Dynamics of the Earth’s fluid core: Implementation of a Clairaut coordinate system

Abstract

If the reference state of a rotating and self gravitating fluid body is one of hydrostatic equilibrium then the figure of the body is a spheroid such that a cross sectional area parallel to the equatorial plane of the body is a circle while that parallel to a meridional plane is an ellipse. The effects of the fluid body’s flattened (spheroidal) figure is small on the frequencies of the body’s short-period (shorter than a few hours in the case of the Earth) normal modes. For the long-period normal modes, however, these effects must be considered. Furthermore, the body’s wobble and nutation modes owe their existence to its ellipsoidal figure. In the conventional approach to computing these frequencies, an orthogonal coordinate system is usually considered. It is then necessary to have the knowledge of the derivatives of the material properties of the body, such as the density and Lamé parameters, in order to include the effects of the ellipticity in the dynamical equations. In the available Earth models, however, these derivatives are not well defined. In order to minimize the effects of these derivatives in the treatment of the dynamical problems we use a non-orthogonal (Clairaut) coordinate system. Using this approach, we compute the frequencies and displacement eigenfunctions for some of the inertial modes of a realistic spheroidal model of the Earth’s fluid core and compare them to the known results for an Earth model with a homogeneous and incompressible fluid core.