Finite Prandtl number 2-D convection at high Rayleigh numbers

Finite Prandtl number thermal convection is important to the dynamics of planetary bodies in the solar system. For example, the complex geology on the surface of the Jovian moon Europa is caused by a convecting, brine-rich global ocean that deforms the overlying icy “lithosphere”. We have conducted a systematic study on the variations of the convection style, as Prandtl numbers are varied from 7 to 100 at Rayleigh numbers 106 and 108. Numerical simulations show that changes in the Prandtl number could exert significant effects on the shear flow, the number of convection cells, the degree of layering in the convection, and the number and size of the plumes in the convecting fluid. We found that for a given Rayleigh number, the convection style can change from single cell to layered convection, for increasing Prandtl number from 7 to 100. These results are important for determining the surface deformation on the Jovian moon Europa. They also have important implications for surface heat flow on Europa, and for the interior heat transfer of the early Earth during its magma ocean phase. Electronic Supplementary Material is available if you access this article at . On that page (frame on the left side), a link takes you directly to the supplementary material. Electronic Supplementary Material is available if you access this article at . On that page (frame on the left side), a link takes you directly to the supplementary material.     

Algebraic Solution of GPS Pseudo-Ranging Equations

Several procedures for solving, in a closed form the GPS pseudo-ranging four-point problem P4P in matrix form already exist. We present here alternative algebraic procedures using Multipolynomial resultant and Groebner basis to solve the same problem. The advantage is that these algebraic algorithms have already been implemented in algebraic software such as “Mathematica” and “Maple”. The procedures are straightforward and simple to apply. We illustrate here how the algebraic techniques of Multipolynomial resultant and Groebner basis explicitly solve the nonlinear GPS pseudo-ranging four-point equations once they have been converted into algebraic (polynomial) form and reduced to linear equations. In particular, the algebraic tools of Multipolynomial resultant and Groebner basis provide symbolic solutions to the GPS four-point pseudo-ranging problem. The various forward and backward substitution steps inherent in the clasical closed form solutions of the problem are avoided. Similar to the Gauss elimination techniques in linear systems of equations, the Multipolynomial resultant and Groebner basis approaches eliminate several variables in a multivariate system of nonlinear equations in such a manner that the end product normally consists of univariate polynomial equations (in this case quadratic equations for the range bias expressed algebraically using the given quantities) whose roots can be determined by existing programs (e. g., the roots command in MATLAB). © 2002 Wiley Periodicals, Inc.     

GPS Solutions: Closed Forms, Critical and Special Configurations of P4P

P4P is the pseudo-ranging 4-point problem as it appears as the basic configuration of satellite positioning with pseudo-ranges as observables. In order to determine the ground receiver/satellite receiver (LEO networks) position from four positions of satellite transmitters given, a system of four nonlinear (algebraic) equations has to be solved. The solution point is the intersection of four spherical cones if the ground receiver/satellite receiver clock bias is implemented as an unknown. Here we determine the critical configuration manifold (Determinantal Loci, Inverse Function Theorem, Jacobi map) where no solution of P4P exists. Four examples demonstrate the critical linear manifold. The algorithm GS solves in a closed form P4P in a manner similar to Groebner bases: The algebraic nonlinear observational equations are reduced in the forward step to one quadratic equation in the clock bias unknown. In the backward step two solutions of the position unknowns are generated in closed form. Prior information in P4P has to be implemented in order to decide which solution is acceptable. Finally, the main target of our contribution is formulated: Can we identify a special configuration of satellite transmitters and ground receiver/satellite receiver where the two solutions are reduced to one. A special geometric analysis of the discriminant solves this problem. ? 2002 Wiley Periodicals, Inc.     

Elektronenmikroskopische Untersuchungen an mikritischen Kalken

Zusammenfassung Es werden die Entstehungsmöglichkeiten von mikritischen Kalken diskutiert und an Hand von elektronenmikroskopischen Aufnahmen Hinweise auf genetische Interpretationen vorgeführt.

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The origin of micritic limestones is discussed and genetic interpretation of micrites from different environments, studied by electron microscope, is given.

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Résumé L'origine des calcaires micritic est discutée et une interprétation génétique des micrites de milieux différents est donnée à l'aide de la microscopie électronique.

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Vorgetragen auf der 56. Jahresversammlung der Geologischen Vereinigung in Wien im Februar 1966.     

Reconstructing the historical changes in Daphnia mean size and planktivorous fish abundance in lakes from the size of Daphnia ephippia in the sediment

The zooplankton community structure in lakes is highly influenced by size-selective predation by fish, with small zooplankton species dominating at high predation pressure. Remains of cladocerans are preserved in the sediment and may be used to trace historical changes in fish predation. We determined how contemporary data on planktivorous fish were related to the size of Daphnia ephippia (dorsal length) in the surface sediment (0-1 cm) of 52 mainly shallow lakes with contrasting densities of fish and nutrients (TP: 0.002-0.60 mg P l^-1). Density of fish expressed as catch per unit effort, in terms of numbers in multiple mesh-sized gill nets (CPUE_n), decreased significantly with increasing mean size of ephippia. The relationship was improved by adding TP as an independent variable, now explaining 90% of the variation in CPUE_n on the full data set covering lakes in Denmark, Greenland and New Zealand, and 78% if only data on Danish lakes were used. CPUE by weight of planktivorous fish and mean weight of Daphnia in the pelagial during summer were also related to ephippial size. By including contemporary data on established relationships between the sizes of egg-bearing female Daphnia and ephippia, we inferred changes in the CPUE_n, mean size of ephippia-bearing Daphnia and summer mean body weight of Daphnia from ephippial size in four lakes during the past 1-2 centuries. In a hypertrophic lake subject to periodic fish kills, Daphnia mean body weight was high and CPUE_n was low compared with those in two eutrophic lakes, while CPUE_n was low and Daphnia body weight was high in the least eutrophic, clearwater lake. Estimated CPUE_n and Daphnia mean weight in the surface sediment of these four lakes corresponded well with contemporary data. Only small changes in ephippial size with time were observed in the clearwater lake and in one of the lakes that had suffered early eutrophication, while major changes occurred in the two other lakes that had been subjected to a major increase in nutrient input or fish kills. We conclude that Daphnia ephippia preserved in the surface sediments of lakes may be a useful and efficient method to quantify the present-day abundance of planktivorous fish and Daphnia mean size. The method is particularly valid in surveys aimed to give a general picture of the fish stock and the ecological state in a set of lakes in a region rather than a precise estimate for a single lake. Though some evidence is provided, more work is needed to evaluate whether the equations are valid for hind-casting in down-core palaeoecological studies.     

Time series tapering for short data samples

We explore the effect of applying tapered windows on atmospheric data to eliminate overestimation inherent in spectra computed from short time series. Some windows are more effective than others in correcting this distortion. The Hamming window gave the best results with experimental data. The Hanning window, which performed almost as well with the same data set, was used to demonstrate the effectiveness of the window in a numerical simulation of the process. We recommend that either of these windows be applied to short data samples prior to spectral analysis.     

Statistics and geometry of the eigenspectra of three-dimensional second-rank symmetric random tensors

Geoscientists have undertaken mapping of the Earth's crustal strain (or stress) fields using a great variety of field data. The output can be represented by a 3-D second-rank symmetric random strain tensor. The random principal strains-land rotations of the random tensor are frequently computed. The accuracy is calculated using a first-order approximation. The distribution aspects of the random principal strains and rotations have received almost no attention in Earth Sciences. A first-order approximation of accuracy may not be sufficient if the signal-to-noise ratio is small, as is often the case for geodetically derived random strain tensors. Therefore, the purpose of this paper is to investigate the distribution and estimation problems of the general 3-D second-rank tensor equation GΛG ^T= T , where T is a given 3-D second-rank symmetric random tensor, Λ a diagonal (3 × 3) random eigenvalue matrix, and G a (3 × 3) random orientation matrix, which is also orthogonal. Λ and G are to be estimated (or solved) from T . If some eigenvalues coincide, additional conditions are imposed on the eigenvectors so that they can be chosen uniquely. The joint probability density function (pdf) of the random eigenvalues and rotations will be worked out, given a joint pdf of the elements of random tensors T. Because the rotations are of special interest in Earth Sciences, we shall also derive the joint marginal pdf of random rotations. The geometry of eigenspectra will be studied. The biases of random eigenvalues and rotations will be derived, which have been neglected in the past. They can be very crucial in interpreting the pattern of a derived strain field, however, when applied to a real Earth Science problem. The variance-covariance matrices will be computed using a second-order approximation.