Internal solitary waves shoaling onto a shelf: Comparisons of weakly-nonlinear and fully nonlinear models for hyperbolic-tangent stratifications

2014 ◽  
Vol 78 ◽  
pp. 17-34 ◽  
Author(s):  
Kevin G. Lamb ◽  
Wenting Xiao
Keyword(s):  
Solitary Waves ◽  
Nonlinear Models ◽  
Internal Solitary Waves ◽  
Hyperbolic Tangent ◽  
Fully Nonlinear ◽  
Weakly Nonlinear

scholarly journals Anomalous width variation of rarefactive ion acoustic solitary waves in the context of auroral plasmas

2004 ◽  
Vol 11 (2) ◽  
pp. 219-228 ◽  
Author(s):  
S. S. Ghosh ◽  
G. S. Lakhina
Keyword(s):  
Solitary Waves ◽  
Nonlinear Theory ◽  
Acoustic Waves ◽  
Large Amplitude ◽  
Magnetized Plasma ◽  
Amplitude Variation ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
Ion Acoustic Solitary Waves ◽  
Ion Acoustic

Abstract. The presence of dynamic, large amplitude solitary waves in the auroral regions of space is well known. Since their velocities are of the order of the ion acoustic speed, they may well be considered as being generated from the nonlinear evolution of ion acoustic waves. However, they do not show the expected width-amplitude correlation for K-dV solitons. Recent POLAR observations have actually revealed that the low altitude rarefactive ion acoustic solitary waves are associated with an increase in the width with increasing amplitude. This indicates that a weakly nonlinear theory is not appropriate to describe the solitary structures in the auroral regions. In the present work, a fully nonlinear analysis based on Sagdeev pseudopotential technique has been adopted for both parallel and oblique propagation of rarefactive solitary waves in a two electron temperature multi-ion plasma. The large amplitude solutions have consistently shown an increase in the width with increasing amplitude. The width-amplitude variation profile of obliquely propagating rarefactive solitary waves in a magnetized plasma have been compared with the recent POLAR observations. The width-amplitude variation pattern is found to fit well with the analytical results. It indicates that a fully nonlinear theory of ion acoustic solitary waves may well explain the observed anomalous width variations of large amplitude structures in the auroral region.

Fully nonlinear internal waves in a two-fluid system

1999 ◽  
Vol 396 ◽  
pp. 1-36 ◽  
Author(s):  
WOOYOUNG CHOI ◽  
ROBERTO CAMASSA
Keyword(s):  
Solitary Wave ◽  
Euler Equations ◽  
Deep Water ◽  
Large Amplitude ◽  
Nonlinear Models ◽  
Weak Nonlinearity ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
The Waves ◽  
Two Fluid

Model equations that govern the evolution of internal gravity waves at the interface of two immiscible inviscid fluids are derived. These models follow from the original Euler equations under the sole assumption that the waves are long compared to the undisturbed thickness of one of the fluid layers. No smallness assumption on the wave amplitude is made. Both shallow and deep water configurations are considered, depending on whether the waves are assumed to be long with respect to the total undisturbed thickness of the fluids or long with respect to just one of the two layers, respectively. The removal of the traditional weak nonlinearity assumption is aimed at improving the agreement with the dynamics of Euler equations for large-amplitude waves. This is obtained without compromising much of the simplicity of the previously known weakly nonlinear models. Compared to these, the fully nonlinear models' most prominent feature is the presence of additional nonlinear dispersive terms, which coexist with the typical linear dispersive terms of the weakly nonlinear models. The fully nonlinear models contain the Korteweg–de Vries (KdV) equation and the Intermediate Long Wave (ILW) equation, for shallow and deep water configurations respectively, as special cases in the limit of weak nonlinearity and unidirectional wave propagation. In particular, for a solitary wave of given amplitude, the new models show that the characteristic wavelength is larger and the wave speed is smaller than their counterparts for solitary wave solutions of the weakly nonlinear equations. These features are compared and found in overall good agreement with available experimental data for solitary waves of large amplitude in two-fluid systems.

Interaction of Fully-Nonlinear Internal Solitary Waves of Opposite Polarity

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

<p>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.  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.<span> </span></p>

scholarly journals A method for computing unsteady fully nonlinear interfacial waves

1997 ◽  
Vol 351 ◽  
pp. 223-252 ◽  
Author(s):  
JOHN GRUE ◽  
HELMER ANDRÉ FRIIS ◽  
ENOK PALM ◽  
PER OLAV RUSÅS
Keyword(s):  
Solitary Waves ◽  
Bottom Topography ◽  
The Body ◽  
Interfacial Waves ◽  
Nonlinear Method ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
Taylor Series Expansions ◽  
Layer Flow ◽  
Fully Nonlinear Method

We derive a time-stepping method for unsteady fully nonlinear two-dimensional motion of a two-layer fluid. Essential parts of the method are: use of Taylor series expansions of the prognostic equations, application of spatial finite difference formulae of high order, and application of Cauchy's theorem to solve the Laplace equation, where the latter is found to be advantageous in avoiding instability. The method is computationally very efficient. The model is applied to investigate unsteady trans-critical two-layer flow over a bottom topography. We are able to simulate a set of laboratory experiments on this problem described by Melville & Helfrich (1987), finding a very good agreement between the fully nonlinear model and the experiments, where they reported bad agreement with weakly nonlinear Korteweg–de Vries theories for interfacial waves. The unsteady transcritical regime is identified. In this regime, we find that an upstream undular bore is generated when the speed of the body is less than a certain value, which somewhat exceeds the critical speed. In the remaining regime, a train of solitary waves is generated upstream. In both cases a corresponding constant level of the interface behind the body is developed. We also perform a detailed investigation of upstream generation of solitary waves by a moving body, finding that wave trains with amplitude comparable to the thickness of the thinner layer are generated. The results indicate that weakly nonlinear theories in many cases have quite limited applications in modelling unsteady transcritical two-layer flows, and that a fully nonlinear method in general is required.

scholarly journals Spectral methods for internal waves: indistinguishable density profiles and double-humped solitary waves

2011 ◽  
Vol 18 (3) ◽  
pp. 351-358 ◽  
Author(s):  
M. Dunphy ◽  
C. Subich ◽  
M. Stastna
Keyword(s):  
Euler Equations ◽  
Solitary Waves ◽  
Dynamic Simulations ◽  
Large Lakes ◽  
Full Spectrum ◽  
Density Profiles ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
First Order ◽  
Spectral And Pseudospectral Methods

Abstract. Internal solitary waves are widely observed in both the oceans and large lakes. They can be described by a variety of mathematical theories, covering the full spectrum from first order asymptotic theory (i.e. Korteweg-de Vries, or KdV, theory), through higher order extensions of weakly nonlinear-weakly nonhydrostatic theory, to fully nonlinear-weakly nonhydrostatic theories and finally exact theory based on the Dubreil-Jacotin-Long (DJL) equation that is formally equivalent to the full set of Euler equations. We discuss how spectral and pseudospectral methods allow for the computation of novel phenomena in both approximate and exact theories. In particular we construct markedly different density profiles for which the coefficients in the KdV theory are very nearly identical. These two density profiles yield qualitatively different behaviour for both exact, or fully nonlinear, waves computed using the DJL equation and in dynamic simulations of the time dependent Euler equations. For exact, DJL, theory we compute exact solitary waves with two-scales, or so-called double-humped waves.

scholarly journals Nonlinear hydroelastic waves on a linear shear current at finite depth

2019 ◽  
Vol 876 ◽  
pp. 55-86 ◽  
Author(s):  
T. Gao ◽  
Z. Wang ◽  
P. A. Milewski
Keyword(s):  
Solitary Waves ◽  
Small Amplitude ◽  
Modulational Instability ◽  
Finite Depth ◽  
Time Dependent ◽  
Mapping Technique ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Constant Vorticity

This work is concerned with waves propagating on water of finite depth with a constant-vorticity current under a deformable flexible sheet. The pressure exerted by the sheet is modelled by using the Cosserat thin shell theory. By means of multi-scale analysis, small amplitude nonlinear modulation equations in several regimes are considered, including the nonlinear Schrödinger equation (NLS) which is used to predict the existence of small-amplitude wavepacket solitary waves in the full Euler equations and to study the modulational instability of quasi-monochromatic wavetrains. Guided by these weakly nonlinear results, fully nonlinear steady and time-dependent computations are performed by employing a conformal mapping technique. Bifurcation mechanisms and typical profiles of solitary waves for different underlying shear currents are presented in detail. It is shown that even when small-amplitude solitary waves are not predicted by the weakly nonlinear theory, we can numerically find large-amplitude solitary waves in the fully nonlinear equations. Time-dependent simulations are carried out to confirm the modulational stability results and illustrate possible outcomes of the nonlinear evolution in unstable cases.

scholarly journals Long-term evolution of strongly nonlinear internal solitary waves in a rotating channel

2009 ◽  
Vol 16 (5) ◽  
pp. 587-598 ◽  
Author(s):  
J. C. Sánchez-Garrido ◽  
V. Vlasenko
Keyword(s):  
Solitary Waves ◽  
Nonlinear Effects ◽  
Kelvin Waves ◽  
Internal Solitary Waves ◽  
Kelvin Wave ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
Petviashvili Equation ◽  
Rotating Channel

Abstract. The evolution of internal solitary waves (ISWs) propagating in a rotating channel is studied numerically in the framework of a fully-nonlinear, nonhydrostatic numerical model. The aim of modelling efforts was the investigation of strongly-nonlinear effects, which are beyond the applicability of weakly nonlinear theories. Results reveal that small-amplitude waves and sufficiently strong ISWs evolve differently under the action of rotation. At the first stage of evolution an initially two-dimensional ISW transforms according to the scenario described by the rotation modified Kadomtsev-Petviashvili equation, namely, it starts to evolve into a Kelvin wave (with exponential decay of the wave amplitude across the channel) with front curved backwards. This transition is accompanied by a permanent radiation of secondary Poincaré waves attached to the leading wave. However, in a strongly-nonlinear limit not all the energy is transmitted to secondary radiated waves. Part of it returns to the leading wave as a result of nonlinear interactions with secondary Kelvin waves generated in the course of time. This leads to the formation of a slowly attenuating quasi-stationary system of leading Kelvin waves, capable of propagating for several hundreds hours as a localized wave packet.

Internal solitary waves in a two-fluid system with a free surface

2016 ◽  
Vol 804 ◽  
pp. 201-223 ◽  
Author(s):  
Tsubasa Kodaira ◽  
Takuji Waseda ◽  
Motoyasu Miyata ◽  
Wooyoung Choi
Keyword(s):  
Free Surface ◽  
Solitary Wave ◽  
Surface Waves ◽  
Solitary Waves ◽  
Silicone Oil ◽  
Internal Solitary Wave ◽  
Internal Solitary Waves ◽  
Weakly Nonlinear ◽  
Two Fluids ◽  
Strongly Nonlinear

Internal solitary waves in a system of two fluids, silicone oil and water, bounded above by a free surface are studied both experimentally and theoretically. By adjusting an extra volume of silicone oil released from a reservoir, a wide range of amplitude waves are generated in a wave tank. Wave profiles as well as wave speeds are measured using multiple wave probes and are then compared with both the weakly nonlinear Korteweg–de Vries (KdV) models and the strongly nonlinear Miyata–Choi–Camassa (MCC) models. As the density difference between the two fluids in the experiment is relatively small (approximately 14 %), but non-negligible, special attention is paid to the effect of the boundary condition at the top surface. The nonlinear models valid for rigid-lid (RL) and free-surface (FS) boundary conditions are considered separately. It is found that the solitary wave of the FS model for a given amplitude is consistently narrower than that of the RL model and it propagates at a slightly lower speed. Due to strong nonlinearity in the internal-wave motion, the weakly nonlinear KdV models fail to describe the measured internal solitary wave profiles of intermediate and large wave amplitudes. The strongly nonlinear MCC-FS model agrees better with the measurements than the MCC-RL model, which indicates that the free-surface boundary condition at the top surface is crucial in describing the internal solitary waves in the experiment correctly. Leaving the top surface free in the experiment allows us to observe small and relatively short wave packets on the top surface, particularly when the amplitude of the internal solitary wave is large. Once excited, the wave packet is located above the front half of the internal solitary wave and propagates with a speed close to that of the internal solitary wave underneath. A simple resonance mechanism between short surface waves and long internal waves without and with nonlinear effects is examined to estimate the characteristic wavelength of modulated short surface waves, which is found to be in good agreement with the observed wavelength when nonlinearity is taken into account. Using ray theory, the evolution of short surface waves in the presence of a background current induced by an internal solitary wave is also investigated to examine the location of the modulated surface wave packet.

scholarly journals Estimate of energy loss from internal solitary waves breaking on slopes

2021 ◽  
Author(s):  
Kateryna Terletska ◽  
Vladimir Maderich ◽  
Tatiana Talipova
Keyword(s):  
Energy Loss ◽  
Solitary Waves ◽  
Wave Breaking ◽  
Numerical Experiments ◽  
Wave Amplitude ◽  
Three Dimensional ◽  
Bottom Layer ◽  
Internal Solitary Waves ◽  
Weakly Nonlinear ◽  
Shelf Slope

Abstract. Internal solitary waves (ISW) emerge in the ocean and seas in different forms and break on the shelf zones in a variety of ways. Their breaking on slopes can produce intensive mixing that produces such process as biological productivity and sediment transport. Mechanisms of ISW of depression interaction with the slopes related to breaking and changing polarity as they shoal. We assume that parameters that described the process of interaction of ISW in a two-layer fluid with the idealised shelf-slope are: the non-dimensional wave amplitude α (wave amplitude normalized on the upper layer thickness), the ratio of the height of the bottom layer on the shelf to the incident wave amplitude β and angle γ. Based on three-dimensional αβγ classification diagram with four types of interaction with the slopes it was discussed: (1) ISW propagates over slope without changing polarity and wave breaking; (2) ISW changes polarity over slope without breaking; (3) ISW breaks over slope without changing polarity; (4) ISW both breaks and change polarity over the slope. Relations between the parameters α,β,γ for each regime were obtained using the empirical condition for wave breaking and weakly nonlinear theory for the criterion of changing the polarity of the wave. In the present paper the α,β,γ diagram was validated for idealised real scale topography configurations. Results of the numerical experiments that were carried out in the present paper and results of field and laboratory experiments from other papers are in good agreement with proposed classification and estimations. Based on 85 numerical experiments ISWs energy loss during interaction with slope topography with 0.5° < γ < 90° was estimated. Hot spots zones of high levels of energy loss were shown for idealized configuration that mimics continental shelf at Lufeng Region SCS.

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