Modelling Rogue Waves Wake of Destruction Meghan Kennealy, Chelsea - PowerPoint PPT Presentation

Modelling Rogue Waves Wake of Destruction Meghan Kennealy, Chelsea Bright, Saul Hurwitz, Roy Gusinow, Kendall Born Supervised by Thama Duba MISG 2019 12 Jan 2019 1 / 43

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Types of oceanic waves Tsunamis - generated by earthquakes Surface-gravity waves - Wind-generated Rogue waves - ? 3 / 43

Rogue waves Rogue waves - also known as “freak”, “monster” or “abnormal” waves - are waves whose amplitude is unusually large for a given sea state. Unexpected and known to appear and disappear suddenly. Also occur in optical fibers, atmospheres and plasmas. 4 / 43

Size Comparison Figure: The size comparison between a large rogue wave, a seven storey building, a giraffe and an average human being. 5 / 43

The Draupner wave, New Year’s Day 1995. Using a laser, the Draupner oil platform in the North Sea measured a wave with height of 25.6m In February 2000, an oceanographic research vessel recorded a wave of height 29m in Scotland 3-4 large oil tankers are badly damaged yearly when traveling the Agulhas current off the coast of South Africa. 6 / 43

Causes of rogue waves Wave-wave interaction Wave-current interaction Spatial focusing Focusing due to nonlinearity 7 / 43

Recap - Linear Causing Mechanisms Geometrical or Spatial Focusing Wave-Current Interaction Focusing due to Dispersion 8 / 43

Solution Ψ = − Hg cosh( k ( h + z )) sin( kx − σ t ) 2 σ cosh( kh ) η = H 2 cos( kx − σ t ) at z = 0 σ 2 gk tanh( kh ) 9 / 43

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Characteristics Gaussian bell shaped Higher amplitude than normal Travels long distances without breaking Breaks inside ocean 12 / 43

Solitary Waves Solitary Waves are solutions to these equations, occurring when there is a balance of the dispersive and nonlinear effects. We are dealing with the Nonlinear Shrödinger Equation, which is considered a Non Linear Evolution Equation . 13 / 43

Korteweg-de Vries (KdV) equation Consider the Korteweg-de Vries (KdV) equation, ∂ t + ∂ 3 u ∂ u ∂ x 3 + 6 u ∂ u ∂ x = 0, (1) where ∂ 3 u ∂ x 3 is the dispersive term, which causes the wave to "spread out", whilst the nonlinear term, 6 u ∂ u ∂ x effectively causes the wave to resist this effect. This balance creates a solitary wave. 14 / 43

What is a Soliton? A soliton is a solitary wave which maintains its shape when it moves at a constant speed and conserves amplitude, shape, and velocity after a collision with another soliton. Solitons differ from breathers, which are waves that oscillate in time (breathe). Kinks represent waves with a steep inclination. All these are also solutions to nonlinear wave equations. 15 / 43

Soliton Types In deep water conditions there are three accepted solutions to the Nonlinear Schrödinger Equation in the form of solitons. Solitons are the accepted rogue wave models. The deep water condition for a rogue wave is that kh > 1.36 where k is the wavenumber and h is the water depth. 16 / 43

Assumptions and Approximations Constant density ρ - fair assumption Wavelength λ > amplitude A - fair until the wave is rogue Negligible viscosity - fair for ocean water Irrotational flow - perhaps an approximation Only body force is gravity - fair The water is deep and the bed flat -fair 17 / 43

Homogenous Nonlinear Schrödinger Equation ∂ 2 ψ i ∂ψ ∂ t + 1 � 2 ψ = 0 � � ∂ξ 2 + k � ψ 2 2 t : Time ξ : Distance 18 / 43

Coupled System - Manakov System ∂τ + ∂ 2 ψ i ∂ψ ∂ x 2 + 2 k ( | ψ | 2 +| φ | 2 ) ψ = 0 ∂τ + ∂ 2 φ i ∂φ ∂ x 2 + 2 k ( | ψ | 2 +| φ | 2 ) φ = 0 19 / 43

Coupled non-linear Shrödinger equation Figure: Collision of two waves at angel θ 20 / 43

Coupled non-linear Shrödinger equation α∂ 2 A ∂ x 2 + β∂ 2 A � � ∂ A ∂ A ∂ A ∂ y 2 + γ ∂ A ζ | A | 2 A + 2 ζ | B | 2 A � � ∂ t = − C x ∂ x − C y ∂ y + i − i ∂ x ∂ y α∂ 2 B ∂ x 2 + β∂ 2 B ∂ B ∂ B ∂ B � ∂ y 2 + γ ∂ B � � ζ | B | 2 B + 2 ζ | A | 2 B � ∂ t = − C x ∂ x − C y ∂ y + i − i ∂ x ∂ y 21 / 43

Collision of Waves The angular frequency, � ω = gk , and the magnitude of the wave vector, � k 2 x + k 2 κ = y . where, k x = κ cos( θ ), 22 / 43

Collision of Waves and, k y = κ sin( θ ). ω ( k 5 x − k 3 x k 2 y − 3 k x k 4 y − 2 k 4 x κ + 2 k 2 x k 2 y κ + 2 k 4 y κ ) ζ = 2 κ 2 ( k x − 2 κ ) 23 / 43

Collision of Waves Once the waves collide, the angular frequency becomes, � � � � � 2 + 16 ζ 2 A 2 � A 2 0 + B 2 � ξ 2 � A 2 0 + B 2 0 B 2 τ K 2 + τ K 2 ) ± Ω = ± ( ξ 0 0 0 24 / 43

Solution To Nonlinear Schrödinger Equation � � 1 + 2(1 − 2 a )cosh( b ξ ) + i b sinh( b ξ ) e i ξ , ψ ( ξ , t ) = � 2 a cos( wt ) − cosh( b ξ ) where � � b = 8 a (1 − 2 a ) w = 2 1 − 2 a 25 / 43

Waves Types by Varying a 0 < a < 0.5 − Akhmediev Breather a → 0.5 − Peregrine Soliton 0.5 < a < ∞− Kuznetsov-Ma Soliton 26 / 43

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An Upgrade: The Dysthe Model Still assumes negligible viscosity and incompressibility, solutions are still solitons and breathers that go to plane waves at ±∞ ∂ 2 φ 1 ∂ 2 φ 1 ∂ 3 φ 1 ∂ 3 φ 1 ∂ 4 φ 1 ∂φ 1 ∂ t + 1 ∂φ 1 ∂ x + i ∂ x 2 − i ∂ y 2 − 1 ∂ x 3 + 3 ∂ x ∂ y 2 − 5 i ∂ x 4 2 8 4 16 8 128 ∂ 4 φ 1 ∂ 4 φ 1 ∂ 5 φ 1 ∂ 5 φ 1 ∂ 5 φ 1 − 15 i ∂ x 2 ∂ y 2 − 3 i ∂ y 4 + i 2 | φ 1 | 2 φ 1 + 7 ∂ x 5 − 35 ∂ x 3 ∂ y 2 + 21 ∂ x ∂ y 4 32 32 256 64 64 ∂φ ∗ + 3 2 | φ 1 | 2 ∂φ 1 ∂ x − 1 ∂φ 0 1 4 φ 2 ∂ x + i φ 1 ∂ x = 0 1 30 / 43

One dimensional, nonlinear, Korteweg-de Vries The KdV equation for a soliton in shallow water in one dimension, ∂ x − h 2 gh ∂ 3 v � g ∂ v ∂ t − 3 h v ∂ v � ∂ x 3 = 0 (2) 2 6 Using the following relations to remove the dimensions, � g u = v t → 1 x → x h t h − t h 6 31 / 43

One dimensional, nonlinear, Korteweg-de Vries After making the variables dimensionless, the standard form of the KdV is, ∂ t + ∂ 3 u ∂ u ∂ t + 6 u ∂ u ∂ x 3 = 0. (3) The solution for a soliton is, � � c u ( x , t ) = − c � 2 sech 2 2 ( x − ct − x 0 ) (4) 32 / 43

Figure: The plot of the solution to the one dimensional KdV equation 33 / 43

Two dimensional, nonlinear, Korteweg-de Vries ∂ t + h ∂ 3 u = f ∂ 2 u � ∂ u � ∂ + ∂ F ( u ) (5) ∂ x 3 ∂ x 2 ∂ x 1 ∂ x 1 1 2 Where the stream function is represented by, F ( u ) = au + b 2 u 2 − 1 3 du 3 (6) 34 / 43

Two dimensional, nonlinear, Korteweg-de Vries The solution is then � u ( t , x 1 , x 2 ) = b 2 d ± k 1 6 h � d � t ( − 4 adk 2 1 − b 2 k 2 1 + 4 dfk 2 2 + 8 dhk 4 1 ) � tanh + k 1 x 1 + k 2 x 2 + k 3 (7) 4 dk 1 35 / 43

Two dimensional, nonlinear, Korteweg-de Vries The following boundary conditions are imposed on equation 7 ∂ 3 u ∂ 3 u u 0,0,0 = 0, | (0,0,0) = 0, | (0,0,0) = 0 ∂ x 2 ∂ x 2 2 2 t > 0 L 1 > 0 L 2 > 0 x 1 ∈ (0, L 1 ) x 2 ∈ (0, L 2 ) where u has the parameters u ( t , x 1 , x 2 ). 36 / 43

Two dimensional, nonlinear, Korteweg-de Vries This allows us to obtain 3 equations to solve for k 1 , k 2 , k 3 � b + 2 6 dhk 1 tanh( k 3 ) = 0 4 k 3 1 tanh 2 ( k 3 ) sech 2 ( k 3 ) − 2 k 3 1 sech 4 ( k 3 ) = 0 4 k 3 2 tanh 2 ( k 3 ) sech 2 ( k 3 ) − 2 k 3 2 sech 4 ( k 3 ) = 0 37 / 43

The 2D, nonlinear KdV equation Figure: The plot of the solution to the two dimensional KdV equation 38 / 43

The 2D, nonlinear Kadomtsev-Petviashvili equation Particular case of the KdV equation, with a term γ which is dependent on the dispersion medium ∂ t + ∂ 3 u = 3 γ 2 ∂ 2 u � ∂ u � ∂ − 6 u ∂ u (8) ∂ x 3 ∂ x 2 ∂ x 1 ∂ x 1 1 2 The solution is a hyperbolic function, u ( t , x 1 , x 2 ) = α + β tanh 2 ( k 1 x 1 + k 2 x 2 + k 0 + t ω ) (9) 39 / 43

The 2D, nonlinear Kadomtsev-Petviashvili equation Where, α = − (3 γ 2 k 2 2 − k 1 ω − 8 k 4 1 )/6 k 2 β = − 2 k 2 1 , (10) 1 The k constants can then be determined as above, with the boundary conditions 40 / 43

The 2D, nonlinear Kadomtsev-Petviashvili equation Figure: The plot of the solution to the two dimensional KP equation 41 / 43

Animations and Acknowledgements 42 / 43