Rigorous Link Research Articles

A Phosphorus Index transport factor based on variable source area hydrology for New York State

The Phosphorus (P) Index concept is used in many states to help develop nutrient management plans for livestock agriculture to protect water quality. Although many P indices conceptually incorporate variable source area (VSA) runoff processes, they generally assume proximity to a water course is an adequate proxy of runoff risk. Here we propose a VSA-based transport factor that uses the topographic index concept to indicate runoff risk. We compared both transport factors based on the current New York State P Index and our VSA-based P Index to field measures of runoff probability across an abandoned agricultural site in upstate New York. We also compared transport factors and P indices using the current and VSA-based approaches on a farm in the Catskill Mountains of New York to evaluate differences at whole-field and farm scales. Field runoff-risk measurements were better correlated with VSA-based transport factor (<i>r</i>² = 0.63, α = 0.05) than with the current dissolved-P transport factor (<i>r</i>² = 0.40, α = 0.05). Although these point-scale differences in transport factor values translated into field-scale differences in transport factor, the net differences at the farm scale and in P Index were not very large. On a field-by-field basis, 12 out of 21 fields had different transport factor categories with the VSA method. However, the total land area classified as high risk changed very little between the two methods. There was more land classified as moderate risk using the VSA-based approach than using the current methods, due to some low risk areas being classified as higher risk and some high-risk areas being classified as lower risk. The VSA approach allows managers and producers to more easily manage farm units (e.g., fields) at finer resolutions both spatially and temporally, which will increase the options for managing nutrients on fields. These types of more rigorous links between management tools and physical hydrology provide valuable, more scientifically defensible information for improving our ability to control nonpoint source pollution.

Transmission of 640-Gb/s RZ-OOK Channel Over 100-km SSMF by Wavelength-Transparent Conjugation

We report the first demonstration of 640-Gb/s return-to-zero ON-OFF keying channel transmission over a 100-km standard single-mode fiber link employing midspan phase conjugation. A frequency-degenerate conjugate field spanning more than 20 nm is created in a low-birefringence parametric mixer for the first time. Physical separation of the conjugate field from the original field is enabled by utilizing pump polarization nondegeneracy. Rigorous link characterization using a high-quality 640-Gb/s transmitter and a high-sensitivity receiver revealed error-free (bit error ratio <;10-9) performance, eliminating the need for impractical fiber length control or electronic signal processing. The compatibility of wavelength-transparent conjugation with spectrally inefficient channel implies that channels with higher bit rates and better spectral efficiency can also be compensated.

Renormalization of tracer turbulence leading to fractional differential equations

For many years quasilinear renormalization has been applied to numerous problems in turbulent transport. This scheme relies on the localization hypothesis to derive a linear transport equation from a simplified stochastic description of the underlying microscopic dynamics. However, use of the localization hypothesis narrows the range of transport behaviors that can be captured by the renormalized equations. In this paper, we construct a renormalization procedure that manages to avoid the localization hypothesis completely and produces renormalized transport equations, expressed in terms of fractional differential operators, that exhibit much more of the transport phenomenology observed in nature. This technique provides a first step toward establishing a rigorous link between the microscopic physics of turbulence and the fractional transport models proposed phenomenologically for a wide variety of turbulent systems such as neutral fluids or plasmas.

Statistical-mechanical description of classical test-particle dynamics in the presence of an external force field: modelling noise and damping from first principles

Aiming to establish a rigorous link between macroscopic random motion (described e.g. by Langevin-type theories) and microscopic dynamics, we have undertaken a kinetic-theoretical study of the dynamics of a classical test-particle weakly coupled to a large heat-bath in thermal equilibrium. Both subsystems are subject to an external force field. From the (time-non-local) generalized master equation a Fokker-Planck-type equation follows as a "quasi-Markovian" approximation. The kinetic operator thus defined is shown to be ill-defined; in specific, it does not preserve the positivity of the test-particle distribution function $f(\mathbf{x}, \mathbf{v}; t)$. Adopting an alternative approach, previously proposed for quantum open systems, is proposed to lead to a correct kinetic operator, which yields all the expected properties. A set of explicit expressions for the diffusion and drift coefficients are obtained, allowing for modelling macroscopic diffusion and dynamical friction phenomena, in terms of an external field and intrinsic physical parameters.

Rotation and anisotropy of galaxies revisited

The use of the tensor virial theorem (TVT) as a diagnostic of anisotropic velocity distributions in galaxies is revisited. The TVT provides a rigorous global link between velocity anisotropy, rotation and shape, but the quantities appearing in it are not easily estimated observationally. Traditionally use has been made of a centrally averaged velocity dispersion and the peak rotation velocity. Although this procedure cannot be rigorously justified, tests on model galaxies show that it works surprisingly well. With the advent of integral-field spectroscopy it is now possible to establish a rigorous connection between the TVT and observations. The TVT is reformulated in terms of sky-averages, and the new formulation is tested on model galaxies.

Towards a rigorous derivation of the fifth order KP equation

The Kadomtsev–Petviashvili equations (KP) are universal models for dispersive, weakly nonlinear, almost one-dimensional long waves of small amplitude. In [L. Paumond, A rigorous link between KP and a Benney–Luke equation, Differential Integral Equations 16 (9) (2003) 1039–1064], we proved a rigorous link between KP and a Benney–Luke equation. Here, we derive a new model still valid when the Bond number is equal to 1/3. Following the work [R.L. Pego, J.R. Quintero, Two-dimensional solitary waves for a Benney–Luke equation, Physica D 132 (4) (1999) 476–496], we show the existence of solitary waves for this equation and their convergence towards the solitary waves of the fifth order KP equation. We also link rigorously the dynamics of the two equations.

Gyroscopic effect in compact toroidal plasmas

The gyroscopic effect in compact toroidal plasmas is investigated using a minimum energy principle in which the angular momentum is added as a constraint. If attention is limited to incompressible plasmas, the flow and field vectors can be expanded in eigenfunctions of the curl (Beltrami functions). Then the flow and field appear, in general, as a spectrum of Beltrami vectors. Adding the angular momentum as a constraint has a strong effect on the energy ordering of the system. Without this constraint, the state with lowest “ordered” energy has a fine structure element that would, in practice, decay, leading to a force-free state. However, if a nonzero angular momentum is specified, then the lowest-energy state has finite pressure and significant flow. For angular momentum of sufficient magnitude, this is a “smooth” state that should have good magnetic confinement. These effects may be an indicator that the gyroscopic effect improves the stability. However, this is by no means certain since a rigorous link between minimum energy and stability in a flowing plasma has not yet been established.

Computational Evaluation of Strain Gradient Elasticity Constants

Classical effective descriptions of heterogeneous materials fail to capture the influence of the spatial scale of the heterogeneity on the overall response of components. This influence may become important when the scale at which the effective continuum fields vary approaches that of the microstructure of the material and may then give rise to size effects and other deviations from the classical theory. These effects can be successfully captured by continuum theories that include a material length scale, such as strain gradient theories. However, the precise relation between the microstructure, on the one hand, and the length scale and other properties of the effective modeling, on the other, are usually unknown. A rigorous link between these two scales of observation is provided by an extension of the classical asymptotic homogenization theory, which was proposed by Smyshlyaev and Cherednichenko (J. Mech. Phys. Solids 48:1325‐1358, 2000) for the scalar problem of antiplane shear. In the present contribution, this method is extended to three-dimensional linear elasticity. It requires the solution of a series of boundary value problems on the periodic cell that characterizes the microstructure. A finite element solution strategy is developed for this purpose. The resulting fields can be used to determine the effective higherorder elasticity constants required in the Toupin-Mindlin strain gradient theory. The method has been applied to a matrix-inclusion composite, showing that higher-order terms become more important as the stiffness contrast between inclusion and matrix increases.

Equilibrium behavior of sessile drops under surface tension, applied external fields, and material variations

This article describes the equilibrium shape of a liquid drop under applied fields such as gravity and electrical fields, taking into account material properties such as dielectric constants, resistivities, and surface tension coefficients. The analysis is based on an energy minimization framework. A rigorous and exact link is provided between the energy function corresponding to any given physical phenomena, and the resulting shape and size dependent force term in Young’s equation. In particular, the framework shows that a physical effect, such as capacitive energy storage in the liquid, will lead to 1/R “line-tension”-type terms if and only if the energy of the effect is proportional to the radius of the liquid drop: E∝R. The effect of applied electric fields on shape change is analyzed. It is shown that a dielectric solid and a perfectly conducting liquid are all that is needed to exactly recover the Young–Lippmann equation. A dielectric liquid on a conducting solid gives rise to line tension terms. Finally, a slightly resistive liquid on top of a dielectric, highly resistive solid gives rise to contact angle saturation and accurately matches the experimental data that we observe in our electro-wetting-on-dielectric devices.

A rigorous link between KP and a Benney-Luke equation

Kadomtsev-Petviashvili (KP) equations model weakly nonlinear dispersive waves, which are essentially unidimensional, when weak transverse effects are taken into account. These equations can be formally obtained by asymptotic methods from the Euler equations, but there is no rigorous justification yet. In this paper we consider an intermediate equation (BL) derived first by Benney and Luke. (BL) reduces formally to (KP) if we seek waves propagating essentially in one direction, with a weak variations in time and in the transverse direction measured by a small parameter ${\varepsilon}$. We show rigorously that the $L^2(\mathbb R^2)$-norm of the difference between the amplitude of the wave given by (KP) and the one given by (BL) is of order $\mathcal{O}({\varepsilon}^{3/4})$ during a growing with ${\varepsilon}$ time.

Lagrangian coherent structures and mixing in two-dimensional turbulence

We introduce a Lagrangian definition for the boundaries of coherent structures in two-dimensional turbulence. The boundaries are defined as material lines that are linearly stable or unstable for longer times than any of their neighbors. Such material lines are responsible for stretching and folding in the mixing of passive tracers. We derive an analytic criterion that can be used to extract coherent structures with high precision from numerical or experimental data sets. The criterion provides a rigorous link between the Lagrangian concept of hyperbolicity, the Okubo–Weiss criterion, and vortex boundaries. We apply the results to simulations of two-dimensional barotropic turbulence.

$N$ -Body Quantum Scattering Theory in Two Hilbert Spaces: $N$ -Body Integral Equations

Scattering for a nonrelativistic system of \(\) distinguishable and spinless particles interacting via short-range pair potentials is considered. Half-on-shell integral equations (the CG equations) are proposed, the solutions of which determine approximate scattering amplitudes that converge to the exact scattering amplitude. It is proved, under mild Holder integrability assumptions, that these apparently singular equations actually have a compact kernel for real energies and, consequently, a unique solution. The CG equations have a structure that is much simpler than the Yakubovskii equations and similar to that of coupled-reaction-channel equations. The driving terms look like distorted-wave Born integrals and nonorthogonality integrals. However, there is no restriction to channels with only two asymptotic bound clusters and for all channels, no matter how many bound clusters, appropriate boundary conditions are exactly satisfied. This work completes the establishment of a rigorous mathematical link between the solutions of the half-on-shell CG equations and the on-shell transition operators defined in time-dependent multichannel scattering theory, and it provides for the first time a rigorous theoretical basis for practical calculations of scattering amplitudes for certain problems with \(\).

Rigorous link between the conductivity and elastic moduli of fibre-reinforced composite materials

We derive rigorous cross-property relations linking the effective transverse electrical conductivity cr* and the effective transverse elastic moduli of any transversely isotropic, two-phase ‘fibre-reinforced’ composite whose phase boundaries are cylindrical surfaces with generators parallel to one axis. Specifically, upper and lower bounds are derived on the effective transverse bulk modulus k* in terms of cr* and on the effective transverse shear modulus //* in terms of cr*. These bounds enclose certain regions in the ct*-ac* and cr*-/r* planes, portions of which are attainable by certain microgeometries and thus optimal. Our bounds connecting the effective conductivity cr* to the effective bulk modulus ft* apply as well to anisotropic composites with square symmetry. The implications and utility of the bounds are explored for some general situations, as well as for specific microgeometries, including regular and random arrays of circular cylinders, hierarchical geometries corresponding to effective-medium theories, and checkerboard models. It is shown that knowledge of the effective conductivity can yield sharp estimates of the effective elastic moduli (and vice versa), even for infinite phase contrast.

Rigorous Link Between the Electrical and Mechanical Properties of Composite Materials

ABSTRACTCross-property relations that link rigorously the effective electrical conductivity (or dielectric constant) and the effective elastic moduli of two-phase, isotropic composite materials are discussed. The cross-property relations can be optimal in some cases, i.e., they are realized by particular microstructures. The relations are applied to specific two-phase composites as well as to cracked media.

Comments on ‘‘Rigorous link between fluid permeability, electrical conductivity, and relaxation times for transport in porous media’’

Recently, Avellaneda and Torquato [Phys. Fluids A 3, 2529 (1991)] derived several expressions for both the static and dynamic permeability for flow through porous media, in terms of the characteristic viscous relaxation times. In this Brief Communication the focus is on the physical interpretation, Darcy’s law is explicitly obtained, and a slightly misleading statement (which has no effect on the mathematics but may induce erroneous interpretations) is corrected.

Performance of a simulation model describing protozoa-induced mineralization of bacterial C and N in a sandy loam

Mathematical modelling of a microcosm experiment provided a rigorous link between soil organism populations and the processes in the C and N cycles for a clay loam soil. In order to check the internal consistency of the model, we tested its performance with the data set obtained for a sandy loam. We empirically lowered the value of the half saturation constant for the protozoan consumption of bacteria from 175 to 50 (μg C g−1 soil) and initialized the state variables with the appropriate values. The model was also used to quantify the effect of texture on bacterial-protozoan interactions and flows of C and N using the simulated data for a clay loam and a sandy loam. The CO2-C evolved during the first 12 d was mainly due to glucose addition (180 μg C g−1 soil) and cycling of C in the soil (130 μg C g−1 soil). During this interval, C inputs into bacteria were 14-fold greater than the initial pool size. Protozoa increased total CO2-C evolution by 16% and increased net NH4-N mineralization 3.5-fold compared with the protozoa-minus treatment. Mineralization of bacterial C and N was more rapid in the sandy loam than in the clay loam. Key words: N mineralization-immobilization, bacteria, model, protozoa, Typic Cryoboroll, sandy loam

Diffusion and reaction among traps: some theoretical and simulation results

Diffusion and reaction in heterogeneous media arise in a host of phenomena in the physical and biological sciences. The determination of the mean survival timeτ (i.e., inverse trapping rate) and relaxation timesTn, n=1,2,3,... (i.e., inverse eigenvalues), associated with diffusion among partially absorbing, static traps with surface rate constantk are problems of long-standing interest. The limitsk=∞ andk=0 correspond to the diffusion-controlled case (i.e., perfect absorbers) and reaction-controlled case (i.e., perfect reflectors), respectively. This paper reviews progress we have made on several basic aspects of this problem: (i) the formulation of rigorous bonding techniques and computational methodologies that enable one to estimate the mean survival time τ and principal relaxation timeT1 (ii) the quantitative characterization of the microstructure of nontrivial continuum (i.e., off-lattice) models of heterogeneous media; and (iii) evaluation of τ and T1 for the same models. We also describe a rigorous link between the mean survival time t and a different effective parameter of the system, namely the fluid permeability tensork associated with Stokes flow through the same porous medium.

Rigorous link between fluid permeability, electrical conductivity, and relaxation times for transport in porous media

A rigorous expression is derived that relates exactly the static fluid permeability k for flow through porous media to the electrical formation factor F (inverse of the dimensionless effective conductivity) and an effective length parameter L, i.e., k=L2/8F. This length parameter involves a certain average of the eigenvalues of the Stokes operator and reflects information about electrical and momentum transport. From the exact relation for k, a rigorous upper bound follows in terms of the principal viscous relation time Θ1 (proportional to the inverse of the smallest eigenvalue): k≤νΘ1/F, where ν is the kinematic viscosity. It is also demonstrated that νΘ1≤DT1, where T1 is the diffusion relaxation time for the analogous scalar diffusion problem and D is the diffusion coefficient. Therefore, one also has the alternative bound k≤DT1/F. The latter expression relates the fluid permeability on the one hand to purely diffusional parameters on the other. Finally, using the exact relation for the permeability, a derivation of the approximate relation k≂Λ2/8F postulated by Johnson et al. [Phys. Rev. Lett. 57, 2564 (1986)] is given.

Adiabatic elimination in nonlinear dynamical systems

We reconsider the problem of the adiabatic elimination of selected dynamical variables in the description of nonlinear systems, with a special emphasis on the identification of suitable criteria for the global validity of this procedure. The problem is analyzed in detail using as a guideline the one-mode homogeneously broadened laser model, with an injected signal and an arbitrary population difference for added flexibility. We propose five conditions for the global validity of the adiabatic limit, after consideration not only of the relative size of the time scales involved, but also of the magnitude of all parameters, of the physical variables, and of their fluctuations. The adiabatic elimination is formulated in the context of the dressed-mode description, leading to a generalization of the procedure adopted in earlier studies of multimode absorptive bistability. In addition, we establish a rigorous link between the adiabatic-elimination procedure and the multiple-time-scale approach to dynamic systems. The scaling behavior of the variables, of the eigenvalues of the linearized equations, and of the dressed-mode amplitudes and eigenvectors with respect to the smallness parameter of the problem is studied in detail. The relevance of what we shall define as "normal" and "anomalous" scaling in the text to the applicability of the adiabatic-elimination process is clarified. We also develop an alternative adiabatic-elimination scheme for the special case where all the rate constants of the system have the same order of magnitude, but another smallness parameter can be identified. From our analysis, it should be clear that our main conclusions are model independent, and not at all restricted to the specific features of the dynamical system selected as a test case for our discussion.