shallow rogue waves: observations, laboratory experiments, theories and modelling efim pelinovsky,...
TRANSCRIPT
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
16 October, 2005 Trinidad 3 m
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
Nonlinear Coordinates and velocity
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)
Nonlinear Coordinates and velocity
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
0 2 00 4 00co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 00 4 00co o rd in a te
-0 .2
0
0 .2
0 2 00 4 00co o rd in a te
-0 .2
0
0 .2
0 2 00 4 00co o rd in a te
-0 .2
0
0 .2
0 2 00 4 00co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 00 4 00co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 00 4 00co o rd in a te
-0 .2
0
0 .2
0 2 00 4 00co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 00 4 00co o rd in a te
-0 .2
0
0 .2
0 2 00 4 00co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 00 4 00co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 00 4 00co o rd in a te
-0 .2
0
0 .2
0 2 00 4 00co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 00 4 00co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 00 4 00co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 0 0 4 0 0co o rd in a te
-0 .2
0
0 .2
0 2 00 4 00co o rd in a te
-0 .2
0
0 .2
0 2 00 4 00co o rd in a te
-0 .2
0
0 .2
Demodulation: no freak wave Recurrence
0 . 0 0 . 4 0 . 8 1 . 2 1 . 6 2 . 0
wave height
0 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0d
istr
ibu
tio
n f
un
cti
on shallow
deep
Wind Wave Distribution
H
meanH
H
HF
ˆ1
2
2/ˆ1(4exp
h
HH meanˆ
0 .4 0 .8 1 .2 1 .6 2k
1 e -0 0 5
0 .0 0 0 1
0 .0 0 1
0 .0 1
0 .1
1
S (k ) K = 0 .2 7 K = 0 .2 4 K = 0 .2 0 K = 0 .1 8
Gaussian Shape
Wave Trajectories, small Ur
Wave Trajectories, Ur = 1
1 .6 2 2 .4 2 .8 3 .2
A m a x
0
0 .2
0 .4
0 .6
0 .8
1P (A m ax)
lin ea rn o n lin ea r
30
33 M
m
Moment Calculations
340
44
s
Mm
skewness
kurtosis
skewness
Relaxationto
steady state
0 2 0 4 0 6 0 8 0tim e
-0 .2
0
0 .2
0 .4
0 .6
0 .8
1
1 .2
m 3
U r= 0 .7
U r= 0 .5
U r= 0 .2
U r= 0 .9 5
kurtosis
Relaxationto steady state
0 2 0 4 0 6 0 8 0tim e
-0 .4
-0 .2
0
0 .2
0 .4
0 .6
0 .8
1
1 .2
m 4
U r= 0 .7
U r= 0 .5
U r= 0 .2
U r= 0 .9 5
0 0 .4 0 .8U r
-0 .6
-0 .4
-0 .2
0
0 .2
0 .4
0 .6
0 .8
1m
m 3
m 4
Steady State
Ur = 0.2
0 .1 1k
0 .0 0 0 1
0 .0 0 1
0 .0 1
0 .1
1S (k )
t = 0 t = 2 0t = 8 0
(a )
Ur = 1
0 .1 1k
0 .0 0 0 1
0 .0 0 1
0 .0 1
0 .1
1S (k )
t = 0 t = 1 8t = 9 0
(d )
0 .1 1k
0 .0 0 0 1
0 .0 0 1
0 .0 1
S (k )K = 0 .2 7K = 0 .2 4K = 0 .2 0K = 0 .1 8
relative spectrum width
Crest distribution
0 1 2 3A
0 .0 0 0 1
0 .0 0 1
0 .0 1
0 .1
1P (A )
R ay le ig hU r = 0 .9 5U r = 0 .5U r = 0 .0 7
North Sea (depth 7 m, Hs = 2.5 m, Ts = 8.4 sec)
0 0 .1 0 .2 0 .3 0 .4 0 .5k , m -1
0
1
2
3S (k ), m 3
(a )
0 .0 1 0 .1 1k , m -1
0 .0 0 0 1
0 .0 0 1
0 .0 1
0 .1
1
S (k ), m 3
(b )
Initial Spectra Steady State
0 0 .2 0 .4 0 .6 0 .8 1U r
-0 .4
0
0 .4
0 .8m
m o d e lle d
N o rth S ea
k u rto s is
sk ew n essUniversal
curves
Pelinovsky, E., and Sergeeva, A. Numerical modeling of the KdV random wave field. European Journal of Mechanics – B/Fluid. 2006, vol. 25, 425-434.
Sergeeva, A., Pelinovsky, E., and Talipova T. Nonlinear Random Wave Field in Shallow Water: Variable Korteweg – de Vries Framework. Natural Hazards and Earth System Science, 2011, vol. 11, 323-330
Soliton Turbulence V.E. Zakharov, Kinetic equation for solitons, Soviet JETP 60 (1971) 993-1000 A. Salupere, P. Peterson, and J. Engelbrecht, Long-time behaviour of soliton ensembles. Part 1 – Emergence of ensembles, Chaos, Solitons and Fractals, 14 (2002) 1413-1424.• Salupere, P. Peterson, and J. Engelbrecht, Long-time behaviour of soliton ensembles. Part 2 – Periodical patterns of trajectorism, Chaos, Solitons and Fractals 15 (2003a) 29-40.• Salupere, P. Peterson, and J. Engelbrecht, Long-time behavior of soliton ensembles, Mathematics and Computers in Simulation 62 (2003b) 137-147.• Salupere, G.A. Maugin, J. Engelbrecht, and J. Kalda, On the KdV soliton formation and discrete spectral analysis, Wave Motion 123 (1996) 49-66.M. Brocchini and R.Gentile. Modelling the runu-up of significant wave groups. Continental Shelf Research, 2001, vol. 21, 1533-1550.
Statistical Characteristics of Solitons: No Interaction – Linear Approach
iiii
N
i
xtKxKAtxu
22
1
4sech),( 22 ii KA
• Soliton amplitudes and phases are random and statistically independent
• Phases are uniformly distributed in domain -L/2 < x < L/2
Moments of N-Soliton Ensembles
2/1224A
L
NK
L
Nu
2/33
23
8
3
16A
L
NK
L
N
2/32/3
2/54/3
2/33
5
5
32
5
32
A
A
N
L
K
K
N
LSk
3
2/7
6
7
35
218
35
18
A
A
N
L
K
K
N
LKu
Two-soliton Interaction: elementary act of soliton turbulence
8.22
1 A
A 8.22
1 A
A
«Exchange» «Overtake»
Analysis for various ratio of soliton amplitudes
Two “first” moments (mean value and dispersion) are constant (KdV invariants)“Skewness” and “Kurtosis” are reduced when solitons interact
Extreme characteristics versus A2/A1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
10
12
A2
max
(Kur
tosis
)-m
in(K
urto
sis)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
A2
max
(Ske
wnes
s)-
min(
Skew
ness
)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.65
0.7
0.75
0.8
0.85
0.9
0.95
1
A2
min
(Y)
Skewness difference Kurtosis difference:
Wave amplitude at collision time
Not the same point
A1/A2 = 2.8!
Numerical Simulation within KdV equation
In progress
0 0 .2 0 .4 0 .6 0 .8 1U r
-0 .4
0
0 .4
0 .8m
m o d e lle d
N o rth S ea
k u rto s is
sk ew n ess
Universalcurves
Soliton Turbulence, Ur >>1
4/1~UrSk 2/1~UrKu
Conclusions:1.No rogue waves in unidirectional hyperbolic field2. Rogue Waves in Interacted Riemann Waves3. Positive skewness growing with Ur in KdV4. Sign-Variable Kurtosis via Ur in KdV5. Highest Probabilities for large Ur6. Universal curves for 3d and 4th moments7. Soliton turbulence is not Gaussian process with small variations of moments
1. Didenkulova I., and Pelinovsky E. Rogue waves in nonlinear hyperbolicsystems (shallow-water framework). Nonlinearity, 2011, 24, R1-R18. 2. Slunyaev A., Didenkulova I., Pelinovsky E. Rogue waters. ContemporaryPhysics. 2011, 52, No. 6, 571 – 590.3. Pelinovsky, E.N., and Rodin, A.A. Nonlinear deformation of a large-amplitude wave on shallow water. Doklady Physics, 2011, 56, 305-308.4. Didenkulova, I., Pelinovsky, E., and Sergeeva, A. Statistical characteristics of long waves nearshore. Coastal Engineering, 2011, 58, 94-102.5. 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 (experiment).6. Didenkulova I. Shapes of freak waves in the coastal zone of the Baltic Sea (Tallinn Bay). Boreal Environment Research, 2011, 16, 138-149.7. Nikolkina I., Didenkulova I. Rogue waves in 2006-2010. Natural Hazardsand Earth System Sciences, 2011, 11, 2913-2924.8. Sergeeva, A., Pelinovsky, E., and Talipova, T. Nonlinear random wave field in shallow water: variable Korteweg – de Vries framework. Natural Hazards and Earth System Science, 2011, 11, No. 1, 323-330.
Recent References: