A new nutation series for a more realistic model earth

The frequency-dependent correction coefficients with respect to the forced nutations of a rigid earth are computed using the complex scalar gravitational-motion equations for an earth model with an anelastic mantle. Oceanic loads and tidal currents enter the model via outer boundary conditions. The ellipticity of the core-mantle boundary and the dynamical ellipticity are adjusted to observations. This requires the behaviour inside the model earth to be regarded as non-hydrostatic. Some relevant equations for the evaluation of boundary conditions and some terms in the equations of motion are expanded to second order in ellipticity. The computation of the equipotential-surface ellipticity profile is carried to second order as well. These second-order expansions lead to increased accuracy of the results in general. Moreover, one achieves a better reliability for the integration at frequencies close to a resonance. This allows the integration of the equations of motion at any relevant nutation period without the need for a normal-mode expansion. A complete new nutation series for a realistic model earth is presented.     

Some terms of nutation derived from the BIH data

Summary. The z -term of latitude and the w -term of UT, as computed by the BIH, are suitable for investigating imperfections in the representation of the real nutation in space. They have been previously used for deriving only the amplitude of the principal term of nutation (Feissel & Guinot). In this study they are used more generally for deriving the amplitudes of two terms of nutation and searching for a possible nutation in space due to the nearly-diurnal wobble.
Two kinds of data are used for this purpose: the values of z and w from 1962.0 to 1980.0 at 0.05 yr intervals and the values of z from 1967.0 to 1980.0 at 5 day intervals. The first ones have been used previously by Feissel & Guinot (using the values up to 1974.0 and 1976.95 respectively). The second ones have not previously been used for such studies.
The derived amplitudes of the principal and the fortnightly terms of nutation are in good agreement with other values deduced from observations, and with the theoretical ones corresponding to the best models of a non-rigid earth. The values obtained are not yet sufficiently precise to distinguish between these models, but could become so in the near future.
The least-squares fits, performed among the z and w data in order to look for any error in the representation of nutation, show the existence of a retrograde term, of a 434 day period for z and 444 day for w and an amplitude of the order of 0.01 arcsec. It may be due to the nearly-diurnal wobble.     

Ellipticity corrections for seismic phases

The advent of broad-band seismology has meant that use is being made of a wide range of seismic phases, for many of which ellipticity corrections have not been readily available. In particular, when many seismic phases are used in location schemes, it is important that the systematic effects of ellipticity are included for each phase.
An efficient and effective procedure for constructing ellipticity corrections is to make use of the ray-based approach of Dziewonksi & Gilbert (1976), as reformulated by Doornbos (1988), in conjunction with the rapid evaluation of traveltimes and slownesses for a given range using the tauspline procedure of Buland & Chapman (1983).
Ellipticity coefficients have been tabulated for a wide range of seismic phases and are available in electronic form. The ellipticity correction procedures have been extended to include an allowance for diffraction phenomena, for example P _diff, S _diff diffracted along the core-mantle boundary. Corrections for additional phases can be generated by building the ellipticity coefficients from suitable combinations of the coefficients for different phase segments.     

Extended formulation for Rayleigh-wave computations at high frequencies in a spherical layered earth

Summary. The earlier formulation of dispersion function, amplitude response and surface ellipticity for Rayleigh waves in a spherical layered earth has been extended by the reduced-delta matrix to avoid computational instability at high frequencies. We derive a modified formulation, where the layer thickness is kept separated during matrix multiplications. The present formulations are valid for an unlimited frequency range. A numerical experiment with single precision (six decimal digits accuracy) on an IBM 360/44 computer is performed at frequencies from 0.4 to 12.5 Hz on a Shield structure down to a depth of 1090 km below which a complete sphere is assumed. The experiment shows that both reduced-delta matrix and modified formulations generate roots of the dispersion equation without any reduction of layers. However, to avoid unnecessary computation, a layer-reduction procedure has been derived from a modified formulation when the contribution of deep layers becomes insignificant on surface waves.     

Could the energy near the FCN and the FICN be explained by luni-solar or atmospheric forcing?

The observed time-series of precession/nutation show residuals with respect to an empirical model based on the rigid Earth theoretical nutations and a frequency dependent transfer function with resonances to the Earth's normal modes. These residuals display energy mainly in the frequency domain around 430 and 500 days in the inertial frame. In this frequency band, the energy is possibly related to two normalmode frequencies: the free core nutation (FCN) and the free inner core nutation (FICN). In this paper, we examine the possibility of obtaining this energy from the resonance effect induced by a luni-solar (or planetary) forcing, or by an atmospheric forcing at a frequency very close to these Earth free nutations. The amplification factor due to the resonance is computed from an analytical formula expressed in the case of a simplified three-layer ellipsoidal rotating earth (with an elastic inner core, a liquid outer core and an elastic mantle), as well as the empirical formula based on the analysis of VLBI observations. For the tidal forcing, the theoretical results do not show any resonance at the level of precision we have examined but it is still possible to find one frequency near the FCN or FICN frequencies which could be excited. In contrast, for the atmospheric pressure the level of energy needed could be obtained from the diurnal pressure, depending on the noise level of the Earth's global pressure. We also show that the combination of three waves can explain the observed decrease of energy with time. While the tidal potential amplitudes are too small, a pressure noise level of 0.5 Pa would be sufficient to excite these waves.     

Extension of the Thomson-Haskell method to non-homogeneous spherical layers

Summary. Amplitude spectra of Rayleigh and Love waves in a layered non-gravitating spherical earth have been obtained using as a source, displacement and stress discontinuities. In each layer elastic parameters and density follow specified functions of radial distance and the solutions of the equations of motion are obtained in terms of exponential functions. The Thomson—Haskell method is extended to this case. The problem reduces to simple calculations as in a plane-layered medium. Numerical results of phase and group velocities up to periods of 300 s in various earth models when compared with earlier results (obtained by numerical integration) show that the present method can be used with sufficient accuracy. The differences in phase velocity, group velocity and amplitude (also surface ellipticity in the case of Rayleigh waves) between spherical- and flat-earth models have been investigated in the range 20–300–s period and expressed in polynomials in the period.     

A Hamiltonian approach to dissipative phenomena between the Earth's mantle and core, and effects on free nutations

The Hamiltonian formalism was recently applied by Getino (1995a,b) for the study of the rotation of a non-rigid earth with a heterogeneous and stratified liquid core. That earth model is generalized here by including the effect of the dissipation arising from the mantle-core interaction, using a model similar to that of Sasao, Okubo & Saito (1980), which includes both viscous and electromagnetic coupling. First, a solution for the free nutations is obtained following a classical approach, which in our opinion is more familiar to most of the readers than the Hamiltonian treatment. This solution provides a theoretical basis clear enough to study both the qualitative and quantitative effects of the dissipations considered in the hypotheses. The main qualitative features are, besides the delays, that the free core nutation (FCN) suffers an exponential damping, while the chandler wobble (CW) is not damped at first order, by the dissipation considered. The numerical values obtained for the complex compliances agree with the most recent experimental computations.
Next, the problem is studied under a Hamiltonian formalism, and a solution equivalent to the above is obtained. Besides its interest from a theoretical point of view, this formalism is necessary in order to apply canonical perturbation methods in order to obtain analytical nutation series.     

基于空间自相关的区域经济极化结构演化研究——以江苏省为例

从空间自相关视角厘清经济发展水平较高和较低地区的空间组织关系演化规律对区域经济健康发展具有重要意义。在界定集极、孤极和外围等概念基础上,运用地统计分析方法,按两级三区和三级五区结构体系分别构建江苏省全局和局部经济空间极化结构,分析其演化过程,结果显示：①全局空间集极范围不断扩大,孤极数量减少,逐渐形成南集极北孤极、南极北外的空间结构,其间空间自相关作用逐渐增强,区位是极化动因;② 局部空间演化走势不尽相同：沿江和沿京杭运河地区经济空间极化结构逐渐升级,而沿东陇海和沿海地区却相对恶化。因此认为当前江苏省全局空间中南北显著分化和局部空间相背演化现象值得关注。     

The classical analysis of the response of a long baseline radio interferometer

summary . A general derivation of the classical response of a long baseline interferometer is presented including the effects of finite bandwidth of the receiver systems, instrumental delays and phase shifts, instabilities in the separate local oscillators, as well as the effects of the retarded baseline. The general theory is then applied to a pair of antennae attached to the Earth and discussed in the light of a number of geophysical phenomena including the effects of precession, nutation, rotation, polar motion, annual aberration and Earth tides.     

The forced nutations of an elliptical, rotating, elastic and oceanless earth

Summary. We compute the luni-solar forced nutations of an elliptical, rotating, self-gravitating, elastic, hydrostatically prestressed and oceanless earth. Several recent structural models are considered, each possessing a fluid outer core and solid inner core. Complete results are given for the nutation of the 'axis of figure for the Tisserand mean surface' which best represents the observational effects of the Earth's nutational motion. Differences between results for different structural models are observationally insignificant. Differences between our results and Molodensky's are as large as ∼ 0.002 arcsec at six month and at 18.6 yr.     

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.     

Palaeomagnetism of the Macaronesian Insular Region: The Cape Verde Islands

Palaeomagnetic pole positions have been determined for a collection of igneous rocks, comprising nearly five hundred samples, from the Cape Verde Islands of Santa Antao, Sao Vicente, Sao Nicolao and Sao Tiago. Limited data from the islands of Sal, Maio and Fogo are also presented. Stratigraphic control suggusts that the lavas are overwhelmingly Miocene in age on Sao Tiago and Sao Nicolao. Similarity in the palaeomagnetic pole positions indicates that Miocene lavas are also dominant on Santa Antao and Sao Vicente.
Substantial areas within two of the islands are of reversed polarity only, suggesting either a rapid extrusion rate, or the existence of a long reversed polarity epoch during the Miocene period. The palaeomagnetic pole positions for each island are close to the present geographic pole, excluding the possibility of Post-Miocene differential crustal spreading (or rotation about a vertical axis) in this part of the Atlantic. The palaeomagnetic pole position for the entire survey is consistent with the Miocene geographic pole being removed from, but close to, the present geographic pole; and is in harmony with the European polar wandering curve.     

On imperfection of elasticity in the Earth's interior

Summary. Attempts are made to estimate the distribution of damping with depth, based on the observed damping of P waves, of free vibrations, and of surface waves and partly on a discussion by D. L. Anderson and R. S. Hart. Results are consistent in order of magnitude, but could easily be wrong by a factor of 2. Comments are added on the need for further work. Stress is laid on the damping of the free nutation, which indicates that α in the modified Lomnitz law is about 0.2 on an average.     

THE PALAEOMAGNETISM OF THE KAINOZOIC BASALTS OF VICTORIA

Summary. The average geomagnetic pole position consistent with the magnetization of the Newer Volcanics of Victoria, which are for the most part of Pleistocene age, coincides with the present geographic pole just as do all pole determinations from Miocene and later rock formations from the Northern Hemisphere. During this time it appears that the Earth's magnetic field has approximated, on average, to that of a geocentric axial dipole and that the land masses concerned have maintained their present positions. In contrast, the pole determined from Lower Tertiary Volcanics of Victoria does not agree with the geographic pole nor with equivalent determinations from the Northern Hemisphere, suggesting that the relative positions of the continents may have been different in the past.     

Calculation of poles of instantaneous rotation from poles of finite rotation

Summary. If the assumption can be made that the track of the plate tectonic pole of finite rotation is smooth, then the track of the pole of instantaneous rotation can be calculated. The method is illustrated by the example of the Indian/Pacific pole positions.     

Magnetic and viscous coupling at the core–mantle boundary: inferences from observations of the Earth's nutations

Dissipative core–mantle coupling is evident in observations of the Earth's nutations, although the source of this coupling is uncertain. Magnetic coupling occurs when conducting materials on either side of the boundary move through a magnetic field. In order to explain the nutation observations with magnetic coupling, we must assume a high (metallic) conductivity on the mantle side of the boundary and a rms radial field of 0.69 mT. Much of this field occurs at short wavelengths, which cannot be observed directly at the surface. High levels of short-wavelength field impose demands on the power needed to regenerate the field through dynamo action in the core. We use a numerical dynamo model from the study of Christensen & Aubert (2006) to assess whether the required short-wavelength field is physically plausible. By scaling the numerical solution to a model with sufficient short-wavelength field, we obtain a total ohmic dissipation of 0.7–1 TW, which is within current uncertainties. Viscous coupling is another possible explanation for the nutation observations, although the effective viscosity required for this is 0.03 m² s⁻¹ or higher. Such high viscosities are commonly interpreted as an eddy viscosity. However, physical considerations and laboratory experiments limit the eddy viscosity to 10⁻⁴ m² s⁻¹, which suggests that viscous coupling can only explain a few percent of the dissipative torque between the core and the mantle.     

The late Maastrichtian palaeomagnetic pole of the Pacific Plate

Summary. The Pacific plate's late Maastrichtian (∼ 69 Ma) palaeomagnetic pole, which constrains the northward motion of the Pacific plate during the Cainozoic and latest Cretaceous, was studied. A recently proposed method for obtaining oceanic plate palaeomagnetic poles by combining dissimilar data was extended to accept, as input, the relative amplitudes of magnetic lineations with different azimuths or widely separated sites or both. Combining late Maastrichtian palaeomagnetic data-the relative amplitudes and skewness of magnetic lineations, palaeolatitudes from a palaeomagnetic study of basalt and sediment in vertical cores, a pole from the inversion of the magnetic anomaly over a seamount, and present locations of equatorial sediment facies—yielded a best fit pole of 71°N, 9°E and a 95 per cent confidence ellipse with the major semiaxis of 6° striking 91° clockwise from north and the minor semiaxis of 2° striking 1° clockwise from north. This best fit pole, when compared to the pole expected if the hotspots have been fixed with respect to the spin axis, demonstrates that the hotspots in the Pacific Ocean have shifted ∼ 10° south with respect to the spin axis during the Cainozoic. This best fit pole, when compared to the best fit Campanian pole of the Pacific plate, demonstrates that the pole wandered rapidly, 1.1° Ma^-1, with respect to the Pacific plate during the latest Cretaceous.     

Core precession: flow structures and energy

Experiments using a precessing liquid-filled oblate spheroid with ellipticity ( a − b )/ a =1/400 extend and clarify earlier research. They yield flow data useful for estimating flows in the Earth's liquid core. Observed flows illustrate and confirm a nearly rigid liquid sphere with retrograde drift and lagging a cavity (mantle) axis in precession. The similarities of the observed lag angle with that computed for a rigid sphere, and earlier energy dissipation research both support the use of a rigid sphere analytical model to predict energy dissipation and first-order flow within the core–mantle boundary (CMB). Second-order boundary layer and interior cylindrical flow structures also are photographed and measured. Interior flows are never turbulent or unstable at near-Earth parameters, although complex and transient flow patterns are observed within the boundary layer. Other mechanisms proposed to explain net heat loss from the Earth and maintenance of the geodynamo typically require acceptance of some critical but unproven premise. Precession and CMB configuration are known with certainty and precision. Analytical difficulties have been the obstacle. Experiments illustrate the consequences of precession and ellipticity, provide criteria for validating analytical and numerical models, and may yield direct knowledge of the Earth's deep interior with careful scaling.     

A geomagnetic secular variation study (31000–19 500 bp) in Western Canada

Summary. Palaeomagnetic results from 212 horizons spread evenly through an 18 m sedimentary sequence in southern British Columbia are reported. Radiocarbon ages suggest that the sequence spans the interval from 31 200 to 19 500 yr bp. No evidence for any large geomagnetic excursions (such as the so-called Mono Lake Excursion) is found, but a distinctive pattern of 'normal' secular variation is observed with declination and inclination swings of 45° and 25° peak to peak amplitude respectively. For the most part the secular variation consists of low amplitude oscillations about the field vector of a geocentric axial dipole expected at the site latitude, but three relatively large perturbations occur at approximately 4000 yr intervals. These perturbations systematically bias the overall mean to shallow inclinations and easterly declinations in a manner reminiscent of the spatially non-isotropic secular variation model proposed by Cox. The bias involved is about 6° in declination and 3° in inclination (overall mean D = 5.8°E, I = 64.2°, α95 = 0.9°, N = 212 horizons), which leads to a pole which is both 'far-sided' and 'right-handed'. If the horizons involved in the three major perturbations are eliminated the mean direction ( D = 1.2°E, I = 67.2°, α95 = 0.8°, N = 125 horizons) does not differ from that of a geocentric axial dipole despite the small cone of confidence.     

Rotational and elliptical splitting of the free oscillations of the Earth

Summary. We present a table of rotational and elliptical splitting parameters for earth model 1066A, including all terms through second order in rotation and first order in ellipticity. An algorithm for calculating the second-order Coriolis splitting by summing over all modes which are coupled to first order is given in detail. Coupling to secular (or zero frequency) modes, as well as the usual seismic modes, can provide significant contributions to these splitting parameters.