Introduction to Wind Wave Model Formulation

  

Igor V. Lavrenov

 Prof. Dr., Head of Oceanography Department

 State Research Center of the Russian Federation

Arctic and Antarctic Research Institute,Bering 38, 199397, St.Petersburg, Russia, e-mail: lavren@aari.nw.ru.

(AARI)

Contents

Introduction

1. Hydro dynamical (Classical) Problem Formulation of Wind Wave Modelling 1.1 Main equations of self-consistent motion of the water-air system 1.2 Equations for the water motion 1.3 Air motion description 1.4 Boundary conditions in the interface air-water 1.5 Difficulties of obtaining deterministic problem solution for wind wave modeling

2. Wind Wave Energy Balance Equation (Spectral Approach)

2.1 Probability description of sea surface 2.2 Wind wave spectrum description 2.3 Energy balance equation for wind wave spectrum evolution 2.4 Parameterisations of different physical mechanisms forming wind wave spectrum 3. Wind Wave Model Input and Output Parameters

Conclusion

Part 1. Hydro dynamical Problem Formulation of Wind Wave Modelling

1.1 Main equations of self-consistent motion of the water-air system The law of mass conservation and momentum:

ddiv( ) 0

di

i it

U

, (1.1)

where i is air density (i=1) or water density (i=2) respectively; iU is a velocity of medium motion. If the fluid density remains constant, the equation (1.1) is simplified as:

div( ) 0i U . (1.2)

The equation of momentum conservation, referred to the axes immovably connected with the rotating Earth, has a form of:

d[ ] grad( )

di

i i i i i iPt

U

ΩU g F (1.3)

The first term is an inertia force of the mass acceleration; the second one containing a rotation vector Ω or a double angular velocity of the Earth's rotation is the Coriolis force.

The term iF in (1.3) is a resultant of all forces acting onto a unit fluid volume. The viscosity effects being significant, water may be considered as isotropic incompressible fluid and a stress tensor may be written in the form of:

2ij ij ijP p e (1.4)

where ij is a single tensor ( 1ij at i=j, otherwise 0ij ); is the viscous fluid coefficient:

1

2ji

ijj i

UUe

x x

, (1.5)

where ije is a deformation velocity tensor, the incompressibility condition (1.2) is realised the friction force per unit volume is equal to:

2 ij ijij

ij ij

e UF

x x

. (1.6)

The two-layer model with discontinuities in the density and the kinematic coefficient of viscosity at the mobile interface surface ( , )t r is considered as:

3 -3 1 2 1a a

-3 2 2 1w w

1.2 10 g sm 1.5 10 sm s with;

1.0 g sm 1.0 10 sm s with

z

z

.

(1.7) The lower fluid is assumed to be immovable at the initial time moment:

( , , 0) 0, ( , 0) 0.z t t U r r (1.8)

Due to significant difference in the a, a and w, w values (ww / aa >100) , the natural simplifications in the general equations (1.1)–(1.3) at z > and z < are different.

1.2 Equations for the water motion (classical theory of potential waves) The velocity field should be presented in the form of grad( ) , U V where is

the velocity potential and rot ( )V A is its solenoidal (vorticity) component

rot ( ) ( ) U A . Then div ( ) ( ) 0 and ( ) ( ) U U V , i.e. a viscous force is determined only by the vorticity component. It is important only in thin boundary layers above the water surface and near bottom. The water motion can be considered as potential and the dynamic equations take the following form at z < :

2

210

2

Pgz

t z

; (1.9)

2

20

z

, (1.10)

where and are the horizontal differential operators.

1.3 Air motion description

Unlike the water motion description, the viscous terms and flow vorticity are essential in the motion equations of atmospheric boundary layer. The air flow velocity U is represented in the form of three items:

1 2 3 U U U U , (1.11)

where 1U is an averaged flow velocity; 2U is a deviation from 1U

created by waves at the water surface; 3U is random turbulent velocity fluctuations determined by equations of closing (Phillips, 1980). It is assumed that the air flow at the initial stage of its development is the usual turbulent boundary layer above a rigid wall. Further, the usual assumptions for the theory of logarithmic boundary layer on a wall should be considered to be fulfilled, so that the friction velocity *U can be assigned at a distance from the mobile underlying surface.

1.4 Boundary conditions in the interface air-water The problem of a self-consistent motion of the water-air, (i.e. two-layer medium) is solved using a kinematic marginal condition and the condition of normal stress continuity at the interface z =

1/ 22

a w 1U Ut

, (1.12)

1/ 22

a w 1P P

, (1.13)

where ~ 10 cm3 s-2 is the coefficient of surface tension at the water-air interface normalised by . The value Pа (at z=) should be determined in (1.13) using the solution of equations for random hydrodynamic fields Uа and Pа of boundary layer in atmospheric. But the pressure Pw (at z=) can be directly expressed by the potential velocity derivatives (1.9).

1.5 Difficulties of obtaining deterministic problem solution for wind wave modeling

A complete system of the equations (1.3), (1.9)–(1.13) for determining the surface evolution at the initial conditions (1.8) presents considerable difficulty for the analysis. Unlike to the usual classical theory of potential waves with the given pressure distribution Pа at the surface , either the surface itself or the pressure are not determined in the wind wave theory. These two unknown functions are not independent, that is why a co-ordinated solution of both equations (1.9)–(1.12) for wave disturbances at z< and complicated equations of vortex current above the wave boundary at z> is required for surface determining problem.

Part 2. Wind Wave Energy Balance Equation

(Spectral Approach)

0 5 10 15 20 25

tim e(s)

-10

-5

0

5

10

W ave period (s)

W ave height (m)

Wind wave time series

2.1 Probability description of sea surface

An obvious feature of the wind waves is their random character. Probability wind wave model Longuet-Higgins M.S. (1960)

i

iiyixii tykxkatyx )sin(),,( (2.1)

where ),,( tyx is Gaussian probability process with spectral density

i

iyxyx adkdkkkS 2),( 2 (2.2)

Wave height distribution for narrow case is described by Rayleigh law:

4exp)8exp()(2

02 hhmhhF (2.3)

where

0

0 ),( yxyx dkdkkkSm

0 1 2 3 4 5

h/h(m ean)

1e-009

1e-008

1e-007

1e-006

1e-005

0.0001

0.001

0.01

0.1

1

F(h

)

Rayleigh function of wave height distribution

I t m e a n s t h a t m e a n w a v e h e i g h t o f N1 l a r g e s t w a v e f o r 1N i s e q u a l t o

2/lnln22 3101 NhNmh N ( 2 . 4 )

w h e r e 0031 4005.4 mmh

T h e l a r g e s t o n e o f t h o u s a n d w a v e i s a c c e p t e d a s m a x i m u m w a v e

hhhh 0.35.19.1 %33110001 ( 2 . 5 )

2.2 Spectrum Wind Wave Description

The wind waves present a non-stationary probabilistic hydrodynamic

process. The displacement of the water-air interface ( , )t r is considered as a random moving surface. Probabilistic distributions of the values at the

finite space and time sets ,n ntr (n =1,2) is a study objects. The

distribution of probabilities for value at a fixed point is approximated by the Gaussian distribution.

The correlation function is introduced as :

( , ) ( , ) ,K t t t r r r (2.6)

where cornered parentheses mean averaging by a statistical ensemble.

T h e s p a t i a l - t e m p o r a l c o r r e l a t i o n f u n c t i o n ( , )K t r i s c o n n e c t e d w i t h

t h e s p e c t r u m ( , )S k o f a r a n d o m p r o c e s s b y t h e F o u r i e r t r a n s f o r m a t i o n :

td ΔΔt rkrk dtiKS )(exp

2

1),(

3

r, ( 2 . 7 )

T h e d i s p e r s i o n o f s u r f a c e w a v e s < 2 > i s c a l c u l a t e d i n t e g r a t i n g t h e

s p e c t r u m ( , )S k o v e r t h e t w o - d i m e n s i o n a l w a v e v e c t o r k a n d t h ef r e q u e n c y .

A t w o - d i m e n s i o n a l w a v e s p a t i a l s p e c t r u m ( )S k i s d e t e r m i n e d b y t h ee q u a t i o n ( 2 . 7 ) a c c o r d i n g t o t h e f o r m u l a :

2

1( ) ( , ) d ( , 0 ) e x p ( ) d

( 2 )S S K i

k k r k r r ( 2 . 8 )

a n d t h e f r e q u e n c y s p e c t r u m S ( ) i s d e t e r m i n e d a s :

1( ) ( , ) d ( 0 , ) e d

2i tS S K t t

k k . ( 2 . 9 )

T h e s e c o n d m o m e n t s o r t h e i r c o r r e s p o n d i n g s p e c t r a a r e k n o w n t o g i v ec o m p l e t e s t a t i s t i c a l i n f o r m a t i o n a b o u t a r a n d o m f i e l d , i n c a s e o f t h eG a u s s i a n p r o c e s s ( D a v i d a n e t a l . , 1 9 7 8 ) . T h i s d e t e r m i n e s t h e i m p o r t a n c e o ft h e s p e c t r a l w a v e c h a r a c t e r i s t i c i n f o r m a t i o n , a s s o o n a s t h e e x p e r i m e n t a ld a t a o f d i s t r i b u t i o n f u n c t i o n a l l o w t o c o n s i d e r t h e r a n d o m f i e l d o f l e v e ld i s t u r b a n c e s a s b e i n g a p p r o x i m a t e l y G a u s s i a n . F o r t h e g i v e n s p e c t r u m t h eG a u s s i a n s u r f a c e m o d e l c a n b e a b a s i s f o r o b t a i n i n g s t a t i s t i c a l i n f o r m a t i o na b o u t t h e g e o m e t r i c c h a r a c t e r i s t i c s o f a m o v i n g r a n d o m s u r f a c e : m e a nn u m b e r o f s t a t i o n a r y p o i n t s ( m a x i m u m s , m i n i m u m s , h y p e r b o l i c p o i n t s , e t c . )p e r u n i t s u r f a c e , s t a t i s t i c a l d i s t r i b u t i o n s o f m a x i m u m a n d m i n i m u m h e i g h t se t c . ( W . P i e r s o n , Y u . K r y l o v , M . L o n g u e t - H i g g i n s i n t h e 1 9 6 0 s a n d l a t e r b yV . R o z h k o v a n d Y u . T r a p e z n i k o v i n 1 9 9 0 ) .

2.3 Typical form of wind wave spectrum JONSWAP Spectrum Approximation

The most typical frequency-angular approximations ( , )S given in the following form:

( , ) ( ) ( , )S S Q , (2.10)

where ( )S is the frequency spectrum; ( , )Q is the energy angular distribution.

The frequency approximation in the form of the JONSWAP spectrum (Hasselmann et al.,1973) is used as:

4max

2 2 2max max

5exp ( ) (2 )2 5 4( ) e JS g

, (2.11) where

0.07 at 1 ;

0.09 at 1 ;J

max

.

71 is spectrum peakness

1 2 3 4

frequency

0

0.2

0.4

0.6

0.8

1

Sp

ect

rum

JO N SW AP spectrum

S~1/w **5

JONSWAP Spectrum approximation

1

frequency

1e-007

1e-006

1e-005

0.0001

0.001

0.01

0.1

1

10

Spe

ctru

m

JO N SW AP spectrum

S~1/w **5

JONSWAP Spectrum Approximationalgorithmic scale

T h e e n e r g y a n g u l a r d i s t r i b u t i o n i s u s e d c o n s e c u t i v e l y i n t h e f o r m o f t w oa p p r o x i m a t i o n s , o n e o f t h e m b e i n g a n o r d i n a r y c o s i n e e n e r g y d i s t r i b u t i o n :

1

2

( 1 )c o s ( ) a t 2 ;

( , ) 2 ( 2 1 )

0 a t 2 .

n

n

n

Q n

( 2 . 1 2 )

T h e s e c o n d a n g u l a r d i s t r i b u t i o n i s u s e d i n t h e f o r m o b t a i n e d a c c o r d i n g t o t h eJ O N S W A P e x p e r i m e n t a l d a t a ( H a s s e l m a n n D . e t a l . , 1 9 8 0 ) :

2 1 2 2( , ) 2 ( 1 ) ( 2 1 ) c o s ( ( ) 2 )s sJQ s s , ( 2 . 1 3 )

w h e r e m a xs s ; m a x 9 . 7 7 4s ;

4 . 0 6 f o r 1 a n d 2 . 3 4 i n a l l o t h e r c a s e s .

a )

b )

JONSWAP Spectrum approximationwith cos angular distribution with n=12

a)

b )

JONSWAP Spectrum approximationwith cos angular distribution with n=2

a )

b )

JONSWAP Spectrum approximationwith JONSWAP angular distribution

2.4 Kinetic Equation of Wind Wave Spectrum Evolution

An attempt to solve the problem of wind wave numerical simulation in the real ocean scales with a help of the deterministic hydrodynamic formulation is practically non-realizable. The freedom degrees number of the system is practically unlimited. Significant achievements in wind wave numerical simulation are connected with the kinetic equation describing the wave spectrum evolution. The statistically space homogeneous and stationary and random water-air

interface ( , )t r . For describing the random field evolutions the "slow" co-ordinates and time are introduced. Field variability scales exceed essentially the typical lengths and periods of waves. The corresponding generalization of statistically homogeneous and stationary field can be achieved by considering the local spectra, depending on the slow

coordinates er and the time te (the index "e" is omitted below):

( , , , )S S t k r . (2.16)

Similarly, the wave action spectrum can be written as:

( , , , ) / .N N t S k r (2.17)

d( ) ( ) ( )

d

N NN N N G

t t

r k

r k . ( 2 . 1 8 )

I n c a s e i f t h e d e r i v a t i v e s , , r k c a n b e p r e s e n t e d i n t h e f o r m o f t h eH a m i l t o n e q u a t i o n s :

d d d; ; ,

d d dt t t t

r k

k r

H H H ( 2 . 1 9 )

t h e e q u a t i o n ( 2 . 1 8 ) c a n b e r e - w r i t t e n i n t h e f o r m o f t h e f u l l t i m e d e r i v a t i v e :

d d d d

d d d d

N N N N NG

t t t t t

r k

r k . ( 2 . 2 0 )

T h e e q u a t i o n s ( 2 . 1 8 ) a n d ( 2 . 2 0 ) a r e c a l l e d k i n e t i c . T h e y a r e w e l l - k n o w n i nt h e o r e t i c a l p h y s i c s . I t i s a g e n e r a l i z a t i o n o f t h e J . L i u w i l ' s t h e o r e m o f t h e g a sd i s t r i b u t i o n f u n c t i o n c o n s e r v a t i o n a s a p a r t i c l e s y s t e m m o v i n g i n t h e p h a s es p a c e ( L a n d a u & L i f s h i t s , 1 9 7 3 ) . T h e v a l u e i n t h e r i g h t - h a n d s i d e o f t h ee q u a t i o n ( 2 . 1 8 ) , i s c a l l e d t h e " c o l l i s i o n i n t e g r a l " . T h e i n t e g r a l - d i f f e r e n t i a le q u a t i o n ( 2 . 1 8 ) w i t h t h e c o l l i s i o n i n t e g r a l d e s c r i b i n g a m o l e c u l e c o l l i s i o n i nt h e p h a s e s p a c e , i s c a l l e d t h e B o l t s m a n e q u a t i o n . H e p r o p o s e d i t i n 1 8 7 2 .

Picture for problem formulation of wind wave modelling in the global scale

Main equation. The evolution of a two-dimensional sea wave spectrum

( , , , , )S t being a function of the frequency , direction (measuredcounterclockwise from the parallel), latitude , longitude and time t isdescribed by the equation:

1 ( cos ) ( ) ( )( )

cos

S S S SBS G

t

, (3.1)

where B(S) is the differential operator; and G is the source function.

The equation (3.1) is written in a flux form.

The motion equations for a wave packet along the arc of the great circle canbe written as follows:

sin;gC

R

(3.2)

cos;

cosgCR

(3.3)

costan ,gC

R

(3.4)

where gC is the group velocity; and R is the Earth's radius.

2.4 Parameterisations of different physical mechanisms forming wind wave spectrum The function in the right side of the equation (2.18) is assigned the sense of to different physical mechanisms forming the wind wave spectrum (K.Hasselmann’s 1960,1966,1979) . The right side of the equation (2.18) is called "a source function". It is presented as a sum of different physical mechanism approximations:

i iGG . (2.21)

The source function should be assumed to include, at least, the following components (Davidan et al., 1985): G1 is the mechanism describing the wind-to-wave energy flux due to influence of the turbulent pressure variation field; G2, G3, G4 are the energy flux to waves due to wave interaction (G2 -linear, G3 -nonlinear) with averaged air flow and atmospheric turbulence (G4); G5 is the energy exchange due to the wave interaction with water turbulence; G6 is the energy dissipation due to bottom friction; G7 is the energy dissipation due to wave breaking; G8 is the nonlinear energy transfer in wind wave spectrum.

Wind wave energy input

Miles model of wind wave energy input. The component Gin of the wind wave energy input is usually determined with a help of the relation based on the model of averaged air flow interaction with waves proposed by J.Miles (1960). In spite of the fact that this model is proposed in 1957, it describes accurately enough the mechanism of wind wave energy input. It is used even nowadays. This mechanism specified by using full-scale observation data (Snyder et al., 1981) can be described as follows:

where U10 is the wind speed at a 10 m level; is the angle between the wind speed and the direction of wave spectral component propagation; a1 and a2 are the parameters assumed to be about 1. Nowadays more sofiticated approximations are developed (Chalikov&Belevich, 1995; Makin&Kudriavtcev, 2003)

a 101 2

W

( , ) max 0; 0.25 cos( ) 1 ( , ) ,in U

UG a a S

c

Non-Linear Energy Transfer in Wind Wave Spectrum

• The problem of non-linear energy transfer in wind wave spectrum was formulated by K.Hasselmann (1960,1962,1963,1965,1966) and V.Zakharov (1968) in 1960s. As a result of non-linear interaction, the

wave spectrum evolution equation can be presented as follows:

1 2 3 1 2 3 1 2 3

( )( , , , ) ( ) ( )

NT

t

k

k k k k k k k k

2 3 1 1 2 3 1 2 3d d dN N N N N N N N k k k

Four-wave interaction diagram (Hasselmann, 1963).

Non-linear transfer function for the JONSWAP spectrum with =7: 1 - according to results (Hasselmann S. and Hasselmann K.,1981); 2 – by the Lavrenov’s

a )

b )

JONSWAP Spectrum approximationwith non-linear energy transfer

Wave Energy Dissipation

K.Hasselmann (1974) suggested the wave energy dissipation parameterisation, connected with wave breaking. In his opinion it be considered as random distribution of perturbing forces, making up pressure pulsations with small scales in space and time in comparison with the proper wave length and period.. The wave dissipation used in the WAM model (The WAM model, 1988; Komen et al., 1994) connected with wave breaking is accepted in the form of quasi-linear approximation, as suggested by G.Komen (1984) on the basis of the Hasselmann model:

where c, n and m are the model parameters; is the mean frequency of the wave spectrum; PM is the constant of the Pierson-Moskovits spectrum .

1PM

( , ) ( , )mn

dsG c S

Wind wave interaction with non-uniform current and bottom

Freak wave generation in non-uniform current

Freak wave collision with the “Taganrorsky Zaliv”.  

The “Taganrogsky zaliv” is a ship of the unrestricted sailing radius. The vessel’s length is 164.5 m, the largest width is 22 m, the displacement during accident is 15 000

tons, the board height above water is 7 m.

Map of the Southeast Coast (South Africa). 1 – location of abnormal wave accidence (Mallory, 1974); 2 – Location of the "Taganrogsky Zaliv” tanker-refrigerator.

Synoptic charts of the southern Indian Ocean on April 27, 1985.

Wave rays arriving to given point in Agulhas current with frequency : 1– 0.20 rad s1; 2-0.38 rad s-1 (at angle 0= -30°); 3 – 0.76 rad s-1; 4 – 0.93 rad s-

1 (at angle 0=30°). .

Wave breaking in shallow water

 

Wave Transformation in Water covered with Ice Cover

Ice field distribution in South ocean around

Antarctic (February, 1985)

Ice field distribution in South ocean around

Antarctic (August, 1973)

3. Wind Wave Model

Input and output parameters

Input parameters:• Wind speed and its direction in every grid point

and in 3-6 hours time step;• Current speed and its directions in every grid

point;• Water depth (in shallow water) in every grid

point);• Ice cover (mainly as a movable boundary)

Output parameters:

• Wind wave spectrum (two dimensional: frequency –angular)

• Wave height for swell and wind sea (significant or mean value);

• Wave period (mean, spectrum maximum);

• General direction of wave propagation;

Russian Global Wind Wave Modelexample of global forecast for 09.06.03

www.hydromet.ru

New Book about Wind Wave Modeling by Igor V. Lavrenov, Springer, 2003, 386p.

http//www.aari.nw.ru