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1.
Summary. An existing experimentally verified model for energy dissipation in a processing spherical cavity filled with liquid assumed to be in a semirigidized state except for a viscous Ekman boundary layer is applied to the Earth's liquid core to assess energy dissipation magnitudes. Application of the model to the best available Earth data occurs at the derived energy dissipation maximum for the model. Other existing research showing that the Earth's atmosphere appears to adjust to a state of maximum dissipation led to generic models for systems of maximum dissipation. The maximum dissipation mantle—core model with core motion driven by Earth precession alone, coupled to the mantle only by viscous shear stresses, and with a spherical mantle—core boundary leads to energy dissipation rates on the order of 104 times those necessary for an Earth dynamo. The maximum dissipation model also leads to excessive magnetic field drift rates and to excessive retardation of the Earth's rotation rate. Effects of the mantle—core ellipticity and of magnetic field coupling are briefly discussed and are used to help develop a less than maximum dissipation model also driven by precession alone but using the additional coupling to yield a model more consistent with observed phenomena.  相似文献   

2.
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.  相似文献   

3.
Summary. The basic equations describing the dynamical effects of the Earth's fluid core (Liouville, Navier-Stokes and elasticity equations) are derived for an ellipsoidal earth model without axial symmetry but with an homogeneous and deformable fluid core and elastic mantle.
We develop the balance of moment of momentum up to the second order and use Love numbers to describe the inertia tensor's variations. The inertial torque takes into account the ellipticity and the volume change of the liquid core. On the core—mantle boundary we locate dissipative, magnetic and viscous torques. In this way we obtain quite a complete formulation for the Liouville equations.
These equations are restricted in order to obtain the usual Chandler and nearly diurnal eigenfrequencies.
Then we propose a method for calculating the perturbations of these eigenfrequencies when considering additional terms in the Liouville equations.  相似文献   

4.
We use the Direct Solution Method (DSM) together with the modified operators derived by Geller & Takeuchi (1995) and Takeuchi, Geller & Cummins (1996) to compute complete synthetic seismograms and their partial derivatives for laterally heterogeneous models in spherical coordinates. The methods presented in this paper are well suited to conducting waveform inversion for 3-D Earth structure. No assumptions of weak perturbation are necessary, although such approximations greatly improve computational efficiency when their use is appropriate.
An example calculation is presented in which the toroidal wavefield is calculated for an axisymmetric model for which velocity is dependent on depth and latitude but not longitude. The wavefield calculated using the DSM agrees well with wavefronts calculated by tracing rays. To demonstrate that our algorithm is not limited to weak, aspherical perturbations to a spherically symmetric structure, we consider a model for which the latitude-dependent part of the velocity structure is very strong.  相似文献   

5.
This article investigates the effects of a mushy inner core boundary on the eigenperiods of the Slichter modes for a simple, but realistic, earth model (rotating, spherical configuration, elastic inner core and mantle, neutrally stratified, inviscid, compressible liquid core). It is found that the influence of the mushy boundary layer is substantial compared with some other effects, such as those from elasticity of the mantle, non-neutral stratification of the liquid outer core and ellipticity of the Earth and centrifugal potential. The results obtained here may set a lower bound on the eigenperiods of the Slichter modes for a realistic earth model. For example, for a PREM model, the lower bound of the central period of the Slichter modes should be about 5.3 hr.  相似文献   

6.
We present a 'pseudo-bending' approach to 3-D ray tracing in a spherical earth with discontinuities. This method is based on a three-point perturbation associated with a first-order approximation, while Snell's law in curvilinear coordinates is applied at the discontinuities. We demonstrate the computational accuracy and efficiency of the pseudo-bending method in tracing rays for various velocity models by comparing results with analytical solutions and with results from the bending method. The improvement of efficiency is significant, but is reduced as the number of discontinuities increases. Since the bending approach may be computationally unstable in some situations, even though it is exact, the pseudo-bending approach is preferable for automatic calculation of rays.  相似文献   

7.
A new nutation series for a more realistic model earth   总被引：1，自引：0，他引：1  
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.  相似文献   

8.
Polar motion is modelled for the large 2004 Sumatra earthquake via dislocation theory for an incompressible elastic earth model, where inertia perturbations are due to earthquake-triggered topography of density–contrast interfaces, and for a compressible model, where inertia perturbation due to compression-dilatation of Earth's material is included; density and elastic parameters are based on a multilayered reference Earth. Both models are based on analytical Green's functions, propagated from the centre to the Earth's surface. Preliminary and updated seismological solutions are considered in elucidating the effects of improving earthquake parameters on polar motion. The large Sumatra thrust earthquake was particularly efficient in driving polar motion since it was responsible for large material displacements occurring orthogonally to the strike of the earthquake and to the Earth's surface, as imaged by GRACE gravity anomalies over the earthquake area. The effects of earthquake-induced topography are four times larger than the effects of Earth's compressibility, for l = 2 geopotential components. For varying compressional Earth properties and seismic solution, modelled polar motion ranges from 8.6 to 9.4 cm in amplitude and between 117° and 130° east longitude in direction. The close relationship between polar motion direction, earthquake longitude and thrust nature of the event, are established in terms of basic physical concepts.  相似文献   

9.
Summary. Shear-wave travel times in a spherically averaged earth are estimated using 'differential' S minus P ( S – P ) travel-time measurements and detailed statistical procedures. Fourteen earthquakes and 48 stations are specially selected, yielding 302 S - P times for 6° < Δ < 111°. Analysis of variance techniques are used to estimate simultaneously azimuthally varying source and station adjustments while constructing an S – P travel-time model. A method of weighting the equations of condition based on the distribution of stations and epicentres is developed to reduce the effects of systematic errors due to non-random sampling of the Earth. The resulting S - P travel times are added to the 1939 Jeffreys–Bullen and the 1968 Herrin P travel times as a function of distance to obtain shear-wave travel-time models. Confidence intervals for the models are estimated from the variance of the observed S – P travel times.
The standard error for a single observed S – P travel time (6° < Δ < 111°) is 2.1 s and the residual distribution is not significantly different from a normal distribution at the 95 per cent confidence level. For 30° < Δ < 80° the mean S travel time is 1.3 s later than the corresponding mean for Jeffreys–Bullen tables, which is significant at the 95 per cent confidence level.  相似文献   

10.
Body tides on an elliptical, rotating, elastic and oceanless earth   总被引：17，自引：0，他引：17  
Summary. The Earth's deformation caused by the luni-solar tidal force is defined as the 'body tide'. We compute the effects of the Earth's rotation and elliptical stratification on the body tide for a number of modern elastic structural models. Rotation and ellipticity within the mantle are found to affect tidal observations by about 1 per cent. A consequence is an improved estimate for the fluid core resonance in the diurnal tidal band. Agreement between results for the different structural models is very good. As a result, the results computed here can be used to model the tidal effects of a globally averaged, oceanless, rotating, elliptical and elastic earth to an accuracy of at least one part in 300.  相似文献   

11.
We consider the two coupled differential equations of the two radial functions appearing in the displacement components of spheroidal oscillations for a transversely isotropic (TI) medium in spherical coordinates. Elements of the layer matrix have been explicitly written—perhaps for the first time—to extend the use of the Thomson-Haskell matrix method to the derivation of the dispersion function of Rayleigh waves in a transversely isotropic spherical layered earth. Furthermore, an earth-flattening transformation (EFT) is found and effectively used for spheroidal oscillations. The exponential function solutions obtained for each layer give the dispersion function for TI spherical media the same form as that on a flat earth. This has been achieved by assuming that the five elastic parameters involved vary as r p and that the density varies as r p-2, where p is an arbitrary constant and r is the radial distance. A numerical illustration with p = - 2 shows that, in spite of the inhomogeneity assumed within layers, the results for spherical harmonic degree n , versus time period T , obtained here for the Primary Reference Earth Model (PREM), agree well with those obtained earlier by other authors using numerical integration or variational methods. The results for isotropic media derived here are also in agreement with previous results. The effect of transverse isotropy on phase velocity for the first two modes of Rayleigh waves in the period range 20 to 240 s is calculated and discussed for continental and oceanic models.  相似文献   

12.
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.  相似文献   

13.
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.  相似文献   

14.
Summary. This paper extends an earlier study (Sengupta & Julian) of travel times of P waves of deep-focus earthquakes to include shear waves. Primary advantage of deep-focus earthquakes is the reduction of anomalies caused by complex structures near the source. The standard deviations of travel times and station anomalies of this study are about half as large as those determined from the data of shallow-focus earthquakes (e.g. Herrin et al.; Hales & Roberts). Spherically-symmetric velocity models derived from the travel times by a linearized inverse technique have resolving lengths of about 70 km for standard errors in velocity of about 0.02 km/s. No pronounced reversal of either compressional or shear velocity was required at the base of the mantle to satisfy the data, though a small velocity decrease could not be entirely ruled out. Some anomalous rapid changes in compressional velocity gradient were, however, found centred around the depths of 2400 and 2600 km. The models derived in this study agree most closely with that of Herrin et al . for compressional velocity and the model 1066B of Gilbert & Dziewonski for shear velocity.  相似文献   

15.
The free oscillations of an anelastic aspherical earth   总被引：1，自引：0，他引：1  
Summary. Some century-old results, due to Rayleigh and Routh, have been adapted to investigate the normal mode eigenfrequencies and eigenfunctions of an earth with laterally variable anelasticity and to determine the transient response of such an earth to earthquakes. Using degenerate perturbation theory, the eigenfrequencies are found to first order and the associated eigenfunctions to zeroth order in the small deviations of the Earth away from a spherical perfectly elastic reference earth model. Both the eigenfrequencies and the eigenfunctions are complex and, in addition, the latter are not mutually orthogonal, reflecting the non-Hermitian character of the normal mode eigenvalue problem. The effect of laterally heterogeneous attenuation on the shape of an unresolved multiplet spectrum has been investigated in the surface-wave geometrical-optics limit. Singlet cancellation leads in that limit to the appearance of a single resonance peak whose decay rate or apparent Q −1 depends only on the average attenuation structure underlying the source—receiver great-circle path.  相似文献   

16.
Summary. This note reports on the remarkable focusing of seismic body waves at or near the antipode (Δ= 180°) of an earthquake's epicentre. The particular seismic velocity structure and sphericity of the Earth cause body-wave phases such as P (diff), PKP, PP, PPP, PcPPKP, SKSSKS, SS , etc. to converge individually at antipodal distances after being diffracted, reflected or refracted at discontinuities. This focusing strongly amplifies each signal up to almost one order of magnitude with respect to the normal phase recorded two or more degrees away. Since the signal/noise ratio is enhanced in the same proportion, seismograms at antipodal distances provide clear and strong arrivals of otherwise weak phases. Antipodal monitoring of seismic waves is suggested as a powerful means of exploring the Earth's interior. The study of these 'seismic images' generated at focal points of seismic rays will yield information on the departures from lateral homogeneity and sphericity of the core, as well as stronger constraints on earth models. To interpret the observations correctly, the data must be compared with theoretically generated seismograms. Since the appropriate ray theory equations (see, e.g. Scholte; Gilbert & Helmberger; Richards) are singular at Δ=180°, a corrective measure is taken which provides a formal expression for the wave amplitude that remains finite at the antipode, and reproduces the usual expressions at other distances.  相似文献   

17.
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18.
Summary A technique is presented for calculating geoid height anomalies over two-dimensional models of Earth structure. The method consists of convolving gravity anomalies over the structure with filters which take into account the finite size of the structure in the third dimension and the curvature of the Earth. Similar filters are also developed for a flat earth case. The method is applied to a sea-surface gravity profile crossing the Tonga-Kermadec trench and is found to give good agreement with a Geos-3 radar altimetry profile in the same region. The example demonstrates that introducing arbitrary offsets in computing gravity anomalies can result in spurious long-wavelength effects in the computed geoid. Comparison of the results obtained using flat earth and spherical earth filters suggests that the effects of the curvature of the Earth only become significant for wavelengths in the gravity field greater than about 1000 km.  相似文献   

19.
Summary. A simplified model of the solar quiet-time ionospheric current system is used to calculate the induced currents in a model earth. The conductivity is assumed to be constant below a depth of about 400 km and zero above that depth. The current induced in the north—south conductivity anomaly under the Rocky Mountains is then estimated from the time-varying potential difference between points at 30 and 45° latitude at the surface of the conducting sphere. The purpose of these calculations is to investigate whether variations in the latitude of the northern hemisphere current system vortex will substantially alter the relationship between the observed magnetic field components at the Earth's surface and the local magnetic field gradient caused by the conductivity anomaly. We find that a 10° shift in the latitude of the ionospheric current focus causes a change of 6 per cent or less in the transfer function from the field components to the gradient in the total field. Thus such latitude shifts cannot explain much of the magnetic field gradient variation at periods near 24 hr that has been observed near Boulder, Colorado.  相似文献   

20.
Summary . We present a variety of examples, showing systematic fluctuations as a function of angular order of measured eigenfrequencies for given normal modes of the Earth. The data are single station measurements from the GEOSCOPE network. Such fluctuations are attributed to departures from the lowest order asymptotic expression of the geometrical optics approximation. We derive first-order asymptotic expressions for the location parameter for all three components of the Earth's motion, by a method based on the stationary phase approximation and geometric relations on the unit sphere.
We illustrate the sensitivity of the fluctuations to the different parameters involved (source parameters, epicentral distance, laterally heterogeneous earth model) with synthetic examples corresponding to GEOSCOPE observations. Finally, we show the results of first attempts at inversion, which indicate that, when the fluctuations are taken into account, more accurate estimates of the great circle average eigenfrequencies can be obtained, and additional constraints put on the structure in the neighbourhood of the great circle.  相似文献   

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