Spectral, initial value approach for viscoelastic relaxation of a spherical earth with a three-dimensional viscosity - I. Theory

We present a spectral, initial value approach to the forward modelling of the viscoelastic response of a spherical earth with a 3-D viscosity structure. It represents an alternative to a variety of numerical methods for 2-D and 3-D postglacial rebound modelling used recently (the finite element method, the perturbation method and the semi-analytical approach). We employ surface spherical harmonics up to second rank to parametrize the spatial dependence of material and field quantities. The spectral parametrization converts the balance momentum equation, Poisson's equation and the constitutive equation to a system of eight simultaneous ordinary differential equations in radial variables that may be solved by a method of numerical integration. In contrast with the spherically symmetric problem, the presence of lateral viscosity variations causes the spectral equations to be coupled and the system cannot be solved separately for individual angular degree and order. The time dependence of the problem is treated directly in a time domain (not in the Laplace domain) as a time evolution problem. Approximating time derivatives by forward Euler differencing leads to an explicit time differencing scheme with time splitting of the div operator in the balance momentum equation. The central point of this paper is to present the theory as transparently as possible. We hope to report on numerical results soon.