An analytical solution for the elastic response to surface loads imposed on a layered, transversely isotropic and self-gravitating Earth

E. Pan, J.Y. Chen, M. Bevis, Andrea Bordoni, Valentina Roberta Barletta, A. Molavi Tabrizi

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Abstract

We present an analytical solution for the elastic deformation of an elastic, transversely isotropic, layered and self-gravitating Earth by surface loads. We first introduce the vector spherical harmonics to express the physical quantities in the layered Earth. This reduces the governing equations to a linear system of equations for the expansion coefficients. We then solve for the expansion coefficients analytically under the assumption (i.e. approximation) that in the mantle, the density in each layer varies as 1/r (where r is the radial coordinate) while the gravity is constant and that in the core the gravity in each layer varies linearly in r with constant density. These approximations dramatically simplify the subsequent mathematical analysis and render closed-form expressions for the expansion coefficients. We implement our solution in a MATLAB code and perform a benchmark which shows both the correctness of our solution and the implementation. We also calculate the load Love numbers (LLNs) of the PREM Earth for different degrees of the Legendre function for both isotropic and transversely isotropic, layered mantles with different core models, demonstrating for the first time the effect of Earth anisotropy on the LLNs.
Original languageEnglish
JournalGeophysical Journal International
Volume203
Issue number3
Pages (from-to)2150-2181
ISSN0956-540X
DOIs
Publication statusPublished - 2015

Bibliographical note

This article has been accepted for publication in Geophysical Journal International ©: 2015 The authors. Published by Oxford University Press on behalf of the Royal Astronomical Society. All rights reserved.

Keywords

  • Composition of the planets
  • Elasticity and anelasticity
  • Dynamics
  • Gravity and tectonics
  • Mechanics, theory and modelling
  • Planetary interiors

Cite this

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title = "An analytical solution for the elastic response to surface loads imposed on a layered, transversely isotropic and self-gravitating Earth",
abstract = "We present an analytical solution for the elastic deformation of an elastic, transversely isotropic, layered and self-gravitating Earth by surface loads. We first introduce the vector spherical harmonics to express the physical quantities in the layered Earth. This reduces the governing equations to a linear system of equations for the expansion coefficients. We then solve for the expansion coefficients analytically under the assumption (i.e. approximation) that in the mantle, the density in each layer varies as 1/r (where r is the radial coordinate) while the gravity is constant and that in the core the gravity in each layer varies linearly in r with constant density. These approximations dramatically simplify the subsequent mathematical analysis and render closed-form expressions for the expansion coefficients. We implement our solution in a MATLAB code and perform a benchmark which shows both the correctness of our solution and the implementation. We also calculate the load Love numbers (LLNs) of the PREM Earth for different degrees of the Legendre function for both isotropic and transversely isotropic, layered mantles with different core models, demonstrating for the first time the effect of Earth anisotropy on the LLNs.",
keywords = "Composition of the planets, Elasticity and anelasticity, Dynamics, Gravity and tectonics, Mechanics, theory and modelling, Planetary interiors",
author = "E. Pan and J.Y. Chen and M. Bevis and Andrea Bordoni and Barletta, {Valentina Roberta} and Tabrizi, {A. Molavi}",
note = "This article has been accepted for publication in Geophysical Journal International {\circledC}: 2015 The authors. Published by Oxford University Press on behalf of the Royal Astronomical Society. All rights reserved.",
year = "2015",
doi = "10.1093/gji/ggv432",
language = "English",
volume = "203",
pages = "2150--2181",
journal = "Geophysical Journal International",
issn = "0956-540X",
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An analytical solution for the elastic response to surface loads imposed on a layered, transversely isotropic and self-gravitating Earth. / Pan, E.; Chen, J.Y. ; Bevis, M.; Bordoni, Andrea; Barletta, Valentina Roberta; Tabrizi, A. Molavi .

In: Geophysical Journal International, Vol. 203, No. 3, 2015, p. 2150-2181.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - An analytical solution for the elastic response to surface loads imposed on a layered, transversely isotropic and self-gravitating Earth

AU - Pan, E.

AU - Chen, J.Y.

AU - Bevis, M.

AU - Bordoni, Andrea

AU - Barletta, Valentina Roberta

AU - Tabrizi, A. Molavi

N1 - This article has been accepted for publication in Geophysical Journal International ©: 2015 The authors. Published by Oxford University Press on behalf of the Royal Astronomical Society. All rights reserved.

PY - 2015

Y1 - 2015

N2 - We present an analytical solution for the elastic deformation of an elastic, transversely isotropic, layered and self-gravitating Earth by surface loads. We first introduce the vector spherical harmonics to express the physical quantities in the layered Earth. This reduces the governing equations to a linear system of equations for the expansion coefficients. We then solve for the expansion coefficients analytically under the assumption (i.e. approximation) that in the mantle, the density in each layer varies as 1/r (where r is the radial coordinate) while the gravity is constant and that in the core the gravity in each layer varies linearly in r with constant density. These approximations dramatically simplify the subsequent mathematical analysis and render closed-form expressions for the expansion coefficients. We implement our solution in a MATLAB code and perform a benchmark which shows both the correctness of our solution and the implementation. We also calculate the load Love numbers (LLNs) of the PREM Earth for different degrees of the Legendre function for both isotropic and transversely isotropic, layered mantles with different core models, demonstrating for the first time the effect of Earth anisotropy on the LLNs.

AB - We present an analytical solution for the elastic deformation of an elastic, transversely isotropic, layered and self-gravitating Earth by surface loads. We first introduce the vector spherical harmonics to express the physical quantities in the layered Earth. This reduces the governing equations to a linear system of equations for the expansion coefficients. We then solve for the expansion coefficients analytically under the assumption (i.e. approximation) that in the mantle, the density in each layer varies as 1/r (where r is the radial coordinate) while the gravity is constant and that in the core the gravity in each layer varies linearly in r with constant density. These approximations dramatically simplify the subsequent mathematical analysis and render closed-form expressions for the expansion coefficients. We implement our solution in a MATLAB code and perform a benchmark which shows both the correctness of our solution and the implementation. We also calculate the load Love numbers (LLNs) of the PREM Earth for different degrees of the Legendre function for both isotropic and transversely isotropic, layered mantles with different core models, demonstrating for the first time the effect of Earth anisotropy on the LLNs.

KW - Composition of the planets

KW - Elasticity and anelasticity

KW - Dynamics

KW - Gravity and tectonics

KW - Mechanics, theory and modelling

KW - Planetary interiors

U2 - 10.1093/gji/ggv432

DO - 10.1093/gji/ggv432

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SN - 0956-540X

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