On the role of the Jeffreys'sheltering mechanism in the sustain of extreme water waves

The effect of the wind on the sustain of extreme water waves is investigated experimentally and numerically. A series of experiments conducted in the Large Air-Sea Interactions Facility (LASIF) showed that a wind blowing over a strongly nonlinear short wave group due to the linear focusing of a modulated wave train may increase the life time of the extreme wave event. The expriments suggested 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. Based on a Boundary-Integral Equation Method and a pressure distribution over the steep crests given by the Jeffreys'sheltering theory, similar numerical simulations have confirmed the experimental resultsComment: accept\'{e} pour publication 200

A nonlinear Schr\"odinger equation for water waves on finite depth with constant vorticity

A nonlinear Schr\"odinger 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 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

Rogue waves in the atmosphere

The appearance of rogue waves is well known in oceanographics, optics, and cold matter systems. Here we show a possibility for the existence of atmospheric rogue waves.Comment: 2 pages, 1 figur

Black Reaction to Segregation and Discrimination in Post-Reconstruction

Equality of protection under the laws, as guaranteed by the Fourteenth Amendment to the United States Constitution, implies that in the administration of criminal justice no person, by reason of his race or color, shall be subjected for the same offense to any greater or different punishment than that to which persons of another race or color are subjected. It also suggests that all citizens are entitled to protection of their civil rights and against discriminatory practices based upon race, color, creed, or religion. Unfortunately, in October 1883 when the United States Supreme Court declared the Civil Rights Acts of 1875 unconstitutional, the legislative framework requiring states to provide for civil rights in public places of accommodation and transportation was dismantled. It further had the effect of nullifying the civil rights act passed by Florida lawmakers in 1873

Approximate rogue wave solutions of the forced and damped Nonlinear Schr\"odinger equation for water waves

We consider the effect of the wind and the dissipation on the nonlinear stages of the modulational instability. By applying a suitable transformation, we map the forced/damped Nonlinear Schr\"odinger (NLS) equation into the standard NLS with constant coefficients. The transformation is valid as long as |{\Gamma}t| \ll 1, with {\Gamma} the growth/damping rate of the waves due to the wind/dissipation. Approximate rogue wave solutions of the equation are presented and discussed. The results shed some lights on the effects of wind and dissipation on the formation of rogue waves.Comment: 10 pages, 3 figure

Physical Mechanisms of the Rogue Wave Phenomenon

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin-Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrodinger equation, the Davey - Stewartson system, the Korteweg - de Vries equation, the Kadomtsev - Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon

Numerical Simulations of Water Waves\u27 Modulational Instability Under the Action of Wind and Dissipation

Since the work of Benjamin & Feir (1967), water waves propagating in infinite depth are known to be unstable to modulational instability. The evolution of such wave trains is well described through fully nonlinear simulations, but also by means of simplified models, such as the nonlinear Schrödinger equation. Segur et al. (2005) and Wu et al. (2006) studied theoretically and numerically the evolution of this instability, and both concluded that a long term restabilization occurs in these conditions. More recently, Kharif et al. (2010) considered wind forcing and viscous dissipation within the framework of a forced and damped nonlinear Schrödinger equation, and discussed the range of parameters for which this behavior is still valid. This work aims to demonstrate how numerical simulations are useful to analyze their theoretical predictions. Since we are dealing with long term stability, results are especially complicated to obtain experimentally. Thus, numerical simulations of the fully nonlinear equations turn out to be a very useful tool to provide a validation for the model. Here, the evolution of the modulational instability is investigated within the framework of the two-dimensional fully non linear potential equations, modified to include wind forcing and viscous dissipation. The wind model corresponds to the Miles theory. The introduction of dissipation in the equations is briefly discussed. The marginal stability curve derived from the fully nonlinear numerical simulations coincides with the curve obtained by Kharif et al. (2010) from a linear stability analysis. Furthermore, the long term evolution of the wave trains can be obtained through the numerical simulations, and it is found that the presence of wind forcing promotes the occurrence of a permanent frequency-downshifting without invoking damping due to breaking wave phenomenon