Wind Wave and Storm Surge Workshop Halifax 16-20/06/03 Wind Wave and Storm Surge workshop-Halifax 16-20/06/03 WAVES IN SHALLOW WATER Jean-Michel Lefèvre

Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

Wind Wave and Storm Surge workshop-Halifax 16-20/06/03

WAVES IN SHALLOW WATER

Jean-Michel Lefèvre

Meteo-FranceMarine and Oceanography

[email protected]

Aknowledgement to Jaak Monbaliu from UKL for having provided some of the material


water particle trajectories in progressive waves

of different relative depth (Dean & Dalrymple, 1991; their fig 4.3)

INTRODUCTION

Swell propagation is a « quasi » linear process outside the surf zone



For practical purposes, one considers that a wave is moving in deep water when its wave length is less than twice the depth and that a wave is moving in shallow water when its wave length is more than 1/20 of the depth. In those two cases the dispersion relations can be approximated (with a error less than a few%) with the following relations : in deep water : in shallow water :

gk2 dgk 22

d</20

T=10s, =156m d>78 m

=1OOm d<5 m T=/sqrt(gd)=14s


water particle velocities (Dean & Dalrymple, 1991; their fig 4.1)

So long waves (large period) fell the bottom at larger depth than short waves, and the longer the swell is, the more important the wave transformation is.


)sin(sinh

)(cosh

2 11 tkx

kh

zhkH

)cos(sinh

)(sinh

2 11 tkx

kh

zhkH

water particle trajectory (Dean & Dalrymple, 1991; their fig 4.2)


)(cos2

tkxH

Particle kinematics progressive waves

)sin(sinh

)(cosh

2tkx

kh

zhkC

H


horizontal and vertical velocity

)cos(sinh

)(cosh

2tkx

kh

zhkH

xu

)sin(sinh

)(sinh

2tkx

kh

zhkH

zw

Water particles in a wave over shallow water move in an almost closed circular path near the surface. The orbits become progressively flattened with depth as shown on the figure below.


tkxH coscos2

tkxkh

zhkHg

sincoscosh2

)(cosh

STANDING WAVE

In shalow water, the wave period T necessary to produce a seiche in a closed basin of dimension L is T=2xL/sqrt(gd) or with wave lenght of two times L.

For an open basin on one edge, T=4xL/sqrt(gd)

Can be written as the superposition of two progressive waves propagating in the opposite directions


distribution of water particle velocities in a standing wave

(Dean & Dalrymple, 1991; their fig 4.6)

There is no crest propagation.


particle displacement in a standing wave

tkxkh

zhkH cossinsinh

)(cosh

2 11

tkxkh

zhkH coscossinh

)(sinh

2 11

mean position (x1,z1)


tkxkh

zhkH

xu sinsin

sinh

)(cosh

2

particle velocities in a standing wave

tkxkh

zhkH

zw sincos

sinh

)(sinh

2


potential energy waves


Engineering wave properties

Starting from a flat surface, it is the energy necessary to move the water in order to get a wavy surface, because of the gravity force.


zdmgPEd )(

2

hz

Lx

x

Lx

x

T dxh

gL

PEdL

PE2

)(1)(

1)(

2

162)(

22 Hg

hgPE T

potential energy

16)(

2HgPE waves

N

nnwaves H

gPE

1

2

16)(


kinetic energy

22)(

2222 wudxdz

wudmKEd

Lx

xh

dzdxwu

LKE

2

1 22

16

2HgKE

mean energy, over one wave lenght or one wave period, due to the orbital velocities of the water particles


8

2HgPEKEE

total energy


wave power

power transmitted to a vertical surface of 1m width from the surface to the bottom.


energy flux

h

D dzupF

dtdzupT

Fh

D

Tt

t

1

work done by the dynamic pressure

averaged over one wave period

dtdztkxkh

zhkgHk

kh

zhkg

TF

h

Tt

t

)cos(cosh

)(cosh

2cosh

)(cosh1


)cos(cosh

)(cosh

2tkx

kh

zhkHggzp

)(zKggzp p

kh

zhkK p cosh

)(cosh

hydrostatic dynamic

pressure response factor


group velocity Cg

)(cos2

)(cos2 221121 txk

Htxk

H

2,

2 11

kkk

2,

2 22

kkk

tk

xktkxH

2

1cos)cos(

kCg

group velocity


wave group characteristics



kh

kh

kgHF

2sinh

21

2

1

8

1 2

integrate up to mean water surface

neglect 2-nd order terms in wave height

ECnF

kh

kh

C

Cn g

2sinh

21

2

1


tkx

k

define

t

0

t

k

temporal variation in the wave number vector is balanced by spatial changes in the wave angular frequency

Conservation of waves

scalar phase function

waves entering shallow water


wave number vector


conservation of waves


beach

bottom


0 k

0)sin(

dx

kd

0

0sinsin

CC

ykconstk sin

refraction

(Snell’s law)

0)cos()sin(

y

k

x

k

no alongshore variations :


Snell’s law states that the component of the wave number vector parallel to the isobath is conserved along trajectories. Also since =kc, and because is conserved along trajectories the Snell’s law reads:

The component of the phase speed parallel to the isobath is also conserved along trajectories

.constc

Sin


In shallow water this expression simplifies to:

Let us consider the case of a wave propagating perpendicular to the isobaths. From Snell’s law we have:

Initially and the direction of wave

propagation is not modified.

.Constd

Sin

0cossin dt

dk

dt

dk

0 0dt

d


wave rays

idealized bathymetry



Conservation of energy

2211 bECnbECnF

2

1

2

112 b

b

C

CHH

g

g

rsKKHH 12

shoaling coeff.

refraction coeff.

2

0

2

012 2 b

b

C

CHH

g

2

1

2

112 cos

cos

g

g

C

CHH

in case or regular isobaths


REFRACTION

a cb

d fe

g

h


ddCg

k

1

s

d

dCgs

k

1

d

dCgks

1

These equations can be numerically integrated in order to determine ray paths. Manual methods have also been developed.


refraction parallel contours (Dean & Dalrymple, 1991; their fig 4.18)

h=8m

T=10s

0=40°

h/gT2

=0.0082

=20°

KR=0.905




calculated rays beach with rip channels (Dean & Dalrymple, 1991; their fig 4.21)


Spectral approach

Numerical spectral wave model can deals with depth refraction

Modern wave prediction is based on the solution of the so called energy balance equation. This equation is derived from the principle of the conservation of wave action. In this theory, it is assumed that:

•Amplitudes, frequencies and wave lengths of individual waves vary slowly with respect to the intrinsic space scale (the wave length) and time scale (the wave period), and that linear theory is valid.


•This means that in regions where characteristics of waves are varying rapidly, the theory presented here is not valid and others formulations are required. In the surf zone where waves are breaking, one cannot apply this theory, in « theory ». However the breaking effect can be taken into account in spectral models. The balance equation for the action density spectrum N reads:

(1)

izt

N

SN

dti

dz


Where: =( )

denotes the propagation velocity of the wave group in the four dimensional phase space of (spatial coordinates) and (wave number vector),

is the density action,

F is the energy spectrum,

is the intrinsic angular frequency,

S is the source term.

dt

zd

dt

dz i

x

k

txk

txkFtxkN

,,

),,(,,


The velocity of a wave group is given by the propagation equations:

Where:

is the absolute frequency, (in the fixe frame)

W denotes the dispersion relation, U(x,t) is the current velocity

i

i

k

W

dt

dx

i

i

x

W

dt

dk

txkW ,, Uk


The intrinsic frequency depends on k(x,t) and the water depth d(x,t) through the dispersion relation:

which is the wave frequency in a frame moving with the current

dkkg tanh


Spectral approach

Equation (1) holds in any coordinate system (x,l,t) where x denotes spatial co-ordinates (spherical or Cartesian) and l denotes the two independent spectral variables (chosen from , k, , , and ). In Cartesian coordinates, without current and source terms the energy balance equation reduces to the energy spectrum propagation equation:

k

0

F

dt

dFCg

xt

Fi

i


where,

denotes the group velocity and ,

denotes the rate of change of

direction of a wave component following the path of the wave component. The second term of the equation:

represents the change in the energy spectrum due to the divergence of energy density flux and is termed the shoaling contribution.

)()(i

i

k

W

dt

dxgC

dt

d

FCgx

i

i

FgC


The last term:

describes the redistribution of spectral energy density over the spectrum. It corresponds to a change of direction of a spectral component and is termed the angular refraction. In the general case of a two dimensional bathymetry, the distinction between shoaling and angular refraction is not useful for quantitative studies since most of the time it is not possible to separate the two processes: The consideration of simple cases can help us understanding these processes.

F

dt

d


Shoaling

In the simple situation of waves initially propagating perpendicular (for instance in the x direction) to parallel isobaths we have seen that the direction of the rays remains constant Equation the propagation equation becomes:

The energy flux normal to shore remains constant.

0dt

d

0

FCgdx

dx


As waves propagate from deep water into shallow water, their velocity initially increases slightly and then decreases. Therefore, energy increases after a slight decrease. Energy growth is however limited by wave breaking. In the surf zone, the linear approximation is not longer valid and other approximate equations are needed. While, wave length decreases, waves are becoming more and more steep until they break. What happens to the absolute frequency (or period) in the stationary case?. We have: Stationarity implies:

0

t

k


and since

we have:

So the absolute absoluteperiod remains constant along the path.In terms of energy E instead of spectral energy density F(k), the balance equation in stationary conditions reads:

0

.gCdt

d

0dt

d

0

ECg


This means that the energy flux remains constant between two ray paths. If denotes the length of a crest (section) between two ray paths, then:

s


In the situation where waves approach a straight beach (with straight isobaths), this is equivalent to:

where is the local angle of incidence of the waves.

.constsEC g

22111 2sCEsCE gg

.constCosEC g


h (mètres) k (m-1) (mètres) H/H0

1005025151052

0,0400,0410,0460,0550,0650,0900,142

157153137114977044

10,950,910,94

11,121,36

APPLICATION

For a wave of 10 s period, and k0 = 0.04 m-1 with a

height of H0 off shore, one can compute the wave transformation, λ et H/H0 at several water depths:

H/H0=(1+2kh/sinh 2kh)

k~ k0/(tanh k0 h)


CURRENT REFRACTION

Strong currents are normally present in shallow water areas. Modelling waves in such places often requires taking into account wave-current interactions. This can be done with the use of a coupled wave-current-surge model but this is beyond the scope of this workshop. We will only discuss the effect of slowly varying and limited current speed on waves. Wave kinematics along rays is described through the following relations:

t

Uk

t

d

ddt

d

.

s

UkCgdU

t

d

ddt

d

...

s

Uk

s

d

ddt

dk

.

U

kd

dkdt

d

.1


For A steady medium with a constant depth these equations simplify to: 0

dt

d

s

UkCg

dt

d

..

sU

kdtdk

.

U

kkdt

d

.1


The absolute frequency (which is wave frequency in a fixed frame) remains constant along ray paths in a steady medium, not the intrinsinc frequency (which is wave frequency in a frame moving with the current) unless the current is uniform.Consider the example of a long shore current decreasing towards the shore and waves initially propagating in the direction of the current. Equations implies that waves will turn towards the shore, similarly to what would happen without current but with depth refraction. Another simple example is a wave propagating in the direction of a current which is variable in the direction of propagation, in a steady medium. Since the absolute frequency remains constant along the ray path, we have:

=const. 0 kU


where 0 is wave frequency at the location where there is no current. According to the dispersion relation in deep water, one can show that the above relation reduces to the quadratic equation in , where c denotes the phase

velocity (c=/k) namely;

which has one realistic solution if is positive.

The critical situation appears when this term is equal to zero. That is if the current is equal and opposite to the local group velocity, but waves break shortly before this kinematic limits is reached.

0c

c

002

0

2

c

U

c

c

c

c

041

c

U


WAVE GROWTH INTO SHALLOW WATER

In the theory of the equilibrium in the high frequency range of the wind wave k-spectrum, it was found according to dimensional arguments that its shape is universal and is k-3. Since in deep water the dispersion relation reduces to: 2=gk, we have

k-3 dk=2-5 /g d The frequency spectrum F() which is related to the wave number spectrum F(k) through the relation: F(k)dk=F()d, has therefore a -5 tail. A new universal spectrum similar to the JOWSWAP spectrum and called the TMA spectrum was proposed (Bows et al. 1985):


FTMA(,d) = r(,d) FJONSWAP () where r(,d) is an empirically determined function

The growth is limited by the depth in shallow water. Formula have been derived in order to compute the wave height as a function of depth, wind speed and fetch.


In the case of an idealized flat bottom, a formulation for the maximum significant wave height (Hm) have been established by CERC (1973) based on various measurements. Once simplified, the formulation becomes for shallow water:

))(3tanh(03.0 75.0

210

210

U

dUH m

d (m) U10(m/s) Hm(m)

1 10 0.3

5 20 1.5

10 30 2.7

20 50 6

1 30 0.5


BOTTOM DISSIPATION

In shallow water, the ocean bottom is responsible for energy dissipation by:

- bottom friction- percolation - wave induced bottom motion

Other wave bottom interactions are also important in certain circumstances such as:

- bottom scattering - bottom elasticity.


The first process is conservative, the second is not. Outside the surf zone, bottom friction is usually the most relevant process. A version commonly used for the bottom friction source term comes from the JONSWAP experiment (Hasselmann et al. 1973):

where is an empirically determined coefficient. Several values have been proposed The typical value for is 0.05 m2s-3.

),()2sinh(

2),(, kF

kdg

kkS fricbot


For the above relation reduces to :

The typical time scale for decreasing the energy by a factor 2.7 and therefore the wave height by a factor 1.65 is 200 times the depth d. During this time, the wave has travelling over a distance

Examples:

1kd

),(200

1),(),(, kF

dkF

gdkS fricbot

gd

d

d

3

200

d (m) (hours) (Km)

10 0.5 17.

20 1. 50.

40 2. 150.


Depending on the nature of bottom sediment, percolation can contribute significantly or even dominates bottom dissipation (in cases of porous bottom) and the corresponding source term can be parametrized as:

where a is also an empirically determined coefficient (a=0.01 m/s for coarse sand with a 1mm diameter and a=0.0006 m/s for fine sand with a .25 mm diameter). For the above relation reduces to :

),()2sinh(

2),(

2

. kFkdg

akS perc

1kd

),(2

),(. kFakS perc


The typical time scale for decreasing the energy by a factor 2.7 and therefore the wave height by a factor 1.65 depends on the size of the sand and on the wave lenght :

d3

2.0

(m) (hours)

10 1.

100 10.

for coarse sand:

(m) (hours)

10 0.05

100 0.5

for fine sand:


WAVE BREAKING

In deep water wave breaking is controlled by wave steepness. A theoretical upper limit for wave steepness is 0.14 or 1/7. In the surf zone, because of bottom friction, the upper part of the wave tends to increase its speed relative to the lower part. At some time the crest overtakes the preceding trough and the wave breaks. Several kinds of breaking can occur, depending on the steepness of the beach and on the steepness of the waves before entering shallow water areas;


Spilling breaking The water from the crest falls along the forward face of the wave.

Plunging breaking

The water from the crest rapidly overtakes the wave base and plunges foward into the wave trough.

Surging breaking

The base of the wave surges up the beach before the crest can plunge forward

Collapsing breaking

This is between plunging and surging breaking


types of wave breaking (Dean & Dalrymple, 1991; their fig 4.22)


Wave Breaking

•Spilling when S<0.4, le plus fréquent : H/d=3/5 (gently and regular slopes)

•Plunging when 0.4<S<2 H/d>3/5 (steep slopes, obtacles)

•Surging when S>2 le plus rare: pentes très fortes

SH

tan

/ )

0 0

: beach slope

tan

H0/ 0: steepness in deep water


Wave breaking

bb hH

where possibly depends on a number of parameters (e.g. Weggel, 1972)

2)()(

gT

Hmamb b

To make things simples, =0.6=3/5


wave breaking dimensionless parameters (Dean & Dalrymple, 1991; their fig 4.23)

Steepness=kH off shore

Breaking depth


The evolution of the waves in the surf zone is strongly nonlinear. However, it has been shown (Battjes and Janssen 1978), that the source term for wave breaking in the spectral energy balance can be parametrized with

where is the mean frequency, Q is the fraction of breaking waves determined with

Hm is the maximum possible wave height :

Etot is the total energy.

tot

mbreaking E

kFHQkS

),(

8),(

2

28

ln

1

m

tot

H

E

Q

Q

dHm 6.0


DIFFRACTION

When waves propagate towards an obstacle, waves are diffracted behind the obstacle, towards the sheltered area so that the energy gradient does not become infinite.

Diffraction cannot be described in the framework of the linear wave theory we presented because rapid changes in the wave characteristics occur when diffraction arises. This phenomenon is essentially relevant for coastal engineering applications. However, the cumulative effects of refraction in areas with irregular bathymetry may also cause locally large variations so that diffraction may arise, preventing energy from increasing unrealistically


diffraction (Dean & Dalrymple, 1991; their fig 4.24)


diffraction

)(),()(),,,( tTyxFzZtzyx

0),(22

2

2

2

yxFky

F

x

F

Laplace equation becomes Helmholtz equation

khkg tanh2 tiekhyxF

g

i cosh),(

tieyxFzhktzyx ),())(cosh(),,,(


diffraction pattern - waves normal to breakwaterfull line is an exact solution; dashed line is an approximate solution



diffraction pattern behind a breakwater for normal incident

waves (Dean & Dalrymple, 1991; their fig 4.29)


Wave set up

• Variations of the wave energy (significant wave height) due to shallow water transformation (refraction, shoaling, breaking) cause a horizontal variation of the wave momentum

Gradients of the mean sea level and wave-driven currents


Incident wave propagating towards de beach

(x direction)

dxdMxx

hgdxd

)(1

Mxx=(2-0.5)E shoreward transport

=0.5+kh/sinh(2kh)

In deep water =0.5 and Mxx=E/2

In shalow water =1 and Mxx=E+E/2

- : mean surface elevation above still water

-h: water depth


• The wave set up is about 15-20% Hrms (about Hs/1.4)

• Replace H by Hrms, not by H1/3, when applyng a formula with a single sinusoidal wave of height H to the actual sea state (also for wave power, wave energy…)


Currents generated by breaking waves Long shore currentsWhen waves approach the coastline at an oblique angle, a longshore current is established which flows parallel to the shoreline in the nearshore region with speed between 0.5 and 2 nm.


Rip currents Rip current are strong, narrow currents with speeds up to 4 nm which flow seawards from the surf zone.


L=1m, d=0.5, assume shalow water (!) to make it simple

Compute resonant wave period and fish speed to create such wave, so the fish on the left can get out of the tank where his fish mate is located

1. Seiche (an application from Steeve W. Lyons)


2.Wave power

• Compute the power of a wave of 20 m height, 100m crest lenght, and 20 second period.

• Same with respectively 2m, 100m, 7s

• And 2m, 100m, 14 s


3. Wave lenght computation

• T=14 s• Compute for depth of 5, 3,

2 m


4. Tsunami

• 2m height (H), 200 km wave lenght, 5 km water depth (d)

• Is it deep or shalow water!• Compute wave height just before

breaking, depth where breaking starts (assuming Hb=0.6d and no angular refraction ).


5 Shoaling

• Compute wave height just before breaking, depth where breaking starts, and no angular refraction (assuming Hb=0.6d and no angular refraction ) for incoming deep water swell of:– 5 m height (H), 7s wave

period – 5 m height (H), 14s wave

period

S1

S2

crête

crête

P1.s1 = P2.s2 E1.c1. s1 = E2.c2. S2


6. Bottom refraction

1=45°

d1=5m

d2=2.5m

2=?



h=8m

T=10s

0=40°

h/gT2

=0.0082

=20°

KR=0.905




7. Dissipation due to bottom friction

),(200

1),(),(, kF

dkF

gdkS fricbot

gd

d

d

3

200

d (m) (hours) (Km)

10

20

40 .

Documents

Wind Wave and Storm Surge Workshop Halifax 16-20/06/03 Wind Wave and Storm Surge workshop-Halifax 16-20/06/03 WAVES IN SHALLOW WATER Jean-Michel Lefèvre