Freak Waves and Analytical

Theory of Wind-Driven Sea

V.E. Zakharov

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

“New Year” wave – 1995 year

Extreme wave in the Black sea – 2002 year

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.

),( zrZ ),( yxr

V 0divV 0

hz |

Ht

Ht

UTH

sdsdssssGdzdrTr

)()(),(2

12

),(),( ssGssG - Green function of the Dirichlet-Neuman problem

hz | 0

zz

...210 HHHH432

k -- average steepness

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:

),,,(),,,(

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:

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

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

Interaction coefficients and resonance

conditions

4-wave resonance curves

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

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

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

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

`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

),( 11/12

0

11/2 tUtn

Sea swell - no input

and dissipation

Easy to get in simulations

Growing wind seaSelf-similarity in an explicit form

Zakharov-Zaslavskii

inverse cascade

Direct cascade of

Zakharov and Filonenko

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

`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

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

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

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

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

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)

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