SCIENTIFIC PROGRAMS AND ACTIVITIES

Nail Akhmediev (Presentation slides)
Rogue waves - higher order structures

Ken Melville
The Equilibrium Dynamics and Statistics of Wind-Driven Gravity-Capillary Waves

Akhmediev, Nail
Australian National University, Canberra

Rogue waves - higher order structures

Peregrine breather being the lowest order rational solution of the nonlinear Schroedinger equation is commonly considered as a prototype of a rogue wave in the ocean. Higher-order rational solutions are far from being as simple as the Peregrine breather itself. They are not as simple as a nonlinear superposition of solitons either. Only recently, the complexity of their spatio-temporal structures started to be revealed.

Basic thoughts on their classification will be presented in this talk.

Alazard, Thomas
Ecole Normale Superieure, Paris

The Cauchy problem for the water waves equations, local and global aspects

This talk will present recent results on the analysis of the Cauchy problem for the water gravity waves. This includes firstly a discussion of the regularity thresholds for the initial conditions : the initial surfaces we consider turn out to have unbounded curvature and no regularity is assumed on the bottom. An application is given to 3D water waves in a canal or a basin. Secondly, normal form methods will be discussed. This corresponds to the analysis of three and four-wave interactions. These are joint works with Nicolas Burq and Claude Zuily, and joint work with Jean-Marc Delort.

Some fundamental issues in internal wave dynamics

One of the simplest physical setups supporting internal wave motion is that of a stratified incompressible Euler fluid filling the domain between two rigid horizontal plates. This talk will present asymptotic models capable of describing large amplitude wave propagation in this environment, and in particular of predicting the occurrence of self-induced shear instability in the waves' dynamics. Some curious properties of the Euler setup for laterally unbounded domains revealed by the models will be discussed.

We study both experimentally and numerically the evolution of nonlinear wave packets. We solve numerically a system of nonlinear evolution equations for the propagation of wave packets with various orders of approximation and validate the numerical solutions with experimental measurements. In the presence of wave breaking, a new parameterization is introduced to account for energy dissipation due to wave breaking and its capability to capture breaking wave characteristics is examined in comparison with laboratory experiments.

A new method, based on a relaxed variational principle, is presented for deriving approximate equations for water waves. It is particularly suitable for the construction of approximations. The advantages will be illustrated on numerous examples in shallow and deep water. Using carefully chosen constraints in various combinations, several model equations are derived, some being well-known, others being new. These models are studied analytically, exact travelling wave solutions are constructed, and the Hamiltonian structure unveiled.

This is a joint work with Didier Clamond, University of Nice Sophia Antipolis.

On the spectral shape of the source terms of the radiative transfer equation

Currently, large domain wave forecast models are not phase-resolving but are based on the so-called 3rd generation spectral models. The basis of these models is the radiative transfer equation, which relates the change of spectral energy to the sum of three source terms: energy input from the wind, energy transfer between different wave scales and energy dissipation, mainly due to wave breaking (in deep water) and bottom friction (in shallow water). The net effect of these sources in a developing sea is an increase of energy and a downshift of the spectral peak. However, this does not provide sufficient constraints on the spectral shape of the individual source terms.

I will present various field observations to shed light on the spectral shape of the source terms, with emphasize on the contributions by breaking waves.

Shoaling of nonlinear water waves

We review the classical theory for shoaling of a solitary wave, and extend this to consider the propagation of an undular bore over a gentle monotonic bottom slope connecting two regions of constant depth, in the framework of the variable-coeficient Korteweg-de Vries equation. We show that, when the undular bore advances in the direction of decreasing depth, its interaction with the slowly varying topography results, apart from an adiabatic deformation of the bore itself, in the generation of a sequence of isolated solitons, that is an expanding large-amplitude modulated solitary wavetrain propagating ahead of the bore. Using nonlinear modulation theory we construct an asymptotic solution describing the formation and evolution of this solitary wavetrain. Our analytical solution is supported by direct numerical simulations.

Surface signature of internal waves

Based on a Hamiltonian formulation of a two-layer ocean, we consider the situation in which the internal waves are treated in the long-wave regime while the surface waves are described in the modulational regime. Using Hamiltonian perturbation theory, we derive an asymptotic model for surface-internal wave interactions, in which the nonlinear internal waves evolve according to a KdV equation while the smaller-amplitude surface waves propagate at a resonant group velocity and their envelope is described by a linear Schrodinger equation. In the case of an internal soliton of depression for small depth and density ratios of the two layers, the Schrodinger equation is shown to be in the semi-classical regime and thus admits localized bound states. This leads to the phenomenon of trapped surface modes which propagate as the signature of the internal wave, and thus it is proposed as a possible explanation for bands of surface roughness above internal waves in the ocean. Numerical simulations taking oceanic parameters into account are also performed to illustrate this phenomenon.

This is joint work with Walter Craig and Catherine Sulem.

Wind turbulence over ocean waves and air-sea momentum flux

We present our recent LES (large eddy simulation) results of wind turbulence modified by ocean surface waves.

In the constant stress layer above a smooth non-breaking surface wave train, a wave-induced momentum flux (stress) reduces the turbulent stress and the turbulent kinetic energy (TKE) dissipation rate inside a very thin layer (inner layer), when the wind speed is much larger than the wave phase speed. This leads to an increased equivalent surface roughness (or the drag coefficient) for the wind. Since the inner layer height is often smaller than the wave amplitude, it is necessary to introduce a wave-following coordinate and redefine the wave-induced stress when the LES results are analyzed.

When a surface waves breaks (or is sufficiently steep) airflow separates and exerts a large force on the wave. The effects of breaking waves on near-surface wind turbulence and drag coefficient are investigated using LES. The impact of intermittent and transient wave breaking events is modeled as localized form drag, which generates airflow separation bubbles downstream. The simulations are performed for very young sea conditions under high winds, comparable to previous laboratory experiments in hurricane-strength winds. In such conditions more than 90 percent of the total air-sea momentum flux is due to the form drag of breakers; that is, the contributions of the non-breaking wave form drag and the surface viscous stress are small. Detailed analysis shows that the breaker form drag impedes the shear production of the TKE near the surface and, instead, produces a large amount of small-scale wake turbulence by transferring energy from large-scale motions (such as mean wind and gusts). This process shortcuts the inertial energy cascade and results in large TKE dissipation (integrated over the surface layer) normalized by friction velocity cubed. Consequently, the large production of wake turbulence by breakers in high winds results in the small drag coefficient obtained in this study.

Breaking of the internal tide

Nonlinear steepening of low-mode internal tides and the subsequent arrest of steepening by non-hydrostatic dispersion is a common mechanism for the generation of internal solitary waves in the ocean. However, it is known that the earth's rotation can retard the steepening process and in some cases prevent the emergence of the solitary waves. The Ostrovsky equation, the Korteweg-de Vries equation with a nonlocal integral term representing the effects of rotation, is introduced as model for these processes. Recent work on a breaking criteria for the reduced Ostrovsky equation (in which the linear non-hydrostatic dispersive term with a third-order derivative is eliminated) is discussed. This equation is integrable provided a certain curvature constraint is satisfied. It is demonstrated, through theoretical analysis and numerical simulations, that when this curvature constraint is not satisfied at the initial time, then wave breaking inevitably occurs. The breaking criteria is
applied to several oceanic examples including internal tides in the South China Sea and radiation of the internal tide from the Hawaiian Island chain.

Surface waves and dissipation

Surface waves at an air-water interface are usually modeled using inviscid dynamics. However, water is a viscous fluid, and resulting dissipative effects, though small, can play an important role in the wave dynamics when the waves propagate over long distances. Previous experiments have shown that the dissipation rate of waves is strongly affected by conditions at the free surface. So to derive a model that predicts dissipation rate, one usually allows for weak viscosity and assumes one of three types of boundary conditions at the surface: (i) the surface is shear-free, also referred to as ``clean'', (ii) the surface admits no tangential velocities, also referred to as ``fully contaminated'', or (iii) the surface is elastic. Here we discuss experiments within the context of these three models in an effort to better understand the boundary condition at the air-water interface and the ranges of applicability of these models.

Josserand, Christophe
Institut D'Alembert

Wave turbulence in vibrating plates

The concept of wave turbulence that has been introduced originally for ocean waves applies in fact in very different domains. We have recently shown theoretically and numerically that wave turbulence could be observed on elastic plates. Experiments performed by different groups have however shown discrepancies with the theory. I will discuss here first the general framework of wave turbulence on plates. Then I'll discuss how we can explain the differences between theory and experiments. Finally, I will show how inverse cascades could be present in the dynamicsm although it is a priori not possible.

Generally, in coastal and ocean waters, the velocity profiles are typically established by bottom friction and by surface wind stress and so are varying with depth. Currents generate shear at the bed of the sea or of a river. For example ebb and flood currents due to the tide may have an important effect on waves and wave packets. In any region where the wind is blowing there is a surface drift of the water and water waves are particularly sensitive to the velocity in the surface layer. We consider the effect of constant non zero vorticity on the Benjamin-Feir instability of 2D surface gravity waves on arbitrary depth.

Very recently, Thomas, Kharif & Manna [1] using the method of multiple scales derived a nonlinear Schroedinger equation in finite depth and in the presence of uniform vorticity. They demonstrated that vorticity modifies significantly the modulational instability properties of weakly nonlinear plane waves, namely the growth rate and bandwidth. Furthermore, it was shown that these plane wave solutions may be linearly stable to modulational instability for an opposite shear current independently of the dimensionless parameter kh, where k and h are the carrier wavenumber and depth respectively. Within the framework of the fully nonlinear water wave equations, Francius & Kharif, have recently investigated the modulational instability of a uniform wave train on a shearing flow of constant vorticity and extended to steeper waves the results of Thomas, Kharif & Manna [1].

Logarithmic scaling of wave collapse

The dynamics of quasi-monochromatic wave packet on the free surface of infinite depth fluid is described by the focusing two-dimensional (2D) nonlinear Schrodinger equation (NLSE) for short enough wavelength (when the capillary force is significant). The dynamics of similar wave packet in the case of finite depth is given by 2D Davey-Stewartson (Benney-Roskes) equation (DSE). Both NLSE and DSE have generic solutions in the form of finite-time singularity accompanied by the contraction of the spatial scale of solution to zero which is called by wave collapse.These collapses are responsible for the formation of the strongly nonlinear waves on a fluid surface.
We study the universal self-similar behaviour near collapse time t_c, i.e. the spatial and temporal structures near singularity. Collapses in both NLSE and DSE share a strikingly common feature that the collapsing solutions have a form of a rescaled soliton. The time dependence of the rescaled soliton width L(t) determines also the solution amplitude ~1/L(t). At leading order L(t)~ (t_c-t)^{1/2} for both NLSE and DSE. Collapse of NLSE requires the modification of that scaling which has a well-known loglog form ~ (\ln|\ln(t_c-t)|)^{-1/2}. Loglog scaling for NLSE was first obtained asymptotically in 1980's and later proven by Merle and Raphael in 2006. However, it remained a puzzle that this scaling was never clearly observed in simulations or experiment. Here solved that puzzle by developing a perturbation theory beyond the leading order logarithmic corrections for NLSE. We found that the classical loglog modification NLSE requires double-exponentially large amplitudes of the solution ~10^10^100, which is unrealistic to achieve in either physical experiments or numerical simulations. In contrast, we found that our new theory is valid starting from quite moderate (about 3 fold) increase of the solution amplitude compare with the initial conditions. New scaling is in excellent agreement with simulations.

Nonlinear inviscid damping in 2D Euler

We prove the global asymptotic stability of shear flows close to planar Couette flow in the 2D incompressible Euler equations. Specifically, given an initial perturbation of the Couette flow which is small in a suitable regularity class we show that the velocity converges strongly in L2 to another shear flow which is not far from Couette. This strong convergence is usually referred to as "inviscid damping" and is roughly analogous to Landau damping in the Vlasov equations. Joint work in progress with Jacob Bedrosian

Nonlinear long waves over a muddy beach

Abstract

The Equilibrium Dynamics and Statistics of Wind-Driven Gravity-Capillary Waves

Recent field observations and modeling of breaking surface gravity waves suggest that air-entraining breaking is not sufficiently dissipative of surface gravity waves to balance the dynamics of wind-wave growth, nonlinear interactions and dissipation for the shorter gravity waves of O(10) cm wavelength. Theories of parasitic capillary waves that form at the crest and forward face of shorter steep gravity waves have shown that the dissipative effects of these waves may be one to two orders of magnitude greater than the viscous dissipation of the underlying gravity waves. Thus the parasitic capillaries may provide the required dissipation of the short wind-generated gravity waves. This has been the subject of speculation and conjecture in the literature. Using the nonlinear theory of Fedorov & Melville (1998), we show that the dissipation due to the parasitic capillaries is sufficient to balance the wind input over some range of wave ages and wave slopes. The range of wavelengths over which these parasitic capillary waves are dynamically significant approximately corresponds to the range of wavelengths that are suppressed by oil on water, as measured by Cox & Munk (1954), who also found that these waves contributed significantly to the mean square slope of the ocean surface, which they measured to be proportional to the wind speed. Here we show that that the mean square slope predicted by the theory is proportional to the square of the friction velocity of the wind, u*2, for small wave slopes, and to u* for larger slopes.

The Direct Numerical Simulation (DNS) of the Navier-Stokes equation for a two-phase flow (water and air) is used to study the dynamics of the modulational instability of free surface waves and its contribution to the interaction between ocean and atmosphere. If the steepness of the initial wave is large enough, we observe a wave breaking and the formation of large scale dipole structures in the air. Because of the multiple steepening and breaking of the waves under unstable wave packets, a train of dipoles is released and propagate in the atmosphere at a height comparable with the wave length. The amount of energy dissipated by the breaker in water and air is considered and, contrary to expectations, we observe that the energy dissipation in air is comparable to the one in the water.

Rogue Waves in Shallow Waters

An overview on the problem of rogue or freak wave formation in shallow waters is given. A number of huge wave accidents, resulting in damages, ship losses and people injuries and deaths, are known and summarized in recent catalogues and books. This presentation addresses the nature of the rogue wave problem from a general viewpoint based on non-dispersive and weakly dispersive wave process ideas. We start by introducing some primitive elements of sea wave physics with the purpose of paving the way for further discussion. We discuss linear physical mechanisms which are responsible for high wave formation, at first. Nonlinear effects which are able to cause rogue waves are emphasized. In conclusion we briefly discuss the generality of the physical mechanisms suggested for the rogue wave explanation; they are valid for rogue wave phenomena in geophysics and plasma.
(in collaboration with Alexey Slunyaev, Ira Didenkulova, Christian. Kharif, Irina Nikolkina, Anna Sergeeva, Tatiana Talipova and Ekaterina Shurgalina)

Nonlinear Energy Transfers in a Wind Wave Spectrum

Nonlinear wave-wave interactions involving quadruplets constitute the basis for modern wave modeling and wave forecasts. In most modern operational wave models such as WAM, or WAVEWATCHIII, quadruplet wave-wave interactions are simulated by the Discrete Interaction Approximation, commonly referred to as the DIA, as formulated by WAMD1 (1988). We give a description of DIA, and we introduce a new approximation, the Two-Scale Approximation (TSA), based on the separation of a spectrum into a broad-scale component and a local-scale (perturbation) component. TSA uses a parametric representation of the lower-order or "broad-scale" spectral structure, while preserving the degrees of freedom essential to a detailed-balance source term formulation, by including the second order scale in the approximation. We present tests using idealized wave spectra, including JONSWAP spectra (Hasselmann et al., 1973) with selected wave hypothetical peakednesses, and perturbation cases, as well as well as additional tests for fetch-limited wave growth, and storm waves generated by hurricane Juan (2003). Generally, TSA is shown to work well when its basic assumptions are met, when its first order, broad-scale term represents most of the spectrum, and its second order term is a perturbation-scale residual term representing the rest of the spectrum. These conditions are easily met for test cases involving idealized JONSWAP-type spectra and in time-stepping cases when winds are spatially and temporally constant. To some extent, they also appear to be met in more demanding conditions, when storms move through their life cycles, with winds that change in speed and direction, and with complex wave spectra, involving swell-windsea interactions, multiple peaks fp1, fp2, …and directional shears. In these cases, we show that TSA can be generalized (e.g. double, or multiple TSAs) and work reasonably well when the spectrum is partitioned according to individual spectral peaks. In this situation, TSA's basic assumptions are met in each segment of the spectrum (each spectral peak region), in terms of its first order broad-scale, and second order perturbation-scale terms. Comparisons will be made with integrations of the full Boltzmann integral (FBI) for quadruplet wave-wave interactions.

Enhanced transfer of wind energy into surface waves

Asymptotic multi-layer analyses and computation of solutions for turbulent flows over steady and unsteady monochromatic surface wave are reviewed, in the limits of low turbulent stresses and small wave amplitude. The structure of the flow is defined in terms of asymptotically-matched thin-layers, namely the surface layer and a critical layer, whether it is 'elevated' or 'immersed', corresponding to its location above or within the surface layer. The results particularly demonstrate the physical importance of the singular flow features and physical implications of the elevated critical layer in the limit of the unsteadiness tending to zero. These agree with the variational mathematical solution of Miles [J. Fluid Mech., 3, 185-204 (1957)] for small but finite growth rate, but they are not consistent physically or mathematically with his analysis in the limit of growth rate tending to zero. As this and other studies conclude, in the limit of zero growth rate the effect of the elevated critical layer is eliminated by finite turbulent diffusivity, so that the perturbed flow and the drag force are determined by the asymmetric or sheltering flow in the surface shear layer and its matched interaction with the upper region. But for groups of waves, in which the individual waves grow and decay, there is a net contribution of the elevated critical layer to the wave growth. Critical layers, whether elevated or immersed, affect this asymmetric sheltering mechanism, but in quite a different way to their effect on growing waves. These asymptotic multi-layer methods lead to physical insight and suggest approximate methods for analyzing higher amplitude and more complex flows, such as flow over wave groups.

Most of approximate models for surface and internal waves are derived from some asymptotic expansions (with respect to "small" parameters) in various regimes of amplitudes, wavelenghts,.. Their solutions are not supposed to be good approximates for all times but only on some relevant "long"" time scales. Proving such long time existence is not an easy task for most of water waves systems. This talk will present such results for Boussinesq type systems and for a "full dispersion" system.

The nonlinear Schrödinger equation, dissipation and ocean swell

The focus of this talk is less about how to solve a particular mathematical model, and more about how to find the right model of a physical problem. The nonlinear Schrödinger (NLS) equation was discovered as an approximate model of wave propagation in several branches of physics in the 1960s. It has become one of the most studied models in mathematical physics, because of its interesting mathematical structure and because of its wide applicability – it arises naturally as an approximate model of surface water waves, nonlinear optics, Bose-Einstein condensates and plasma physics. In every physical application, the derivation of NLS requires that one neglect the (small) dissipation that exists in the physical problem. But our studies of water waves (including freely propagating ocean waves, called “swell”) have shown that even though dissipation is small, neglecting it can give qualitatively incorrect results. This talk describes an ongoing quest to find an appropriate generalization of NLS that correctly predicts experimental data for ocean swell. As will be shown, adding a dissipative term to the usual NLS model gives correct predictions in some situations. In other situations, both NLS and dissipative NLS give incorrect predictions, and the “right model” is still to be found.

Tataru, Daniel
University of California, Berkeley

Two dimensional water waves

TBA