Time-domain approach to linearized rotational response of a three-dimensional viscoelastic earth model induced by glacial-isostatic adjustment: I. Inertia-tensor perturbations

Abstract

SUMMARY For a spherically symmetric viscoelastic earth model, the movement of the rotation vector due to surface and internal mass redistribution during the Pleistocene glaciation cycle has conventionally been computed in the Laplace‐transform domain. The method involves multiplication of the Laplace transforms of the second‐degree surface‐load and tidal‐load Love numbers with the time evolution of the surface load followed by inverse Laplace transformation into the time domain. The recently developed spectral finite‐element method solves the field equations governing glacial‐isostatic adjustment (GIA) directly in the time domain and, thus, eliminates the need of applying the Laplace‐domain method. The new method offers the possibility to model the GIA‐induced rotational response of the Earth by time integration of the linearized Liouville equation. The theory presented here derives the temporal perturbation of the inertia tensor, required to be specified in the Liouville equation, from time variations of the second‐degree gravitational‐potential coefficients by the MacCullagh's formulae. This extends the conventional approach based on the second‐degree load Love numbers to general 3‐D viscoelastic earth models. The verification of the theory of the GIA‐induced rotational response of the Earth is performed by using two alternative approaches of computing the perturbation of the inertia tensor: a direct numerical integration and the Laplace‐domain method. The time‐domain solution of both the GIA and the induced rotational response of the Earth is readily combined with a time‐domain solution of the sea level equation with a time‐varying shoreline geometry. In a follow‐up paper, we derive the theory for the case when GIA‐induced perturbations in the centrifugal force affect not only the distribution of sea water, but also deformations and gravitational‐potential perturbations of the Earth.

The rotational feedback on linear-momentum balance in glacial isostatic adjustment

“…In terms of the zeroth-and second-degree spherical harmonics, this expression, correct to a first order in m i , is (e.g. Martinec & Hagedoorn 2005, eqs 94 and 95)…”

“…The Chandler wobbling of the rotation vector can be removed by moving-average filtering of m(t) over the Chandler-wobble period [as conventional, we will use a complex notation and define m(t) = m 1 (t) + i m 2 (t)]. As a result, long-term motion of the rotation vector is expressed as (Wu & Peltier 1984;Vermeersen & Sabadini 1996;Martinec & Hagedoorn 2005) …”

“…The six conditions on vanishing a rigid-body translation and rotation are derived in (Martinec & Hagedoorn 2005, section 2.4). These conditions ensure that the position of the centre of mass is fixed during GIA process and the relative angular-momentum vector h(t) vanishes.…”

“…The rotational response of the Earth to the surface load is described by the instantaneous inertia tensor C R (t) of the configuration κ(t). Correct to first order in u , the tensor C R (t), as viewed in the rotating frame O(x 1 , x 2 , x 3 ), can be linearized as (Martinec & Hagedoorn 2005, eqs 13 and 14)…”

“…Eqs (41) are derived by applying the incremental field theory (e.g. Wu & Peltier 1982;Wolf 1991), which allows us to adopt the assumption of a spherical approximation (Martinec & Hagedoorn 2005, section 4.2). This means that (i) the Earth is represented by a sphere B with the unit normal to the external surface ∂V and an internal discontinuity coinciding with the unit radial vector e r , n = e r , and (ii) the unperturbed mass density 0 is radially dependent only, that is 0 = 0 (r).…”