Some remarks about the degree-one deformation of the Earth

SUMMARY The degree-one deformation of the Earth (and the induced discrepancy between the figure centre and the mass centre of the Earth) is computed using a theoretical approach (Love numbers formalism) at short timescales (where the Earth has an elastic behaviour) as well as at long timescales (where the Earth has a viscoelastic or quasi-fluid behaviour). For a Maxwell model of rheology, the degree-one relaxation modes associated with the viscoelastic Love numbers have been investigated: the M, mode does not exist and there is only one transition mode (instead of two) generated by a viscosity discontinuity. The translations at each interface of the incompressible layers of the earth model [surface, 670 km depth discontinuity, core-mantle boundary (CMB) and inner-core boundary (ICB)] are computed. They are elastic with an order of magnitude of about 1 mm when the excitation source is the atmospheric continental loading or a magnetic pressure acting at the CMB. They are viscoelastic when the earth is submitted to Pleistocene deglaciation, with an order of magnitude of about 1 m. In a quasi-fluid approximation (Newtonian fluid) because of the mantle density heterogeneity their order of magnitude is about 100 m (except for the ICB, which is in quasi-hydrostatic equilibrium at this timescale). Degree-one deformation has particular characteristics related to geodesy as well as to mechanics: it is related to the centre of mass of the Earth and is strongly dependent on the choice of the origin of the reference frame. We have first to provide some precise definitions concerning these reference frames. We consider an initial undeformed reference sphere that is radially stratified. The geometric centre of the external spherical surface is coincident with that of each layer, and is identical to the centre of mass of the whole sphere. For a degree-one deformation, the external surface of the planet remains spherical but is translated with respect to the initial configuration; similarly, each internal layer of the planet is translated but the centre of mass does not change and remains fixed relative to the centre of the initial reference sphere. We define the figure centre as the geometric centre of the translated external spherical surface. The geometric centres of the various internal layers move in the deformed configuration and are different to the figure centre; they do not play any geodetic role and will not be given precisely (but they may be easily computed). The sources of degree-one deformation are essentially internal or external redistributed loads (including the atmosphere and the oceans), that is without net change in the total mass of the Earth. With respect to the reference configuration, a load is the expression of a mass transfer from a given hemisphere to the other one. This is observed for atmospheric and oceanic loading. For solid-mass anomalies, such as ice sheets, cool internal zones, surface tectonics, etc., the conservation of the centre of mass imposes a displacement of the particles in a direction opposite to that of the mass anomalies. This displacement is distributed within the whole planet.