Physical mechanisms of the Rogue Wave phenomenon Final Report Manuel A Andrade Mentor Dr Ildar Gabitov Math 585 1 We were in a storm and the tanker was running before the sea This amazing wave came from the aft and broke over the deck I didn t see it until it was alongside the vessel but it was special much bigger than the others It took us by surprise I never saw one again Philippe Lijour first mate of the oil tanker Esso Languedoc describing the huge wave that slammed into the ship o the east coast of South Africa in 1980 5 1 Summary In this project the rogue wave phenomenon is introduced along with its importance The main equations governing both linear and nonlinear theory are presented The three main linear theories proposed to explain the rogue rave phenomenon are presented and a linear model reproducing rogue waves due to dispersion is shown A nonlinear model for rogue waves in deep and shallow water is also exhibit 2 Introduction Seafarers speak of walls of water or of holes in the sea or of several successive high waves three sisters which appear without warning in otherwise benign conditions But since 70s of the last century oceanographers have started to believe them 4 Storm wave height can reach up to 8 10 m in the deep sea under extreme wind conditions Nevertheless observations were indeed reported for suddenly emerged huge waves on an otherwise quiet and calm background wave field in deep water Such waves are called rogue waves or freak waves monster waves giant waves steep waves etc These waves can easily reach a wave height over 10 m without any warning and thus pose great dangers to ships 6 Naval architects have always worked on the assumption that their vessels are extremely unlikely to encounter a rogue Almost everything on the sea is sailing under the false assumption that rogue waves are at worst vanishingly rare events The new research suggest that s wrong and has cost lives Between 1969 and 1994 twenty two super carriers were lost or severely damaged due to the occurrence of sudden rogue waves a total of 542 lives were lost as a result 5 Freak rogue or giant waves correspond to large amplitude waves surprisingly appearing on the sea surface Such waves can be accompanied by deep troughs holes which occur before and or after the largest crest 4 There are several definitions for such surprisingly huge waves but the one that is more popular now is the amplitude criterion of freak waves which define them as waves such height exceeds at least twice the significant wave height 2 Figure 1 New year wave in the northern sea 4 AI Hf r 2 Hs 1 Where AI abnormality index Hf r height of the freak wave and Hs significant wave height which is the average wave height among one third of the highest waves in a time series usually of length 10 30 min In that way the abnormality index AI is the only parameter defining whether the wave is rogue or not 1 3 Rogue Waves observations According to orthodox oceanography rogue waves are so rare that no ship or oil platform should ever expect to encounter one But as the shipping lanes fill with supercarriers and the oil and gas industry explores ever deeper parts of the ocean rogue waves are being reported far more often than they should The most spectacular sighting of recent years is probably the so called New Year Wave which hit Statoil s Draupner gas platforms in the North Sea on New Year s Day 1995 The significant wave height at the time was around 12 metres But in the middle of the afternoon the platform was struck by something much bigger According to measurements made with a laser it was 26 metres from trough to crest 5 Hundreds of waves satisfying condition 1 have been recorded by now and several waves with an abnormality index larger than three Ai 3 are known 1 As an example Figure 1 shows the New Year Wave with an AI 3 19 3 4 Water waves equations Force acting over most of the real fluid is composed of three contributions namely pressure force body force and viscous force Among these only viscous forces have shear forces that change the rotational status of fluid particles It is this shear forces that change the rotational status of fluid particles Therefore when viscous forces are neglected the vorticity will be neither created nor destruyed 6 In the coastal region where the water depth is from a few meters to a few tens of meters the boundary layer region is much smaller than the entire flow region Therefore it is justified to assume that the water waves can be governed by the Laplace equation based on the potential flow theory 6 Outside the boundary layer the viscous e ect diminishes toward the far field This implies that in a location away from the solid body the fluid gradually loses the driving mechanism that changes its vorticity status If the fluid flow is initially irrotational it will remain so This type of flow is called irrotational flow 6 If the flow is irrotational there exists a scalar velocity potential function that can be expressed as follows u i j k x y z 2 We can consider that water waves have been generated from a fluid that was initially at rest that is from an irrotational motion and the irrotationality condition implies that the flow satisfies the Laplace equation 2 2 2 2 2 2 0 x2 y z 3 Where is the Laplacian operator x y z cartesian coordinates velocity potential of the flow To solve 3 conditions on boundaries are needed The fluid domain that is considered is bounded by two kinds of boundaries the interface which separates the air from the water and the wetted surface of an impenetrable solid the sea bottom for instance 1 The kinematic boundary condition states that the normal velocity of the surface is equal to the normal velocity of the fluid at the surface this condition can be represented with the following equation 0 t x x y y z z 4 Where x y t represents the free surface elevation Since and are both unknown on the free surface a second boundary condition is needed the dynamic 4 boundary condition This condition is derived from the Bernoulli equation and can be expressed as follows 1 gz 0 t 2 z 5 Where g gravity acceleration 1 P gz 0 t 2 Considering the sea bottom equation z h x y 4 takes the form of the sea bottom boundary condition h h 0 x x y y z z h x y 6 Where h is the water depth The water wave problem reduces to solve the system of equations consisting of the Laplace equation 3 kinematic boundary condition 4 dynamic boundary condition 5 and sea bottom boundary condition 6 1 Although the