Abstract

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.

Highlights

  • 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

  • There are many important topics related to internal waves (IWs) in the ocean

  • We develop here an analytic model for long IWs of finite amplitude for the three-layer fluid with an arbitrary combination of the layers’ thicknesses

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Summary

Introduction

There are many important topics related to internal waves (IWs) in the ocean. Large IWs are highly significant for sediment resuspension and transport (Bogucki and Redekopp, 1999; Stastna and Lamb, 2008; Reeder et al, 2011) and for the biology on the continental shelf (Sandstrom and Elliott, 1984). The associated strong distortion of the density field has a severe impact on acoustic signaling (Apel et al, 2007; Chin-Bing et al, 2009; Warn-Varnas et al, 2009; Sridevi et al, 2011) Their capacity to break and impact the local microstructure has major consequences for the understanding of interior ocean mixing (Muller and Briscoe, 2000). The danger from IWs is considered so critical that, similar to the systems of tsunami warning, the potential for automated detection systems for large-amplitude IWs (internal soliton early warning systems) is being discussed Such systems were even tested to support drilling campaigns and guarantee the safety of drilling platforms (Stober and Moum, 2011)

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