The equilibrium configuration of the earth involving minimum stress difference

Abstract

From a theoretical point of view it is possible to consider the earth as s selfgravitating fluid (i.e., subject to gravitational and rotational potential). The equilibrium configuration of the earth, known as its hydrostatic equilibrium, may be sought in this scheme. Hydrostatic equilibrium agrees very well with the long-wavelength shape and gravitational potential of the planet: the discrepancy between the observed and the theoretical values is generally less than 0.5%. The term stress difference is used here to mean the non-hydrostatic part of the stress tensor. The equilibrium configuration computed in the hydrostatic case is clearly an equilibrium configuration in which the stress difference is zero. In this paper, the principal aim is to perform the further step of considering the external crust of the earth as a perfect solid which is bounded by the external observed topography and the internal computed (by an isostatic model) mohorovc̆ić discontinuity. The internal part of the earth is treated in such a way as to minimize the stress difference. We consider a radial homogeneous model of density and a number of different shells in the interior of the earth, each one with constant density. The shape of each shell is then sought that is as close as possible to the equilibrium shape in a hydrostatic configuration. We consider three cases: 1. the case in which the stress difference is a minimum, 2. the case in which the configuration is as close as possible to the spherical case and produces the minimum difference between the observed and computed external gravitational fields, 3. the mixed case, i.e., that in which the stress difference is a minimum and produces the minimum difference between the observed and computed external gravitational fields. From each of these cases we find an internal configuration of the earth, starting from a radial homogeneous model of density. We work out the shape of each surface at different radii within the earth. The most interesting of these is the Core Mantle Boundary (CMB), i.e., the most relevant discontinuities within the earth. Here we compare our version of the shape of the CMB with that obtained by other methods such as the seismically inferred method. The correlation coefficient seems to be encouraging.