Stony Brook AMS - Downloadable Preprints, 1992

In the Rayleigh-Taylor problem, a gravitationally or acceleration driven instability occurs at the interface between two fluids of different densities, as the light fluid accelerates the heavy one. As a result of this interfacial instability, the two phases interpenetrate, producing a mixing zone, which is the object of study here.

Computational solutions to the Rayleigh-Taylor fluid mixing problem are formulated in terms of the two fluid two dimensional Euler equations. Data from these solutions are analyzed from the point of view of Reynolds averaged equations. The computations, which were carried out with the front tracking method, are highly resolved, but relatively inexpensive, so that statistical convergence of ensemble averages can be achieved. The computations are consistent with the experimentally observed growth rates for the size of the mixing zone for nearly incompressible flows. The analysis here is aimed partly at understanding the turbulent structure in the interior of the mixing zone. For the time interval over which the use of two-dimensional physics is valid, the dynamics of the interior portion of the mixing zone is simplified by the use of scaling variables, but does not confirm the notion of a fixed point. The outer edge of the mixing zone suggests fixed-point behavior.

A second goal of this paper is to study the effect of compressibility. We find that, for even moderate compressibility, the mixing rate (defined by the location of the mixing boundary) fails to satisfy universal scaling, and moreover, has a significant dependence on compressibility.

This numerical study was performed on Intel iPSC/860 parallel computers. The parallel efficiency is 90\%.

Exploiting the symmetry of the molecule $C_{60}$, we obtain the precise algebraic and numerical expressions for the eigenvalues and eigenfunctions of the H\"uckel problem for $C_{60}$.

In Proceedings of the International Conference on Computational Methods in Water Resources, Denver Colorado, June, 1992. Computational Mechanics Publications, Southampton, UK.

In single phase flow of a passive tracer through heterogeneous porous media, a mixing layer develops between the tagged and untagged regions of the fluid. The mixing region expands as the time evolves. We have developed a multi- length scale theory for the growth of the mixing region induced by a general random velocity field (multi-fractal field). The theory relates the statistics of the mixing layer to the statistics of the random field and derives an effective equation which governs the statistical properties of the mixing layer. The theory provides an analytic prediction for the growth of the size of the mixing layer. The scaling behavior of the mixing layer in a general random velocity field is determined over all length scales.

The analysis of the multi-length scale theory shows that the growth rate (i.e. the scaling exponent) of the mixing layer at length scale l depends on the statistical properties of the random field on all length scales smaller than l. In general, the sca ling exponent of the mixing layer is non-Fickian on all finite length scales. The asymptotic diffusion is non-Fickian when the correlation function of the random field decays slowly at large length scales, and Fickian when the correlation function of the random field decays rapidly at large length scales. Let b be the asymptotic scaling exponent of the mixing layer, and r be the asymptotic scaling exponent of the correlation function of the velocity (or permeability) field. Then r = max { 1 / 2 , 1 + b / 2}. Furthermore the theory explains why the effective macroscopic diffusion is a non-decreasing function of the length scale.

The asymptotic scaling behavior of the mixing region induced by a random velocity field is determined by applying Corrsin's hypothesis in both the Eulerian and Lagrangian pictures. Both pictures lead to the same results for the asymptotic scaling exponent of the mixing region. Both longitudinal and transverse diffusion are asymptotically non-Fickian (Fickian) when the correlation function of the random field decays more slowly (rapidly) than 1/r at large length scales.

We study a general multiple layer Rayleigh-Taylor instability problem for both compressible and incompressible inviscid fluids. Our main result is that many features of a multilayer system are universal, in the sense they do not depend on such details as the number of layers, their thicknesses, equations of state for the fluids, and equilibrium density distributions. It is shown that a compressible system is always more unstable than the corresponding incompressible one. We give a universal upper bound for the growth rate for a given perturbation wave number. Necessary and sufficient conditions of stability are determined. For compressible fluids, it is possible for the system to be unstable even if there is no density inversion anywhere. General Rayleigh-Taylor unstable modes are characterized, and the range of unstable wave numbers is determined.

It is known that every positive integer is expressible as the sum of at most eight pyramidal numbers $P(m) = m(m+1)(m-1)/6$. We conjecture that five pyramidal numbers always suffice, and that four always suffice for sufficiently large integers. This paper reports on a computer search which demonstrates this conjecture up to $1,000,000,000$, or one billion.

A recently developed theory of fluid mixing is summarized here. It describes linear advection of a passive scalar by a random velocity field. Asymptotic scaling laws are given, which are consistently derived from leading order results of primitive and renormalized perturbation theory in both the Eulerian and Lagrangian pictures. The theory has been verified by numerical simulation, for moderately large perturbation parameters, and by comparison to an exactly solvable case. The theory is suitable for the description of the passive transport of pollutants in ground water. In this context, the time independent velocity $v$ is a random field, of nonzero mean $\bar v$, with a fluctuation $\delta v = v - \bar v$ which is used as the perturbation expansion parameter. For the purposes of deriving analytic asymptotics, $v$ is assumed to be a Gaussian random field. For the simulation studies of this theory, $v$ is derived from Darcy's law, with heterogeneous geology specified by a log--normal random permeability field.

The perturbation theory distinguishes three qualitatively distinct regimes. The infra--red finite case corresponds to a Fickean, or classical, diffusion process. The super--renormalizable case corresponds to anomalous diffusion where the dominant divergences in lowest order are of diffusion type. Field and experimental data appear to fall into this case. The infra--red nonrenormalizable case results in infinite scaling exponents, or non unique scaling behavior, which depends essentially on an infra--red regularization (cutoff scale). Practical applications of this theory to Kriging and conditional simulation are proposed.

The collision of a shock with a material interface produces a variety of complicated refraction patterns. In this article, transitions from self-similar refractions to more complicated configurations are studied using an approximate scattering analysis. This analysis suggests that there are five different regimes for the transition from a regular self-similar wave to a composite irregular wave. Two of these five cases have been incorporated into a front tracking code to provide enhanced resolution computations of such flows.

The interpolation of bivariate functions with discontinuous values is performed using a hybrid method of cubic spline interpolation and interface tracking. The resulting method retains the advantages of the high order of accuracy provided by the spline interpolation in regions where the function is smooth, while providing an accurate resolution of the discontinuities in the solution and minimizing oscillation in the interpolated solution due to interpolation across regions of high gradients.

The method of front tracking is a numerical method designed for the enhanced resolution of a distinguished set of waves in the solution to a system of partial differential equations. The article discusses the structure of a front tracking code that has been successfully applied to a variety of different interface dominated flows.

The region of $\W$ comprising admissible shock waves is bounded by the loci of structurally unstable dynamical systems. Explicit formulae are presented for the loci associated with saddle-node, Hopf, and Bogdanov-Takens bifurcation, and with straight-line heteroclinic connections. Using Melnikov's integral analysis, we calculate the tangent to the homoclinic part of the admissibility boundary at Bogdanov-Takens points of $\W$. Furthermore, using numerical methods, we explore the heteroclinic loci corresponding to curved connecting orbits and the complete homoclinic locus.

We find the region of admissible waves for a generic, two-dimensional slice of the fundamental wave manifold, and compare it with the set of shock points that comply with the Lax admissibility criterion, thereby elucidating how this criterion differs from viscous profile admissibility.

We analyze linearized Euler equations of a general stationary multiple layer stratified system for both compressible and incompressible inviscid fluids. Our main result is that many features of a multilayer system are universal, in the sense they do not depend on such details as the number of layers, their thicknesses, equations of state for the fluids, and equilibrium density distributions. Necessary and sufficient conditions of stability are determined. For compressible fluids, it is possible for the system to be unstable even if there is no density inversion anywhere. It is shown that a compressible system is always more unstable than the corresponding incompressible one. We give a universal upper bound for the growth rate for a given perturbation wave number. General \rt unstable modes are characterized, and the range of unstable wave numbers is determined. Properties of stable modes are discussed. Numerical algorithms for solving the eigenvalue problem of the set of linearized Euler equations are given.

Spatially heterogeneous permeability data from several sites is reviewed. The spatial correlations in this data are statistically significant for short distances only. Multifractal power-normal distributions fit this data about as well as the more commonly used exponential models, while better for long distance correlations inferred indirectly from hydrological studies, with the data combined between distinct sites. The data reviewed, when interpreted in the power-normal, multifractal model, is consistent with our earlier theoretical prediction that the fractal scaling exponent $\beta$ satisfies $\beta \in [0 , 1]$.

We study the macroscopic diffusion induced by a random velocity field on all time scales. We show that the transient growth rate of the mixing layer is nonfractal even for a fractal velocity field. For a velocity field with an exponentially decaying correlation function, the comparisons of the exact results under Corrsin's hyperthesis with the results from the linear and quasi-linear theories are presented.

An analytic structural reliability analysis is performed for first contact miscible, two phase flow in a one dimensional porous medium. The sensitivity of the probability for time-to-breakthrough is examined for two common models of the joint probability distribution between the porosity and the permeability of the medium. First order analysis is essentially exact for the normal--normal model. The accuracy of first order analysis for the normal--lognormal model is determined using a specific example of field data.

A cyclical dependence on the breakthrough probability to the statistical nature of the porosity and permeability is found in the normal--normal model. In the normal--lognormal model, this cyclical variation is reduced, if not entirely eliminated, in favor of dependence on the statistical nature of the permeability.

Dispersion is the result, observable on large length scales, of events which are random on small length scales. When the length scale on which the randomness operates is not small, relative to the observations, then classical dispersion theory fails. The scale up problem refers to situations in which randomness occurs on all length scales, and for which classical dispersion theory necessarily fails. The purpose of this article is to present non Fickian, theories of dispersion, which do not assume a scale separation between the randomness and the observed consequences, and which do not assume a single length scale.

Porous media flow properties are heterogeneous on all length scales. The geological variation on length scales below the observational length scale can be regarded as unknown and unknowable, and thus as a random variable.

We develop a systematic theory relating scaling behavior of the geological heterogeneity to the scaling behavior of the fluid dispersivity. Three qualitatively distinct regimes (Fickian, non Fickean and nonrenormalizable) are found. The theory gives consistent answers within several distinct analytic approximations, and with numerical simulation of the equations of porous media flow.

Comparison to field data is made. The use of Kriging to generate constrained ensembles for conditional simulation is discussed.

Fortunato A. Ascioti, Woods Hole Oceanographic Institute, Woods Hole, MA 02543, USA, and IMAAA CNR, Via S. Loja, Tito Scalo (PZ) 85050, Italy.

Edward Beltrami, Dept. of Applied Mathematics and Statistics and The Marine Sciences Research Center, SUNY at Stony Brook, Stony Brook, NY 11794, USA.

T. Owen Carroll, Institute for Pattern Recognition and The Harriman School of Managment, SUNY at Stony Brook, Stony Brook, NY 11794, USA.

Creighton Wirick, Oceanographic Sciences Division, Brookhaven National Laboratory, Upton, NY, 11973, USA.

A controversial issue in ecosystem modeling is whether the irregular fluctuations that one observes in nature are due solely to random environmental factors or whether, at least partially, a deterministic mechanism is responsible for the unpredictable behavior. This second alternative is called deterministic chaos and the issue in this paper is to decide if actual plankton time series can vindicate the hypothesis of chaotic dynamics. The near neighbor forecasting method is a recent technique for detecting determinism in a time series and we apply it to measurements of phytoplankton and zooplankton biomass obtained at a single station in the Middle Atlantic Bight. Although the results do not conclude the presence of chaos, they do give some support to the idea that deterministic nonlinear trophic dynamics may account for at least some of the variability that is seen in the data, particularly in terms of inferring zooplankton oscillations from those of phytoplankton.

We present a hybrid method for the minimization of the effective free energy of certain physical systems. The three stages of this method approximate the system interaction potential in three forms: (1) square-well, (2) quadratic, and (3) Lennard-Jones. The first stage uses a geometric method to minimize the free energy, based on a square-well potential. The second stage, assuming the pair interaction is quadratic, gives an analytical form for the minimization. The last stage, introducing more realistic physics, {\it i.e.}, a Lennard-Jones pair interaction, uses the Monte Carlo method to perform minimization.

The first stage is less accurate, but much more efficient, and eliminates most of the local minima of the free energy from further consideration. The refinement done by the latter steps reduces the error and delivers accurate results.

We present a hybrid method for the minimization of the effective free energy of a three-dimensional biological system. Three types of pair potential formulations are considered: (1) square-well, (2) quadratic, and (3) Lennard-Jones. The first part of the hybrid method uses a geometric method to match patterns approximately, based on a square well potential. The second part of the method is a closed form solution for minimization based on a quadratic potential. A Monte Carlo method, as a third step, is used to further prioritize the approximately matched patterns. Application of this method to the binding of proteins on DNA is discussed in this paper.

The essential mathematical issue addressed here is the efficient location of the global minimum and of all local minima which are approximate global minima for an objective function with a very large number of local minima.

A multinephron model of the inner medullary urinary concentrating mechanism in the mamalian kidney is described. A procedure to represent the distribution of nephrons with a finite number of different nephron types is introduced. The model is compared with a shunt model. These studies show that, for uniform transport permeabilities: (1) 40 properly weighted nephron types are sufficient to represent the distribution of nephrons; (2) the two models give essentially the same results.