Littoral drift and the prediction of shoreline changes

I r.,
-i,~ p tifljpn

Florida Sea Grant

Technical Paper-

LITTORAL DRIFT AND THE PREDICTION OF SHORELINE CHANGES

Richard L. Pfeffer
Principal Investigator

The Geophysical Fluid Dynamics Institute
Florida State University
TPllahassee, Florida

The information contained in this paper was developed Liji-r the
auspices of the Florida Sea Grant College Program, with support from
the liOAA'Office of Sea Grant, U. S. Department of Commerce, :grant
number n4-6-158-44 This document- is a Technical Paper of the'State
University System .of Florida Sea Grant College Program, 2001 McCarty
Hall, University of Florida, Gainesville, FL 32611. Technical Papers"
are duplicated in limited quantities for specialized audiences requiring
rapid-access to information, which may be unedited.

Deceriber 1976

PREFACE

This report consists of two parts. Part I contains

5 technical papersk.by Barcilon, Lau, Miller, Tam and Travis,

all of which are published in scientific journals with high

standards of review. The first four of these cover results

of our early work under the-Sea Grant program. These papers

provide insight into the mechanisms of formation of transverse

sand bars, submarine longshore bars and rip currents. The

fifth paper is a thorough review paper in which theoretical and

experimental modelling of physical processes pertaining to the

near shore are discussed and compared with field observations.

Part II, by.Christopher Miller, presents the final computer model

for predicting changes in the plan shape of shorelines due to

the littoral drift component. This part of the report includes

a discussion of how the sediment transport rate is related empiri-

cally to the water flow; the range of incident wave angles for

which the governing equations are stable, the finite difference

scheme, as well as a listing of the computer program. It also

includes a test of the model on specific coastal sites in Florida.

PART I: The study of longshore bars has led to the con-

viction that the dredging of certain submarine longshore :bar

systems may actually lead to severe erosion of the shoreline.

A further suggco-ion has, been made concerning the possibility

of reversing erosion trends in some beach locations by properly

*Editorialnote: The five papers are included by reference only, ds they
appear in the literature already.

References

Barcilon, A., aid J. P. Lau. 1973. A Model for Formation of Transverse
Bars. J. Geophys. Res. 78(15):2656-2664.

Lau, J. P., and B. Travis. 1973. Slowly-varying Stokes Waves and
Submarine Longshore Bars. J. Geophys. Res. 78(21):4489-4497.

Lau, -J. P., and A. Barcilon. 1973. "Harmonic Generation of Shallow.
Water Waves over Topography. J. Phys. Ocean. 2(4):405-410.

Miller, C., and A. Barcilon. 1976. The Dynamics of the Littoral
Zone. Rev. Geophys. Space Phys. 14(1):81-91.

Tam, C. K. W. 1973. The Dynamics of Rip Currents. J. Geophys. Res.
78(12):1937-1943.

contouringl the bottom topography to conform with wave-reso-

nant equilibrium patterns determined by the theory. It remains,

however, to determine the logistics and economics of such a

venture. Specific abstracts of each of the 5 individual papers

in this part of the final report are as follows:

The paper by Lau and Barcilon (1972) investigates the
reflection and non-linear interaction between the first and
second harmonics of a two-dimensional BousSinesq wave train.
Effects of topography are included, with the depth departing
from a constant in a finite region. It is'found that topogra-
phy can speed up or reward energy transfer between the first
and second harmonics. The reflection coefficient is signifi-
cantly different from the one obtained by using linear theory.

In the paper by Barcilon and Lau (1973).an extension
of Kennedy's potential model is used to investigate the forma-
tion of sand bars normal to a gently sloping beach. The results
show that the spacing between the transverse bars depends upon
the inverse of the beach slope and upon th& square of the drift
velocities across the bars.- In spite of certain drawbacks the
theoretical predictions compare well with several observational
studies.

The paper by Lau and Travis (1973) investigates the mass
transport velocity in the Stokes boundary layer due to slowly
var :. Sto 'av.-s 1ipingrl ingr on and reflecting from a plane-

tion is interpreted to indicate the possible locations of sub-
marine longshore sand-bar formation. It is found that the num-
ber of bars is likely to increase when the;bottom gradient is
slight and that the spacing between the crests of the bars in-
creases seaward. for some distance offshore. These results are in
qualitative agreement with field observations.

The.paper by Tam (1973) investigates the dynamics of -
rip currents using shallow water equations with a horizontal
eddy viscosity term. In this paper similarity solutions of the
model equations are found which appear to..give reasonable repre-
sentations of the velocity profile and other characteristics of
rip currents.

The comprehensive reviauwpaper by. Miller and Barcilon
(1976) covers the present knowledge concerning the dynamics of
rip currents, longshorecurrents and computer modelZling of
beach deformation due. to wave-induced erosion and accretion.

PART II: As mentioned above, this part of the Final

Report (by Miller) contains'the final predictive computer model

including a listing of the computer.program and.a.test of the

model on specific coastal sites in Florida. .The numerical model

is.based upon recent developments in the theory of longshore

currents and Lagrangian description of shoreline deformation.

The computer program requires as input data the breaker character-

istics (i.e., height, angle and duration for each wave consider-,

ed) computed from raw data. The choice of sites for testing.the

predictive characteristics of this program:was dependent on the

availability of wave and bathymetric data. During 1975 deep

water ship wave data for the Gulf of Mexico were analyzed for

long-term (100 years) study of St. George Island. In addition,

a beach nourishment project at Jupiter Island'provided the.

opportunity for a short-term. (8 months to 5 years) study. The

field observations on .wve climate .and transverse profiles

gathered by our collaborators at the University of 11crida were

provided to our group for analysis during the course of the year.

Observed changes in the St. George and Jupiter Island plan

profiles were compared with test predictions of the computer

model and are presented in this part of the Final Report. More-

over, the computer prediction was .extended.beyond the present time

for these beaches by 20 years and 5.years, respectively, and will

require further monitoring of the beach.morphology to verify the

future predictions.

One result of the study was the definition of a

reasonable range for the empirical coefficient linking the

sediment and water motions. It was found for St. George

Island that, over the time period of interest, the longshore

mode of sand transport dominated and therefore, good predic-

tions could be made if the nearshore wave field were known well.

For Jupiter Island, on the short term, the onshore-offshore

component of sand movement predominated, thus making it possi-

ble to model only general.trends due to the re-working of the

strandline by the longshore drift. Special attention is affor-

ded the endpoint condition in each case.

The major conclusion of this study is that the present

numerical-model is a viable predictor of shoreline movement

if (1) the predominant direction of sand transport is longshore.

(2) the nearshore wave climate can be adequately resolved (3)

SSpo in boundaries are t-eated in a physically -ealistic t:;r

The Numerical Prediction of Shoreline Changes Due

to Wave-Induced Longshore Sediment Transport

Christopher D. Miller

Geophysical Fluid Dynamics Institute

Florida State University

Tallahassee, Florida

1975

vii

Acknowledgements

The author would like to extend his, appreciation to the

following people:

Dr. Charles Quon of the Bedford Institute of Oceanography,

Dr. Paul Schwarztrauber of NCAR for helpful discussions; Todd L.

Walton of the University of Florida for the use of his wave

modification program; Ziya Ceylanli of the University of Florida

for valuable advice and assistance on the Jupiter Island project;

Dr. Richard Pfe.ffer for making available the facilities and

resources of the Geophysical Fluid Dynamics Institute, Florida

State University.

viii

Abstract

We have attempted to quantify numerically the changes

which occur in ..the plan shape of beaches due to.wave-induced

longshore sand transport. The approach of this study has been

to draw upon recent developments in the theory of longshore

currents, -beach .deformation, and sediment transport to -synthe-

size a numerical model which can be calibrated in accordance

wifh field observation and laboratory studies and, subsequently,

used to make predictions of.shifts in a shoreline, given cer-

tain bathymetric and wave data as input. The work of Longuet-

Higgins (1970a,b) on longshore currents and that of LeBlond

(1972) on shoreline evolution constitute the fram:ewor: within

which we build our numerical model. The model is applied to

two Florida coastal regions, the Apalachicola Bay region in the

Panhandle and Jupiter Island on the southeast coast.

Table of Contents

ckno ledgeents . . .

Abstract. . . . . . .

SectLion

1. introduction . . . .. .

II. The Theory of Longshore Currents and the

Equations Describing Strandline Movement ...

III. Discussion of How the Volume Transport Rate

of Sediment is Related Empirically to the

Volume Transport Rate of Water . . .

IV. Linear Stability Analysis of Governing Equations

V. Finite Difference Form of Governing Equations and

and Numerical Scheme for their Integration .-.

VI. Structure and Listing of Computer Program. .

VII. Application of Model to St. George Island .

... r..t,. ation of Modei to Jupiter TIsland '

'ccr ; ; 3 S es o .oa .....

Page
S. -xiii

. ix

* 1.

.* 5.

* 19.

. .. 24.

* 30.

* 35.

* 57.

* 6,6.

. T .

108.

I. Introduction

A beach-face can only achieve a state of quasi-

equilibrium. Acted on by varying.wave climate, wave-induced

.circulations, tidal currents, wind-generated currents, etc.,

it undergoes modification on both short and .long time scales.

Movements of sediment are induced in both the onshore-offshore

direction and the longshore direction. There is much evidence

that the sediment motion normal to shore is cyclical in nature

(e.g., the classic winter-summer variations in the transverse

profile) and that over the course of a year the net loss or

gain of sand to the beach system in this direction approximates

zero. Exceptions to this tenuous rule occur when the sand moved

offshore is made unavailable for eventual transport shoreward,

for instance, when a nearshore canyon acts as a sink for the

sand flow (as.in Southern California) or when storm waves remove

the sand to such a depth-that the 'summer' accretivee) waves

c~..n..-. ef.c- *- shor-ward --iration of the resul.t.ng semi--

perman.ent offshore sand bars (e.g., off the west coast of

Florida), etc. In this study we.are concerned with time scales

of, at least, 1 year and longer. We assume implicitly that over

the period of a year there is no net displacement of the shore-

line due to the onshore-offshore shifting of sand. If this were

not the case then we would have to establish criteria, both

theoretical and empirical, governing the movement of sand normal

to the beach; examples of such approaches are provided in

se-ction III.

There are several agents which can be responsible for

the introduction of s-ind into or the removal of sand from a

beach system. A river can discharge enormous quantities of

sediment into the coastal zone replenishing the beaches con-

tinuously. A tidal inlet with its delicate balance between

currents and sand transport can act as an impasse to the long-

shore 'river of sand' flowing by its mouth, trapping a substantial

amount in shoals both outside and inside of the inlet channel.

Violent storms (e.g., hurricanes) with their associated surge

and high waves can carry beach sand landward ('washover') or

far seaward making it inaccessible to the normal accretive pro-

cesses. The refraction and diffraction of waves around barriers,

man-made (jetties, groins) and natural (tips of islands and

spits), conveys sand into quiescent 'shadow' regions.where it is

sheltered from wave attack. In other words, in order to model

c.rre.tl. the ch.g in C coastal ar'C.., one. must. be famr iliar

;,iL.-;" the h:yr.a: yT i .L: .d marine g-omorphology peculiar to rhac.

locale.

Confinin ou attention! to t Ihe longhore drift of sand

which we view in this study to be the principal means by which the

particulate matter at the coast is redistributed we seek to

apply recent developments in the theory of longshore currents

(Longuet-Higgins, 1970a,b) and shoreline deformation (LeBlond,

1972) to the practical problem of predicting the change in the

shape of a shoreline over a period of time. The procedure.entails

choosing a coastal site for which adequate data on wave climate

and shoreline development exist. With wave and bathymetric data

3

serving as inputs the distribution of breaker wave characteristics

along the shore is computed; using this information a forcing,

function for the longshore flux of.water and sand can be derived.

The longshore divergence of these flows leads to local accumu-

lations or deficits of sediment and the subsequent movement in

time of the strandline can be monitored. A predictive computer

model is developed based on these simple concepts and is applied

to St George Island, a barrier island fronting the Apalachicola

Bay, Florida, and Jupiter Island on the south-east coast of

Florida. St. George Island is presently undergoing developmental

pressures. Jupiter Island, plagued by erosion problems, is the

site of a recent beach fill project; the question naturally arises

as to whether or not the local beach system can retain this

artificially deposited sand. Our approach to each site differs

because of the nature of the inputs (source and analysis of wave

data, bathymetry), the scale of the motions, and the time period

o intc-r.ast. We elaborate on these points in-sections VII and

VIII.

Previous numerical studies on the molding of a coastline

composed of loose material by wave-induced.forces include Price, et al.

(1972), Komar (1973), and LeBlond (1972). Price, et al. and Komar

formulated one-dimensional Lagrangian descriptions for beach

change, i.e., the translation of the coordinate points which

define the shoreline was restricted totbo-and-fro motion along

a line parallel to one of the fixed coordinate axes. Figure 1

from Komar indicates how the shoreline is represented .and its

movement normal to itself. In LeBlond's model the beach points

.t I

*( e.c di i-e of 71r

;-----i

.

{I

' Y

- L L ~ W ~

L 9,

Discretizat ion of shoreline according to KoImar (1973)

Fig-re 1.-

5

are free to move in the entire horizontal-plane (a two-dimensional

Lagrangian formulation); this allows.for a more accurate tCra;cking

of the evolving beach shape as well as a truer modeling of end

point boundary conditions (see Figure 4). Among the disad-

vantages are the possibility .of very irregular spacing between

points and the merging of adjacent points (note: our computer

model allows for 're-setting' the beach if this:is warranted).

The general format for this study involves a discussion

of the equations governing the fluid and sediment motions, the.

finite-difference form of these equations and a scheme for their

.integration, a treatment of the empiricism which links the magni-

tude of the sand flow to the longshore current, an explanation

and listing of the computer program, application of this numerical

model to specific beaches and conclusions.

II. Longshore Currents and Beach Deformation

Over the years various approaches have been adopted.in

attempts to describe how the orbital motion of water waves is

converted into the circulation velocities found in and near.the

surf zone. Longshore currents, which are prominent when wave

crests break skew to the bathymetric contours, have been treated

theoretically by considering the balances of mass, momentum, and/or.

energy in the wave-breaking region. Galvin (1967) has provided

a summary of longshore current theory and supporting lab and

ffeld data up to 1967. Galvin'.s conclusion that both theory

6

and data are wholly inadequate led to more sophisticated theo-

retical models by Thornton (1969), Bowen (1969) and Longuet-

Higgins (1970a,b). Based on the conservation of momentum and

the "radiation stress" concepts introduced by Longuet-iIiggins

and Stewart (1960,1961,1962,1964) these three researchers

independently developed models to explain how the longshore

current is generated and what accounts for its cross-stream

(shore-normal) profile. Each postulated that the main forcing

for this current is the oblique approach of a long-crested

breaking wave front. The steady state balance was taken to be

between the shore-normal gradient of the component of excess

momentum flux (radiation stress) parallel to shore and retarding

bottom and lateral friction. Bowen used a bottom friction term

proportional to the longshore current, v, and a constant hori-

zontal eddy viscosity coefficient.. Thornton and Longuet-Higgins

e1riv.d e. :oressions .fr the bot.toi_ stress and l-aeral couple n

:-::.. .ifer froi. o n' a d .. icn ae ore plausible

physically. They..showed that the bottom friction is proportional

to the product uv where u is the ampiitude of the local orbital

velocity perpendicular to the shoreline. The eddy .coefficient

was assigned an offshore dependence, tending to zero at the shore

and increasing monotonically toward the breaker line, where

maximum mixing is to be expected. In Thornton's model the mixing

coefficient was allowed to realistically decay seaward of the

breaker line, whereas in Longuet-Higgins' model it increased

continuously from shore seaward. Thornton solved his.equations

7

numerically; Bowen and Longuet-Higgins obtained analytical

solutions. The longshore current profiles in each study were

supported by the available data, such as the laboratory results

of Galvin and Eagleson (1965).

For our purposes the equations of Longuet-Higgins seem

the most appropriate because of the .physical bases for his

derivation and the ease of applying.his results. To be specific,

his expression for the longshore current as a function of the

non-dimensional offshore coordinate, x* = x/xb (modified slightly

by inverting the x-axis so :as to have a positive depth gradient)

is

V +0

C:)

.,here x. -: width of surf zone
5- a 1/2- 1/2 3/2
K g s
K 8. C g

a = ratio between the wave amplitude and the local mean

depth in the surf zone, 2C.4

C = bottom stress coefficient -0.01

g = gravitational constant

s = beach slope tan. where 8 .is :the beach angle

= angle b ween the gradient of the local depth con-

tours, and the wave propogation vector at the' breaker

line

P = TNSS/C(. aC) i a measure of the strength of lateral

mixing relative to bottom friction where N is a :Iumner

indicating the magnitude of the eddy coefficient.

P equal to 0.4 is a special case in which (1) is rodif.ied t

include a. logarithmic term.

-- 1 3,, :

Equation (1) is strictly valid only when. w b is small so that

cos b, ~-1 It is worthwhile to take note of the assumptions and

simplifications which lead to (1), To enumerate:

1. Linear shallow water theory is employed in the surf

zone and immediately seaward.

2. The- wave amplitude in the surf zone is taken to'be a

... The e.ch is p.ane and constant' sloping and _s act,<

on by a monochromatic wave train.,

4.. The angle of wave incidence is assumed to vary little

across the surf zone (due to .uch influence as re-

fraction, wave-current interaction, etc.)

5. The horizontal eddy viscosity coefficient is set pro-

portional to puL where p is the density., u a. charc--

teristic velocity taken to be the wave velocity and. L a.

9

characteristic length .taken to be the horizontal

coordinate, x, i.e.,

where :h is the local depth.. .N is dependent on the level

.of turbulence in the water, -a -reasonable r.n.e based on

field measurements (Inman, et al,. 1971) being 0
If the .condition of small b is relaxed (as it. must be for any

practical study) then the expressions in (1) are multiplied by

cos Ob. We will return to some of these points later. as they

affect our model.

The merit of Longuet-Higgins' model is that it removes

much-of the previous dependence on empiricism. Battjes (1972.)

and Earle (1974) have extended Longuet-Higgins' analysis to

include a wave field characterized by a Rayleigh wave amplitude

distribution.

Under special circumstances non-unikforit ies of the wave

el.{ .n t:e --., ..re dir.ecti (-,n c- .- rQ',uc norn--n ligi- e

gradients in the radiation stress at-the breaker line and force

a longshore current. O'Rourke and LeBlond (1972) have studied

the nature .of these .additional functions in .the. setting .of an

idealized semi-circular bay.and concluded .that, whereas the stress

due to.the obliqueness of long-crested waves is dominant, the

contribution made by a 'lngshcre modulation in the.wave height

can be significant, with a longshore variation in the angle of.

wave inciden-ce playing a minor role. LeSlond (1972) has expanded

10

on Longuet-Higgins' (1970b) analytical expressions for a long-

shore current to take into account all three types of drive.

The quantities of interest to us are the volume transport

rates 'of the longshore current and its sediment load. We assume

that the major portion of this transport is confined to the surf

zone.. Outside the turbulent wave breaking region the wave-

induced bottom stresses exerted on the sand grains decrease

rapidly as does the mean longshoree) current (Thornton, 1969).

We expect, therefore, for the sediment transport rate to decay

rapidly seaward of the breakerline. Furthermore, Longuet-Higgins '

formulation tends to overestimate the magnitude of lateral friction

in the seaward zone (due to the artificially high mixing coef-

ficient) as well as the role of bottom friction (since shallow

water theory magnifies the true orbital velocities in this

region). Multiplying (1) by x* and cos Ob and integrating

across the surf zone, O
? orei. -pr unit time

If, as with LeBlond (1972), we assume that the volume transport

of sand which accompanies this longshore flow is simply a :fraction

of the total water transport then we have

Qj C- (3).

II

The determination .of T in terms of meaningful physical quantities

measured in the laboratory and field is discussed in section III.

2. Shoreline Movement

There is no rigid boundary separating the beach from the.

ocean. Where they meet at any moment defines the instantaneous

shoreline. A function of space and time this line undulates in

response to wave run-up, the presence of edge waves, wind and

wave-induced set-up, the tidal cycle, surge, etc. For.our pur-

poses we consider it to be the mean water level with respect to

the local tidal conditions. Mathematically at any time, .t, this

line can be described by an equation of the form (see Figure 2)

F(.y,0,) (4)

We can establish the following relations for the local normal,
A
n, and tangential, t, unit vectors:

ae .o'r n -(
', '" y' c

A
and since n and t- are orthogonal (A t 0)

/ Z /OF -' (6)
1 };j

A

S XL

S7 A

Figure 2.

Definition of relation between fixed and local

coordinate systems.

13

The movement of the shoreline normal to itself will be

a function of both the magnitude and the longshore variability

of the littoral drift. The beach.will prograde if more sand

is deposited in an area than removed over some time interval

and will retreat if the sand extracted exceeds .that supplied.

In other words, local erosion and accretion depend on the sign

of the longshore divergence of the sand transport; this quantity

is expressed- in terms of the.longshore coordinate, y, as

d V d X dL K ,.
P(; 3JFJ 7 >- (7)
L t' .,; ^ .i /

The displacement of the shoreline in the x-direction

depends solely on the variation of Q in the y-direction and any

displacement in the y-direction depends only on the x-coimponent

oF vith reference to the right-hand coordinate system of

Fiurce 2 .e see that th e projection of on the y axis is
Uy
given by

a) it xrF
-/ Y / Id- F I Z V2
/'_. L-, [fL / 8

and its x-component resolution is

JF
CY i ,~\ y1 F jf '. (9)
[\C / (y/ j

where Ois positive counterclockwise.

14

Therefore we can express the temporal change in the

horizontal coordinates of any beach point (y,x) as a balance

.. jF'j)) C ) dx a
JrJ

We follow the approach of LeBlond (1972) and cast (10) and (11)

into forms more appropriate for application to arbitrarily-shaped

shorelines. Referring to the vertical cross-section of the

plane-sloping beach in Figure 3 it is assumed that the profile

remains unchanged in time, i.e., the slope at a particular point

along the beach is constant. In response to erosion or accretion

the entire profile shifts laterally inward or outward, respec-

tively. This is a convenience and implies that either: (1) Thce
distribution of sand transport capacity across the surf zone is

such as to maintain the profile, or (2) there may be a smoothing

effect normal to shore due to waves re-working the sediment into

an 'equilibrium' profile. Neither of these hypothesized factors

is considered explicitly here. The amount of sand gained or
lost is proportional to the area of the parallelogram EFGH.

'D' represents the depth beyond which there is little or no sand

transport (in this study, D is the depth at which waves begin to

break). The initial plan shape of the beach is specified by a

set of discrete points (see Figure 4) whose movement in the
set of discrete point's (see Figure 4) whose movement in the

<-- -.

Cross-sectional profile of lateral movement of a beach

point.

Figure 3.

f-'o'S .i- -f i- 1 h.ihA

r-^-r Y

4: _~~
d-fZ
A;~ ) li

J41 -;.-'

Figure 4.

Representation of strandline in LeBlond (1972).

17

horizontal plane is-determ-ined by the net-amount of sediment

transported into or out of the control volumes. These control

volumes are bounded at the mean shoreline by line s:;n.:-t.-:

joining adjacent beach points, by -the plane-sloping bottom, and

by a line parallel to shore at an offshore depth, D. A simple

continuity equation relates the translation of the beach points,

normal to the local shoreline, to the longshore .divergence of

sand transport, i.e.,

+ -- (12.)

In terms of the fixed coordinate system (y,x) and in

view of (10) and (11), (12) becomes

oiv c,-_<; 0 G.
"; P J. (13a)

J t o.s't-- -aL '

where D is a function of position along the beadc, i )e., D = D(-)

[Note: There is a typographical error in LeBlond's (1972) equation

14b.] Equation (13) is valid for. a right-hand coordinate system

with 0 positive counterclockwise or a.left-hand coordinate system

with 6 positive clockwise.

It is obvious that there will be seaward discharcres of

water and sediment (e.g., in rip currents) interrupting the

lonq.hore flow. In a strict 'control volume' approach these trais-

ports have to .be accounted for to' satisfy .mass balances. We are

18

assuming that on large spatial and time scales their contri-

bution is minor.

III. Longshore Tjai-.sport of..Sediment

In -and near the surf zone the waves provide a-.large

part of the stress required to dislodge sand particles: an.d .make

them available for transport by' the mean currents. There are

two modes of sediment movement which can result, ''suspended'

transport or 'bedload' transport. Suspension of sand particles

in the fluid column can occur in response to the turbulent

action of the breaking waves and the presence bf a small cur-

rent is sufficient to advect these sand grains. 'Bedload'

motion is the treep of sediment particles in constant or

intermittent contact with the bed and requires a threshold

shear stress .to overcome static friction and initiate motion.

The dominance of one mode over the other is largely dependent

on incident wave type and to a smaller extent on the sand

characteristics. Suspended material is more likely to be

associated w ith plunging breakers reas ereas bedload movement

often predominates when the breakers are spilling or surging.

Spilling breakers invariably occur when large waves break on

a mild slope. As the incident wave height decreases and/or

t"eh, heAci- slope increases in a continuous fashiCion the .spillinrg

breaker evolves successively into a pl-unging, collapsing, and

surging breaker. Galvin (1972) has provided both descriptive

and parametric classifications for that's breaker types. The

transition in the: direction spilling--plungi ng-collaspsing--

surgin. -is -a inverse Ifunction of the .deep--.-:ater wa.e steepness

(or breaker height) and.a. direct -function -if the beach slope.

For.some time it has been recognized that.a parameter

critical to the question of whether sand is moved onshore or

offshore (resulting in a 'summer-swell' profile or a 'winter-storm'

profile, respectively) is the deep-water wave steepness, H /L

where Ho is the deep-water wave height-and L is the deep-

water wavelength. Laboratory experiments.by Johnson (1949),

Rector (1954), Scott (1954), Saville (1959) point to a value of

H /Lo = 0.025 as marking the transition between winter and

summer profiles. Values greater than this correspond to erosion

and values less than this to deposition, although this is not

a stringent rule. Saville's (1950) experiments suggest that

the suspended mode of sand transport dominates over the bedload

mode for large wave steepness and that this relation is

reversed when the wave steepness is low. However, it cannot

be stated that there exists general agreement as to which mode

is predominant in the surf zone.

Presently there are no definitive experimental studies

relating deep-water wave steepness and breaker type to the sand

transport mode and its direction.

We are restricting our attention to the littoral drift

component. Dean (1973) has formulated a relation between the

longshore transport of suspended material and the longshore

component of energy flux. By assuming that a fraction (empirical)

of the energy flux is consumed by the falling sand grains,

determining a volumetric suspended concentration and using.

Longuet-Higgins' (1970a) expression for the average longshore

velocity he obtains

o D f- ,

(14)

where Q

CD

s

ps w
w

Hb

b
E
a

= volume transport rate of. sand

= bottom drag coefficient

= beach slope

=,density of sand and water respectively

= fall velocity of sand grains which is a function

of grain diameter

Save height at breaking

=wave angle at breaking

= longshore flux of wave energy = C bEbsin

where Eb is the wave energy density and Cgb the

group velocity at the breaker line.

In contrast, by considering only bedload motion, Komar

and Inm.an (1970) followed Inman and Bagnold (1963) and expressed

lte onshoree transport rate as an immersed- weightI transport.

1 ,'

(15)

Q Q

where g = gravitational constant

a = correction factor for pore space

Laboratory and field studies indicated that I could be set

proportional to the product FbCb where Fb = lateral wave thrust

at breaker line = b (see Longuet-Higgins (1972));
-- -sin 29b

cb = phase velocity at breaker line, :i.e.,

o (16)

22

where o is the empirically determined non-dimensional constant

of proportionality. If we re-arrange (15) we get an expression

for Q

Qa. ( (17)

If we expand equations (14) and. (17) and use the

theoretical-empirical breaking criterion

b/ 'h (18)

where hb = breaking depth
y =0.8

we obtain, respectively,

~ GHC,,s
G1 S )(19)

and QG G(, ,. )20)

Equation (19) in comparison to (20) contains the additional

parametric dependencies on cD, s and w and therefore offers the

possibility of modeling the effect of these parameters. There
is also to be noted the difference in the exponents of H It

is instructive to compare the expression for Q given.in

section II, equation (3), with the above results. Equation (3)

can be re-written as

/ C / (21)

where
7- Z I

We note that the functional dependence on the breaker.height,.

Hb, and angle-of breaking,
Furthermore, the product Fbcb, contained in (20), is the quantity

against which many laboratory and field observations are taken

(see Shore Protection Manual Vol. 1, U.S. Army Coastal

Engineering Research Center, 1973). Equating (20) and (21) we

obtain an expression for our unknown proportionality coefficient,

T, i.e.,

5 P ; -a(22)

If we insert typical values

cD = 0.01

K = 0.5 'see Das (1972))

py = 1.02 g/cmr
3
s = 2.65. g/cm (quartz)

a = 0.6 (packed sand)

S= -0.2 corresponding to P = 0.13

we get T = -0.0026. Physically this means that for each cubic

Smelter of -water transported across .a plane perpendicular to the

shoreline 0.002.6 cubic'meters of -sand will accompany it. There

is considerable scatter.in :the data and, therefore, in the

estimates of :K ::easurements h-ave been made by various means
O

24

under a variety of test conditions (e.g., differing wave 'spectra,

beach bathymetry, duration of record, instrumentation, in-

terpretation, etc.). Das (1971, 1972) has described several of

the methods employed in the lab and field for determining the

rate of sediment transport and summarized much of the data on

Ko. Noda (1971) has reviewed the techniques presently available

for measuring littoral drift in the field.

In view of the order-of-magnitude uncertainty in the

value of K we treat this quantity as a control variable subject

to adjustment over a reasonable range. It seems unlikely that'

K will assume a single value appropriate for all beaches'since

there are beach parameters (such as w. in (19)) whose significance

has not been guaged.

IV. Stability Analysis of Governing Equations

It is customary and worthwhile to determine if one's

Working equations are subject to any intrinsic instabilities

for some range of the parameters involved. If the instabilities

of the analytic form of the equations can be identified, then,

spurious results appearing in their.numerical integration can

be labeled and/or avoided. We consider, again, equations (13)

in the coordinate system defined in Figure 4. To avoid unneces-

sary and lengthy computations we postulate that the original

(unperturbed) shoreline is straight and lies parallel with' the

horizontal axis. By superposing small perturbations on this

configuration and examining'under what conditions these

25

disturbances grow, decay, or .remain unchanged we can detr;rmin

!when our equations will behave peculiarly, i.e., admit

oscillatory solutions that "blow up".

We express Q, the transport rate, as

-- ( C^(^7- ) ^"(23)

where B contains implicitly all empirical constants, as well'as

the functional form of Hb which is assumed independent of the

longshore coordinate; -~o =. b where :"o is some constant initial

value for -b adjusted due to changes in beach orientation (6).

Upon expansion this becomes

r = i /SSv ( ,,-, -6 t- e4 ;- E)s(r 'Q *
Q0L = r ( (24)

We note that (see Figure 2)

-l st, d (25)

= C~'Id. :0 (26)
.y

so that we can write the lovngshore derivative ,:of Q as

--- (27)

where

oY o "d
y 0 (28)

Cs)S I- S ra (2-9)

Now, let y be a function of the original (t = 0) arc

length 's' (linear) and the time 't', i.e.,

7,

Then we can evaluate the derivatives in (27 ) as

SY /) s
y) / o

and

2
9K
-^j

d d 2 )'!
( __J_

(31)

92y ^Y/ d{ ^
y Ss Y/,
0 .^.J.,_~._o~. s._ / ~ .__
Jsl '* ^k

In view of the folloing relations
S7-

d 3 / S S
^s^ ^..s

(32)

(33)

we can rewrite (30) and (31) as

)

)x a^ 7 y.4

(34)

(30)

Sdx7d 5

d?; f;:(X/

* -_ J/S 0
C)Y Y/,

LI i~ ~4 .~ i
i"~~)L~,
3s,.: Js, I
i Js 35 z `7 -3-
it jir
3 3 i

S2/ d y J .-
Y/ 5- o., -Z

1 ~

Our original governing equations (13a) (13b) become

- cJ 41

.o-t
*o~

where 3 yQ/y..is defined by (27), (28), (29), (34) -and (35).

We now introduce expansions of the form

x f b ot .- ing elana to 0 (.)
a(l ~os-,b l so cy,:t, ), o l w,.n v i,,- :.) +....

ana establish the following r-elations valid to O(E)

c)? : _

dsir .4

0 0

.1-J
c-2" ,**J" ,
,^ *. -.......
< : & -

-and

Zy (d y)
***.1 <._
- ~~ [

(3,5)

(36)

(37)

(3;8)

(39)

~____ iliL_____~___

d~s

"Y

dZo \
(e),

28

Substituting (39) into the x equation, (36), we obtain

C / d 5 J'i o (

0 L

(40)

The 0(c) equation extracted is

ta P 0%

or

d 'S (41)

This is of the form of a one-dimensional heat (diffusion) equation

which is a well-studied linear second-order partial differential

equation. A fundamental property of this equation is that initial

value information can only be propagated in one direction, i.e.,

it is not possible to integrate this equation backwards in time

to determine the initial distribution of xl. For this reason the

coefficient on the right-hand side of (41) must always be ,positive...

We identify the regions of stability and instability according

to
/ i (-0- Z o .i .
1, .,. (42)

I .0, / ; 5

- ( I 2 U<12d. < 0

29

Thus, for breaker angles greater than +450 the shoreline will

be unstable to perturbations of all wavelengths and will undergo

oscillations of increasing -amplitude. This has been confirmed

numerically. A small disturbance of arbitrary wavelength is

imposed on an initially straight beach such that b assumes values

greater than +450. It is found that the shoreline is stable to

the disturbance if gb does not exceed +450 and is unstable

otherwise.

It is interesting to speculate whether such an instability

occurs in the field. Bowen (personal communication) has noted

that wind-generated waves in small lakes can break on the beach

at very acute angles and cause the shoreline to deform in a wave-

like manner, i.e., be responsible for periodically spaced shore-

line protuberances. Aerial photographs of coastal areas

frequently show undulations of the shoreline with.definite

wavelengths. However, Dolan (1970,71) and Vincent (1973) have

correla ed the existcnc. of these wanders with- inner and outer

submarine bar rhythms. These bar systems may represent in

themselves an instability of the submarine bed to longshore

currents as suggested by Sonu (1972) and theorized by Barci.lon

and Lau (1973).

In our model we exclude angles, #b, which fall within the

unstable range of (42). It is conceivable that a second-order

term in Q added to the right-hand side of (36) and (37) might

damp the growing oscillation. .Rowever, the-physical justifi-

cation of such an "artificial viscosity" term is -not clear. The

30

omission of | b >45 is not considered serious. It is usually

true, especially for swell waves on mild slopes, that refraction

will limit the breaking angle to the stable regime. This may

not be the case for local sea on steep slopes. An analysis of

our study results indicates that over 95% of the %b's generated

satisfied the stability criterion.

V. Finite Difference Form of Equations

We wish to express our governing equations in a form

appropriate for numerical integration. Equations (13a) and

(13b) are discretized according to LeBlond (1972) as

t J 73 v A (43a)

d \ /

S(43b)

where the superscript n denotes the time level and the subscript j

the space level. Referring to Figure (4) we make the following

comments:

1. 0. is the orientation of the beach segment, j, in

the fixed coordinate system (y,x). The O's, of course, are

altered as the beach points migrate.

31

2. B. is the 'effective' (averaged) angle at a point, j,

given by (Qo._. Q_)

3. A. is the sum of the distances between point j and

adjacent points j-1 and j+1, i.e.,

4. The transports Q are evaluated at mid-segment points

and are characteristic of a segment (not a point).

At endpoints these definitions of B, A, and Q need to be modified.

The general class of integration schemes we adopt is

'predictor-corrector'. A.predictor-corrector method represents

an iterative approximation to a fully implicit scheme.

Kurihara (1965), Lilly (1965), and Baer and Simons (1970) have.

discussed the performance (e.g., stability, conservation

properties, accuracy, phase errors, etc.) of several of the

more widely used predictor-corrector schemes (leapfrog-trape-

zoidal, Adams-Moulton, Mlilne). The advantage of such multi-

step methods lies in their ease of application.and speed

(provided the proper step size, At, is chosen). We employ

Hamming's (1962) predictor-corrector method which consists of

the fourth-order Milne predictor and Hamming corrector. The

Hamming corrector is favored over more traditional correctors

(Milne, Moulton) because it exhibits stronger stability,

although at the price of an increase in the magnitude of the

truncation error.

32
"+' -"- h -fJ
Predictor: 4- (44)

< *' ? .7 . -C ]
Corrector: 7 (45)

where z = (y,x) and f represents the right-hand side of (43a,b).

It is obvious that, in addition to-the initial datum, 3 values

of z and the corresponding f's are required at the n-1, n-2,

and n-3 time levels. Since these are not available a special

method is required to generate them. We revert to a numerical

method based on a Lagrangian interpolation formula (Ralston, 1965,

p. 191) which yields estimates for zl,z2,z3, given zo, namely

-Z- /Y ^fJ tf ) (46)

Z -o 2 / (47)

The error term is 0(At5). We guess values for zl1z2,z3, calculate

the corresponding flf2,f3 and use (46), (47), and (48) to com-

pute new values of zl,z2,z3. This procedure is then iterated to

convergence. (An alternative method for furnishing starting values

is the Runge-Kutta scheme.) Ralston (1965) has provided a careful

analysis of the properties of predictor-corrector methods as well

as their merit in relation to other schemes.

The step size, At, must satisy several criteria. In a

physical sense.it is controlled by the spatial increment Ayj

33
(the distance between neighboring beach points) and by the

average speed of the sand particles, vs (a function of the long-

shore. current strength and the grain {characteristics).. Linear

computational stability requires that

nt a v A/ : (49)

A rough estimate for vs can be had by noting that the triangular

wedge through which the longshore current flows has a cross-

sectional area of 1/2 Dbxb and therefore the sand transport

rate equals z

v = zTV/SX (50)

A.more rigorous requirement than (49) is that the increment,

At, be small enough to meet the convergence condition on the

corrector equation, (45), preferably small enough so as to

achieve convergnc_ inn one or two interations; it must also.be

sufficiently small .to satisfy any restrictions on the najgnitude

of the local truncation error which is given approximately by

(Ralston, p. 189)

(51)

where z + is the predicted value and Zn+l the corrected value.

In addition, the step size should be large enough so that round-

.off errors and the number of derivative evaluations is minimized;

otherwise, the multi-step method loses its chief advantage,

34

namely, speed. Ideally one would like to adjust At so that

only one application of the corrector equation is necessary.

Equation (51) is helpful in two ways: 1. knowledge of E, as

the integration proceeds, can suggest in which direction At

should be adjusted for efficiency; 2. E can be used to actually

modify the solution of the corrector equation. The proper

choice of the step size is a function of the geometry of a beach

site and the incident wave energy levels.

VI. Computer Program

Structure

The program is divided into 11 sections--a -core and ten

subroutines. We note below the designation and function of each

part:

1. Main Program

a. Read input parameters (which run program)
b. Establish shoreline
c. Call working subroutines
d. Execute Hamming predictor-corrector (repeat)

2. Subroutine EMPIRCL

a. Set value of constants appearing in expression for
longshore current
b. Compute coefficient, T, the ratio between the sand
and water transport rates

3. Subroutine ADJUST

a. Read in values of breaker height, angle, .and dura-
tion (fractional) of a particular wave type for
each beach segment; compute transport rates
b. We expect the angle of wave attack to change as
the beach orientation is altered. An adjustmentt"
angle, the difference between the old and new beach
angles, is added to the original (b and a revised
transport figure is calculated. This is done at
time intervals chosen by the user. Any accompany-
ing refractive modification of wave height is
considered secondary and is -neglected.

4. Subroutine INITL

Generate all necessary starting values for use by the
Hamming :scheme as outlined.in section V.

5. Subroutine DERIV

a. Given the beach coordinates compute the beach seg-
ment angles and the spacing between adjacent points.
b. Given the volume transport rates of sand along the
beach compute the incremental change in position of
each beach point :over a time interval, At. A Fortran
ENTRY statement links DERIV with that part of sub-
routine ADJUST that re-computes the incident angles
on some regular basis because of the re-shaping of
the shoreline.

6. Subroutine AREA

a. The surface area 'df the beach is an important

quantity. Its change can be monitored by co.p:uting
the area difference between two successive strand-
lines.. In Figure 4a the calculation is straightfor-
ward since the y coordinate of each endpoint remains
constant. Figure 4b represents the more general
case wherein the endpoints are allowed to move
freely. A rough estimate of the net areal change
(additions due to accretion minus depletions due
to erosion) can be had in the following way: (i)
connect the endpoints A, B and E, F as shown, (ii)
compute area under curves AF and BE (summations
over a series of trapezoids); these are the exact
areas under a discrete beach which is itself an
approximation to the real strandline, (iii) compute
the areas of trapezoids ABCD and EFGH, (iv) sub-
tract the two numbers in (ii) and, then, from this
result subtract the areas computed in (iii); this
number represents crudely the increase or decrease
in beach area. If the positive or negative con-
tribution near an endpoint is desired we can esti-
mate this at the left end to be ABA' where A' is
the point on curve AF at.which a line dropped from
B parallel to the vertical axis intersects. The
x coordinate of point A' is determined by linearly
interpolating between the beach points on either
side. The area, then, is just the area under AA'
minus the area ABCD. Similarly, the area EFE' can
be computed.
b. Approximate volumetric changes can be obtained by
multiplying the discrete trapezoidal areas by the
local value of Db.(see Figure 2).

7. Sub -utine -.iSC

f, for some reason, i- 'is desirable to have ;he spacing

between neighboring beach points more or less equal, it is possible

to reset the beach points to accomplish this. The circumstances

which might dictate this action are-many: (i) a more rational con-

trol over the size of At would result; (ii) a few beach points may

be moving at an anomalous rate compared to their neighbors (e.g.,

the point.at the tip of a rapidly expanding spit); (iii) eqqal

increments might be more compatible with the longshore resolution

of the wave field, etc.

With reference to Figure 4c we shift.only interior points;

endpoints must retain their positions if the beach shape is not to

be disturbed. All or only part of the shoreline can be reset.

A 4,vAT Tf

LAND

Figure 4.a. The difference. in area between successive strandclines

whose endpoints have the same abscissal coordinate.

WATER .

L AN D

,I

S' 1

Figure 4b. The difference in area between successive strandlines

of arbitrary shape and orientation.

I ,

^-^--- f-xecl

Figure 4c. Schematic diagram defining beach points and their

movement in the 're-setting' process.

39

Over that portion of the beach which is to be re-defined the lengths

of the discrete longshore segments are summed over and divided by

the total number of segments to yield an "average" spatial incre-

ment. The first interior point is moved along the segment immediately

to its left (like a bead on a string) until the distance between it

and the fixed point on its left side is the "average" increment.

This interior point now becomes the fixed point for the next interior

point, i.e., the second interior point is moved along the line seg-

ment joining it to the new fixed point until the distance between

them is, again, the average increment. (note: movement along .these

segments can be forward or backward). This process is repeated

until the fixed point on the right hand.side is reached (either

the right endpoint or the point defining the right boundary of that

portion of beach to be reset). Because the right boundary point is

not allowed to move'this procedure must be iterated 4 or 5 times

before all the beach increments converge toward one value. This

method must be applied thoughtfully; otherwise, the resultant shore-

line mayv deviate too much from its former shape.

8. Subroutine RESULTS

Display results of computationsin print-out form

9. Subroutine PLOTTER

Use the Florida State University plotting package (Fortran

callable, calcomp-like routines) for displaying the shoreline evo-

lution graphically.

10. Subroutine ROTATE

Rotate the N-S, E-W axes if beach points are desired in a

new coordinate system. This is .used for plotting purposes, i.e.,

to show:direction of maximum beach change.

40

11. Subroutine ERROR

Calculate error between actual and predicted quantities.

Below we provide a listing of the program with accompanying

comments

PROGRAM SiORLIN(TAPEi,INPUT,OUTPUT,TAPE5=INPUT,TAPES=OUTPUT,PLOT)
fr** THIS PROGRAM MONITORS THE CHANGE IN THE PLAN SHAPE OF A SHORELINE
r DUE TO THE DIFFERENTIAL LONGSHORE TRANSPORT OF SELIENT INDUCED BY
r WAVES BREAKING AT A ANGLE TO THE SHORE.NECESSARY INPUTS ARE THE
r HAVE CHARACTERISTT.S (H IGHT,ANGLE OF INCIDENCE DURATION)AT THE
C 3REAKER LINE AS A -JN3TION OF LONGSHORE POSITION AND THE -ORIZONT&L
( COORDINATES OF THE P-OTNTS WHICH DEFINE THE SHORELINE AS RECORDED
f AT VARIOUS TIMES.
r THESE QUANTITIES ARE ASSUMED TO HAVE 3EEN GENERATED IN ANOTHER
C PROGRAM.IN ADDITION IT IS ASSUMED THAT THE BREAKER ANGLES WHOSE
r ABSOLUTE VALUE IS SRE TER THAN 45 DEGREES HAVE BFEN EXCLUDED.
f AB HAS UNITS OF DE0SEES.
f
C
C THIS PROGRAM IS WRITTEN IN FORTRAN IV LANGUAGE .FOR THE CDC 6500
r COMPU.TER.SAMPLE V0LJES FOR MANY iF THE PROGRAM PARAMETERS APE USED.
C PLOTTING ROUTINES ARE WRITTEN FOR THE GOULD PLOTTER.
r
r THE LETTERS A,..,I.T. AS THEY APPEAR THROUGHOUT STAND FOR THE PHRASE
r EAS DEFINED IN TEXT-..
C
DIMENSION FRAC(50)
DIMENSION OXX(90),DYY(90),HI(90),H2(90),Zl(90),Z2(9 ) ,X(90),Y(90)
COMMON DEL(0) ,ErA(90) THETA(9"0)
COMMON X0(9 ) 0),X1(90),X2(90)X3(90),Y (90),YI(90) Y2(90),Y3(90)
COMMON DOY (90) ,DX3(90). ,OY (90),rXi(90),DY?290),DX2(u) ,oDY3(90),
TDX3(93)
CMMON/ LOC/f-I (5 30) '
C MM ON/ L LK/ ( 50 -)
C 0 ;. , '/ '' L ; C 1
COMMON/d 1/KOUNT. Th117,(50)
COM'ON/3LK5/L (10) M( IC)
COMMON/BLOCO/II
COMMDN/3LOCC/DI
COMMON/3LOC2/LIMITLI 1,LIM2
COMMON/3LOC3/CV PT V"T2
COMMON/3L0C/OIF DIFV, DIF1, 0IFVi
COMMON/3LOC7/IVT Dr
COMMONM/LOC11/COEFF
50 FORMAT(IX,5F .2)
1I FORMAT(2I 2213,I5, =5 0)
52 FORHAT(iH ,2(2X,TI), X,2(2X,I3),?XI5,2X,F4.0)
<5 FORMAT(1X,10T3)
5n FORMAT(2X-,7(3XTI) ,5X, F5.l)
61 FORMAT(3XI4)
C+*Y- THE PLOTTING MODE IS ENTERED INTI-PLOTLIS AND GOULD LIBRARY
r ROUTINES ARE CALLED.
CALL PLOTS( 0. 0,0.0, 4HPLOT,P)
r-** THE INTEGER VARIABLE-S L AND M ARE USED TN SUBROUTTNE ADJUST
r TO PICK OUT THE BE-ASH POINTS OF INTEREST IN COMPUTING NET
C TRANSPORT RATES F-R VARTABLE STRETCHES OF SHORELINE.
REAO(5,35) (L(I) M( I) I= 1.0)
0 3OEFF IS USED. IN SUI3PUTINE ADJUST.

COEFF=0.5

L ... R P.cA D ROGRAir PAPA'1ETEDS:NM1 iT THE_ iJ'i*JM E- Or WAVE DATA S.TS,
N2 TS T 3 N!"U:B;E? OF TTIES THE CORRErCTOR EQUATO1 S. AR- APPL-LIEO
SLI-' TS THEi NUi'!H EF OF DOTNiS W'HIG4 1AKE UP THE BAS: S'HOr-ELTNE,-
M MAX IS THE MBEF C- -TNT~ S WrIC-T MAKe UPI THE SHOELTNE AS
Cr 0-E.zr.'VD AT A LATER TI1M't-IrNT IS THE TI!TERVAL(EXP ESSO s.
r NUIA3-FP OF TTM'". ST=PS) IT WHICH RESULTS A.E DISPLAYr.P3T T IS THE
C TIME STE>P.
rE rU( C, si) N1,M2, TM, M)IN, IFI T, DT
t .I T (o, Z) i i, 2 LTI MA IP'-'L IT TT
r '4 SO-;F Wr.;KINS DEFI"7ITIOTn S:
LI IT=LIM4-1
LI 1= L T -1 i
LiM 2= LTM- 2
C :" FRAC IS THE FRACTION OF T-r1 TOTAL TTME OF RECORD(EXPRESSED AS
r THE UNIT 1) DURING W-1HTH A PARTICULAR i;4PWAE ACTS.H AND A3 aE THE.
r YRE.AKEr< H-EIHT AN" AtGL,-,RSPECTIVELYFOP THAT WAVJE AS
C CnlPUTED FOi< EACH SEC4 -SEGMENT.
READ( ) (F aC(I) =1, Ni .. .

READ(l) ( ((H3(II J),3J), J), II=1,Ni) ,J=i, LIMI)-
r
Cp"* THE DURATION OF A PARTICULAR tWA9E TYPE OVER THE PERIOD OF A YEAR
C IS COMPJTED TN TFM3S OF NUMBER 0; TIME STEPS. IF THE I -TEGRATION
.( PROCEEDS LESS THAN A YER.-THEN T-IS CALCULATION MUST Sc3 '10DIFIEl.
.0DO t- I .i,N
GU N41C(I)=FRC(I) E03e. ? ./lT
ry.x THE' TOTAL NU'4.ER 3F TIME STEPS IS COMPUTED.YFAR IS THE NU3E, O'
V ARS OVER WHICri TTH NTEG.- YE ,R=, .
NUMnT=YEAR36:->" 2+. /CT
C ,f-llFINE FACTORS FOP O0\NVERTIN5 FROM RADIANS TO DEGREES AND. ACGKY--
PI =,. ATAN( .)
CVRT1=PIT/180.
CVRT2=i./CVPTi
C- REAi INITIAL VFRTICtAL-HCORiZnTAL COORDINATES OF :EACr- POINTS.
S ALSO RE\D THE COPr.DINTTES :S REOROE3O AT SOME LATER TIME IF THESE
r ARE AjAILA.rL.E(FOP -C3 ARISC-N WIT PREDICTED VALUES).
FirEAD (5k ) (XO (T) ,YC(I) I=1, LIM)
i A:i- ( c^ n'0 ) X(YX-(i) ,YY(T r T= ':'AX,
0-- C V: CONVERTS OUTSTAN3DI IS UNITS TO METERS-IN THIS CASE,FEET TO
C ME TER S .
CV=1. /3.28
DO i I=1,.LIM
XO (I) = V^X' r(I)
YO (I)= CVlYC(I)
r, THE INITIAL VERTIIA ANN HORIZONTAL COORDINATES ARE DESIGNATED
c X AND Y,RESPECTTVELY.
X( I) =X0(I)
1 Y(T)=YO(I)
DO 12 I=1,M ,X

YX(I)=cM*XX(i)
12 YY(I) =C YY(T
r*A THE ATAN2 FUNCTION aLLOS FOP UNAMBIGUOUS EVALUATION OF
r THE ,NGULIP ORIE T4TION OF EACH SEGMENTSS IPPESPECTiV' OF
(t UADRANT LOCATTn'.T HIT TA AND Bc.TA ARE A...I.T.
00 17 I=1,LIM1
17 THETA(T)=ATAV -(X(1+1)-X(I)., Y +1)-Y(T) )
S,, THETA.(0) .,TH: T (LTi') t:-RE DFFFIMIP AS S.THEY .!ERE FOR THE JU.PITEP
r ISLAND RD'J::GT,-A Z C SU i-' r.PT IS PEtL TTTED IF A PROPER STGOAGE
r LO.ATIO.N IS A&.LLOTFT- TE ZE-RO-SU3SCRiPT .' AR.TCLE.
THE T ( TTA,.TA( I ) (T -HE A (2)- T r T ( 1))
THLT T( LIK) :HF (Ll l ('T- ET, (LI 2)-THE't LIM ))
DO 18 I=1,LTl "
18 BETA (I)= (THE:TA I) +T c)-TA(T-1) )/2.

' TE EH-T."IR.Ijr SEPTM--:T T'RPORT OEFFI iENMT IS COMniTED
CALL EF IPCGL
r
C"' tIE CALL AOJUST TO ESTA3LISh THE FIRST SET OF BREAKER H.ETGHTS. AN n
( AJG ESL .
r 4 I TS T4-1 V-- -D TA S-T NUJY- E-,R.
IT=1
CALL AOJUST(LT-)
r
K' -N KOUN IS. A FLAG USF3 IN SU2B OUTINr-. E F;I !
KOIIN 1
r
E7 FPRIV CO(PUTES Th- CHANGE IN THE (Y,X) COOROINATES ')73VE 1 TTIE ST P.
CALL DERId (Xo, YCfn X ,. nYO -'LT );')
KOU- T 0
r', ALL REQJTP.ED STARTING VALUES FOR THE COORDINA TS (Y,X) ARE
C GENFATED IN INTTL,
CALL INITL( LT., IT D)
C
rC" IND IS AM ITNICATOR CF CONVER-GEN"E OP L.CK OF CON'ERSENCE. TN TNTTL.
IF(IND.EOD.)30 TO- 10 C
WRITE (. 2 C0)
230 FORMAT( x A CrON IER EN"E*)
Gn TO 102
10, CALL DEIV(X1,Y1,3YXt1, 1,LTM)
CALL DERIV (X? Y2, ,X? LI M)
CALL DERII (X3, 3,DO3,DY3,LI;-)

COPU-TE THE' DIFFERENCCF I rLAN AREA OF THE EACH AS OcSERJ-EO
r INITIALLY AND AS O3SFRVEO AT SOME LATER TIME (OIFA1) AS -ELL AS

S H V THN E T P I,-,LU1%TNIC F HN TF \i )
SLL. A ( Y X IY, LT, Y NXX, MAX)
DIFAi=nIFA
DIF iD=OTFV .
WHtTE (o,67) DIFA1, 0IF l
57 FOPMAT(9X,TH OF-ERIVE CHANGc I\ BEACH ARFA OVEP StME TIME
SINTERVAL=*,Fl.l//2'?,"THE -3SERVED ChANGE IN BEACH VOLUME OVER
SOME TIME INTFRVA_=l,F!0.1)
C'**. INITIALTZE THE TIIE STED COUNTERS III,IV:I III REGULATES THE

;AlTTIO i'TWFEl iA.'F D 3AT*. SETS IV M''lITORS THE TOTT!. NUiJER
r-LL T E TEPfE
Sf T T = "

G' TO 3
2 II=1I+1
CALL ADJUST (LTIM)
r
f- ~ IF ALL ,DATA, SETS -IAIF E3EN USLc~n~, RETURN TO DATA SET NO,..
TF(IT..E ,ML 1) II=I
III=0
3 III=1II+1
IVrTV+I
IF(IV.GEoNUMDT)GO TO3
IF(III.EQ. Nd(II)+1) GO TO 2
r
rx IP OFTE1;NFES THE INTERVALS AT HHTCH RESULTS ARE PRINTED OUT AND
r PLOTS MADE.

TP=(IV/IPRINT) *IPRI'IT-IV
IF(IP) )S, 4,5
h CALL AREA(Y,X,LIM,Z1,Z2,LT-M)
WRIT= (,ol6) DIFA PF'!
5 FO P'lT(td *3HAlN.G I7 TdE SURFACE EREr. OF THF SUBAERI'AL BEACH"I,
iF10. //ZX, VD3LtI'JtM PIC CHANGE IN THE SIj.AERIAL 9EACH:=,Fi..:F1)
CALL RESULTS(X, 7,2,Z1 i,LITH~ 'AX)
LH-L PLOTTER(X,Y,Z2,ZlLTM)
IF AN INSPECTION OF THE RESULTS LEADS TO THE CONCLUSION T-IAT
THiE BEACH POINTS S-3ULL0 E RESET THr-, Wz- S-T THE INODTCATOR IN01
EQUAL TO 1. J' ANI J? ARE THE LEFT ANO EIGHTT O3UNDARY POTNTS FOR
TH- SECTION OF EAOC TO BE RESET.
Ji=1E
T N 1 -1
J2 = [ (
iF(TN i.n D 1) CAL_ RFSF, (Z2,Z1,Z J J .i)
IF(TIV ."'-f. -'In-1 ) G. TO 102

r
C

i
r THr PRFDIrfli EIOUATIrN"7 ACi APPLTnq
b O ..J=l),'1
Pi. (J)= YO(J +'-./3. 3TT (2. *nY (J)-DYZ (J)+2.*Yv l(J))
7 H (J) XJ) )++G./3, 7T (2. ~' X3 J)X-DX2 ).2.t X .(J))
CALL,
CALL Li .Iv (Hi i XX,tuYYt LIM)
S" LCOUiNT IS A CQOUiTr W4ICK ,PECTFTESc NUM:'ER OF TIMES TiHE Cno-hE3T
-., 5PQUA I O S ACE APPFLIE .
r'
Lo C U iN\T 3
r THLP THL CPO-C TC. ErUSTITCNS A3K APPLIT-L
6 DO c J=L, LIri
71. (J)-= -1./ .8 .*Y3( -(J)+3. /8, OTT (OYY (J)+2 ."DY3 (J) -.Y2( J) )
8 Z2( J) = 93./B8.XZ7(J ./8. x!(J)+3./8.0T* (XX(J)+2.IOX(J) -rX2(J))
LCOU E T=LCOC!I)T+ G C i
TF(LCOU!JT.Ef.N?)G3 Tr0 i
9 CALL DERIV( Z2 ,Z.i-, XX ,'nY,LiM)
GO TO 5
r
L.X" AS ThE ITTNTRATICN IS ADVANCED 1 TIME STEP ALL PERTINENT VaRIrLES
c ARE UPDATED.

10 DO 11 J=1iLIM
DYOfJ)=OYi(J)
DX. (J) =0Xi(J)
DY1(J) =5Y2 (J)
DX1 (J) =9X2(J)
OY2(J) =9Y.3(J)
DX2(J)=DX3(J)
KO (J) Xl(J)
0 (J)..=Y1 )f J)
XI (J) =X2(J)
Y (J).=Y2(J)
Y2(.J) =Y3(J)
X2 (J) =X3(J)
SI )U =71(J)
CALL DERIv (Z2 Ziy,33J 3 lY ,LTM)
GO TO 3
1I2 PRINT 103
i 3 FORPAT(1H ,*THIS TS THE END OF THE rURR-NT RUNS)
END

>U .O TIN 1 ,'-i CL
" PIDCLS ASSTSINS VALJS ALL P.AR'CETcRS WhiCH APPEAR IN TtHE EXPRESSICi;
r FOR THE- S,_'IMENT TRAPF'F.ORT COEFFICIENT* *
REAL KON
COM. N/ L 'OCn / I
CO H i'!ON/3LL.C9/T9 T *V
1 H .; i N\13i / 3 L C r''/ A iA
r"-DlEFITNITIONS, A=ALLOLaACF FOR OPRE SPACE
r C~j= 30TTOM C.RA'- COFFICIENT
SKO=_3Oi'cTAT Op PROPORTIONALITY BETWEEN THE IMMERSED
r NETGi1T TRANSPORT RATE ANO THE QUANTITY F~)C(3) AS
r OEFI'JED Tr' TnE TEXT
SN=A MFNSUP- 'F T.- EFFECT IVENESS OF LATER L MI'YTNG
G= RAV IT".TI ONAL O.NST IT (MKS)
r G"AMIMA=ATT) 3ET'-Ec-N WA-VE HEIGHT AND MEAN JEPTi :AT BREAKING
r ALPHA=PATIJ 3FThEEN WAVE 4AMPLITUnE AND LOCAL DEPTr TM THE
r SURF ZO3L. ALPHAAS SUPPORTED. BY -MEASUREH ENTS,IS
r USU_.LY SET EOUAL TO) G.-MM/2.
r S=AN A.JERAE S.LOP_ FOR THE COASTAL SIT.E 3EING CONSTOERcD
C RHOWRH-OS=DcNSITY OF WATER AN) SAND :ESPErTIVELY

r Tv=TNAPOTT CFFcTET FOR THE IATER MOTION
t=SEDIM.N TANSPORT COEFFICIENT
C P, 7l2, 2, ST ARE A.,.I.T.
wC o-. 3
N 0.01 -
K0=0.3
N=J.0 :3
GjAMMA 78
ALPHA=30.3
S= u 2
P=SPI N /( A.-ALPhA kP)
Pi=-, 75tSORT(9,/15, +1./ )
P2=-. 73 -SQPT( 3. /15 .T 1./P)
P. T= (1./ (P ( l+2. )'(P -PI 2) (Pi- )) +2./( ( P-2.) 3.))
P OW=1. 02;]
PHOS=2. ;1
TV =PSI'5.*"PT*uLP: r. SO M T(G) L) G/ ( c n MM *5./2.))
T = KO u MMA D" P 0 / ( P I PSI* A ALH 5. ( HC S- 3 ))
:"-"NOTE: IF,IN THE COO "31 NV T :" SYSTEM ESTA L LTSHEn.,THF OE PTH- Gr-.AdIENT TS
C NEGATIVE, THE THE SIGN OF TV AN3 T IS REVERSE
WRITE( ,1) NP,PSITV,T
1 FORi-IAT(2X,TN= ,F7. //2X, p=-,F 7.4//2X,'oSI= *,F7.4//2 +TV= *,Flu .3
2//2X, T=:+, FI/.')
PE TU I','
END n

SU6?OUTINE AOJUST(N)
r'xw ADJUST SERVES T.FEE PUP3SES:
C (1) IT UrOATES T-IE INCIDENT WAVF CLIMATE.
( (2) AS THE S-0RELIINM FVLVES IT ADJUSTS THF INPJT BREAKER
C ANGLe-S ANu RE COMDUJTES TF LONG cPOKiE TRANSPORT )RATFS.
r (3) A NET TPNISPO-T FIGURE IS CALCULATED FOR PRE-SPECIIFIE
STRETCHES 3F SHOR-LIN.FT.HIS HAS SIGNIFICANCE WITH RESPECT
STO A DETRMIAETHATION OF WHETHER 1HE SEDIMENT FL.OU TS
T CON IN-t T3 MORE O3 LESS CLOSED CELLS oR IS A CONTINUOUS
r STREAM INTERRUPTED ONLY OCCASIONALLY.
DIMENSION CHI(9C) ,O FT(25)
COMMONi DEL(93 ) ETAt 2r ) ,T.HETA(90)
CO'M ON XG 90),41(o ) X2(99 1,X3(9n) Y (9Y) ,Yi(9)),Y?(90),Y3(90)
COMMON aYO (90) , ? (9 ) ,DY1(93) ,D 1 (0qoI r1v2(96) ,0X2(0O) ,nY3(00),
*D13(90)
COMinN/BLOC/-i (5l,90)
COMMON/3LK/A' (5 Cy, C)
CJdM i-mO",/3LOC ,/ l I
CO' M ON/3LOC / I/LIMIT ,L Mi,! LTi!M2
C Q'; N/ i Ott 3 C V KXTir I T L T t'
CC--n.CN/dL C /VOVf "9 5

CO 0 -N/1LO 10G / GA 'M-i A
COM,0 0N3/3L OC 1/C 0 FFF
L.nMMON/3LK /PHIO ~-I (90)
COC'1lON!/3LK3 1/D30, ,C(C)
COMMi ? N/ LKr iL ( 1, ) M( 1 )
f (.1)
DO 7 J=I,L-TITi
C3(J) =:B(I11 J)/3AMMA
7 PH T J) -=VRT 1iA1 (II,J)
C+*i' WE MAj E THE, FOLLOWI0., ASSUMPTIONS;
03(C) =03(i)
D3 (LIM)=D03(LT 41)
PHI(C,)=PHI(i)-(PHI (?)1-FI ) 1))
PHI(LIM)=PHI( LIMi ) -(F-HI(LiM2) -PI (LTM1))
r*'X-oTE;IF TH. ,-ND L-OUkNOA0v CONDTTTtnS WE''ERE C=O ., WE WOULD) SET .PHI(0)
r. Pi-I(LIM) =PIT//., Cr.PFSPGONDIIN'G TO THE INCIDENT WA.VE CRESTS BEING
r PcRPrENDIULA: TO THE SHORt-LINE.
p- T!jMnN
FENT0Y -Ti.ANSPT

)CUFFF TS A SO'F'A-'.T A IT lrAEILY CHOSE F-rACTIONAL .0=FFICT::NT
H'-TICH i ANSLA TSF CH'AN',. IN L3CAL '3.EAC Or NT'ATI ON INTO A
CHANG IN LqCL 3F.AkR A NiLELE'3LONJ il972) SET IT EOUAL TO 1.i
no 3 J= LIM:IT
I=J-1 i
3 PHI (I) =o I( .) +CCLrF (GTI(L -3T( T 3 ))
P A ~u 3F r'AUTTON. TH' AjOV\- ADJISTMLNT OC .THE BREKE<- AN3; L. UST F
-ADE 1THOU -'IH TIF LL' I TlE- S:'3-ELN EAHIRI.TS Si.HAR -i.T TAL t'U .T
AN IS A RG1ON C" rA'rI CHANE. TrHEN TH-- APPLICATION OF- .fH'S A-F-'0jL'A
tMAY. L[~ TO iNST A3lL TT-S.TN T-I-S CA-S TT IS 3ETTEP TO oD OMrPUT- THE
N':.iK ALTO.GFT ET- .
'..* t ANn I ARE A.n. .,lT.

DO 1 j=-,LI Tr
I-- J-1
V(T) -TV (GAHA' OP(f ) ) ) (5 ./2,) ST N(2.,PHI(T))
1 0CI) =T* (I)
4L- SET -HI EtQU'L TO PFSN~,IT V-LLUE OF FBT .
00 J=1,,L.IMIT
-I=J-1

.'* C' I1.1I)=JE! tA(1I)

L(T) AND M(I) KEP.FS E"T THE LEFT ANO EIGHTT .dOUNCA7Y POINTS
ON AN I.lTERVAL D. WMHTCH HE WISH TO CuiMtPUTE A .NET TRAiNsORT. RATE.
POINT C
DO t; I=1,10
'iFT ( T) = .
L1=L (I)
Mi= M(T)
DO 6 J=L1,M
ONET (I)=QNET( I)+ 4-( J) (.J-1))
COiT I NUZ
W.RITF (8,8)-LlI I",CMFT.(T)
CONTINN.lU"
FORP A ( XIb T !? ( 'iu 1 XF7. 1)
FORMAT(1H leftET "D3I!T',5Sx, RIGHT POINT .-,5X, NET RATEr)
RETURN
END

SU-3ROUTINz INITL(N,INn)
T N TTL TS A S LF -< If.A. TOr s- NT L'iS F T-_ C iORJINATES, Y,V) ,F THF -3.7: POINTS FO USE:
c;r .IEPt F o T-Oui -r S .i If Q -ATIONS. -ri NEHO 1 SC T D I E TTzXEL-

i S i N X>1 (90 N X 2(9Y) Y 3(9 C)L YYi{9C) YY2(9C) ,YY3( 9 )
CO"i,' N 'EI(3 US) .3ET.( 9') T.FTAC('O)
CO 'L MO C X ( 1( r) ,X2(90) 3 ( .u) (9 ) Y1(9.) ,Y2 (1 ) ,V3.(90)
.C OMM DYl (9u) (9 1 OYi(' ) ,0X (9") YYT(90) ,aX2 O) ,Y3(93) ,
OUMMU,/3LOC2/LIMITT ,LIM1,LTM2.
.CO iMON/ L3C2/ TIV D .
...* IMA IS .lT, MaXIHJM NUI m.FR OF. ITERATIONS ALLOWED
T MA X= 2 .
r-*>- ED S 1S TH, L`'SSEST PFPPPITTED OIFFEF'ENuE bETWErCN 000OTDINITE VALUES
r GE NERATr0 ON SUCLESSIIVE ITLRATIONS CC'ONVcRGENCE CpITER'ION)q.
SEPS =' ~-
r, G cSSS IS A A R .APiTTA Cr CO ORDIN TE V1, LU ES.
GUESS= .3
CALL DE'-IV( 1 A ,1 C,3YG 0 Y 3, N)
PO0 2 T= -N ,
SYl.( T.).=YO (I) i S U E' S
2 X1(I) =XO(T)- JESS
CALL nERj ( X1 ,Yi. 1,DY 1,N)
PO z I=t, N
Y ) .( Y I f. !)-.-3 U ES'F
:=3 X I I -I f l-T)-) UE SS
C r; LL ''4 1)X/. Y X.- Y? ,N) .
0 0 i ,1 I, .N
Y3.(. )=Y? (T) UFS I1-
'-/ 4 XJ(I>=X2(I)-E, SS
CALL %E1I'f X3, Y-'.JX3 YY -,N)
IF .F.T=- T .^ 1 A
.*IFr..T F.i.E-., O I -'-iAX) S-3 .T 1.5 .. "

If

U.:-IJ-t ,l ) ,--Jr-; A [1 ;

r

00 -. T=1,N
XXI(I) = I< ( I)
XY 2(T) ( 2 ( T
XXs() =X3( I)
YY1(I )=YilfI
SY (I ) =Y 1 (I)
6 YY AR(I)= Y3(I)
DO "7 '=1
Xi (I) =XJ (T) I-T/2L'.* (9,"X^( I) +19.c DX1(TC -53.*ODX (I)+XD 3fI))
Yi (I)=-Y3 (I)+ T/C?t, '( vC( ) +i19. D' Yl (I)-5. DY2( I) +3Y3(T))
X? (I) = < (I) +rf/3. ,(' (T) +u." nx (I) UO, (I))
Y2 (I) =Y1 (1) DoT/-. + (" Y. ( I)+-. Y1( I)+DY2( I))
X3 (I) =X'(I) D3iT/ .'( 3,+U. 7 Y3(I)=Yj(1) t rT/8. ^(3. DY (I) +9.*)Yil(T_ -9.*OY2(I)+3.40'3(T))
8 I=1
9 IF(AR.SC(Xx (1I)-Xi(If;) LE, EPS.Ai.D.A3S(.YilC)-Yl (I) ) .LE.EPS)GO TO 1'
GO TO 13
10 IF(AQS(XX2(I)-X2(I)) )LE.EPS.ANJ.4i3S(YY?vI)-'2(I)).LE.FPS)30 TO 11
,0 TO 13
11 I (AbS(XX3 ) -X3(I) ).LF.EPS.AN,.ABS( YY3(I)-7r3(I)) LE.EPS)GO TO 12

GO TO 13

12 I. EQ.LIMIT)GO TO '3
rO TO 9
1.3 CALL DERIV(Xli,Y1,JX1, Y1l,N)
CALL DERI/V (X2,Y2, XZ2, yY ,N)
CALL DERIR(X ,Y73,3X3 ,DY3,)
GO TO 5
1- TND=1
0S TO 17
I IND=0
270 FORMAT(1H ,8(4X,FiC.3))
17 RE TURN
END

SUBROUTINE DERIV(XYyOX,DY,LIM)
,E ORIV COMPUTES THE INCPEMENTAL CHANGFS9OY AND DX,IN THE (Y,X)
COORDINATECS SIVEN BE-CM SEGMENT ANGLESTHE DISTANCE BETWEEN ADJACENT
-EACH POINTSGiEL./?, THE SPEAKER DEPTHS AND THE T:?ANSPORT RATS
ODISENSIC N DX(j-0) Y( 9?)) ,X(9 ) ,Y19.0),
ODINMNSION FACTOR(9 )
COMMON X0(90),X1(90),X2(90),X3( 90) Y.(90), Y(90) Y2 (90) Y3(90)
COMMON DYO(90),DX0 (90) ,5Yi(90)0o 1(90O ,DY2(90),OX2(13),DY3(90),
.0X3(90)
COMMON ED L(90) ,ErT(9C),THETA(90)-
COMMON/3LOCi/PT
COMiiHON/3LOC 2/LIMIT ILI 1,LIM2
COMMON/BLOC5/Q ,Q(90)
COMMON/3LOC'/IVDT
COMMON/3LK3/OB0q03 (0)
COMMON/3LK,/KOUNTIII, NDC(50)
DO 1 I=~jLIMI
i THETA(I)=ATAN2(X(Iii)-X(I) Y(I+i)-Y() )
THETA(G)=THETA ()-(THETA(2)-THET(1) )
THETA(LIM)TA(LI-ALI) ((LI)(THETA(LIM2)-THETA(LIMi))
00 2 I=iLTM
2 BETA(I)=(THETA(I)f tETA(I-1))/2.
.>$ IF WE ARE AT THE ST4PT OF THE PROGRAM(KOUNT=1) OP THE WAVE DATA
SET IS CHANGING W= .ENTER SUBROUTINE ADJUST TO RECOMoUTE TRANSPORT
FIGURES.
IF(KOUNT.EQ.i.OR.III.EQ.NDC(II)) CALL -TRANSPT.LI')

47

4 1 i T;?
3 i)L L .i 1. S .r ( (i t ) X ( +1 ) ) + (Y(I+ Y y( T I+1) ) *- 2)+S 0 r (.( y (.IT +1- I). -
'X(T)) ?+ (Y.( +I )-Y (T)) 2)
.-.. PEL(1) AN1 uEL(Ll ) Ab nO FIN- TN A SPECIAL W AY.
[ZL ( ) SO T ( ( ( ) -X ( 1 )) + (Y )-Y (.)) ".?) 2
[ I L( I ) =SOr 'T A ( (L l' L) -X t L.l ) + (Y(LIM) -Y(L.I l) )" 2) (Y .
DO 3 i1=,LIT
FACT OR (I)=2 ./1-3 (T) ( ( T) -0(1- ) ) /D L (I:)
DY(I)= STN( t _TA(T)) FA.CTOr(r T)
r-: DX(i) -COrS(?ET.A(3 ) 'F CTOR(I)
r r^- O -:IF, TIN T-A (O Rn I'JT!ATL SYSTEM CIOI tUSci T DEPTH G-tADIT-N- IS
r NE ATTV- THLi! Tu- IGNS PR)SELLIN THE SIN AN COS2 FUNiCTIONS
SAOVc RE E.'d.>F FrFc-D.H -itWVV: THE- SIGN OF V (EOUI I LENJTLY,T) ALSO
G CHANGES. THF -'NT EFFECT -S TO LEAVE TF-.QE-OUJATTONS .U4NCAhLSE..
OF TJ r-' .
rNr

SU 9"ROUTINE APF EA (, Y, LTM,A, 3,N)
r
r I GIVEN HE( COORRiN1TES OF THE oFA.i POINTS ?T 2 DIFFERENT TIES TiHE
r CHANGE IN SUBAFPIAL SURFACE A-REA OJER THAT TIME TNTERVAL IS COMPUTED.
r COOi c COORDTNATES (A, ) REFFF TO THE MQC RE RECENT POSI Tni LC.URVE 2).
r
r AN APPROXIM'ATF CA_DUL'ATION IS PERF3R"^ED ALSO FOR THE CHANGEE .IN THE
S. VOLUME OF MATERIAL.
f NOT' .TH'T .ABSOLUT- L'AULURTS OF ADEA AND VOLUME APE NOT..C O'UJ TED .UT
C RATHER THE RELATIVE C- NGE IN EACH.
r THIS SUBROUTINE C.,N BE MODIFIED TO COtiPUTE..CHANGES OVER A\Pt DS.CRETE
C LENGTH OF BEACH.
T ,
C THE SIGN CONVENTIONS I.STABLTSiED IN THIS SUoROUTTNE ARE FJO A
C COORDiN'ATE SY-STEM I1 WHTC- THE DEPTH GRADIENT IS POSIT VE.
( TF THE.JEPTH GRPUTENT IS NEAT:IVE.T4HE RESULTS CITED .3ELOR NEED
r TO 3E MOOIFIn.
DIMIEN'SION YDEL ( 9 ) (9 ) ,Y (90) ,9(90) ,A(9q ).
L3GICAL A1,A2,A3,A. bi,32,5s 3 .B
r0 1C'N .DEL ( 9j) BET.A( ) ,THTA T A(9i)
COMiMOi N XJ(9i.) 1(90) ,X2(90),X3(9G),YO(90) ,Y (9 .- ) Y;(
COM ON iJYr ( ) DX'!( 9r) Y1 ( 90) X 1 ( 90) Y 2(9 ) ,X2 ( ) 0 ODY3(9 ) ,
SDX ( 3 [ 1)

Qii *- J

VO LI =1 .
no 1 1i=i LT '1,I
ARE 1 =AR- E -+.r'- (Y1+ 1+1) -Y(T) ) ( I ) + ( I) ) ( D
1 VOL1..-.VOLi+. 5 (Y (I+ 1 -f1) ) (X {I+ 1) +X(I) .D9(1)
COJMoUTE AREA UNDJR CUPVE -
Ni =N-I
,&RE.".. 2=0. .
VO L 2 = G ,
nO 2 T=1, N1
AREA2= Ao~A-2+.-t5(A(I+1) A(I)) ( l+1) +3(T))
2 VOL2=.'-'OL2+. (A(I+1) -. ( i ) ( ( ;(I)I-1) +9 (.I) ) n3 (I)
rOVDUTTE THE A-EA OF THE FAP LEFT- TRAPE70IO A..T,.T.,
To A 1=A3S (. ("(1)-A (1) )" (X (1) -(.1) ) )
SVOLS3=TPAPIfrn( 1)
r' PIITE T HE H \RE' F0 TH Fi P.- PIG-HT T r.. ZOE A ,. D. I .T .
TRA P2 =A3S ** (Y(. TM) -A ( ')) Y"'-(X (LIMF-)'3 (N ) )
VOL4 TRAP20 r3 (LI M)
rOMPUT. THE CONTRI3'JTIC I U. TO TiH EX'T NSF TnN OF THE EN: IN.I T -LEPT
IF ( Y S T, J. ( oM -. ). -'-To. 3
D0O .I=1,2
.F ("Y ,' I l)-, T. .1 ) Yr:3" TOn 5
l- C') IT I % )_
5 KI +1
r *' A .N'E X(K) IS DEFINEDl (SOLELY :`F TH.E "PUPPOSES O0C THIS
r S.U3ROUUTINE) 3v LT\!EAR INTERPOLATYIN. A NeW Y.(K) IS ALSO DEFINED.
THE OLO VALUES AFE STORE.

K-Y=X (K)
X(K)=X(K) +(A(K-I)-X(K)) (A(1) -Y(K) /(Y ('<-!)-Y(K))
YK=Yf K)
Y(.K)=A(1)
Kl=K-I
ARE A=0.
p tLA=0.
O 6 T=1I, K
REA3=ARES3+.5 (Y( I+1) -v(I)) x(X(I+1) +X(T)
F VOL inL +. (Y (I+1)- (T)) (:< (1+1) +X(I) )'*DB(I)
Xt K) =XK
Y( )= YK
3 00. T = ,2?
IF(A ( 1). 3T.Y(1) SO TO 8
COUNT NU-
8 K- I+
rI* A 1EW 3(K+ IS DEFINED "Y LINER INTEPPOLATION.A MEH 1 (K) IS ALSO
c DLFINEO.THe OLD VALJES ARE STORED.
K=B3(K)
b(K.)= (K) (9=(K-i)- (K) ) (Y(i)-A(K))/(A(K-l)-A(K))
A(K) =Y (1)
K1=K-
AoEA3=0.
VO L= 0.
DO 9 I=1 K1l
AR .A3=A- A3+.5 (A(I+1) -A(I))* (3(I+1) + (T))
a VOL5 = VOL5+... (A I+1) -A (I)) (3 T+1) +3(1 )) 09(i)
B(K) =9K
A(K) = AK
COMPUTE THE CONTRIBUTION DUE TO THE PXTENSTOiN OF THE ENrDOOTMTS(RIGHT).
IF(Y(LI5 ).GT. A(N))30 TC 11
00- 12 I=1,2
IF(A(N-1).LT.Y(LTM)) GO TO 13
12 CONTT NU
13 K=N- I
Pv-U N, NEW K(K) IS DEFINED "Y LINEAR INTERPOLATION.A NEW ;(K) IS ALSO
C nEFINEO,.TE OLO VALJES ARE STOPEB.
cK=B( K)
3(K)=r3(K) +(='(K+i)-3(K)) (Y(LI M) -A (K))/ (A(K+1) -A (K))
AK=A(K)
A(K)=Y(LIK)
AREA=O,.
VOL6=0.
0 19 T=KN1
AREA4 AEA- t (A( +)A( ) -A(I)) (1i+1)+5(I))
!"+ VOLTA:VOLS'o5: [A(T+1.)-8tl)) (3(I i )+J! ))x03(Ir)
( K ) =: K
.(K) =,K
11 DO 15 I=1,23
IF( '(LI;-I)).LT.A(,M)) GO T0 16
1i CONTITNUI
16 K=LIM -I

fr* A NW X(EK) S OFIN F PY INEAR FNTERPOLATION.A NEW Y(K) IS ALSO
r DCFTNED .FH- OLD 4 ItS AL J STOPED.
XK=X (K)
X(K)=X(K) (X(K+ ) -X(K ) ) (^(N)-Y(K))/(Y(K+l)-Y(K))
YKY (K)

VOL= 0.
DO 17. =K,LTLMI
AR: A = AREA4+. 5 (Y(I i)-Y (I) ) I (X(I 1) +Y( I))
17 VOL =VOLo+.5 (Y(T1+1) -Y( I)) (X If t) +X(I) ) D (I)
X(K>=XK
T(. K =YK
r
L A.RL A2-REA1E
VOL VOL -J OL1
"' WE .NOW CO'PUTL THE A(.AtL (PIFA) ANn VOLUPETRIC (DTF/) C.IANGES.

.- x Al ,A A,AL ,j ?,I P ,3 A:E LCOG, ''L V iA.R LES 'HiC- DEFTN F HE
POSSI-L :r'':L NATION; "HI r 'L1 ,i T-E N IlPDTINTb OF 2 ST RA N LINES
A1=A ( i) .G T. Y i ) .A AP ) T iL 1)
A2=,(1) G". Y(1) .A4? ') (N) L1.. Y' (LTCI
A3=, (1) .L f .v( 1) AN t ( ) LT.Y(LIM)
A- A( l) .LE.Yl I).AMP. A(N) .GT .,Y-(LIM)
b[ =3 (1) GT.X( 1 ) X \PU, ,. r ( N ) T .X (LTI)
I2=: (1) .GT. X(1) A.f r, M (N) .L X (LIM)
i3=' (I).LToX i) 1 P. (N).LT.Y(LIM)
94 =Jt ) .LE. X( ) .ANDi. F (M) .GT.X (LIM)
r 4-. TH T- FINNITIONS C7 PTFA ANC. nFiJ 'ED'tn .n W I' CH COM31 I;',TION
r OF TH-iL 1~VE I I2J._ ST'TEMENIS IS TRUE.
" -

TF( A. AIU .9?)
IF (A 2 A24U. 92)
IF (A I-.A ,i -)
IF (A AND. D r)
IF( A ANn. 31)
IF (AL. AND. Bl)
IF (A2 AND. 3)
IF (A2 4 ND. 3)
IF(A.! ANr.84)
TF(AI .ANn. 4)
IF (A3. VNO.32)
IF (A3.AND.. 02)
IF(A1 .AND. 1)
IF(A1 ANDO. !)
IF (A3 .ANO. 3)
IF (A .ANO, 3)
TF(A A- Ni .8 2)
IF(AL.AND. 2)
IF (A 2 ANO. -)
IF(A2 ANOD. )
IF( A3 AND l)
iF(A3. AND. B1)
IF(Ai. AND. t7)
IF(A1 AND. 3)
TF A1 .AN.. 2)
IF(A AN 32))
IF (A3. AND. P4)
IF(A3.AND..31)
IF(AL. AN B3)
IF (AL AND. 3)

iF(AT ' 3.Si)
L- :.- 4"Ld a s

DIFA=C
DIF:-/=VOL
nTFA=-t
DIFd =- VOL
DTFA=A-TRAOT-TRAP2
niF FJ=J OL-Y OL3- V OL-;
nIFA=i, +TR Pl+T AP2
0IFJ =- CL+VOL3+ OL4
DIFA=4+TRAPl-TAP2
DIFV=VCL+ JOL3- OL4
DIF-=A-TRAP1+TRtP2
DIFV=VCL-JOL.j+4OL4
DIFA=4+?. "AREA3-TR4Pi-TRAP2
01FV= /O L+ 2. 0 L 5-V DL3-VOL
DIF =^.-2 .AoE-(3+TR.n4i+TR^P2
DITF 4=L-2. VJOL5L+VOL 3-VOL
DTFA= -2. "AREAA-Ta P 1 -TPAP
I FJ =J OL- ?. -V OL 6-VOL 3+V\OLb-
DIF A +Z.. AREALA+TRPl- TR'VP2
DIF J= CL+2. VOL5 +-O L3-VOL,
DIFA=A+?. *ARPA-TA-TPP-TRAP2
DIFJ= OL+ ?. OL6-VOL3-VOL4
DIFA=4-2. AREA4+TK l+ TRP..P2
DIFi=VOL-2. VOLot-V3L 3-VOL.
DIFA=4-2. ABrCAA+2.*AREA3-TRAPi+ToAP2
nIFO=JOL-2,* OL6+Z.*VOLS-JOL3+VOL4
JFA+2. 'AAR"AST2.*AREA3+TRAPi-TpAP?
DIF!=VOL+2. *VOL-2. *VOL +VOL3-VOL4
DTFA=?-2. ArkA3 2.*AR A. + TRAPi+TRAP2
DIFV=JCL-2., JOL 5-2, 'VOL6+VGL3+VOLL

., i v A :- I,-A 1rR ..-r P2

FND

SU30OUTINE RESET x:,Yy. L, J)
r r -ES-ET RE-AP AN GES THE EAC- H POiNTS FOR ANY SECTION. OF THE SHORELINE
C OP ALONG THE ENTIRE SHnRELINE SUCH THAT 1HE SPACING .BETWEEN ADJACENT
r POINTS IS MADE MrF:E OP LESS EQUALANO.,AT THE SAME. TIME-,DISTORTITON
r OF THE oAlGH SrHAPE IS MINIMTZED.
C L IS THE FAR RIS-T FIXED POINT A.D.T.T.
r J IS THE FAR LEFT F7TXE POINT A.D.I.T.
OItcNSION X(90) (90),n ELXY(90)
COMMON OEL(90) ,bETA(9" ),THETA(90)
r+x 'JICOUNT IS A 30UNT-R FOP THE NUMBER OF TTFES THE FOLLOWING PR.OCEt'.IRE
C IS ITEfAT0D.
IC OU NT =
C'+W. NJ'IT. IS NUiMB3R OF "ERMITTED ITERATIONS.
NUMIT=5
Ji= J+ 1
L2`= L-2
r THE PEACH A 'EShNt-T A4ISL.S AND THE OISTa N.I- 3ET ELTN SJ-OCE SSrIVE :
C POINTS TS CC-'1PIlTF D.
11 DO 1 I=J,L1
THFTA(T)-=ATAN?(Y((T+i)-XfT),Y(I.+i)-Y(T))
1 DELXY (L)=SO(T-((X(I+1) v (I)) -2+(v (I+l)f-v (Ji)-)"2)
[,vMPUTE A MEAN DISTANCE BETWEEN EACH POINTS.

r

(
(
(.

C.

C'J i' J
n0 2 I=J,L..1
2 SUM=SU.i -' LYY I)
DE LME L SUM/( L -J+1 )
T=J
*" EXECUTE TTE PROCEDULF AS D7SCRIEUD IN Ti;F TEXT FOR THL
TRA!4SLATTON O- TH- 3FArP- POINTS E. i- O!r THE FIKRT POINT
ALONG THE 3I~:H SESIENT TO ITS LEFT 'JUTTL THE DISTANCE 8ETWcFEN
I.T AND THE P*Tt!T JN ITS LEFT IS D~3 LNLW, cOEFTIN Ti:. SEGMENT
c.TWI-CEN T-iTq NEW POSITION AND Tri PTDO NT I'.:.IEDIATELr TJ ITS FIP mT
AND NOV= THIS S-COND POINTT ALO'G TH- NE' SEGMNT... TC.
?3 IF(OFLNEW- EL-Y (T) )1CI ?0 ,3
i1 X(I+1) =<( T+ ) C- (CF NE -DELAY(I)) SIN( 'H TA (I))
Y(I+1)= (T+1) (- OFLNE -OD ELY ( I ) 0 S(TnETA(I))
GO TO ?I
30 Y(X =:X I+i 1) F(DEL nNEt -n LXY (I)) -SIN( TiETFA(T+l))
Y( I+ =Y(t +1) + (EL N -i FL-F Yf ) fStTOS~T Tf, I+ 1)
1 DELXY (I f =S1 RT(( .rT+>?)-X(It)) F+^( I+2)-) +L))-2)
20 I.=I+
TF(T.EO.L) GO TO 22
GO TO 23
22 1CoJNT=ICO.UN!T+
IF(TiCOUNT.NE.NUMTT) :0 TO 11
*CK TO SEE IF TE' DISTANCE S-TWEEN ADJACENT POINTS IS APPROXIMATELY
WRITE(i,25) ( ELXY(I) I= ,Li)
25 FORMAT (2X,5 (3X 13.1 ))
RETURN
END

SUBROUTINE RESULT (X Y Z2, ZI L MAX)

C"*,* RESULTS.PRINTS OUT SOME OF THE MORE IMPORTANT NUMBERS GENERATED
C BY SHORLTN.
DIMENSION ZI(90),Z2(90),X(90),Y(90)
COMMON OEL(90) ,BETA(90) ,T-ETA(9'0)
COMM3N X0(90),Ki(90),X2(90),X3(90) Yf(90) Yi(90),Y2(9U),Y3(90)
COMMON OY0(99) ,0X3 (0) ,Y1(90) DX1(90),O0Y2(96) ,X2(90) tY'3(90),
;DX3(90)
COMMON/3L COMMON/BLOC21LI MITLI 1 LTM2
COMMON/3LOCS/QO,Q(90)
COMMON/3LOC7'/IVDT
COMMON/lLK2/P-IO,iT (0)
TIME=IV*DT
DAYS=TIME/24.
YEAOS=OAYS/355.
PRINT ivIV D,TIMEo.DAYS,YEARS
1 FOPMAT(3X,*D O.TT^ STEPS(TV)=,h4,5X,4TIME INOPEHENT (T)=.,F4,~I
~iy HRS-. ;Rf EL SED7 T E-( I4XUTi= Fil.1,ix, HRS ^,F3.:i x,
'(0 D AYS-= '" F 2, 1X4 '*Y EA', 7 )
PRINT 2
2 FO RMAT(//SX POINT13 X X(TI ME=0) ,10X,*Y{TTME=0)*,10X,X (TII E=
'TVXOT)* itOX, Y(T14E=YTVXDT)P,5XT*X( PRESENT) ,5X, Y(PRESENT) /17X,.
ZMEIERS*,13X, *METERS*,3X, METERS ,1X,*METERS*,15X,METRS*,
RiOX, M METERS) *
IF(MAX.GT.LI A) K=MA <
IF(MAX.LE.LIM) K=LIM.
DO 4 .I=i,K
4 WRITE(5 3)I,X(T),Y(I),Z2(I) ,Z (I),XX(I),YY(I)
3 FORMAT(X,1-i I3,3X,3(7X, Fi.2),16X,F1i.2,8X,F1G.?,5X,F1a.2)
PRINT q
5 FOPMAT(//X,*POINT*,i0X,3EACH ANGLEIOX'0XWAVE ANGLE*,0X,*9EACH
POINT SEPARATION* ,SX,'TRANSPORTS*/17X,^*DEGREES*,l4X,.*OEGR-ES*,13X
*METERS*~22X *UB3I METERS/-R*)
r- OONVETT. FROM RADINS TO DEGREES.
DO 8 1=1,LIMIT
J=I-i
PHI(J)=CVCT2*PHI(J)
8 THETA(J) =VRT2*THETA(J)
DO 6 I=1,LIMIT
J=I-I
b WRITE(6,7)J,THETA(J),PHI(J),OEL(J),Q(J)
**** N RI=1RDLM GPEE& 9ACK TO RADIANS.
J=I-1
PHI(J) =CVRTiPHI(J)
10 THETA(J)=CVRTI*THETA(J)
7 FORMAT(iH y ?XI3 79,FO.2,9XFI0.2,13XFit0.2,25X,F13,2)
CALL. ERROR (X, Y,ZZL,LIM,MAX)
RFET'RN
ENO

rSU OU 1TIN E OI Z 2 Zi, LI ,M4X)
r+-4 rRPn, 304N '3E 'nL MiNFF "FDSIATILE THAN IT IS NOW. T PPESET TT
r COMPUTES THE FoC-:NT L_ .OR OF THE 'REOTCTED AREAL ANDJ VOLUMETRIC
C CHANGES AS CuMA"PF.j TO THO.) OSERkIn. IT ALSC CAN CO'PARE OBSEnED
r ANn PRFiCTE.0 SHL KLT4E PUJSITIONS POINT -Y POTNT.
DIMENSION Zi T'),"Y(9 ), 3 LTT (3O) xaLT y (30)x(t nlv y(o)
CO_,lM N rEiL(3),) FTA( .G ),TH7TA(90)
k4- IM N 9 q

COH-MOr-/3LK 1/ X (Qr) ,Y .)
COMMU N/ LOC(/ DIFA, DI FJ nIF-r., TF'
r' *" SHORELINE 3N c ,O-.1^RE1 ITHT TFTSELF LT VARIOUS STAGES IF THE
c THE NIi',I3ER OF POTIlTS HHICH CONSTITIUT; THE SHORELINE IS A4LAAYS THE
r SAH(iHIS CthLO 6: FF-CTEO 3Y A SUROUJTINE HICH D[ECREASES O-R
INCREcAS-S THq NU 3 OF POINTS W-ILE RETAINING THE _EACH S-APE-
S SUCH A SU30QJTINE 13 NOT PPOVIDED HERETN).
IF(LIM.NE.AX) GO TD
PO 1 I=1, LIi-
.DELTAX(T)=?Z (I) -y (I)
1 DELTA.Y (I) =Z1 I) YY (I)
WRITE(b,2)
2 FORMAT(IH ,'-OELTAr (T) ~" -' LTAy(T),)
WRITE (6,4) ( I.,9 O LTAX(T) DELT 7Y (I) I=i, LIM)
FORM1AT (3RX, TL.,2 ?X, Fl.E 5 P9. 2)
5 PCTI= A3S((DIFA-DIF' )/OIF*10 0.)
PCT2= A3S(fO.IFV-CIF/1)/OIFv100o.)
WRITe(6to) PCT1,P-T2?
6 FO~MA-T(iH ,PER C.E-NT E"OR IN REAL CHANGE PREDICTION= FF4 1-//IX
*PEP. CENT El.ROR IN JOLIUMETRIS CHANGE PREDISTIOiN=-,Ft.)-
FE TU P N
FNn

SUEBOUTINE PLOTTE?(-(,Y,Z2,Z1,LIM)
^l- PLOTTER CAN MAKF AS IMAY SEPARATE PLOTS AS DESIRED ON A SINGLE
F FRAME.
r., I TITLE,LA:U,-LA9V ARE LLErn-Th LA3ELS.
?: ,i'rN iTLF (.') LAdUL(?) LA8V(2)

r U AN2 AF E THEI- LOCATIONS FOP ALL THE. HORIZONTAL AND-3
r VERTICAL COO INAT ,PESPECTIVELYTO BE USED IN A SINGLE FRAME.
DI'MENSTON U(120), (10)
CO:4 MMON O ( 90) ET -( -) 3 ),Y (- IVY2(9 ) ,Y3(90)
COlON YO(90)Y (9) ,DYi(93),DXi (90) r2(913) X2()(9 ;,3(9 ),
tOX3 (0C)
SCOM ON/3LKt/ XX( X 36) ,vOY( AS) ", .....)-
C"'" ,.QP IS I LESS THAJ rh NU iER OF PLOTS ON A SINLE F
IUSS(I) SPrIFIS TiE POINTS N THE AiRPAYS U AND W AT WHICH THE
ONLOING PLOT IS T PMNATED AND A NEW PLOT IS BEGUN,

I FOP THE SAKE OF C3ML-TENESS -WE INCLUDE THE OTHER SiUROUTINES
r THAT COMPLEMENT PLCTT~R,NAMEL EASY, SCALING GRAPH ANO O3ORDEP 0
S THESE SU3ROUTINFSIN TURNINTRFACF WITH ROUTINES(EG, PLOTSYMBOL,
S Nti.iBiEPETC. ) ROM TIE PLOTLI3 AN@ GOULD LI3rARIES.
A SAMPLE FOLLOWS.
COMiMON/IU3S/IUBS(I) ,TP
TJ 3'S (2 )=5
TUL'(: 2)=15
ITrTTLE(I) =oHi ST.

ITI TL () = H rH
IT TLE(3) -ISL N
ITTTLE() =7H 1i73,
ITTTLE(5) =7 1970
LAU( 1) ='HY 3O0 0 ,
LA L U(2) ="H ()
LABt(1)=7HX OCORO
LA (2) = ri (M)
PO 1 T=z iL
SW(I) =X(I)
1 U(I)=Y(I)
DO 2 T=1,1.
h(I+f~4)=XX( T)
2 (I+i- ) =YY(I)
CALL POTATE(WU,28)
C LL EASY(U, ,28,ITITL.FL 3U,L. 3V, 0)
CALL PLOT( 0,0.999)
100 IQ =*
IUPS(3)=19
DO 3 .T=1,1-,
W(I)=Z2(I)
3 U(I =Zi>(I)
DO I=1, 14
W( I+1 ) =X( I)
L U(I+i4)=Y(I)

CALL EIASY(0,q,2ATTITLELA3LAV0)
CALL PLOT(C,0,9999)
PETUN .
END

SUPROUTITNE ROTATE(K,Y,M)
r
C~**ROTATE RE-EVALUqWES THF BEACH POINT COORDINATES TN A SYSTEM THAT
r IS ROTATED THROUGH N AN3LE -N; WITH RESP-CT TO THE OLD AXES
C OEF MOTIVATION FOR THIF OPERATION TS TO MAKE CLEARER THE DIRECTION
r AND MAGNITUDE OF SHORELINE CHANGE I.E. ENHANCE VISUALLY THTiS CHANGE
DIMENSION X(N),Y(I)
COMMON/ 3LOC3/C VRT RT 1, VRT
rfl*ANG TS THE POTATION AlGLF IN RADIANS POSITIVE COUNTERCLOCKWISE
ANG=2u. z CVoT1
COSA=COS ANG)
SIN =SIN (ANS)
o \ T = K )
v0 i T=ieN!
Y( i)=Y ( ) COSA+X(I) INA
1 X(i)=-YSAVExSINA-X'(I) *OSZ 0
FETUPN
EN

SUO1,OUi NE AA.;Y (U,V,N,I i i TL ,-LA3U Ail-AiV,IDASH)
C THI S aU ROUTINE- PLOTS A STN'LE G. RAPh ON A E-ACKGROUNC THAT IS
C LAR TESAN. X I[ THE HORIZONTAL ARRAY, Y THE VERTICAL ARRAY,. N THE
C OF POI-NiS TO BE fLOT TED, TITLE A hOLLkERITh TITLE L A5X A HOLLERITH
C LADiL FOR X, AN9 LABY A HOLLERii H LABEL FOR Y. NAME IS A FILE
C NAME IN HIOLLERIi;H FOi;,MAf
C -A.JY AUJ3OUTItN4E -- CHECK .ON CALL FORVHOLLERITH VARIABLE
u-..i-lNSION U(N).,V(N> ,ITI LE(5 ,.LABUl2) ,LABV(2)
U MAX -U ()
U Mi NU (1)
VA ,AX=V( 1)
V iIN=V 1)
DO lu I=2,N
if (U(I) GT1.UMAX) UMAX=U(I)
IF(U(I) .LT.U'IIN) UHIN=U(I)
IF(V(I) GT.VMAX) VMAX=V(I)
IFI ( (1.L) .LT .VMIN) VMIN=V I)
lG CO-Ni NUt
SIFFX=UMAX-UMIN
01FFY=VMAX-VMIN
UMAAX=+UAX+J .03 DIFFX
U..UNI =Ut- N-0, .03ODIFFX
jVMAX =VMNAX+0 ..03* OIFFY
S/VMIN=VH ilT;- 0 3 OIFFY
DX=DIFFX/2C
XX=ALOGC' (DX) -+10.
KX=XX
KX=KX-luO
RX=OX/(,dJ.U **KX)
.P X 1i= '
IF(RX.GT.1..4) PX=2.0
IF(RX.GT.3.3) PX=5.0
IF(RX.GT.7.1) PX=10.0
DELU=PX'*OX/iX .
LAGEL=5
IFiRX.G T.3 .3.AN.O.RX,LE.7.1) LABEL=4
OY=-DIFFY/. .
YY=ALOb 0i DY) +1 0.
KY=YY
KY=KY-IGJ
RY=0Y/( 10. 0**KY)
PY=1.0
IF(RYY.GT.i .4) PY=2.G
IF(RYCbT.3.3) PY=5.0
IF (RY.GT.7 .1) PY==10.b
DOLV=PY*DY/RY
D -L V P Y 0 i.< / Y
CALL SCALNG(Ui-INUMAX, VMIN,.VMAX)
CALL eO :E-U.IT" U'AXVMIHn VMAX ,DLL bOELV ITTTLE, LAlULABV.LARE..
-; AL GL L S G L;N1 i I N U A V M I .MIT A X, ;)

END

SUBROUTINE SCAL.NG(UMINI, UMAX,VMINV, VAX)
CO H ON/ CALE/A ,7C Oly-, Y-, XR,XL
C i-IS SUEROUUrNE SES U! SE CALI;i. BEr'EEN (U,V) AND (X,Y)
C fUV ) ArLE (ABSC I2SSA,OR-I- JTJTE) AND 0(XY) -:ARE IN INCHES ON CRT.
C THE RLAIIONS ARE X-A*U+U, Y.=C*V+D,
C EQUAL SALE FACTORS (A AND C) ARE USEO IF THE GRAPH IS NEARLY SQUARE.
A=6.5/ (UM AXUMIN)
C=4.75/tMAX-VMIN)
R=A/C
IF(R.GT.1.5.0R .R.LT.L..7) GO. TO 10
A=AMIN1IA,C)
C=A
10 CONTINUE
.B=4.0-G .5*A*(,U AX+'UMI N)
D=2.9-CG,.5*(VMAX+VM'IN)
Y i=C.VM.AX-+D
Y0V=CVMIN+q
XL A U.lIN' +
X 1( MUHAX+B
RETUr t N
END

bU-1ROUU INL E RAP- kU,V,N,IDAS h)
COMMON/IUtj S/-U S(iO, [(
COMMON/CA LE/A, C CDXR, XL
DIMENSION U,N) ,,V(N) (6) (66
X=A VU(1)+B
Y=C v (1) +0
CALL PLOTir ,Y,3)
JO =S,-=IA3S ( IASH)
IF(JuASHiLE .9) GO TO 2C0
C COUNT NUfINBE, OF DIGITe IN 1LASH
NbIGIf=6
IF (JOASfH.LE9999-9 NDIGIT=5
I JDAS,-i.LE.9999) NDIIlz=4
If (JDAsi.LE. 999) NDIGiT=3
IF(JOASrH.LE.99) NCIGIT=2
C OECOMPOSE INTO FOUR INTEGERS
NJEN= 1LU3 0
NUM=JD AH
00 99 K=1,o
I K)= NUHI/NDEN
NU;l=NUM-I( K) NDEN

NhEN=NOFN/11
99 CuNTINUC
C MINIMUM SPACING IS 1/128 INCH
00 100 K=1,6
IF(I(K) .GT.7) i (K)=7
100 CONTINUE
NFIRST=7-NOIGIT
DO 101 NI=NFIRST,6
H (Ni)=,1./ (2.0 I(NI))
101 CONTINUE
C INITIALIZE
LEVEL=NFIRST
EXCESS=r(NFIRST)
00 103 K=2,N
C DRAW JASHEO LINE TO NEXT POINT IN ARRAYS.
XOLO=A*U(K-1) +8
YOLO=C*V(K-1)+D
XNE=AMU(K) +B
YNE:=C*V(K) -+0
X CIFF=ANEt-XOLO
YDIFF=Y NEW-YOD
TOTAL=SQRT(XOIFF XDIFF+YOIFF YOIFF+3 .00001)
ACOS=XDIFF/TOTAL
YSIN=YOIFF/TOTAL
TCGO=TOTAL
X=XOLD
Y=YOLD
C NcXT SECTION OF LINE
10' CONTINUE
ITE S T.=LEVEL+NDIGIT
IPEN=2
IF(( (ITEST/.2) *2) .EQ.ITEST) IPEN=3
P=AMiN1 (EXCESS TOGO)
X=X+P*XCOS
Y=Y+P*YSIN
CALL PLOT(X,Y,IPEN)

TOGO=TOGO-P
*EXCESS=EXCESS-P
F (EXCESS.LT.O. 0J5) LEVEL=LEVEL +1
IF(LEVEL.Gr.6) LEVEL=NFiRST
IF(EXCESS LT.0.005) EXCESS=H(LEVEL)
IF(TOGO.GT .0,C5) bO TO 102
C OTHtRWISE GO TO NEX[ POINT IN ARRAY AN3 CONTINUE
1l3 CONTINUE.
RETURN
C SOLID LINE GRAPH WHEN IOASH HAS ONLY ONE DIGIT.
2u0- CONTINUE
IQB=1
00 10 K=2,N '
X=A*U(K)+6;
Y=C*V(K)+D
IF(K.EQ.IU 3 (IQB)) 1C00 1001
10C 'CALL PLOT(X,Y,3)
IQ3=IQu+1
bO TO iJ
10-1 CONTINUE
CALL PLCT(X,Y,2)
10 CONTINUE
RETURN
END

SUiROUl NE or 00RDE (U-lINI,lMAX,VMIN,.-VMAX,;U,DV, TITLE LAUUtLAB:RVINC)
LIM N 0NS I 'L,. A (5 ,LA.U.(2) LAf3BV 2)
*CCMMON/SCA E/A, r_,C,'J, r 1,YBA R,-X L
C THIS .SUROUTINE- DRA;3 A kEC ANGU L R OR ER T IC. -MARKS AT ..DI. V
C INCREMt- t AlI ALON.- ArJCIS A, ORDINATE, iHEiN LABELS EVERY ING T tCK Ifi KS.
C lIT IIIT S ,RITfEN ALONu T.-E TOP, LAEU,LABV ALONG THE ABSCISSA,
C UxUDINATl AFcIS. THC FIRS FOJU A..GJMLNT ARE 1RANSFERED 'TO SET.
CALL aCALNG(UINI, uMAX,VMIN, V1AX)
U A LL r i L N G- U i-i I -N 11. AX I VM I V :'l A X N
ULAE=OU*INC
VLA3=Dv-INC
FLAG=
C OPAi 'LO:.uER (XR,Yd) iO (XPYi) TO (XL YT) (XLXL,YB) .TO (XR,Y3)
CALL PLOT(AR,YB,3)
CALL PLOT( XR,YT,2J
CALL- PLOr( XL,YT, 2)
CALL PLOT(XLYB, 2)
..CALL PLOT( .ARYB,2)
C NUMBER OF FIGURES TO RIGHT OF DECIMAL POINT
N;xU=1.5-ALOG10 (ULAb)
NRI=1.5-AL OG10 (VLAB)
IEU=0
IEV=-u
IF(NRU.GE.4) IE U=2-NRJ
IF NhKVr.GE.. 4) IEV=2-NRV
IF(NPU,.GE.4-) NRU=2
IF(NPV .E. 4) NRV=2
QU=AMAXI(A3S (UMAX) ,A3 (UMIN))
MU= ALOG 10i (.U)
IF (MU.GE.L4) IEU=;-iU
IF(U.U.GE.4) NRU=2
Q =AMAX1(ABS( 'VMAX),ABS (HMIN)
MV=ALOolO(QVJ
IF(MV.GE.4) IEV=MV
IF(MtV.GE.4) NRV=2
C SCALING FAGTORa FOR U AND \ .
FU=1O0.0J*IEU
.,!-- *F =v'1.F=i .6**IEV
N CHU=N MRG U
IF(NRU.LT. ) NCHU=-I
N CH V =Nr, v
IF (NRV.LT. 0) NCH =-
C TIC KMARKS ALO.U A oCi,:;A

U:U+ -, LQJ J51'D
20 CO i INUE
U=U-cU
I=1-- 1
IF(U.LE.UM'IN) GO TO 30
C OTf:-i HIE uKiA C TICK MARKS
X=A*U+B
C .- T ICK-= 0.084
IF ( ((I/INC- IN C) .EQ. I TICK=2. 5*TICK
Y B= Y +Ti ICK
CALL PLGCT(XYB, 3)
CALL PLOT(X,Yu5,2)

C DRAW .LIN' F:F'O; X X,Y ) TO (X,YT-0.1)
Si =YT- '-K
CALL PLOT(X.YT 3)
CALL PL:O.T(XYTtI,2)
IF ( l/IiNC) >:INC) .NE.I) ,0 TO 20
C O HEKwISL, -h RiTE U BE.LOi- TICK MNArK
C NUMBER OF FIGURES IC LE.F- OF DECIMAL POINT

: iL.U= 1 *
SUIFU
I- (A:I.;(UU) ..LT..OG]l) GO 10 25
N-LU= LO 10( ABS(UU)) -1
25 C.nrt r JJlln

IF(NLU.LE. ) NLU=I
NT=N LU +NC-I U
XLA=X-J. 07*NT
YLAJ=Y-0.2
IF(NCHU..GT .-i.ANJ.NChU.LT. 1) NCHU=G
CALL NUH3ER(XLAB, LAB, .14,UU, .,NCiiU)
GO 1 0 2
30 CuNTINUE
IF(XLAB.LT.XL) IFLAG=1
C TICK MARKS ALON- ORDINATE
J= (MAX+OV)/DV
S=J*CV
V=/j+3 .i000IODV
4l CONTINUE
J=J-1
V=V-OV
IF(V.LE.VMIN) GO TO 50
C OTdERWISE DRAW TICK MARKS
Y=C V+0
C ULRA'W LINE FROM (XL,Y) TO (XL+0.1,Y)
TICK=:. 04
IF(((J/INC) *INC> EQ.J) TICK=2. 5-TICK
XLL=XL+TICK
CALL PLOT(XL-,Y3)
CALL PLOT(XLLY,2)
C ORAr LINE FROM (XR,Y) TO (XR-0.1,Y)
XRR=XR-TICK
CALL PLOT(XRY,3)
CALL PLOT(XRR,Y,2)
IF(((J/INC)*INC).Nrc.J) GO TO 40
C OTHERWISE, RITE V NEXT TO TICK MARK.
C NUMBER OF FIGURES TO LEFT OF DECIMAL POINT
NLV=1
Vv='//FV
IFtABS(vV) .LT.G.j1) GO 10 26
NLV=ALOGi-l (ABS(VJ))+1
26 CONI NUE
IF(NLV.LE.1) NLV=1
NT=NLV+N-1V
XLAB=XL-0.06
YLAd=Y-u.07-NT
C ELIMINATE LAST LABEL NHEN OVERLAP CCULD OCCUR AT CORNER.
IF(IFLAG.EQ.I.ANJ.YLAB.LT.YB) GO TO 50
SF(NCHV.GT .-l.ANO.NCHV.LT.1) NCFV=0
CALL NU;'1BER(XLA=!YLAD G14, V,9G.,NCHV)
GO TO L40

C i IN''I FLE ON TOP BORDE~, AESCISSA AND CROINATE LABELS
HT=0j.14
XJ=3.4
Y=YT -0. 2
CALL SYM3OL X,Y ,HT,IITLE 0. ,50)
ISCL=8H( X10 )
X-XL+-0 4
Y=zY-G-.5
CALL SYMBOL(X,Y,HTLABU,0.,20)
IF(IEU.EQ.0') GO TO 60
C OTHERWISE, WRITE SCALING FACTOR
X=XL+3.2
CALL SYM30L (X ,Y ,HT,ISCL,G. ,8)
X=XL+3.8
Y=Y+0 .1
Q=-IEU
CALL NUMBER(XY,3.07,Q,C.,-i)
60 CONTINUE
X=XL-G.3
Y=YB+0.2
GALL SYMBOL. (X,Y, HTLABV,90. ,2J)
IF(IEV.EQ.0) GO rO 6.1
C OHLR ISE, I RITE SCALING FACTOR
Y=:fz+3. J
CALL SYI30L (X ,Y-,HT,ISCL,90. ,8)
y =A-+-. 1
.Y Y+0.6
Q=-IEV
CALL NUMBER (X,Y ,G. 7,Q,9 ,-1)
61 CUNilINUE
RETURN

VII. Apalachicola Bay

1. Nature of Inputs

St. George Island is part of a barrier island chain in

the Apalachicola Bay region of northwest Florida (see Figure 5).

Because of its location it is subject on the average to low-

to-moderate wave energy levels. Waves propagating from deep

water toward the island pass over a broad, shallow continental

shelf region and experience bottom friction damping, the degree

of which depends directly on the wave height and period

(equivalently, wavelength) i.e., the higher, longer waves are

attenuated more rapidly. The net energy loss can be substantial

when integrated over the total travel time from intermediate

depth water to the point of incipient breaking. Other means by

which energy.can be subtracted from a wave train are the pre-

s-.nc of adverse winds and shear currents, and non-linear wave-

ay inl-eractionc (including dissipation due co capillary wves)

In shallow water the'wave energy density.increases, competing

effectively against bottom friction to enhance the wave height

and induce breaking. An exception may occur on very-mild slopes

where bottom damping is sufficient to extinguish the wave.

Depending upon the bathymetry,refraction and diffraction ,can

augmentt or reduce -the local wave height.

Walton "(1973) in a study on the distribution of littoral

drift,along the entire Florida shoreline considered deep-water

,wave .-data. ,as -shis sour-e .of wave information and incorporated .in

Figure 5. The Apalachicola Bay region.

/6

G** *L
9Y

I5

his model the influences of bottom friction, shoaling and

refraction. We .have chosen to utilize Walton's model on wave

modification in shallow water.as the means by which we generate

the breaker data essential to our model, b(y) and H, b(), the

breaker angle and height as functions of longshore position.

We will not detail Walton's work, since he has provided a

thorough explanation of his methodology, but rather outline his

general approach and the changes we introduce.

The wave data source is the U.S. Naval Weather Service.

Command, Summary of -Synoptic Meterological Observations available

from the NOAA Environmental Data Service, National Climatic

Center, Asheville, N.C. These are shipboard observations of

meteorological and sea conditions made by ships.in passage. The

drawbacks inherent in such data are many (we shall not en,-ulmerate)

but they represent the best general compilation of marine data

at present. The record extends through the years 1865-1971 with

eic -ty percent of the observat-ions occurrin-g during the period

1954-1971. The .pert inent annually averaged tables are Table 18,

which gives the percent frequency of wind direction versus sea

heights, and Table 19, which gives the percent frequency :of wave

height versus wave period. Using these tables several bits of

information are computed. The frequency of occurrence of a wave

of a given height, period and direction of propagation is de-

termined and expressed as'a fraction of the total time of record

and is subsequently adjusted according to the following formula:

the geographical oceanic region which is assumed to contribute,.

waves to a specific coastal area is divided into "data squares";

this necessitates that data from adjacent squares be weighted.

Walton used 20-40'data squares as shown in Figure 6 to blanket

the Florida coastline and linearly interpolated the wave climate

between adjacent squares. We chose a set of finer resolution 10

squares in the Gulf of Mexico (Figure 7) because of the high

density of data in each square and the coverage of the Florida

Panhandle. Our method of weighting, somewhat different from

Walton's, is illustrated in Figure 8. A reference line is drawn

due south of.St. George Island. Additional lines are drawn to

the center of each 10 square and the angle, 6, between these lines

and the reference line is measured. A weighting term, d,,with

respect to.62, is determined from the formula

2 .i 1 (52)

where 6. 283, 8., --17, -36>, -51., -58, -6 i ,

so that d = 0.417. The individual weighting factors,

are then applied to each wave type in the respective squares to

ascertain the contribution from each square to the mean frequency

(the fraction of time over which a specific wave endures):. At

this point a set of deep-water input data has been established.

The next step is to track each wave component into shore

monitoring its change in direction due to refraction and its

change in amplitude due to shoaling, refraction, bottom friction

and percolation through the sand grains. Walton's numerical

model to accomplish this has the following structure:

Data 'sauarcs used in Walton (1973).

*- *

Figure 6.

.I '

; I -3 Q I i 2 *. i

__ I __I 1

-..... "--- "-- --
4 ...i i _.__ _
I i

I
..- .. .i

-- . s- .. i r -

S I u d i n i

Figure 7. Data squares used in present study.

g I
S3

tc~*re c

Figure 8.

Geometric weighting of annually-averaged data from

each square.

1. The orientation of the wave fronts as they approach

shore is computed using Snell's law of refraction for a bottom,

topography composed of straight and parallel bottom contours.

The refraction and shoaling coefficients are calculated con-

comitantly.

2. The computation of the coefficients of bottom friction

and percolation follows the work of Bretschneider and Reid (1954).

Required inputs are the lengths and slopes of a series of bottom

sections comprising a bottom profile normal to the stretch of

shoreline being considered (rather than the true profile over

which the waves pass).. In our model we consider seven profiles

coincident with the seven lines in Figure 8 and weight the

results in the same manner as before.

The product of these calculations is the breaker height,

the breaker angle and the fractional duration of each deep-water

wave type for a segment of beach. Repeating this process for

each beach segment of interest we obtain a longshore distribu-

tion of breaker heights and angles, each of these quantities

contributing independently to the magnitude of the longshore

current, e.g., a decreasing angle of incidence longshore could

be offset by an increasing wave height and vice-versa. The

reader is referred to Walton (1973) for a more complete discus-

sion of the assumptions, approximations and limitations under-

lying the above data reduction and analyses.

The application of our numerical model requires that we

discretize the strandline in a coordinate system established

with reference to some semi-permanent landm-rk. A feature which

is present on all the bathymetric sheets.of the U.S. Hdrographic

Office that we have used (which provide us with a. progressive

history of the St. George shoreline) is the St. George' light-

house on Cape St. George (see Figure 5). Stapor (1971) has

indicated that the lighthouse, constructed in 1847, has a margin

of error associated with its position on the charts which falls

within accepted map standards. The lighthouse is the origin of

our coordinate system which, for convenience, has its ordinate

running due noxth and its abscissa due east. The shoreline of

1873 as depicted on smooth sheet No. 1184 is divided.into 57

segments; the northwest tip of the island is beach point 1, the

northeast tip beach point 58. The points are irregularly

spaced, being packed more closely where the.beach exhibits

large horizontal curvature; the maximum spatial increment is

840 m (in the mildly concave middle section), the minimum about

320 m (in the area of Cape St. George). Resolving -each beach

point into vertical (x) -and horizontal (y) coordinates, the

separation between points and the angular orientation of each

segment is straightforward to calculate. This information, when

fed into the wave modification program previously discussed,

ultimately determines the longshore variation of breaker height

and angle.. Supplementary information on the Apalachicola Bay

region is provided by smooth sheets:H1265 (1974), H5794-5 (1935),

H5319 (1935)., 2265 (1896) 6788 (1943). The 1873 .strandline

serves as the baseline for the predictive model. If the

present (circa 1970) strandline can be generated, even

qualitatively, then the model could be used, albeit cautiously,

for future projections.

2. Long-time Integration of Predictive Equations

The integration of (43a,b), as they apply to St. George

Island, cannot be accomplished blindly. One must be aware of

any special features that contribute to the dynamic balance of

the island.

St. George Island in 1873 was composed of three-parts

separated by two hurricane-cut inlets (the bay side of the island

is marked by hurricarwashover deposits). In Figure 9 is a

schematic diagram identifying the major sections and the inlets.

The stability of tidal inlets is a complex.problem which we do

not treat here (see O'Brien and Dean, 1972; Dean and Walton, 1975).

The littoral drift past an inlet can be interrupted and sand

deposited, leading to closure of the inlet. The question of

closure rests on knowledge of the inlet cross-sectional .area,

the tidal velocities, the wave climate, the magnitude of bottom

and side friction and the level of littoral drift. We assume

that the rate of sediment transport across the inlet is reduced

relative to its upstream value by some fraction. Since Sand Island

Pass and New Inlet Pass both eventually close, an estimate of the

volume of material contained in these inlets, the period over-which-

closure progresses (assumed to be unidirectional) and an average

upstream littoral drift rate can yield a value for this fraction.,

An alternative method for determining the rate at which the longshore

SA; AC HI C" .
"1. .

5AAYf

i SLe A5D
...J .

\- ---c-

Sy/D of' r~ .xico

Figure 9, The southwest portion of st, Geirge sand, 1873.

drift is trapped in the vicinity of an inlet is to obtain an inde-

pendent measure of the growth rate of the shoals insideand outside

the channel. In the absence of such information we simply.assume

that the transport rate across the mouth of the inlets is'the

average of the upstream and downstream values. This is an unwar-

ranted assumption if the inlet does not bypass a substantial por-

tion of the longshore sediment load for, then, the downstream

shore is likely to be cut back due to sand deprivation.

Due south of Cape St. George is an extensive series of

shoals projecting some 8 kilometers into the Gulf of Mexico. These

shoals are focus areas for incoming wave energy and, consequently,

thebreaker energy expended on the shore to move sediment is re-

duced. This submarine relief will attenuate, re-direct, or even

block waves propagating toward the Cape. We expect that the level

of wave activity in the vicinity of the Cape, as computed previously,

will tend to be an overestimate, at least, in relation to the energy

levels at connticuous portions of the beach. We, therefore, ~reduc

the energy input to this region, due to waves from the south and

southwest, by about 25%; this figure is arrived at by considering

the degree of wave damping over this special bottom relief and the

percentage of waves that are likely to break far from shore.

Off the northwest tip of Sand Island is a rather permanent

shallow, submarine feature, the East Bank. The tidal ebb flow

through West Pass has transported local material seaward-'and the

wave levels have been too low to reverse this trend and confine

the sediment to the littoral zone. This shoal, which sweeps to

the south and west, almost attaches itself to the shoreline. With

the Cape St. George Shoals intercepting waves from the southeast

and the East Bank doing the same for waves from the southwest, it

is anticipated that the levels and periods of wave activity in the

.area of Sand.Island will be diminished compared to those values

computed in disregard of these prominent shoals. As Iefore, we

reduce the transport figures, accordingly. In addition, much of

the longshore drift toward West Pass is likely to be diverted to

the East Bank by tidal currents in the presence of low incident

wave levels, i.e., only small quantities of sand will be deposited

at the tip of Sand Island. This situation will prescribe the boun-

dary condition at the northwest end of St. George Island in our

model.

It should be noted that there is an overall bias toward

low wave energy in this study. Since ships tend to avoid bad

weather our deep-water wave observations are on the low'side.

Also, major storms, such as hurricanes, can cause rapid and marked

fluctuations in a beach system. We are presuming that over a long

period (e.g., greater than 50 years) there is a "smoothing" effect

line responds to long-term forcing (e.g., hurricane breaches in

a barrier island on a tidal sea are usually repaired on a relatively

short time scale).

To be specific, the integration of the governing equations

for St. George Island over the period 1873-1970 was carried

out according to the following procedure:

I. The wave characteristics at the breaker line (height,

angle, fraction of a year over which a particular wave acts), as

generated by Walton's program,' serve as input to our model. A

cumulative frequency of occurrence of all waves yields the time,

expressed as a fraction of a year, during which onshore waves

are expected, e.g., for St.. George Island it was found that

on-shore (breaking) waves are present about half the time, the

exact number being 0.51. Thus over a 98 year interval relatively

calm periods prevailed for approximately 48 years. The sets

of breaker data are inserted in random order into the model to

compute the forcing function for the longshore motions. Equations

(23a,b) are integrated for a number of time steps equivalent.to

1 year. This process is repeated for as many years.as desired,

i.e., one year does not differ from any other year inasmuch as

the deep-water wave climate remains unchanged (although the

breaker angles do change in response to the evolving beach

shape).

2. In (24a,b) it is found by trial that choosing a

step size of At 50 hours and applying the corrector twice is

the most efficient compromise, i.e., the truncation error is

kept small and the integration proceeds fairly rapidly.

3. As mentioned in section V special care must be taken

at endpoints. We see in Figure 4 that for point jr Qj-_' j

and A. must be defined differently than the same quantities as

they apply to the interior points. By endpointss" we mean those

points at the extremities at or near which the longshore trans-

port approaches zero (i.e., Q = 0). This definition is offered

in lieu of more detailed information about the tidal, wave, ,and

current dynamics in these areas. A more formal consideration

of the sediment flux at the tips of St. George Island would

entail finer wave refraction and diffraction computations, a

knowledge of the magnitude of the tidal streams and of the

leakage of sand from the island to offshore shoals. By refer-

ring to the boat sheets and noting where the-shoreline beiains

to curve.inward away from the. predominant wave direction we

choose the endpoint positions and measure manually the effective

beach angles,, 0, at these points. These angles are important

in that they determine the direction in which the ends pro-

grade or recede; they are adjusted as the integration proceeds.

The'initial at the northeast tip of.the island is 2T,/3 (radians),

*at the northwest tip it is 31/2. The endpoint A is twide the

distance between the endpoint and. its .neighboring point.

4. Other parameters in the model assume the following

values:

K = 0.4

.N = 0.05

It was found that the magnitude of the sand transport was not

particularly sensitive to variations in N.

Results are displayed in Figures 10-15. St. George

Island is viewed in 3 sections: 1) the Cape St. George region,

2) the long, arc-like middle section, and 3) the elongating

northeast tip. bapporting evidence is pro-ided by Stapor (1971)

who determined the areas of erosion and accretion for shorelines

in the Apalachicola Bay .region. He considered the redistribution

of the bathvmetric contours to be indicative of.the direction and

Figure 10. Southwest St. George Island as recorded in 1873 and

1970.

'- ,, E
r:JV i E

.T .:2 L ,I U

-. The letter

has under

*s 'A' and 'E' signify that the more recent of the
once local accretion or erosion,respectively.

two strandlines

A

N AN!) (a ~AS9

R*

\ \
N
\% "% fF7

OCEAN

C-,
n- r-,

,. ~I..

II '4 I

V r3L ? -

\ '\

Ai 70 \ -
( ^ *; \...7< >
\ '; r^ .*

A L-A----^
.. "*^

0 1. -ri-c__

( .',

lc / 0

-t

E. .

: I

!_ j
,,

-L; i

Piguid 11. A comp. rion of southwest St. George Island in 1873

and 1970 (predicted).

r -* .

"-. .

'ANP IS. PASS

A

OC EA J

,o.

_I.. '^ -. '. _

1'/13

A

( g c. ),

1-I-~.~I:-
A. l i

--' C-

Figure 12.

The middle section of St. George Island as recorded

in 1873 and 1970.

r-. -
D i

I L.} I N L

(c t0

L

L

r

"i
1

AX

-~

'i

lII

}pL~
,~~~~1

1 7 1/ 9

' OCEAN

c.

---~-----~-I----~---~ ------

---

-- -- -, --- n~ f -- -- -- --

o}"7 ,

I/

,.. )

Figure 13. A comparisor- of the middle. sectich of St. George Island

in 1873 aid 197C predictedd).

T -- --
L-- u -

r r

A,'
h -

-: (73
E ."

I-

I-

I-

*- -

A
S- flNEW INLET

C R'D

*Y CYGORD

~1 .Z4L

1. Ff

l XI ij

--
r.

mL

~3Lfl

OCEAN

I- -,

-L

1.

Figure 14. Norlihcast St. George Island as recorded in 1873 and

1970.

0- r'- r

U-

Ir

r7 ti

I' r

J.~~~~~ ~~~ ..; ,1-...~-- l~.~.: ,_~__ J

I-.-

OCEAN

-.

i-I
- ,*; 1

A

K~ /7 -

/F3

A ,:.

/ /;

//

t ./

.__J.. _. ._L.
-I
L ,r C:

ti r'i

Figure 15. A comrari::on of northeast St. George Island in 1873

and 1970 (predicted).

j 9 -

-r,- i c ,, M

' 0 ,/

/

--

C I

C
i "

J-

2.. -

Z;L

3 20

O2.8 3 OQ4

iF73

-4' '-

vY C,0 0IRD

.L.
"- i
x
I

%-O
1-

- -~-----~--~--~ 1

...--L.

( M.

( X I 1

and volume of sediment transport. By subtracting isobaths as

given on old and new boat sheets and contouring the results

Stapor was able to compute and. approximate sand budget for

St. George Island and adjacent areas over a 70-80 year period.

The salient differences between the actual plan profiles

of 1970 and those of 1873 (Figures 10,12,14) are:

1. The Cape St. George area undergoes a lateral shift

of its strandline to the north and west, i.e., there is

erosion to the east of the'Cape and accretion to the west. The

northwest tip of the island is cut back slightly. Sand Island

Pass is closed. These features are confirmed by Stapor's

computations as seen in Figures 16 and 17, where, in addition,

sand deposition is observed on the East Bank and the Cape St.

George Shoals. Our predictions (Figure 11) show the same trends.

as Figure 10 with the exception of the inlet closure. (note: since

we have not modelled thee inlet dynamics it is not expected that

the mou t of the inlet will enlarge, diminish, or migrate

appreciably.

2. The long, middle section of the island, over most

of its length, experiences erosion (Figure 12). Stapor (Figure 18)

indicates a sand transport direction away from the concave
3 3 3 3
middle: 76 x 10 m3/yr to the southwest and 60 x 10 m./yr to

the northeast. By expanding our concept of 'control volume' to

include any length of shoreline we can compute a net volume drift

rate for a particular stretch of beach acted on by a particular

wave climate. By summing over the difference.between successive

transport..rates, i.e.,

Cape St. Ceorge

ST. JOSEPH'S

POINT
1 8 74. --
l3-- V

0 3 10 .( 70 &a
C_=^<'Ki~~Ipl~

CAPE SAN
S1B74---

5 DOG ISLAhD
ICi3---

C? // -Je

1' E ST. GECF
ISLAND

. lighthouse

1 0 1 2 3 l.

, WEST PASS AREA

S1GE 6 ---

S1942-
> U C^ ," 1 90 ? -- -----

Shoreline changes, Apalachicola Bay (Stapor 1971).

/.

Figure 16.

i/

... .

/ \ \ ,.L .\C .-
*. \" ... .[{ "

-'

Bathym.try, 1942, 6 foot C.. \
Shoreline, 1942 \
Deposit;oaal sitea '
Erosioaal site C.L
Ligihtlh.- e Aad dat, t1'
LiliI~~:e ~ CAPE
ST. GCEO6
SHO AL .
f t i

Figure 17. Erosion and- deposition patterns, West Pass Area
(Stapor 1971).
-" . X 7" *' .

.. .,f tU ,- .{

isyiU 55

.' 18. T- fig e ..-'Gr Is and aart

Figure 18. Transport figures for St. George Island and adjacent

coastline (Stapor 1.971)

(53)

(see SUBROUTINE ADJUST in computer program), for our range of

wave parameters, we obtain figures, corresponding to Stapcr's,
3 3 3 3
of 4-25 m /hour (approximately 35 x 10 220 x 10 m3/ye.r).

Figure 13 displays an.overall erosive pattern. but differs from

Figure 12 in the degree of shoreline retreat.and the position

of maximum change.

3. According to Figures 14 and 16 the northeast end of

the island has been growing (a spit-like feature). Figure 14

indicates :that areas adjacent to the expanding tip ha-ve advanced

seaward interrupted only by a. few smaller pockets of erosion.

Figure 15 predicts a substantial growth at the endpoint although

not the eventual sharp veering to the north (away from the

dominant direction of wave approach) seen in Figure 14. In

addition the prediction shows the shoreline being cut back

along the entire stretch of beach upstream of the endpoint,

i.e., the tip of the island is being fed sand from nearby

beaches as well as from the island's middle section. This con-

trasts with Figure 14 which shows the middle section to be the

major contributor of sand.

Because the wave data is averaged on an annual basis and

extraordinary wave conditions (e.g., hurricane-produced) are

filtered ;out it is not expected that future trends will differ

significantly from those occurring in the past. Using the

observed strandline of 1970 the nearshore wave field is re-computed

and a 20-year projection is made. The results (Figures 19,20,21)

exhibit patterns similar to those computed before,such as the

erosion of the concave middle)which makes it susceptible to

breaching. The exception is the northeast tip of the island

which now turns inward as well as prograding northward, i.e.,

the beach reacts to the incident waves in such a way as to

minimize their erosive effects. In the presence of an ample

sand supply and low wave energy it is unlikely that the end of

the island will curl in appreciably; rather, its excursions in

a westerly direction should be intermittent, being counteracted

by sand deposits sufficient to direct the tip's advance to the

north and east. Figure 21 again points to erosion immediately

upstream of the tip with accretion farther west.

The foregoing predictions in most cases are in reasonable

agreement qualitatively with observed strandline changes. For

the results to be more pleasing quantitatively certain improve-

ments are obvious: (1) the quality of the wave observations could

be enhanced by in situ recording of swell and local sea (shallow

water wave generation is.not accounted for in this study);

(2) rather than considering the depth contours to be straight

and parallel offshore of each beach segment, a formal refraction

analysis using the actual bathymetry should lead to better fore-

casting of nearshore wave conditions; .(3) an attempt could be.

made to model the effects of severe storms (hurricanes).' Strictly

speaking, this falls outside the outline of the present report

Figure 19. Southwest St. George Island, 1970 and 1990 (predicted)

r*
C CE

S L rI i i

*

~N

"*. "\ /970 (oCeirv ef! )
A \

\ "\

'S

.e dcoe 4

5AND 35.
PASS

1-J-- I
'--

!-

V I

^ L

S-r-

l E70

OCEAN

4j'.., ,'i.-J-- L I-

~---~---~~-~-----------T--7------------- -T 1- i ~7

LL ~----i---~---------.

1 '^ *,r

Figure 20. Middle section of St. George Island, 1970 and 1990

(predicted)

7' 71 V

.I L R Li

.I C'3

j/I .^ '

.-r
.' y :

!-- ._
/

/!rcd;'''d '

L-^
/r

1 /"

/'

/

42
A-
A

p ~

OCC AN

1 '

v, -' f
I F -

i', '

i .
I. .. I

f s'frSt F*P -

1T

'm--.'

Li

-i
I

.-.--- 1
._ J id -

.A I

r

'/ -.i

i i L

I*~~C~I (Il~VI*R ~ m"Vv C"~

Figure, 21. Northeast St. George Island, 1970 and 1990 (predicted).

,T'-; ) A

(pred;c^cZ ed
_. r,- E ...-."

.i ^: .... '. .

2 .... 2 8 3 O 0 '

.S i .Mi X*C -

-- ,-

. ... : .OC E
"" ,. 0 :
1 '
'*,~ z "s ~ J"
C 0 R )( X t .

because longshore drift would then only be one of the important

components of motion--the offshore-onshore movement of sediment

associated with high water levels and high waves would be very

significant. The present model could be used to monitor a beach

after it has incurred heavy damage to ascertain what contribution

the longshore transport makes to restoration.

In addition, for a coastal region with complex endpoint

boundary conditions (e.g., islands, spits, capes, penisulas) a

packing of thebeach points at the ends is advantageous if it is

accompanied by a finer resolution of the wave field. Since

nearshore wave measurements are usually lacking this would have

to be accomplished by more careful refraction and diffraction-

analyses. The diffraction analysis could be based on experi-

mental data or be an approximation to the existing mathematical

theory for ideally shaped barriers. Furthermore, in such regions

ere there is sharp curvature of the wavefront the longshore

gradients in the breaker wave amplitude and incident angle may

drive a non-negligible longshore current. Shoreline change is

often manifested most dramatically at endpoints; this dictates

that we treat these-boundaries in special ways.

VIII. Jupiter Island

The Jupiter Island phase of this research was undertaken

in cooperation with the Coastal Engineering Laboratory of the

University of Florida. Jupiter Island, about 15 miles north of

West Palm Beach, is the site of a recent beach restoration project.

During its duration wave, wind, and beach.profile data were

recorded by a University of Florida contingent. Figure 22 shows

the island.bordered on the north by :St. Lucie Inlet and on the

south by Jupiter Inlet. The project limits are marked. Figure

23 is a photograph of the construction site and Figure 24 a

diagram of the beach nourishment area showing the location of the

sand fill which is placed on alternate sides of the public beach.

Its movement and redistribution within the project limits is

monitored by beach profiling (a sample is shown in Figure 25).

The wave height and period are recorded at only one point

along the beach in 20 feet of water. The breaker angle, in the

absence of more than one wave pressure sensor for directional

resolution, .is logged visually at approximately the same longshore

site. Shore-normal profiles of the beach are taken before the

placement of the fill, at the time of fill, and from 6 months to

1 year afterward at varying points along the beach.
In this study we have the benefit of in situ \wave

observations, albeit at one point, and finer bathymetric data.

In order to generate the necessary longshore breaker data we

employ a refraction program (Dobson, 1967) and, .by working out-

ward from the position of the one wave guage, a deep-water wave

climate is established. The details of such an approach are.

contained in Mogel,. et al. (1970). Suffice it to say, a fan.of

wave rays of different periods is tracked-seaward from this one

nearshore point across a bathymetry which is represented by a

fine inner grid of depth points and :a larger coarse outer grid.

The inner grid width is determined by the longshore distance

rN

~:--A

figure 22. Juoi ter Is;land.

SjuCh tJ~r~fr*it;.h L\~It

'- ~ U--.44- ..-- LI-- .~ -- .

.L_ 4 ---i ri -r _.L _..-.: .4

:ij

I1

-*-

r:

a'... 1. *.4 a C 4t444 .44 *4 44 : 4 4

rl ~Figure 23; An aer;L~ai phlOtograpLh of the f&1J. area, Jupiter Isizndi.

~- -
)~
I
i_
c .~~ ~
I

il_:
I .

4` LC ~ ~
1Lj~
.1

fi.

-r
1
I' .
i

i.
r ..:..':

-4-- 4..

.3~

/ -'-4,-

,~ .4.j
Sj r 7 ~ 'c,

-~~. -.44... K

X~~ ... '
4/ .
.119

4. .4 ,,-
i :i:v--~

-~~~-Srr~ ----- -------------

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