A numerical wave prediction model DOLPHIN: Theory and test results

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Delft University of Technology Department of Civil Engineering Group of Fluid Mechanics

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A numerical wave prediction model DOLPHIN: Theory and test results

s.

Mandal Report No. 3-85

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PROJECT REPORT Delft University of Technology Department of Civil Engineering Group of Fluid Mechanics

Project title NORAD IND-013

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_{Project description Transfer of ocean modelling capability to}

the National Institute of Oceanography of India

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Customer Ship Research Institute of Norway Marine Technology Centre

Trondheim, Norway

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represented by O.G. Roumb

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Project leader dr. L.R. Rolthuijsen

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work carried out by S. Mandal

dr. L.R. Rolthuijsen

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Conclusion This report gives the basic theory of ocean waves and ocean wave prediction and a description of the physical background and of tests of the wave hindcast model DOLPRIN

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Status of report progress report

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City/date: Delft, February 20, 1985

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Contents

page

I lntroduction ₁

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11 Basic wave theory

11.1 lntroduction ₃ 11.2 Basic equations 11.2.1 Condition of incompressibility ₅ 11.2.2 Dynamic equation ₅

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11.2.3 Boundary conditions ₅ 11.2.4 Condition of irrotationality ₆ 11.2.5 Linearization ₇

11.3 Linear wave theory

11.3.1 Surface profile ₈

11.3.2 Velocity potential 8

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11.3.3 Particle velocities 9

11.3.4 Particle paths 10

11.3.5 Dispersion equation, wave length and

phase velocity 12

11.3.6 Pressure ₁₂

11.3.7 Wave energy ₁₃

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11.3.8 Wave groups 15

III Wind generated ocean waves 111.1 Introduction I

17

111.2 Description of ocean waves 17

111.3 Observation of waves 19

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IV Ocean wave prediction theories

lV.1 Wave prediction in the standard wind-field 21 lV.2 Wave prediction in a varying windfield ₂₄

lV.2.1 Spectral approach ₂₄

lV.2.2 Parametric approach ₂₈

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V Physical background of the model DOLPHIN 29

VI Swamp cases study ₄₁

VII Summary ₆₄

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Acknowledgements ₆₅ References ₆₆

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I INTRODUCTION

The subject of damages of offshore platforms, disappearences of ships, beach erosion etc. forces us to assess the amount of energy carried Over ocean surface by waves. To predict the extreme sea-state, one must have wave records available for a long period (more than 10 years). We have hardly any information for such a long period, but we can have wind

information (more than 10 years) from the meteorological centres. Based on historic wind fields and using forecasting techniques, one can find out the required longterm wave statistics.

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The study of the wave forecasting techniques began over 37 years ago for supporting the planning and operation of offshore operations. The wave prediction method developed by Svendrup and Munk (1947) is based on the significant wave height approach. This method has been revised many times and is still applied widely today, largely because of its simplicity and efficiency. The applications of the spectral concepts is introduced by Pierson and Marks (1952). Later on the universal wave spectra for fully developed windsea developed by Pierson and Moskowitz (1964) has been widely implemented. The wi~ely accepted energy balance equation is first introduced by Hasselmann (1960). This model is based upon the numerical solution of the spectral energy balance equation. The source functions of the spectral balance equation represent the process of wave generation, wave breaking and dissipation. The non-linear wave-wave interactions, theoretical aspects, is first studied by K. Hasselmann (1962, 1963). The models developed by Barnett (1968) and Ewing (1971) have introduced approximate representation of the non-linear interactions. Based on the JONSWAP (Joint North Sea Wa~e Project) experiments. Hasselmann et al.

(1973) have concluded about the importance of the nonlinear wave-wave interaction. Inclusion of this term in the source function needs bigger computer facility which limits our interpretation. Later on Hasselmann et al. (1976) have proposed

a

simplified parametric wave model where a dimensionless peak frequency is used to characterize the stage of wave development.

Very recently many wave prediction models are studied and compared (SWAMP '82). Each model is having its own strong points and also

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weaknesses in some areas.

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1

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The numerical wave prediction model DOLPHIN is being developed very recently. It is a hybrid point model as the combination of the parametric wind-sea and spectrally treated swells. It is directionally decoupled. The development of this model is based on the study of directionally decoupled energy distribution of wind generated waves (Holthuijsen, 1983). A

comparative study with SWAMP '82 report shows good results. It will very soon be in operational use for the study of North Sea waves to verify the model and for forecasting the wave for specific requested locations. Presently DOLPHIN is a deepwater model with untested shallow water ~odifications.

For understanding of the wave prediction techniques, the au thor started this study with basic wave theory (Chapter II, a brief review). Chapter III describes how the ocean waves can be for formulated in the form of numerical expressions of Energy spectrum. Chapter IV describes the wave prediction theory. The physics of the model is explained in

Chapter V. Chapter VI describes the comparative study with SWAMP cases.

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Further development of DOLPHIN model and conclusions are summarized 1n the last chapter.

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2

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₊

11 BASIC WAVE THEORY

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11.1 IntroductionThe waves can broadly be classified into two categories, depending on the ratio of the wave leng th (L) to the mean water dep th (h). If L/h is much greater than 1 (Le. L/h » 1), we speak of "long waves". Tides, and flood-waves in rivers are examples of long waves. If L/h is not much greater than 1, we speak of gravity waves or short waves. Wind-generated waves and ship-waves are examples of gravity waves.

There are certain differences in the properties of long waves and gravity waves which have led to different mathematical theories for which reaons the two categories are usually treated separately. The present study is restricted to short waves. The wind-generated waves are more complicated in structure and appearance. Their description requires spectra! and statistica! methods, which wi11 be treated in the "Wave prediction theory" chapter. Here the mathematical theory of sinusoidal progressive waves is presented in some detail, since that is basic for an understanding of wave phenomena.

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For elementary theory of short waves, all effects which are not essential to the phenomenon will be ignored. This leads to the following set of assumptions:

a) Non-viscous fluid of constant density (incomprehensive and homogeneous) b) Stress-free upper surface

c) No surface tension

d) Rigid, impermeable, horizontal bottom e) Periodic, long crested waves

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Parameters

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'

According to the preceding assumptions, the independent parameters are listed below: • mass density (p) • gravity acceleration (g) • mean depth (h) • wave height (H) • wave length (L)

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+

bas ed on: "Sbortwaves", Int. Inst.Hydr.Envir.Engng., Delft J.A. Bat tj es ₃

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•

z

-L--

z = n(x,t)

•

~ ...•-pressure

,

p

•

; )} 17; 771;;;,', 'j'»' " , 1 " » ; > ; » , ,; , > _{; »}

z

=

₇-h Fig. U-1.

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In order to describe the wave motion in a unique manner, we must specify a reference system. We will choose it such that relative to that system, the horizontal velocity below the level of the wave troughs has an average value equal to zero. We use the orthogonal axes, with x-axis horizontal, positive in direction of wave advance and with the z-axis positive

upwards, with z=O in the me~n water level (MWL) as indicated in fig. 11-1.

The free surface elevation can be expressed as

•

z ""

n(x - ct) (1.1 )

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where c

=

celerity.

The time between passage of successive wave crests past a point (x = const.) is called wave period (T) which is related to L

&

C by the identity

L

OK

CT

(1.2)

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Here L is the independent parameter, whereas C

&

T are not. The dependent variables characterizing the flow field are the x- and z-components of the flow velocity and the pressure denoted as u, wand p respectively.

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4

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11.2 Basic equations

In this chapter, the basic equation governing the wave motion under the assumptions listed above will be formulated.

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The condition of incompressibility of the water is the net volume flow through an arbitrary, submerged closed control surface must be zero. In other words, we can say that the volumetric strain rate must be zero. This leads to the following constant on the velocity field

(2D):

•

(2.1)

A statement of Newton's second law of motion should in the case

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considered here include only pressure gradients and gravity as force terms. This leads to

du

_{- 1.}

_2.P.

dt

-=

p ax

•

dw

_{- 1. ~}

-= -g dt p az (2.2a)

(2.2b)

du dw

The total acceleration (dt' dt) can also be expressed into alocal contribution and a convective contribution i.e.

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du -""_dt -+u-+w-.au au au at ax az and

•

w -.a_azw

Equations (2.2) can be written as (using p

=

const.)

•

aau + ut aau + wx aazu

+l(.E.

ax p + gz) - 0

(2.3 )

aw aw awan

- + u - + w - + _(oL. + gz) = 0

at ax az az p

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We will state only boundary conditions at the free surface and at the bottom. Kinematic boundary conditions for a non-viscous fluid merely state that no fluid particles cross a surface bounding the fluid i~e.

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w = 0 at Z = -h and w = dn _{at Z = n(x,t)}

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dt or w =~+ u~ at Z = n(x,t) dt dX (2.4 ) (2.5 )

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The condition of a stress-free upper surface can be expressed as

p

=

0 at Z n(x,t) (2.6 )

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The fact that the shear stresses at the surface are zero need not be stated since the fluid is assumed to be non-viscous, which is to say that shear stresses are zero everywhere.

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Due to the velocity gradients, the fluid particles can rotate about their axes as they move. The velocity of rotation in the x-z plane is

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rly

=

\(dW - ~)dX

d

Z

Under the conditions assumed, it can be shown that the rotation of each fluid particle is constant i.e.

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drl • 0

ëftY

Therefore, motio~starting from rest will be irrotational

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rly

=

\(dW _ dU) "" 0

dX

d

Z

(2.7)

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The condition of irrotationality ensures the existence of a scalar

function, the so-called velocity potential (~), such that its derivative in any one direction equals the component of the flow velocity in that direction:

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u "" dd~X' W _-

äZ

d~ (2.8 )

Hence from equation (2.1) we can write

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6

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(2.9)

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This is Laplace equation.

Substitution of equations (2.7) and (2.8) into equation (2.3) gives

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This implies that the quantity in the brackets has the same value throughout the field. In an undisturbed region, it equals zero, because each of the three terms in parenthesis is zero there. Therefore the quantity in brackets is zero everywhere:

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(2.10)

This is the so-called Bernouilli equation for unsteady flow.

•

which givesFinally equation (2.8) can be substituted into the boundary conditions,

~

=

0

d

Z

at z == -h (2.11 )

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~=~+~

dn

d

Z

d

t

dX dX at z

=

n(x,t) (2.12) and d<l>

+

~{{d<l»

+

(~<I>z)2}

+ sn

=

0

dt

dX 0 at z

=

n{x,t) (2.13)

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11.2.5 Linearization

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Despite the simplification obtained by the introduction of a velocity potential, it has not been possible to obtain an exact solution to the proceding equations to closed form for a periodic wave, due to the

non-linear character of the free surface conditions (equations (2.12) and (2.13».

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The simplest approximate solutions may be based on the assumptions of relatively low waves

(H/L

«

1 and

H/h

«

1), in which case the non-linear quadratic terms in the free surface condition are small compared to the linear ones. Hence equations (2.12) and (2.13) can be rewritten as

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7

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a<jJ_ an at z 0

(2.14)

az--ät

and ~+ _gn

₌

_{0 at z}

₌

₀

_(2.15)

•

at

A solution of the linearized equations, representing a progressive wave, will be investigated in chapter

11.3.

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11.3

Linear wave theory

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Here we assume the sinusoidal waves progressing at a constant speed in the positive x-direction. Hence, the equation of free-surface

(1.1)

can be written as

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n(x,t)

= -~

sin{~x - ct)}

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=

-a sin(kx - wt)

=

+a sin(wt - kx)

(3.1)

where elevation amplitude wave number a

=

_2".

H 27T/L

=

27T/T (3.2a) (3.2b) (3.2c)

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k -=

angular wave frequency w

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The wave number k represents the phase change per unit propagation

distance at a given instant, and the wave frequency ~ represents the phase change per unit time at fixed point.

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A resume of the kinematic equations is given below.

z

=

0 _Ol:a<jJ an

(2.14)

ät

I az a

2

<jJ a2<jJ I.

(2.9)

-- + -- =0· ax2 az

2

'

I ·1 a<jJ_ 0 (2.11 )

äZ-•

z

=

-h

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8

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Using these equations we want a solution for velocity potential (~) for a progressive wave representeq by equation (3.1). The solution for ~ must vary sinusoidally with x and t, although not necessarily in phase with

n

,

and we must allow for a variation of the amplitude of ~ (say ~(z» with z. Therefore we assume a solution of the form

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q,(x,z,t) ~(z) sin(Wt - kx

+

a) (3.3 )

where ~(z) and

a

are to be determined.

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Using the boundary conditions and Laplace equation, we find out the value of ~ and

a

as ~(z)

=

wa cosh k(h + z) k sinh kh

(3.4a)

and

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7T

a

= -.

2

(3.4b)

Substituting these values in equation (3.3), we get the expression for the velocity potential

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cosh k(h + )

~(x,z,t)

=

wka Z (wt k)

~ sinh kh • cos - x (3.5 )

11.3.3 Particle velocities

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Partial differentiation of

q,

with respect to x and z,'

respectively gives

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Cl~ u

= -~

= Clx

cosh

k(h +. ) [wa kh Z ] [sin(wt - kx)] sinh

(3.6a)

=

11

sin s

(3.6b)

and

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Clq, sirthk(h + )

w =

äZ

= [wa sinh kh z ] [cos(wt - kx)]

(3.7a)

,..

.. w.cos s

(3.7b)

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In order to investigate the variation of the velocity amplitudes with z, we first consider their values near the surface (z=O) and the bottom

(z=-h) :

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9

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'l:l = wa/tanh kh and

0

= wa at z=O (3.8a)

ü = wa/sinh kh and

0

o

at z=-h (3.8b)

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The relative magnitudes of Ü are dependent on (kh) or h/L at boundaries. If kh »1 (deep water) and if also k(h+z) » 1 (upper region of the deep water), we can approximate the hyperbolic functions in equations (3.6a) and (3.7a) and then

~" kz

u ~ w ~ wa.e (3.9)

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The condition "deep water" or kh » 1 is of ten taken to be kh ~ 3 or h/L ~ ~. If kh «1 (shallow water), then cosh x~ 1 and sinh x ~ x for x «1. Hence

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tl

=

Wa = ckh .!h

and _(3.10)

~ = wa(l

+

.!) h

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The conditionh/L

<:

210. "shallow water" or kh « 1 is of ten taken to be kh ~ ~, or z Q z z

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z=-h z=-h

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Shallowkh

«

1 Intermediate kh=O(l) Deep kh

»

1 Fig. II-2.

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The equations .given above enable us to calculate how the velocity in a fixed point varies with time. We consider a given partiele. In mathemati-cal description, we must label the partiele which we want to follow, e.g.

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10

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by its coordinates (x ,z ) at some previous time (mean position of o 0

particles). The displacements in the x- and z-direction from the mean partiele position (x ,z ) are denoted as X(t) and ç(t).

o 0 z I I I I I I!;(t) I (xo,Zo) I X(t) I --- 1 diLsplacement vector p08ition of particle at time t

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Z o

:mean poai tion of particle

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I I

x

Fig. U-3.

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The displacements are equal to the time integral of the partiele velooity components. t'

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x(t') ""

J

u.(x ""xo'z = zo,t)dt. or x(t) = - XCos(wt - kX ) o (3.11)

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and ç( t) eSin(wt - kx ) o where

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"

cosh k(h + zo) u X ,.,

-

= a w sinh kh and sinh k(h + z )

e

~ 0 =-=a sinh kh w

(3.12)

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11

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\

.

~..:.,..... • t '

'd:;;i

I :

,

I :

,

~ I I I i'jï I, I lil } 1 / / /1i'ti/ / /1/1/ / / / /11 / / / / / / /1 // /11 / 11 (x , z ) o 0

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Fig.

U-4.

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It can be seen from equations (3.11) that X and

ç

of any particle are 900

different in phase, and that the partiele path during one wave cycle is an ellipse, with its axes as shown in fig.

1

1-4

.

In deep water

G=~ ,

so that

Xz

e

which means that the partiele paths there are circular.

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Substitution of equations (3.1) and (3.5) into equation (2.15) gives the dispersion equation for gravity surface waves.

w

2 ...gk tanh kh _(3.13)

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The wave length can be written as

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2 L - (gT ) tanh kh

27T

27T

h

=

L tanh- -o L

(3.14)

The phase velocity (c) can be expressed as

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c ...gT tanh kh

27T

•

...co t

h

27Th

an --L (3.15)

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12

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In deep water (kh

»

1) c

=

c

=

gT

•

0 2n In shallow water (kh « 1) c

=

Vih

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(3.16a) (3.16b) 11.3.6 Pressure

---The pressure can be calculated from the Bernouilli equation (2.10). Neglecting the term ~(u2+w2), this can be written as

•

p

+

pgz

=

-

p-

dq,

dZ (3.17a)

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where wave induced pressure (p+) is -p

at.

dq,

In the absence of wave, the pressure (p ) is hydrostatic i.e. o

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po

=

-Pgz

Then

(3.17b)

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waveFiguretrough.11-5 shows the pressure distribution under a wave crest and under a

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z - MWL (z=ü)

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I I

I

\

I, \

\,(PO

\,; Po , I,

I/J/IJ/lJIII/ 1/

lij

11/1/ 111111II7}1I 1111/ 1110', ,I" I" 1j/ I/lil z=-h

Fig. II-5

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13

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Substitution of equation (3.5) gives

and use of dispersion equation (3.13)

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cash k(h + z)

p+

=

pga cosh kh sin(wt - kx) (3.18a)

=

p+

sin (wt - kX) (3.18b)

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In deep water (kh » 1) kz

~+

=

pga e (3.19a)

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In shallow water (kh

«

1)

f\

= pga (3.19b) and

•

(3.20)

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waves per unit horizontal area is defined asThe time averaged, vertically-integrated kinetic energy (Ek) of the

•

n

Ek -

f

-h

Using the values of u and w from equations (3.6) and (3.7), restricti~g

2

2

~p(u

+

w )dz (3.21)

the waves as free gravity surface waves (3.13) dispersion equation, and retaining only second-order terms, we find

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(3.22)

The time averaged, vertically integrated potential energy (Ep) of the waves per unit horizontal area is defined as

•

_"2

E

:a ~pgn

p (3.23)

Substitution of equation (3.1) for

n

gives

•

₂ E =

t

pga p (3.24)

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14

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In linear approximation, it is a general property of free waves (E

=

P Ek)· The total time-averaged, vertically integrated wave energy per unit horizontal area is

•

(3.25)

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The energy transfer (FT) through a vertical plane of unit width, normai to the propagation direction (x

=

const.) is defined as

n

ET

=

J

{p

+

pgz

+

~p(}

-h

2

+

w )}udz (3.26)

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Retaining only second-order terms, ET can be approximated as o ET =

f

p+ u dz -h "" ~pga2.[! + (3.27a)

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kh sinh 2kh] • c (3.27b)

=

E.n , c where ~< n < 1 (3.27c) I

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In deep water n

= ~

In shallow water n

=

1.

•

A

particular kind of·wave groups is obtained by adding two periodic wave systems of slightly different frequency and wave number, travelling

in the same direction:

•

(3.28)

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The phase difference

o

s

=

•

= (ow)t - (Ok)x

•

15

•

Points of phase reinforcement (maximum amplitude) alternate with points of phase cancellation (minimum amplitude) as shown in figure 11-6.

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•

.

.

.

.

.

.

.

.. _

.

. n

1

--- 1il2

(souree: Groen and Dorrestein, 1976)

Fig. II-6.

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The velocity of propagation of the groups, the so-called group velociqy (Cg) can be determined from the phase difference, i.e.

•

_ät(

a

_os)

₊

_{c -}

a

_(os)

₌

₀ g

dX

ow

dw c

₌

_ok O! dk g

(3.29)

•

Substitution of the dispersion equation and carrying out the differentiation gives

•

cg

=

n.c

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The factor n is the same as in energy transfer equation. In other words, the group velocity Cg' calculated on the basis of purely kinematic considerations has the same value as the propagation velocity P/E which was calculated on the basis of energy considerations. Hence a wave group can be considered as an energy packet. The equation of energy transfe~ rate is then expressed as

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P

=

E.c g

(3.30)

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16

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111 WIND GENERATED OCEAN WAVES

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111.1 IntroductionIn the description of ocean waves it is useful to distinguish three time scales. In the first time scale, the sea surface can be described as a stationary Gaussian with zero-mean process. In the second time scale, the wave characteristics are considered as slowly varying functions of time and place. Here the waves travel for a period of a few days. The third time scale is very large, this is required for the design of structures which protect large communities from the sea.

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Initially the waves are considered as a stationary Gaussian, zero-mean process. The relationship between sea surface elevation and wave spectrum are explained below.

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111.2 Description of ocean waves

The sea surface elevation on a time scale of about one hundred characteristic wave periods can be expressed as a stationary, Gaussian process with zero-mean. This implies that the joint probability of the sea

,

surface elevation n(t) at a·number of n arbitrary chosen different moments is fully described with the auto-covariance function

C(T)

C(T)

=

<n(t)n(t

+

T»

•

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The theoretical basis for equating the sea surface elevation to a Gaussian process is the central limit theorem which states that sum of a large number of statistically independent variables is Gaussian distributed. In the case of ocean waves, the sea surface elevation n at a moment t is

I

considered to be the sum of an infinite number of Fourier components, 'each with its own frequency and direction. Each of these components is

generated independently from the other components. They are therefore

I

statistically independent in origin and remain independent during their journey across the ocean because they travel independently according eo linear wave theory. If the waves become very steep (e.g. in a severe storm or in a very shallow water), the cent ral limit theorem no longer applies

I

due to non-linear effects. Here the auto-covariance function

C(T)

describes the sea surface elevation completely in a statistical sense. The variance density spectrum can be expressed as

•

•

•

+

_{source: "Oc ean wave theory", Int. Lns}

t , Hydr. Envir. Engng., Delft L.H. Holthuijsen

17

•

00

E'(f)

=

f

C(T)

-21TifTdT

e (2.1)

•

This spectrum has an additional advantage over the auto-covariance

function for ocean waves and many other physical phenomena. Which is that the hydrodynamic processes are well described in time of Fourier

components. As the relationship (2.1) is reversible, no information

•

00

f

21TifTdf

C(T)

=

E' (f) e (2.2)

•

is lost in the transformation,exactly the same information as the auto-covarianceindicating that the spectrum containsfunction, only the format of the information is changed. In the field of ocean wave theo~y it is conventional to define a spectrum E(f) slightly different from E'(f).

•

E(f)

=

2E' (f) for f ~ 0 (2.3a)

and

E(f)

=

0 for f < 0 (2.3b)

•

The variance

<n

2

>

of the sea surf ace elevation is then expressed as

00

<n

2

>

=

c(o)

=

J

E(f)df o

(2.4 )

•

which indicates that the spectrum distributes the variance over

frequencies. E(f) should therefore be interpreted as a variance density with

•

f2

f

E(f)df fJ

equal to the contribution of the frequencies between f1 and f2 to the total varianee

<n

2>. The dimension of E(f) is m2/Hz if the elevation is given in mand the frequencies in Hz.

•

•

The varianee

<n

2> is equal to the total energy ETOT of the waves per unit surface area if multiplied with a properly chosen coefficient

2 E = \ P

g<n

> TOT (2.5 )

•

18

•

•

In many wave problems it is not sufficient to define the energy density as a function of frequency alone. It is sometimes required to distribute the wave energy over directions (8) as weIl. Then the wave energy over

frequencies and directions may be denoted with E(f,8). It will suffice to state that the integ ral

f2 82

J J

E(f,8)dfd8 (2.6)

fJ

is equal to the contribution to the total variance

<n

2> of the components with frequencies between f1 and f2 and travelling in directions between 8

1

and

8

2• This is analogous to the interpretation of E(f) (fig. III-~). As the total energy density at a frequency f is distributed over the directions 8 in E(f,8) it follows that

2n E (f) ..

J

E(f ,e ) de o

•

•

•

•

•

E(f

•

•

Fig. 111-1 Contribution of wave components on (f1,f2) and

(8~,82) to the total variance.

111.3 Observation of waves

•

Visual observations are still being carried out in many situations. If observations of waves are to be used in quantitative manner, it should be carried out with instruments whenever possible. Usually only two

parameters are obtained from these observations, the significant wave height Hand significant wave period T •

s s

•

•

•

•

•

The measurement of sea surface e1evation at one horizontal position is usua11y carried out with a wave gauge or a buoy. The measurement with a wave gauge is common1y based on an e1ectrica1 resistance measurement. Two wires run through the sea surface to a certain depth be10w the water

surface. The e1ectrica1 current through the wires encounters a resistance, which can be measured and the resu1t is stored on an ana10g or digita1 data carrier (e.g. paper tape or magnetic tape). The measurements with the Waverider buoy are based on acce1erations, which are integrated twice thus reproducing the sea surface e1evation as a function of time. A radio

transmitter on-board the buoy sends the data to a shore-based station where the data are stored on a paper tape or magnetic tape.

The time series thus obtained are ana1yzed to estimate the energy density spectrum E(f). This procedure is based on the Fast Fourier Tr~ns-form, which indicates the conventiona1 va1ues for the spectra1 reso1ution, the Nyquist frequency and the statistical reliability of the spectral estimates. These va1ues are determined from desired characteristics on the one hand and practical considerations on the other. High reso1ution

combined with high statistica1 confidence of the estimates require 10qg periods of observations during which the waves shou1d be stationary. rn I practice a wave record is between 15 and 30 minutes long. This 1ength is primarily based on assumed stationary of the wave field. Generally a resolution of about 0.01 Hz is chosen (actua11y 1/128 Hz due to the fact that most FFT procedures require series with a number of samples equal to a power of 2). The Nyquist frequency is usually 0.5 Hz or 1.0 Hz

corresponding to a sample interval in time of 1.0 to 0.5 seconde

•

•

•

•

•

•

The methods to determine the two dimensional spectrum E(f,8) are mostly based on the observation of a limited number of spatial characte-ristics of the sea surface. A method which is used is the measurement :of

the slope of the surface in two orthogonal directions with pitch-and-roll buoy. It is possible with such a buoy to determine at each frequency a mean direction and a directional spreading of the wave energy (Kuik and Holthuijsen, 1981). With the marketing of a pitch-and-roll version of Ithe Waverider (Wavec) these observations may weIl become routine. More

advanced methods which provide details of the two-dimensional spectrum, require sophisticated equipment (e.g. stereophotography, radar). These advanced methods are not yet suited for routine observations.

•

•

•

20

•

IV OCEAN WAVE PREDICTION THEORY

+

•

Many techniques are available to forecast or hindcast waves, vary~ng from simple rules to highly complex computer modeIs. All models consider the generation of waves by wind and the propagation of waves over the I

ocean. Based on the meteorological information, the prediction or

I

hindcasting of wave conditions by two methods. The first is one where the windfield and the wavefield can be completely described with only a few parameters in a generally accepted standard situation. The relationships

I

between the wind-parameters and the wave parameters in this situation are

I

relatively weIl known from observations. In the second method, the wave characteristics can be determined in a wind field varying arbitrarily in time and place. This gives more reliable results.

•

•

•

IV.l Wave prediction in the standard wind-field

The wave growth is best described in an ideal situation where a

I

constant wind (U) blows perpendicularly off a long and straight coast over deep water. The most obvious development in the wavefield is the increase

•

of significant wave height (H ) and the significant wave period with

s

I

increasing distance from the coast (fetch, F). Initially the wave growth is quite rapid but the rate of growth decreases as the waves grow higher

•

and af ter some distance the waves do not grow anymore, then the waves are said to be fully developed. The empirical relationship between the wind parameters and the wave parameters are expressed in terms of a dimension-less fetch

F

(= gF/u2), a dimensionless wave height

H

(=gH

lu

2) and a

s

dimensionless wave period

T

(= gT

IU).

The observed dependencies can be s

•

approximated analytically with

•

Expressions which fit most of the available data weIl are given in the Shore Protection Manual (SPM, 1973).

•

H

=

0.283 tanh (0.0125 FO.42) (l.1 )

T

=

7.54 tanh (0.077 FO.25)

•

+ bas ed on:" Oceanwave theory", Int.Inst.Hydr.Envir.Engng., Delft L.R. Holthuijsen

21

•

•

A more advanced description of the wave field is based on the notion that the undulating sea surface is composed of a large number of individual wave components, each characterized by a frequency.f and a direction 8. The two-dimensional spectrum E(f ,8) gives the distribution of wave ene.rgy over the frequencies and directions. In the above ideal situation, the frequency spectrum, obtained by integrating the two-dimensional spectrum over all directions, develops from high frequencies to the low frequencies

(fig. IV-i). During an extensive and detailed study of the wave growth (the _:!ointB_orth ~ea Wave R,roject

=

JONSWAP , Hasselmann et al., 1973),I it was found that the shape of the spectrum was constant during most of the

I

growth of the waves, and only in fully developed stage does the peak flatten somewhat.

•

•

•

8 E (fI (mI/HZ) 6

•

4

•

U.IO mis 2

•

0,1 0,2 0.3 f (Hz) 0,4

•

Fig. IV-l The development of the spectrum in the ideal wind field.

•

To formulate this shape in the analytical expression, the investigato~s in JONSWAMP adapted an expression, which was formulated in an independent

study of the fully developed sea state:

•

22

•

E(f)

=

E'(f).y(f) _{(1.2 )}

•

where E'(f) is a spectrum with the shape of the fully developed spectrum I proposed by Pierson and Moskowitz (1964) (fig. IV-2), but with arbitrary values for the energy scale coefficient a and the frequency f •

m

•

E(f)

=

ag2(2TI)-4f-SeXp[-; (~)-4] m (1.3 ) E(f) JOIISWA' (M.... l""' ••.•.• 1111) T.' 1.1 a•• 0.07 IT, • (lOt

•

•

•

•

Fig. IV-2 Shapes of the PM-spectrum and the mean JONSWAP spectrum.

•

As the spectra of the growing waves were much more peaked than the

Pierson-Moskowitz spectrum for fully developed waves, it was necessary to use a peak enhancement function y(f)

•

y(f)'

=

yexp{- _} (1.4 )

•

where y is the maximum enhancement at the peak frequency and cr is the width of this function. The shape of the Pierson-Moskowitz (PM) spectrum is equal to that of the JONSWAP spectrum for y - l.O. From the observation in JONSWAP (1973)

•

23

•

•

y ;: _3.3 0 0 ;: _0.07 for f < f (1.5 ) a m 0 ;: ob ;: 0.09 for f ~ f_m

The scale parameters were observed in JONSWAP and the best-fit express~ons through observations are

•

ex;: 0.076

'F

0•22

'"

4

for F < 10 (young sea state) (1.6 )

1 ;:

3.5 F-0.33 m

•

where the dimensionless peak frequency f is defined as

m

'"_{f ==}

_U

_f

_/g

m m

•

The values of ex and f for the PM spectrum are m

•

ex== 0.0081 and '"_{f == 0.13} m y za 1.0

'"_F

_>

₁₀5 _{(fully developed sea state) (1.7)}

•

For many engineering problems, the frequency spectrum gives sufficient wave information but many problems require additional directional information as provided by the two-dimensional spectrum E(f,8). The

two-dimensional energy spectrum E(f,8) was constructed from equation (1.3) by assuming a fixed eosine square spreading factor defined relative to the local wind direction i.e.

•

E(f,8) ...E(f)

.

-

2 cos (6 - 6 )2 for (8 - 8 ) <-·TI

TI W W

=

2

(1.8)

•

=

0

for (8 - 8 )

>!

w 2

•

IV.2 Wave prediction in a varying windfield

In this method, the sea surface is considered to be the sum of a large number of individual wave components, each of which propagates over the ocean surface with a constant frequency in accordance with linear wave theory. To determine the energy density of one such component at one time

•

24

•

•

and one place, conventional computations are used to determine the path of wave component from its origin (usually a coastline) to the point of

interest. Af ter leaving the coast, the wave component encounters the influence of wind, bottom, wave breaking and also wave-wave interactions.

•

I

By considering at regular intervals the energy gained or released by wave component, one can determine its evolution till it reaches the point of forcast. This procedure can be repeated for a larger number of wave

components of different frequencies and directions which finally arrive in the point of forcast at a given time. This process can be carried out !or a large number of different times and locations, thus providing the final

I

two-dimensional wave spectrum as a function of time over the entire ocean.

•

•

The above process i.e. the evolution of the surface wave field in I

space and time is based on the transport or energy balance equation fo.r

dE

~

~

dt (f,8,x,t)

=

S(f,8,x,t) (2.9a)

•

or ~ ~ ~ ~

ät

(f,8,x,t)

+

Cg(f,8).VE(f,e,x,t)

=

S(f,8,x,t) (2.9b)

•

where E is the energy density of the wave field described as a functi~n of

~

frequency f, direction e, position x and time t. c is the deep water group velocity. S is the source function. The source function represents

1

E(f)

•

•

E(f) 5(f)

•

Fig. IV-3.

•

25

•

•

•

•

•

•

•

•

•

I

•

•

•

•

three types of energy transfer (Fig. IV-3) i.e.

S

=

S._1n

+

S 1_n

+

Sd_s (2.10)

where Sin represents the energy input from the atmosphere, Snl the

transfer"from one wave component to another (non-linear interaction) artd Sds represents processes that serve to dissipate wave energy.

The growth of the energy density is described in the existing models for wave prediction as the sum of a linear growth and an exponential growth. For linear growth, Phillips (1957) explained the initial generation of gravity waves on an undisturbed sea surface through a resonant excitation by incoherent atmospheric turbulent pressure

fluctuations convected by the mean wind. This theory is commonly referred to as the resonance theory of Phillips. The corresponding source term ~s (as part of Sin) a constant

dE

dt

for a given combination of frequency, direction and wind speed.

The exponential growth may be contributed by the following process~s. a)

A

feed-back mechanism proposed by Miles (1957) - the wave induced

pressure differences across a non-breaking wave crest (no flow

separation) transfers energy from the wind to the wave component at a rate proportional to the energy density of the wave component.

b)

A

sheltering mechanism first proposed by Jeffreys (1925) - Wind energy

is transferred to the waves by wave induced pressure differences across a breaking wave. In this situation flow separation occurs and the

energy transfer is more effective than in the mechanism proposed by Miles (1957). But this mechanism has not been considered seriously for a long time due to the negative results of old experiments. Later on experiments by Banner and Milville (1976) have revived the interest in this mechanisme

c) Non-linear energy transfer between wave components - This is a complex mechanism where groups of wave components interact with single wave components. During the exponential growth the effect for a wave component gives a net ga in of energy. The non-linear wave-wave

interactive has been studied theoretically by Hasselmann (1962, 1963).

•

The above first two mechanisms are not considered exp1icit1y in existing wave prediction. The third mechanism is inc1uded on1y in a few advanced mode1s in a parameterized form. The exponentia1 growth is therefore usua11y expressed as

•

dE

- - B.E dt

•

where B is determined from observations.

•

To stop the energy density from growing beyond a certain limit E (Pierson-Moskowitz spectrum limit), the growth in truncated by mu1tip1Ying the souree function by r, i.e.

dE ..(0:

+

BE)r dt

•

with r= 1 for E

=

0 and r40 when E + Eoo. The factor r represents the dissipative mechanism (Sds) of wave breaking. In the fina1 stage, the energy density is a constant and is determined by a ba1ance between ene rgy input (from wind and non-1inear interaction) and energy dissipation due to wave breaking.

•

•

E(f,')

•

•

•

t

Fig. IV-4 The wave growth in constant wind field.

•

•

•

•

To obtain a review of the energy transfers, consider first the growth of a number of frequencies from one direction (fig. IV-4). At lower

frequencies (fS)' the wave energy is still increasing. At high

frequencies the energy density has already obtained the saturation level where the energy input is balanced by energy dissipation. The overall picture for the spectrum at one moment in time is shown in fig. IV-S.

•

E(f)

growth

wind

•

dissipation

•

~

inte;ractiCJD

f

•

Fig. IV-S The energy balance of a wave spectrum in a constant wind field.

•

In shallow areas of the ocean, bottom effects are included in the models by adding a term for bottom dissipation to the source function and by using conventional wave refraction and shoaling theory to account for bottom effects on the energy propagation.

•

The parametric models, introduced by Hasselmann et al. (1973, 19761), assume that the spectrum can be given an analytical shape defined by five

I

•

parameters (a, fm, Y, 0a' Ob). These models highlight the role played by the non-linear wave-wave interaction in the distribution of energy across the spectrum. One important effect of the non-linear wave-wave interaction is that they tend to force the shape of the spectrum into a universal shape. From numerical experiments and observations it appears that this shape is close to that of the mean JONSWAP spectrum (y

=

3.3,

•

Ga = 0.07, ob

=

0.09); i.e. if the spectrum bas a shape different from

the universal shape, the non-linear wave-wave interactions will tend to remove these differences. Hasselmann (1980) explained that the non-linear

•

28

•

•

interactions a1so impose a more or 1ess universa1 re1ationship between

a and f

_m

in a varying wind field. The wave forecast is thus reduced from predicting a large number of individua1 wave components to the prediction of one spectra1 sca1e parameter. It shou1d be emphasized that this

approach is on1y app1icab1e during wave growth. If the wind speed decreases be10w the phase speed of the wave components or if the wind I direction differs marked1y from the main wave direction, the waves do not grow and non-1inear interaction disappears. The wave model shou1d then on1y transport the swe11 energy over the ocean surface. Therefore a parametric wave model shou1d a1ways be combined with a pure swe11 modet The combination is called a coupled hybrid model.

•

•

•

•

•

•

•

•

•

29

•

v

PHYSICAL BACKGROUND OF THE MODEL 'DOLPHIN'

•

One of the conclusions of the JONSWAP experiments is that if for some reason the shape of the spectrum differs from some standard shape, the non-linear interactions will redistribute the energy in the spectrum such that the standard shape is attained. This process will only be operative during the generation phase of wave development (i.e. Young Sea-state). Depending on the magnitude of the shape discrepancy, the frequency and wind-speed, the ra te of change varies to form the standard shape. The rate of change seems to be largest for highest frequency (Hasselmann et al., 1980). The observed spectra studied by Holthuijsen, (1983) and some directional chracteristics of these spectra suggest that the non-linear interactions only partly force the spectral shape into a standard shape. If non-linear interactions do not dominate the directional charateristics of the wave spectrum, we can hindcast wave components independently from different directions. A directionally decoupled model (explained below~ has been originally suggested by Seymour (1977) to hindcast the wave spectrum in a stationary situation.

•

•

•

•

Fig. V-l.

•

•

•

Fig. V-1 shows a deep water basin with an arbitrary geometry of coastline. Here a stationary homogeneous wind blows in the direction as shown in the figure. P is the hindcast point. The hindcast is first concentrated on wave components from an arbitrarily chosen direction (8). These wave components are assumed to be generated independently from other wave components travelling in other directions (81, 82, 83, etc.). For these

I

wave-components (in the direction 8), the independent variables are 8 and R(8) (distance to shore in direction 8). The wind speed (U) and

•

•

30

•

•

gravitational acceleration (g) are constant for this problem. The

directional decoupling means that another water basin with same values at independent variables (R,8) will give identical results. As the geometry of the water basin is irrelevant as long as R(8) is not affected, we can choose the ideal fetch limited situation (fig. V-2).

•

_,

-,

-/ / ( \ "-\ I ...~~.l..f...jI.L.olr..L._

,

I

\ \ I

•

\ \ p \/

•

Fig. V-2.

,

'

\ - I

,/"

_...

/

•

Here the fetch F - R(8) cos 8. Then the estimated spectral density Emodel for the direction 8 and an arbitrary coastline is equal to the spectrum density Eideal with fetch F

=

R(S) cos S.

Emodel(f,S)

=

Eideal(f,S, F

=

R(S)cos S)

•

For hindcasting the spectrum in an ideal situation, it is assumed a standard spectrum, JONSWAP spectrum (Hasselmann et al. 1973). An ideal generation fetch-limited frequency spectra can be written as

•

2 -4 -5 5 f -4

E(f)

=

ag (2n) f exp[-

4(~)

+

lny

m 2 (f -.f .) exp{ - m} 22] 20 f m (5.1)

where, a c Phillips' constant

fm - peak frequency

y

=

Ratio of maxima 1 spectral energy to the maximum of ~he corresponding PM-spectrum (called Peak enhancement factor)

a

=

width of spectral peak

•

•

•

for f (; fm for f

>

f m

•

31

•

•

The fully developed Pierson-Moskowitz spectrum

(1964)

is

•

2 -4 -5 5 f-4

Epif)

=

ag (2rr) f exp[4 (T) ]

m

(5.2)

For

y

=l,

equation

(5.1)

reduces to the PM spectral shape. The scale parameters for the PM-spectrum are

•

_a

₌

_0.0081

(5.3a)

I

fm

=

0.14 g/U19•5 ~ 0.13 g/U10

(5.3b)

•

where UlO is the wind speed at height z

=

10

m above mean sea level. The dimensionless energy

(E)

and peak frequency

(f )

for PM spectrum are

m

•

E

=

3.6

x

10-3

_(5.3c)

f

=

0.13

m

(5.3d)

•

From the

JONSWAP

experiments

(1973),

the observed mean values of shapel

parameters are

•

y ...

3.3

o

-

0.07

a

c

==

0.09

b

(5.4)

I

From Hasselmann et al.

(1976)

gives non-dimensional relations~ip as

•

E

=

7.39 x 10-6 ~3.05

m or

•

1 =

2.078 x 10-2~0.328

m

a

...

3.628

x

10-21

0.87

m y

=

4.124

1

0.32

m (5.5) I

•

From Mitsuyasu et al.

(1980)

the nondimensional relationships are

E

=

6.84 x 10-6y-3

m

•

32

•

•

•

I

·

•

•

•

•

•

•

•

•

•

1 =

m ct

1.8983 x 10-2~1/3

3.26 x 10-2

1

6/7

m

4.42

1

3/7

m

(5.6)

y

=

I

Based on these studies along with PM-limits the following non-dimensional

relationships are used for DOLPHIN model (young sea state to fully

developed sea-state).

(Fig. V-3)

(5.7a)

(5.7b)

I

a

=

1.068 x 1013

(Fig. V-4)

b

=-0.87/c

=

+

17.4

c

=

-0.05

and from Yamaguchi (1982)

"" ~4 .E

(ZW)

.

f

1.245

Y ""

30.08 [

m -

0.135]

(Fig. V-5)

ct

(5.7c)

a =

0.07

a

Ob

=

0.09

(5.7d)

(5.7e)

I I

The dimensionless wind sea energy (E) and peak frequency can be expressed

as

"" 2 4

E

=

gE/UlO

(5.8) :

and

_(5.9)

I

The two-dimensional energy spectrum can be expressed as

E(f,6)

=

E(f) • ~os

(6 - 6w) if (6 - 6w) ~

I

(5.10a)

""0

if

(6 - 6 )

> ~

(S.10b)

w 2

where

.

'

ff

m (-) w ~

•

•

•

•

•

•

•

•

•

•

•

1.0

0.5

Hasselmann et al.(1976)

and Dolphin july 5 1984

0.1

-_

(1980) -

.

_

.

_

I I

I

I

I

0.0001

Fig.

V.3

f

m vs.

Ê

0.001

3.6 10-

3.

_0.01

E (-)

-fu

11

y deve

1

oped

•

•

l..U VI

•

•

•

•

•

_•

•

•

•

•

Q.1

SWELL

--- Dolphin

j

uly 5 1984

---

Hasselmann et al. 1976

l

a

Mitsuyasu et al. 1980

0.01

PM I __ -::-. -I

_-

_-

-

.-' 1-'

0.1

0.13

0.2

0.3

0

.

4

0.5

0.6

0.7

0

.

8

0

.

9

1.0

--

_fp

----.

-Fig. V.4

a

vs. fp

•

•

w (j\

•

•

•

•

•

_•

•

•

1~

SWELL

10

9

8'"

T

6

5

Dolphin july 5, 1984

Hasselmann et al. (1976)

Mitsuyasu et al. (1980)

4

_---.

-.--.

---=----_

-

__:-:=:; -_- _ - -

--l

3

:;;.

-_

-_"

2

,--

1-·-_

1~---~---;---~~---~----~----~--~--~--_r~

0.1

0.13

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1.0

....

f

P •

-Fig. V.5 Y

vs.

fp

•

•

-•

8

=

loca1 wind direction w

00

•

E(8 )

₌

_f

E(f,8)df (5.11)

0

From the universa1 growth curve [SWAMP '82, fig. 7.6], we assume the dimension1ess wind wave energy as the function of dimension1ess time.

•

r;.< ",c D

~ =

A(tanh (Bt » (5.12)

•

=

function of~. (5.13) I

•

Either the growth curve of the SAlL model or the BMO model are used to obtain values of A, B, C and

n,

these values are:

•

SAlL A 3.6 x 10-3 B

=

0.867 x 10-15 C == 3.25 D == 0.4

•

BMO A = 3.6 x 10-3 -21 B == 0.21 x 10 C == 4.667 D ..0.3

•

There are two types of wave fieids considered in this study. The Hrst one, where all spectra1 components propagate along the characteristic with

I

•

group velocity of the peak frequency is called the wind-waves (E1(f,8). The second one, in which all spectral wave components propagate with speeds which are determined for each components separately by linear ~ave theory is called swell (E2(f,8». The tota1 energy density of any

spectra1 wave component is the sum of its energy density in each wave I

field for that component. That is

•

•

37

•

•

In this model, the directionally decoupled energy balance equations are evaluated for wind wave and for swell separately along a straight characteristic which extends from the point of forecast to a point on a coast or an open boundary in each of all chosen directions. The energy balance equation for wind waves from direction (8) can be expressed as

•

(5.14)

where _{E1(8) is the directional energy density of wind waves} propagating in the direction considered (m2/rad.) I

51(8) is the ra te of change of E1(8) induced by wind. I f El is less than some upper limit, it represents,

growth, otherwise decay (m2/rad/s).

S2(f,8) is the ra te of transfer of 2-dimensional swell energy E2 to the wind wave field at the same frequency (m2/rad/Hz/s).

is the direction from which wave component approach the point of forecast (Rad)

f is the frequency (Hz)

•

•

•

•

For underdeveloped waves 51(8) can be calculated from equation (5.13). For overdeveloped sea, E1(8) is more than some upper limit, then 51 represents decay and it relaxes to local PM-spectrum. 52(8) can be calculated from the expression

(5.15)

•

•

In the ocean, the wind blows arbitrarily in all directions. When the wind

I

blows with a constant speed for a long period, we get a fully developed wave spectra. If the wind suddenly drops, a part of the wind-sea ener$y drops to swell. And again ff the wind speed increases, the growing

wind-sea absorbes swell. This phenomena can be explained as below. Itlis shown in fig.

V-6.

•

•

38

•

•

•

•

•

•

•

•

•

•

•

•

I

I

[. E (b) at tl - t2 20

I

E (c) at t2 - t3 m s

_-

"-_

-

"- ,--, / /' ...

....

-

I

'"

...

"

I 5 mis

-/ I I I I Fig. V-6. 10 mis (a) E f

As an example we assume, initially (tl) the wind blows with speed I

UlO

=

20 mis. At t2, before the wind-sea is fully developed, the wind I

drops to 5 mis. Here af ter t2 the sea is overdeveloped and the wind-ses is dropping to swell during the time interval (t2 - t3). Fig. V-6c shows how much wind-sea drops to swells (shaded area). Again wind-speed

increases to 10 mis, where swells return to wind-sea. This is shown in fig. V-6d. At t4 for wind speed 10 mis, it is a fully developed sea with some swell which originated in the U

=

20 mis-area. For a constant wind speed, the wind-sea grows as shown in fig. VI-3. This happens at the young sea state.

The energy balance equation for swell from the direction (8) is expressed as

(5.16)

=

the two-dimensional energy density of a swell component with frequency (f), propagating in the direction (8) (m2/rad/Hz).

•

B

=

1 if Sl < 0

=

0 if Sl > 0

•

the rate of change of E2 corresponding to Sl when

Sl is negative (frequency distribution of Sl with sign reversed)

S2

=

the rate of change of transfer of swell energy

E

2 to the wind wave field at the same frequency, (assumecf to

• be zero or positive) (m2/rad/Hz/s).

S3

=

the rate of change of swell energy due to

swell-bottom interaction, used only for shallow water.

•

The generation of swells from wind-sea decay is shown in fig. V-6a I

and V-6c. Similarly return of swells to wind-sea is shown in fig. V-6d~

•

effects, we have to introducePresently the DOLPHIN is a deep water model. For shallow watershallow water propagation as described in chapter 11.3. Then by introducing bottom dissipation for swell and by replacing the generating source term by one, we can run this model up to

I

the water depth of 20 m for shallow water wave generation. From Jansen et al. (1984) for swell

•

S = S =

-r.

ds 3 (5.17)

•

(Refer to equation (4.10) in chapter IV) where

r

=

0.027 m2/33

d

=

water depth

•

•

For shallow water wave generation(Shore Protection Manual, 1973) replace A and B in expression (5.13) by

and 2 f1

A

=

A

tanh (edf )

B

=

B/

tanh(ed 1)2/d e

=

0.53 f1 ""0.75

cr

=

gd/UlO (5.18)

•

with

•

40

•

VI SWAMP CASES STUDY

•

The ~ea Wave !!odelling froject (SWAMP) was initiated for wave mode~s intercomparison study to test the present understanding of the physics .of

wind generated surface waves. The strategy chosen in SWAMP to reveal the basic differences between the 10 participating models is to run these models for a series of hypothetical, idealised wind fields separately bo critical influences such as asymmetrical boundary conditions, suddenly changing wind direction, discontinuous spatial wind distributions etc. I

This approach is adopted for all test cases. All case studies in SWAMP '

•

•

were carried out in cartesian(deep water). DOLPHIN is put through the same tests as in SWAMP (ascoordinates and for an infinite depth oceanI explained below) except the hurricane case.

•

A set of seven hypothetical test cases is designed for intercomparison,

of all models behaviour. All the models are run an identical grids. A

I

comparison study is made to validate the model. Case I (advection test) is

I

•

a pure swell propagation experiment. This case study is not made in SWAMP report for intercomparison results. In the present report, the following cases are studied.

Case II

•

The geometry of the wind field and boundary conditions are shown i~ figure VI-l. A stationary uniform wind (UlO = 20

mis)

blows perpendi-cular to off-shore as shown in figure VI-2. The wind field is assumed infinite in the downstream direction. Initially (at t=O) the wave energy is zero and the spectrum at coastline remains zero for t> O. The DOLPHIN model is run until a stanary state is reached.

The shape of the one-dimensional wind-sea spectra can be character~zed by the peak frequency (fp)' a steep low frequency forward face (f < fp) and a more slowly decreasing high frequency (f > f ). The fully developed

p

spectra for infinite depth and duration is shown in figure VI-2. The DOLPHIN model when fully developed gives the Pierson-Moskowitz (PM)

spectra (figure VI-2).

The fetch-limited growth curve is shown in figure VI-3. The overshoot phenomenon is clearly seen below 400 km fetch. At 1000 km fetch, it is almost fully developed sea-state. The duration limited one-dimensional spectral growth curves are similar to the fetch limited growth curves.

•

•

•

•

41

•

•

•

'"

Figure VI-4 and VI-5 show the non-dimensional energy (E*) and peak frequency

1)

respectively as a function of fetch X* for stationary

p

state. Similarly duration limited growth curves are shown in figure VI-6 and VI-7. In fetch limited case, DOLPHIN (E*) shows a very closed to HYPA results in the growing stage (fig. VI-4) and then af ter 1000 km fetch, 'it is fully developed (PM-spectra). For peak frequency (fig. VI-5) it is j

initially very closed results to TOHOKU model and later on it is limited by PM-peak frequency. For duration limited Eenes growth curves, it is very closed results to the BMO model (since coefficients (A, B, C, D) in

euqation (5.13) are taken from BMO model) as shown in figure VI-6.

•

•

•

Though all models are calibrated against fetch and duration limited data, there is a lot of variation among the different models. Much of this

I variation may be due to the indeterminacy of the drag law relating to wind speed Ux or UlO and due to gustiness of the wind or background

turbulence in the ocean.

Case 111

•

The geometry of the wind-field for the slanting fetch case is shown in figure VI-a. Initially (t=O) the ocean is calm and then a uniform wind (UlO

=

20

mIs)

blows diagonally (45°) as shown in figure (VI-a). The wave field is zero on the boundaries x=O and y=0 and open boundaries are at x

=

1000 km, y

=

1000 km. The model is run for different points (A,'B, Dl' B2' Bl' Cl' D2' C2, C). Here we can observe that energy density

increases from point B to A and two-dimensional distributions is similjar to BMO model. The results of spectra between points (Bl, and B2), (Dl and D2), (C and C2) give the behaviour of the model for slanting case. It shows symmetric, (i.e. energy density and mean direction are the same for Bl and B2). Table 6.1 shows the comparison of nondimensional

energies and frequency distributions of various models at three selected I locations (A, B, C) of figure VI-a. Table 6.2 shows the comparison of the energies and frequencies with respect to case 11.·

The model experiences a reduced growth ra te relative to case 11 a]ong the diagonal, because in this case the fetch width is narrower than

case 11.

•

•

•

•

•

42

•

•

Case IV

•

The geometry of the half-plane field is shown in figure VI-9. Assum~ a stationary front (dotted line), divide the area into two half-planes. A

I

constant wind (UlO

=

20 mIs) blows as shown in figure VI-9 over the left half-plane. The other half is calm. The energy is zero at y=0 (land boundary) and other sides are open boundaries. The main purpose of this I

study is to test the windsea-swell transition algorithms (The DOLPHIN shows that wind-sea is released to swells (at Point C) where no wind exists.

•

•

The influence of the front can be studied by considering the steady state total energy at the points A and B as shown in figure VI-9a and 9b. The wind-sea increases similarly to case 11 study. In this model the comparison of EIV and ElI for some points A and Bare shown in table 6.3. Very close to the coast there is same value (i.e. EIV ElI) but influence of front is seen clearly at point B, where a part of EIV is transmitted to right side. A more clear picture of swell field in calm sea (at point C) can be observed in two-dimensional spectral distributions.'

This shows that the swell components have radiated from the wind-sea. Table 6.4 shows the relative energies, mean frequencies and mean

directions of swell at point C. These data (table 6.4) are graphically I

shown in figure VI-9c which shows there is no swell dissipation of the model DOLPHIN, hence ratio of EC an~ EB is high.

•

•

•

Case V

•

A diagonal front (45°) is shown in figure VI-10. Initially the sea is calm, then the wind blows to the north (U

=

20

mIs)

SE of the front aJd

y

to west (U

=

-20 mIs) NW of the front. The wind blows to the north for

x

grid points directlyon the diagonal as considered by SWAMP report. The

I

purpose of this case test is to determine the response of the model for changing wind direction. At y

=

800 km (s-line) a series of test results are determined and the total energy and mean wave directions are plotted as shown in figure VI-lOb and VI-lOc. This shows the decrease in wave energy as the waves cross the front and then again slowly increases. The changes of total wave energy and other wave field properties (NW) may be attributed to difference in wind-sea swell transfer algorithms, different slanting fetch windsea growth rates for the newly generated windsea

region, different swell dampoing rates and the chosen numerical finite difference schemes.

•

•

•

43

•