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

2009 ◽  
Vol 16 (1) ◽  
pp. 33-42 ◽  
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.

Transformation of the first mode internal solitary wave over topography in three-layer flow

2020 ◽  
Author(s):  
Kateryna Terletska ◽  
Tatiana Talipova ◽  
Roger Grimshaw ◽  
Zihua Liu ◽  
Vladimir Maderіch
Keyword(s):  
Solitary Wave ◽  
Wave Field ◽  
Solitary Waves ◽  
Internal Solitary Wave ◽  
Wave Transformation ◽  
Internal Solitary Waves ◽  
Wave Characteristics ◽  
Second Mode ◽  
Layer Flow ◽  
Bottom Step

<p>Transformation of the first mode internal solitary wave over the underwater bottom step in three-layer fluid is studied numerically. In the three layer flow two modes (the first and the second) of the internal waves are existed. It is known that interaction of the first mode internal solitary wave with an underwater obstacle is the mechanisms of second-mode internal solitary waves generation. Different scenarios of transformation are realized under different wave characteristics: wave amplitude, position of the step and thickness of the layers as is the two layer case [1]. Formation of the second mode internal solitary waves during interaction of the first mode internal solitary waves occurs only for special range of wave characteristics and thickness of the layers that was defined in this investigation. The second mode internal solitary waves appear as in the reflected wave field as well as in the transmitted wave field. Transfer of energy from incident mode one wave into reflected and transmitted waves (the first and the second modes) during transformation is also studied. Dependence of the amplitudes of generated solitary waves (transmitted and reflected) from amplitude of the incident wave is obtained.  Comparison of numerical results (reflected and transmitted coefficients) with the theoretical calculations [2] shows good agreement in the range of wave characteristics that corresponds to the weak interaction.  </p><p> </p><p>1. Talipova T., Terletska K., Maderich V., Brovchenko I., Pelinovsky E., Jung K.T., Grimshaw R. Internal solitary wave transformation over a bottom step: loss of energy. Phys. Fluids. 2013. № 25. 032110; doi:10.1063/1.4797455</p><p>2.    Liu Z., Grimshaw R. and Johnson E.  The interaction of a mode-1 internal solitary wave with a step and the generation of mode-2 waves Geophysical & Astrophysical Fluid Dynamics 2019, N 4, V 113, https://doi.org/10.1080/03091929.2019.1636046</p><p> </p>

Statistical properties of the internal solitary wave ensemble

2021 ◽  
Author(s):  
Tatiana Talipova ◽  
Ekaterina Didenkulova ◽  
Anna Kokorina ◽  
Efim Pelinovsky
Keyword(s):  
Wave Propagation ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Internal Solitary Wave ◽  
Statistical Moments ◽  
Nonlinear Propagation ◽  
Positive Sign ◽  
Cubic Nonlinearity ◽  
Gardner Equation ◽  
Water Stratification

<p>Internal solitary wave ensembles are often observed on the ocean shelves. The long internal baroclinic tide is generated by a barotropic tide on the shelf edges, and then transforms into the soliton-like wave packets during the nonlinear propagation to the beach. The tide is a periodic process and the solitary wave ensemble appears on the shelf usually each semi-diurnal period of 12.4 hours. This process is very sensitive to the variation of the tide characteristics and the hydrology.</p><p>We study the propagation of the soliton ensembles numerically in the framework of the spatial form of the Gardner equation (i.e., the Korteweg-de Vries equation with both, quadratic and cubic nonlinearities) assuming horizontally uniform background and applying periodic conditions in time. The water stratification and the local depth are taken similar to the conditions of the north-western Australian shelf, where the stratification admits the existence of solitons but not breathers. The numerical simulation is performed using the Gardner equation with the negative sign of the cubic nonlinearity. For the study of the statistic properties of the solitary waves we use the ensemble of 50 realizations with the same set of 13 solitary waves which are located randomly. The histograms of the wave amplitudes change as the waves travel. The histogram variations become significant after 50 km of the wave propagation. The third (skewness) and the fourth (kurtosis) statistical moments are computed versus the travel distance. It is shown that the both moments decrease by 20% when the solitary wave groups travel for about 150 km.</p><p>A similar simulation is conducted for a variable background within the framework of the variable-coefficient Gardner equation. At some location the water stratification corresponds to the positive sign of the local coefficient of the cubic nonlinearity, and then internal breathers may exist. The wave propagation in horizontally inhomogeneous hydrology leads to the occurrence of complicated patterns of solitons and breathers; in the course of the transformation they can disintegrate or form internal rogue waves. Under these conditions the statistical moments of the wave field are essentially different from case when the breather-like waves cannot occur.</p><p>The research was supported by the RFBR grants No 19-05-00161 (TT and EP) and 19-35-60022 (ED). The Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS” (№ 20-1-3-3-1) is also acknowledged by ED</p>

Cure Monitoring of Adhesive for Composite/Metal Bonded Structure Based on Highly Nonlinear Solitary Waves

2019 ◽  
Author(s):  
Wu Bin ◽  
Li Mingzhi ◽  
Liu Xiucheng ◽  
Wang Heying ◽  
He Cunfu ◽  
...  
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Curing Process ◽  
One Dimensional ◽  
Granular Chain ◽  
Highly Nonlinear ◽  
Highly Nonlinear Solitary Waves ◽  
Simulation Results ◽  
The One ◽  
Nonlinear Solitary Waves

Abstract In this paper, a nondestructive evaluation technique based on highly nonlinear solitary waves (HNSWs) is proposed to monitor the curing process of adhesive for composite/metal bonded structure. HNSWs are mechanical waves with high energy intensity and non-distortive nature which can form and propagate in a nonlinear system, such as a one-dimensional granular chain. In the present study, a finite element model of the one-dimensional granular chain is established with the commercial software Abaqus, to study the reflection behavior of HNSWs at the interface between the particle at the end of chain and the sample. The simulation results show that the time of flight (TOF) of the primary reflected solitary wave decreases with the stiffness of the sample increases, and the amplitude ratio (AR) between the primary reflected solitary wave and the incident solitary wave increases. An HNSWs transducer based on the one-dimensional granular chain is designed and fabricated. The relationship between the characteristic parameters of the primary reflected solitary wave (TOF and AR) and the curing time of adhesive for a composite/metal bonded structure is experimentally investigated. The experiment results suggest that the TOF decreases and the AR increases as the epoxy cures. The experimental results are in good agreement with the simulation results. This study provides a new characterization method for monitoring the curing process of adhesive for composite/metal bonded structure.

scholarly journals Transformation of internal solitary waves at the "deep" and "shallow" shelf: satellite observations and laboratory experiment

2013 ◽  
Vol 20 (5) ◽  
pp. 743-757 ◽  
Author(s):  
O. D. Shishkina ◽  
J. K. Sveen ◽  
J. Grue
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Laboratory Data ◽  
Water Layer ◽  
Internal Solitary Waves ◽  
Field Observations ◽  
Edge Waves ◽  
Sar Images ◽  
Time Periods ◽  
Bottom Step

Abstract. An interaction of internal solitary waves with the shelf edge in the time periods related to the presence of a pronounced seasonal pycnocline in the Red Sea and in the Alboran Sea is analysed via satellite photos and SAR images. Laboratory data on transformation of a solitary wave of depression while passing along the transverse bottom step were obtained in a tank with a two-layer stratified fluid. The certain difference between two characteristic types of hydrophysical phenomena was revealed both in the field observations and in experiments. The hydrological conditions for these two processes were named the "deep" and the "shallow" shelf respectively. The first one provides the generation of the secondary periodic short internal waves – "runaway" edge waves – due to change in the polarity of a part of a soliton approaching the shelf normally. Another one causes a periodic shear flow in the upper quasi-homogeneous water layer with the period of incident solitary wave. The strength of the revealed mechanisms depends on the thickness of the water layer between the pycnocline and the shelf bottom as well as on the amplitude of the incident solitary wave.

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>

The stability of internal solitary waves

1983 ◽  
Vol 94 (2) ◽  
pp. 351-379 ◽  
Author(s):  
D. P. Bennett ◽  
R. W. Brown ◽  
S. E. Stansfield ◽  
J. D. Stroughair ◽  
J. L. Bona
Keyword(s):  
Solitary Wave ◽  
Internal Waves ◽  
Solitary Waves ◽  
Model Equation ◽  
Focus Of Attention ◽  
Internal Solitary Waves ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
The Stability ◽  
The One

A theory is developed relating to the stability of solitary-wave solutions of the so-called Benjamin-Ono equation. This equation was derived by Benjamin (5) as a model for the propagation of internal waves in an incompressible non-diffusive heterogeneous fluid for which the density is non-constant only within a layer whose thickness is much smaller than the total depth. In his article, Benjamin wrote in closed form the one-parameter family of solitary-wave solutions of his model equation whose stability will be the focus of attention presently.

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.

Some comparisons between heterogeneous and homogeneous layers for nonlinear SH waves in terms of heterogeneous and nonlinear effects

2020 ◽  
pp. 108128652094635 ◽  
Author(s):  
Dilek Demirkuş
Keyword(s):  
Solitary Wave ◽  
Large Scale ◽  
Multiple Scales ◽  
Nonlinear Effects ◽  
Small Scale ◽  
Solitary Wave Solutions ◽  
Nls Equation ◽  
Sh Waves ◽  
Wave Solutions ◽  
The One

This paper aims to make some comparative studies between heterogeneous and homogeneous layers for nonlinear shear horizontal (SH) waves in terms of the heterogeneous and nonlinear effects. Therefore, with this aim, two layers are defined as follows: on the one hand, one layer consists of hyperelastic, isotropic, heterogeneous, and generalized neo-Hookean materials; on the other hand, another layer is made up of hyperelastic, isotropic, homogeneous, and generalized neo-Hookean materials. Moreover, it is assumed that upper boundaries are stress-free and lower boundaries are rigidly fixed. The method of multiple scales is used in both analyses, in addition to using the known solutions of the nonlinear Schrödinger (NLS) equation, called bright and dark solitary wave solutions; these comparisons are made, numerically, and then all results are given for the lowest branch of both dispersion relations, graphically. Moreover, these comparisons are observed both on a large scale and on a small scale, not only in terms of the bright and dark solitary wave solutions but also in terms of the heterogeneous and nonlinear effects.

scholarly journals The modified Korteweg - de Vries equation in the theory of large - amplitude internal waves

1997 ◽  
Vol 4 (4) ◽  
pp. 237-250 ◽  
Author(s):  
R. Grimshaw ◽  
E. Pelinovsky ◽  
T. Talipova
Keyword(s):  
Solitary Wave ◽  
Internal Waves ◽  
Solitary Waves ◽  
Large Amplitude ◽  
Nonlinear Term ◽  
Vries Equation ◽  
Nonlinear Effects ◽  
Wave Generation ◽  
Density Stratification ◽  
De Vries

Abstract. The propagation of large- amplitude internal waves in the ocean is studied here for the case when the nonlinear effects are of cubic order, leading to the modified Korteweg - de Vries equation. The coefficients of this equation are calculated analytically for several models of the density stratification. It is shown that the coefficient of the cubic nonlinear term may have either sign (previously only cases of a negative cubic nonlinearity were known). Cubic nonlinear effects are more important for the high modes of the internal waves. The nonlinear evolution of long periodic (sine) waves is simulated for a three-layer model of the density stratification. The sign of the cubic nonlinear term influences the character of the solitary wave generation. It is shown that the solitary waves of both polarities can appear for either sign of the cubic nonlinear term; if it is positive the solitary waves have a zero pedestal, and if it is negative the solitary waves are generated on the crest and the trough of the long wave. The case of a localised impulsive initial disturbance is also simulated. Here, if the cubic nonlinear term is negative, there is no solitary wave generation at large times, but if it is positive solitary waves appear as the asymptotic solution of the nonlinear wave evolution.

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