57 research outputs found
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
Spectral modification of seismic waves propagating through solids exhibiting a resonance frequency: a 1-D coupled wave propagation-oscillation model
A 1-D model is presented that couples the microscale oscillations of non-wetting fluid blobs in a partially saturated poroelastic medium with the macroscale wave propagation through the elastic skeleton. The fluid oscillations are caused by surface tension forces that act as the restoring forces driving the oscillations. The oscillations are described mathematically with the equation for a linear oscillator and the wave propagation is described with the 1-D elastic wave equation. Coupling is done using Hamilton's variational principle for continuous systems. The resulting linear system of two partial differential equations is solved numerically with explicit finite differences. Numerical simulations are used to analyse the effect of solids exhibiting internal oscillations, and consequently a resonance frequency, on seismic waves propagating through such media. The phase velocity dispersion relation shows a higher phase velocity in the high-frequency limit and a lower phase velocity in the low-frequency limit. At the resonance frequency a singularity in the dispersion relation occurs. Seismic waves can initiate oscillations of the fluid by transferring energy from solid to fluid at the resonance frequency. Due to this transfer, the spectral amplitude of the solid particle velocity decreases at the resonance frequency. After initiation, the oscillatory movement of the fluid continuously transfers energy at the resonance frequency back to the solid. Therefore, the spectral amplitude of the solid particle velocity is increased at the resonance frequency. Once initiated, fluid oscillations decrease in amplitude with increasing time. Consequently, the spectral peak of the solid particle velocity at the resonance frequency decreases with tim
Quantifying Diapir Ascent Velocities in Power‐Law Viscous Rock Under Far‐Field Stress: Integrating Analytical Estimates, 3D Numerical Calculations and Geodynamic Applications
Diapirism is crucial for heat and mass transfer in many geodynamic processes. Understanding diapir ascent velocity is vital for assessing its significance in various geodynamic settings. Although analytical estimates exist for ascent velocities of diapirs in power-law viscous, stress weakening fluids, they lack validation through 3D numerical calculations. Here, we improve these estimates by incorporating combined linear and power-law viscous flow and validate them using 3D numerical calculations. We focus on a weak, buoyant sphere in a stress weakening fluid subjected to far-field horizontal simple shear. The ascent velocity depends on two stress ratios: (a) the ratio of buoyancy stress to characteristic stress, controlling the transition from linear to power-law viscous flow, and (b) the ratio of regional stress associated with far-field shearing to characteristic stress. Comparing analytical estimates with numerical calculations, we find analytical estimates are accurate within a factor of two. However, discrepancies arise due to the analytical assumption that deviatoric stresses around the diapir are comparable to buoyancy stresses. Numerical results reveal significantly smaller deviatoric stresses. As deviatoric stresses govern stress-dependent, power-law viscosity, analytical estimates tend to overestimate stress weakening. We introduce a shape factor to improve accuracy. Additionally, we determine characteristic stresses for representative mantle and lower crustal flow laws and discuss practical implications in natural diapirism, such as sediment diapirs in subduction zones, magmatic plutons or exhumation of ultra-high-pressure rocks. Our study enhances understanding of diapir ascent velocities and associated stress conditions, contributing to a thorough comprehension of diapiric processes in geology.ISSN:1525-202
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
Relationship between tectonic overpressure, deviatoric stress, driving force, isostasy and gravitational potential energy
We present analytical derivations and 2-D numerical simulations that quantify magnitudes of deviatoric stress and tectonic overpressure (i.e. difference between the pressure, or mean stress, and the lithostatic pressure) by relating them to lateral variations in the gravitational potential energy (GPE). These predictions of tectonic overpressure and deviatoric stress associated with GPE differences are independent of rock rheology (e.g. viscous or elastic) and rock strength. We consider a simple situation with lowlands and mountains (plateau). We use a numerical two-layer model consisting of a crust with higher Newtonian viscosity than that in the mantle, and also a three-layer model in which the two-layer lithosphere overlies a much less viscous asthenosphere. Our results (1) explain why estimates for the magnitude of stresses in Tibet, previously published by different authors, vary by a factor of two, (2) are applied to test the validity of the thin sheet approximation, (3) show that the magnitude of the depth-integrated tectonic overpressure is equal to the magnitude of the depth-integrated deviatoric stress if depth-integrated shear stresses on vertical and horizontal planes within the lithosphere are negligible (the thin sheet approximation) and (4) show that under thin sheet approximation tectonic overpressure is required to build and support continental plateaus, such as in Tibet or in the Andes, even if the topography and the crustal root are in isostatic equilibrium. Under thin sheet approximation, the magnitude of the depth-integrated tectonic overpressure is equal to the depth-integrated horizontal deviatoric stress, and both are approximately 3.5 × 1012 N m−1 for Tibet. The horizontal driving force per unit length related to lateral GPE variations around Tibet is composed of the sum of both tectonic overpressure and deviatoric stress, and is approximately 7 × 1012 N m−1. This magnitude exceeds previously published estimates for the force per unit length required to fold the Indo-Australian Plate south of India, and hence the uplift of the Tibetan plateau could have folded the Indian Plate. We also discuss the mechanical conditions that are necessary to achieve isostasy, for which the lithostatic pressure is constant at a certain depth. The results show that tectonic overpressure can exist at a certain depth even if all deviatoric stresses are zero at this depth, because this tectonic overpressure is related to horizontal gradients of vertical shear stresses integrated across the entire depth of the lithosphere. The magnitude of the depth-integrated tectonic overpressure of 3.5 × 1012 N m−1 implies that the pressure estimated from observed mineral assemblages in crustal rocks is likely significantly different from the lithostatic pressure, and pressure recorded by crustal rocks is not directly related to depth. In case of significant weakening of the entire lithosphere by any mechanism our analytical and numerical studies provide a simple estimation of tectonic overpressure via variations in GP
Melt Migration and Chemical Differentiation by Reactive Porosity Waves
Melt transport across the ductile mantle is essential for oceanic crust formation or intraplate volcanism. However, mechanisms of melt migration and associated chemical interaction between melt and solid mantle remain unclear. Here, we present a thermo-hydro-mechanical-chemical (THMC) model for melt migration coupled to chemical differentiation. We consider melt migration by porosity waves and a chemical system of forsterite-fayalite-silica. We solve the one-dimensional (1D) THMC model numerically using the finite difference method. Variables, such as solid and melt densities or MgO and SiO2 mass concentrations, are functions of pressure (P), temperature (T), and total silica mass fraction (urn:x-wiley:15252027:media:ggge22741:ggge22741-math-0001). These variables are pre-computed with Gibbs energy minimization and their variations with evolving P, T, and urn:x-wiley:15252027:media:ggge22741:ggge22741-math-0002 are implemented in the THMC model. We consider P and T conditions relevant around the lithosphere-asthenosphere boundary. Systematic 1D simulations quantify the impact of initial distributions of porosity and urn:x-wiley:15252027:media:ggge22741:ggge22741-math-0003 on the melt velocity. Larger perturbations of urn:x-wiley:15252027:media:ggge22741:ggge22741-math-0004 cause larger melt velocities. An adiabatic or conductive geotherm cause fundamentally different vertical variations of densities and concentrations, and an adiabatic geotherm generates higher melt velocities. We quantify differences between melt transport (considering incompatible tracers), major element transport and porosity evolution. Melt transport is significant in the models. We also quantify the relative importance of four porosity variation mechanisms: (a) mechanical compaction and decompaction, (b) density variation, (c) compositional variation, and (d) solid-melt mass exchange. In the models, (de)compaction dominates the porosity variation. We further discuss preliminary results of 2D THMC simulations showing blob-like and channel-like porosity waves
Modeling of craton stability using a viscoelastic rheology
Archean cratons belong to the most remarkable features of our planet since they represent continental crust that has avoided reworking for several billions of years. Even more, it has become evident from both geophysical and petrological studies that cratons exhibit deep lithospheric keels which equally remained stable ever since the formation of the cratons in the Archean. Dating of inclusions in diamonds from kimberlite pipes gives Archean ages, suggesting that the Archean lithosphere must have been cold soon after its formation in the Archean (in order to allow for the existence of diamonds) and must have stayed in that state ever since. Yet, although strong evidence for the thermal stability of Archean cratonic lithosphere for billions of years is provided by diamond dating, the long-term thermal stability of cratonic keels was questioned on the basis of numerical modeling results. We devised a viscoelastic mantle convection model for exploring cratonic stability in the stagnant lid regime. Our modeling results indicate that within the limitations of the stagnant lid approach, the application of a sufficiently high temperature-dependent viscosity ratio can provide for thermal craton stability for billions of years. The comparison between simulations with viscous and viscoelastic rheology indicates no significant influence of elasticity on craton stability. Yet, a viscoelastic rheology provides a physical transition from viscously to elastically dominated regimes within the keel, thus rendering introduction of arbitrary viscosity cutoffs, as employed in viscous models, unnecessary
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