A Numerical Study of the Exact Evolution Equations for Surface Waves in Water of Finite Depth

We describe a pseudo-spectral numerical method to solve the systems of one-dimensional evolution equations for free surface waves in a homogeneous layer of an ideal fluid. We use the method to solve a system of one-dimensional integro-differential equations, first proposed by Ovsjannikov and later derived by Dyachenko, Zakharov, and Kuznetsov, to simulate the exact evolution of nonlinear free surface waves governed by the two-dimensional Euler equations. These equations are written in the transformed plane where the free surface is mapped onto a flat surface and do not require the common assumption that the waves have small amplitude used in deriving the weakly nonlinear Korteweg–de Vries and Boussinesq long-wave equations. We compare the solution of the exact reduced equations with these weakly nonlinear long-wave models and with the nonlinear long-wave equations of Su and Gardner that do not assume the waves have small amplitude. The Su and Gardner solutions are in remarkably close agreement with the exact Euler solutions for large amplitude solitary wave interactions while the interactions of low-amplitude solitary waves of all four models agree. The simulations demonstrate that our method is an efficient and accurate approach to integrate all of these equations and conserves the mass, momentum, and energy of the Euler equations over very long simulations.     

A Formulation for Water Waves over Topography

Many interesting free-surface flow problems involve a varying bottom. Examples of such flows include ocean waves propagating over topography, the breaking of waves on a beach, and the free surface of a uniform flow over a localized bump. We present here a formulation for such flows that is general and, from the outset, demonstrates the wave character of the free-surface evolution. The evolution of the free surface is governed by a system of equations consisting of a nonlinear wave-like partial differential equation coupled to a time-independent linear integral equation. We assume that the free-surface deformation is weakly nonlinear, but make no a priori assumption about the scale or amplitude of the topography. We also extend the formulation to include the effect of mean flows and surface tension. We show how this formulation gives some of the well-known limits for such problems once assumptions about the amplitude and scale of the topography are made.     

Particle trajectories beneath small amplitude shallow water waves in constant vorticity flows

We investigate the particle trajectories in a constant vorticity shallow water flow over a flat bed as periodic waves propagate on the water’s free surface. Within the framework of small amplitude waves, we find the solutions of the nonlinear differential equations system which describes the particle motion in the considered case, and we describe the possible particle trajectories. Depending on the relation between the initial data and the constant vorticity, some particle trajectories are undulating curves to the right, or to the left, others are loops with forward drift, or with backward drift, others can follow some peculiar shapes.     

Exact steady periodic water waves with vorticity

We consider the classical water wave problem described by the Euler equations with a free surface under the influence of gravity over a flat bottom. We construct two‐dimensional inviscid periodic traveling waves with vorticity. They are symmetric waves whose profiles are monotone between each crest and trough. We use bifurcation and degree theory to construct a global connected set of such solutions. © 2003 Wiley Periodicals, Inc.     

A Higher-Order Internal Wave Model Accounting for Large Bathymetric Variations

A higher-order strongly nonlinear model is derived to describe the evolution of large amplitude internal waves over arbitrary bathymetric variations in a two-layer system where the upper layer is shallow while the lower layer is comparable to the characteristic wavelength. The new system of nonlinear evolution equations with variable coefficients is a generalization of the deep configuration model proposed by Choi and Camassa [ 1 ] and accounts for both a higher-order approximation to pressure coupling between the two layers and the effects of rapidly varying bottom variation. Motivated by the work of Rosales and Papanicolaou [ 2 ], an averaging technique is applied to the system for weakly nonlinear long internal waves propagating over periodic bottom topography. It is shown that the system reduces to an effective Intermediate Long Wave (ILW) equation, in contrast to the Korteweg-de Vries (KdV) equation derived for the surface wave case.     

Nonlinear Wave–Current Interactions in Shallow Water

We study here the propagation of long waves in the presence of vorticity. In the irrotational framework, the Green–Naghdi equations (also called Serre or fully nonlinear Boussinesq equations) are the standard model for the propagation of such waves. These equations couple the surface elevation to the vertically averaged horizontal velocity and are therefore independent of the vertical variable. In the presence of vorticity, the dependence on the vertical variable cannot be removed from the vorticity equation but it was however shown in 1 that the motion of the waves could be described using an extended Green–Naghdi system. In this paper, we propose an analysis of these equations, and show that they can be used to get some new insight into wave–current interactions. We show in particular that solitary waves may have a drastically different behavior in the presence of vorticity and show the existence of solitary waves of maximal amplitude with a peak at their crest, whose angle depends on the vorticity. We also show some simple numerical validations. Finally, we give some examples of wave–current interactions with a nontrivial vorticity field and topography effects.     

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.     

Stokes waves with constant vorticity: I. Numerical computation

Periodic traveling waves are numerically computed in a constant vorticity flow subject to the force of gravity. The Stokes wave problem is formulated via a conformal mapping as a nonlinear pseudodifferential equation, involving a periodic Hilbert transform for a strip, and solved by the Newton‐GMRES method. For strong positive vorticity, in the finite or infinite depth, overhanging profiles are found as the amplitude increases and tend to a touching wave, whose surface contacts itself at the trough line, enclosing an air bubble; numerical solutions become unphysical as the amplitude increases further and make a gap in the wave speed versus amplitude plane; another touching wave takes over and physical solutions follow along the fold in the wave speed versus amplitude plane until they ultimately tend to an extreme wave, which exhibits a corner at the crest. Touching waves connected to zero amplitude are found to approach the limiting Crapper wave as the strength of positive vorticity increases unboundedly, while touching waves connected to the extreme waves approach the rigid body rotation of a fluid disk.     

Two-dimensional travelling waves over a moving bottom

Two-dimensional travelling waves on an ideal fluid with gravity and surface tension over a periodically moving bottom with a small amplitude are studied. The bottom and the wave travel with a same speed. The exact Euler equations are formulated as a spatial dynamic system by using the stream function. A manifold reduction technique is applied to reduce the system into one of ordinary differential equations with finite dimensions. A homoclinic solution to the normal form of this reduced system persists when higher-order terms are added, which gives a generalized solitary wave—the homoclinic solution connecting a periodic solution.     

A note on the wave propagation in water of variable depth

In the present work, utilizing the two dimensional equations of an incompressible inviscid fluid and the reductive perturbation method we studied the propagation of weakly nonlinear waves in water of variable depth. For the case of slowly varying depth, the evolution equation is obtained as the variable coefficient Korteweg-de Vries (KdV) equation. Due to the difficulties for the analytical solutions, a numerical technics so called “the method of integrating factor” is used and the evolution equation is solved under a given initial condition and the bottom topography. It is observed the parameters of bottom topography causes to the changes in wave amplitude, wave profile and the wave speed.     

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.     

Quasiperiodic Solutions in Weakly Nonlinear Gas Dynamics. Part I. Numerical Results in the Inviscid Case

We exhibit and study a new class of solutions for the one-dimensional inviscid Euler equations of Gas Dynamics in a bounded domain with reflecting boundary conditions, in the weakly nonlinear regime. These solutions do not present the usual wave breaking leading to shock formation, even though they have nontrivial acoustic components and operate in the nonlinear regime. We also show that these 'Non Breaking for All Times' (NBAT) solutions are globally attracting for the long time evolution of the equations.
The Euler equations of Gas Dynamics (in the weakly nonlinear regime with reflecting boundary conditions) can be reduced to an inviscid Burgers-like equation for the acoustic component, with a linear integral self-coupling term and periodic boundary conditions. The integral term arises as a result of the nonlinear resonant interactions of the sound waves with the entropy variations in the flow. This integral term turns out to be weakly dispersive. The NBAT solutions arise as a result of the interplay of this dispersion with the 'standard' wave-breaking nonlinearity in the Burgers equation.
In addition to the previously known weakly nonlinear standing acoustic wave NBAT solutions, we found a family of new, never-breaking, attracting solutions by direct numerical simulation. These are quasiperiodic in time with two periods. In phase space these solutions lie on a surface 'centered' around the standing waves. Only two standing-wave solutions (the maximum amplitude and the trivial vanishing wave) are in the attracting set. All of the others are quasiperiodic in time with two periods.     

Global bifurcation of steady gravity water waves with critical layers

We construct families of two-dimensional travelling water waves propagating under the influence of gravity in a flow of constant vorticity over a flat bed, in particular establishing the existence of waves of large amplitude. A Riemann–Hilbert problem approach is used to recast the governing equations as a one-dimensional elliptic pseudodifferential equation with a scalar constraint. The structural properties of this formulation, which arises as the Euler–Lagrange equation of an energy functional, enable us to develop a theory of analytic global bifurcation.     

广义缓坡方程

运用表面波Hamilton方法和缓坡逼近假定,分析缓变三维流场和非平整海底对波浪传播的影响,推导出广义缓坡方程。海底地形由两个分量组成:慢变分量,其水平长度尺度大于表面波的波长;快变分量,其振幅与表面波的波长相比为一小量,但是其频率与表面波频率相当。该广义缓坡方程具有广泛的适用范围,以下著名的缓坡方程成为它的特例:经典的Berkhoff缓坡方程;包含环境流效应的Kirby缓坡方程;描述波状海底特征的Dingemans缓坡方程。另外,同时也得到了描述环境流场和快变海底效应的广义浅水方程。     

Remarks on a Strongly Nonlinear Model for Two‐Layer Flows with a Top Free Surface

We revisit in this paper the strongly nonlinear long wave model for large amplitude internal waves in two‐layer flows with a free surface proposed by Choi and Camassa [1] and Barros et al. [2]. Its solitary‐wave solutions were the object of the work by Barros and Gavrilyuk [3], who proved that such solutions are governed by a Hamiltonian system with two degrees of freedom. A detailed analysis of the critical points of the system is presented here, leading to some new results. It is shown that conjugate states for the long wave model are the same as those predicted by the fully nonlinear Euler equations. Some emphasis will be given to the baroclinic mode, where interfacial waves are known to change polarity according to different values of density and depth ratios. A critical depth ratio separates these two regimes and its analytical expression is derived directly from the model. In addition, we prove that such waves cannot exist throughout the whole range of speeds.     

The Interaction of a Strong Bore with Small Disturbances in Shallow Water

This paper gives an extension of previous work [2] on weakly nonlinear shallow water waves over a variable bottom to include the effects of strong bores and small surface disturbances. We first consider the interaction of a strong bore with quiescent water over an isolated bottom disturbance to highlight some of the modifications that are introduced in our results for both noncritical and transcritical Froude numbers. We also exhibit the secular effect on the bore trajectory of a bottom disturbance that has a nonzero average. In a second example, we consider the interaction of a strong bore with a small amplitude periodic surface disturbance upstream. We show that downstream of the bore, the wave length of this disturbance increases, whereas its amplitude increases (decreases) depending on whether the bore speed is larger (smaller) than a critical value. We also use this example to illustrate the derivation of the solution and bore trajectory to second order accuracy. All our asymptotic results, obtained in the form of multiple scale expansions, are compared with numerical solutions for a number of illustrative cases.     

The effect of surface topography on weakly nonlinear roll waves

The propagation of short waves in turbulent single layer flows forming on inclined surfaces has received considerable interest in the past. It is well known that such flows on flat surfaces are unstable if the Froude number of the unperturbed uniform state exceeds a critical value. In the initial linear stage disturbances grow exponentially with propagation distance but it has been shown that weakly nonlinear effects may limit the maximum wave amplitude under strictly periodic conditions leading in turn to a train of permanent roll waves. The present study investigates how the flow behaviour is affected if the slope of the bounding surface is no longer constant but changing slowly in the streamwise direction. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear Stability of Baroclinic Fronts in a Channel with Variable Topography

The governing equations describing baroclinic bottom-trapped fronts in a channel with variable bottom topography are shown to be a noncanonical Hamiltonian system. The Hamiltonian formalism is exploited to derive a variational principle for arbitrary steady solutions based on an appropriately constrained energy functional. The variational principle is exploited to obtain formal and nonlinear stability conditions. In the infinitesimal amplitude limit, these stability conditions reduce to previously obtained normal mode results for the transverse gradient of the mean frontal potential vorticity.     

Local bifurcation and regularity for steady periodic capillary–gravity water waves with constant vorticity

We study periodic capillary–gravity waves at the free surface of water in a flow with constant vorticity over a flat bed. Using bifurcation theory the local existence of waves of small amplitude is proved even in the presence of stagnation points in the flow. We also derive the dispersion relation. Moreover, we prove a regularity result for the free surface.