Weakly Nonlinear Internal Waves in Shear

An evolution equation in a finite depth fluid for weakly nonlinear long internal waves is derived in a stratified and sheared medium. The equation reduces to the Korteweg-deVries equation when the depth is small compared to the wavelength, and to the Benjamin-Ono equation when the depth is large compared to the wavelength. Both the cases with and without critical levels are investigated. Numerical solutions to the evolution equation are presented to illustrate the effect of shear on the evolution of a waveform.     

On linear and nonlinear Rossby waves in an ocean

This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a β-plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode (n=1) analysis. An extension of the higher modes (n?2) of this work will be made in a subsequent paper.     

Evolution Equations for Long,Nonlinear Internal Waves in Stratified Shear Flows

Evolution equations for long, nonlinear internal waves are derived when the basic stratified shear flow has a slow temporal and spatial variation as well as the usual dependence on the vertical coordinate. When the horizontal waveguide has a limited vertical extent the evolution equation is a variable coefficient Korteweg-deVries equation, while in the deep fluid case the evolution equation is a variable coefficient Benjamin-Davis-Ono equation. Explicit expressions are obtained for the coefficients of these equations.     

Propagation of weakly nonlinear waves in fluid-filled thick viscoelastic tubes

In the present work, we studied the propagation of small-but-finite-amplitude waves in a prestressed thick walled viscoelastic tube filled with an incompressible inviscid fluid. In order to include the dispersion, the wall's inertial and shear effects are taken into account in determining the inner pressure–inner cross-sectional area relation. Using the reductive perturbation method, the propagation of weakly nonlinear waves in the long-wave approximation is investigated. After obtaining the general evolution equation in the long-wave approximation, by a proper scaling, it is shown that this general equation reduces to the well-known evolution equations such as the Burgers, Korteweg-de Vries (KdV), Koteweg-de Vries–Burgers (KdVB) and the generalized Burgers' equations. By proper re-scaling of the perturbation parameter, the modified form of the evolution equations is also obtained. The variations of the travelling wave profile with initial deformation and the viscosity coefficients are numerically evaluated and the results are illustrated in some figures.     

Nonlinear modulation of SH waves in an incompressible hyperelastic plate

Propagation of nonlinear shear horizontal (SH) waves in a homogeneous, isotropic and incompressible elastic plate of uniform thickness is considered. The constituent material of the plate is assumed to be generalized neo-Hookean. By employing a perturbation method and balancing the weak nonlinearity and dispersion in the analysis, it is shown that the nonlinear modulation of waves is governed asymptotically by a nonlinear Schr?dinger (NLS) equation. Then the effect of nonlinearity on the propagation characteristics of asymptotic waves is discussed on the basis of this equation. It is found that, irrespective of the plate thickness, the wave number and the mode number, when the plate material is softening in shear then the nonlinear plane periodic waves are unstable under infinitesimal perturbations and therefore the bright (envelope) solitary SH waves will exist and propagate in such a plate. But if the plate material is hardening in shear in this case nonlinear plane periodic waves are stable and only the dark solitary SH waves may exist.     

Nonlinear Hamiltonian waves with constant frequency and surface waves on vorticity discontinuities

Waves with constant, nonzero linearized frequency form an interesting class of nondispersive waves whose properties differ from those of nondispersive hyperbolic waves. We propose an inviscid Burgers‐Hilbert equation as a model equation for such waves and give a dimensional argument to show that it models Hamiltonian surface waves with constant frequency. Using the method of multiple scales, we derive a cubically nonlinear, quasi‐linear, nonlocal asymptotic equation for weakly nonlinear solutions. We show that the same asymptotic equation describes surface waves on a planar discontinuity in vorticity in two‐dimensional inviscid, incompressible fluid flows. Thus, the Burgers‐Hilbert equation provides an effective equation for these waves. We describe the Hamiltonian structure of the Burgers‐Hilbert and asymptotic equations, and show that the asymptotic equation can also be derived by means of a near‐identity transformation. We derive a semiclassical approximation of the asymptotic equation and show that spatially periodic, harmonic traveling waves are linearly and modulationally stable. Numerical solutions of the Burgers‐Hilbert and asymptotic equations are in excellent agreement in the appropriate regime. In particular, the lifespan of small‐amplitude smooth solutions of the Burgers‐Hilbert equation is given by the cubically nonlinear timescale predicted by the asymptotic equation. © 2009 Wiley Periodicals, Inc.     

Amplitude Modulation of Nonlinear Waves in Fluid-Filled Tapered Tubes

We study the modulation of nonlinear waves in fluid-filled prestressed tapered tubes. For this, we obtain the nonlinear dynamical equations of motion of a prestressed tapered tube filled with an incompressible inviscid fluid. Assuming that the tapering angle is small and using the reductive perturbation method, we study the amplitude modulation of nonlinear waves and obtain the nonlinear Schrödinger equation with variable coefficients as the evolution equation. A traveling-wave type of solution of such a nonlinear equation with variable coefficients is obtained, and we observe that in contrast to the case of a constant tube radius, the speed of the wave is variable. Namely, the wave speed increases with distance for narrowing tubes and decreases for expanding tubes.     

Instability of power-law fluid flows down an incline subjected to wind stress

In this paper we investigate the effect of a prescribed superficial shear stress on the generation and structure of roll waves developing from infinitesimal disturbances on the surface of a power-law fluid layer flowing down an incline. The unsteady equations of motion are depth integrated according to the von Kármán momentum integral method to obtain a non-homogeneous system of nonlinear hyperbolic conservation laws governing the average flow rate and the thickness of the fluid layer. By conducting a linear stability analysis we obtain an analytical formula for the critical conditions for the onset of instability of a uniform and steady flow in terms of the prescribed surface shear stress. A nonlinear analysis is performed by numerically calculating the nonlinear evolution of a perturbed flow. The calculation is carried out using a high-resolution finite volume scheme. The source term is handled by implementing the quasi-steady wave propagation algorithm. Conclusions are drawn regarding the effect of the applied surface shear stress parameter and flow conditions on the development and characteristics of the roll waves arising from the instability. For a Newtonian flow subjected to a prescribed superficial shear stress, using an analytical theory, we show that the nonlinear governing equations do not admit roll waves solutions under conditions when the uniform and steady flow is linearly stable. For the case of a general power-law fluid flow with zero shear stress applied at the surface, the analytical investigation leads to a procedure for calculating the characteristics of a roll waves flow. These results are compared with those yielded by the numerical procedure.     

A model equation for nonlinear wavelength selection and amplitude evolution of frontal waves

Summary We study a model equation describing the temporal evolution of nonlinear finite-amplitude waves on a density front in a rotating fluid. The linear spectrum includes an unstable interval where exponential growth of the amplitude is expected. It is shown that the length scale of the waves in the nonlinear situation is determined by the linear instabilities; the effect of the nonlinearities is to limit the amplitude's growth, leaving the wavelength unchanged. When linearly stable waves are prescribed as initial data, a short interval of rapid decrease in amplitude is encountered first, followed by a transfer of energy to the unstable part of the spectrum, where the fastest growing mode starts to dominate. A localized disturbance is broken up into its Fourier components, the linearly unstable modes grow at the expense of all other modes, and final amplitudes are determined by the nonlinear term. Periodic evolution of linearly unstable waves in the nonlinear situation is also observed. Based on the numerical results, the existence of low-order chaos in the partial differential equation governing weakly nonlinear wave evolution is conjectured.     

Well-posedness of the Ostrovsky–Hunter Equation under the combined effects of dissipation and short-wave dispersion

The Ostrovsky–Hunter equation provides a model for small-amplitude long waves in a rotating fluid of finite depth. It is a nonlinear evolution equation. In this paper we study the well-posedness for the Cauchy problem associated with this equation in presence of some weak dissipation effects.     

On Periodic Wave Solutions to (1+1)-Dimensional Nonlinear Physical Models Using the Sine-Cosine Method

We investigate two interesting (1+1)-dimensional nonlinear partial differential evolution equations (NLPDEEs), namely the nonlinear dispersion equation with compact structures and the generalized Camassa–Holm (CH) equation describing the propagation of unidirectional shallow water waves on a flat bottom, and arising in the study of a certain non-Newtonian fluid. Using an interesting technique known as the sine-cosine method for investigating travelling wave solutions to NLPDEEs, we construct many new families of wave solutions to the previous NLPDEEs, amongst which the periodic waves, enriching the wide class of solutions to the above equations.     

The Evolution of Nonlinear Wave Trains in Stratified Shear Flows

The propagation of an internal wave train in a stratified shear flow is investigated for a Boussinesq fluid in a horizontal channel. Linear effects are primarily reflected in the dispersion relation for the various modes. The phenomenon of Eckart resonance occurs for more realistic stratification profiles. The evolution of nonlinear internal wave packets is studied through a systematic perturbation analysis. A nonlinear Schrodinger equation for the envelope of the internal wave train is derived. Depending on the relative sign of the dispersive and nonlinear terms, a wave train may disperse or form an envelope soliton. The analysis demonstrates the existence of two types of critical layers: one the ordinary critical point where ū=c, while the other occurs where ū=c_g. In order to calculate the coefficients of the nonlinear Schrodinger equation a numerical code has been developed which computes the second-harmonic and induced mean motions. The existence of these envelope solitons and their dependence on environmental conditions are discussed.     

Stability of contact discontinuity for the Boltzmann equation

The Boltzmann equation which describes the time evolution of a large number of particles through the binary collision in statistics physics has close relation to the systems of fluid dynamics, that is, Euler equations and Navier-Stokes equations. As for a basic wave pattern to Euler equations, we consider the nonlinear stability of contact discontinuities to the Boltzmann equation. Even though the stability of the other two nonlinear waves, i.e., shocks and rarefaction waves has been extensively studied, there are few stability results on the contact discontinuity because unlike shock waves and rarefaction waves, its derivative has no definite sign, and decays slower than a rarefaction wave. Moreover, it behaves like a linear wave in a nonlinear setting so that its coupling with other nonlinear waves reveals a complicated interaction mechanism. Based on the new definition of contact waves to the Boltzmann equation corresponding to the contact discontinuities for the Euler equations, we succeed in obtaining the time asymptotic stability of this wave pattern with a convergence rate. In our analysis, an intrinsic dissipative mechanism associated with this profile is found and used for closing the energy estimates.     

Three-Dimensional Water Waves

We study the evolution of small-amplitude water waves when the fluid motion is three dimensional. An isotropic pseudodifferential equation that governs the evolution of the free surface of a fluid with arbitrary, uniform depth is derived. It is shown to reduce to the Benney-Luke equation, the Korteweg-de Vries (KdV) equation, the Kadomtsev-Petviashvili (KP) equation, and to the nonlinear shallow water theory in the appropriate limits. We compute, numerically, doubly periodic solutions to this equation. In the weakly two-dimensional long wave limit, the computed patterns and nonlinear dispersion relations agree well with those of the doubly periodic theta function solutions to the KP equation. These solutions correspond to traveling hexagonal wave patterns, and they have been compared with experimental measurements by Hammack, Scheffner, and Segur. In the fully two-dimensional long wave case, the solutions deviate considerably from those of KP, indicating the limitation of that equation. In the finite depth case, both resonant and nonresonant traveling wave patterns are obtained.     

Large Amplitude Internal Waves of Permanent Form

In this paper the weakly nonlinear theory of long internal gravity waves propagating in stratified media is extended to the fully nonlinear case by treating Long's nonlinear partial differential equation for steady inviscid flows without restriction to small amplitudes and long wavelengths. The existence of finite amplitude solutions of “permanent form” is established analytically for a large class of stratification profiles, and properties are calculated numerically for the case of a hyperbolic tangent density profile in a large range of fluid depths. The numerical results agree well with the experimental data of Davis and Acrivos over the full range of wave amplitudes measured; such agreement is not obtainable with existing weakly nonlinear theories.     

The Evolution in Space and Time of Nonlinear Waves in Parallel Shear Flows

It is shown, using a quite general formulation, that the amplitude evolution equation for slowly varying finite amplitude waves is usually first order in both space and time. One advantage of the present formulation is that it becomes possible to easily identify, from their linear eigensolutions, interesting exceptional cases in which the amplitude evolves according to a partial differential equation that is second order in either space or time. The theory is applied to a number of specific problems, including flows with broken line profiles, and inviscid shear flows having nonlinear critical layers.     

On the interaction of deep water waves and exponential shear currents

A train of periodic deep-water waves propagating on a steady shear current with a vertical distribution of vorticity is investigated by an analytic method, namely the homotopy analysis method (HAM). The magnitude of the vorticity varies exponentially with the magnitude of the stream function, while remaining constant on a particular streamline. The so-called Dubreil–Jacotin transformation is used to transfer the original exponentially nonlinear boundary-value problem in an unknown domain into an algebraically nonlinear boundary-value problem in a known domain. Convergent series solutions are obtained not only for small amplitude water waves on a weak current but also for large amplitude waves on a strong current. The nonlinear wave-current interaction is studied in detail. It is found that an aiding shear current tends to enlarge the wave phase speed, sharpen the wave crest, but shorten the maximum wave height, while an opposing shear current has the opposite effect. Besides, the amplitude of waves and fluid velocity decay over the depth more quickly on an aiding shear current but more slowly on an opposing shear current than that of waves on still water. Furthermore, it is found that Stokes criteria of wave breaking is still valid for waves on a shear current: a train of propagating waves on a shear current breaks as the fiuid velocity at crest equals the wave phase speed. Especially, it is found that the highest waves on an opposing shear current are even higher and steeper than that of waves on still water. Mathematically, this analytic method is rather general in principle and can be employed to solve many types of nonlinear partial differential equations with variable coefficients in science, finance and engineering.     

On the interaction of deep water waves and exponential shear currents

A train of periodic deep-water waves propagating on a steady shear current with a vertical distribution of vorticity is investigated by an analytic method, namely the homotopy analysis method (HAM). The magnitude of the vorticity varies exponentially with the magnitude of the stream function, while remaining constant on a particular streamline. The so-called Dubreil–Jacotin transformation is used to transfer the original exponentially nonlinear boundary-value problem in an unknown domain into an algebraically nonlinear boundary-value problem in a known domain. Convergent series solutions are obtained not only for small amplitude water waves on a weak current but also for large amplitude waves on a strong current. The nonlinear wave-current interaction is studied in detail. It is found that an aiding shear current tends to enlarge the wave phase speed, sharpen the wave crest, but shorten the maximum wave height, while an opposing shear current has the opposite effect. Besides, the amplitude of waves and fluid velocity decay over the depth more quickly on an aiding shear current but more slowly on an opposing shear current than that of waves on still water. Furthermore, it is found that Stokes criteria of wave breaking is still valid for waves on a shear current: a train of propagating waves on a shear current breaks as the fiuid velocity at crest equals the wave phase speed. Especially, it is found that the highest waves on an opposing shear current are even higher and steeper than that of waves on still water. Mathematically, this analytic method is rather general in principle and can be employed to solve many types of nonlinear partial differential equations with variable coefficients in science, finance and engineering.     

Weakly Nonlinear Waves in Nonideal Fluids

We study the propagation of weakly nonlinear waves in nonideal fluids, which exhibit mixed nonlinearity. A method of multiple scales is used to obtain a transport equation from the Navier–Stokes equations, supplemented by the equation of state for a van der Waals fluid. Effects of van der Waals parameters on the wave evolution, governed by the transport equation, are investigated.