Interaction of Fully-Nonlinear Internal Solitary Waves of Opposite Polarity

Mapping Intimacies ◽

10.5194/egusphere-egu21-3045 ◽

2021 ◽

Author(s):

Kevin Lamb

Keyword(s):

Wave Field ◽

Solitary Waves ◽

Kdv Equation ◽

Nonlinear Coefficient ◽

Opposite Polarity ◽

Internal Solitary Waves ◽

Linear Wave ◽

Gardner Equation ◽

Fully Nonlinear ◽

The Kdv Equation

Previous studies have suggested that fully nonlinear internal solitary waves (ISWs) are very soliton-like as the interaction of two ISWs results in only very small changes in amplitude of the interacting ISWs and in the production of a very small amplitude wave train. Previous studies have, however, considered ISWs with the polarity predicted by the sign of the quadratic nonlinear coefficient of the KdV equation. The Gardner equation, which is an extension of the KdV equation that includes a cubic nonlinear term, has ISWs of two polarities (i.e., waves of depression and elevation) when the cubic coefficient of the Gardner equation is positive. These waves are soliton solutions of the Gardner equations. &#160;In this talk I will discuss the interaction of ISWs of opposite polarity in continuous asymmetric three layer stratifications. Regions in parameter space where ISWs of opposite polarity exist will be discussed and I will demonstrate via fully nonlinear numerical simulations that the interaction of ISWs of opposite polarity waves are far from soliton-like: their interaction can result in very large changes in wave amplitude and may produce a very complicated wave field with multiple large ISWs, a large linear wave field and breather-like waves.&#160;

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Multiscaled Solitary Waves

10.5194/npg-2017-11 ◽

2017 ◽

Author(s):

Oleg G. Derzho

Keyword(s):

Solitary Waves ◽

Multiple Scales ◽

Kdv Equation ◽

Core Area ◽

Internal Solitary Waves ◽

Density Profiles ◽

The Core ◽

The Kdv Equation

Abstract. It is analytically shown how competing nonlinearities yield new multiscaled (multi humped) structures for internal solitary waves in shallow fluids. These solitary waves only exist for large amplitudes beyond the limit of applicability of the KdV equation or its usual extensions. Multiscaling phenomenon exists or do not exist for almost identical density profiles. Trapped core inside the wave prevents appearance of such multiple scales within the core area. It is anticipated that multiscaling phenomena exist for solitary waves in various physical origins.

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Interaction of linear modulated waves and unsteady dispersive hydrodynamic states with application to shallow water waves

Journal of Fluid Mechanics ◽

10.1017/jfm.2019.534 ◽

2019 ◽

Vol 875 ◽

pp. 1145-1174 ◽

Cited By ~ 6

Author(s):

T. Congy ◽

G. A. El ◽

M. A. Hoefer

Keyword(s):

Water Waves ◽

Large Scale ◽

Kdv Equation ◽

Linear Wave ◽

Mean Flow ◽

Undular Bore ◽

Expansion Waves ◽

Flow Interaction ◽

The Kdv Equation ◽

New Type

A new type of wave–mean flow interaction is identified and studied in which a small-amplitude, linear, dispersive modulated wave propagates through an evolving, nonlinear, large-scale fluid state such as an expansion (rarefaction) wave or a dispersive shock wave (undular bore). The Korteweg–de Vries (KdV) equation is considered as a prototypical example of dynamic wavepacket–mean flow interaction. Modulation equations are derived for the coupling between linear wave modulations and a nonlinear mean flow. These equations admit a particular class of solutions that describe the transmission or trapping of a linear wavepacket by an unsteady hydrodynamic state. Two adiabatic invariants of motion are identified that determine the transmission, trapping conditions and show that wavepackets incident upon smooth expansion waves or compressive, rapidly oscillating dispersive shock waves exhibit so-called hydrodynamic reciprocity recently described in Maiden et al. (Phys. Rev. Lett., vol. 120, 2018, 144101) in the context of hydrodynamic soliton tunnelling. The modulation theory results are in excellent agreement with direct numerical simulations of full KdV dynamics. The integrability of the KdV equation is not invoked so these results can be extended to other nonlinear dispersive fluid mechanic models.

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The transformation of an interfacial solitary wave of elevation at a bottom step

Nonlinear Processes in Geophysics ◽

10.5194/npg-16-33-2009 ◽

2009 ◽

Vol 16 (1) ◽

pp. 33-42 ◽

Cited By ~ 23

Author(s):

V. Maderich ◽

T. Talipova ◽

R. Grimshaw ◽

E. Pelinovsky ◽

B. H. Choi ◽

...

Keyword(s):

Solitary Wave ◽

Solitary Waves ◽

Numerical Study ◽

Nonlinear Effects ◽

Ocean Model ◽

Gardner Equation ◽

Fully Nonlinear ◽

Layer Flow ◽

The One ◽

Bottom Step

Abstract. In this paper we study the transformation of an internal solitary wave at a bottom step in the framework of two-layer flow, for the case when the interface lies close to the bottom, and so the solitary waves are elevation waves. The outcome is the formation of solitary waves and dispersive wave trains in both the reflected and transmitted fields. We use a two-pronged approach, based on numerical simulations of the fully nonlinear equations using a version of the Princeton Ocean Model on the one hand, and a theoretical and numerical study of the Gardner equation on the other hand. In the numerical experiments, the ratio of the initial wave amplitude to the layer thickness is varied up one-half, and nonlinear effects are then essential. In general, the characteristics of the generated solitary waves obtained in the fully nonlinear simulations are in reasonable agreement with the predictions of our theoretical model, which is based on matching linear shallow-water theory in the vicinity of a step with solutions of the Gardner equation for waves far from the step.

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Dust acoustic solitary waves in non-thermal plasmas consisting of negatively charged dust grains and isothermal electrons

Journal of Plasma Physics ◽

10.1017/s0022377809008083 ◽

2009 ◽

Vol 75 (4) ◽

pp. 455-474 ◽

Cited By ~ 1

Author(s):

ANIMESH DAS ◽

ANUP BANDYOPADHYAY

Keyword(s):

Solitary Wave ◽

Solitary Waves ◽

Acoustic Waves ◽

Kdv Equation ◽

Thermal Parameter ◽

Charged Dust ◽

Average Temperature ◽

The Kdv Equation ◽

Dust Acoustic Solitary Waves ◽

Dust Acoustic

AbstractA Korteweg–de Vries (KdV) equation is derived here, that describes the nonlinear behaviour of long-wavelength weakly nonlinear dust acoustic waves propagating in an arbitrary direction in a plasma consisting of static negatively charged dust grains, non-thermal ions and isothermal electrons. It is found that the rarefactive or compressive nature of the dust acoustic solitary wave solution of the KdV equation does not depend on the dust temperature if σdc < 0 or σdc > σd*, where σdc is a function of β1, α and μ only, and σd*(<1) is the upper limit (upper bound) of σd. This β1 is the non-thermal parameter associated with the non-thermal velocity distribution of ions, α is the ratio of the average temperature of the non-thermal ions to that of the isothermal electrons, μ is the ratio of the unperturbed number density of isothermal electrons to that of the non-thermal ions, Zdσd is the ratio of the average temperature of the dust particles to that of the ions and Zd is the number of electrons residing on the dust grain surface. The KdV equation describes the rarefactive or the compressive dust acoustic solitary waves according to whether σdc < 0 or σdc > σd*. When 0 ≤ σdc ≤ σd*, the KdV equation describes the rarefactive or the compressive dust acoustic solitary waves according to whether σd > σdc or σd < σdc. If σd takes the value σdc with 0 ≤ σdc ≤ σd*, the coefficient of the nonlinear term of the KdV equation vanishes and, for this case, the nonlinear evolution equation of the dust acoustic waves is derived, which is a modified KdV (MKdV) equation. A theoretical investigation of the nature (rarefactive or compressive) of the dust acoustic solitary wave solutions of the evolution equations (KdV and MKdV) is presented with respect to the non-thermal parameter β1. For any given values of α and μ, it is found that the value of σdc completely defines the nature of the dust acoustic solitary waves except for a small portion of the entire range of the non-thermal parameter β1.

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Three-Soliton Interaction and Soliton Turbulence in Superthermal Dusty Plasmas

Zeitschrift für Naturforschung A ◽

10.1515/zna-2018-0452 ◽

2019 ◽

Vol 74 (9) ◽

pp. 757-766 ◽

Cited By ~ 3

Author(s):

Rustam Ali ◽

Prasanta Chatterjee

Keyword(s):

Wave Field ◽

Kdv Equation ◽

Perturbation Technique ◽

Great Influence ◽

Dusty Plasmas ◽

Second Order ◽

Soliton Interaction ◽

Bilinear Method ◽

The Kdv Equation ◽

Skewness And Kurtosis

AbstractPropagation and interaction of three solitons are studied within the framework of the Korteweg-de Vries (KdV) equation. The KdV equation is derived from an unmagnetised, collision-less dusty plasma containing cold inertial ions, stationary dusts with negative charge, and non-inertial kappa-distributed electrons, using the reductive perturbation technique (RPT). Adopting Hirota’s bilinear method, the three-soliton solution of the KdV equation is obtained and, as an elementary act of soliton turbulence, a study on the soliton interaction is presented. The concavity of the resulting pulse is studied at the strongest interaction point of three solitons. At the time of soliton interaction, the first- and second-order moments as well as the skewness and kurtosis of the wave field are calculated. The skewness and kurtosis decrease as a result of soliton interaction, whereas the first- and second-order moments remain invariant. Also, it is observed that the spectral index κ and the unperturbed dust-to-ion ratio μ have great influence on the skewness and kurtosis of the wave field.

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ON THE SHOALING OF SOLITARY WAVES IN THE KDV EQUATION

Coastal Engineering Proceedings ◽

10.9753/icce.v34.waves.44 ◽

2014 ◽

Vol 1 (34) ◽

pp. 44 ◽

Cited By ~ 1

Author(s):

Zahra Khorsand ◽

Henrik Kalisch

Keyword(s):

Solitary Waves ◽

Kdv Equation ◽

The Kdv Equation

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Mode-2 internal solitary waves offshore Central America discovered by seismic oceanography method

10.5194/egusphere-egu2020-9318 ◽

2020 ◽

Author(s):

Wenhao Fan ◽

Haibin Song ◽

Yi Gong ◽

Shaoqing Sun ◽

Kun Zhang

Keyword(s):

Phase Velocity ◽

Central America ◽

Solitary Waves ◽

Kdv Equation ◽

Seismic Survey ◽

Internal Solitary Waves ◽

Acquisition Time ◽

Apparent Velocity ◽

Seismic Oceanography ◽

Mode 2

In the past, most of the internal solitary waves (ISWs) found by seismic oceanography (SO) method were mode-1 ISWs. We discover many mode-2 ISWs in the Pacific coast of Central America by using SO method for the first time. These mode-2 ISWs are convex mode-2 ISWs with the maximum amplitudes of about 10 m, and most of them are ISWs with smaller amplitudes. The pycnocline for the mode-2 ISWs on the shelf (ISW3) is displaced 6.4% of the total seawater depth from the mid-depth of the total seawater. The deviation is large, and it shows a strong asymmetry feature of the peaks and troughs on the seismic profile. This is consistent with the results of previous numerical simulation. Observing the changes in the fine structure of mode-2 ISWs packet through pre-stack migration, it was found that the overall waveform of the three mode-2 ISWs (ISW1, ISW2, and ISW3) on the shelf during the acquisition time period of about 40 seconds is stable. The apparent phase velocity of these mode-2 ISWs calculated by the pre-stack migration section using the Common Offset Gathers is about 0.5 m/s, and their apparent propagation directions are from SW to NE along the seismic line (44 &#176; N, 0&#176; pointing north). The vertical amplitude distribution and estimated apparent velocities of these mode-2 ISWs are basically consistent with the theoretical values &#8203;&#8203;calculated from the KdV equation. By analyzing the apparent velocities of the three mode-2 ISWs (ISW1, ISW3, and ISW5) with relatively small apparent velocity errors, it is found that the apparent velocity of mode-2 ISWs generally increases with the increasing depth of seawater. In addition, the apparent phase velocity of the mode-2 ISWs with a larger maximum amplitude is generally larger. Based on the analysis of hydrological data in the study area, it was found that a strong anticyclone developed on the northwest side of the seismic survey line and a weaker anticyclone developed on the southeast side. These anticyclones will increase the depth of the thermocline in the surrounding seawater. According to previous studies, the deepening of the thermocline (pycnocline) maybe conducive to the generation of mode-2 ISWs.

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