Rotational modes of Poincaré Earth models

ABSTRACT We study the following rotational modes of Poincaré Earth models: the tilt-over mode (TOM), the spin-over mode (SOM) and free core nutation (FCN), using first a simple Earth model with a homogeneous and incompressible liquid core (LC) and a rigid mantle (MT). We obtain analytical solutions for the periods of these modes as well as that of the Chandler wobble (CW). We show analytically the distinction between the TOM and the SOM and that the FCN is indeed the same mode as the SOM of a wobbling Earth. The reduced pressure, in terms of which the vector momentum equation is known to reduce to a scalar second-order partial differential equation called the Poincaré equation, is used as the independent variable. Analytical solutions are then found for the displacement eigenfunctions in a meridional plane of the liquid core for the aforementioned modes. We next consider a three-layer Earth model similar to above which also includes a rigid inner core (IC). We first show that analytical solutions exist for the period and eigenfunctions of the CW if the IC is locked to the MT, i.e. they have the same wobbling motion. We show that this is significant as it shows that the CW manifests itself for a Poincaré (incompressible and inviscid LC) wobbling Earth model. We further allow for the inner core to wobble independently and compute numerically the periods and displacement eigenfunctions of the TOM, SOM and FCN, as well as those for still another rotational mode, the inner-core wobble (ICW). Next we show that the presence of the characteristic surfaces intercepted by the inner-core, when computing the period and eigenfunctions of the free inner-core nutation (FICN), may be the reason for the slow (or lack of the) convergence of this mode. Finally, we show that even though the wobbling motion of the mantle is ignored when solving for the frequencies of the ICW and the FICN when Sasao's approximation is used, the analytical solutions for both these modes yield periods nearly identical to those in the literature for a similar Earth model with mantle allowed to wobble as well. We infer that the Sasao's approximation, or the severe truncation of the series solution of the field variables, the pressure, the gravitational potential and the components of the displacement vector, may not be adequate to accurately describe the motion in the liquid core during the excitation of the FICN.