Analytical Interpolation Research Articles

Topographic accessibility and the tectonic interpretation of gravity data

During recent gravity surveys in Patagonia, we were unable to acquire data on the North Patagonian Icecap, which is completely inaccessible for land surveying. The Icecap is clearly of tectonic significance, but, because of its tectonic history, it is uplifted and rugged, and consequently inaccessible. Therefore the distribution of geophysical observations which can be acquired in a rugged field area is dependent on the tectonic history. This suggests that a tectonic history which is reconstructed from geophysical data may be systematically biased.We use models of local and flexural isostasy to estimate the gravity errors associated with interpolation across inaccessible topography. The gravity error is largest for Pratt isostasy, where the mass deficit which supports the topography is at relatively shallow depths. The gravity error is least for flexural isostasy, because in this case the inaccessible topography is supported regionally by a mass deficit which extends beyond the inaccessible region.An analytical flexural interpolation scheme is proposed for interpolation across data gaps associated with inaccessible topography. Flexural theory and Gauss’s theorem are used to predict the gravity anomaly due to the mass excess of the inaccessible topography. We apply this scheme to the North Patagonian Icecap, to demonstrate that flexural interpolation predicts a relative gravity low at the site of the Icecap, which would not be predicted by purely geometrical interpolation schemes.

Double screening in polyelectrolyte solutions: Limiting laws and crossover formulas

We have derived general expressions for the Helmholtz free energy of a polyelectrolyte solution and the average size of a labeled chain as functions of polyelectrolyte concentration, salt concentration, and the strengths of excluded volume and electrostatic interaction all varying from very low to very high. We have shown that the bare excluded volume interaction and the electrostatically screened Coulomb interaction between any two segments are both screened by the presence of polyelectrolyte chains at nonzero concentrations. The screening length associated with this excluded volume screening is a function of Debye length and the entropy of the connected polymer and consequently depends on the polyelectrolyte concentration differently at different salt concentrations. By treating the correlation of the monomer density fluctuations, we have derived integral equations coupling the effective interaction between any two segments and the effective step length of a labeled chain. In the limit of the excluded volume screening length and the effective step length being independent of wave vectors, they are related by algebraic equations. These equations reduce to scaling laws with numerical prefactors for semidilute and concentrated polyelectrolyte solutions. Our development provides limiting laws and analytical interpolation formulas for the excluded volume screening length, effective step length, free energy of the polyelectrolyte solution, and the average size of a chain for various polyelectrolyte concentrations in an isotropic solution and for arbitrary strengths of excluded volume and electrostatic interactions. The excluded volume screening leads to an attractive component in the effective potential interaction at intermediate distances between two segmental charges of the same sign. The strength and range of the attractive potential are determined by the concentrations of the polyelectrolyte and the various ions.

Operator Method in the Problem of Quantum Anharmonic Oscillator

The problem of quantum anharmonic oscillator is considered as a test for a new nonperturbative method of the Schrödinger equation solution-the operator method (OM). It is shown that the OM zeroth-order approximation permits us to find such analytical interpolation for eigenfunctions and eigenvalues of the Hamiltonian which ensures high accuracy within the entire range of the anharmonicity constant changing and for any quantum numbers. The OM subsequent approximations converge quickly to the exact solutions of the Schrödinger equation. These results are justified for different types of anharmonicity (double-well potential, quasistationary states, etc.) and can be generalized for more complicated quantum-mechanical problems.

London equation of state for a quantum-hard-sphere system.

The London analytical interpolation equation between zero and packing densities for the ground-state energy of a many-boson hard-sphere system is corrected for the reduced mass of a pair of particles in a ``sphere-of-influence'' picture. It is thus brought into good agreement with computer simulations and with experimental results extrapolated out to close packing.

Interpolation scheme for dynamic electron correlations in the electron gas

An analytical interpolation scheme is proposed to include correlation effects in the local-field correction G( q, ω) of the electron gas. Rigorous asymptotic limits of G( q, ω) for large and small wave vectors and frequencies are combined with the known frequency dependent behaviour of G( q, ω), as revealed by the time-dependent HF-approximation. The correct interpretation of the known sum rules in the limits of small and large wave vectors and frequencies is of crucial importance for estimating the correlation contributions.

Calculation of cross-slip parameters in f.c.c. crystals

The activation energy for cross-slip in f.c.c. crystals is calculated. The usual ambiguity due to the uncertain inner cut-off radius for dislocations is avoided by including explicitly the core energy. The numerical results are described by appropriately scaled analytical interpolation formulae.

Experimental characterization of steady two-dimensional vortex couples

We study the evolution of unsteady two-dimensional vorticity structures surrounded by fluid at rest. The flow is initiated by a short fluid impulse in a horizontal layer of mercury and is constrained to be two-dimensional by a vertical uniform magnetic field. The impulse is generated by an electric pulse between two electrodes, and a flow circulation can be produced by diverting part of the current through the external frame. The velocity field is measured from the streaks of small particles floating on the free upper surface, and the vorticity is then obtained by means of an analytical interpolation and differentiation. The flow always evolves toward a set of independent steady structures with symmetry which are either circular vortices (monopoles) or couples (dipoles). The latter have a linear or circular steady motion depending on the flow circulation around them. The region of non-zero vorticity is always close to a circle. The steadiness is confirmed by plotting the vorticity versus the stream function in the frame of reference moving with the couple. We obtain a curve, as appropriate for a steady solution of the Euler equation. The slope of this curve is either constant or has no maximum. We suggest that this result could correspond to a general stability condition. The interaction between two symmetric couples at various angles of incidence yields two new couples by exchange of their vortices. Oscillations of the resulting couples are often damped by releasing a circular vortex.

Limiting energy spectrum of a saturated radiation belt

Through a further application of the condition for magnetospheric wave growth in the presence of anisotropic charged‐particle distributions, the Kennel‐Petschek theory that traditionally imposes an upper bound on the integral flux of charged particles at energies above a certain threshold is extended to provide a limit on the differential flux at any energy above this threshold. Thus, a modest reformulation of the nonrelativistic Kennel‐Petschek problem for electrons and protons enables a limiting energy spectrum to be derived, such that (for specified pitch‐angle anisotropy s of the energetic particle population) electromagnetic‐cyclotron waves at each frequency less than a fraction s/(s + 1) of the equatorial gyrofrequency are marginally stable against spontaneous generation. The limiting spectrum is given in closed form for integer values of s (>0) and computed numerically or by analytical interpolation for non‐integer values of s. Asymptotic expansions for energies E barely above and much greater than the minimum resonant energy E* provide estimates of the limiting energy spectrum J4π*(E) in these extremes, regardless of whether s is an integer. A reconsideration of the original Kennel‐Petschek problem, in which the differential energy spectrum is not calculated but specified as a certain power law (J4π ∝ E1‐l), enables both the wave frequency ω*/2π corresponding to maximum spatial growth rate and the limiting integral flux I4π* above the minimum resonant energy E* to be calculated in closed form as functions of l and s.

A potential energy surface for the Li+HCl reaction

The potential energy surface for the Li+HCl reaction has been calculated by performing UHF/CI ab initio computations for a large number of molecular geometries. To make the surface suitable for dynamical studies, the analytical interpolation of the CI points has been performed using a multi-body expansion of the potential and bond order variables with constraints on diatom asymptotic and transition state energies. To test the quality of the potential surface an investigation of its dynamical properties has been carried out using quasiclassical trajectories. Calculated scattering quantities have been compared to experiment.

Multi‐Dimensional Flow Analysis of Solid‐Liquid Mixtures

A two‐phase one‐equation turbulence model applied to dense slurry flow is suggested. The differential equations originate from the spatial averaging of the conservation equations, which reveals the phase interaction terms. The turbulence parameter (effective eddy‐viscosity) is determined by solving a transport type equation. Small scale experiments and microscopic analysis provide the constitutive equations. Finite volume techniques with local analytical interpolation are employed in the computer code written for two ‐ and three‐dimensional bounded domains. The numerical simulation for water/sand and water/glass sphere mixture flow illustrates the model application. The approach permits to verify some model assumptions and to predict suspension flow parameters for applicative situations.

Effects of non-locality and surface diffuseness on the electromagnetic response of a vacuum metal interface

A characteristic feature of the response of a metal to an external probe, such as photons or electrons is an induced surface change. Allowing for a non-local response and a smooth electron density profile in the surface region the classical singular surface charge will relax spatially and the center of gravity of the induced charge will be an important physical quantity characterizing the response. We propose a simple analytical interpolation formula for this, having the correct static limit as well as giving a relatively good description of the dynamical response.

A new method of computing geopotential fields

Summary. A new method is proposed for the geopotential field computation and gravitational attraction modelling. The usual method is to use a uniform density discrete numerical integration to represent either the gravitational potential or the gravitational attraction from a given density configuration. In this paper, an interpolation scheme is explained, using a piecewise continuous basis function to represent the arbitrarily varying density configuration in one, two and three dimensions. This new approach greatly simplifies the potential integrations and, in certain cases where symmetry exists, analytical evaluation of the integrals is also possible. Numerical tests and examples are given for a hypothetical salt dome, a vertical dyke with varying density structure and the hydrostatic ellipticity of earth model 1066B. The numerical error in this method is limited to the analytical approximation and interpolation errors in each case. This new approach can also be used as efficiently for other potential field studies.

Calculation of the electrostatic potential at any point of an ionic crystal

Abstract The electrostatic potential inside a cavity of an ionic crystal is determined by means of an analytical interpolation, without the use of the reciprocal lattice. The degree of approximation is discussed with regard to the dimension and shape of the cavity. The total crystalline potential can be calculated at any point of the lattice. The method is applied to NaCl, CaCO3, NaN3 and CsCl.

Zur freien Energie von Quantenplasmen kleiner Dichte

AbstractA general explicit formula for the free energy of low density plasmas at temperatures in the range of 103 ≲ T ≲ 108 °K is proposed. This formula is obtained by an analytical interpolation between the result for the free energy of low temperature plasmas and that for high temperature plasmas, which are known from other papers [1–7].