scholarly journals The Propagation of Internal Solitary Waves over Variable Topography in a Horizontally Two-Dimensional Framework

2018 ◽  
Vol 48 (2) ◽  
pp. 283-300 ◽  
Author(s):  
Chunxin Yuan ◽  
Roger Grimshaw ◽  
Edward Johnson ◽  
Xueen Chen
Keyword(s):  
Solitary Wave ◽  
Wave Field ◽  
Solitary Waves ◽  
Variable Coefficient ◽  
Transverse Direction ◽  
Density Stratification ◽  
Internal Solitary Waves ◽  
Two Dimensional ◽  
Petviashvili Equation ◽  
Background Density

AbstractThis paper presents a horizontally two-dimensional theory based on a variable-coefficient Kadomtsev–Petviashvili equation, which is developed to investigate oceanic internal solitary waves propagating over variable bathymetry, for general background density stratification and current shear. To illustrate the theory, a typical monthly averaged density stratification is used for the propagation of an internal solitary wave over either a submarine canyon or a submarine plateau. The evolution is essentially determined by two components, nonlinear effects in the main propagation direction and the diffraction modulation effects in the transverse direction. When the initial solitary wave is located in a narrow area, the consequent spreading effects are dominant, resulting in a wave field largely manifested by a significant diminution of the leading waves, together with some trailing shelves of the opposite polarity. On the other hand, if the initial solitary wave is uniform in the transverse direction, then the evolution is more complicated, though it can be explained by an asymptotic theory for a slowly varying solitary wave combined with the generation of trailing shelves needed to satisfy conservation of mass. This theory is used to demonstrate that it is the transverse dependence of the nonlinear coefficient in the Kadomtsev–Petviashvili equation rather than the coefficient of the linear transverse diffraction term that determines how the wave field evolves. The Massachusetts Institute of Technology (MIT) general circulation model is used to provide a comparison with the variable-coefficient Kadomtsev–Petviashvili model, and good qualitative and quantitative agreements are found.

Asymptotic expansions for solitary gravity-capillary waves in two and three dimensions

2009 ◽  
Vol 465 (2109) ◽  
pp. 2725-2749 ◽  
Author(s):  
M. J. Ablowitz ◽  
T. S. Haut
Keyword(s):  
Surface Tension ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Asymptotic Series ◽  
High Order ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Small Surface ◽  
Petviashvili Equation ◽  
Dimensional Series

High-order asymptotic series are obtained for two- and three-dimensional gravity-capillary solitary waves. In two dimensions, the first term in the asymptotic series is the well-known sech 2 solution of the Korteweg–de Vries equation; in three dimensions, the first term is the rational lump solution of the Kadomtsev–Petviashvili equation I. The two-dimensional series is used (with nine terms included) to investigate how small surface tension affects the height and energy of large-amplitude waves and waves close to the solitary version of Stokes’ extreme wave. In particular, for small surface tension, the solitary wave with the maximum energy is obtained. For large surface tension, the two-dimensional series is also used to study the energy of depression solitary waves. Energy considerations suggest that, for large enough surface tension, there are solitary waves that can get close to the fluid bottom. In three dimensions, analytic solutions for the high-order perturbation terms are computed numerically, and the resulting asymptotic series (to three terms) is used to obtain the speed versus maximum amplitude curve for solitary waves subject to sufficiently large surface tension. Finally, the above asymptotic method is applied to the Benney–Luke (BL) equation, and the resulting asymptotic series (to three terms) is verified to agree with the solitary-wave solution of the BL equation.

Internal solitary waves with subsurface cores

2019 ◽  
Vol 873 ◽  
pp. 1-17 ◽  
Author(s):  
Yangxin He ◽  
Kevin G. Lamb ◽  
Ren-Chieh Lien
Keyword(s):  
Numerical Simulation ◽  
South China Sea ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Internal Solitary Waves ◽  
Two Dimensional ◽  
Background Current ◽  
Near Surface ◽  
Nonlinear Solutions ◽  
Surface Background

Large internal solitary waves with subsurface cores have recently been observed in the South China Sea. Here fully nonlinear solutions of the Dubreil–Jacotin–Long equation are used to study the conditions under which such cores exist. We find that the location of the cores, either at the surface or below the surface, is largely determined by the sign of the vorticity of the near-surface background current. The results of a numerical simulation of a two-dimensional shoaling internal solitary wave are presented which illustrate the formation of a subsurface core.

scholarly journals Two-dimensional numerical simulations of shoaling internal solitary waves at the ASIAEX site in the South China Sea

2015 ◽  
Vol 22 (3) ◽  
pp. 289-312 ◽  
Author(s):  
K. G. Lamb ◽  
A. Warn-Varnas
Keyword(s):  
South China Sea ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Gravity Waves ◽  
South China ◽  
Internal Solitary Waves ◽  
Two Dimensional ◽  
Shoaling Waves ◽  
The Waves ◽  
Inertia Gravity Waves

Abstract. The interaction of barotropic tides with Luzon Strait topography generates some of the world's largest internal solitary waves which eventually shoal and dissipate on the western side of the northern South China Sea. Two-dimensional numerical simulations of the shoaling of a single internal solitary wave at the site of the Asian Seas International Acoustic Experiment (ASIAEX) have been undertaken in order to investigate the sensitivity of the shoaling process to the stratification and the underlying bathymetry and to explore the influence of rotation. The bulk of the simulations are inviscid; however, exploratory simulations using a vertical eddy-viscosity confined to a near bottom layer, along with a no-slip boundary condition, suggest that viscous effects may become important in water shallower than about 200 m. A shoaling solitary wave fissions into several waves. At depths of 200–300 m the front of the leading waves become nearly parallel to the bottom and develop a very steep back as has been observed. The leading waves are followed by waves of elevation (pedestals) that are conjugate to the waves of depression ahead and behind them. Horizontal resolutions of at least 50 m are required to simulate these well. Wave breaking was found to occur behind the second or third of the leading solitary waves, never at the back of the leading wave. Comparisons of the shoaling of waves started at depths of 1000 and 3000 m show significant differences and the shoaling waves can be significantly non-adiabatic even at depths greater than 2000 m. When waves reach a depth of 200 m, their amplitudes can be more than 50% larger than the largest possible solitary wave at that depth. The shoaling behaviour is sensitive to the presence of small-scale features in the bathymetry: a 200 m high bump at 700 m depth can result in the generation of many mode-two waves and of higher mode waves. Sensitivity to the stratification is considered by using three stratifications based on summer observations. They primarily differ in the depth of the thermocline. The generation of mode-two waves and the behaviour of the waves in shallow water is sensitive to this depth. Rotation affects the shoaling waves by reducing the amplitude of the leading waves via the radiation of long trailing inertia-gravity waves. The nonlinear-dispersive evolution of these inertia-gravity waves results in the formation of secondary mode-one wave packets.

Simulation of the Transformation of Internal Solitary Waves on Oceanic Shelves

2004 ◽  
Vol 34 (12) ◽  
pp. 2774-2791 ◽  
Author(s):  
Roger Grimshaw ◽  
Efim Pelinovsky ◽  
Tatiana Talipova ◽  
Audrey Kurkin
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Variable Coefficient ◽  
Asymptotic Theory ◽  
Vries Equation ◽  
Internal Solitary Waves ◽  
Variable Depth ◽  
North West ◽  
Wave Parameters ◽  
De Vries

Abstract Internal solitary waves transform as they propagate shoreward over the continental shelf into the coastal zone, from a combination of the horizontal variability of the oceanic hydrology (density and current stratification) and the variable depth. If this background environment varies sufficiently slowly in comparison with an individual solitary wave, then that wave possesses a soliton-like form with varying amplitude and phase. This stage is studied in detail in the framework of the variable-coefficient extended Korteweg–de Vries equation where the variation of the solitary wave parameters can be described analytically through an asymptotic description as a slowly varying solitary wave. Direct numerical simulation of the variable-coefficient extended Korteweg–de Vries equation is performed for several oceanic shelves (North West shelf of Australia, Malin shelf edge, and Arctic shelf) to demonstrate the applicability of the asymptotic theory. It is shown that the solitary wave may maintain its soliton-like form for large distances (up to 100 km), and this fact helps to explain why internal solitons are widely observed in the world's oceans. In some cases the background stratification contains critical points (where the coefficients of the nonlinear terms in the extended Korteweg–de Vries equation change sign), or does not vary sufficiently slowly; in such cases the solitary wave deforms into a group of secondary waves. This stage is studied numerically.

Blow up and instability of solitary wave solutions to a generalized Kadomtsev–Petviashvili equation and two-dimensional Benjamin–Ono equations

2007 ◽  
Vol 464 (2089) ◽  
pp. 49-64 ◽  
Author(s):  
Jianqing Chen ◽  
Boling Guo ◽  
Yongqian Han
Keyword(s):  
Initial Data ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Blow Up ◽  
Positive Energy ◽  
Two Dimensional ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Petviashvili Equation ◽  
Special Case

Let with p being the ratio of an even to an odd integer. For the generalized Kadomtsev–Petviashvili equation, coupled with Benjamin–Ono equations, in the form it is proved that the solutions blow up in finite time even for those initial data with positive energy. As a by-product, it is proved that for all , the solitary waves are strongly unstable if . This result, even in a special case , improves a previous work by Liu (Liu 2001 Trans. AMS 353 , 191–208) where the instability of solitary waves was proved only in the case of .

scholarly journals The Effect of a Variable Background Density Stratification and Current on Oceanic Internal Solitary Waves

Fluids ◽  
2018 ◽  
Vol 3 (4) ◽  
pp. 96
Author(s):  
Zihua Liu ◽  
Roger Grimshaw ◽  
Edward Johnson
Keyword(s):  
Internal Waves ◽  
Variable Coefficient ◽  
Vries Equation ◽  
State Density ◽  
Density Stratification ◽  
Internal Solitary Waves ◽  
Climatological Data ◽  
Horizontal Variation ◽  
Background Density ◽  
Background State

Large amplitude, horizontally propagating internal waves are commonly observed in the coastal ocean. They are often modelled by a variable-coefficient Korteweg–de Vries equation to take account of a horizontally varying background state. Although this equation is now well-known, a term representing non-conservative effects, arising from horizontal variation in the underlying basic state density stratification and current, has often been omitted. In this paper, we examine the possible significance of this term using climatological data for several typical oceanic sites where internal waves have been observed.

scholarly journals Numerical Simulation of Shoaling Internal Solitary Waves in Two-layer Fluid Flow

MATEMATIKA ◽  
2018 ◽  
Vol 34 (2) ◽  
pp. 333-350 ◽  
Author(s):  
Mun Hoe Hooi ◽  
Wei King Tiong ◽  
Kim Gaik Tay ◽  
Kang Leng Chiew ◽  
San Nah Sze
Keyword(s):  
Numerical Simulation ◽  
Fluid Flow ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Variable Coefficient ◽  
Method Of Lines ◽  
The Other ◽  
Internal Solitary Waves ◽  
The Method Of Lines ◽  
Different Types

In this paper, we look at the propagation of internal solitary waves overthree different types of slowly varying region, i.e. a slowly increasing slope, a smoothbump and a parabolic mound in a two-layer fluid flow. The appropriate mathematicalmodel for this problem is the variable-coefficient extended Korteweg-de Vries equation.The governing equation is then solved numerically using the method of lines. Ournumerical simulations show that the internal solitary waves deforms adiabatically onthe slowly increasing slope. At the same time, a trailing shelf is generated as theinternal solitary wave propagates over the slope, which would then decompose intosecondary solitary waves or a wavetrain. On the other hand, when internal solitarywaves propagate over a smooth bump or a parabolic mound, a trailing shelf of negativepolarity would be generated as the results of the interaction of the internal solitarywave with the decreasing slope of the bump or the parabolic mound. The secondarysolitary waves is observed to be climbing the negative trailing shelf.

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>

A regularized model for strongly nonlinear internal solitary waves

2009 ◽  
Vol 629 ◽  
pp. 73-85 ◽  
Author(s):  
WOOYOUNG CHOI ◽  
RICARDO BARROS ◽  
TAE-CHANG JO
Keyword(s):  
Stability Analysis ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Wave Solution ◽  
Wave Amplitude ◽  
Solitary Wave Solution ◽  
Shear Instability ◽  
Internal Solitary Waves ◽  
Strongly Nonlinear ◽  
Regularized Model

The strongly nonlinear long-wave model for large amplitude internal waves in a two-layer system is regularized to eliminate shear instability due to the wave-induced velocity jump across the interface. The model is written in terms of the horizontal velocities evaluated at the top and bottom boundaries instead of the depth-averaged velocities, and it is shown through local stability analysis that internal solitary waves are locally stable to perturbations of arbitrary wavelengths if the wave amplitudes are smaller than a critical value. For a wide range of depth and density ratios pertinent to oceanic conditions, the critical wave amplitude is close to the maximum wave amplitude and the regularized model is therefore expected to be applicable to the strongly nonlinear regime. The regularized model is solved numerically using a finite-difference method and its numerical solutions support the results of our linear stability analysis. It is also shown that the solitary wave solution of the regularized model, found numerically using a time-dependent numerical model, is close to the solitary wave solution of the original model, confirming that the two models are asymptotically equivalent.

scholarly journals Enhanced diapycnal mixing by polarity-reversing internal solitary waves in the South China Sea

2021 ◽  
Author(s):  
Yi Gong ◽  
Haibin Song ◽  
Zhongxiang Zhao ◽  
Yongxian Guan ◽  
Kun Zhang ◽  
...  
Keyword(s):  
South China Sea ◽  
Solitary Wave ◽  
Solitary Waves ◽  
South China ◽  
The South China Sea ◽  
Internal Solitary Wave ◽  
Internal Solitary Waves ◽  
Diapycnal Mixing ◽  
Dongsha Atoll ◽  
Diapycnal Diffusivity

Abstract. Shoaling internal solitary waves near the Dongsha Atoll in the South China Sea dissipate their energy and thus enhance diapycnal mixing, which have an important impact on the oceanic environment and primary productivity. The enhanced diapycnal mixing is patchy and instantaneous. Evaluating its spatiotemporal distribution requires comprehensive observation data. Fortunately, seismic oceanography meets the requirements, thanks to its high spatial resolution and large spatial range. In this paper, we studied three internal solitary waves in reversing polarity near the Dongsha Atoll, and calculated the spatial distribution of resultant diapycnal diffusivity. Our results show that the average diffusivities along three survey lines are two orders of magnitude larger than the open-ocean value. The average diffusivity in the internal solitary wave with reversing polarity is three times that of the non-polarity-reversal region. The diapycnal diffusivity is higher at the front of one internal solitary wave, and gradually decreases from shallow to deep water in the vertical direction. Our results also indicates that (1) the enhanced diapycnal diffusivity is related to reflection seismic events; (2) convective instability and shear instability may both contribute to the enhanced diapycnal mixing in the polarity-reversing process; and (3) the difference between our and previous diffusivity profiles is about 2–3 orders of magnitude, but their vertical distribution is almost the same.

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