Rogue Waves Alex Andrade Mentor Dr Ildar Gabitov Physical Mechanisms of the Rogue Wave Phenomenon Christian Kharif Efim Pelinovsky Friday April 16 2010 Rogue Waves in History Rogue 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 2010 Significant Wave Height Hs Significant Wave Height Hs Is 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 2010 A Rogue Wave is not a Tsunami Tsunamis 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 2010 Then 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 Hf AI 2 Hs AI Abnormality Index Rogue Wave in the North Sea AI 3 19 Hf 18 04 m Hf Friday April 16 2010 Hs Why is important its study a Norwegian dreamer 2005 c Sinking of tanker Prestige in 2002 b Norwegian tanker Wilstar 1974 d Sinking of the World Glory tanker in 1968 Friday April 16 2010 Recognition of the phenomenon b The New Year Wave AI 2 24 Hf 26 m c A hole in the sea AI 2 46 Hf 9 3 m d A freak group AI 2 23 Hf 13 71 Kharif et al 2009 Friday April 16 2010 Possible physical mechanisms of Rogue Wave Generation 1 Linear mechanisms 1 Geometrical or Spatial Focusing 2 Wave current Interaction 3 Focusing Due to Dispersion 2 Nonlinear mechanisms 1 Weakly nonlinear rogue wave packets in deep and intermediate depths Friday April 16 2010 The Water Wave problem The 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 velocity potential water surface elevation g gravity h Water depth Z position in the vertical axis Friday April 16 2010 Linear Mechanisms Linear 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 velocity potential water surface elevation g gravity h Water depth Z position in the vertical axis Friday April 16 2010 Geometrical Focusing of Water Waves Coast 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 1976 Friday April 16 2010 Geometrical Focusing of Water Waves If the water depth h h x the shallow water wave is described by the ordinary differential equation 2 d d 2 h x gh x k 0 g dx dx in the vicinity of caustics it has the form of the Airy equation k2 d2 x 0 2 dx L And its solution is described by the Airy function xk 2 3 x const Ai 1 3 L Where g gravity h Water depth water surface elevation x distance wave frequency k wave number Ai Airy function Friday April 16 2010 Wave Current Interaction Extreme 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 8 U x 1 x const Ai 3 k x x0 exp ik t k Where U velocity of the current Wave frequency water surface elevation x0 position of the blocking point wave frequency k wave number at the blocking point Ai Airy function t time Friday April 16 2010 Dispersion 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 2010 Dispersion enhancement of transient wave groups The wave amplitude satisfies the energy balance equation A2 cgr A2 0 t x and its solution is found explicitly A0 x cgr t A x t 1 t dc0 d x cgr t At each focal point the wave becomes extreme having infinite amplitude A 1 Tf t Kinematic 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 Where A Amplitude A0 Initial amplitude Cgr Velocity of the group c0 initial velocity x0 position of the blocking point Tt Focusing time Friday April 16 2010 Dispersion enhancement of transient wave groups Generalizations 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 x t k exp i kx t dk This integral can be calculated for smooth freak waves initial data for instance for a Gaussian pulse with amplitude A0 in the long wave approximation x t k A0 3 1 exp 2 2h2 ctk 2 h ct 2 6 x ct 77h2 ctk 4 x ct Ai 3 9 77h2 ctk4 h2 ct 2 This equation model the freak wave formation in a dispersive wave packet on shallow water 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 function Friday April 16 2010 Dispersion enhancement Friday April 16 2010 Nonlinear Mechanisms When 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 valid 1 ka 1 Wave steepness an important measure in deep water 2 a h 1 Important measure in shallow water Friday April 16 2010 Weakly nonlinear rogue wave packets in deep and intermediate depths Simplified nonlinear model of 2D quasi periodic deep water wave trains is based on the nonlinear Schr dinger equation i A A cgr t x 0 2 A 0 k02 2 2 A A 2 8k0 x 2 where the surface elevation is given by 1 x t A x t exp i k0 x w0 t c c 2 One solution to the nonlinear Schr dinger equation corresponds to the so called algebraic breather 4 1 2i 0 t A x t A0 exp i 0 t 1 2 2 1 16k0 x 4 0 2 t2 Where A