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Freak Waves and Wave Breaking Catastrophic Events in Ocean
V.E. Zakharov [1,2,3,4]
in collaboration with Alexander Dyachenko [3,4], Andrey Pushkarev [2,4],
Alexander Prokofiev [3,4] and Alexander Korotkevich [1,3]
1.
2.
3.
4.
University of Arizona
Lebedev Institute for Physics
Landau Institute for Theoretical Physics
Waves and Solitons LLC, W. Sereno Dr., Gilbert, AZ,
85233, USA
Let  ( r , t ) elevation of ocean surface. In the first approximation this is quasistationary, quasi homogeneous random process, close to Gaussian.
There are two types of rare catastrophic events on
the ocean surface:
1. Freak waves (major catastrophic event)
2. Wave breaking (minor catastrophic event)
Freak waves are responsible for ship-wreaking, loss of boats, cargo and lives.
Wave breaking is the most important mechanism of wave energy dissipation
and for transport of momentum from wind to ocean.
Analytic theory is for both of these are not developed
“New Year” wave – 1995 year
Extreme wave in the Black sea – 2002 year
Freak waves – program for future.
We are planning to perform massive numerical simulation of different
stationary wave spectra to determine dependence of probability of
freak wave formation on energy spectrum.
Measuring of energy spectrum is a relatively easy problem. It will
Make possible to estimate a danger of freak wave appearance
in a given place in a given time…
Wave
breaking
Evolution of the surface waves spectra is described by the
Hasselmann kinetic equation
N k
 Sin  S nl  S diss
t
Here S nl nothing but the standart quantum kinetics equation for Boson-type
Quasiparticles in the limit of large occupation numbers.
Sin
- nothing but the standart quantum kinetics equation for Boson-type
quasiparticles in the limit of large occupation numbers.
S diss
- dissipation due to wave breaking.
To find Sin we should have an adequate theory of atmospheric boundary layer
over ocean. This boundary layer is badly turbulent.
To find
S diss
we need the theory of wave breaking. It is not developed yet.
But we can perform a numerical experiment
k
R
 i ( R /U  RU / )   k R
t
V
 i (UV /  RB / )  g ( R  1)   kV
t
should be the same in both equations. The instability growth rate was
k
Magnified surface shape Y(x) for time t=2994
(start of run-off)
0.025
0.02
0.015
0.01
0.005
0
-0.005
-0.01
-0.015
-4
-3
-2
-1
0
1
2
3
4
Surface shape Y(x) for time t=2994
Similar scale for both axes
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
0
0.2
0.4
0.6
0.8
1
H(t)
0.00018
0.00016
0.00014
0.00012
0.0001
8e-05
6e-05
4e-05
2e-05
0
0
5000
10000
15000
20000
25000
30000
35000
Steepness μ(t), smoothed
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
5000
10000
15000
20000
25000
30000
35000
H (t )
t
Particular experiment
0
-2e-12
-4e-12
-6e-12
-8e-12
-1e-11
-1.2e-11
-1.4e-11
-1.6e-11
-1.8e-11
-2e-11
0
5000
10000
15000
20000
t
25000
30000
35000
Approximation of function:
H
(    0 ) 
H
  0  0.02 : f ( x)  6 106  4 102 x 2
  0  0.02 : f ( x)  2 105  5 107 x8
(numerical values obtained by least squares method)
0.00025
0.0002
0.00015
0.0001
5e-05
0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Program
Development of these experiments will make possible to determine
way and improve essentially quality of wave-forecasting models.
S diss in a reliable
Then from comparison with experimental data we will be able to estimate which
model of Sin is closer to reality.