Topological properties of disjoint channel networks within enclosed regions

Enclosure of some portion of one or more natural stream-drainage basins by superposition of a rectangle on a map of drainage network results in fragmentation of the natural basins into a set of disjoint channel networks. Each of these may have some channel links and forks of the natural network plus truncated links intersected by the enclosure boundary. The topological properties of the network elements in the enclosure are used to set up a model of random network patterns, in which the number of disjoint channel networks is expressed as a function of the number of links and forks in the enclosures. This function is shown to be a multiplicative constant times the square root of the number of links or forks. Empirical data on square and rectangular enclosures of several sizes from the Inez (Kentucky)Quadrangle map showed that the predicted multiplicative constants do not agree with observation, but that the square-root relation seems to hold at least to a first approximation. The models thus can be used as a norm against which deviations of real-world enclosures from network pattern randomness can be studied.     

Trajectoires rectilignes du problème des 3 corps,lorsque la constante des forces vives est nulle

This problem is reducible to the integration of the EulerChazy's differential equation. We give a new simplified version of the equation, and we interpret in the corresponding diagram the study of two extreme degenerate cases.

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Presenté à la Conference du Mécanique Céleste, Oberwolfach, Germany, août 27–septembre 2, 1972.     

Applications of thermodynamics to fundamental earth physics

New fundamental thermodynamic relationships of complete generality and absolute rigour of derivation are not to be expected, because the subject has such a secure and complete basis in classical physics. There is, however, still scope for original, fundamental work based on recognised assumptions and approximations which may be obviously acceptable in particular situations. Clarification of relationships between thermodynamic parameters for materials within the Earth is particularly important because there is so little possibility of measuring them individually. This survey first summarises the established relationships in a very condensed form and then concentrates on some recent developments which have direct bearing on the thermal and mechanical states of the Earth's mantle and core. Considerable use is made of the thermodynamic Grüneisen parameter, which is a dimensionless quantity of order unity for almost all materials, solid, liquid and gaseous, and is directly related to the pressure dependences of elastic constants. This allows its value to be estimated for the different regions of the Earth from seismological data. The thermodynamic (heat engine) efficiency of convection in a homogeneous medium, driving tectonic activity or the geomagnetic dynamo, is found to be the ideal (Carnot) efficiency corresponding to adiabatic temperature differences between the heat source and sink, within the assumption that the thermal expansion coefficient is not strongly temperature dependent. The use of this conclusion to infer tectonic stresses is indicated. The thermodynamic basis for Lindemann's melting law is restated and the reasons for supposing it to be valid for materials at megabar pressures reaffirmed. Application to the inner core boundary gives a fixed point on the Earth's temperature profile. Use of thermodynamic relationships in the interpretation of shock wave compressions is indicated.     

Literatur: Neue Bücher

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