The Earth's rotational modes revisited

In the conventional treatment of the Earth's rotational dynamics using the Earth's angular momentum description (AMD), it is customary to assume that the velocity/displacement of a mass element in the liquid core (LC) has a displacement as well as an explicit rigid rotation component in addition to the uniform (solid-body) rotation. This makes for a very complex set of non-linear differential equations in the treatment of the dynamics of this body. In this work, I will use a simple three-layer Earth model with a rigid mantle (MT), a rigid inner core (IC), and an incompressible and homogeneous LC to show that in the alternative linearised dynamics of this body, it is redundant to assign a rigid rotation component to the motion. Next, I will use an approximation commonly used in dealing with the Earth's rotational dynamics, and further assume that the MT rotates uniformly, to show that the linearised equations yield identical analytical results to those in the literature for the periods of the inner-core wobble (ICW) and the free inner-core nutation (FICN).