A mathematical study of finite-amplitude rock-folding

The problem of the finite-amplitude folding of an isolated, linearly viscous layer under compression and imbedded in a medium of lower viscosity is treated theoretically by using a variational method to derive finite difference equations which are solved on a digital computer. The problem depends on a single physical parameter, the ratio of the fold wavelength, L, to the "dominant wavelength" of the infinitesimal-amplitude treatment, L_d. Therefore, the natural range of physical parameters is covered by the computation of three folds, with L/L_d = 0, 1, and 4.6, up to a maximum dip of 90°.

Significant differences in fold shape are found among the three folds; folds with higher L/L_d have sharper crests. Folds with L/L_d = 0 and L/L_d = 1 become fan folds at high amplitude. A description of the shape in terms of a harmonic analysis of inclination as a function of arc length shows this systematic variation with L/L_d and is relatively insensitive to the initial shape of the layer. This method of shape description is proposed as a convenient way of measuring the shape of natural folds.

The infinitesimal-amplitude treatment does not predict fold-shape development satisfactorily beyond a limb-dip of 5°. A proposed extension of the treatment continues the wavelength-selection mechanism of the infinitesimal treatment up to a limb-dip of 15°; after this stage the wavelength-selection mechanism no longer operates and fold shape is mainly determined by L/L_d and limb-dip.

Strain-rates and finite strains in the medium are calculated f or all stages of the L/L_d = 1 and L/L_d = 4.6 folds. At limb-dips greater than 45° the planes of maximum flattening and maximum flattening rat e show the characteristic orientation and fanning of axial-plane cleavage.

Oxygen, carbon and hydrogen isotope studies of contact metamorphism

The O¹⁸/O¹⁶, C¹³/C¹², and D/H ratios have been determined for rocks and coexisting minerals from several granitic plutons and their contact metamorphic aureoles in northern Nevada, eastern California, central Colorado, and Texas, with emphasis on oxygen isotopes. A consistent order of O¹⁸/O¹⁶, C¹³/C¹², and D/H enrichment in coexisting minerals, and a correlation between isotopic fractionations among coexisting mineral pairs are in general observed, suggesting that mineral assemblages tend to approach isotopic equilibrium during contact metamorphism. In certain cases, a correlation is observed between oxygen isotopic fractionations of a mineral pair and sample distance from intrusive contacts. Isotopic temperatures generally show good agreement with heat flow considerations. Based on the experimentally determined quartz-muscovite O¹⁸/O¹⁶ fractionation calibration curve, temperatures are estimated to be 525 to 625°C at the contacts of the granitic stocks studied.

Small-scale oxygen isotope exchange effects between intrusive and country rock are observed over distances of 0.5 to 3 feet on both sides of the contacts; the isotopic gradients are typically 2 to 3 per mil per foot. The degree of oxygen isotopic exchange is essentially identical for different coexisting minerals. This presumably occurred through a diffusion-controlled recrystallization process. The size of the oxygen isotope equilibrium systems in the small-scale exchanged zones vary from about 1.5 cm to 30 cm. A xenolith and a re-entrant of country rock projecting into on intrusive hove both undergone much more extensive isotopic exchange (to hundreds of feet); they also show abnormally high isotopic temperatures. The marginal portions of most plutons have unusually high O¹⁸/O¹⁶ ratios compared to "normal" igneous rocks, presumably due to large-scale isotopic exchange with meta-sedimentary country rocks when the igneous rocks were essentially in a molten state. The isotopic data suggest that outward horizontal movement of H₂O into the contact metamorphic aureoles is almost negligible, but upward movement of H₂O may be important. Also, direct influx and absorption of water from the country rock may be significant in certain intrusive stocks.

Except in the exchanged zones, the O¹⁸/O¹⁶ ratios of pelitic rocks do not change appreciably during contact metamorphism, even in the cordierite and sillimanite grades; this is in contrast to regional metamorphic rocks which commonly decrease in O¹⁸ with increasing grade. Low O¹⁸/O¹⁶ and C¹³/C¹² ratios of the contact metamorphic marbles generally correlate well with the presence of calc-silicate minerals, indicating that the CO₂ liberated during metamorphic decarbonation reactions is enriched in both O¹⁸ and C¹³ relative to the carbonates.

The D/H ratios of biotites in the contact metamorphic rocks and their associated intrusions show a geographic correlation that is similar to that shown by the D/H ratios of meteoric surface waters, perhaps indicating that meteoric waters were present in the rocks during crystallization of the biotites.

Metasomatism and magmatic assimilation at a gabbro-limestone contact, Christmas Mountains, Big Bend region, Texas

A composite stock of alkaline gabbro and syenite is intrusive into limestone of the Del Carmen, Sue Peake and Santa Elena Formations at the northwest end of the Christmas Mountains. There is abundant evidence of solution of wallrock by magma but nowhere are gabbro and limestone in direct contact. The sequence of lithologies developed across the intrusive contact and across xenoliths is gabbro, pyroxenite, calc-silicate skarn, marble. Pyroxenite is made up of euhedral crystals of titanaugite and sphene in a leucocratic matrix of nepheline, Wollastonite and alkali feldspar. The uneven modal distribution of phases in pyroxenite and the occurrence' of nepheline syenite dikes, intrusive into pyroxenite and skarn, suggest that pyroxenite represents an accumulation of clinopyroxene "cemented" together by late-solidifying residual magma of nepheline syenite composition. Assimilation of limestone by gabbroic magma involves reactions between calcite and magma and/or crystals in equilibrium with magma and crystallization of phases in which the magma is saturated, to supply energy for the solution reaction. Gabbroic magma was saturated with plagioclase and clinopyroxene at the time of emplacement. The textural and mineralogic features of pyroxenite can be produced by the reaction 2( 1-X) CALCITE + AN_XAB_l-X = (1-X) NEPHELINE+ 2(1-X) WOLLASTONITE+ X ANORTHITE+ 2(1-X) CO₂. Plagioclase in pyroxenite has corroded margins and is rimmed by nepheline, suggestive of resorption by magma. Anorthite and wollastonite enter solid solution in titanaugite. For each mole of calcite dissolved, approximately one mole of clinopyroxene was crystallized. Thus the amount of limestone that may be assimilated is limited by the concentration of potential clinopyroxene in the magma. Wollastonite appears as a phase when magma has been depleted in iron and magnesium by crystallization of titanaugite. The predominance of mafic and ultramafic compositions among contaminated rocks and their restriction to a narrow zone along the intrusive contact provides little evidence for the generation of a significant volume of desilicated magma as a result of limestone assimilation.

Within 60 m of the intrusive contact with the gabbro, nodular chert in the Santa Elena Limestone reacted with the enveloping marble to form spherical nodules of high-temperature calc-silicate minerals. The phases wollastonite, rankinite, spurrite, tilleyite and calcite, form a series of sharply-bounded, concentric monomineralic and two-phase shells which record a step-wise decrease in silica content from the core of a nodule to its rim. Mineral zones in the nodules vary 'with distance from the gabbro as follows:

0-5 m CALCITE + SPURRITE + RANKINITE + WOLLASTONITE
5-16 m CALCITE + TILLEYITE ± SPURRITE + RANKINITE + WOLLASTONITE
16-31 m CALCITE + TILLEYITE + WOLLASTONITE
31-60 m CALCITE + WOLLASTONITE
60-plus CALCITE + QUARTZ

The mineral of a one-phase zone is compatible with the phases bounding it on either side but these phases are incompatible in the same volume of P-T-X_CO2.

Growth of a monomineralio zone is initiated by reaction between minerals of adjacent one-phase zones which become unstable with rising temperature to form a thin layer of a new single phase that separates the reactants and is compatible with both of them. Because the mineral of the new zone is in equilibrium with the phases at both of its contacts, gradients in the chemical potentials of the exchangeable components are established across it. Although zone boundaries mark discontinuities in the gradients of bulk composition, two-phase equilibria at the contacts demonstrate that the chemical potentials are continuous. Hence, Ca, Si and CO₂ were redistributed in the growing nodule by diffusion. A monomineralic zone grows at the expense of an adjacent zone by reaction between diffusing components and the mineral of the adjacent zone. Equilibria between two phases at zone boundaries buffers the chemical potentials of the diffusing species. Thus, within a monomineralic zone, the chemical potentials of the diffusing components are controlled external to the local assemblage by the two-phase equilibria at the zone boundaries.

Mineralogically zoned calc-silicate skarn occurs as a narrow band that separates pyroxenite and marble along the intrusive contact and forms a rim on marble xenoliths in gabbro. Skarn consists of melilite or idocrase pseudomorphs of melili te, one or two . stoichiometric calcsilicate phases and accessory Ti-Zr garnet, perovskite and magnetite. The sequence of mineral zones from pyroxenite to marble, defined by a characteristic calc-silicate, is wollastonite, rankinite, spurrite, calcite. Mineral assemblages of adjacent skarn zones are compatible and the set of zones in a skarn band defines a facies type, indicating that the different mineral assemblages represent different bulk compositions recrystallized under identical conditions. The number of phases in each zone is less than the number that might be expected to result from metamorphism of a general bulk composition under conditions of equilibrium, trivariant in P, T and u_CO2. The "special" bulk composition of each zone is controlled by reaction between phases of the zones bounding it on either side. The continuity of the gradients of composition of melilite and garnet solid solutions across the skarn is consistent with the local equilibrium hypothesis and verifies that diffusion was the mechanism of mass transport. The formula proportions of Ti and Zr in garnet from skarn vary antithetically with that of Si Which systematically decreases from pyroxenite to marble. The chemical potential of Si in each skarn zone was controlled by the coexisting stoichiometric calc-silicate phases in the assemblage. Thus the formula proportion of Si in garnet is a direct measure of the chemical potential of Si from point to point in skarn. Reaction between gabbroic magma saturated with plagioclase and clinopyroxene produced nepheline pyroxenite and melilite-wollastonite skarn. The calcsilicate zones result from reaction between calcite and wollastonite to form spurrite and rankinite.

Interaction of aqueous inorganic orthophosphate and phosphate rock

An experimental study was made of the interaction of phosphate rock and aqueous inorganic orthophosphate, calcium, and hydroxyl ions. A model of the reaction was developed by observing electron diffraction patterns in conjunction with concentration changes of chemical components. The model was applied in explaining the performance of batch suspensions of powdered phosphate rock and packed columns of granular phosphate rock. In both cases the reaction consisted initially of a rapid nucleation phase that occurred in a time period of minutes. In the batch system the calcium phosphate nuclei then ripened into larger micro-crystals of hydroxyapatite, which eventually became indistinguishable from the original phosphate rock surface. During column operation the high supersaturation ratio that existed after the rapid nucleation phase resulted in a layer of small nuclei that covered a slowly growing hydroxyapatite crystal.

The column steady-state rate constant was found to increase with increasing temperature, pH, and fluoride concentration, and to decrease with increasing concentrations of magnesium sulfate, ammonium chloride, and bicarbonate ion.

An engineering feasibility study indicated that, based on economic considerations, nucleation of apatite on phosphate rock ore has a potential use as a wastewater phosphate removal treatment process.

Reconstructing Past Depositional and Diagenetic Processes through Quantitative Stratigraphic Analysis of the Martian Sedimentary Rock Record

High-resolution orbital and in situ observations acquired of the Martian surface during the past two decades provide the opportunity to study the rock record of Mars at an unprecedented level of detail. This dissertation consists of four studies whose common goal is to establish new standards for the quantitative analysis of visible and near-infrared data from the surface of Mars. Through the compilation of global image inventories, application of stratigraphic and sedimentologic statistical methods, and use of laboratory analogs, this dissertation provides insight into the history of past depositional and diagenetic processes on Mars. The first study presents a global inventory of stratified deposits observed in images from the High Resolution Image Science Experiment (HiRISE) camera on-board the Mars Reconnaissance Orbiter. This work uses the widespread coverage of high-resolution orbital images to make global-scale observations about the processes controlling sediment transport and deposition on Mars. The next chapter presents a study of bed thickness distributions in Martian sedimentary deposits, showing how statistical methods can be used to establish quantitative criteria for evaluating the depositional history of stratified deposits observed in orbital images. The third study tests the ability of spectral mixing models to obtain quantitative mineral abundances from near-infrared reflectance spectra of clay and sulfate mixtures in the laboratory for application to the analysis of orbital spectra of sedimentary deposits on Mars. The final study employs a statistical analysis of the size, shape, and distribution of nodules observed by the Mars Science Laboratory Curiosity rover team in the Sheepbed mudstone at Yellowknife Bay in Gale crater. This analysis is used to evaluate hypotheses for nodule formation and to gain insight into the diagenetic history of an ancient habitable environment on Mars.

I. Studies on alkaline phosphatase activity in developing sea urchin embryos. II. Studies on the activation of protein biosynthesis in sea urchin eggs at fertilization

I. Alkaline phosphatase activity in the developing sea urchin Lytechinus pictus has been investigated with respect to intensity at various stages, ionic requirements and intracellular localization. The activity per embryo remains the same in the unfertilized egg, fertilized egg and cleavage stages. At a time just prior to gastrulation (about 10 hours after fertilization) the activity per embryo begins to rise and increases after 300 times over the activity in the cleavage stages during the next 60 hours.

The optimum ionic strength for enzymatic activity shows a wide peak at 0.6 to 1.0. Calcium and magnesium show an additional optimum at a concentration in the range of 0.02 to 0.07 molar. EDTA at concentrations of 0.0001 molar and higher shows a definite inhibition of activity.

The intracellular localization of alkaline phosphatase in homogenates of 72-hour embryos has been studied employing the differential centrifugation method. The major portion of the total activity in these homogenates was found in mitochondrial and microsomal fractions with less than 5% in the nuclear fraction and less than 2% in the final supernatant. The activity could be released from all fractions by treatment with sodium deoxycholate.

II. The activation of protein biosynthesis at fertilization in eggs of the sea urchins Lytechinus pictus and Strongylocentrotus purpuratus has been studied in both intact eggs and cell-free homogenates. It is shown that homogenates from both unfertilized and fertilized eggs are dependent on potassium and magnesium ions for optimum amino acid incorporation activity and in the case of the latter the concentration range is quite narrow. Though the optimum magnesium concentrations appear to differ slightly in homogenates of unfertilized and fertilized eggs, in no case was it observed that unfertilized egg homogenates were stimulated to incorporate at a level comparable to that of the fertilized eggs.

An activation of amino acid incorporation into protein has also been shown to occur in parthenogenetically activated non-nucleate sea urchin egg fragments or homogenates thereof. This activation resembles that in the fertilized whole egg or fragment both in amount and pattern of activation. Furthermore, it is shown that polyribosomes form in these non-nucleate fragments upon artificial activation. These findings are discussed along with possible mechanisms for activation of the system at fertilization.