Nonlinear Wave Model Research Articles

Dispersion management for solitons in a Korteweg-de Vries system.

The existence of "dispersion-managed solitons," i.e., stable pulsating solitary-wave solutions to the nonlinear Schrodinger equation with periodically modulated and sign-variable dispersion is now well known in nonlinear optics. Our purpose here is to investigate whether similar structures exist for other well-known nonlinear wave models. Hence, here we consider as a basic model the variable-coefficient Korteweg-de Vries equation; this has the form of a Korteweg-de Vries equation with a periodically varying third-order dispersion coefficient, that can take both positive and negative values. More generally, this model may be extended to include fifth-order dispersion. Such models may describe, for instance, periodically modulated waveguides for long gravity-capillary waves. We develop an analytical approximation for solitary waves in the weakly nonlinear case, from which it is possible to obtain a reduction to a relatively simple integral equation, which is readily solved numerically. Then, we describe some systematic direct simulations of the full equation, which use the soliton shape produced by the integral equation as an initial condition. These simulations reveal regions of stable and unstable pulsating solitary waves in the corresponding parametric space. Finally, we consider the effects of fifth-order dispersion. (c) 2002 American Institute of Physics.

Profile Position Control of Batch Distillation Based on a Nonlinear Wave Model

Batch distillation poses challenging operation and control problems caused by a great change in process gain and a very nonlinear response in distillate composition during the operation. A nonlinear wave model captures an essential dynamic characteristic of the batch distillation column. A profile position control strategy based on the nonlinear wave model is proposed for the control of batch distillation columns. The profile position and distillate composition are estimated from selected tray temperature measurements by observers, which are based on the combination of the nonlinear wave propagation model and the quasi-dynamic model by Quintero-Marmol and Luyben (Chem. Eng. Sci. 1992, 47, 887). This control strategy has been applied to binary and multicomponent separation of ideal and nonideal mixtures. The proposed control scheme is shown to deal with severe nonlinear and varying process gains of batch operation and to be robust to model−plant mismatches.

Nonlinear strain waves and densities of defects induced in metal plates by external energy fluxes

A model of nonlinear periodic traveling strain waves induced in laser-irradiated metal plates with quadratic nonlinearity is proposed. Interaction between the elastic strain fields and the concentration of point defects is taken into account. The effect of generation-recombination processes on the evolution of nonlinear localized waves is considered. An equation for the amplitudes of the nonlinear waves is derived. It is employed to analyze the attenuation of the waves with allowance for low-and high-frequency losses.

Nonlinear Stationary Wave Maintenance and Seasonal Cycle in the GFDL R30 GCM

Abstract In this study, the climatological stationary wave maintenance is examined from nonlinear perspective using the GFDL R30 GCM outputs, a fully nonlinear stationary wave model, and a linear stationary wave model. The primary focus of the study is on the nature of the stationary nonlinearity and relative contribution to the total nonlinearity by various factors, such as heating, orography, and the interaction between flows forced by heating and orography. It is found that both the nonlinear effect of the diabatic heating and the nonlinear interaction between flows forced by orography and diabatic heating are important contributors toward the total stationary nonlinearity in northern winter and summer. Some regional features, such as the anticyclone off the northwest coast of North America in winter and the southwestern U.S. summer anticyclone, are entirely due to the nonlinear interaction between flows forced by heating and orography. Consistent with the linear stationary wave maintenance, the diabat...

Profile position control of distillation columns based on a nonlinear wave model

A nonlinear wave model captures an essential dynamic chanctcristic of a variety of distillation columns. A profile position control strategy based on the non1inear wave model is proposed for the control of a variety of distillation colunms such as high purity distillation column, complex distillation configurations, batch distillation configurations and for the startup of the columns. In this paper, we are mainly concerned with the control of batch distillation without and with chemical reaction among them. The controller has been developed by incOIpOrating nonlinear wave model into the generic model control framework. An observer based on the nonlinear wave model is also proposed to determine the profile position. Tight control of profile position, which leads to fast stabilization of product compositions. has been achieved. Since the profile position control alone produces a slight offset from the desired product concentration, the composition/profile position cascade system has been used to remove the offset.

Improvement of Parabolic Nonlinear Dispersive Wave Model

Improvements to a previously published nonlinear parabolic wave model are developed and implemented. A second-order correction to a free-surface boundary condition used to develop the original model is formulated. The correction takes into account the complete second-order transformation between amplitudes of the velocity potential and those of the free-surface elevation. Additionally, wide-angle propagation terms are included in the model. It is shown that the model with the second-order correction retains the properties of third-order Stokes theory quite well in deep water. Comparisons of model behavior to data reveal that both nonlinearity and wide-angle propagation effects need to be included in the model for general wave transformation problems in shallow water. Skewness predictions are considerably improved by using both the second-order correction and by retaining a greater number of frequency components in the calculation. Asymmetry calculations are aided by incorporation of frequency-squared weighting for distribution of the dissipation function. Further improvement may entail a different form of the breaking model.

Linear stability of solitary waves of the Green‐Naghdi equations

AbstractWe investigate the eigenvalue problem obtained from linearizing the Green‐Naghdi equations about solitary wave solutions. Unlike weakly nonlinear water wave models, the physical system considered here has nonlinearity in its highest derivative term. This results in more detailed asymptotic analysis of the eigenvalue problem in the presence of a large parameter. Combining the technique of singular perturbation with the Evans function, we show that for solitary waves of small amplitude, the problem has no eigenvalues of positive real part and the Evans function is nonvanishing everywhere except the origin. This fact then leads to the linear stability of these solitary waves. © 2001 John Wiley & Sons, Inc.

Thomson scattering measurements of saturated ion waves in laser fusion plasmas.

We have measured the characteristics of saturated ion-acoustic waves in inertial confinement fusion plasmas. A 263-nm probe laser has been applied to simultaneous Thomson scatter on both ion-acoustic waves excited by thermal electrostatic fluctuations and by stimulated Brillouin scattering of a kilojoule laser beam of varying intensity. The Thomson scattering spectra show saturated ion-wave amplitudes for intensities above 5x10(14) W cm(-2) consistent with three dimensional nonlinear wave modeling.

Approximation models for nonlinear thermal waves

Approximation models for nonlinear thermal waves

Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method

In this paper, we present a reliable algorithm to study the known model of nonlinear dispersive waves proposed by Boussinesq. The modified algorithm of Adomian decomposition method is used with an emphasis on the single soliton solution. New exact periodic solutions and polynomial solutions are obtained. The results of numerical examples are presented and only few terms are required to obtain accurate solutions.

Simulating vertical mixing in a shelf-break region: addition of a shear instability model, accounting for the overall effect of internal tides, on top of a one-dimensional turbulence closure mixed layer model

A simple parameterization of eddy diffusivity is used to simulate the shear mixing from tidally induced internal waves generated in the continental slope region southwest of Brittany. Near the edge of the shelf, the seasonal thermocline oscillates under the forcing of the barotropic tide which propagates over the shelf break. A composite model is constructed to simulate the mixing of the upper ocean from both external (wind stress) and internal (internal waves) sources. A simple one-dimensional eddy kinetic energy model, which predicts the temperature profile from heat flux and wind stress inputs, is validated with respect to regional hydroelimatic conditions, then coupled with a two-layer model of non-linear internal waves, to simulate the mixing encountered in shelf break fronts submitted to tidal forcing. Numerical runs on a transect perpendicular to the shelf break show the formation of a spot of cool water over the edge of the continental margin. The one-dimensional eddy kinetic energy model has been validated over a decade with temperature profiles over the abyssal plain adjacent to the continental slope. An annual validation experiment has also been conducted for the combined models, beginning on 1st January 1985, as well as a short-term validation experiment, using a set of high-frequency temperature measurements at two stations near the shelf edge in September and early October 1985. The simulation has also been spatially validated against three sets of infrared satellite images. The one-dimensional model is calibrated for the minimum turbulent kinetic energy, whereas the best fit to the high-frequency measurements in autumn 1985 above the slope provides the optimum values for the initial thermal content and for the parameterization constant of internal wave diffusivity. The combined model reproduces successfully the seasonal and the high-frequency (neap-spring tidal cycle) variation of the temperature field in the upper ocean along a transect perpendicular to the shelf break. Horizontal advection and mesoscale turbulence somewhat limit the performance of the model at low-amplitude tides and over the shallower part of the shelf, but the satisfactory overall agreement between the results and the measurements is consistent with the formation of a shelf break front in the northern Bay of Biscay and southern Celtic Sea, mainly as a result of mixing enhanced by tidally induced internal waves.

Errors in bathymetric retrievals using linear dispersion in 3-D FFT analysis of marine radar ocean wave imagery

The phenomenon of ocean wave-shoaling, and the associated reduction of ocean wave phase speed with decreased water depth, provides useful information for inferring water depth D (bathymetry) in coastal environments. One strategy for relating D to phase speed C and wave-vector K of long wavelength ocean waves involves using the one-dimensional (1-D) linear (gravity wave) dispersion relationship C/sup 2/=g*tanh(KD)/K. In principle, this approach has limitations because the approach is based on a WKB approximation, so it cannot be applied when D varies appreciably over the wavelength of a shoaling wave. Also, the approach is restricted to waves that have small wave height. The author uses a set of marine radar image sequences and applies this linear approximation, using a 3-D FFT analysis of 88 sets of image sequences spaced half an hour apart. The author inverts the dispersion relation to solve for D. Depths between 3.6 and 5.8 m were tested, for root mean square (RMS) wave heights offshore between 8 and 3 m. The author shows that for low to moderate wave heights, the approach does generally retrieve the correct depth in water depths of 5 m and greater for moderate wave RMS heights. However, an increase in the RMS wave height from 1 m to 3.5 m produced a much poorer depth estimate, proving the need for an application of a nonlinear wave model to the problem. The errors also increase with shallower depths as expected, as the error dependence on depth and wave height is determined.

Analysis of the intensification of tsunami waves near the South Coast of Crimea

We perform the numerical analysis of the intensification of tsunami waves in the course of their propagation from the open part of the Black Sea to the shelf zone. For this purpose, we use a one-dimensional model of nonlinear long waves taking into account the effect of bottom friction. We study four profiles of the bottom corresponding to the south coast of the Crimean Peninsula and establish the predominant role of the bottom pattern and insignificant contribution of nonlinearity to the transformation of waves in the process of their propagation in the direction of the coast. Down to depths of 50 m, all changes in the height of waves are described by the Green law. For the evaluation of vertical run-up of waves, it is important to take into account nonlinear effects. The highest vertical run-ups of waves are observed in the parts of the shelf zone located near Yalta and Alushta.

Stability of a class of solitary waves in compressible elastic rods

We prove that the solitary waves of a model for nonlinear dispersive waves in cylindrical compressible hyperelastic rods are orbitally stable. This establishes that the shape of the wave is stable.

Illustration of a nonlinear wave model of the spiral structure of galaxies with different types of rotation curves

The results of a nonlinear wave theory of spiral structure that we have developed are compared with observational data on the galaxies M81, M51, UGC 2259, and NGC 3883, which are characterized by different types of rotation curves.

A practical method for analysis of river waves and for kinematic wave routing in natural channel networks

The behaviour of river waves is described using a simplified dimensionless form of the momentum equation in conjunction with the continuity equation. Three dimensionless parameters were derived based on a quantitative linear analysis. These parameters, which depend on the Froude number of the steady uniform flow and the geometric characteristics of the river, permit quantification of the influence of inertia and pressure in the momentum equation. It was found that dynamic and diffusion waves occur mainly on gentle channel slopes and the transition between them is characterized by the Froude number. On the other hand, the kinematic wave has a wide range of applications. If the channel slope is greater than 1%, the kinematic wave is particularly suitable for describing the hydraulics of flow. Since slopes in natural channel networks are often greater than 1%, an analytical solution of the linearized kinematic wave equation with lateral inflow uniformly distributed along the channel is desirable and was therefore derived. The analytical solution was then implemented in a channel routing module of an existing simple rainfall–runoff model. The results obtained using the analytical solution compared well with those obtained from a non-linear kinematic wave model. Copyright © 2000 John Wiley & Sons, Ltd.

A practical method for analysis of river waves and for kinematic wave routing in natural channel networks

The behaviour of river waves is described using a simplified dimensionless form of the momentum equation in conjunction with the continuity equation. Three dimensionless parameters were derived based on a quantitative linear analysis. These parameters, which depend on the Froude number of the steady uniform flow and the geometric characteristics of the river, permit quantification of the influence of inertia and pressure in the momentum equation. It was found that dynamic and diffusion waves occur mainly on gentle channel slopes and the transition between them is characterized by the Froude number. On the other hand, the kinematic wave has a wide range of applications. If the channel slope is greater than 1%, the kinematic wave is particularly suitable for describing the hydraulics of flow. Since slopes in natural channel networks are often greater than 1%, an analytical solution of the linearized kinematic wave equation with lateral inflow uniformly distributed along the channel is desirable and was therefore derived. The analytical solution was then implemented in a channel routing module of an existing simple rainfall–runoff model. The results obtained using the analytical solution compared well with those obtained from a non-linear kinematic wave model. Copyright © 2000 John Wiley & Sons, Ltd.

Concurrent complementary operator boundary conditions for optical beam propagation

We have demonstrated the superb performance of the complementary operators method (COM) and its concurrent variant in suppressing reflection from computation boundaries with the finite-difference beam propagation method. We have applied them in modeling linear and nonlinear wave propagation in multidimension. Numerical examples show that C-COM performs better and requires less storage with slightly faster speed in most cases.

Non-linear propagation of unidirectional wave fields over varying topography

The present study aims at investigating the non-linear triad interaction process affecting shoaling surface gravity wave fields. The triad interaction phenomenon being enhanced towards the shore, the domain of study is extended up to the surf zone. Three 1D non-linear wave models (one phase-resolving and two phase-averaged spectral models) have been implemented and compared to laboratory experiments performed in a wave flume. This set of models includes two existing models and a new one which has been developed in the frame of this work. The models include a breaking dissipation term based on the parametrical model of Battjes and Janssen [Battjes, J.A., Janssen, P.A.E.M., 1978. Energy loss and set-up due to breaking of random waves. Proc. 16th Int. Conf. Coastal Eng. (ASCE), Vol. 1, pp. 569–587.]. The investigations concern the evolution of variance spectra, spectral significant wave height and mean period over a barred bathymetric profile. In addition, the performances of the different models are analysed by computing the spectral source term for triad interactions. We found that all models are able to reproduce the main features of non-linear mechanisms affecting a wave field in the near-shore zone. The phase-resolving model gives the most accurate results for non-breaking situations. It correctly reproduces the non-linear coupling effect in decreasing water depths due to wave–wave interactions, as well as the harmonic release after a bar. However, the model is computationally time-consuming. The CPU time is considerably reduced using phase-averaged models. They give satisfactorily results on harmonic generation. However, they do not reproduce the release of harmonics as water depth increases. In breaking conditions, the variance spectra undergo significant changes under the combined effects of non-linear energy transfers and dissipation. The depth-induced wave breaking model included in the equations provides a good estimate of the energy decay in the surf zone.

A reduced model for nonlinear dispersive waves in a rotating environment

The simplest model for geophysical flows is one layer of a constant density fluid with a free surface, where the fluid motions occur on a scale in which the Coriolis force is significant. In the linear shallow water limit, there are non-dispersive Kelvin waves, localized near a boundary or near the equator, and a large family of dispersive waves. We study weakly nonlinear and finite depth corrections to these waves, and derive a reduced system of equations governing the flow. For this system we find approximate solitary Kelvin waves, both for waves traveling along a boundary and along the equator. These waves induce jets perpendicular to their direction of propagation, which may have a role in mixing. We also derive an equivalent reduced system for the evolution of perturbations to a mean geostrophic flow.