Analytical Solutions for the Periods of the Chandler Wobble and Free-Core Nutation of a Three-Layer Poincaré Earth Model

Abstract:

It is well established that the Poincaré problem, a second-order hyperbolic partial differential equation subject to boundary conditions, describing the dynamics of rotating, incompressible and inviscid fluids is ill-posed in the sense that the existence of its analytical solutions depends on the geometry of the rotating container. Analytical solutions exist if the fluid is contained in a uniformly rotating spheroid with a rigid boundary. However, except for a set of purely toroidal normal modes, these solutions are not known if the fluid is contained within a uniformly rotating thick rigid spheroidal shell. We define a Poincaré Earth model as a rotating Earth model with a rigid mantle (MT), a homogeneous, incompressible and inviscid liquid core (LC) and, for a three-layer Earth model, a rigid inner core (IC). We show in this work that if the IC and the MT of a Poincaré Earth model rotate at the same rate then analytical solutions exist for the Chandler wobble (CW) and the free core nutation (FCN) of the model. The existence of analytical solutions for the FCN is not surprising as this mode is purely toroidal. We show that the flow in the LC is nearly geostrophic during the CW and the dynamics of the core is no longer described by the Poincaré equation. We infer from these results that it is reasonable that numerical solutions may be found for the periods of FCN, CW and the inner-core wobble (ICW) of a more realistic Earth model with inner-core allowed to wobble independent of the mantle. We also present the displacement eigenfunctions for the CW in a three layer Earth model.