Large-amplitude Internal Solitary Waves Research Articles

High-level Green–Naghdi model for large-amplitude internal waves in deep water

In this paper, a high-level Green–Naghdi (HLGN) model for large-amplitude internal waves in a two-layer fluid system, where the upper-fluid layer is of finite depth and the lower-fluid layer is of infinite depth, is developed under the rigid-lid free-surface approximation. The equations of the present HLGN model follow Euler's equations under the sole assumption that the horizontal and vertical velocity distributions along the vertical column are presented by known shape functions for each layer. The linear dispersion relations of the HLGN model for different levels are presented and compared with those obtained by other strongly nonlinear models for deep water, including the fully nonlinear models that include the dispersion effects $O(\mu )$ (where $\mu$ is the ratio of the upper-fluid layer depth to a typical wavelength) derived by Choi & Camassa (Phys. Rev. Lett., vol. 77, 1996, pp. 1759–1762) and $O(\mu ^2)$ derived by Debsarma et al. (J. Fluid Mech., vol. 654, 2010, pp. 281–303). It is shown that the HLGN model has a wider application range than other models. Solutions of travelling large-amplitude internal solitary waves in the absence and presence of background shear-current are then investigated by using the HLGN model. For the no-current cases, results obtained by the HLGN model show better agreement with Euler's solution on wave profile, velocity profile at the maximum interface displacement and wave speed compared with those obtained by other models. For the background shear-current cases, results obtained by the HLGN model also show good agreement with those obtained by solving the Dubreil-Jacotin–Long equation.

Internal solitary wave in the Lombok Strait: Satellite-observed spatiotemporal characteristics and their propagations modulated by the Indonesian Throughflow

The Lombok Strait shows active large-amplitude internal solitary waves (ISWs) and is also a primary gateway for the Indonesian Throughflow (ITF), which is a critical ocean current affecting the ocean ecosystem. This study collected 858 satellite images from 2018 to 2022, and ISW wave crests were extracted. Analysis shows that ISW 5-year cumulative occurrences reached the highest/ lowest of 136/28 days, with a 29 %/6 % frequency in October/ June. Satellite and reanalysis data revealed that ISW occurrence and propagation speed correlate to ITF variations. Enhanced southward ITF corresponds to less ISW occurrence and decreased/increased northward/southward ISW propagation speed. To understand how ITF modulates ISWs, three-dimensional MITgcm simulations were employed. Results show that when southward ITF decreases, ISWs tend to generate earlier/later, thus leading to a longer/shorter propagation distance for ISWs in the north/ south direction, revealing a suppressive effect on northward ISW generation.

Correction to: Error Calculation of Large-Amplitude Internal Solitary Waves Within the Pycnocline Introduced by the Strong Stratification Approximation

Correction to: Error Calculation of Large-Amplitude Internal Solitary Waves Within the Pycnocline Introduced by the Strong Stratification Approximation

Error Calculation of Large-Amplitude Internal Solitary Waves Within the Pycnocline Introduced by the Strong Stratification Approximation

Error Calculation of Large-Amplitude Internal Solitary Waves Within the Pycnocline Introduced by the Strong Stratification Approximation

A high-order spectral method for effective simulation of surface waves interacting with an internal wave of large amplitude

In this study, we propose a new method for simulating complex surface waves interacting with a large amplitude internal solitary wave. Our model is based on a high-order spectral method for surface waves with the bottom boundary conditions computed from an internal wave solver. The convergence of the model is first tested using an internal wave case without surface waves. We then perform simulations of a canonical nonlinear wave interaction case involving two surface wave components and a prescribed internal wave component. The initial wave energy growth rate of the resonant wave component is found to agree with the analytical value predicted by the perturbation theory. We also use the model to simulate the interaction between surface waves and a weakly nonlinear internal wave, and the result is nearly identical to that calculated using a two-layer model. Finally, we show the application of our model in a multiscale nested modeling framework, where the internal wave parameters are extracted from the mesoscale simulation using a nonhydrostatic ocean model. The evolution of the spatial variations of the surface roughness is captured and the surface wave orbital velocity also changes in space due to the surface motions induced by the internal wave. Our phase-resolved model provides a computationally efficient tool for simulating complex surface wave fields in the background of internal waves of large amplitudes.

Applicability of high-order unidirectional internal solitary wave theoretical model

Internal solitary waves exist widely in the oceans, and their generations, propagation evolutions, and dissipations have profound effects on the ocean environment, topography, and marine structures. Typically, two basic theoretical models are now being developed to govern the evolutions of internal solitary waves at the interface of two immiscible inviscid fluids. One is a unidirectional wave propagation model described by the KdV (Korteweg-de Vries) equation, and the other is a bidirectional wave propagation model depicted by the Miyata-Choi-Camassa (MCC) equation. Neither of them, however, can effectively characterize the course of the evolution of large-amplitude internal solitary wave. In this paper, a modified unidirectional internal solitary wave model is established by adjusting the coefficients of the original unidirectional model. The adjusted coefficients are determined through asymptotic analysis by matching with the MCC model. The efficacy of the modified coefficients is investigated by comparing the modified model with the original model. The experiments on the generation of internal solitary waves with varying amplitudes are carried out by comparing the internal solitary wave solution of the modified equation. It is shown that the modified model is suitable for describing the waveform of internal solitary waves with small, medium, and large amplitudes within the limiting amplitude of the MCC model. By quantitatively analyzing the agreement of the effective wavelength, wave speed, and waveform of steady-state internal solitary waves between the unidirectional model and the MCC model, the applicability of the modified model in characterization of the properties of steady-state internal solitary waves is further investigated. In addition, the stability of unidirectional theoretical model is analyzed for simulating the propagation of large-amplitude internal solitary wave under flat bottom condition. It is found that the unidirectional model is suitable for initiating its own internal solitary solution provided that the numerical scheme is stable. It is shown that the modified unidirectional model can be used to characterize large-amplitude internal solitary waves, and is also expected to be applied to the study of marine structure hydrodynamics.

Theoretical and experimental investigation on strongly nonlinear internal solitary waves moving over slope-shelf topography

The variable-coefficient modified Gardner equation is proposed to describe large amplitude internal solitary waves (ISWs) propagating over slope-shelf topography. The model is adapted from weakly nonlinear Gardner equation by adjusting its original coefficients such that the model is capable of representing the crucial characteristics of large amplitude ISWs, including the effective wavelength and wave speed. To numerically solve the nonlinear equation, the fourth-order Runge-Kutta method is applied for temporal discretization, second-order central finite difference numerical scheme is implemented for the space discretization and the iterative scheme is applied to treat the coupling term of time and space. The numerical method is then verified by comparing fairly with laboratory experiments. It's found the variable-coefficient modified Gardner model is a reliable theoretical tool to describe the behavior of large amplitude internal solitary waves passing through bottom topography.

Direct measurements reveal instabilities and turbulence within large amplitude internal solitary waves beneath the ocean

Internal solitary waves are ubiquitous in coastal regions and marginal seas of the world’s oceans. As the waves shoal shoreward, they lose the energy obtained from ocean tides through globally significant turbulent mixing and dissipation and consequently pump nutrient-rich water to nourish coastal ecosystem. Here we present fine-scale, direct measurements of shoaling internal solitary waves in the South China Sea, which allow for an examination of the physical processes triggering the intensive turbulent mixing in their interior. These are convective breaking in the wave core and the collapse of Kelvin–Helmholtz billows in the wave rear and lower periphery of the core, often occurring simultaneously. The former takes place when the particle velocity exceeds the wave’s propagating velocity. The latter is caused by the instability induced by the strong velocity shear overcoming the stratification. The instabilities generate turbulence levels four orders of magnitude larger than that in the open ocean.

Experimental and numerical studies on internal solitary waves with a free surface

Abstract

An interesting oddity in the theory of large amplitude internal solitary waves

An interesting oddity in the theory of large amplitude internal solitary waves

High-order unidirectional model with adjusted coefficients for large-amplitude long internal waves

To describe large amplitude internal solitary waves in a two-layer system, we consider the high-order unidirectional (HOU) model that extends the Korteweg–de Vries equation with high-order nonlinearity and leading-order nonlinear dispersion. While the original HOU model is valid only for weakly nonlinear waves, its coefficients depending on the depth and density ratios are adjusted such that the adjusted model can represent the main characteristics of large amplitude internal solitary waves, including effective wavelength, wave speed, and maximum wave amplitude. It is shown that the solitary wave solution of the adjusted HOU (aHOU) model agrees well with that of the strongly nonlinear Miyata–Choi–Camassa (MCC) model up to the maximum wave amplitude, which cannot be achieved by the original HOU model. To further validate the aHOU model, numerical solutions of the aHOU model are presented for the propagation and interaction of solitary waves and are shown to compare well with those of the MCC model. The aHOU model is further extended to the case of variable bottom and is solved numerically. In comparison with the MCC model for variable bottom, it is found that the aHOU model is a simple, but reliable theoretical model for large amplitude internal solitary waves, which would be useful for practical applications.

Feature identification in time-indexed model output.

We present a method for identifying features (time periods of interest) in data sets consisting of time-indexed model output. The method is used as a diagnostic to quickly focus the attention on a subset of the data before further analysis methods are applied. Mathematically, the infinity norm errors of empirical orthogonal function (EOF) reconstructions are calculated for each time output. The result is an EOF reconstruction error map which clearly identifies features as changes in the error structure over time. The ubiquity of EOF-type methods in a wide range of disciplines reduces barriers to comprehension and implementation of the method. We apply the error map method to three different Computational Fluid Dynamics (CFD) data sets as examples: the development of a spontaneous instability in a large amplitude internal solitary wave, an internal wave interacting with a density profile change, and the collision of two waves of different vertical mode. In all cases the EOF error map method identifies relevant features which are worthy of further study.

Potential generation sites of internal solitary waves and their propagation characteristics in the Andaman Sea-a study based on MODIS true-colour and SAR observations.

The presence of large-amplitude Internal Solitary Waves (ISWs or solitons) is quite common in the Andaman Sea, located in the north-eastern Indian Ocean basin. ISWs are known to induce strong vertical velocities which can play an essential role in the mixing transport of nutrients and are proven hazardous to offshore oil platforms. The surface signatures of ISWs can be detected using remote sensing instruments like Synthetic Aperture Radar (SAR) and sunglint true-colour images. The present study makes an effort to delineate as well as detect the possible potential generation locations of mode-1 long living ISWs in the Andaman Sea using remote sensing observations. To accomplish this, the Moderate Resolution Imaging Spectroradiometer (MODIS) true-colour images of Terra/Aqua satellites for the months of March and April during 2014-2016 are used to map the distribution and propagation characteristics of ISWs. These maps along with SAR imgaes from ENVISAT and TerraSAR-X are used to detect the possible generation locations of ISWs. The study considers the possible generation location of ISW as the circumcentre of each wave packet as they radially propagate along a two-dimensional frame. The analysis reveals five potential ISW generation hotspots that are distributed along the Northern Andaman Sea, as well as locations in the discontinuities off the Nicobar Islands and the great passage. The ISWs that form over these regions are hitting the continental shelf within the Andaman Sea. Interestingly, the waves from two potential generation sites between the Nicobar Islands appear to radiate waves in two opposite directions, towards the Andaman Sea and the southern Bay of Bengal.

Combined effects of topography and bottom friction on shoaling internal solitary waves in the South China Sea

A numerical study to a generalized Korteweg-de Vries (KdV) equation is adopted to model the propagation and disintegration of large-amplitude internal solitary waves (ISWs) in the South China Sea (SCS). Based on theoretical analysis and in situ measurements, the drag coefficient of the Chezy friction is regarded as inversely proportional to the initial amplitude of an ISW, rather than a constant as assumed in the previous studies. Numerical simulations of ISWs propagating from a deep basin to a continental shelf are performed with the generalized KdV model. It is found that the depression waves are disintegrated into several solitons on the continental shelf due to the variable topography. It turns out that the amplitude of the leading ISW reaches a maximum at the shelf break, which is consistent with the field observation in the SCS. Moreover, a dimensionless parameter defining the relative importance of the variable topography and friction is presented.

Optimal transient growth in thin-interface internal solitary waves

The dynamics of perturbations to large-amplitude internal solitary waves (ISWs) in two-layered flows with thin interfaces is analysed by means of linear optimal transient growth methods. Optimal perturbations are computed through direct–adjoint iterations of the Navier–Stokes equations linearized around inviscid, steady ISWs obtained from the Dubreil-Jacotin–Long (DJL) equation. Optimal perturbations are found as a function of the ISW phase velocity $c$ (alternatively amplitude) for one representative stratification. These disturbances are found to be localized wave-like packets that originate just upstream of the ISW self-induced zone (for large enough $c$) of potentially unstable Richardson number, $Ri<0.25$. They propagate through the base wave as coherent packets whose total energy gain increases rapidly with $c$. The optimal disturbances are also shown to be relevant to DJL solitary waves that have been modified by viscosity representative of laboratory experiments. The optimal disturbances are compared to the local Wentzel–Kramers–Brillouin (WKB) approximation for spatially growing Kelvin–Helmholtz (K–H) waves through the $Ri<0.25$ zone. The WKB approach is able to capture properties (e.g. carrier frequency, wavenumber and energy gain) of the optimal disturbances except for an initial phase of non-normal growth due to the Orr mechanism. The non-normal growth can be a substantial portion of the total gain, especially for ISWs that are weakly unstable to K–H waves. The linear evolution of Gaussian packets of linear free waves with the same carrier frequency as the optimal disturbances is shown to result in less energy gain than found for either the optimal perturbations or the WKB approximation due to non-normal effects that cause absorption of disturbance energy into the leading face of the wave. Two-dimensional numerical calculations of the nonlinear evolution of optimal disturbance packets leads to the generation of large-amplitude K–H billows that can emerge on the leading face of the wave and that break down into turbulence in the lee of the wave. The nonlinear calculations are used to derive a slowly varying model of ISW decay due to repeated encounters with optimal or free wave packets. Field observations of unstable ISW by Moum et al. (J. Phys. Oceanogr., vol. 33 (10), 2003, pp. 2093–2112) are consistent with excitation by optimal disturbances.

Laboratory investigation on internal solitary waves interacting with a uniform slope

Internal solitary waves (ISWs) propagating in a two layer stratified fluid system are studied by laboratory experiments through the standard lock-release method. We investigated the generation, propagation, and breaking phases of large amplitude internal solitary waves of depression, propagating horizontally in a wave tank. ISWs main features depend on the geometrical parameters that define the initial experimental setting. Relations between ISWs geometric and kinematic features and the initial setting parameters are analyzed and empirical relations are developed. The approach of the ISWs towards a uniform slope is investigated. Depending on both wave properties and slopes values, different physical processes characterize the shoaling phase, causing different breaker type. Following a qualitatively analysis of the different breaking mechanisms, collapsing-plunging breakers show the larger contribution in terms of mixing. The characteristics of the breaking events affecting internal wave packets propagating towards the North of the Messina Strait (Mediterranean Sea) are discussed. Additional laboratory experiments are performed to investigate how the pycnocline thickness is affected by the breaking of plunging ISWs. The consequent increase of the pycnocline thickness results to be nonlinearly related with the Iribarren number.

Modelling and observations of oceanic nonlinear internal wave packets affected by the Earth’s rotation

The large-amplitude internal solitary waves commonly observed in the coastal ocean can propagate for long distances for long times, so that it may be necessary to take account of the effects of the Earth’s background rotation. In this case an appropriate model wave evolution equation is the Ostrovsky equation, whose typical solutions indicate that internal solitary waves will evolve into envelope wave packets. Unlike the more usual Korteweg-de Vries solutions which are typically rank-ordered wave packets, these are centred with the largest waves in the middle. This qualitative feature, together with certain key quantitative parameters such as the envelope carrier wavenumber and speed, can be sought in oceanic observations. Hence we have examined many SAR images of internal solitary waves with the general aim of finding features indicating that rotational effects have become significant. From these we report in detail on six typical cases of which four give indications of rotational effects. In addition we use a two-layer fluid model to estimate how the rotational parameters depend on the background stratification and topography.

Numerical simulations of shoaling internal solitary waves of elevation

We present high-resolution, two- and three-dimensional direct numerical simulations of large amplitude internal solitary waves of elevation on the laboratory scale, shoaling onto and over a small-amplitude shelf. The three-dimensional, mapped coordinate, spectral collocation method used for the simulations allows for accurate modelling of both the shoaling waves and the bottom boundary layer. The shoaling of the waves is characterized by the formation of a quasi-trapped core which undergoes a spatially growing stratified shear instability at its edge and a lobe-cleft instability in its nose. Both of these instabilities develop and three-dimensionalize concurrently, leading to strong bottom shear stress. We explore significant regions of Schmidt and Reynolds number space and demonstrate that the formation of shear instabilities during shoaling is robust and should be readily observable in a number of standard laboratory setups. In the experiments with a corrugated bottom boundary, boundary layer separation is found inside each of the corrugations during shoaling. This more complex boundary layer phenomenology precludes the formation of the lobe-cleft instability almost completely and hence provides a different mechanism for fluid and material exchange across the bottom boundary layer. Our analyses suggest that all of these wave-induced instabilities can lead to enhanced turbulence in the water column and increased shear stress on the bottom boundary. Through the generation and evolution of these instabilities, the shoaling of internal solitary waves of elevation is likely to provide systematic mechanisms for material mixing, cross-boundary layer transport, and sediment resuspension.

Effects of large-amplitude internal solitary waves on ROV operation—A numerical study

Remotely operated vehicles (ROVs) are unique tools for underwater industrial exploration and scientific research of offshore areas and the deep ocean. With broadening application of ROVs, the study of factors that affect their safe operation is important. Besides the technical skills to control ROV movement, the dynamical ocean environment may also play a vital role in the safe operation of ROVs. In this paper, we investigate the influence of large-amplitude internal solitary waves (ISWs), focusing on the forces exerted on ROVs by ISWs. We present a methodology for modeling ISW-induced currents based on KdV model and calculate the ISW forces using the Morrison equation. Our results show that an extremely considerable load is exerted by the ISW on the ROV, resulting in a strong disturbance of the vehicle’s stability, affecting the ROV control. The numerical results of this work emphasize the importance of considering dynamical conditions when operating underwater vessels, such as ROV. Further laboratory and field investigation are suggested to gain more understanding of this subject.

Characteristics of Mach effect induced by oblique wave-wave interactions of internal solitary waves in ocean

The Mach effect induced by the oblique wave-wave interactions of internal solitary waves is a special phenomenon of the ocean and is also the main reason of generating a large amplitude internal solitary wave, while they can cause it damage and even crash as the ocean engineering structures or underwater vehicles encounter the internal solitary wave with a large ampiltude in the ocean. In order to simulate the oblique interaction of internal solitary waves in the thermocline marine environment and control the Mach effect in the laboratory, two vertical bottomless barrels are placed in the fresh/saline water’s two-layer fluid. So the mixed fresh/saline water region is called the density thermocline of the marine. Keeping exchange of water under the barrel’s bottoms is to facilitate the formation of the mixed region height difference in the inner and outer parts of barrels. And then remove the two barrels smoothly and quickly, so that the mixed fresh/saline water region’s gravitational collapse motion can be made use of to generate two rows of internal solitary waves. They will occur the oblique wave-wave interaction in their propagating intersection area. The experimental results showed that two rows of solitary wave oblique interaction depends on the stratification environment, the distance between the two barrels and the amplitude and phase of two solitary waves. It has been vertfied that the above method can effectively control the generation of Mach effect in the stratified fluid tank. Further analysis showed that the Mach effect induced by the oblique interaction of internal solitary waves will lead to remarkably increasing amplitude in correspondence with its raising propagation velocity, whose direction is controlled by the amplitude difference between two internal solitary waves, and the greater amplitude difference will bring on the larger Mach effect’s critical angle. Through testing the theoretical models including KdV’, BO’ and MCC’s equations to compare with the experimental results , the adaptability of the Mach effect to the nonlinear theoretical models is verified by experiments.