ON PRINCIPAL MODES OF POINCARE'S OPERATOR IN A SPHERE

Abstract

Even the simplest cases of rotating fluids in a sphere or a spherical shell, as in the earth's core, are not yet completely studied; in particular, this is the case of Poincar\'{e}'s operator. Namely, the following problem arises: To choose simple, or least oscillatory, fields from a family of vector or tensor fields in some space. We introduce a functional defined on such fields and use it to compare their oscillatory behavior. Particularly, we used this functional to classify known eigenfunctions of Poincar\'{e}'s operator in a sphere. It is also possible to use the functional in a more complicated problem where a spherical shell is taken instead of a sphere; it is possible to choose a basis, arranged in ascending order of values of this functional, in the space of smooth solenoidal fields, and construct successive Galerkin subspaces in this basis.

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Computational Seismology, Vol. 3.