Water Waves

The following sections are included:

• PREFACE

• ACKNOWLEDGMENT

• CONTENTS

Using techniques from the theory of oscillatory integrals, we prove rigorous estimates which show that the linearization of the vortex sheet equations of motion about a quiescent state disperse under certain circumstances. Such dispersion is only possible only through the joint effects of surface tension (which damps high frequency modes) and gravitation (which damps low frequency modes).

A laboratory study of the nonlinearity of random waves (the primary waves) and low-frequency waves induced by the primary waves on inclined beaches is presented. The experiments were carried out in a wave flume that is 65 m long, 1.2 m wide with a water depth of 0.7 m. Three plane beaches with different slopes (1/20, 1/30 and 1/40) were used in separated experiments. Incident primary waves with a Person-Moskowitz (PM) spectrum were mechanically generated with significant wave height ranging from 0.047 to 0.125 m. The time series of surface elevations at different water depths along the beach were simultaneously recorded. Low-frequency waves are obtained from the measured wave data with a low-pass filter. The results show that the primary waves on the beach are highly nonlinear. The surface skewness and kurtosis, two statistical measures of wave nonlinearity, are functions of a nondimensional parameter H_s/d, where H_s is the local significant wave height and d is the water depth. The spectra of low-frequency waves in shoaling region are affected by beach slope and the energy of incident primary waves. The energy ratio between low-frequency waves and primary waves strongly correlates to the local surface skewness. The growth (dissipation) rate of low-frequency waves on beaches is controlled by the Iribarren number, ξ = β/(H₀/L₀)^1/2, where β is the beach slope, H₀ and L₀ are the significant wave height and wavelength of incident primary waves, respectively.

This article applies some statistical theories to the Euler equations to approximate equilibrium solutions for initial data resulting from a vortex patch. Bifurcation diagrams are presented for each statistical theory. The predictions of the statistical theories are then compared with some direct numerical simulations. All computations are done in a doubly periodic domain.

The upper most layers of the ocean, along with the lower atmospheric boundary layer, play a crucial role in the air-sea fluxes of momentum, heat, and mass, thereby providing important boundary conditions for both the atmosphere and the oceans that control the evolution of weather and climate. In particular, the fluxes of heat and gas rely on exchange processes through the molecular layers, which are usually located within the viscous layer, which is in turn modulated by the waves and the turbulence at the free surface. The understanding of the multiple interactions between, molecular layers, viscous layers, waves and turbulence is therefore paramount for an adequate parameterization of these fluxes.

We present evidence of a clear coupling between the surface waves, the surface temperature, and the surface turbulence. The modulation of the surface temperature by the waves lead to a measurable wave-coherent air-sea heat flux. When averaged over time scales longer than the wave period, the coupling between the surface temperature and turbulence leads to a spatial relationship between the temperature, divergence and vorticity fields that is consistent with spatial patterns of Langmuir turbulence. On time scales for which the surface wave field is resolved, we show that the surface turbulence is modulated by the waves in a manner qualitatively consistent with rapid distortion theory.

The text and figures presented here are compiled from papers by Veron et al. (2008a, 2008b, 2009) ©Copyright 2008, 2009, American Meteorological Society (AMS).

We study the stabilizing effect of rotational forcing in the nonlinear setting of two-dimensional shallow-water equations. The pressureless version of these equations admit global smooth solutions for a large set of sub-critical initial configurations. But what happens with more realistic models, in the presence of pressure? It is shown that when rotational forcing dominates the pressure, it prolongs the life-span of such sub-critical solutions, for a time period ln(1/δ) ≫ 1 dictated by the ratio δ =Rossby number/squared Froude number. Our study reveals a "nearby" periodic-in-time approximate solution in the small δ regime, upon which hinges the long time existence of the exact smooth solution. These results are in agreement with the close-to-periodic dynamics observed in the "near inertial oscillation" (NIO) regime which follows oceanic storms.

This paper and the corresponding presentation at the workshop discuss practical numerical wind wave modeling. It reviews the history, common numerical models and approaches as well, as problems and unresolved issues that can be the subject of further research.

This article explains the nonexistence of double periodic continuous solutions for semilinear wave equations with smooth data and no resonance. This phenomenon is due to having a nonlinearity interacting with a linear operator with infinite dimensional kernel.

This note discusses the recent development on the existence and stability of two-dimensional and three-dimensional gravity-capillary waves on water of finite-depth. The Korteweg-de Vries (KdV) equation is derived formally from the exact governing equations and the solitary-wave solutions are obtained. The recent results on the existence of solutions of the exact equations near the solitary-wave solutions of the KdV equation are presented and various two-and three-dimensional wave solutions are given. Then, the stability of these solutions using exact equations is discussed. The ideas and methods to obtain these results are briefly mentioned.

Tsunami sources created by subduction-type earthquakes are usually confined to strips of a large but finite aspect ratio. We analyze such tsunamis that propagate across a region of finite depth. Based on classical shallow-water-wave theory, an accurate and convenient form of the asymptotic solution is obtained for comprehensive descriptions of the non-dispersive aspects of the waves. The directivity of the energy radiation and the domain of pulse persistence are quantitatively described. Our analysis gives an estimate of the initial water-surface displacement at the tsunami source of the 2004 Indian Ocean Tsunami, which turns out to be in excellent agreement with the result computed from the seismic wave inversion.

For autonomous and nonautonomous flow fields, we employ a Lagrangian diagnostic to computationally distinguish the Lagrangian coherent structures that govern the field's underlying dynamics. The Lagrangian diagnostic we use is the method of relative dispersion. As with all finite-time methods, a termination time for the relative dispersion algorithm must be determined. With the aid of statistics, a major result in this work is finding such a termination time.

A system of Coupled Nonlinear Schroedinger (CNLS) equations is studied. In particular, we study soliton and soliton like solutions. We show under what conditions exact soliton solutions exist and how they gave rise to initial value numerical results. Furthermore, we used variational technique to explore analytical results.

A highly efficient and accurate numerical scheme for initial and boundary value problems of a two-dimensional Boussinesq system which describes three-dimensional water waves is used to study in details the oblique interaction of a solitary wave and the evolution of solitary waves coming out of a narrower channel.

The purpose of this note is to present a simple and concise presentation of the a priori nonlinear estimates for the water-wave problem, based on the analysis in Coutand & Shkoller.¹

A hands-on, portable laboratory was set-up near the CBMS conference room in order to observe and demonstrate some of the concepts described in the lecture series. Here we describe the apparatus, the demonstrations that were done, and how they fit into the lecture series.