Stochastic perturbation analysis of groundwater flow. Spatially variable soils,semi-infinite domains and large fluctuations

As is well known, a complete stochastic solution of the stochastic differential equation governing saturated groundwater flow leads to an infinite hierarchy of equations in terms of higher-order moments. Perturbation techniques are commonly used to close this hierarchy, using power-series expansions. These methods are applied by truncating the series after a finite number of terms, and products of random gradients of conductivity and head potential are neglected. Uncertainty regarding the number or terms required to yield a sufficiently accurate result is a significant drawback with the application of power series-based perturbation methods for such problems. Low-order series truncation may be incapable of representing fundamental characteristics of flow and can lead to physically unreasonable and inaccurate solutions of the stochastic flow equation. To support this argument, one-dimensional, steady-state, saturated groundwater flow is examined, for the case of a spatially distributed hydraulic conductivity field. An ordinary power-series perturbation method is used to approximate the mean head, using second-order statistics to characterize the conductivity field. Then an interactive perturbation approach is introduced, which yields improved results compared to low-order, power-series perturbation methods for situations where strong interactions exist between terms in such approximations. The interactive perturbation concept is further developed using Feynman-type diagrams and graph theory, which reduce the original stochastic flow problem to a closed set of equations for the mean and the covariance functions. Both theoretical and practical advantages of diagrammatic solutions are discussed; these include the study of bounded domains and large fluctuations.     

Relations between metamorphism and orogeny in a typical section of the Indian Himalayas

Summary A comprehensive geological and petrological investigation has been undertaken in an area of about 10 000 km² in the Indian Himalaya (S-Lahul, Himachal Pradesh). The development of mineral assemblages in metamorphic rocks of medium grade is considered to be a dominant Alpine event, although almost exclusively Paleozoic and Precambrian rocks have been involved. The Barrowian type of this metamorphism, ranging from the anchi- to the sillimanite zone, took place under the elavated T-gradient of about 4°C/100 m. It is suggested that normal geothermal conditions prevailed only in the outermost zone of this orogenic belt. In the Late Tertiary this metamorphic series has been moved as a huge nappe upon the Lower Himalaya. During this process a unique feature of reverse metamorphism has been formed. It can be shown that this feature was caused by a combination of metamorphism and very rapid tectonic movements.

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Beziehungen zwischen Metamorphose und Orogenese in einem typischen Querschnitt des Indischen Himalaya

Zusammenfassung Ein etwa 10 000 km² großes Gebiet des Indischen Himalaya (S-Lahul, Himachal Pradesh) wurde einer umfassenden geologischen und petrographischen Bearbeitung unterzogen. Die Ausbildung der in den mittelgradig metamorphen Gesteinen vorliegenden Mineralparagenesen wird als ein dominant alpines Ereignis betrachtet, obwohl fast nur paläozoische und präkambrische Gesteine einbezogen wurden. Die Metamorphose selbst ist von Barrow-Typus und reicht von der Anchi- bis zur Sillimanitzone. Sie fand unter einem erhöhten Temperaturgradienten von etwa 4°C/100 m statt. Nur in der äußersten Zone des Orogens entsprach der Gradient etwa normalen geothermischen Verhältnissen. Im Spättertiär wurde diese alpin-metamorphe Serie in Form einer mächtigen Kristallindecke dem Niedrigen Himalaya aufgeschoben. Dabei wurde eine einzigartige Zone inverser Metamorphose ausgebildet. Es konnte gezeigt werden, daß diese inverse Metamorphose durch das Zusammenwirken von Metamorphose und sehr rasch ablaufender Tektonik entstand.

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With 13 Figures

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Dedicated to Prof. Dr. Dr. h. c.B. Sander on the occasion of his 90th birthday.     

Late Quaternary glacial and climatic history of northern Cumberland Peninsula,Baffin Island,N.W.T., Canada

Radiocarbon dates on molluses in marine facies associated with glacial deposits in northern Cumberland Peninsula indicate both main fiord (Laurentide) ice and local glaciers remained at their late Wisconsin maxima until ca. 8000 BP. Essentially continuous deglaciation followed; local corrie glaciers melted out by 7100 BP and by 5500 BP fiord glaciers had receded behind the present margin of the Penny Ice Cap. The Hypsithermal warm interval probably lasted from ca. 8000 to 5000 BP. Lichenometry and radiocarbon dates on peat and buried organic horizons delimit a detailed Neoglacial chronology. Of 46 outlet and corrie glaciers investigated, the oldest Neoglacial moraines are dated lichenometrically at 3200 ± 600 BP. Subsequent advances terminated immediately prior to ca. 1650, 780, 350, and 65 yr BP, the most recent of which marked the most extensive ice coverage during the Neoglacial. The highest occurrence of lateral moraines from late Wisconsin advances of local and Laurentide ice suggest that at the late Wisconsin glacial maximum, depression of snowline varied from 450 m below present at the coast to 350 m below present level in the vicinity of the Penny Ice Cap. Moraines, surrounded by glacial ice and lying above the present steady-state ELA, suggest that during the Hypsithermal snowline was up to ca. 200 m above its present elevation. A radiometrically controlled reconstruction of relative summer paleotemperatures for the postglacial derived independently of lichenometry agrees well with the lichenometric age dating of moraines. The data suggest that between ca. 1650 and 900 BP climatic conditions were unfavorable for glacier growth, whereas the period ca. 800-65 yr BP was one of general glacial activity. During the last decade permanent snow cover has been increasing in the area. Previously reported data on climatic trends in the Canadian Arctic based on palynological analyses are similar to the chronology reported here.     

Carbon monoxide on the primitive earth

The reaction of CO + OH^? in aqueous solution to give formate was studied as a carbon monoxide sink on the primitive earth and in the present ocean. The reaction is first order in OH^? and first order in the molar CO concentration. The second order rate constant is given by log k(M^?1hr^?1) = 15.83?4886/T between 25°C and 60°C. Using the solubility of CO in sea water, and assuming a pH of 8 for a primitive ocean of the present size, the halflife of CO in the atmosphere is calculated to be 12 × 10⁶ yr at 0°C and 5.5 × 10⁴ yr at 25°C.Three other CO sinks would have been important in the primitive atmosphere: CO + H₂ → H₂CO driven by various energy sources, CO + OH → CO₂ + H, and the Fischer-Tropsch reaction of CO + H₂ → hydrocarbons, etc. It is concluded that the lifetime of a CO atmosphere would have been very short on the geological time scale although the relative importance of these four CO sinks is difficult to estimate.The CO + OH^? reaction to give formate is a very minor CO sink on the earth at the present time.     

Numerical experiments on the stability of preplanetary disks

Gravitational stability of gaseous protostellar disks is relevant to theories of planetary formation. Stable gas disks favor formation of planetesimals by the accumulation of solid material; unstable disks allow the possibility of direct condensation of gaseous protoplanets. We present the results of numerical experiments designed to test the stability of thin disks against large-scale, self-gravitational disruption. The disks are represented by a distribution of about 6 × 10⁴ point masses on a two-dimensional (r, φ) grid. The motions of the particles in the self-consistent gravity field are calculated, and the evolving density distributions are examined for instabilities. Two parameters that have major influences on stability are varied: the initial temperature of the disk (represented by an imposed velocity dispersion), and the mass of the protostar relative to that of the disk. It is found that a disk as massive as 1M_⊙, surrounding a 1M_⊙ protostar, can be stable against long-wavelength gravitational disruption if its temperature is about 300°K or greater. Stability of a cooler disk requires that it be less massive, but even at 100°K a stable disk can have an appreciable fraction ($\text{～}\text{1}\text{3}$) of a solar mass.     

A BME solution of the inverse problem for saturated groundwater flow

In most real-world hydrogeologic situations, natural heterogeneity and measurement errors introduce major sources of uncertainty in the solution of the inverse problem. The Bayesian Maximum Entropy (BME) method of modern geostatistics offers an efficient solution to the inverse problem by first assimilating various physical knowledge bases (hydrologic laws, water table elevation data, uncertain hydraulic resistivity measurements, etc.) and then producing robust estimates of the subsurface variables across space. We present specific methods for implementing the BME conceptual framework to solve an inverse problem involving Darcys law for subsurface flow. We illustrate one of these methods in the case of a synthetic one-dimensional case study concerned with the estimation of hydraulic resistivity conditioned on soft data and hydraulic head measurements. The BME framework processes the physical knowledge contained in Darcys law and generates accurate estimates of hydraulic resistivity across space. The optimal distribution of hard and soft data needed to minimize the associated estimation error at a specified sampling cost is determined. This work was supported by grants from the National Institute of Environmental Health Sciences (Grant no. 5 P42 ES05948 and P30ES10126), the National Aeronautics and Space Administration (Grant no. 60-00RFQ041), the Army Research Office (Grant no. DAAG55-98-1-0289), and the National Science Foundation under Agreement No. DMS-0112069.     

On the Dynamics of the Jovian Ionosphere and Thermosphere: III. The Modelling of Auroral Conductivity

Recent work has been concerned with calculating the three-dimensional ion concentrations and Pedersen and Hall conductivities within the auroral region of Jupiter for varying conditions of incident electron precipitation. Using the jovian ionospheric model, we present results that show the auroral ionospheric response to changing the incoming flux of precipitating electrons (for constant initial energy) and also the response to changing the initial energy (for both constant flux and constant energy flux). The results show that, for expected energy fluxes of precipitating particles, the average auroral integrated Pedersen conductivity attains values in excess of 1 mho. In addition, it is shown that electrons with an initial energy of around 60 keV are particularly effective at generating auroral conductivity: Particles of this energy penetrate most effectively to the layer of the jovian ionosphere at which the auroral conductivity is at a maximum.     

First Results from Viper: Detection of Small-scale Anisotropy at 40 GHz

Using all available major samples of Seyfert galaxies and their corresponding closely matched control samples of nonactive galaxies, we find that the bar ellipticities (or axial ratios) in Seyfert galaxies are systematically different from those in nonactive galaxies. Overall, there is a deficiency of bars with large ellipticities (i.e., "thin" or "strong" bars) in Seyfert galaxies compared to nonactive galaxies. Accompanied with a large dispersion due to small number statistics, this effect is strictly speaking at the 2 sigma level. To obtain this result, the active galaxy samples of near-infrared surface photometry were matched to those of normal galaxies in type, host galaxy ellipticity, absolute magnitude, and, to some extent, redshift. We discuss possible theoretical explanations of this phenomenon within the framework of galactic evolution, and, in particular, of radial gas redistribution in barred galaxies. Our conclusions provide further evidence that Seyfert hosts differ systematically from their nonactive counterparts on scales of a few kiloparsecs.     

A spatially and temporally adaptive solution of Richards’ equation

Efficient, robust simulation of groundwater flow in the unsaturated zone remains computationally expensive, especially for problems characterized by sharp fronts in both space and time. Standard approaches that employ uniform spatial and temporal discretizations for the numerical solution of these problems lead to inefficient and expensive simulations. In this work, we solve Richards’ equation using adaptive methods in both space and time. Spatial adaption is based upon a coarse grid solve and a gradient error indicator using a fixed-order approximation. Temporal adaption is accomplished using variable order, variable step size approximations based upon the backward difference formulas up to fifth order. Since the advantages of similar adaptive methods in time are now established, we evaluate our method by comparison with a uniform spatial discretization that is adaptive in time for four different one-dimensional test problems. The numerical results demonstrate that the proposed method provides a robust and efficient alternative to standard approaches for simulating variably saturated flow in one spatial dimension.