New extended thin-sheet approximation for geodynamic applications—II. Two-dimensional examples

The potential of the new extended thin-sheet approximation (ETSA) has been investigated by application to a representative range of 2-D problems. The system of governing equations presented by Medvedev & Podladchikov (1999) for 3-D modelling was reduced to two dimensions and tested on problems involving one- and two-layer systems of Newtonian viscous materials. The application of ETSA in each case included (1) setting boundary conditions, (2) completion of equations by evaluation of coefficients, (3) comparison of equations with governing equations of existing thin-sheet approximations, and (4) linear analysis of small perturbations and determination of their dominant wavelengths. It is shown that most previous approaches can be derived by simplification of an extended system under specified boundary conditions. Linear analyses compare well with exact analytical solutions over a wide range of wavelengths for modelling isostatic adjustment, Rayleigh-Taylor instabilities and the development of instabilities due to lateral compression and extension. These problems cannot be described by the previous generation of thin-sheet approximations. Our results suggest that the new extended thin-sheet approximation (ETSA) will be a powerful tool for the realistic modelling of complicated 2- and 3-D geodynamic structure

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

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 ɛ (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 ETS

Modelling of viscoelastic plume-lithosphere interaction using the adaptive multilevel wavelet collocation method

Modelling of mantle flows with sharp viscosity contrasts in a viscoelastic medium is a challenging computational problem in geodynamics because of its multiple-scale nature in space and time. We have employed a recently developed adaptive multilevel wavelet collocation algorithm to study the dynamics of a small rising diapir interacting with a stiff lithosphere in a Maxwell viscoelastic mantle. In this kinematic model we have prescribed the upward velocity of the diapir and then we need to integrate in time onlythe momentum equation governing the temporal evolution of the pressure, stress andvelocity components, which together constitute a sixth-order system in time. The total number of collocation points did not exceed 104, compared to more than 106 gridpoints using conventional evenly spaced grid methods. The viscosity of the diapir is10−4 times lower than that of the surrounding mantle, while the viscosity of the thinlithosphere, about 5-10 per cent of the entire layer depth, is 104-108 times stiffer than the ambient mantle. Our results demonstrate the efficacy of wavelets to capture thesharp gradients of the stress and pressure fields developed in the diapiric impingement process. The interaction of the viscoelastic lithosphere with therisingviscoelastic diapir results in the localization of stress within the lithosphere. The magnitude of the stress fields can reach around 100-300 MPa. Our simple kinematic model shows clearly that viscoelasticity canpotentially play an important role in the dynamics of thelithosphere, especially concerning the potential severage of the lithosphere by mantle upwelling

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

ISSN:1365-246XISSN:0956-540

New extended thin-sheet approximation for geodynamic applications—II. Two-dimensional examples

ISSN:1365-246XISSN:0956-540

Density variations in the thickened crust as a function of pressure, temperature, and composition

Constraints on density as a function of pressure, temperature, and composition are crucial to understand isostatic movements during geodynamic processes. Here, we provide a systematic series of density diagrams extracted from thermodynamic calculations for a variety of crustal compositions within a wide P–T range. We quantify systematic density changes in collisional settings for relevant compositional variations and attempt to simplify the density–composition relationships. Rock densities depend strongly on pressure, temperature, and composition. Densities at some selected pressure–temperature conditions increase linearly with increasing Al2O3 as well as MgO/FeO contents in pelitic rocks. Al- and Fe-rich pelites yield the highest densities, which is mostly due to the formation of garnet but also depends on other minerals and changes of reactions. The effect of loading on densities is investigated, and we show that for deep burial, a meta-pelite rich in Fe and Mg yields much larger density changes than a dry basalt and that the burial of such a rock with a composition close to typical lower crust may result in significant negative buoyancy. Metamorphism of hydrous lower crust due to pressurization and heating thus leads to densification of thickened lower crust, while heating of dry crust leads to a decrease in density. Hence, water-loaded isostatic subsidence due to metamorphism of water-saturated lower crust is substantial and increases with the thickness and depth of the reacting layer, while dry compositions show much less or only transient densification and subsidence. The density change due to thermal expansion, an extensively used concept in geodynamic models, predicts uplift under the same P–T conditions and is an order of magnitude smaller than the density variation calculated from petrologically consistent diagrams

Modelling of viscoelastic plume–lithosphere interaction using the adaptive multilevel wavelet collocation method

ISSN:1365-246XISSN:0956-540

Modeling of Compaction Driven Flow in Poro-Viscoelastic Medium Using Adaptive Wavelet Collocation Method

. Different regimes of compaction driven flow have been studied within the framework of a poro-viscoelastic medium. A single dimensionless parameter, the Deborah number De, has been identified, which enables the portrayal of the solution from the purely viscous matrix limit (De ø 1) to the poro-elastic (De AE 1) matrix limit. In viscous limit the evolution of a porosity disturbance (porosity wave) is governed by nonlinear convection-diffusion equation, while in the poro-elastic limit it evolves according to a Burgers-like non-linear advection equation. In both regimes porosity waves of higher amplitude propagate faster. However in the viscous limit porosity waves go through each other in soliton-like fashion, while in poro-elastic limit they coalesce and thus enhance melt segregation. The introduction of other variables, such as chemistry, would induce different responses in the flow for low and high De, allowing for diverse feedback situations. 3 Introduction Compaction-driven flow..