# Ocean tide loading on a self-gravitating, compressible, layered, anisotropic, viscoelastic and rotating earth with solid inner core and fluid outer core

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## Abstract

A novel ocean tide loading model is developed which allows the earth to be self-gravitating, compressible, layered, anisotropic, viscoelastic and rotating, with solid inner core and fluid outer core.
The deformation equations of the earth are developed, following the analytical mechanics approach. The standard-linear-solid-type rheology, as well as the grain-boundary relaxation model for the dissipation mechanism within the earth are adopted in this study. The thermodynamic state of the earth is accounted for, through its absolute temperature, Gibbs free activation energy, viscosity and Q profiles.
For the numerical integration of the equations of deformation, the following models are considered: a) PREM for the elasticity parameters of the earth appropriately modified at tidal frequencies, using dispersion relations, b) SL8 model for the Q profile of the earth, c) viscosity profile with the following viscosities: 2.5 x 10[to the power of 22] poise for the lower mantle, 10[to the power of 22] poise for the transition zone and 10[to the power of 17] poise for the LVZ, d) SAMMIS ET. AL, [1977] model for the Gibbs free activation energy profile (for the transition zone and lower mantle), with an adiabatic temperature gradient of 0.3 K/km. The value of 125 kCal/Mole for the LVZ is considered, and e) STACEY'S [1977] thermal model for the temperature profile of earth.
Complex load number h'[subscript n], k'[subscript n] and l'[subscript n], are calculated and the results are the following:
a) The rotation of the earth has an effect on the load numbers that can be as much as 1.8%, 3.1% and 3.3% respectively, depending on the degree of expansion. There is a weak latitude dependence of the load numbers for n ≤ 4; when latitude varies from 0° to ±45°, its effect is of the order of 0.4%.
b) The effect of anisotropy in the upper mantle can be as much as 1.9%, 2.3% and 2.5% respectively, depending on the degree of expansion.
c) At the semidiurnal periods, the load numbers on a viscoelastic earth are about 0.2% larger than their corresponding values on an elastic earth. At fortnightly periods, viscoelastic h'[subscript 100], k'[subscript 100] and l'[subscript 100] are larger than their corresponding elastic values by 0.5%, 1.5% and 1.3%, respectively/ For other values of n, the effect of viscosity is smaller.
Complex Green's functions are determined for displacements, gravity and tilt; they are given in the same form a those of FARRELL [1972], for easy implementation with existing software. The predictive power of the model is tested against accurately determined M[subscript 2] gravity tide residuals at 10, globally distributed, tidal stations. It is shown that the difference between observed residual gravity and predicted load gravity tide amplitudes is reduced for all tested stations as much as 63%, when compared to predictions on an elastic, isotropic and nonrotating earth. There is also an improvement in the phases of the predicted load gravity tide.
All the novel features of this research are included in the new version of the LOADSDP software package [PAGIATAKIS, 1982]. LOADSDP software can be used to evaluate displacements, gravity perturbations and tilt at arbitrary locations on the surface of the earth with an accuracy better than 1%.