Polar wandering of a dynamic earth

Abstract

S U M M A R YThe rotational behaviour of a stratified visco-elastic planet submitted to changes in its inertia tensor is studied in a viscous quasi-fluid approximation. This approximation allows for large displacements of the Earth rotation axis with respect to the entire mantle but is only valid for mass redistribution within the planet occurring on the time scale of a few million years. Such a motion, called true polar wander (TPW), is detected by palaeomagneticiens assuming that the Earth's magnetic field remains on average aligned with the spin axis. Our model shows that a downgoing cold slab induces a TPW which quickly brings this slab to the pole for a mantle of uniform viscosity. The same slab is slowly moved toward the equator when a large viscosity increase with depth takes place in the mantle. Our model is also suitable to investigate the effects of a non-steady-state convection on the Earth's rotation. We discuss these effects using a simple mass redistribution model inspired by the pioneering paper of Goldreich & Toomre (1969). It consists of studying the TPW induced by a random distribution of slabs sinking into the mantle. For such a mass redistribution, only a strongly stratified mantle can reduce the Earth's pole velocity below l"Ma-', which is the upper bound value observed by palaeomagnetic investigations for the last 200 Ma. Our model also shows that when corrected for the hydrostatic flattening, the Earth's polar inertia generally corresponds to the maximum inertia, as it is presently observed. However, this may not be the case during some short time periods. We also discuss The amount of excess polar flattening that can be related to tidal deceleration. This frozen component is found to be negligible. The equation of motion of a rotating body in a rotating frame is the well known Euler dynamic equation. When no external torque is applied, it reads d -( J -w ) + w A J . w = O . dt where J is the second-order symmetric inertia tensor and o is the angular velocity. Both are expressed in a rotating Earth-tixed coordinate system. This equation also holds when J is a time-dependent function (Munk & MacDonald 1960). In this case, it takes the name of Liouville equation.The inertia tensor J is traditionally divided into three contributions of decreasing amplitudes. The first is the tensor of a spherical non-rotating Earth. We write it as IS, where 6 , is the Kronecker symbol. This term is close to 284 0.33Ma2 where M and a are the mass and the radius of Earth. A second term is due to the centrifugal potential that deforms the Earth. It can be shown that this potential is proportional to wiwi -$02Sii where wi are the components of o in the geographical frame (e.g. Lambeck 1980). Any change in rotation is therefore equivalent to a new potential applied to the Earth's surface. Under such a boundary condition, the planet evolves toward a new configuration corresponding to an inertia tensor equal to the convolution of k T ( t ) , the tidal Love number of harmonic degree 2, with the time history of the cha...

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