On computing viscoelastic Love numbers for general planetary models: the \texttt{ALMA${}^3$} code

TL;DR

This paper introduces ALMA^3, an open-source Fortran code that efficiently computes viscoelastic Love numbers for planetary models, including time and frequency-dependent responses, using a novel Laplace inversion method as an alternative to traditional techniques.

Contribution

It presents a new numerical scheme based on Laplace inversion for calculating Love numbers, and implements it in ALMA^3, supporting complex rheologies and dynamic planetary responses.

Findings

Validated Love numbers for Earth and other planets with realistic rheologies.

Demonstrated the effectiveness of the Laplace inversion scheme as an alternative to normal modes.

Provided time derivatives of Love numbers for geodetic signal modeling.

Abstract

The computation of the Love numbers for a spherically symmetric self-gravitating viscoelastic Earth is a classical problem in global geodynamics. Here we revisit the problem of the numerical evaluation of loading and tidal Love numbers in the static limit for an incompressible planetary body, adopting a Laplace inversion scheme based upon the Post-Widder formula as an alternative to the {traditional viscoelastic normal modes method. We also consider, whithin the same framework, complex-valued, frequency-dependent Love numbers that describe the response to a periodic forcing, which are paramount in the study of the tidal deformation of planets. Furthermore, we numerically obtain the time-derivatives of Love numbers, suitable for modeling geodetic signals in response to surface loads variations. A number of examples are shown, in which time and frequency-dependent Love numbers are evaluated for the Earth and planets adopting realistic rheological profiles. The numerical solution scheme is implemented in ALMA${}^3$ (the plAnetary Love nuMbers cAlculator, version 3), an upgraded open-source Fortran 90 program that computes the Love numbers for radially layered planetary bodies with a wide range of rheologies, including transient laws like Andrade or Burgers.