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.     

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.     

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 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.     

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.     

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.     

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.     

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.     

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.     

Normal modes of the viscoelastic earth

Summary A uniformly valid linear viscoelastic rheology is described which takes the form of a 'generalized' Burgers' body and which appears capable of reconciling the behaviour of the Earth's mantle across the complete spectrum of geodynamic time-scales. This spectrum is bracketed by the short time-scales of body wave and free oscillation seismology on which anelastic effects are dominant, and the long time-scale of mantle convection on which the Earth behaves viscously. The parameters of the model which control the viscous response are fixed by post-glacial rebound data whereas those which govern the anelasticity are to be determined by fitting the model to observations of seismic Q. The paper is concerned primarily with a discussion of the normal mode spectrum of the Earth as a generalized Burgers' body. Focusing upon the homogeneous model, it includes an initial analysis of the accuracy of first-order perturbation theory as a method of calculating the respective Q s of the elastic gravitational free oscillations. Also considered are the quasi-static modes of relaxation which only exact eigenanalysis can reveal. The importance of these modes is assessed within the context of a discussion of the effect of viscoelasticity upon the efficiency of Chandler wobble excitation.     

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.     

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.     

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.     

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.     

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.     

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.     

Orthogonal intersections of megatrends in the Western Pacific ocean basin: a case study of the Mid-Pacific mountains

Present-day tectonic concepts of events on the western paleo-Pacific lithosphere must be assessed relative to new data. Data collected by the newer techniques of geophysical surveying reveal leaky fracture zones, trending NNW–SSE and WSW–ENE; non-sequential in-line ages on most seamount chains; and orthogonal intersections of fracture zones. The fracture zones meander, braid, merge, splay, start and stop at any place, and are generally aligned with, or contain, linear chains of seamount. The combination of these in-line features is called megatrends. When the GEOSAT data are compared to the available bathymetry, this seemingly jumbled tectonic structure is verified. As the pole of rotation changes, the stress field changes alignment to agree with the Chandler wobble of Earth. Younger megatrends, propagating ever eastward, cross the older, already imprinted megatrends. During the Cretaceous, the voluminous outpouring of igneous rock created the large Pacific plateaus and rises where the megatrends, active and inactive, orthogonally intersected. The magma floods at the intersections flowed outwardly, and the outward flooding accounts for the fanning magnetic lineations around the Manihiki, Magellan, Shatskiy, and Hess rises. A case study of the Mid-Pacific Mountains (MPM), lying in the north-central Pacific Ocean basin, shows that the MPM formed about 125–110-Ma by overprinting the orthogonal intersections of fracture zones at the Molokai and Easter/Krusenstern–Emperor megatrends and the Murray and Tubai/Mamua megatrends. The MPM have been undergoing distortion into a vortex structure, a feature which has been confirmed by updated bathymetry, GEOSAT altimetry data, and drillsite information.     

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.     

Period and Qw of The Chandler Wobble

A splined ILS/IPMS data set (1900–1973) from the most homogeneous values available has been analysed by the maximum entropy method of Burg. Principal conclusions are: (1) the spectral character of the Chandler wobble is a single broad peak, (2) the period is 432·95 ± 1·02 mean solar days and, (3) the Q _w is 36±10. Measurements indicate that Q _w is non- stationary in time.