A new numerical method to calculate inhomogeneous and time-dependent large deformation of two-dimensional geodynamic flows with application to diapirism

Abstract

SUMMARY A key to understand many geodynamic processes is studying the associated large deformation fields. Finite deformation can be measured in the field by using geological strain markers giving the logarithmic strain f = log 10 (R), where R is the ellipticity of the strain ellipse. It has been challenging to accurately quantify finite deformation of geodynamic models for inhomogeneous and time-dependent large deformation cases. We present a new formulation invoking a 2-D marker-in-cell approach. Mathematically, one can describe finite deformation by a coordinate transformation to a Lagrangian reference frame. For a known velocity field the deformation gradient tensor, F, can be calculated by integrating the differential equation DtFij = LikFkj, where L is the velocity gradient tensor and Dt the Lagrangian derivative. The tensor F contains all information about the minor and major semi-half axes and orientation of the strain ellipse and the rotation. To integrate the equation centrally in time and space along a particle’s path, we use the numerical 2-D finite difference code FDCON in combination with a marker-in-cell approach. For a sufficiently high marker density we can accurately calculate F for any 2-D inhomogeneous and time-dependent creeping flow at any pointforadeformationfupto4.Comparisonbetweentheanalyticalandnumericalsolutionfor the finite deformation within a Poiseuille‐Couette flow shows an error of less than 2percent for a deformation up to f = 1.7. Moreover, we determine the finite deformation and strain partitioning within Rayleigh‐Taylor instabilities (RTIs) of different viscosity and layer thickness ratios. These models provide a finite strain complement to the RTI benchmark of van Keken et al. Large finite deformation of up to f = 4 accumulates in RTIs within the stem and near the compositional boundaries. Distinction between different stages of diapirism shows a strong correlation between a maximum occurring deformation of f = 1, 3 and 4, and the early, intermediate and late stages of diapirism, respectively. Furthermore, we find that the overall strain of a RTI is concentrated in the less viscous regions. Thus, spatial distributions and magnitudes of finite deformation may be used to identify stages and viscosity ratios of natural cases.