Directions and possibilities of mathematical geology

This paper considers the present state of mathematical geology. Three directions are recognized: applied, theoretical, and mathematical. Applied mathematical geology includes formal use of mathematics to solve problems and computer processing of data. Success is achieved by a correspondence of mathematical methods used to the nature of geological data. This correspondence can be demonstrated by purely mathematical means. Theoretical mathematical geology uses mathematics as a language of geology; however, a number of methodological problems must be solved: formalization of initial geological concepts and creation of a strict conceptual basis, substantiation of initial principles of mathematical simulation, creation of theoretical geological models, problems of elementary and coincidence in geology, and methodological substantiations of possibilities of any mathematical model to approximate geological models. The essense and significance of these problems are considered. The main task of mathematical geology is to prove its correspondence to the nature of the geological objects studied, geological data obtained, and geological problems solvable. Finally, the main problems of mathematical geology are not so much mathematical as geological and methodological.     

The effect of bulk rock composition on the stability of amphibole in the upper mantle: Implications for solidus positions and mantle metasomatism

Summary The stability of pargasitic amphibole in the upper mantle is a function of water content and bulk rock composition, and under water-undersaturated conditions, the stability of amphibole controls the solidus position. Experiments in the system Tinaquillo peridotite +0.2% H₂O, a refractory peridotite under water-undersaturated conditions, show that amphibole is stable to 1030°C and 26 kb. In contrast, pargasitic amphibole is stable to 1150°C and 30 kb in Hawaiian pyrolite, a more fertile peridotite composition. This indicates that under water-undersaturated conditions, the most fertile part of a crystallizing mantle diapir with an inhomogeneous composition will solidify first while a more refractory component will contain an alkali-rich melt which will have the ability to metasomatize adjacent regions. The relative stabilities of amphibole in refractory and fertile bulk compositions may result in increasing rather than diminishing chemical contrasts in high temperature lherzolite, i.e. a process of metamorphic differentiation. Ti, Fe, Al and Na metasomatism can therefore be considered a normal occurrence associated with the upward migration and solidification of an H₂O-bearing mantle diapir.

---

Der Einfluß der Gesamtgesteins-Zusammensetzung auf die Stabilität von Amphibol im oberen Mantel: Bedeutung für Solidus-Positionen und Mantel-Metasomatose

Zusammenfassung Die Stabilität pargasitischer Amphibole im oberen Mantel ist eine Funktion von Wassergehalt und Gesamtgesteins-Zusammensetzung. Unter wasser-untersättigten Bedingungen, kontrolliert die Stabilität von Amphibol die Solidus-Position. Experimente in dem System Tinaquillo Peridotit +0,2% H₂O, einem refraktären Peridotit unter wasser-untersättigten Bedingungen, zeigen daß Amphibol bis 1030°C und 26 Kb stabil ist. Im Gegensatz dazu ist pargasitische Hornblende in einem Hawaii-Pyrolit, von mehr fertiler Peridotit-Zuammensetzung, bis 1150°C und 30 Kb stabil. Das zeigt, daß bei wasser-untersättigten Bedingungen der am meisten produktive Teil eines kristallisierenden Mantel-Diapirs mit inhomogener Zusammensetzung sich zuerst verfestigen wird, während eine mehr refraktäre Komponente eine alkali-reiche Schmelze enthalten wird, die wiederum die Fähigkeit hat, umliegende Bereiche metasomatisch zu beeinflussen. Die relativen Stabilitäten von Amphibol in refraktären und fertilen Gesamtzusammensetzungen können dazu führen, daß die chemischen Gegensätze in Hochtemperaturlherzoliten eher zunehmen als abnehmen, d. h. ein Prozeß metamorpher Differentiation. Ti, Fe, Al und Na Metasomatose können deshalb als ein verbreiteter Vorgang, der mit der Aufwärtsbewegung und Verfestigung eines H₂O-führenden Mantel-Diapirs assoziiert ist, betrachtet werden.

---

---

With 4 Figures     

An internal variable model for the creep of rocksalt

Summary The creep strain rate of rocksalt, like that of other ductile crystalline materials, can be described by a power law equation of the type ( ) ⁿ , where the active stress is the difference between the total deviatoric applied stress and an internal stress _i . In this paper, the origin and the nature of this internal stress, which develops during inelastic deformation of the material, are discussed. It is shown that this internal stress can serve as an internal (or state) variable in the constitutive model of rocksalt, which reflects the microstructure evolution of the material under the competitive action of hardening and recovery mechanisms.An analysis of experimental data, both our own and those taken from the literature, demonstrates that such a law is able to correctly reproduce rocksalt creep test results in the steady-state domain. The proposed model is in accordance with the macroscopic and microscopic behavior of salt, and with direct measurements of the internal stresses made by others on this material.     

Hawaiian basalt and Icelandic rhyolite: Indicators of differentiation and partial melting

In spite of the voluminous basaltic volcanism on the island of Hawaii, rhyolite is not produced. Iceland, on the other hand, exhibits common rhyolitic volcanism amounting to some 10–12% of its surface rocks. This contrast is investigated using the fundamental igneous processes exhibited by sheet-like Hawaiian lava lakes and Shonkin Sag laccolith in Montana. Highly differentiated, residual melts normally reside within inwardly advancing solidification fronts and are generally inaccessible to eruptive processes. Only when a large initial phenocryst population is present, from which a thick basal cumulate can rapidly form, is it possible to supply highly differentiated melt into the active (i.e., eruptable) portion of the magma chamber. Although there is protracted control of differentiation at Hawaii by settling of olivine, further differentiation occurs within the solidification fronts. Only by repeated transport and holding is it possible to differentiate beyond the critical composition of the leading edge of the solidification front ( 7% MgO and 51.5% SiO₂). Crystal size distributions (CSDs) for Hawaii and Shonkin Sag are used to demonstrate the inferred physical and chemical processes of solidification, including the kinetics of crystallization.A ubiquitous feature of these basaltic bodies is the formation of coarse veins and segregations of refined melt and granophyres within the upper solidification front. It is this fundamental bimodal feature which is the key to understanding Icelandic silicic volcanism.Rhyolites in Iceland occur mainly as a bimodal population with basalts associated with central volcanoes. Rhyolites, granophyres, and felsites are common, with the intrusions often being layered. Ash flows and true granite-like intrusions are rare. The voluminous silicic lavas at Torfajokull central volcano contain disequilibrium phenocryst assemblages. This, and the disagreement in oxygen isotopic values between rhyolites and basalts, reflects extensive partial melting of the heterogeneous basaltic crust of Iceland to produce these rhyolites. Relatively small, chemically distinct, and spatially intimate silicic bodies are formed by concentrating granophyric segregations from earlier cycles of solidification. This process is also reflected in the layered granophyric instrusion of Slaufrudalur in eastern Iceland. Slaufrudalur is an unvented subterranean caldera, equivalent in igneous processes and style to the subaerial Torfajokull caldera.Hawaii is dominated by fractional crystallization due to crystal settling and does not produce rhyolite. Iceland's tectonics allow continual and extensive reprocessing of thin, hot basaltic crust which produces rhyolite by concentrating original silicic segregations and veins and by partially melting intermediate extrusives, which have subsided deep into the crust.

---

Zusammenfassung Auf Hawaii treten, trotz intensiven Basalt-Vulkanismusses, keine Rhyolithe auf. Auf Island dagegen ist Rhyolith, mit 10–12% des anstehenden Gesteins, verbreitet. Dieser Kontrast wurde anhand grundlegender magmatischer Prozesse untersucht, wie sie in flachen Lava-Seen Hawaiis und im Shonkin Sag Laccolith Montanas auftreten. Hochdifferenzierte Restschmelzen verbleiben innerhalb langsam nach innen vorrückender Erstarrungsfronten und sind meist unerreichbar für eruptive Prozesse. Nur wenn anfänglich bereits große Mengen von Einsprenglingen vorhanden sind, die rasch am Boden der Magmenkammer akkumulieren, kann eine hochdifferenzierte Schmelze in den aktiven (d.h. eruptiven) Teil der Magmenkammer gelangen. Obwohl auf Hawaii die Differentiation durch die Kristallisation von Olivin anhaltend kontrolliert wird, findet an der Erstarrungsfront weitere Differentiation statt. Nur durch wiederholten Transport und zeitweiliges Verharren ist es möglich, über die kritische Zusammensetzung der vordersten Erstarrungsfront hinaus zu differenzieren (ca. 7% MgO und 51,5% SiO₂). An Kristallgrö-ßenverteilungen (CDS) von Hawaii und Shonkin Sag können die angenommenen physikalischen und chemischen Prozesse der Kristallisation und die Kristallisationskinetik gezeigt werden. Ein weit verbreitetes Merkmal dieser Basaltkörper ist die Bildung grobkristalliner Gänge und Absonderung von stark differenzierten Schmelzen und Granophyren innerhalb der oberen Erstarrungsfront. Diese ausgeprägt bimodale Charakteristik ist der Schlüssel zum Verständnis des sauren isländischen Vulkanismus.Isländische Rhyolithe treten meist in bimodaler Verbreitung mit Basalten in Zusammenhang mit zentralen Vulkanen auf. Rhyolithe, Granophyre und Feisite sind häufig, in oft geschichteten Intrusionen. Ignimbrite und echte Granitintrusionen sind selten. Die großen Mengen SiO₂-reicher Laven am Torfajokull-Zentralvulkan enthalten Ein-sprenglinge, die sich nicht im Gleichgewicht mit der Matrix befinden. Dies, und die unterschiedlichen delta-¹⁸O-Werte von Rhyolithen und Basalten, zeigen, daß ausgeprägtes teilweises Aufschmelzen der heterogenen Basaltkruste von Island zur Produktion dieser Rhyolithe führte. Relativ kleine, nahe benachbarte saure Körper, die aber deutliche Unterschiede in ihrem Chemismus aufweisen, werden gebildet durch die Konzentration granophyrischer Teilschmelzen aus früheren Kristallisationszyklen. Dieser Vorgang wird auch widergespiegelt in der »layered intrusions« von Slaufrudalur in Ostisland. Slaufrudalur ist eine geschlossene unterirdische Kaldera, deren magmatische Prozesse und Baustil der subaerischen Torfajokull-Kaldera entsprechen.Die Prozesse in Hawaii sind dominiert von gravitativer Kristallisationsdifferentiation und es werden keine Rhyolithe produziert. Die isländische Tektonik führt zu kontinuierlicher starker Wiederaufarbeitung von dünner, heißer basaltischer Kruste. Dabei wird, durch die Konzentration ursprünglicher saurer Teilschmelzen und Gänge und durch die teilweise Aufschmelzung intermediärer Intrusiva, die tief in die Kruste abgesunken sind, Rhyolith produziert.

---

Résumé En dépit du volcanisme basaltique volumineux des îles Hawaï, il n'y existe pas de rhyolite. En Islande, par contre, le volcanisme rhyolitique est commun et représente 10 à 12% des roches de la surface. Ce contraste est examiné sur la base des processus ignés fondamentaux présentés par les lacs de lave d'Hawaï et le laccolite de Shonkin Sag au Montana. Normalement, les liquides résiduels hautement différenciés résident à l'intérieur des fronts de solidification qui progressent vers l'arrière et sont généralement à l'abri des processus éruptifs. Ce n'est que dans le cas d'une population initiale abondante de phénocristaux, qui se rassemblent dans un cumulat basai épais, que des liquides hautement différenciés peuvent être fournis à la portion active (c'est-à-dire »éruptible«) de la chambre magmatique. A Hawaï, bien que la différenciation soit continuellement régie par la cristallisation d'olivine, la poursuite du processus a lieu à l'intérieur des fronts de solidification. Ce n'est que par la répétition d'actions de transport et de stagnation qu'il est possible de différencier audelà de la composition critique du front de solidification (±7% MgO et 51,5% SiO₂). A partir de la distribution de la taille des cristaux à Hawaï et à Shonkin Sag, on peut déduire les processus physique et chimique de la solidification, y compris la cinétique de la cristallisation.Une particularité courante de ces corps basaltiques est la formation de veines grenues et de ségrégations de liquides très différenciés et de granophyres à l'intérieur du front supérieur de solidification. Cette manifestation bimodale est la clé qui permet de comprendre le volcanisme siliceux islandais.En Islande, les rhyolites constituent d'ordinaire une population bimodale avec les basaltes centraux. Les rhyolites, les granophyres et les felsites sont fréquents, et souvent sous forme d'intrusions litées. Les coulées ardentes et les vraies intrusions de type granitique sont rares. Les volumineuses laves siliceuses du volcan central de Torfajokull contiennent des assemblages de phénocristaux en déséquilibre. Ce fait, ainsi que la non concordance des isotopes de l'oxygène entre rhyolites et basaltes, traduisent, à l'origine de ces rhyolites, une fusion partielle extensive de la croûte basaltique hétérogène d'Islande. Des corps siliceux relativement petits et chimiquement distincts bien que d'emplacements très voisins se sont formés par concentration de fusions partielles granophyriques lors des premiers cycles de solidification. Ce processus s'exprime également dans l'intrusion granophyrique litée de Slaufrudalur, en Islande orientale. Slaufrudalur est une caldeira souterraine fermée, équivalente par son style et son processus igné à la caldeira subaérienne de Torfajokull.A Hawaï, le phénomène dominant est la cristallisation fractionnée gravitative, sans production de rhyolite. La tectonique de l'Islande permet la régénération continue et extensive d'une mince croûte basaltique chaude. Les rhyolites y sont engendrées par la concentration des veines et ségrégations siliceuses originelles et par la fusion partielle de masses extrusives intermédiaires descendues profondément dans la croûte.

---

, . , 10–12% . , Shonkin Sag Laccolith Montanas. . , , . , . . ( 7% MgO 51,5% SiO₂). (CDS) Shonkin Sag , , . . . . , , . . , Torfajokull , ., ¹⁸O , . , , , , « » («layered intrusions») Slaufrudalur, . , , Torfajokull. . . , , , .

     

Sublethal abnormalities in embryos of winter flounder,Pseudopleuronectes americanus,from Long Island Sound

Sublethal abnormalities were examined in developing embryos of the winter flounder,Pseudopleuronectes americanus, as part of a study of its early reproductive success in Long Island Sound and two sites in Boston Harbor. These sites represented varying levels of anthropogenic contamination that were possibly affecting reproduction, as ascertained by chromosomal abnormalities and lowered developmental rates in the embryos. Effort was focused on examination of blastula and tail-bund embryos from field-captured females spawned at the laboratory. Abnormalities observed included evidence of cytotoxicity and chromosome damage. Embryos of fish from New Haven were usually the most aberrant, while embryos from other sites, notably Hempstead, Shoreham, and both Boston Harbor stations, showed subtle indications of abnormality.     

Unsteady hydromagnetic flow through a porous medium between two infinite parallel porous plates with time-varying suction

Unsteady two-dimensional flow of a viscous incompressible and electrical-conducting fluid through a porous medium bounded by two infinite parallel plates under the action of a transverse magnetic field is presented when there is time-varying suction at the plates. The lower plate is at rest while the upper plate is oscillating in its own plane about a constant mean velocity. Expressions for the velocity, fluctuating parts of the velocity, amplitude, and phase of the skin-friction are obtained. The flow phenomenon has been characterized by the parametersK (permeability of the porous medium),N(magnetic parameter) (frequency parameter), andA(variable suction parameter) and the role of these parameters on the flow characteristics has been studied.     

Magnetic reconnection in high-temperature plasma of solar flares

Quasi-steady high-temperature current sheets are an energy source during the main or hot phase of solar flares. Such sheets are shown to be stabilized with respect to the tearing instability by a small transverse component of magnetic field existing in the sheets.     

Implications of liouville's theorem on the apparent brightness temperatures of solar radio bursts

Liouville's theorem for radiation, of which the generalized étendue is a consequence, implies ² d² d² A = constant along the ray path, where is the refractive index and d² and d² A are the ranges, respectively, of solid angle and of area that define a ray (actually a bundle of rays). Implications of this concept on the propagation of radio waves from the actual to the apparent source in the solar corona (i.e., the scatter image of the true source) are discussed. The implications for sources of fundamental plasma radiation include: (1)The observed solid angle (defining the directivity) and apparent area A of the source are compatible with Liouville's theorem only if the apparent source (the scatter image of the true source) corresponds to the envelope of subsources with a small filling factor f. (2) The brightness temperature T _Bof the actual source is greater than that of the apparent source by f ^-1. (3) For sources of fundamental plasma radiation the factor f is very small ( 10^-2). (4) A long-standing discrepancy between the observed low value of T _B at meter/decameter wavelengths for the quiet Sun and the known coronal temperature may be resolved by noting that the implied coronal temperature is given by T _B f and that the factor f must be significantly less than unity.A brief discussion is included of the relation between Liouville's theorem, the generalized étendue, Milne's laws, occupation numbers, extension in phase, and suppression of emission by a medium with refractive index unequal to unity.     

The possible formation of a hydrogen coma around comets at large heliocentric distances

An observational test--the detection of a hydrogen coma around comets at large heliocentric distances--is proposed for determining whether comets were formed by the agglomeration of unaltered, ice-coated, interstellar grains. Laboratory experiments showed that amorphous water ice traps H2, D2, and Ne below 20 K and does not release them completely until the ice is heated to 150 K. Gas/ice ratios as high as 0.63 are obtainable. Thus, if the ice-coated interstellar grains were not heated above approximately 110 K, prior to their agglomeration into cometary nuclei, the inward propagating heat waves should release from the comets a continuous flux of molecular hydrogen. This flux would exceed that of water molecules at approximately 3 AU preperihelion and approximately 4 AU postperihelion.     

Tepev Mons on Venus: Morphology and elastic bending models

Tepev Mons is a large volcanic structure of about 250 km in diameter with an elevation of 5 km above the surroundings, located at the southwestern edge of Bell Regio. It is surrounded by a moat with a depth of about 0.5 km. If this moat is considered to be caused by bending of the lithosphere due to the load of the volcano, then elastic bending models give limits for the effective flexural rigidity FR and the effective elastic thickness of the lithosphere L: 2 x 10²³ Nm FR 3 x 10²⁴ Nm and 30 km L 100 km. High flexural rigidities are associated with small depressions and large thicknesses of the lithosphere and vice versa.Contribution No 345, Institut für Geophysik der Universität Kiel, F.R.G.