Influence Of Currents On Freak Waves Based On Peregrine Breather

Freak waves or rogue waves,which appear sporadically and subsequently disappear without a trace,can endanger ships and marine structures due to their extraordinarily large wave heights.One of the most likely mechanisms is the modulational instability,the modulational instability of waves can be modeled well by the nonlinear Schrodinger(NLS)equations.Peregrine(1983)obtained an analytical solution of the NLS equation,which is called Peregrine Breather(PB)solution,the PB is localized in both space and time and the carrier wave surface elevation should be amplified by a factor of 3,so it can be considered as a prototype of freak waves.Because waves nearly always coexist with currents in the ocean,wave-current interactions are considered as an important research direction of ocean engineering studying.The currents are always not uniform with depth,however,very little research has been done about the freak waves on the shear currents,many problems remain to be solved.The main objective of the present study is to research the influence of currents on the freak waves by combining theoretical derivation and physical experiments.Firstly,a nonlinear Schrodinger equation for the propagation of two-dimensional surface gravity waves on linear shear currents in finite water depth(SCNLS)is derived.Next,using the equation,the properties of the modulational instability of gravity waves on linear shear currents are investigated.It is shown that shear currents significantly modify the modulational instability properties of waves:for waves propagating in the same direction as the uniform current,the current was found to decrease the growth rate of the instability,for a uniform opposing current,the current was found to increase the growth rate of the instability.For waves on shearing currents,positive vorticity tends to enhance the sideband instabilities,whereas negative vorticity is prone to suppress the instability.In some cases,the vorticity of a linear shear current can counteract the effects of the corresponding depth-uniform current.Furthermore,the influence of linear shear currents on the PB is also studied.It is demonstrated that depth-uniform opposing currents can reduce the breather extension in both the time and spatial domain in intermediate water depth,but following currents have the adverse impact,suggesting that wave packets with freak waves formed on following currents contains more hazardous waves in finite water depth.Lastly,a nonlinear Schrodinger equation for the propagation of three-dimensional surface gravity waves on linear shear currents in finite water depth(TSCNLS)is derived,and the horizontal surface current is assumed to be stationary and slowly varying spatially.Using the equation,the properties of the modulational instability of gravity waves on linear shear currents are investigated,a formula of the instability growth rate of the perturbations is obtained,results suggesting that the instability region decreases with decreasing water depth;for a uniform following current,the current was found to decrease the instability region;for a uniform opposing current,the current was found to increase the instability region,but the vorticity has the inverse inpact.Next,in order to verify the effectiveness of the theoretical derivation,a series of laboratory experiments were performed to study the PB evolution in a wave flume of finite depth and deep water.The experimental results show that the maximum wave amplification is closely related to the initial Ursell number and the water depth.If the initial Ursell number larger than 0.05,the maximum crest amplification can exceed three.For the cases with nearly initial Ursell number 0.05,the distance that the group has to travel to achieve its maximum wave height will decrease significantly as the water depth increases.Additionally,the experimental results are compared with computations based on both the nonlinear Schrodinger(NLS)equation and the Dysthe equation,both with a dissipation term.It is found that both models with a dissipation term can predict the maximum amplitude amplification of the primary waves.However,the Dysthe equation also can predict the group horizontal asymmetry.A series of laboratory experiments were conducted to study the evolution of PB in a wave flume in finite depth,and wave trains were initially generated in a region of quiescent water and then propagated into an adverse current region where the current velocity strength gradually increased from zero to an approximately stable value.The experimental results showed that the characteristic spectrum of the PB persists even on strong currents,thus making it a viable characteristic for prediction of freak waves(the specific triangular spectra of freak waves can be detected at early stages of their development in a chaotic wave field).It was also found that the adverse currents tend to shift the focusing point upstream compared to the cases without currents.The evolution of PB on depth-sheared adverse currents and depth-uniform adverse currents was experimentally investigated.The wave train was generated on quiescent water and then entered a region with adverse currents.According to the SCNLS equation,a qualitative analysis about the effect of the adverse current on the evolution of the PB is present.Experiment results show that in the presence of sheared adverse currents and depth-uniform currents,the dynamics of the wave groups both are modified significantly,currents can accelerate the development of wave evolution,and the extreme waves occurred.The contribution of the higher harmonics to the extreme wave is significant for the cases with the adverse sheared currents.The experimental results are compared with the PB of the SCNLS equation.Due to the effect of the nonlinearity and the dissipation,it is found that the maximum amplification in the experiment is attained at a longer distance compared with the PB prediction,if the initial steepness is small or in deeper water depth,the theoretical results compares reasonably well with the experimental results.