Higher-order (2 + 4) Korteweg-de Vries-like equation for interfacial waves in a symmetric three-layer fluid

The nonlinear deformation of long internal waves in the ocean is studied using the dispersionless Gardner equation. The process of nonlinear wave deformation is determined by the signs of the coefficients of the quadratic and cubic nonlinear terms; the breaking time depends only on their absolute values. The explicit formula for the Fourier spectrum of the deformed Riemann wave is derived and used to investigate the evolution of the spectrum of the initially pure sine wave. It is shown that the spectrum has exponential form for small times and a power asymptotic before breaking. The power asymptotic is universal for arbitrarily chosen coefficients of the nonlinear terms and has a slope close to –8/3

Section: Dispersionless Gardner Equation For Long Internal Wavesmentioning

confidence: 99%

Fourier spectrum and shape evolution of an internal Riemann wave of moderate amplitude

Kartashova

Pelinovsky

Talipova

2013

“…In the case of IWs in a three-layer fluid, it is necessary to produce a secondorder weakly nonlinear theory to resolve some of such situations. The attempts in this direction have so far been limited to the specific case of symmetric stratification (Kurkina et al, 2011b).…”

Section: O E Kurkina Et Al: Propagation Regimes Of Interfacial Solmentioning

confidence: 99%

“…The properties of mode-1 IWs in a symmetric three-layer fluid (when the undisturbed state is symmetric about the mid-depth) were analyzed up to the second order in nonlinearity by Grimshaw et al (1997). An extension involving up to the fourth-order nonlinear terms was derived in Kurkina et al (2011b). The model of Yang et al (2010) involved both modes of IWs in a general three-layer ocean, the first order in nonlinearity and dispersion (Korteweg-de Vries (KdV) approximation; only mode-2 waves were analyzed).…”

Section: O E Kurkina Et Al: Propagation Regimes Of Interfacial Solmentioning

confidence: 99%

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Propagation regimes of interfacial solitary waves in a three-layer fluid

Kurkina¹,

Kurkin²,

Rouvinskaya³

et al. 2015

Self Cite

Abstract. Long weakly nonlinear finite-amplitude internal waves in a fluid consisting of three inviscid layers of arbitrary thickness and constant densities (stable configuration, Boussinesq approximation) bounded by a horizontal rigid bottom from below and by a rigid lid at the surface are described up to the second order of perturbation theory in small parameters of nonlinearity and dispersion. First, a pair of alternatives of appropriate KdV-type equations with the coefficients depending on the parameters of the fluid (layer positions and thickness, density jumps) are derived for the displacements of both modes of internal waves and for each interface between the layers. These equations are integrable for a very limited set of coefficients and do not allow for proper description of several near-critical cases when certain coefficients vanish. A more specific equation allowing for a variety of solitonic solutions and capable of resolving most near-critical situations is derived by means of the introduction of another small parameter that describes the properties of the medium and rescaling of the ratio of small parameters. This procedure leads to a pair of implicitly interrelated alternatives of Gardner equations (KdV-type equations with combined nonlinearity) for the two interfaces. We present a detailed analysis of the relationships for the solutions for the disturbances at both interfaces and various regimes of the appearance and propagation properties of soliton solutions to these equations depending on the combinations of the parameters of the fluid. It is shown that both the quadratic and the cubic nonlinear terms vanish for several realistic configurations of such a fluid.

“…To the best of the author's knowledge, neither specific nonlinearity in terms of power series of wave amplitudes necessary to reveal a two-humped structure nor regions of density profiles with a single pycnocline at which such structures exist have been examined in the literature. Kurkina et al (2011) derived a KdV-like equation with quadratic and quartic nonlinear terms for interfacial transient waves for the specific three-layer geometry. Assumption of a small albeit finite wave amplitude was essential to balance nonlinearity and dispersion in that study.…”

Section: Introductionmentioning

confidence: 99%

Brief communication: Multiscaled solitary waves

Derzho

2017

Abstract. It is analytically shown how competing nonlinearities yield multiscaled structures for internal solitary waves in stratified shallow fluids. These solitary waves only exist for large amplitudes beyond the limit of applicability of the Korteweg-de Vries (KdV) equation or its usual extensions. The multiscaling phenomenon exists or does not exist for almost identical density profiles. The trapped core inside the wave prevents the appearance of such multiple scales within the core area. The structural stability of waves of large amplitudes is briefly discussed. Waves of large amplitudes displaying quadratic, cubic and higher-order nonlinear terms have stable and unstable branches. Multiscaled waves without a vortex core are shown to be structurally unstable. It is anticipated that multiscaling phenomena will exist for solitary waves in various physical contexts.