Fast viscous flow in the upper mantle: numerical stability in finite element models

Stress changes , such as those imposed by to earthquakes or ice mass loss, lead to viscous relaxation in the Earth’s interior. Stress relaxation is often modelled using steady-state rheological behaviours, based on linear diffusion creep or power-law stress dependence creep. However, rock mechanical experiments and microphysical models show that steady-state flow is always preceded by a transient phase, during which resistance to shear stress can be orders of magnitude lower than at eventual steady state, which leads to higher strain rates than predicted by steady-state flow laws. We are interested in the effects of transient upper-mantle deformation on surface deformation of the Earth. Deformation at the grain scale can be accommodated by different types of defects in the crystal lattice. We focus on the role of dislocations and their elastic interactions in olivine grains. We use a flow law for dislocation creep that includes the effect of dislocation interactions on strain rates and evolution of dislocation density with strain and time. This flow law is based on new experimental and theoretical work on olivine. It has two main elements: 1) dislocation interactions reduce the amount of available stress driving motion of dislocations and thus of the rate of dislocation creep; 2) evolution of dislocation density is affected by viscous creep. This model leads to transient high strain rates in environments where stress is changing and steady-state (approximately power-law) behaviour sufficiently long after a stress change. We use the finite element platform (GTECTON) to model the viscoelastic response to surface loads, such as hydrological loading or the loads of melting glaciers. The transient deformation involved may result in fast and slow deformation at different time scales, so numerical stability can become an issue. The sharp non-linearity of the flow laws plays an important role in this instability. The size of time steps in the models is a crucial factor in stability, and leads to a trade-off between accuracy and efficiency. In this study we explore implicit time marching strategies to improve the numerical stability and accuracy of the solutions. This allows us to run efficient models of solid earth deformation for problems in which loads are rapidly changing, where we aim at building a better understanding of the time dependent strength of the upper mantle.