METHODS FOR COMPUTING INTERNAL FLATTENING, WITH APPLICATIONS TO THE EARTHS STRUCTURE AND GEODYNAMICS

After general comments (Section 1) on using variational procedures to compute the oblateness of internal strata in the Earth and slowly rota ting planets, we recall briefly some basic concepts about barotropic e quilibrium figures (Section 2), and then proceed to discuss several ac curate methods to derive the internal flattening. The algorithms given in Section 3 are based on the internal gravity field theory of Claira ut, Laplace and Lyapunov. They make explicit use of the concept of a l evel surface. The general formulation given here leads to a number of formulae which are of both theoretical and practical use in studying t he Earth's structure, dynamics and rotational evolution. We provide ex act. solutions for the figure functions of three Earth models, and app ly the formalism to yield curves for the internal flattening as a func tion of the spin frequency. Two more methods, which use the general de formation equations, are discussed in Section 4. The latter do not rel y explicitly on the existence of level surfaces, They offer an alterna tive to the classical first-order internal field theory, and can actua lly be used to compute changes of the flattening on short timescales p roduced by variations in the LOD. For short durations, the Earth behav es elastically rather than hydrostatically. We discuss in some detail static deformations and Longman's static core paradox (Section 5), and demonstrate that in general no static solution exists for a realistic Earth model. In Section 6 we deal briefly with differential rotation occurring in cylindrical shells, and show why differential rotation of the inner core such as has been advocated recently is incompatible wi th the concept of level surfaces. In Section 7 we discuss first-order hydrostatic theory in relation to Earth structure, and show how to der ive a consistent reference Earth model which is more suitable for geod ynamical modelling than are modern Earth models such as 1066-A, PREM o r CORE11. An important result is that a consistent application of hydr ostatic theory leads to an inertia factor of about 0.332 instead of th e value 0.3308 used until now. This change automatically brings 'hydro static' values of the flattening, the dynamic shape factor and the pre cessional constant into much better agreement with their observed coun terparts than has been assumed hitherto. Of course, we do not imply th at non-hydrostatic effects are unimportant in modelling geodynamic pro cesses. Finally, we discuss (Sections 7-8) some implications of our wa y of looking at things for Earth structure and some current problems o f geodynamics. We suggest very significant changes for the structure o f the core, in particular a strong reduction of the density jump at th e inner core boundary. The theoretical value of the free core nutation period, which may tie computed by means of our hydrostatic Earth mode ls CGGM or PREMM, is in somewhat better agreement with the observed va lue than that based on PREM or 1066-A, although a significant residue remains, We attribute the latter to inadequate modelling of the deform ation, and hence of the change in the inertia tensor, because the stat ic deformation equations were used, We argue that non-hydrostatic effe cts, though present, cannot explain the large observed discrepancy of about 30 days.