Bending of spherical lithosphere–axisymmetric case

Effects of sphericity are commonly ignored in the lithospheric bending problem. In order to examine its effects, I solve a simple axisymmetric spherical-shell model. The full solution and the asymptotic solution are derived from the basic equations, and their relationship to the flat-plate solution is examined. For displacement, effects of sphericity are small, and use of the flat-plate solution produces results that are numerically indistinguishable from those of the spherical solution. The most significant effect of sphericity appears in the stress, in particular the normal stress along the strike direction of the trench. This stress is approximately given by Eu_r/R , where E is Young's modulus, u_r is the vertical deformation of the shell and R is its radius of curvature. If the shell (lithosphere) is bent downwards and reaches 30 km, this stress can become about 5 kbar in the Earth. While plastic behaviour may set in under such high pressure conditions and analysis beyond elasticity theory may be required, sphericity may be a cause of large compressive stress in the trench strike direction. This stress may play an important role in forming the overall shape of the Earth's subduction zones.     

Orthogonality and mean squares of the vector fields given by spherical cap harmonic potentials

It is well known that the vector fields derived from spherical harmonics are orthogonal over the sphere. It is now shown that the vector fields derived from spherical cap harmonics are orthogonal over the cap, to the same extent as the cap potentials are, and expressions are given for their mean squares.     

A three-dimensional spherical mesh generator

A new spherical mesh generator is described. It represents an efficient, deterministic packing of tetrahedra into a solid sphere, a spherical shell, or both. The mesh can be used for finite-element solutions to a wide variety of global numerical modelling problems in the geosciences. The nodes within the mesh are distributed uniformly, and long, thin tetrahedra are avoided. The method proposed here offers several advantages over 3-D Delaunay algorithms for finite-element mesh generation. For the related problem of trivariate scattered data interpolation, which is not considered here, the 3-D Delaunay algorithms are the method of choice.     

The growing spherical seismic source

Summary. A solution is found for the seismic radiation from an arbitrarily growing spherical source in an inhomogeneously prestressed elastic medium. The general problem of the growing seismic source in a prestressed medium is formulated as a boundary value problem. For the special case of the growing spherical source, an expansion in vector spherical harmonics reduces the problem to a set of one-dimensional Volterra integral equations. The equations can be easily formed through the use of Bessel function recursion relations. The integral equations for a growing spherical cavity are solved numerically. Waveforms are then computed for homogeneous and inhomogeneous stress fields for several growth histories. The resulting waveforms are similar to the waveforms of the corresponding instantaneous problem, but stretched out in time and reduced in amplitude. The effects of diffraction and the overshoot of equilibrium are reduced with a reduction in growth rate. The effects caused by inhomogeneity of the stress field are quite strong for the growing as well as for the instantaneous seismic source.     

World population in a grid of spherical quadrilaterals

"We report on a project that converted subnational population data to a raster of cells on the earth. We note that studies using satellites as collection devices yield results indexed by latitude and longitude. Thus it makes sense to assemble the terrestrial arrangement of people in a compatible manner. This alternative is explored here, using latitude/longitude quadrilaterals as bins for population information.... The results to date of putting world boundary coordinates together with estimates of the number of people are described. The estimated 1994 population of 219 countries, subdivided into 19,032 polygons, has been assigned to over six million five minute by five minute quadrilaterals covering the world."