On the Kaup–Broer–Kupershmidt systems

Consideration is given to three different full dispersion Boussinesq systems arising as asymptotic models in the bi-directional propagation of weakly nonlinear surface waves in shallow water. We prove that, under a non-cavitation condition on the initial data, these three systems are well-posed on a time scale of order O(1ε) , where ɛ is a small parameter measuring the weak nonlinearity of the waves. For one of the systems, this result is new even for short time. The two other systems involve surface tension, and for one of them, the non-cavitation condition has to be sharpened when the surface tension is small. The proof relies on suitable symmetrizers and the classical theory of hyperbolic systems. However, we have to track the small parameters carefully in the commutator estimates to get the long time well-posedness. Finally, combining our results with the recent ones of Emerald provide a full justification of these systems as water wave models in a larger range of regimes than the classical (a, b, c, d)-Boussinesq systems.

Exact solutions of various Boussinesq systems

Direct numerical simulation (DNS) of a droplet-laden, turbulent Couette airflow over a waved water surface is performed modeling the marine atmospheric boundary (MABL) layer carrying idealized spume droplets. Both the instantaneous and mean flow properties, the characteristics of the vortex structures and the momentum exchange between air turbulence and waved water surface and droplet-mediated momentum transfer are investigated. A Eulerian–Lagrangian approach is employed in DNS where full, 3D Navier–Stokes equations for the carrier air are solved in a Eulerian frame, and the trajectories of individual droplets are simultaneously tracked in a Lagrangian frame. The impact of the droplets on the carrier air flow is modeled via a point force approximation. The droplets size is considered in the range of spume droplet sizes observed in MABL. Various water surface roughness and droplet injection scenarios are considered, and both instantaneous and phase-averaged flow fields, the Reynolds stresses and the eigenvalues of the local air velocity gradient tensor are evaluated in DNS. Numerical results show a strong dependence of the droplet-mediated airflow modification on-the-droplet injection mechanism. Droplets injected with the surrounding air velocity effectively mitigate the vortex structures by reducing their swirling strength and suppress the momentum flux from air turbulence to water surface by weakening both ejections and sweeping events, and thus accelerating the mean flow as compared to the droplet-free flow. On the other hand, droplets injected with the velocities of the Lagrangian particles of the water surface enhance both the swirling strength of the vortex structures and air-flow turbulent stresses and decelerate the mean wind. The results also show that these effects of droplet-mediated flow modification become less pronounced as the water surface wave steepness increases.

Airborne ultrasound reflection from water surface waves is modelled to advance uses of acoustic signals to measure water surface waves and apply the measurements to explore interactions of water waves with rigid structures in a laboratory setting. When the ultrasound is incident on a moving periodic water surface wave, the reflected signal can be treated as diffraction from a moving corrugated reflection grating. Under the condition that the amplitude of the water surface waves is much less than the incident acoustic wavelength, diffraction theory leads to analytical formulas for the spectra of the acoustic signal relating to the water wave amplitudes and frequencies. Complementary modeling based on ray theory and wave superposition illustrates the diffraction and validates formulas of water wave reflection from a surface-piercing barrier structure, where two counter-propagating water waves are involved.

Under consideration in this paper is a variable-coefficient generalized Boussinesq (gBq) system, which can model the propagation of long weakly nonlinear and weakly dispersive surface waves in shallow water. With the aid of the Darboux transformation and symbolic computation, soliton solutions of the gBq system are obtained that do not have singularities under a selection of the spectral parameters. Interactions of the solitons with the elastic properties are discussed. The outcome of this paper might be of some help for the investigation of nonlinear and dispersive problems in fluid dynamics.

Because of the enormous earthquake in Sumatra on December 26, 2004, and the devastating tsunami which followed, I have chosen the focus of my mini-course lectures at this year’s PASI to be on two topics which involve the dynamics of surface water waves. These topics are of interest to mathematicians interested in wave propagation, and particularly to Chilean scientists, I believe, because of Chile’s presence on the tectonically active Pacific Rim. My first lecture will describe the equations of fluid dynamics for the free surface above a body of fluid (the ocean surface), and the linearized equations of motion. From this, we can predict the travel time of the recent tsunami from its epicenter off of the north Sumatra coast to the coast of nearby Thailand, the easy coasts of Sri Lanka and south India, and to Africa. In fact, the signal given by ocean waves generated by the Sumatra earthquake was felt globally; within 48 h distinguishable tsunami waves were measured by wave gages in Antarctica, Chile, Rio di Janeiro, the west coast of Mexico, the east coast of the United States, and at Halifax, Nova Scotia. To describe ocean waves, we will formulate the full nonlinear fluid dynamical equations as a Hamiltonian system [19], and we will introduce the Greens function and the Dirichlet-Neumann operator for the fluid domain along with the harmonic analysis of the theory of their regularity. From an asymptotic theory of scaling transformations, we will derive the known Boussinesq-like systems and the KdV and KP equations, which govern the asymptotic behavior of tsunami waves over an idealized flat bottom. When the bottom is no longer assumed to be perfectly flat, a related theory [6, 13] gives a family of model equations taking this into account. My second lecture will describe a series of recent results in PDE, numerical results, and experimental results on the nonlinear interactions of solitary surface water waves. In contrast with the case of the KdV equations (and certain other integrable PDE), the Euler equations for a free surface do not admit clean (‘elastic’) interactions between solitary wave solutions. This has been a classical concern of oceanographers for several decades, but only recently have there been sufficiently accurate and thorough numerical simulations which quantify the degree to which solitary waves lose energy during interactions [3, 4]. It is striking that this degree of ‘inelasticity’ is remarkably small. I will describe this work, as well as recent results on the initial value problem which are very relevant to this phenomenon [14, 18].

Validity of the resonant four-wave interaction system in a model for surface water waves on an infinite deep sea

The ability to sense water surface waves has been described in only a few species, but across a wide taxonomic range. Water surface waves are typically used to localize prey or to avoid predators, and in some cases also for sexual communication. Here we add to the sparse knowledge of the use of this sensory modality by reporting observational and experimental evidence that wood frogs (Lithobates sylvaticus) respond to water surface waves generated by conspecifics; that there are pronounced differences in response between males and females; and that they use surface waves in a behavioural context not previously reported for anuran reproductive behaviour: sexual eavesdropping. Because the water waves that elicit the described responses are incidental by-products of calling and locomotion behaviour, we consider this an example of sexual eavesdropping rather than sexual communication. Males quickly and accurately approach a surface wave source, thus aiding in mate acquisition which in this species is mainly achieved by scramble competition. By contrast, females move away from a surface wave source. This may help them avoid sexual harassment by mate-searching males. Because it assures that only the fastest, strongest, and potentially fittest males can amplex them, it may also be a strategy for indirect mate choice by females.

For this work an optical method for measuring surface acoustic waves was adapted to acoustics using airborne ultrasound for measurements on water surface waves. With ultrasound incident on periodic traveling water surface waves, the reflected signal can be treated as the diffraction pattern from a moving corrugated reflection grating as long as the amplitude of the water surface waves is much less than the incident acoustic wavelength. The acoustic signal received at the first-order diffraction maxima is amplitude modulated at the frequency of the water surface wave. The intensity of this modulation is directly proportional to the amplitude of the water surface wave. Using spectral decomposition of the signal, the water surface wave amplitudes are precisely determined at sum and difference frequencies around the source peak. The transmission of water surface waves incident on a solid piercing boundary was measured using this method to understand capillary-gravity wave interactions with boundaries. [Work supported by NASA.]

The vertebrate-catching behaviour of the fishing spider Dolomedes triton (Araneae, Pisauridae)

This paper is devoted to a theoretical investigation on the wave amplitude enhancement of surface sea water waves with under-sea periodic arrays of cylinders. A two-dimensional shallow water wave equation is derived and solved by using the plane-wave expansion method. The lattice types studied here include triangular, square and hexagonal lattices. These under-sea structures alter the sea bottom topography and induce constructive interference on the surface water waves. Given that the wave potential energy is dependent on the square of the wave amplitude, this mechanism can thus be used to increase the potential energy. It is shown that the enhancement factor depends on two geometric parameters and the maximum wave amplitude can be found by adjusting the two geometric parameters. Among the lattice types, the triangular and square lattice structures can induce more wave amplitude enhancement (and thus potential energy) than the hexagonal structures. Guided by numerical simulations, we have performed a reduced-scale water tank experiment to demonstrate the feasibility of the proposed idea. Preliminary experimental results show promising evidence of the predicted wave amplitude enhancement, suggesting perspective real-scale nearshore deployment and test.

As a usual component in virtual scenes, water surface plays an important role in various graphical applications, including special effects, video games, and virtual reality. Although recent years have witnessed significant progress based on Navier–Stokes equations and simplified water models, large‐scale water surface waves with high‐frequency visual details remain computationally expensive for interactive applications. This article proposes a novel frequency‐aware neural network to synthesize consistent and detailed water surface waves from low‐resolution input. At its core, our approach leverage the wavelet transformation theory over space, frequency and direction, and incremental supervision to decompose the 4D amplitude function into multiple smaller subproblems. Specifically, we first customize four subnetworks and corresponding loss functions for super‐resolution of spatial resolution, temporal evolution, wave direction subdivision, and wave number, respectively. Then, to enforce the upsampling along each dimension orthogonal to each other, we introduce a cooperative training scheme to fine‐tune and integrate the proposed subnetworks with carefully designed training dataset. Our method can visually enhance high‐resolution spatial details, temporal coherence, interactions with complex boundaries, and various wave patterns with flexible control along multiple dimensions. Through extensive experiments, our method arrives at 13 speedup for 32 upsampling of various simulation scenarios. We also validate the effectiveness and robustness of our method to produce realistic water surface waves toward artistic innovation.

The study of a turbulent air flow over capillary-gravity water surface waves by direct numerical simulation

Optimal strategies to steer and control water waves

Boussinesq systems in two space dimensions over a variable bottom for the generation and propagation of tsunami waves