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

TL;DR

This paper discusses the application of the periodic inverse scattering transform (PIST) to analyze internal solitary waves in the ocean, introducing improved numerical methods and demonstrating insights into wave phenomena through nonlinear Fourier spectra.

Contribution

It presents new numerical techniques for computing the PIST spectrum and applies the method to real ocean data, advancing the analysis of nonlinear wave dynamics.

Findings

Enhanced spectral eigenvalue bracketing methods

A new root-finding algorithm for PIST spectrum calculation

Insight into soliton-induced acoustic resonances in ocean data

Abstract

The periodic, inverse scattering transform (PIST) is a powerful analytical tool in the theory of integrable, nonlinear evolution equations. Osborne pioneered the use of the PIST in the analysis of data form inherently nonlinear physical processes. In particular, Osborne's so-called nonlinear Fourier analysis has been successfully used in the study of waves whose dynamics are (to a good approximation) governed by the Korteweg--de Vries equation. In this paper, the mathematical details and a new application of the PIST are discussed. The numerical aspects of and difficulties in obtaining the nonlinear Fourier (i.e., PIST) spectrum of a physical data set are also addressed. In particular, an improved bracketing of the "spectral eigenvalues" (i.e., the +/-1 crossings of the Floquet discriminant) and a new root-finding algorithm for computing the latter are proposed. Finally, it is shown how the PIST can be used to gain insightful information about the phenomenon of soliton-induced acoustic resonances, by computing the nonlinear Fourier spectrum of a data set from a simulation of internal solitary wave generation and propagation in the Yellow Sea.