Spectral analysis of internal solitary waves propagating over a stepped bottom topography via the Koopman operator

In this paper, we present a study about the frequency characteristics of the process of internal solitary waves (ISWs) interacting with a stepped bottom topography. We perform experimental measurements of the waveforms and flow fields under various wave-making conditions by considering the degree of subsequent breaking. The piecewise dynamic mode decomposition (PDMD) method, which we have proposed, is introduced to construct the Koopman operator, linearize the process, and extract spectral information of the interaction. Furthermore, the universality of this method and the physical meaning of segmentation points are discussed for the ISW problem. The innovative part of this study lies in that to suit the precondition of PDMD, the energy formula of a Koopman mode is modified with emphasis on the damping rate. The spectra calculated by the modified modal energy are more in line with the physical phenomenon of the evolution. Through the spectral analysis, we infer that the occurrence of breaking may limit the main energy part of waveforms into a relatively low-frequency range, instead of generating high-frequency rapid oscillations. In contrast, the flow fields will contain more high-frequency information during the breaking process. The specific performance is that the spectra of vorticity fields have high-frequency sidebands that are clearly separated from the main energy part. Finally, to understand the flow behavior of ISWs, we extract and analyze the spatial information of the decomposed modes at dominant or distinctive frequencies. The modes corresponding to the oscillations of trailing edges and the early breaking phenomenon of vorticity fields are observed.