Rogue WavesAlex AndradeMentor: Dr. Ildar GabitovPhysical Mechanisms of the Rogue Wave PhenomenonChristian Kharif, Efim PelinovskyFriday, April 16, 2010Rogue Waves in HistoryRogue Waves have been part of marine folklore for centuries.Seafarers speak of “walls of water”, or of “holes in the sea”, which appears without warning in otherwise benign conditions.Friday, April 16, 2010Significant Wave Height HsSignificant Wave Height, HsIs the average wave height (through to crest) of the one-third largest waves.It is commonly used as a measure of the height of ocean waves.Friday, April 16, 2010A Rogue Wave is not a TsunamiTsunamis are a specific type of wave not caused by geological effects.In deep water, tsunamis are not visible because they are small in height and very long in wavelength.They may grow to devastating proportions at the coast due to reduced water depth.Friday, April 16, 2010Then, what is a Rogue Wave?Also called “Freak” or “Giant Waves”, Rogue Waves are waves whose height, Hf is more than twice the significant wave height, Hs: HsHfRogue Wave in the North SeaAI = 3.19, Hf = 18.04 mAI =HfHs> 2AI= Abnormality IndexFriday, April 16, 2010Why is important its study?d) Sinking of the World Glory tanker in 1968.c) Sinking of tanker Prestige in 2002b) Norwegian tanker “Wilstar”, 1974a) “Norwegian dreamer”, 2005Friday, April 16, 2010Recognition of the phenomenon(b) The “New Year Wave”AI = 2.24, Hf = 26 m(c) A “hole” in the seaAI = 2.46, Hf = 9.3 m(d) A freak groupAI = 2.23, Hf = 13.71Kharif et.al. , 2009Friday, April 16, 2010Possible physical mechanisms of Rogue Wave Generation1. Linear mechanisms1. Geometrical or Spatial Focusing2. Wave-current Interaction3. Focusing Due to Dispersion2. Nonlinear mechanisms1. Weakly nonlinear rogue wave packets in deep and intermediate depthsFriday, April 16, 2010The Water Wave problemThe water wave problem reduces to solve the system of equations:The difficulty in solving water wave problems arises from the nonlinearity of kinematic and dynamic boundary conditions.Where: ∆=Laplace Operator; Φ=velocity potential; η=water surface elevation; g= gravity; h= Water depth, Z= position in the vertical axis.Friday, April 16, 2010Linear MechanismsLinear theory is constructed on the assumptions:1. ka<<1 (Wave steepness; an important measure in deep water)2. a/h<<1 (Important measure in shallow water).With these assumptions, the nonlinear terms can be neglected and the corresponding system of equations to be solved is linear:Where: ∆=Laplace Operator; Φ=velocity potential; η=water surface elevation; g= gravity; h= Water depth, Z= position in the vertical axis.Friday, April 16, 2010Geometrical Focusing of Water WavesCoast shape or seabed directs several small waves to meet in phase. Their crest heights combine to create a freak wave.The result is spatial variations of the kinematic and dynamic variables of the problem.Coast of Finnmark, Norway. 1976Friday, April 16, 2010Geometrical Focusing of Water WavesIf the water depth, h=h(x), the shallow water wave is described by the ordinary differential equation:in the vicinity of caustics, it has the form of the Airy equationAnd its solution is described by the Airy functionWhere: g= gravity; h= Water depth; η=water surface elevation, x=distance, ω=wave frequency, k=wave number; Ai()=Airy function.gddx�h(x)dηdx�+�ω2− gh(x)k2�η =0d2ηdx2−k2Lxη =0η(x)=const · Ai(−xk2/3L1/3)Friday, April 16, 2010Wave-Current InteractionExtreme waves often occur in areas where waves propagate into a strong opposing current.The first theoretical models of the freak wave phenomenon considered wave current interaction.Generalizing the Airy function used for the Geometrical Focusing of Water Waves:η(x)=const · Ai�(8∂U/∂xΩ(k∗))13k∗(x − x0)�exp(ik∗− ωt)Where: U= velocity of the current; Ω=Wave frequency; η=water surface elevation, x0=position of the blocking point, ω=wave frequency, k*=wave number at the blocking point; Ai()=Airy function; t=time.Friday, April 16, 2010Dispersion enhancement of transient wave groups Waves with similar frequency will group together and separate from other wave groups. This process of self-sorting is called dispersion.Trains of waves traveling in the same direction but at different speeds pass through one another. When they synchronize, they combine to form large waves.Friday, April 16, 2010Dispersion enhancement of transient wave groups Where: A=Amplitude; A0=Initial amplitude; Cgr=Velocity of the group; c0=initial velocity; x0=position of the blocking point, Tt= Focusing time.The wave amplitude satisfies the energy balance equation∂A2∂t+∂∂x(cgrA2)=0and its solution is found explicitly,A(x, t)=A0(x − cgrt)�1+t(dc0/d(x − cgrt))At each focal point, the wave becomes extreme, having infinite amplitude A ∼1�Tf− tKinematic approach assumes slow variations of the amplitude and frequency along the wave group, and this approximation is not valid in the vicinity of the focal points.Friday, April 16, 2010Dispersion enhancement of transient wave groups η(x,t)=Water displacement; A0= Initial wave amplitude; k= wavenumber (spatial frequency of the wave in radians per unit distance); h= Water depth; c= Phase velocity; x= distance; t= time; Ai= Airy functionGeneralizations of the kinematic approach in linear theory can be done by using various expressions of the Fourier integral for the wave field near the caustics.η(x, t)=+∞�−∞η(k)exp( i( kx − ωt))dkThis integral can be calculated for smooth “freak waves” (initial data), for instance for a Gaussian pulse with amplitude, A0, in the long wave approximationThis equation model the freak wave formation in a dispersive wave packet on shallow water.η(x, t)=A0k3�h2ct2exp�12h2ctk2�x − ct +677h2ctk4��Aix − ct +977h2ctk43�h2ct2Friday, April 16, 2010Dispersion enhancementFriday, April 16, 2010When wave amplitude increases beyond a certain range, the linear wave theory may become inadequate. The reason is that those higher order terms that have been neglected in the derivation become increasingly important as wave amplitude increases. The linear theory assumptions are no longer valid1.ka<<1 (Wave steepness; an important measure in deep water)2.a/h<<1 (Important measure in shallow water).Nonlinear MechanismsFriday, April 16, 2010Weakly nonlinear rogue wave packets in deep and intermediate depthsA(x, t)=A0· exp(iω0t)�1 −4(1