Analysis of a mogi-type model describing surface deformations induced by a magma chamber embedded in an elastic half-space

Andrea Aspri, Elena Beretta, Corrado Mascia

Research output: Contribution to journalArticle

Abstract

Motivated by a vulcanological problem, we establish a sound mathematical approach for surface deformation e ects generated by a magma chamber embedded into Earth’s interior and exerting on it a uniform hydrostatic pressure. Modeling assumptions translate the problem into classical elasto-static system (homogeneous and isotropic) in a half-space with an embedded cavity. The boundary conditions are traction-free for the air/crust boundary and uniformly hydrostatic for the chamber boundary. These are complemented with zero-displacement condition at infinity (with decay rate). After a short presentation of the model and of its geophysical interest, we establish the well-posedness of the problem and provide an appropriate integral formulation for its solution for cavity with general shape. Based on that, assuming that the chamber is centered at some fixed point z and has diameter r > 0, small with respect to the depth d, we derive rigorously the principal term in the asymptotic expansion for the surface deformation as ε = r/d → 0+. Such a formula provides a rigorous proof of the Mogi point source model in the case of spherical cavities generalizing it to the case of cavities of arbitrary shape.

Original languageEnglish (US)
Pages (from-to)223-255
Number of pages33
JournalJournal de l'Ecole Polytechnique - Mathematiques
Volume4
DOIs
StatePublished - Jan 1 2017

Fingerprint

Half-space
Cavity
Hydrostatic Pressure
Elastostatics
Hydrostatics
Point Source
Decay Rate
Well-posedness
Model
Asymptotic Expansion
Interior
Fixed point
Infinity
Boundary conditions
Formulation
Zero
Arbitrary
Term
Modeling

Keywords

  • Asymptotic expansions
  • Double layer potentials
  • Lamé operator
  • Single

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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AB - Motivated by a vulcanological problem, we establish a sound mathematical approach for surface deformation e ects generated by a magma chamber embedded into Earth’s interior and exerting on it a uniform hydrostatic pressure. Modeling assumptions translate the problem into classical elasto-static system (homogeneous and isotropic) in a half-space with an embedded cavity. The boundary conditions are traction-free for the air/crust boundary and uniformly hydrostatic for the chamber boundary. These are complemented with zero-displacement condition at infinity (with decay rate). After a short presentation of the model and of its geophysical interest, we establish the well-posedness of the problem and provide an appropriate integral formulation for its solution for cavity with general shape. Based on that, assuming that the chamber is centered at some fixed point z and has diameter r > 0, small with respect to the depth d, we derive rigorously the principal term in the asymptotic expansion for the surface deformation as ε = r/d → 0+. Such a formula provides a rigorous proof of the Mogi point source model in the case of spherical cavities generalizing it to the case of cavities of arbitrary shape.

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