On the anomalies in gravity associated with lateral inhomogeneities in temperature: 2. Numerical results

摘要

In the first part of the paper, we obtained the analytical relationships determining the changes in the topography of the geoid and the component of horizontal displacements of the Earth's surface, which appear under the action of the point heat source located at the arbitrary depth in the mantle. For the real model of radially heterogeneous Earth with hydrostatic distribution of the initial stresses, the solution of the problem on thermoelastic deformations is represented in the form of spherical expansions with the coefficients determined by the corresponding coefficients of spherical expansions of the product of temperature, bulk modulus, and bulk compression modulus with the same indices. As mentioned in part I, the variation in the external potential is contributed by three effects: the reduction in density in the heated area, the increase in density in the external (not heated) area due to its elastic compression, and the attraction of the near-surface simple layer that is formed due to the change in the shape of the external surface under its elastic deformation. The total effect of these three factors is represented in the form of a spherical series expansion. It is shown that in the limiting case of the high-order spherical functions, the ratios of the radial displacements of the geoid to the radial displacements of the external surface tend to zero. Since at high orders of the spherical functions, the effects of sphericity are negligible, this statement means that at any thermoelastic deformations of the uniform elastic halfspace, the three effects listed above exactly compensate each other. Due to this compensation, the question of the interpretation of the observed relationships between the coefficients of series expansions of the temperature and geoid can only be solved after the detailed numerical calculations, since the arbitrarily small radial inhomogeneities of the medium (e.g., those associated with the depth changes of its rheological properties) are not only capable of significantly changing the magnitude of the radial displacements of the geoid but also altering their sign. Moreover, even in the uniform Earth's model, the effects of sphericity of its external surface and self-gravitation can also provide a noticeable contribution, which determines the signs of the coefficients in the expansion of the geoid's shape in the lower-order spherical functions. In order to separate these effects, below we present the results of the numerical calculations of the total effects of thermoelastic deformations for the two simplest models of spherical Earth without and with self-gravitation with constant density and complex-valued shear moduli and for the real Earth PREM model (which describes the depth distributions of density and elastic moduli for the high-frequency oscillations disregarding the rheology of the medium) and the modern models of the mantle rheology. Based on the calculations, we suggest the simplest interpretation of the present-day data on the relationship between the coefficients of spherical expansion of temperature, velocities of seismic body waves, the topography of the Earth's surface and geoid, and the data on the correlation between the lower-order coefficients in the expansions of the geoid and the corresponding terms of the expansions of horizontal inhomogeneities in seismic velocities. The suggested interpretation includes the estimates of the sign and magnitude for the ratios between the first coefficients of spherical expansions of seismic velocities, topography, and geoid.