The Static Deformation of an Earth with a Fluid Core: A Physical Approach

Chinnery, Michael A.

doi:10.1111/j.1365-246x.1975.tb05872.x

Cited by 38 publications

(22 citation statements)

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“…Thus, equations (6)–(7) and (10)–(11) become In view of the regularity conditions at the center of the Earth, r = 0, we know that with A and B being two constants. Then, the gravitational fluxes q ℓ m (1) and q ℓ m (2) , equations (28)–(29), have to satisfy the following regularity conditions where we have utilized This way, following Longman [1963], Smylie and Mansinha [1971], and Chinnery [1975], we impose the core‐mantle boundary conditions where I C and C 4 are the 8 × 4–matrix, that we report in equation (A4), and the 4‐vector of constants that must be determined by means of the Earth surface boundary conditions.…”

Section: Core‐mantle Boundary Conditionsmentioning

confidence: 99%

“…The solution of equation (36) can be expressed as where A j and B j , with j = 1, 2, are four constants of integration and Because, for the seismic problem, the top surface of the ocean is a free surface on which the radial displacement follows the geoid, we impose the following boundary conditions at the top of the ocean r = b (6371 km) Thus, from equations (39)–(40) evaluated at r = b , we obtain where for brevity we have defined α as Then equations (37)–(40) yield In order to determine B 1 and B 2 , we must consider proper boundary conditions at the bottom of the ocean, namely at the interface r = a (6368 km) between the solid Earth and the ocean. Following Longman [1963], Smylie and Mansinha [1971], and Chinnery [1975], we impose …”

Section: The Self‐consistent Global Oceanmentioning

confidence: 99%

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GRACE gravity data help constraining seismic models of the 2004 Sumatran earthquake

et al. 2011

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The analysis of Gravity Recovery and Climate Experiment (GRACE) Level 2 data time series from the Center for Space Research (CSR) and GeoForschungsZentrum (GFZ) allows us to extract a new estimate of the co‐seismic gravity signal due to the 2004 Sumatran earthquake. Owing to compressible self‐gravitating Earth models, including sea level feedback in a new self‐consistent way and designed to compute gravitational perturbations due to volume changes separately, we are able to prove that the asymmetry in the co‐seismic gravity pattern, in which the north–eastern negative anomaly is twice as large as the south–western positive anomaly, is not due to the previously overestimated dilatation in the crust. The overestimate was due to a large dilatation localized at the fault discontinuity, the gravitational effect of which is compensated by an opposite contribution from topography due to the uplifted crust. After this localized dilatation is removed, we instead predict compression in the footwall and dilatation in the hanging wall. The overall anomaly is then mainly due to the additional gravitational effects of the ocean after water is displaced away from the uplifted crust, as first indicated by de Linage et al. (2009). We also detail the differences between compressible and incompressible material properties. By focusing on the most robust estimates from GRACE data, consisting of the peak‐to‐peak gravity anomaly and an asymmetry coefficient, that is given by the ratio of the negative gravity anomaly over the positive anomaly, we show that they are quite sensitive to seismic source depths and dip angles. This allows us to exploit space gravity data for the first time to help constraining centroid‐momentum‐tensor (CMT) source analyses of the 2004 Sumatran earthquake and to conclude that the seismic moment has been released mainly in the lower crust rather than the lithospheric mantle. Thus, GRACE data and CMT source analyses, as well as geodetic slip distributions aided by GPS, complement each other for a robust inference of the seismic source of large earthquakes. Particular care is devoted to the spatial filtering of the gravity anomalies estimated both from observations and models to make their comparison significant.

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Section: Core‐mantle Boundary Conditionsmentioning

confidence: 99%

Section: The Self‐consistent Global Oceanmentioning

confidence: 99%

GRACE gravity data help constraining seismic models of the 2004 Sumatran earthquake

et al. 2011

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“…These conditions can be expressed in the form (e.g. Chinnery 1975;Tromp & Mitrovica 1999), and eq. (39):…”

Section: Extended Differential-equation Formulationmentioning

confidence: 99%

The rotational feedback on linear-momentum balance in glacial isostatic adjustment

Martinec

Hagedoorn

2014

Geophysical Journal International

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S U M M A R YThe influence of changes in surface ice-mass redistribution and associated viscoelastic response of the Earth, known as glacial isostatic adjustment (GIA), on the Earth's rotational dynamics has long been known. Equally important is the effect of the changes in the rotational dynamics on the viscoelastic deformation of the Earth. This signal, known as the rotational feedback, or more precisely, the rotational feedback on the sea level equation, has been mathematically described by the sea level equation extended for the term that is proportional to perturbation in the centrifugal potential and the second-degree tidal Love number.The perturbation in the centrifugal force due to changes in the Earth's rotational dynamics enters not only into the sea level equation, but also into the conservation law of linear momentum such that the internal viscoelastic force, the perturbation in the gravitational force and the perturbation in the centrifugal force are in balance. Adding the centrifugal-force perturbation to the linear-momentum balance creates an additional rotational feedback on the viscoelastic deformations of the Earth. We term this feedback mechanism, which is studied in this paper, as the rotational feedback on the linear-momentum balance.We extend both the time-domain method for modelling the GIA response of laterally heterogeneous earth models developed by Martinec and the traditional Laplace-domain method for modelling the GIA-induced rotational response to surface loading by considering the rotational feedback on linear-momentum balance. The correctness of the mathematical extensions of the methods is validated numerically by comparing the polar-motion response to the GIA process and the rotationally induced degree 2 and order 1 spherical harmonic component of the surface vertical displacement and gravity field. We present the difference between the case where the rotational feedback on linear-momentum balance is considered against that where it is not. Numerical simulations show that the resulting difference in radial displacement and sea level change between these situations since the Last Glacial Maximum reaches values of ±25 and ±1.8 m, respectively. Furthermore, the surface deformation pattern is modified by up to 10 per cent in areas of former or ongoing glaciation, but by up to 50 per cent at the bottom of the southern Indian ocean. This also results in the movement of coastlines during the last deglaciation to differ between the two cases due to the difference in the ocean loading, which is seen for instance in the area around Hudson Bay, Canada and along the Chinese, Australian or Argentinian coastlines.Key words: Sea level change; Geopotential theory; Earth rotation variations. I N T RO D U C T I O NThe redistribution of ice and water over the Earth's surface during glaciation and deglaciation cycles due to changes in climate induces 3-D crustal motion, gravity-field variations and changes in sea level, which is known collectively as glacial isostatic adjustment (GIA). The surface-m...

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“…(16a). Particularly, the first and third columns of the CMB matrix C describe the perturbation of the equipotential surface at the core radius and the buoyancy of the mantle into the core due to any departure of the CMB topography from the equipotential surface, respectively, while the second column accounts for the fact that the tangential displacement at the bottom of the mantle is not constrained by the inviscid core (Longman, 1963;Dahlen,1974;Chinnery, 1975;. Within this framework, the spheroidal vector solution can be written in terms of an homogeneous solution, y 0 m , which satisfies the boundary conditions at the CMB and at the Earth surface, and a particular solution, y 1 m , which accounts for the displacement dislocation,…”

Section: Spheroidal Perturbationsmentioning

confidence: 99%

“…Within the assumption of an inviscid core, the core-mantle boundary (CMB) matrix is given by (Longman, 1963;Farrell, 1972;Dahlen,1974;Saito, 1974;Chinnery, 1975;)…”

Section: Appendix C: Momentum and Poisson Equations And Boundary Condmentioning

confidence: 99%

On the response of the Earth to a fault system: its evaluation beyond the epicentral reference frame

Cambiotti

Sabadini

2015

Geophys. J. Int.

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SUMMARYPrevious formalisms for determining the static perturbation of spherically symmetric selfgravitating elastic Earth models due to displacement dislocations deal with each infinitesimal element of the fault system in its epicentral reference frame. In this work we overcome this restriction and present novel and compact formulas for obtaining the perturbation due to the whole fault system in an arbitrary and common reference frame. Furthermore, we show that, even in an arbitrary reference frame, it is still possible to discriminate the contributions associated to the polar, bipolar and quadrupolar patterns of the seismic source response, as well as their relation with the along strike, along dip and tensile components of the displacement dislocation. These results allow a better understanding of the relation between the static perturbation and the whole fault system, and find direct applications in geodetic problems, like the modelling of long-wavelength geoid or gravity data from GRACE and GOCE space missions and of the perturbation of the deviatoric inertia tensor of the Earth.

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The Static Deformation of an Earth with a Fluid Core: A Physical Approach

Cited by 38 publications

References 10 publications

GRACE gravity data help constraining seismic models of the 2004 Sumatran earthquake

GRACE gravity data help constraining seismic models of the 2004 Sumatran earthquake

The rotational feedback on linear-momentum balance in glacial isostatic adjustment

On the response of the Earth to a fault system: its evaluation beyond the epicentral reference frame

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