The time evolution emanating from "internal dam-break" initial conditions is studied for a class of models of stratified Euler fluids in configurations close to two-homogeneous layers separated by a thin diffused interface. Direct numerical simulations and experiments in wave tanks show that such initial conditions eventually give rise to coherent structures that are close to solitary-wave solutions moving ahead of a region of dispersive wave motion and turbulent mixing close to the location of the initial dam step. A priori theoretical predictions of the main features of these solitary waves, such as their amplitudes and speeds, appear to be unavailable, even for simplified models of wave evolution in stratified fluids. With the aim of providing estimates of the existence, amplitude and speed of such solitary waves, an approach based on Inverse Scattering Transform (IST) for completely integrable models is developed here and tested against direct numerical simulations of Euler fluids and some of their models based on long wave asymptotics. In particular, a novel regularized version of the Kaup model is developed to cure its mathematical ill-posedness, thereby making possible the use of its analytical predictions on the solitary-waves "content" emanating from step-like initial conditions. The a priori estimates of the new regularized model are found to be in reasonable agreement with the direct numerical simulations for weakly nonlinear regimes, this significantly improves predictions based on unidirectional models such as the Korteweg-de Vries equation.
A fluid system bounded by a flat bottom and a flat surface with an internal wave and depth-dependent current is considered. The Hamiltonian of the system is presented and the dynamics of the system are discussed. A long-wave regime is then considered and extended to produce a KdV approximation. Finally, a solitary wave solution is obtained.
We present a numerical study of essentially nonlinear dynamics of surface gravity waves on deep water with constant vorticity using governing equations in conformal coordinates. The dispersion relation of surface gravity waves on shear flow is known to have two branches, one of which is weakly dispersive for long waves. Weakly nonlinear evolution of the waves of this branch can be described by the Benjamin-Ono equation, which is integrable and has soliton and multi-soliton solutions. Currently, the extent to which the properties of such solitary waves obtained within the weakly nonlinear model…
We derive a new hyperbolic model describing the propagation of internal waves in a stratified shallow water with a non-hydrostatic pressure distribution. The construction of the hyperbolic model is based on the use of additional `instantaneous' variables. This allows one to reduce the dispersive multi-layer Green--Naghdi model to a first-order system of evolution equations. The main attention is paid to the study of three-layer flows over uneven bottom in the Boussinesq approximation with the additional assumption of hydrostatic pressure in the intermediate layer. The hyperbolicity conditions …
Physics-Informed Neural Networks for the Korteweg-de Vries Equation for Internal Solitary Wave Problem: Forward Simulation and Inverse Parameter Estimation
In this paper, we investigate mode-2 solitary waves in a three-layer stratified flow model. Localised travelling wave solutions to both the fully nonlinear problem (Euler equations), and the three-layer Miyata-Choi-Camassa equations are found numerically and compared. Mode-2 solitary waves with speeds slower than the linear mode-1 long-wave speed are typically generalised solitary waves with infinite tails consisting of a resonant mode-one periodic wave train. Herein we evidence the existence of mode-2 embedded solitary waves, that is, we show that for specific values of the parameters, the am…
Interaction of a solitary wave with a long background wave is studied within the framework of rotation modified Benjamin-Ono equation describing internal waves in a deep fluid. With the help of asymptotic method, we find stationary and nonstationary solutions for a Benjamin-Ono soliton trapped within a long sinusoidal wave. We show that the radiation losses experienced by a soliton and caused by the Earth rotation can be compensated by the energy pumping from the background wave. However, the back influence of the soliton on the background wave eventually leads to the destruction of the cohere…
The dynamics of perturbations to large-amplitude Internal Solitary Waves (ISW) in two-layered flows with thin interfaces is analyzed 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 …
We consider the dynamics of internal envelope solitons in a two-layer rotating fluid with a linearly varying bottom. It is shown that the most probable frequency of a carrier wave which constitutes the solitary wave is the frequency where the growth rate of modulation instability is maximal. An envelope solitary wave of this frequency can be described by the conventional nonlinear Schrodinger equation. A soliton solution to this equation is presented for the time-like version of the nonlinear Schrodinger equation. When such envelope soliton enters a coastal zone where the bottom gradually line…
We study the flow of water waves over bathymetry that varies periodically along one direction. We derive a linearized, homogenized model and show that the periodic bathymetry induces an effective dispersion, distinct from the dispersion inherently present in water waves. We relate this dispersion to the well-known effective dispersion introduced by changes in the bathymetry in non-rectangular channels. Numerical simulations using the (non-dispersive) shallow water equations reveal that a balance between this effective dispersion and nonlinearity can create solitary waves. We derive a KdV-type …
The motion of an interface separating two fluids under the effect of electric fields is a subject that has picked the attention of researchers from different areas. While there is an abundance of studies investigating the free surface wave properties, very few works have examined the associated velocity field within the bulk of the fluid. Therefore, in this paper, we investigate numerically the flow structure beneath solitary waves with constant vorticity on an inviscid conducting fluid bounded above by a dielectric gas under normal electric fields in the framework of a weakly nonlinear theory…
An unstructured-grid, nonhydrostatic, GVC ocean model Part I: Model description and application of vertical hybrid coordinates to internal solitary waves
This paper presents specific features of solitary wave dynamics within the framework of the Ostrovsky equation with variable coefficients in relation to surface and internal waves in a rotating ocean with a variable bottom topography. For solitary waves moving toward the beach, the terminal decay caused by the rotation effect can be suppressed by the shoaling effect. Two basic examples of a bottom profile are analyzed in detail and supported by direct numerical modelling. One of them is a constant-slope bottom and the other is a specific bottom profile providing a constant amplitude solitary w…
We study solitary wave solutions of the fifth-order Korteweg - de Vries equation which contains, besides the traditional quadratic nonlinearity and third-order dispersion, additional terms including cubic nonlinearity and fifth order linear dispersion, as well as two nonlinear dispersive terms. An exact solitary wave solution to this equation is derived and the dependence of its amplitude, width and speed on the parameters of the governing equation are studied. It is shown that the derived solution can represent either an embedded or regular soliton depending on the equation parameters. The no…
Evolution of weakly nonlinear and slowly varying Rossby waves in planetary atmospheres and oceans is considered within the quasi-geostrophic equation on unbounded domains. When the mean flow profile has a jump in the ambient potential vorticity, localized eigenmodes are trapped by the mean flow with a non-resonant speed of propagation. We address amplitude equations for these modes. Whereas the linear problem is suggestive of a two-dimensional Zakharov-Kuznetsov equation, we found that the dynamics of Rossby waves is effectively linear and moreover confined to zonal waveguides of the mean flow…
Formation and evolution of turbulence in convectively unstable internal solitary waves of depression shoaling over gentle slopes in the South China Sea
Cornell University（康奈尔大学）；University of Washington（华盛顿大学）；University of Waterloo（滑铁卢大学）；Wandering Wakhs Research（Wandering Wakhs 研究）
Strongly nonlinear models of internal wave propagation for incompressible stratified Euler fluids are investigated numerically and analytically to determine the evolution of a class of initial conditions of interest in laboratory experiments. This class of step-like initial data severely tests the robustness of the models beyond their strict long-wave asymptotic validity, and model fidelity is assessed by direct numerical simulations (DNS) of the parent Euler system. It is found that the primary dynamics of near-solitary wave formation is remarkably well predicted by the models for both wave a…
This paper presents a theoretical and experimental study of the long-standing fluid mechanics problem involving the temporal resolution of a large, localised initial disturbance into a sequence of solitary waves. This problem is of fundamental importance in a range of applications including tsunami and internal ocean wave modelling. This study is performed in the context of the viscous fluid conduit system-the driven, cylindrical, free interface between two miscible Stokes fluids with high viscosity contrast. Due to buoyancy induced nonlinear self-steepening balanced by stress induced interfac…
The article reflects the author's point of view on the disasters that occurred with American atomic submarines Thresher and Scorpion in 1960th and Indonesian diesel submarine Nanggala-420 in April 2021. A possible role of giant internal waves in such tragic events is discussed. Parameters of large-amplitude internal solitary waves, their shapes and speeds of propagation are presented. The author's reconstruction of the disasters is given.
Laboratory experimental results are presented for nonlinear Internal Solitary Waves (ISW) propagation in deep water configuration with miscible fluids. The results are validated against direct numerical simulations and traveling wave exact solutions where the effect of the diffused interface is taken into account. The waves are generated by means of a dam break and their evolution is recorded with Laser Induced Fluorescence (LIF) and Particle Image Velocimetry (PIV). In particular, data collected in a frame moving with the waves are presented here for the first time. Our results are representa…
We develop a weakly nonlinear model to study the spatiotemporal manifestation and the dynamical behavior of surface waves in the presence of an underlying interfacial solitary wave in a two-layer fluid system. We show that interfacial solitary-wave solutions of this model can capture the ubiquitous broadening of large-amplitude internal waves in the ocean. In addition, the model is capable of capturing three asymmetric behaviors of surface waves: (i) Surface waves become short in wavelength at the leading edge and long at the trailing edge of an underlying interfacial solitary wave. (ii) Surfa…
In this paper, using multiple scale analysis we derive a generalized mathematical model for amplitude evolution, and for calculating the energy exchange in resonant and near-resonant global triads consisting of weakly nonlinear internal gravity wave packets in weakly non-uniform density stratifications in an unbounded domain in the presence of viscous and rotational effects. Such triad interactions are one of the mechanisms by which high wavenumber internal waves lead to ocean turbulence and mixing via parametric subharmonic instability. Non-uniform stratification introduces detuning - mismatc…
We investigate experimentally stratified turbulence forced by waves. Stratified turbulence is present in oceans and it is expected to be dominated by nonlinear interaction of internal gravity waves as described by the Garrett & Munk spectrum. In order to reach turbulent regimes dominated by stratification we use the Coriolis facility in Grenoble (France) which large size enables us to reach regimes with both low Froude number and large Reynolds number. Stratification is obtained by using vertically linearly varying salt concentration and we force large scale waves in a $6\times6\times 1$ m$^3$…
Considered in this paper is a bi-directional model for the propagation of interfacial capillary-gravity waves in a two-layer system of fluids with rigid lid condition for the upper layer and lower layer with a much larger or infinite depth. The system is derived from a reformulation of the Euler equations for internal waves with nonnegligible surface tension effects in the interface and the corresponding asymptotic model under the Benjamin-Ono regime. Another unidirectional model, so-called regularized Benjamin equation, generalizing the Benjamin equation, is also introduced. Well-posedness of…
Finite-amplitude hydromagnetic Rossby waves in the magnetostrophic regime are studied. We consider the slow mode, which travels in the opposite direction to the hydrodynamic or fast mode, in the presence of a toroidal magnetic field and zonal flow by means of quasi-geostrophic models for thick spherical shells. The weakly-nonlinear, long waves are derived asymptotically using a reductive perturbation method. The problem at the first order is found to obey a second-order ODE, leading to a hypergeometric equation for a Malkus field and a confluent Heun equation for an electrical-wire field, and …
The interaction of localised solitary waves with large-scale, time-varying dispersive mean flows subject to nonconvex flux is studied in the framework of the modified Korteweg-de Vries (mKdV) equation, a canonical model for nonlinear internal gravity wave propagation in stratified fluids. The principal feature of the studied interaction is that both the solitary wave and the large-scale mean flow -- a rarefaction wave or a dispersive shock wave (undular bore) -- are described by the same dispersive hydrodynamic equation. A recent theoretical and experimental study of this new type of dynamic s…
We present the results of a laboratory experimental study of an internal wave field generated by harmonic, spatially-periodic boundary forcing from above of a density stratification comprising a strongly-stratified, thin upper layer sitting atop a weakly-stratified, deep lower layer. In linear regimes, the energy flux associated with relatively high frequency internal waves excited in the upper layer is prevented from entering the lower layer by virtue of evanescent decay of the wave field. In the experiments, however, we find that the development of parametric subharmonic instability (PSI) in…
This paper presents our investigation into the modification of a finite-width internal gravity wave beam arising from triadic resonance instability. We present both experimental and weakly non-linear modelling to examine this instability mechanism, in which a primary wave beam generates two secondary wave beams of lower frequencies and shorter length scales. Through a versatile experimental set-up, we examine how this instability develops over hundreds of buoyancy periods. Unlike predictions from previous zero-dimensional weakly non-linear theory, we find that the approach to a saturated equil…
A weakly-nonlinear potential theory is developed for the description of deep penetrating pressure fields caused by single and colliding wave groups of collinear waves due to the second-order nonlinear interactions. The result is applied to the representative case of groups with the sech-shape of envelope solitons in deep water. When solitary groups experience a head-on collision, the induced due to nonlinearity dynamic pressure may have magnitude comparable with the magnitude of the linear solution. It attenuates with depth with characteristic length of the group, which may greatly exceed the …
Chemical transport by weakly nonlinear internal gravity waves in stars
Scripps Institution of Oceanography, University of California, San Diego（加州大学圣地亚哥分校斯克里普斯海洋研究所）；Center for Computational Astrophysics, Flatiron Institute（纽约州立大学平顿研究所计算天体物理中心）；Department of Engineering Sciences and Applied Mathematics Northwestern University（西北大学工程科学与应用数学系）；Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA), Northwestern University（西北大学跨学科天体物理探索与研究中心）
