Some remarks about the rotations of a viscous planet and its homogeneous liquid core: linear theory

110

136

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...

Section: G * [ O I ( T ) O I ( T ) -:O2(t)sii]mentioning

confidence: 99%

Polar wandering of a dynamic earth

Ricard

Spada

Sabadini

1993

110

136

“…(c) , takes into account the fluid-solid boundary conditions at the core-mantle boundary (Chinnery 1975;Crossley & Gubbins 1975), while the 3 x 6 matrix ML(a) is built in such a way as to satisfy the external surface boundary conditions . Effects due to a fluid core rotating relative to a viscoelastic mantle have recently been considered by Lefftz & Legros (1992).…”

Section: Mathematical D E S C R I P T I O N O F T H E F I V E -L a Y mentioning

confidence: 99%

Effects on post-glacial rebound from the hard rheology in the transition zone

Spada¹,

Sabadini²,

Yuen³

et al. 1992

112

We analyse the influences of a viscosity increase in the transition zone between 420 and 670 km on the geophysical signatures induced by post-glacial rebound, ranging from the perturbations in the Earth's rotation to the short wavelength features associated with the migration of the peripheral bulge. A self-gravitating model is adopted, consisting of an elastic lithosphere, a three-layer viscoelastic mantle and an inviscid core.\ud \ud The horizontal displacements and velocities and the stress pattern are extremely sensitive to the viscosity increase and to the chemical stratification of the transition zone. The hardening of the upper and the chemical density jumps in mantle below the 420 discontinuity induces a channel effect which contaminates the horizontal deformation both in the near-field and in the far-field from the ice-sheets. These findings indicate that intraplate geodetic data can be used to put bounds on the viscosity increase in the transition zone and on the amount of chemical stratification in the mantle.\ud \ud The stress field induced in the lithosphere by the Pleistocenic ice-sheet disintegration is a very sensitive function of mantle viscosity stratification. The existence of seismic activity along passive continental margins of previously glaciated areas requires a substantial viscosity increase in the mantle, with the viscosity of the transition zone acting as a controlling parameter. A viscously stratified mantle is responsible for a delayed upward migration of stress in the lithosphere which can account for the seismicity today

“…In the frame of the linear approximation, we can separate the equatorial problem from the axial problem. We have presented a theoretical study of the equatorial problem (Lefftz & Legros 1992, this issue) hereafter referred to as Paper I. We denote B the mean axial rotation of the Earth.…”

Section: Euler Equationsmentioning

confidence: 99%

Influence of viscoelastic coupling on the axial rotation of the Earth and its fluid core

Lefftz

Legros

1992

Self Cite

S U M M A R Y Linear equations governing the axial rotation of the Earth are dcveloped for a model with a Maxwell homogeneous mantle and a homogeneous inviscid fluid core having a differential rotation relative t o the mantle.We find three eigenfrequencies for the axial perturbations in rotation; the first one is relative t o friction acting at the core-mantle boundary and the others are relaxation modes associated with viscoelastic Love numbers.W e point out the differential rotation of the core when the Earth is submitted to zonal geophysical excitations like glaciation and deglaciation, tidal torque or other processes occurring at the core-mantle boundary (CMB). W e show that the tidal deceleration involves an eastward drift of the core with respect t o the mantle. However, for precise values of the mantle viscosity and of the frictional constant at the C M B , we find that the last deglaciation involves a westward drift of the core with respect t o the mantle which may be correlated with the observed westward drift of the geomagnetic field of the Earth.