Truncation effects in computing free wobble/nutation modes explored using a simple Earth model. Issue 3 (9th March 2017) - Seyed-Mahmoud, Behnam; Rochester, Michael G.; Rogers, Christopher M.
Abstract: The displacement field accompanying the wobble/nutation of the Earth is conventionally represented by an infinite chain of toroidal and spheroidal vector spherical harmonics, coupled by rotation and ellipticity. Numerical solutions for the eigenperiods require truncation of that chain, and the standard approaches using the linear momentum description (LMD) of deformation during wobble/nutation have truncated it at very low degrees, usually degree 3 or 4, and at most degree 5. The effects of such heavy truncation on the computed eigenperiods have hardly been examined. We here investigate the truncation effects on the periods of the free wobble/nutation modes using a simplified Earth model consisting of a homogeneous incompressible inviscid liquid outer core with a rigid (but not fixed) inner core and mantle. A novel Galerkin method is implemented using a Clairaut coordinate system to solve the classic Poincaré problem in the liquid core and, to close the problem, we use the Lagrangean formulation of the Liouville equation for each of the solid parts of the Earth model. We find that, except for the free inner core nutation (FICN), the periods of the free rotational modes converge rather quickly. The period of the tiltover mode is found to excellent accuracy. The computed periods of the Chandler wobble and free core nutation are nearly identical to the values cited in the literature for similar Earth models, but that for the inner core wobble is slightly different. Truncation at low-degree harmonics causes the FICN period to fluctuate over a range as large as 90 sd, with different values at different truncation levels. For example, truncation at degree 6 gives a period of 752 sd (almost identical with the value cited in the literature for such an Earth model) but truncation at degree 24 is required to obtain convergence, and the resulting period is 746 ± 1 sd, as more terms are included, with no guarantee that its proximity to earlier values is other than fortuitous. We conclude that the heavy truncation necessitated by the conventional LMD is unsatisfactory for the FICN.