1. Efim Pelinovsky, Shallow Rogue Waves: Observations,
laboratory experiments, theories
and modelling
Efim Pelinovsky,
Alexey Slunyaev,
Ira Didenkulova,
Irina Nikolkina,
Anna Sergeeva,
Tatiana Talipova
Ekaterina Shurgalina
Artem Rodin
Grant No. FP7 GA
Volkswagen
Foundation,
Germany
Department of Nonlinear Geophysical Processes,
Institute of Applied Physics, Nizhny Novgorod, RUSSIA
Wave Engineering Laboratory, Institute of Cybernetics, Tallinn, Estonia
Applied Mathematics Department, State Technical University, Nizhny Novgorod, Russia

2. Shallow water rogue waves in Taiwan
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
Taiwan (Chien et al, 2002)

3. Rogue wave in Kalk Bay on 26 August 2005
The wave over 9 m washed two people off the breakwater
in Kalk Bay (South Africa).
The wave overflows the breakwater.

4. 16 October, Trinidad 3 m

5. 16 October, Trinidad 3 m

6. 16 October, Trinidad 3 m

7. South Korea, May 4, 2008 At least eight people are reported to have
been killed after they
were swept away
by high waves.

8. the day after a very large storm had passed through.
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

9. 1) A rogue wave starts to break low on the shore
25 m

10. 25 m 2) The rogue wave races up the shore
(approximately 2 sec after the first picture)
25 m

11. 3) The rogue wave breaks over the students
who were at least 25 meters above sea level (approx. 2 sec after the previous picture)

12. Rogue Waves 2010
February 14, 2010

14. Rogue waves from mass media in 2006-2010
(Nikolkina & Didenkulova, 2011)
78 true events
106 possible

15. 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.

16. Rogue waves in : 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.

17. Rogue waves in 2006-2010: damage27 6 79 46 5 7 79 90 14

18. Rogue waves in shallow water: possible mechanisms
Wave-current interaction
Wave blocking
Random caustics
Wave-bottom (coast) interaction
Focuses
“Itself” wave dynamics
Nonlinear wave interaction
Dispersive focusing
Modulational instability
Wave-atmosphere interaction
Weak dispersion leads to relatively long life- time of individual waves, which makes them more hazardous!
The most probable mechanism of rogue wave generation in deep water does not work in shallow water!

19. Tallinn Bay, Baltic Sea, depth 2 m
I.Didenkulova,
C.Anderson,
Freak waves of different
types in the coastal zone
of the Baltic Sea.
Natural Hazards and
Earth System Sciences
2010, 10,
I.Didenkulova,
Shapes of freak waves
In the coastal zone of
the Baltic Sea
(Tallinn Bay), Boreal
Env. Research,
2011, 16,
Tallinn Bay, Baltic Sea, depth 2 m

20. Constant Depth, No boundaries
Shallow Water Equations (Hyperbolic System)
Constant Depth, No boundaries
1D Problem
H – total water depth
u – depth averaged velocity
g – gravity acceleration

21. Unidirectional Riemann Wave (right)
Velocity of
points
on wave profile

22. Wave Breaking
Wave amplitude/depth

23. “Right” Wave
“Left” Wave

24. Weak Amplitude – 0.3 m
Depth 1 m

25. Strong Amplitude – 0.9 m
Depth – 1m

26. Sign-Variable Wave

27. Sign-Variable Wave
Sign-variable wave transforms into unipolar pulse

28. No Variation in Statistics!
Random Non-breaking Riemann Wave
No Variation in Statistics!
Probability density function W
One of them
or both
are
Non-Gaussian

29. Nonlinear Shallow Water Waves on plane beach
(Carrier & Greenspan, 1958)

30. Hodograph Transformation
(Carrier & Greenspan, 1958)
Implicit form

31. x(t),r(t) u(t) Explicit solution for the moving shoreline
if an incident wave is given far from the
shoreline where it is linear
Nonlinear Coordinates
and velocity
of moving shoreline
x(t),r(t)
u(t)

32. Nonlinear Coordinates of non-moving shoreline
“Nonlinear” Moving Shoreline
Nonlinear Coordinates
and velocity
of moving shoreline
Linear Coordinates
and velocity
of non-moving shoreline
x=0, R(t), U(t)
x(t), r(t), u(t)

33. Nonlinear Coordinates of non-moving shoreline
First Result: Linear Theory predicts Maxima
for
sine
wave
Nonlinear Coordinates
and velocity
of moving shoreline
Linear Coordinates
and velocity
of non-moving shoreline
x=0, R(t), U(t)
x(t), r(t), u(t)

34. Second result: linear theory “predicts” wave breaking
The same as Jacobian
Transformation
“Linear” Vertical Acceleration = Gravity Acceleration

35. 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

36. Nonlinear “real” velocity of
the moving shoreline

37. Statistical moments for velocity
“Linear” statistics for
“nonlinear” shoreline velocity

38. Nonlinear “real” runup

39. The mean sea level onshore
The mean sea level rises – Set-Up
But rmax = Rmax and rmin = Rmin does not change
Conclusion: Flooding Time exceeds “Dry” Time

40. Standard deviation for the
sea level onshore
Nonlinearity decreases
the standard deviation
Therefore, nonlinear relation between standard
deviation and significant wave height

41. Skewness and kurtosis (> 0)
Shoreline displacement is non Gaussian process,
Shoreline velocity (its derivative) is Gaussian

42. Observations of the wave
field onshore

43. Set-up

44. Universal spectra
Explanation
from kinematics

45. Distribution function
Skewness = 0.2
Distribution function
Kurtosis = - 0.6

46. Mean level Skewness and Kurtosis
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,
Mean level
Skewness and Kurtosis
Experiments in Hannover large flume will be soon

47. KdV model for shallow water
Inverse scattering method
Discrete l - solitons
Continuous l – dispersive tail

48. solitons + evolves dispersive train disturbance
Initial
disturbance
evolves
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

49. Maximal soliton - Ursell parameter Number of solitons
Soliton-like disturbance
- Ursell parameter
Number of solitons
Maximal soliton

50. Large Ursell number max = 2 0
Inverting - no generation of freak wave!

51. Small Ursell number 0(freak) > 21 One small soliton
Freak wave is almost linear wave
in spite of its large amplitude!

52. h(x) < 0 – solitonless potential
Freak Wave as a deep hole (depression)
h(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,

53. 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)
A.R. Osborne, Numerical construction of nonlinear wave-train solutions of the
periodic Korteweg – de Vries equation, Phys. Rev. E 48 (1993)
A.R. Osborne, Behavior of solitons in random-function solutions of the periodic
Korteweg – de Vries equation, Phys. Rev. Lett. 71 (1993)
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
A.R. Osborne, and M. Petti, Laboratory-generated, shallow-water surface waves:
analysis using the periodic, inverse scattering transform. Phys. Fluids 6 (1994)
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)

54. Korteweg – de Vries equation
Modulated wave field:
no Benjamin – Feir instability

104. Demodulation: no freak wave
Recurrence

105. Wind Wave Distribution

106. Gaussian Shape

107. Wave Trajectories, small Ur

108. Wave Trajectories, Ur = 1

110. Moment Calculations
skewness
kurtosis

111. skewness
Relaxation to
steady state

112. kurtosis
Relaxation
to steady state

113. Steady State

114. Ur = 0.2

115. Ur = 1

116. relative spectrum width

117. Crest distribution

118. North Sea (depth 7 m, Hs = 2.5 m, Ts = 8.4 sec)

119. Steady State
Initial Spectra

120. Universal
curves
Pelinovsky, E., and Sergeeva, A. Numerical modeling of the KdV random
wave field. European Journal of Mechanics – B/Fluid. 2006, vol. 25,
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,

123. Soliton Turbulence
V.E. Zakharov, Kinetic equation for solitons, Soviet JETP 60 (1971)
A. Salupere, P. Peterson, and J. Engelbrecht, Long-time behaviour of soliton
ensembles. Part 1 – Emergence of ensembles, Chaos, Solitons and Fractals,
14 (2002)
Salupere, P. Peterson, and J. Engelbrecht, Long-time behaviour of soliton
ensembles. Part 2 – Periodical patterns of trajectorism, Chaos, Solitons and
Fractals 15 (2003a)
Salupere, P. Peterson, and J. Engelbrecht, Long-time behavior of soliton
ensembles, Mathematics and Computers in Simulation 62 (2003b)
Salupere, G.A. Maugin, J. Engelbrecht, and J. Kalda, On the KdV soliton
formation and discrete spectral analysis, Wave Motion 123 (1996)
M. Brocchini and R.Gentile. Modelling the runu-up of significant wave groups.
Continental Shelf Research, 2001, vol. 21,

124. Statistical Characteristics of Solitons:
No Interaction – Linear Approach
Soliton amplitudes and phases are random and
statistically independent
Phases are uniformly distributed in domain
-L/2 < x < L/2

125. Moments of N-Soliton Ensembles

126. Two-soliton Interaction:
elementary act of soliton turbulence
«Overtake»
«Exchange»
Analysis for various ratio
of soliton amplitudes

127. Two “first” moments (mean value and dispersion) are constant (KdV invariants)
“Skewness” and “Kurtosis” are reduced when solitons interact

128. Extreme characteristics versus A2/A1
Wave amplitude at collision time
Not
the same point
A1/A2 = 2.8!
Skewness difference
Kurtosis difference:

129. Numerical Simulation within KdV equation

132. In progress

133. Universal
curves
Soliton Turbulence, Ur >>1

134. Conclusions: No rogue waves in unidirectional hyperbolic field
2. Rogue Waves in Interacted Riemann Waves
3. Positive skewness growing with Ur in KdV
4. Sign-Variable Kurtosis via Ur in KdV
5. Highest Probabilities for large Ur
6. Universal curves for 3d and 4th moments
7. Soliton turbulence is not Gaussian process
with small variations of moments

135. Recent References:
1. Didenkulova I., and Pelinovsky E. Rogue waves in nonlinear hyperbolic
systems (shallow-water framework). Nonlinearity, 2011, 24, R1-R18.
2. Slunyaev A., Didenkulova I., Pelinovsky E. Rogue waters. Contemporary
Physics. 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,
4. Didenkulova, I., Pelinovsky, E., and Sergeeva, A. Statistical
characteristics of long waves nearshore. Coastal Engineering, 2011,
58,
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, (experiment).
6. Didenkulova I. Shapes of freak waves in the coastal zone of the
Baltic Sea (Tallinn Bay). Boreal Environment Research, 2011, 16,
7. Nikolkina I., Didenkulova I. Rogue waves in Natural Hazards
and Earth System Sciences, 2011, 11,
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,