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Thus we can analyse the rotational modes by the seismic method or by the mechanical method (Liouville equation). Either way, the key is the coupling between the fluid core and the solid mantle or the solid inner core. Hereafter we will use the mechanical method. Equation of Rotation The equations governing the angular velocities ω, ω c , ω ic are derived from the angular momentum theorem. Let I m , I c , I ic be the inertia tensors of the mantle, of the fluid core, of the solid inner core. Then their angular momenta are H m = I m ω, H c = I c (ω + ω c ), H ic = I ic (ω + ω ic , and the total angular momentum H t of the whole Earth is H t = H m + H c + H ic or t c c ic ic Hij = Iij ωj + Iij ωj + Iij ωj (79) m c ic where Iij = Iij + Iij + Iij is the inertia tensor of the whole Earth.

The glaciation/deglaciation is modeled by three spherical sheets of ice which are analysed in spherical harmonics with the same time-dependence for the evolution of their respective heights. Glaciation builds up over a long period of T0 = 9 × 104 years but deglaciation is precipitated in a short period of T1 − T0 = 104 years (Fig. 4-a). The variation in the sea level concomitant with the variation in the height of the ice sheets is taken into account and translates into a surface loading, which is analysed in spherical harmonics.

Finally the equations of state, in the form f (P, V, T ) = 0 or in any of the partial forms α(P, T ), KP (P, T ), KT (P, T ), γ(P, T ), η(P, T ), κ(P, T ), f (P, V ) = 0 connect variations of these parameters. To complete this setup of equations, we have to add the equations for the gravitational potential Φ: ∆Φ = −4πGρ Poisson s equation f = ∇Φ (5) As you can see, our setup neglects electromagnetism. The analysis of these equations provides information on: • evolution of the parameters • evolution of the form of the Earth • convection inside the Earth • variations in density and inertia tensor.