Christian Kharif

Extreme Ocean Waves

Preface.- Freak Waves: Peculiarities of Numerical Simulations.- Rogue Waves in High-Order Nonlinear Schrodinger Models.- Non-Gaussian Properties of Shallow Water Waves in Crossing Seas.- Modeling of Rogue Wave Shapes in Shallow Water.- Runup of Long Irregular Waves on Plane Beach.- Symbolic Computation for Nonlinear Wave Resonances.- Searching for Factors that Limit Observed Extreme Maximum Wave Height Distributions in the North Sea.- Extremes and Decadal Variations of the Northern Baltic Sea Wave Conditions.- Extreme Waves Generated by Cyclones in Guadeloupe.- An Analytical Model of Large Amplitude Internal Solitary Waves.

Influence of wind on extreme wave events: Experimental and numerical approaches

The influence of wind on extreme wave events is investigated experimentally and numerically. A series of experiments conducted in the Large Air-Sea Interactions Facility (LASIF) shows that a wind blowing over a short wave group due to the dispersive focusing of a longer frequency modulated wave train (chirped wave packet) may increase the time duration of the extreme wave event by delaying the defocusing stage. A detailed analysis of the experimental results suggests that the air flow separation that occurs on the leeward side of the steep crests may sustain longer the maximum of modulation of the focusing-defocusing cycle. Furthermore it is found that the frequency downshifting observed during the formation of the extreme wave event is more important when the wind velocity is larger. The experiments have pointed out that the transfer of momentum and energy is strongly increased during extreme wave events. Two series of numerical simulations have been performed using a pressure distribution over the steep crests given by the Jeffreyssheltering theory. The first series corresponding to the dispersive focusing confirms the experimental results. The second series that corresponds to extreme wave events due to modulational instability shows that wind sustains steep waves which then evolve into breaking waves. Furthermore, it was shown numerically that during extreme wave events the wind-driven current may play a significant role in their persistence.

Nonlinear-dispersive mechanism of the freak wave formation in shallow water

Abstract The mechanism of the freak wave formation related to the spatial–temporal focusing is studied within the framework of the Korteweg–de Vries equation. A method to find the wave trains whose evolution leads to the freak wave formation is proposed. It is based on the solution of the Korteweg–de Vries equation with an initial condition corresponding to the expected freak wave. All solutions of this Cauchy problem by the reversal of abscissa represent the possible forms of wave trains which evolve into the freak wave. It is found that freak waves are almost linear waves, and their characteristic Ursell parameter is small. The freak wave formation is possible also from the random wave field and the numerical simulation describes the details of this phenomenon. It is shown that freak waves can be generated not only for specific conditions, but also for relative wide classes of the wave trains. This mechanism explains the rare and short-lived character of the freak wave.

Nonlinear wave focusing on water of finite depth

Abstract The problem of freak wave formation on water of finite depth is discussed. Dispersive focusing in a nonlinear medium is suggested as a possible mechanism of giant wave generation. This effect is considered within the framework of the nonlinear Schrodinger equation and the Davey–Stewartson system, describing 2+1-dimensional surface wave groups on water of finite depth. In the 2+1-dimensional case, the dispersive grouping is accompanied with a geometrical focusing. Necessary wave conditions for the occurrence of such a phenomenon are discussed. Influence of non-optimal phase modulation and presence of strong random wave component are found to be weak: they do not cancel the mechanism of wave amplification. The mechanism of dispersive focusing is compared with the wave enhancement due to the Benjamin–Feir instability, which is found to be extremely sensitive with respect to weak random perturbations.

Formation of undular bores and solitary waves in the Strait of Malacca caused by the 26 December 2004 Indian Ocean tsunami

[1]xa0Deformation of the Indian Ocean tsunami moving into the shallow Strait of Malacca and formation of undular bores and solitary waves in the strait are simulated in a model study using the fully nonlinear dispersive method (FNDM) and the Korteweg-deVries (KdV) equation. Two different versions of the incoming wave are studied where the waveshape is the same but the amplitude is varied: full amplitude and half amplitude. While moving across three shallow bottom ridges, the back face of the leading depression wave steepens until the wave slope reaches a level of 0.0036–0.0038, when short waves form, resembling an undular bore for both full and half amplitude. The group of short waves has very small amplitude in the beginning, behaving like a linear dispersive wave train, the front moving with the shallow water speed and the tail moving with the linear group velocity. Energy transfer from long to short modes is similar for the two input waves, indicating the fundamental role of the bottom topography to the formation of short waves. The dominant period becomes about 20 s in both cases. The train of short waves, emerging earlier for the larger input wave than for the smaller one, eventually develops into a sequence of rank-ordered solitary waves moving faster than the leading depression wave and resembles a fission of the mother wave. The KdV equation has limited capacity in resolving dispersion compared to FNDM.

Three-dimensional instabilities of periodic gravity waves in shallow water

A linear stability analysis of finite-amplitude periodic progressive gravity waves on water of finite depth has extended existing results to steeper waves and shallower water. Some new types of instability are found for shallow water. When the water depth decreases, higher-order resonances lead to the dominant instabilities. In contrast with the deep water case, we have found that in shallow water the dominant instabilities are usually associated with resonant interactions between five, six, seven and eight waves. For small steepness, dominant instabilities are quasi two-dimensional. For moderate and large steepness, the dominant instabilities are three-dimensional and phased-locked with the unperturbed nonlinear wave. At the margin of instability diagrams, these results suggest the existence of new bifurcated three-dimensional steady waves.

A model of water wave 'horse-shoe' patterns

The work suggests a simple qualitative model of the wind wave ‘horse-shoe’ patterns often seen on the sea surface. The model is aimed at explaining the persistent character of the patterns and their specific asymmetric shape. It is based on the idea that the dominant physical processes are quintet resonant interactions, input due to wind and dissipation, which balance each other. These processes are described at the lowest order in nonlinearity. The consideration is confined to the most essential modes : the central (basic) harmonic and two symmetric oblique satellites, the most rapidly growing ones due to the class I1 instability. The chosen harmonics are phase locked, i.e. all the waves have equal phase velocities in the direction of the basic wave. This fact along with the symmetry of the satellites ensures the quasi-stationary character of the resulting patterns. Mathematically the model is a set of three coupled ordinary differential equations for the wave amplitudes. It is derived starting with the integro-differential formulation of water wave equations (Zakharov’s equation) modified by taking into account small (of order of quartic nonlinearity) non-conservative effects. In the derivation the symmetry properties of the unperturbed Hamiltonian system were used by taking special canonical transformations, which allow one exactly to reduce the Zakharov equation to the model. The study of system dynamics is focused on its qualitative aspects. It is shown that if the non-conservative effects are neglected one cannot obtain solutions describing persistent asymmetric patterns, but the presence of small non-conservative effects changes drastically the system dynamics at large times. The main new feature is attructive equilibria, which are essentially distinct from the conservative ones. For the existence of the attractors a balance between nonlinearity and non-conservative effects is necessary. A wide class of initial configurations evolves to the attractors of the system, providing a likely scenario for the emergence of the long-lived threedimensional wind wave patterns. The resulting structures reproduce all the main features of the experimentally observed horse-shoe patterns. In particular, the model provides the characteristic ‘crescent’ shape of the wave fronts oriented forward and the front-back asymmetry of the wave profiles.

On two approaches to the problem of instability of short-crested water waves

The work is concerned with the problem of the linear instability of symmetric shortcrested water waves, the simplest three-dimensional wave pattern. Two complementary basic approaches were used. The first, previously developed by Ioualalen & Kharif (1993, 1994), is based on the application of the Galerkin method to the set of Euler equations linearized around essentially nonlinear basic states calculated using the Stokes-like series for the short-crested waves with great precision. An alternative analytical approach starts with the so-called Zakharov equation, i.e. an integrodifferential equation for potential water waves derived by means of an asymptotic procedure in powers of wave steepness. Both approaches lead to the analysis of an eigenvalue problem of the type det IA -IS) = 0 where A and B are infinite square matrices. The first approach should deal with matrices of quite general form although the problem is tractable numerically. The use of the proper canonical variables in our second approach turns the matrix B into the unit one, while the matrix A gets a very specific ‘nearly diagonal’ structure with some additional (Hamiltonian) properties of symmetry. This enables us to formulate simple necessary and sufficient a priori criteria of instability and to find instability characteristics analytically through an asymptotic procedure avoiding a number of additional assumptions that other authors were forced to accept. A comparison of the two approaches is carried out. Surprisingly, the analytical results were found to hold their validity for rather steep waves (up to steepness 0.4) for a wide range of wave patterns. We have generalized the classical Phillips concept of weakly nonlinear wave instabilities by describing the interaction between the elementary classes of instabilities and have provided an understanding of when this interaction is essential. The mechanisms of the relatively high stability of shortcrested waves are revealed and explained in terms of the interaction between different classes of instabilities. A helpful interpretation of the problem in terms of an infinite chain of interacting linear oscillators was developed.

A nonlinear Schrödinger equation for water waves on finite depth with constant vorticity

A nonlinear Schrodinger equation for the envelope of two dimensional surface water waves on finite depth with non-zero constant vorticity is derived, and the influence of this constant vorticity on the well-known stability properties of weakly nonlinear wave packets is studied. It is demonstrated that vorticity modifies significantly the modulational instability properties of weakly nonlinear plane waves, namely the growth rate and bandwidth. At third order, we have shown the importance of the nonlinear coupling between the mean flow induced by the modulation and the vorticity. Furthermore, it is 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.

Modelling of Tsunami Propagation in the Vicinity of the French Coast of the Mediterranean

The problem of tsunami-risk for the French coast of the Mediterraneanis discussed. Historical data of tsunami manifestation on the French coast are described and analysed.Numerical simulation of potential tsunamis in the Ligurian Sea is done and the tsunami wave heightdistribution along the French coast is calculated. For the earthquake magnitude 6.8 (typical value forMediterranean) the tsunami phenomenon has a very local character. It is shown that the tsunami tide-gaugerecords in the vicinity of Cannes–Imperia present irregularoscillations with characteristic periodof 20–30 min and total duration of 10–20h.Tsunami propagating from the Ligurian sea to the west coastof France have significantly lesser amplitudes and they are more low-frequency (period of 40–50min).The effect of far tsunamis generated in the southern Italy and Algerian coast is studied also, thedistribution of the amplitudes along the French coast for far tsunamis is more uniform.