Fast analytical models for texture evolution in anisotropic polycrystals

To use observations of seismic anisotropy to constrain mantle flow patterns, we need a model for how progressive deformation of a rock leads to preferred orientation (CPO) of its constituent crystals. An important class of such models comprises so-called `self-consistent' (SC) models such as VPSC (viscoplastic SC) and SOSC (second-order SC). However, calculations based on SC models are far too costly for use in 3-D time-dependent convection simulations. To overcome this difficulty, we have developed two new analytical models that combine the accuracy of SC models with a greatly enhanced (by orders of magnitude) computational efficiency. The basis of our new models is the discovery that the crystallographic spin predicted by SC models as a function of crystal orientation is always a generalized spherical harmonic of degree 2, regardless of the CPO of the aggregate. This fact allows us to find an analytical expression for the spin to within an arbitrary amplitude, which we then determine by fitting to the predictions of the SOSC model.  Our first new model, ANPAR, uses the analytical expression for the spin to calculate evolving CPO in an aggregate comprising many (typically 2000) individual grains. The resulting CPO is visually indistinguishable from the SOSC predictions, but is ~ 50000 times faster to compute. Our second model, SBFTEX, is based on a more economical representation of CPO as a weighted sum of a small number of  analytical `structured basis functions' (SBFs), each of which represents the virtual CPO that would be produced by one intracrystalline slip system acting alone. The model consists of analytical expressions for the weighting coefficients of the SBFs as functions of the finite strain experienced by the aggregate. While somewhat less accurate than ANPAR, SBFTEX is ~ 2000 times faster, or ~ 10⁸ times faster than SOSC. We will illustrate the predictions of ANPAR and SBFTEX for pure olivine polycrystals, a simple model for the upper 400 km of the mantle.