Interfacial Solitary Waves Research Articles

Highly nonlinear internal solitary waves with a free surface

A new mathematical model is derived for internal wave propagation in a two-layer fluid domain with a free surface in which the lower layer is of finite depth. The mathematical model consists of three coupled nonlinear equations for displacement of the interface and velocity potentials of the two layers, each of which is expressed as the convex combination of their values at two arbitrary depths. Model equations are correct up to O(μ4) terms, where μ is the ratio of harmonic mean of the depths of the two layers to double of a typical wavelength. Finite amplitude internal solitary wave profiles as well as free surface wave profiles are obtained numerically for different wave speeds, density ratios and depth ratios. Interfacial solitary wave profiles obtained here are in very good agreement with recent experimental observations of Zhao et al. (2020). The figures presented here demonstrate how the interfacial wave profile predicted by the free surface model differs from predictions of rigid-lid two-layer fluid models.

Air/water interfacial waves with a droplet at the tip of their crest

In nature, it is common to observe water wave crests with a droplet at their tip. This fascinating configuration remains unexplained from the physical point of view. The present study explores such a unique local configuration numerically. Solitary waves that propagate at the interface between two layers of irrotational fluids are considered. Extending the work of Guan et al. [“A local model for the limiting configuration of interfacial solitary waves,” J. Fluid Mech. 921, A9 (2021)], the density ratio has been decreased to a very small value equal to 0.001, which is close to the air/water density ratio at sea level (0.0013). A highly accurate solution for the limiting configuration of solitary waves is obtained by solving the irrotational Euler equations using the boundary integral method and Newton's iterations. It is confirmed that the limiting configuration consisting of a droplet sitting on a crest with a 120° angle exists for very small density ratios. This limiting configuration obviously does not exist for surface waves with a void on the top, thus stressing the crucial role played by the air. The droplet is stationary in a frame of reference moving with the wave and experiences intense shear at its tip. From the energy point of view, the formation of a crest with a droplet is accompanied by a remarkable drop of kinetic and potential energies of water in the vicinity of the crest. Furthermore, we present a simple set of scaling relations for the fall of the droplet.

Interfacial electrohydrodynamic solitary waves under horizontal electric fields

Interfacial waves between two superimposed dielectric fluid layers under a horizontal electric field are investigated from asymptotic and numerical aspects. The fluid is taken to be inviscid, incompressible and non-conducting in each layer. The competing forces resulting from gravity, surface tension and electric field are all considered. A systematic procedure is proposed to derive model equations of multiple scales in various possible limits from the electrified Euler equations in the framework of the Zakharov–Craig–Sulem formulation. Based on thorough analyses of the Dirichlet–Neumann operators in the long-wave approximation, classic weakly nonlinear models – including the Boussinesq-type system, the Korteweg–de Vries (KdV) equation and its variants, and the Benjamin-type equation – are obtained under different scaling assumptions. In addition, strongly nonlinear models (without the smallness assumption on the wave amplitude) in the Benjamin and Barannyk–Papageorgiou–Petropoulos regimes are derived. In these models, the electric effects are shown to produce dispersive regularisations of long and short waves. A modified boundary integral equation method is developed to compute solitary waves in the original electrified Euler equations. Through comparisons with solitary-wave solutions in the Euler equations, it is found that in various asymptotic regimes, weakly nonlinear models are in overall good agreement when wave amplitudes are small. In contrast, the range of validity of the strongly nonlinear model is much broader. It is shown that the horizontal electric field plays a significant role in the physical system: it expands the range of parameters for the existence of progressive waves, changes the qualitative characteristics of solitary waves and leads to a new type of solitary wave, namely the KdV–wavepacket mixed type.

A local model for the limiting configuration of interfacial solitary waves

Abstract

The role of the free surface on interfacial solitary waves

We investigated theoretically and experimentally internal solitary waves (ISWs) in a two-layer fluid system with a top free surface. Laboratory experiments are performed by lock-release, under Boussinesq and non-Boussinesq conditions. Experimental results are compared with those obtained by the analytical solution of the Korteweg–de Vries (KdV) weakly nonlinear equation and by the strongly nonlinear Miyata-Choi-Camassa (MCC) model. We analyze the initial conditions which allow to find soliton solutions for both rigid-lid (-RL) and free-surface (-FS) boundary conditions. For the MCC-FS model, we employ a new mathematical procedure to derive the ISW-induced free surface displacement. The density structure strongly affects the elevation of the free surface predicted by the MCC-FS model. The free surface maximum displacement increases mostly with the density difference, assuming non-negligible values also for smaller interfacial amplitudes. Larger displacements occur for thinner upper layer thickness. The MCC-FS model gives the best prediction in terms of both internal waves geometric/kinematic features and surface displacements. The existence of a free surface allows the ISW to transfer part of its energy to the free surface: the wave celerity assumes lower values with respect to ISW speed resulting from the MCC-RL model. For ISWs with a very large amplitude, this behavior tends to fade, and the MCC-RL and the MCC-FS model predict approximately the same celerity and interfacial geometric features. For small-amplitude waves also, the predictions of the KdV-RL equation are consistent with experimental results. Thus, ISWs with an intermediate amplitude should be modeled taking into account a free top surface as the boundary condition.

Laboratory Experiments on an Internal Solitary Wave over a Triangular Barrier

Laboratory experiments were conducted to investigate the evolution of interfacial internal solitary waves (ISWs) incident on a triangular barrier. ISWs with different amplitudes were generated by gravitational collapse. The ISW energy dissipation and turbulence processes were calculated as waves passed over the triangular barrier. Experimental results showed that ISWs were reflecting back off the triangular barrier, and shoaling ISWs led to wave breaking and mixing when waves propagated over the obstacle. Wave instability created the dissipation of energy as it was transmitted from waves to turbulence. The rate of ISW energy dissipation, the maximum turbulent dissipation, and the buoyancy diffusivity linearly increased with the increase in the incident wave energy.

Two-Layer Non-Hydrostatic Model for Generation and Propagation of Interfacial Waves

When pycnocline thickness of ocean density is relatively small, density stratification can be well represented as a two-layer system. In this article, a depth integrated model of the two-layer fluid with constant density is considered, and a variant of the edge-based non-hydrostatic numerical scheme is formulated. The resulting scheme is very efficient since it resolves the vertical fluid depth only in two layers. Despite using just two layers, the numerical dispersion is shown to agree with the analytical dispersion curves over a wide range of kd, where k is the wave number and d the water depth. The scheme was tested by simulating an interfacial solitary wave propagating over a flat bottom, as well as over a bottom step. On a laboratory scale, the formation of an interfacial wave is simulated, which also shows the interaction of wave with a triangular bathymetry. Then, a case study using the Lombok Strait topography is discussed, and the results show the development of an interfacial wave due to a strong current passing through a sill.

Prolongation Structure of a Generalised Inhomogeneous Gardner Equation in Plasmas and Fluids

Abstract In this article, the prolongation structure technique is applied to a generalised inhomogeneous Gardner equation, which can be used to describe certain physical situations, such as the stratified shear flows in ocean and atmosphere, ion acoustic waves in plasmas with a negative ion, interfacial solitary waves over slowly varying topographies, and wave motion in a non-linear elastic structural element with large deflection. The Lax pairs, which are derived via the prolongation structure, are more general than the Lax pairs published before. Under the Painlevé conditions, the linear-damping coefficient equals to zero, the quadratic non-linear coefficient is proportional to the dispersive coefficient c(t), the cubic non-linear coefficient is proportional to c(t), leaving no constraints on c(t) and the dissipative coefficient d(t). We establish the prolongation structure through constructing the exterior differential system. We introduce two methods to obtain the Lax pairs: (a) based on the prolongation structure, the Lax pairs are obtained, and (b) via the Lie algebra, we can derive the Pfaffian forms and Lax pairs when certain parameters are chosen. We set d(t) as a constant to discuss the influence of c(t) on the Pfaffian forms and Lax pairs, and to discuss the influence of d(t) on the Pfaffian forms and Lax pairs, we set c(t) as another constant. Then, we get different prolongation structure, Pfaffian forms and Lax pairs.

Transmission and reflection of internal solitary waves incident upon a triangular barrier

We report upon laboratory experiments and numerical simulations examining the evolution of an interfacial internal solitary wave incident upon a triangular ridge whose peak lies below the interface. If the ridge is moderately large, the wave is observed to shoal and break similar to solitary waves shoaling upon a constant slope, but interfacial waves are also observed to transmit over and reflect from the ridge. In laboratory experiments, by measuring the interface displacement as it evolves in time, we measure the relative transmission and reflection of available potential energy after the incident wave has interacted with the ridge. The numerical simulations of laboratory- and ocean-scale waves measure both the available potential and kinetic energy to determine the partition of incident energy into that which is transmitted and reflected. From shallow-water theory, we define a critical amplitude, $A_{c}$, above which interfacial waves are unstable. The transmission is found to decrease from one to zero as the ratio of the incident wave amplitude to $A_{c}$ increases from less than to greater than one. Empirical fits are made to analytic curves through measurements of the transmission and reflection coefficients.

Propagation regimes of interfacial solitary waves in a three-layer fluid

Abstract. Long weakly nonlinear finite-amplitude internal waves in a fluid consisting of three inviscid layers of arbitrary thickness and constant densities (stable configuration, Boussinesq approximation) bounded by a horizontal rigid bottom from below and by a rigid lid at the surface are described up to the second order of perturbation theory in small parameters of nonlinearity and dispersion. First, a pair of alternatives of appropriate KdV-type equations with the coefficients depending on the parameters of the fluid (layer positions and thickness, density jumps) are derived for the displacements of both modes of internal waves and for each interface between the layers. These equations are integrable for a very limited set of coefficients and do not allow for proper description of several near-critical cases when certain coefficients vanish. A more specific equation allowing for a variety of solitonic solutions and capable of resolving most near-critical situations is derived by means of the introduction of another small parameter that describes the properties of the medium and rescaling of the ratio of small parameters. This procedure leads to a pair of implicitly interrelated alternatives of Gardner equations (KdV-type equations with combined nonlinearity) for the two interfaces. We present a detailed analysis of the relationships for the solutions for the disturbances at both interfaces and various regimes of the appearance and propagation properties of soliton solutions to these equations depending on the combinations of the parameters of the fluid. It is shown that both the quadratic and the cubic nonlinear terms vanish for several realistic configurations of such a fluid.

Numerical study of interfacial solitary waves propagating under an elastic sheet.

Steady solitary and generalized solitary waves of a two-fluid problem where the upper layer is under a flexible elastic sheet are considered as a model for internal waves under an ice-covered ocean. The fluid consists of two layers of constant densities, separated by an interface. The elastic sheet resists bending forces and is mathematically described by a fully nonlinear thin shell model. Fully localized solitary waves are computed via a boundary integral method. Progression along the various branches of solutions shows that barotropic (i.e. surface modes) wave-packet solitary wave branches end with the free surface approaching the interface. On the other hand, the limiting configurations of long baroclinic (i.e. internal) solitary waves are characterized by an infinite broadening in the horizontal direction. Baroclinic wave-packet modes also exist for a large range of amplitudes and generalized solitary waves are computed in a case of a long internal mode in resonance with surface modes. In contrast to the pure gravity case (i.e without an elastic cover), these generalized solitary waves exhibit new Wilton-ripple-like periodic trains in the far field.

Laboratory study of flow field for ISW evolution on a flat bottom

Spatial variation of the velocity field induced by propagation of an interfacial solitary wave (ISW) is one of the most important research themes in marine science. This velocity field together with the vortex within the system can affect marine operations as the ISW propagates in the ocean. The governing equations presently available are modified to model the velocity and vortex, and are used in numerical experiments. This paper presents the results of laboratory experiments and numerical modeling of the velocity field induced by the propagation of an ISW in depression and elevation on a flat bottom. Laboratory observations reveal that the water particle orbits in different water layers have different motion trails due to the effect of the ISW. In addition, the numerical and laboratory experiments show a self-generated vortex above the interface of a depression ISW as it propagates in a stratified two-layer fluid system on a flat bottom. Interestingly, the interface itself may limit the vertical transfer of mass between the upper and bottom layers in a stratified fluid system.

Effects of varying pycnocline thickness on interfacial wave generation and propagation

Interfacial solitary waves (ISWs) can be observed in stratified ocean and laboratory flumes which generally comprise a pycnocline in finite thickness. This has led to the adoption of a two-layer or three-layer fluid system for studying the ISWs. However, this assumption has limited success for understanding the effects of variable pycnocline thicknesses on the wave generation and propagation. This paper discusses the broadening of pycnocline thickness on the generation of an ISW in a wave flume and its subsequent effects on wave amplitude and potential energy across a submerged trapezoidal obstacle under laboratory conditions. Three different definitions for the nominal pycnocline thickness are discussed, based on the measured fluid density profiles and the calculated buoyant frequency profiles. As the pycnocline thickness increased after repeated wave-makings, the incident ISW amplitude decreased as expected, as well as the wave speed across the obstacle. Moreover, as the thickness broadened, the fluid system appeared to contain two or multiple peaks rather than single peak when the pycnocline was thin and finite. This implies that the resultant wave might consist of several modes (at least mode-2) in the flume even at its generation stage.

The effect of surface stress on interfacial solitary wave propagation

The propagation of long wavelength disturbances on the surface of a fluid layer of finite depth is considered. Attention is focused on the effect of stress applied at the surface. Constant surface tension leads to a normal stress at the surface, but the presence of a surfactant or the application of an electric field can give rise to tangential stresses. In the large Reynolds number limit, the evolution equation for the surface elevation contains contributions from both boundary layers in the flow; one is adjacent to the free surface while the other lies at the base of the fluid layer.Aweakly non-linear analysis is performed leading to an evolution equation similar to the classic Korteweg-de Vries equation, but modified by additional terms due to the viscosity and to the tangential and normal stress at the surface. It is demonstrated that careful treatment of the boundary layer at the free surface is necessary when the tangential stress at the surface is non-zero. Particular cases of flows with tangential surface stress due the presence of a surfactant or due to an electric field are discussed, and a pseudo-spectral scheme is used in order to obtain some typical numerical results.

Internal solitary wave transformation over a bottom step: Loss of energy

In this paper, we extend the numerical study of Maderich et al. [“Interaction of a large amplitude interfacial solitary wave of depression with a bottom step,” Phys. Fluids 22, 076602 (2010)10.1063/1.3455984] on the interaction of an interfacial solitary wave with a bottom step, considering (i) the energy loss of solitary waves of both polarities interacting with a bottom step and (ii) features of the transformation of a large-amplitude internal solitary waves at the step. We show that the dependence of energy loss on the step height is not monotonic, but has different maximum positions for different incident wave polarities. The energy loss does not exceed 50% of the energy of an incident wave. The results of our numerical modeling are compared with some recent results from laboratory tank modeling.

Effect of frontal slope on waveform evolution of a depression interfacial solitary wave across a trapezoidal obstacle

Waveform evolution associated with a depression interfacial solitary wave (ISW) over uniform slope or slope-shelf with different front slopes has been observed in many laboratory experiments and field studies since the 1980s, and by numerical modeling in more recent time. However, attention was mainly focused on waveform validation with KdV theory, wave run-up, reflection, energy dissipation and evolution on emergent mild uniform slope, rather on wave evolution and inversion across the plateau of a slope-shelf with steep front slopes, which are common in field conditions. Since it is often difficult to measure the entire process of waveform evolution in the field, laboratory experiments are adopted as an alternative to compliment field observations. This paper presents the results of laboratory experiments on ISW evolution across a submerged trapezoidal obstacle with four different front slopes, ranging from mild to very steep and vertical. Our results indicate that not only the likelihood of internal hydraulic jump, waveform inversion and the strength of internal wave breaking reduced, but more wave energy dissipated as well, as the front slope steepened. Upon using Hilbert–Huang transform to analyse the wave profiles recorded at four probe locations along the wave flume, the variability in the instantaneous characteristic wave period can be detected while an ISW propagated across a trapezoidal obstacle.

Interfacial Wave Motion of Very Large Amplitude: Formulation in Three Dimensions and Numerical Experiments

Interfacial gravity driven motion of a two-fluid system bounded above and below by rigid lids, is studied. The interfacial motion is three-dimensional, fully nonlinear and fully dispersive. By the method of successive approximations, various approximations of the method are derived, where the truncated versions are computationally fast, still being highly accurate. The ac- curacy is tested out by numerical calculations and comparisons to accurate interfacial solitary waves in two dimensions wherein the reference computations, the full nonlinearity and dispersive effects are kept. The calculations show that the methods expressed by its quadratic and cubic approximations provide very accurate representation of the waves. Error estimates obtained by Euclidian norm tend to zero when the amplitude goes to zero. The cubic approximation of the normal velocity along the interface has an error of less than 0.02 percent on the side of the lower, deep fluid layer and 0.24 percent on the side of the upper, shallow fluid, respectively, when the wave amplitude is equal to the upper layer depth. The cubic approximation is still very good for solitary waves of very large amplitude; even as large as close to the conjugate flow limit, which in the present computations is 9.66... times the upper layer depth (depth ratio of 20.4, density ratio of 0.986). A downward shift of the reference level improves the calculation of the normal velocity in the thin layer.

Potential hazards and dynamical analysis of interfacial solitary wave interactions

Over the last few decades, a lot of attention has been concentrated on the consequences of marine impacts, especially those caused by the tsunami wave train. Internal solitary waves are similar to the surface waves that commonly occur in the waters of the ocean or large lakes and can have significant effects on oceanic mixing, climate change, the movement of submerged plankton, and the weathering of geological structures. This motion can be severe enough to create natural hazards, such as submarine tsunamis in the ocean. These could also even occur in large lakes. Numerical modeling has shown that the waveform of a soliton that interacts with others of a similar kind would emerge unchanged from the collision, except for a phase shift. However, the results from laboratory experiments are rather limited, despite the successful generation of ISWs using a collapse mechanism in a wave flume. This paper reports on some interesting facts compiled from the results of a series of laboratory experiments on the investigation of the head-on collision of two ISWs. Our results confirm that the waveforms of two depression ISWs will more or less retain their initial shape after a head-on collision. However, the transmitted wavelength will broaden when two elevation ISWs collide, perhaps affected by bottom friction. Overall, the resulting waveforms induced by such head-on collisions agree well with the theoretical predictions for depression ISWs, regardless of their scale of amplitude, but the results are only valid for elevated waveforms of large amplitude.

Assessment of the major hazard potential of interfacial solitary waves moving over a trapezoidal obstacle on a horizontal plateau

Over the last few decades, a lot of attention has been concentrated on the consequences of marine impacts, especially those caused by the tsunami wave train. Internal solitary waves are similar to the surface waves that commonly occur in the waters of the ocean or large lakes and can have significant effects on oceanic mixing, climate change, the movement of submerged plankton, and the weathering of geological structures. This motion can be severe enough to create natural hazards such as submarine tsunamis in the ocean. These could also even occur in large lakes. The present work aims to contribute to this knowledge base by studying internal wave propagation on a shallow continental shelf following a particular marine impact. A series of laboratory experiments were conducted in order to clarify the movement of an interfacial solitary wave across a uniform slope and a horizontal plateau forming a slope-shelf topography. The results obtained from test runs indicate that the wave maintains its strength, having a direct impact on the natural ecology of the local oceanic environment. Comparison with different seabed topographies is also presented to demonstrate the propagation of an internal wave over a trapezoidal barrier. A better fitting and more appropriate model is employed to examine the relationship between the physical parameters for better predicting the evolution of an internal solitary wave as it moves over a trapezoidal obstacle on a horizontal plateau.

Interaction of a large amplitude interfacial solitary wave of depression with a bottom step

This paper is devoted to the study of the transformation of a finite-amplitude interfacial solitary wave of depression at a bottom step. The parameter range studied goes outside the range of weakly nonlinear theory (the extended Korteweg–de Vries or Gardner equation), and we describe various scenarios of this transformation in terms of the incident wave amplitude and the step height. The dynamics and energy balance of the transformation are described. Several numerical simulations are carried out using the nonhydrostatic model based on the fully nonlinear Navier–Stokes equations in the Boussinesq approximation. Three distinct runs are discussed in detail. The first simulation is done when the ratio of the step height to the lower layer thickness after the step is about 0.4 and the incident wave amplitude is less than the limiting value estimated for a Gardner solitary wave. It shows the applicability of the weakly nonlinear model to describe the transformation of a strongly nonlinear wave in this case. In the second simulation, the ratio of the step height to the lower layer thickness is the same as that in the first run but the incident wave amplitude is increased and then its shape is described by the Miyata–Choi–Camassa solitary wave solution. In this case, the process of wave transformation is accompanied by shear instability and the billows that result in a thickening of the interface layer. In the third simulation, the ratio of the step height to the thickness of the lower layer after the step is 1.33, and then the same Miyata–Choi–Camassa solitary wave passes over the step, it undergoes stronger reflection and mixing between the layers although Kelvin–Helmholtz instability is absent. The energy budget of the wave transformation is calculated. It is shown that the energy loss in the vicinity of the step grows with an increase of the ratio of the incident wave amplitude to the thickness of the lower layer over the step.