Abstract
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.
Highlights
An internal solitary wave (ISW) is a common oceanic phenomenon in which the amplitude is possibly extremely large
The piecewise dynamic mode decomposition (PDMD) method we have proposed is believed to be suitable for the ISW problem
If the quasi-linear segments split by PDMD have different performances under various incident conditions, the spectral analysis of the ISW problem is not so meaningful
Summary
An internal solitary wave (ISW) is a common oceanic phenomenon in which the amplitude is possibly extremely large. An ISW generally acts as a soliton in stable propagation, it may contain complex flow characteristics with multi-frequency when interacting with topography. The evolution of large-amplitude ISWs is a strong nonlinear process.. Xu et al presented high-resolution, two- and three-dimensional direct numerical simulations to investigate the instability of large amplitude shoaling ISWs and suggested that all of these wave-induced instabilities can lead to enhanced turbulence in the water column and increased shear stress on the bottom boundary.. Terletska et al studied ISWs shoaling over an idealized slope-shelf topography and proposed a three-dimensional diagram to classify the wave breaking.. Talipova et al extended the above numerical studies by considering the energy loss of both polarities interacting with a bottom step and discussed the features of the transformation of a largeamplitude ISW.. The possibility of the internal breather generation in the range of intermediate wavelengths and in a thin intermediate layer has been demonstrated. Liu et al examined the transformation of mode-1 ISWs incident on a bottom step and the consequent generation of mode-2 ISWs.
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