The propagation of internal solitary waves (ISWs) flowing over the submerged topography is a strongly nonlinear process. To extract the dynamic characteristics of this process, an improved dynamic mode decomposition method is proposed in this paper, which is named piecewise dynamic mode decomposition (PDMD). The innovation of this method is to split the entire evolution process into several quasi-linear segments before modal analyzing to reduce the requirements on the spatial and temporal resolutions of input measured data. A feasible criterion for linearity is introduced by combining the proper orthogonal decomposition method, which is an important basis of PDMD. The data used in the analysis are provided by the experiments conducted in a stratified wave tank. The experimental conditions are set as ISWs flowing over two typical bottom topographies. The interfacial displacement and flow field information are analyzed as the measured data. Through reconstruction and modal analysis of experimental data, the effectiveness and flexibility of PDMD are verified for the ISW problem. The physical meaning of segmentation points can be explained. Based on the results of model decomposition, the main propagation characteristics of ISWs under different conditions are discussed. The evolution of the waveform or local flow phenomena can be simplified to the superposition of linear modes with frequency information.
Waveform deformation and breaking are widespread phenomena when internal solitary waves (ISWs) encounter changing topographies, which have been observed in many parts of oceans. In this study, experiments are performed in a series of combinations of bottom step topographies with different heights and ISWs in different amplitudes within a two-layer stratified fluid system. According to experimental results, the evolution processes of ISWs over the bottom step are classified into four typical regimes as the wave–step interaction varying from weak to strong, which are the transmission regime, transitional regime, breaking regime, and reflection regime, corresponding to the evolution patterns of steady passage, deformation, breaking, and strong reflection, respectively. To describe the intensity of wave–step interaction, a new improved interaction parameter is proposed, which takes both relative amplitude of ISWs and relative topography changes into consideration, and achieved better effectiveness in defining the boundaries between different regimes. In terms of energy properties, with the wave–step interaction becoming stronger, the transmission ratio keeps decreasing throughout all regimes, while the reflection wave starts to appear since the breaking regime and its energy keeps increasing. At the critical point between the breaking regime and reflection regime, the reflection ratio equals the transmission ratio, and the energy loss ratio reaches its maximum.
The propagation of internal solitary waves (ISWs) underwater and their interactions with surface waves can affect the sea surface state and generate distinct surface signatures of bright-dark striped patterns in remote sensing imagery. These patterns have been one of the most important methods for wide coverage detection and observation of ISWs. This study establishes a fully nonlinear numerical model for ISW-surface wave interaction simulation based on the multi-domain boundary element method. By numerical simulation, the formation mechanism, characteristics, and influencing factors of ISW surface signatures are revealed. The physical essence of the ISW surface signature is the spatially inhomogeneous wavelength and roughness changes in the surface waves due to the convergence and divergence effects of the ISW. Based on the surface wave wavelength changes, the surface area affected by an ISW is divided into three regions from front to back-the converging, diverging, and restoring regions-corresponding to the regions where the surface waves are compressed, stretched, and recover, respectively. By comparing different cases, it is found that the ISW amplitude, surface wave amplitude, and surface wave wavelength are the main factors influencing the ISW surface signature. ISWs with larger amplitudes have stronger convergence and divergence effects on surface waves, making the overall scale of the ISW surface signature larger. Larger surface wave amplitudes provide stronger resistance to the ISW divergence effect and lead to smaller diverging regions, and larger surface wave wavelengths make the converging region larger and the diverging and restoring regions smaller.
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.
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