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.     

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.     

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.     

Nonlinear Waves in a Shear Flow with a Vorticity Discontinuity

We consider nonlinear finite-amplitude progressive shear-flow waves on a basic velocity profile consisting of two coflowing layers of inviscid equal-density fluid, each of uniform but different vorticity. The problem is formulated as a nonlinear integral equation describing the shape of the vorticity discontinuity in a frame of reference in which the flow is steady. Numerical solutions to this equation are presented for a range of values of the vorticity ratio Ω. For 1 > © ≥ ? 1 the theoretical maximum wave amplitude occurs when the wave crest forms a 90° corner which just touches the appropriate critical-layer stagnation point. The linearized stability of the progressive wave states to arbitrary subharmonic isovortical disturbances is studied numerically. The results indicate stability at moderate values of the wave amplitude.     

Do peaked solitary water waves indeed exist?

Many models of shallow water waves, such as the famous Camassa–Holm equation, admit peaked solitary waves. However, it is an open question whether or not the widely accepted peaked solitary waves can be derived from the fully nonlinear wave equations. In this paper, a unified wave model (UWM) based on the symmetry and the fully nonlinear wave equations is put forward for progressive waves with permanent form in finite water depth. Different from traditional wave models, the flows described by the UWM are not necessarily irrotational at crest, so that it is more general. The unified wave model admits not only the traditional progressive waves with smooth crest, but also a new kind of solitary waves with peaked crest that include the famous peaked solitary waves given by the Camassa–Holm equation. Besides, it is proved that Kelvin’s theorem still holds everywhere for the newly found peaked solitary waves. Thus, the UWM unifies, for the first time, both of the traditional smooth waves and the peaked solitary waves. In other words, the peaked solitary waves are consistent with the traditional smooth ones. So, in the frame of inviscid fluid, the peaked solitary waves are as acceptable and reasonable as the traditional smooth ones. It is found that the peaked solitary waves have some unusual and unique characteristics. First of all, they have a peaked crest with a discontinuous vertical velocity at crest. Especially, unlike the traditional smooth waves that are dispersive with wave height, the phase speed of the peaked solitary waves has nothing to do with wave height, but depends (for a fixed wave height) on its decay length, i.e., the actual wavelength: in fact, the peaked solitary waves are dispersive with the actual wavelength when wave height is fixed. In addition, unlike traditional smooth waves whose kinetic energy decays exponentially from free surface to bottom, the kinetic energy of the peaked solitary waves either increases or almost keeps the same. All of these unusual properties show the novelty of the peaked solitary waves, although it is still an open question whether or not they are reasonable in physics if the viscosity of fluid and surface tension are considered.     

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.     

Dynamics of Strongly Nonlinear Internal Solitary Waves in Shallow Water

We study the dynamics of large amplitude internal solitary waves in shallow water by using a strongly nonlinear long-wave model. We investigate higher order nonlinear effects on the evolution of solitary waves by comparing our numerical solutions of the model with weakly nonlinear solutions. We carry out the local stability analysis of solitary wave solution of the model and identify an instability mechanism of the Kelvin–Helmholtz type. With parameters in the stable range, we simulate the interaction of two solitary waves: both head-on and overtaking collisions. We also study the deformation of a solitary wave propagating over non-uniform topography and describe the process of disintegration in detail. Our numerical solutions unveil new dynamical behaviors of large amplitude internal solitary waves, to which any weakly nonlinear model is inapplicable.     

Steady solution of solitary wave and linear shear current interaction

The steady solution of a solitary wave propagating in the presence of a linear shear background current is investigated by the Green–Naghdi (GN) equations. The steady solution is obtained by use of the Newton–Raphson method. Three aspects are investigated; they are the wave speed, wave profile and velocity field. The converged GN results are compared with results from the literature. It is found that for the opposing-current case of the solitary wave with a small amplitude, the results of the GN equations match results from the literature well, while for the solitary wave with a large amplitude, results from the literature are seen to be not as accurate. In the following-current case, though the amplitude of the solitary wave is small, the GN results are shown to be accurate. The velocity along the water column at the wave crest and the velocity field for different cases are calculated by the GN equations. The results of the GN equations show obvious differences when compared with the results obtained by superposing the no-current results and linear shear current linearly. We find that for the same current strength, the vortex is stronger for the steep solitary-wave case than that for the small solitary-wave case.     

Exact solitary water waves with capillary ripples at infinity

We prove the existence of solitary water waves of elevation, as exact solutions of the equations of steady inviscid flow, taking into account the effect of surface tension on the free surface. In contrast to the case without surface tension, a resonance occurs with periodic waves of the same speed. The wave form consists of a single crest on the elongated scale with a much smaller oscillation at infinity on the physical scale. We have not proved that the amplitude of the oscillation is actually nonzero; a formal calculation suggests that it is exponentially small.     

Experimental Evidence for Breather Type Dynamics in Freak Waves

Freak or rogue waves on the ocean seemingly appear from nowhere, cause severe damage to ships and offshore structures due to their large crest heights, and disappear at once. Since the Draupner wave measured on New Year's day 1995 finally confirmed the existence of freak waves, different models were developed to describe them. One deterministic model to investigate their occurrence is the nonlinear Schrödinger equation (NLS) describing the nonlinear evolution of wave train envelopes. Due to the modulation instability, also referred to as Benjamin-Feir instability, strong spatial localization of wave amplitude may arise and breather type solutions are hypothesized to form the dynamical back-bone of rogue waves. To test this hypothesis, breather type solutions of the NLS are compared to two extreme wave records: the Draupner wave recorded in central North Sea and the Yura wave recorded in the Sea of Japan. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Internal waves in an unbounded non-Boussinesq flow

Weakly nonlinear internal waves in an unbounded non-Boussinesq flow with uniform stratification are treated with a Laurent-type expansion. The expansion eliminates the problem encountered with a traditional expansion in wave amplitude where higher harmonics grow exponentially faster with higher order. The results show that the second-order wave correction to the linear estimate of the wave speed of internal waves in an unbounded layer is always negative, meaning that higher amplitude waves travel slower.     

Stability properties of steady water waves with vorticity

We present two stability analyses for exact periodic traveling water waves with vorticity. The first approach leads in particular to linear stability properties of water waves for which the vorticity decreases with depth. The second approach leads to a formal stability property for long water waves that have small vorticity and amplitude although we do not use a small‐amplitude or long‐wave approximation. © 2006 Wiley Periodicals, Inc.     

On the pressure transfer function for solitary water waves with vorticity

In this paper we analyse the role which the pressure function on the sea-bed plays in determining solitary waves with vorticity. We prove that the pressure function on the flat bed determines a unique, real analytic solitary wave solution to the governing equations, given a real analytic vorticity distribution. In particular, the pressure function on the flat bed prescribes a unique surface profile for the resulting solitary water wave.     

Rotational waves generated by current‐topography interaction

We study nonlinear free‐surface rotational waves generated through the interaction of a vertically sheared current with a topography. Equivalently, the waves may be generated by a pressure distribution along the free surface. A forced Korteweg–de Vries equation (fKdV) is deduced incorporating these features. The weakly nonlinear, weakly dispersive reduced model is valid for small amplitude topographies. To study the effect of gradually increasing the topography amplitude, the free surface Euler equations are formulated in the presence of a variable depth and a sheared current of constant vorticity. Under constant vorticity, the harmonic velocity component is formulated in a simplified canonical domain, through the use of a conformal mapping which flattens both the free surface as well as the bottom topography. Critical, supercritical, and subcritical Froude number regimes are considered, while the bottom amplitude is gradually increased in both the irrotational and rotational wave regimes. Solutions to the fKdV model are compared to those from the Euler equations. We show that for rotational waves the critical Froude number is shifted away from 1. New stationary solutions are found and their stability tested numerically.     

Nonlinear Alfvén/Beltrami waves—An integrable structure built around the Casimir

We formulate a nonlinear wave equations that describe amplitude and pitch modulations of one-dimensional Alfvén waves propagating on a dispersive nonlinear plasma. The well-known fact that the ideal Alfvén wave can propagate on a homogeneous ambient magnetic field with conserving an arbitrary wave shape of any amplitude is explained by invoking the Casimirs stemming from a “topological defect” (or, a kernel) in the Poisson bracket operator of the ideal magnetohydrodynamic (MHD) system. Including the Hall term, however, the Alfvén waves are affected by the dispersive effect, and the aforementioned simplicity of the ideal Alfvén waves is greatly lost; an arbitrary wave can no longer propagate with a constant shape. Yet, we observe an integrable structure in the nonlinear modulation (induced by a compressible motion) of the Alfvén waves, which is described as nonlinear deformation of “Beltrami vortex” pertaining to the Casimirs.     

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.     

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)     

Solitary wave solutions to the Isobe-Kakinuma model for water waves

We consider the Isobe-Kakinuma model for two-dimensional water waves in the case of a flat bottom. The Isobe-Kakinuma model is a system of Euler-Lagrange equations for a Lagrangian approximating Luke's Lagrangian for water waves. We show theoretically the existence of a family of small amplitude solitary wave solutions to the Isobe-Kakinuma model in the long wave regime. Numerical analysis for large amplitude solitary wave solutions is also provided and suggests the existence of a solitary wave of extreme form with a sharp crest.     

Rossby Waves on a Shear Flow with Recirculation Cores

Large-amplitude Rossby waves riding on a background flow with a weak shear can be calculated up to a critical amplitude for which the meridional velocity, in a frame traveling with the wave, approaches zero at some point. Here we consider waves with an amplitude slightly greater than the critical amplitude by incorporating a region of recirculating fluid (vortex core) near this critical point. The effect of the vortex core is to introduce an extra nonlinear term into the equation for the wave amplitude proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude. The main effect due to the vortex core is a broadening of the wave profile. Furthermore, we show that the vortex core family has a limiting amplitude, with the limiting amplitude corresponding to a semi-infinite bore.     

Exact Free Surface Flows for Shallow Water Equations I: The Incompressible Case

A comprehensive exact treatment of free surface flows governed by shallow water equations (in sigma variables) is given. Several new families of exact solutions of the governing PDEs are found and are shown to embed the well-known self-similar or traveling wave solutions which themselves are governed by reduced ODEs. The classes of solutions found here are explicit in contrast to those found earlier in an implicit form. The height of the free surface for each family of solutions is found explicitly. For the traveling or simple wave, the free surface is governed by a nonlinear wave equation, but is arbitrary otherwise. For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed; in another case, the free surface is a horizontal plane while the flow underneath is a sine wave. The existence of simple waves on shear flows is analytically proved. The interaction of large amplitude progressive waves with shear flow is also studied.