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

Aspri, Andrea; Beretta, Elena; Mascia, Corrado

Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 223-255.

Motivés par un problème volcanologique, nous établissons une approche mathématique solide pour les effets de déformation de surface engendrés par une chambre magmatique contenue à l’intérieur de la Terre et soumise à une pression hydrostatique uniforme. Des hypothèses de modélisation traduisent le problème en un système élasto-statique (homogène et isotrope) classique dans un demi-espace avec une cavité incluse. Les conditions au bord sont sans traction au bord air/croûte, et uniformément hydrostatiques au bord de la chambre. Elles sont complétées par une condition de déplacement nul à l’infini (avec taux de décroissance). Après une courte présentation du modèle et de son intérêt géophysique, nous établissons que le problème est bien posé et proposons une formulation intégrale appropriée pour sa solution dans le cas d’une cavité de forme générale. En conséquence, supposant que la chambre est centrée en un point $z$ et est de diamètre $r>0$ petit par rapport à la profondeur $d$, nous en déduisons rigoureusement le terme principal du développement asymptotique de la déformation de surface sous la forme $\epsilon =r/d\to 0^{+}$. Une telle formule permet de donner une preuve rigoureuse du modèle des points sources de Mogi dans le cas des cavités de forme arbitraire, qui généralise celle dans le cas des cavités sphériques.

Motivated by a vulcanological problem, we establish a sound mathematical approach for surface deformation effects 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 $\epsilon =r/d\to 0^{+}$. 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.

Classification : 35C20, 31B10, 35J25, 86A60
Keywords: Lamé operator, asymptotic expansions, single and double layer potentials
Mot clés : Opérateur de Lamé, développement asymptotique, potentiels de simple et double couche

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