Introduction to numerical methods for geologists, mathematicians and physicists

Textbook on applied numerical methods: plate tectonics, subduction, mantle convection and planetary models

Introduction to Applied Numerical Methods February 3, 2010

Filed under: numerical methods — tgerya @ 11:04 pm

introduction to numerical methods for geologistsThis textbook provides user-friendly introduction to applied numerical methods  for graduate students and researchers working in the fields of geology, mathematics and physics.

Modelling of geological processes numerically was for a long time considered as a complex research domain of high-level mathematicians and geophysicists with strong experience in numerical methods. Now, for the first time, students and researchers in geology and natural sciences can learn this subject from a single, accessible textbook. It only requires minimal mathematical background (derivatives and simple linear algebra).

The first part of this textbook gradually introduces relevant partial differential equations of continuum mechanics and explains in details numerical methods applied for their solution. The second part contains reader-friendly introduction into state-of-the-art visco-elasto-plastic thermo-mechanical geodynamic modelling.

This textbook does not only present numerical methods but also provides insight into the modelling of  major geological processes. Numerical examples include models of subduction, lithospheric extension, continental collision, slab break-off, intrusion emplacement, mantle convection and planetary core formation. 47 exercises and 67 MATLAB programs allow readers to test their understanding and experiment with numerical methods and models.

numerical methods introduction preview::::::: matlab codes for numerical methods


Venus vs. Mars: what planet is more geologically exciting? March 15, 2014

Filed under: Uncategorized — tgerya @ 3:21 pm

Mars is certainly more accessible planet than Venus and its surface is studied in much more details. However, surface of Venus seems to be much more exciting geologically and expose many large-scale features of enigmatic origin, such as coronae and novae

Venus coronae are 513 approximately circular structures that range in diameter from 60 km to 2000  km. Coronae typically have a raised rim superimposed on which is an annulus of closely spaced concentric fractures and/or ridges. Venus novae are ‘‘radially fractured centers’’ 100–300 km in diameter, 64 have been identified. In contrast to volcanoes, novae show a dominance of tectonic activity (e.g., surface fracturing) over volcanic activity. Origin of both novae and coronae is often explained by an interaction of the Venus lithosphere with mantle plumes, however details of this process remain controversial.

New theory supported by 3D themomechanical numerical models suggests that both novae and coronae can result from magma-assisted crustal convection, induced by decompression melting of the hot rising mantle plumes. During the initial stage, nova forms by stellate fracturing of a topographic rise forming atop the growing crustal convection cell. Few million years later, nova can convert to coronae by inward dipping concentric fracturing of the nova rise margins and subsequent outward thrusting of partially molten crustal rocks over the Venus surface.



Porous water enables subduction initiation November 4, 2013

Most of the presently active intra-oceanic subduction zones are relatively young. However, subduction initiation process and controlling factors remain poorly understood. 

Previous models of subduction initiation assumed excessive weakening of tectonic plate boundaries which does not reconcile with laboratory rock strength measurement. New 2D numerical hydro-thermo-mechanical (HTM) model of spontaneous intra-oceanic subduction initiation has been developed, where solid rock deformation and porous water percolation are fully coupled. It is demonstrated that although subduction fails to initiate under fluid-absent conditions, it can naturally start when porous water percolates inside oceanic crust and along the plate boundaries.



3D numerical model of mid-ocean ridge patterns variability May 27, 2013

Filed under: plate tectonics pictures,plate tectonics theory — tgerya @ 6:12 pm

The morphology of natural mid-ocean ridges changes significantly with the rate of extension. Full spreading rate on Earth varies over more than one order of magnitude, ranging from less than 10 mm/yr at the Gakkel Ridge in the Arctic Ocean to 170 mm/yr at the East Pacific Rise.

Recent paper reproduces and investigates the spreading patterns as they vary with spreading rate using 3-D thermomechanical numerical models. With increasing spreading rate, four different regimes are obtained: (a) stable alternating magmatic and amagmatic sections (≈ 10 mm/yr), (b) transient features in asymmetrically spreading systems (≈ 20 mm/yr), (c) stable orthogonal ridge-transform fault patterns (≈ 40 mm/yr) and (d) stable curved ridges (≥ 60 mm/yr). Modeled ultraslow and slow mid-ocean ridges share key features with natural systems. Abyssal hills and oceanic core complexes are the dominant features on the flanks of natural slow-spreading ridges. Numerically, very similar features are produced, both generated by localized asymmetric plate growth controlled by a spontaneous development of large-offset normal faults (detachment faults).



First high-resolution numerical model of compressional thrust wedges February 20, 2013

Plastic thrust wedges such as submarine accretionary wedge systems or compressional thin-skinned fold-and-thrust belts  have been intensely studied over  several decades, in particular since the application of the critical taper theory. 3D models of these wedges were exclusively restricted to analogue modeling approaches.

Recent paper of Ruh et al. (2013) presents first 3D, high-resolution, fully staggered finite difference grid, marker in cell model for thin-skinned fold-and-thrust belts with a visco-brittle/plastic rheology. This model was used to understand the influence of pre-existing backstop offsets on the structural evolution of accretionary wedges and the role of basal frictional strength within these systems.




FD vs FE for Adaptive Mesh Refinement (AMR) for quad-tree meshes February 5, 2013

Filed under: numerical methods — tgerya @ 6:16 pm

In geodynamics, finite elements are believed to be the only suitable choice when adaptive mesh refinement is applied under conditions of large and sharp viscosity variations. New adaptive staggered grid (ASG) formulation challenges this common opinion by demonstrating simplicity and robustness of conservative finite differences constructed on block structured quad-tree meshes.

In the recent paper new staggered grid formulation is described for discretizing incompressible Stokes flow which has been specifically designed for use on adaptive quad-tree type meshes. The key to this new adaptive staggered grid (ASG) stencil is in the form of the stress conservative finite difference constraints which are enforced at the “hanging” velocity nodes between resolution transitions within the mesh. The new computationally inexpensive ASG stencil is (i) stable and does not produce spurious pressure oscillations across regions of grid refinement which intersect discontinuous viscosity structures and (ii) possesses the same order of accuracy as the classical non-adaptive staggered grid discretization.



Modeling of earthquakes with geodynamic codes December 13, 2012

Filed under: numerical methods,subduction zone — tgerya @ 4:32 pm

At present, modeling of earthquakes is mainly performed with specialized seismic rupture codes in which fault surface is prescribed. In contrast, in long-term geodynamic continuum models faults form and evolve spontaneously and self consistently in response to acting stresses and deformations. Can we use geodynamic codes for earthquake modeling?

Based on geodynamic code I2ELVIS, van Dinther and co-workers validate a continuum, visco-elasto-plastic numerical model of spontaneous rupture with a new laboratory approach. The analogous laboratory setup includes a visco-elastic gelatin wedge underthrusted by a rigid plate with defined velocity-weakening and -strengthening regions. Geodynamic simulation approach includes velocity-weakening friction to spontaneously generate a series of fast frictional instabilities that correspond to analog earthquakes. A good match between numerical and laboratory source parameters is obtained. The wide range of observed physical phenomena, including back-propagation and repeated slip, and the agreement with laboratory results demonstrate that visco-elasto-plastic geodynamic models with rate-dependent friction form a new tool that can greatly contribute to our understanding of the seismic cycle at subduction zones.



Precambrian geodynamics: long-standing enigma

The special challenge of Precambrian geodynamics is that presently there is no approved paradigm of global geodynamics and lithosphere tectonics for the early Earth, such as modern-day plate tectonics.

Recent review of Precambrian geodynamics confronts existing concepts to numerical and analogue models and is focused on three critical directions: (1) subduction and plate tectonics, (2) collision and orogeny, (3) cratons formation and stability. It is concluded that future progress requires cross-disciplinary efforts with a special emphasis placed upon quantitative testing of existing geodynamic concepts and extrapolating back in geological time, using both global and regional numerical thermomechanical models validated for modern Earth conditions.



Transition from rifting to spreading (Transform faults revisited) November 4, 2012

The characteristic pattern of mid-ocean ridges, sectioned by transform faults stands as an inherent feature of terrestrial plate tectonics. A fundamental unresolved problem is how this pattern formed and why it is maintained. One common view is that oceanic transform faults are typically inherited from the continental plate breakup.

New high-resolution 3D thermomechanical numerical models of the incipient oceanic spreading investigate nucleation and long-term evolution of ridge-transform spreading patterns. Ridge-transform oceanic spreading patterns form gradually and become fully established several million years after the plate breakup. It is demonstrated on the basis of simple analyses that the ridge-transform system is a long-term plate growth pattern that is generally different from an initial plate rifting pattern. Geometry of the ridge-transform system is governed by geometrical requirements (180o rotational symmetry for open space occupation) for simultaneous accretion and displacement of new plate material within two offset spreading centers connected by a sustaining rheologically weak transform fault. According to these requirements, the characteristic spreading-parallel orientation of oceanic transform faults is the only thermomechanically consistent steady state orientation. Results of numerical experiments compare well with both incipient and mature ridge-transform systems observed in nature.


Eduction = Anti-Subduction July 14, 2012

Plate eduction is a geodynamic process, which is opposite to subduction. It is characterized by normal-sense coherent motion of previously subducted continental plate. Eduction may occur during continental collision after slab detachment (slab breakoff) has separated the negatively buoyant oceanic plate from the positively buoyant orogenic root.

Eduction was hypothesized based on natural observations but has not been modeled. Duretz et al. (2012) presented first systematic thermomechanical numerical modeling study of eduction. The results show that this process can lead to the rapid decompression of the subducted continental crust and a topographic uplift associated with extension of the orogen. Eduction follows slab detachment (slab breakoff) and can occur in combination to other geodynamic processes (e.g. slab retreat) and may play a signicant role in the geodynamic evolution of collision zones and exhumation of ultrahigh-pressure (UHP) rock complexes.


Delamination: when, why and how does it happen?

Delamination is a process of separation of the high-density lithospheric continental mantle from the low-density crust. It is often proposed for the evolution of continental collision zones but is rarely modeled.

Ueda et al. (2012) investigated regimes of mantle delamination in collision zones based on systematic 2D  numerical thermomechanical modeling. Distinct modes of this process are identified, in which orogens undergo mantle delamination concurrently or after collision.  Delamination propagates  along  the Moho of the subducted plate together with the retreating trench.  Topography migrates with the focused and localized separation between crust and lithospheric mantle (delamination front). Delamination can last for hundreds of Myr been associated with repetitive slab  break-offs  (lithospheric dripping offs) producing long-living hot accretionary orogen underplated by convecting asthenospheric mantle.