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1.
Jacques Hinderer Hilaire Legros Martine Amalvict 《Geophysical Journal International》1982,71(2):303-332
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
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
2.
B. Guinot 《Geophysical Journal International》1982,71(2):295-301
Summary. Estimates of the amplitude, of the phase and of the ellipticity of the Chandlerian nutation are given. The ellipticity is larger than that expected from the equilibrium pole tide. On the other hand, in 1980 the amplitude of the Chandlerian nutation again changed when the total motion of the pole was at a minimum. 相似文献
3.
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. 相似文献
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. 相似文献
4.
S. N. Bhattacharya 《Geophysical Journal International》1976,47(3):411-444
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
5.
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. 相似文献
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. 相似文献
6.
Jean-Francois Guéguen Pho Hoang-Trong Hilaire Legros 《Geophysical Journal International》1987,89(3):839-855
Summary. A first-order form of the Euler's equations for rays in an ellipsoidal model of the Earth is obtained. The conditions affecting the velocity law for a monotonic increase, with respect to the arc length, in the angular distance to the epicentre, and in the angle of incidence, are the same in the ellipsoidal and spherical models. It is therefore possible to trace rays and to compute travel times directly in an ellipsoidal earth as in the spherical model. Thus comparison with the rays of the same coordinates in a spherical earth provides an estimate of the various deviations of these rays due to the Earth's flattening, and the corresponding travel-time differences, for mantle P -waves and for shallow earthquakes. All these deviations are functions both of the latitude and of the epicentral distance. The difference in the distance to the Earth's centre at points with the same geocentric latitude on rays in the ellipsoidal and in the spherical model may reach several kilometres. Directly related to the deformation of the isovelocity surfaces, this difference is the only cause of significant perturbation in travel times. Other differences, such as that corresponding to the ray torsion, are of the first order in ellipticity, and may exceed 1 km. They induce only small differences in travel time, less than 0.01s. Thus, we show that the ellipticity correction obtained by Jeffreys (1935) and Bullen (1937) by a perturbational method can be recovered by a direct evaluation of the travel times in an ellipsoidal model of the Earth. Moreover, as stated by Dziewonski & Gilbert (1976), we verify the non-dependence of this correction on the choice of the velocity law. 相似文献
7.
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. 相似文献
8.
In a previous paper, the authors considered the free rotation of an earth model composed of a rigid mantle and a liquid core in the presence of dissipation and under the Hamiltonian formalism, obtaining analytical expressions for the free nutation modes.
In this paper we treat the forced motion. Approximate analytical solutions are worked out by means of Hori's perturbation method, the free solutions obtained in the former paper playing the role of the unperturbed solutions required in the application of the method. These solutions are consistent in the sense that, with the usual terminology, the rigid body solutions and the complex transfer functions are calculated with the same parameters.
Besides in-phase terms, the dissipation at the core–mantle boundary studied in this paper gives rise to out-of-phase terms. From a qualitative perspective, we discuss the issue of the resonance in this context. The presence of dissipation changes dramatically the character of the FCN wobble; that is, it is no longer a regular oscillation but a damped one. A strict resonance phenomenon cannot take place thereby, since the forcing perturbations are oscillations with a real (non-complex) frequency. 相似文献
In this paper we treat the forced motion. Approximate analytical solutions are worked out by means of Hori's perturbation method, the free solutions obtained in the former paper playing the role of the unperturbed solutions required in the application of the method. These solutions are consistent in the sense that, with the usual terminology, the rigid body solutions and the complex transfer functions are calculated with the same parameters.
Besides in-phase terms, the dissipation at the core–mantle boundary studied in this paper gives rise to out-of-phase terms. From a qualitative perspective, we discuss the issue of the resonance in this context. The presence of dissipation changes dramatically the character of the FCN wobble; that is, it is no longer a regular oscillation but a damped one. A strict resonance phenomenon cannot take place thereby, since the forcing perturbations are oscillations with a real (non-complex) frequency. 相似文献
9.
Z. R. Peng 《Geophysical Journal International》1997,131(3):607-617
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
10.
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
11.
John M. Wahr 《Geophysical Journal International》1981,64(3):705-727
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. 相似文献
12.
Wooil Moon 《Geophysical Journal International》1981,67(3):735-746
Summary. A new method is proposed for the geopotential field computation and gravitational attraction modelling. The usual method is to use a uniform density discrete numerical integration to represent either the gravitational potential or the gravitational attraction from a given density configuration. In this paper, an interpolation scheme is explained, using a piecewise continuous basis function to represent the arbitrarily varying density configuration in one, two and three dimensions. This new approach greatly simplifies the potential integrations and, in certain cases where symmetry exists, analytical evaluation of the integrals is also possible. Numerical tests and examples are given for a hypothetical salt dome, a vertical dyke with varying density structure and the hydrostatic ellipticity of earth model 1066B. The numerical error in this method is limited to the analytical approximation and interpolation errors in each case. This new approach can also be used as efficiently for other potential field studies. 相似文献
13.
William P. O'Connor 《Geophysical Journal International》1986,85(1):1-11
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. 相似文献
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. 相似文献
14.
The concept of a deformation of a simple, non-rotating, spherically symmetric earth model with a fluid outer core, although it is a highly artificial physical situation, provides a useful computational algorithm that allows one lo determine analytically modes of vibration without any Love-number theory. In particular, on these analytically determined modes, we impose regularity conditions at the centre and boundary conditions at the surface, as well as conditions of continuity at the inner-core-outer-core boundary and at the core-mantle boundary. They lead to an eigenvalue equation for the frequency of oscillation. The range of frequencies obtained in this way for different earth models gives an indication of the influence of compressibility and non-homogeneity on the spectrum of eigenfrequencies. 相似文献
15.
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. 相似文献
16.
V. Dehant M. Feissel P. Defraigne F. Roosbeek J. Souchay 《Geophysical Journal International》1997,130(2):535-546
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
17.
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 Mo 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). 相似文献
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). 相似文献
18.
Jean-Claude Mareschal 《Geophysical Journal International》1978,54(3):703-710
Summary. A method is outlined to determine the dynamic behaviour of a phase boundary in the Earth when non-uniform time-varying pressure and temperature conditions are assumed at the Earth's surface. An integral equation describing the phase boundary motion is derived and it is solved under a linearizing assumption. The solution is obtained in the form of a double integral transform. Short and long time-expansions of the solution can be obtained from series expansion and integration of the Laplace transform along a branch cut. The method is illustrated by considering a stepwise change in surface pressure conditions.
For short times, the solution exhibits the same type of time dependence (i.e. the first-order term is in t1/2 ) as the one obtained in the one-dimensional case (i.e. uniform pressure perturbation at the Earth's surface).
For long times, it is shown that the time dependence of the phase boundary motion is almost identical to the one derived for the one- dimensional case if the wavenumber kL of the surface excitation is such that κ k 2 L τ≤ 1 (where τ is the relaxation time associated with the one-dimensional phase boundary motion and κ is the thermal diffusivity). If κ k 2 L τ > 1, then the relaxation time for the phase boundary motion in two dimensions is of the order of κ−1 k −2 L .
When considering parameters that would be appropriate for a basalt to eclogite phase transition at Moho depth, the latter situation is met only when the load wavelength is smaller than 35 km. 相似文献
For short times, the solution exhibits the same type of time dependence (i.e. the first-order term is in t
For long times, it is shown that the time dependence of the phase boundary motion is almost identical to the one derived for the one- dimensional case if the wavenumber k
When considering parameters that would be appropriate for a basalt to eclogite phase transition at Moho depth, the latter situation is met only when the load wavelength is smaller than 35 km. 相似文献
19.
F. Luzón F. J. Sánchez-Sesma J. L. Rodríguez-Zúñiga A. M. Posadas J. M. García J. Martin M. D. Romacho M. Navarro 《Geophysical Journal International》1997,129(3):571-578
The diffraction of P, S and Rayleigh waves by 3-D topographies in an elastic half-space is studied using a simplified indirect boundary element method (IBEM). This technique is based on the integral representation of the diffracted elastic fields in terms of single-layer boundary sources. It can be seen as a numerical realization of Huygens principle because diffracted waves are constructed at the boundaries from where they are radiated by means of boundary sources. A Fredholm integral equation of the second kind for such sources is obtained from the stress-free boundary conditions. A simplified discretization scheme for the numerical and analytical integration of the exact Green's functions, which employs circles of various sizes to cover most of the boundary surface, is used.
The incidence of elastic waves on 3-D topographical profiles is studied. We analyse the displacement amplitudes in the frequency, space and time domains. The results show that the vertical walls of a cylindrical cavity are strong diffractors producing emission of energy in all directions. In the case of a mountain and incident P, SV and SH waves the results show a great variability of the surface ground motion. These spatial variations are due to the interference between locally generated diffracted waves. A polarization analysis of the surface displacement at different locations shows that the diffracted waves are mostly surface and creeping waves. 相似文献
The incidence of elastic waves on 3-D topographical profiles is studied. We analyse the displacement amplitudes in the frequency, space and time domains. The results show that the vertical walls of a cylindrical cavity are strong diffractors producing emission of energy in all directions. In the case of a mountain and incident P, SV and SH waves the results show a great variability of the surface ground motion. These spatial variations are due to the interference between locally generated diffracted waves. A polarization analysis of the surface displacement at different locations shows that the diffracted waves are mostly surface and creeping waves. 相似文献