New extended thin-sheet approximation for geodynamic applications-I. Model formulation

Abstract

Thin-sheet approximations are widely used in geodynamics because of their potential for fast computation of 3-D lithospheric deformations using simple numerical techniques. However, this simplicity imposes limits to boundary conditions, rheological settings and accuracy of results. This paper presents a new approach to reduce these restrictions. The mathematical formulation of the model involves the construction of the depth distributions of stress and velocity fields using asymptotic approximations of 3-D force balance and rheological relations. The asymptotic treatment is performed on the basis of a small geometry parameter ε (thickness to width ratio of the thin sheet) with a high accuracy while keeping terms which are capable of generating strong singularities due to possible large variations in material properties in layered systems. The depth profiles are verified by a condition of exact equilibrium in the depth-integrated force balance and by an asymptotic approach to the boundary conditions. The set of analytical depth profiles of velocities and stresses, together with the 2-D equations representing the integrated force balance, result in an extended thin-sheet approximation (ETSA). The potential of the ETSA is demonstrated by applications to problems with different types of boundary conditions and consideration of the types of systems of equations governing each case. These studies have not found any strong limitations to the boundary conditions considered and demonstrate the greater generality and higher accuracy of ETSA in comparison with the previous generation of thin-sheet approximations. The accompanying paper demonstrates the results of 2-D experiments based on ETSA.