Viscoelastic Love numbers and long-period geophysical effects

Abstract

Summary Long term deformations strongly depend on the Earth model and its rheological parameters, and in particular its viscosity. We give the general theory and the numerical scheme to compute them for any spherically non rotating isotropic Earth model with linear rheology, either elastic or viscoelastic. Although the Laplace transform is classically used to compute viscoelastic deformation, we choose here instead, to implement the integration with the Fourier transform in order to take advantage of the Fast Fourier Transform algorithm and avoid some of the Laplace transform mathematical difficulties. We describe the methodology to calculate deformations induced by several geophysical signals regardless of whether they are periodic or not, especially by choosing an adapted time sampling for the Fourier transform. As examples, we investigate the sensitivity of the displacements due to long period solid Earth tides, Glacial Isostatic Adjustment (GIA), and present-day ice melting, to anelastic parameters of the mantle. We find that the effects of anelasticity are important for long period deformation and relatively low values of viscosities for both Maxwell and Burgers models. We show that slight modifications in the rheological models could significantly change the amplitude of deformation but also affect the spatial and temporal pattern of the signal to a lesser extent. Especially, we highlight the importance of the mantle anelasticity in the low degrees deformation due to present-day ice melting and encourage its inclusion in future models.

On computing viscoelastic Love numbers for general planetary models: the ALMA3 code

“…As shown by Spada & Boschi (2006) and Spada (2008), a possible way to circumvent these difficulties is to compute the inverse Laplace transform through the Post-Widder (PW) formula (Post, 1930;Widder, 1934). We note, however, that other viable possibilities exist, as the one recently discussed by Michel & Boy (2021), who have employed Fourier techniques to avoid some of the problems inherent in the Laplace transform method. While Fourier techniques may be more appropriate to take complex rheologies into account, and are clearly more relevant to address Love numbers at tidal frequencies, the motivation of our approach is to address in a unified framework the computation of LNs describing both tidal and surface loads.…”

“…Then, we test numerical results from ALMA 3 by reproducing the viscoelastic LNs for an incompressible Earth model computed within the benchmark exercise by Spada et al (2011). Finally, we discuss the impact of the incompressibility approximation assumed in ALMA 3 by comparing elastic and viscoeastic LNs for a realistic Earth model with recent numerical results by Michel & Boy (2021), which employ a compressible model.…”

“…Here we focus on numerical results recently obtained by Michel & Boy (2021), who employed Fourier techniques to compute frequency-dependent viscoelastic LNs for periodic forcings both of loading and tidal types. They have adopted an Earth model with the elastic structure of PREM (Preliminary Reference Earth Model, Dziewonski & Anderson, 1981) and a fully liquid core, and replaced the outer oceanic layer with a solid crust layer, adjusting crustal density in such a way to keep the total Earth mass unchanged.…”

“…They have adopted an Earth model with the elastic structure of PREM (Preliminary Reference Earth Model, Dziewonski & Anderson, 1981) and a fully liquid core, and replaced the outer oceanic layer with a solid crust layer, adjusting crustal density in such a way to keep the total Earth mass unchanged. Following Michel & Boy (2021), we have built a discretised realization of PREM suitable for ALMA 3 with a fluid core and 28 homogeneous mantle layers, which has been used to obtain the numerical results discussed below.…”

“…Figure 5 compares elastic Love numbers obtained by Michel & Boy (2021) in the range of F5 harmonic degrees between n = 2 and n = 10, 000 with those computed with ALMA 3 . The largest difference between the two sets of LNs can be seen for h n in the loading case (Figure 5a), where the assumption of incompressibility leads to a significant underestimation of deformation across the whole range of harmonic degrees.…”