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.     

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.     

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.     

Wind stress forcing of the North Sea 'pole tide'

We conduct numerical simulations of the wind forcing of sea level variations in the North Sea using a barotropic ocean model with realistic geography and bathymetry to examine the forcing of the 14 month 'pole tide', which is known to be anomalously large along the Denmark–Netherlands coast. The simulation input is the monthly mean surface wind stress field from the National Centers for Environmental Prediction (NCEP) reanalysis for the 40 year period 1958–1997. The ocean model output sea level response is then compared with 10 coastal tide gauge records from the Permanent Service for Mean Sea Level (PSMSL) over the same period of time. Besides the strong seasonal variations, several prominent quasi-periodicities exist near 7 years, 3 years, 14 months, 9 months and 6.5 months. Correlations and spectral analyses show remarkable agreement between the model output and the observations, particularly in the 14 month, or Chandler, period band. The latter indicates that the enhanced pole tide found in the North Sea along the Denmark–Netherlands coast is actually the coastal set-up response to wind stress forcing with a periodicity of around 14 months. We find no need to invoke a geophysical explanation involving resonance enhancement of the pole tide in the North Sea to explain the observations.     

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.     

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.     

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.     

The period and Q of the Chandler wobble

Summary. We have extended our calculation of the theoretical period of the Chandler wobble to account for the non-hydrostatic portion of the Earth's equatorial bulge and the effect of the fluid core upon the lengthening of the period due to the pole tide. We find the theoretical period of a realistic perfectly elastic Earth with an equilibrium pole tide to be 426.7 sidereal days, which is 8.5 day shorter than the observed period of 435.2 day. Using Rayleigh's principle for a rotating Earth, we exploit this discrepancy together with the observed Chandler Q to place constraints on the frequency dependence of mantle anelasticity. If Qμ in the mantle varies with frequency σ as σα between 30 s and 14 months and if Qμ in the lower mantle is of order 225 at 30 s, we find that 0.04 ρ^α≤ 0.11; if instead Qμ in the lower mantle is of order 350 near 200 s, we find that 0.11 ≤α≤ 0.19. In all cases these limits arise from exceeding the 68 per cent confidence limits of ± 2.6 day in the observed period. Since slight departures from an equilibrium pole tide affect the Q much more strongly than the period we believe these limits to be robust.     

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     

On the application of asymptotic analysis to the dynamical theory of the pole tide

Summary. This article examines the effects of boundaries on the pole tide in an ocean of constant depth. The cyclically continuous global ocean solutions to Laplace's tidal equations with the pole tide forcing are used as the particular solutions to the problem. The approach here is to find approximate asymptotic solutions to the homogeneous tidal equations which can be added to the global particular solutions so that the normal component of velocity will vanish at the boundary. At the very long period of the pole tide, the unforced motions are assumed to be non-divergent, and so only the homogeneous vorticity equation must be solved.
The first case considered is a zonal ocean bounded by parallels of latitude equidistant from the equator. Asymptotic solutions are found in order to satisfy the zonal boundary condition, and this gives rise to a narrow zonal boundary current. The contribution of these solutions is exponentially small compared to the forced global pole tide except in the immediate vicinity of the northern and southern boundaries.
Next, the effect of meridional boundaries is considered. When a linear form of bottom friction is assumed, two approximate homogeneous solutions are found to construct a general solution that satisfies the meridional boundary conditions. One solution decays exponentially in longitude and gives rise to a western boundary current, while the other solution is independent of longitude. The meridional boundary conditions are used to match the homogeneous and particular solutions, and so the solution for the interior of the ocean satisfies the eastern boundary condition. The resulting solution for the pole tide has a western boundary current term, while over the majority of the ocean domain the solution has a term varying with the wavelength of the forcing (the global solution) and a zonal motion term used to satisfy the eastern boundary condition. Comparisons are made with the wind-driven ocean circulation problem.     

A new derivation of the ratio Q(wobble)/Qμ (mantle)

Summary An extension of the Love-Larmor theory to a low-loss unelastic earth model, leads to the surprisingly simple approximation
   
where τ_s= 447.4 sidereal day is the static wobble period, τ_R= 306 sidereal day is the rigid-earth wobble period and τ_w= 433 sidereal day is the observed Chandler period. Q _W, Q _μ are the respective average Q values of the wobble and the Earth's mantle at τ_W. The known numerical factor F is only slightly dependent on the Earth structure.     

The effects of the atmosphere and oceans on the Earth's wobble — I. Theory

Summary The theory of wobble excitation for a non-rigid earth is extended to include the effects of the earth's fluid core and of the rotationally induced pole tide in the ocean. The response of the solid earth and oceans to atmospheric loading is also considered. The oceans are shown to be affected by changes in the gravitational potential which accompany atmospheric pressure disturbances and by the load-induced deformation of the solid earth. These various improvements affect the excitation equations by about 10 per cent. Atmospheric and oceanic excitation can be computed using either an angular momentum or a torque approach. We use the dynamical equations for a thin fluid to relate these two methods and to develop a more general, combined approach. Finally, geostrophic winds and currents are shown to be potentially important sources of wobble excitation, in contrast to what is generally believed.     

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.     

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.     

Seismic excitation of the Chandler wobble revisited

Summary. An overview is taken of the last decade of studies of the effect of earthquakes on the polar motion. The treatment of the liquid outer core in static deformation is reviewed and some misconceptions in a number of papers are pointed out. Volterra's formula is generalized to the case of a liquid core which does not obey the highly idealized Adams—Williamson density law. The focal mechanism representation of Smylie & Mansinha (1971) is corrected for neglected terms arising from coordinate curvature, bringing the computed polar shifts into near numerical agreement with those of other workers. On the basis of the comparison of the observed and computed polar shifts for the Chile 1960 and Alaska 1964 events, it is suggested that the observed polar shifts for large earthquakes may be useful as discriminators in selecting focal mechanism parameters. The observed level of Chandler wobble excitation provides a constraint on some of the more extreme values of seismic moment recently proposed, unless these are supposed to depend only weakly on magnitude. The cumulative effect of the 30 largest earthquakes in the period 1901–64, recently examined by O'Connell & Dziewonski, is found to yield a rms Chandler wobble excitation of 0".10, using the random walk theory of Mansinha & Smylie (1967). This is close to the observed level (∼ 0".15). In addition to yielding the solution to a very long-standing geophysical puzzle, the study of the effect of earthquakes on the polar motion over the last decade may have produced a useful tool for the elucidation of seismic mechanism.     

Variationsl solution of long-period oscillations of the Earth

Summary The linearized equation of motion for the slightly elliptical rotating earth is obtained and using Phinney & Burridge's generalized spherical harmonics, the variational principle is derived for the normal mode oscillations of the Earth. The numerical solutions of two earth models 1066B and B1S6 are searched by minimizing the energy functional for the terrestrial spectral range longer than the lowest order free oscillation. The periods of core modes computed for the earth model B1S6, with stably stratified outer core, ranges from about 4 to 13hr and the periods for the 1066B are much more spread without clustering around the periods of 6 and 12 hr as in B1S6. The results for the earth model 1066B indicate that an outer core can support long-period oscillations even when it is not stably stratified. The Chandler wobble periods obtained are 402.3 day for B1S6 and 402.7 day for 1066B.     

The effects of post-seismic motions on the moment of inertia of a stratified viscoelastic earth with an asthenosphere

Summary. We give the analytical formulation for calculating the transient displacement of fields produced by earthquakes in a stratified, selfgravitating, incompressible, viscoelastic earth. We have evaluated the potential of viscous creep in the asthenosphere in exciting the Chandler wobble by a four-layer model consisting of an elastic lithosphere, a two-layer Maxwell viscoelastic mantle, and an inviscid core. The seismic source is modelled as an inhomogeneous boundary condition, which involves a jump condition of the displacement fields across the fault in the lithosphere. The response fields are derived from the solution of a two-point boundary value problem, using analytical propagator matrices in the Laplace-transformed domain. Transient flows produced by post-seismic rebound are found to be confined within the asthenosphere for local viscosity values less than 10²⁰P. The viscosity of the mantle below the low-viscosity channel is kept at 10²²P. For low-viscosity zones with widths greater than about 100 km and asthenospheric viscosities less than 10¹⁸P, we find that viscoelasticity can amplify the perturbations in the moment of inertia by a factor of 4–5 above the elastic contribution within the time span of the wobble period. We have carried out a comparative study on the changes of the inertia tensor from forcings due to surface loading and to faulting. In general the global responses from faulting are found to be much more sensitive to the viscosity structure of the asthenosphere than those produced from surface loading.     

On the amplitude of the Chandler wobble

The change in the inertia tensor of the Earth, due to the mass shift following a seismic event, has been computed by several authors for non-rotating earth models. Rotation is taken into account in the present paper, and the additional change in the inertia tensor is computed for an equivalent earth model, in which the axis of geometrical symmetry becomes tilted instead of the axis of greatest inertia. Rotation is thus seen to produce an increase by a factor 1.4 in the amplitude variation of the Chandler wobble, with respect to the non-rotating case, which, when added to the 1.4 amplitude increase due to the precessional re-adjustment of the equatorial bulge, gives a factor of 2 increase of the Chandler wobble amplitude with respect to the case of a rigid earth model.     

The effect of polar wander on the tides of a hemispherical ocean

Summary. A numerical model is constructed of the tides in a hemispherical ocean driven by the forces corresponding to the Y₂^–2 equilibrium tide. The model is used to study how tidal dissipation is affected by changes in the position of the ocean relative to the Earth's rotational axis and to test a hypothesis concerning the Gerstenkorn event.
As the position of the Earth's axis is varied with respect to the ocean, the model shows changes in the dissipation rate due to the changing position and importance of individual resonances of the ocean. However, a cooperative effect is also observed which results, for an ocean of depth 4400 m, in broad frequency bands near 10 rad day⁻¹ and-6 rad day⁻¹ in which the dissipation rate remains high.
The cooperative effect is found to arise from the existence, in an unbounded ocean, of resonances at these frequencies which match the tidal forces. When ocean boundaries are introduced, the new resonances near these frequencies contain a large component of the underlying resonance and as a result are themselves a good match to the driving forces.
For the real ocean, these findings imply that changes in the position of the pole, and also possibly changes in the shape of the ocean, will on average have little effect on the energy dissipated by the tides. However in the past changes in the mean depth and area of the ocean or the increased rotation rate of the Earth may have resulted in a smaller dissipation rate.