B.S. Daya Sagar,Qiuming Cheng and Frits Agterberg: Handbook of Mathematical Geosciences: Fifty Years of IAMG

Mathematical Geosciences -     

Low-frequency layer-induced dispersion in a weak-contrast vertically heterogeneous orthorhombic medium

In this paper, we consider wave propagation in a periodically layered medium with orthorhombic symmetry. The weak-contrast approximation is utilized to derive the low-frequency dispersion in effective properties for P, S1 and S2 waves. We show that the dispersion term for all effective properties is controlled by the second-order contrasts in elastic properties from the layers. We also compute the sensitivity matrices for second- and fourth-order coefficients from eigenvalues of frequency-dependent system matrix associated with kinematic parameters for individual wave modes.     

On shear‐wave triplications in a multilayered transeversely isotropic medium with vertical symmetry axis

The presence of triplications (caustics) can be a serious problem in seismic data processing and analysis. The traveltime curve becomes multi‐valued and the geometrical spreading correction factor tends to zero due to energy focusing. We analyse the conditions for the qSV‐wave triplications in a homogeneous transversely isotropic medium with vertical symmetry axis. The proposed technique can easily be extended to the case of horizontally layered vertical symmetry axis medium. We show that the triplications of the qSV‐wave in a multilayered medium imply certain algebra. We illustrate this algebra on a two‐layer vertical symmetry axis model.     

Galactic cosmic rays—clouds effect and bifurcation model of the Earth global climate. Part 2. Comparison of theory with experiment

The solution of energy-balance model of the Earth global climate and the EPICA Dome C and Vostok experimental data of the Earth surface palaeotemperature evolution over past 420 and 740 kyr are compared.In the framework of proposed bifurcation model (i) the possible sharp warmings of the Dansgaard-Oeschger type during the last glacial period due to stochastic resonance is theoretically argued; (ii) the concept of climatic sensitivity of water in the atmosphere, whose temperature instability has the form of so-called hysteresis loop, is proposed, and based on this concept the time series of global ice volume over the past 1000 kyr, which is in good agreement with the time series of δ¹⁸O concentration in the sea sediments, is obtained; (iii) the so-called “CO₂ doubling” problem is discussed.     

Experimental investigation of the effect of metasomatism by carbonatic melt on the composition and structure of the deep mantle

The compositions of various transition-zone and lower-mantle phases and coexisting carbonatic melts were determined by exploratory melting experiments in chemically complex CO₂-bearing systems at 20–24.5 GPa and 1600–2000 °C. The melts are highly ultramafic, enriched in K, Na, Ca, Fe, and Mg, and depleted in Al and Si. Melting experiments were also carried out with the compositions on the join Mg₂SiO₄–Na₂CO₃ at 3.7 GPa and 1200–1600 °C. The solidus assemblage of MgCO₃ and Na₂MgSiO₄ melts incongruently to produce forsterite and Na-rich melt. The new results and other recent studies in CO₂-bearing systems suggest that carbonatic melt could be present, either transiently or permanently, in the whole Earth's upper mantle and at least the uppermost lower mantle. Carbonate-melt metasomatism is recognized as a process that could have a major effect on the composition and structure of the deep mantle, and thus play an important role in its evolution. Due to the unique properties of the carbonatic melt, its circulation in an otherwise static mantle could be a more efficient process than the solid-state convection for maintaining equilibrium in most of the mantle not involved directly in plate tectonics.     

The lunar crater Giordano Bruno as seen with optical roughness imagery

Images acquired by the LRO NAC allow 3D phase-ratio imagery of several areas in the crater Giordano Bruno. This is a new optical, remote-sensing technique that allows a determination of the optical roughness of the lunar surface. Our study confirms complicated impact melt movement on the flanks and the floor of the crater. In many cases, however, flow structures seen on the inner wall can be attributed to regolith/debris taluses rather than impact melt flows. It was found that the whirlpool-like formation seen near the western side of the crater Giordano Bruno has a small central depression that can be interpreted as either a vortex cavity or a feature which resembles such a vortex formed by viscous flows coming from the crater flanks. We discuss several features that confirm the young age of the crater, concluding, however, that it was not formed within human history; its age is, likely, on the order of 1 My.     

Gravity signals from the lithosphere in the Central European Basin System

We study the gravity signals from different depth levels in the lithosphere of the Central European Basin System (CEBS). The major elements of the CEBS are the Northern and Southern Permian Basins which include the Norwegian–Danish Basin (NDB), the North-German Basin (NGB) and the Polish Trough (PT). An up to 10 km thick sedimentary cover of Mesozoic–Cenozoic sediments, hides the gravity signal from below the basin and masks the heterogeneous structure of the consolidated crust, which is assumed to be composed of domains that were accreted during the Paleozoic amalgamation of Europe. We performed a three-dimensional (3D) gravity backstripping to investigate the structure of the lithosphere below the CEBS.Residual anomalies are derived by removing the effect of sediments down to the base of Permian from the observed field. In order to correct for the influence of large salt structures, lateral density variations are incorporated. These sediment-free anomalies are interpreted to reflect Moho relief and density heterogeneities in the crystalline crust and uppermost mantle. The gravity effect of the Moho relief compensates to a large extent the effect of the sediments in the CEBS and in the North Sea. Removal of the effects of large-scale crustal inhomogeneities shows a clear expression of the Variscan arc system at the southern part of the study area and the old crust of Baltica further north–east. The remaining residual anomalies (after stripping off the effects of sediments, Moho topography and large-scale crustal heterogeneities) reveal long wavelength anomalies, which are caused mainly by density variations in the upper mantle, though gravity influence from the lower crust cannot be ruled out. They indicate that the three main subbasins of the CEBS originated on different lithospheric domains. The PT originated on a thick, strong and dense lithosphere of the Baltica type. The NDB was formed on a weakened Baltica low-density lithosphere formed during the Sveco-Norwegian orogeny. The major part of the NGB is characterized by high-density lithosphere, which includes a high-velocity lower crust (relict of Baltica passive margin) overthrusted by the Avalonian terrane. The short wavelength pattern of the final residuals shows several north–west trending gravity highs between the Tornquist Zone and the Elbe Fault System. The NDB is separated by a gravity low at the Ringkøbing–Fyn high from a chain of positive anomalies in the NGB and the PT. In the NGB these anomalies correspond to the Prignitz (Rheinsberg anomaly), the Glueckstadt and Horn Graben, and they continue further west into the Central Graben, to join with the gravity high of the Central North Sea.     

Radiogenic trigger and driver for continental rifting and initial ocean spreading

Closure and opening of oceans on time‐scales of a few hundred million years is a fundamental tectonic process on Earth, typically referred to as a “Wilson cycle”. Subduction of oceanic and continental crust leading up to and during continent–continent collision can refertilize and enrich the orogenic continental lithospheric mantle in heat‐producing elements. The resulting thermal anomaly weakens the lithosphere and, along with structural weaknesses (e.g. sutures), make this orogenic lithosphere more prone to rifting given an extensional stress field. Thermal modelling shows that anomalously hot lithosphere can focus asthenospheric upwellings over time‐scales of a few hundred million years. Processes related to closure of oceans thus provide a mechanism for later localization of rifting and an extensional driving force.     

Samovar: a thermomechanical code for modeling of geodynamic processes in the lithosphere—application to basin evolution

We present a new 2D finite difference code, Samovar, for high-resolution numerical modeling of complex geodynamic processes. Examples are collision of lithospheric plates (including mountain building and subduction) and lithosphere extension (including formation of sedimentary basins, regions of extended crust, and rift zones). The code models deformation of the lithosphere with viscoelastoplastic rheology, including erosion/sedimentation processes and formation of shear zones in areas of high stresses. It also models steady-state and transient conductive and advective thermal processes including partial melting and magma transport in the lithosphere. The thermal and mechanical parts of the code are tested for a series of physical problems with analytical solutions. We apply the code to geodynamic modeling by examining numerically the processes of lithosphere extension and basin formation. The results are directly applicable to the Basin and Range province, western USA, and demonstrate the roles of crust–mantle coupling, preexisting weakness zones, and erosion rate on the evolutionary trends of extending continental regions. Modeling of basin evolution indicates a critical role of syn-rift sedimentation on the basin depth and a governing role of Peierls deformation in cold lithospheric mantle. While the former may increase basin depth by 50%, the latter limits the depth of rift basins by preventing faulting in the subcrustal lithosphere.