scholarly journals Retrieval of Internal Solitary Wave Amplitude in Shallow Water by Tandem Spaceborne SAR

2019 ◽  
Vol 11 (14) ◽  
pp. 1706
Author(s):  
Jia ◽  
Liang ◽  
Li ◽  
Fan
Keyword(s):  
Solitary Wave ◽  
Shallow Water ◽  
Layer Thickness ◽  
Kdv Equation ◽  
World Ocean ◽  
Internal Solitary Wave ◽  
Accurate Estimation ◽  
Sar Images ◽  
De Vries

The accurate estimation of the upper layer thickness in a two-layer ocean is a crucial step in the retrieval of internal solitary wave (ISW) amplitude from synthetic aperture radar (SAR) data. In this paper, we present a method to derive the upper layer thickness and the consequent ISW amplitude by combining two consecutive SAR images with the extended Korteweg-de Vries (eKdV) equation. An ISW case observed twice by the Chinese C-band SAR GaoFen-3 (GF-3) and the German X-band SAR TerraSAR-X (TS-X) with a temporal interval of approximately 11 minutes in shallow water to the southeast of Hainan Island in the northwestern South China Sea was used to demonstrate the applicability of the method. Using the in situ measurements of temperature and salinity near the observed ISW, the proposed method yielded an ISW amplitude of −4.52 m, in close proximity to −5.66 ± 1.24 m derived by applying the classic Korteweg–de Vries (KdV) equation based on the continuously stratified theory. Moreover, the climatological dataset of the World Ocean Atlas 2013 (WOA13) was also used with the proposed method in the Hainan case, and the results showed that the method can still provide a reasonable estimate of ISW amplitude in shallow water even when in situ oceanic stratification measurements are absent. The application of our method to derive the ISW amplitude from consecutive SAR images seems highly promising with the increasing emergence of tandem satellites in orbits.

scholarly journals Combined Effect of Rotation and Topography on Shoaling Oceanic Internal Solitary Waves

2014 ◽  
Vol 44 (4) ◽  
pp. 1116-1132 ◽  
Author(s):  
Roger Grimshaw ◽  
Chuncheng Guo ◽  
Karl Helfrich ◽  
Vasiliy Vlasenko
Keyword(s):  
Solitary Wave ◽  
Wave Packet ◽  
Numerical Simulations ◽  
Solitary Waves ◽  
Combined Effect ◽  
Internal Solitary Wave ◽  
Internal Solitary Waves ◽  
De Vries ◽  
Ostrovsky Equation ◽  
Variable Topography

Abstract Internal solitary waves commonly observed in the coastal ocean are often modeled by a nonlinear evolution equation of the Korteweg–de Vries type. Because these waves often propagate for long distances over several inertial periods, the effect of Earth’s background rotation is potentially significant. The relevant extension of the Kortweg–de Vries is then the Ostrovsky equation, which for internal waves does not support a steady solitary wave solution. Recent studies using a combination of asymptotic theory, numerical simulations, and laboratory experiments have shown that the long time effect of rotation is the destruction of the initial internal solitary wave by the radiation of small-amplitude inertia–gravity waves, and the eventual emergence of a coherent, steadily propagating, nonlinear wave packet. However, in the ocean, internal solitary waves are often propagating over variable topography, and this alone can cause quite dramatic deformation and transformation of an internal solitary wave. Hence, the combined effects of background rotation and variable topography are examined. Then the Ostrovsky equation is replaced by a variable coefficient Ostrovsky equation whose coefficients depend explicitly on the spatial coordinate. Some numerical simulations of this equation, together with analogous simulations using the Massachusetts Institute of Technology General Circulation Model (MITgcm), for a certain cross section of the South China Sea are presented. These demonstrate that the combined effect of shoaling and rotation is to induce a secondary trailing wave packet, induced by enhanced radiation from the leading wave.

Analytic Solitary-wave Solutions for Modified Korteweg - de Vries Equation with t-dependent Coefficients

2001 ◽  
Vol 56 (5) ◽  
pp. 366-370 ◽  
Author(s):  
Woo-Pyo Hong ◽  
Myung-Sang Yoona
Keyword(s):  
Solitary Wave ◽  
Symbolic Computation ◽  
Vries Equation ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Modified Kdv Equation ◽  
De Vries ◽  
Truncated Painlevé Expansion

Abstract We find analytic solitary wave solutions for a modified KdV equation with t-dependent coefficients of the form ut - 6α(t)uux + ß (t) uxxx -6γu2ux = 0. We make use of both the application of the truncated Painleve expansion and symbolic computation to obtain an auto-Bäcklund transformation. We show that kink-type analytic solitary-wave solutions exist under some constraints on α (t), ß (t) and γ.

Observations of fine structure changes in shoaling internal solitary waves based on seismic oceanography method

2021 ◽  
Author(s):  
Haibin Song ◽  
Yi Gong ◽  
Yongxian Guan ◽  
Wenhao Fan ◽  
Yunyan Kuang
Keyword(s):  
Fine Structure ◽  
Solitary Wave ◽  
Phase Velocity ◽  
Shallow Water ◽  
Solitary Waves ◽  
Internal Solitary Wave ◽  
Internal Solitary Waves ◽  
Seismic Events ◽  
Velocity Amplitude ◽  
Seismic Oceanography

<p>In the study of shoaling internal solitary waves, the observation and research on the internal fine structure and the effect of the topography are still insufficient. We try to make up for such insufficiency by seismic oceanography method. A first-mode depression internal solitary wave was observed propagating on the continental slope in the northeast South China Sea near Dongsha Atoll. We used common offset gathers (COGs) to obtain a series of images of this internal solitary wave that evolved over time, and studied the changes in internal fine structure by analyzing the seismic events in COG migrated sections. We found that the seismic events were broken during the shoaling, which was caused by the instability induced by internal solitary wave. We picked six events which represent six waveform and analyzed their evolution. It was found that the change in shape of waveform at different depths is different. The waveform in deep water deforms before that in shallow water, and the waveform in shallow water deforms to a greater degree. In addition, we also counted four parameters of phase velocity, amplitude, wavelength, and slopes of front and rear during the shoaling. The results show that the phase velocity and amplitude of waveform in shallow water increases, the wavelength decreases, and the slope of rear gradually becomes larger than that of the front. We have compared the observed changes with previous study made by numerical simulation.</p>

Phenomena on Rogue Waves and Rational Solution of Korteweg-de Vries Equation

2013 ◽  
Author(s):  
Ni Song ◽  
Wei Zhang ◽  
Qian Wang
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Kdv Equation ◽  
Rational Solution ◽  
Rogue Waves ◽  
Nls Equation ◽  
Engineering Structures ◽  
Weakly Nonlinear ◽  
Nonlinear Mechanism ◽  
De Vries

An appropriate nonlinear mechanism may create the rogue waves. Perhaps the simplest mechanism, which is able to create considerate changes in the wave amplitude, is the nonlinear interaction of shallow-water solitons. The most well-known examples of such structure are Korteweg-de Vries (KdV) solitons. The Korteweg-de Vries (KdV) equation, which describes the shallow water waves, is a basic weakly dispersive and weakly nonlinear model. Basing on the homogeneous balanced method, we achieve the general rational solution of a classical KdV equation. Numerical simulations of the solution allow us to explain rare and unexpected appearance of the rogue waves. We compare the rogue waves with the ones generated by the nonlinear Schrödinger (NLS) equation which can describe deep water wave trains. The numerical results illustrate that the amplitude of the KdV equation is higher than the one of the NLS equation, which may causes more serious damage of engineering structures in the ocean. This nonlinear mechanism will provide a theoretical guidance in the ocean and physics.

Exact Periodic Solitary Wave and Double Periodic Wave Solutions for the (2+1)-Dimensional Korteweg-de Vries Equation

2009 ◽  
Vol 64 (9-10) ◽  
pp. 609-614 ◽  
Author(s):  
Changfu Liu ◽  
Zhengde Dai
Keyword(s):  
Solitary Wave ◽  
Function Method ◽  
Kdv Equation ◽  
Periodic Wave ◽  
Jacobi Elliptic Function ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
De Vries ◽  
Trial Function Method ◽  
A New Technique

A new technique, the extended ansatz function method, is proposed to seek periodic solitary wave solutions of integrable systems. Exact periodic solitary wave solutions for the (2+1)-dimensional Korteweg-de Vries (KdV) equation are obtained by using this technique. By using the trial function method, Jacobi elliptic function double periodic solutions are also constructed for this equation. This result shows that there exist periodic solitary waves in the different directions for the (2+1)- dimensional KdV equation

scholarly journals Dispersive dynamics in the characteristic moving frame

2019 ◽  
Vol 475 (2223) ◽  
pp. 20180784 ◽  
Author(s):  
D. J. Ratliff
Keyword(s):  
Shallow Water ◽  
Lagrangian Density ◽  
Kdv Equation ◽  
Moving Frame ◽  
Modulation Equations ◽  
Water Flows ◽  
Shallow Water Flows ◽  
The Kdv Equation ◽  
De Vries ◽  
Klein Gordon

A mechanism for dispersion to automatically arise from the dispersionless Whitham Modulation equations (WMEs) is presented, relying on the use of a moving frame. The speed of this is chosen to be one of the characteristics which emerge from the linearization of the Whitham system, and assuming these are real (and thus the WMEs are hyperbolic) morphs the WMEs into the Korteweg-de Vries (KdV) equation in the boosted coordinate. Strikingly, the coefficients of the KdV equation are universal, in the sense that they are determined by abstract properties of the original Lagrangian density. Two illustrative examples of the theory are given to illustrate how the KdV may be constructed in practice. The first being a revisitation of the derivation of the KdV equation from shallow water flows, to highlight how the theory of this paper fits into the existing literature. The second is a complex Klein–Gordon system, providing a case where the KdV equation may only arise with the use of a moving frame.

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