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Results for “dispersive equations”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Physics-Informed Neural Network Method for Forward and Backward Advection-Dispersion Equations

Advection-dispersion equations (ADEs) are commonly used to describe transport phenomena in porous media. Even though mature discretization-based numerical methods for ADEs exist, some challenges still remain, especially when it comes to solving advection-dominated forward ADEs and diffusion-dominated backward ADEs. The latter problem usually arises in the source identification context and leads to numerically unstable grid-based solutions that require a form of regularization or should be treated as an inverse problem that is computationally more expensive because it requires solving the forward problem multiple times. In this study, we propose a discretization-free approach based on the physics-informed neural network (PINN) method for solving coupled ADE and Darcy flow equations with space-dependent hydraulic conductivity. In this approach, the hydraulic conductivity, hydraulic head, and concentration fields are approximated with deep neural networks (DNNs). We assume that the conductivity field is given by its values on a grid, and we use these values to train the conductivity DNN. The head and concentration DNNs are trained by minimizing the residuals of the flow equation and ADE and using the initial and boundary conditions as additional constraints. The PINN method is applied to one- and two-dimensional forward ADE problems, where its performance for various P\'{e}clet numbers ($Pe$) is compared with the analytical and numerical solutions. We find that the PINN method is accurate with errors of less than 1\% and outperforms some conventional discretization-based methods for $Pe$ larger that 100. Next, we demonstrate that the PINN method remains accurate for the backward ADEs, with the relative errors in most cases staying under 5\% compared to the reference concentration field. Finally, we show that when available, the concentration measurements can be easily incorporated in the PINN method and significantly improve (by more than 50\% in the considered cases) the accuracy of the PINN solution of the backward ADE.

He, Qizhi↗
NASA NTRS2019

Electromagnetic Scattering by Discrete Random Media. I: The Dispersion Equation and the Configuration-Averaged Exciting Field

We consider the scattering of a plane electromagnetic wave obliquely incident on a plane-parallel layer of discrete random medium with non-scattering boundaries. We solve the Lax integral equation for the conditional configuration-averaged exciting field coefficients by assuming a special-form solution, that is, by representing the conditional configuration-averaged exciting field coefficients as a linear combination of the coefficients corresponding to an up-going and a down-going wave. This solution representation is supposed to be valid within the whole domain occupied by the particles, even in the close proximity of the boundaries. By balancing the waves with different propagation directions and wavenumbers we derive two homogeneous systems of equations corresponding to the generalized Lorenz–Lorentz law and two inhomogeneous systems of equations corresponding to the generalized Ewald–Oseen extinction theorem. It is shown that (i) the two homogeneous systems of equations of the generalized Lorenz–Lorentz law reduce to a single homogeneous system of equations corresponding to a semi-infinite discrete random medium at normal incidence; (ii) the dispersion equation is direction and polarization independent; and (iii) the two inhomogeneous systems of equations of the generalized Ewald–Oseen extinction theorem can be reduced to two scalar equations by means of the addition theorem for vector spherical harmonics. It is also shown that the same dispersion equation can be obtained without assuming a special-form solution representation in the proximity of the boundaries.

Adrian Doicu↗
NASA NTRS1987

A quartic dispersion equation for internal gravity waves in the thermosphere

A new quartic dispersion equation in the square of the complex vertical wave number is derived by employing the 'shallow atmosphere' approximation and an ion drag approximation. These approximations allow the coefficients of the quartic equation to be given in terms of the corresponding cubic equation (which neglects the Coriolis force and the zonal ion drag component), but modified to take into account these neglected effects. Coupling between the extraordinary viscosity wave mode and the other three wave modes is highlighted and numerical solutions are compared for this quartic equation, an exact eighth order equation and the cubic equation. For the first time the validity of using the 'shallow atmosphere' approximation to describe internal gravity wave motions is demonstrated.

Hickey, M. P.↗
DOE OSTI2021

Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications

Fractional calculus-based differential equations were found by previous studies to be promising tools in simulating local-scale anomalous diffusion for pollutants transport in natural geological media (geomedia), but efficient models are still needed for simulating anomalous transport over a broad spectrum of scales. This study proposed a hierarchical framework of fractional advection-dispersion equations (FADEs) for modeling pollutants moving in the river corridor at a full spectrum of scales. Applications showed that the fixed-index FADE could model bed sediment and manganese transport in streams at the geomorphologic unit scale, whereas the variable-index FADE well fitted bedload snapshots at the reach scale with spatially varying indices. Further analyses revealed that the selection of the FADEs depended on the scale, type of the geomedium (i.e., riverbed, aquifer, or soil), and the type of available observation dataset (i.e., the tracer snapshot or breakthrough curve (BTC)). When the pollutant BTC was used, a single-index FADE with scale-dependent parameters could fit the data by upscaling anomalous transport without mapping the sub-grid, intermediate multi-index anomalous diffusion. Pollutant transport in geomedia, therefore, may exhibit complex anomalous scaling in space (and/or time), and the identification of the FADE’s index for the reach-scale anomalous transport, which links the geomorphologic unit and watershed scales, is the core for reliable applications of fractional calculus in hydrology.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Rigid-Mode Limit of the Yokoya Matrix Formalism and the Burov-Lebedev Dispersion Equation

Transverse single-bunch instabilities of space-charge-dominated coasting beams with round and flat transverse geometries are studied using a unified dispersion-relation framework. The analysis combines the Burov-Lebedev formalism, which captures space-charge tune spread, Landau damping, and instability threshold behavior, with Yokoya’s projection method for representing coherent transverse mode structure and its dependence on beam aspect ratio. In the rigid-beam limit, the formulation reduces to a scalar dispersion relation of Burov-Lebedev paper. For non-rigid transverse oscillations, truncation of Yokoya’s Hermite-based expansion yields a finite-dimensional matrix eigenvalue problem in which space-charge and coupling impedance effects enter through Burov-Lebedev–type denominators. This approach provides a consistent basis for comparing rigid and non-rigid instability behavior in round and flat beams and for assessing the role of beam ellipticity in modifying coherent mode structure and stability thresholds.

43 PARTICLE ACCELERATORS↗
NASA NTRS1985

New Instabilities in Line Driven Winds

The physical mechanisms which potentially lead to instabilities in line driven winds, the drift instability and the line shape instability, are discussed. A general three dimensional treatment of the stability problem of line driven winds which leads to the general dispersion equation is proposed. From this dispersion equation automatically a third physical mechanism driving instability in stellar winds is deduced; the thermal drift instability which is related to changes in absorption of radiation caused by temperature perturbations. This mechanism results in growing inwardly propagating sound waves.

Martens, P. C. H.↗
NASA NTRS1978

The coupling of Alfven and compressional waves

The article studies the hydromagnetic wave propagation characteristics in a mixture of cold and hot plasma in the presence of an inhomogeneous magnetic field. Electron and ion distribution functions with a temperature anisotropy and a density gradient are used to obtain the dispersion equation by solving the Vlasov equation and Maxwell equations. From the solutions of the dispersion equation we find that the Alfven waves can couple to unstable drift mirror waves under certain conditions. The polarization of the coupled waves is studied for varying parameters of temperature anisotropy and the cold to hot density ratio. From detailed comparison of the theoretical results with the low-frequency wave properties observed in the magnetosphere we propose that the storm-associated magnetic field oscillations with periods of 100-600 s might be caused by the coupling of Alfven waves and the unstable drift mirror waves.

Lin, C. S.↗
NASA NTRS1994

Theory of 2 omega(sub pe) radiation induced by the bow shock

A new radiation emission mechanism is proposed to explain electomagnetic radiation observed at twice the electron plasma frequency, 2 omega(sub pe), in the upstream region of the Earth's bow shock. This radiation had its origin at the electron foreshock boundary where energetic electron beams and intense narrow-band Langmiur waves are observed. The proposed emission mechanism results from the interaction of the electron beam and Langmuir waves that are backscattered off thermal ions. This interaction is described by a nonlinear dispersion equation which incorporates an effect owing to electron trajectory modulation by the backscattered Langmuir waves. Subsequent analysis of the dispersion equation reveals two important consequences. First, a long-wavelength electrostatic quasi-mode with frequency at 2 omega(sub pe) is excited, and second, the quasi-mode and the electomagnetic mode are nonlinearly coupled. The implication is that, when the excited 2 omega(sub pe) quasi-mode propagates in an inhomgeneous medium with slightly decreasing density, the quasi-mode can be converted directly into an electromagnetic mode. Hense the electomagnetic radiation at twice the plasma frequency is generated. Numerical solutions of the dispersion equation with the choice of parameters that describe physical characteristics of the electron foreshock are presented, which illustrates the viability of the new mechanism.

Yoon, Peter H.↗
NASA NTRS1988

Plasma instabilities stimulated by HF transmitters in space

Diffuse incoherent signal returns are often observed on Alouette and ISIS topside ionograms in addition to coherent echoes of electromagnetic and electrostatic waves. These diffuse signals, which at times can be the dominant features on topside ionograms, have been attributed to sounder-induced temperature anisotropies which drive the Harris instability. Previous theoretical investigations were based on the electrostatic approximation to the dispersion equation. The present paper will present calculations indicating that when the electromagnetic terms are retained in the dispersion equation and when the sounder-stimulated perpendicular electron temperature approaches 1 keV, then the whistler mode can have a temporal growth rate larger than the electrostatic electron cyclotron harmonic wave mode central to the diffuse resonance problem. Present sounders lack the power and antenna lengths to generate whistler mode waves in this manner. In addition, such waves would have large group velocities and would quickly leave the vicinity of the sounder. Experiments to investigate the wave growth, propagation, and damping of such stimulated waves are planned for the 1990s using a highly flexible sounder on the Space Shuttle and a receiver on a subsatellite.

Benson, Robert F.↗
NASA NTRS1989

Numerical modeling for primary atomization of liquid jets

In the proposed numerical model for primary atomization, surface-wave dispersion equations are solved in conjunction with the jet-embedding technique of solving mean flow equations of a liquid jet. Linear and approximate nonlinear models have been considered. In each case, the dispersion equation is solved over the whole wavelength spectrum to predict drop sizes, frequency, and liquid-mass breakup rates without using any empirical constants. The present model has been applied to several low-speed and high-speed jets. For the high-speed case (the LOX/H2 coaxial injector of the Space Shuttle Main Engine Preburner), predicted drop sizes and liquid breakup rates are in good agreement with the results of the CICM code, which have been calibrated against measured data.

Przekwas, A. J.↗
NASA NTRS1992

Electrostatic ion cyclotron velocity shear instability

A local electrostatic dispersion equation is derived for a shear flow perpendicular to an ambient magnetic field, which includes all kinetic effects and involves only one important parameter. The dispersion equation is cast in the form of Gordeyev integrals and is solved numerically. Numerical solutions indicate that an ion cyclotron instability is excited. The instability occurs roughly at multiples of the ion cyclotron frequency (modified by the shear), with the growth rate or the individual harmonics overlapping in the wavenumber. At large values of the shear parameter, the instability is confined to long wavelengths, but at smaller shear, a second distinct branch at shorter wavelengths also appears. The properties of the instability obtained are compared with those obtained in the nonlocal limit by Ganguli et al. (1985, 1988).

Lemons, D. S.↗
NASA NTRS1985

Lower-hybrid instabilities driven by an ion velocity ring

The lower-hybrid instabilities in high-beta (ratio of plasma to magnetic pressure) plasmas driven by ring-ion distributions in velocity space are investigated. A dispersion equation including electromagnetic effects is derived. In the low-beta limit, analytic expressions are obtained which illuminate the physical nature of the instabilities. The complete dispersion equation is solved numerically as a function of ring speed and plasma beta for several types of ring distribution. Electromagnetic effects are important for relatively energetic rings even in the low-beta regime, suppressing growth rates and shifting the angle of propagation to more oblique angles. Stabilization by thermal effects is also discussed. Application of these results to the earth's bow shock, AMPTE, comets and solar flares is suggested.

Akimoto, K.↗
NASA NTRS1999

Selection of Optical Glasses Using Buchdahl's Chromatic Coordinate

This investigation attempted to extend the method of reducing the size of glass catalogs to a global glass selection technique with the hope of guiding glass catalog offerings. Buchdahl's development of optical aberration coefficients included a transformation of the variable in the dispersion equation from wavelength to a chromatic coordinate omega defined as omega = (lambda - lambda(sub 0))/ 1 + 2.5(lambda - lambda(sub 0)) where lambda is the wavelength at which the wavelength is calculated and lambda(sub 0) is a base wavelength about which the expansion is performed. The advantage of this approach is that the dispersion equation may be written in terms of a simple power series and permits direct calculation of dispersion coefficients. While several promising examples were given, a systematic application of the technique to an entire glass catalog and analysis of the subsequent predictions was not performed. The goal of this work was to apply the technique in a systematic fashion to glasses in the Schoft catalog and assess the quality of the predictions.

Griffin, DeVon W.↗
NASA NTRS1989

Effects of unequal particle number densities on Alfven waves

Analytic plasma theory and numerical solutions of the dispersion equation are used to show that the assumption that the linear properties of the waves are determined by a charge-neutral plasma in the absence of the nonthermal particles, while the nonthermal particles cause growth or additional damping superposed onto the background, is seriously flawed even for stable plasmas. Even when the nonthermal particles do not contribute significantly to the dispersion equation, unequal thermal electron and ion number densities (due to the presence of the nonthermal particles) may cause fundamental low wave number modifications to the Alfven modes, including the creation of a new resonance and severely modified dispersion. These results are found for both cold and warm plasmas. Previous work on Alfven waves should be reevaluated in view of these results.

Cairns, I. H.↗
DOE OSTI2021

A statistical study of three-second foreshock ULF waves observed by the Magnetospheric Multiscale mission

We perform a statistical study of 3-s ultra-low frequency (ULF) waves using Magnetospheric Multiscale observations in the Earth's foreshock region. The average phase velocity in the plasma rest frame is determined to be anti-sunward, and the intrinsic polarization is right-handed. We further examine the linear instability conditions based on the drift-bi-Maxwellian distribution functions according to the observed plasma conditions. The resulting instability is a solution to the common dispersion equation of the ion/ion right-hand non-resonant and left-hand resonant instabilities. The predicted wave propagation is also predominantly anti-sunward. The cyclotron resonant conditions of the solar wind and backstreaming beam ions are evaluated, and we find that, in some cases, the anti-sunward propagating waves can be resonant with beam ions, which was overlooked in previous studies. The study suggests that the dispersion equation provides the 3-s ULF waves a fundamental explanation that unifies a rich variety of resonant conditions.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1985

Collisionless tearing in a field-reversed sheet pinch assuming nonparallel propagation

The problem of collisionless linear tearing is examined assuming a wave vector with a component normal to the equilibrium field. The geometry is defined and the general form of the linear dispersion equation is calculated. The linear theory results when k is parallel to B are reviewed, and Ampere's law is calculated for the external adiabatic region when k times B does not equal zero, using two-fluid theory. A solution is obtained for the approximate form of the perturbed currents and vector potential assuming quasi-parallel k. The resonant current contributions within the singular layer are calculated, obtaining an estimate of the dispersion equation. The form of the adiabatic currents within the singular layer is calculated, showing that an x-z current system persists even in the limit k perpendicular to B goes to zero. Finally, the perturbed vector potential solutions across the singular layer are matched to obtain the shape of the complete eigenfunction.

Quest, K. B.↗