Three-dimensional surface waves propagating over long internal waves

A non-uniform current, such as may be generated by long internal waves, interacts with short surface waves and causes patterns on the sea surface that are of interest. In particular, regions of steep breaking waves may be relevant to specular radar scattering.A simple approach to modelling this problem is to take a set of short, surface waves of uniform wavenumber on the sea surface, as may be caused by a gust of wind. The direction of propagation of the surface waves is firstly taken to be the same as that of the current, and surface tension and viscous effects are neglected. We have a number of methods of solution at our disposal: linear (one-dimensional) ray theory is simple to apply to the problem, a nonlinear Schrödinger equation for the modulated wave amplitude, modified to include to effect of the current, can be used and solutions can be found using a fully nonlinear irrotational flow solver. Comparisons between the ‘exact’ nonlinear calculations for two dimensions (which are too complicated/ computationally intensive to be extended to three dimensions) compare well with the two approximate methods of solution, both of which can be extended, within their limitations, to model the full three-dimensional problem; here we present three-dimensional results from the linear ray theory.By choosing such a simple (although we consider physically realistic) initial state of uniform wavenumber short waves and assuming a sinusoidal surface current, we can reduce the two-dimensional problem to dependence on three non-dimensional parameters.In three-dimensions, we consider an initial condition with a uniform wavetrain at an angle α say, to the propagating current, thus introducing a fourth parameter into the problem. Extension of the linear ray theory from one space to two space dimensions is numerically quite simple since we maintain uniformity in the direction perpendicular to the current, and the only difficulty lies with the presentation of results, due to the large number of variables now present in the problem such as initial wavenumber, angle of propagation, position in (x, y, t) space etc. In this paper we present just one solution in detail where waves are strongly refracted and form two distinct foci in space-time. There is a collimation of the short waves with the direction of the propagating current.     

Nonlinear dynamics of hydrostatic internal gravity waves

Stratified hydrostatic fluids have linear internal gravity waves with different phase speeds and vertical profiles. Here a simplified set of partial differential equations (PDE) is derived to represent the nonlinear dynamics of waves with different vertical profiles. The equations are derived by projecting the full nonlinear equations onto the vertical modes of two gravity waves, and the resulting equations are thus referred to here as the two-mode shallow water equations (2MSWE). A key aspect of the nonlinearities of the 2MSWE is that they allow for interactions between a background wind shear and propagating waves. This is important in the tropical atmosphere where horizontally propagating gravity waves interact together with wind shear and have source terms due to convection. It is shown here that the 2MSWE have nonlinear internal bore solutions, and the behavior of the nonlinear waves is investigated for different background wind shears. When a background shear is included, there is an asymmetry between the east- and westward propagating waves. This could be an important effect for the large-scale organization of tropical convection, since the convection is often not isotropic but organized on large scales by waves. An idealized illustration of this asymmetry is given for a background shear from the westerly wind burst phase of the Madden–Julian oscillation; the potential for organized convection is increased to the west of the existing convection by the propagating nonlinear gravity waves, which agrees qualitatively with actual observations. The ideas here should be useful for other physical applications as well. Moreover, the 2MSWE have several interesting mathematical properties: they are a system of nonconservative PDE with a conserved energy, they are conditionally hyperbolic, and they are neither genuinely nonlinear nor linearly degenerate over all of state space. Theory and numerics are developed to illustrate these features, and these features are important in designing the numerical scheme. A numerical method is designed with simplicity and minimal computational cost as the main design principles. Numerical tests demonstrate that no catastrophic effects are introduced when hyperbolicity is lost, and the scheme can represent propagating discontinuities without introducing spurious oscillations.      

Nonlinear waves in electromigration dispersion in a capillary

We construct exact solutions to an unusual nonlinear advection–diffusion equation arising in the study of Taylor–Aris (also known as shear) dispersion due to electroosmotic flow during electromigration in a capillary. An exact reduction to a Darboux equation is found under a traveling-wave ansatz. The equilibria of this ordinary differential equation are analyzed, showing that their stability is determined solely by the (dimensionless) wave speed without regard to any (dimensionless) physical parameters. Integral curves, connecting the appropriate equilibria of the Darboux equation that governs traveling waves, are constructed, which in turn are shown to be asymmetric kink solutions (i.e., non-Taylor shocks). Furthermore, it is shown that the governing Darboux equation exhibits bistability, which leads to two coexisting non-negative kink solutions for (dimensionless) wave speeds greater than unity. Finally, we give some remarks on other types of traveling-wave solutions and a discussion of some approximations of the governing partial differential equation of electromigration dispersion.     

Nonlinear dispersion properties of high-frequency waves in the gradient theory of elasticity

The dispersion law ceases to be linear already at ultrasonic frequencies of elastic vibrations of particles as mechanical perturbation waves propagate through the medium. A variant of the continuum model of an elastic medium is proposed which is based on the assumption of pair and triplet potential interaction between infinitely small particles; this allows one to represent the dispersion law with any required accuracy. The corresponding wave equation, which is still linear, can have an arbitrarily large order of partial derivatives with respect to the coordinates. It is suggested that the results of comparing the representations of the dispersion law from the elasticity and solid-state physics viewpoints should be used to determine nonclassical characteristics of the elastic state of the medium. The theoretical conclusions are illustrated with calculations performed for plane waves propagating through aluminum.     

Effect of anomalous dispersion dependences on scattering and generation of internal waves

Novosibirsk. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 3, pp. 47–55, May–June 1994.     

Field of long internal lee waves in a plane-parallel shear layer

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 1, pp. 83–88, January–February, 1993.     

Nonlinear long wave modulation in symmetric waves of a plane magnetic layer

The special features of the distribution of the magnetic field in the photosphere of the Sun and the experimental discovery of waves which propagate along magnetic tubes in the solar atmosphere have brought about the publication recently of a large number of articles which study the wave-conducting properties of media with a magnetic structure. One of the simplest cases was that of a plane magnetic layer, which was studied in detail in the linear approximation [1–3]. Starting from the dispersion properties of such a structure, [4] indicates the possibility of the existence in it of solitons in the approximation of waves of low amplitude which are long in relation to the layer. The present study has used the method of different-scale expansions to obtain the Schrödinger equation describing the propagation of nonlinear modulations of a symmetric harmonic mode over a plane magnetic layer in an incompressible fluid. A similar equation has been deduced, for example, for waves in water [5–9].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, 164–168, March–April, 1985.The author wishes to thank M. S. Ruderman for formulating the problem and for useful discussions, and V. B. Baranov for his attention to the study.     

Analysis of current induced by long internal solitary waves in stratified ocean

An approximate theoretical expression for the current induced by long internal solitary waves is presented when the ocean is continuously or two-layer stratified. Particular attention is paid to characterizing velocity fields in terms of magnitude, flow components, and their temporal evolution/spatial distribution. For the two-layer case, the effects of the upper/lower layer depths and the relative layer density difference upon the induced current are further studied. The results show that the horizontal components are basically uniform in each layer with a shear at the interface. In contrast, the vertical counterparts vary monotonically in the direction of the water depth in each layer while they change sign across the interface or when the wave peak passes through. In addition, though the vertical components are generally one order of magnitude smaller than the horizontal ones, they can never be neglected in predicting the heave response of floating platforms in gravitationally neutral balance. Comparisons are made between the partial theoretical results and the observational field data. Future research directions regarding the internal wave induced flow field are also indicated.     

Hamiltonian long wave expansions for internal waves over a periodically varying bottom

We derive a Hamiltonian formulation for two-dimensional nonlinear long waves between two bodies of immiscible fluid with a periodic bottom. From the formulation and using the Hamiltonian perturbation theory, we obtain effective Boussinesq equations that describe the motion of bidirectional long waves and unidirectional equations that are similar to the KdV equation for the case in which the bottom possesses short length scale. The computations for these results are performed in the framework of an asymptotic analysis of multiple scale operators.     

Propagation of periodic long waves

Summary A method is developed for the computation of the steady solution of the shallow water equations with quasi periodic boundary conditions. Because of dissipation the influence of the initial conditions becomes negligible with increasing time and the solution finally depends on the boundary conditions. The unknowns variables (velocity and surface elevation) and the boundary conditions are developed in power series of a small perturbation parameter. The problem is then transformed in a sequence of linear problems which have the same associated homogeneous problem. By separating the time and space variables in the homogeneous problem we obtain an homogeneous elliptic problem of which we compute the first eigenvalues and eigensolutions. These are related to the characteristic oscillations of the water in the basin. The solution of each linear problem is then obtained as an eigensolution expansion with time dependent coefficients. These coefficients are solutions of ordinary differential equations which can be solved directly without proceeding step by step in time. In this way we are reduced to a stationary problem i.e. the determination of the eigenvalues and eigensolutions of the elliptic problem and to the computation of several integrals needed for the determination of the time dependent coefficients. A first test of the method has been carried out for a one-dimensional problem i.e. the tidal wave in a canal of finite length and constant depth. In this case the various steps of the procedure outlined above can be performed analitically. The results have been compared with those obtained by a step by step numerical integration of the shallow water equations. The agreement between these sets of results is good for the range of values of the parameters currently used in the applications.

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Sommario Viene presentato un metodo per il calcolo della soluzione a regime delle equazioni delle onde lunghe, dipendente dalle sole condizioni al contorno quasi-periodiche, dopo aver mostrato che l'influenza delle condizioni iniziali diventa col tempo trascurabile a causa del termine di resistenza. Il metodo si basa sullo sviluppo in serie di potenze di un piccolo parametro sia delle incognite (velocità ed elevazioni del pelo libero) sia delle condizioni al contorno al fine di trasformare il problema non lineare in una serie di problemi lineari aventi lo stesso problema omogeneo associato. Con la separazione delle variabili spazio, tempo quest'ultimo problema viene ricondotto ad un problema ellittico omogeneo di cui si calcolano i primi autovalori ed autosoluzioni. Da ultimo la soluzione di ciascun problema lineare è ottenuta come sviluppo in serie di autosoluzioni a coefficienti dipendenti dal tempo: questi si ricavano risolvendo analiticamente delle equazioni differenziali ordinarie. Si elimina cosi la necessità di procedere passo passo nel tempo analogamente ai classici metodi armonici di soluzione di sistemi lineari. Riassumendo, l'applicazione del metodo riconduce alla soluzione di un problema stazionario (determinazione di autovalori ed autosoluzione del problema ellittico) e quindi al calcolo dei vari integrali necessari per la determinazione dei coefficienti temporali. Il metodo è stato provato nel caso semplice della propagazione di onde di marea in un canale di lunghezza finita e sezione costante. Per questo esempio i vari passi di calcolo possono essere svolti analiticamente. I risultati sono stati confrontati con quelli ottenuti dalla integrazione numerica delle equazioni col metodo delle caratteristiche, ottenendo un buon accordo.

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Work sponsored by the CNR (National Research Council) in the framework of Project ?Conservazione del suolo? Subproject ?Dinamica dei Litorali? Publ. n° 46.     

Nonlinear elastic shear waves

Summary The dynamics of one-dimensional two-component shear motion in elastic-isotropic homogeneous media is studied assuming isentropic finite displacements. Wave breaking of initially continuous waves on the infinite interval is discussed for weakly nonlinear waves.The problem of a resonating finite-thickness shear layer in primary resonance for single-component motion exhibits jump discontinuities of particle velocity, shear strain and stress in a finite frequency band near primary resonance.Under certain conditions two-component motion can be reduced to a quasi-single-component motion.     

Nonlinear shear-layer instability waves

The effects of critical-layer nonlinearity on spatially growing instability waves on shear layers between parallel streams are discussed. In the two-dimensional incompressible case, the flow in the critical layer is governed by a nonequilibrium (unsteady) nonlinear vorticity equation. The initial exponential growth of the instability wave is converted into algebraic growth during the streamwise aging of the critical layer into a quasi-equilibrium state. A uniformly valid composite formula for the instability wave amplitude, accounting for both nonparallel and nonlinear effects, is shown to be in good agreement with available experimental results. Nonlinear effects occur at smaller amplitudes for the three-dimensional and supersonic cases than in the two-dimensional incompressible case. The instability-wave amplitude evolution is then described by one integro-differential equation with a cubic-type nonlinearity, whose inviscid solution always end in a singularity at finite downstream distance.The US Government has the right to retain a nonexclusive royalty-free license in and to any copyright covering this paper.     

On the theory of long waves

A new method of studying plane steady wave motion of a gravity fluid is elucidated in this paper. This method succeeds in establishing the existence of a solitary wave, for example, and in giving the first complete foundation for the approximate Rayleigh theory [1], which concerns the theory of finite-amplitude long waves. Underlying the method are general boundary properties of univalent functions, used earlier by the author to construct a qualitative theory of jet fluid motions [2].     

Nonlinear internal waves in a fluid of infinite depth in the presence of a horizontal magnetic field

An investigation is made into the propagation of long nonlinear weakly nonone-dimensional internal waves in an incompressible stratified fluid of infinite depth in the presence of a horizontal magnetic field. It is shown that such waves are described by an equation representing the extension of the Benjamin-Ono equation to the weakly nonone-dimensional case. The equation obtained differs from that obtained in [4], which is attributable to the anisotropy of the medium resulting from the presence of a magnetic field. The stability of a soliton with respect to flexural perturbations is investigated. A particular case of the variation of the density with height at constant Alfvén velocity is examined in detail.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 65–72, November–December, 1987.     

Nonlinear waves in diatomic crystals

A possible improvement of a continuum model for diatomic crystals is examined using continuum limit of a discrete diatomic model. For this purpose, various discrete models of diatomic lattice are compared at the linearized and weakly nonlinear levels. The suitable numbering of the atoms in the lattice is found which is better adopted for continualization than the familiar pair numbering introducing two sub-lattices. The coupled governing partial nonlinear differential equations for longitudinal strain and relative distance between the atoms are obtained in the continuum limit that allows us to describe localization of the strains due to the presence of the atoms of two kinds. It is found, that the equations obtained possess two kinds of localized wave solutions, one related to the acoustical branch and the other one related to the optical branch.     

Rotationally symmetric internal gravity waves

Many mathematical models formulated in terms of non-linear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating non-linear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is to provide the group theoretical modeling of internal waves in the ocean. The approach is based on a new concept of conservation laws that is utilized to systematically derive the conservation laws of non-linear equations describing propagation of internal waves in the ocean. It was shown in our previous publication that uni-directional internal wave beams can be obtained as invariant solutions of non-linear equations of motion. The main goal of the present publication is to thoroughly analyze another physically significant exact solution, namely the rotationally symmetric solution and the energy carried by this solution. It is shown that the rotationally symmetric solution and its energy are presented by means of a bounded oscillating function.     

A novel internal waves generator

We present a new kind of generator of internal waves which has been designed for three purposes. First, the oscillating boundary conditions force the fluid particles to travel in the preferred direction of the wave ray, hence reducing the mixing due to forcing. Second, only one ray tube is produced so that all of the energy is in the beam of interest. Third, temporal and spatial frequency studies emphasize the high quality for temporal and spatial monochromaticity of the emitted beam. The greatest strength of this technique is therefore the ability to produce a large monochromatic and unidirectional beam.     

Asymptotic models of internal stationary waves

Equations of stationary long waves on the interface between a homogeneous fluid and an exponentially stratified fluid are considered. An equation of the second-order approximation of the shallow water theory inheriting the dispersion properties of the full Euler equations is used as the basic model. A family of asymptotic submodels is constructed, which describe three different types of bifurcation of solitary waves at the boundary points of the continuous spectrum of the linearized problem. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 151–161, July–August, 2008.