New aspects of the equilibrium pole tide

Summary. We have developed a new spherical harmonic algorithm for the calculation of the loading and self-gravitating equilibrium pole tide. Based on a suggestion of Dahlen, this approach minimizes the distortions in tide height caused by an incomplete representation of the ocean function. With slight modification our approach easily could be used to compute self-gravitating and loading luni-solar tides as well.
Using our algorithm we have compared the static pole tide with tide observations at a variety of locations around the world. We find statistically significant evidence for pole tide enhancements in mid-ocean as well as the shallow seas.
We have also re-investigated the effect of the static tide on the Chandler wobble period. The difference between the wobble period of an oceanless, elastic earth with a fluid core (Smith & Dahlen) and the period of an earth minus static oceans yields a 7.4-day discrepancy. We conclude from tide observations that much of the discrepancy can probably be accounted for by non-equilibrium pole tide behaviour in the deep oceans.     

Atmospheric excitation of the annual wobble

Summary. The excitation of the annual wobble due to the atmosphere is computed on the basis of modern global homogeneous atmospheric pressure and temperature fields of 5°× 5°.
The contribution of the oceans is estimated using two hypotheses of the response to the atmospheric load.
The results are compared with estimates by other authors and with data from astronomical observations. The atmospheric excitation computations have now reached their maximum accuracy and the discrepancies with the observations demonstrate the large errors in the estimates of the non-atmospheric contributions for which constraints are given.     

Meteorological excitation of the annual polar motion

Summary. Numerous studies have indicated that the annual term in the polar motion cannot be explained in any detail by meteorological/hydrological excitation and no reasonable alternative excitations have been put forward. Part of the problem has been that the hydrostatic adjustment of the oceans to the atmospheric pressure changes has traditionally been computed using the inverse barometer approach. This approach does not properly model the gravitational interaction between the atmosphere and oceans, and the inverse barometer theory is modified in this paper to account for this properly. The information necessary to compute the ocean tide and polar excitation caused by any change in the atmospheric pressure pattern is presented. The results of the application of this theory to two global atmospheric pressure data sets are examined and compared to results of other workers.
It is concluded that the atmosphere is observed well enough to answer the question of the annual excitation of polar motion and it is argued that the ground water excitation is the component with the largest error and remains the chief obstacle to the successful solution of this problem.     

Detailed gravity-tide spectrum between one and four cycles per day

Summary. The results of previous work by the authors is used to remove most of the effects of ocean and atmospheric loading from an 18-month Earth gravity-tide record. The remaining signal is examined for additional influence of ocean and atmosphere and for evidence of the frequency-dependence of the response of the solid earth. Variations in time of the measured tides are shown to result from the atmospheric tide at S ₂ and appear to result from variations in ocean tides at other frequencies. The frequency-dependence of the solid earth response near 1 cycle per siderial day is found to be consistent with the nearly diurnal free wobble. However, the influence of the ocean on the small but crucial Ψ₁ tide is uncertain. Anomalous responses are observed at several other frequencies but except for the case of ρ₁ it is argued that anomalous ocean tides are plausible and could therefore explain the observations.     

Modelling the pole tide and its effect on the Earth's rotation

Summary. The pole tide is the response of the ocean to incremental centrifugal forces associated with the Chandler wobble. The tide has a potentially important effect on the period and damping of the wobble, but it is at present not well constrained by observations. Here, we construct both analytical and numerical models for the pole tide. The analytical models consider the tide first in a global ocean and then in an enclosed basin on a beta-plane. The results are found to approach equilibrium linearly with decreasing frequency and inversely with increasing basin depth. The numerical models solve Laplace's tidal equations over the world's oceans using realistic continental boundaries and bottom topography. The results indicate that the effects of the non-equilibrium portion of the deep ocean tide on the Chandler wobble period and damping are negligible.     

The Passive Influence of the Oceans upon the Rotation of the Earth

A general theory is developed which allows the exact numerical computation of the static equilibrium response of a non-rotating spherically symmetric Earth model covered by thin oceans with geometrically irregular coastlines to the action of an imposed static tidal or centrifugal potential. The theory is self-consistent, and takes into account the gravitational self-attraction of the oceans and the elastic-gravitational response of the Earth model to both the applied potential and the equilibrium oceanic tidal load on the surface. The results are used to determine the influence of an equilibrium pole tide on the free period and the associated rotational eigenfunction of the Chandler wobble. If the pole is globally well represented by this equilibrium approximation, its effect is to increase the Chandler wobble period by 27·6 days. It is shown that a fully self-consistent theory of the rotation of an Earth model with oceans predicts that changes in spin and wobble will be coupled, and that the Chandler wobble should, as a result, be accompanied by an associated periodic change in the length of day. The consequences of spin-wobble coupling are explored quantitatively, and found to be slight.     

Oscillation, Nutation and Wobble of an Elliptical Rotating Earth with Liquid Outer Core

We have constructed a general first-order theory describing those small oscillations of a rotating elliptical earth that are affected by the presence of a liquid outer core. The theory is applicable to free core oscillations and earth tides. Care has been taken to include the effects of the wobble or nutation due to the rotation of the outer core relative to the solid earth. On the basis of the theory the free spheroidal modes of degree 2 and order 1 have been investigated. We have searched for and listed undertones with periods less than 28 hr. No upper limit to the eigenperiods has been detected. It is shown that stable, unstable and neutral polytropic cores are capable of free oscillation. At a period close to the sidereal day the spheroidal mode is accompanied by rigid rotation of the liquid outer core with respect to the solid earth. This is the well-known diurnal wobble of the Earth. It appears probable that the diurnal wobble is one of a class of similar wobbles that involve large toroidal motions in the outer core. Finally, the amplitudes of the 18·6-yr principal nutations has been computed. Excellent agreement is found with observed values.     

Further evidence for oceanic excitation of polar motion

While the role of the atmosphere in driving variations in polar motion is well established, the importance of the oceans has been recognized only recently. Further evidence for the role of the oceans in the excitation of polar motion is presented. To estimate the equatorial excitation functions, χ ₁ and χ ₂ , for the ocean, we use velocity and mass fields from a constant-density ocean model, driven by observed surface wind stresses and atmospheric pressure, for the period 1993–1995; comparison with similar functions derived from a more complex density-stratified ocean model indicates the effectiveness of the simple constant-density modelling approach. Corresponding atmospheric excitation functions are computed from NCEP/NCAR re-analyses. Results indicate significant improvements in the agreement with the observed polar motion excitation when the simulated oceanic effects are added to atmospheric excitation. Correlations between the polar motion and the geophysical signals at periods of 15–150 days increase from 0.53 to 0.80 and from 0.75 to 0.88 for χ ₁ and χ ₂ , respectively. The oceanic signals are particularly important for seasonal variations in χ ₁ (correlation increases from 0.28 to 0.85 when oceanic excitation is included). A positive impact of the oceans on more rapid polar motion is also observed, up to periods as short as 5 days. The sensitivity of the results to different forcing fields and different amounts of friction in the oceans is also discussed.     

Random walk of the Earth's pole related to the Chandler wobble excitation

Summary. The equation governing the polar motion shows that the polar secular drift and the Chandler wobble amplitude are related to each other. In particular, a drift of the mean pole position comes out as a consequence of the maintenance of the Chandler wobble by possible step perturbations of the Earth's inertia tensor.
The minimum excitation functions necessary to explain the Chandler wobble amplitude variations for the period 1901–84 are derived from the Chandler term, with the hypothesis that the excitations follow a uniform random distribution in time. It is shown that they have the statistical properties of the steps of a two-dimensional random walk. These functions are then used to derive, from a statistical simulation, a lower limit of the secular drift which may result from the excitation of the Chandler wobble.
The drift generated by the random walk is of the same order of magnitude as the observed secular drift for the period 1901–84, but their time dependence is different. This indicates that the observed secular drift cannot be explained as the consequence of an excitation of the Chandler wobble by random steps of the Earth's inertia tensor. However, the possible contribution of the Chandler wobble excitation to the polar drift has to be taken into account when other mechanisms, such as lithospheric rebound related to deglaciation, are proposed.     

Refined correlations between atmospheric and rapid polar motion excitations

In this work we employ Monte Carlo experiments to explore reports by others of a statistically significant correlation between atmospheric angular momentum variations and polar motion on timescales of days to months. Our experiments verify that the correlation between atmospheric and geodetic excitation is statistically different from zero at the 0.997 confidence level, and demonstrate that the correlations improve with more recent recreations of the older data sets. Additional Monte Carlo experiments reveal that, during the previous decade, about 60 per cent of the atmospheric excitation was effective in exciting rapid polar motion, and nearly 80 per cent of the geodetic excitation was atmospheric in origin on these timescales; with the older data sets, barely 50 per cent of the geodetic excitation could be ascribed to an atmospheric origin. Possible sources of the remaining polar motion excitation are briefly discussed. Our work implies that simply subtracting atmospheric angular momentum from geodetic data may not be the best way to remove the atmospheric contribution. We also present the first correlation results employing atmospheric excitation data corrected for the dynamic response of the oceans to barometric pressure forcing.     

The influence of earthquakes on the Chandler wobble during 1977–1983

Summary. A direct calculation is made of the effect on the Chandler wobble of 1287 earthquakes that occurred during 1977–1983. The hypocentral parameters (location and origin time) and the moment tensor representation of the best point source for each earthquake as determined by the 'centroidmoment tensor' technique were used to calculate the change in the Chandler wobble's excitation function by assuming this change is due solely to the static deformation field generated by that earthquake. The resulting theoretical earthquake excitation function is compared with the 'observed' excitation function that is obtained by deconvolving a Chandler wobble time series derived from LAGEOS polar motion data. Since only 7 years of data are available for analysis it is not possible to resolve the Chandler band and determine whether or not the theoretical earthquake excitation function derived here is coherent and in phase with the 'observed' excitation function in that band. However, since the power spectrum of the earthquake excitation function is about 56 dB less than that of the 'observed' excitation function at frequencies near the Chandler frequency, it is concluded that earthquakes, via their static deformation field, have had a negligible influence on the Chandler wobble during 1977–1983. However, fault creep or any type of aseismic slip that occurs on a time-scale much less than the period of the Chandler wobble could have an important (and still unmodelled) effect on the Chandler wobble.     

Asymptotic eigenfrequencies of the Earth's normal modes

We derive asymptotic formulae for the toroidal and spheroidal eigenfrequencies of a SNREI earth model with two discontinuities, by considering the constructive interference of propagating SH and P-SV body waves. For a model with a smooth solid inner core, fluid outer core and mantle, there are four SH and 10 P-SV ray parameters regimes, each of which must be examined separately. The asymptotic eigenfrequency equations in each of these regimes depend only on the intercept times of the propagating wave types and the reflection and transmission coefficients of the waves at the free surface and the two discontinuities. If the classical geometrical plane-wave reflection and transmission coefficients are used, the final eigenfrequency equations are all real. In general, the asymptotic eigenfrequencies agree extremely well with the exact numerical eigenfrequencies; to illustrate this, we present comparisons for a crustless version of earth model 1066A.     

Some remarks about the degree-one deformation of the Earth

The degree-one deformation of the Earth (and the induced discrepancy between the figure centre and the mass centre of the Earth) is computed using a theoretical approach (Love numbers formalism) at short timescales (where the Earth has an elastic behaviour) as well as at long timescales (where the Earth has a viscoelastic or quasi-fluid behaviour). For a Maxwell model of rheology, the degree-one relaxation modes associated with the viscoelastic Love numbers have been investigated: the M_o mode does not exist and there is only one transition mode (instead of two) generated by a viscosity discontinuity.
The translations at each interface of the incompressible layers of the earth model [surface, 670 km depth discontinuity, core-mantle boundary (CMB) and inner-core boundary (ICB)] are computed. They are elastic with an order of magnitude of about 1 mm when the excitation source is the atmospheric continental loading or a magnetic pressure acting at the CMB. They are viscoelastic when the earth is submitted to Pleistocene deglaciation, with an order of magnitude of about 1 m. In a quasi-fluid approximation (Newtonian fluid) because of the mantle density heterogeneity their order of magnitude is about 100 m (except for the ICB, which is in quasi-hydrostatic equilibrium at this timescale).     

Comments on the Chandler wobble Q

Summary. The Chandler wobble Q _w, as obtained from the astronomical data cannot be equated with the Q _m of the source of damping, as an examination of Chandler wobble energetics reveals. We find that if dissipation occurs in the mantle then Q _w≃ 9 Q _m, implying that either the mantle Q is frequency dependent or the wobble Q is much larger than 100. If the dissipation is in the oceans then Q _w≃ 20 Q _o, and the pole tide must be far from equilibrium.     

Estimation of period and Q of the Chandler wobble

The period P and Q -value of the Chandler wobble are two fundamental functional of the Earth's internal physical properties and global geodynamics. We revisit the problem of the estimation of P and Q , using 10.8 yr of modern polar motion as well as contemporary atmospheric angular momentum (AAM) data. We make full use of the knowledge that AAM is a major broad-band excitation source for the polar motion. We devise two optimization criteria under the assumption that, after removal of coherent seasonal and long-period signals, the non-AAM excitation is uncorrelated with the AAM. The procedures lead to optimal estimates for P and Q. Our best estimates, judging from comprehensive sets of Monte Carlo simulations, are P = 433.7 ± 1.8 (1σ) days, Q =49 with a la range of (35, 100). In the process we also obtain (as a by-product) an estimate of roughly 0.8 for a 'mixing ratio' of the inverted-barometer (IB) effect in the AAM pressure term, indicating that the ocean behaves nearly as IB in polar motion excitation on temporal scales from months to years     

Estimation of the pole tide gravimetric factor at the chandler period through wavelet filtering

Wavelet analysis for filtering is used to improve estimation of gravity variations induced by Chandler wobble. This method eliminate noise in superconducting gravimeter (SG) records with bandpass filters derived from Daubechies wavelet. The SG records at four European stations (Brussels, Membach, Strasbourg and Vienna) are analysed in this study. First, the earth tidal constituents are removed from the observed data by using synthetic tides, then the gravity residuals are filtered into a narrow period band of 256–512 d by a wavelet bandpass filter. These data are submitted to three regression analysis methods for estimating the gravimetric factor of the Chandler wobble. After processing by wavelet filtering, SG records can provide amplitude factors δ and phase lags κ of the Chandler wobble with much smaller mean square deviation (MSD) than these provided by former studies. It is mainly because the wavelet method can effectively eliminate instrumental drift and provide smoothed data series for the regression analysis.     

A physical explanation of the static core paradox

Summary. The introduction of Rayleigh friction and Newtonian cooling into the dynamical problem of determining the excitation of the normal modes of oscillation of an earth model with a fluid core by a transient earthquake source is shown to provide a fully satisfactory resolution and a clear physical explanation of the difficulties and paradoxes which have arisen in previous treatments of the corresponding static deformation problem. The source of the previous difficulties is that the dissipation-free limit is associated with an essential singularity in the static response, unless the stratification in the core is neutral. This singularity, in turn, exists because the eigenfrequency spectrum of any earth model with a non-neutrally stratified core has an accumulation point at zero frequency.     

A comparison of Sq analyses with model calculations

Summary. Three analyses of Sq are examined in detail. The analyses separate Sq into external and internal parts. The external part is used as an inducing field in a model earth consisting of a perfectly conducting inner sphere (con-ductosphere) surrounded by a thin non-uniformly conducting shell representing in some detail the distribution of oceans and continents. The calculated induced part is compared to the internal part deduced in the analysis. The comparison is poor and the reason is shown to be that the magnetic observatories are not located in regions influenced by the main induced oceanic currents. It is further shown that some Sq analyses are unacceptable on physical grounds.     

Quadrangle-grid velocity–stress finite difference method for poroelastic wave equations

A quadrangle-grid velocity–stress finite difference method, based on a first-order hyperbolic system that is equivalent to Biot's equations, is developed for the simulation of wave propagation in 2-D heterogeneous porous media. In this method the velocity components of the solid material and of the pore fluid relative to that of the solid, and the stress components of three solid stresses and one fluid pressure are defined at different nodes for a staggered non-rectangular grid. The scheme uses non-orthogonal grids, allowing surface topography and curved interfaces to be easily modelled in the numerical simulation of seismic responses of poroelastic reservoirs. The free-surface conditions of complex geometry are achieved by using integral equilibrium equations on the surface, and the source implementations are simple. The algorithm is an extension of the quadrangle-grid finite difference method used for elastic wave equations.     

An optimal complex AR.MA model of the Chandler wobble

summary . The character of the Chandler wobble suggests that the optimal representation of the observed polar motion is a complex AR.MA model. This paper develops the theory of such a model and presents a modification of the scalar AR.MA computer program of Akaike, Arahata & Ozaki. The complex AR.MA model is applied to the ILS data covering the period 1900–1975. An optimal AR.MA (1,4) model is obtained. The model parameters are interpreted to give a Chandler frequency of 0.8400 ± 0.0039 cpy a Q value of 50 < Q < 300, with the most probable value of 96 and the power density of the excitation to be about 0.51 × (0".01)²/cpy. This result leads to the conclusion that the contribution of meteorological sources to the excitation of the Chandler wobble is about 11 to 19 per cent.