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The Earth is a rapidly rotating body. The centrifugal pull makes its shape resemble a flattened ellipsoid and Coriolis forces support waves in its fluid core, known as inertial waves. These waves can lead to global oscillations, or modes, of the fluid. Periodic variations of the Earth's rotation axis (nutations) can lead to an exchange of angular momentum between the mantle and the fluid core and excite these inertial modes. In addition to viscous torques that exist regardless of the shape of the boundaries, the small flattening of the core–mantle boundary allows inertial modes to exert pressure torques on the mantle. These torques effectively couple the rigid-body dynamics of the Earth with the fluid dynamics of the fluid core. Here we present the first high-resolution numerical model that solves simultaneously the rigid body dynamics of the mantle and the Navier–Stokes equation for the liquid core. This method takes naturally into account dissipative processes in the fluid that are ignored in current nutation models. We find that the Free Core Nutation mode, mostly a toroidal fluid flow if the mantle has a large moment of inertia, enters into resonance with nearby modes if the mantle's moment of inertia is reduced. These mode interactions seem to be completely analogous to the ones discovered in 2006 by D. Schmitt in a uniformly rotating ellipsoid with varying flattening. [ABSTRACT FROM AUTHOR]