shallow rogue waves: observations, laboratory experiments, theories and modelling efim pelinovsky,...

Shallow Rogue Waves: Observations, Shallow Rogue Waves: Observations, laboratory experiments, theories laboratory experiments, theories and modellingand modelling Efim Pelinovsky,

Alexey Slunyaev,Ira Didenkulova, Irina Nikolkina,Anna Sergeeva, Tatiana TalipovaEkaterina ShurgalinaArtem Rodin

Department of Nonlinear Geophysical Processes,Institute of Applied Physics, Nizhny Novgorod, RUSSIA

Grant No. FP7 GA-234175

Volkswagen Foundation, Germany

Wave Engineering Laboratory, Institute of Cybernetics, Tallinn, Estonia

Applied Mathematics Department, State Technical University, Nizhny Novgorod, Russia

Taiwan (Chien et al, 2002)

Historical “Rabid-Dog Waves” 50-years Data Base:140 eyewitness reports from 1949 – 1999

More than 496 people lost their lives and more than 35 crafts were capsized due to nearshore freak waves

Shallow water rogue waves in Taiwan

The The wave over 9 m washed two people off the breakwaterwave over 9 m washed two people off the breakwaterin Kalk Bay (South Africa). in Kalk Bay (South Africa).

The wave overflows the breakwater.The wave overflows the breakwater.

Rogue wave in Kalk Bay on 26 August 2005

16 October, 2005 Trinidad 3 m

At least eight people are reported to have been killed after they were swept away by high waves.

South Korea, May 4, 2008South Korea, May 4, 2008

In October, 1998, thirteen students in the Bamfield Marine Station Fall Program were taken on a field trip to Kirby Point, a wave-beaten peninsula on the southwest corner of Dianna Is. (Barkley Sound, Vancouver Island, British Columbia),

to view the large open-ocean swell breaking on the shore the day after a very large storm had passed through. The students split into two groups and sat atop two adjacent rock outcrops, at least 25 meters above sea level

After about 45 minutes of wave watching, one student tried to capture the feel of these huge waves thundering onto the shore by taking three pictures in quick succession of what looked to be a nice example of a large wave as it started to break

1) A rogue wave starts to break low on the shore1) A rogue wave starts to break low on the shore

25 m

2) The rogue wave races up the shore2) The rogue wave races up the shore (approximately 2 sec after the first picture)

25 m

3) The rogue wave breaks over the students3) The rogue wave breaks over the students who were at least 25 meters above sea level

(approx. 2 sec after the previous picture)

Rogue Waves 2010 February 14, 2010

Rogue waves from mass media in 2006-2010(Nikolkina & Didenkulova, 2011)

78 true events

106 possible

Rogue waves in 2006-2010: shallow water

14 of 30 events led to the damage of the vessel, 7 events – to its loss (in deep waters only 5 ships were damaged). These events are also

associated with extremely high number of human fatalities (79 persons) and injuries (90 persons). For comparison, the number of

human losses in the deep water area is significantly less: 6 fatalities and 27 injuries.

In August 2010 the ship carrying 60 people (only 21 rescued) capsized and sank minutes before arriving in harbour.

Another large loss of lives (11 fatalities) occurred in this area when a fishing boat “Jaya Baru” was engulfed by 6 m waves in May, 2007.

Rogue waves in 2006-2010: coast

Totally, during 2006–2010, 39 such events were

reported, which caused 46 fatalities and 79 injuries. Usually such waves appear

unexpectedly in calm weather conditions and result in the washing the person off to the sea.

Rogue waves in 2006-2010: damage

75

149079

7946

6

27

Rogue waves in shallow water: possible mechanisms

The most probable mechanism of rogue wave generation in deep water does not work in shallow water!

Weak dispersion leads to relatively long life-time of individual waves, which makes them more hazardous!

0 10 20 30 40 50 60-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (sec)

Sea

leve

l (m

)

0 10 20 30 40 50 60-0.4

-0.2

0

0.2

Time (sec)

Sea

leve

l (m

)

Tallinn Bay, Baltic Sea, depth 2 m

I.Didenkulova,Shapes of freak wavesIn the coastal zone ofthe Baltic Sea (Tallinn Bay), Boreal Env. Research,2011, 16, 138-148

I.Didenkulova, C.Anderson,Freak waves of differenttypes in the coastal zoneof the Baltic Sea.Natural Hazards and Earth System Sciences

2010, 10, 2021-2029

Shallow Water Equations (Hyperbolic System)

Constant Depth, No boundaries

0

Huxt

H0

2

)( 22

gHHu

xt

uH

1D Problem

H – total water depthu – depth averaged velocityg – gravity acceleration

Unidirectional Riemann Wave (right)

0)(

x

HHV

t

H

ghgHu 2

ghgHu

ghV 232

3

Velocity ofpoints

on wave profile

tHVxHtxH )(),( 0

Wave Breaking

dx

xdH

dH

dVTbr )(

max

1

0

0 0.2 0.4 0.6 0.8 1Àì ï ë è òóä à

0

2

4

6

8

10

X

Wave amplitude/depth

/brcTX

0 1 2 3 4 5

Ãë óá è í à, H/h

-2

-1

0

1

2

3

4

5V

(H)/

c “Right” Wave

“Left” WavehHH cr 9

4

ghgHu

ghV 232

3

Weak Amplitude – 0.3 mDepth 1 m

Strong Amplitude – 0.9 mDepth – 1m

Sign-Variable Wave

Sign-Variable Wave

Sign-variable wave transforms into unipolar pulse

Random Non-breaking Riemann Wave

ghgHu 2

tHVxHtxH )(),( 0

No Variation in Statistics!

Probability density function W

|/|)()( dudWuW One of them

or bothare

Non-Gaussian

Nonlinear Shallow Water WavesNonlinear Shallow Water Waveson plane beachon plane beach

0

uxxt

0

xg

x

uu

t

u

xxh (Carrier & Greenspan, 1958)

Hodograph Transformation

01

2

2

2

2

1

u

2

2

1u

g

22

1 22

u

gx

11

gt

0)(2 hg

Implicit form

(Carrier & Greenspan, 1958)

L(t)

r(t) = L(t)

R(t)

h

Explicit solution for the moving shoreline if an incident wave is given far from the

shoreline where it is linear

x(t),r(t)x(t),r(t)u(t)u(t)

Nonlinear Coordinates and velocity

of moving shoreline

“Nonlinear” Moving Shoreline

Linear Coordinates and velocity

of non-moving shoreline

x=0, R(t), U(t)x=0, R(t), U(t)

g

utUtu

)(

g

u

g

utRtr

2)(

2


of moving shoreline

x(t), r(t), u(t)x(t), r(t), u(t)

First Result: Linear Theory predicts Maxima

maxmax Uu

maxmax Rr 00

22

L

H

R

forsinewave

Linear Coordinates and velocity

of non-moving shoreline

x=0, R(t), U(t)x=0, R(t), U(t)


of moving shoreline

x(t), r(t), u(t)x(t), r(t), u(t)

Second result: linear theory “predicts” wave breaking

g

utUtu

)(

gUU

t

u

1

“Linear” Vertical Acceleration = Gravity Acceleration

1max1

2

2

2

dt

Rd

gBr

0),(

),(

II

xt

The same as Jacobian Transformation

For irregular runup:Linear theory for nonlinear

extreme runup characteristics

• Nonlinearity does not contribute in

Distribution Functions of Heights

For narrow-band Gaussian Process –

Rayleigh Distribution

• Significant Wave Height does not

change

Nonlinear “real” velocity of the moving shoreline

g

utUtu

)(

gUU

t

u

1

1max1

2

2

2

dt

Rd

gBr

t

U(t

), u

(t)

dtg

utU

Tdttu

Tu

Tn

Tnn

00

11

Statistical moments for velocity

dd

dU

gU

Tu

Tnn

0

11

1

nn Uu“Linear” statistics for “nonlinear” shoreline velocity

Nonlinear “real” runup

g

u

g

utRtr

2)(

2

t

R(t

), r

(t)

g

Ud

d

dUR

TgdR

T

g

udt

g

utR

Tr

T T

T

2

11

2

1

2

0 0

0

2

)()(

The mean sea level onshore

gg

Ur U

22

22

The mean sea level rises – Set-UpBut rmax = Rmax and rmin = Rmin does not change

Conclusion: Flooding Time exceeds “Dry” Time

Standard deviation for the sea level onshore

222

2222

4

rR

g

URr

Nonlinearity decreases the standard deviation

22222 2 rrr Rr

Therefore, nonlinear relation between standard deviation and significant wave height

2/322

3

3

3

32

8

r

rrr

Rr

Skewness and kurtosis

222

222

4

4

42

2343

r

rrrr

R

R

r

Shoreline displacement is non Gaussian process,Shoreline velocity (its derivative) is Gaussian

(> 0)

Observations of the wave field onshore

Set-up

t

R(t

), r

(t)

Universal spectra

4~)( S

Explanationfrom kinematics

Distribution functionSkewness = 0.2

Kurtosis = - 0.6

Denissenko P., Didenkulova I., Pelinovsky E., Pearson J. Influence of the nonlinearity on statistical characteristics of long wave runup. Nonlinear Processes in Geophysics, 2011, 18, 967-975.

Experiments in Hannover large flume will be soon

Mean level Skewness and Kurtosis

t

ch x

ch

x

13

2 60

2 3

3

KdV model for shallow water

Inverse scattering method

0),(2

2

txU

dxd

323h

U

Discrete - solitons

Continuous – dispersive tail

Initialdisturbance

evolves solitons +dispersive train

Delta-function (as model of the freak wave) evolves in one soliton and dispersive train

Inverted (in x) dispersive train + soliton will evolve in delta-function

But delta-function is not weak nonlinear and dispersive wave

Soliton-like disturbance

Url

h 0

2

3 - Ursell parameter

N EUr

3

2

1

4

1

21Number of solitons

m Ur

Urm

4

3

3

2

1

4

1

20

2

Maximal soliton

Large Ursell number

max = 2 0

Inverting - no generation of freak wave!

Small Ursell number

1 03 Ur One small soliton

Ur 1

6

Freak wave is almost linear wave in spite of its large amplitude!

0(freak) > 21

Freak Wave as a deep hole (depression)

(x) < 0 – solitonless potential

Only dispersive tail

Similar to linear problem

No limitations on characteristics of deep holes!

Pelinovsky, E., Talipova, T., and Kharif, C. Nonlinear dispersive mechanism of the freak wave formation in shallow water. Physica D, 2000, vol. 147, N. 1-2, 83-94.

Korteweg – de Vries equation Periodic boundary conditions – cnoidal waves

Representation of the solutions through Theta-Function (Matveev, Osborne et al)

A.R. Osborne, E. Segre, and G. Boffetta, Soliton basis states in shallow-water ocean surface waves, Phys. Rev. Lett. 67 (1991) 592-595.A.R. Osborne, Numerical construction of nonlinear wave-train solutions of the periodic Korteweg – de Vries equation, Phys. Rev. E 48 (1993) 296-309. A.R. Osborne, Behavior of solitons in random-function solutions of the periodicKorteweg – de Vries equation, Phys. Rev. Lett. 71 (1993) 3115-3118.A.R. Osborne, Solitons in the periodic Korteweg – de Vries equation, the -function representation, and the analysis of nonlinear, stochastic wave trains, Phys. Review E 52 (1995) N. 1 1105-1122.A.R. Osborne, and M. Petti, Laboratory-generated, shallow-water surface waves:analysis using the periodic, inverse scattering transform. Phys. Fluids 6 (1994) 1727-1744.A.R. Osborne, M. Serio, L. Bergamasco, and L. Cavaleri, Solitons, cnoidal waves and nonlinear interactions in shallow-water ocean surface waves, Physica D 123 (1998) 64-81.

Korteweg – de Vries equation

Modulated wave field:no Benjamin – Feir instability

)sin(1)0,( 0 KxmAxA

062

31

3

32

y

ch

yhc