Internal solitary waves in the ocean: Analysis using the periodic, inverse scattering transform

The Korteweg-de Vries (KdV) equation is arguably the most famous "soliton-bearing" equation, governing phenomena as seemingly disparate as the motion of lattices, the collective behavior of plasmas and the evolution of hydrodynamic waves (see, e.g., [1,2,18,20,23,26,28] and the references therein). From a mathematical point of view, the most interesting property of the KdV equation, discovered by the celebrated numerical experiment of Zabusky and Kruskal (ZK) [26], is the ability of its (localized) traveling-wave solutions to retain their "identity" (shape, speed) after colliding with each other. This has been a topic subject to much research over the last few decades, giving birth to the idea of integrable equations and leading to the discovery of the inverse scattering transform. In particular, it has been shown that the KdV equation is "fully-integrable" on both the real line and periodic intervals; therefore, it can be solved exactly by the (periodic) inverse scattering transform [1]. What is more, Salupere et al. [23] have shown that the ZK experiment features more than just the particle-like interactions of the localized solutions of the KdV equation: there exist hidden solitons, and soliton ensembles emerge on long time scales in the "wave soup" created by the ZK sine wave initial condition.

It is this richness of the solution of the periodic KdV equation and its integrability by the scattering transform that form the basis of Osborne’s nonlinear Fourier analysis [12]. The latter has been successfully employed in the Fourierlike decomposition of data from inherently nonlinear physical phenomena such as shallow-water ocean surface waves [22], laboratory-generated surface waves [20] and internal gravity waves in a stratified fluid [28]. In the present work, we focus on the implementation and application of Osborne’s nonlinear Fourier analysis to yet another nonlinear-wave phenomenon governed by the KdV equation: internal solitary waves in the ocean.

In their 1980 article, Osborne and Burch [18] put forth a model of internal solitary waves in the ocean based on the KdV equation. Since then, their model has been applied to a variety of internal solitary wave phenomena (see, e.g., [2] and the references therein). In particular, the latter model was employed by Zhou et al. [27] to explain the anomalous acoustic signal loss in the presence of internal solitary waves in the ocean. More recently, Chin-Bing et al. [6] and Warn-Varnas et al. [24,25] have provided evidence of such anomalous signal losses via coupled oceanacoustic simulations. However, though there is ample evidence of the effects of internal solitary waves on acoustic signals, there remain unanswered (theoretical) questions regarding, e.g., energy transfer (mode conversions) in the acoustic field (see, e.g., [24] and the references therein). Thus, it is a goal of the present work to lay the foundation for the use of the nonlinear Fourier analysis of internal solitary wave trains in the ocean in obtaining more accurate and “natural” wavenumber (or mode) ranges than the ordinary Fourier transform. These, in turn, can be used to prove or disprove conjectures regarding energy transfer in the acoustic field.

This paper is organized as follows. In Section 2, the KdV-based mathematical model of internal solitary waves in the ocean is described. In Section 3, the scattering transform for the KdV equation is presented and its interpretation as a nonlinear generalization of the Fourier transform is discussed. In Section 4, the numerical implementation of the PIST is addressed, some of the shortcomings of previous algorithms are pointed out, and improvements are suggested. In Section 5, the PIST is applied to the analysis of simulated internal solitary wave data and the results are presented. Finally, in Section 6, we discuss the future of the present technique.

In this section, we summarize Osborne and Burch’s [18] model of internal solitary waves (or solitons, as the case might be) in the ocean. It is based upon the assumption of a two-layer stratification in the ocean, where the two layers are separated either by the thermocline or by the pycnocline, i.e., a region of rapid change in the temperature or the density of the stratification, respectively. Then, it is supposed that solitary waves traveling on the interface between the two layers are governed by the ubiquitous KdV equation, whose solitary wave solutions are solitons (so, the terms ‘solitary wave’ and ‘soliton’ become interchangeable henceforth). The (displaced) interface is a surface of constant temperature or constant (potential) density, and it is thus referred to either as an isotherm or as an isopycnal, depending on the type of stratification. This physical set up is illustrated in Fig. 1 (see also Fig. 3 in [18]). Moreover, for the remainder of this paper, we restrict ourselves to the case of a desnity-stratified ocean, so we use the terms ‘pycnocline’ and ‘isopycnal’ instead of ’ther

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