The Multiple-Frequency Chandler Wobble

J. Phys. Earth, 30, 389-419, 1982

THE MULTIPLE-FREQUENCY CHANDLER WOBBLE

Cheh PAN

Hunter Geophysics, 188 South Whisman Road, Mountain View, California, U.S.A. (Received April 5, 1982; Revised, October 7, 1982)

The problem of the excitation and damping of the Chandler wobble is reviewed and the conventional models analyzed. Using an axially near-symmetric and slightly triaxial Earth Model, a solution of the Liouville equation reveals that the Chandler wobble has multiple frequency and is rather a quasi- permanent phenomenon in the Earth. The multiple Chandler frequency consists of a fundamental frequency attributable to the Earth's triaxiality, as well as series of small feedback frequencies arising from the Earth's instantaneous inertia,. relative angular momentum, and inertia variation. After the introduction of the anelasticity of the mantle into the model, the Chandler wobble is found to be a combination of the Kelvin-Voigt and Maxwell oscillations with the former mechanism dominating the motion. The Chandler feedback comes mainly from the anelastic upper mantle, but it does not rule out fractional contributions from the oceans and atmosphere, the mantle-core coupling, or even large earthquakes; the Chandler wobble may behave like a multiple-coupling differential oscillation with the mantle as the major coupler. The feedback mechanism or frequency modulation gives the Chandler wobble a "beat" appearance. The quasi-permanency or the relaxation time of a multiple-frequency Chandler wobble is of the order of 104 to 106 years or higher, up to the contribution from the Maxwell feedback. The multiple-frequency model explains the Chandler amplitude and frequency variations as well as the persistence of the wobble beyond the damping relaxation time predicted by the single-frequency model.

1. Introduction

Since its discovery in 1891, the Chandler wobble has been an unsolved mystery either for its excitation or for its damping mechanism. The wobble is conventionally known as the free Eulerian nutation of the Earth's figure axis around the rotation axis; it must, therefore, subject to darning and would even- tually be damped completely by the natural dissipative processes within the Earth. Analysis of observed data shows that the relaxation time of the Chandler wobble is between 14 and 73 years with a most probable value of 23 years (JEFFREYS, 1972a). However, systematic observation of the latitude since the turn of the century have shown essentially no noticeable changes of the wobble amplitude besides exhibiting an envelope characteristic of "beat" as that shown in Fig. 1. The persistence of the Chandler wobble implies that it must be maintained by

389 390 C. PAN

Fig. 1. The X and Y components of the polar motion of the ILS homogeneous series. The X component represents polar motion along the Greenwich meridian, and the

Y component represents motion along 90° East Longitude.(After WILSON and VICENTE, 1980) some mechanism of excitation. The coupling between the mantle and the core, either dynamically or electromagnetically, is proven inadequate to maintain the Chandler wobble (BUSSE, 1970; ROCHESTER,1970a). There are yet arguments that, if the cross-coupling of precession and wobble is introduced, the core-mantle coupling may play a more important role (STACEY,1970), as well as that, the impulsive torques resulting in core-mantle coupling may be the common geophysical cause of the excitation of the Chandler wobble and the irregular changes in the length of the day (RUNCORN,1970). In the late 60's, the postulate that large earthquakes are the excitation of the Chandler wobble was reopened (MANSINHA and SMYLIE,1967, 1968, 1970; SMYLIEand MANSINHA,1968, 1971). However, later studies (HAUBRICH,1970; CHINNERYand WELLS, 1972; DAHLEN, 1971, 1973; ISRAELet al., 1973) demonstrate that earthquakes are insufficient to maintain the Chandler wobble. Nevertheless, PINESand SHAHAM(1973a, b) contend that the Chandler wobble and the earthquakes may have a common excitation source, the release of the elastic energy in the Earth; whereas, O'CONNELLand DZIEWONSKI (1976) argue that the cumulative effect of large earthquakes makes a noticeable contribution to the excitation of the wobble. After an overview of the studies of the seismic excitation of the Chandler wobble in the last decade, MANSINHAet al. (1979) state that earthquakes can provide sufficient excitation to maintain the Chandler wobble. On the other hand, WILSONand HAUBRICH(1976) and WILSON and GABAY(1981) find a 0.5 coherence between the Chandler excitation and the seasonal variations of the oceans and atmosphere, while similar analyses (GRABER, 1976; WILSON and GABAY,1981) do not support a correlation between the Chandler excitation and large earthquakes. The Multiple-Frequency Chandler Wobble 391

The spectral analysis of latitude variations shows mostly that the Chandler wobble has a broad peak and a single period somewhere around 420-440 days (MUNK and MACDONALD,1960, pp. 144-153; PEDERSENand ROCHESTER,1972; JEFFREYS,1972a; ROCHESTER,1973; CURRIE,1974; GRABER,1976). The variable single broad peak of the Chandler wobble results in a diversity of Q-value estimates from the wobble, ranging from as low as 30-60 (ROCHESTER,1970b, 1973), 72±20 (CURRIE, 1974, 1975), to 60-600 (GRABER, 1976), and 170 and 47-1,000 (WILSON and VICENTE, 1980),depending upon the data and analysistechniques used by the differentauthors. The disparityof the Q-value estimates com- Plicatesthe interpretationof the damping mechanism of the wobble. If the Q- value is as low as 30-60, then the damping of the wobble cannot be attributable entirelyto the mantle,because the Q-value from seismicwaves and freeoscillations is about 80-100 for the upper mantle and 2,000 for the lower mantle (ANDERSON and ARCHAMBEAU, 1964; ANDERSON et al.,1965; JEFFREYS, 1972b). There is a suggestionthat the ocean may be the energy sink for the Chandler wobble (MUNK and MACDONALD, 1960, pp. 171-173; ROCHESTER, 1970b, 1973). Recent study of the pole tide (MILLER and WUNSCH, 1973; WUNSCH, 1974) indicatesthat the ocean does not seem likelyto be the primary energy sink of the wobble, though it still cannot be ruled out. However, DICKMAN(1979) contends that, if the pole tide is enhanced, then the oceans are capable of damping the Chandler wobble between 2 to 80 years. On the other hand, if the Q-value is 170 or higher as those recently evaluated from the improved polar motion data, then the mantle alone is more than adequate to accomplish the damping of the Chandler wobble. How- ever, STACEY(1970) argues that if the Chandler wobble is damped by the mantle alone, the Q-value from the wobble should be 7.5 times greater than the Q-value of the mantle. Whereas, O'CONNELLand DZIEWONSKI(1976) state that the damping of an episodically excited oscillation such as the Chandler wobble cannot be estimated from its spectrum unless specific assumptions about the nature of the excitation are made. Owing to the inhomogeneities in the observed data (WELLSand CHINNERY, 1973), the results of the spectral analysis of the Chandler wobble vary with the method used. There is evidence that not only the amplitude, but also the period of the Chandler wobble vary with time (MUNK and MACDONALD,1960, pp. 150-153; YUMI, 1970; JIJIMA,1971; PROVERBIOet al., 1972; GRABER,1976; VICENTE and CURRIE, 1976; CARTER,1981). Harmonic analyses of the observed data have also shown that the Chandler wobble consists of split or multiple frequency (GAPOSCHKIN,1972; PEDERSENand ROCHESTER,1972; MILLERand WUNSCH,1973; MCCARTHY,1974; O'CONNELLand DZIEWONSKI,1976; CARTER,1981; DICKMAN, 1981; CHAO,1982). The split or multiple Chandler frequency is compatible with the envelope characteristic of the wobble appearance, which has led COLOMBO and SHAPIRO(1968) to propose a "beat" model for the Chandler wobble. Co- lombo and Shapiro suggest that the Chandler wobble may be attributable to a mechanical interaction between the upper layer with the remainder of the Earth, 392 C. PAN

and, thus, exhibits the envelope characteristic of "beat" phenomena. PAN (1972, 1975) has shown theoretically, by the solution of the equation of polar motion on a slightly triaxial, axially near-symmetric and quasi-rigid Earth model, that the Chandler wobble has two frequency components and is a quasi-permanent phenomenon in the Earth. Its motion is similar to the resonant oscillation of two coupled-oscillators having the same natural frequency plus or minus a small feedback frequency and indeed exhibits beat. However, the feedback frequency arising from a triaxial, axially near-symmetric and quasi-rigid Earth is about four orders of magnitude smaller than that of the observed value. A quasi-rigid Earth model is, therefore, unable to account for the observed Chandler frequency modulation. In the present paper, the assumption of a quasi-rigid Earth is relaxed, and the apparent two-frequency Chandler model derived by PAN (1972, 1975) is further developed into a physically more legitimate multiple-frequency model. The hinge of the solution of the mystery of the Chandler wobble is on the perturbation scheme and the Earth model. The conventional model of free Eulerian nutation of a rigid body of revolution lacks the perturbation to account for the excitation and damping of the wobble, while the MUNK and MACDONALDscheme (1960, pp. 38-39) has overlooked the contribution of the perturbation to the Chandler frequency modulation. Using an improved perturbation scheme and Earth model, this paper presents a more realistic approach to the problem, which gives a multiple frequency solution that explains not only the variations of the Chandler amplitude and frequency but also the observed persistence of the wobble beyond the damping relaxation time predicted by the conventional single-frequency model, or the mystery of the Chandler excitation and damping that has long been obscured by the first-order approximation of the conventional Earth model.

2. The Perturbation and the Earth Model

The Chandler wobble has long been cited in classical mechanics textbooks as a typical example of the free Eulerian nutation of a biaxial rigid body whose rotation axis is slightly misaligned with the symmetrical or principal axis, and such a free nutation is in an equilibrium state. The observation of the nearly 14-month Chandler period instead of the 10-month period predicted by the rigid model had little impact on the fundamental dynamics and geometry the model represents; the lengthening of the Chandler period is attributable to the non- rigidity of the Earth which can be additive to the model. The discovery of the Earth's slight triaxiality from satellite observations (GOLDREICHand TOOMRE, 1969) did not affect the Eulerian model either; there is readily the free rotation of an asymmetrical top explained in mechanics textbooks (SOMMERFELD,1952, pp. 146-150; LANDAUand LIFSHITZ,1960, pp. 116-122) that can be borrowed The Multiple-Frequency Chandler Wobble 393 to account for such a case. However, the rigid model does not explain how the Earth's rotation axis can be misaligned from the figure axis in an equilibrium state of rotation and how the free nutation can be preserved in a rotating body as deformable and dissipative as the Earth. That is to say, the Eulerian equation of rigid body rotation, in fact, lacks the perturbation to account for the excitation and damping mechanism of the Chandler wobble; but excitation and damping are precisely the mechanisms that have mystified the Chandler wobble since its discovery. The Liouville equation (MUNK and MACDONALD,1960, pp. 9-10) is a gen- eralized Eulerian equation which allows particles in the rotating system to move among themselves. A Liouville rotation or the rotation described by the Liouville equation is therefore no longer a rigid rotation because of its tolerance of relative motion and redistribution of mass in the rotating system. However, a complete solution of the Liouville equation is extremely difficult; all inquiries concerning irregularities in rotation take the form of special solutions to the equation through a perturbation scheme to simplify the equation. In the study of the rotation of the Earth, the most widely used perturbation scheme is due to MUNK and MACDONALD(1960, pp. 38-39). In the scheme, the part of the Earth's inertia that is subject to motion is assumed to be additive to the principal inertia of a biaxial rigid Earth. This variable part of the Earth's inertia, as well as the relative angular momentum induced by the relative motion in this part of inertia, are assumed negligibly small in comparison to the principal inertia and relative angular momentum of the biaxial Earth. The principal inertia of the biaxial Earth are thus referred to as the principal inertia of the Earth (MUNK and MACDONALD, 1960, p. 38). The reference frame in the scheme is chosen such that the reference axis or the major axis of the frame is aligned with the principal axis of the biaxial Earth and also nearly parallel to the rotation axis (MUNK and MACDONALD, 1960, pp. 13-14). Under such a scheme, the Liouville equation reduces to a simple form that separates, similar to that of the rigid model, the two wobble components from the diurnal component of the Earth's rotation, while the variable part of the Earth's inertia and the relative angular momentum, as well as their rates of change, constitute the perturbation that excite the polar motion. The scheme, therefore, introduces polar excitation into the Eulerian equation but still, keeps the simplicity of the rigid model to permit expression of the Chandler wobble neatly by a linear complex system. Moreover, its reference frame is convenient as a geographic frame for the observation of polar motion or latitude variation. The Munk and MacDonald scheme explains the slight misalignment between the Earth's figure and rotation axes through polar excitation. Because of polar excitation, however, the instantaneous figure axis around which the rotation axis revolves is no longer the principal or symmetric axis as that in the rigid model, but is an excitation axis which comes into being due to the superimposition of the products of inertia and relative angular momentum onto the biaxial Earth. This instantaneous figure axis or excitation axis deviates slightly from the principal or 394 C. PAN symmetrical axis (MUNK and MACDONALD,1960, p. 44, Fig. 6.1); it is therefore not totally symmetrical but near-symmetrical to the Earth, and referring to this axis the Earth becomes slightly triaxial even if the principal inertia are biaxial (PAN, 1975; 1983a, Fig. 1). However, the reference axis in the scheme is not aligned with this instantaneous figure axis but with the symmetrical or principal axis about which the rotation axis no longer revolves due to polar excitation. On the other hand, the near-parallelism between the rotation and reference axes the scheme assumes is valid only at the moment of incipient polar excitation. After incipient excitation this parallelism deteriorates cyclically in the order twice the Chandler amplitude, because during polar motion the reference axis is rigidly fixed in the Earth while the rotation axis traces out a body cone around the instantaneous figure axis. The choice of the reference frame in the scheme has ignored the axial near-symmetry of the polar motion and thus gives a Chandler frequency identical to that of a free Eulerian nutation (MUNK and MACDONALD,1960, p. 38, Eq. 6.1.4). In the real Earth the rotation axis is actually observed to revolve around the axis of reference (MUNK and MACDONALD,1960, p. 5) which is equivalent to the instantaneous figure axis or excitation axis in the scheme. The wobble frequency about such a near-symmetrical axis cannot be exactly the same as that about a symmetrical axis. This is true even for a rigid Earth, as long as the instantaneous figure axis is near-symmetrical, because referring to such an axis the products of inertia do not vanish (PAN, 1975, 1983a). The Munk and MacDonald scheme has left out the frequency modulation in a Liouville rotation because of its neglect of the axial near-symmetry of the polar motion. From the above analysis we can see that an Earth under polar excitation is axially near-symmetirc and slightly triaxial. That is to say, under excited rotation the Earth possesses not only one but two figure axis; one is the principal or symmetrical axis about which the products of inertia vanish, and the other is the instantaneous figure axis or excitation axis which comes into being due to polar excitation or the relative motion and redistribution of mass in the Earth. The instantaneous figure axis is not really a generalization of the principal axis as MUNK and MACDONALD(1960, p. 41, footnote) have suggested, because it is not a symmetrical axis about which the products of inertia vanish. PAN (1983a) has a detailed discussion on the Earth's axial near-symmetry and slight triaxiality, as well as the distinction between the principal and instantaneous figure axes in the Earth. In the present paper, the Munk and MacDonald scheme is improved with the introduction of the Earth's axial near-symmetry and slight triaxiality, and consequently the wobble frequency excitation into the perturbation. The Earth model used in the paper is fundamentally the same as that in PAN (1975, 1978, 1983a, b). The slight triaxiality of the Earth's principal inertia, A

(1)

where for the present Earth, the values of A=8.0108×1044g-cmcm2, B=8.0110× 1044g-cmcm2, and C=8.0372×1044g-cmcm2 are used for calculation in the paper. A, B, and C are the moments of inertia referring to the Earth's principal axes about which the rotation would reach complete equilibrium; they are therefore not the instantaneous inertia of an Earth under polar excitation. The instantaneous inertia of an excited Earth can be defined through the Earth's axial near-symmetry, which, as illustrated in Fig. 2, represents the deviation of the instantaneous inertia axes from the principal axes. The coordinate systems and notation in Fig. 2 are adapted from space dynamics (THOMSON,1963, pp. 201-202). The (x, y, z) system in the figure represents the Earth's instantaneous inertia axes. The z-axis in the system is the in-

Fig. 2. The (x, y, z) coordinate system is the frame of reference in the Earth, which coincides with the Earth's axes of instantaneous moments of inertia. 396 C. PAN stantaneous figure axis around which the rotation axis revolves, and it is equivalent to the excitation axis in the Munk and MacDonald scheme. The (x, y, z) system is used as the frame of reference in the Earth. That is, the frame of reference in this new scheme is chosen such that it coincides always with the Earth's instantaneous inertia axes instead with the principal axes as that in the conventional model. Here, "always" means the reference frame keeps fixed to the instantaneous inertia axes even if the latter change with time. The instantaneous figure axis can be referred to also as the reference figure axis because it is in fact equivalent to the reference axis in the real Earth around which the rotation axis re- olves. The (a, b, c) system in Fig. 2 is the Earth's principal axes upon which the products of inertia vanish and the Earth's rotation would reach complete equilibrium. The c-axis in this system can be called the equilibrium or principal figure axis of the Earth, and it is equivalent to the principal axis or reference axis in the Munk and MacDonald scheme. But here one should note that in this new scheme neither the rotation axis nor the reference axis is assumed to be nearly parallel to the principal figure axis as that in the Munk and MacDonald scheme. The slight deviation between the (x, y, z) and (a, b, c) systems depicted by the angles (θ,φ) in Fig. 2 represents the Earth's axial near-symmetry. With the Earth's axial near-symmetry defined and the reference frame speci- fied, then, the Earth's instantaneous moments and products of inertia are, respectively (THOMSON,1963, pp. 201-202; PAN, 1975, 1983a, b),

(2)

where for the present Earth, θ=0.8° or 0.014rad* according to PINES and SHAHAM

(1973), and φ=27.3° or 40° (PAN, 1975). Table 1 shows the calculated values of the instantaneous inertia. The Earth's instantaneous inertia, as defined in Eq. 2, are not small additives superimposed upon the principal inertia as those defined in the Munk and MacDonald scheme, but instead expressed through the slight deviation of the instantaneous inertia axes from the principal axes or the axial near-symmetry of the Earth. Moreover, under a Liouville rotation the inertia are not invariables as those in a rigid rotation but subject to vary with time. This means we can dif- ferentiate Eq. 2 with respect to time. The rates of change of the Earth's instantaneous moments and products of inertia are then, with (・) designating d/dt,

* θ=0 .8° is the only value available to the author in the literature. The Multiple-Frequency Chandler Wobble 397

Table 1.2. The Earth'srates of instantaneous change of the moments Earth's and instantaneous products of inertia.inertia at θ=0.014rad. 398 C. PAN

(3)

where (θ,φ) represent the change of the Earth's axial near-symmetry. Physically, θ is equivalent to the rate of polar wandering or secular polar shift, and φ denotes the rate of major tectonic plate movements and mantle flows. PAN (1983a) has a detailed discussion of the physical implication of this rate pair. Equation 3 expresses the Earth's inertia variation through the variation of the deviation between the instantaneous and principal inertia axes. However, this does not necessarily imply that the principal inertia, A, B, and C, are those of a rigid Earth; in a Liouville rotation the principal inertia of the rotating body are also subject to vary. In this paper, however, the variation of the Earth's principal inertia with respect to time is assumed to be very small and thus neglected from Eq. 3. Table 2 lists the values of the inertia variation in the Earth calculated from Eq. 3, assuming θ=10cm/year=5.377×10-16 rad/sec, and φ=5cm/year= 2.535×10-16rad/sec. Figure 3 presents a geometric interpretation of the relationships between the rigid model, the Munk and MacDonald model, and the Earth model used in the present paper. The figure is a plane projection about the north pole but its scale is not proportional to that on the real Earth. The coordinates (a, b, c) in the figure are the plane projection of the (a, b, c) system shown in Fig. 2, and (x, y, z) are the projection of the (x, y, z) system. The angle between the b- and y-axis in the figure is namely the azimuth angle φ in Fig. 2, while the projection of the deviation angle θ is shown by the distance cz. According to GOLDREICH and TOOMRE (1969), the a-axis in the present Earth is in the meridian plane 15°W- 165°E, and the b-axis in the meridian plane 105°W-75°E. This gives a φ-angle

of 40° if the secular polar shift is along the meridian 65°W (MARKOWITZ, 1968), or a φ-angle of 27.3° if the secular polar shift is along 77.7°W (YUMI and WAKO, 1968), because the y-axis follows always the direction of secular polar shift or polar wandering (PAN, 1975). On the other hand, (x1, x2, x3) are the projection of the reference frame used in the Munk and MacDonald model, which is coincidental with the reference frame of a rigid rotation. Because the Munk and MacDonald model is fundamentally biaxial, their x1-axis is through the meridian of Greenwich, and their x2-axis towards 90° east of Greenwich, the same as the orientation of the rigid system. Also shown in the figure is the Chandler motion of the rotation pole on a non-rigid Earth, which describes a deformed circle around the z-pole, while the rigid nutation, shown by the dashed curve in the figure, describes a full circle around the x3-pole. The Multiple-Frequency Chandler Wobble 399

Fig. 3. The diagrammatic depiction of two-dimensional projections of the different coordinate systems in the Earth. The (x, y, z) and (a, b, c) systems, and the angle

pair (θ,φ) are the same as those defined in Fig. 2. The (x1, x2, x3) system cor- responds to the reference frame used in the Munk and MacDonald scheme, which is coincidental with that of the rigid model. The deformed circle represents the polar motion on a rheological Earth and the dashed circle is the polar motion on a rigid Earth; the two models are not consistent.

3. The Equation of Polar Motion

Using the perturbation scheme and Earth model previously defined and with

(・) denoting d/dt, the Liouville equation simplifies, fundamentally the same as that in PAN (1975, Eq. 4), to become a first order, three-component simultaneous system of linear differential equations,

(4) 400 C. PAN

where, the same as those defined in PAN (1975), (mx, my, mz) are small dimensionless quatities describing the Earth's rotation, and (mx, my, mz)are their rates of challge; whereas Ω is the mean rotation speed of the Earth, 7.292×10-5 rad/sec. (hx, hy, hz) is the relative angular momentum in the Earth which can be defined relative to an inertial system fixed in space and coinciding with the (x, y, z) system at the moment of motion; i.e.,

(5)

where (ux, uy, uz) is the relative motion, M is the mass involved in the relative motion, and ΔIxz, ΔIyz, and ΔIz are the additional inertia induced in the Earth due to the relative motion or the redistribution of the mass M. σx and σy in Eq. 4 are the fundamental frequency constituents of the Chandler wobble arising due to the Earth's triaxiality and are defined as:

(6)

As we shall see later, the square root of the product of these two frequency con- tituents constitutes the fundamental frequency of the Chandler wobble. On the Other hand, (ψx, ψy, ψz) is the dimensionless excitation function of the polar motion free of external torques; the three components of the function are defined as:

(7)

where (hx, hy, hz) is the rate of change of the relative angular momentum. Here one should note that, however, the excitation function defined in Eq. 7 is not exactly the same as that in MUNK and MACDONALD(1960, p. 38, Eq. 6.1.5) due to the difference in schemes. In the Munk and MacDonald scheme the excitation is exterior to a biaxial and axially symmetric Earth, because it is due to, as was previously expounded, additional inertia superimposed upon the principal The Multiple-Frequency Chandler Wobble 401 inertia of a biaxial rigid Earth. Whereas, the excitation in Eq. 7 is interior to a triaxial and axially near-symmetric Earth, because the instantaneous moments and products of inertia as well as the relative anglar momentum in the function are those of the whole Earth and therefore are inseparable from the Earth model. The solution of the equation of three-component polar motion in Eq. 4 is yet very complicated. A solution of the equation has been shown in PAN (1975) on a slightly triaxial, axially near-symmetric and quasi-rigid Earth. However, for the study of the Chandler wobble, as was previously mentioned, the assumption of quasi-rigidity has to be relaxed. Then, after some algebraic manipulation to successively eliminate unknowns, the system of three-component, first order simultaneous linear differential equations in Eq. 4 are reduced to a single third-order linear differential equation for each polar component. The thrid- order equations of the three polar components have complicated common coef- ficents consisting of long series of terms. For the sake of mathematical simplicity, small-order terms are neglected from these coefficient series. Then, the coefficient of the third-order derivative of the polar components becomes unit, and the coefficients of the acceleration, velocity, and displacement terms are respectively,

(8)

(9)

(10) These coefficients give a second- to third-order resolution to the polar motion instead of the first-order approximation of the conventional model. Then, by following the standard procedure of the solution of a cubic equation, and assuming an initial condition such that, mx=my=mz=mx=my=mz=0, at t=0, after complicated algebraic manipulation to determine the arbitrary coefficients and regrouping of terms, the solution of Eq. 4 finally becomes,

(11) 402 C. PAN

(12)

(13)

In the solution, the first terms are the incipient polar motion, the second terms are the secular polar shift or polar wandering, and the periodic terms are the Chandler wobble components that are under study in the present paper. (Kx, Ky, Kz) and (Hx, Hy, Hz) in the equations are complicated amplitude parameters determined from the initial conditions of the polar motion; they consist of the Earth's instantaneous inertia, rotation speed, wobble frequency, as well as the excitation function. The detailed definition of these parameters for an incipient solution of the equation of polar motion on a triaxial, axially near- symmetric, and quasi-rigid Earth is shown in PAN (1975). In the present paper these amplitude parameters are not specifically defined, because the main concern in the paper is not the amplitude but the frequency constituents of the Chandler wobble; i.e., the parameters α,β, and γ in Bqs. 11, 12, and 13, which consist of the coefficients shown in Eqs. 8, 9, and 10. With small-order terms neglected, these three frequency parameters are, respectively, defined as follows,

(14)

(15)

(16) where Eqs. 14 and 15 constitute the frequency components of the Chandler wobble, while Eq. 16 is the damping frequency of the wobble. However, to call the parameter γ in Eq, 16 the "damping" frequency at this moment is not accurate. Such a terminology implies friction within the rotating body, which is not the case in a Liouville rotation. In a Liouville rotation the particles in relative motion in the rotating system are only under the central forces; otherwise the angular momentum of the system cannot be conserved (BECKER,1954, pp. 162- 164). Prior to the conversion of the model into a deformable continuum through a transfer function, the parameter γ represents, as implied in Eqs. 11, 12, and 13 and related equations, the attempt of an excited non-rigid earth to reach its rotation equilibrium through the readjustment of its instantaneous inertia by means of relative motion and redistribution of mass in its interior. Friction comes into being only after the non-rigid Earth is converted into a deformable continuum. This, consequently, implies that, the rotation of a deformable body involving non- central forces can hardly be in an equilibrium state (PAN, 1983a). The conventional belief derived from the rigid body rotation that the Earth's rotation must be in an equilibrium state is not really valid. PAN (1983a) has a detailed The Multiple-Frequency Chandler Wobble 403

discussion in this regard. The parameter γ gives a relaxation time for the wobble of a non-rigid Earth in the order of 107 years or higher, about a geologic epoch, the same as that of a quasi-rigid Earth (PAN, 1975). The dissipation of the wobble energy in a non-rigid Earth is, therefore, very slow. Moreover, as shown in Eqs. 11, 12, and 13, γ gives also the relaxation time for the secular polar shift; i.e., the secular motion would accelerate drastically as

soon as the time t reaches 1/γ. Such motion, however, is relative to a reference frame rigidly fixed to the Earth at the moment of incipient excitation and under the condition that all the parameters in the equations are constant with respect to time, which is not what that follows from the perturbation scheme. The

perturbation scheme is based on that (mx, my, mz)≪1, and the reference frame is always to coincide with the instantaneous inertia axes while the latter vary with time. This is a mathematical pitfall of the incipient solution of the polar equation; MUNK and MACDONALD(1960, pp. 270-273) have met analogous situation in their incipient solution based upon their own perturbation scheme, which gives a polar wandering as a linear function of time. Munk and MacDonald were therefore forced to state that their solution is valid for the polar displacements up to 1,000km. In our case, however, the main concern is not the relaxation of the secular motion but the Chandler frequency modulation, which has a time scale orders-of-magnitude smaller and is therefore valid within the secular relaxation time. The choice of the reference frame in the scheme, on the other hand, prevents the reference pole to wander away from the instantaneous figure pole and is ex- pedient for the study of the Chandler frequency modulation.

4. The Multiple-Frequency Wobble

Equations 11, 12, and 13 describe polar motion. These equations manifest that the wobble components are not readily separable from the diurnal component of the Earth's rotation; i.e., the Chandler wobble is in fact three-dimensional. Moreover, the equations show also that the Chandler wobble apparently has two frequency components, α and β. This leads PAN (1975) to conclude that, the Chandler wobble exhibits indeed the "beat" phenomena, because the wobble frequency, analogous to that of the resonant oscillation of two coupled- oscillators, consists of a natural frequency,

(17)

plus or minus a small feedback frequency,

(18)

Through the manipulation of the trigonometric functions of sine and cosine, we can deduce that the feedback frequency gives the wobble an effect of modulation; i.e., 404 C. PAN

(19) where D1 and D2 represent the amplitude components of the wobble. However, the beating mechanism of the Chandler wobble is different from the simple resonant oscillation of two coupled-oscillators in three aspects: (1) The solution of Eq. 4 for each polar component involves a third-order linear differential equation, while the oscillation of coupled-oscillators involves only a second-order differential equation. (2) The excitation function defined in Eq. 7 that appears at the right side of Eq. 4 is not a linear function of the polar motion (mx,my, me), it therefore does not directly feed back to the wobble as the coupling of the oscillators does. (3) The "beats" of the wobble, components, as shown by Eqs. 11, 12, and 13 and also in Fig. 1, are "in phase" with each other; whereas, those of the resonant oscillation, as shown in Fig. 4, are "out of phase" with each other. The feedback frequency, Δσr, calculated from a triaxial, axially near-symmetric, and quasi-rigid Earth model, on the other hand, is about 4.8×

10-14 to 9.5×10-14sec-1 (PAN,1975), which is four orders of magnitude smaller that the observed value, about 2.94×10-10sec-1, based upon GAPOSCHKIN'S analysis of the observed data (1972). This great discrepancy between the theoretical and observed values suggests that a quasi-rigid Earth is not adequate for

Fig. 4. The resonant oscillation of two coupled-oscillators having the same natural frequency. The Multiple-Frequency Chandler Wobble 405

the interpretation of the Chandler wobble; however, such a great discrepancy is also not likely to be totally accounted for by the Earth's non-rigidity alone. The interpretation of the Chandler beating mechanism through Eqs. 17, 18, and 19 is based upon the assumption that the parameters α and β, respectively defined in Eqs. 14 and 15, are the two frequency components of the Chandler

wobble. This interpretation of α and β is, unfortunately, hasty and misleading. α and β are, in fact, two parameters defined solely for mathematical convenience in the solution of the third-order differential equation for each polar component; they group together during the solution long series of terms representing different physical mechanisms in the Earth that contribute to the Chandler frequency. As two identities these two parameters bear little physical significance. Con- sequently also are the apparent natural frequency σr and the feedback Δ σr, both of which comprise long series of terms arising due to different physical mechanisms in the Earth.

The natural frequency, σr, as defined in Eq. 17, can be further expanded as,

(20)

That is, the natural frequency of the Chandler wobble is, in fact, to consist of a fundamental frequency, √ σxσy, and also series of small feedback frequencies. Now let,

(21) and

(22)

(23)

(24) where μ is the dimensionless rigidity of the Earth, and μ/(1+μ) is the transfer function (MUNK and MACDONALD, 1960, p.42) that converts a non-rigid Earth into deformable continuum. σ is the fundamental frequency of the Chandler 406 C. PAN

wobble attributable to the Earth's triaxiality; either of its constituents, σxcand σy defined in Eq. 6, is equivalent to the single wobble frequency of a biaxial and axially symmetric Earth (MUNK and MACDONALD, 1960, p. 38). Whereas, Δσa

is a feedback frequency due to the Earth's instantaneous inertia, while Δσq is due

primarily to the relative motion and Δσs due mainly to the inertia variation in the Earth. Substituting Eqs. 21-24 in Eq. 20, we then have, for a deformable Earth,

(25)

Equation 25 manifests that, the natural frequency of the Chandler wobble can be further separated into a fundamental frequency and three series of small feedback frequencies. The major contribution to the wobble's natural frequency, as shown in Eq. 17, is from the velocity coefficient q, which associates itself with the rate of change of the polar motion in the differential equations of the polar components. This means, the presence of Δσa, Δσq, andΔ σs in the wobble's natural frequency is equivalent to adding small spring and dashpot in parallel with the main oscillator (MORSE,1948, pp. 60-61). The Earth behaves, therefore, like a Kelvin-Voigt body (MUNK and MACDONALD,1960, pp. 33-34) in this case. On the other hand, the feedback frequency Δ σr, as defined in Eq. 18, is due mainly to the displacement coefficient s, which, as defined in Eq. 10, can be further separated into three series of terms involving respectively the Earth's instantaneous inertia, relative angular momentum, and inertia variation. That is,

(26)

(27)

(28)

Because the coefficient s is proportional to the displacement of the oscillation, the feedback mechanism of Δ σI, ΔσM, and ΔσV is equivalent to adding small spring and dashpot in series with the main oscillator (MORSE,1948, pp. 60-61). The Earth wobbles like a Maxwell body (MUNK and MACDONALD,1960, pp. 33-34) in this case.

Substituting Eqs. 25-28 for the parameters α and β, the Chandler frequency components finally becomes, in a third-order resolution,

α0=σ+Δ σa+Δ σq+Δ σs+(Δ σI+Δ σM+Δ σV), (29) The Multiple-Frequency Chandler Wobble 407

β0=σ+Δ σa+Δ σq+Δ σs-(Δ σI+Δ σM+Δ σV). (30) Equations 29 and 30 tell us that, the Chandler wobble is not really a two-frequency beating as PAN (1975) assumed, because the two apparent frequency components α0 and β0 are physically to consist of a fundamental frequency attributable to the Earth's triaxiality as well as series of freedback frequencies arising due to the Earth's instantaneous inertia, relative angular momentum, and inertia variation. The feedback frequencies act as small springs and dashpots either in parallel or in series with the fundamental oscillation. The Earth wobbles therefore like a combination of a Kelvin-Voigt body and a Maxwell body. Such a combination gives rise to a complicated Chandler feedback mechanism. The observed beating characteristics of the Chandler wobble shown in Fig. 1 or the Chandler spectra shown in Fig. 5 are compatible with such a feedback model. However, because a multiple Chandler frequency is in conflict with the conventional model of free single-frequency nutation, most investigators in the past discredited the split and sidebands of the Chandler spectra as due to the inhomogeneities of the data, and sought to eliminate them through analysis techniques (PEDERSENand ROCHESTER, 1972; CURRIE,1974, 1975). The attempt to derive a single frequency from a multiple-frequency motion accounts for, besides the data inhomogeneities, the diversity of the Chandler period and Q-value estimates. Nevertheless, the Chandler frequency modulation shown by Eqs. 29 and 30 look more complicated than the Chandler spectra of 70-year observation in Fig. 5. This apparent dis-

Fig. 5. Unsmoothed power spectrum of the polar motion from ILS-IPMS monthly means, December 1899-October 1970. (After PEDERSENand ROCHESTER,1972) 408 C. PAN

crepancy between the model and the observation can be resolved if we look further at the Chandler feedback mechanism in a quantitative sense. The Chandler feedback frequencies, as shown in Eqs. 22-24 and 26-28, can be grouped into three categories. Δ σI and Δ σa arise due solely to the instantane-

ous inertia of a slightly triaxial and axially near-symmetric Earth, ΔσM and Δ σq involve the relative angular momentum, while ΔσV and Δσs attribute to the inertia variation in the Earth. Among the six feedbacks, ΔσM and Δ σq need further speci- fication on the relative motion in the Earth, while the other four can be readily calculated through Eqs. 22, 24, 26, and 28. Adopting the Earth's instantaneous inertia and their rates of change shown respectively in Tables 1 and 2 and taking the average observed Chandler frequency of 0.844 cycle/year (2.6754×10-8sec-1) as the fundamental Chandler frequency, and from Eq. 21 the transfer function is,

(31)

Table 3 lists the values of ΔσI, Δσa, ΔσV, and Δσs calculated. However, if we assume that the fundamental Chandler frequency is not the average but the dominant spectral peak in Fig. 5, or about 0.835 cycle/year (2.6455×10-8sec-1), then, the transfer function becomes 0.695. Table 4 lists the values of Δ σI, Δσa, ΔσV, and Δσs calculated with this new transfer function. These values vary slightly from those listed in Table 3. From Tables 3 and 4 we can find that: (1) The contribution to the Chandler feedback mechanism from the inertia variation in the Earth is negligibly small if the variation is due mainly to the mantle flow, plate motion, and polar wandering; (2) the Maxwell feedback frequency ΔσI is less than 1% of the Kelvin-Voigt

Table 3. The Chandler feedback frequencies not involving relative motion under transfer function=0.702; θ=0.014 rad.

Table 4. The Chandler feedback frequencies not involving relative motion under transfer function=0.695; θ=0.014 rad. The Multiple-Frequency Chandler Wobble 409 feedback frequency Δσa; and (3) the variation of the feedback frequencies with the fundamental frequency is small. However, based upon Gaposchkin's analysis, the observed feedback frequency is about 2.94×10-10 sec-1. The largest feedback frequency listed in Tables 3 and 4, Δσa, is only about 1.8% of the observed value. The main contribution to the Chandler feedback mechanism is, therefore, from neither the instantaneous inertia nor the inertia variation. On the other hand, the relative angular momentum in the Earth, as shown in Eq. 5, consists of two parts. The first part arises due to the motion relative to the (x, y, z) system (MUNKand MACDONALD,1960, p. 9, Eq. 3.1.5), and the second part is due to the mass redistribution accompanying the motion relative to an inertial system fixed in space and coinciding with the rotating (x, y, z) system at the moment of the motion. Past studies (MANSINHAand SMYLIE,1967, 1968, 1970; SMYLIEand MANSINHA,1971; DAHLEN,1971; O'CONNELLand DZIEWONSKI,1976; MANSINHAet al., 1979) concern only the effect of the mass redistribution on the excitation of the Chandler amplitude. Based upon a further simplified version of the Munk and MacDonald scheme, these studies treat the mass redistribution arising from the relative motion in the Earth as additional inertia superimposedupon the principal inertia of an axially symmetric and biaxial Earth to substitute for the instantaneous inertia of the whole Earth in their excitation function, and the relative angular momentum is totally neglected from the excitation. A comparison of the excitation function in this paper with that in the past studies can see such an oversimplificationin the past studies. Sub- stituting Eq. 5 in Eq. 7, one can find that the polar excitation due to the mass redistribution in the relative angular momentum is practically identical with the excitation due to large earthquakes defined in the past studies except a sign reversal and a differencein Earth model. MUNKand MACDONALD (1960, pp. 51-54) have discussed the polar excitation due to changes in relative angular momentum as well as in instantaneous inertia (products of inertia). In a preliminary analysis under the present perturbation scheme on a quasi-rigid Earth, PAN (1975) has quantitatively demonstrated that the effect of relative motion on polar excitation, even in a transient sense, is not negligible. In the modulation of the Chandler frequency, the relative angular momentum appears, as respectively shown in Eqs. 23 and 27, in the feedback frequencies Δ σq and Δ σM. The component hz of the relative angular momentum dominates Δσq while the components hx and hy dominate ΔσM. If the three components of the relative angular momentum, hx, hy, and hz, are of the same magnitude, then, from Eqs. 23 and 27 we can find that Δ σM is only about 1% of Δσq. This implies that, the Chandler wobble is dominantly a Kelvin-Voigt

oscillation. The observed feedback frequency is, as already cited, about 2.94× 10-10sec-1, which needs a relative angular momentum of 3.5×1035g-cmcm2/sec, or about 1/170,000 of the Earth's total angular momentum, to accounted for in a Kelvin-Voigt oscillation. On the other hand, for a dominantly Maxwell feedback, the angular momentum required is 80 times higher, to reach about 2.8× 410 C. PAN

1037g-cmcm2/sec,or 1/2,000 of the Earth's total angular momentum. Such a magnitude is very difficult to attain by the relative motion in the Earth. This further supports that the Chandler wobble is predominantly a Kelvin-Voigt oscillation. The Chandler wobble, even if it is predominantly Kelvin-Voigt as we have already demonstrated, yet involves relative motion or motions in the Earth that can give rise to a relative angular momentum equivalent to 1/170,000 of the Earth's total angular momentum. In order to continuously feed back to the Chandler wobble, moreover, the relative motion needs to be a perpetual rather than a transient phenomenon in the Earth. The motions in the oceans and atmosphere are of considerable speeds, but the mass redistribution involved is not large enough to totally account for such a great relative angular momentum. It is yet likely such motions could contribute a fraction to the Chandler excitation, as suggested by WILSONand HAUBRICH(1976) and WILSONand GABAY(1981) of the seasonal variations of the oceans and atmosphere, and by DICKMAN(1979) and CARTER(1981) of the enhanced pole tide. Whereas, the relative motion due to earthquakes is transient. Such a transient motion, even occurs episodically and involves considerable mass redistribution, is not likely to provide a con- tinuous feedback to the Chandler wobble, though O'CONNELLand DZIEWONSKI (1976) report that the cumulative effect of large earthquakes can contribute, as shown in Fig. 6, a modulation to the Chandler excitation. The main contribution

Fig. 6. Polar motion synthesized from excitation by large earthquakes (top three curves) and the smoothed observed polar motion with annual term removed (bottom curve, ILS). The damping for each synthetic curve is determined by the value of Qs. (After O'CONNELLand DZIEWONSKI,1976) The Multiple-Frequency Chandler Wobble 411

to the Chandler feedback is more likely to come from the anelastic mantle; the relative motion such as the viscous coupling between the lithosphere and the remainder of the Earth through the upper mantle (COLOMBOand SHAPIRO,1968; PAN, 1975) or even the coupling between the mantle and the core (STACEY,1970; RUNCORN,1970) would involve a mass redistribution greater than all other possible relative motions in the Earth. PAN (1975) has made a preliminary study on the viscouscoupling through the upper mantle. BUSSE (1970) and ROCHESTER (1970a) discredit the adequacy of the coupling between the mantle and the core to excite the Chandler wobble; whereas, SMITH(1977, 1980) and DAHLEN(1980) attribute the lengthening of the Chandler period mainly to the effects of the core and of the yielding of the mantle. However, all the past studies except that of PAN(1975) are based upon the Munk and MacDonald scheme, a remodeling of the problem under the new scheme presented in this paper may not give the same results. The present paper, as a continuation of the study initiated in PAN(1975), concentrates on the Chandler frequency modulation from the mantle; i.e., the feedback from the viscous coupling between the lithosphere and the remainder of the Earth through the upper mantle. The feedback mechanism from the mantle can be evaluated through the direct introduction of the anelasticity Q of the mantle into the perturbation. The quantity Q is useful because it does not depend on the detailed mechanism by which energy is dissipated (MUNKand MACDONALD,1960, pp. 21-22). The

predominant Kelvin-Voigt feedback frequency of the Chandler wobble, Δσq, consists of, as shown in Eq. 23, three terms. The major feedback is from the first term involving the component hz of the relative angular momentum, while the components hx and hy of the relative angular momentum contribute to the feedback through the second and third terms that are orders-of-magnitude smaller. For instance, if the three components of the relative angular momentum, hx, hy, and hz, are of the same magnitude, the second term is less than 0.005%, and the third term is only about 2% of the first term in Eq. 23. This means, the relative angular momentum component hz dominates the Kelvin-Voigt feedback mechanism. The first term in Eq. 23 can be rewritten such as,

(32)

where the fundamental Chandler frequency σ is defined in Eq. 21. A comparison of the terms within the parenthesis at the right side of Eq. 32 with the terms involving relative angular momentum in the excitation function defined in MUNK and MACDONALD(1960, p. 38, Eq. 3.1.5), one can find that they are practically identical except a difference in Earth model as well as hz taking the places of hx and hy. This contribution of the relative angular momentum to the Chandler frequency excitation is totally neglected in the Munk and MacDonald scheme. Equation 32 can be physically interpreted in correlation with the viscous coupling in the mantle. The equation, together with the relative angular mo- 412 C. PAN mentum component hz shown in Eq. 5, represents an equatorial or east-west coupling of the lithosphere about the z-axis with the remainder of the Earth. If such a viscous coupling is indeed to occur in the mantle, then, the anelastic upper mantle will contribute a dashpot effect to the Chandler wobble such that,

(33) where Q is the anelasticity in the upper mantle. From Eqs. 32 and 33, we then obtain the Kelvin-Voigt feedback of the Chandler wobble, approximately,

(34)

Equation 34 presents a convenient way to evaluate the Chandler feedback from the mantle. The Q-value in the mantle, as has been previously noted, is estimated from the attenuation of seismic waves and free oscillations. The Q-value in the upper mantle is about 80-100 according to ANDERSON and ARCHAMBEAU (1964) and ANDERSONet al. (1965),but in the low velocity layer it can be as low as 50 (ANDERSONet al., 1965;JEFFREYS, 1972b; HELMBERGER, 1973; OLSEN et al., 1980). With these Q-values and for a fundamental Chandler frequency of 0.844 cycle/year, Eq. 34 gives a Δσq about 1.34×10-10sec-1 to 1.67×10-10sec-1 for the whole upper mantle to involve in the coupling, and about 2.67×10-10sec-1 if the coupling occurs strictly along the low velocity layer. On the other hand, if the fundamental frequency is 0.835 cycle/year, then Δ σq is 1.32×10-10sec-1 to

1.65×10-10sec-1 involving the whole upper mantle and 2.65×10-10sec-1 along the low velocity layer. The variation of Δσq with the change of the fundamental Chandler frequency, like that of other feedback frequencies shown in Tables 3 and 4, is small. The above calculation shows that, if the previous cited observed value of the feedback frequency of 2.94×10-10sec-1 and the observation of the anelasticity Q in the upper mantle are accurate, then the viscous coupling involving the whole upper mantle makes up 46 to 57% of the Chandler feedback mechanism, while a coupling strictly along the low velocity layer covers as much as 91% of the Chandler frequency modulation. This consequently suggests that, if the viscous coupling through the upper mantle is indeed to occur in the Earth, it is well capable of providing a mechanism to modulate the Chandler wobble. However, the compatibility of the model depends upon, as already noted, the observation of the feedback frequency and of the Q-value in the upper mantle, as well as the contribution to the modulation from the Maxwell and other feedback frequencies. CARTER(1981) reports that the maximum variation of the Chandler frequency caused by modulation is 0.06 cycle/year or 1.9×10-9sec-1 per 0.1sec of arc decrease in the magnitude of the polar motion. Carter's value is the rate of the Chandler amplitude decrease which is not equivalent to the feedback frequency defined in the present paper. However, in case the observed feedback frequency is The Multiple-Frequency Chandler Wobble 413 of such a magnitude, then, even the coupling along the low velocity zone will contribute only 14% to the Chandler feedback mechanism. CARTER(1981) cites another value of 0.036 cycle/year per 0.1sec of arc decrease of the Chandler amplitude, which increases the coverage of the Chandler feedback by the coupling along the low velocity zone only to 24%. On the other hand, as Eq. 34 implies, a Q- value of 45 in the upper mantle will cover the entire Chandler feedback mechanism. If the Q-value in the upper mantle is lower than 45, then the observed Chandler feedback frequency should be greater than the 2.94×10-10sec-1 cited in the present paper, otherwise the modulation would be overdone. For instance, if the Q-value in the upper mantle is as low as 20, then the feedback frequency has to be doubled. The Chandler feedback, therefore, gives a lower bound to the Q-value in the upper mantle that is compatible with the present observation. Based upon the above calculation of the Kelvin-Voigt feedback of the Chan- dler wobble from the anelasticity Q in the upper mantle, it is of interest to make a reverse calculation for the relative angular momentum arising due to such a coupling mechanism. From Eq. 33, we can calculate that the relative angular momentum arising from the coupling involving the whole upper mantle is 1.57× 1035 to 1.96×1035g-cmcm2/sec, about 1/400,000 to 1/300,000 of the Earth's total angular momentum, while a coupling strictly along the low velocity layer gives a greater relative angular momentum, about 3.14×1035g-cmcm2/sec, nearly 1/200,000 of the Earth's total angular momentum. The calculation is compatible with the relative angular momentum of 3.5×1035g-cmcm2/sec calculated from the observed feedback frequency and, therefore, supports the intuitive assumption made through Eq. 33. So far we have demonstrated that the viscous coupling through the upper mantle can be the main feedback to the Chandler modulation, but it is not likely that the mantle is the sole source of the feedback mechanism. As was previously mentioned, motions in the oceans and atmosphere, or even that due to large earthquakes, may contribute fractions to the Chandler modulation. Though BUSSE (1970) and ROCHESTER(1970a) discredit the contribution from the mantle-core coupling to the Chandler excitation, it cannot yet be ruled out that such a coupling may help to feed back to the Chandler wobble. However, because the outer core is liquid, the coupling, if indeed to occur, would involve much more complicacy and uncertainty than that through the upper mantle. The Chandler wobble may behave like a multiple-coupling differential oscillation involving the Kelvin-Voigt and Maxwell mechanisms and with the main feedback from the mantle. After a quantified analysis of the Chandler feedback mechanism, we are now in a position to look at the modulation effects of the feedback frequencies on the Chandler wobble. From Tables 3 and 4 as well as the evaluation of the feedbacks Δ σq and Δ σM, we can see that the feedback frequencies Δ σV and Δ σs can be totally neglected from Eqs. 29 and 30, because their contribution to the Chandler modulation is extremely small. On the other hand, Δ σI is at least two orders of magnitude smaller than Δ σM, while Δ σM and Δσa are both about two orders of magnitude 414 C. PAN

smaller than Δ σq, For the sake of simplicity, Δ σI can also be neglected if the observation period is of the order of 100 years. Then, the Chandler wobble becomes, following the analytical interpretation of the beating of coupled oscillators (MORSE,1948, pp. 52-66) and through the manipulation of the trigonometric functions of sine and cosine,

(35) The Multiple-Frequency Chandler Wobble 415

The beating mechanism of the Chandler wobble shown in Eq. 35 is very much complicated. However, in case that the relative angular momentum components and hy in the Earth are very small and the Chandler wobble behaves hx predominantly like a Kelvin-Voigt oscillation, we can further neglect the effect of the

Maxwell feedback ΔσM from the oscillation. This means, the Maxwell mechanism or the apparent feedback Δσr has no effect on the Chandler beating. The Chandler wobble then approximates,

(36)• Though Eq. 36 is much more simplified than Eq. 35, the beating mechanism it exhibits is yet more complicated than the simple two-frequency model shown in Fig. 4. The Chandler wobble shown by Eq. 36 exhibits an explicit double modula-

tion due to Δ σq and Δσa. Before concluding the multiple-frequency model, it is of interest to make an evaluation of the relaxation time of the Chandler wobble from the Chandler feedback mechanism. Substituting Eqs. 29 and 30 into Eq. 16, the Chandler damping frequency becomes,

(37) where ΔσI and Δ σV, as shown in Tables 3 and 4, are respectively in the order of

3×10-14sec-1 and 1×10-20sec-1. Whereas, from previous calculation we know that, if the relative angular momentum components hx and hy are of comparable magnitude of that of hz, Δ σM is about two orders of magnitude smaller than Δ σq, or about 3×10-12sec-1. The magnitude of γ0 is, therefore, dependent mainly on that of Δ σM. If ΔσM is two order of magnitude smaller than Δ σq, then the relaxation time of the Chandler wobble is nearly 104 years. However, in case the relative angular momentum components hx and hy in the Earth are negligibly small in comparison with hz, then the relaxation time of the Chandler wobble could reach the order of 106 years or higher, in agreement with the quasi-permanency of the Chandler wobble based upon a quasi-rigid Earth model.

5. Conclusions

An analysis of the conventional Earth model and perturbation scheme finds that if the Earth is under polar excitation it becomes naturally axially near-symmetrical and slightly triaxial even if its principal inertia are biaxial; and the rotation of such an Earth is not in an equilibrium state. With the improvement from 416 C. PAN the lack of perturbation of the rigid model of free Eulerian nutation and the oversimplification of the Munk and MacDonald scheme, a perturbation scheme that allows relative motion and mass redistribution in the interior of an axially near-symmetric and slightly triaxial Earth gives a three-dimensional solution to the equation of polar motion or the linearized Liouville equation free of external torques. The solution represents a third-order approximation of polar motion which reveals: (1) The Chandler wobble is not readily separable from the diurnal component of the Earth's rotation and has multiple frequency. The multiple Chandler frequency consists of a fundamental frequency attributable to the triaxiality of the Earth as well as series of small feedback frequencies arising from the Earth's instantaneous inertia, relative angular momentum and inertia variation. (2) The Chandler wobble behaves like a complicated combination of Kelvin-Voigt and Maxwell oscillations, with the Kelvin-Voigt mechanism dominating the motion. (3) The feedback mechanism of the Chandler wobble is most likely to come from the anelastic mantle, though relative motions in other parts of the Earth, such as the seasonal variations of the oceans and atmosphere, the enhanced pole tide, the mantle-core coupling, or even large earthquakes, may contribute fractions to the Chandler modulation. The Chandler wobble may behave like a multiple-coupling differential oscillation with the anelastic mantle as the major coupler. (4) Due to its complicated feedback mechanism, the Chandler wobble exhibits beating characteristics of double or treble modulation in appearance in the observation period of the order of 100 years. (5) The quasi- permanency or the relaxation time of the Chandler wobble ranges from 104 to 106 years or higher, up to whether the Maxwell feedback plays an important role in the damping mechanism or not. A multiple-frequency Chandler wobble explains not only the time variation of the Chandler amplitude and period but resolves also the persistence of the wobble beyond the damping relaxation time predicted by the single-frequency model. The multiple-frequency Chandler wobble yet needs an incipient excitation to start or restart with, but there is no need for a frequent excitation to maintain the wobble as the single-frequency model should; the concept of quasi-permanency interprets the damping of the wobble. The hinge of the solution of the Chandler mystery is on the perturbation scheme and the Earth model. The more realistic perturbation scheme developed in the present paper enables a higher- order resolution of the Chandler motion that was previously obscured by the conventional model of first-order approximation.

The work was initiated at the Institute of Geophysics, Academica Sinica, Beijing, China.

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