Gravity Water Waves Research Articles

Quantum Mechanical and Optical Analogies in Surface Gravity Water Waves

We present the theoretical models and review the most recent results of a class of experiments in the field of surface gravity waves. These experiments serve as demonstration of an analogy to a broad variety of phenomena in optics and quantum mechanics. In particular, experiments involving Airy water-wave packets were carried out. The Airy wave packets have attracted tremendous attention in optics and quantum mechanics owing to their unique properties, spanning from an ability to propagate along parabolic trajectories without spreading, and to accumulating a phase that scales with the cubic power of time. Non-dispersive Cosine-Gauss wave packets and self-similar Hermite-Gauss wave packets, also well known in the field of optics and quantum mechanics, were recently studied using surface gravity waves as well. These wave packets demonstrated self-healing properties in water wave pulses as well, preserving their width despite being dispersive. Finally, this new approach also allows to observe diffractive focusing from a temporal slit with finite width.

Time-Reversal Analogy by Nonlinear Acoustic–Gravity Wave Triad Resonance

Time reversal of free-surface water (gravity) waves due to a sudden change in the effective gravity has been extensively studied in recent years. Here, we show that an analogy to time-reversal can be obtained using nonlinear acoustic-gravity wave theory. More specifically, we present a mathematical model for the evolution of a time-reversed gravity wave packet from a nonlinear resonant triad perspective. We show that the sudden appearance of an acoustic mode in analogy to a sudden vertical oscillation of the liquid film, can resonate effectively with the original gravity wave packet causing energy pumping into an oppositely propagating (time-reversed) surface gravity wave of an almost identical shape.

The NLS approximation for two dimensional deep gravity waves

This article is concerned with infinite depth gravity water waves in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. Our goal is to study this problem with small wave packet data, and to show that this is well approximated by the cubic nonlinear Schrodinger equation (NLS) on the natural cubic time scale.

Amplitude and Phase of Wave Packets in a Linear Potential.

We theoretically study and successfully observe the evolution of Gaussian and Airy surface gravity water wave packets propagating in an effective linear potential. This potential results from a homogeneous and time-dependent flow created by a computer-controlled water pump. For both wave packets we measure the amplitudes and the cubic phases appearing due to the linear potential. Furthermore, we demonstrate that the self-acceleration of the Airy surface gravity water wave packets can be completely canceled by a linear potential.

Wellposedness of the 2D full water wave equation in a regime that allows for non- $$C^1$$ C 1 interfaces

We consider the two dimensional gravity water wave equation in a regime where the free interface is allowed to be non-$C^1$. In this regime, only a degenerate Taylor inequality $-\frac{\partial P}{\partial \bf n}\ge 0$ holds, with degeneracy at the singularities. In \cite{kw} an energy functional $\mathcal E(t)$ was constructed and an a-prori estimate was proved. The energy functional $\mathcal E(t)$ is not only finite for interfaces and velocities in Sobolev spaces, but also finite for a class of non-$C^1$ interfaces with angled crests. In this paper we prove the existence, uniqueness and stability of the solution of the 2d gravity water wave equation in the class where $\mathcal E(t)<\infty$, locally in time, for any given data satisfying $\mathcal E(0)<\infty$.

Surface Waves Enhance Particle Dispersion

We study the horizontal dispersion of passive tracer particles on the free surface of gravity waves in deep water. For random linear waves with the JONSWAP spectrum, the Lagrangian particle trajectories are computed using an exact nonlinear model known as the John–Sclavounos equation. We show that the single-particle dispersion exhibits an unusual super-diffusive behavior. In particular, for large times t, the variance of the tracer ⟨ | X ( t ) | 2 ⟩ increases as a quadratic function of time, i.e., ⟨ | X ( t ) | 2 ⟩ ∼ t 2 . This dispersion is markedly faster than Taylor’s single-particle dispersion theory which predicts that the variance of passive tracers grows linearly with time for large t. Our results imply that the wave motion significantly enhances the dispersion of fluid particles. We show that this super-diffusive behavior is a result of the long-term correlation of the Lagrangian velocities of fluid parcels on the free surface.

Low Regularity Solutions for Gravity Water Waves

We prove local well-posedness for the gravity water waves equations without surface tension, with initial velocity field in $H^s$, $s > \frac{d}{2} + 1 - \mu$, where $\mu = \frac{1}{10}$ in the case $d = 1$ and $\mu = \frac{1}{5}$ in the case $d \geq 2$, extending previous results of Alazard-Burq-Zuily. The improvement primarily arises in two areas. First, we perform an improved analysis of the regularity of the change of variables from Eulerian to Lagrangian coordinates. Second, we perform a time-interval length optimization of the localized Strichartz estimates.

Waves and Structures in the Boussinesq Equations

The classical Boussinesq equation describing gravity waves in shallow waters is under consideration. Hirota’s bilinear representation is used to construct exact solutions describing wave packets, waves on solitons, and “dancing” waves. The principle of multiplying the solutions of the Hirota equation is formulated, which helps constructing more complex structures made of solitons, wave packets, and other types of waves.

Global solution for the 3D gravity water waves system above a flat bottom

Given any suitably small, localized, and smooth initial data, in this paper, we prove global regularity for the 3D finite depth gravity water wave system. As a byproduct, we rule out the small, localized traveling waves in 3D, which do exist for the same system in 2D.

Two-dimensional gravity water waves with constant vorticity, I: Cubic lifespan

This article is concerned with the incompressible, infinite depth water wave equation in two space dimensions, with gravity and constant vorticity but with no surface tension. We consider this problem expressed in position-velocity potential holomorphic coordinates, and prove local well-posedness for large data, as well as cubic lifespan bounds for small data solutions.

Sufficiently strong dispersion removes ill-posedness in truncated series models of water waves

Truncated series models of gravity water waves are popular for use in simulation. Recent work has shown that these models need not inherit the well-posedness properties of the full equations of motion (the irrotational, incompressible Euler equations). We show that if one adds a sufficiently strong dispersive term to a quadratic truncated series model, the system then has a well-posed initial value problem. Such dispersion can be relevant in certain physical contexts, such as in the case of a bending force present at the free surface, as in a hydroelastic sheet.

PREDICTING THE BREAKING STRENGTH OF GRAVITY WATER WAVES FROM DEEP TO SHALLOW WATER

We revisit the classical but as yet unresolved problem of predicting the breaking strength of 2-D and 3-D gravity water waves.Our goal is to find a robust and local parameterization to predict the breaking strength of 2-D and 3-D gravity water waves. We use a LES/VOF model described by Derakhti &amp; Kirby (2014) to simulate nonlinear wave evolution, breaking onset and post-breaking behavior for representative cases of focused wave packets or modulated wave trains. Using these numerical results, we investigate the relationship between the breaking strength parameter b and the breaking onset parameter B proposed by Barthelemy et al. (2018). While the results are potentially applicable more generally, in this paper we concentrate on breaking events due to focusing or modulational instability in wave packets over flat bottom topography and for conditions ranging from deep to intermediate depth, with depth to wavelength ratios ranging from 0.68 to 0.13.

On the Motion of a Compressible Gravity Water Wave with Vorticity

We prove a priori estimates for the compressible Euler equations modeling the motion of a liquid with moving physical vacuum boundary with unbounded initial domain. The liquid is under influence of gravity but without surface tension. Our fluid is not assumed to be irrotational. But the physical sign condition needs to be assumed on the free boundary. We generalize the method used in Lindblad and Luo (Commun Pure Appl Math, 2008) to prove the energy estimates in an unbounded domain up to arbitrary order. In addition to that, the a priori energy estimates are in fact uniform in the sound speed $$\kappa $$ . As a consequence, we obtain the convergence of solutions of compressible Euler equations with a free boundary to solutions of the incompressible equations, generalizing the result of Lindblad and Luo (2008) to when you have an unbounded domain. On the other hand, we prove that there are initial data satisfying the compatibility condition in some weighted Sobolev spaces, and this will propagate within a short time interval, which is essential for proving long time existence for slightly compressible irrotational water waves.

Coexistence of solitons and extreme events in deep water surface waves

We study experimentally, in a large-scale basin, the propagation of unidirectional deep water gravity waves stochastically modulated in phase. We observe the emergence of nonlinear localized structures that evolve on a stochastic wave background. Such a coexistence is expected by the integrable turbulence theory for the nonlinear Schr{\"o}dinger equation (NLSE), and we report the first experimental observation in the context of hydrodynamic waves. We characterize the formation, the properties and the dynamics of these nonlinear coherent structures (solitons and extreme events) within the incoherent wave background. The extreme events result from the strong steepening of wave train fronts, and their emergence occurs after roughly one nonlinear length scale of propagation (estimated from NLSE). Solitons arise when nonlinearity and dispersion are weak, and of the same order of magnitude as expected from NLSE. We characterize the statistical properties of this state. The number of solitons and extreme events is found to increase all along the propagation, the wave-field distribution has a heavy tail, and the surface elevation spectrum is found to scale as a frequency power-law with an exponent --4.5 $\pm$ 0.5. Most of these observations are compatible with the integrable turbulence theory for NLSE although some deviations (e.g. power-law spectrum, asymmetrical extreme events) result from effects proper to hydrodynamic waves.

Sobolev estimates for two dimensional gravity water waves

Sobolev estimates for two dimensional gravity water waves

Resonant interactions of rotational water waves in the equatorial f-plane approximation

We investigate here the resonance phenomenon in periodic unidirectional water waves in flows of constant vorticity governed by the equatorial f-plane approximation. The relevance of such water waves displaying a one dimensional wave vector is also underlined in the paper—in the context of equatorial capillary-gravity water waves—and serves as the basis for the resonance analysis which is carried out by means of dispersion relations for equatorial water waves that were quite recently derived [see the work of Constantin, Differ. Integr. Equations 26(3-4), 237–252 (2013) and Martin, Nonlinear Anal.: Theory, Methods Appl. 96, 1–17 (2014)]. We show that, while gravity water waves do not exhibit three-wave resonance, the four-wave resonance occurs irrespective of the vorticity.

Spectrum of shallow water gravity waves generated by confined two-dimensional turbulence

Two-dimensional turbulence can generate shallow water waves. These waves carry information about the structure of their turbulent source. A theoretical calculation of the wave spectrum is proposed, which may lead to an insightful way to characterize turbulence.

Strichartz Estimates and the Cauchy Problem for the Gravity Water Waves Equations

This paper is devoted to the proof of a well-posedness result for the gravity water waves equations, in arbitrary dimension and in fluid domains with general bottoms, when the initial velocity field is not necessarily Lipschitz. Moreover, for two-dimensional waves, we can consider solutions such that the curvature of the initial free surface does not belong to $L^2$. The proof is entirely based on the Eulerian formulation of the water waves equations, using microlocal analysis to obtain sharp Sobolev and H\older estimates. We first prove tame estimates in Sobolev spaces depending linearly on H\older norms and then we use the dispersive properties of the water-waves system, namely Strichartz estimates, to control these H\older norms.

On the phase dependence of the soliton collisions in the Dyachenko–Zakharov envelope equation

Abstract. We study soliton collisions in the Dyachenko–Zakharov equation for the envelope of gravity waves in deep water. The numerical simulations of the soliton interactions revealed several fundamentally different effects when compared to analytical two-soliton solutions of the nonlinear Schrodinger equation. The relative phase of the solitons is shown to be the key parameter determining the dynamics of the interaction. We find that the maximum of the wave field can significantly exceed the sum of the soliton amplitudes. The solitons lose up to a few percent of their energy during the collisions due to radiation of incoherent waves and in addition exchange energy with each other. The level of the energy loss increases with certain synchronization of soliton phases. Each of the solitons can gain or lose the energy after collision, resulting in increase or decrease in the amplitude. The magnitude of the space shifts that solitons acquire after collisions depends on the relative phase and can be either positive or negative.

Time quasi-periodic gravity water waves in finite depth

We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions - namely periodic and even in the space variable x - of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the quasi-linear nature of the gravity water waves equations and the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these problems by first reducing the linearized operators obtained at each approximate quasi-periodic solution along the Nash-Moser iteration to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme which requires very weak Melnikov non-resonance conditions (losing derivatives both in time and space), which we are able to verify for most values of the depth parameter using degenerate KAM theory arguments.