The Propagation of Nonlinear Wave Envelopes

Benney, D. J.; Newell, Alan C.

doi:10.1002/sapm1967461133

Cited by 548 publications

(435 citation statements)

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“…It is thus concluded that Stokes waves are unstable to certain side-band disturbances and they cannot propagate over long distances without changing form. This work was further generalized by Benney and Newell (1967) for general weakly nonlinear dispersive systems and by Benney and Roskes (1969) for the study of three dimensional instability of Stokes waves.…”

Section: Survey Of Literaturementioning

confidence: 97%

On slowly-varying Stokes waves

1970

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ON SLOWLY VARYING STOKES WAVES by VINCENT HWA-HING CHUIn this thesis investigations are made on the theory of a train of slowly modulated gravity waves propagating over uneven bottom topography. The primary object is to study the interplay of amplitude dispersion the frequency dispersion in waves on the surface of water where the depth is not too shallow compared to a typical wave length.The solution of the wave train is expressed in expansions of the WKB type with a small parameter which is proportional to the wave steepness and the rate of modulation. A systematic perturbation scheme that may be carried out for all orders is presented. It is found that new terms directly representing modulation rate must be included to extend the scope of the Whitham's theory based on an averaged variational principle. Several specific examples are discussed and the following main conclusions are reached:1. The Stokes waves of constant amplitude are unstable for all depths under three dimensional disturbances. 2. For quasi-steady waves normally incident on a mild beach, the local rate of depth variation is found to affect the wave phase which in turn gives additional dispersion effect as compared to Stokes waves of constant amplitude. 3. In deep water, there exist solutions of permanent wave envelopes which represent exact balance between amplitude dispersion and frequency dispersion. 4. Numerical computation on the transient development of the wave envelopes reveals that any wave group deviated from permanent form will eventually disintegrate into a sequence of peaks separated by nodal points (where amplitude is zero) and locally the wave group tends to approach the dynamically stable permanent form.

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Section: Survey Of Literaturementioning

confidence: 97%

On slowly-varying Stokes waves

1970

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“…In the limit in which the time derivatives in Eq. (37b) may be neglected, this system reduces to the nonlinear Schrödinger equation (Benney and Newell, 1967), which in 1D and in the absence of linear damping is…”

Section: Zakharov Equationmentioning

confidence: 99%

Fundamental statistical descriptions of plasma turbulence in magnetic fields

Krommes¹

2002

Physics Reports

267

397

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A pedagogical review of the historical development and current status (as of early 2000) of systematic statistical theories of plasma turbulence is undertaken. Emphasis is on conceptual foundations and methodology, not practical applications. Particular attention is paid to equations and formalism appropriate to strongly magnetized, fully ionized plasmas. Extensive reference to the literature on neutral-fluid turbulence is made, but the unique properties and problems of plasmas are emphasized throughout. Discussions are given of quasilinear theory, weak-turbulence theory, resonance-broadening theory, and the clump algorithm. Those are developed independently, then shown to be special cases of the direct-interaction approximation (DIA), which provides a central focus for the article. Various methods of renormalized perturbation theory are described, then unified with the aid of the generating-functional formalism of Martin, Siggia, and Rose. A general expression for the renormalized dielectric function is deduced and discussed in detail. Modern approaches such as decimation and PDF methods are described. Derivations of DIA-based Markovian closures are discussed. The eddy-damped quasinormal Markovian closure is shown to be nonrealizable in the presence of waves, and a new realizable Markovian closure is presented. The test-field model and a realizable modification thereof are also summarized. Numerical solutions of various closures for some plasmaphysics paradigms are reviewed. The variational approach to bounds on transport is developed. Miscellaneous topics include Onsager symmetries for turbulence, the interpretation of entropy balances for both kinetic and fluid descriptions, self-organized criticality, statistical interactions between disparate scales, and the roles of both mean and random shear. Appendices are provided on Fourier transform conventions, dimensional and scaling analysis, the derivations of nonlinear gyrokinetic and gyrofluid equations, stochasticity criteria for quasilinear theory, formal aspects of resonance-broadening theory, Novikov's theorem, the treatment of weak inhomogeneity, the derivation of the Vlasov weak-turbulence wave kinetic equation from a fully renormalized description, some features of a code for solving the direct-interaction approximation and related Markovian closures, the details of the solution of the EDQNM closure for a solvable three-wave model, and the notation used in the article.

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“…The coupled nonlinear Schrödinger equations govern the evolution of two interacting wave packets in nonlinear and dispersive physical systems [10]. These equations are particularly important in nonlinear optics as they govern the pulse propagation in birefringent nonlinear optical fibers [11].…”

Section: Coupled Nonlinear Schrödinger Equationsmentioning

confidence: 99%