coastal engineering - unimasr€¦ · 1 waves knowledge of waves and he forces they generate are...

Coastal Engineering

Prof. K. A. Rakha

Cairo University

Faculty of Engineering

2013

Table of Contents

1 Waves ...................................................................................................... 1-1

1.1 Description of Waves ........................................................................................ 1-1

1.2 Wind and Waves ............................................................................................... 1-2

1.3 Sea and Swell ..................................................................................................... 1-4

1.4 Small Amplitude Wave Theory ....................................................................... 1-5

1.4.1 Solving the Dispersion Equation .................................................................. 1-10

1.5 Reflected Waves .............................................................................................. 1-15

1.6 Short Term Wave Analysis ............................................................................ 1-16

1.6.1 Time Domain Analysis ................................................................................. 1-16

1.6.2 Short-Term Wave Height Distribution ......................................................... 1-17

1.6.3 Frequency Domain Analysis ........................................................................ 1-18

1.7 Wave Generation ............................................................................................ 1-20

1.7.1 Wave Hindcasting ......................................................................................... 1-21

1.8 Wave Transformation .................................................................................... 1-23

1.8.1 Refraction and Shoaling ............................................................................... 1-24

1.8.2 Wave Diffraction. ......................................................................................... 1-27

1.8.3 Wave Breaking ............................................................................................. 1-29

1.9 Wave Models ................................................................................................... 1-33

2 Water Level Variations ......................................................................... 2-1

2.1 Astronomic Tides .............................................................................................. 2-1

2.1.1 Equilibrium Tide (Moon) ............................................................................... 2-2

2.1.2 Daily Inequality .............................................................................................. 2-5

2.1.3 Spring/Neap Tides .......................................................................................... 2-8

2.1.4 Other Effects ................................................................................................. 2-10

2.1.5 Tide Analysis and Prediction ........................................................................ 2-10

K.A.Rakha Jan. 2013 2

2.1.6 Datums .......................................................................................................... 2-13

2.2 Storm Surge..................................................................................................... 2-14

2.3 Barometric Surge ............................................................................................ 2-15

2.4 Seiche ............................................................................................................... 2-17

2.5 Tsunami ........................................................................................................... 2-18

2.6 Eustatic (Sea) Level Change .......................................................................... 2-18

2.7 Isostatic (Land) Rebound and Subsidence ................................................... 2-18

2.8 Global Climate Change .................................................................................. 2-20

3 Currents in the Marine Environment ................................................. 3-1

3.1 Tidal Currents................................................................................................... 3-1

3.2 Wind Generated Currents ............................................................................... 3-1

3.3 Stratification and Density Currents ................................................................ 3-1

3.4 Wave Induced Currents ................................................................................... 3-2

3.4.1 Shore-normal currents .................................................................................... 3-2

3.4.2 Shore-parallel currents .................................................................................... 3-2

3.4.3 Two-dimensional Currents ............................................................................. 3-6

3.5 Hydrodynamic Models ..................................................................................... 3-8

4 Nearshore Sediment Transport ............................................................ 4-1

4.1 Longshore Sediment Transport ...................................................................... 4-3

4.1.1 Predicting Potential Littoral Drift ................................................................... 4-3

4.1.2 Littoral Drift Budget ....................................................................................... 4-4

4.2 On/Offshore Sediment Transport ................................................................... 4-6

4.3 Coastal Sediment Cells ..................................................................................... 4-7

4.4 Sediment Transport Models ............................................................................ 4-8

4.4.1 Morphology and Shoreline Change Models ................................................. 4-10

4.5 One-line Models .............................................................................................. 4-10


4.5.1 Analytical Solution ....................................................................................... 4-12

4.5.2 Model Classification according to Time and Space ..................................... 4-15

4.5.3 Reducing Uncertainty ................................................................................... 4-17

5 References ................................................................................................. 1

1 Waves

Knowledge of waves and he forces they generate are essential for the design of coastal

projects since they are the major factor that determines the geometr of beaches, the

planning and design of marinas, waterways, shore protection measures, hydraulic

structures, and other coastal works.

1.1 Description of Waves

The subject of water waves covers phenomena ranging from capillary waves that have very

short wave periods (order 0.1 seconds) to tides, tsunamis (earthquake generated waves) and

seiches (basin oscillations), where wave periods are expressed in minutes or hours

(Kamphuis, 2000). Wave heights also vary in height from a few millimeters for capillary

waves to 10’s of meters for long waves. A classification by wave frequency of the various

types of waves is given in Fig. 1.1. In the middle of the range of frequencies are the waves

that are the focus of this chapter. They are normally known as gravity waves or wind-

generated waves. Their periods range from 1 to 20 (to 30) seconds and their wave heights

are seldom greater than 10 m. Yet, because of their prevalence, these waves account for

most of the total available wave energy.

Mangor (2004) divides waves into short waves and long waves with short waves of periods

less than 20 second. Long waves are defined as the waves with periods ranging from 20 sec

to 40 min and are divided into surf-beats, harbour resonance, seiche and tsunamis. Water

level oscillations with periods or recurrence intervals larger than an hour such as

astronomical tides and storm surge are referred to as water-level variations.

The shape of a water surface subjected to wind is so complex that it almost defies

description. Even when the first puffs of wind impact an otherwise flat water surface the

resulting distortions present non-linearities that make rigorous analysis impossible. When

the first ripples generated by these puffs are subsequently strengthened by the wind and

interact with each other, the stage has been set for what is known as a confused sea. The

waves will continue to grow ever more complex through processes known only to the sea

itself. It is necessary to simplify the confusion and to use these simplified concepts in

design. This chapter will establish a bridge from the confusing and complex sea state to

theoretical expressions that are simple and can be used for most design purposes.

Waves

K.A.Rakha Jan. 2013 1-2

Figure 1.1: Wave Classification by Frequency (after Kinsman, 1965).

1.2 Wind and Waves

For theoretical analysis of wave generation, the reader is referred to more extensive

references on this subject such as Dean and Dalrymple (1991), Dingemans (1997),

Horikawa (1988), Ippen (1966), Kinsman (1965), Sarpkaya and Isaacson (1981), who

discuss various theoretical models at length. In general, it may be said that wind speed and

wave activity are closely related. There are other important variables to consider such as

depth of water, duration of the storm and fetch (the distance over which the wind blows

over the water and generates waves). At this stage only wind is considered and water

depth, wind duration and fetch are assumed to be unlimited. The resulting waves are often

called Fully Developed Sea and these conditions are approximated only in the deep, open

sea.

The relationship between wind and waves in the open sea is so predictable that sailors have

for centuries drawn a close parallel between wind and waves. The Beaufort Scale in Table

1.1 is a formalized relationship between sea state and wind speed that can be used to obtain

an estimate of waves in the open sea when wind speed is known.

24 h 12 h 5 min 30 s 1 s 0.1 sPeriod

Wave

band

Primary

disturbing

force

Primary

restoring

force

TranstidalLong-period

InfragravityGravity

Ultragravity Capillary

Storm systems, tsunamis

Sun, Moon

Coriolis forceGravity

Wind

Surface tension

Time (s)

En

ergy

(L2)

24 h 12 h 5 min 30 s 1 s 0.1 s

24 h 12 h 5 min 30 s 1 s 0.1 sPeriod

Wave

band

Primary

disturbing

force

Primary

restoring

force

TranstidalLong-period

InfragravityGravity

Ultragravity Capillary

Storm systems, tsunamis

Sun, Moon

Coriolis forceGravity

Wind

Surface tension

Time (s)

En

ergy

(L2)

24 h 12 h 5 min 30 s 1 s 0.1 s

Waves


Table 1.1: Beaufort Scale of Wind And Sea State1)

Beaufort

Wind

Force

Wind

Speed

(knots)2)

Description of

wind Description of Sea

Approx

Hs (m)

ApproxT

(sec)

0 0-1 Calm Sea like a mirror. 0 1

1 1-3 Light airs Ripples are formed. 0.025 2

2 4-6 Light breeze

Small wavelets, still short but more

pronounced; crests have a glassy

appearance, but do not break

0.1 3

3 7-10 Gentle breeze Large wavelets, crests begin to break.

Perhaps scattered white caps. 0.4 4

4 11-21 Moderate

breeze

Small waves, becoming larger; fairly

frequent white capping. 1 5

5 17-21 Fresh breeze

Moderate waves, taking a more

pronounced long form; many white caps

are formed (chance of some spray).

2 6

6 22-27 Strong breeze

Large waves begin to form; the white

foam crests are more extensive

everywhere (probably some spray).

4 8

7 28-33 Moderate gale

Sea heaps up and white foam from

breaking waves begins to be blown in

streaks along the direction of the wind

(spindrift).

7 10

8 34-40 Fresh gale

Moderately high waves of greater length;

edges of crests break into spindrift. The

foam is blown in well-marked streaks

along the direction of the wind. Spray

affects visibility.

11 13

9 41-47 Strong gale

High waves. Dense streaks of foam along

the direction of the wind. Sea begins to

roll. Visibility affected.

18 16

10 48-55 Whole gale3)

Very high waves with long overhanging

crests. The resulting foam is in great

patches and is blown in dense white

streaks along the direction of the wind.

On the whole, the surface of the sea takes

a white appearance. The rolling of the sea

becomes heavy and shocklike. Visibility

is affected.

25 18

11 56-63 Storm3)

Exceptionally high waves (small and

medium sized ships might for a long time

be lost to view behind the waves). The

sea is completely covered with long white

patches of foam lying along the direction

of the wind. Visibility affected.

354) 204)

12 64-71 Hurricane3)

Air filled with foam and spray. Sea

completely white with driving spray;

visibility very seriously affected.

404) 224)

1) Fully developed sea - unlimited fetch and duration. 2) 1 knot 1.8 km/hr 0.5 m/s 3) Required durations and fetches are seldom attained to generate fully developed sea. 4) Really only a 30-40 m deep interface between sea and air.

Waves


1.3 Sea and Swell

Waves generated locally by wind are generally known as sea, which consists of waves of

many different wave heights and periods as shown in the time series in Fig. 1.2 (irregular

waves). The waves in Fig. 1.2 form what is called a wave train. The waves, on average

propagate more or less in the wind direction.

Fig. 1.2: Record of Locally Generated Sea

On large bodies of water, the waves will travel beyond the area in which they are

generated. For example, waves generated by a storm off the French coast may travel

southward and eventually arrive to Tunisia. While the waves travel such long distances, the

energy of the individual waves is dissipated by internal friction and wave energy is

transferred from the higher frequencies to lower frequencies. The resulting waves arriving

in Tunisia will be more orderly than the initial sea, generated off France, with longer wave

periods (10-20 sec) and smaller wave heights. Such waves, which are generated some

distance away and travel into an area, are called swell (see Fig. 1.3).

On most coasts, sea and swell occur simultaneously. The exceptions are enclosed bodies of

water such as lakes, reservoirs and inland seas, where swell cannot arrive from long

distances away.

WL (m)

Time (sec)

WL (m)WL (m)

Time (sec)

Waves


Fig. 1.3: Sea and Swell

1.4 Small Amplitude Wave Theory

In this section a simplified method of representing wave motion will be introduced. It is

called Small Amplitude Wave Theory. It may appear to be almost impossible to adequately

represent locally generated, confused sea as in Fig. 1.2. It might also be expected that any

simple theory would be more applicable to the more regular swell conditions. Yet over the

years, it was found that for most problems there is no need to differentiate between sea and

swell or to use a more complicated wave theory. Small Amplitude Wave Theory can be

confidently applied to both sea and swell (Kamphuis, 2000). More complex wave theories

have been developed, but they are normally used only for research and complex designs.

For most straightforward designs small amplitude wave theory has been found sufficient.

A wave is periodic if its motion and surface profile recur in equal intervals of time termed

the wave period. A wave form that moves horizontally relative to a fixed point is called a

progressive wave and the direction in which it moves is termed the direction of wave

propagation. A progressive wave is called wave of permanent form if it propagates without

experiencing any change in shape.

The basis for small amplitude wave theory is the sinusoidal wave, shown in Fig. 1.4.

Furthermore it is assumed that the ocean waves are two dimensional, small in amplitude,

and progressively definable by their wave height and period for a given water depth.

A right hand system of coordinates is used with its origin at still water level (SWL). The

SWL is defined as the water surface that would exist in the absence of any wave action.

The x axis is horizontal and parallel to the direction of wave propagation. The y axis is

horizontal and perpendicular to the x axis. The z-axis is vertically up and therefore the

Waves


position of the bottom is at z = -d. The highest point of the wave is the crest and the lowest

point is the trough. The sinusoidal water surface η may be described by,

T

t

L

xat) - (kx a =

22coscos (1.1)

where a is the amplitude of the wave, x is distance in the direction of wave propagation, t is

time, k is the wave number (the angular frequency at which the wave pattern repeats itself

in space), is the angular wave frequency (the angular frequency of repetition in time), L

is the wave length and T is the wave period. The values of k and are calculated from,

T

2 =

L

2 = k

; (1.2)

The maximum vertical distance between crest and trough of the wave is called the wave

height, H(=2a). Since in an actual wave train, such as in Fig. 1.2, the wave heights and

lengths are not all the same, statistical representative values are used. The ratio of wave

height to wave length (H/L) is called wave steepness. The wave form moves forward and

the velocity of propagation of the wave (or phase speed) is calculated from,

T

L = C (1. 3)

Mean water level (MWL) is defined as the level midway between wave crest and trough.

In small amplitude wave theory, MWL is the same as SWL, but for higher order wave

theories MWL will be above SWL Further, waves are differentiated as Long-Crested or

Short-Crested which refers to the length of the wave crest perpendicular to the wave shape

and its velocity of propagation. Swell is normally long crested (the wave is recognizable as

a single crest over a hundred meters or so) and Sea is normally short crested. Waves are

considered to be in deep water when d/L > 0.5 and in shallow water when d/L > .0.0.

Between these limiting conditions, the water depth is called transitional.

The Small Amplitude Wave Theory expressions are summarized in Table 1.2. Equation [1]

(equation numbers in square brackets refer to those in Table 1.2) describes the water

surface fluctuation as shown in Fig. 1.4. Equation [2] calculates the velocity of

propagation, C, assuming the wave retains a constant form. The 'tanh' term has two

asymptotic values. For large depths, kd (or d/L) is large resulting in,

1 L

d = kd

2tanhtanh (1.4)

For small depths,

)L

d2( )

L

d2( = kd

tanhtanh (1. 5)

Thus, it is possible to give deep and shallow water asymptotic values for C as in Table 1.2.

It has been customary to define deep water as d/L>0.5 (tanh kd = 0.996) and shallow water

is usually defined as d/L<0.05 (kd = 0.592, while tanh kd = 0.531).

Waves


Fig. 1.4: Sinusoidal Wave and Wave Parameters

Waves propagate at velocity C, but the individual water particles do not propagate; they

move in particle orbits as shown in Fig. 1.5. For small amplitude wave theory, such particle

orbits are elliptical and if the water is 'deep', they become circular. Their size decreases

with depth. Horizontal and vertical orbital velocity components, u and w, and orbit

semi-axes, A and B, are given in Eqs. [4] to [7].

SWL

L

x

z

c

Ha=H/2

Trough

Crest

d

z = -d

SWL

L

x

z

c

Ha=H/2

Trough

Crest

d

z = -d

Waves


Table 1.2: Common Expressions for Linear Progressive Waves

Parameter General Deep

(d/L > 0.5)

Shallow

(d/L < 0.05)

1. Water Surface

t)(kx-

cos 2

H =

w

2. Velocity of

Propagation

(Dispersion

Equation) kd tanh

2

gL =

kd tanh 2

gT =

k =

T

L = C

2

gT = Co gd = C

3. Wave Length kd tanh 2

gT = CT = L

2

2

gT = L

2

o CTL

4. Horizontal

Orbital Velocity w

cos

kdsinh

d)k(z+cosh

T

H =u

wT

cos e H

= uzko

oo

wd

gcos

2

H =u

5. Vertical

Orbital Velocity w

sin kdsinh

d)k(z+sinh

T

H = w

wT

sin e H

= wzko

oo

wd

z

T

sin 1

H = w

6. Horizontal

Semi- Axis kdsinh

d)(z+kcosh

2

H =A e

2

H = A

zkoo

o

4

H = A

d

gT

7. Vertical Semi-

Axis kdsinh

d)(z+ksinh

2

H = B

A = B oo 1

2

H = B

d

z

8. Pressure

Kz+- = g

pp

9. Pressure

Response Factor kdcosh

d)(z+coshk = K p

e = Kzk

po

1 = Kp

10. Energy

Density

2

8

1gHE

11. Wave Power EC = P G

2

EC = P

oo EC = P

12. Group

Velocity Cn = CG

2

C = C

o

G o C = CG

13. Group

Velocity

Parameter

kd2sinh

kd2 + 1

2

1 =n

2

1 = n o 1 =n

Waves


Fig. 1.5: Orbital Motion of Particles.

The pressure fluctuations at any point below the water surface are related to the water level

fluctuations at the surface. If the wave were infinitely long, the water level would be

horizontal at any time and the pressure fluctuations would be hydrostatic. The pressure

fluctuation would be (gH), where is the fluid density and g is the gravitational

acceleration. For waves of limited length the pressure fluctuations are smaller than (gH).

The ratio of the actual pressure fluctuations to (gH), is called the pressure response factor,

Kp, and it is a function of wave length (or wave period) and depth below the surface. For

longer waves or for locations close to the water surface, the pressure response factor

approaches 1. For shorter waves or for locations far below the water surface, the pressure

response factor approaches zero. Eqs. [8] and [9] quantify the pressure response.

Elliptical Orbits

A > B

A A

B

B

SWL

Bottom z = -dw = 0

u > 0

Elliptical Orbits

A > B

A A

B

B

SWL

Bottom z = -dw = 0

u > 0

Waves


Wave Energy is expressed per unit surface area as Energy Density, E, in joules/m2

as in Eq.

[10]. It is made up of half Potential Energy and half Kinetic Energy. Eq. [11] gives Wave

Power, P, arriving at any location. Its units are watts/m of wave crest.

Eq. [2] indicates that longer period waves travel faster than shorter period waves. A real

wave train, as in Fig. 1.2, contains many different wave periods and therefore it would

stretch out (disperse) as it traveled. The longest waves would lead and run further and

further ahead with time and distance, while the shortest waves would lag further behind.

Hence Eq. [2] is called the Dispersion Equation.

Equation [2] also means that waves of roughly the same period tend to travel together.

Waves of almost the same period interfere to form beats or wave groups, resulting in two

wave speeds involved: the speed of the individual waves given by Eq. [2] and the speed of

the wave group, which is C multiplied by the factor n, given in Eq. [13]. In deep water n

approaches ½ and in shallow water n approaches 1. Thus CG<C, but in very shallow water

CG approaches C.

1.4.1 Solving the Dispersion Equation

To solve Eq. [2] and all the other equations in Table 1.2, it is necessary to know the wave

length, L, which may be calculated using Eq. [3]. However, Eq. [3] is implicit and can only

be solved numerically. Tables of solutions have been prepared that yield L as well as other

important wave characteristics (see Table 1.3). Such tables are known as Wave Tables and

have been published in Shore Protection Manual (l984) and Wiegel (l964). To use the

wave tables, the deep water approximation of wave length is first calculated as given by

Eq. [3]. Then using the depth of water, d, it is possible to enter the wave tables with d/Lo to

evaluate all the remaining wave parameters.

The use of the wave table is suitable for only a few calculations. For a large number of

calculations, L or C may be calculated using a root finding technique such as

Newton-Raphson, but such a technique requires iteration. To speed up such computations,

approximations may be used such as the one proposed by Hunt (1979),

)y0.0675 + y0.0864 + y0.4622 +0.6522y + (1 +y = gd

C 1-542 1-2

(1. 6)

where

L

d2 =y

o

(1. 7)

Waves


Table 1.3: Wave Table

0L

d tanh kd

L

d kd sinh kd cosh kd

kd2sinh

kd2 Ks

0.000 0.000 0.0000 0.000 0.000 1.00 1.000 ∞

002 112 0179 112 113 01 0.992 2.12

004 158 0253 159 160 01 983 1.79

006 193 0311 195 197 02 975 62

008 222 0360 226 228 03 967 51

0.010 0.248 0.0403 0.253 0.256 1.03 0.958 1.43

015 302 0496 312 317 05 938 31

020 347 0576 362 370 07 918 23

025 386 0648 407 418 08 898 17

0.030 0.420 0.0713 0.448 0.463 1.10 0.878 1.13

035 452 0775 487 506 12 858 09

040 480 0833 523 548 14 838 06

045 507 0888 558 588 16 819 04

0.050 0.531 0.0942 0.592 0.627 1.18 0.800 1.02

055 554 0993 624 665 20 781 1.01

060 575 104 655 703 22 762 0.993

065 595 109 686 741 24 744 981

070 614 114 716 779 27 725 971

0.075 0.632 0.119 0.745 0.816 1.29 0.707 0.962

080 649 123 774 854 31 690 955

085 665 128 803 892 34 672 948

090 681 132 831 929 37 655 942

095 695 137 858 0.968 39 637 937

0.10 0.709 0.141 0.886 1.01 1.42 0.620 0.933

11 735 150 940 08 48 587 926

12 759 158 0.994 17 54 555 920

13 780 167 1.05 25 60 524 917

14 800 175 10 33 67 494 915

0.15 0.818 0.183 1.15 1.42 1.74 0.465 0.913

16 835 192 20 52 82 437 913

17 850 200 26 61 90 410 913

18 864 208 31 72 1.99 384 914

19 877 217 36 82 2.08 359 916

Waves


0L

d tanh kd

L

d kd sinh kd cosh kd

kd2sinh

kd2 Ks

0.20 0.888 0.225 1.41 1.94 2.18 0.335 0.918

21 899 234 47 2.05 28 313 920

22 909 242 52 18 40 291 923

23 918 251 57 31 52 271 926

24 926 259 63 45 65 251 929

0.25 0.933 0.268 1.68 2.60 2.78 0.233 0.932

26 940 277 74 75 2.93 215 936

27 946 285 79 2.92 3.09 199 939

28 952 294 85 3.10 25 183 942

29 957 303 90 28 43 169 946

0.30 0.961 0.312 1.96 3.48 3.62 0.155 0.949

31 965 321 2.02 69 3.83 143 952

32 969 330 08 3.92 4.05 131 955

33 972 339 13 4.16 28 120 958

34 975 349 19 41 53 110 961

0.35 0.978 0.358 2.25 4.68 4.79 0.100 0.964

36 980 367 31 4.97 5.07 091 967

37 983 377 37 5.28 37 083 969

38 984 386 43 61 5.70 076 972

39 986 395 48 5.96 6.04 069 974

0.40 0.988 0.405 2.54 6.33 6.41 0.063 0.976

41 989 415 60 6.72 6.80 057 978

42 990 424 66 7.15 7.22 052 980

43 991 434 73 7.60 7.66 047 982

44 992 443 79 8.07 8.14 042 983

0.45 0.993 0.453 2.85 8.59 8.64 0.038 0.985

46 994 463 91 9.13 9.18 035 986

47 995 472 2.97 9.71 9.76 031 987

48 995 482 3.03 10.3 10.4 028 988

49 996 492 09 11.0 11.0 026 990

0.50 0.996 0.502 3.15 11.7 11.7 0.023 0.990

∞ 1.000 ∞ ∞ ∞ ∞ 0.000 1.000

Waves


Example 1.1

In this example the small amplitude wave parameters given in Table 1.2 are calculated for

a wave of period, T = 10 sec, with a wave height, H = 1.5 m in a depth of water, d = 9.4 m.

First, it is necessary to calculate the deep water wave length and relative depth:

060.0156

4.9;156)100(56.156.1

2

22

o

oL

dmT

gTL

The wave table (Table 1.3) yields the following:

0.881=n; 1.22kd; 0.703=kd

; 0.575=kd; 0.104=L

d

)762.01(*5.0coshsinh

tanh

From the value of L

d, the wave length in 9.4 m of water and wave number, k, may now be

calculated:

069.02

;4.90104.0

L

kmd

L

From these, the following parameters may be computed; is assumed to be 1035 kg/m3 for

sea water.

crestwaveof w/mECPj/m gH

E

m/s CnCm/s =T

L=C

G

G

730,22;28548

;96.7)04.9(881.0;04.9

2

At the bottom:

0.1)(cosh;0)(sinh;0)(; dzkdzkdzkdz

and the horizontal component of orbital velocity is:

)cos(67.0)cos(703.

1

10

)5.1()cos(

sinh

1tkxtkxtkx

kdT

HuB

Thus, at the bottom, uB has a maximum value uB = 0.67 m/s and the vertical velocity

component of orbital motion at the bottom is zero. The amplitude of the orbital motion at

the bottom is

Waves


m. 1.07 = (0.703) 2

1.5 =

kd 2

H = AB

sinh

and the orbital diameter is 2AB = 2.14 m. The pressure response factor Kp at the bottom is:

0.82 = 1.22

1 =

kd = K Bpcosh

1)(

which means that the maximum pressure fluctuation caused by the wave height H = 1.5 m

is:

water)of (m)( 1.23 = 1.50.82 = H K p

Waves


1.5 Reflected Waves

When a wave reaches a rigid, impermeable vertical wall the wave is completely reflected.

After some time, under well controlled conditions, the reflected waves and the incident

waves together form a system of waves whose form no longer moves forward in space,

commonly known as a standing wave. A theoretical expression for such a standing wave,

as shown in Fig. 1.6, may be obtained by superposition of the equations for an incident and

a reflected wave. It may be seen that the pattern repeats itself every half wave length and

that the first location of the maximum wave height (antinode) is at the structure, while the

first location of zero wave height (node) is located L/4 from the wall. The maximum wave

height is twice the height of the original incident wave.

Fig. 1.6: Standing Waves

L

Node Node

Anti-Node Anti-Node

At t = 0, T, ...

At t = T/2,

3T/2, ...

L

Node Node

Anti-Node Anti-Node

At t = 0, T, ...

At t = T/2,

3T/2, ...

Waves


Partial wave reflection will result if the reflecting surface is sloping, flexible or porous and

yields a variation in wave height. The partial antinodes (Hmax) are less than twice the

incident wave height, while the partial nodes (Hmin) are greater than zero.

1.6 Short Term Wave Analysis

In the first part of this chapter, waves on the sea surface were assumed to be nearly

sinusoidal with constant height, period and direction (i.e., monochromatic waves). Visual

observation of the sea surface and measurements indicate that the sea surface is composed

of waves of varying heights and periods moving in differing directions. In the first part of

this chapter, wave height, period, and direction could be treated as deterministic quantities.

Once we recognize the fundamental variability of the sea surface, it becomes necessary to

treat the characteristics of the sea surface in statistical terms. This complicates the analysis

but more realistically describes the sea surface. The term irregular waves will be used to

denote natural sea states in which the wave characteristics are expected to have a statistical

variability in contrast to monochromatic waves, where the properties may be assumed

constant. Monochromatic waves may be generated in the laboratory but are rare in nature.

In analysis of wave data, it is important to distinguish between Short-Term and Long-Term

wave analysis. Short-Term analysis refers to analysis of waves that occur within one wave

train or within one storm; Long-Term analysis refers to the derivation of distributions that

cover many years. This section deals with short term wave analysis.

Two approaches exist for short term analysis of irregular waves: spectral methods and

wave-by-wave (wave train) analysis. Spectral approaches are based on the Fourier

Transform of the sea surface. This analysis is usually called frequency domain analysis

since the wave spectra is used rather than a time series. Indeed this is currently the most

mathematically appropriate approach for analyzing a time-dependent, three-dimensional

sea surface record. Unfortunately, it is exceedingly complex and at present few

measurements are available that could fully tap the potential of this method. However,

simplified forms of this approach have been proven to be very useful.

The other approach used is wave-by-wave analysis. In this analysis method, a time-history

of the sea surface at a point is used, the undulations are identified as waves, and statistics

of the record are developed. This method is used called the time domain analysis since it

deals with a time series of the water surface. The primary drawback to the wave-by-wave

analysis is that it cannot tell anything about the direction of the waves. Indeed, what

appears to be a single wave at a point may actually be the local superposition of two

smaller waves from different directions that happen to be intersecting at that time.

Disadvantages of the spectral approach are the fact that it is linear and can distort the

representation of nonlinear waves.

1.6.1 Time Domain Analysis

In the time-domain analysis of irregular or random seas, wave height and period,

wavelength, wave crest, and trough have to be carefully defined for the analysis to be

performed. The definitions provided earlier in the regular wave section of this chapter

assumed that the crest of a wave is any maximum in the wave record, while the trough can

be any minimum. However, these definitions may fail when two crests occur within an

Waves


intervening trough lying below the mean water line. Also, there is not a unique definition

for wave period, since it can be taken as the time interval between either two neighbouring

wave troughs or two crests. Other more common definitions of wave period are the time

interval between successive crossings of the mean water level by the water surface in a

downward direction called zero down-crossing period or zero up-crossing period for the

period deduced from successive up-crossings.

Using these definitions of wave parameters for an irregular sea state, the periods and

heights of irregular waves are not constant with time, changing from wave to wave. Wave-

by-wave analysis determines wave properties by finding average statistical quantities (i.e.,

heights and periods) of the individual wave components present in the wave record. Wave

records must be of sufficient length to contain several hundred waves for the calculated

statistics to be reliable.

Average statistical representations for an irregular sea state may be defined in several

ways. These include the mean height H , the root-mean-square height, and the mean height

of the highest one-third of all waves known as the significant height. Among these, the

most commonly used is the significant height, denoted as Hs or H1/3. Significant wave

height has been found to be very similar to the estimated visual height by an experienced

observer (Kinsman, 1965). The average of the highest 10% (H0.1) or the highest 1% (H0.01)

is also sometimes used for design purposes. The average statistical period could be the

mean period, or average zero-crossing period, etc.

1.6.2 Short-Term Wave Height Distribution

The heights of individual waves may be regarded as a stochastic variable represented by a

Probability Distribution Function (PDF). From an observed wave record, such a function

can be obtained from a histogram of wave heights normalized with the mean heights in

several wave records measured at a point. The Rayleigh distribution was found to be the

most suitable distribution for representing wave heights within a storm (short term). Figure

1.7 shows the Rayleigh distribution (p curve) together with the cumulative Rayleigh

distribution (P curve). Equation (1.8) provides the equation for the Rayleigh PDF,

2

4exp

2 H

H

H

H

H

Hp

(1. 8)

The Cumulative Distribution Function (CDF) of wave heights based on the Rayleigh

distribution (the probability that any individual wave of height H' is not higher than a

specified wave height H) can be written as,

2

4exp

H

H

H

HP

(1. 9)

The Rayleigh distribution is generally adequate, except in shallow water in which it may

overestimate the number of large waves. Investigations of shallow-water wave records

from numerous studies indicate that the distribution deviates from the Rayleigh, and other

distributions have been shown to fit individual observations better (SPM, 1984). The

Waves


primary cause for the deviation is that the large waves suggested in the Rayleigh

distribution break in shallow water.

Using the Rayleigh distribution the following relationships can be derived:

H 0.707 = H H 0.63 = H

H 1.67 = H H 1.27 = H

srmss

s0.01s0.1 (1. 10)

Fig. 1.7: Rayleigh distribution

1.6.3 Frequency Domain Analysis

Considering a single-point time-history of surface elevation, spectral analysis proceeds

from viewing the record as the variation of the surface from the mean and recognizes that

this variation consists of several periodicities. In contrast to the wave-by-wave approach,

which seeks to define individual waves, the spectral analysis seeks to describe the

distribution of the variance with respect to the frequency of the signal. By convention, the

distribution of the variance with frequency is written as S(f) with the underlying

assumption that the function is continuous in frequency space (see Fig. 1.8). The reason for

this assumption is that all observations are discretely sampled in time, and thus, the

analysis should produce estimates as discrete frequencies which are then statistically

smoothed to estimate a continuum. S(f) is known as the Wave Variance Spectral Density

Function or Wave Spectrum. Variance is a statistical term and it is preferable to develop a

physical explanation for the wave spectrum. This is attained by using the frequency energy

spectrum E(f). Assuming linear wave theory valid, the energy of the wave field may be

estimated by multiplying S(f) by ρg.

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

H / H

Pro

ba

bilit

y

p

P

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

H / H

Pro

ba

bilit

y

p

P

Waves


The surface can be envisioned not as individual waves but as a three-dimensional surface,

which represents a displacement from the mean and the variance to be periodic in time and

space. The simplest spectral representation is to consider E(f,θ), which represents how the

variance is distributed in frequency f and direction θ. E(f,θ) is called the 2-D or directional

energy spectrum because it can be multiplied by ρg to obtain wave energy. The advantage

of this representation is that it tells the engineer about the direction in which the wave

energy is moving.

The different wave height statistics (e.g. significant wave height) can be determined from

the moments of the wave spectra. The moments of the wave spectrum are defined as,

df S(f) f = mn

0n

f

f (1. 11)

The zero moment is therefore the area under the spectrum

2

0o = df S(f) = m f

f

f

(1. 12)

From the area under the wave spectrum, assuming the wave height distribution to be

Rayleigh, the various wave heights may be estimated. To distinguish between significant

wave height (derived from time domain analysis) and its counterpart, derived from

frequency analysis, the latter is called the Characteristic Wave Height or Zero Moment

Wave Height.

fmoch 4 = H = H (1. 13)

The representation of the wave energy distribution with frequency is a large improvement

over the time-domain analysis methods discussed earlier. With this information it is

possible to study resonant systems such as the response of drilling rigs, ships' moorings,

etc. to wave action, since it is now known in which frequency bands the forcing energy is

concentrated. It is also possible to separate sea (shorter period waves) and swell (longer

period waves) via the wave spectrum, when both occur simultaneously.

Since there are many wave frequencies (or wave periods) represented in the spectrum it is

usual to characterize the wave spectrum by its peak frequency fp, the frequency at which

the spectrum displays its largest variance (or energy). The peak period may be defined as,

f

1 = T

p

p (1. 14)

Since the measured spectra show considerable similarity, a number of attempts have been

made to formulate parametric expressions. One commonly used spectrum in wave

hindcasting and forecasting projects is the single-parameter spectrum of Pierson-

Moskowitz PM (Pierson and Moskowitz 1964). An extension of the PM spectrum is the

JONSWAP spectrum (Hasselmann et al. 1973, 1976).

Waves


Fig. 1.8: Wave spectra.

1.7 Wave Generation

When a gentle breeze blows over water, the turbulent eddies in the wind field will

periodically touch down on the water, causing local disturbances of the water surface. The

wind speed must be in excess of 0.23 m/s to overcome the surface tension in the water.

Theory (Phillips, 1957) shows wind energy is transferred to waves most efficiently when

they both travel at the same speed. But wind speed is normally greater than the wave

speed. For this reason the generated waves will form as an angle to the wind direction so

that the component of wind speed in the direction of wave propagation approaches the

wave speed. The generated wave crests are short crested, irregular waves.

Once the initial wavelets have been formed and the wind continues to blow, energy is

transferred from the wind to the waves. Much of the wind energy is transferred to the

higher frequency waves, i.e., the wind causes more ripples to form on top of existing

waves, rather than increasing the size of the larger waves directly by shear and pressure

differences. This pool of high frequency energy is then transferred to lower frequencies by

the interaction of the high frequency movement with the adjacent slower moving water

particles. This wave-wave interaction transfers wave energy to the lower frequencies of the

wave spectrum.

Earlier we stated that wave height and wave period is closely related to wind speed. It

should therefore be possible to derive wave conditions from known wind conditions. In

fact, it should be possible to reconstruct a wave climate at a site from historical, measured

wind records. Such a computation is known as Wave Hindcasting.

f (Hz)fp

S(f)

(m2/Hz)

Area = 2f

Waves


1.7.1 Wave Hindcasting

For most locations, it is difficult to find long term wave data that is essential for the design

of any coastal project. Hindcasted wave data is usually used for such purposes.

1.7.1.1 Parametric Methods

The theory of wave generation has had a long and rich history. Beginning with some of the

classic works of Kelvin (1887) and Helmholtz (1888) in the 1800's, many scientists,

engineers, and mathematicians have addressed various forms of water wave motions and

interactions with the wind. In the early 1900's, the work of Jeffreys (1924, 1925)

hypothesized that waves created a "sheltering effect" and hence created a positive feedback

mechanism for transfer of momentum into the wave field from the wind. However, it was

not until World War II that organized wave predictions began in earnest. During the

1940's, large bodies of wave observations were collated and the bases for empirical wave

predictions were formulated. Sverdrup and Munk (1947) presented the first documented

relationships among various wave-generation parameters and resulting wave conditions.

The method was later extended by Bretschneider (e.g., Bretschneider, 1958) to form the

empirical method, now known as the SMB Method. The method is described fully in

Shore Protection Manual (1977). In Shore Protection Manual (1984) this method was

replaced by the Jonswap Method, based on research on wave spectra in growing seas by

Hasselmann et al (1973).

The Jonswap, SMB and similar methods are called parametric methods because they use

wind parameters to produce wave parameters, rather than develop a detailed description of

the physics of the processes. Although, these methods produce only Significant Wave

Height (Hs) and Significant Wave Period (Ts), they may be extended to provide estimates

of the parametric wave spectra.

Waves are not only a response to wind speed (U). Wind direction (θ) determines the

general direction of wave travel (wind and wave directions are defined as the directions

from where they come). Fetch (F), the distance over which the wind blows over the water

to generate the waves, is important. Storm duration (t) is important and finally the depth of

water in the generating area (d) influences the wave conditions through bottom friction.

Parametric wave hindcasting derives H and T from U, F, t and d. The wave direction is

usually assumed to be the wind direction. This assumption can be a source of substantial

errors in wave direction, that will result in large errors in the computation of responses

such as alongshore sediment transport rate. If F, t and d are all infinite, the result is a Fully

Developed Sea. The waves are fully developed so that any added wind energy is balanced

by wave energy dissipation rate resulting from internal friction and turbulence. In that case,

the resulting wave conditions are a function of wind speed only, as described by the

Beaufort Scale (Table 1.1). When F, t or d are limited, the resulting waves will be smaller.

The Jonswap method of wave hindcasting uses the following dimensionless expressions.

U

gd=d ,

U

gt=t ,

U

gT=T ,

U

gH=H ,

U

gF=F

2

**p

p

*

2

momo

*

2

* (1. 15)

Waves


These are dimensionless versions of fetch length, characteristic (zero moment) wave

height, peak period of the spectrum, storm duration and depth of water. Note that F, H,

and d are in metres, t and T are in seconds and U is in m/sec.

The Jonswap relationships are:

2

1

** )(F 0.0016Hmo (1. 16)

3

1

** )(F 0.286=Tp (1. 17)

and

3

2

** )(8.68 Ft (1. 18)

Three different conditions must be distinguished for waves generated in deep water. They

can be Fetch Limited, Duration Limited or Fully Developed Sea. On a small water body,

the waves would be limited by a short fetch and Hmo and Tp can be calculated directly from

Eqs. 1.14 and 1.15. On a larger body of water, the same equations apply, but wind duration

may limit the size of waves. Eq. 1.16 is then used to calculate an effective fetch (the fetch

needed to produce the same wave height if the duration had been infinite)

68.8

t=F

*3/2

eff

* (1. 19)

When F* < Feff

*, the waves are fetch limited and Eqs. 1.14 and 1.15 are used with F

*; when

Feff* < F

* the waves are duration limited and Eqs. 1.14 and 1.15 are used with Feff

* . Finally,

for a large body of water and a large duration a fully developed sea exists, which is

calculated using the following upper limits:

71,500=t ; 8.134=T ; 0.2433=H**

p*mo (1. 20)

The procedure of computing Hmo and Tp by Jonswap has been published as a nomogram in

the Shore Protection Manual (1984).

1.7.1.2 Numerical Models

For many applications, the above simplistic hindcast methods are good enough for first

estimates especially of maximum conditions. However, for many applications, it is

necessary to have a long-term hindcast wave climate relating waves to wind at hindcast

intervals which usually are 1 hour, 3 hours or 6 hours. For this purpose, numerical models

are used. These models can be one dimensional 1D as explained in Kamphuis (2000) or

two dimensional 2D.

Waves


Two dimensional models calculate the spectral wave fields over large areas. The WAM

model (WAMDI, 1988), and the Wavewatch (Tolman, 1991) are examples of such models.

Figure 1.9 provides a sample of the wave field calculated over the Mediterranean Sea at a

certain instant using the Wavewatch model (Eldeberky et al., 2002).

Fig. 1.9: Sample of Wavewatch results over the Mediterranean (Eldeberky et al.

2002).

These models can be run in forecast mode using wind forecasted over the water body.

Many centers world wide sell hindcasted data obtained from advanced offshore wave

models (e.g. British Met Office BTO). Many other centers provide wave forecasts based on

advanced offshore wave models (e.g. https://www.fnmoc.navy.mil). These forecasts range

from global forecasts to local forecasts.

1.8 Wave Transformation

Coastal engineering considers problems near the shoreline normally in water depths of less

than 20 m. The study of shoreline change and beach protection frequently requires analysis

of coastal processes over entire littoral cells, which may span over tens of kilometres.

https://www.fnmoc.navy.mil/

Waves


Wave data are generally not available at the site or depths required. Often a coastal

engineer will find that data have been collected or hindcast at sites offshore in deeper water

or nearby in similar water depths. Thus it is essential in such case to transform the waves

from offshore or nearby locations to nearshore locations.

Waves propagating through shallow water are strongly influenced by the underlying

bathymetry and currents. A sloping or undulating bottom, or a bottom characterized by

shoals or underwater canyons, can cause large changes in wave height and direction of

travel. Shoals can focus waves, causing an increase in wave height behind the shoal. Other

bathymetric features can reduce wave heights. The magnitude of these changes is

particularly sensitive to wave period and direction and how the wave energy is spread in

frequency and direction. In addition, wave interaction with the bottom can cause wave

attenuation.

Wave height is often the most significant factor influencing a project. Designing with a

wave height that is overly conservative can greatly increase the cost of a project and may

make it uneconomical. Conversely, underestimating wave height could result in

catastrophic failure of a project or significant maintenance costs. Approaches for

transforming waves are numerous and differ in complexity and accuracy.

Processes that can affect a wave as it propagates from deep into shallow water include:

Refraction.

Shoaling.

Diffraction.

Dissipation due to friction.

Dissipation due to percolation.

Breaking.

Additional growth due to the wind.

Wave-current interaction.

Wave-wave interactions

The first three processes are propagation effects because they result from convergence or

divergence of waves caused by the shape of the bottom topography, which influences the

direction of wave travel and causes wave energy to be concentrated or spread out.

Diffraction also occurs due to structures that interrupt wave propagation. The dissipation

and breaking processes are sink mechanisms because they remove energy from the wave

field through dissipation. The wind is a source mechanism because it represents the

addition of wave energy if wind is present. The presence of a large-scale current field can

affect wave propagation and dissipation. Wave-wave interactions result from nonlinear

coupling of wave components and result in transfer of energy from some waves to others.

1.8.1 Refraction and Shoaling

Wave shoaling is the change in wave height due to the change in water depth. Refraction is

the turning of the direction of wave propagation when the wave front travels at an angle

with the depth contours in shallow water. The refraction is caused by the fact that the

Waves


waves propagate more slowly in shallow water than in deep water. A consequence of this

is that the wave fronts tend to become aligned with the depth contours.

The wave-propagation problem can often be readily visualized by construction of wave

rays. If a point on a wave crest is selected and a wave crest orthogonal is drawn, the path

traced out by the orthogonal as the wave crest propagates onshore is called a ray (Fig.1.10).

Fig. 1.10: Wave Rays for straight and parallel contours.

Wave Refraction causes the waves to be focused on headlands or over shoals (Fig. 1.11).

In bays or submarine canyons the wave energy is reduced due to refraction.

Wave RayWave Crest

Shoreline

Wave RayWave Crest

Shoreline

b

Waves


Fig. 1.11: Wave refraction at headlands and in bays.

ContoursOrthogonals

Bay BayHead land

ContoursOrthogonals

Bay BayHead land

Waves


Assuming energy flux is conserved between the wave rays,

constantnCEb (1. 21)

This equation can be reduced to the following (see Dean and Dalrymple, 1991),

ors HKKH (1. 22)

Where the shoaling coefficient Ks can be calculated from,

kdnnC

CnK oo

stanh2

1 (1. 23)

The refraction coefficient Kr is calculated from,

b

bK o

r (1. 24)

1.8.1.1 Straight and Parallel Contours

For straight and parallel contours Snell’s law can be used to determine the wave direction α

at any depth based on the deep water wave direction,

o

o

CC

sinsin (1. 25)

Where the subscript o denotes deep water conditions. Equation (1.24) can also be

simplified to be,

cos

cos orK (1. 26)

1.8.2 Wave Diffraction.

Wave diffraction is a process of wave propagation that can be as important as refraction

and shoaling. The classical introduction to diffraction treats a wave propagating past the tip

of a breakwater (see Fig. 1.12). In Fig. 1.12 Region I would not include any waves if

diffraction did not occur. The spilling of energy across the wave rays into the shadow zone

is termed as diffraction.

Any process that produces an abrupt or very large gradient in wave height along a wave

crest also produces diffracted waves that tend to move energy away from higher waves to

the area of lower waves. Thus initial wave energy is reduced as diffracted waves are

produced. Refraction and diffraction of course take place simultaneously in most cases and

therefore the above distinction is an academic separation of two closely related processes.

Waves


Fig. 1.12: Wave diffraction at the tip of a breakwater.

Region II

Wave crest

Region III

Breakwater

Region I

(Perfect calm)

L

No diffraction

With Diffraction Effects

Breakwater

Region II

Wave crest

Region III

Breakwater

Region I

(Perfect calm)

L

No diffraction

With Diffraction Effects

Breakwater

Waves


Figure 1.13 shows the diffraction of irregular waves in a port obtained using a numerical

model. Such models are important tools for the design of new ports.

Fig. 1.13: Wave diffraction in a port using a short wave model (Mangor, 2004).

For simple harbours with small changes in depth it is possible to use diffraction templates

(SPM, 1984). For more complex situations numerical models that include refraction and

diffraction need to be used as discussed later.

1.8.3 Wave Breaking

Wave shoaling causes wave height to increase to infinity in very shallow water. There is,

however, a physical limit to the height of the waves: the ratio of wave height to wave

length or the wave steepness (H/L). When this physical limit is exceeded, the wave breaks

and dissipates its energy. At this point Eq. (1.21) is no longer valid. Wave shoaling,

refraction and diffraction transforms waves from deep water to the point where they break

and then their wave height begins to decrease markedly, because of energy dissipation. The

sudden decrease in the maximum value of wave height defines the breaking point and

determines the breaking parameters (Hb, and db).

The breaker type is a function of the beach slope (m) and the wave steepness (H/L).

Spilling breakers, occur on flat beach slopes as shown in Fig. 1.14. In spilling breakers

(Fig. 1.15), the wave crest becomes unstable and cascades down the shoreward face of the

wave producing a foamy water surface. Several wave crests may be breaking

simultaneously, giving the appearance of several rows of breaking waves throughout the

breaking zone.

Waves


Plunging breakers occur on steeper beaches. In plunging breakers, the crest curls over the

shoreward face of the wave and falls into the base of the wave, resulting in a high splash.

They are, for example, predominant when swell breaks on flat sandy beaches. They are

also the most common breaker type in hydraulic model studies, in which the beach

steepness is often exaggerated.

Collapsing breakers occur on steep beaches. In collapsing breakers the crest remains

unbroken while the lower part of the shoreward face steepens and then falls, producing an

irregular turbulent water surface.

Surging breakers occur on very steep beaches. The waves simply surge up and down the

beach and there is very little or no breaking.

Many studies have been performed to develop relationships to predict the wave height at

incipient breaking Hb. Several of these formulas are available in Kamphuis (1991)

including criterion for irregular waves. The simplest of these formulas is the solitory wave

criterion,

78.0b

bb

d

H (1. 27)

Where γb is the breaker index.

Waves


Fig. 1.14: Breaker types.

Air entrainmentSpilling

breaker

Very flat beach slope

Plunging

breaker

Steep beach slope

Surging

breaker

Very steep beach slope

Air entrainmentSpilling

breaker

Very flat beach slope

Plunging

breaker

Steep beach slope

Surging

breaker

Very steep beach slope

Waves


Spilling

Plunging

Fig. 1.15: Photos of spilling and plunging breakers.

Waves


1.9 Wave Models

Several types of short wave models exist and are applied to different applications. These

models include some of the processes discussed earlier. These processes will not all

dominate or exist at a certain location as shown in Fig. 1.16.

Thus, different models exist that include the relevant physical processes for certain

applications. It is essential to select the most suitable model for a certain application by

determining the important physical processes involved. Then the suitable model is selected

to perform the calculations accurately and efficiently.

Battjes (1994) classified wave models into phase-averaged and phase-resolving models.

Figure 1.17 provides a chart of the different types of models used in the coastal

environment. The Boussinesq type of models are usually used for harbour agitation studies

(see Fig.1.13). Such models include refraction, diffraction and wave-wave interaction in

shallow water. The FUNWAVE Model is an example of a 2D Bousinesq model available

at the University of Delaware

The Mild Slope Equations MSE include refraction and diffraction (the elliptic form

includes reflection also) and are thus commonly used for modeling areas where both

refraction and diffraction are important. The REFDIF Model is an example of a parabolic

MSE model available at the University of Delaware.

Spectral wave models are used for the transformation of wave spectra from deep water to

the shallow area. Such models do not include diffraction. The SWAN model, STWAVE

part of CEDAS package), and NSW (part of the MIKE21 package) models are examples of

such models.

Waves


Fig. 1.16: Different wave processes relevant for different marine and coastal

applications.

Reflection

Bottom friction

Wave-wave interaction

Depth-Breaking

White capping

Wind Input

Refraction and Shoaling

Diffraction

HarboursNear-

shore

Shelf

SeasOceansProcess

Reflection

Bottom friction

Wave-wave interaction

Depth-Breaking

White capping

Wind Input

Refraction and Shoaling

Diffraction

HarboursNear-

shore

Shelf

SeasOceansProcess

dominant

significant but not dominant

of minor Importance

Blank negligible

Waves


Fig. 1.17: Different types of wave models used for different applications.

Rcpwave

Approximation

Hyperbolic

Approximation

Parabolic

Approximation

OthersBoussinesqMild Slope

Equation

Domain

Refraction

Ray

Tracing

Snell's

Law

Phase ResolvingPhase Averaged

Spectral

Rcpwave

Approximation

Hyperbolic

Approximation

Parabolic

Approximation

OthersBoussinesqMild Slope

Equation

Domain

Refraction

Ray

Tracing

Snell's

Law

Phase ResolvingPhase Averaged

Spectral

2 Water Level Variations

Although the design of structures is normally considered to be a function of wave

conditions, water levels are also very important. A structure close to shore that is subject to

waves will be exposed to larger waves for higher water levels because the water depth

determines where waves break. This results in increased forces on the structure and

overtopping of water that will damage the structure and areas behind it. Conversely, when

the water level drops, the same structure may not be exposed to waves at all.

Thus most damage to structures occurs when the water levels are high. Similarly, high

water levels cause retreat of sandy shores, even if they are backed by substantial dunes.

The higher water levels allow larger waves to come closer to the shore. These waves will

erode the dunes and upper beach and deposit the sand offshore. If the water level rise is

temporary, most of this loss will be regained at the next low water. Permanent water level

rise, however, will result in permanent loss of sand. Shorelines consisting of bluffs or cliffs

of erodable material are continuously eroded by wave action. High water levels, however,

will allow larger waves to attack the bluffs directly, causing a temporary rapid rate of

shoreline recession.

According to Kamphuis (2000), there are several types of water level fluctuations and they

can be classified according to their return period as:

Short Term

Astronomic Tides

Storm Surge

Seiche

Long Term

Eustatic (Sea) Level Rise

Isostatic (Land) Emergence and Subsidence

Climate Change

Other short term water level changes such as wave setup will be discussed in the next

chapter.

2.1 Astronomic Tides

Astronomic tides are observed as the periodic falling and rising of the water surface for

major water bodies on the earth. Astronomic tides are the result of a combination of forces

acting on individual water particles. The main forces are:

Gravitational attraction of the earth,

Water Level Variations


Centrifugal force generated by the rotation of the earth – moon combination,

Gravitational attraction of the moon,

Gravitational attraction of the sun.

Because of its relative closeness, the moon induces the greatest effect on the tides.

2.1.1 Equilibrium Tide (Moon)

Kamphuis (2000) considered only the first three forces (neglecting the force of the sun)

and assumed that the whole earth is covered with water to describe the tidal movement.

The resultant force on the water particles can be shown to be a small horizontal force that

moves the water particle A in Fig. 2.1 toward the moon and particle B away from the

moon, resulting in two bulges of high water, (Defant, 1961; Ippen, 1966). As we turn with

the earth’s angular velocity ωE around the earth's axis at CE in the direction of the arrow,

we turn through this deformed sphere of water and experience two high water levels and

two low water levels per day. The resulting tidal period would be 12 hrs. However, the

moon-earth system also rotates around CME with velocity ωME in the same direction as the

earth's rotation. The bulges move with the moon and hence the tidal period is 12.42 hrs (12

hr & 25 min).

The tide in Fig. 2.1 is called Equilibrium Tide since it results from the assumption that the

tidal forces act on the water for a long time so that equilibrium is achieved between the tide

generating force and the slope of the water surface.

The sun's gravity forms a second, smaller set of bulges toward the sun and away from the

sun. Since our day is measured with respect to the sun, the period of the tide generated by

the sun is 12 hrs.

Both these equilibrium tides occur at the same time and they will add up when the moon

and sun are aligned (at new moon and full moon). At those times, the tides are higher than

average. At quarter moon, the forces of the sun and moon are 90° out of phase and the

equilibrium tides subtract from each other. At such a time, the tides will be lower than

average. The higher tides are called Spring Tides and the lower ones Neap Tides. Fig. 2.2

demonstrates this. The phases of the moon are shown at the bottom of the figure and it is

seen that, except for some phase lag, the maximum tides (spring tides) in Fig. 2.2

correspond to new and full moon, while the neap tides correspond to the quarter moon.



Fig. 2.1: Equilibrium Tide.

Moon

Earth

Equilibrium Tide

MEE

CE

CME

AB

Moon

Earth

Equilibrium Tide

MEE

CE

CMEMoon

Earth

Equilibrium Tide

MEE

CE

CME

AB



Fig. 2.2: Tide Predictions for Stations in the Arabian/Persian Gulf.

-0.5

0

0.5

1

1.5

2

2.5

3

0 48 96 144 192 240 288 336 384 432 480 528 576 624 672 720

Time (Hr)

Le

ve

l (m

)

-0.5

0

0.5

1

1.5

2

2.5

3

0 48 96 144 192 240 288 336 384 432 480 528 576 624 672 720

Time (Hr)

Le

ve

l (m

)

-0.5

0

0.5

1

1.5

2

0 48 96 144 192 240 288 336 384 432 480 528 576 624 672 720

Time (Hr)

Le

ve

l (m

)

Bushehr, Iran

Al-Ahmadi, Kuwait

Khasab, Hormuz

(Oman)



2.1.2 Daily Inequality

Fig. 2.1 was drawn looking down on the earth’s axis. Since the equilibrium tide is three

dimensional in shape (it forms a distorted sphere), the picture is the same when the earth is

viewed from the side, as shown in Fig. 2.3. An observer, C, travelling along a constant

latitude would experience two tides of equal height per day. However, the moon or sun is

seldom in the plane of the equator. When the moon or sun has a North or South

Declination with respect to the equator, as shown in Fig. 2.4, one bulge of the equilibrium

tide will lie above the equator and one below the equator. An observer moving along

constant latitude would now experience two tides per day of unequal height. This is called

Daily Inequality. The daily inequality is most pronounced when the moon or sun is furthest

North or South of the equator. It generally increases with latitude and there is no daily

inequality at the equator. Daily inequality is demonstrated in Fig. 2.2.

The daily inequality cycle generated by the moon repeats itself every 29.3 days. For the

tide generated by the sun, the daily inequality is greatest shortly after mid-summer and

mid-winter, causing higher tides in early January and early July.



Fig. 2.3: Equilibrium Tide (from side)

MoonEarth

Equator

N

Latitude

Equilibrium Tide

C

E

MoonEarth

Equator

N

Latitude

Equilibrium Tide

MoonEarth

Equator

N

Latitude

Equilibrium Tide

C

E



Fig. 2.4: Daily Inequality.

Moon

Earth

Equator

N

Latitude

Declination

C

E

Moon

Earth

Equator

N

Latitude

Declination

Moon

Earth

Equator

N

Latitude

Declination

C

E



2.1.3 Spring/Neap Tides

The semidiurnal rise and fall of tide can be described as nearly sinusoidal in shape,

reaching a peak value every 12 hr and 25 min. This period represents one-half of the lunar

day. Two tides are generally experienced per lunar day because tides represent a response

to the increased gravitational attraction from the (primarily) moon on one side of the earth,

balanced by a centrifugal force on the opposite side of the earth. These forces create a

"bulge" or outward deflection in the water surface on the two opposing sides of the earth.

The magnitude of tidal deflection is partially a function of the distance between the moon

and earth. When the moon is in perigee, i.e., closest to the earth, the tide range is greater

than when it is furthest from the earth, in apogee. Conversely, when the moon is in apogee,

the potential term is at a minimum value. This difference may be as large as 20 percent.

The tidal force envelope produced by the moon's gravitational attraction is accompanied by

a tidal force envelope of considerably smaller amplitude produced by the sun. The tidal

force exerted by the sun is a composite of the sun's gravitational attraction and a centrifugal

force component created by the revolution of the earth's center-of-mass around the center-

of-mass of the earth-sun system, in an exactly analogous manner to the earth-moon

relationship. The position of this force envelope shifts with the relative orbital position of

the earth in respect to the sun. Because of the great differences between the average

distances of the moon (238,855 miles) and sun (92,900,000 miles) from the earth, the tide

producing force of the moon is approximately 2.5 times that of the sun.

Spring tides occur when the sun and moon are in alignment. This occurs at either a new

moon, when the sun and moon are on the same side of the earth, or at full moon, when they

are on opposite sides of the earth. Neap tides occur at the intermediate points, the moon's

first and third quarters. Figure 2.6 is a schematic representation of these predominant tidal

phases. Lunar quarters are indicated in the tidal time series shown in Fig. 2.2.

When the moon is at new phase and full phase, the gravitational attractions of the moon

and sun act to reinforce each other. Since the resultant or combined tidal force is also

increased, the observed high tides are higher and low tides are lower than average. This

means that the tidal range is greater at all locations which display a consecutive high and

low water. Such greater-than-average tides results are known as spring tides - a term which

merely implies a "welling up" of the water and bears no relationship to the season of the

year.

At first- and third-quarter phases (quadrature) of the moon, the gravitational attractions of

the moon and sun upon the waters of the earth are exerted at right angles to each other.

Each force tends in part to counteract the other. In the tidal force envelope representing

these combined forces, both maximum and minimum forces are reduced. High tides are

lower and low tides are higher than average. Such tides of diminished range are called neap

tides, from a Greek word meaning "scanty".



Fig. 2.6: Spring and Neap Tides.



2.1.4 Other Effects

So far we have explained the characteristics of tides, based on four influences, the

gravitational attraction of the sun and moon, and the declination of the sun and moon.

There are many other, secondary effects. For example, we have assumed that the sun and

the moon travel in circular orbits relative to the earth. These orbits are actually elliptical

and therefore the distances between the earth and the sun and moon change in a periodic

fashion. This effect (and many others) can be viewed as a separate tide generator (like the

moon in Fig. 2.1). Each such tide generator has its own strength, frequency and phase

angle with respect to the others. The resulting tide is, therefore, a complex addition of

effects of the moon, the sun and many secondary causes. Each component is called a tide

constituent (Dronkers, 1964).

Until now we have assumed that the earth is completely covered with water and that the

same forces act everywhere continuously. It was seen that the tide moves relatively slowly,

while the earth turns more rapidly through the tide. In reality, the earth’s large land masses

will not turn through the tide, but will move the water masses along with them, disrupting

our picture. The only place where an equilibrium tide can possibly develop is in the

Southern Hemisphere, where the earth is circled by one uninterrupted band of water. An

equilibrium tide can form there and it will progress into the various oceans. It takes time to

travel along those oceans and hence the actual tide constituent (water level fluctuation) lags

behind its related theoretical tide constituent (from equilibrium theory), causing high water

to occur after the moon crosses the local meridian and causing spring tide some time after

full (or new) moon.

The earth’s geography not only confines the water and moves it along with the surface of

the earth, it also causes certain tidal constituents to resonate locally in the various oceans,

seas, bays and estuaries. Thus some constituents are magnified in certain locations, while

others simply disappear, making the tide at each location quite unique. One aspect that is

often magnified by the land mass is the daily inequality, increasing the difference between

the larger and smaller daily tides so that the small tides become virtually non-existent. The

Semi-Diurnal (twice per day) tides then become Diurnal (once per day). An example of

this is shown in Fig. 2.2.

2.1.5 Tide Analysis and Prediction

The equilibrium theory of tides is a hypothesis that the waters of the earth respond

instantaneously to the tide-producing forces of the sun and moon. For example, high water

occurs directly beneath the moon and sun, i.e., at the sublunar and subsolar points. This

tide is referred to as an equilibrium tide. The tide-producing forces can be written in a

polynomial expansion approximation. These expansion terms involve astronomical

arguments describing the location of the sun and moon as well as the location of the

observer on the earth. Although several variational forms of the series expansion have been

published, the development presented in Schureman (1924) is given below. Alternate

forms of expansion are discussed in Dronkers (1964).

According to equilibrium theory, the theoretical tide can be predicted at any location on the

earth as a sum of a number of harmonic terms contained in the polynomial expansion



representation of the tide-producing forces. However, the actual tide does not conform to

this theoretical value because of friction and inertia as well as differences in the depth and

distribution of land masses of the earth.

Because of the above complexities, it is impossible to exactly predict the tide at any place

on the earth based on a purely theoretical approach. However, the tide-producing forces

(and their expansion component terms) are harmonic; i.e., they can be expressed as a

cosine function whose argument increases linearly with time according to known speed

criteria. If the expansion terms of the tide-producing forces are combined according to

terms of identical period (speed), then the tide can be represented as a sum of a relatively

small number of harmonic constituents. Each set of constituents of common period are in

the form of a product of an amplitude coefficient and the cosine of an argument of known

period with phase adjustments based on time of observation and location. Observational

data at a specific time and location are then used to determine the coefficient multipliers

and phase arguments for each constituent, the sum of which are used to reconstruct the tide

at that location for any time. This concept represents the basis of the harmonic analysis,

i.e., to use observational data to develop site-specific coefficients that can be used to

reconstruct a tidal series as a linear sum of individual terms of known speed.

Tide Analysis consists of separating a measured tide into as many of its constituents as can

be identified from the length of record available. The tide is assumed to be represented by

the harmonic summation,

nnonnno uVtaHfHtH cos)(

( 2.1 )

where

H(t) = Water level at time t (t is measured from start of the year)

Ho = Mean water level above some defined datum

Hn = Mean amplitude of tidal constituent n

fn = Factor for adjusting mean amplitude (for each year)

an = speed of constituent n ( 2π / T where T is the tidal period )

(Vo+u)n = Equilibrium argument (for each year)

κn = Phase shift of tidal constituent n



For any location the tide can be calculated provided that the values of Ho, Hn and κn are

known. These values are computed from observed tidal time series data, usually from a

least squares analysis. The time-specific arguments (fn and Vo + u) are determined from

formulas or tables.

Most of the constituents listed in Table 2.1 are associated with a subscript indicating the

approximate number of cycles per solar day (24 hr). Constituents with subscripts of 2 are

semidiurnal constituents and produce a tidal contribution of approximately two high tides

per day. Diurnal constituents occur approximately once a day and have a subscript of 1.

Symbols with no subscript are termed long-period constituents and have periods greater

than a day; for example, the Solar Annual constituent Sa has a period of approximately 1

year.

There are also constituents that describe interactions between other constituents. One year's

record will comfortably provide the amplitudes and phase angles of 60 such tide

constituents. One important tidal constituent has a period of 18.6 years. It cannot be

calculated from a reasonable record length and is therefore introduced by formulas. Factors

are computed (fn and Vo + u) that adjust the amplitude and phase shift as function of time

relative to this 18.6 year cycle.

For many construction projects, local tidal information will not be available and tides need

to be measured and analysed specifically for a project. In that case, it is usual to collect

rather short tidal records. For record lengths of a month or so, tide analysis can only yield

the lunar and solar, semi-diurnal tides, daily inequality, lunar declinational tides and at

most five or six other constituents that can readily be separated. But that is often sufficient

for approximate predictions.



2.1.6 Datums

Water level and its change with respect to time have to be measured relative to some

specified elevation or datum in order to have a physical significance. In the fields of

coastal engineering and oceanography this datum represents a critical design parameter

because reported water levels provide an indication of minimum navigational depths or

maximum surface elevations at which protective levees or berms are overtopped. It is

therefore necessary that coastal datums represent some reference point which is universally

understood and meaningful, both onshore and offshore. The following are some of the

commonly used datums,

HAT Highest Astronomical Tide

MHWS Mean High Water Springs

MHWN Mean High Water Neaps

MSL Mean Sea Level

MLWN Mean Low Water Neaps

MLWS Mean Low Water Springs

LAT Lowest Astronomical Tide

Mean sea level (MSL) was widely adopted as a primary datum on the assumption that it

could be accurately computed from tidal elevation records measured at any well-exposed

tide gauge. MSL determinations are based on the arithmetic average of hourly water

surface elevations observed over a long period of time. The ideal length of record is

approximately 19 years, a period that accounts for the 18- to 19-year long-term cycle in

tides and is sufficient to remove most meteorological effects. When estimates of MSL are

required, but less than 19 years of data are available, computations should be based on an

integral number of tidal cycles, for example, an integral number of years or 29-day

spring/neap cycles. For gauges where hourly data are not available, or their use is

impractical, MSL can be approximated as the tidal datum midway between MHW and

MLW. This datum, referred to as Mean Tide Level (MTL), may differ from MSL

depending on the local relative importance of the diurnal components of the tide.

Table 2.1: Tidal period and speed for some important Constituents.

Constituent Tidal Period (hr) Speed (deg/hr)

M2 12.421 28.984

S2 12.000 30.000

K1 23.935 15.041

O1 25.819 13.943

Sa 8780.488 0.041

Ssa 4390.244 0.082



Mm 661.765 0.544

Msf 354.680 1.015

Mf 327.869 1.098

S1 24.000 15.000

Q1 26.870 13.398

P1 24.067 14.958

N2 12.659 28.439

υ2 (NU2) 12.626 28.512

K2 11.967 30.082

L2 12.192 29.528

(2N)2 12.906 27.895

μ2(MU2) 12.872 27.968

T2 12.017 29.958

M4 6.2103 57.968

(MS)4 6.103 58.984

(2MS)6 4.092 87.968

2.2 Storm Surge

Storms are atmospheric disturbances characterized by low pressures and high winds. A

storm surge represents the water surface response to wind-induced surface shear stress and

pressure fields. Storm-induced surges can produce short-term increases in water level that

rise to an elevation considerably above mean water levels.

The water level fluctuation due to storm surge is an increase in water level resulting from

shear stress by onshore wind over the water surface (Fig. 2.7). This temporary water level

increase occurs at the same time as major wave action and it is the cause of most of the

world's disastrous flooding and coastal damage. Parts of Bangladesh are flooded regularly

by storm surge resulting from passing cyclones with the loss of thousands of lives. In a

1990 cyclone, the water levels rose by 5-10 m and it was estimated that more than 100,000

lives were lost. The shorelines along the southern borders of the North Sea, particularly the

Netherlands, were flooded in 1953, because storm surge caused dike breaches.

During storm surge the water level at a downwind shore will be raised until the slope of the

water surface counteracts the shear stress from the wind. Computations of storm surge are

carried out using the same depth-averaged two dimensional equations of motion and

continuity that are used for tidal computations. In this case wind-generated shear stress is

the main driving force. For simple problems, the equations can be reduced to a one-

dimensional computation,



gD

U

dx

dS2

cos

( 2.2 )

where S is the storm surge (the setup of the water level by the wind), x is the direction over

which the storm surge is calculated, ζ is a constant (=3.2x10-6

), U is the wind speed, φ is

the angle between the wind direction and the x-axis and D is the new depth of water

(=d+S). Equation 2.2 clearly shows that storm surge is greatest in shallow water.

2.3 Barometric Surge

Since strong winds are the result of large pressure fluctuations, a barometric surge will

accompany storm surge. Suppose there is a difference in barometric pressure Δp between

the sea and the shore, then an additional water level Δh rise will be generated:

g

ph

( 2.3 )

where ρ is the density of water. Equation 2.3 results in a water level rise of about 0.1 m for

each kPa of pressure difference. A major depression can easily generate a pressure

difference of 5 kPa, resulting in a potential barometric surge of 0.5 m.



Fig. 2.7: Storm surge in closed and open seas.

s

W

d

D sW

Closed Basin

Open Sea

s

W

s

W

d

D sW

d

D sW

Closed Basin

Open Sea



2.4 Seiche

Seiches are standing waves or oscillations of the free surface of a body of water in a closed

or semiclosed basin. These oscillations are of relatively long period, extending from

minutes in harbors and bays to over 10 hr in the Great Lakes. Any external perturbation to

the lake or embayment can force an oscillation. In harbors, the forcing can be the result of

short waves and wave groups at the harbor entrance.

The oscillations will continue for some time because friction forces are quite small. The

wave length of the fundamental mode of the oscillation (a standing wave) for a closed

basin (Fig. 2.8) is twice the effective basin length (B). In general, the wave length is

2B/(1+nh) for the nh harmonic. For an open ended basin (open coast), the fundamental

wave length is 4 times the effective length of the shelf (B) over which the storm surge was

initially set up. In general, for the nh harmonic it is 4B/(1+2nh). The period of oscillation

(T=L/C) for a closed basin may be calculated as:

Fig. 2.8: Seiche wavelengths.

L/2

B

2L1/2

3L1/4

B

L/4

Open Basin Closed Basin

L/2

B

2L1/2

3L1/4

B

L/4

Open Basin Closed Basin



2.5 Tsunami

Tsunami is a single wave generated by sub-sea earthquakes and typically has a period of 5

to 60 minutes. Tsunami waves can travel long distances and is normally not very high in

deep water. In shallow water the wave shoaling can reach a height more than 10 m.

Tsunamis are rare and coastal structures seldom take them into account.

2.6 Eustatic (Sea) Level Change

The term Eustatic refers to a global change in ocean water levels; the result of melting or

freezing of the polar ice caps and the thermal expansion of the water mass with

temperature change. The water levels 25,000 years ago were 150 m below the present level

(Kamphuis, 2000). Between then and 3,000 years ago, water level rose at a more-or-less

steady rate of about 7 mm/yr to almost the present water level. The present average rate of

eustatic rise is small and therefore difficult to measure. Estimate range from 1 to 3 mm/yr.

This relatively small rate of rise, nevertheless, causes the ocean shores to be submerging

and is at least partly responsible for the fact that most beaches around the world are

eroding over the long term.

2.7 Isostatic (Land) Rebound and Subsidence

The common natural cause for isostatic (land) elevation change is a result of the

adjustment of the earth's crust to the release of pressure exerted by the 1 to 2 km thick ice

sheet that covered it during the last glaciation. Typically, the earth’s crust was severely

depressed by the ice and a rise (forebulge) was formed in the earth’s crust ahead of the

glaciers. When the ice retreated, the earth's surface rebounded (upward) where the glaciers

had been and lowered where the forebulge had occurred. This process still takes place

today, but at a much reduced rate. Most areas in the higher latitudes experience isostatic

rebound and areas at more intermediate latitudes experience some subsidence.

Although subsidence does occur naturally, often it is man-made. Pumping groundwater,

petroleum and natural gas are common causes. Subsidence exacerbates the effects of

eustatic sea level rise since the relative sea level rise with respect to the land will now be

greater.



Fig. 2.9: Tsunami generation and propagation.



2.8 Global Climate Change

The final and potentially most dangerous water level change results from trends in global

climate. In the discussion of eustatic sealevel rise, we have already seen that global

warming after the last glaciation has resulted in a sealevel rise of 100 to 150 m through

melting of the polar ice caps and thermal expansion of the water in the ocean. The present

rate has slowed down to 1 to 1.5 mm/yr, but any additional warming would increase this

rate of sealevel rise.

Concern is centered around the production of the so-called greenhouse gases. These

combustion products are thought to act as an insulating blanket over the earth, decreasing

the net longwave radiation from the earth into space and thus trapping the sun's heat to

cause global warming. It is a controversial subject and indeed there is a contingent of

respected scientists that disputes the whole idea.

According to Kamphuis (2000), predicted rise in water level for the year 2025 varies from

0.1 to 0.2 m. For 2050, the estimates vary from 0.2 to 1.3 m and for 2100 the estimates are

0.5 to 2 m.

3 Currents in the Marine Environment

Various types of currents exist in the marine environment. These currents may exist in the

open sea or in the nearshore area. In the open sea the currents are mainly tidal and wind

driven currents. In the nearshore area wave induced currents can also exist.

3.1 Tidal Currents

Tidal currents are induced by the gravity forces of the sun, the moon, and the planets. Tidal

currents are oscillatory currents with typical periods of about 12 or 24 hours (semi-diurnal

and diurnal). Tidal currents are influenced by the sea bottom contours and by coastal

morphology. They are strongest at large water depths and in estuaries or straits where the

current is forced into a narrow area. The most important tidal currents for coastal

morphology are the currents generated at tidal inlets.

In the deep, open ocean, the fluid velocity (tidal current or horizontal tide) is in phase with

the tidal water level fluctuations (vertical tide). At high water there is a maximum current

velocity in the direction of tide propagation. When the tide approaches land, however, the

phase relationship between horizontal and vertical tide changes. In the case of a tidal inlet

or bay, the water level fluctuations in the bay are driven by the tidal water level in the sea.

Rising water levels in the sea cause a current to flow into the bay, raising its water level.

This inflow of water is called flood and the outflow current during the other half of the

tidal cycle is called ebb.

3.2 Wind Generated Currents

Wind generated currents are caused by the wind shear stress along the sea surface. These

currents are normally located in the upper layer of the water body and are thus not very

important from a morphological point of view. Wind currents can have an important role

however in the movement of pollutants and oil spills. In shallow waters and in lagoons,

wind generated currents can be important. Wind generated currents are typically less than 5

per cent of the wind speed.

3.3 Stratification and Density Currents

An estuary is defined as a tidal area where a river meets the sea. It has salt water on its

downstream limit (sea) and fresh water on the upstream limit (river). The salt sea water

normally has a salinity in the vicinity of 35 parts per thousand (ppt) and a density of 1035

kg/m3. The fresh water has a density of 1000 kg/m3.

The way the transition from salt to fresh water takes place depends on the amount of

mixing that takes place in the estuary. In a well-mixed estuary (an estuary with much

turbulence), salt and fresh water are thoroughly mixed at any location. Salinity simply

varies along the estuary from 35 ppt in the sea to zero ppt in the river and at any specific

location, salinity and density will vary with the tide.

If there is little mixing in the estuary, the lighter fresh water will lie over the heavier salt

water, resulting in a stratified estuary. The density differences will induce currents. In deep

Currents in the Marine Environment


oceans, density currents also exist due to stratification caused by temperature and salinity

differences over the water depth.

3.4 Wave Induced Currents

Besides the wave-induced oscillatory currents the breaking of waves induces other non-

oscillatory currents. These currents are directed in the on/offshore and alongshore

directions.

3.4.1 Shore-normal currents

The breaking phenomena and the asymmetric wave form in the nearshore area results in a

mass transport of water with in the upper layers. This results in a water surplus in the surf

zone (wave setup). This surplus water returns to the sea via rip and undertow currents.

3.4.1.1 Rip Currents

At certain intervals along the coastline, the longshore current will form a rip current. The

rip currents are directed in the offshore direction. The rip opening in the bars will often

form the lowest section of the coastal profile with a local setback in the shoreline opposite

the rip opening (see Fig. 3.1). Field observations showed that the rip current velocity might

exceed 1.0 m/sec and occasionally extend more than 500 m from the breaker line.

Presently there is no proven method to predict rip current generation and the spacing

between rips.

Figure 3.2 shows the typical nearshore currents for different wave incidence (Harris, 1969).

3.4.1.2 Undertow

The undertow current is a return flow concentrated near the bed (see Fig. 3.3). This current

is important in the formation of bars. The mass transport carried toward the beach due to

waves is concentrated between the wave trough and crest elevations. Because there is no

net mass flux through the beach, the wave-induced mass transport above the trough is

largely balanced by a reverse flow or undertow below the trough. The undertow current at

the bottom may be relatively strong, generally 8-10 percent of the wave celerity.

The wave setup at the still water line is about 0.15db.

3.4.2 Shore-parallel currents

Longshore currents are the dominating current in the nearshore zone, generated by

obliquely approaching breaking waves. This current has its maximum close to the breaker

line (see Fig. 3.2). During storms the longshore current can reach values of 2.5 m/s. The

longshore current carries sediment along the shoreline (littoral drift) as explained later.



Fig. 3.1: Photo of rip currents observed along a beach.



Fig. 3.2: Nearshore Circulation systems (Harris, 1969)

Vl

b

Rip

Current

Breaker

Typical

current

distribution

A. Oblique (b large)

B. Normal (b ~ 0)

C. Slightly Oblique (b small)

b

Vl

b

Rip

Current

Breaker

Typical

current

distribution

A. Oblique (b large)

B. Normal (b ~ 0)

C. Slightly Oblique (b small)

b



Fig. 3.3: Schematic of the undertow current.

Undertow

MWL

SWL

Wave Setup

Undertow

MWL

SWL

Wave Setup



3.4.3 Two-dimensional Currents

Along a straight coastline the above mentioned shore parallel and shore normal currents

exist. When, combined they are three-dimensional in nature. For complex bathymetries

two dimensional currents exist. These currents occur due to irregular bathymetries or due

to the existence of structures in the nearshore zone (such as groins or breakwaters).

Coastal structures influence the current pattern in two ways: by obstructing the shore-

parallel currents and by setting up secondary circulation currents. The nature of the

obstruction to the shore-parallel currents depends on the extent and geometry of the coastal

structure. If the structure is located within the breaker zone, the obstruction leads to

offshore directed currents that will cause loss of beach material. If the structure is a

harbour, the current will follow the upstream breakwater and reach the entrance area (see

Fig. 3.4). These currents will cause sedimentation and will affect the navigation. It is thus

important to provide smooth currents that will be acceptable for the navigation and will

reduce the sedimentation. A smooth layout of the main and secondary breakwaters with a

narrow entrance is the best alignment, rather than the alignment provided in Fig. 3.4.

At the leeward side of coastal structures, special current patterns can develop caused by the

sheltering effect of the structure in the diffraction area. The wave setup in the sheltered

areas will be lower than that in the adjacent exposed areas generated a gradient in water

level that will drive currents (e.g. see Fig. 3.5). These circulation currents can be dangerous

to swimmers who might swim in the sheltered areas during storms.

If the structure extends beyond the breaker zone, the shoe parallel current will be directed

along the structure where the increasing depth will cause the currents to be reduced. This

will cause the deposition of sediment forming a shoal off the breaker zone. In the lee of a

major coastal structure the effect of return currents towards the sheltered area will also be

pronounced. In this case however, the current patterns will be smoother and less dangerous

for swimmers.



Fig. 3.4: Schematic of the currents at the SUMED Harbour at Sedi Kerir, Egypt

(Rakha and Abul-Azm, 2003)



Fig. 3.5: Measured flow field behind a breakwater (Mory and Hamm, 1995)

3.5 Hydrodynamic Models

The physical understanding and the mathematical modeling of hydrodynamic processes in

the nearshore zone is important for coastal engineers and for those involved in coastal zone

management. Hydrodynamic (HD) computations represent the core of any simulation for

water quality, siltation, and morphology studies. The development of a robust and flexible

HD model could be regarded as the first step towards the development of a fully integrated

modeling system capable of modeling hydrodynamics, sediment transport, morphological

processes and water quality.

HD models vary from fully three dimensional 3D to simpler one dimensional 1D models.

For such models they may differ in the choice of the numerical grid, the discretization



method, the time difference scheme, the solution technique, and the treatment of boundary

conditions.

Many HD models have been developed for the prediction of tidal and wind induced

currents. These models have been applied to oceans and seas or smaller areas such as tidal

inlets and harbour. These models are usually based on the hydrostatic pressure assumption

resulting in the shallow water equations.

Wave-driven currents describe the mean motion that is generated in coastal areas where

wind-generated short waves refract, shoal, diffract or break. The spatial variation of wave

momentum, the gradients of the radiation stresses, is the main driving force. The wave-

induced current diminishes rapidly away from the coast. This current is important in

limited areas where the water depth is much smaller than the wave-length and intensive

wave deformation is taking place. Generation of currents by waves includes longshore,

undertow, and rip currents.

Models for wave-driven currents vary from simple models that predict the longshore

currents over the beach profile to more detailed 2D models that predict the depth averaged

currents. Other models have been developed that calculate the undertow distribution over

the water depth. More detailed models have been developed that impose the vertical profile

of the longshore current and the undertow on the 2D depth integrated currents resulting in

the so called Quasi-3D models.

4 Nearshore Sediment Transport

The breaking waves and the nearshore currents transport beach sediments. Sometimes this

transport results only in a local rearrangement of sand into bars and troughs, or into a series

of rhythmic embayments cut into the beach. At other times there are extensive longshore

displacements of sediments, possibly moving hundreds of thousands of cubic meters of

sand along the coast each year.

A distinction is made between two modes of sediment transport: suspended sediment

transport, in which sediment is carried above the bottom by the turbulent eddies of the

water, and bed-load sediment transport, in which the grains remain close to the bed and

move by rolling and saltating. Although this distinction may be made conceptually, it is

difficult to separately measure these two modes of transport on prototype beaches.

The longshore sediment transport rate is directed parallel to the coast and is among the

most important nearshore processes that control the beach morphology, and determines in

large part whether shores erode, accrete, or remain stable. An understanding of longshore

sediment transport is essential to sound coastal engineering design practice.

Sediment is also transported in the on/offshore direction causing changes in the beach

profile. The seasonal changes in the beach profile are caused by the on/offshore transport

of sediment. Figure 4.1 provides a sketch for different definitions along a beach profile.

Sediment transport can also result from the currents generated by alongshore gradients in

breaking wave height due to the sheltering effect. This transport is manifest as a movement

of beach sediments toward the sheltered area. The result is transport in the “upwave”

direction on the downdrift side of the structure. This, in turn, can create a buildup of

sediment on the immediate, downdrift side of the structure.

Nearshore Sediment Transport


Fig. 4.1: Definition of coastal terms (SPM, 1984).

MHWL

MLWLMSL

Dunes

Coastal

hinterland Coastal Area

Littoral ZoneShore or

beach

Coast

Backshore Foreshore or

Beach Face

Beaker Zone

Closure

depth

Nearshore Zone Offshore

Zone

MHWL

MLWLMSL

Dunes

Coastal

hinterland Coastal Area

Littoral ZoneShore or

beach

Coast

Backshore Foreshore or

Beach Face

Beaker Zone

Closure

depth

Nearshore Zone Offshore

Zone



4.1 Longshore Sediment Transport

The movement of beach sediment along the coast is referred to as littoral transport or

longshore sediment transport, whereas the actual volumes of sand involved in the transport

are termed the littoral drift. This longshore movement of beach sediments is of particular

importance in that the transport can either be interrupted by the construction of jetties and

breakwaters (structures which block all or a portion of the longshore sediment transport),

or can be captured by inlets and submarine canyons. In the case of a jetty, the result is a

buildup of the beach along the updrift side of the structure and an erosion of the beach

downdrift of the structure. The impacts pose problems to the adjacent beach communities,

as well as threaten the usefulness of the adjacent navigable waterways.

4.1.1 Predicting Potential Littoral Drift

Littoral transport is often described under the assumption that the shoreline is nearly

straight with nearly parallel depth contours. This assumption is very often valid for

relatively short sections of the shore and for smooth transitions assumed between such

sections. If unlimited amounts of sand are assumed to exist over the active beach profile,

the potential sediment transport rates can be estimated. The longshore sediment transport

rates can be estimated using bulk or detailed formulations.

4.1.1.1 Bulk Formulations

In these types of formulations the longshore sediment transport rate Qs is calculated from a

semi-empirical formula. The well know CERC formula (SPM, 1984) is an example of such

a formula and can be written as,

bsb

s

HQ

2sin10*2.2

2/56

(4.1)

Where Hsb is the significant wave height at the breaker, b is the breaker wave angle (see

Figs. 3.2 and 4.2), is the breaker index (Hb/db) and the subscript b denotes the breaker

line. It can be seen that this formula does not include some parameters that will definitely

have an effect on the littoral drift such as the grain size. The Kamphuis (1991) formula

includes more parameters and can provide better estimates for the littoral drift. This

formula can be written as,

bbopsbs DmTHQ 2sin10*4.6 6.025.075.05.124 (4.2)

where m is the beach slope, D is the grain size, and the subscript p denotes the peak wave

period.

Figure 4.2 shows the effect of the wave height and the angle of incidence on the littoral

drift as predicted from the Kamphuis (1991) formula.

The GENESIS model (part of the CEDAS package) uses such a bulk formulation for

calculating the sediment drift.



4.1.1.2 Detailed Formulations

In these formulations, the sediment transport rates are calculated over the beach profile and

integrated over the profile to determine the littoral drift. Many different models exist for

the calculations at a single location ranging from simpler models based on equations for

calculating the sediment transport rates (e.g. UNIBEST) to models that calculate the

suspended sediment concentration over the full water depth (e.g. LITPACK model).

4.1.2 Littoral Drift Budget

A littoral drift budget is the sum of littoral transport contribution by all the possible

combinations of wave heights and directions (determined from the nearshore wave rose).

At a particular beach site the transport might be to the right (to East as in Fig. 4.2) during

part of the year and to the left during the remainder of the year. If the left and right

transports are denoted respectively QsL and QsR, with QsR being assigned a positive

quantity and QsL assigned a negative value for transport direction clarification purposes,

then the net annual transport is defined as Qs NET = QsR + QsL. The net longshore sediment

transport rate is therefore directed right and positive if QsR > QsL, and to the left and

negative if QsR < |QsL|. The net annual transport can range from essentially zero to a large

magnitude, estimated at a million cubic meters of sand per year for some coastal sites.

The gross annual longshore transport is defined as Qs GROSS = QsR + |QsL|, the sum of the

temporal magnitudes of littoral transport irrespective of direction. It is possible to have a

very large gross longshore transport at a beach site while the net transport is effectively

zero.

These two contrasting assessments of longshore sediment movements have different

engineering applications. For example, the gross longshore transport may be utilized in

predicting backfilling rates in navigation channels and uncontrolled inlets, whereas the net

longshore transport more often relates to the deposition versus erosion rates of beaches on

opposite sides of jetties or breakwaters.

An important parameter in relation to the littoral drift conditions is the variation of the net

transport with any variation in the coastline orientation. If for example a groin is

constructed along a beach, it will initially block the sediment resulting in a zero net

transport rate at the groin. Thus sand will accrete upstream of the groin forming a coastline

with an orientation that produces a zero net transport rate. The efficiency of the groin

depends very much on the angle between the present coastline and the zero net littoral drift

coastline orientation. If this angle is small, the groin will be efficient as it will be able to

create a long accreting area. If the angle is large, the groin will influence a very short part

of the coastline and thus groins will not be useful for such areas.



Fig. 4.2: Effect of different parameters on the littoral drift.

b

Normal to

Coastline

Coastline

-ve +veWave fro

nt

-90 -60 -30 0 30 60 90

Qs

0 1 2 3 4

Hs (m)

Qs

To

East

To

West

b

Normal to

Coastline

Coastline

-ve +veWave fro

ntb

Normal to

Coastline

Coastline

-ve +veWave fro

nt

-90 -60 -30 0 30 60 90

Qs

0 1 2 3 4

Hs (m)

Qs

To

East

To

West



4.2 On/Offshore Sediment Transport

Varying the wave conditions, results in varying onshore and offshore transport rates over

the beach profile. These transports are to some extent reversible and therefore irrelevant in

terms of longshore sediment transport.

When the beach profile is exposed to high wave conditions, sediment near the shoreline

will be transported offshore and typically be deposited in a bar resulting in an overall

flattening of the shoreface (winter profile). The foreshore however will get steeper and the

shoreline will recede. During following periods of lower wave conditions and swell, the

bar will slowly move towards the shoreline and the profile will be rebuilt (summer profile).

Figure 4.3 provides a schematic of the summer and the winter beach profiles.

It is important to take into account these temporary profile changes to avoid erosion of the

coast. Thus, a sufficient setback is essential when designing any structures in the coastal

area.

The on/offshore sediment transport is closely related to the form of the coastal profile.

Several investigations showed that a coastal profile possesses an average, characteristic

form referred to as a theoretical equilibrium profile. A well known equation for

representing the equilibrium profile was proposed by Dean (1977),

3

2

Axd (4.3)

where A is a constant that depends on the mean grain size, and x is the offshore direction.

The equilibrium profile is valid till the closure depth that can be calculated from

(Hallemeyer, 1978),

2

2

12,

12, 5.6828.2s

s

scgT

HHd

(4.4)

where Hs,12 is the significant wave height exceeded 12 hour per year, and Ts is the

corresponding significant wave period.



Fig. 4.3: Schematic of summer and winter beach profiles.

4.3 Coastal Sediment Cells

The coastline is a series of interlinked physical systems, comprising both offshore and

onshore elements. Sediment (clay, silt, sand, gravel etc.) is moved around the coast by

waves and currents in a series of linked systems (sediment transport cells). Simple cells

comprise an arrangement of:

Sediment source areas (e.g. eroding cliffs, rivers, the sea bed);

Areas where sediment is moved by coastal processes; and

Sediment sinks (e.g. beaches, estuaries or offshore sinks).

Along a particular stretch of coast there may be a series of such cells, often operating at

different scales. In contrast to river catchments, coastal systems have no obvious

boundaries. Suspended sediments, for example, may be carried thousands of miles around

the coast. Although headlands can be identified which appear to mark the limits of coarse

sediment transport, they are often not permanent boundaries.

Winter Bar Profile

Summer Berm Profile

Winter Bar Profile

Summer Berm Profile



4.4 Sediment Transport Models

As discussed earlier sediment is brought in suspension by the waves and the currents and

transported by the currents. For fine sediment the advection dispersion model described

above is used to predict the movement of sediment by tidal currents. The settling of

sediment and the erosion of sediment from the bed must be included in such models.

For sediment transported by the wave-induced currents, different types of models exist.

These models range from simple formulas used to calculate the longshore sediment

transport rates (as discussed earlier) to more detailed models. The detailed models calculate

the vertical variation in turbulent energy, currents, and suspended sediment. The STP

model of DHI Water and Environment is an example of such a model (see Fig. 4.4).

The LITPACK model includes many different models that obtain data from the STP

model. These models are tailored for different applications. The LITDRIFT model is used

to calculate the littoral drift. LITLINE is a shoreline model use to calculate the shoreline

changes as explained later. LITTREN is used to calculate the filling of a trench and finally

LITPROF calculated the beach profile changes due to on/offshore sediment transport.

Other wave-induced sediment transport models, calculate the sediment transport rates from

formulas based on the depth averaged currents (from a 2D HD model) and the wave

conditions (from a wave model).



Fig. 4.4: Different models in the LITPACK package of DHI.

STP

LITDRIFT

LITLINE

LITTREN

LITPROF

STP

LITDRIFT

LITLINE

LITTREN

LITPROF



4.4.1 Morphology and Shoreline Change Models

Morphology models calculate the changes in the bathymetry due to the waves and currents.

De Vriend et al. (1993) classified the different types of morphology models into four

categories; coastline models, coastal profile models, coastal area models, and local models.

A flow chart of a typical process type coastal area model is shown in Fig. 4.5. In such

models several modules are required. A wave module, a hydrodynamic module, a sediment

transport module, and a sediment balance module. The accuracy of the morphology model

will thus depend on the accuracy of each module. Thus, it is essential to choose modules

that are compatible.

Such process type morphology models will require extensive computer time and can not be

applied to simulate long term changes in the morphology. Thus, shoreline models (as

explained later) are commonly used for such purposes. The LITLINE model of Fig. 4.4 is

an example of such models. These models simulate the movement of the shoreline

assuming no changes in the beach profile.

4.5 One-line Models

One-line models used to estimate longshore sand transport rates and long-term shoreline

changes generally assume that the profile is displaced parallel to itself in the cross-shore

direction. The profile may include bars and other features but is assumed to always

maintain the same shape. This assumption is best satisfied if the profile is in equilibrium.

The one-line model is formulated on the conservation equation of sediment in a control

volume or shoreline reach, and a bulk longshore sand transport equation. It is assumed that

there is an offshore closure depth dC (Fig. 4.6) at which there are no significant changes in

the profile, and the upper end of the active profile is at the berm crest elevation dB. The

constant profile shape moves in the cross-shore direction between these two limits. This

implies that sediment transport is uniformly distributed over the active portion of the

profile. The incremental volume of sediment in a reach is simply (dB + dC)∆x∆y, where ∆x

is the reach of shoreline segment, and ∆y is the cross-shore displacement of the profile.

Conservation of sediment volume may be written,

0)(

1

q

x

Q

ddt

y s

cB

(4.5)

where, y is the shoreline position, t is time, dB is the berm height, dc is the closure depth, q

is a source or sink in sediment and Qs is the longshore sand transport rate (m3/sec). The co-

ordinate system x and y-directions are defined in Fig. 4.6. The value of Qs is calculated

from formulas similar to those described earlier.



Fig. 4.5: Flow chart of typical process type morphology model.

Sediment

Balance

Wave

Field

Bed

Topography

Sediment

Transport

Current

Field Time

Stepping

Sediment

Balance

Wave

Field

Bed

Topography

Sediment

Transport

Current

Field Time

Stepping



Fig. 4.6: Schematic for terms used in One-line models.

4.5.1 Analytical Solution

If the angle of the shoreline is small with respect to the x axis and simple relationships

describe the waves, analytical solutions for shoreline change may be developed. As an

example, assuming that the breaking wave angle αb is small, the following planform

shoreline change equation can be derived,

02

2

x

yD

t

y (4.6)

Where

d

qD

2 (4.7)

where d = dc + dB and q is determined from,

dx

dyqqQ be 2sin2sin (4.8)

where is defined as the angle between the wave crest and the x-axis.

y

dB

Q t

dC

Y x

X

tXX

QQ



A number of researchers have employed this equation or slight variations of it to provide

analytical solutions to shoreline change under certain assumptions (the boundary

conditions and initial conditions of the problem).

Pelnard-Considére (1956) first presented an analytical solution to this simplified shoreline

change equation for the case of an impermeable groin or jetty impounding the longshore

sand transport on the updrift side of the structure under a stationary (constant) wave

climate (Fig. 4.7).

The following equation was derived,

)(tan4 2

uerfcueDt

y u

b

(4.9)

where, t is the time and

Dt

xu

4 (4.10)

the erfc can be approximated from the following equation,

2

54321)( ueTAATATATATuerfc (4.11)

where

A1= 0.254829592

A2= -0.284496736

A3= 1.421413741

A4= -1.453152027

A5= 1.061405429

and

PuT

1

1 (4.12)

where P = 0.3275911.

The time required for the structure to fill to capacity t = tf can be found by setting x = 0 and

y to the barrier length S,

b

fD

St

2

2

tan4 (4.13)



Fig. 4.7: Schematic for case of complete barrier.

Yo

S

bbdx

dy tan

Shoreline at time t

Original Shoreline X

Y



4.5.2 Model Classification according to Time and Space

According to Kamphuis (2000), models can be classified according to the time and space

(area) covered. Figure 4.8 identifies Short Term Small Area Models (S), Medium Term

Medium Area Models (M) and Long Term Large Area Models (L). The exact definitions

of S, M and L will obviously be a function of the problem to be solved. Some typical

definitions are: S-models cover prototype durations of hours (or less) and areas of 1 to 100

m2. Coastal applications are models of bedforms (ripples and dunes), breakwater cross-

sections, local scour near structures, water intakes, sewerage outfalls and diffusers.

M-models typically cover prototype areas of several km2 and durations of years. Coastal

applications are models of shore sections (littoral cells), harbors, inlets, estuaries or

portions of estuaries, and shore protection with offshore structures. This category also

includes fluid flow models (waves and currents) that cover medium areas, although they

may only represent a short duration. Examples are refraction and diffraction of a single

wave condition. We include them under M models because their outcome is normally

applied to medium term problems such as wave agitation in a harbor and coastal

morphology.

L-models typically cover areas greater than 100 km2 over centuries. Examples are models

representing the shoreline evolution of sections covering thousands of years, or the

development of all or a portion of river deltas over centuries.



0.01 0.1 1 10 100 1000

0.1 1 10 100 1000 10000

Time (yrs)

Time (hrs)

100

1000

0.01

10

0.1

1

10

100

1000

Are

a (

km

2)

Are

a (

m2)

Short Term

Small Area

Medium Term

Medium Area

Long Term

Long Area

S

M

L

0.01 0.1 1 10 100 1000

0.1 1 10 100 1000 10000

Time (yrs)

Time (hrs)

100

1000

0.01

10

0.1

1

10

100

1000

Are

a (

km

2)

Are

a (

m2)

Short Term

Small Area

Medium Term

Medium Area

Long Term

Long Area

S

M

L

Fig. 4.8: Model Classification (reproduced from Kamphuis, 2000).



Contrary to physical models, in numerical modeling, a problem must be clearly understood

before a model can be properly formulated so that it produces valid solutions. Equations

governing the processes, numerical methods, transfer functions and calibration coefficients

must all be known, at least approximately, from the outset. In many cases many constants

and coefficients used in numerical models are not so well known. For example, we only

know dispersion coefficients within one or two orders of magnitude, most of the time.

Thus, because of uncertainties in the equations and the coefficients, and because of

approximations made in the numerical simulation of the equations, numerical modeling

results can only produce qualitative results, at best. Interpretation of such qualitative results

into quantitative estimates is the major task of numerical modeling. It requires a thorough

understanding of the coastal processes, the applicable equations, the various interactions

between variables and the shortcomings in evaluation of the coefficients. In addition, just

as the physical modeler needs to know about scaling, scale effects and laboratory effects,

the numerical modeler needs to understand the implications of simplifications brought into

the model, and numerical modeling methods with its pitfalls, such as instabilities,

numerical diffusion and dispersion, smoothing, etc.

For some problems we know enough about the equations and coefficients to model them

numerically with some confidence. Such tractable problems as fluid flow with relatively

simple boundary conditions can be modelled using either physical models or numerical

models, combining long waves, short waves and currents. Once numerical models can be

successfully applied to solve a type of problem, the use of physical models for such a

problem will decline.

4.5.3 Reducing Uncertainty

Physical and numerical models produce results with some degree of uncertainty. The

uncertainties vary from model to model depending on the capability of the model to

accurately represent the physical processes at hand. Wave models for example have less

uncertainty than sediment transport models.

The uncertainties in model results can be reduced through proper model validation, which

consists of

Benchmarking,

Calibration,

Verification.

Any numerical model, regardless of its sophistication, should be properly benchmarked.

The model must be run for simple boundary conditions and with simplified equations to be

compared with analytical solutions. For example, a 1D shoreline model must be able to

simulate diffusion-type solutions of Pelnard Considère (1956), before it can be successfully

applied to more complex situations.

In calibration, the model parameters are adjusted so that the model reproduces measured

prototype values. The concept of model calibration is based on the fact that directly

measured prototype parameters contain less uncertainty than the output of the model,



which is based on the combined uncertainties of the input data and the model. Thus it

makes good sense to calibrate the model sediment transport rate using prototype

bathymetry measurements. For example a hydrodynamic model that calculates the tidal

currents in a coastal lagoon can be calibrated by comparing the predicted currents with

actual field measurements.

Calibration can consist of several intermediate stages. A coastal morphology model can

first be calibrated for wave heights, current directions and magnitudes, velocity

distributions, and sediment transport rates and their distributions, before being calibrated

for final changes in morphology. If the model is only calibrated for the currents, then the

uncertainty in the morphology calculations will still remain. This is often the case for

advection dispersion modeling, where the hydrodynamic model is calibrated against

measurements of the currents whereas the advection dispersion model is not calibrated.

This results in uncertainties in the dispersion coefficients used and is often compensated by

performing a sensitivity study.

Model calibration inherently assumes that the model extrapolates existing conditions. For

example, a coastal model that was calibrated against shoreline change data can predict

future shoreline change. An estuary model calibrated with changes in shoals and channels

can predict changes in shoals and channels. But, consider a beach that was only interrupted

by some shore-perpendicular structures, such as groins or jetties. A model calibrated with

the historical conditions along such a beach when used to design offshore breakwaters to

protect this shore will have another uncertainty. Even a carefully calibrated model will not

include wave diffraction, the major new influence introduced by the shore-parallel

structures. Such a model can only be useful for the design of offshore breakwaters, if it is

benchmarked against simple diffraction solutions and compared to known, similar

prototype situations.

To gain further confidence in the generality of a model, the calibrated model must be

verified against additional prototype data that were not used in the calibration.

Thus, one key to success is extensive prototype monitoring to obtain as much information

as possible, about the input parameters (waves, tides, currents), but especially also about

the output parameters (new wave and current patterns that resulted from the design,

shoreline change, sediment volume changes).

In the past, calibration and verification of physical models consumed most of the time

required for a model study. It was a major cost item, but it was, in fact, only a fraction of

the total cost, considering the other large costs of physical modeling. The cost of a

numerical model study is more directly related to the running time of the study. Thus

lengthy validation will not just increase the cost of a model study by some fraction, but by

a factor (perhaps 2 to 10). This makes adequate validation of numerical models

problematic and in the recent past we have moved away from proper validation.

Verification is often degraded to quick and simple comparisons of model results with

sparse field data.

5 References

Battjes, J.A. (1994). Shallow water wave modelling. International Symposium Waves -

Physical and Numerical Modelling, Vancouver, Canada, pp. 1-23.

Bretschneider, C. (1952). Revised Wave Forecasting Relationships. Proceedings of the 2nd

Coastal Engineering Conference, American Society of Civil Engineers, pp 1-5.

Dean, R. G. (1977). “Equilibrium Beach Profiles: U.S. Atlantic and Gulf Coasts,”

Department of Civil Engineering, Ocean Engineering Report No. 12, University of

Delaware, Newark, DE.

Dean, R. G. and Dalrymple, R. A. (1991). Water Wave Mechanics for Engineers and

Scientists, World Scientific Pub. Co., Teaneck, NJ.

Defant, A. (1961). Physical Oceanography, Vol II, Pergamon Press, New York.

De Vriend, H.J., Zyserman, J.A., Nicholson, J., Roelvink, J.A., Péchon, P., and Southgate,

H.N. (1993). Medium-term 2DH coastal area modelling. Coastal Eng., Vol.21, pp.193-224.

Dronkers, J. J. (1964). Tidal Computations in Rivers and Coastal Waters, North-Holland

Publishing Company - Amsterdam, John Wiley & Sons, Inc., New York.

Eldeberky, Y., Metwally, A., Rakha, K.A., and Cavaleri, A. (2002). Wave hindcast in the

eastern Mediterranean Sea. 21st International Conference on Offshore Mechanics and

Arctic Engineering, June 23-28, 2002, Oslo, Norway.

Jeffreys, H. (1924). On the Formation of Waves by Wind. Proc. Roy. Soc. Lond., Vol 107,

pp 189-206.

Jeffreys, H. (1925). On the Formation of Waves by Wind. Proc. Roy. Soc. Lond., Ser. A.,

Vol 110, pp 341-347.

Harris, T.F.W. (1969). Nearshore Circulations, Field observation and experimental

investigation of an underlying cause in wave tanks. Proceedings of Symposium on Coastal

Engineering, Stellenbosch, South Africa, 11 p.

Hasselmann et al. (1973). Measurements of Wind-Wave Growth and Swell Decay During

the Joing North Sea Wave Project (JONSWAP). Deutsche Hydrograph. Zeit.,

Erganzungsheft Reihe A (80), No. 12.

Hasselmann, K. et al. (1976). A Parametric Wave Prediction Model. Jour. Phys. Ocean.,

Vol 6, pp 200-228.

Helmholtz, H. (1888). Uber Atmospharische Bewwgungen. S. Ber. Preuss. Akad. Wiss.

Berlin, Mathem. Physik Kl.

References


Horikawa, K. (Editor) (1988). Nearshore Dynamics and Coastal Processes, University of

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coastal engineering - unimasr€¦ · 1 waves knowledge of waves and he forces they generate are...

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