Title: Longshore current and sediment transport
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Title: Longshore current and sediment transport
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Copyright Date: 1970
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LONGSHORE CURRENT AND SEDIMENT TRANSPORT
















By
EDWARD BENNETT THORNTON


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY














UNIVERSITY OF FLORIDA
1970



























This work is dedicated to my wife, Sandra, to whom I owe the most.













ACKNOWLEDGE IENTS


The author wishes to thank Drs. B. A. Christensen and

R. G. Dean who provided invaluable guidance and inspiration

during the course of this investigation. Dr. Dean was es-

pecially helpful in providing insight into the hydromechanics

of the nearshore region, and Dr. Christensen was particularly

helpful in providing assistance in the sediment transport

aspects of this study. The author also wishes to thank

Dr. Per Bruun who initiated the field study.

Thanks are due a great many people who assisted in the

field measurements which were conducted under all weather con-

ditions. The author wishes to thank Mrs. Mara Lea Hetherington

who typed the manuscript.

Financial support for this study was supplied by the Depart-

ment of Interior, Federal Water Pollution Control Administration,

under contract with the Department of Coastal and Oceanographic

Engineering, University of Florida.















TABLE OF CONTENTS


ACKNOWLEDGMENTS

LIST OF TABLES

LIST OF FIGURES

LIST OF SYMBOLS

ABSTRACT

CHAPTER

I INTRODUCTION

A. Introductory Note

B. Historical Summary

1. Longshore Currents
2. Littoral Drift

C. Purpose and Scope of the Investigation

II CONSERVATION RELATIONSHIPS AND SPECIFICATION
OF WAVE FIELD

A. Introduction

B. Conservation Equations

1. Conservation of Phase
2. Conservation of Mass
3. Conservation of Momentum
4. Conservation of Energy for Fluctuating
Motion
5. Conservation of Energy for the Mean
Motion

C. Description of the Wave Field

1. Waves outside the Surf Zone
2. Breaking Waves
3. Waves inside the Surf Zone


Page

iii

vii

viii

xi

xvi







Page

III LONGSHORE CURRENT THEORY 48

A. Statement of the Problem 48

B. Currents outside the Surf Zone--Neglecting 50
Bottom Friction

1. Mass Transport Velocity 51
2. Distribution of Energy outside the Surf 51
Zone
3. Distribution of Currents outside the 58
Surf Zone
4. Set-down of Mean Water Elevation outside 59
the Surf Zone

C. Currents outside the Surf Zone--Including 60
Bottom Friction

1. Bottom Shear Stress Due to Combined 61
Waves and Currents
2. Changes in Wave Height outside the Surf 68
Zone Due to Bottom Friction and Per-
colation
3. Distribution of Currents 72

D. Longshore Currents inside the Surf Zone 73

1. Wave Set-up inside the Surf Zone 74
2. Velocity Distribution--Constant 81
Sloping Bottom
3. Mean Longshore Current--Constant 87
Sloping Bottom
4. Longshore Current--General Profile 89
5. Longshore Current Distribution-- 92
Including Internal Shear Stress and
Bottom Friction

IV LITTORAL TRANSPORT--ENERGY PRINCIPLE 103

A. Introduction 103

B. Bed Load Transport 104

C. Total Sand Transport outside the Surf Zone 110

D. Total Sand Transport inside the Surf Zone 112

V FIELD EXPERIMENTS 114

A. Description of Experiments 114











1. Sand Transport Measurements
2. Current Measurements
3. Wave Measurements
4. Wind Measurements
5. Tide Record

B. Error Analysis

C. Results and Comparison with Theory

VI CONCLUSIONS


A.

B.

APPENDIX A -

APPENDIX B -


REFERENCES


Longshore Currents

Littoral Transport

BATHYMETRY AT FERNANDINA BEACH, FLORIDA

PREDICTED AND MEASURED DISTRIBUTIONS OF
BED LOAD TRANSPORT














LIST OF TABLES


Table Page

I Longshore Current Formulas 12

II Sediment Transport 130

III Summary of the Littoral Transport Predictive Equation 137
Formulation

IV Longshore Currents 142


vil













LIST OF FIGURES


Figure Page

1 Distribution of Longshore Velocity and Sediment 3
Transport across the Surf Zone (after Zenkovitch [1])

2 Rip Current System 10

3 Uniform Longshore Current System 10

4 Shear Stresses Acting on the Faces of an Elemental 36
Water Column

5 Schematic of the Surf Zone 49

6 Amplification of Wave Energy Density Due to a Shear 57
Current as a Function of the Incident Wave Angle

7 Resolution of Currents and Waves into Respective 66
Velocity and Shear Stress Components

8 Relative Changes in Wave Set-up inside the Surf Zone 79
as a Function of the Incident Wave Angle

9 Laboratory Measurements of Wave Set-down and Set-up 82
(after Bowen [55])

10 Longshore Velocity Distribution across the Surf Zone 86
for a Constant Sloping Beach

11 Comparison of Predicted and Measured Mean Longshore 88
Currents

12 Longshore Velocity Distribution across the Surf Zone 91
for a Beach Described by an nth Degree Polynomial

13 Comparison of Predicted and Measured Velocity 93
Distributions for Natural Beaches

14 Velocity Distribution across the Surf Zone for Low 98
Velocities Including Internal and Bottom Shear Stresses

15 Velocity Distribution across the Surf Zone for Moderate 100
Wave Conditions Including Internal and Bottom Shear
Stresses







Figure Page

16 Velocity Distribution across the Surf Zone for Field 101
Conditions Including Internal and Bottom Shear
Stresses

17 Schematic Diagram Representing Stresses Acting to 106
Cause Bed Load

18 Plan of Pier Showing Instrument Locations and 116
Typical Bottom Profile at Fernandina Beach, Florida

19 Typical Sand Grain Size Analysis Taken at Fernandina 117
Beach, Florida

20 Bathymetry Adjacent to the Fishing Pier at Fernandina 120
Beach, Florida, December 17, 1968

21 Sand Trap with Doors Open 121

22 Lowering Sand Trap from the Pier, Showing View of 121
Crane

23 Current Meter and Pressure Transducer(Mounted on 124
Tripod)

24 Attaching instruments to Tripod 124

25 Variation of Measured Sediment Transport with Time, 132
Test Number 26

26 Distribution of Bed Load Transport outside the Surf 135
Zone, Test Number 19

27 Distribution of Bed Load Transport across the Surf 136
Zone, Test Number 22

28 Comparison of Measured and Predicted Bed Load 139
Transport outside the Surf Zone

29 Comparison of Measured and Predicted Bed Load 141
Transport inside the Surf Zone

30 Bathymetry, November 22, 1966 150

31 Bathymetry, May 26, 1967 151

32 Bottom Profiles Adjacent to Pier, July 24, 1964 152
April 14, 1965

33 Bottom Profiles Adjacent to Pier, May 13, 1965 153
May 26, 1967








Figure

34

35

36

37

38

39

40

41

42

43

44

45


Distribution of

Distribution of

Distribution of

Distribution of

Distribution of

Distribution of

Distribution of

Distribution of

Distribution of

Distribution of

Distribution of

Distribution of


Transport,

Transport,

Transport,

Transport,

Transport,

Transport,

Transport,

Transport,

Transport,

Transport,

Transport,

Transport,


Test Number

Test Number

Test Number

Test Number

Test Number

Test Number

Test Number

Test Number

Test Number

Test Number

Test Number

Test Number


Page

155

156

157

158

159

160

161

162

163

164

165

166












LIST OF SYMBOLS


a = wave amplitude

B = proportionality factor associated with bed load transport

B = proportionality factor associated with bed load transport
inside the surf zone

c = wave celerity

cb = wave celerity at wave breaking

c = speed of wave energy propagation

c = wave celerity in deep water

Cf = Chdzy's coefficient

d = mean sand grain size

D = n + h, total depth of water

Db = total depth of water at location of breaking waves

eh = efficiency coefficient associated with bed load transport

e = efficiency coefficient associated with suspended load transport

E = energy density

E = energy density in deep water

f = friction factor

fh = Th/Nh, dynamic friction factor associated with sediment transport

f = friction factor associated with wave motion
w
fwb = friction factor associated with wave motion at breaking

F = energy density flux

g = acceleration due to gravity

h = depth below still water level








h = depth below still water level at wave breaking

h = depth below still water level in deep water

H = wave height

Hb = wave height at wave breaking

Ho = wave height in deep water

H = significant wave height

i = index corresponding to horizontal coordinate in the x-direction

j = index corresponding to horizontal coordinate in the y-direction

k = wave number

K(a) = ratio between set-up slope and beach slope

K = refraction coefficient
r

K = shoaling coefficient

L = wave length

L = wave length in deep water

mh = mass of bed load sediments

m = mass of suspended sediments

Ii = Mi + Mi, total mass transport per unit width or total mean
momentum per unit area

M = mass transport per unit width associated with mean motion

M. = mass transport per unit width associated with fluctuating motion
I

n = transmission coefficient

Nh = normal stress at the bed

Ns = normal stress at level of suspended sediment

p = pressure

P = power/unit area

Ph = power/unit area expended on the bed

P = power/unit area available for suspended sediment transport
5







q = net mass transport rate of sediment per unit width

qh = bed load transport rate per unit width

qs = suspended load transport rate per unit width

Qs = total longshore sand transport rate

r = roughness parameter

R. = total average resistance force
1

s = fraction of incoming energy flux available to the longshore
current

S = proportionality factor associated with suspended load transport

S = proportionality factor associated with suspended load transport
inside the surf zone

S.. = excess momentum flux tensor (radiation stress tensor)

t = time

T = wave period

Th = sediment shear stress

T. = net horizontal force per unit area due to slope of free water
surface

u, = shear velocity


uhw = water particle velocity due to wave motion at the bed

u. = total water particle velocity
1

u! = fluctuating water particle velocity component
1

Umh = maximum water particle velocity due to wave motion at the bed

ush = velocity of a sediment particle on the bed

u = water particle velocity due to wave motion

U = mean velocity component normal to the beach

U. = U + U., total mean transport velocity

U. = mean transport velocity associated with mean motion

Ui = mean transport volodity associated with flucttuating motion

v = resultant velocity vector of combined wave and current motion

ziii







V = mean velocity component parallel to the beach

Vb = mean velocity component parallel to the beach at location of
breaking waves

Vsh = mean velocity of bed load transport in the longshore direction

V = mean velocity of suspended load transport in the longshore
direction

w = vertical velocity component

w = fall velocity of sand grain

x = horizontal coordinate perpendicular to the beach

X = width of the surf zone
s
y = horizontal coordinate parallel to the beach

z = vertical coordinate

a = incident wave angle

ab = incident wave angle at breaking

a = incident wave angle in deep water

B = bottom slope

6.. = kronecker delta

At = unit length of the beach

e = energy dissipation function

f = energy dissipation due to bottom friction

E = energy dissipation due to percolation

E = eddy viscosity

S= water surface elevation

= mean water surface elevation

nb = mean water surface elevation at breaking position

n = mean water surface elevation for zero angle of wave incidence

< = ratio between breaking wave height and the depth of water
at breaking

= rip current spacing






V' = mixing length

v = kinematic viscosity

a = wave frequency

S= velocity potential

S= permeability coefficient

p = density of fluid

Ps = density of sediments

a = local wave frequency

r = shear stress

TF = fluid shear stress

Th = shear stress at the bed

hw = shear stress at the bed in the direction of wave motion


Thy = shear stress at the bed in the direction of mean motion

Ts = inter-granular fluid shear stress

e = phase lag

X = phase function

an = water particle excursion due to wave motion at the bed


Subscripts

b = breaker line

h = bottom

o = deep water

s = sediment

ss = suspended sediment

w = wave








Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy


LONGSHORE CURRENT AND SEDIMENT TRANSPORT

By

Edward Bennett Thornton

March, 1970



Chairman: B. A. Christensen
Major Department: Civil Engineering

On the basis of a simplified model, an attempt is made to com-

pletely describe the littoral processes including both the longshore

current and sand transport. The basic assumptions are that the con-

ditions are steady, the bottom contours are straight and parallel

but allow for an arbitrary bottom profile, the waves are adequately

described by linear theory, and that spilling breakers exist across

the surf zone. The equations of motion arc employed to the second

order in wave amplitude (first order in energy and momentum). Empha-

sis in the analysis is placed on formulating usable predictive equations

for engineering practice. The wave-induced currents generated parallel

to shore are investigated first. The littoral sand transport, which is

dependent on the strength of the longshore current and the intensity of

the wave action, is subsequently examined.

Conservation equations of mass, momentum, and energy, separated

into the steady and unsteady components, are used to describe second

order wave-induced phenomena of shoaling waves approaching at an angle







to the beach. An expression for the longshore current is developed,

based on the alongshore component of excess momentum flux due to

the presence of unsteady wave motion. This approach is parallel to

but differs quantitatively from recent work by Bowen. The wave-

induced currents are shown to be primarily confined to the area inside

the breaker line, that is, the surf zone. The longshore current ex-

pression is investigated for varying bottom profiles to evaluate the

physical significance of including different frictional resistance

terms. Wave set-down and set-up have been included in the formulation.

Comparison with experimental results from the laboratory and field show

that if the assumed conditions are approximately fulfilled, the pre-

dicted results compare quite favorably.

The wave-induced sand transport alongshore is investigated from

energy considerations in which the quantity of sand transported is

expressed as a function of the energy utilized in bottom friction,

viscous dissipation, and turbulence. Although an energy approach has

been used before, this is the first application to predicting the

distribution across the surf zone of the alongshore sand transport.

Sand transport data were collected in the field using unique bed

load traps. Wave, tide, wind, and current information was collected

simultaneously in order to check the derived predictive equations for

longshore current and sediment transport. Although the absolute values

of sand transport arc not well predicted, quite reasonable predictions

are obtained for the relative distribution of bed load transport both

inside and outside the surf zone, and could be used as a qualitative

predictive relationship for engineering applications.


xvii













CHAPTER I

INTRODUCTION


A. Introductory Note


Waves gather their energy from the wind at sea and then propagate

until eventually reaching a shoreline. It is here that a wave, having

perhaps obtained its energy from the wind over hundreds of miles and

many hours, is dissipated over a very short distance in a matter of

seconds. The area between where the waves first start to break and the

beach is called the surf zone. The incessant dissipation of wave energy

against the shoreline maintains the bottom sand and water within the

surf zone in a constant state of motion, the processes of which are for-

ever striving to reach a state of dynamic equilibrium.

Waves in the open ocean (outside storm areas) tend to be fairly

regular and lend themselves to analytical description. As the waves

move into shallower water, they undergo transformation due to shoaling

and refraction, and their heights and wave lengths change. Theories,

which have been developed to describe waves for various wave heights,

periods, and depths, are reasonably valid up to conditions of near-

breaking. The waves reach a maximum height at breaking. Upon breaking,

they lose their ordered character, and can no longer be described

analytically. Breaking waves are classified as spilling, plunging, or

surging. The manner in which waves break determines the distribution

of energy dissipation and the energy and momentum processes in the

littoral zone.




2

Waves traveling toward the beach at an angle to the shore have

a momentum flux component directed parallel to shore. This momentum

flux can result in the generation of longshore currents. These currents

have been found to be largely confined to the area inside the surf zone,

with the wave-induced currents seaward of this zone being relatively

weak.

Longshore currents are an agent for transpoorting material already

"loosened" from the bottom by the more intense wave action. The long-

shore movement of sand, termed "littoral drift," is then the result of

the combined action of sediment bed agitation by waves, acting primarily

perpendicular to the beach, and transport by the longshore current. The

littoral drift consists of suspended and bed load components. Inside

the surf zone, where the turbulence is great due to breaking waves, both

of these components are significant and vary across the width of the

surf zone. Outside the surf zone, the turbulence and wave agitation are

reduced, and the sand transport is principally in the form of bed load.

The distribution of longshore current and sediment transport across

the surf zone is the subject of this report. Unfortunately, there is

very little published data on this subject. Zenkovich [1] used fluo-

rescent sand tracers and collected suspended sand samples across the surf

zone to determine these distributions. Figure 1 shows the variation of

the sand transport as related to the longshore current and bottom profile.

During the test, the waves were relatively high, and the slope of the

beach small so that spilling breakers prevailed. The waves spilled over

the first bar, and then broke again on the inner bar. The longshore

current distribution is fairly uniform within the surf zcnc. Tlie sand


























Oulor Brcakcr
Line


Figure 1. Distribution of Longshore Velocity and Sediment
Transport across the Surf Zo\c after r ZOnkcvich [1])


I I_~ I ----i--~-----~Y-lO--i~-------~~P~






4

transport is greatest over the bars where the energy dissipation is a

maximum due to the breaking waves.

A knowledge of the variation and extent across the surf zone of

the longshore current and sand transport is important in design con-

siderations of structures placed in the littoral area. This kind of

information is particularly important for groins or similar structures

designed to impede sand movement. On the other hand, it is often

desirable to have natural bypassing about jetties constructed for navi-

gational purposes at inlets and harbors. The distribution of effluent,

introduced onto beaches and into the littoral zone, is also influenced

by the currents in the surf zone. There is a very real need for a more

complete understanding of the littoral zone so that further improvements

and preservation of our beaches can be based on more rational and con-

crete approaches.

This study logically breaks down into considering, first the w3ves

and wave-induced currents separately and, then, the combined action on

the sediments to determine the sand transport distribution. The

following is a review of the literature pertinent to this study, both

to serve the reader as a review of the past accomplishments and also

to gain considerable insight into the problem at hand.


B. Historical Summary

1. Longshore Currents

a. Field experiments

During World War 11, it became apparent there was an urgent need

to better understand the surf zone and litloral environment in order

to improve amphibious landings. This was the inpetus for initiating a






5

thorough study of the mechanics of the surf zone which was later carried

on after the war. Munk and Traylor [2] gathered data available at that

time, conducted additional field experiments, and developed the first

empirical formula relating the longshore current to the incident waves

and bottom topography. They found that the longshore current was a

function of the angle of wave incidence, the height of the breakers, the

wave period, and the slope of the foreshore.

Shepard and colleagues at Scripps Institution of Oceanography con-

tinued this work after the war in a series of field measurements along

a 100--mile stretch of the Southern California coast. From an accumu-

lation of over 1,000 measurements emerged the first rather complete

qualitative picture of the surf zone. The ideas expressed by Shepard [3]

in the original sumnary of the data have changed very little to this day.

The study showed that the most important parameter governing the long-

shore current is the angle of wave approach. It was also found that the

strength of the current was related to the breaker height.

All the field experiments showed that the longshore current was

largely confined to the surf zone and that a substantial velocity variation

could exist across the surf zone. Shepard and Sayner [4] compiled the

results of five years (comprising 800 days of measurements) taken at three

locations along the 1,000-foot pier at Scripps Institution of Oceanography.

The three measurement locations were at the end of the pier outside the

surf zone, just outside the breakers, and inside the surf zone. They

established that the currents inside and outside the surf zone were

governed by the same mecianjsim, that is, tchy ,s-e correlated, and that

the currents could be related to the wave height and direction. hlle

currents varied c3s;ideiebly frLm offshore to the beach; Lhe average






6

current velocity inside the surf zone was 1.0 foot per second compared

to an average current just outside the breakers of about 0.2 foot per

second.

Ingle [5] measured the distribution of longshore currents across

the surf zone in conjunction with sand tracing experiments on a number

of beaches along the California coast. All of these beaches, with the

exception of one, were moderately steep. These tests showed that the

maximum longshore current most often occurred midway between the breaker

line and the swash zone.

The surf zone does not always present such a simple picture as

first envisioned. On many occasions, there are strong currents of

variable direction during times when large breakers approach almost

normal to the coast. These currents are often associated with circu-

lation cells and attendant rip currents which constitute a definable

system. It has been found that these rip currents are lest common

along straight beaches with parallel contours and best developed in

areas of irregular bottom topography.

Shepard and Inman [6] measured the real variation of currents

over a stretch of bench south of Scripps pier. They found that a rip

current system was prevalent in this area and that the rips were con-

trolled,to a large degree,by the wave convergence and divergence

resulting froi refraction of waves over the rather irregular bathymetry

fronting the area. The areas of wave divergence were particularly

evident over the two submarine canyons just offshore of the bench, and

wave convergence occurred over submarine ridges. In areas of wave

convergence, there was an influx of water cross the surf zone, resulting






7

in an accumulation and a hydrostatic head causing a current away from

the area of high breakers.

These currents flow toward areas of divergence and then turn

seaward in the form of rip currents. This outflowing current then

can turn again toward the point of onshore flow, thereby forming a

complete circulation cell. The effect of wave convergence and di-

vergence on the littoral circulation was found to be a maximum during

times of long period waves which are more affected by the bottom

topography.

Bowen [7] presented the results of an analytical investigation

of the case of waves normally incident to the beach and substantiated

some of the observed phenomena. He was able to show that regular

variations in the wave height parallel to the beach could induce

regular circulation patterns. He further showed that the gradients

of wave heights could be possibly associated with edge waves or regular

undulations in the bottom topography.

Sonu et al. [8] found a considerable variation in longshore

currents during field experiments along the coast of North Carolina,

and attributed this variation to the undulations in the bottom and

the bar-trough system that prevailed. He concluded that the currents

under stable conditions appeared to be generated by the momentum trans-

fer from the plunging breakers over the bar and the mass transfer from

the spilling breakers over the shoal. The hydrostatic head potential

arising from the mass transfer into the surf zone is discharged seaward

by rip currents.

Calvin and Savage [9] conducted field e::periments at the same

location and found that,during the four days of testing, a fairly






8

uniform current system prevailed with no evidence of a rip current

system. This demonstrates that the littoral system is not constant

and is governed by the conditions that prevail at the time of obser-

vations.

b. Laboratory studies

Much of the laboratory longshore current data have been found

in conjunction with related littoral drift studies. Saville [10]

measured longshore currents on a model sand beach having an initial

slope of 1:10. The measurements were made after the bottom profile

had achieved equilibrium. It was found that the maximum current always

occurred along the nearshore bar where the waves break and was neatly

uniform across the surf zone, decreasing slightly toward shore. The

currents inside the surf zone were found to be approximately five times

the current just outside the breakers which is in very close agreement

with field measurements mentioned earlier.

Galvin and Eagleson [11] and Brobner and Kamphuis [12] performed

very similar experiments on fixed plane beaches with a slope of 1:10.

These studies showed that the maximum velocity is located between the

still water line on the beach and the breaker position, usually closer

to the still water line than to the breaker position. As the current

flowed down the beach, the local maximum had a tendency to migrate

toward the breaker position. It was also found that the currents were

not uniform in the longshore direction. The currents increased down-

stream from an initial velocity at the retaining wall to a point beyond

which a uniform distribution prevailed. The occurrence of rip-like

currents was not observed in any of these laboratory experiments.








Two rather general types of littoral systems emerge--one, in which

the longshore currents are fairly uniform in the longshore direction,

and, the other, in which circulation cells and attendant rip currents pre-

dominate. These systems are depicted in Figures 2 and 3.

Where rip currents do not exist, there is a general return of water

to the offshore to compensate for the shoreward mass transport of water

by the incident waves. Longuet-Higgins [13] theoretically showed that

for two-dimensional flow the incoming waves produce an onshore drift at

the surface and bottom and an offshore flow at intermediate depths.

This distribution was substantiated in the laboratory by Russell and

Osorio [14] and in the field by Miller and Zeigler [15] whose measure-

ments included waves breaking close to shore. Measurements taken at

Fernandina Beach using an electromagnetic flowneter further substantiate

the occurrence of offshore and onshore flow simultaneously over depth.

Thus, at one section perpendicular to the beach, transport can occur

simultaneously onshore and offshore so that there is little net trans-

port across the breakers.

A complete description of the surf zone requires considering a

system that is three-dimensional and unsteady in time and space. The

tenporal variability appears to be small compared to the spatial vari-

ability (Sonu et al., oo. cit.) and can be attributed partially to the

nonstationarity and stochastic nature of the waves. The area variation

of longshore currents can often be related to the variations in the near-

shore bathymetry and also to coastal structures. Jetties, groins, and

other structures that protrude into the ocean interrupt the normal lon--

shore current, and rip currents and large scale eddies are often located




















BREAKING WAVE /


RIP FEEDFR
CURREn \


LONG SHORE CURRENT


! ..' . ** ," .V. A H


Figure 2. Rip Current System


Figure 3. Uriifor.n Longshore Curre-nL SysLtem


t I _______ CURR
I / VECTOR

o L 0 N G S HO 0 T E


* r C H ",'


C UR RE N T









at the ends of such structures. Bowen (op. cit.) demonstrated that in

the laboratory circulation cells can be induced by edge waves caused

by gradients in the longshore breaker heights. The general complexity

of the surf zone necessitates that simplifying assumptions be made

in order to meaningfully describe particular phenomenon.

c. Theory

Most of the longshore current formulas predict only the mean of

the longshore current velocity. At present, there are at least fifteen

such formulas, none of which satisfactorily predicts all conditions in

nature. This is not too surprising in light of the rather drastic

assumptions that are required to obtain an analytical formulation.

There are four basic approaches to the development of predictive

equations for currents in the surf zo.ie: (1) conservation of mass, (2)

conservation of momentum, (3) conservation of energy, and (L) empirical

correlation. A review of the more familiar equations is presented below,

and Table I lists eleven of these formulas to facilitate co::arison.

Some of the equations have been expressed in slightly different forms

than originally published in order that they all incorporate the same

parameters; however, these changes deal only with the geometric relation

for the breaking depth and slope of the beach and the relationship be-

tween the breaking depth and breaking wave height. The changes do not

alter the currents predicted by the equations. Symbols, utilized in all

equations, are listed in the preface.

Momentum considerations.--Putnam, :hunk, and Traylor [16] used both

the energy and momentum equations to derive the first rational equations

for describing longshore currents. They considered the flux of mass and

momentum into a control volume of differential length bounded by the









TABLE I. LONGSHORE CURRENT FORMULAS


Authors Mean Longshore Current, V Formulation Eq.
No.

Putnam-Munk- 2 sin2b 1/3 Energy Conservation, 1.1
Traylor (1949) f T Solitary waves

E3 sinBsinabsin2a 1/2 Momentum Conservation,
Eagleson (1965) [ gePHb ] Asymmetric-periodic 1.2
waves

[(1 + 2.28gHbsin%)1 /2-
Putnam-Munk- Momentum Conservation,
Traylor (1949) tSolitary waves
A = 20.88 cos


Galvin-Eagleson
(1965)a gTtansin2ab Mass Conservation 1.4


Inman-Bagnold Mass Conservation,
(1963) 2.31 T cosbsinab Rip currents included



Bruun (1963) C [0.95 3/2 tansin2 Mass Conservation 1.6
f IT Mass Conservation 1.6



Bruun (1963) 2.31 b T Rip currents included


[( + 2.28gHsinab)12 2
Inman-Quinn Empirical-based on 1.8
(1951) tansHcosab momentum analysis
A = 108.3
T

H 2/3

Brebner-Kamphius 8.0 sin /3 3[sin 1.65o Empirical-based on
(1963) momentum analysis
+ 0.1 sin 3.30a

H 3/4
14.0 sin 1/28 [sin 1.65 a
Brebner-Kamphius T1/2 Empirical-based on .
(1963) energy analysis
+ 0.1 sin 3.30 ao]


0.241 Hb + 0.0318 T + 0.0374 ab Empirical-least
Harrison (1968) r cal-least 1.1
+ 0.0309 tanR 0.170 square analysis









breaker line and the shore. The change in momentum flux across the

breakers directed parallel to shore is balanced by the bottom shear

stress. Solitary wave theory was used to calculate the momentum of

the breaking waves. In this manner, they obtained an expression for

the mean longshore velocity related to the angle of wave incidence ab,

breaking wave height Hb, bottom slope tanB, wave period T, and friction

factor f. Embodied in all of the momentum analyses is a friction factor

that relates the velocity to the bed shear stress and represents an em-

pirical coefficient. This equation was subsequently revised by Inman

and Quinn [17] who found that a better fit to the data originally

collected by Putnam et al., and additional field data collected by the

authors, was obtained if the constant friction coefficient in the

original equation was changed to be a function of the velocity.

Eagleson [18], using the same control volume approach, developed

a mathematical model to represent the growth of a longshore current

downstream of a barrier. In the associated laboratory experiments, it

was found that a large percentage of the fluid composing plunging

breakers (most coma-on laboratory breaker type) is extracted from the

surf zone. This fluid already has a longshore velocity to which is

added the longshore component of the breaking wave. This argument

provides a mechanism for growth of the longshore current downstream.

The asymptotic solution to the differential equations showed that the

system is stable and that the growth of the currents was bounded.

These results agreed qualitatively with laboratory results and demon-

strated that unless there are perturbations inducing gradients in the

wave e!-rgy in the longshore direction, tbp current system tends to be

uniform alongshore and stable for stationary wave conditions.






14

Bowen [19], in a very recent investigation, used a conservation

of momentum approach to determine the longshore velocity distribution

across the surf zone for the case of a plane beach. Reasonable results

were obtained when compared to laboratory data. This is the same

approach that will be used in this dissertation to develop equations

describing the longshore current distributions for more general situ-

ations. A more complete discussion will be reserved for Chapter III.

Energy considerations.--Putnam et al. (on. cit.) also derived a

mean longshore current equation from energy considerations alone. The

derivation equates the changes in energy flux to the frictional energy

losses parallel to the beach. A difficulty, with the resulting equation,

is that it involves two undetermined constants, the friction factor f

and the percentage of the wave energy available to the longshore current s,

which makes the equation very difficult to apply.

Continuity considerations.--Bruun [20] and Inman and Bagnold [21]

derived similar expressions using the continuity approach. These formu-

lations are based on the fact that the incident waves introduce a mass

flux of water into the surf zone which is then manifested as a spatial

gradient in the longshore current. Both developments consider a plane

beach of infinite length, implying that mass is uniformly introduced into

the surf zone along the beach. The current will grow (sinre mass is

continually being supplied to the surf zone), and, at intervals, it is

necessary that there be outflow from the surf zone unless the current

becomes unbounded. It is postulated that this outflow occurs in the

form of rip currents which are evenly spaced along the coast. Thus, the

equations cuntrin an unknown parameter--the sp-cing of the rip cii'rets.

Unfortunately, few measurements have been made of rip current spa ings







15

so that the use of these equations requires additional experimental

data.

It should be noted that,due to the mass flux of waves, there

is always transport of fluid into the surf zone and that, in all the

physical models whether considering a mass, energy, or momentum

approach, the mass flux must be accounted for in order to obtain a

bounded solution. Thus, the assumption, that the mass transport is

uniformly returned across the surf zone, is at least implied in all

the developments which do not include concentrated return flow by rip

currents.

Bruun (op. cit.) also considered the case where rip currents are

absent, and the return flow is distributed uniformly over the vertical

plane containing the breaker line. He reasoned that waves breaking at

an angle to the beach contribute mass to the surf zone and locally

raise the mean water level as the breaking wave crest propagates down

the beach. This results in a slope of the water surface between crests

which creates a longshore current. The longshore current is balanced

by bottom shear stress related to the velocity through the Chdzy formula.

Galvin and Eagleson (op. cit.), reasoning from the continuity approach,

equated a hypothetical mass flux across the breaker line proportional to

the mass contained in the longshore current. Using both field and labora-

tory data, the two mass fluxes were correlated.

Empirical correlations.--Two types of empirical equations have been

developed. The first type employs physical reasoning to determine the

form and grouping of the important parameters which are then correlated

with experimenntal data. Brebner and Kamphuis (on. cit.) used both the

energy and monmentu equations to obtain reasonable groupings by dimensional






16

analysis of the important parameters. Linear regression was then used

to find the best fit for the longshore velocity to a large number of

data that they had measured in the laboratory.

The second type of analysis employs multiple regression techniques.

Sonu et al. (oD. cit.) used this method to weight the various independent

variables collected in their field studies. They found that the most im-

portant variable affecting the mean longshore current velocity was the

angle of wave incidence, and the second most important, although much

less, was the wind. These results are conflicting with those of a

similar analysis reported by Harrison and Krumbein [22] who, using data

collected at Virginia Beach, Virginia, found the most important variable

to be the wave period which proved to be insignificant in Sonu's analy-

sis. Sonu also performed a multiple quasi-nonlinear regression analysis

which showed the most important variable affecting the mean longshore

current velocity to be the wave height. In a later study, Harrison [23],

using another set of data collected at Virginia Beach, found the incident

wave angle to be the most important, followed by the wave period, height,

and beach slope, respectively. Harrison points out that the use of such

empirical equations is necessarily limited in application to "similar"

situations; it is not possible to extrapolate to any particular case

with confidence. The problem with using such techniques is that they

are devoid of any physical basis and, as such, can give spurious corre-

lation and conflicting results. The physics of the problem predict a

nonlinear combination of the independent variables where linear multiple

regression is inherently a linear combining or additive process.

Evaluation of thori es.--Galvin [2] critically reviewed twelve

mean longshore current theories and tested six of them that were





17

applicable to the "best of the published experimental data" from both

the laboratory and field. In this way, he hoped to determine which

equation most accurately represents the experimental measurements.

Galvin concluded that there is still no completely satisfactory pre-

dictor of mean longshore currents.

Sonu et al. (op. cit.) conducted field experiments and found poor

correlation when compared to six of the above equations. Their

experiments did point out the importance of the nearshore topography

on the current system, and how this may affect the outcome of such

results. Shepard [4] earlier pointed out that another reason for lack

of agreement between theory and field experiments could be the vari-

ation of current across the surf zone; each field data point is usually

based on only a single location in the surf zone.

Many simplifying assumptions are necessary in developing the

theories. Since exact expressions are not available, it is necessary,

inside the surf zone, to select approximate expressions for the wave

speed, wave shape, water particle velocity, partitioning of energy in

the wave field, alongshore variation of waves and currents, and velocity

and energy distributions across the surf zone. It is possible that im-

proved theories for longshore currents will require a better under-

standing of the highly nonlinear waves in the vicinity of the surf zone.

However, it would seem that an improvement in existing formulations could

result by including the distribution of these quantities across the surf

zone rather than considering only the mean values.

The importance of considering other factors, such as the wind, was

demonstrated by Sonu's empirical correlations. He found the wind to be






18

the second most important variable in his set of field data. This

shows the difficulty in comparing tests, particularly field data

where information concerning the effects of bottom topography and

Winds often is not included. The extrapolation of data from one

particular location to another without accounting for the importance

of these effects can lead to invalid results.

All of the equations involve unknown coefficients to be deter-

mined experimentally. Generally, the friction factor in the momentum

and energy equations is evaluated in the same manner as in open

channel hydraulics. The validity of utilizing results from steady flow

situations certainly needs to be investigated further and could hope-

fully result in a refinement.

Sonu points out that another possible improvement might derive

from consideration of the dynamic processes of energy dissipation in

the surf zone environment. The difficulty is that the flux of energy

used in generating longshore currents is only a small fraction of the

total available energy and, as such, represents a second order phenome-

non (Galvin, o2. cit.). On the other hand, it appears that the mass

flux into the surf zone represents a primary feature of the surf zone

so that conservation of mass might be a better basis for longshore

current theories. One difficulty with using the continuity approach

is that, although it allows a description of the mean current, no

procedure has yet been developed, based on continuity considerations,

which provides a prediction of the variation of the current across

the surf zone.








2. Littoral Drift

a. Field experiments

Field experiments have played a very important role in the study

of littoral processes. One of the difficulties encountered in sedi-

ment transport studies is that the similitude laws relating field to

laboratory conditions have not been adequately established and verified.

Thus, field studies are necessary for the understanding of littoral

problems, either to supplement analytical or laboratory work, or as a

means of solution by itself.

The measurement of littoral drift along a particular beach and

the meaningful correlation with the wave environment is an extremely

difficult task. Hence, only very limited data are available. The

formula, most frequently used for determining total littoral drift, is

attributed to Caldwell [25]. Dimensional arguments show that the rate

of sediment transport can be related to the longshore component of wave

power per unit beach length. Caldwell combined the results of field

studies conducted in Anaheim, California, and those of Watts [26] at

South Lake Worth, Florida, and proposed the following formula for the

total longshore transport per unit time due to waves


Qs = C[pgHb2 cb sin2a At] 0.8 (1.12)


where AZ is a unit length of beach, and C is a constant of proportionality.

This relctonrship has been tested against laboratory data by Savage [27]

and shows reasonable correlation.

The earliest published information on the distribution of sand trans-

port along rne beach profile for prototype cornirions was by the Beach

Erosion Board [28]. The distribution of longshore current and suspended





20

sand, obtained from water samples, was measured from piers extending

across the surf zone. These measurements showed that the greatest

sand transport occurred at the breaker line, where the turbulence was

a maximum, and decreased shoreward with another peak in the swash zone--

another area of high turbulence. Seaward of the breakers, the sand trans-

port decreased with increasing depth. There have been surprisingly few

other field experiments of this type. Watts [29] conducted similar

studies using a more elaborate continuous suspended sediment sampler.

The results were qualitatively similar and showed that the amount of

sand in suspension was related to the wave height, or energy, of the

waves for a particular test. In these experiments, and some by Fukushina

and Mizoguchi [30] using suspended samplers made of bamboo poles, the

vertical distribution of suspended sediments was also measured. These

data showed that the amount of suspended sediment in the swash zone and

near the breaker line can be fairly evenly distributed over the vertical

due to the high degree of turbulence throughout the water column. This

is particularly true at the breaker line where a large vertical velocity

component can be present in the case of plunging breakers. The greater

portion of the transport in the surf zone is due to suspended load with

the highest concentration near the bed; outside the suif fore, bed load

is the predominate mode of transport.

Improvements in tracer techniques, particularly using either radio-

active or fluorescent sand tracers, have increased the intensity of

littoral drift studies in the field. A great number of studies have been

conducted in recent years anI have been summarized in a book by Ingle

(op. cit_.). Ingle also conducted a rumbec of studies using fluorescent

tracers on several Southern California beaches. The results of these






21

and some conducted by this researcher, and also earlier investigators,

are all generally similar to that given in Figure 1 in describing the

variation of sand transport across the surf zone. The fluorescent sand

grains are found in greatest concentrations along points of high turbu-

lence. In a bar-trough profile, the sand moves predominantly along the

bar or in the swash zone. There is a minimum of tracer transport in

the trough.

Tracer studies have been conducted using both the Eulerian and

Lagrangian approaches. Host investigators have used the Lagrangian

approach in which the tracer is introduced at a particular location,

and the concentration distribution of the tracer is determined by ob-

taining bottom samples at various sample points. The concentration of

tracers is then determined by counting the tracer grains in the sand

samples. The Eulerian approach is to sample in time along a particular

line across the surf zone, traversing the path of the tracers. A stable

platform or other work facility is generally required in this method.

This was the approach used by Zenkovitch, working from a tramway tra-

versing the surf zone and by Bruun and Battjes [31], working from a pier.

The experiments by Bruun were continued by this author.

The inherent difficulty of fluorescent tracer studies is that

quantitative measurements require recovery of most of the tracer. Un-

fortunately, the recovery level is generally very low, amounting to only

a few percent. This requires that accompanying measureelvents of the

quantity of sand in suspension, or moving on the bed, also be determined.

However, as a too], or aid, for solving engineering problems in which

qualitative information can be extremely important, fluorescent tracer

techniques alone can be of great value.








b. Laboratory studies

A number of laboratory experiments have been conducted to deter-

mine the mechanisms causing sand transport in the surf zone. This

discussion is limited to the three-dimensional studies simulating

conditions in the prototype. Krumbein [32] conducted one of the first

of these experiments and concluded that the mean littoral drift was a

function of the deep water wave steepness, H /L Subsequent studies
0 o
by Saville (o_. cit.) showed that a maximum transport occurred for a

wave steepness of 0.025, and,for steepnesses greater or less than this

value, the transport was less. This was further verified by Shay and

Johnson [33] who also showed the transport to be a function of the wave

angle. A maximum transport was found to occur for a deep water wave

angle of 30 degrees.

Although these results imply tlat the transport is a function of

wave steepness, the reason for this dependence has not been established.

Galvin [34] conducted a series of experiments of breaking waves on labo-

ratory beaches. He developed a classification for determining whether

the waves develop into plunging or spilling breakers as related to wave

steepness and the beach slope. These results, when compared to the

littoral drift studies in the laboratory, indicate that maximum trans-

port occurs for a plunging-type breaker and that the rate of transport

may be more a function of the manner in which the waves break than the

wave steepness.

The study by Saville and a subsequent study by Savage [35] also

provide information concerning the distribution of littoral transport

across the surf zone. A series of traps wn. uFod to traverse the down-

stream profile of a model beach so that both the total and distribution






23

across the surf zone could be determined. It was found that approxi-

mately 90 per cent of the transport occurred inside the surf zone.

The distribution of transport across the surf zone was in qualitative

agreement with field studies.

c. Theory

Although many predictive formulas have been proposed for the

problem of sediment transport in alluvial channels, there have been

relatively few attempts to explain the transport phenomenon due to

combined wave and current action. This is undoubtedly due to the

limited success found using the predictive formulas for unidirectional

flow and also because of the added complexity of dealing with oscil-

lating flow.

A study of the mechanics of the forces acting on a sand grain under

the combined action of waves and currents was made by Eagleson and Dean

[36]. This study considered the combined gravity, friction, current and

wave forces acting on an individual spherical particle on the bed. From

this deterministic approach, accompanied by considerable laboratory

studies, conditions for incipient motion were found. The forces on a

suspended particle were also considered in the same study.

For a given sand grain size and density and beach slopes, there

exists an equilibrium position, the so-called "null point," along the

bottom profile where all the forces on the particle averaged over a wave

cycle are balanced. Eagleson et al. [37] extended the study of the

incipient motion condition mentioned above to better define the "null

point" of a particular sand grain and were able to qualitatively explain

sorting of sand along the profile according to size and weight charac-

teristics. The forces acting on a sand grain can be elucidated, and








considerable insight gained into the nature of the problem from this

deterministic approach to the sediment transport problem.

Difficulty is encountered in extrapolating from a single particle

analysis based upon deterministic mechanics to the many particle analy-

sis which in reality must be considered as a statistical mechanics

problem. Kalkanis [38] used the same arguments as embodied in Einstein's

approach to sediment transport in alluvial channels and extended it to

the case of simple harmonic motion with a superimposed nean current. He

assumed that the distribution of the turbulent water particle fluctu-

ations are Gaussian over the wave period and, on this basis, derived a

probabilistic model for the bed load movement. Laboratory experiments

accompanied this study in an attempt to determine the necessary co-

efficients in Einstein's development. Unfortunately, the complexity of

the formulation requires the determination of five parameters which must

be based on considerably more data than have been examined to date.

Iwagaki and Sawaragi [39] developed a formula for predicting the

total littoral transport rate applicable to long term average values.

The basic assumption was that the average littoral transport is pro-

portional to the shear velocity of the mean longshore current. The mean

longshore current is determined by the Putnam, Munk, and Traylor energy

formula for longshore currents (see Table 1). The formula, presented by

Iwagaki and Sawaragi, in terms of measured quantities at the breaker line,

is

tan8 sin2ab 4/3 cosab 11/3
Q A [ (1.13)


where A is a constant of proportionality, d is the mean grain size, and

the other terms are as previously defined.






25

Le M1hautd and Brebner [40] proposed an equation for the total

mean littoral transport rate based on physical reasoning guided by

empirical information. The equation assumes the form


Qs cbgH2 T sin aab (1.14)


The exponents and constants in this equation were selected to provide

a best fit to the measured data.

A more straightforward approach is given by Bagnold [41]. This

approach is based on considering the work required to move the sedi-

ments as related to the available energy of the waves and currents.

He considers both the bed load and suspended sediment transport. Since

this is essentially the method of analysis being employed in this dis-

sertation, the formulation is presented completely in Chapter IV.


C. Purpose and Scope of the Investigation

The aim of the present research is to investigate the distribution

of wave-induced longshore currents and sediment transport from offshore,

across the surf zone, to the beach. Where possible, formulas are de-

veloped to predict quantitative information. A simplified physical

model is considered which has wide application to nature. The basic

assumptions are that steady-state conditions prevail and that the bottom

contours are straight and parallel to the beach.

The study naturally breaks down into examining, first, the distri-

bution of the shear stresses and longshore currents. Having obtained

this information, it Is then possible to describe the distribution of

sediment transport.

Tools for the analysis are developed in Chapter II. A general

set of conservation equations are set down, and the wave field, inside






26

In Chapter III, theoretical models for longshore current and

shear stress distributions are developed governing the region from

deep water onto the beach. A momentum approach is utilized in which

the changes in momentum flux are balanced by frictional forces in

the direction parallel to shore. The resolution of the shear stresses

into wave and current components is demonstrated. The theoretical

models for longshore currents are checked against existing field data.

Chapter IV utilizes the results of the predicted current and

shear stress distributions to theoretically describe the variation of

sediment transport across the surf zone.

Chapter V describes field experiments conducted which tend to

substantiate the theoretical descriptions of the sediment transport

distributions. The results of the field experiments and theory are com-

pared and discussed.













CHAPTER II

CONSERVATION RELATIONSHIPS AND
SPECIFICATION OF WAVE FIELD


A. Introduction

Past investigations have attempted to relate the mean longshore

currents to wave-induced momentum, energy, or mass flux into the surf

zone. The distribution of longshore currents can be similarly investi-

gated by considering the changes in the momentum, energy, or mass flux

across the surf zone. The present analysis utilizes the momentum

principle to describe the variation of wave-induced longshore currents

from deep water, across the surf zone, to the beach.

It is known that, due to the fluctuating water particle motion of

the waves, there is a momentum flux component. If the waves have a

direction component parallel to shore, a longshore current can be gener-

ated due to changes in the longshore momentum flux component of the

shoaling waves. It is also known that there must be a displacement of

the mean water surface elevation to balance the changes in the onshore

momentum flux component of the shoaling waves.

In waves, the momentum flux is the sum of the pressure and the

product of two velocities. It can be shown that the average momentum

flux is nonlinear in wave height. In order to specify the excess

momentum flux of the waves, it therefore becomes necessary to consider

nonlinear, or higher order, effects of the wave motion. Since the

waves are the mechanism responsible for the generation of the longshore






28

currents, much of the ensuing discussion is devoted to describing the

waves.

The treatment of periodic gravity waves is generally developed

by perturbation schemes built on the exact solution for the linearized

equations of motion with specified boundary conditions. Difficulty,

with this type of analysis,is encountered when considering higher order

theory including a variation in the bottom boundary condition. For

waves traveling in water of varying depth, two different approaches can

be employed: the analytical method and the energy flux method. The

analytical method involves solving the boundary value problem accounting

for the slope of the bottom to the desired degree of approximation.

This technique includes the slope,explicitly,in the perturbation expansion.

The solution for the waves on a plane sloping bottom has bccr worked out

to the second order by several authors. Miei et al. [42] have given a

systematic approach for carrying the analysis to higher orders.

The analytical method is generally more consistent because it

accounts for the bottom slope in the bottom boundary condition to the

same order of approximation as is maintained in the free surface boundary

conditions. The smallness of the bottom slope as well as the smallness

of the amplitude is incorporated in the perturbation analysis. Although

more attractive from an analytical point of view, this method has the

inherent difficulty of requiring tedious computations for each particular

case. Also, and more importantly for our needs, the more rigorous

boundary value problem approach is not readily extended across the surf

zone.

Ite energy method consists of solving the wave problem for a hori-

zontal bottom and then extending thfse results to a sloping bottom by







29

means of energy flux considerations. Hence, for short distances, in

the region outside the surf zone, it is assumed that waves on a sloping

bottom can be considered the same as on a horizontal bottom. Then,

adjacent increments of distance are connected together by means of the

energy flux conservation equation. This results in a prediction of the

wave height at any location outside the surf zone. Inside the surf zone,

it will be assumed that the wave height is governed by the local water

depth.

The method employed allows for the inclusion of dissipative effects,

such as bottom friction, and the focusing of energy due to refraction by

changes in bathymetry and currents. The accounting for these effects is

often more important than the deformation of the water particle motion

due to the bottom slope as given in the analytical procedure [43].

The validity of the energy method can be proven by comparison with

the results of the exact analytical solution. To the first order of

approximation in wave energy,for the case of small bottom slopes, identi-

cal results are obtained for the propagation of energy. This confirms

the use of the energy flux equation to connect the solution for various

depths for small bottom slopes, at least to the first order in wave

energy. In is assumed that the method can then be extended to higher

order nonlinear waves.

The treatment here is to use the energy method approach to describe

the wave field to the first order in energy (second order in wave ampli-

tude). This solution is then substituted into generalized conservation

equations to extract the desired information.




30

B. Conservation Equations

A convenient starting point for this analysis is a statement of

the general conservation equations of mass, momentum, and energy fluxes

applicable to unsteady flow. The analysis is not concerned with the

internal flow structure of the fluid; hence, the derivation can be

simplified by integrating the conservation equations over depth. Conser-

vation equations which have already been developed by Phillips [44]

will be used and are presented below.

The conservation equations will be applied to wave motion, but

they are equally applicable to general turbulent motion. The unsteady

velocity field of the wave motion can be expressed in the same manner

as in the treatment of turbulent motion as the sum of its mean and

fluctuating parts


u = (U (x,y,t) + u!(x,y,z,t), w(x,y,z,t)) i = 1,2 (2.1)

where (1,2) refer to the horizontal coordinates (x,y), respectively,

and z is the vertical coordinate. The tensor notation is used only for

horizontal components of water particle motion. The mean current is

assumed uniform over depth for simplicity. The pressure term can be

stated similarly. These expressions can be substituted into the mass,

momentum, and energy equations, and the mean and fluctuating contri-

butions identified.

The conservation equations are to be averaged over depth and time

(one can consider averaging over a few wave periods). For the case of

waves superposed on a mean current, all the wave motion would be identi-

fied with the fluctuating quantity which, when integrated over the total

depth, can contain a mean contribution due to the waves. The time

averaging of the equations for a general davilopment, being over a short





31

interval compared to the total time, does not preclude long term un-

steadiness in the mean motion.

The utility of the conservation equations derived by Phillips is

that the terms involving the mean and fluctuating quantities have been

separated. This facilitates the understanding of the effect of the un-

steady wave motion on the total flow phenomenon. They are a parti-

cularly useful aid in gaining physical insight into the complicated

mechanisms taking place in the surf zone.

In addition to the "normally considered" conserved quantities of

mass, momentum, and energy, a fourth condition is available to aid this

analysis--conservation of phase, or more simply, conservation of waves.

This condition, as applied to gravity waves, was first stated by

Whitham [45].

1. Conservation of Phase

The conservation of phase will be derived for simplicity in relation

to simple harmonic motion. The expressions to be derived are also appli-

cable to more general wave motion which can be formed by the superposition

of individual Fourier components. The expression for the water surface

profile for harmonic motion is given by


= (xi) cos(kixi Wt) i = 1,2 (2.2)

where (xi) is the amplitude, and the quantity (k.x. at) is called
2 i
the phase function X. The wave-number k. and frequency u can be defined

in terms of the phase function, such that


k = x_ (2.3)
I








u= aX (2.4)
at


An important property of the wave-number can be seen immediately by

vector identity


V x k = V x VX L 0 (2.5)


so that the wave-number is irrotational. From Equations (2.3) and

(2.4), the kinematical conservation equation for the wave-number can

be written


ak.
S = 0 (2.6)
at ax.


For a single wave component, the conservation of phase says that the

rate of increase of the number of waves in a fixed length is balanced

by the net inward flux of waves per unit time. If a can be expressed

as a function of k. and possibly x, from local arguments (such as

assuming a uniform simple harmonic wave train locally),and further
->
allowing a mean current U, then,


a = o(k,xi) + kiUi (2.7)


a is the local wave frequency and is the apparent frequency to an ob-

server moving with the current. Substituting into Equation (2.6),

Whitham (2op. cit.) shown


ak. ak. ak.
+ c + U. = G. (2.8)
at g x. j ax.

where
au
CG - k --1
i ax. j ax.
1 1









and

c 30 (2.9)
gj 3k.
J

c is the speed at which the energy or values of k. are propagated,
gJ J
commonly called the group velocity. The individual crests propagate

with the local phase velocity c = The notation shall be adopted

where, if c or k appear without vector notation, it is understood that

they represent the modulus of the vector quantity.

Equation (2.8) says that the rate of change of the wave-number

ki, following a point moving with the combined group and convective

velocity, is equal to G.. Changes in G. are due to variations in the
1 1
mean current and bottom configuration. These geometrical equations,

which state the kinematical conservation for the wave-number, hold for

any kind of wave motion (Phillips, op. cit.).

2. Conservation of Mass

The general conservation of total mass per unit area can be ex-

pressed


p D + I- = 0 i = 1,2 (2.10)
at 3x. 1
1

D is the total averaged depth of water which can include a mean elevation

n, above (or below) the still water depth h, so that


D(x,y,t) = (n + h) (2.11)


The overbar shall be used to signify time averages. The total mass flux

M. can be partitioned into its mean and fluctuating components
1


M. = H. + i.
1 1 1


(2.12)






34

The mass flux per unit width of the mean flow is

n

~i pUi dz = pDUi (2.13)

-h


The mass transport of the wave motion is



M. pu' dz (2.14)

-h


3. Conservation of Momentum

The equation defining the conservation of horizontal momentum is

derived by integrating the momentum equation, including shear stresses,

over depth and averaging in time. The balance of total momentum per

unit area can be expressed



at ( + S..) = T. + R. (2.15)



Here M. denotes the total horizontal momentum per unit area. Hence,
1
the first term on the left represents the rate of change of the total

mean momentum per unit area which includes both the current momentum

and wave momentum.

The total mean transport velocity can be expressed in terms of the

mass flux per unit area

M. M.
U U + -- (2.16)
i pD i pD

The second term on the left of Equation (2.15) expresses the momcntur

flux of a stL:dy stream having the same mass flux M. and mean transport

velocity IU. as the actual flow together with an express momentum flux
1









term S.. arising from the superposed wave motion, where
1]

n
=fM.M.
Si (puu.u + P6j) dz pgD26 (2.17)

-h


and 6.. is the Kronecker delta.
iJ

The term T. is given by
1


T. = -- pg(n + h) (2.18)
1

and represents the net horizontal force per unit area due to the slope

of the free water surface.

R. is the time mean averaged shear stress which must be included
1

in any realistic treatment of the surf zone where dissipative effects

occur. The form of the shear stress was not explicitly stated by

Phillips in his derivation so this term will be expanded in detail.

Also, insight can be gained into the manner in which the momentum

equation was integrated over depth in arriving at the generalized

momentum conservation equation. Considering the shear stresses on a

column of water (see Figure 4) and integrating over the depth, R. can
1
be expressed


P aT. f PT.
dz + i dz i,j = 1,2 (2.19)
1 ;x. az
-h -h


where T.. includes the combined stresses of waves and currents. The

shear stress on the horizontal plane can be integrated over depth, such

that







z


y x







L N.




















ix
.: ,*.' ...' .. -














rFtrc 4I. Shear SLresses Acting on the raceM of an
.E cmntal .ater Col :,in
L .... _____ _ ."_ ." __"_" __"_' " " : v -.












Szdz = T.i hi (2.20)

-h

where the subscripts refer to surface and bottom.

The Leibnitz rule of integration must be employed to integrate

over depth the stresses occurring on the vertical faces of the water

column. Evaluating the shear stress on the vertical faces and taking

the time average of the terms

11 n
f ^-* a f an 3 h
d dz = T .dz .. (2.21)
dx. x- 31 n 31xn i. h ax.
-h -h


Thus, the general shear stress term is given by



a an h -
ST..dz T + -- (2.22)
i 3x-. n jin 3x. jih ax. ni -hi
-h i,j = 1,2


4. Conservation of Energy for Fluctuating Mlotion

The equation for the conservation of total energy can be partitioned

into the energy contributions from the mean and fluctuating parts with the

aid of the conservation of mass and momentum equations. In the following

development, it is convenient to work with the energy balance for the

fluctuating motion alone, which can be stated


M. 2 2U.M.2 ;U.
(E 2 (UE + F -) + S = (2.23)
at 2Dh ax. i i 2pD Sij x.
1 1

In separating tle mean and fluctuating contributions, the mean energy

density has been represented as the total energy density of an "equivalent"






38

uniform flow with the same depth and mass flux as the actual flow, plus

the energy density of the fluctuating motion, minus an additional term

representing the difference between the energy density of the actual

mean motion and that of the equivalent uniform flow. Then, the first

term in Equation (2.23) is the rate of change of the energy density of

the fluctuating motion minus the correction term. The second term

represents the convection of the fluctuating energy E, by the mean flow,

the energy flux of the fluctuating motion F, minus the correction term.

The transport of energy by the fluctuating motion is given by




S[(u12 + w'2) + g(z ) + P] dz (2.24)
1j J p
-h

which represents the rate of work done by fluctuating water particle

motion (for turbulence, this would be the work done by the Reynolds

stresses) throughout the interior region of the flow, plus the work

done by the pressure and gravity forces. The last term on the left of

Equation (2.23) represents the rate of working by the fluctuating motion

against the mean rate of shear. Dissipative effects of turbulence have

also been allowed where e is the rate of energy dissipation per unit area.

5. Conservation of Energy for the Mean Motion

The energy budget for the mean motion is given by



-[! UiMi + g(n2 h2) + i 2 + g) + = .
2t ax. x. I I
1 i

(2.25)

The terms of this equation represent, respectively, the rate of change

of the kiretic and potential energy of the mean motion, the transport







39

of the total mean energy, and the rate of work by mean motion on the

fluctuating motion and bottom shear stresses.

In the development of the conservation equations, no restrictions

were placed on the wave slopes or amplitudes. Also, no restrictions

were placed on the fluctuating motion so that the equations are equally

applicable to wave or turbulent motion.


C. Description of the Wave Field

1. Waves outside the Surf Zone

The descriptive equations for the wave field are derived by, first,

solving the linearized boundary value problem over a horizontal bed. The

solution can, then, be extended to a higher order by perturbation tech-

niques. The development presented here will retain terms to the second

order in amplitude (first order in energy and momentum) and neglect all

higher order terms. The wave solution can then be substituted directly

into the conservation equations providing a means for describing the

wave-induced mean motions. In making this substitution and dropping all

terms of orders higher than the second, only knowledge of the first order

(linear) wave water particle velocities and surface elevation is neces-

sary. This is because, in expanding and then averaging over the period,

the terms involving higher order quantities in velocity and surface

elevation go to zero. The pressure must be known to the second order in

wave height; however, the average second order pressure component can be

determined from the first order water particle velocities and surface

elevation terms. Thus, only the linear wave solution is required.

In utilizing linear theory, it is assumed thar the motion is irro-

tational and that the fluid is incompressible and inviscid. These have








been shown to be good assumptions, due to the fact that viscosity plays

only a very minor role in determining the hydrodynamics of the wave

field, and, thus, the vorticity is very weak [44]. The fact that the

linear theory is a good approximation is demonstrated by its success in -

describing many observed phenomena.

Assuming simple harmonic motion, the surface elevation is restated

as in Equation (2.2)


H cos(k.x. ot) i = 1,2


where the wave-number components are


k = k eosa
x
(2.26)
k = k sina
y

The arbitrary angle of wave incidence a is measured between a line

parallel to the contours and the wave crests. The velocity potential

for the first order solution is given by


4 - O_ csosh h -sin(K.x. ot) (2.27)
2 ck cosh 1h a

The velocity is related to the gradient of the velocity potential


u.= -V4 (2.28)


k
H L i cosh k(h + z)
u. = cosh kfh ) cos(k.x. ot) (2.29)
i 2 c k cosh kh i 1i

The frequency equation relates the frequency to the local wave-number

anl water depth


o2 = gk tanlh 1-h


(2.30)






41

The pressure can be determined as a function of the depth to the second

order by integrating Euler's vertical momentum equation over the depth

and retaining terms of second order. Only knowledge of the first order

water particle velocities and surface elevation is required to deter-

mine the mean second order pressure term. The time-mean pressure is

given as


p g 1 H2 H2 sinh2 k(z + h) (2.31)
p = sinh2 kh (2.31)


where the first term is the hydrostaric contribution, and the second

term is the second order mean dynamical pressure component.

The group velocity of the waves can be determined from Equation

(2.9)

k k.
i do o i
(c ) = n = c.n (2.32)
gi k dk k k i


where the transmission coefficient is


1 2kh
n (1 + 2(2.33)
2 sinh 2kh

1
For deep water (infinite depth), n = and, in shallow water, n

approaches 1.

The energy of the waves is proportional to the wave height

squared and is equally partitioned between the potential and kinetic

energy, such that the total energy is


E pg2 (2.34)

Recalling that the group velocity expresses the speed of energy propa-

gation, the energy flux of the wave motion is


F. = Ec .
l gl


(2.35)





42

Assuming negligible wave reflection and a mildly sloping bottom,

Equations (2.32) and (2.34) can be substituted directly into

Equation (2.35) to obtain the flux of wave energy.

The mass transport of the waves can be determined by recalling

Equation (2.14) in which the fluctuating velocity was integrated from

the bottom to the surface. Since the water surface elevation is un-

known, the integrand is expanded in a Taylor series about the mean

water surface level n

n n

M. = pu dz + p[u.(n) + z + ...] dz (2.36)

-h n

where the time mean average of the mass transport below the trough is

zero (first integral), and only terms of second order in amplitude

will be retained. Substituting for the wave profile and water particle

velocity of the wave motion, Equations (2.2) and (2.29), the mass

transport of the waves is

E i
.E = p u (2.37)
i ui c k

Simple harmonic motion has been selected to describe the waves because

experience has shown that this solution gives fairly good results for

the deep water case. Inside the surf zone, however, the approximation

is not good, but the assumption of simple harmonic wave trains will be

retained with the exception of certain modifications on the celerity

and amplitude. The wave field could just as easily (in principle, anyway)

have been specified using the cnoidal or other higher order wave theories.

It is felt that, for the present investigation, the accuracy gained would

not justify the much increased complexity of the resulting equations.








2. Breaking Waves

The seaward edge of the surf zone is usually delineated by the

point where the waves first start to break. Inside the surf zone,

the waves are unstable, and the fluid motion tends to lose its ordered

character. Waves break in different ways, depending primarily upon

the wave steepness and slope of the beach. The manner in which they

break has a very definite influence on the hydrodynamics inside the

surf zone which, in turn, affects such quantities as the sediment

transport, longshore currents, and wave runup.

Based on observations by Wiegel [46], breaking waves are usually

classified as spilling, plunging, or surging. Spilling occurs when

the wave crests become unstable, curl over slightly at the top, creating

a foamy advancing face. Plunging occurs on steeper beaches when the

wave becomes very asymmetric; the crest curls over, falling forward of

the face resulting in the creation of considerable turbulence, after

which a bore-like wave front develops. Surging occurs when the wave

crest remains unbroken while the base of the front face of the wave,

with minor turbulence generated, advances up the beach.

There is a continuous gradation in the type of breaking, and

Galvin [34], in a more recent classification, found it convenient to

add a fourth category of collapsing to describe a type intermediate to

plunging and surging. He performed extensive laboratory investigations

to quantitatively classify breaking waves according to the wave and

beach characteristics. Combining his results with earlier works, he

grouped the breaker type depending on beach slope tanS, wave period T,

and either deep water or breaker height H. The breaker type can be

sorted by either of two dimensionless combinations of these variables,






44

an offshore parameter, H /(L tan2B), or an inshore parameter,

Rb/(gT2 tanS). As either of these parameters increases, breaker

type changes from surging to collapsing to plunging to spilling.

Spilling breakers are associated with steep, relatively short period

waves and flat beaches; plunging breakers are associated with waves

of intermediate steepness and the steeper beaches; and surging breakers

are associated with waves of small height and steep beaches.

On natural beaches, breakers classed as spilling are most commonly

observed, followed in decreasing order of frequency by plunging, col-

lapsing, and surging. In the laboratory tests, spilling breakers are

relatively rare compared with collapsing and surging breakers, because

slopes used in laboratory tests are usually steeper than slopes commonly

found in nature due to the physical limitations of space.

The breaking index curve provides a relationship between the breaking

depth Db, the breaking wave height Hb, and the wave period T. In shallow

water, the relationship simplifies to Hb/Db = K, a constant. Reid and

Bretschneider [47] compiled breaking wave data from several sources in-

cluding both laboratory and field data. They found a fairly good corre-

lation for breaking waves as compared to the breaking wave criteria

predicted by the solitary wave theory. The solitary wave theory pre-

dicts a value for < of 0.78, while other theories predict slightly

different values. Theoretical values range from 0.73, found by Laitone

[48], for enoidal wave theory, to a value of 1.0, found by Dean [49]

using a numerical stream function theory. Experiments on steeper

laboratory beaches show that the value of < can be much larger. All

the theoretical values have been calculated, assuming a flat bottom,

and correspond to beaches with very gentle slopes. The important point

is that the bre;ilcr height is governed by the depth of water.









3. Waves inside the Surf Zone

Inside the surf zone, energy is dissipated due to the generation

of turbulence in wave breaking, bottom friction, percolation, and

viscosity. The waves in the surf zone constitute a non-conservative

system in which the use of potential flow theory is no longer valid.

In fact, there is no analytical description available for the waves

in the surf zone. Hence, one is required to make rather gross as-

sumptions and then to test these assumptions experimentally. The

linear wave theory will be retained as the input to the conservation

equations, but with modification to the wave amplitude and speed. The

wave height inside the surf zone is controlled by the depth and is of

the same order of magnitude. Thus, even the second order theory in

wave height is a rather poor assumption, but seems to agree surprisingly

well with measurements of some phenomena.

Spilling breakers lend themselves to a physical treatment since

the potential energy and momentum flux of the waves inside the surf

zone can be expressed approximately in analytical form. If the beach

slope is very gentle, the spilling breakers lose energy gradually, and

the height of the breaking wave approximately follows the breaking

index curve [50]. The height of the wave is then a function of the depth.

In the breaking process, plunging breaker heights may also be inter-

mittently described by the breaking index; however, due to rapid dissi-

pation of energy in breaking, the height may fall below the breaking

index curve, and the residual wave energy may later reform and then break

again in shallower water. This was generally noted for moderately gentle

slopes in laboratory experiments by Nakamura et al. [51].






46

In the present analysis, it will be assumed that the waves act
v
as spilling breakers inside the surf zone and that they follow the

breaking index, K = 0.78, as predicted by the solitary wave theory.

The wave height inside the surf zone is then given by


H = KD (2.38)

It is further assumed that the kinetic and potential energy are

equally partitioned so that the total wave energy can be described in

terms of the wave height which is a function of the depth


E = pgK2D2 (2.39)
8

This is a non-conservative statement of the energy distribution within

the surf zone.

The waves inside the surf zone are assumed to retain their simple

harmonic character so that the wave profile and water particle velocity

are described by

II
n = cos(k xi ot)
2 ii

k (2.40)
u = c k- cos(kixi ct)


The expression for the horizontal water particle velocity is based on

the Airy wave theory and has been simplified for shallow water.

In very shallow water, the waves are non-dispersive with the wave

speed being only a function of the depth. It has been found experi-

mentally that a reasonable approximation to the wave speed in the surf

zone is that predicted by the solitary wave theory [52].


c ,= i + K) =- (i'+ <) .n


(2.41)







47

The wave field has been completely specified. These results may

now be substituted into the general conservation equations to describe

wave-induced phenomena inside and outside the surf zone.













CHAPTER III

LONGSHORE CURRENT THEORY


A. Statement of the Problem

The study of the area in and about the surf zone presents a diffi-

cult problem due to its very complex nature. A proper treatment of the

surf zone must consider a three-dimensional problem of unsteady fluid

motion and is further complicated by moving interfaces at the upper and

lower boundaries, that is, at the water surface and sediment bottom.

It was also noted earlier that more than one longshore current system

could occur for seemingly similar conditions. Thus, it is necessary to

make definite and simplifying assumptions in order to make the problem

tenable to a theoretical approach.

This analysis considers the steady-state distribution of quantities

on a line normal to the shoreline. A schematic of the surf zone area is

shown in Figure 5. The analysis is restricted to the case of an arbitrary

bottom profile with straight and parallel contours in the y-direction

(parallel to the beach). The water depth is then a function of the

x-direction only (perpendicular to the beach). Since the distribution of

mean properties of the wave field is a function of the depth, this elimi-

nates any y-dependence.

An exception to this was found by Bowcen [7]. Using the fact that

incident waves can excite transversal waver, commonly called edge waves,

he showed that if these ,'raves are standing a\.e';, or only slowly pro-

gressive, gradients in the mean water surf, c can be developed in the

48





































BREAKER LINE





WAVE CREST


LONGSHORE
CURRENT
DISTRIBUTION


Figure 5. Scheme of the Surf Zone






50

longshore direction which in turn can result in circulation cells. Thus,

a more exact formulation has to assume suitable spatial averaging in

the longshore direction so as to preclude the effects of any transversal

waves.

It is assumed that wave reflection is negligible. This assumption

is justifiable outside the surf zone for gently sloping bottoms. The

present analysis is shown to be most valid for spilling breakers which

implies a gently sloping bottom. The wave reflection is least for this

type of breaker condition and is assumed negligible inside the surf zone

as well.

Shear stresses at the surface due to the wind will be neglected.

Summarizing, this is a steady-state problem (except for periodic

wave motion) with no y-dependence such that


3h D_ E V U (3.1)
= 0 (3.1)
Sy 3y 3y ay

The notation U = U and U = V will be used henceforth. No restrictions
x y
will be placed on the direction of wave approach for, indeed, it is the

wave motion in the y-direction that is the driving mechanism in the

equations of motion.

The problem can be conveniently discussed by considering separately

the areas outside and inside the surf zone. Since conditions inside the

surf zone are dependent on the incoming waves, the case seaward of the

breakers will be examined first.


B. Currents outside the Surf Zone--Neelecting Bottom Friction

A determination cf the distribution of mass transport and energy

of the waves is frsat necessary in oicer to solve for the wave-induced

currents outside the surf zone.









1. Mass Transport Velocity

Due to the absence of any y-dependence, the mass conservation

Equation (2.10) reduces to


3M
ax
X=0 (3.2)


Integration gives


M = constant = 0 (3.3)


which must be equal to zero since the beach forms a boundary in the

x-direction. This then says


M E
U=- cos = cosa (3.4)
pD pcD

which states that there is a mean reverse current balancing the mass

transport onshore due to the wave motion. This must be true everywhere,

both inside and outside the surf zone, to ensure that there is no accumu-

lation of mass or growth of currents in the y-direction in order to

maintain steady-state conditions in accordance with the original as-

sumptions.

2. Distribution of Energy outside the Surf Zone

The determination of the energy distribution is necessary as a means

of relating the wave heights at various depths. The energy, in turn,

must be related to the local angle of wave incidence. The angle of wave

incidence is affected by refraction which can occur since the waves are

allowed to approach at an arbitrary angle of incidence to the bottom

contours. The problem is further complicated since a shear flow is

allowed which can also produce wave refraction.






52

The general statement of the phase relation gives an expression

relating the frequency, mean current motion, and wave angle. Equation

(2.7) can be expanded to give


w = o + Vk sina + Uk cosa (3.5)

The assumption of steady-state conditions requires that a be uniform

and constant.

The wave-number was shown to be irrotational


3k 3k
x = 0 (3.6)
3x 3y

Since the wave length and amplitude of the waves are independent of the

y-direction, the gradient of the local wave-number in the y-direction

is zero.


ak 3k
S= 0 = -- (3.7)
'y ax

Integrating the gradient of k in the x-direction gives


k sina = constant = k sina (3.8)
o o

where the subscript "o" refers to conditions in deep water. This is a

statement that, for straight and parallel contours, the projection of

the wave-number on the beach is a constant. Substituting into Equation

(3.5), a general expression for the celerity of the waves can be de-

rived where


S= k( -+ V sina + U cosa) (3.9)


or util]inng LquatL.ion (3.8)








c
0
c = ( _-- V) sina U cosa (3.10)
sina
o

For U = V = 0, this expression simplifies to Snell's law for wave re-

fraction.

The conservation of energy for the fluctuating motion, Equation

(2.23), can be expanded for steady-state conditions to give


U 1. 2 U 1 2
a (EU +Ec + SE + S (3.11
Dx x gx 2pD 2pD xy ax xx x (3.11)

where the gradients in the y-direction are zero. This equation states

that the change in energy flux, due to currents and waves plus the work

done by the excess momentum flux on the straining motion, is equal to

the energy dissipated by turbulence and work done on the bottom. The

product of a stress times rate of strain is a quantity that can be

associated with power per unit volume. The last two terms on the left

of the energy equation can be interpreted in this context where the

excess momentum flux tensor then represents a stress and the velocity-

shear a rate of strain. Longuet-Higgins and Stewart [53] were the first

to take cognizance of this, and named the excess momentum flux tensor

a radiation stress.

The excess momentum flux tensor can be determined by substituting

the wave expressions into Equation (2.17). In general terms of energy,

group velocity, and wave speed, an expression applicable to both inside

and outside the surf zone is given by


r c 2c c
E -- cos2a + (-s- 1) E -- sin2a
c 2 c 2 c .
S 1___-
ij pD
C -g sin2a E -& sin2e + E (2 1
Sc 2 c(3.12)






54

The effects of turbulence and surface tension have not been included.

Referring to Equations (3.4) and (3.12), it can be seen that U and

S are of order E. The first and last terms of Equation (3.11) in-
xx

volving the product of these terms are of order (E2) and, hence, will

be ignored in this analysis. From the result of the conservation of

mass, Equation (3.4), Ux = 0, so that the third and fourth terms of the

energy equation are zero. Substituting for Sxy and F and retaining

only terms of first order in energy, the energy equation reduces to


3 sin2a 3V
-(Ecn cosa) + En = E (3.13)
ax 2 3x


where the substitution, c = en, has been made. Substituting for the
g
wave celerity,as given by Equation (3.10), yields



En [(-0-- V) sina cosa U cos2a] + En cosa sina V -
Dx sine ;x

(3.14)

Again recalling that U is of order E, and that all higher order terms

involving the product of U can be neglected in this analysis, the

energy equation can be written


c
En (s---- V) sina cosa + En cosa sina = E (3.15)
3x sina ax


Expanding and cancelling terms, gives

c
o _
S - V) En sina cosa = E (3.16)
sina 3x


Far outside the surf zone, energy dissipation due to bottom effects

can be ignored, and the energy losses due to turbulence can be assumed

negligible. Since the term in the brackets of Equation (3.16) is






55

nonzero, the result of the energy equation outside the surf zone,

assuming no energy dissipation, is


En sin2a = constant (3.17)


The relative amplification of the wave energy is then given by


n sin2a
E-- o o (3.18)
o
E n sin2a (3.18)
0


Since the term involving U in the celerity equation resulted in the

product of higher order terms that are not included, Equation (3.10)

can be written, consistent with this analysis

c
c = (sin V) sina (3.19)
0

or in terms of the wave angle


c 1
sin =-- sina [- ] (3.20)
o 1 - sina
c O
o


Equations (3.16) and (3.18) give a complete description of the wave

amplitude and direction outside the surf zone.

For no current, Equation (3.10) reduces to Snell's law for wave

refraction and can be combined with Equation (3.18) to give

c
E c cos = E cosa = const (3.21)
g o2 o

where the local quantities are related to the conditions in deep water.

The changes in energy density E, or wave height, can be determined from

this equation as a function of the local wave-number and water depth.

For decreasing depth as the waves approach the shore, the local wave

length and the angle of incidence decrease. The effect of shoaling is








determined by the group velocity. The group velocity initially in-

creases slightly so the energy density decreases; the group velocity
-4
then decreases resulting in a continual increase in the energy density

towards shore. A maximum wave height occurs at breaking. Due to the

change in wave angle, which is the result of refraction, the wave crests

become more nearly parallel to the beach. The energy density is less

for waves approaching at an angle to a constant sloping beach than for

waves whose orthogonals are normal to the beach because wave refraction

results in divergence of wave energy.

The effect of a shear current on the energy density can be illus-

trated most simply by considering the case of waves in deep water where

the depth is noc involved as a parameter. The relative amplification of

the energy density as a function of V/c is shown graphically in Figure 6

for various values of the initial angle a It can be seen that for the

case of currents with a component in the direction of wave propagation

(positive ratio of V/c ) there is a decrease in energy density. For

angles greater than zero, there is a point at which the amplification

suddenly becomes very large and tends to infinity. As a -t 90 degrees,

the infinity occurs for all currents in the positive direction. These

infinities represent caustics, or crossings of the wave rays. In reality,

the theory is no longer valid near such points.

As has been shown, there is an excess of momentum flux due to the

presence of the waves. For conservation of momentum flux, there must be

a force exerted in the opposite direction such as a hydrostatic pressure

force or bottom shear stress to b:'lance this excess momentum flux. It

will be prCusentco that oul.si.d the smuf zole the component of excess

mormentum flux directed perpendicuiair to the contours is balanced by mean

waPtr levell set-down.




















1.5-


1.4-


1.3
20














.2 -
0.8 -










-1.0 -0.8 -O.G -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0

00




Figure 6. Amrnlification of Wave Energy Density Due to a
Shear Current as a Function of the Incident
Wave Angle






58

3. Distribution of Currents outside the Surf Zone

There is a component of excess momentum flux directed parallel to

the shore due to the oblique wave approach. The question of whether

this radiation stress can induce a current can be investigated by con-

sidering the general y-momentun equation. Neglecting changes in the

slope of the free water surface, Equation (2.15) can be written



T (Mx + S = R (3.22)
ax x xy y

where the gradients in the y-direction are zero. The conservation of

mass equation showed that Mx = 0; hence, the first term is zero. Com-

parison of the energy equation solution, Equation (3.17), and the excess

momentum flux tensor, Equation (3.12), shows that

c
S = E -_ sina cosa = constant (3.23)
xy c

Therefore, the change of momentum flux due to the waves and mean motion

in the y-direction is zero--there is no driving force for generating a

current outside the surf zone. The only wave-induced current far out-

side the surf zone is then due to the mass transport velocity.

It shall be assumed V = 0 since there is no driving force for

generating currents. The mean current in the y-direction then reduces

to the mass transport velocity

M
U E- sin (3.24)
y pD pcD


The nmas transport velocity generally decreases as the depth in-

creases so that the longshore currents in deep water are very weak. This

is in ag'rement w'ith obsc vantons outside the surf zone. However, very

near the surf zone, the nmacu transport velocity tends to becou e first






59

order in amplitude and is no longer negligible; but, at the same time,

bottom effects become important making the derivation inaccurate.

Thus, very near the surf zone, dissipation of energy must be considered.

This will be discussed in Section C.

4. Set-down of Mean Water Elevation outside the Surf Zone

It is interesting to investigate the balance of momentum perpen-

dicular to the beach. The x-momentum flux equation outside the surf

zone is given by


aS
Sxx- a
x= pg(h + n) (3.25)


where dissipative effects have been neglected. The excess momentum

tensor was shown in Equation (3.12) to be a function of the wave energy.

As waves shoal, the amplitude, and hence the energy, increases (except

over -a short distance in shallow water); thus, the excess momentum flux

tensor must also increase. The changes in the flux of excess momentum

is balanced by a slope of the mean water level. It can be seen from

Equation (3.25) that a positive gradient of the excess momentum flux

(increasing) results in a negative gradient of the mean water surface.

This results in an increasing set-down as the water becomes shallower.

Longuet-lliggins and Stewart [54] have solved Equation (3.25) for the

case of waves approaching perpendicular to an arbitrary plane bottom.

They found

H2 k
n 8 nh 2 (3.26)
8 sinh 2kb

where it is seen that the set-down is a function of the local conditions

only. Thus, for the case of waves approaching at an angle, if the curva-

ture of the wave rays is small, one would expect the same solution for






60

the local set-down. Mei (o_. cit.) solved the problem of waves shoaling

over a plane bottom by means of a perturbation expansion, taking into

account the sloping bottom boundary condition. He derived an expression

relating the local set-down to the deep water conditions


H 2 cosa
2 o o k coth kh
8 cosa (2kh + sinh 2kh (3.27)


This can be expressed by the local conditions using the solution to the

energy equation which relates deep water conditions to shallow water

conditions. Equation (3.27) then reduces to the same expression as

Equation (3.26) for the local set-down.

This set-down outside the surf zone has been verified experimentally

by Bowen et al. [55] in the laboratory; although close to breaking,the

experimental values for set-down were less than the predicted values. As

breaking is approached, and the wave height becomes larger, the second

order theory becomes less valid and would be expected to give less accurate

results. No energy dissipation has been accounted for, which can become

important in shallower water. The effect of including energy dissipation

would be to decrease the energy density as the waves shoal, which pre-

sumably would decrease the energy density gradient. This would result

in a decrease in the excess momentum gradient, decreasing the set-down.

This would help explain the discrepancy between theory and e:perim;nt.


C. Currents uutside the Surf Zone--Including Bottom Friction

The energy equation has been used to relate the changes in wave

height dun to changes in the wave direction, current velocity, and depth

at a prrtLJculal location. Having determined the wave height distribution,

the morarLtu and continuity equations were used to solve for currants







61

generated parallel to the shore. Dissipative effects were neglected in

the above derivation of the longshore currents, and the evaluation of

the energy equation was rather straightforward. Near the surf zone, in

shallower water, the bottom effects can no longer be neglected, and the

dissipation term must be included in the derivation. It will be assumed

that all the energy dissipation outside the surf zone is due entirely to

bottom friction and percolation. Turbulent and viscous energy dissi-

pation above the boundary layer is very small and will be neglected.

Energy utilized in transporting sediment, either along the bed or in

suspension, will be assumed to be included in the bottom friction loss

term and will not be considered separately. This aspect of the problem

will be considered in Chapter IV.

The lnss of mechanical energy is due to work done by the turbulent

shear stresses acting on the bottom. To evaluate the dissipation due to

bottom friction, it is first necessary to investigate the shear stresses

occurring at the bottom.

1. Bottom Shear Stress Due to Combined Waves and Currents

The determination of the bottom bed shear stress for uniform steady

flow has been fairly well established. For oscillatory flow, and particu-

larly combined waves and currents, the bed shear stress is not so well

formulated. This is due primarily to the lack of good empirical data.

Jonsson [56J compiled the available laboratory data and conducted ad-

ditional experiments dealing with turbulent boundary layers in oscilla-

tory flow. From this study, he developed a classification of the flow

regimes similar to that for steady flow. The classification is based on

a roughness parameter and characteristic Peynolds number. It is found

that the wave boundary layer in nature can always be considered in the

"hydrniauli cally rough" Uinrhulcnt regime.






62

Unlike the boundary layer in open channel flow, which essentially

extends over the entire depth of flow, the boundary layer under wave

motion constitutes only a very small fraction of the vertical velocity

distribution. This is because the boundary layer does not have an

opportunity to develop under the unsteady velocity field of the wave

action. Above the boundary layer, the free stream is well described

by the potential flow theory (at least for small waves).

Jonsson used an oscillating flow tunnel to simulate prototype wave

conditions in the laboratory. In the experiments, he was able to measure

the vertical velocity profile in the boundary layer and determine the

bottom shear stress. He found that for simple harmonic motion the velo-

city profile in the boundary layer could be approximated by a logarithmic

distribution and that the instantaneous bottom shear stress was related

to the velocity by the quadratic shear stress formula



hw fw 2 uj hl (3.28)




hw w 2 nih
;hw = fw tUth2 cos( t 8)Cos(U t 0) I (3.29)

where u-h is the velocity amplitude of the oscillating flow just above

the boundary layer, 6 is a phase lag, and f is a friction factor

associated with the wave motion.

He found, further, that the friction factor was practically constant

over an oscillation period. The c.nrtancy of the frictJon factor ior

particular flow conditions is an important result which allow for a

better analytical determination of the corbinrcd shear stIess dWu to w'aves

and currents. Userng the availahbl datt front several source s, he found









that the friction factor for wave motion alone for rough turbulent

boundary layers could tentatively be represented by


1 1 h
+ log = 0.08 log -- (3.30)
4Wr
w w


where r is a measure of roughness, and 4h is the maximum water particle

excursion amplitude of the fluid motion at the bottom as predicted by

linear wave theory

H 1
C H 1 (3.31)
'h 2 sjnh kh (3.31)


Equation (3.30) is based on the roughness parameter r being a

measure of the ripple height. The wave friction factor is seen to be

a function of the wave characteristics. This is because, for granular

beds consisting of a particular grain size, the ripples adjust their

dimensions according to the wave motion, and it is the ripple geometry

that determines the effective roughness.

Unfortunately, there is only, at best, a qualitative understanding

of the ripple geometry as related to sand wave characteristics. Generally,

the ripples are much more symmetrical in shape and much longer crested as

compared to those found in alluvial channels. Inman [57] collected a

large number of observations of ripple geometry and wave conditions from

Southern California beaches. These observations extended from a depth of

170 feet to the shore. Since the wave and sand characteristics vary from

deep water to the beach, the ripple geometry would be expected to vary

also. These observations showed that the size of the sand is the most

important factor in determining the geometry of the ripple. In general,

the coarser the sand, the la-ger the ripples. Also, there was a general






64

correspondence of decreasing ripple height with decreasing water depth.

The ripples were smallest in the surf zone where the higher orbital

velocities of the waves tended to plane the ripples off; the ripples

were almost nonexistent for surf zones with fine sands. The ripple

wave length was related to the orbital excursion. As the bottom orbital

velocity of the shoaling waves increased, the ripple length decreased,

increasing the effective roughness, but, at the same time, the ripple

height is decreased, decreasing the effective roughness. Since quanti-

tative relationships among ripple geometry, sand grain size, and wave

characteristics have not been determined, one is forced to rely on ob-

servations to determine the effective roughness, and, for this reason,

Inman's data are especially relevant.

It is now desired to superpose a mean current on the wave notion

and determine the total shear stress due to wavas and currents combined.

The difficulty in using the quadratic shear stress formula is in stipu-

lating the friction factors. The wave friction factor was seen to be a

function of the wave properties for a deformable bed, that is, the fluid

motion; whereas, the friction factor for steady currents for rough turbu-

lent boundary layers is only a function of the system geometry. It would

seem reasonable to expect that for weak currents, as compared to the water

particle motion of the waves, that the wave dynamics would dominate the

hydrodynamical system. For this reason, it is desirable to derive the

combined bottom shear stress in terms of the wave friction factor alone,

even though it is less well defined than the friction factor for steady

cul rents.

I: is assumed that the Loral instancan'oius bed shlar stress for

combined wives aid currents is re]aled to the velocity by








Th = p 2 V V (3.32)


where v is the resultant instantaneous velocity vector of the combined

wave and current motion, and f is the friction factor. Since the problem

has been formulated as a combination of wave and current motion, it

proves convenient to resolve the shear stresses into components in the

direction of the wave and current components. To do so, the velocity

is first resolved into components in the direction of the current and

wave motion

)(3.33)
|vl= (u + V2 + 2u V sin) (333
w w

where u is the instantaneous velocity of the wave motion measured just
w
above the frictional boundary layer near the bottom. V is the mean

motion which was assumed uniform over the depth and in the longshore

direction only. Resolving the component shear stresses in the direction

of the velocity vectors results in the shear stress and velocity vectors

being proportional to each other as can be seen from the geometrical

representation in Figure 7. The shear stress component for the wave

motion can then be written
->-
- -> w f i- i
I T= P -IV U (3.34)
hw h 2 wU
v

and, similarly, for the shear stress in the direction of the mean current


y = p P J V (3.35)
v

If friction factors for the wave and current motion are now defined,


f = f -I (3.36)
w wl







































\2 \


S+1....











Figure 7. Resolution of Currents and Waves into
Respective Velocity and Shear Stress
Components








f = f (3.37)
y

the shear stress components can then be written

f
hw = u (3.38)
hw 2 w| w

f
S = p V2 (3.39)
hy 2

The shear stress for waves, as defined in Equation (3.38), is in the

same form as given by Jonsson (op. cit.) for which information of the

4 friction factor was found experimentally. Knowing f and the velocity

of the waves and currents, the other friction factors and, hence, shear

stresses can be determined. The total shear stress, in terms of the

shear stress components, is


h= ( w2 + by2 + 2Thw hy sina)2 (3.40)
l h' hw bhy hw hy

Combining Equations (3.32) and (3.36), the total shear stress can also

be written

Sf
Th= p vU (3.41)


or substituting Equation (3.33)

f -2
S- U (uw2 -I V2 + 2u V sina) (3.42)

Since only the steady-state conditions are being considered, it

is desired to find the time nean values. It will be assumed a prior

that near and inside the surf zone the mean currents are much less than
-v
the maximum velocity of the wave motion. Only the case of just outside

the surf zone is considered here, hut it will be shown later that the






68

equations developed are valid for inside the surf zone as well. The

time mean value of the absolute value of the water particle velocity at

the bottom due to the waves is found by integrating over time Equation

(2.29) to give

2 H gk (3.43)
uwl n 2 o cosh kh

Assuming relatively small incident wave angles a, and that V << lu,

as a first approximation


v| = I (3.44)

The total bed shear stress is then given by

- w-i-*; (3.45)
Th = p -2- .u (3.45)
b 2P wjw

This result will be used later in calculating frictional energy dissi-

pation at the bottom.

2. Changes in Wave Height outside the Surf Zone Due to Bottom
Friction and Percolation

The energy equation, Equation (3.11), can now be written to include

the dissipation term. Some of the higher order terms previously neglected

become first order in energy in shallow water and might be included.

Collectively, the contribution of these higher ordc.r terms is only approxi-

mately 4 per cent at the breaker line and, considering the. inaccuracies

in calculating the energy dissipation terms, can be neglected. This leads

to a simplified energy equation



(Ecn coha) = (c + e ) = e (3.4b)


where Ff is the mlean rate of energy di!;sipnt.on doe to bottom fiction,









and eq represents the mean rate energy dissipation due to percolation.

It is assumed that the instantaneous rate at which energy is dissi-

pated per unit bed area due to bottom friction is given by

4
Ef = 'h'vh (3.47)


The dissipation function is to be substituted into the energy equation
-I
concerned only with the fluctuating motion, so that v = u + The
w pD
second velocity term being of second order is much less than the mean

of the absolute value of the velocity of wave motion and will be

neglected. The bottom shear stress is given by Equation (3.45). Jonsson

(oo. cit.) evaluated the energy losses experimentally using the dissi-

pation function as given above. He found that the phase shift in the

shear stress term could be ignored, and the instantaneous dissipation

term evaluated in terms of the wave particle motion at the bottom.

f
f = p I u mh3 s3c wt (3.48)


The energy friction factor f would not necessarily be expected to be

equal to the wave shear stress friction factor f since a phase shift

was introduced in the shear stress term. Indeed, it was found experi-

mentally by Jonsson that they were not equal for laminar flow. However,

for rough turbulent boundary layers, f and f were, for all practical

purposes, the same and constant over the wave period for the experiments

conducted. Thus, the energy friction factor can also be given by Equation

(3.30).

Since the frictional energy dissipation is defined in terms of the

absolute value of the bottio velocity, and theL velocity-shear stress

product is a symmetrical odd-function, the time mean value is equal to






70

the average of Equation (3.48) over one-half the wave cycle. The mean

frictional energy dissipation is then given by


Pfw oH 3 3
Cf =12s (sinh kh) (3

This is the same dissipation term found earlier by Putnam and Johnson

[58] who made similar assumptions in their derivation.

Putnam [59] also theoretically examined the energy losses due to

percolation. If the bed consists of permeable material, pressure

variations due to the wave motion induce currents in the permeable -

layer, and these currents will dissipate some of the mechanical energy

of the wave.

For a permeable bed whose depth is greater than approximately one-

third the wave length L, the amount of mean energy dissipated,egby

viscous forces in the permeable bed per unit area of the bottom per unit

time is given by


c 2 pk112 ( )
I 2v (cosh kh)2 (3.50)

where jis the permeability coefficient of Darcy's law, and v is the

kinematic viscosity of the water. If the depth of the permeable bed is

less than 0.3 L, a more complicated expression must be used so that it

will be assumed here that the bed layer is always greater than this

depth.

It is necessary to evaluate Equation (3.46) numerically due to the

nonlinear dissipation terms. This can be written in terms of the wave

height in a difference form


I = H K K [1 8 n Ax]2 (3.51)
n n-1 s r p. (112c cosca)> .
g









where


K (C ) n (3.52)
s (c )n
gn


is the shoaling coefficient reflecting the changes in the group velocity,

and

cosa
K = ( n-1)- (3.53)
r cosa
n


is the refraction coefficient for a constant sloping bottom. The term

in the brackets of Equation (3.51) includes the effect of energy dissi-

pation. This equation is equivalent to Equation (3.18) for no dissi-

pation, E = 0. The wave height can then be evaluated by starting at a

known value and proceeding to any desired position where the changes

in the wave height compared to the initial wave height are found by

integrating the dissipation term.

Savage [60] conducted laboratory studies to test the above equations

and found reasonable agreement for the changes in wave height due to

frictional dissipation. He further found that the energy losses by perco-

lation were less than the theoretical values by a factor of between 4 and

10. The tests indicated that for sand sizes less than 0.5 millimeters,

there is practically no loss due to percolation. On the other hand, for

sand grain sizes larger than 2 millimeters, the percolation losses could

become very large.

Bretschneider [61] tested the above equations with field data from

the Gulf of Mexico. Percolation losses were not considered in his analy-

sis since the bottom consisted of relatively impermeable fine sediments.

Comparison of field measurements showed that using friction factors that







72

were close to those predicted by Equation (3.30) gave results in reason-

able agreement with the theory. Thus, it is assumed that the use of

Equation (3.51) for predicting the wave height at a particular position

is valid and will be used in this analysis.

3. Distribution of Currents

Having determined the wave height as a function of the incident wave

angle, wave period, and local depth, the wave-induced currents parallel

to the shoreline can be determined from the y-momentum equation


as
xy = L 1 (
x = ( pgH n cosa sina) = hy (354)


This equation states that the change of the y-component of momentum in

the x-direction is balanced by the bottom shear stress directed in the

y-direction. Only the shear stress acting on the bottom is taken into

account. Since the wave height could not be expressed in an explicit

form, it becomes necessary to solve for the changes in the excess

momentum flux numerically. The longshore velocity is then found from

the calculated bottom shear stress as related by the quadratic shear

stress formula

V2
S= pf 2 (3.55)
hy y 2

where the time nean value of f can be determined from Equations (3.36),
y
(3.37) and (3.43)


"h fw 1 fw H (3.56)
fy V w V o cosh kh (3.


and f is determined by Equation (3.30). Therefore, the velocity outside

the surf zone, taking bottom friction into account, is




73


V o cosh kh 3 (H2n sin2a) (3.57)
8 f kH ax
w


This gives the velocity in terms of measurable quantities.


D. Longshore Currents inside the Surf Zone

It has been shown that longshore currents could be induced by

changes in the momentum flux of the waves directed parallel to the

shore outside the surf zone. The wave height inside the surf zone

and, hence, the water particle motion of the waves decreases due to

intense energy dissipation. It might be expected that, due to the more

rapid changes in momentum flux inside the surf zone, faster currents

can be generated.

A simplified approach will be investigated first, and, then,

additional refinements will be included to illustrate various aspects

of the problem and to obtain more accurate results. A uniformly

sloping bottom is initially assumed so that the physical principles in-

volved are not lost in the algebraic details. A more general profile

is then considered later. Also, the first solution considers the re-

sisting force parallel to shore to be only due to shear on the bottom,

and internal shear stresses are neglected.

Again, it is first necessary to specify the energy distribution

across the surf zone in order to determine the distribution of momentum

flux. The energy, as related to the wave height, was shown to be a

function of the local depth. Outside the surf zone, it was shown that,

due to the excess momentum flux directed perpendicular to shore, there

was a set-down of the mean water surface. The changes in the mean water

surface elevation inside the surf zone are investigated next.





74

1. Wave Set-up inside the Surf Zone

Both the wave height and speed are a function of the total local

depth of water. Since there is an unknown change in the mean water

level to balance the excess momentum flux of the waves, the total

depth of water is unknown and must be derived to specify these quanti-

ties.

Conservation of momentum flux in the x-direction states



x(U xx + S ) = Tx (3.58)
ax xx xx x

where there are no gradients in the y-direction. It was explicitly

assumed that the net mass flux perpendicular to the beach was zero so

that there is no contribution from the mean motion to the momentum

flux perpendicular to shore. This is necessary to comply with the

steady-state assumption. If a net mass transport were allowed into

the surf zone and there were no rip currents or other offshore return,

the currents would grow unbounded. Dissipative effects, such as bottom

friction and turbulence, have not been explicitly stated but are

accounted for in the potential energy decay. Equation (3.58) reduces to



S = T = pgD (3.59)
ax xx x ax

which states that the change of excess momentum flux due to wave action

is balanced by a change in the mean water level.

It is assumed that the excess momentum flux tensor inside the surf

zone can be expressed in terms of the energy and wave speed in the same

form as in shallow water. This assumption implies that even under the

breaking waves, water particle motion retains much of its organized

character as described by linear wave theory.






75

Experiments conducted in the field by Miller and Zeigler (op. cit.)

show that this assumption is not as extreme as one might first expect.

Velocity meters were used in these experiments to measure the internal

velocity fields in breaking waves. The results showed that the internal

velocity field of the near breaking and breaking waves, corresponding to

spilling-type breakers, could be qualitatively described by Stokes'

third order wave theory. The tests also showed that different types of

breakers had different internal velocity fields as would be expected,

and that the spilling-type breaker was the most organized.

From Equation (3.12)


S = E(cos2a + ) cos2a (3.60)
xx 2 pc2D


where S is a function of the energy density which decreases inside the
xx

surf zone. Hence, both the momentum flux and the balancing force have a

negative gradient so that the change in the mean water level inside the

surf zone will be positive, representing a wave set-up.

The phase relation, as given by Equation (3.20), relates the velo-

city and wave speed to the incident angle relative to the conditions at

the breaker line, and can be written

(Vb V)
sin = sinb [1 + ----- sinab] (3.61)
cb Cb


In this investigation, waves having a range of periods of approximately

4-18 seconds are considered. Waves of these periods are greatly re-

fracted before reaching the depth at which they break. Thus, the angle

of wave incidence at the edge of the surf zone is generally small--

usually much les;n than 30 d-egrees. Assuming the maximum value of the






76

longshore current velocity would be approximately equal to the component

of the wave speed in the longshore direction, then


IVb V < V < cb sinab (3.62)

so that


Ivb vl (cb sinab)
cVb sinab < cb i sinab << 1 (3.63)
cb b c b


Thus, an approximation for Equation (3.61), using the binomial expansion,

can be conveniently employed.

(V V)
sina sin [1 - sino + ...] (3.64)
bb Cb

The cos2a term is given by

2 (\(V V)
cos2 = 1 - sin2ab [1 2 b sinab + ...] (3.65)
Cb b cb b

or as first approximations


since = c sin (3.66)



cosa = 1 ( -) sin2 (3.67)
b

Putnam et al. [16] used the same assumption as in Equation (3.63)

and verified it in the laboratory and field. le found that the long-

shore current never exceeded the longshore component of the wave

celerity.

Galvin [24] makes a special point to take exception with this

assumption, pod, thus, it is necessary to justify its use hore. In

his laboraLory experiments, he found that Lhe moan longshore velocity









often exceeded the component of the wave speed in the longshore di-

rection. This was also found in a similar laboratory study by Brebner

and Kanphuis (og. cit.). Galvin points out that this is because much

of the water composing the breaking waves has been extracted from the

surf zone and that this water already has a longshore velocity which

is added to the breaking wave component.

This seeming discrepancy in experimental results is clarified by

examining the kinematics of breaking waves. The waves broke by plunging

in Galvin's experiments; whereas, in the experiments of Putnam et al.,

it is implied the waves broke by spilling. Iversen [62] conducted labo-

ratory studies of breaking waves on various beach slopes. He found that

the backwash velocity in spilling breakers was much less than in plunging

breakers so that the fluid contribution to the spilling breaker is rela-

tively less. Also, surf zones with spilling breakers are much wider with

the backwash having much less effect. Galvin's argument does not apply

to spilling breakers on flat beaches which derive little fluid from the

surf zone at incipient breaking. Thus, the assumption given by Equation

(3.63) appears to be valid for spilling waves.

Changes in the total water depth can now be solved by substituting

Equations (2.39), (2.41), (3.60) and (3.67) into (3.59), resulting in


- 1 2 i_) 1 3
SpgK2D2 [( 8( --) (1 sin ) + = ogD (D h)

(3.68)


Expanding and combining terms, this eq oation can be easily inte-

grated to give

2 22 sin2:
[3 -- (1 - .) +1 D U2 i-i bD- = h + constant
[3 (1 Tb
12(1 + ) -S 1 (+ )9) 2D
(3.69)






78

Imposing the boundary condition that Db = hb + b at the edge of the

surf zone, and using the equivalent approximate form of Equation (3.66),


Db
sin2 = sin2ab b (3.70)


The total depth variation is


D = ( K() Db + (1 K(a)) (h hb) (3.71)
(1 K(ab))

where K(a) is a function of the local wave angle such that


K(1)=1 1 (3.72)

12(1 + <) (1 + <) 2


This form corresponds to an earlier formulation by Longuet-Higgins and

Stewart [54]. It should be noted that this equation is applicable to

an arbitrary bottom profile and that the wave set-up depends only on

the local still water depth. The only restriction on the bottom profile

is that it monotonically decreases shoreward in order that the assumption

of spilling breakers be fulfilled across the surf zone, that is, that the

wave height must always be a function of the local depth.

The effect of oblique wave incidence on the set-up can be investi-

gated most simply by assuming a constant sloping beach. Figure 8 shows

the relative changes in the local wave set-up for various angles of wave
-Y
incidence. It can be seen that the wave set-up decreases for increasing

angles, as would be expected.

For all practical angles of wave incidence, as shown in Figure 8,

the change in the total local depth is less than 2 per cent. The set-up

is ipiportant in this analysis for calculations of the wave height. Re-

calling that the wave height i; assumed proportional to the total depth,






























0.05




0.03




0.03


D
0.02




0.01


201'-

. 10 ------


-1.0 -0.8 --0.G -0. 0.2 0.0




Figure E. Relative Changes in Wave Set-up insiJe the Surf Zone
as a Function of the Incident Wave Angle


_~ _~_ __ ~







80

an error of less than 2 per cent in the wave height calculations is

introduced by neglecting the angle of wave incidence. This approximation

is certainly within the accuracy of the basic assumption that the wave

height is proportional to the depth. Hence, for purposes of this

analysis, the effect of oblique wave incidence on the wave set-up in-

side the surf zone will be neglected to simplify calculations. The

equation for wave set-up then reduces to the case of waves normal to

the beach. This is essentially the same solution obtained by Longuet-

Higgins and Stewart ibidd.) where they considered the case of waves

normal to a beach only. The difference in the solution given here is

due to the inclusion of the momentum flux of the mean wave motion in

the radiation stress term.

The depth inside the surf zone simplifies to


D = (1 K) h + Khb + b (3.73)


and the wave set-up inside ihe surf zone is given by


n = K(a) (h. h) + 0b (3.74)


where the set-down at the breaker line nb can be determined from

Equation (3.26).

Bowen et al. [55] conducted laboratory studies to verify the

theory predicting the change. in mnan sea level for waves normally

incident to the shoreline. In his experiments, a plane beach was used

with a slope of 1:]2. The. brciain", Iv'vcs wc:-e of the plunging type,

and, after breaking, the residual energy propagated shoreward in the

torm of a bore.






81

Well outside the breaking point, the prediction of the set-down

by Equation (3.26) and the measured values compared very well. Figure 9

shows a typical observation from Bowen's experiments of the wave height

envelope and mean water level.

Just outside the breaking point, the assumptions of linear theory

are no longer justified, and the measured values of the set-down are

less than predicted by the theory. Between where the waves start to

break and the plunging point, the set-down was found experimentally to be

rather constant. The measurements showed that inside the plunging point

the wave height is very nearly proportional to the mean water depth.

This supports the basic assumption of this analysis that H = KD. As

Bowen points out, it is surprising that the set-up is so well described

by theory using the approximation for the wave momentum flux since linear

theory is essentially assumed valid inside the surf zone.

2. Velocity Distribution--Constant Sloping Bottom

With the mean water profile and energy distribution specified, the

variation of the longshore current across the surf zone can be deter-

mined. The y-momentum equation inside the surf zone can be written

similarly to that outside the surf zone.


aS
---= R (3.75)
@x y


Again, recall that the net mass transport perpendicular to the beach is

zero, and that there is no y-dependence. This equation says that the

change in the momentum flux parallel to the shore due to the waves is

balanced by the force required to overcome bottom friction and internal

lateral shear stresses which must maintain a longshore current. A loss










WAVE PERIOD 1.14 sees.

WAVE HEIGHT H,06.45 cms. Hb
BEACH SLOPE Tan = 0.082


8.55 ciis.


MEAN WATER LEVEL, 7)




THEORY

S .W.L.


EXPERT II!ENT







ENVELOPE OF WAVE HEIGHT


WAVE CREST
.nsta.tW'arg.rar'


r .
400 300 200
DISTANCE FRO' STILL IATER!


100 0

LINE O01 PEAhl. x (ims).


Figure- 9. Larboratory :leasure%,..nts 0o !1V'..-c Set-dow,. and Cet-un
(after Bower. [55])


BEACH -





0

0
o

o




o


POINT
PLUNGE
POINT


-2.0



-1.5

c rms.
-1.0



-0.5


L0.










4


2

Z ces.
---- -


. m I I XK KI( X I


-- ----- -p0






83

of momentum due to potential energy decay across the zone is also implied.

It is assumed that the changes in momentum flux are balanced solely

by the bottom shear stress parallel to shore, R = Thy Substituting

for S from Equation (3.12) gives
xy

a E2
x [E sina cosa 7- coso sinal = Thy (3.76)
ax pc D hy (3.76)

The height, celerity, refracted angle of the waves, and, hence,

changes in momentum flux inside the surf zone can be expressed in terms

of the local depth of water. Substituting Equations (2.38), (2.41),

(3.66), and (3.67) into Equation (3.76) gives


3 1 2 1 <2 sin.b 5/2 D sin2 b
x[8 pg <2 (1- D-1)+] (1
ix 8 8 U( + <) Db 2 hy

(3.77)

which can be written

3/2 D aD
AD3/2( 0.7 sin2b) n= hy (3.78)
Db b x hy

where

S2 sinab
A pg .2 (I
16 1 + )

This is a general equation expressing the changes in momentum flu:: across

the surf zone in terms of the total local depth of water. Included in

the formulation is the set-up of water and the effects of wave refraction

inside the surf zone. This equation is subject to the restriction that

the waves be described as spilling breakers, and, hence, the depth con-

tinuously decreases shoreward from the breaker line. Thus, the momentum

flux decreases monotonically inside the sulf zone since both the energy

and wEve angle decrease with doerea; ;ing dep.h.




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