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On rotational normal modes of the Earth: Resonance, excitation, convolution, deconvolution and all that
Geodesy and Geodynamics xxx (2017) 1e6
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On rotational normal modes of the Earth: Resonance, excitation, convolution, deconvolution and all that Benjamin Fong Chao Institute of Earth Sciences, Academia Sinica, Taipei, Taiwan, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 31 October 2016 Received in revised form 4 March 2017 Accepted 10 March 2017 Available online xxx
Earth's Coriolis force profoundly alters the eigen frequencies, eigen functions, and excitation of rotational normal modes. Some rotational modes of the solid mantle-fluid outer core-solid inner core Earth system are confirmed observationally and some remain elusive. Here we bring together from literature assertions about an excited resonance system in terms of the Green's function and temporal convolution. We raise caveats against taking the face values of the oscillational motion which have been “masqueraded” by the convolution, necessitating deconvolution for retrieving the excitation function which reflects the true variability. Lastly we exemplify successful applications of the deconvolution in estimating resonance complex frequencies. © 2017 Institute of Seismology, China Earthquake Administration, etc. Production and hosting by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction The subjects of small oscillations and rigid-body rotation are among the central topics treated in the standard textbooks of classical mechanics [1e3]. In this paper the Earth is regarded as a classical mechanical body, a rather complex one indeed, that is perpetually undergoing such physical motions in variant forms. Historically both two types of motion of the Earth, namely the elasto-gravitational free oscillation and the rotational variations pertaining to various Earth components, have been studied theoretically dating back to around the turn of the 20th century [4]. Their modern investigations keep pace with advances in observation techniques and are marked, respectively, by the great 1960 Chile earthquake [5] and the advent of the space geodesy in the 1980s [6].
E-mail address: [email protected]. Peer review under responsibility of Institute of Seismology, China Earthquake Administration.
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Remarkably, the two subject matters can in fact be unified, physically and hence mathematically, by formulating them in the form of eigen value problems. Historically attributed to Poincare, Hough, etc. [7], this approach in modern times has led to the successful solutions (see below) of rotational normal modes of resonance with eigen frequencies and eigen functions that belong to a realistic Earth model that consists of the solid mantle-fluid outer core-solid inner core system. Normal modes exist intrinsically to the body in question, irrespective of whether or not they are excited. In the normal-mode perspective the excitation sources and the excitation processes that set the modes in finite oscillation are external to and act independent of the modes themselves. In the case of the Earth the elastogravitational, or seismic, modes are most naturally excited by earthquake events, likened to the ringing of a struck bell. Only during the active faulting (in action possibly up to minutes) are the modes being excited; the subsequent Earth ringing is just free oscillations. On a rather low level, the seismic modes were also found to be excited continually by disturbances attributable to the atmosphereocean-seafloor coupling, manifesting as Earth “hums” [8,9]. The situation with the Earth's rotational modes is the reverse: the rotational modes are evidently subject to continual excitation, aside perhaps from episodic but relatively minor disturbances caused by occasional large earthquakes [10]. The best studied case is the length-of-day variation and the Chandler wobble in the polar motion, for which the exchange of angular-momentum among the
http://dx.doi.org/10.1016/j.geog.2017.03.014 1674-9847/© 2017 Institute of Seismology, China Earthquake Administration, etc. Production and hosting by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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geophysical fluids provides the torquing on the mantle [11]. Less understood cases include the observed as well as supposed core oscillations [12,13], the excitation sources of which stay unknown at present. In this paper were view the basic physics of the excitation of normal modes belonging to the rotating Earth in general, with focuses on the rotational modes. We do so pedagogically to provide aqualitative perspective of the subject matter from the classical mechanics point of view. We also raise subtleties and caveats one should attend to in practical analyses of the observational data. We shall not replicate the mathematical derivations explicitly, the details of which can be found in the cited literature. Nor do we pursue the identity of the possible excitation sources in reality for the rotational modes; they are simply addressed in the context by ageneric excitation function. 2. Excitation of normal modes: convolution A resonance system of N degrees of freedom gives rise to N normal modes upon transformation into the normal coordinates via the eigenvectors [1]. Each normal mode, with the associated natural frequency, undergoes a simple harmonic oscillation, which follows the linearization of the (infinitesimal) perturbation to a minimum-potential equilibrium state. Labeling the spatial dependence by r, the eigen value equation governing normal mode has the simple form [14,15]:
L u2n un ðrÞ ¼ 0
n ¼ 1; 2; …N
(1)
where L is a linear, self-adjoint (or Hermitian), positive-semi definite operator, there of Lun gives the negative of the restoring (e.g. the elasto-gravitational) force per unit mass acting on un, the nth eigenvector or discrete eigen function that satisfies the prescribed boundary conditions. The self-adjointness of L ensures that the eigen frequencies are real and the eigen functions are orthogonal to one another, i.e. the inner product of any distinct pair of eigen functions vanishes, hence the designation normal modes. In the presence of mechanical energy dissipation obeying Newtonian viscosity as conventionally assumed for the realistic Earth, un is to be replaced by the complex eigen frequency sn ¼ un(1 þ i/2Qn), where Q denotes the (intrinsic) quality factor characterizing the decaying harmonic oscillation. The actual oscillation amplitude as a function of time is simply the linear combination of the individual contributions from all the modes (that form a complete set). The simplest case for Eq. (1) is one of single mode with one degree of freedom, for example a simple pendulum. In what follows we focus on such a single-mode system and its temporal behavior, recognizing that, simple as is, its physics can be readily extended to conceive of any number of (isolated and mutually orthogonal) normal modes in more complex systems expressible in the normal coordinates. Suppose we activate onto this single-mode system an external forcing to be described by an excitation function E(t) per unit mass, setting off a motion for which the governing equation can be written as:
L s20 uðtÞ ¼ EðtÞ
(2)
The general solution of Eq. (2) is the linear superposition of two parts:
uðtÞ ¼ u0 ðtÞ þ GðtÞ*EðtÞ
(3)
The first term u0(t) is the homogeneous solution to Eq. (1) that accounts for the initial condition, in the present context a free simple-
harmonic oscillation at eigen frequency s0. This term vanishes if we set the system at rest at t ¼ 0, or let the initial epoch be t ¼ ∞ so that the free oscillation had long died out due to the finite Q. The second term represents the particular solution that has the general form of the temporal convolution, denoted by *, of E(t) with the (temporal) Green's function G(t):
Zt GðtÞ*EðtÞ ¼
Gðt tÞEðtÞdt
(4)
0
The Green's function itself is the solution to the equation of motion in response to a unit impulse forcing expressible as the delta function of time d(t). As such the Green's function can always be recovered from the convolution (4) if E(t) happens to be d(t). It is thus equivalent to the impulse response function in the realm of system engineering; its Laplace (or Fourier) transform is equated to the system transfer function. The G(t) belonging to the simple resonance system takes on the form of a decaying harmonic motion exp(is0t), here stipulated to have unity amplitude without loss of generality. If the excitation E(t) is itself a harmonic function of a given frequency u, then convolution (4) reduces to exp(iut) multiplied by the well-known resonance factor 1/(s20 u2) [2,3,16]. A resonant amplification of oscillation will come in playas u approaches s0(on the complex Laplace domain). By the same token, the convolution (4) for an excitation E(t) of an arbitrary time dependence will magnify the portion of the power of E(t) that resides within the frequency band near resonance compared to that at other non-resonance frequencies. A corollary is that an excited mode does not necessarily imply an excitation source strictly at the specific eigen frequency; rather, as is more often the case, any broad-band excitation function that contains non-vanishing power around the resonance frequency will do. The thought experiment raised in Chao [17], Chao and Chung [18] can well illustrate the essence of a convolution process. Imagine a lone swing in a playground subject to a turbulent wind. The swing sways in the wind, largely following its own rhythm with certain degree of regularity but not quite fully so. Upon scrutiny, the amplitude and phase of the swaying undoubtedly reacts to the wind, but not in a direct or instantaneous fashion that can be discerned. In fact it becomes unlikely to be able to predict the amplitude or phase beyond several cycles of sway. Physically what happens is that at any given moment the swing “remembers” and executes its existing momentum through inertia, while being perturbed by the wind at that particular moment the onset of which would continue to impact the future for a length of time depending on the Q value. Mathematically that is where the Green's function and convolution enter. Fig. 1 depicts a schematic of the temporal convolution of Eq. (4): A computer-generated white noise (playing the role of the turbulent wind as the input) is numerically convolved with the Green's function of a decaying sinusoid (playing the role of the swing as the resonant system function). The outcome of the convolution (the sway of the swing) is a quasi-periodic oscillation with “modulated” amplitude and phase. We observe the following: (i) Upon excitation the oscillation would in general appear fairly smooth irrespective of the “ruggedness” of the excitation function, or whether it is continuous or episodic. The resultant apparent amplitude, albeit smooth, would undulate with time considerably and masquerade as a slow “modulation” of sorts. This amplitude modulation show no indications, explicitly or implicitly, of the exponential decay whatsoever.
Please cite this article in press as: B.F. Chao, On rotational normal modes of the Earth: Resonance, excitation, convolution, deconvolution and all that, Geodesy and Geodynamics (2017), http://dx.doi.org/10.1016/j.geog.2017.03.014
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Fig. 1. A typical convolution scenario of a simple resonance system.
(ii) The apparent phase would also fluctuate slightly and in slow undulation with time, but is kept largely steady with the critical exception of (iv) below. (iii) The apparent period, for example that estimated by means of a Fourier spectral analysis, of any temporal segment of the observational oscillation is never far from the resonance period, but would slowly fluctuate somewhat with time or, as is often, show multiple period “splitting”. The reason can be traced to (i) and (ii) above, in light of the following fact. The superposition of multiple sinusoids of neighboring periods will give rise to beating phenomena in both amplitude and phase. Hence, conversely, a “modulated” sinusoid would masquerade as multiple periods or apparently varying period in a Fourier spectrum given the limited spectral resolution. (iv) The phase would invariably undergo significant, often drastic, changes whenever the amplitude becomes nearzero, for the following reason. At the time when the oscillation has near-zero amplitude and hence near-zero momentum, which could happen fortuitously from time to time under the random excitation, the system has little or nothing to remember through its inertia. So the phase of the oscillation is waiting to be “reset” by the excitation that is to happen in the next moment, hence the drastic phase change. We reemphasize that the features (ieiv) transpire even when the excitation function is random, or downright white noise, as shown in Fig. 1 and confirmed in the Monte-Carlo experiments of Chao and Chung [18]. 3. Let it spin: rotational modes Now we let the system, originally adhering to the inertia frame, undergo a uniform rotation at a fixed angular velocity in space. Case in point is an idealized diurnally rotating Earth at constant angular rate U ¼ 2p radians per sidereal day around a fixed axis in space, called the z axis. In this U-spinning reference frame all normal-mode matters are altered in a fundamental way owing to the introduction of the Coriolis force, one of the two pseudo-forces along with the centrifugal force when the observer is placed in the spinning frame. Mathematically the Coriolis force is represented by an anti-selfadjoint (or skew-Hermitian) operator [14,19]; whereas the centrifugal force is self-adjoint and can be blended into the original L), so that the self-adjoint operator L in Eqs. (1) and (2) are augmented with an anti-self-adjoint part L0 , i.e. L/L þ L0 . In the familiar example of the Earth's seismic normal modes, the spin breaks the presumed spherical symmetry (becoming axisymmetric) with a perturbation of magnitude in proportion to the ratio un/U. That leads to shifting of eigen frequencies and splitting of otherwise degenerate frequency multiplets, in a fashion analogous to the magnetic Zeeman effect in atomic spectrum [20]. We mention in passing that the Coriolis-split seismic triplet 1S1 have eigen periods predicted to be of several hours, by far the longest among seismic modes. Known as the Slichter triplet [21],
they correspond to 3 rigid translational modes of the solid inner core. They have not been detected in seismic or gravimetric data despite decades of active search [22,23]. By the same token, the eigen functions are altered by the rotation as well e they no longer satisfy the orthogonality relationship but rather a generalized orthogonality [14,19]. Moreover, couplings among different modes take place subject to certain sphericalharmonic selection rules [5], again akin to the atomic eigen-state couplings under quantum-mechanical perturbations [20]. The excitation formula also stands to be modified by the introduction of the Coriolis force; the modified formulas were derived by Chao [14], see also Wahr [24]. The convolution relation (4) still stands, while the resonance amplification to a sinusoidal excitation becomes 1/[2sn(sn u)], no longer 1/(s2n u2) as in the inertia frame (although approaching it as u approaches sn asymptotically). In contrast to the seismic normal modes that are in general only slightly perturbed by Earth's spin, a rotational mode can be reformed completely by the spinor even owes its existence to the Earth's spin itself. Let's consider a rigid-body and examine the simplest form of normal modes, the rigid-body modes. A 3dimensional rigid-body has 6 degrees of freedom in its motion, 3 for translation and 3 for rotation. The rotational degree of freedom is D(D 1)/2 in D-dimensional space, which assumes the value of 3 in our 3-dimensional space [25]. Mathematically they are trivial zero-frequency eigen-solutions to Eq. (1) in the inertia frame, and can thus be viewed as 6 trivial, degenerate, secular normal modes [15,24,26]. Referenced to the spinning reference frame, however, the new rigid-body modes under spin are eigen-solutions to the spinning counterpart of Eq. (1) where L is replaced by L þ L0 as stated above. The z-axial translational mode remains invariant w.r.t. the transformation of reference frame. The two xey translational modes become intercoupled with (degenerate) eigen frequency U, which are simply the same motion as before but viewed in a kinematically different standpoint. The above-mentioned Slichter triplet is the quasi-rigid translational modes belonging to the solid inner core that is immersed in the fluid outer core. By contrast, the set of 3 rotational modes, originally those trivial ones about the 3 orthogonal principal axes in the inertia frame, is altered profoundly by the spin. The rotational mode about the rotation z axis, termed the axial spin mode [26], remains invariant still, but the two xey rotational modes are intercoupled to form two new modes, namely the tilt-over mode at(retrograde) eigen frequency U, and the Eulerian wobble (related to which the Chandler wobble is the quasi-rigid version belonging to the Earth, see below) with (prograde) eigen frequency equal to þU divided by the dynamical oblateness of the body [15,26]. The above attests to the equivalence of the rotational normalmode approach to the Eulerian dynamics of rigid-body rotation. It also attests to the general assertion that a small (3-D) rotational variation can be separated to first order into two non-interfering parts [27], the (1-D) axial component and the (2-D) equatorial component.
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3.1. Axial rotational modes: DLOD Physically an axial rotational mode signifies the (1-D) z-component of the angular momentum. It obeys simple arithmetic scalar addition in both the inertia and spinning frames for the equation of motion is invariant with respect to the coordinate transformation (as exemplified in the above rigid-body modes). The mathematical statement is that the Green's function for DLOD is a delta function, equivalent to the trivial transfer function of unity, a direct consequence of the eigen frequency being zero. In the thought experiment analogy of Section 2, this is likened to a fictitious swing of zero inertia hence zero “memory”. Specific to the Earth, the most obvious axial spin mode is that of the mantle representing the z-component for the rotational speed or the length-of-day variation DLOD that has been under study for decades. A more subtle axial spin mode is the supposed axial torsional libration of the solid inner-core within the fluid outer core. It owes its existence to the mantle-inner core gravitational (MICG) torque [28,29], provided of the spherical-harmonic degree2 order-2 density anomalies in the Mantle and inner core. In this mode, the conservation of angular momentum dictates that the inner core librates not alone, but accompanied at the same time by an opposite mirrored libration of the mantle which has an amplitude ~1200 times (the ratio of the moments of inertia of the mantle to the inner core) smaller and manifests as DLOD oscillation that can be observed. The MICG torque also determines the libratinal natural frequency. By equating the MICG axial libration to the observed 6-year DLOD oscillation, the equatorial ellipticities of the inner core and the lower mantle can be constrained, the latter pertains directly to the density anomalies of the Large Low-ShearVelocity Provinces (LLSVP; see references cited in Chao [29]. Indeed this proposition is corroborated by the fact that the 6-year DLOD oscillation does exhibit moderate fluctuations in the amplitude and the period [30], the main traits of a resonance mode under continual excitation. Further investigations should be warranted.
3.2. Equatorial rotational modes: wobbles and nutations The equatorial xey components forms a 2-D vector conveniently expressed in the complex form m ¼ mx þ imy [27]. Denoting the fractional perturbation of the angular velocity, m is the parameter used to quantify the rotational axis orientation (such as for the Chandler wobble and the free-core nutation, see below, and the equatorial Slichter translations for that matter). The positive frequency domain of m corresponds to the prograde rotation, and the negative to retrograde. Chao and Hsieh [31] has demonstrated that, for an individual (isolated) rotational mode in the U-spinning frame, the excitation of m is governed by the following set of equations:
mðtÞ ¼
1 E expðiutÞ 2Uðs0 uÞ
i mðtÞ ¼ e 2U
Zt
exp½is0 ðtetÞEðtÞdt
(6)
∞
i dmðtÞ 1 þ mðtÞ ¼ EðtÞ dt 2Us0
s0
(5)
(7)
where the excitation function, whatever the physical sources may be, is also expressed in the complex form: E ¼ Ex þ iEy.
Eqs. (5)e(7) are actually equivalent to one another. Eq. (5) describes the response of the system to a harmonic excitation with frequency u, which would feel the resonance with the amplification factor 1/(s0 u) as asserted above. Upon integration over u along the Bromwich contour under causality and invoking the Cauchy's theorem of residues one gets Eq. (6) for an E(t) of arbitrary time dependence. Eq. (6) is indeed duly expressed in terms of the temporal convolution of E(t) with the Green's function exp(is0t) as is Eq. (4). Eq. (7) is simply the differential form of Eq. (6). Just as Eq. (6) means convolution, Eq. (7) constitutes the deconvolution where the observed m can be processed numerical to recover the excitation function E; we shall return to this in the next section. We mention that Munk and MacDonald [27] reached the equivalent set of formulas for the Chandler wobble excitation based on the EulerLiouville equation (which is formulated in the s frame of course). Moreover, the excitation Eqs. (5)e(7) can be readily shown to be invariant w.r.t. the kinematic rotational transformation of reference frame, as required by physics [31]. Specific to the Earth, the relevant and interesting physics arises pertaining to an oblate body, for example as one lets aspherically symmetric Earth deform and become oblate subject to the gravitationehydrostatic equilibrium with gravitation now including the centrifugal potential. The oblateness is represented mathematically by the zonal spherical harmonic of degree 2 and order 0, with Earth surface value of ~1/300 and somewhat smaller internal values dictated by Clairaut's equation [5,32]. Suppose in addition we consider an idealized three-layered Earth model comprised of a solid mantle, a fluid outer core and a solid inner core. Then the equatorial rotational motion has 4 degrees of freedom, giving rise to 4 corresponding rotational modes as eigen-solutions [32], namely the mantle Chandler wobble, the free-core nutation (FCN), the free inner-core nutation (FICN), and the inner-core wobble. At the present time the first two have been observed in astrometric and space geodetic measurements of Earth rotation variations, the FICN has been implied in data processing [33,34], whereas the inner-core wobblere mains undetected. We might add here that while classified as nutations on phenomenological grounds [26], FCN and FICN are dynamically distinct from the familiar astronomical precession and nutations resulting from the external luni-solar tides. In addition, there are a host of fluid “core mode” oscillations that are supposed to exist in the rotating fluid outer core as eigen-solutions to the magnetohydrodynamic equations of motion (e.g., Smith [26]; for a review see Holme [13]). We further point out that the steady phase in (ii) above is also observed between the x and y components of m, they would undergo essentially the same undulation of amplitude over time, maintaining the (near-) circular 2-D motion. For the Chandler wobble the x component would lead the y component by a quarter cycle, and the opposite sense for FCN. We stress that, while the nature and the response to excitation are altered by spin as described above, the end behavior of Earth's rotational modes is found to observe loyally all of the above qualitative features (ieiv) that are dictated by convolution. The case of the Chandler wobble has been elucidated by Chao and Chung [18] by virtue of synthetic Monte-Carlo experiments, where a convolution by white-noise excitation well mimics all of (ieiv), particularly for the drastic Chandler-wobble phase jump (iv) that occurred during the 1920s. The FCN is likewise examined by Chao and Hsieh [31] using data obtained from the very-long-baseline interferometry (VLBI) technique. The apparent amplitude as well as the apparent period had clearly been fluctuating depending on the time segment of the data analyzed, and a scenario (iv) of phase jump did occur during the 1990s.
Please cite this article in press as: B.F. Chao, On rotational normal modes of the Earth: Resonance, excitation, convolution, deconvolution and all that, Geodesy and Geodynamics (2017), http://dx.doi.org/10.1016/j.geog.2017.03.014
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4. De-convolution The above in fact leads to some practical caveats that should be drawn as corollaries: It would be misleading to dwell on the face values of the motion amplitude of the normal modes, which is subject to time convolution (4) and have been processed or “masqueraded” by the convolution. Cases in point are the wobbles and the free nutations. For one thing, the changes seen in their oscillation amplitude do not natural mean certain corresponding peculiar behavior in the excitation at the time. Similarly, on account of (iii), it is groundless to assert (albeit unable to rule out either) the existence of a time-variable period or multiple periods for the resonance system upon detecting such from the face values of the oscillation. Moreover, it is futile to attempt to estimate the rate of decay (or Q factor) via the half-width of the apparent spectral peak of the resonance. Another potential, but less recognized, pitfall on account of (iv) [18] is the inclination to proclaim and then search for peculiarity in the excitation at the times when the phase of the oscillation experiences apparent drastic changes. Thus, to the goal of understanding properly the system and its excitation processes, one ought to resort to the deconvolution process which enables the dissociation of the effect of the excitation from the system Green's function. Numerical deconvolution, pioneered by Wiener [35], has been widely applied in signal and image processing, and algorithms abound for treating the variety of geophysical data in particular (e.g., Robinson and Treitel [36]). In our case of the deterministic Green's function exp(is0t), the deconvolution is readily performed in an analytical fashion as Eq. (7), a procedure that is equivalent to a notch filter at the resonance frequency s0 (e.g., Wilson [37], Brzezinski [38]). Such “direct” approach (as opposed to the “integration” approach; see Chao [17]), thereby the excitation function is deconvolved from the observational data, has been rightfully the approach taken in polar-motion excitation studies reported in the literature. Of relevance are the strength of the angular momentum of the geophysical fluids and their exchange with the solid Earth within the resonance frequency band of the Chandler wobble, i.e. the (prograde) period of 434 days in the U-spinning frame. It has become well known(e.g., Gross [11]) that the majority of the polarmotion excitation (as well as the DLOD excitation for that matter) is provided by the tidal and the atmospheric angular momentum variations in a continual fashion, plus secondary contributions due to the oceans, the land hydrology, and presumably the cores. Among them, an ample amount of excitations is at work to excite the Chandler wobble to the observed amplitude of a few hundred mill arc seconds (on the order of several meters on the Earth's surface). Similar deconvolution has been undertaken in studying the FCN and its excitation using VLBI data (e.g., Brzezinski [38,39]); the results however turn out in conclusive thus far. The main reason can be attributed to the fact that the FCN resonance period, some 400þ days (retrograde) in the inertia frame, is nearly-diurnal in the spinning terrestrial frame (in fact FCN is also known in the literature as the Nearly Diurnal Free Wobble in that frame). That alone entails the difficult tasks of making global, sub-half-daily monitoring of the geophysical fluid fluctuations. More critically, at the diurnal timescale the power in the geophysical fluid fluctuations are in general relatively minor, exciting FCN only to the observed amplitude of a fraction of milliarc second (on the order of ~1 cm on Earth's surface). It was proclaimed that sufficient atmospheric power does exist in the right frequency band to excite the FCN [38e40], but no quantitative evidences have been confirmed in the available data even after deconvolution. The deconvolution procedure has proved effective in another application w.r.t. the rotational modes e namely estimating the
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complex eigen frequency s0, or the period P and the Q factor of the resonance Green's function, by a procedure first devised and employed by Furuya and Chao [41]. Suppose, not knowing the true value, one had chosen a s0 somewhat off the true value, i.e. a somewhat erroneous set of (P, Q), in performing the deconvolution (7). The deconvolution notch filter would then somewhat miss the target and unable to completely remove the extra signal power attributable to the resonance. The acquired E(t) would then contain some remnant power that is present only as artefact, elevating the resultant spectral power especially around the resonance band. Conversely, the acquired E(t) would assume the least power, especially in the resonance band, had one chosen the correct (P, Q) in deconvolution. As such, a search of the minimum power of E(t) in the parameter (P, Q) space constitutes an optimum means of estimating (P, Q). This deconvolution method of estimating (P, Q) has been demonstrated to be robust and unbiased through Monte-Carlo simulations [41]. It serves well for the Chandler wobble, yielding P ¼ 433.7 ± 1.8 days (w.r.t. the rotating frame) and Q ¼ 49 (35100) [41], see also Ref. [11] for parallel estimates by other means). Thanks to the fact that the majority of the excitation sources is known (such as the atmospheric angular momentum, see above), they can be favorably removed to leave the residual excitation essentially uncorrelated with the random noise whose variances are to be minimized together, as required by stipulation in the optimization procedure. Such, however, is not entirely the case when one is to apply the technique on the FCN whose excitation sources stay unknown. As such, the remnant spectral power said above cannot be characterized adequately by a random process, and statistical bias may result. Moreover, the effect of the decay (Q factor) in the oscillation amplitude is less conducible to numerical separation from the excitation effect numerically, especially when the data timespan is short. Further compounded by the relatively low signal-to-noise ratio of the FCN data in the first place, the deconvolution method leaves something to be desired for FCN: It yields an optimal estimation of P ¼ 441 ± 4.5 sidereal days (sd, w.r.t. the inertia frame), but is totally insensitive to and hence unable to reach an estimate for the Q value [30]. The 1-sigma uncertainty (±4.5 sd) in P is estimated based on Monte Carlo simulations under white-noise excitations with realistic signal-to-noise ratios. The retrograde diurnal tides, themselves incapable of exciting the FCN (lacking power at all but the tidal frequencies), experience anomalous amplification in amplitude depending on how close they are to the FCN resonance. The observations of such amplifications have been utilized to estimate the FCN's resonance parameters P and Q (see review by Rosat [42]), yielding estimates of P ¼ 425e435 sd. Both sets of the estimates are significantly distinct from the theoretical value P ¼ 460 sd predicted by idealized Earth models under the assumption of rotation-gravitational hydrostatic equilibrium [43]. Aside from the proposition to attribute the lower observed P to anomalously larger oblateness of the outer core [33,44], the causes of the sediscrepancies among the three sets of estimates are yet to be fully understood. 5. Concluding remarks The rotations of various Earth components vary slightly under respective torques on all timescales. These small variations can be duly represented as superposition of normal modes, or eigensolutions to the fundamental equations of motion posed as eigen value problems. Each normal mode can be excited to undergo a simple harmonic oscillation with specific resonance eigen frequency; the excitation process is formulated as a temporal convolution of the Green's function with the respective excitation function.
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The convolution is a notion well established in classical mechanics as well as in modern signal analysis. However, less recognition has been lain on the consequences of the convolution when the excitation is by a (more or less) random, continual process, as is the case for the Earth's rotational normal modes. The said consequences masquerade as apparent fluctuations in the resonance period as well as the undulations in the excited amplitude and phase that are observed. That has regrettably led to unjustified proclamations and likely futile pursuits of peculiarities in the excitation processes or even the nature of the resonance system itself. The process of deconvolution retrieves the nominal excitation function from the observations, yielding vital information to reveal the identification of the excitation sources. However, other than the Chandler wobble and DLOD, little has been confirmed thus far w.r.t. the excitation of the Earth's rotational modes that involve the cores and that are only weakly excited if at all. In the meantime, the deconvolution has been utilized as an effective method of estimating the resonance complex frequencies of the rotational modes including the Chandler wobble and the FCN. Taking the normal-mode point of view thus proves to be constructive and productive toward the practical treatment and understanding of the Earth's rotational motions. We have seen successful applications and useful results for the Chandler wobble and FCN; we anticipate it to likewise elucidate on the supposed rotational modes that are yet to be detected in observations. Acknowledgments This work is supported by the Taiwan Ministry of Science and Technology via grant #104-2116-M-001-006. References [1] H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, Mass, 1950. [2] L.D. Landau, E.M. Lifshitz, Mechanics, Volume 1 of a Course of Theoretical Physics, Pergamon Press, 1969. [3] S. Thornton, J. Marion, Classical Dynamics of Particles and Systems, Thomson Learning, Inc., Belmont, 2004. [4] H. Jeffreys, The Earth, Its Origin, History and Physical Constitution, Cambridge Univ. Press, 1924. [5] F.A. Dahlen, J. Tromp, Theoretical Global Seismology, Princeton Univ. Press, Princeton, 1998. [6] Monog. #24D.E. Smith, D. Turcotte (Eds.), Contributions of Space Geodesy to Geodynamics: Earth Dynamics, Amer. Geophy. Union, Washington, DC, 1993. [7] H. Lamb, Hydrodynamics, 6th ed., Dover Publications, New York, 1932. [8] T. Tanimoto, J. Um, K. Nishida, N. Kobayashi, Earth's continuous oscillations observed on seismically quiet days, Geophys. Res. Lett. 25 (1998) 1553e1556, http://dx.doi.org/10.1029/98GL01223. [9] J. Rhie, B. Romanowicz, Excitation of Earth's continuous free oscillations by atmosphereeoceaneseafloor coupling, Nature 431 (2004) 552e556, http:// dx.doi.org/10.1038/nature02942. [10] B.F. Chao, R.S. Gross, Changes in the Earth's rotation and low-degree gravitational field induced by earthquakes, Geophys. J. Roy. Astron. Soc. 91 (1987) 569e596. [11] R.S. Gross, Earth Rotation Variations e Long Period, Chap. 3.9, in: G. Schubert (Ed.), Treatise on Geophysics, 2nd ed., Elsevier, Amsterdam, The Netherlands, 2015. [12] D. Dehant, P.M. Mathews, Earth Rotation Variations, Chap. 3.10 in Treatise on Geophysics, in: G. Schubert (Ed.), 2nd ed., Elsevier, Amsterdam, The Netherlands, 2015. [13] R. Holme, Large-Scale Flow in the Core, Chap. 8.04, in: G. Schubert (Ed.), Treatise on Geophysics, 2nd ed., Elsevier, Amsterdam, The Netherlands, 2015. [14] B.F. Chao, Excitation of normal modes on non-rotating and rotating earth models, Geophys. J. Roy. Astron. Soc. 68 (1982) 295e315. [15] B.F. Chao, Normal mode study of the Earth's rigid-body motions, J. Geophys. Res. 88 (1983) 9437e9442. [16] P.M. Morse, H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953. [17] B.F. Chao, On the excitation of the Earth's polar motion, Geophys. Res. Lett. 12 (1985a) 526e529. [18] B.F. Chao, W.Y. Chung, Amplitude and phase variations of Earth's Chandler Wobble under continual excitation, J. Geodyn. 62 (2012) 35e39, http:// dx.doi.org/10.1016/j.jog.2011.11.009.
[19] F.A. Dahlen, M.L. Smith, The influence of rotation on the free oscillation of the Earth, Phil. Trans. R. Soc. A 279 (1975) 583e629. [20] B.F. Chao, Symmetry and terrestrial spectroscopy, Geophys. J. Roy. Astron. Soc. 66 (1981) 285e312. [21] L.B. Slichter, The fundamental free mode of the Earth's inner core, Proc. Natl. Acad. Sci. 47 (1961) 186e190. [22] S.A. Rosat, Review of the Slichter modes: an observational challenge, in: Jon M. Phillips (Ed.), The Earth's Core: Structure, Properties and Dynamics, Nova Science Publishers, Inc., 2011, pp. 63e77. [23] H. Ding, B.F. Chao, The Slichter mode of the Earth: revisit with optimal stacking and autoregressive methods on full superconducting gravimeter dataset, J. Geophys. Res. (2015), http://dx.doi.org/10.1002/2015JB012203. [24] J.M. Wahr, A normal mode expansion for the forced response of a rotating earth, Geophys. J. R. Astron. Soc. 64 (1981) 651e675. [25] B.F. Chao, As the world turns, EOS Trans. Amer. Geophys. Union 46 (1985b) 766e770. [26] M.L. Smith, Wobble and nutation of the Earth, Geophys. J. R. Astron. Soc. 50 (1977) 103e140. [27] W.E. Munk, G.J.F. MacDonald, The Rotation of the Earth: a Geophysical Discussion, Cambridge Univ. Press, New York, 1960. [28] B. Buffett, Gravitational oscillations in the length of day, Geophys. Res. Lett. 23 (1996) 2279e2282. [29] B.F. Chao, Dynamics of axial torsional libration under the mantle-inner core gravitational interaction, J. Geophys. Res. (2016) 121, http://dx.doi.org/ 10.1002/2016JB013515. [30] B.F. Chao, W.Y. Chung, Z.R. Shih, Y.K. Hsieh, Earth's rotation variations: Awavelet analysis, Terra Nova 26 (2014) 260e264, http://dx.doi.org/10.1111/ ter.12094. [31] B.F. Chao, Y.K. Hsieh, The Earth's free core nutation: formulation of dynamics and estimation of eigen period from the very-long-baseline interferometry data, Earth Planet. Sci. Lett. (2015), http://dx.doi.org/10.1016/ j.epsl.2015.10.010. [32] F.D. Stacey, Physics of the Earth, Brookfield Press, Brisbane, 1992. [33] P.M. Mathews, T.A. Herring, B.A. Buffett, Modeling of nutation-precession: new nutation series for non-rigid Earth, and insights into the Earth's interior, J. Geophys. Res. 107 (2002), http://dx.doi.org/10.1029/2001JB000390. [34] L. Koot, M. Dumberry, A. Rivoldini, O. de Viron, D. Dehant, Constraints on the coupling at the coreemantle and inner core boundaries inferred from nutation observations, Geophys. J. Int. 182 (2010) 1279e1294, http://dx.doi.org/ 10.1111/j.1365-246X.2010.04711.x. [35] N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series, MIT Press, Cambridge, Mass, 1964. ISBN 0-262-73005-7. [36] E. Robinson, S. Treitel, Geophysical Signal Analysis, Prentice-Hall, Eaglewood Cliffs, 1980. [37] C.R. Wilson, Discrete polar motion equations, Geophys. J. R. Astron. Soc. 80 (1985) 551e554. [38] A. Brzezinski, A simple digital filter for the geophysical excitation of nutation, J. Geod. 81 (2007) 543e551. [39] A. Brzezinski, H. Dobslaw, M. Thomas, Atmospheric and oceanic excitation of the free core nutation estimated from recent geophysical models, in: C. Rizos, P. Willis (Eds.), Earth on the Edge: Science for a Sustainable Planet, SpringerVerlag, Heidelberg, 2014, http://dx.doi.org/10.1007/978-3-642-37222-3_61. [40] T.S. Sasao, J.M. Wahr, An excitation mechanism for the free core nutation, Geophys. J. R. Astron. Soc. 64 (1981) 729e746. [41] M. Furuya, B.F. Chao, Estimation of period and Q of the Chandler wobble, Geophys. J. Int. 127 (1996) 693e702. [42] S. Rosat, N. Florsch, J. Hinderer, M. Llubes, 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 (2009) 331e339, http://dx.doi.org/10.1016/j.jog.2009.09.027. [43] T.S. Sasao, S. Okubo, M. Saito, A simple theory on dynamical effects of a stratified fluid core upon nutational motion of the Earth, in: E. Fedorov, M. Smith, P. Bender (Eds.), Proceedings of the IAU Symposium 78, Nutation and the Earth's Rotation, Reidel, Hingham, Mass, 1980, pp. 165e183. [44] C.R. Gwinn, T.A. Herring, I.I. Shapiro, Geodesy by radio interferometry: studies of the forced nutations of the Earth. Part II: interpretation, J. Geophys. Res. 91 (1986) 4755e4765.
Benjamin Fong Chao. Current Positions. (1) Distinguished Research Fellow, Institute of Earth Sciences, Academia Sinica; (2) Adjunct Professor, Physics Department, National Taiwan University; (3) Adjunct Professor, Department of Earth Sciences, National Central University; (4) Adjunct Professor, Department of Geosciences, National Cheng-Kung University, Taiwan; (5) Emeritus, NASA Goddard Space Flight Center, Maryland, USA. Education. (1) 1973 B.S. Physics, National Taiwan University; (2) 1981 Ph.D. Earth Sciences, Scripps Institution of Oceanography, University of California, San Diego (thesis adviser: Prof. Freeman Gilbert).
Please cite this article in press as: B.F. Chao, On rotational normal modes of the Earth: Resonance, excitation, convolution, deconvolution and all that, Geodesy and Geodynamics (2017), http://dx.doi.org/10.1016/j.geog.2017.03.014
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On rotational normal modes of the Earth: Resonance, excitation, convolution, deconvolution and all that Benjamin Fong Chao Institute of Earth Sciences, Academia Sinica, Taipei, Taiwan, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 31 October 2016 Received in revised form 4 March 2017 Accepted 10 March 2017 Available online xxx
Earth's Coriolis force profoundly alters the eigen frequencies, eigen functions, and excitation of rotational normal modes. Some rotational modes of the solid mantle-fluid outer core-solid inner core Earth system are confirmed observationally and some remain elusive. Here we bring together from literature assertions about an excited resonance system in terms of the Green's function and temporal convolution. We raise caveats against taking the face values of the oscillational motion which have been “masqueraded” by the convolution, necessitating deconvolution for retrieving the excitation function which reflects the true variability. Lastly we exemplify successful applications of the deconvolution in estimating resonance complex frequencies. © 2017 Institute of Seismology, China Earthquake Administration, etc. Production and hosting by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction The subjects of small oscillations and rigid-body rotation are among the central topics treated in the standard textbooks of classical mechanics [1e3]. In this paper the Earth is regarded as a classical mechanical body, a rather complex one indeed, that is perpetually undergoing such physical motions in variant forms. Historically both two types of motion of the Earth, namely the elasto-gravitational free oscillation and the rotational variations pertaining to various Earth components, have been studied theoretically dating back to around the turn of the 20th century [4]. Their modern investigations keep pace with advances in observation techniques and are marked, respectively, by the great 1960 Chile earthquake [5] and the advent of the space geodesy in the 1980s [6].
E-mail address: [email protected]. Peer review under responsibility of Institute of Seismology, China Earthquake Administration.
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Remarkably, the two subject matters can in fact be unified, physically and hence mathematically, by formulating them in the form of eigen value problems. Historically attributed to Poincare, Hough, etc. [7], this approach in modern times has led to the successful solutions (see below) of rotational normal modes of resonance with eigen frequencies and eigen functions that belong to a realistic Earth model that consists of the solid mantle-fluid outer core-solid inner core system. Normal modes exist intrinsically to the body in question, irrespective of whether or not they are excited. In the normal-mode perspective the excitation sources and the excitation processes that set the modes in finite oscillation are external to and act independent of the modes themselves. In the case of the Earth the elastogravitational, or seismic, modes are most naturally excited by earthquake events, likened to the ringing of a struck bell. Only during the active faulting (in action possibly up to minutes) are the modes being excited; the subsequent Earth ringing is just free oscillations. On a rather low level, the seismic modes were also found to be excited continually by disturbances attributable to the atmosphereocean-seafloor coupling, manifesting as Earth “hums” [8,9]. The situation with the Earth's rotational modes is the reverse: the rotational modes are evidently subject to continual excitation, aside perhaps from episodic but relatively minor disturbances caused by occasional large earthquakes [10]. The best studied case is the length-of-day variation and the Chandler wobble in the polar motion, for which the exchange of angular-momentum among the
http://dx.doi.org/10.1016/j.geog.2017.03.014 1674-9847/© 2017 Institute of Seismology, China Earthquake Administration, etc. Production and hosting by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article in press as: B.F. Chao, On rotational normal modes of the Earth: Resonance, excitation, convolution, deconvolution and all that, Geodesy and Geodynamics (2017), http://dx.doi.org/10.1016/j.geog.2017.03.014
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geophysical fluids provides the torquing on the mantle [11]. Less understood cases include the observed as well as supposed core oscillations [12,13], the excitation sources of which stay unknown at present. In this paper were view the basic physics of the excitation of normal modes belonging to the rotating Earth in general, with focuses on the rotational modes. We do so pedagogically to provide aqualitative perspective of the subject matter from the classical mechanics point of view. We also raise subtleties and caveats one should attend to in practical analyses of the observational data. We shall not replicate the mathematical derivations explicitly, the details of which can be found in the cited literature. Nor do we pursue the identity of the possible excitation sources in reality for the rotational modes; they are simply addressed in the context by ageneric excitation function. 2. Excitation of normal modes: convolution A resonance system of N degrees of freedom gives rise to N normal modes upon transformation into the normal coordinates via the eigenvectors [1]. Each normal mode, with the associated natural frequency, undergoes a simple harmonic oscillation, which follows the linearization of the (infinitesimal) perturbation to a minimum-potential equilibrium state. Labeling the spatial dependence by r, the eigen value equation governing normal mode has the simple form [14,15]:
L u2n un ðrÞ ¼ 0
n ¼ 1; 2; …N
(1)
where L is a linear, self-adjoint (or Hermitian), positive-semi definite operator, there of Lun gives the negative of the restoring (e.g. the elasto-gravitational) force per unit mass acting on un, the nth eigenvector or discrete eigen function that satisfies the prescribed boundary conditions. The self-adjointness of L ensures that the eigen frequencies are real and the eigen functions are orthogonal to one another, i.e. the inner product of any distinct pair of eigen functions vanishes, hence the designation normal modes. In the presence of mechanical energy dissipation obeying Newtonian viscosity as conventionally assumed for the realistic Earth, un is to be replaced by the complex eigen frequency sn ¼ un(1 þ i/2Qn), where Q denotes the (intrinsic) quality factor characterizing the decaying harmonic oscillation. The actual oscillation amplitude as a function of time is simply the linear combination of the individual contributions from all the modes (that form a complete set). The simplest case for Eq. (1) is one of single mode with one degree of freedom, for example a simple pendulum. In what follows we focus on such a single-mode system and its temporal behavior, recognizing that, simple as is, its physics can be readily extended to conceive of any number of (isolated and mutually orthogonal) normal modes in more complex systems expressible in the normal coordinates. Suppose we activate onto this single-mode system an external forcing to be described by an excitation function E(t) per unit mass, setting off a motion for which the governing equation can be written as:
L s20 uðtÞ ¼ EðtÞ
(2)
The general solution of Eq. (2) is the linear superposition of two parts:
uðtÞ ¼ u0 ðtÞ þ GðtÞ*EðtÞ
(3)
The first term u0(t) is the homogeneous solution to Eq. (1) that accounts for the initial condition, in the present context a free simple-
harmonic oscillation at eigen frequency s0. This term vanishes if we set the system at rest at t ¼ 0, or let the initial epoch be t ¼ ∞ so that the free oscillation had long died out due to the finite Q. The second term represents the particular solution that has the general form of the temporal convolution, denoted by *, of E(t) with the (temporal) Green's function G(t):
Zt GðtÞ*EðtÞ ¼
Gðt tÞEðtÞdt
(4)
0
The Green's function itself is the solution to the equation of motion in response to a unit impulse forcing expressible as the delta function of time d(t). As such the Green's function can always be recovered from the convolution (4) if E(t) happens to be d(t). It is thus equivalent to the impulse response function in the realm of system engineering; its Laplace (or Fourier) transform is equated to the system transfer function. The G(t) belonging to the simple resonance system takes on the form of a decaying harmonic motion exp(is0t), here stipulated to have unity amplitude without loss of generality. If the excitation E(t) is itself a harmonic function of a given frequency u, then convolution (4) reduces to exp(iut) multiplied by the well-known resonance factor 1/(s20 u2) [2,3,16]. A resonant amplification of oscillation will come in playas u approaches s0(on the complex Laplace domain). By the same token, the convolution (4) for an excitation E(t) of an arbitrary time dependence will magnify the portion of the power of E(t) that resides within the frequency band near resonance compared to that at other non-resonance frequencies. A corollary is that an excited mode does not necessarily imply an excitation source strictly at the specific eigen frequency; rather, as is more often the case, any broad-band excitation function that contains non-vanishing power around the resonance frequency will do. The thought experiment raised in Chao [17], Chao and Chung [18] can well illustrate the essence of a convolution process. Imagine a lone swing in a playground subject to a turbulent wind. The swing sways in the wind, largely following its own rhythm with certain degree of regularity but not quite fully so. Upon scrutiny, the amplitude and phase of the swaying undoubtedly reacts to the wind, but not in a direct or instantaneous fashion that can be discerned. In fact it becomes unlikely to be able to predict the amplitude or phase beyond several cycles of sway. Physically what happens is that at any given moment the swing “remembers” and executes its existing momentum through inertia, while being perturbed by the wind at that particular moment the onset of which would continue to impact the future for a length of time depending on the Q value. Mathematically that is where the Green's function and convolution enter. Fig. 1 depicts a schematic of the temporal convolution of Eq. (4): A computer-generated white noise (playing the role of the turbulent wind as the input) is numerically convolved with the Green's function of a decaying sinusoid (playing the role of the swing as the resonant system function). The outcome of the convolution (the sway of the swing) is a quasi-periodic oscillation with “modulated” amplitude and phase. We observe the following: (i) Upon excitation the oscillation would in general appear fairly smooth irrespective of the “ruggedness” of the excitation function, or whether it is continuous or episodic. The resultant apparent amplitude, albeit smooth, would undulate with time considerably and masquerade as a slow “modulation” of sorts. This amplitude modulation show no indications, explicitly or implicitly, of the exponential decay whatsoever.
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Fig. 1. A typical convolution scenario of a simple resonance system.
(ii) The apparent phase would also fluctuate slightly and in slow undulation with time, but is kept largely steady with the critical exception of (iv) below. (iii) The apparent period, for example that estimated by means of a Fourier spectral analysis, of any temporal segment of the observational oscillation is never far from the resonance period, but would slowly fluctuate somewhat with time or, as is often, show multiple period “splitting”. The reason can be traced to (i) and (ii) above, in light of the following fact. The superposition of multiple sinusoids of neighboring periods will give rise to beating phenomena in both amplitude and phase. Hence, conversely, a “modulated” sinusoid would masquerade as multiple periods or apparently varying period in a Fourier spectrum given the limited spectral resolution. (iv) The phase would invariably undergo significant, often drastic, changes whenever the amplitude becomes nearzero, for the following reason. At the time when the oscillation has near-zero amplitude and hence near-zero momentum, which could happen fortuitously from time to time under the random excitation, the system has little or nothing to remember through its inertia. So the phase of the oscillation is waiting to be “reset” by the excitation that is to happen in the next moment, hence the drastic phase change. We reemphasize that the features (ieiv) transpire even when the excitation function is random, or downright white noise, as shown in Fig. 1 and confirmed in the Monte-Carlo experiments of Chao and Chung [18]. 3. Let it spin: rotational modes Now we let the system, originally adhering to the inertia frame, undergo a uniform rotation at a fixed angular velocity in space. Case in point is an idealized diurnally rotating Earth at constant angular rate U ¼ 2p radians per sidereal day around a fixed axis in space, called the z axis. In this U-spinning reference frame all normal-mode matters are altered in a fundamental way owing to the introduction of the Coriolis force, one of the two pseudo-forces along with the centrifugal force when the observer is placed in the spinning frame. Mathematically the Coriolis force is represented by an anti-selfadjoint (or skew-Hermitian) operator [14,19]; whereas the centrifugal force is self-adjoint and can be blended into the original L), so that the self-adjoint operator L in Eqs. (1) and (2) are augmented with an anti-self-adjoint part L0 , i.e. L/L þ L0 . In the familiar example of the Earth's seismic normal modes, the spin breaks the presumed spherical symmetry (becoming axisymmetric) with a perturbation of magnitude in proportion to the ratio un/U. That leads to shifting of eigen frequencies and splitting of otherwise degenerate frequency multiplets, in a fashion analogous to the magnetic Zeeman effect in atomic spectrum [20]. We mention in passing that the Coriolis-split seismic triplet 1S1 have eigen periods predicted to be of several hours, by far the longest among seismic modes. Known as the Slichter triplet [21],
they correspond to 3 rigid translational modes of the solid inner core. They have not been detected in seismic or gravimetric data despite decades of active search [22,23]. By the same token, the eigen functions are altered by the rotation as well e they no longer satisfy the orthogonality relationship but rather a generalized orthogonality [14,19]. Moreover, couplings among different modes take place subject to certain sphericalharmonic selection rules [5], again akin to the atomic eigen-state couplings under quantum-mechanical perturbations [20]. The excitation formula also stands to be modified by the introduction of the Coriolis force; the modified formulas were derived by Chao [14], see also Wahr [24]. The convolution relation (4) still stands, while the resonance amplification to a sinusoidal excitation becomes 1/[2sn(sn u)], no longer 1/(s2n u2) as in the inertia frame (although approaching it as u approaches sn asymptotically). In contrast to the seismic normal modes that are in general only slightly perturbed by Earth's spin, a rotational mode can be reformed completely by the spinor even owes its existence to the Earth's spin itself. Let's consider a rigid-body and examine the simplest form of normal modes, the rigid-body modes. A 3dimensional rigid-body has 6 degrees of freedom in its motion, 3 for translation and 3 for rotation. The rotational degree of freedom is D(D 1)/2 in D-dimensional space, which assumes the value of 3 in our 3-dimensional space [25]. Mathematically they are trivial zero-frequency eigen-solutions to Eq. (1) in the inertia frame, and can thus be viewed as 6 trivial, degenerate, secular normal modes [15,24,26]. Referenced to the spinning reference frame, however, the new rigid-body modes under spin are eigen-solutions to the spinning counterpart of Eq. (1) where L is replaced by L þ L0 as stated above. The z-axial translational mode remains invariant w.r.t. the transformation of reference frame. The two xey translational modes become intercoupled with (degenerate) eigen frequency U, which are simply the same motion as before but viewed in a kinematically different standpoint. The above-mentioned Slichter triplet is the quasi-rigid translational modes belonging to the solid inner core that is immersed in the fluid outer core. By contrast, the set of 3 rotational modes, originally those trivial ones about the 3 orthogonal principal axes in the inertia frame, is altered profoundly by the spin. The rotational mode about the rotation z axis, termed the axial spin mode [26], remains invariant still, but the two xey rotational modes are intercoupled to form two new modes, namely the tilt-over mode at(retrograde) eigen frequency U, and the Eulerian wobble (related to which the Chandler wobble is the quasi-rigid version belonging to the Earth, see below) with (prograde) eigen frequency equal to þU divided by the dynamical oblateness of the body [15,26]. The above attests to the equivalence of the rotational normalmode approach to the Eulerian dynamics of rigid-body rotation. It also attests to the general assertion that a small (3-D) rotational variation can be separated to first order into two non-interfering parts [27], the (1-D) axial component and the (2-D) equatorial component.
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3.1. Axial rotational modes: DLOD Physically an axial rotational mode signifies the (1-D) z-component of the angular momentum. It obeys simple arithmetic scalar addition in both the inertia and spinning frames for the equation of motion is invariant with respect to the coordinate transformation (as exemplified in the above rigid-body modes). The mathematical statement is that the Green's function for DLOD is a delta function, equivalent to the trivial transfer function of unity, a direct consequence of the eigen frequency being zero. In the thought experiment analogy of Section 2, this is likened to a fictitious swing of zero inertia hence zero “memory”. Specific to the Earth, the most obvious axial spin mode is that of the mantle representing the z-component for the rotational speed or the length-of-day variation DLOD that has been under study for decades. A more subtle axial spin mode is the supposed axial torsional libration of the solid inner-core within the fluid outer core. It owes its existence to the mantle-inner core gravitational (MICG) torque [28,29], provided of the spherical-harmonic degree2 order-2 density anomalies in the Mantle and inner core. In this mode, the conservation of angular momentum dictates that the inner core librates not alone, but accompanied at the same time by an opposite mirrored libration of the mantle which has an amplitude ~1200 times (the ratio of the moments of inertia of the mantle to the inner core) smaller and manifests as DLOD oscillation that can be observed. The MICG torque also determines the libratinal natural frequency. By equating the MICG axial libration to the observed 6-year DLOD oscillation, the equatorial ellipticities of the inner core and the lower mantle can be constrained, the latter pertains directly to the density anomalies of the Large Low-ShearVelocity Provinces (LLSVP; see references cited in Chao [29]. Indeed this proposition is corroborated by the fact that the 6-year DLOD oscillation does exhibit moderate fluctuations in the amplitude and the period [30], the main traits of a resonance mode under continual excitation. Further investigations should be warranted.
3.2. Equatorial rotational modes: wobbles and nutations The equatorial xey components forms a 2-D vector conveniently expressed in the complex form m ¼ mx þ imy [27]. Denoting the fractional perturbation of the angular velocity, m is the parameter used to quantify the rotational axis orientation (such as for the Chandler wobble and the free-core nutation, see below, and the equatorial Slichter translations for that matter). The positive frequency domain of m corresponds to the prograde rotation, and the negative to retrograde. Chao and Hsieh [31] has demonstrated that, for an individual (isolated) rotational mode in the U-spinning frame, the excitation of m is governed by the following set of equations:
mðtÞ ¼
1 E expðiutÞ 2Uðs0 uÞ
i mðtÞ ¼ e 2U
Zt
exp½is0 ðtetÞEðtÞdt
(6)
∞
i dmðtÞ 1 þ mðtÞ ¼ EðtÞ dt 2Us0
s0
(5)
(7)
where the excitation function, whatever the physical sources may be, is also expressed in the complex form: E ¼ Ex þ iEy.
Eqs. (5)e(7) are actually equivalent to one another. Eq. (5) describes the response of the system to a harmonic excitation with frequency u, which would feel the resonance with the amplification factor 1/(s0 u) as asserted above. Upon integration over u along the Bromwich contour under causality and invoking the Cauchy's theorem of residues one gets Eq. (6) for an E(t) of arbitrary time dependence. Eq. (6) is indeed duly expressed in terms of the temporal convolution of E(t) with the Green's function exp(is0t) as is Eq. (4). Eq. (7) is simply the differential form of Eq. (6). Just as Eq. (6) means convolution, Eq. (7) constitutes the deconvolution where the observed m can be processed numerical to recover the excitation function E; we shall return to this in the next section. We mention that Munk and MacDonald [27] reached the equivalent set of formulas for the Chandler wobble excitation based on the EulerLiouville equation (which is formulated in the s frame of course). Moreover, the excitation Eqs. (5)e(7) can be readily shown to be invariant w.r.t. the kinematic rotational transformation of reference frame, as required by physics [31]. Specific to the Earth, the relevant and interesting physics arises pertaining to an oblate body, for example as one lets aspherically symmetric Earth deform and become oblate subject to the gravitationehydrostatic equilibrium with gravitation now including the centrifugal potential. The oblateness is represented mathematically by the zonal spherical harmonic of degree 2 and order 0, with Earth surface value of ~1/300 and somewhat smaller internal values dictated by Clairaut's equation [5,32]. Suppose in addition we consider an idealized three-layered Earth model comprised of a solid mantle, a fluid outer core and a solid inner core. Then the equatorial rotational motion has 4 degrees of freedom, giving rise to 4 corresponding rotational modes as eigen-solutions [32], namely the mantle Chandler wobble, the free-core nutation (FCN), the free inner-core nutation (FICN), and the inner-core wobble. At the present time the first two have been observed in astrometric and space geodetic measurements of Earth rotation variations, the FICN has been implied in data processing [33,34], whereas the inner-core wobblere mains undetected. We might add here that while classified as nutations on phenomenological grounds [26], FCN and FICN are dynamically distinct from the familiar astronomical precession and nutations resulting from the external luni-solar tides. In addition, there are a host of fluid “core mode” oscillations that are supposed to exist in the rotating fluid outer core as eigen-solutions to the magnetohydrodynamic equations of motion (e.g., Smith [26]; for a review see Holme [13]). We further point out that the steady phase in (ii) above is also observed between the x and y components of m, they would undergo essentially the same undulation of amplitude over time, maintaining the (near-) circular 2-D motion. For the Chandler wobble the x component would lead the y component by a quarter cycle, and the opposite sense for FCN. We stress that, while the nature and the response to excitation are altered by spin as described above, the end behavior of Earth's rotational modes is found to observe loyally all of the above qualitative features (ieiv) that are dictated by convolution. The case of the Chandler wobble has been elucidated by Chao and Chung [18] by virtue of synthetic Monte-Carlo experiments, where a convolution by white-noise excitation well mimics all of (ieiv), particularly for the drastic Chandler-wobble phase jump (iv) that occurred during the 1920s. The FCN is likewise examined by Chao and Hsieh [31] using data obtained from the very-long-baseline interferometry (VLBI) technique. The apparent amplitude as well as the apparent period had clearly been fluctuating depending on the time segment of the data analyzed, and a scenario (iv) of phase jump did occur during the 1990s.
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B.F. Chao / Geodesy and Geodynamics xxx (2017) 1e6
4. De-convolution The above in fact leads to some practical caveats that should be drawn as corollaries: It would be misleading to dwell on the face values of the motion amplitude of the normal modes, which is subject to time convolution (4) and have been processed or “masqueraded” by the convolution. Cases in point are the wobbles and the free nutations. For one thing, the changes seen in their oscillation amplitude do not natural mean certain corresponding peculiar behavior in the excitation at the time. Similarly, on account of (iii), it is groundless to assert (albeit unable to rule out either) the existence of a time-variable period or multiple periods for the resonance system upon detecting such from the face values of the oscillation. Moreover, it is futile to attempt to estimate the rate of decay (or Q factor) via the half-width of the apparent spectral peak of the resonance. Another potential, but less recognized, pitfall on account of (iv) [18] is the inclination to proclaim and then search for peculiarity in the excitation at the times when the phase of the oscillation experiences apparent drastic changes. Thus, to the goal of understanding properly the system and its excitation processes, one ought to resort to the deconvolution process which enables the dissociation of the effect of the excitation from the system Green's function. Numerical deconvolution, pioneered by Wiener [35], has been widely applied in signal and image processing, and algorithms abound for treating the variety of geophysical data in particular (e.g., Robinson and Treitel [36]). In our case of the deterministic Green's function exp(is0t), the deconvolution is readily performed in an analytical fashion as Eq. (7), a procedure that is equivalent to a notch filter at the resonance frequency s0 (e.g., Wilson [37], Brzezinski [38]). Such “direct” approach (as opposed to the “integration” approach; see Chao [17]), thereby the excitation function is deconvolved from the observational data, has been rightfully the approach taken in polar-motion excitation studies reported in the literature. Of relevance are the strength of the angular momentum of the geophysical fluids and their exchange with the solid Earth within the resonance frequency band of the Chandler wobble, i.e. the (prograde) period of 434 days in the U-spinning frame. It has become well known(e.g., Gross [11]) that the majority of the polarmotion excitation (as well as the DLOD excitation for that matter) is provided by the tidal and the atmospheric angular momentum variations in a continual fashion, plus secondary contributions due to the oceans, the land hydrology, and presumably the cores. Among them, an ample amount of excitations is at work to excite the Chandler wobble to the observed amplitude of a few hundred mill arc seconds (on the order of several meters on the Earth's surface). Similar deconvolution has been undertaken in studying the FCN and its excitation using VLBI data (e.g., Brzezinski [38,39]); the results however turn out in conclusive thus far. The main reason can be attributed to the fact that the FCN resonance period, some 400þ days (retrograde) in the inertia frame, is nearly-diurnal in the spinning terrestrial frame (in fact FCN is also known in the literature as the Nearly Diurnal Free Wobble in that frame). That alone entails the difficult tasks of making global, sub-half-daily monitoring of the geophysical fluid fluctuations. More critically, at the diurnal timescale the power in the geophysical fluid fluctuations are in general relatively minor, exciting FCN only to the observed amplitude of a fraction of milliarc second (on the order of ~1 cm on Earth's surface). It was proclaimed that sufficient atmospheric power does exist in the right frequency band to excite the FCN [38e40], but no quantitative evidences have been confirmed in the available data even after deconvolution. The deconvolution procedure has proved effective in another application w.r.t. the rotational modes e namely estimating the
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complex eigen frequency s0, or the period P and the Q factor of the resonance Green's function, by a procedure first devised and employed by Furuya and Chao [41]. Suppose, not knowing the true value, one had chosen a s0 somewhat off the true value, i.e. a somewhat erroneous set of (P, Q), in performing the deconvolution (7). The deconvolution notch filter would then somewhat miss the target and unable to completely remove the extra signal power attributable to the resonance. The acquired E(t) would then contain some remnant power that is present only as artefact, elevating the resultant spectral power especially around the resonance band. Conversely, the acquired E(t) would assume the least power, especially in the resonance band, had one chosen the correct (P, Q) in deconvolution. As such, a search of the minimum power of E(t) in the parameter (P, Q) space constitutes an optimum means of estimating (P, Q). This deconvolution method of estimating (P, Q) has been demonstrated to be robust and unbiased through Monte-Carlo simulations [41]. It serves well for the Chandler wobble, yielding P ¼ 433.7 ± 1.8 days (w.r.t. the rotating frame) and Q ¼ 49 (35100) [41], see also Ref. [11] for parallel estimates by other means). Thanks to the fact that the majority of the excitation sources is known (such as the atmospheric angular momentum, see above), they can be favorably removed to leave the residual excitation essentially uncorrelated with the random noise whose variances are to be minimized together, as required by stipulation in the optimization procedure. Such, however, is not entirely the case when one is to apply the technique on the FCN whose excitation sources stay unknown. As such, the remnant spectral power said above cannot be characterized adequately by a random process, and statistical bias may result. Moreover, the effect of the decay (Q factor) in the oscillation amplitude is less conducible to numerical separation from the excitation effect numerically, especially when the data timespan is short. Further compounded by the relatively low signal-to-noise ratio of the FCN data in the first place, the deconvolution method leaves something to be desired for FCN: It yields an optimal estimation of P ¼ 441 ± 4.5 sidereal days (sd, w.r.t. the inertia frame), but is totally insensitive to and hence unable to reach an estimate for the Q value [30]. The 1-sigma uncertainty (±4.5 sd) in P is estimated based on Monte Carlo simulations under white-noise excitations with realistic signal-to-noise ratios. The retrograde diurnal tides, themselves incapable of exciting the FCN (lacking power at all but the tidal frequencies), experience anomalous amplification in amplitude depending on how close they are to the FCN resonance. The observations of such amplifications have been utilized to estimate the FCN's resonance parameters P and Q (see review by Rosat [42]), yielding estimates of P ¼ 425e435 sd. Both sets of the estimates are significantly distinct from the theoretical value P ¼ 460 sd predicted by idealized Earth models under the assumption of rotation-gravitational hydrostatic equilibrium [43]. Aside from the proposition to attribute the lower observed P to anomalously larger oblateness of the outer core [33,44], the causes of the sediscrepancies among the three sets of estimates are yet to be fully understood. 5. Concluding remarks The rotations of various Earth components vary slightly under respective torques on all timescales. These small variations can be duly represented as superposition of normal modes, or eigensolutions to the fundamental equations of motion posed as eigen value problems. Each normal mode can be excited to undergo a simple harmonic oscillation with specific resonance eigen frequency; the excitation process is formulated as a temporal convolution of the Green's function with the respective excitation function.
Please cite this article in press as: B.F. Chao, On rotational normal modes of the Earth: Resonance, excitation, convolution, deconvolution and all that, Geodesy and Geodynamics (2017), http://dx.doi.org/10.1016/j.geog.2017.03.014
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B.F. Chao / Geodesy and Geodynamics xxx (2017) 1e6
The convolution is a notion well established in classical mechanics as well as in modern signal analysis. However, less recognition has been lain on the consequences of the convolution when the excitation is by a (more or less) random, continual process, as is the case for the Earth's rotational normal modes. The said consequences masquerade as apparent fluctuations in the resonance period as well as the undulations in the excited amplitude and phase that are observed. That has regrettably led to unjustified proclamations and likely futile pursuits of peculiarities in the excitation processes or even the nature of the resonance system itself. The process of deconvolution retrieves the nominal excitation function from the observations, yielding vital information to reveal the identification of the excitation sources. However, other than the Chandler wobble and DLOD, little has been confirmed thus far w.r.t. the excitation of the Earth's rotational modes that involve the cores and that are only weakly excited if at all. In the meantime, the deconvolution has been utilized as an effective method of estimating the resonance complex frequencies of the rotational modes including the Chandler wobble and the FCN. Taking the normal-mode point of view thus proves to be constructive and productive toward the practical treatment and understanding of the Earth's rotational motions. We have seen successful applications and useful results for the Chandler wobble and FCN; we anticipate it to likewise elucidate on the supposed rotational modes that are yet to be detected in observations. Acknowledgments This work is supported by the Taiwan Ministry of Science and Technology via grant #104-2116-M-001-006. References [1] H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, Mass, 1950. [2] L.D. Landau, E.M. Lifshitz, Mechanics, Volume 1 of a Course of Theoretical Physics, Pergamon Press, 1969. [3] S. Thornton, J. Marion, Classical Dynamics of Particles and Systems, Thomson Learning, Inc., Belmont, 2004. [4] H. Jeffreys, The Earth, Its Origin, History and Physical Constitution, Cambridge Univ. Press, 1924. [5] F.A. Dahlen, J. Tromp, Theoretical Global Seismology, Princeton Univ. Press, Princeton, 1998. [6] Monog. #24D.E. Smith, D. Turcotte (Eds.), Contributions of Space Geodesy to Geodynamics: Earth Dynamics, Amer. Geophy. Union, Washington, DC, 1993. [7] H. Lamb, Hydrodynamics, 6th ed., Dover Publications, New York, 1932. [8] T. Tanimoto, J. Um, K. Nishida, N. Kobayashi, Earth's continuous oscillations observed on seismically quiet days, Geophys. Res. Lett. 25 (1998) 1553e1556, http://dx.doi.org/10.1029/98GL01223. [9] J. Rhie, B. Romanowicz, Excitation of Earth's continuous free oscillations by atmosphereeoceaneseafloor coupling, Nature 431 (2004) 552e556, http:// dx.doi.org/10.1038/nature02942. [10] B.F. Chao, R.S. Gross, Changes in the Earth's rotation and low-degree gravitational field induced by earthquakes, Geophys. J. Roy. Astron. Soc. 91 (1987) 569e596. [11] R.S. Gross, Earth Rotation Variations e Long Period, Chap. 3.9, in: G. Schubert (Ed.), Treatise on Geophysics, 2nd ed., Elsevier, Amsterdam, The Netherlands, 2015. [12] D. Dehant, P.M. Mathews, Earth Rotation Variations, Chap. 3.10 in Treatise on Geophysics, in: G. Schubert (Ed.), 2nd ed., Elsevier, Amsterdam, The Netherlands, 2015. [13] R. Holme, Large-Scale Flow in the Core, Chap. 8.04, in: G. Schubert (Ed.), Treatise on Geophysics, 2nd ed., Elsevier, Amsterdam, The Netherlands, 2015. [14] B.F. Chao, Excitation of normal modes on non-rotating and rotating earth models, Geophys. J. Roy. Astron. Soc. 68 (1982) 295e315. [15] B.F. Chao, Normal mode study of the Earth's rigid-body motions, J. Geophys. Res. 88 (1983) 9437e9442. [16] P.M. Morse, H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953. [17] B.F. Chao, On the excitation of the Earth's polar motion, Geophys. Res. Lett. 12 (1985a) 526e529. [18] B.F. Chao, W.Y. Chung, Amplitude and phase variations of Earth's Chandler Wobble under continual excitation, J. Geodyn. 62 (2012) 35e39, http:// dx.doi.org/10.1016/j.jog.2011.11.009.
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Benjamin Fong Chao.
Please cite this article in press as: B.F. Chao, On rotational normal modes of the Earth: Resonance, excitation, convolution, deconvolution and all that, Geodesy and Geodynamics (2017), http://dx.doi.org/10.1016/j.geog.2017.03.014