Constraining the parameters of the Andrade rheology in Earth's mantle with Love numbers of 12 tidal constituents

Abstract

We use estimates of Earth's tidal response at 12 frequencies (M2, Mqm, Msqm, Mtm, Mstm, SN, Mf, Msf, Mm, Msm, Ssa, Sa) to constrain the values of the Andrade parameters in Earth's mantle. The viscoelasticity and anelasticity of the planet are modeled using the Andrade rheology which depends on the frequency of the tidal forcing, the viscosity profile of Earth's interior, and two parameters ($\alpha$ and $\zeta$ that are unknown for planetary interiors. We compute Earth's tidal deformation and its Love numbers for the 12 mentioned frequencies, for 3 viscosity profiles, and for a total of 2850 different combinations of values of $\alpha$ and $\zeta$ in the ranges $\alpha \in \left(\mathrm{0.13,0.37}\right)$ and $\zeta \in \left({10}^{-2},{10}^5\right)$.

By comparing the computed complex Love numbers k₂, h₂ and l₂ at each tidal frequency with their available estimates, we obtained the values of the Andrade parameters $\alpha$ and $\zeta$ that successfully reproduce Earth's observed anelastic behavior. Our best estimates for these parameters in Earth's interior are $\alpha \in \left(0.218,0.238\right)$ and $\zeta \in \left(0.5,133\right)$. However, values within the broader ranges of $\alpha \in \left(0.200,0.256\right)$ and $\zeta \in \left(0.2,485\right)$ may also be suitable, with the majority of the variability stemming from uncertainties in Earth's viscosity profile. Not all combinations of $\alpha$ and $\zeta$ within these ranges are acceptable and every value of $\alpha$ has its own appropriate range of $\zeta :\alpha$ values closer to the upper bound 0.256 are only suitable if 0.2$<\zeta <20$, while $\alpha$ values closer to the lower bound 0.2 must be accompanied by 3$<\zeta <485$.

Here we emphasize the importance of using an Andrade rheology with two free parameters when modeling planetary interiors, instead of the common practice of always setting $\zeta =1$. Depending on the used viscosity profile, $\zeta =1$ may not adequately explain the observed anelasticity of the Earth within the explored tidal frequency range.