Three-dimensional Water Waves Research Articles

Rigidity of three-dimensional steady internal waves with constant vorticity governed by the [formula omitted]-plane approximation

This paper investigates the structural implications of constant vorticity for steady three-dimensional internal water waves governed by the equatorial f-plane approximation. Under the assumption that the first components of the non-zero constant vorticity vectors in the upper and lower layers are zeros, it is proved that every three-dimensional traveling internal wave with bounded velocity is either two-dimensional or a trivially three-dimensional shear flow with the pressures in both layers are identically equal.

Liouville-type results for time-dependent stratified water flows over variable bottom in the β-plane approximation

We consider here time-dependent three-dimensional stratified geophysical water flows of finite depth over a variable bottom with a free surface and an interface (separating two layers of constant and different densities). Under the assumption that the vorticity vectors in the two layers are constant, we prove that bounded solutions to the three-dimensional water waves equations in the β-plane approximation exist if and only if one of the horizontal components of the velocity, as well as its vertical component, are zero; the other horizontal component being constant. Moreover, the interface is flat, the free surface has a traveling character in the horizontal direction of the nonvanishing velocity component, being of general type in the other horizontal direction, and the pressure is hydrostatic in both layers. Unlike previous studies of three-dimensional flows with constant vorticity in each layer, we consider a non-flat bottom boundary and different constant vorticity vectors for the upper and lower layer.

On some recently derived exact solutions to the Euler equations

We present some explicit solutions with radial character in the setting of Eulerian framework to the three-dimensional nonlinear water waves in arctic regions. In particular, we focus on the motion of water waves with a moving bed and the pressure varying on free surface.

Liouville-type results for the time-dependent three-dimensional (inviscid and viscous) water wave problem with an interface

Under consideration here are time-dependent three-dimensional stratified water flows of finite depth with a free surface and an interface (playing the role of an internal wave and separating two layers of constant and different densities). The main outcome of the paper is that, under the assumption that the vorticity vectors in the two layers are constant, non-vanishing and parallel to each other, we prove that bounded solutions to the three-dimensional equations are essentially two-dimensional. More precisely, the free surface, the interface, the pressure and the velocity field present no variations in the direction orthogonal to the direction of motion. In addition, one of the horizontal components of the velocity field is constant throughout the flow and the vorticity also points in the direction orthogonal to the wave propagation direction. The rigid lid case is analyzed from the perspective of relaxed assumptions concerning the vorticity vectors. We also consider the viscous water wave problem case with normal and tangential stress conditions at the free surface and find that under similar conditions on the vorticity vectors, the conclusion about the two-dimensionality remains in place. Along the way, we also provide an explicit solution for the time-dependent three-dimensional inviscid water wave problem with an interface.

An exact DtN artificial boundary condition for motion analysis of water wave with undulated seabed

This paper is concerned with three-dimensional water wave motion over undulated seabed. For the wave motion problem in seawater of finite depth, the fluid domain is usually unbounded horizontally but bounded vertically by free water surface and seabed which have their own boundary conditions. To study local wave motion over undulated region only, a novel Dirichlet-to-Neumann (DtN) boundary condition is suggested on an artificial circular cylindrical surface by which the fluid domain is divided into an interior region with finite volume and an exterior region. A unique solution of wave motion in the interior region is guaranteed by the novel boundary condition on the artificial cylindrical surface and the other boundary conditions on mean free water surface and undulated seabed. The solution is obtained by using Boundary Integral Equation (BIE) with all boundary conditions including the novel artificial boundary condition. Upon verification for a submerged square cylinder, the present model is extended to the case of an array of rounded-rectangular cylinders or circular paraboloidal shoals.

Scattering of surface waves by a vertical truncated structured cylinder.

This paper describes the solution to the problem of scattering of plane incident waves on water of constant depth by a bottom mounted circular cylinder, extending partially through the depth, which has an internal structure comprised of closely spaced thin vertical barriers between which fluid is allowed to flow. The problem is solved under full depth-dependent linearized water wave theory using an effective medium equation to describe the fluid motion in cylinder and effective boundary conditions to match that flow to the fluid region outside the cylinder. The interest in this problem lies in the development of novel solution methods for fully three-dimensional water wave interaction with bathymetric plate arrays. Results computed using this theory are compared with a shallow water approximation based on the recent work of Marangos & Porter (2021 Shallow water theory for structured bathymetry. Proc. R. Soc. A 477, 20210421.) and with accurate computations of an exact representation of the geometry using a discrete set of plates. Other results highlight the resonant directional lensing effects of this type of cylindrical plate array device.

On three-dimensional free surface water flows with constant vorticity

<p style='text-indent:20px;'>We present a survey of recent results on gravity water flows satisfying the three-dimensional water wave problem with constant (non-vanishing) vorticity vector. The main focus is to show that a gravity water flow with constant non-vanishing vorticity has a two-dimensional character in spite of satisfying the three-dimensional water wave equations. More precisely, the flow does not change in one of the two horizontal directions. Passing to a rotating frame, and introducing thus geophysical effects (in the form of Coriolis acceleration) into the governing equations, the two-dimensional character of the flow remains in place. However, the two-dimensionality of the flow manifests now in a horizontal plane. Adding also centripetal terms into the equations further simplifies the flow (under the assumption of constant vorticity vector): the velocity field vanishes, but, however, the pressure function is a quadratic polynomial in the horizontal and vertical variables, and, surprisingly, the surface is non-flat.</p>

On flow simplification occurring in viscous three-dimensional water flows with constant non-vanishing vorticity

We show that a viscous steady water flow of constant non-vanishing vorticity situated below a free surface (assumed to have the shape of a two-dimensional time dependent graph) and above a flat bottom has to be two-dimensional; that is, while satisfying the viscous three-dimensional water wave equations, it turns out that the free surface, the velocity field and the pressure present no variation in one of the horizontal directions. Moreover, the vorticity must have only one non-zero component which points in the horizontal direction orthogonal to the direction of the surface wave propagation. Compared with previous works, our study here has the advantage that it considers the Navier–Stokes equations and also takes into account the normal and tangential stress boundary conditions.

On a two dimensional nonlocal shallow-water model

In the present study we describe the asymptotic perturbation method to derive a two-dimensional nonlocal shallow-water model equation in the context of full water waves. Starting from the incompressible and irrotational governing equations in the three-dimensional water waves, we show that such a equation arises in the modeling of the propagation of shallow water waves over a flat bed. The resulting equation is a two dimensional Camassa-Holm equation-type with weakly transverse effect for the horizontal velocity component. The equation captures stronger nonlinear effects than the classical dispersive integrable equations like the Korteweg-de Vries and Kadomtsev-Petviashvili equations. We also address some properties of this model equation and how it relates to the surface wave. Analytically, we establish the local well-posedness of this model in a suitable Sobolev space. We then investigate formation of singularities and existence of traveling-wave solutions to this quasi-linear model equation with an emphasis on the understanding of weak transverse effect. Finally, we provide a Liouville-type property to obtain unique continuation result for the strong solution.

Some Explicit Solutions to the Three-Dimensional Nonlinear Water Wave Problem

We present some explicit solutions (given in Eulerian coordinates) to the three-dimensional nonlinear water wave problem. The velocity field of some of the solutions exhibits a non-constant vorticity vector. An added bonus of the solutions we find is the possibility of incorporating a variable (in time and space) surface pressure which has a radial structure. A special type of radial structure of the surface pressure (of exponential type) is one of the features displayed by hurricanes, cf. Overland (Earle, Malahoff (eds) Overland in ocean wave climate, Plenum Pub. Corp., New York, 1979).

Geophysical water flows with constant vorticity and centripetal terms

We consider here three-dimensional water flows governed by the geophysical water wave equations exhibiting full Coriolis and centripetal terms. More precisely, assuming a constant vorticity vector, we derive a family of explicit solutions, in Eulerian coordinates, to the above-mentioned equations and their boundary conditions. These solutions are the only ones under the assumption of constant vorticity. To be more specific, we show that the components of the velocity field (with respect to the rotating coordinate system) vanish. We also determine a formula for the pressure function and we show that the surface, denoted z=eta (x,y,t), is time independent, but is not flat and can be explicitly determined. We conclude our analysis by converting to the fixed inertial frame, the solutions we obtained before in the rotating frame. It is established that, in the fixed frame, the velocity field is non-vanishing and the free surface is non-flat, being explicitly determined. Moreover, the system consisting of the velocity field, the pressure and the surface defining function represents explicit and exact solutions to the three-dimensional water waves equations and their boundary conditions.

On reference solutions for ship waves

The classic problem of ship waves of infinitely small amplitudes is studied within the recently developed approach of reference solutions [V. G. Gnevyshev and S. I. Badulin, Moscow Univ. Phys. Bull. 72, 415 (2017)]. The evolution of narrow-banded Gaussian wavetrains of a finite volume is considered as an alternative to the conventional inherently point-wise tracking of wavetrains within the asymptotic methods of the stationary phase or the steepest descent. The approach allows for avoiding general problems of the methods: occurrence of singularities of wave fields. The non-singular solutions for a stationary ship wake are presented in an analytical form in terms of two key dimensionless quantities: the Froude number and the aspect ratio of the initial (boundary) domain of wave generation. Systems of transverse and diverging ship waves as well as amplitude and phase effects of their transformation can be analyzed separately within this approach for small Froude numbers. The essential role of dispersion of three-dimensional water waves is emphasized and detailed.

One-Dimensional Symmetry for the Solutions of a Three-Dimensional Water Wave Problem

We prove a one-dimensional symmetry result for a weighted Dirichlet-to-Neumann problem arising in a model for water waves in dimensions 2 and 3. More precisely, we prove that stable solutions in dimension 2 and minimizers and monotone solutions in dimension 3 depend on only one Euclidean variable. Monotone solutions in the 2-dimensional case without weights were studied in de la Llave and Valdinoci (Math Res Lett 16(5):909–918, 2009). In our paper, a crucial ingredient in the proof is given by an energy estimate for minimizers obtained via a comparison argument.

A variational principle for three-dimensional water waves over Beltrami flows

We consider steady three-dimensional gravity–capillary water waves with vorticity propagating on water of finite depth. We prove a variational principle for doubly periodic waves with relative velocities given by Beltrami vector fields, under general assumptions on the wave profile.

A Variational Reduction and the Existence of a Fully Localised Solitary Wave for the Three-Dimensional Water-Wave Problem with Weak Surface Tension

Fully localised solitary waves are travelling-wave solutions of the three-dimensional gravity-capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence has been predicted on the basis of numerical simulations and model equations (in which context they are usually referred to as `lumps'), and a mathematically rigorous existence theory for strong surface tension (Bond number $\beta$ greater than $\frac{1}{3}$) has recently been given. In this article we present an existence theory for the physically more realistic case $0<\beta<\frac{1}{3}$. A classical variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle featuring a perturbation of the functional associated with the Davey-Stewartson equation. A nontrivial critical point of the reduced functional is found by minimising it over its natural constraint set.

Model Equations for Three-Dimensional Nonlinear Water Waves under Tangential Electric Field

We are concerned with gravity-capillary waves propagating on the surface of a three-dimensional electrified liquid sheet under a uniform electric field parallel to the undisturbed free surface. For simplicity, we make an assumption that the permittivity of the fluid is much larger than that of the upper-layer gas; hence, this two-layer problem is reduced to be a one-layer problem. In this paper, we propose model equations in the shallow-water regime based on the analysis of the Dirichlet-Neumann operator. The modified Benney-Luke equation and Kadomtsev-Petviashvili equation will be derived, and the truly three-dimensional fully localized traveling waves, which are known as “lumps” in the literature, are numerically computed in the Benney-Luke equation.

Scattering of oblique waves by permeable vertical flexible membrane wave barriers

The interaction of obliquely incident surface gravity waves with a vertical flexible permeable submerged membrane wave barrier is investigated in the context of three-dimensional linear water wave theory. From the general formulation of the submerged membrane barrier, the performance of bottom-standing, surface-piercing and fully extended membrane wave barriers are analyzed for various values of wave and structural parameters. The analytic solution of the physical problem is obtained using eigenfunction expansion method and a coupled boundary element-finite difference method has been used to get the numerical solution. In the boundary element method, since the boundary condition on the membrane barrier is not known a priori, the membrane response and velocity potentials are solved simultaneously using appropriate discretization with the help of finite difference scheme. The convergence of the analytic and numerical solution techniques is discussed. The study reveals that for suitable combination of wave and structural parameters, approximately (45–50)% incident wave energy can be dissipated irrespective of membrane barrier configurations. Further, in certain situations, nearly full wave reflection and zero transmission occur for all barrier configurations. The study will be useful in the design of flexible permeable membrane to act as an effective wave barrier for creation of tranquility zone in the marine environment.

A simple criterion for transverse linear instability of nonlinear waves

We prove a simple criterion for transverse linear instability of nonlinear waves for partial differential equations in a spatial domain Ω×R⊂Rn×R. For stationary solutions depending upon x∈Ω only, the question of transverse (in)stability is concerned with their (in)stability with respect to perturbations depending upon (x,y)∈Ω×R. Starting with a formulation of the PDE as a dynamical system in the transverse direction y, we give sufficient conditions for transverse linear instability. We apply the general result to the Davey–Stewartson equations, which arise as modulation equations for three-dimensional water waves.

Detection and analysis of coherent groups in three-dimensional fully-nonlinear potential wave fields

We investigate the emergence of coherent groups in three-dimensional fully-nonlinear potential deep water waves whose initial spectrum is assumed to be of the JONSWAP type with directional distribution given by cos nθ, where n is the integer varying from 1 to 16. The analysis is based on the results of long-term wave simulations performed using a numerical solution of a three-dimensional Laplace equation for the velocity potential subject to nonlinear kinematic and dynamic boundary conditions at the free surface. The main characteristics of wave groups such as their average velocity, maximum group wave height, lifetime and length are analysed. The statistics of extreme waves occurring in the detected groups are discussed. Spatial and temporal scale characteristics of wave groups are compared to the previous results.

Diffraction of oblique water waves by small uneven channel-bed in a two-layer fluid

Obliquely incident water wave scattering by an uneven channel-bed in the form of a small bottom undulation in a two-layer fluid is investigated within the frame work of three-dimensional linear water wave theory. The upper fluid is assumed to be bounded above by a rigid lid, while the lower one is bounded below by a bottom surface having a small deformation and the channel is unbounded in the horizontal directions. Assuming irrotational motion, perturbation technique is employed to calculate the first-order corrections to the velocity potentials in the two fluids by using Fourier transform approximately, and also to calculate the reflection and transmission coefficients in terms of integrals involving the shape function representing the bottom deformation. Consideration of a patch of sinusoidal ripples shows that the reflection coefficient is an oscillatory function of the ratio of twice the component of the wave number along x-axis and the ripple wave number. When this ratio approaches one, the theory predicts a resonant interaction between the bed and interface, and the reflection coefficient becomes a multiple of the number of ripples. High reflection of incident wave energy occurs if this number is large.