (vi) The phase would almost always undergo significant, sometimes drastic, changes whenever the amplitude becomes near-zero. The same physics applies to the FCN, so all of the behavior (i–vi) listed above are loyally observed by FCN in the VLBI data as evident in Fig. 1, just as they appear in the CW data (Chao and Chung, 2012). In particular, the amplitude and phase undulations found in the FCN series would masquerade as apparent fluctuations of the period (cf. Hinderer et al., 2000; Herring et al., 2002; see also Jiang and Smylie, 1995; Roosbeek et al., 1999), depending on the time segment used (see (ii) above). To highlight this apparent period fluctuation, Fig. 3 presents the Fourier power spectra of m(t ), here zero-padded to 214 = 16 384 points to enhance spectral sampling, of the 11-year segment datasets, each segment advanced in time by 2.8 yr (hence with 75% overlap consecutively). We see spectral peaks at (apparent) periods varying widely between 420–468 days. Fig. 3 also shows the Fourier spectrum of the 23-year full dataset that peaks at the apparent dominant period of ∼455 days with surrounding “fine structures”. The point here

is that these apparent periods do not represent the precise true eigenperiod, neither do they carry information about Q . In that sense we maintain that, while serving certain practical purposes, the empirical “models” of the FCN amplitude variability segment by time segments (e.g., Lambert, 2007) do not address the root physics of the apparent variability. The reason for behavior (vi) above is that at the time when the amplitude becomes near-zero, which could well happen fortuitously or “aimlessly” anytime under the random excitation, the phase is easily “reset” in the absence of momentum by the “next” excitation. To inspect such behavior in the FCN data, we carry out a complex demodulation on the FCN time series m(t ) (see Brillinger, 1973; Chao and Chung, 2012), by simply multiplying the function exp(+i2π t /440 days) to m(t ) to “un-do” the FCN oscillation at periodicity chosen to be 440 days (see below) w.r.t. F 0 . The resultant demodulated time series is shown in Fig. 4: Its absolute value now gives the running amplitude envelope of m(t ); its argument gives the running phase of m(t ) (the zero reference phase is arbitrary). The phase in effect reveals the cumulative deviation of the zero crossing of the time series from the zero crossings that a steady free oscillation would have. Here we have performed a low-pass (Butterworth) filtering at cutoff period of 200 days beforehand to remove high-frequency noises. A clear phase jump of more than ∼2 rad is evident over the years of ∼1998–2000 as previously reported by Shirai et al. (2005). More significant is the temporal concurrence of this phase jump with the pronounced minimal, near-zero amplitude during the same time (see also Fig. 1), in striking resemblance to the behavior of CW during the 1920s (Chao and Chung, 2012; see also Zotov and Bizouard, 2012). The point is that, despite the overlap in time with geophysical occurrences of the strong 1997–98 ENSO event or certain geomagnetic jerk event (Shirai et al., 2005), neither this concurrence nor the phase jump itself implies certain unique or peculiar behavior of the excitation process, but rather due to the simple physics of (vi) above. 7. Deconvolution of FCN (Equation (11)) Equation (11) is the inverse operation to the convolution (10), constituting the deconvolution where one inputs the observed m(t ) and outcomes the excitation function E (t ). In effect, the deconvolution acts to disassociate the free FCN from the excitation function. To realize the deconvolution numerically (the left side of Equation (11)) we apply the (discrete) notch filter of Wilson (1985, his Equation (4a)); see also Brzezinski (2007). Exploiting the deconvolution we now seek optimal estimation of the complex eigenfrequency (i.e. P and Q ) of FCN.

JID:EPSL

6

AID:13515 /SCO

[m5G; v1.161; Prn:20/10/2015; 13:51] P.6 (1-10)

B.F. Chao, Y. Hsieh / Earth and Planetary Science Letters ••• (••••) •••–•••

Fig. 4. The complex demodulation (at a chosen period of 440 days) of the complex FCN series m(t ) = sin ε0 dψ(t ) + idε (t ) from the VLBI data, after a low-pass (Butterworth) filtering at cutoff period of 200 days. Its absolute value (a) gives the running envelope of the amplitude; its argument (b) gives the running phase (the zero reference phase is arbitrary). The phase jump coincides in time with the minimum near-zero amplitude during ∼1998–2000, marked by the shade.

7.1. Estimation of the complex eigenfrequency of FCN by the deconvolution method Alternative to the often-used resonance method, we perform the deconvolution method devised by Furuya and Chao (1996, henceforth FC96, their Criterion I) to estimate FCN’s eigenfrequency. The FC96 estimator works as follows. Suppose, not knowing its true value, one had chosen a complex eigenfrequency somewhat off the true value, i.e. a somewhat erroneous set of ( P , Q ), in setting the parameters for the deconvolution. The deconvolution notch filter would then be somewhat off the target and unable to remove the resonance power completely. The acquired E (t ) would then contain, in addition to the true E (t ), some extra power that is not supposed to be present. This would elevate artificially the resultant spectral power especially around the resonance band, provided that the excitation is statistically independent of the observation noises. Conversely, the acquired E (t ) would assume the least power in the resonance band had one chosen the correct ( P , Q ). As such, the FC96 deconvolution method boils down to the search for the minimum power of E (t ) as a function of the input ( P , Q ). The FC96 approach has successfully led to highly-constrained, unbiased estimates of ( P , Q ) for CW. For CW such estimation is highly facilitated by the removal of the known excitation, namely the atmospheric angular momentum (AAM) (see FC96) and the additional oceanic and hydrological angular momentum (see Nastula and Gross, 2015) available from meteorological data or models so that the coherent power is greatly reduced. However, that is not the case with FCN, where at present no excitation source is known quantitatively, although prior studies have supported the supposition that the nearly diurnal AAM variations (w.r.t. F  ), or more precisely the associated pressure loading exerted on the core–mantle boundary, possess sufficient power in the right frequency band to excite the FCN (e.g., Sasao and Wahr, 1981; Dehant et al., 1996; Gegout et al., 1998; Brzezinski et al., 2002, 2014). In short of such knowledge our present approach in fact becomes equivalent to that proposed by Jeffreys (1972) for CW in the early years. Thus, we conduct a series of deconvolution on m(t ) while varying the input combination ( P , Q ): P in the range of 350–550

days with steps of 1 day, and Q in the range of 3–100 000 at 50 logarithmic intervals. For each input set ( P , Q ) we calculate the spectral power of the resultant E (t ) within the nutation band of 420–468 days (5 elementary Fourier frequency bins for the VLBI full dataset). In order to reduce possible interference of the rather large high-frequency noises, we filter the m(t ) time series prior to the deconvolution with a complex-valued bandpass filter in the band of −350 ∼ −500 days. This filter is constructed by a “shift” of a real-valued Butterworth low-pass filter (centered at the zero frequency) to one that is centered at the negative FCN period of nominal −440 days (see below) with the same bandwidth, by multiplying the filter impulse response with the factor exp(−i2π t /440 days). The resultant contour pattern of our (deconvolved) excitation power as a function of the input ( P , Q ) is shown in Fig. 5. From Fig. 5, our optimum estimate of P , the one that minimizes the excitation power, is determined to be 441 sd (converted from unit of solar day). For the Q value estimation, on the other hand, no minimum w.r.t. Q is manifested in Fig. 5, meaning no sensitivity toward constraining Q (we shall come back to this in Section 7 below). At this point we shall assess the uncertainties associated with our P estimates acquired from the deconvolution process. Following FC96, we do so by performing a series of Monte Carlo simulations. We generate according to Equation (10) a synthetic 23-year FCN m(t ) by convolving a randomly-generated white “noise” (representing the excitation function E (t ) with unity power) with the FCN resonance function exp(i σ0 t ) having the chosen P = 440 days and the Q of infinity. The latter, meaning pure sinusoids without damping, is in keeping with the complete insensitive to Q in reality found above. We repeat this procedure 1000 times with different randomlygenerated white excitation functions. It turns out that, instead of recovering precisely the input value of 440 days every time, we get a statistical distribution or histogram of the recovered P shown in Fig. 6(a). It displays a near-Gaussian distribution with an essentially unbiased peak value of 441 days with a standard deviation (or the “1-sigma” estimation uncertainty) of ∼4.5 days. Additional Monte Carlo tests have confirmed that this result stays essentially

JID:EPSL

AID:13515 /SCO

[m5G; v1.161; Prn:20/10/2015; 13:51] P.7 (1-10)

B.F. Chao, Y. Hsieh / Earth and Planetary Science Letters ••• (••••) •••–•••

Fig. 5. The spectral power of the (de-convolved) excitation function for FCN (from the 23-year VLBI full dataset bandpass filtered) as contours in the ( P , Q ) plane where P and Q are the input parameters to the deconvolution. The objective is to locate the minimum power, which occurs at P = 441 days but provides no constraint on Q .

unchanged with increased number of repeat trials (upwards from 1000). We repeat the same process but now adding to the synthetic m(t ) a white “observational” noise series prior to the deconvolution, starting with a low noise level and going up with higher noise levels by steps. That way we obtain a set of standard deviations in P estimates as a function of the signal-to-noise ratio (SNR), which is defined as the ratio of the integrated Fourier amplitude (over the narrow spectral band of 420–468 days as before) to the background noise level. Such empirical relation for 23-year long noisy synthetic data, presented in Fig. 6(b), constitutes the basis of our assessment of estimation uncertainties quoted below. The curve asymptotically rests toward the value of ∼4.5 days (as with the noise-free case above), which is in fact rather insensitive to SNR once the signal level exceeds the noise level (i.e. an SNR larger than ∼1) in the spectrum within the nutation band. Note that the deconvolution estimator would work reasonably well even when the said SNR is below 1. As can be read readily form Fig. 6 given the SNR of FCN in the full dataset shown in Fig. 3, the associated 1-sigma uncertainty

7

for our P estimate is ±4.5 days. Thus, we state that our optimal estimate for FCN’s eigenperiod P using the deconvolution method from the VLBI full dataset is 441 ± 4.5 days. We can do the P estimation the same way for the 11-year VLBI segment datasets, with 75% overlap as above in Fig. 3, and we obtain Fig. 7. The associated uncertainties are assessed just as above for the full dataset: We do a large number of additional Monte Carlo simulations as above for different data timespans, and find that the uncertainty is roughly proportional to the inverse of the square-root of the total timespan, as expected for a consistent estimator. In particular, the plot (not shown) for the 11-year synthetic data corresponding to Fig. 6(b) shows essentially the same empirical relation but resting on the value of ∼8.0 days. Hence we assign the 1-sigma uncertainty of ±8.0 days to the P estimates from the 11-year segment datasets (the error bars in Fig. 7). From Fig. 7 we see that the 11-year estimates of P fluctuate considerably from segment to segment. Barring those from the noisy pre-1992 segments (shown as dashed lines), these P estimates fall in a range of ∼440–460 sd, which lies within the associated 2-sigma estimation uncertainty. Furthermore, Fig. S1 of the Supplementary Material reports on additional numerical experiments that we have conducted w.r.t. Fig. 7. By replacing the bandpass filter used above in the pre-treatment of VLBI full dataset with a real-valued low-pass filter at cutoff of 200 days (thus retaining not only the retrograde FCN band but also the prograde counterpart), we obtain the corresponding P estimate to be 444 sd, within the 1-sigma uncertainty of the above estimate of 441 sd. Fig. S1 shows the results for this low-pass filtered data as well as those for the un-filtered data obtained by the same procedure as in Fig. 7. While the (moderate) deviations among them are evident, the three sets are well consistent within the associated estimation uncertainties. 7.2. Comparison with prior results from the resonance method Many prior studies have estimated the complex eigenfrequency of FCN using the resonance method, and obtained more or less converged estimates for the eigenperiod P and, to a lesser extent, the quality factor Q (for a review see Rosat et al., 2009). We shall make a comparison with our present results. Using VLBI data Gwinn et al. (1986) examined the near-FCNresonance amplification of the retrograde annual nutation in F 0 . Numerous subsequent studies took on the amplification of certain retrograde diurnal tidal components in F  , particularly 1, based

Fig. 6. Monte Carlo statistics of the P estimation determined by the deconvolution method on 23-year worth of synthetic FCN data: (a) Histogram of the 1000 recovered P estimation (relative to the input 440 days), for noise-free synthetic data. (b) The 1-sigma standard deviation of the recovered P , as a function of the spectral amplitude SNR, for noisy synthetic data. The arrow indicates the SNR of FCN in the real VLBI full dataset.

JID:EPSL

8

AID:13515 /SCO

[m5G; v1.161; Prn:20/10/2015; 13:51] P.8 (1-10)

B.F. Chao, Y. Hsieh / Earth and Planetary Science Letters ••• (••••) •••–•••

Fig. 7. FCN eigenperiod P estimated by the deconvolution method from the VLBI 11-year segment datasets, along with that from the 23-year full dataset (thick line, at 441 ± 4.5 sd) (the dashed lines indicate those based on the noisy pre-1992 data). The horizontal lines indicate the timespan of the data segment used, the error bars are the associated 1-sigma estimation uncertainties assessed from Monte Carlo simulations.

on superconducting gravimeter records (to name a few, Neuberg et al., 1987; Defraigne et al., 1994; Florsch and Hinderer, 2000; Rosat et al., 2009) or long-period seismograms (Cummins and Wahr, 1993). These result in P estimates clustered around 425–435 sd, significantly shorter than the theoretical value of 460 sd predicted by idealized Earth models under the rotation–gravitation hydrostatic equilibrium (Sasao et al., 1980; Wahr, 1981b). In comparison, our estimate of P = 441 ± 4.5 sd lies between the above values, while still on the shorter side of the theoretical period. The resonance and the deconvolution methods of estimation are distinct in fundamental ways. The resonance method is indirect in the sense that resonance effects exist as part of the intrinsic physics of the Earth; whether the eigen-mode in question (FCN here) is actually excited in reality is irrelevant. That is how FCN was first detected as early as in the 1980s, long before the VLBI observations became good enough to reveal the actual presence of FCN, or even in the complete absence of FCN in the tidal observations. Another advantage of the resonance method is the relative ease of quantifying the affected nutational/tidal components to high enough precisions, for they are pure sinusoids of known periods. Yet the resonance method relies on the existence of suitable, and hence limited few, nutational/tidal components that are in close proximity to FCN in frequency. More importantly, it faces two complications: The estimation algorithm finds substantial correlations among the parameters to be estimated ( P , Q and the resonance strength), so that sensitivity and trade-off issues become of concern (Cummins and Wahr, 1993; Defraigne et al., 1994; Rosat et al., 2009). Further, errors in the ocean and atmospheric tidal loading effects (to be accounted for beforehand) inject substantial biases directly into the tidal admittance estimates (e.g., Neuberg et al., 1987). In contrast, the deconvolution method performed in this paper is applied directly onto the VLBI data containing FCN signals that are excited well above observable level. The estimation algorithm is “clean” in the sense that the presence of FCN is independent of the nutational/tidal components, as long as the timespan is long enough to be free from spectral interferences by the tides. Neither does the algorithm involve other error-prone parameters; it only focuses on P and Q without practical tradeoff between them – in fact our result can well constrain P but is insensitive to Q . On the other hand, the success of the deconvolution method relies on the assumption that the excitation function be random or spectrally “flat” at least in the spectral band taken for evaluating the excitation power. That is what we assume in our Monte Carlo simulations. In the study of CW of FC96, the removal of the major systematic excitation function (namely AAM) essentially assures such randomness of the residual excitation. However, lacking knowledge about the excitation source(s) for FCN, strictly speaking we cannot rule out possible systematic errors, even though we

have removed the nutations/tides to our best. The influences of such systematic errors may explain the apparently systematic fluctuations of P estimates seen in Fig. 7 (and Fig. S1). The estimation of the decay rate β of a slightly-damped harmonic signal, or the corresponding quality factor Q , is inherently not as well constrained as with P : Its estimation uncertainty is in general 2Q time larger than that of the frequency (or period), i.e., δβ/β ≈ 2Q δ ω/ω (see Dahlen, 1976; Chao and Gilbert, 1980). The estimates of FCN’s Q value by prior studies using the resonance method range from a few thousand to essentially infinity. One should recognize that these reported values refer to the Q  w.r.t. F  (see the paragraph following Equation (6) above). It is numerically larger than the Q w.r.t. F 0 by a factor of ∼440, but both in fact describe the same damping history which is dictated by the true physical decay rate β . Here we maintain that, as one is more accustomed to quote the FCN period in terms of P (w.r.t. F 0 ) than P  (w.r.t. F  ), Q is the more appropriate parameter than Q  to describe the physical phenomenon of energy dissipation. For example, a Q  of, say, 20 000 is the same as a Q of about 45. After Q cycles of free oscillation with each cycle being ∼440 days long, the amplitude would be reduced to e −π , or ∼4%. Our deconvolution procedure pertains to the Q of FCN w.r.t. F 0 . At this time we can only rule out the likelihood of Q being lower than ∼30 (or Q  lower than 13 000), below which no minimum pattern (cf. Fig. 5) exists even w.r.t. P . The insensitivity of the excitation spectral power to the input value of Q , seen in Fig. 6, means no practical information for Q can be extracted via our deconvolution estimator. It presumably stems from the fact that the length of our data timespan is insufficient to discriminate the decay effects in the amplitude. Yet another conceivable reason relates to what was noted before: In the case of CW in FC96, the excitation function is largely known so that the amplitude-decay behavior can be better dissociated leading to higher sensitivity to Q even with only 1–2 decades of data. In contrast, lacking such knowledge for FCN, the damping tends to get absorbed numerically into the excitation solution. 8. Conclusions Basing on Chao (1982, 1983) we formulate the equations of motion for the excited FCN in terms of the temporal convolution of a free-rotational mode of resonance with an excitation function. We obtain Equations (6), (7), (8) w.r.t. the uniformly rotating reference frame F  , and the equivalent Equations (9), (10), (11) w.r.t. the inertia space F 0 . We show that the two sets of equations are invariant in form under the kinematic transformation of the two reference frames. The convolution formulation of FCN well explains the dynamic behavior of FCN (w.r.t. F 0 ) observed by the VLBI technique since

JID:EPSL

AID:13515 /SCO

[m5G; v1.161; Prn:20/10/2015; 13:51] P.9 (1-10)

B.F. Chao, Y. Hsieh / Earth and Planetary Science Letters ••• (••••) •••–•••

the 1980s. Clearly consistent with the response of a resonance system under continual excitation, these behaviors include the undulation of the FCN amplitude and the apparent fluctuations in the period and phase over time, and the temporal concurrence of a phase jump with the minimal, near-zero amplitude (during ∼1998–2000). All are in complete analogy to the dynamic behavior of CW (w.r.t. F  ) as demonstrated by Chao and Chung (2012). The convolution formulation is then exploited to obtain optimal estimates for FCN’s complex eigenfrequency, or the eigenperiod P and quality factor Q , following the deconvolution method employed successfully by Furuya and Chao (1996) for CW. While the method is insensitive to Q under the present condition, we obtain the optimal estimate of P = 441 ± 4.5 sd for FCN using the 23-year (1992–2014) VLBI full dataset. The 1-sigma estimation uncertainty is assessed via extensive sets of Monte Carlo simulations. The P estimates from 11-year segment datasets are found to fluctuate considerably between ∼440–460 sd with 1-sigma uncertainty of ±8.0 sd. Our P estimate lies in between those in the range 425–435 sd determined by prior studies using the resonance method and the theoretical value of ∼460 sd predicted by idealized Earth models under the rotation–gravitation hydrostatic equilibrium. Insofar as our present objective is to estimate the complex eigenfrequency of the FCN, the excitation function solution E (t ) is a by-product of the deconvolution procedure (11) and employed here only in locating its minimum power numerically. Nevertheless, physically the excitation function is of fundamental importance: it gives the time evolution of the forcing and, as such, elucidates on the identity of the physical sources that excite FCN in the first place. More elaborate studies of it will await further investigations. Acknowledgements We thank Dan MacMillan, Xiang’E Lei, Jose Ferrandiz, and Chengli Huang for helpful discussions. The VLBI data series are provided graciously by the International Earth Rotation and Reference Systems Service (IERS) via website http://hpiers.obspm.fr/ eop-pc/products/combined/C04.html. This work is supported by the Ministry of Science and Technology, Taiwan via grant #103-2116-M-001-024. Appendix A. Supplementary material Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.epsl.2015.10.010. References Bizouard, Ch., Gambis, D., 2009. The combined solution C04 for Earth orientation parameters consistent with International Terrestrial Reference Frame 2008. Tech. Note. Observatoire de Paris. Brillinger, D.R., 1973. An empirical investigation of the Chandler wobble and two proposed excitation processes. Bull. Int. Stat. Inst. 45 (3), 413–434. Brzezinski, A., 2007. A simple digital filter for the geophysical excitation of nutation. J. Geod. 81, 543–551. Brzezinski, A., Capitaine, N., 1993. The use of the precise observations of the Celestial Ephemeris Pole in the analysis of geophysical excitation of earth rotation. J. Geophys. Res. 98, 6667–6675. Brzezinski, A., Bizouard, Ch., Petrov, S., 2002. Influence of the atmosphere on Earth rotation: what new can be learned from the recent atmospheric angular momentum estimates? Surv. Geophys. 23, 33–69. Brzezinski, A., Dobslaw, H., Thomas, M., 2014. Atmospheric and oceanic excitation of the free core nutation estimated from recent geophysical models. In: Rizos, C., Willis, P. (Eds.), Earth on the Edge: Science for a Sustainable Planet. In: Intern. Assoc. Geodesy Symposia, vol. 139. Springer-Verlag, Heidelberg. Buffett, B.A., Herring, T.A., Mathews, P.M., 2002. Modeling of nutation–precession: effects of electromagnetic coupling. J. Geophys. Res. 107. http://dx.doi.org/ 10.1029/2001JB000056.

9

Chao, B.F., Gilbert, F., 1980. Autoregressive estimation of complex eigenfrequencies in low frequency seismic spectra. Geophys. J. R. Astron. Soc. 63, 641–657. Chao, B.F., 1982. Excitation of normal modes on non-rotating and rotating earth models. Geophys. J. R. Astron. Soc. 68, 295–315. Chao, B.F., 1983. Normal mode study of the Earth’s rigid-body motions. J. Geophys. Res. 88, 9437–9442. Chao, B.F., Chung, W.Y., 2012. Amplitude and phase variations of Earth’s Chandler Wobble under continual excitation. J. Geodyn. 62, 35–39. http://dx.doi.org/ 10.1016/j.jog.2011.11.009. Chao, B.F., Chung, W.Y., Shih, Z.R., Hsieh, Y.K., 2014. Earth’s rotation variations: a wavelet analysis. Terra Nova 26 (4), 260–264. http://dx.doi.org/10.1111/ ter.12094. Cummins, P.R., Wahr, J.M., 1993. A study of the Earth’s free core nutation using IDA gravity data. J. Geophys. Res. 98, 2091–2104. http://dx.doi.org/10.1029/ 92JB01956. Dahlen, F.A., Smith, M.L., 1975. The influence of rotation on the free oscillations of the Earth. Philos. Trans. R. Soc. A 279, 583–629. Dahlen, F.A., 1976. Models of the lateral heterogeneity of the Earth consistent with eigenfrequency splitting data. Geophys. J. R. Astron. Soc. 44, 77–105. Defraigne, P., Dehant, V., Hinderer, J., 1994. Stacking gravity tide measurements and nutation observations in order to determine the complex eigenfrequency of the nearly diurnal free wobble. J. Geophys. Res. 99, 9203–9213. http://dx.doi.org/ 10.1029/94JB00133. Dehant, V., Bizouard, Ch., Legros, H., Lefftz, M., Hinderer, J., 1996. The effects of atmospheric pressure perturbations on precession and nutations. Phys. Earth Planet. Inter. 96 (1), 25–40. Dehant, D., Mathews, P.M., 2007. Earth rotation variations. In: Dziewonski, A., Romanowicz, B. (Eds.), Treatise on Geophysics. Elsevier. Ed.-in-chief G. Schubert (Chap. 3.10). Feynman, R.P., Leighton, R.B., Sands, M., 1964. Feynman Lectures on Physics. Addison–Wesley, Boston. Florsch, N., Hinderer, J., 2000. Bayesian estimation of the free core nutation parameters from the analysis of precise tidal gravity data. Phys. Earth Planet. Inter. 117, 21–35. Furuya, M., Chao, B.F., 1996. Estimation of period and Q of the Chandler wobble. Geophys. J. Int. 127, 693–702. Gegout, P., Hinderer, J., Legros, H., Greff, M., Dehant, V., 1998. Influence of atmospheric pressure on the Free Core Nutation, precession and some forced nutational motions of the Earth. Phys. Earth Planet. Inter. 106, 337–351. http:// dx.doi.org/10.1016/S0031-9201(97)00100-3. Gross, R.S., 2007. Earth rotation variations – long period. In: Dziewonski, A., Romanowicz, B. (Eds.), Treatise on Geophysics. Elsevier. Ed.-in-chief G. Schubert (Chap. 3.9). Gwinn, C.R., Herring, T.A., Shapiro, I.I., 1986. Geodesy by radio interferometry: studies of the forced nutations of the Earth. Part II: interpretation. J. Geophys. Res. 91, 4755–4765. Herring, T.A., Dong, D., 1994. Measurement of diurnal and semidiurnal rotational variations and tidal parameters of Earth. J. Geophys. Res. 99, 18051–18072. Herring, T.A., Mathews, P.M., Buffet, B.A., 2002. Modeling of nutation–precession: very long baseline interferometry results. J. Geophys. Res. 107. http://dx.doi.org/ 10.1029/2001JB000165. Hinderer, J., Boy, J.P., Gegout, P., Defraigne, P., Roosbeek, F., Dehant, V., 2000. Are the free core nutation parameters variable in time? Phys. Earth Planet. Inter. 117, 37–49. IERS Conventions 2010, 2010. In: Petit, G., Luzum, B. (Eds.), IERS Technical Note #36. Verlag des Bundesamts für Kartographie und Geodäsie, Frankfurt am Main. Jeffreys, H., 1972. The variation of latitude. In: Melchoir, P., Yumi, S. (Eds.), Rotation of the Earth: IAU Symp. #48. D. Reidel, Dordrecht, pp. 39–42. Jiang, X., Smylie, D.E., 1995. A search for free core nutation modes in VLBI nutation observations. Phys. Earth Planet. Inter. 90, 91–100. Lambert, S.B., 2007. Empirical modeling of the retrograde Free Core Nutation. Tech. Note. Obs. de Paris. ftp://hpiers.obspm.fr/eop-pc/models/fcn/notice.pdf. Lambert, S.B., Dehant, V., 2007. The Earth’s core parameters as seen by the VLBI. Astron. Astrophys. 469, 777–781. Mathews, P.M., Herring, T.A., Buffett, B.A., 2002. Modeling of nutation–precession: new nutation series for non-rigid Earth, and insights into the Earth’s interior. J. Geophys. Res. 107. http://dx.doi.org/10.1029/2001JB000390. Melchoir, P., 1983. The Tides of the Planet Earth. Pergamon Press, Oxford. Morse, P.M., Feshbach, H., 1953. Methods of Theoretical Physics. McGraw–Hill, New York. Munk, W.E., MacDonald, G.J.F., 1960. The Rotation of the Earth: A Geophysical Discussion. Cambridge Univ. Press, New York. Nastula, Y., Gross, R., 2015. Chandler wobble parameters from SLR and GRACE. J. Geophys. Res. 120 (6), 4474–4483. Neuberg, J., Hinderer, J., Zurn, W., 1987. Stacking gravity tide observations in central Europe for retrieval of the complex eigenfrequency of the nearly diurnal free wobble. Geophys. J. R. Astron. Soc. 91, 853–868. Roosbeek, F., Defraigne, P., Feissel, M., Dehant, V., 1999. The Free Core Nutation period is between 431 and 434 sidereal days. Geophys. Res. Lett. 26, 131–134.

JID:EPSL

10

AID:13515 /SCO

[m5G; v1.161; Prn:20/10/2015; 13:51] P.10 (1-10)

B.F. Chao, Y. Hsieh / Earth and Planetary Science Letters ••• (••••) •••–•••

Rosat, S., Florsch, N., Hinderer, J., Llubes, M., 2009. Estimation of the Free Core Nutation parameters from SG data: sensitivity study and comparative analysis using linearized least-squares and Bayesian methods. J. Geodyn. 48, 331–339. http:// dx.doi.org/10.1016/j.jog.2009.09.027. Sasao, T.S., Okubo, S., Saito, M., 1980. A simple theory on dynamical effects of a stratified fluid core upon nutational motion of the Earth. In: Fedorov, E., Smith, M., Bender, P. (Eds.), Proceedings of the IAU Symposium 78, Nutation and the Earth’s Rotation. Reidel, Hingham, MA, pp. 165–183. Sasao, T., Wahr, J.M., 1981. An excitation mechanism for the free core nutation. Geophys. J. R. Astron. Soc. 64, 729–746. Shirai, T., Fukushima, T., 2001. Construction of a new forced nutation theory of nonrigid Earth. Astron. J. 121, 3270–3283. Shirai, T., Fukushima, T., Malkin, Z., 2005. Detection of phase disturbances of free core nutation of the Earth and their concurrence with geomagnetic jerks. Earth Planets Space 57, 151–155.

Smith, M.L., 1977. Wobble and nutation of the Earth. Geophys. J. R. Astron. Soc. 50, 103–140. Toomre, A., 1974. On the nearly diurnal wobble of the Earth. Geophys. J. R. Astron. Soc. 38, 335–348. Wahr, J.M., 1981a. The forced nutations of an elliptical, rotating, elastic and oceanless Earth. Geophys. J. R. Astron. Soc. 64, 705–727. Wahr, J.M., 1981b. A normal mode expansion for the forced response of a rotating earth. Geophys. J. R. Astron. Soc. 64, 651–675. Wallace, P.T., Capitaine, N., 2006. Precession–nutation procedures consistent with IAU 2006 resolutions. Astron. Astrophys. 459, 981–985. http://dx.doi.org/ 10.1051/0004-6361:20065897. Wilson, C.R., 1985. Discrete polar motion equations. Geophys. J. R. Astron. Soc. 80, 551–554. Zotov, L.V., Bizouard, C., 2012. On modulations of the Chandler wobble excitation. J. Geodyn. 62, 30–34. http://dx.doi.org/10.1016/j.jog.2012.03.010.