Optimized formulas for the gravitational field of a tesseroid

Various tasks in geodesy, geophysics, and related geosciences require precise information on the impact of mass distributions on gravity field-related quantities, such as the gravitational potential and its partial derivatives. Using forward modeling based on Newton’s integral, mass distributions are generally decomposed into regular elementary bodies. In classical approaches, prisms or point mass approximations are mostly utilized. Considering the effect of the sphericity of the Earth, alternative mass modeling methods based on tesseroid bodies (spherical prisms) should be taken into account, particularly in regional and global applications. Expressions for the gravitational field of a point mass are relatively simple when formulated in Cartesian coordinates. In the case of integrating over a tesseroid volume bounded by geocentric spherical coordinates, it will be shown that it is also beneficial to represent the integral kernel in terms of Cartesian coordinates. This considerably simplifies the determination of the tesseroid’s potential derivatives in comparison with previously published methodologies that make use of integral kernels expressed in spherical coordinates. Based on this idea, optimized formulas for the gravitational potential of a homogeneous tesseroid and its derivatives up to second-order are elaborated in this paper. These new formulas do not suffer from the polar singularity of the spherical coordinate system and can, therefore, be evaluated for any position on the globe. Since integrals over tesseroid volumes cannot be solved analytically, the numerical evaluation is achieved by means of expanding the integral kernel in a Taylor series with fourth-order error in the spatial coordinates of the integration point. As the structure of the Cartesian integral kernel is substantially simplified, Taylor coefficients can be represented in a compact and computationally attractive form. Thus, the use of the optimized tesseroid formulas particularly benefits from a significant decrease in computation time by about 45 % compared to previously used algorithms. In order to show the computational efficiency and to validate the mathematical derivations, the new tesseroid formulas are applied to two realistic numerical experiments and are compared to previously published tesseroid methods and the conventional prism approach.     

Paleo- and Neoproterozoic magmatic and tectonometamorphic evolution of the Isla Cristalina de Rivera (Nico Pérez Terrane, Uruguay)

The Isla Cristalina de Rivera crystalline complex in northeastern Uruguay underwent a multistage magmatic and metamorphic evolution. Based on SHRIMP U–Pb zircon, Th–U–Pb monazite (CHIME-EPMA method) and K–Ar age data from key units several events can be recognized: (1) multistage magmatism at 2,171–2,114?Ma, recorded on zircon of the granulitic orthogneisses and their 2,093–2,077?Ma overgrowths; (2) a distinct amphibolite facies metamorphism at ~1,980?Ma, recorded by monazite; (3) greenschist facies reworking and shearing at ca. 606?Ma (monazite and K–Ar on muscovite) along the Rivera Shear Zone, and finally (4) intrusion of the post-tectonic Sobresaliente and Las Flores granites at around 585?Ma. Lithological similarities, geographic proximity and coeval magmatic and metamorphic events indicate a similar tectonometamorphic evolution for the Isla Cristalina de Rivera, the Valentines Block in Uruguay and the Santa María Chico Granulitic Complex in southern Brazil, since at least 2.1?Ga.     

The influence of recultivation technique and seed mixture on erosion stability after restoration in mountain environment

Control of erosion, and all of its after effects, from increased surface drainage and erosion to the formation of karst, is one of the essential problems when undertaking recultivation following necessary interventions in the sub-alpine and alpine vegetation stage (high zones). Average slope inclinations of 30–45% in the vicinity of ski runs, and far above in areas of natural erosion and avalanche zones, make restoration processes with sufficient erosion protection the prerequisite for success. Only a sufficient vegetation development of more than 70% ground cover stabilises the topsoil in the long term and reduces soil erosion to an acceptable degree. From 1999 to 2002, an international EU project with the participation of research groups and private firms from Austria, Italy and Germany was carried out under the direction of the Agricultural Research and Education Centre Raumberg-Gumpenstein (AREC) on five different Alpine sites at altitudes from 1,245 to 2,350 m above sea level. The aim of the work was the formulation of practice-relevant requirements for recultivation following intervention in high zones, especially following constructional measures in the vicinity of ski runs and lifts, torrent- and avalanche barriers. In a statistical comparison, the relationship between restoration techniques, seed mixtures of differing ecological value and vegetation cover was observed. The influence of application technique on erosion processes after restoration was obvious for the first two vegetation periods. Only with the additional use of mulch covers could increase surface drainage and noticeable soil loss be avoided. At high altitudes, the choice of seed mixture, irrespective of whether rapid or slow growing and independent of the extent of accompanying fertilisation, had no significance in the first two vegetation periods following sowing. In the following growing seasons, however, higher cover values were obtained with site-specific seed mixtures at three of the five experimental sites. While few species of the commercial seed mixture showed satisfactory persistency, most of the grasses and in particular the alpine leguminosae of site-specific seed mixtures increased their share during the observation period. In the long-term, sufficient protection against erosion is only guaranteed by the use of stable, enduring and ecologically adapted species.     

Oxygen isotope exchange between H2O and CO2 at elevated CO2 pressures: Implications for monitoring of geological CO2 storage

Traditionally, the application of stable isotopes in Carbon Capture and Storage (CCS) projects has focused on δ¹³C values of CO₂ to trace the migration of injected CO₂ in the subsurface. More recently the use of δ¹⁸O values of both CO₂ and reservoir fluids has been proposed as a method for quantifying in situ CO₂ reservoir saturations due to O isotope exchange between CO₂ and H₂O and subsequent changes in δ¹⁸O_H2O values in the presence of high concentrations of CO₂. To verify that O isotope exchange between CO₂ and H₂O reaches equilibrium within days, and that δ¹⁸O_H2O values indeed change predictably due to the presence of CO₂, a laboratory study was conducted during which the isotope composition of H₂O, CO₂, and dissolved inorganic C (DIC) was determined at representative reservoir conditions (50 °C and up to 19 MPa) and varying CO₂ pressures. Conditions typical for the Pembina Cardium CO₂ Monitoring Pilot in Alberta (Canada) were chosen for the experiments. Results obtained showed that δ¹⁸O values of CO₂ were on average 36.4 ± 2.2‰ (1σ, n = 15) higher than those of water at all pressures up to and including reservoir pressure (19 MPa), in excellent agreement with the theoretically predicted isotope enrichment factor of 35.5‰ for the experimental temperatures of 50 °C. By using ¹⁸O enriched water for the experiments it was demonstrated that changes in the δ¹⁸O values of water were predictably related to the fraction of O in the system sourced from CO₂ in excellent agreement with theoretical predictions. Since the fraction of O sourced from CO₂ is related to the total volumetric saturation of CO₂ and water as a fraction of the total volume of the system, it is concluded that changes in δ¹⁸O values of reservoir fluids can be used to calculate reservoir saturations of CO₂ in CCS settings given that the δ¹⁸O values of CO₂ and water are sufficiently distinct.     

Monazite CHIME/EPMA dating of Erinpura granitoid deformation: Implications for Neoproterozoic tectono-thermal evolution of NW India

The in-situ “chemical” Th–U–Pb dating of monazite with the electron microprobe is used to unravel the Neoproterozoic tectono-thermal history of the “Erinpura Granite” terrane in the foreland of the Delhi Fold Belt (DFB) in the NW Indian craton. These granitoids are variably deformed and show effects of shearing activity. Monazites from the Erinpura granitoids recorded two main events; (1) protolith crystallization at 863 ± 23 Ma and (2) recrystallization and formation of new Th-poor monazite at 775 ± 26 Ma during shear overprint. Some components of the Erinpura granitoids, such as the Siyawa Granite and granites exposed near Sirohi town, show evidence of migmatization. This migmatization event is documented by anatexis and associated monazite crystallization at 779 ± 16 Ma. The age data indicate an overlap in timing between anatectic event and ductile shear deformation. The end of the tectono-thermal event in the Sirohi area is constrained by a 736 ± 6 Ma Ar–Ar muscovite age data from the ductile shear zone.     

Discovery of Ordovician–Silurian metamorphic monazite in garnet metapelites of the Alpine External Aiguilles Rouges Massif

The pre-Mesozoic, mainly Variscan metamorphic basement of the Col de Bérard area (Aiguilles Rouges Massif, External domain) consists of paragneisses and micaschists together with various orthogneisses and metabasites. Monazite in metapelites was analysed by the electron microprobe (EMPA-CHIME) age dating method. The monazites in garnet micaschists are dominantly of Variscan age (330–300 Ma). Garnet in these rocks displays well developed growth zonations in Fe–Mg–Ca–Mn and crystallized at maximal temperatures of 670°C/7 kbar to the west and 600°C/7–8 kbar to the east. In consequence the monazite is interpreted to date a slightly pressure-dominated Variscan amphibolite-facies evolution. In mylonitic garnet gneisses, large metamorphic monazite grains of Ordovician–Silurian (~440 Ma) age but also small monazite grains of Variscan (~300 Ma) age were discovered. Garnets in the mylonitic garnet gneisses display high-temperature homogenized Mg-rich profiles in their cores and crystallized near to ~800°C/6 kbar. The Ordovician–Silurian-age monazites can be assigned to a pre-Variscan high-temperature event recorded by the homogenised garnets. These monazite age data confirm Ordovician–Silurian and Devonian–Carboniferous metamorphic cycles which were already reported from other Alpine domains and further regions in the internal Variscides.     

The sinkhole enigma in the Alpine Foreland, Southeast Germany: Evidence of impact-induced rock liquefaction processes

Sudden collapse of the Quaternary soil to form sinkholes on the order of meters and tens of meters has been a geologic phenomenon within living memory in a localized area north of Lake Chiemsee in Southeast Germany. Failing a satisfying explanation, a relation with an undefined glaciation process has always been proposed. Excavations and geophysical measurements at three newly affected sites show underground features such as prominent sandy-gravelly intrusions and extrusions typical of rock liquefaction processes well known to occur during strong earthquakes. Since strong earthquakes can reasonably be excluded to have affected the area under discussion, it has been suggested that the observed widespread liquefaction is related with the recently proposed Holocene Chiemgau meteorite impact event. Except for one earlier proposed but unassertive relation between impact and liquefaction, the obviously direct association of both processes in the Chiemgau area emphasizes that observed paleoliquefaction features need not necessarily have originated solely from paleoseismicity but can provide a recognizable regional impact signature.     

Lateral fold growth and linkage in the Zagros fold and thrust belt (Kurdistan,NE Iraq)

The Zagros fold and thrust belt is a seismically active orogen that has accommodated the N–S shortening between the Arabian and Eurasian plates since the Miocene. Whereas the southeast parts of the belt have been studied in detail, the northwest extent has received considerably less attention, being part of the Republic of Iraq. In this study, we investigate fold growth in the area NE of Erbil (Kurdistan, Iraq). In particular, we focus on the interaction of the transient development of drainage patterns along growing antiforms, as this directly reflects the kinematics of progressive fold growth. Detailed geomorphological studies of the Bana Bawi‐, Permam‐ and Safeen‐fold trains show that these anticlines did not develop from a single embryonic fold but by lateral linkage of several different fold segments. These segments, with length between 5 and 25 km, have been detected by mapping ancient and modern river courses; these initially cut the nose of growing folds until eventually defeated, leaving curved wind gaps behind. Depending on the alignment of the initial embryonic folds, the segments can either record a linear‐ or an en‐echelon linkage. Comparison of natural examples from the Zagros fold and thrust belt in Iraq with published numerically modelled fold growth suggests that both linear‐linkage and en‐echelon linkage are mechanically feasible and are common processes during progressive shortening and fold growth.     

Mesozoic (Late Jurassic) palynomorphs and charcoal from a sandstone pebble from Quaternary glacial deposits of the North Sea

The palynological contents as well as macroscopic charcoal fragments from a calcareous sandstone pebble from Quaternary glacial deposits of the North Sea (80 km SW of the Dogger Bank) are described. The taxonomic composition of the palynoflora points to a Late Jurassic (or Late Jurassic/Early Cretaceous) age of the sandstone. The charcoal is interpreted as direct evidence of palaeo‐wildfire occurring during this period in the source area of the sediments of this particular sandstone. This finding, together with already published data on Mesozoic deposits, allows to conclude that wildfires were obviously widespread during this period in the area of the modern day North Sea and adjacent areas. Copyright © 2012 John Wiley & Sons, Ltd.     

Trophic interactions of the pelagic ciliate Stentor spp. in North Patagonian lakes

The zooplankton of oligotrophic lakes in North Patagonia is often dominated by mixotrophic ciliates, particularly Stentor amethystinus and Stentor araucanus. Therefore, we tested whether Stentor spp. (i) is an important food for juvenile endemic (Cheirodon australe, Galaxias maculatus, Odontesthes mauleanum, Percichthys trucha) and introduced (Oncorhynchus mykiss) fish species, and (ii) represents a remarkable grazer of bacteria. Ingestion rates of fish estimated by disappearance of Stentor in feeding experiments ranged between 8 (G. maculatus) and 53 (C. australe) ciliates per fish and day, and assimilation rates measured by using radioactively labelled Stentor ranged between 3 (P. trucha) and 52 (C. australe) ciliates per fish and day. However, although we detected the consumption of Stentor by fish, the daily consumption amounted to at most 0.2% of the fish biomass which can not cover the energy requirement of the fish. Furthermore, the daily consumption was equivalent to a maximum of 1.6% of the Stentor standing stock so that fish predation does not seem to be an important mortality factor for the ciliates. The clearance rate of Stentor sp. on natural bacteria was on average 3.8 μl cil⁻¹ h⁻¹. The daily ingestion (mean 3.9 ng C cil⁻¹ d⁻¹) was about 3.5% of the individual biomass of Stentor sp. Therefore, bacteria ingestion might explain a ciliate growth rate of appr. 1% d⁻¹, which was about 17% of the photosynthesis of endosymbiotic algae. The maximum density of Stentor sp. in the lake could ingest about 1 μg C L⁻¹ d⁻¹ bacteria which is only 3% of average bacterial production. Thus, grazing by Stentor sp. does not seem to be a main loss factor for the bacteria.