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Freak Waves and Analytical
Theory of Wind-Driven Sea
V.E. Zakharov
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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 for both of them is not developed
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“New Year” wave – 1995 year
Extreme wave in the Black sea – 2002 year
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Old is always gold
Sir Francis Beaufort, FRS, FRO, FRGS,
1774, Ireland — 1857, Sussex
The Beaufort Scale is an
empirical measure for
describing wind speed
based mainly on
observed sea conditions.
Its full name is the
Beaufort Wind Force
Scale.
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),( zrZ ),( yxr
V 0divV 0
hz |
Ht
Ht
UTH
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sdsdssssGdzdrTr
)()(),(2
12
),(),( ssGssG - Green function of the Dirichlet-Neuman problem
hz | 0
zz
...210 HHHH432
k -- average steepness
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Normal variables:
*
*
||2
2
kkk
k
kkk
k
aak
i
aag
*a
Hi
t
ak
][̂]ˆ[])ˆ[̂(ˆ]ˆ[̂))((ˆ 2
212
21 kkkkkkkkt
]ˆ[]ˆ[̂]ˆ[])ˆ()[( 22
21 kkkkkgt
Truncated equations:
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),,,(),,,(
2
1
321
3
321
***
321321321
kkkkTkkkkT
bbbbTdkbbH kkkkkkkkkkkkkkk
)( 4
1233210
*
3
*
2
*
1
)4(
012312332103
*
2
*
1
)3(
0123
123321032
*
1
)2(
01231233210321
)1(
0123
12210
*
2
*
1
)3(
012122102
*
1
)2(
0121221021
)1(
012
00
bOdkbbbBdkbbbB
dkbbbBdkbbbB
dkbbAdkbbAdkbbA
ba
Canonical transformation - eliminating three-wave
interactions:
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2
4132
323241412
41
2
3131
424231312
31
2
2121
434321212
21
32324141
42423131
43432121
323241414132
2
41
424231313142
2
31
434321212143
2
21
43214
1
4321
21234
)(
))(()(4
)(
))(()(4
)(
))(()(4
))((
))((
))((
)()()(2
)()()(2
)()()(2
12)(
1
32
1
q
qqkkqqkk
q
qqkkqqkk
q
qqkkqqkk
qqkkqqkk
qqkkqqkk
qqkkqqkk
qqkkqqkk
qqkkqqkk
qqkkqqkk
qqqqqqqq
T
|| kq where
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Statistical theory of wind-driven seas
• The Hasselmann equation (1962) - kinetic
equation for water waves
nldissin SSSdt
dE
dissin SS ,
nlS
- empirical functions
- the `first principle' term
32132103210
310210321320
2
0123
)()(
)(||2
kkkkkkk ddd
nnnnnnnnnnnnTSnl
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Interaction coefficients and resonance
conditions
4-wave resonance curves
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Is there a chance
for an analytical theory?
Homogeneity properties
Exact stationary solutions
)()( 4/193kk NSNS nlnl
431
34
PgCE p
311
31
34
QgCE p
- direct cascade
Zakharov & Filonenko 1966
- inverse cascade
Zakharov & Zaslavskii 1982
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Phillips, O.M., JFM. V.156,505-531, 1985.
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The nonlinearity gives a chance
for the analytical theory
The nonlinear
relaxation rate is one
(or more) orders
higher than wind
wave pumping rate
Thus, an asymptotic model can be developed
where effect of wave-wave resonant interactions
is a dominating mechanism
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Self-similar solutions
)()( 24 q
p
qp xbaxE kk
Power-like dependencies for total energy E
and a characteristic frequency s
To check in simulations?
q
p
p
x
xE
0
0
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`Magic links' for the SS-solutions
2
110;
2
19
qp
qp
31
2
3
2
4
g
dtdE
g
E p
ss
p
Linear links of exponents
Kolmogorov-like link of energy and its flux
- pre-exponents
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),( 11/12
0
11/2 tUtn
Sea swell - no input
and dissipation
Easy to get in simulations
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Growing wind seaSelf-similarity in an explicit form
Zakharov-Zaslavskii
inverse cascade
Direct cascade of
Zakharov and Filonenko
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0
0
p
q
10
10
2
2 4
10
/ ;
/ ;
/p
xg U
g U
U g
0.6 < p < 1.1; 0.68 < 1070 < 18.6;
0.23 < q < 0.33; 10.4 < 0 < 22.6
`Scientific curves' of wave growth: wind speed scaling
Our thanks to Paul Hwang
qp,,~,~
are not universal
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`Magic links' in sea experiments
Black Sea
Babanin et al., 1996
US coast, N.Atlantic
Walsh et al 1989
Bothnian Sea, unstable
Kahma & Calkoen 1992
Bothnian Sea, stable
Kahma & Calkoen 1992
10 1
2
qp
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The `most analytical' theory
`Magic links' of our power-law self-similar
solutions can be re-written in a form of
simple algebraic relationship
3
0
4
a universal constant
pak - wave steepness
)2( xkt pp - number of waves in periods
or wave lengths
7.00
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Invariant of wave growth
• Does not contain wind speed (?!);
• Does not contain parameters of self-
similar solutions (parameter of adiabaticity
if we assume the slowly varying wave
growth conditions);
• Does not refer to initial state. Waves forget
their history
3
0
4
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How to treat the invariant?
• Lifetime is proportional to the instant
nonlinear relaxation rate
• In fact, we change a concept:
`Waves evolve on their own'
instead of
`Wind rules waves'
3
0
4
nl ~~ 4
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Does the invariant work?Collection of Paul A. Hwang of sea experiments
and his `empirical invariant'
varies e-times for 5 orders of dimensionless fetch !!!
empirical ln039.054.04 )(
empirical
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Does the invariant work?Our collection of simulations of duration-limited growth
Somewhat eclectic presentation: wind-free invariant
(ordinate) vs wind speed scaled variables (absise)
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Our wind-free invariant implies
wind-free scaling of wave growth dependencies
)8(
~;
~2Fetch
gTT
FetchH
H s
25~
~~
TH
Waves in a sector
to the off-shore direction
for up to 15 years of
measurements
030