Constraining Earth's mantle rheology with Love and Shida numbers at the M2 tidal frequency

We use measurements of Earth's tidal response at the $M_2$ frequency to constrain the rheology of its mantle. The viscoelasticity and anelasticity of the planet are modeled with an Andrade rheology that depends on two parameters: $\alpha$ and $\zeta$. In this paper, we propose an improved algorithm to compute Earth's tidal deformation. Its Love and Shida numbers $k_2$, $h_2$, $k_3$ and $l_2$ as well as the tidal lag $ϵ$ were calculated for two viscosity profiles and for a wide range of values of $\alpha$ and $\zeta$. By comparing our results with geodetic measurements we obtain the range of values of $\alpha$ and $\zeta$ that successfully describes Earth's viscoelastic behavior. Values of $\zeta$ as high as ${10}^5$ can not be excluded. For $\zeta =1$, $\alpha$ should be in the range from 0.19 to 0.33, while for a $\zeta ={10}^5$, $\alpha$ is most likely between 0.11 and 0.17. We believe that a similar rheology should be used in geophysical models of other rocky planets and satellites. The obtained results are mostly representative of the lower mantle.

We have also shown that for several combinations of the two parameters $\alpha$ and $\zeta$ we could obtain nearly identical values of Earth's $\Re \left(k_2\right)$, $\Re \left(h_2\right)$, $\Re \left(l_2\right)$ and $\Re \left(k_3\right)$ with considerably different values of the associated tidal lag. This shows that the approach of always setting $\zeta =1$ might be too simplistic and an Andrade rheology with two free parameters is needed to constrain both the real and imaginary parts of Love and Shida numbers.