Earth's long‐period wobbles: a Lagrangean description of the Liouville equations

SUMMARY The two low‐frequency wobble modes of the Earth are the Chandler wobble (CW) and the inner core wobble (ICW). The CW period is about 400 d for an ocean‐free model, and the ICW period has previously been estimated to be about 6.6 yr. The latter mode is particularly difficult to compute due to approximations that must be introduced to describe the response of the fluid core at long periods. Here we construct a Lagrangean formulation of the Liouville equations for the solid parts of the Earth, with disturbances from hydrostatic equilibrium referred to a uniformly rotating reference frame. Gravitational coupling, elastic deformation of the solid parts and compressibility of the fluid core are included, but not dissipative mechanisms. We show that at these low frequencies the Poisson equation for the gravitational perturbation can be solved without explicit knowledge of the outer core flow. Deformation of the solid parts can be assumed to be described by static Love numbers. With these assumptions, we find that the CW solution is essentially the same as in classic results using the linear momentum equations. On the other hand, the ICW is here found to have a period of about 7.5 yr for PREM, some 15 per cent longer than previous studies as a consequence of elastic deformation of the inner core. The existence of the ICW depends crucially on the gravitational torque exerted by the mantle on the inner core.