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Abstract.For a self-gravitating viscoelastic compressible sphere we have shown that unstable modes can exist by means of the linear viscoelastic theory by both initial-value and normal-mode approaches. For a uniform sphere we have derived analytical expressions for the roots of the secular determinant based on the asymptotic expansion of the spherical Bessel functions. From the two expressions, both the destabilizing nature of gravitational forces and the stabilizing influences of increasing elastic strength are revealed. Fastest growth times on the order of ten thousand years are developed for the longest wavelength. In contrast, a selfgravitating incompressible viscoelastic model is found to be stable. This result of linear approximation suggests that a more general approach, e.g., non-Maxwellian rheology or a non-linear finite-amplitude theory, should be considered in global geodynamics.

We have developed the numerical algorithm for the computation of transient viscoelastic responses in the time domain for a radially stratified Earth model. Stratifications in both the elastic parameters and the viscosity profile have been considered. The particular viscosity profile employed has a viscosity maximum with a contrast of O(10²) in the mid lower mantle. The distribution of relaxation times reveals the presence of a continuous spectrum situated between O(10²) and O(104) years. The principal mode is embedded within this continuous spectrum. From this initial‐value approach we have found that for the low degree harmonics the non‐modal contributions are comparable to the modal contributions. For this viscosity model the differences between the time‐domain and normal‐mode results are found to decrease strongly with increasing angular order. These calculations also show that a time‐dependent effective relaxation time can be defined, which can be bounded by the relaxation times of the principal modes.

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