Elastic stress and deformation near a finite spherical magma body: Resolution of the point source paradox

Abstract

Approximate solutions are obtained for the stress and displacement fields due to a pressurized spherical cavity in an elastic half‐space. The solutions take the form of series expansions in powers of ε = a/d, where a is the cavity radius and d is the depth. The leading‐order term in the expression for the surface uplift, which arises at O(ε3), recovers the well‐known result of Mogi for the response to a point dilatation. The first higher‐order correction accounts for a cavity of finite size and thus offers the possibility of fitting leveling data for not only the depth but also the radius and pressure increment. However, this correction is of O(ε6) and, consequently, is weak. The result provides a formal explanation for the success of the point dilatation model in representing uplift data even when it is known independently that ε is not small. The higher‐order correction causes the surface uplift to fall off more rapidly in the radial direction, implying that a fit of the point source solution tends to underestimate the depth d. In contrast to the surface displacement, the stress field near the cavity is affected profoundly by the proximity of the free surface. Three higher‐order corrections to the stress field are obtained, which result in a uniformly valid approximation to O(ε5). The hoop stress at the cavity exhibits a tensile maximum at the circle of tangency with a cone with its apex at the free surface. This result appears to be consistent with the locus of fractures radiating outward from the magma body inferred by seismic methods in Long Valley, California.

Physics-based models of ground deformation and extrusion rate at effusively erupting volcanoes

“…Kinematic models of ground deformation, although useful, do not have predictive capability, cannot easily model additional data sets such as extrusion rate, gravity change, or gas emissions, and cannot in isolation uniquely estimate both the volume of a magma chamber and its pressure change [e.g., McTigue, 1987].…”

“…For a spherical chamber in an elastic full-space, b ch = 3/4m where m is the shear modulus of the elastic medium [McTigue, 1987]; however, this result does not hold for nonspherical shapes [Amoruso and Crescentini, 2009]. Using the finite element method we numerically compute b ch for chambers with aspect ratios w ranging from 0.05 to 20 in an elastic half-space with n = 0.25 and m = 20 GPa.…”

Dynamic deformation of Etna Volcano observed by satellite radar interferometry

“…In the half-space model the shape of the uplift curve is dependent on the depth of the source, while the properties of the source (change in pressure and cavity radius) are inseparable [McTigue, 1987] wards the center (Fig. 4B) and the elliptical nature of the uplift is further revealed in its residual range displacements (Fig 3B, bottom).…”

Asymptotic expansion for harmonic functions in the half‐space with a pressurized cavity

“…Additionally, from a geological point of view, sometimes it is reasonable to consider the magma chamber small compared to the distance from the boundary of the half-space, see [6,15,17]; by adding these hypotheses and fixing the geometry of the cavity, some efforts have been done during the last decades to find some explicit or approximate solutions to the mathematical model. The simplest approximate solution obtained by asymptotic expansions is due to McTigue when the cavity is a sphere, see [15]. The other few solutions available concern ellipsoidal shapes, dike and faults, see [6,17].…”