A Hamiltonian model for the propagation of internal water waves interacting with surface waves, a current and an uneven bottom is examined. Using the so-called Dirichlet-Neumann operators, the water wave system is expressed in the Hamiltonian form, and thus the motions of the internal waves and surface waves are determined by the Hamiltonian formulation. Choosing an appropriate scaling of the variables and employing the Hamiltonian perturbation theory from Hamiltonian formulation of the dynamics, we derive a KdV-type equation with variable coefficients depending on the bottom topography to describe the internal waves.
We derived consistently, according to the second order perturbation approach, the extended KdV equation for an uneven bottom for the case of $α=O(β)$ and $δ=O(β^2)$. This equation can be obtained only when the bottom is given by a piecewise linear function. For the case of $α=O(β)$ and $δ=O(β)$, a unidirectional wave equation is derived in the first order approach with the same limitation for the bottom profile.
The propagation of surface water waves interacting with a current and an uneven bottom is studied. Such a situation is typical for ocean waves where the winds generate currents in the top layer of the ocean. The role of the bottom topography is taken into account since it also influences the local wave and current patterns. Specific scaling of the variables is selected which leads to approximations of Boussinesq and KdV types. The arising KdV equation with variable coefficients, dependent on the bottom topography, is studied numerically when the initial condition is in the form of the one soli…
In this study, we give a survey of derivations of KdV-type equations with an uneven bottom for several cases when small (perturbation) parameters $α, β, δ$ are of different orders. Six different cases of such ordering are discussed. Surprisingly, for all these cases the Boussinesq equations can be made compatible only for the particular piecewise linear bottom profiles, and the correction function has a universal form. For such bottom relief, several new KdV-type wave equations are derived. These equations generalize the KdV, the extended KdV (KdV2), the fifth-order KdV (KdV5) and the Gardner …
The paper's main goal is to compare the motion of solitary surface waves resulting from two similar but slightly different approaches. In the first approach, the numerical evolution of soliton surface waves moving over the uneven bottom is obtained using single wave equations. In the second approach, the numerical evolution of the same initial conditions is obtained by the solution of a coupled set of the Boussinesq equations for the same Euler equations system. We discuss four physically relevant cases of relationships between small parameters $α,β,δ$. For the flat bottom, these cases imply t…
We study the evolution of nonlinear surface gravity water-wave packets developing from modulational instability over an uneven bottom. A nonlinear Schrödinger equation (NLSE) with coefficients varying in space along propagation is used as a reference model. Based on a low-dimensional approximation obtained by considering only three complex harmonic modes, we discuss how to stabilize a one-dimensional pattern in the form of train of large peaks sitting on a background and propagating over a significant distance. Our approach is based on a gradual depth variation, while its conceptual framework …
The regularisation of nonlinear hyperbolic conservation laws has been a problem of great importance for achieving uniqueness of weak solutions and also for accurate numerical simulations. In a recent work, the first two authors proposed a so-called Hamiltonian regularisation for nonlinear shallow water and isentropic Euler equations. The characteristic property of this method is that the regularisation of solutions is achieved without adding any artificial dissipation or ispersion. The regularised system possesses a Hamiltonian structure and, thus, formally preserves the corresponding energy f…
In this note we look at the influence of a shallow, uneven riverbed on a soliton. The idea consists in approximate transformation of the equation governing wave motion over uneven bottom to equation for flat one for which the exact solution exists. The calculation is one space dimensional, and so corresponding to long trenches or banks under wide rivers or else oceans.
We study the effects of an uneven bottom on the internal wave propagation in the presence of stratification and underlying non-uniform currents. Thus, the presented models incorporate vorticity (wave-current interactions), geophysical effects (Coriolis force) and a variable bathymetry. An example of the physical situation described above is well illustrated by the equatorial internal waves in the presence of the Equatorial Undercurrent (EUC). We find that the interface (physically coinciding with the thermocline and the pycnocline) satisfies in the long wave approximation a KdV-mKdV type equat…
The authors of the paper "The third-order perturbed Korteweg-de Vries equation for shallow water waves with a non-flat bottom" [1] claim that they have derived the full third order perturbed KdV equation for the case of uneven bottom. We show that the authors' derivation is not consistent due to the fact that they took into account only some of the third order corrections but not all of them. Moreover, we show that a consistent third order perturbed Korteweg-de Vries equation for shallow water waves with a non-flat bottom cannot be derived for a general form of bottom function.
A two-dimensional water wave system is examined consisting of two discrete incompressible fluid domains separated by a free common interface. In a geophysical context this is a model of an internal wave, formed at a pycnocline or thermocline in the ocean. The system is considered as being bounded at the bottom and top by a flatbed and wave-free surface respectively. A current profile with depth-dependent currents in each domain is considered. The Hamiltonian of the system is determined and expressed in terms of canonical wave-related variables. Limiting behaviour is examined and compared to th…
Convection over a wavy heated bottom wall in the air flow has been studied in experiments with the Rayleigh number $\sim 10^8$. It is shown that the mean temperature gradient in the flow core inside a large-scale circulation is directed upward, that corresponds to the stably stratified flow. In the experiments with a wavy heated bottom wall, we detect large-scale standing internal gravity waves excited in the regions with the stably stratified flow. The wavelength and the period of these waves are much larger than the turbulent spatial and time scales, respectively. In particular, the frequenc…
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.
In this paper, we study Bragg resonance, i.e. the triad interaction between surface and/or interfacial waves with bottom ripple, in presence of background velocity. We show that when one of the constituent waves of the triad has negative energy, the amplitudes of all the waves grow exponentially. This is very different from classic Bragg resonance in which one wave decays to cause growth of the other. The instabilities we observe are `explosive', and are different from normal mode shear instabilities since our velocity profiles are linearly stable. Our work may explain the existence of large a…
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…
The KdV equation can be derived in the shallow water limit of the Euler equations. Over the last few decades, this equation has been extended to include both higher order effects (KdV2) and an uneven river bottom. Although this equation is not integrable and has only one conservation law, exact periodic and solitonic solutions exist for the even bottom case. The method used to find them assumes the same function forms as for KdV solutions. KdV2 equation imposes more constraints on parameters of solutions. For soliton case KdV2 solution occurs for particular ratio of wave amplitude to water dep…
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 …
Free surface, axially symmetric shallow flow is analysed in both the centrifugal and centripetal directions. Referring to the inviscid steady flow over a flat plate characterised by a unique value of specific energy, the analytical sub- and supercritical solutions are determined. Furthermore, the sub- and supercritical steady solutions for the flow with friction over a flat plate are determined, provided that inertial terms are important compared with frictional terms. As the inviscid case, the sub- and supercritical solutions over a bottom topography are determined, provided that the bottom u…
We study experimentally the propagation of internal waves in two different three-dimensional (3D) geometries, with a special emphasis on the refractive focusing due to the 3D reflection of obliquely incident internal waves on a slope. Both studies are initiated by ray tracing calculations to determine the appropriate experimental parameters. First, we consider a 3D geometry, the classical set-up to get simple, 2D parallelogram-shaped attractors in which waves are forced in a direction perpendicular to a sloping bottom. Here, however, the forcing is of reduced extent in along-slope, transverse …
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…
Coupled mixed convective and stratified systems are common in natural flows. To study experimentally the associated dynamics, we use a singular property of water: its non-linear equation of state is characterised by a maximum density close to $4^{\circ}$C. By heating the top of a tank at $35^{\circ}$C and cooling the bottom at $0^{\circ}$C, a two-layer configuration spontaneously appears. The convective motion in the bottom layer consists mostly of a large-scale circulation and rising cold plumes. This turbulent flow generates propagating internal gravity waves in the upper stably-stratified l…
Interfacial internal waves in a stratified fluid excited by periodic free-surface perturbations in a closed tank are studied experimentally. Barotropic-baroclinic energy conversion is induced by the presence of a bottom obstacle. The connection between horizontal surface velocities and internal wave amplitudes is investigated, the developing flow patterns are described qualitatively, and the wave speeds of internal waves are systematically analyzed and compared to linear 2- and 3-layer theories. We find that, despite of the fact that the observed internal waves can have considerable amplitudes…
We report on a numerical study of turbulent convection driven by a combination of internal heat sources and sinks. Motivated by a recent experimental realisation (Lepot et al. 2018), we focus on the situation where the cooling is uniform, while the internal heating is localised near the bottom boundary, over approximately one tenth of the domain height. We obtain scaling laws $Nu \sim Ra^γPr^χ$ for the heat transfer as measured by the Nusselt number $Nu$ expressed as a function of the Rayleigh number $Ra$ and the Prandtl number $Pr$. After confirming the experimental value $γ\approx 1/2$ for t…
A fully nonlinear, three-dimensional numerical model is developed for the simulation of tidal flow over arbitrary bottom topography in an ocean with realistic stratification. The model is capable of simulating accurately the generation of fine-scale internal wave tidal beams, their interaction with an ocean thermocline and the subsequent generation of solitary internal waves that propagate on this thermocline. Several preliminary simulation results are shown for uniform and non-uniform flow over an idealized two-dimensional ridge, which are compared with linear theory, and for flow over an ide…
When shallow water flows over uneven bathymetry, the water surface is modulated. This type of problem has been revisited numerous times since it was first studied by Lord Kelvin in 1886. Our study analytically examines currents whose unperturbed velocity profile $U(z)$ follows a power-law $z^q$, flowing over a three-dimensional uneven bed. This particular form of $U$, which can model a miscellany of realistic flows, allows explicit analytical solutions. Arbitrary bed shapes can readily be imposed via Fourier's theorem provided their steepness is moderate. Three-dimensional vorticity-bathymetry…
Quantifying the physical mechanisms responsible for the transport of sediments, nutrients and pollutants in the abyssal sea is a long-standing problem, with internal waves regularly invoked as the relevant mechanism for particle advection near the sea bottom. This study focuses on internal-wave induced particle transport in the vicinity of (almost) vertical walls. We report a series of laboratory experiments revealing that particles sinking slowly through a monochromatic internal wave beam experience significant horizontal advection. Extending the theoretical analysis by Beckebanze et al (2018…
Boundary layer turbulence in coastal regions differs from that in deep ocean because of bottom interactions. In this paper, we focus on the merging of surface and bottom boundary layers in a finite-depth coastal ocean by numerically solving the wave-averaged equations using a large eddy simulation method. The ocean fluid is driven by combined effects of wind stress, surface wave, and a steady current in the presence of stable vertical stratification. The resulting flow consists of two overlapping boundary layers, i.e. surface and bottom boundary layers, separated by an interior stratification.…
The internal dynamics during the coalescence of a sessile droplet and a subsequently deposited impacting droplet, with either identical or distinct surface tension, is studied experimentally in the regime where surface tension is dominant. Two color high-speed cameras are used to capture the rapid internal flows and associated mixing from both side and bottom views simultaneously by adding an inert dye to the impacting droplet. Given sufficient lateral separation between droplets of identical surface tension, a robust surface jet is identified on top of the coalesced droplet. Image processing …
In this article, we propose an inverse technique that accurately reconstructs the ocean's density stratification profile simply from free surface elevation data. Satellite observations suggest that ocean surface contains the signature of internal tides, which are internal gravity waves generated by the barotropic tides. Since internal tides contain the information of ocean's density stratification, the latter can in principle be reconstructed from the free surface signature. We consider a simple theoretical model that approximates a continuously stratified ocean as discrete layers of constant …
An unstructured-grid, nonhydrostatic, GVC ocean model Part I: Model description and application of vertical hybrid coordinates to internal solitary waves
