rogue waves - university of arizona

Rogue WavesAlex Andrade

Mentor: Dr. Ildar Gabitov

Physical Mechanisms of the Rogue Wave PhenomenonChristian Kharif, Efim Pelinovsky

Friday, April 16, 2010

Rogue 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.


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.


A 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.


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:

HsHf

Rogue Wave in the North SeaAI = 3.19, Hf = 18.04 m

AI =Hf

Hs> 2 AI= Abnormality Index


Why is important its study?

d) Sinking of the World Glory

tanker in 1968.

c) Sinking of tanker Prestige in 2002

b) Norwegian tanker “Wilstar”, 1974

a) “Norwegian dreamer”, 2005


Recognition 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 = 13.71

Kharif et.al. , 2009


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


The 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.


Linear 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.


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


Geometrical Focusing of Water Waves

If 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 equation

And its solution is described by the Airy function

Where: g= gravity; h= Water depth; η=water surface elevation, x=distance, ω=wave frequency, k=wave number; Ai()=Airy function.

gd

dx

h(x)

dη

dx

+

ω2 − gh(x)k2

η = 0

d2η

dx2− k2

Lxη = 0

η(x) = const ·Ai(−xk2/3

L1/3)


Wave-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/∂x

Ω(k∗))

13 k∗(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.


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.



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(cgrA

2) = 0

and 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 ∼ 1Tf − 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.



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 function

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.

η(x, t) =

+∞

−∞

η(k)exp(i(kx− ωt))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

This equation model the freak wave formation in a dispersive wave packet on shallow water.

η(x, t) =A0

k 3

h2ct2

exp

1

2h2ctk2

x− ct+

6

77h2ctk4

Ai

x− ct+ 977h2ctk4

3

h2ct2


Dispersion enhancement


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).

Nonlinear Mechanisms


Weakly nonlinear rogue wave packets in deep and intermediate depths

A(x, t) = A0 · exp(iω0t)

1− 4(1 + 2iω0t)

1 + 16k02x2 + 4ω0

2t2

Where: A=Amplitude; A0=Initial amplitude; k0= Wavenumber; ω0=Frequency of the carrier wave; h= Water depth; cgr= Group phase velocity; c.c=Complex conjugate; (...)=weak highest armonics of the carrier wave.

i

∂A

∂t+ cgr

∂A

∂x

=

ω0

8k20

∂2A

∂x2+

ω0k202

|A|2 A

Simplified nonlinear model of 2D quasi-periodic deep-water wave trains is based on the nonlinear Schrödinger equation:

where the surface elevation is given by

η(x, t) =1

2(A(x, t)exp(i(k0x− w0t)) + c.c+ · · · )

One solution to the nonlinear Schrödinger equation corresponds to the so-called algebraic breather


Weakly nonlinear rogue wave packets in deep and intermediate depths

(Algebraic breather graph)


Future WorkWeakly nonlinear rogue waves in shallow water.

Conclusions1. Precise physical mechanisms causing the rogue

waves phenomenon are still unknown.

2. Rogue Waves should be considered to reduce the number of ships sinked worldwide.

3. Main physical mechanisms believed to produce rogue waves were presented.

4. The dispersion enhancement mechanisms has been used to create Rogue Waves in lab conditions.


References

1. A. S. Christin Kharif, Efim Pelinovsky. Rogue Waves in the Ocean. Springer Berlin Heidelberg, 2009.

2. C. Kharif and E. Pelinovsky. Physical mechanisms of the rogue wave phenomenon. European Journal of Mechanics - B/Fluids, 22(6):603 – 634, 2003.

3. http://en.wikipedia.org/wiki/Freak_wave. Last checked: 02/18/2010

4. http://en.wikipedia.org/wiki/Significant_wave_height Last checked: 04/7/2010


http://en.wikipedia.org/wiki/Freak_wave

http://en.wikipedia.org/wiki/Freak_wave

http://en.wikipedia.org/wiki/Significant_wave_height

http://en.wikipedia.org/wiki/Significant_wave_height

rogue waves - university of arizona

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