Love numbers, the “sea-level equation” and the glacial isostatic adjustment problem

**Giorgio Spada, ***Dip. di Fisica e Astronomia (DIFA), Settore Geofisica, Università di Bologna, Bologna, Italia.*

*Web Lecture on March 18, 2021 at 3:00 PM (Paris Time)*

*Figure caption :** Solution of the Sea Level Equation, showing the present-day rate of relative sea-level change in response to the late-Pleistocene deglaciation, obtained using the open-source code SELEN4 (Spada & Melini, 2019). The map reflects the loading Love numbers sensitivity to the rheological structure of the Earth, gravitational and rotational effects, and the history of melting. *

Since their introduction by AEH Love (1909), the “Love numbers” became a fundamental tool in the context of various problems in global geodynamics, involving the Earth’s response to various types of forcing (surface loading, tidal or seismic excitation). From a physical standpoint, the Love numbers represent suitably normalized displacements and gravitational potential variations induced by localized sources (e.g., surface loads), and corresponding to a given spatial wavelength. Their application is limited, however, to problems in which it is assumed that the Earth has a “one-dimensional” structure, which means that all the physical parameters (density, elastic constants) are only dependent upon the planetary radius, and the rheology (the stress-strain relationship) obeys a linear law. In this particularly important case, a source characterized by a given wavelength shall only induce deformations and gravity changes of the same wavelength.

A meaningful, fully analytical expression for the Love numbers can only be obtained for the so-called elastic Kelvin sphere, i.e, the simplest Earth model, characterized by a uniform constant density and a constant rigidity. Such analytical solution can be easily generalized to the case of the viscoelastic Kelvin sphere, characterized by either transient or steady-state rheological laws. For two-layer Earth models composed by a uniform mantle and a core, the algebraic complexity explodes and meaningful Love numbers expressions, showing analytically the dependence upon the model parameters, are not possible anymore. For this reason, starting since the seventies, a semi-analytical method (called “Viscoelastic Normal Modes” method, and introduced by Peltier, 1974) has been designed, in order to compute the Love numbers for realistic elastic or viscoelastic planetary models, in a range of harmonic degrees suitable for the solution of global or regional geodynamical problems.

In the first part of this talk, I shall focus on the meaning and on the importance of the Love numbers in geodynamics and on the problem of their evaluation, especially in the case of complex planetary structures. Attention shall be devoted to the traditional theory of the Viscoelastic Normal Modes but also to new approaches in which some of the limitations of the classical approach has been surpassed (Spada, 2008). Furthermore, in the second part, I shall discuss a very important application of the Love numbers technique, in the framework of the so-called “Sea Level Equation” which represents the fundamental non-linear integral equation that governs the Glacial Isostatic Adjustment (GIA) process (Farrell & Clark, 1976). In this context, Love numbers are exploited due to their sensitivity to the rheological profile of the mantle, which is inferred by comparing modelled with observed histories of relative sea-level change throughout a wide range of time scales (Spada & Melini, 2019).

References :

Farrell, W. E., & Clark, J. A. (1976). On postglacial sea level. Geophysical Journal International, 46(3), 647-667.

Love, A. E. H. (1909). The yielding of the Earth to disturbing forces. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 82(551), 73-88.

Peltier, W. R. (1974). The impulse response of a Maxwell Earth. Reviews of Geophysics, 12(4), 649-669.

Spada, G. (2008). ALMA, a Fortran program for computing the viscoelastic Love numbers of a spherically symmetric planet. Computers & Geosciences, 34(6), 667-687.

Spada, G., & Melini, D. (2019). SELEN 4 (SELEN version 4.0): a Fortran program for solving the gravitationally and topographically self-consistent sea-level equation in glacial isostatic adjustment modeling. Geoscientific Model Development, 12(12), 5,055-5,075.