Equations governing modulations of weakly nonlinear water waves are described. The modulations are coupled with wave-induced mean flows except in the case of water deeper than the modulation length scale. Equations suitable for water depths of the order the modulation length scale are deduced from those derived by Davey and Stewartson [5] and Dysthe [6]. A number of ases in which these equations reduce to a one dimensional nonlinear Schrodinger (NLS) equation are enumerated.Several analytical solutions of NLS equations are presented, with discussion of some of their implications for describing the propagation of water waves. Some of the solutions have not been presented in detail, or in convenient form before. One is new, a “rational” solution describing an “amplitude peak” which is isolated in space-time. Ma's [13] soli ton is particularly relevant to the recurrence of uniform wave trains in the experiment of Lake et al.[10].In further discussion it is pointed out that although water waves are unstable to three-dimensional disturbances, an effective description of weakly nonlinear two-dimensional waves would be a useful step towards describing ocean wave propagation.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0334270000003891

Water waves, nonlinear Schrödinger equations and their solutions

Transition to turbulence in plane channel flow occurs even for conditions under which modes of the linearized dynamical system associated with the flow are stable. In this paper an attempt is made to understand this phenomena by finding the linear three‐dimensional perturbations that gain the most energy in a given time period. A complete set of perturbations, ordered by energy growth, is found using variational methods. The optimal perturbations are not of modal form, and those which grow the most resemble streamwise vortices, which divert the mean flow energy into streaks of streamwise velocity and enable the energy of the perturbation to grow by as much as three orders of magnitude. It is suggested that excitation of these perturbations facilitates transition from laminar to turbulent flow. The variational method used to find the optimal perturbations in a shear flow also allows construction of tight bounds on growth rate and determination of regions of absolute stability in which no perturbation growth is possible.

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Three‐dimensional optimal perturbations in viscous shear flow

In this note we use the method of multiple scales to derive the two coupled nonlinear partial differential equations which describe the evolution of a three-dimensional wave-packet of wavenumber k on water of finite depth. The equations are used to study the stability of the uniform Stokes wavetrain to small disturbances whose length scale is large compared with 2π/ k . The stability criterion obtained is identical with that derived by Hayes under the more restrictive requirement that the disturbances are oblique plane waves in which the amplitude variation is much smaller than the phase variation.

On three-dimensional packets of surface waves

This book describes the remarkable connections that exist between the classical differential geometry of surfaces and modern soliton theory. The authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Backlund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Backlund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory. It is with these transformations and the links they afford between the classical differential geometry of surfaces and the nonlinear equations of soliton theory that the present text is concerned. In this geometric context, solitonic equations arise out of the Gaus-Mainardi-Codazzi equations for various types of surfaces that admit invariance under Backlund-Darboux transformations. This text is appropriate for use at a higher undergraduate or graduate level for applied mathematicians or mathematical physics.

/pdf/backlund-and-darboux-transformations-geometry-and-modern-4k80wq4g1i.pdf

Bäcklund and Darboux transformations : geometry and modern applications in soliton theory

Four-wave interactions are shown to play an important role in the evolution of the spectrum of surface gravity waves. This fact follows from direct simulations of an ensemble of ocean waves using the Zakharov equation. The theory of homogeneous four-wave interactions, extended to include effects of nonresonant transfer, compares favorably with the ensemble-averaged results of the Monte Carlo simulations. In particular, there is good agreement regarding spectral shape. Also, the kurtosis of the surface elevation probability distribution is determined well by theory even for waves with a narrow spectrum and large steepness. These extreme conditions are favorable for the occurrence of freak waves.

/pdf/nonlinear-four-wave-interactions-and-freak-waves-360c27qd68.pdf

Nonlinear Four-Wave Interactions and Freak Waves

The equations governing the nonlinear development of a centred three-dimensional disturbance to plane parallel flow at slightly supercritical Reynolds numbers are obtained, In contrast to the corresponding equation for two-dimensional disturbances, two slowly varying functions are needed to describe the development: the amplitude function and a function related to the secular pressure gradient produced by the disturbance. These two functions satisfy a pair of coupled partial differential equations. The equations derived in Hocking, Stewartson & Stuart (1972) are shown to be incorrect, Some of the properties of the governing equations are discussed briefly.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112074001765

On the nonlinear evolution of three-dimensional disturbances in plane Poiseuille flow

We consider the propagation of a weak nonlinear wave whose energy is concentrated in a narrow band of wavenumbers in a fluid which is both dispersive and dissipative. We use the small amplitude equations of Whitham's theory of slowly varying wave trains, modified slightly to include dissipation, to show that the modulation of the wave may be described by a nonlinear Schrodinger equation. For long waves which are purely dispersive we obtain the Kortewegde Vries equation, and for long waves which are dissipative we obtain Burgers’ equation by suitable transformations of the nonlinear Schrodinger equation. We mention the problem of Stokes waves in deep water and comment briefly upon invariant far-field theory.

The propagation of a weak nonlinear wave

This paper is concerned with a general study of the modal structure for stratified viscous plane Couette flow with a constant buoyancy frequency. When the overall Richardson number Ri is zero, the velocity and temperature modes are distinct but as Ri is increased there is an intricate interaction between them. Some simple analytical results are obtained for large and small values of the Reynolds number and more detailed results are given for and ½. The present theory would appear to be reasonably complete for 0 [les ] Ri [les ] ¼; for Ri > ¼, however, an important open question concerns the relationship between the limiting form of the viscous modes as the Reynolds number tends to infinity and the spectrum of internal gravity waves.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112077001815

On the stability of stratified viscous plane Couette flow. Part 1. Constant buoyancy frequency

The linear stability of elliptic pipe flow is considered for finite aspect ratios thereby bridging the gap between the small-aspect-ratio analysis of Davey & Salwen (1994) and the large-aspect-ratio asymptotics of Hocking (1977). The flow is found to become linearly unstable above an aspect ratio of about 10.4 to the spanwise-modulated analogue of the Orr-Sommerfeld mode to which plane Poiseuille flow first loses stability. This disturbance is found to possess a series of intense vortices along its critical layer at lateral stations far removed from the central minor axis. The critical Reynolds number appears to fall from infinity as the aspect ratio increases above 10.4, ultimately approaching Hocking's (1977) asymptotic result at much larger aspect ratios.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112096000559

A. Davey

Papers

On three-dimensional packets of surface waves

On the nonlinear evolution of three-dimensional disturbances in plane Poiseuille flow

The propagation of a weak nonlinear wave

On the stability of stratified viscous plane Couette flow. Part 1. Constant buoyancy frequency

On the linear instability of elliptic pipe flow