RASA, 8-9 ноября 2014.

1. Freak Waves and Analytical
Theory of Wind-Driven Sea
V.E. Zakharov

2. 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

3. “New Year” wave – 1995 year
Extreme wave in the Black sea – 2002 year

21. 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.

22.   Z (r, z) r  (x, y)
V  divV  0   0
zh  |





H
t
H

t   
H T U

23. 1
          
T dr dz G s s s s dsds

r
( , ) ( ) ( )
2
 2
G(s, s) G(s, s) - Green function of the Dirichlet-Neuman problem

z
zh   | 0

z
... 0 1 2 H  H  H  H 
2 3 4   
  k -- average steepness

24. ˆ ( ( )) ˆ[ ˆ ] ˆ( ˆ[ ˆ ]) [ ˆ ] ˆ[ 2 ]
  k    k k  k k k  1  k  k   t
g 1 k k k k k t
Normal variables:
 *


 
 * 

2 | |
2
k k
k
k
k k
k

k
a a
k
i
a a
g
  
2 1
H

* a
i
ak
t




2
2
     [( ) ( ˆ ) ][ ˆ ] ˆ[ ˆ ][ ˆ ] 2 2
2
Truncated equations:

25. Canonical transformation - eliminating three-wave
interactions:
a b
 
A b b dk A b b dk A b b dk
  
  
  
     
B b b b dk B b b b dk
   

 
     
 
 
1
* * *
0 0
H b b dk T b b b b k k k kk k k k k k k k k k k
     
  
2
( , , , ) ( , , , )
1 2 3
3
1 2 3
1 2 3 1 2 3 1 2 3
T  k  k  k  k 

T k k k k
( ) 4
0 1 2 3 123
*
3
*
2
*
1
(4)
3 0 1 2 3 123 0123
*
2
*
1
(3)
0123
2 3 0 1 2 3 123
*
1
(2)
1 2 3 0 1 2 3 123 0123
(1)
0123
0 1 2 12
*
2
*
1
(3)
2 0 1 2 12 012
*
1
(2)
1 2 0 1 2 12 012
(1)
012
B b b b dk B b b b dk O b
  
     

26. 
1
1
  
q q q q
q q q q
 k k q q k k q q

 
 
    
2(   )   ( )  
( )
k k q q k k q q
    
2(   )   ( )  
( )
    
2( ) ( ) ( )
  
     
k k q q k k q q
  
( )( )
1 2 1 2 3 4 3 4
k k q q k k q q
    
( )( )
1 3 1 3 2 4 2 4
k k q q k k q q
    
( )( )
1 4 1 4 2 3 2 3
k k q q k k q q
(  )( 
)
2 1 2 1 2 3 4 3 4

q
k k q q k k q q
(  )( 
)
2 1 3 1 3 2 4 2 4

q
k k q q k k q q
(  )( 
)
2 1 4 1 4 2 3 2 3
 
 
 
 
 
 

2
2 3 1 4
1 4
2
1 3 1 3
1 3
2
1 2 1 2
1 2
2 3 1 4 1 4 1 4 2 3 2 3
2
1 4
2 4 1 3 1 3 1 3 2 4 2 4
2
1 3
3 4 1 2 1 2 1 2 3 4 3 4
2
1 2
1 2 3 4
4
1
1 2 3 4
1234 2
( )
4( )
( )
4( )
( )
4( )
12
( )
32
 
 
 
 
 
 

q
k k q q k k q q
T
where q | k |

27. Statistical theory of wind-driven seas
• The Hasselmann equation (1962) - kinetic
equation for water waves
dE   
in diss nl S S S
dt
in diss S , S
nl S
- empirical functions
- the `first principle' term
S   T n n n  n n n  n n n 
n n n nl
      
2 | | ( )
0 2 3 1 2 3 0 1 2 0 1 3
2
0123
k k k k dk dk dk

     
( ) ( )
0 1 2 3 0 1 2 3 1 2 3

28. Interaction coefficients and resonance
conditions
4-wave resonance curves

29. Is there a chance
for an analytical theory?
Homogeneity properties
 ( )  ( ) 3 19/ 4 S N k S N k nl nl    
Exact stationary solutions
3 4
1
3
4
 E  C g P  p
4 
11
3
1
3
3
E  C g Q  p
- direct cascade
Zakharov & Filonenko 1966
- inverse cascade
Zakharov & Zaslavskii 1982

30. Phillips, O.M., JFM. V.156,505-531, 1985.

31. 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

32. Self-similar solutions
p q E k  ax  bkx 
( ) ( ) 4 2q
p
Power-like dependencies for total energy E
and a characteristic frequency s
q
E 

x
 
p
p
0

x
 0
To check in simulations?

33. `Magic links' for the SS-solutions
Linear links of exponents
9 1 
10 1
2
;
2


 



q
p
q
p
Kolmogorov-like link of energy and its flux
- pre-exponents
3
1
2
3
E p
2
4









g
dt
dE
g
ss
p




34. Easy to get in simulations
Sea swell - no input
and dissipation
2 /11 n  t U  t 
( , ) 2 1/11
0

35. Growing wind sea
Self-similarity in an explicit form
Zakharov-Zaslavskii
inverse cascade
Direct cascade of
Zakharov and Filonenko

36. `Scientific curves' of wave growth: wind speed scaling
0
0
p
q
 
 
 
  
0.6 < p < 1.1; 0.68 < 1070 < 18.6;
0.23 < q < 0.33; 10.4 < 0 < 22.6
~,~, p,q
are not universal
2
xg U
g U
U g

 
 
/ ;
10
/ ;
2 4
10
/ p
10



Our thanks to Paul Hwang

37. `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
q
10 1
2
p



38. The `most analytical' theory
`Magic links' of our power-law self-similar
solutions can be re-written in a form of
simple algebraic relationship
 4 
3
0
p   ak - wave steepness
t (2k x) p p   - number of waves in periods
or wave lengths
 a universal constant
0.7 0  

39. Invariant of wave growth
 4 
3
0
• 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

40. How to treat the invariant?
 4 
3
0
• Lifetime is proportional to the instant
nonlinear relaxation rate
• In fact, we change a concept:
nl  ~  ~ 4
`Waves evolve on their own'
instead of
`Wind rules waves'

41. Does the invariant work?
Collection of Paul A. Hwang of sea experiments
and his `empirical invariant'
   empirical   4 0.540.039ln ( )
varies e-times for 5 orders of dimensionless fetch !!!
empirical 

42. 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)

43. Our wind-free invariant implies
wind-free scaling of wave growth dependencies
(8 )
~
;
~
2Fetch
g
T T
Fetch
H
H s

 
2
~
H T
~5
~
0 30
Waves in a sector
to the off-shore direction
for up to 15 years of
measurements