Physical mechanisms of the Rogue WavephenomenonFinal ReportManuel A. Andrade.Mentor: Dr. Ildar Gabitov.Math 585.1"We were in a storm and the tanker was running before the sea. This amazing wavecame from the aft and broke over the deck. I didn’t see it until it was alongside thevessel but it was special, much bigger than the others. It took us by surprise. I neversaw one again." Philippe Lijour, first mate of the oil tanker Esso Languedoc, describingthe huge wave that slammed into the ship off the east coast of South Africa in 1980. [5]1 SummaryIn this project, the rogue wave phenomenon is introduced along with its importance. The main equationsgoverning both linear and nonlinear theory are presented. The three main linear theories proposed to explainthe rogue rave phenomenon are presented and a linear model reproducing rogue waves due to dispersion isshown. A nonlinear model for rogue waves in deep and shallow water is also exhibit.2 IntroductionSeafarers speak of “walls of water”, or of “holes in the sea”, or of several successive high waves (“threesisters”), 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 back-ground wave field in deep water. Such waves are called rogue waves (or freak waves, monster waves, giantwaves, steep waves, etc.). These waves can easily reach a wave height over 10 m without any warning andthus pose great dangers to ships.[6]Naval architects have always worked on the assu mp tion that their vessels are extremely unlikely toencounter 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 1969and 1994 twenty-two super carriers were lost or severely damaged du e to the occurrence of sudden roguewaves; 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 su ch surprisingly huge waves, but the one that is more popular now isthe amplitude criterion of freak waves, which define them as waves such height exceeds at least twice thesignificant wave height:2Figure 1: New year wave in the northern sea[4]AI =HfrHs> 2 (1)Where AI=abnormality index, Hfr=height of the freak wave, and Hs= significant wave height, which isthe 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 observationsAccording to orthodox oceanography , rogue waves are so rare that no ship or oil platform should ever expectto encounter one. But as the shipping lanes fill with supercarriers and the oil and gas industry exploresever-deeper parts of the ocean, rogue waves are being reported far more often than they should. The mostspectacular sighting of recent years is probably the so-called New Year Wave , which hit Statoil’s Draupnergas platforms in the North Sea on New Year’s Day 1995. Th e significant wave height at the time was around12 metres. But in the middle of the afternoon the platform was struck by something much bigger. Accordingto measurements made with a laser, it was 26 metres from trough to crest.[5]Hundreds of waves s atisfying condition (1) have been recorded by now, and several waves with an abnor-mality index larger than three (Ai > 3) are known. [1] As an example, Figure 1 shows the New Year Wave,with an AI =3.19.34 Water waves equationsForce 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 th at change the rotational status offluid particles. It is this shear forces that change the rotational status of fluid particles. Therefore, whenviscous 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 boundarylayer region is much smaller than the entire flow region. Therefore, it is justified to assume that the waterwaves can be governed by the Laplace equation based on the potential flow theory. [6]Outside the boundary layer, the viscous effect diminishes toward the far field. This implies that in alocation away from the solid body, the fluid gradually loses the driving mechanism that changes its vorticitystatus. If the fluid flow is initially irrotational, it will remain so. This type of fl ow is called irrotationalflow.[6]If the flow is irrotational, there exists a scalar velocity potential function φ that can be expressed asfollows:u = −∇φ = −�∂φ∂xi +∂φ∂yj +∂φ∂zk�(2)We can consider that water waves have been generated from a fluid that was initially at rest -that is, froman irrotational motion and the irrotationality condition implies that the flow satisfies the Laplace equation∂2φ∂x2+∂2φ∂y2+∂2φ∂z2= ∇2φ = �φ =0 (3)Where � is the Laplacian operator � = ∇·∇, (x, y, z)=cartesian coordinates, φ =velocity potential ofthe flow.To solve (3), conditions on bou nd aries are needed. The fluid domain that is considered is bounded bytwo kinds of bou ndaries: the interface, which separates the air from the water; and the wetted surface of animpenetrable solid (the sea bottom, for instance). [1]The kinematic boundary condition states that the normal velocity of the surface is equal to the n ormalvelocity of the fluid at the surface, this condition can be represented with the following equation∂η∂t+∂φ∂x∂η∂x+∂φ∂y∂η∂y−∂φ∂z=0,z= η (4)Where η(x, y, t) represents the free surface elevation.Since η and φ are both unknown on the free surface, a second boundary condition is needed: the dynamic4boundary condition. Th is condition is derived from the Bernoulli equation and can be expressed as follows∂φ∂t+12∇φ ·∇φ + gz =0,z= η (5)Where g=gravity acceleration.∂φ∂t+12∇φ ·∇φ +Pρ+ gz