Modelling anisotropic viscosity

Many of Earth’s layers – from the crust to the inner core – are mechanically anisotropic. Anisotropic (i.e. direction-dependent) behaviour of rocks can derive from intrinsic properties of rock forming minerals or from microscopic or macroscopic layering of rocks and/or melts with different composition (extrinsic anisotropy). The Earth Science community often discusses the phenomena of seismic anisotropy, which results from the direction dependent propagation of seismic waves. However, materials that are characterized by elastic (seismic) anisotropy often exhibit viscous anisotropy as well, which is less explored. In geodynamics we are primarily interested in anisotropic viscosity in the crust and the mantle, where both intrinsic and extrinsic anisotropy are present. To model anisotropic viscous behaviour, we have to handle the viscosity as a 4th order tensor while also thinking about the re-orientation of anisotropy (or evolution of texture) in time. In the upper mantle the main source of anisotropy derives from the lattice preferred orientation (LPO) of olivine. Under deformation olivine grains rotate into the deformation direction (we often refer to this as texture evolution), resulting in a texture where some – or many – olivine grains are aligned with each other. Furthermore, because single olivine crystals are mechanically anisotropic – which means they deform more easily along some slip systems than others – then LPO that is developed in the upper mantle will yield anisotropic viscosity on a macroscopic scale. The foundation of our modelling approach is the Modified Director Method, which includes texture evolution and micromechanical models, both deriving from rock mechanic laboratory experiments on olivine aggregates (Hansen et al., 2016a, 2016b). The micromechanical model allows us to calculate the stress needed to achieve a certain strain rate on an aggregate, while the texture evolution model calculates the rotation of grains under a given deformation. To integrate these models into a geodynamic code, or use it to model the evolution of texture and anisotropic viscosity under specific deformation paths, we have to characterize our texture with a rank 4 viscosity or fluidity tensor (Király et al., 2020). It has been shown that the anisotropy related to olivine textures can be characterized by the Hill coefficients (Hill, 1948; Signorelli et al., 2021). Here we show that by building a large database of different textures derived from geodynamic models, we can define a linear model between simple texture parameters and the Hill coefficients with a reasonable cost. This is advantageous for integrating anisotropic viscosity into 4D geodynamic models because it allows for a direct determination of the viscosity tensor from the evolving rock texture, saving a large amount of computational time.