CIES Army
Coast. Ere Res. Ctr
MR 80-6
WHO]
DOCUMENT
COLLECTION
A Numerical Model for
Predicting Shoreline Changes
by
Bernard Le Mehaute and Mills Soldate
MISCELLANEOUS REPORT NO. 80-6
JULY 1980
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A NUMERICAL MODEL FOR PREDICTING SHORELINE
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Bernard Le Mehaute and Mills Soldate DACW72-7T-C-0002
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KEY WORDS (Continue on reverse side if necessary and identify by block number)
Currents Numerical model Shoreline evolution
Great Lakes Shore structures Wave diffraction
Holland Harbor, Michigan Shoreline changes Wave refraction
20. ABSTRACT (Continue om reverse side if necessary and identify by block number)
A mathematical model for long-term, three-dimensional shoreline evolution
is developed. The combined effects of variations of sea level; wave refrac-
tion and diffraction; loss of sand by density currents during storms, by rip
currents, and by wind; bluff erosion and berm accretion; effects of manmade
structures such as long groin or navigational structures; and beach nourish-
ment are all taken into account. A computer program is developed with various
subroutines which permit modification as the state-of-the-art progresses. The
program is applied to a test _case at Holland Harbor, Michigan.
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PREFACE
This report is published to provide coastal engineers with a mathematical
modeling procedure for predicting shoreline evolution resulting from the con-
struction of navigation and shore structures. The model is calibrated with a
test case at Holland Harbor, Michigan. This report is a continuation of an
investigation by Le Mehaute and Soldate (1977) to determine the feasibility
of applying numerical models to real case situations. The work was carried
out under the coastal structures program of the Coastal Engineering Research
Center (CERC).
The report was prepared by Bernard Le Mehaute and Mills Soldate, Tetra
Tech, Inc., Pasadena, California, under CERC Contract No. DACW72-7T-C-0002.
The authors acknowledge the assistance of Dr. J.R. Weggel, CERC, and
C. Johnson, U.S. Army Engineer District, Chicago, for their pertinent comments
during this investigation. Informal discussions with Dr. R. Dean, University
of Delaware, were very constructive. The contributions of Dr. I. Collins,
Dr. C. Sonu, and J. Nelson are also acknowledged, as well as A. Ashley's
analysis of the input data.
Dr. J.R. Weggel was the CERC contract monitor for the report, under the
general supervision of N.E. Parker, Chief, Engineering Development Division.
Comments on this publication are invited.
Approved for publication in accordance with Public Law 166, 79th Congress,
approved 31 July 1945, as supplemented by Public Law 172, 88th Congress,
approved 7 November 1963.
Colonel, Corps of Engineers
Commander and Director
CONTENTS
Page
CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI). ......
SHAMS) JANI) DIS TIN IITIONSS, 65.6110 6 6 bo 6 A oo bo 6.0 6 610 9 6 7
I INDE OIDONCHEHON 5 eo. BMG No Had Ola) 6! dio YONG og Oo GG lla o. oO. 66, 6 9
II THE OREM CALS DEVELO PMENMS ME wircimten Meanreiiirey retire irises hey irey ules ar Nile arene oeabal
III INU IDM GG. Si 6 5 A ai fee Uahiusie “ay) cee cal Sea ee
1. Description and naistomy of Nevaleneion Project at Holland
Ete nobel Mean aint GemcEMG MO SONU Gul G nM ECE MOON ECOG 20)0'.0 Bal
Zo GOOMOMNONOME' Golo 6 66 (a oa 6 6 6 4 5 00 61619 0 oo 0) (OS
See laktetOmaluMart ers italles kim oun ueWen reriuren ety ew etot Pcie tas Pau) Palla fatal < Yh verti aaa area
ALTON eh ere(oneto)) Loven vaio ey inl ioulocateh Vow ida NOs a Dae Om OMEGA IME A A Gua laiiol tac. oo
Soi labcbrollOraye G5 0 emcniteecuio A .o) 2S)
6. Littoral Drift Soe aimeree Sao. hee Sinuiseles., 66 0 0 a oo ES
7. Analysis of Aerial Photos... 6 Would 1G) oll so 0 0 4S
8. Comparisons of Profiles from 1945 ail 1975 SwaaresrS- ee ao. (2S
a! Sorcerers (6 oo 6 oo 6 6 oO oO 0 6 6 0 16, 06 | oo) 5
IV CONCHLUSTONSWANDERE COMMEND AMON Sincere seco icles uu reuncimizen enn THe ESS)
LITERATURE: GETED:0) 4 pita cannon iosnite usa heestuces ee ol eprep age at eae Gy)
NPS COMUNE PROGR AG Sb gid odie 1h velo oldies (a) id) elo eo lo oo, 6 0 68
TABLES
1 History of shoreline positions at Holland, Michigan, from 1950 to
17S wacthy andawaithowtmllalkemlen ell (coms; e CtelOMSiipy.) venue pletlelt siete) Miele aon LY
2 Summary of cross-sectional changes and computed volumetric changes
scoe NGS ging) IDPS SUES 6 6.6 66 6b 0 6 6 06 6 6 BG O10 6 oO 6 0 6 §M
3 Summary of shoreline accretion rates north of Holland Harbor
GINS) Sake aN) enue etn ice laai san crrennl IMim Mn MENG Io \ducol co ooo jo Go SQ
4Sunnary of beach area’ Ghanses south) of Holland Harbor v5) 9) ae eS
5 SSO eyes sxoie Jeol iemyel oksdaese, g 6 6 6 oo 16. 6 oa oo 6 6 00 60 6. §9
FIGURES
LY MCOASt aM ZONe r/o Ne inp icy. ellie!) ou, lovatsiyhceuanlsc eh monll-sitibet icouireel (emiacl) volte REN st Mo Met ican em mela
2; Her eh OL DINERO T abe TMy 4 oy Ley ley May well den viey deinieitliceiipeuian Wolk SOc ney ine] Mei vaveMi mt alte tame
Sr) Chanigevot ibeach prosiwlieny matoe -uuainien emccd ach nenite | cliesiite: Vetitoireniom esi l/cih har atone mm
4 Changes of sand volume due to translation of one beach profile. .... 14
SOREL CESNOR WAVelreEnactdonssonway cum ved beach mri menisci no) nc mnrsnn wit TmneLes
(Sj WWhisererceleealov PACINE: TNSIEERCSLOM Go 6 5 6 6 O10 Of do 61a a d10.0 00 0 o,0 20
7 Noeehesiom, tose Chiskirereealon Seth, 5 5566 66 0,8 6 60,010 010 of 0 6 6 o Bd
28
BY
CONTENTS
FIGURES--Continued
Page
Initial shoreline and incoming wave direction (A); shorelines at
SUCCESSIVE wills Ore ies Aes Sites. MOMeS einel ZOMG (3) 6 6 6.46 06 5 «6 Za
Shorelines at successive times of 5At, 10At, and 15At (A); shorelines
KOTMEMME A NET (Be eat ne Wvenutel peers era eaincae 2 nes
Initial shoreline and incoming wave direction (A); computed and
minimal shorelines for finite-difference scheme of time 1At (B) ... 27
Transport function Q(a,) = cos a, sin za, for selected values of z.. 28
Layout of shoreline reaches characterized by different beach
COMMONS 4s Tessa aula ere mi taal nC CU rains Vet ipas llear Megane sa pentivern er Ne" Cerys paltenicetalet Melee at MES
Nearshomesbathymeitaayy a olelamcetl aac naire mrinan eile) elec ei cleey nena nS.
Beach profile summaries) tom holland vHanbor sy 2). ). «= 1 20.) eee eee NOD
HolstandsHarhoreneciairshonrembiatany ne Casveuemrcuitely ope carted ill chl a stent een-mneuinar an ee
Conditions at Holland Harbor entrance before dredging in 1973
aniGeO7Aey Rs welts) Si ieluis 40
Wind rose for an average 12-month period and for an average 9-month
Ce rree speTlod ror Souchexnlralce Ma chaiearlen) |) riicuen one -amCaye acini ume nnEraL
Wave rose for an average 12-month period and for an assumed 9-month
LESAGE) SILO! ieOEAe SOEs Iba WolelyyeinG 6 5 56605060000 0 Ail
Wave height exceedance probabilities (9-month ice-free period) .... 42
Variations of probability of various wave heights with months of
VELOVGH SVEN eS Ton We MRGe Meet esE No nbd ea ANN VEO M ON ic ateMmL ates Pein bg taal Baie MN taal Leh eNeM Uen, weala. aay
Neem iWonemly lakes leweils simee 1205.6 6965656 556 60 oo ooo A
Monthly variations of computed gross and net littoral drifts for
Elo gel rani eaMakc hae oan ve) ar ieeetimnre nine) imac uel, eeinete st sumed eile Mathai cy ie er Salle ome i rial 37 /R SRE MEETS
Shoreline positions at Holland, Michigan, in 1973, as determined
by aerial photos. aes : 48
Smoothed average rates of shoreline erosion and accretion at
Foliar dimen al Chioameensr nee e reid ie cect Vel ee epee Prem y icine ibsemlenaiice, Maco une ws, aie! recreate AC)
Historical shorelines north of Holland, Michigan, from 1849 to 1945. . 51
SuMmMaiyaOLesanda bud getmnorsehi oa Holellatyd pa Matchlgany cue mnonnr ini tn te lnnnre nD S)
Historical shorelines immediately south of Holland, Michigan, between
TSANS) eeiravele INCU toes o lay Lvaiiio: GERM (a enWia\ ual or colts Me: bpiiben nl ag coy al No oar eaeaan et yA
Historical shorelines farther south of Holland, Michigan, between
USVI a@mel ISH o 6 6 6 6 6 6 6 9 6 56.8 OOo 68 Oo a oo 8 oo 6 Oo a oo, S55
Conpurcecl .ckeita, wersms actual datas .o5 56 8 oo oo oot USD
CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) UNITS OF MEASUREMENT
U.S. customary units of measurement used in this report can be converted to
metric (SI) units as follows:
Multiply by To obtain
inches 25.4 millimeters
2.54 centimeters
square inches 6.452 Square centimeters
cubic inches 16.39 cubic centimeters
feet 30.48 centimeters
0.3048 meters
square feet 0.0929 square meters
cubic feet 0.0283 cubic meters
yards 0.9144 meters
square yards 0.836 square meters
cubic yards 0.7646 cubic meters
miles 1.6093 kilometers
square miles 259.0 hectares
knots 1.852 kilometers per hour
acres 0.4047 hectares
foot-pounds 1.3558 newton meters
millibars OMS lOme kilograms per square centimeter
ounces 28.35 grams
pounds 453.6 grams
0.4536 kilograms
ton, long 1.0160 metric tons
ton, short 0.9072 metric tons
degrees (angle) 0.01745 radians
Fahrenheit degrees 5/9 Celsius degrees or Kelvins!
Ifo obtain Celsius (C) temperature readings from Fahrenheit (F) readings,
use formula:
Cr= (5/9) (E32)
To obtain Kelvin (K) readings, use formula:
K= (6/9) (E32) oer
SYMBOLS AND DEFINITIONS
proportionality constant (8 = aa)
height of bluff (in case of erosion) or berm (in case of accretion)
phase velocity at breaking
deepwater phase velocity
water depth at ye
water depth
breaking depth
variation of sand volume due to horizontal displacement during time, dt
variation of sand volume due to beach slope variation during time, dt
group velocity at breaking
deepwater wave group velocity
gravity acceleration
breaking wave height
deepwater wave height
diffraction coefficient
refraction coefficient from deep water to the line of breaking inception
percentage of silt
time-constant parameter
wavelength at breaking
deepwater wavelength
length of groin perpendicular to shore
loss of sand by rip currents
beach slope at shoreline
horizontal axis parallel to the average initial beach profile
horizontal axis perpendicular to x, oriented positively seaward
vertical axis, positive upward
longshore energy flux
dimensionless littoral drift discharge
littoral drift discharge in cubic yards per year
loss of bluff volume (in case of erosion) due to silt
loss of bluff volume (in case of erosion) due to silt
SYMBOLS AND DEFINITIONS--Continued
loss of sand by wind
sand discharge in a direction perpendicular to the beach
average bottom slope to depth of no motion
horizontal displacement at depth, D characterizing slope variation
GQ?
shoreline
wave period
time
volume of sand over a stretch of shoreline, Ax, unity
shoreline coordinates
dimensionless shoreline
deepwater limit of the beach
function of x, y, t defining the bottom topography
shoreline elevation above a datum z = o
iD
158 08) SLi
deepwater wave angle with the ox-axis
angle of breaking with the shoreline
deepwater wave angle with the shoreline
angle of the beach limit with the ox-axis in front of a groin
horizontal displacement of the beach profile during time, dt
function characterizing beach-slope variation as function of x
function characterizing beach-slope variation as function of t
function characterizing beach-slope variation with time due to rapid
change of sea level
function characterizing beach-slope variation with time due to manmade
structures such as groin or navigation works
a symptotic (in axium) value of Ay) when t >
variation of wave direction by diffraction
angle of wave ray with a perpendicular to shore in a diffraction zone
water density
A NUMERICAL MODEL FOR PREDICTING SHORELINE CHANGES
by
Bernard Le Mehaute and Mills Soldate
I. INTRODUCTION
This report establishes a mathematical model for shoreline evolution and
calibrates the model with a test case at Holland Harbor, Michigan. Even though
the mathematical model is general and could be applied to a number of situa-
tions, the emphasis is on the Great Lakes, and more specifically, on shoreline
evolution near navigation structures.
An interim report by Le Mehaute and Soldate (1977) reported on the feasi-
bility of applying existing mathematical models to real case situations. This
report is a continuation of that first investigation.
The present mathematical model includes many of the characteristics already
covered in the literature. In addition, the model presents an integrated ap-
proach on a large number of phenomena previously neglected. Its main purpose
is to develop a practical numerical scheme which could be used to predict shore-
line evolution. The model would then be able to point out the shortcomings in
the present state of knowledge. Therefore, the mathematical model covers some
aspects of shoreline evolution which cannot be quantified with the data obtained
from the considered test case of Holland Harbor. The mathematical model then
has to be regarded as a research guide for the future.
One important aspect is the effect of sand size and density. It is well
known that the rate of shoreline erosion and sediment loss is largely affected
by these parameters. The fine sand is transported offshore, while larger size
sand tends to proceed alongshore according to a littoral drift formula. This
effect could not presently be quantified; therefore, it is introduced in the
mathematical model as a constant.
The present mathematical model can continuously be upgraded as the state-
of-the-art progresses, and as the model is tested for a large number of cases.
It is important to remember that the model deals with long-term shoreline evo-
lution as defined in Le Mehaute and Soldate (1977). Short-term evolution must
be considered as local perturbations which are superimposed on the presently
defined topography.
Three time scales of shoreline evolution which can be distinguished are
(a) geological evolution over hundreds and thousands of years, (b) long-term
evolution from year-to-year or decade, and (c) short-term or seasonal evolu-
tion during a major storm.
Associated with these time scales are distances or ranges of influence over
which changes occur. The geological time scale deals, for instance, with the
entire area of the Great Lakes. The long-term evolution deals with a more
limited stretch of shoreline and range of influence; e.g., between two head-
lands or between two harbor entrances. The short-term evolution deals with
the intricacies of the surf zone circulation; e.g., summer-winter profiles,
bar, rhythmic beach patterns, etc.
For the problem under consideration, long-term evolution is of primary
importance, the short-term evolution appearing as a superimposed perturbation
on the general beach profile. Evolution of the coastline is characterized by
low monotone variations or trends on which are superimposed short bursts of
rapid development associated with storms.
The primary cause of long-term evolution is water waves or wave-generated
currents. Three phenomena intervene in the action which waves have on shore-
line evolution:
(a) Erosion of beach material by short-period seas versus accre-
tion by longer period swells;
(b) effect of water level changes on erosion; and
(c) effect of breakwaters, groins, and other structures.
Although mathematical modeling of shoreline evolution has inspired some
research, it has received only limited attention from practicing engineers.
The present methodology is based mainly on (a) the local experience of engi-
neers who have a knowledge of their sectors, understand littoral processes,
and have an inherent intuition of what should happen; and (b) movable-bed
scale models that require extensive field data for their calibration.
In the past, theorists have been dealing with idealized situations, rarely
encountered in engineering practice. Mathematical modelers apparently have
long been discouraged by the inherent complexity of the phenomena encountered
in coastal morphology. The lack of well-accepted laws of sediment transport,
offshore-onshore movement, and poor wave climate statistics have made the
task of calibrating mathematical models very difficult. Considering the im-
portance of determining the effect of construction of long groins and naviga-
tion structures, and the progress made in determining wave climate and littoral
drift, a mathematical approach now appears feasible.
The complexity of beach phenomena could, to a large extent, be taken into
account by a numerical mathematical scheme (instead of closed-form solutions),
dividing space and time intervals into small elements in which the inherent
complexity of the morphology could be taken into account. Furthermore, better
knowledge of the wave climate, a necessary input, will allow a better calibra-
tion of coastal constants (such as those in the littoral drift formula).
In past investigations (Le Mehaute and Soldate, 1977), many important ef-
fects have been neglected, such as combined effects of wave diffraction around
littoral obstacles, change of sea level, height of berm and bluff, beach slope,
etc. The present investigation attempts to include all the important factors
associated with long-term shoreline evolution. In the case of the Great Lakes,
importance must be given to variations in lake level. The coastal zone is de-
fined by a three-dimensional bottom topography instead of a two-dimensional
shoreline (or two lines) as in the previous cases.
Mathematical modeling is essentially approached by:
(a) Understanding the phenomenology of shoreline evolution
quantitatively.
(b) Calibrating, determining empirical constants, and separating
various effects, such as due to change of lake level, navigation
Structures, etc.
(c) Predicting long-term future evolution.
(d) Assessing the effect of future construction.
The mathematical model presented in this report is just a first step in this
direction. Much of the model may be modified after the application of the mathe-
Matical model to other cases which could be used for further calibration.
Theoretical developments for the model are presented in Section II. Section
III describes the shoreline evolution recorded at Holland Harbor, Michigan, and
compares the mathematical model with the test case investigated. Section IV
provides recommendations and conclusions. An Appendix describes a computer
program to investigate shoreline behavior.
Ii. THEORETICAL DEVELOPMENTS
Consideration is given to a coastal zone limited by boundaries at a small
distance from the surf zone (Fig. 1). The bottom topography is defined in a
three-coordinate system, oxyz, by a function zp = f(x,y,t) where the ox-axis
is parallel to the average shoreline direction, the oy-axis is perpendicular
seaward, and oz is positive upward from a fixed horizontal datum. The angle
of the shoreline with the ox-axis is small. The shoreline is defined by y = y,,
Z = Zg = Zph(x,yg,t) which also defines the sea level as a function of time.
Waterline
Yb Ys Ye Yq y
Figure 1. Coastal zone.
The deepwater limit of the beach is y = Ye: (This limit defines the con-
tour line where the sand is no longer moved by wave action.) The water depth
at y= y, is D,. It will be assumed that D, remains constant as sea level
and beach profiles change. Therefore, dzg/dt = 3z,/odt.
B is the height of the bluff in case of erosion, i.e., when dV¥o/at < 0,
and) the jhetght of the berm incase) of accretion aes), when ioye/.ot On (Ealoeme))
EROSION ACCRETION
Figure 2. Height of bluff or berm.
The quantity of sand over a stretch of shoreline, Ax = unity and bounded by
theydatums9z 5" 0) sy) —) 0, andiithelbeach) profile iz7 yy atetime sy stamps
Y
We] | ater) ae
0
Assume that, for some reason, the beach profile changes during an infini-
tesimal amount of time, dt. A further assumption is that the initial beach
profile which is considered at time, t = t;, could be the normal "equilibrium
profile." (The equilibrium profile may never exist under varying prototype
conditions (similarly two-dimensional waves never exist), but it is a conven-
ient idealized concept which could be approached in two-dimensional wave tank
experiments. In this case, it could be defined as the statistical long-term
average beach profile which exists under a given wave climate. The model here
is actually independent from this definition. )
The departure and modification from this initial beach profile can be
characterized by (see Fig. 3):
(a) A translation in the yz plane defined by an elementary
vector of components.
IV dD
re ake
where dD/dt is the rate of change of sea level. Note that this
translation is independent from the beach profile and, in particular,
if the beach profile normally exhibits a number of significant bar
formations, under normal conditions the translation will reproduce
this characteristic at the same water depth.
(b) A perturbation characterizing the departure or variation
from the initial profile. Since the rate of the vertical component
12
y,(t+dt) vg (t) Yq (t + dt) Yo (0) 1
Figure 3. Change of beach profile.
of translation is dD/dt, the perturbation can be defined only by a
horizontal displacement. At the deepwater limit, D,, the horizontal
displacement characterizing the rate of change of the average beach
slope is defined by
95, (x,t)
ot
so that
The quantity of sand within the considered domain at time (t + dt) is
vp
V(t + dt) = ih apy (Epp pe \ cle). chy (GD)
0
and the variation
Y, dz
V 1 6 eas
ae i = dy YY, and 0 are fixed limits)
VI OZ OZ OZ
Ge [es en
dt 0 at Oy cle 9 Bee” eke
where 9z,/0x is the variation of beach elevation along the ox-axis. Since
the angle of the beach with the ox-axis is small
13
dZp OZp (2
bans aye m is the beach slope),
Se Ovane
i.e., the change in z occurring along the beach is small relative to change
across profile and the beach profile variation over a distance dx is small
but remains an infinitesimal of higher order than over a distance dy.
For the same reason, the velocity of beach variation along the ox-axis
dx/dt is small as compared to the beach variation along oy; therefore
IZp Cbs IZp dy
OxeMGit Oven ae
and could be neglected. To evaluate the other terms it is simpler to refer to
Figure 3. It is seen that the integral is equal to the sum of:
(a) The change of volume of sand, dv/dt, due to the translation,
dyg/dt, is the sum of the change of volume due to the horizontal
translation (Fig. 4)
(B + Cc) — (2)
and the change of volume due to the vertical displacement
(ve - yp) (3)
where Vi is the ordinate characterizing the location of the bluff.
c
Aas Yb
dD
B
_——
_-—
_——
~~
_-—
6 ———= Cc
AY Ye
Figure 4, Changes of sand volume due to translation
of one beach profile.
14
the
The sea level displacement, dD, being small, the difference in
the quantity of sand represented by the two triangles ABC and A'B!C!
in Figure 4 is considered as a difference of infinitesimals and there-
fore is neglected. So, combining equations (2) and (3)
dv 8Yg dD
—_—_— = —-_—___-=— - 4 _—__ 4
ony oe SET Ole: Bb. ide (4)
It is seen that when dV/dt = 0
Via cnaty
b dD
dyna = oe
Vela BeSTDP ioe Ss
where S is the average bottom slope. Note that this expression is
independent of the beach profile and, therefore, implicitly takes into
account bar formation.
(b) The change of volume due to the perturbation and departure
from the equilibrium profile. This departure is characterized by
a change of slope. Therefore, the corresponding variation of sand
volume (area AEG in Fig. 3) is
Se ene) 3)
t BTS aha on TOL eel
in accordance with the previous assumption. Therefore, the total
variation of sand volume, dV/dt, is (adding eqs. 4 and 5):
dV _ 5 any. 5 oS
qe Oo Ue) ae = Cle lee ay Mane (6)
This variation of volume is due to the variation of littoral drift along
ox-axis and the onshore-offshore motion. The following terms are included:
(a) The discharge of sand leaving the beach per unit of width
includes:
(1) Qyw due to loss of sand by wind.
(2) Qs 09D due to the quantity of silt contained in the
bluff and which tends to move offshore by suspension. This
loss occurs only in case of erosion (dy,/at < 0) and is equal
to Qs =UKAB oye/ ot), where) Ka) iisiithespercentage) of silt ain
the bluff.
(3) QF due to the loss of sand from the beach by density
current during a storm. QF is a function of the size dis-
tribution and density of material. A beach of fine material
(<0.1 tm) tends to erode more rapidly than a beach of coarse
material (>1 mm). The coarse material tends to move along
the shore while the fine sand moves offshore.
15
The determination of these three quantities is given from sand
budget investigations.
(b) A general term, M(x,t), expressing the local variation in
the sand budget due to loss of sand by rip currents along groins,
and to sudden dumping of sand in case of beach nourishment or flood.
(c) The variation of littoral drift along the ox-axis which is
re)
OQa(S) > One 2 Cbg) So ae dx (7)
Many formulas in literature sources express the rate of longshore transport,
Q,, as a function of the incident wave energy along a straight shoreline. Long-
shore transport is also a function of the sand characteristics (size distribu-
tion and density), wave steepness, beach slope, etc.
The form of this formulation on shoreline evolution is of paramount impor-
tance. In particular, determination of the relative rate of sediment transpor-
ted in suspension and by bedload is very important since this ratio influences
the loss of sediment by rip currents.
Such evaluation is beyond the state-of-the-art, and any improvement would
require a major effort beyond the scope of the present investigation. Any im-
provement in the longshore transport rate formula could eventually be introduced
in the model at a later date. Therefore, it is assumed that the rate of sedi-
ment transport is independent of density and size and depends solely on the
longshore energy flux, P,, by the empirical relationship
On) Shad, sO ID, (8)
where Q, is in cubic yards per year.
P,» is in foot-pounds per second per foot of shoreline and is expressed by
the relationship (U.S. Army, Corps of Engineers, Coastal Engineering Research
Center, 1977)
2
- OF BND ae
Po 64 TASK) sin 2c, (9)
where
Kp = refraction coefficient from deep water to the line of breaking
inception
T = wave period
H, = deepwater wave height
a, = angle of the deepwater wave with the shoreline
ap = angle of breaking with the shoreline
Note that this equation expresses implicitly the rate of littoral drift
along a straight shoreline in terms of breaking wave characteristics (except
for the deepwater wave height, H,). The equation is assumed to hold in case
16
of gentle beach curvature. The refraction coefficient, Kp, amd angle, ap,
can be determined as functions of the deepwater wave characteristics H,, T, a
(or ap) and the angle of the shoreline at breaking, dy,/dx.
In the case of a groin perpendicular to the shore, at x > - », the deep-
water wave angle, a,, with bottom contours is equal to the angle a of the
wave with the ox-axis since the shoreline has the same direction as the ox-axis;
at x = 0 (i.e., at the groin), the shoreline becomes parallel to the incident
wave crest very rapidly. Therefore, a, = o and
i
(10)
8X |x = 0
Qo = -tan™
In the general case, for any value of x
oy.
Be =il § 11
ao a + tan ARE ( )
The breaking wave characteristics of wave height, Hp, water depth dz,
and the angle of breaking op, can be obtained from the deepwater wave charac-
ieriisiGLes SHA ah, and a5.) a5) vse eiven by ‘equation (11)))an) texms) of a) ;and
dy,/dx which takes into account the curvature of the shoreline. The following
equations permit a determination of Hp, dp, and ap, provided the bottom contours
are parallel along a wave ray (Le Mehaute, 1961), but could be curved along the
shoreline (Fig. 5):
PAR ALLEL
CONTOURS
7
Figure 5. Effects of wave refractions
on a curved beach.
(a) Snell's law.
ee ee 1
CO Sani: ST /25 (2)
(b) Wavelength (dispersion relation).
L 27d
2 = tan n (+) (13)
O Db
(c) Conservation of energy flux between wave orthogonals (G is
the group velocity).
2 = H2 14
HS cos a,G, = He cos a,G, (14)
(d) Breaking criteria.
Hy, 21d
— = 0.14 tanh (15)
Lp Lp
From these equations, it has been shown (Le Mehaute and Koh, 1967) that when
a, remains smaller than 50°, ap could be approximated as
O
9.25 + 5.5 Mo 16
=~ Jie a 5 —
On =A cs (16)
where
aie
lL. =
@ 27
Therefore, the refraction coefficient
ae COS Ap 1/2
A COS Ob
could be written, inserting oa, as given by equation (11), and ap as given
by equation (16)
2)
cos(a - tan7! 9Y, ae
Tee ae fe ee ee a Nd = (G7)
fi ( ois: 2aH2
cos |(a - tan Ye/, ) (0.25 SS Oloc hy of, 2)
( Be 2
Now it is possible to formulate the variation of littoral drift over a distance
Ax = unity. From equation (9)
2Q¢
ox
‘Db 2 dKp if
5 + AH52Kp ae Zap (18)
= AH2K% 2 cos 2a ee
oR Dia
where
On the other hand, since (see eq. 16)
0p 30p ( Ho
ARTE PST, 0.25 + 5.5 —= (19)
and by taking into account equation (11)
lor 1 one
—o = (20)
a bie ( Ys/,.) iH
Finally, by inserting equation (20) into equation (19)
O25 = 5.5 1o/, 32
Jone) y t Lo 9 Ys
(21)
= 1 6/5 oo
In case of wave diffraction, the wave height varies significantly along a wave
crest. Then, the previous refraction coefficient, Kps has to be replaced by
a combined diffraction-refraction coefficient, such as K,K,. (A diffraction
current in opposite direction to a longshore current takes place at a distance
from the end of the groin.) The variation 9Kp/dx is much larger than dKp/dx.
Therefore, in analogy with equation (18)
3Q¢ dp OS 2
— = nix (2x2 cos 2a, =— + 2Kp == sin 20, (22)
In a diffraction zone, az, is due to the sum of variation of shoreline direc-
tion, tan7! dY¢/9x, and because of diffraction the rotation of the wave crest
around the end of the groin, §& (Fig. 6). 6 is the angle which has the end
of the groin as apex and extends from the limit of the shaded area to the con-
Sidered location defined by x. Figure 6 shows that © = ajo - 8' and 6' =
tan7! x/2, where 2% is the length of the groin. Therefore,
= em! 28 Ss wenel & 23
apy waean = + A an (23)
1 Oeste Change Une er eL (24)
OR rox et (x 2)
and, differentiation equation (23) and inserting equation (24)
30 a*y
SDAIN tens, San Sepp live Tce wale. (25)
Q
Se a C2)
where dKp/ ox is the coefficient of variation of wave height due to combined
effects of wave diffraction and wave refraction. It also includes the effect
of diffraction current.
An empirical formulation for determining the combined effect of diffraction
and refraction is more suitable to quantitative analysis of a real sea spectrum
19
Figure 6. Diffraction zone notation.
than more exact theories of wave diffraction which are valid for periodic waves
over a horizontal bottom and are represented by a Fresnel integral. For this,
it will be assumed that the energy travels laterally along a wave crest as well
as along a wave ray. The lateral speed of propagation is assumed to be equal
to the group velocity of a periodic wave of average period. (Actually, since
the problem is confined to very shallow-water waves, and the longest waves of
the spectrum diffract most, the limit of the diffraction zone is defined by an
angle such as the velocity of propagation of wave energy along the crest and
is simply VgD, where D is the water depth.) This lateral transmission of
energy results in a decrease of wave energy from the exposed area to the shaded
area, such that (Fig. 7)
D
H2dx = i H? ds (26)
A 0
where s is the distance along a crest from the end of the groin. Figure 7
shows from simple geometrical relationships that
Q V2
OD ee ea PET
Gp @, 25) 2
oA = -% tan (45° - a,) (27)
oC) = Litany (4570+ oF)
therefore,
: ‘ an Uecann (4 eac)) us
He ee ee H*dx (28)
SO ley A) eh cae) ISO Say)
20
Figure 7. Notation for diffraction study.
In the case where the long groin is in the previously defined wave diffrac-
tion zone as in Figure 6, it is assumed that the wave energy which reaches the
groin is absorbed by friction. It is also assumed that the combined effects of
diffraction and refraction of a wave spectrum can simply be represented by a
sinusoidal variation of wave height along the breaking line (Mobarek and Wiegel,
1966) (Fig. 7).
Assuming that Kp(x) is of the form sin k(x - X5) as shown in Figure 7,
Such as Kp at A= 0, Kp at C = 1, then to satisfy these conditions
T 1 1
oe oe (ean CS Ss) En Gers a) (29)
which after some simple trigonometric transformations can still be written
k = 7 cos 2a (30)
mats O
to satisfy the equation expressing the conservation of energy flux, such as
given by equation (29), then
and Kp(x) = H(x)/H, is given by
os (2 1/2 2
Kp (x) = (2 een) sin (ee ee bso & can CS> = 2031) (31)
sin Ot Ay
2 |
By differentiation equation (31)
dKp (x) a 2 cos ae a Tm cos (20,)
EI | NIA sin Gs AL
7 Cos (2a,)
COS \ aa erm [ocacteecae ani (4S 201) (32)
Inserting this value in the littoral drift equation (22) permits the mathemat-
ical model to completed.
In summary, the variation of volume of sand dV/dt given by equation (6)
is equal to the sum of:
(a) The loss of sand by wind Que
(b) The loss of silt Qs = K,B dy,/at where K, is the percentage
of silt in the bluff.
(c) The loss of sand by density current during storn, Qf. and
loss by rip current (or brought upon by nearby river).
(d) The quantity of sand dredged or (at the opposite) deposited
during beach nourishment.
(e) The variation of littoral drift along the ox-axis is 9Q,/9x.
In a refraction zone alone, 9Q,/dx is given by equation (18) where K,,
is given by equation (17). op is given by equation (16) as a function of
A>, and a, by equation (11), with respect to the ox-axis. Also, 9a,/9x
is given by equation (21).
In a diffraction zone 9Q,/3x is given by equation (22) instead of equa-
tion (18) and the diffraction coefficient Kp by equation (31), and dKp/dx
by equation (32) for a wave spectrum. ap is now given by equation (23) and
dap / ax by equation (25).
Since all the phenomenological equations have been established, it is more
convenient to express them in dimensionless form. The ''general" equation ex-
pressing the sand budget balance can still be written. (The loss terms have
been dropped for simplicity and can easily be included if necessary.)
Ys Gb) a ey Oe
Golestehe We) aare ore Vb) mes a Do ae ae (33)
For an analysis, this equation is considered in dimensionless form. Let
e length
B, + De
where Bo is chosen later
+ At
—————— (35
G34 Dae
BS DD
Q = 5 KAKs sin 2op (36)
a2
The general equation is thus transformed to
A ag Dy) a
B+ Di. Me ee yh dd, 1 Dy, dAy nis KpKe sin 2a,
Mo eo
n~
The hats will be dropped from all variables at this point in all discus-
Sions except those relating to observed results. Also, the subscript s for
y and Q will be omitted. Hence, the general equation used in the following
is
DO AS
B + De. dy 1 De day 9 Knkp sin 20,
* (We - ¥p) = + 5 eS sO)
Bo + De at dt 2 Bo Dende dX 2
Recall
> _ COS A
Rumcosmap
where a5 = a + tan-! dy/ax and ap, = function of ap, as f(a.) (the function
f depends only on H,/Lg as previously shown). Therefore,
Ne
22 . 12 9
Kikp sin 202, = Ks cos a, Sin ap,
Note
ee cr ec) Bets
ax COS % Sin ap = (cos A cos ap Jes = Sill Cy) Sala «p) aa (a5) aa
The general equation (37) then becomes (after some rearrangements)
BceaeD) 2
dy O C 1 a“y
ae 2 SS PCy), Se oe eee Rei) (38)
ot Be De Tae ( Lax) axe
where
By ap 1D)
O Cc Ja, dD
RES yait) = B+ Ds F (a,) yx Ye 5 Yp) dt
D oK
1 C dAy D :
Ss =— 9
7 By © A ce + 2Kp 5, COS Op Sin ap (39)
and
a(x) in the diffraction zone
a
The above equation is the general dimensionless form which gives the time-
dependent sand budget.
The general equation is nonlinear and appears to be impossible to solve ana-
lytically. As it is also difficult to solve computationally in the most univer-
sal case (where lake level, beach slope, and bluff-berm height vary as functions
of x, y, t), several simple cases are described in detail which indicate the
behavior of the equations. Although numerical results are presented, detailed
descriptions of the numerical methods used are given in a later discussion.
23
The simplest examples are those of beaches having negligible or identically
constant bluff-berm, uniform beach slope, and uniform lake level. In these
instances, the general equation (38) reduces to
2 aK
aa F(a) ieee oy 2Kp TES Ks GSN Cy 2 PGs) ee (40)
ot : (C7, ) ox ox ox
i+ ax
subject to the appropriate boundary conditions.
Suppose (as shown in Fig. 8) the beach is initially zero, bounded by a
breakwater which is a complete littoral barrier, and subjected to waves having
uniform positive deepwater direction. The boundary conditions are
oy
= 1S > TEIN Ou ke 0)
OX
e)
avi Olay xd hls
ax
where L is a distance along ox, large enough so as not to be influenced by
the groin.
The shoreline is calculated for a fixed Hj, T, at intervals of equal At
(shown in Fig. 8) for selected times. The general shape of these curves is
Similar to the type formulations obtained by Pelnard-Considere (1956).
INCOMING WAVE DIRECTION
INCOMING WAVE DIRECTION
INITIAL SHORELINE \ INITIAL SHORELINE
Figure 8. Initial shoreline and incoming wave direction (A);
shorelines at successive times of 1At, 2At, SAt,
10At, and 20At (B) (vertical scale exaggerated).
24
An extension of the above example is shown in Figure 9 where both sides of
a breakwater are considered. The initial position of the shoreline and wave
conditions in deep water are as before (Fig. 9). The boundary conditions
become
ox
oY = O at x = -O (simplification of more exact condition)
Oyen a
— = -tana at x = +0
ox
The uniform depth theory of Penny and Price (1952) is used as an approxi-
mation (not substantiated) for diffraction about the end of the breakwater.
The shoreline is calculated for various multiples of a fixed At (Fig. 9,A).
INCOMING WAVE
DIRECTION
\ INITIAL SHORELINE
INCOMING WAVE INCOMING WAVE
DIRECTION DIRECTION
\
INITIAL SHORELINE
Figure 9. Shorelines at successive times of SAt, 10At, and 15At (A);
shorelines for time 2At (B) (vertical scale exaggerated).
25
Allowing the wave direction to alternate between +a and -a at a fixed
rate gives an interesting variant to the previous example. The boundary condi-
tions are as before for a> 0. The results are shown in Figure 9(B) for the
same values as the physical parameters used in the problem of Figure 9(A). Of
interest is that the undulatory patterns of the shoreline in Figure 9(A) dis-
appear in Figure 9(B). Hence, diffraction-induced undulations in a natural
shoreline probably rarely appear since offshore wave climates are often
multidirectional.
The numerical scheme used to generate the preceding examples was based on
the use of implicit finite differences. Such schemes, whether implicit or ex-
plicit (or both), are commonly used to efficiently solve parabolic problems.
However, even in the case where only refraction is considered (Fig. 8), the
boundary condition
numerically gives a solution which initially may not conserve mass, i.e., the
integrated transport equation
L
-- I ydx = Q(L)
may not be satisfied.
Unfortunately, this feature is unavoidable for most such schemes (the ex-
ceptions are discussed below) as the following demonstrates. Figure 10(A)
shows an initially straight shoreline. In any finite-difference scheme, after
one time increment At, the shoreline is bounded below by the solid shoreline
in Figure 10(B). This shoreline has the least possible area, A, where
A= as tan a (41)
The conservation of mass equations requires
ACQ (LS TAtcosmaksanwicn =A
Thus, At must satisfy the inequality
Ne > Hist D | (42)
2 Sin) Gp Cos= -a
Since the accuracy (and in explicit schemes, stability as well) depends on the
ratio } = At/Ax*, the above inequality places a lower bound on the accuracy of
the solution which may be unacceptable in practice. The finite-difference form
of the equation for the conservation of mass may be incorporated directly into
the numerical scheme. In this case a solution exists which is similar to the
previous case but shows a small erosion throughout the reach. For engineering
applications, the primary quantity of interest is the amount of sand on a given
shoreline. Then it is more important to conserve mass than to satisfy the shore-
line boundary condition as written in the present form. The general equation is
26
a= INCOMING WAVE DIRECTION —— MINIMAL SHORELINE
——-—-— COMPUTED SHORELINE
Ax 2Ax Ax 2Ax x
Figure 10. Initial shoreline and incoming wave direction (A); computed and
minimal shorelines for finite-difference scheme of time 1At (B)-
now used to derive an equivalent equation for the transport Q which, although
subject to similar numerical problems, will satisfy the transport boundary con-
ditions exactly.
In a situation where only refraction is important, the general equation
then becomes
OF, Ow
dt Ox
where Q is cos J Sin op, Ap 18 f(a), and a, is a - tan! dy/ox. Differen-
tiating by x gives
as i eS (43)
The transport function Q can be considered as a function of o, which may
be solved for a,, such as a = g(Q). Thus, the transport equation (43)
becomes
ane SO, Dior
a ania ic) = op an(g(Q) - a) = aD
therefore,
BO) ls 37Q
st Cosi (g(a) ao
but
3g(Q _ dg(Q) 2Q
at dol) 3c
therefore,
aQ__ cos? (g(Q) - a) 37Q
ot dg(Q /ag ax?
(44)
27
Assuming a solution for this equation is known, the shoreline y can be calcu-
lated from the equation
t
y(t,» = y(o,x) + [ = (t,x) dt
9)
In practice, the equation for Q is not solved in the above form. Implicit in
the above formulation is the assumption that the function g exists. However,
as shown in Figure 11, g is not single-valued if the maximum range of the
angle a, is greater than approximately 41°. This difficulty may be removed
by considering the equation for Q and y as a system subject to the boundary
conditions for Q. Note that
cos? (2(@) = 5) 40 j[, | (2 j
L
dg (Q)/aQ da, ox
Hence, the equation for Q becomes
a0) al) 1 92Q
STE Ra EINE MRD // AULT NOE WARE (45
Oe cy is (2¥/ax) ax )
0.6
z = 0.25+ 55H /Ly
0.5 eis
os z = 0.75
g
£
ae)
is]
8 z = 0.50
0.2
z = 0.25
0.1
ao
10 20 30 40 50
INCOMING WAVE ANGLE (°)
Figure 11. Transport function Q(a,) = cos a sin za,
for selected values of z.
28
This, together with the equation dy/dt = 3Q/dx, is solved in a cyclic
scheme. One possible method is the centered Crank-Nicolson type implicit-
explicit scheme discussed below. Suppose y is given for all x at a given
time, t, and that from t the wave climate is specified by the (constant)
triple (a5H), 1). Let
Ny ie mi EQ Ot Eh
( xd) = dag ee (8¥/ax)°
At
eres
L(t,x) = an approximation to L(t,x)
Integrating the Q equation gives
At 92Q 97Q
OG <2 Neyo) = OlGeno) mal L(t,x) aoe IGE av Att, x) aa (46)
it ANE
where
OQ OCs hd = 206) = WGe= iss
oS eee ee ee ee (47)
ox Ax
Integrating the y gives
At [3Q IQ
QB S ING559) = 37(E5) Sea] Oe (48)
2 \\ee aX
t + At
where
30 _ Q(x + Ax) = Qe = 4x) (49)
OX 2Ax
These equations are solved numerically, subject to the appropriate boundary
conditions, in steps using the cyclic algorithm:
(a) Wee 16Ge 2 Wes) S Ges Wee,
(b) Calculate Q(t + At,x) Vx subject to the appropriate boundary
conditions.
(c) Calculate y(t + At,x) Vx; calculate L(t + At,x) and set this
equal to L(t + At,x); then calculate new Q.
(d) If new Q compares with old Q, stop; if not go to step (c).
Tests with this scheme have shown that it converges to its limit after one
application of step (c).
This method can easily be modified to solve the equation where both dif-
fraction and variations in lake level are allowed; i.e., m is the slope
(refer to eqs. 37 to 40).
OB 1 0
= ED cos a, sin ap mars (50)
(28)
For convenience, let Q = coS dp sin ap
Ty ce ee
Q = K2Q (51)
As before, referring to equations (43) and (44)
CON “heyy apace a uals (CQ) OQ es ule as
ot ox dt cco hie COSS Ce (O)Mena) dOMN ots icoscmcs(@) momo more
Also,
2Q _ 2 && oND
0 Be OD
from equation (51).
The second term in each of the above two equations is negligible in physi-
cal situations of usual interest where the distance between the shoreline and
the tip of the breakwater is large compared to the distance the shoreline
changes during a time At.
Therefore, the transport equation becomes
aq a2Q
— = eos (BOQ) > &) == (52)
at dE /y9 ox
and
oY. 2 eR ly, (53)
ot ox mdt
This system is solved using the same type of algorithm as used previously.
In the present situation where only refraction is important, several approx-
imations are possible which produce problems having analytic solutions. The
most direct approximation, and essentially the assumption of Penard-Considere
(1956), is to approximate
a2y
p) i
ung (2 cos A cos ap - sin 9 sin of) ——7> 7-5 (54)
We CVax)
at
where z = 0.25 + 5.5 H,/Lo (see eq. 16), subject to the boundary conditions
OL
ox
= anino
tad
i
°
by
where a is aconstant. For the standard breakwater problem, the most logical
choice for this constant is given by
Z
1
= ———;— = |(Z cos a, cos op - Sin a, Sin a,)——77 a (55)
eee yan aa ( Oo b O Or, BAY a
since the shoreline in this case is principally governed by its behavior at the
breakwater. This problem (defined by eq. 55) has the solution
Vat
2
VAeent) = 2) canoe lems /4at
V0
K
- tan a x erfc{— == (56)
Gr
which is the same as that of Penard-Considere except that the constant a has
been changed. This problem, however, doesn't conserve mass since
co
3 P :
een y(x,t) dx = z sin a cos a sin za cos a
ae 46
When this approximation is used in the transport equation for Q, the
problem becomes
3Q 2
Ns cia
dt ax?
subject to the boundary conditions
Q(x = 0) = 0
Oise 2) GS G Sin oH,
which has the solution
cone
Q(x,t) = cos a sin ap erf leas) (57)
Integrating the equation,
dy _ 2Q
ot ox
gives
VR 2 x x
= i Moai? (ere ene | es woPe ae
y(X%,t) = cos o sin Oy [2 e = Sree ze) (58)
which is of the same form as the previous solution (eq. 56).
III. INPUT DATA
1. Description and History of Navigation Project at Holland Harbor.
The input data, which are pertinent only to the present investigation, deal
exclusively with the area near Holland Harbor, Michigan (U.S. Army Engineer
District, Detroit, 1975). Construction at Holland Harbor began about 1856 when
the city of Holland cut through a narrow tongue of land between Lake Michigan and
Lake Macatawa (Fig. 12). The present dimensions of the navigation project were
established in 1909. The existing navigation structures have been constructed,
3|
LAKE MICHIGAN
REACH 4 REACH 3. REACH REACH 1
°
tS SF ee
BAV4UI9bI
BAV WL
3NI7T ALNNOD
‘ay 1$3993919V3
LEGEND am
28-29x——_
1944-——.
7U.S.6.8. 1000 C) 1000___ 2000 3000 FT
Figure 12. Layout of shoreline reaches characterized
by different beach conditions.
reconstructed, and repaired by segments during different periods over the past
117 years. In general, the north side shows accretion while the south side
shows accretion from the breakwater to about 1,200 feet south and then general
erosion farther south.
The following areas (Fig. 12) appear to be affected by the navigation struc-
tures as evidenced by aerial photos, condition surveys, and plat maps for the
period of record (1849 to 1945):
(a) North of Holland Harbor (Eaglecrest Road to the North Pier of
Holland Harbor). It is considered that the accretion fillet north of
the piers has been relatively stable since 1933. After that date, the
predominantly southward-moving littoral drift has been diverted lake-
ward resulting in rising nearshore elevations north of the breakwater
and deposition of material in the entrance channel. This reach of
shoreline, about 5,000 feet in length, is characterized by increasing
accretion from north to south (Fig. 12, reach 1).
(b) South of Holland Harbor.
(1) Holland Harbor entrance channel to a point 200 feet south
of the Ottawa-Allegan County line (Fig. 12, reach 2). Reach 2 extends
about 2,000 feet south of Holland Harbor, and consists of a sand beach
about 50 feet wide backed by low sand dunes. The shoreline adjacent to
the south pier accreted at the rate of 9.6 feet per year from 1871 to
1944, a total of 700 feet. This area of accretion diminishes progres-
Sively southward from the piers to a point 200 feet south of the county
line where the shoreline before major construction (1871) coincides
with that of the 1929 and 1944 surveys.
(2) From 200 feet south of the Ottawa-Allegan County line to
147th Avenue (Fig. 12, reach 3). This reach is about 2,400 feet in
length and is characterized by high, stabilized sand dunes (up to 120
feet above low water datum) which are being undercut by wave action.
ge
(3) From 147th Avenue to a point approximately 1,900 feet south
of 146th Avenue (Fig. 12, reach 4). Reach 4, about 4,500 feet long,
is characterized by eroding sand dunes up to 225 feet high. Figure
12 shows that erosion is greater at the northern end of the reach and
decreases progressively southward. At the southern end of the reach
the shoreline before construction of the navigation structures coin-
cides closely with that of the 1929 and 1944 surveys. Erosion appears
to be greatest at a point about 800 feet south of 147th Avenue where
the shoreline receded about 220 feet from 1866 to 1945, or an average
of 3 feet per year.
The rates of movement of the shoreline near the harbor have decreased
(1945-73) but the trends of accretion on the north side and erosion on the
south side have continued. The present lower rates of shoreline movement
are due to several factors:
(a) The rate of shoreline movement due to the navigation structures
decreases with time.
(b) Local interests have built more shore protection structures as
their property was threatened. The consequence of these structures is
to reduce the apparent rate of erosion locally, but this is accomplished
at the expense of steepening of the offshore bottom profiles and in-
creased erosion downdrift. The shoreline protective structures now
extend almost continuously from 1,000 feet to about 4,600 feet south
of the harbor.
(c) The bluffs, currently under erosive wave attack between 5,000
and 7,000 feet south of the harbor, are very high so that erosion rates
measured in feet per year are low but the corresponding volume rates
(in cubic yards per foot) are high.
The shorelines apparently have become comparatively stabilized since about 1933.
2. Geomorphology.
Figure 13 shows the nearshore bathymetry determined by a survey in April
1973. Beach profiles from this survey and one in May 1945 are summarized in
Figure 14. Three or four lines of longshore bars are prominent most of the
time and occur in water depths to approximately 30 feet. The first bar, in
depths of 1 to 4 feet, can change rapidly from storm to storm and is often
broken into short segments from 200 to 1,000 feet long by rip channels. Adja-
cent to the breakwaters, the rip channels tend to elongate lakeward as far as
the second bar. The second and third bars, occurring in depths of 6 to 14 feet,
appear to be affected only by larger storm waves and are relatively continuous
along the shore. Aerial photos show that these bars tend to lose their promi-
nence as they approach the breakwaters from both sides, suggesting that pre-
vailing erosion exists across the bar crests near the breakwaters. Tie fourth
bar occurs at depths of 15 to 20 feet about 1,500 feet offshore and is charac-
terized by crescentic crests displaying wavelengths between 3,000 and 5,000
feet. A crescentic fourth bar with an average wavelength of about 5,000 feet
existed north of Holland Harbor, but none existed on the south side.
Although the general bathymetry follows the coastline, significant departure
from this trend occurs in a region offshore of the Holland Harbor breakwaters
(Fig. 15). The 24-, 27-, and 29-foot-depth contours meander sharply off the
33
“(SL6T ‘2T0TI0q ‘3OTIISTC TeouTSug Away *S* WOLF) TOGteYH pueTTOH “AxzoUAYyIeG eLOYSIVeN “ET eAndTYy
Be T
|
F i ; 7
| 5 z z
3 0008 ¢ +
Gan 213NI7
fan :
a) | Sa a ee
eS SS SS ee: =
Se = ~ Saat ==
30009
34
HO,LAND APR 73 SEC 111
ea € ;
Lan 18 873 0 1a howe ELevano® —— ora, ---
ReOG 7e00 ©3000 8200-3400
G00 800 FOOU FrOO PAC
“oc CCC‘ CC
FEET
Beach profile summaries for Holland Harbor (from U.S. Army Engineer
Figure 14.
District, Detroit, 1975).
59)
HOLLAND APR 73 SEC 111
" '
{eo a: sre 6 reo
FEET
° 20
“200 0 200 400 600 800 00 1200 00 00 00 2000 7700 7400 2600 7000 3000 S200 34a0
FEET
1 1
© 100 «0 600 800 WOO 1700 00 00 1800 7000 2700 7400 2600 72800 3000 HOO B00
cla
(wD in 87Re 1D
3
0 600 800 00 FD 00 1600 1900 OOD ROC r400 7400 FeoD SOOO 3200 B400
FEEy
Figure 14. Beach profile summaries for Holland Harbor (from U.S. Army Engineer
District, Detroit, 1975) .--Continued
36
“(SZ6T ‘2T01290q £39TIISTG TooUTSUq AWAY *S*p WoTZ) ATJOWAYIEG STOUSTeSU TOGIeH pueTIOH “ST oansty
Si
harbor entrance, delineating a ridgelike formation which extends obliquely
lakeward from north to south. Deeper contours to the southwest of this forma-
tion suggest that it may extend as deep as 50 feet. The ridge has a maximum
height of approximately 7 feet, and occurs due west of the harbor entrance.
Between the ridge and the harbor entrance is a steep trough about 34 feet deep.
A lack of survey data before the harbor construction prevents a reliable con-
clusion as to whether or not this underwater ridge was caused by the presence
of the breakwaters. However, the possibility cannot be ruled out that it was
formed by the littoral drift arriving predominantly from the north and flushed
lakeward by the breakwater. This material, unable to return shoreward due to
the absence of significant swell activity in this region, could have accumulated
to form such a formation on the offshore bed over a long period. Implications
of this interpretation are significant because it may mean that little, if any,
bypassing material across the harbor entrance could be expected to reach the
southern shore of Holland Harbor.
Many of the dunes reach a height of 150 feet, and in a few places exceed
200 feet above the lake level. The highest dunes are confined to a belt about
1 mile wide but lower ones occur for a width of several miles near Holland.
Dune building by wind action is still active. North of Holland Harbor some
low dunes are actively migrating inland or leeward from the bluff onto rela-
tively level upland. In this area the dunes are 40 to 120 feet high. At nu-
merous places, slumping of vegetation is evident on the bluff slopes. South
of Holland the dunes are higher, frequently more than 200 feet above lake level.
The dunes are well vegetated except for occasional blowouts on larger dunes.
At many places, dunes descend directly to the lake or to a narrow strip of
beach up to approximately 50 feet in width. The only area near Holland Harbor
where a substantial beach exists is about 2,000 feet immediately north and about
500 feet immediately south of the breakwaters. Maximum beach width in this area
is about 500 feet on the north side and 150 feet on the south side, indicating
the predominance of littoral drift from the north. Berm development is gener-
ally poor for many miles of shoreline north of Holland. Berms develop more fre-
quently south of Holland, growing to an average height of 1 foot in the swash
zone. These berms are usually truncated at positions of a recessed shoreline
where a rip current has gouged a deep channel across the surf zone.
3. Littoral Materials.
The dominant littoral material which comprises dunes, beaches, and nearshore
lake bottom is glacial sediment belonging to a fine sand category (less than
0.42 millimeter in median diameter). Scattered pebbles are found both on the
dunes and the lake bottom. Sand-sized material is dominantly quartz {80 to 85
percent) with up to 12 to 15 percent feldspar. Heavy materials represent only
ZEON Se PeXcenit.
a. North of Holland Harbor. An analysis of bluff, foreshore (above low
water datum) and bottom samples indicates that the majority of these sands
belong to the fine sand category. (The Unified Soil Classification System
defines fine sand as ranging in size from 0.42 to 0.074 millimeter (1.25 to
3.75 phi), and medium sand as ranging from 0.42 to 2.0 millimeters or 1.25 to
-1.00 phi). North of the harbor, the average bluff and foreshore sand consists
of 88.0 percent fine sand and 11.7 percent medium sand; the average lake bottom
sand consists of 95.0 percent fine sand and 5.0 percent medium sand. A general
38
trend exists for the sand size to become progressively smaller from foreshore
to offshore. Average median diameters are 0.34 millimeter at the foreshore,
0.28 millimeter at depths of 5 and 10 feet, 0.23 millimeter at depths of 15 and
20 feet, and 0.18 millimeter at depths of 25 and 30 feet. Dune and bluff sand
is somewhat finer than the foreshore sand, but coarser than the lake bottom
sand. There is little systematic variation in sand size with distances from
the breakwater. Sorting ranges between 0.28 and 0.40 phi, indicating very good
sorting; sorting varies little between dune, foreshore, and the 5-foot depth
and becomes poorer toward deeper water, reaching approximately 0.40 phi at
depths of 20 and 30 feet.
b. South of Holland Harbor. The majority of the sands from the dune, the
bluff, the foreshore, and the lake bottom on the south side of Holland belong
to the fine sand category. Generally, only minor differences exist between
average sands from the north and the south sides of Holland. The average bluff
and foreshore sand consists of 85.1 percent fine sand (finer than 0.42 milli-
meter) and 14.9 percent medium sand. The average lake bottom sand consists of
94.7 percent fine sand and 5.3 percent medium sand. Median diameters decrease
from foreshore toward offshore, about 0.25 millimeter at depths of 5 and 10
feet, about 0.20 millimeter at depths of 15 and 20 feet, and about 0.16 milli-
meter at depths of 25 and 30 feet. Dune and bluff sand is finer than the fore-
shore sand but coarser than the lake bottom sand. The lake bottom sands are
distinctly coarser immediately south of the harbor and also along lines 11 and
12 (Fig. 13), about 6,000 feet south of the harbor. Sorting displays a wider
scatter than on the north side of Holland, ranging between 0.3 and 0.5 phi.
As on the north side, sorting becomes poorer from foreshore toward offshore.
The principal sources of littoral material are the beach and the bluff.
Generally along eastern Lake Michigan, and particularly near Holland Harbor,
the supply of littoral material from major inland water runoff is negligible.
For this reason, the long-term geologic trend appears to be for sediment from
shore erosion to fill in the lake.
c. Shoaling and Dredging. Shoaling and dredging records indicate that the
annual shoaling in the entrance channel of Holland Harbor averaged about 25,000
cubic yards for the period 1965-70. The pattern of accretion shows some yearly
variation and indicates that material encroaches into the entrance area from
both north and south. In addition, the relative amount of drift from north
and south appears to vary from year to year. Figure 16 shows conditions at
the harbor entrance before dredging in 1973 and 1974.
4. Meteorology.
The dominant factors are winds, waves, and water level variations. The
winds and waves are directly responsible for sediment movements, and fluctua-
tions of water levels separate the regimes of the two areas affected, i.e.,
the backshore above the waterline and the foreshore below the waterline.
a. Winds. Winds are a dominant force affecting the Holland Harbor area;
they produce a number of major effects by (a) generating a force for waves,
(b) causing lake level changes, and (c) transporting sand across the beaches,
particularly the finer sediment sizes. Wind data (available from ship reports)
over the southern half of Lake Michigan (south of 44° N.) show that more than
39
DEPTH IN FEET.
ALL DEPTHS ARE REFERRED
TO LOW WATER DATUM
576.8 ft. ABOVE 1.G.L.D. 1955
MAR. 1973 MAR. 1974
Figure 16. Conditions at Holland Harbor entrance before dredging in 1973
and 1974 (from U.S. Army Engineer District, Detroit, 1975).
21,000 observations were taken from 1963 to 1973. These data have been sum-
marized for a 12-month period (Fig. 17,A) and for a 9-month ice-free period
(Fig. 17,B). A 3-month ice period would correspond to a severe winter.
b. Waves. Sources of wave data for Lake Michigan were (a) Summary of
Synoptic Meteorological Observations (SSMO) (National Oceanic and Atmospheric
Administration (NOAA), 1963-73); (b) Saville (1953); (c) Cole and Hilfiker
(1970); and (d) Littoral Environment Observation (LEO) program data from the
U.S. Army Coastal Engineering Research Center (CERC).
Figure 18 shows deepwater wave roses for an average 12-month period and for
an assumed 9-month ice-free period, respectively, in southern Lake Michigan.
Comparison of these wave and wind roses reveals a close agreement with the wind
and wave statistics. Dividing wave heights into two groups (i.e., by heights
lower and higher than 4.1 feet), the low waves occur most frequently from the
south, while the high waves occur predominantly from the north.
Figure 19 shows exceedance probabilities of deepwater waves based on SSMO
and hindcast data of Saville (1953) and Cole and Hilfiker (1970). Only those
waves which occur during a 9-month ice-free period and are traveling shoreward
are included in the statistics. The three statistics generally agree quite
well for wave heights up to about 8 feet. For wave heights larger than 8 feet
(which occur only about 1 percent of the time), a discrepancy between the three
is evident. The SSMO data show a less frequent occurrence of higher waves, but
4
(oe)
10% 10 %
NW. NE NE
1 Pron i Oe
{| ly - 7. ” “
g ‘J / Lx ; rs Oy
\ a ES eh LES
KE oc, 03 ,§
= toy, SESS w Se Ie Sa
100 S PERIOD (~,
- aN SSS —— ae
Z WSL SRE!
s LW
Lf aan
ap VS /
bs \\\ SSS,
5% a
vy
SW: SE
1Q%o
s WIND SPEED 2
(7) 0-10 kn
1-21 kn
Sone) 22-33 kn
WSIGESET 34-47kn
Figure 17.
Wind rose for an average 12-month period (A) and for an
average 9-month ice-free period (B) for southern Lake
Michigan (data from SSMO observations, 1963-73).
WAVE HEIGHT
Co) < 0.25 meter
GEE] 0.25 -1.25
Figure 18.
1.25-2.25
GEEZER 2.25 <
( <0.82 foot )
(082-410 " )
(410-738 " )
(7.38< mi)
Wave rose for an average 12-month period (A) and for an
assumed 9-month ice-free period (B) for southern Lake
Michigan (data from SSMO observations, 1963-73).
WAVE HEIGHT, (ft)
100
EXCEEDANCE PROBABILITY, ( pct.)
0.1
—— SSMO (1963-73)
—-— SAVILLE (1953)
-+—+ COLE and HILFIKER (1970)
0.01
Figure 19, Wave height exceedance probabilities
(9-month ice-free period).
this may be partly due to the fact that ships tend to avoid storms. This may
also be less true for the higher windspeeds; hence, the frequency of occurrence
of high waves may tend to be overestimated. The disagreement between Saville!'s
and Cole and Hilfiker's hindcast data probably results because Saville's data
are based on coastal winds which are generally weaker than those on the deep-
water lake surface. Figure 20 summarizes the probability of occurrence of
various wave heights as a function of the month of the year.
100
1.64 ft OR HIGHER
3.28 ft OR HIGHER
4.92 ft OR HIGHER -
PROBABILITY (Pct)
cé,)
oO
c JAN FEB MAF APR MAY JUNE JULY AUG SEPT OCT NOV DEC JAN
Figure 20. Variations of probability of various
wave heights with months of the year.
42
5. Hydrology.
a. Water Level Variations. Figure 21 summarizes the observed mean monthly
lake levels since 1920. Records are available back to 1860.
b. Currents. Currents in the Holland area are variable. They generally
appear to take a northward set during the summer and a southward set during the
winter in response to the prevailing wins although reversals are frequent.
There also appears to be evidence of a rotational current system in Lake
Michigan which is not very well understood. In the immediate nearshore area
it is likely that the currents are wave induced and will propagate alongshore
in the direction of the dominant waves at the time. Inshore of the first bar,
rip currents are extremely well developed and can add a significant amount of
offshore water movement to the general longshore current. Intensity of the rip
currents is evident from a number of rip channels which are gouged 1 to 2 feet
below the adjacent ground. Longshore currents approaching Holland Harbor are
deflected lakeward along its breakwaters. Aerial photos indicate that the de-
flected current, after bypassing the harbor entrance, will move a considerable
distance lakeward before turning parallel to and toward the shore off the down-
drift coast.
c. Ice. Floating ice is common in the Holland Harbor area during the
winter months. Ice accumulates on the beach by wind and wave action and re-
portedly often extends in a solid mass from the shoreline to more than 1,000
feet offshore. Wave heights are substantially decreased within an ice field
and ice along the shore blocks wave action to some extent. However, the action
of ice can accelerate erosion processes; e.g., ice being pushed upon the beach
will loosen consolidated beach and bluff material and some of the sediment be-
comes embedded in the ice at the shore and over the longshore bars to be carried
elsewhere if the ice drifts away during the spring thaw. The average ice season
at Holland Harbor extends from late December to late March.
6. Littoral Drift Estimate from Wave Statistics.
Littoral drift has been computed from the available wave statistics dis-
cussed earlier. The governing relationship used was (U.S. Army, Corps of
Engineers, Coastal Engineering Research Center, 1977)
Qy= 01000133 RER
where Q is the potential littoral drift in cubic yards per day, and E the
longshore momentum flux in feet per pound per day per foot of beach. The re-
lationship between of, and a, is determined from linear theory in assuming
that the bottom contours are straight and parallel. The number of waves per
day for each wave height period-direction class is given by
_ BS Ie
be Sanu Os Sa
where p is the probability of occurrence of that wave class. Computations
were made using the wave statistics from SSMO data, Saville (1953), and Cole
and Hilfiker (1970). The duration of computation is for the 9-month ice-free
period from late March to late December.
43
*(SL6I
*}T0130q
¢
JOTIZSTG To9uTsug Away
S
NM WOlF) QOZ6T 9duTS
STOAST
oyeT ATYJUOW UPA
‘TZ eins ty
44
All of the littoral drift computations show a large gross drift of 300,000
to 500,000 cubic yards, defined as the sum of all north and south movement,
regardless of direction, whereas the net drift is much smaller. Individual
yearly predictions show a wide variability in net littoral drift quantities.
The littoral drift is predicted to be from south to north with a mean value
of more than 265,000 cubic yards. The SSMO data and Coles and Hilfiker's wave
statistics lead to the prediction of 61,000 and 76,000 cubic yards, respectively,
from north to south. This direction is consistent with the observed growth of
the fillets on each side of the harbor, the north side showing a much greater
accretion.
Figure 22 summarizes the monthly fluctuations of littoral drift based on
the SSMO data. The figure shows that the monthly drift is large during late
fall when the activity of extratropical cyclones intensifies in these regions,
and small during the summer months, especially in June, when wave heights are
generally smaller.
MONTHLY GROSS DRIFT
x10" (yd?/mo)
MONTHLY NET DRIFT
s
&
io 0 JAN BEDE MAR APR MAY JUNE UG SEPY OCT NOV DEC
es iI‘ JULY
ee yj
ted 10 / 7
°
251210
imal AVG. ICE-FRE® PERIOD
Y AVG. ICE PERIOL
Figure 22. Monthly variations of eo of computed
gross and net littoral drifts
for Holland, Michigan.
7. Analysis of Aerial Photos.
A sequence of eight sets of aerial photos of Holland Harbor taken over the
interval 27 July 1950 to 5 October 1973 was analyzed to assess the long-term
evolutionary development of the shoreline. A regression analysis was verformed
to correct the shoreline for lake level variations and to determine the long-
term erosion-accretion rates along the coast.
45
Table 1 presents shoreline positions (with and without lake level correc-
tions) as a function of distance along the coast, as determined from the aerial
photos. The corrected shoreline positions are referenced to a common lake level
of 578.5 feet, as determined by a regression against lake level. Figure 23
Shows the 1973 Holland Harbor shoreline. The origin of the coastline coordinate
is taken at the harbor, positive in the north direction. The shoreline position
was measured at 290-foot increments along the coast.
A regression of shoreline position against lake level and time provided an
estimate of the long-term evolutionary trend of the shoreline at each station
along the coast. The regression study revealed general erosion extending south
of the harbor and accretion immediately north of the harbor. Over an 8,410-
foot span south of the harbor, the average beach erosion rate was 0.75 foot
per year. Over a 4,060-foot span immediately north of the harbor, the average
accretion rate was 1.65 feet per year. A span of 10,585 feet, starting 4,930
feet north of the harbor, had an average erosion rate of 1.28 feet per year.
These evolutionary trends of the shoreline include natural effects as well as
any effects which can be attributed to the harbor.
Figure 24 shows the evolutionary smoothed trend (defined as the average
accretion or erosion rate between 1950 and 1973) along the coast near Holland
Harbor. Erosion dominates south of the harbor, accretion immediately north,
and erosion again farther north of the harbor. The smoothed curve in Figure 20
demonstrates that the accretion resulting from the harbor extends about 5,000
feet north of the harbor. These estimates of the evolutionary trend of the
shoreline are subject to large annual and spatial fluctuations.
8. Comparisons of Profiles from 1945 and 1975 Surveys.
The surveys of May 1945 and April 1973 (lake levels at 578.4 and at 580.1
feet, International Great Lakes Datum (IGLD), respectively) permit a comparison
of profile changes from which sediment volume changes can be estimated. Since
the actual profile locations in the two surveys did not coincide (in general),
the 1945 survey was interpolated on the lines of the 1973 survey to allow a
comparison. Table 2 summarizes the cross-sectional area changes and the volume
computations in two parts (above and below low water datum). The last colum
in the table gives the erosion or accretion rates determined from the aerial
photos.
The average volumetric accretion rate on the north side (from ground survey
data) is 2.13 cubic yards per year per foot of shoreline for 1,950 feet (450 to
2,400 feet north of the breakwater) compared with the average shoreline accre-
tion rate of 1.92 feet per year for this same region determined from Figure 20.
The average volumetric erosion rate on the south side of the harbor from the
survey comparisons is 1.41 cubic yards per year per foot of beach, whereas the
values from Figure 20 for the south stretch (470 to 5,550 feet south) give 0.82
foot per year as an average rate of shoreline erosion. The agreement on the
north side indicates that the approximation of 1 square foot of beach per cubic
yard is reasonably valid, whereas on the south side this is less true. The dis-
crepancy on the south side is related to the bluff erosion contribution. The
main erosion rate below the low water datum level is only 0.16 cubic yard per
year per foot. Hence, the bluffs contribute at least an additional 1.25 cubic
yards per year per foot, which exceeds the conditions for the approximation of
1 square foot of beach to correspond to 1 cubic yard of material.
46
Table 1.
History of shoreline positions at Holland, Michigan,
from 1950 to 1973, with and without lake level
corrections. |
Coastal
position
(ft)
27 June 1950
Distance lakeward from base line
(ft)
[11 oct. 1955] 8 June 1960 [19 Oct, 1962
Uncorrected lake levels
5 Sept.
1967 [9 May 1965] § Oct, 1973]
f 0.00 1219.79 1088.80 1113.36 1146.11 1188.68 1187.04 1178.86 1146. 32
-290.00 982.38 949.38 982.38 982.38 998.78 966.01 941.45 921.15
\ -580.00 851,40 835.02 851.40 851.40 884.14 835.02 835.02 757.39
-870.00 769.53 736.78 769.53 766.26 785.90 744.97 783.16 716.45
-1160.00 720.41 671.29 687.67 687.67 705.68 671.29 687.67 655.04
-1450,00 671.29 630. 36 622.17 630.36 654.92 632.00 627,09 624.33
-1740.00 630.36 638.55 613.99 613.99 632,00 618.90 622.17 624,33
-2030.00 597.61 605. 80 605.80 597.61 $99.25 605.80 $89.43 583.39
-2320.00 $73.05 $07.56 $23.94 $15.78 $40.31 523.94 $15.75 $11.75
-2610.00 $23.94 474.82 491,19 483.00 499,38 491.19 474.82 460.57
-2900.00 483.00 433. 88 425.70 442.07 442,07 442.07 435.52 388.93
-3190.00 450.26 392.95 376.58 425.70 409.32 442.07 422.42 317.28
&} -3480.00 412.60 327.46 327.46 384.77 343.83 360.21 392.95 307.05
i= 2 -3770.00 376.58 319.27 311.09 376.58 343.83 327.46 343.83 296.81
ts -4060.00 376.58 343.83 335.65 376.58 353.66 330.73 352.02 327.52
ove -4350.00 376.58 352.02 343.83 376.58 360.21 356.93 335.65 286.58
2 2| -4640.00 373.30 343.83 343.83 386.40 409. 32 386.40 348.74 266.11
ea -4930.00 363.48 340.56 343.83 392.95 401.14 381.49 343.83 276.34
-5220.00 376.58 376.58 368.39 376.58 376.58 376.58 352.02 307.05
-$510.00 384.77 360.21 368.39 384.77 396.23 392.95 368.39 276,34
-5800.00 392.95 360.21 343.83 376.58 402.78 409. 32 376.58 307.05
-6090.00 412.60 343.83 376.58 384.77 409, 32 409.32 368.39 307.05
- 6380.00 425.70 343.83 368.39 384.77 425.70 392.95 376.58 276. 34
-6670.00 419.15 352.02 343.83 392.95 474.82 409.32 392.95 286.58
-6960.00 399.50 360.21 352.02 389.68 419.15 417.51 417.51 276.34
-7250.00 389.68 368.39 376.58 392.95 373.30 409.32 409.32 307.05
-7540.00 376.58 368.39 352.02 368.39 360.21 396.23 376.58 307.05
-7830.00 343.83 368.39 327.46 360.21 376.58 379.85 345.83 286.58
-8120.00 327.46 360.21 297.99 327.46 286.53 343.83 320.91 245.64
-8410.00 327.46 343.83 294.71 311.09 283.25 327.46 294.71 255.87
-870C6.00 327.46 1498.13 311.09 294.71 270.15 294.71 278.34 245.64
580.00 1473.57 1399. 89 1555.43 1449.01 1612.74 1540.70 1542.34 1498.40
870.00 1342.59 1309.84 1391.70 1358.96 1424.45 1358.96 1375.33 1310.08
&#; 1160.00 1288.28 1236.16 1318.03 1303.29 1375.33 1290.19 1304.93 1228.20
3B] 1450.00 1238.16 1170.67 1219.79 1236.16 1309.84 1224.70 1260.72 1136.08
$&&| 1740.00 1203.42 1129.74 1187.04 1195.23 1260.72 1208.33 1219.79 1166.79
3*| 2030.00 1178.86 1090.44 1154.30 1195.23 1211.60 1196. 87 1211.60 1115.61
2 2 2320.00 1146.11 1096.99 1105.18 1162.48 1244.35 1195.23 1195.23 1043.97
Ea} 2610.00 1113.36 1096.99 1088.80 1137.92 1227.97 1155.93 1162.48 1043.97
2900.00 1074.07 1064.24 1090.44 1105.18 1146.11 1146.11 1137.92 1023.50
3190.00 1031.50 1018.13 1064.24 1083.89 __1126.46 1096.99 1080.62 1003.
Corrected lake levels
0.00 1211.21 1096.31 1113.36 1150.40 1161.86 1178.46 1189.23
~290.00 974.29 956,72 982.38 986.43 973.46 957.91 961.62
580.00 827.42 847.25 851.40 858.38 840.47 821.05 827.27
870.00 761. 86 743.50 769.53 770.09 761.93 737.30 754.81
-1160.00 714.08 676.86 687.67 690.84 685.81 664.94 679.72 686.83
-1450.00 667.60 633.59 622.17 632.21 643.37 628.30 622.47 642.81
1740.00 629.99 638.87 613.99 614.17 630. 84 618.53 621.71 626.19
-2030.00 $95.91 607.29 605.80 598.47 593.93 604.10 $87.30 $91.91
-2320.00 $68.75 $11.33 $23.94 $17.90 $26.85 $19.63 $10.37 $33.28
-2610.00 518.61 479.47 491.19 485.66 482.75 485.87 468.17 487.18
-2900.00 475.24 440.68 425.70 445.95 417.80 434.30 425.81 427.77
-3190.00 436.12 405.33 376.58 432.77 365.13 427.93 404.75 387.99
5 -3480.00 404.36 334.67 327.46 388.89 318.07 351.96 382.65 348.27
AG -3770.00 370.06 324.98 311.09 379.84 323.45 320.94 335.68 329.42
23 -4060.00 373.38 346.63 335.65 378.18 343.67 327.54 343.50
8 a -4350.00 367.70 359.79 343.83 381.02 332.45 348.05 331.00
ae -4640.00 256.78 358.29 343.83 394.67 357.68 369.88 348.74
& |] -4930.00 349.45 352.84 343.83 399.97 357.28 367.46 346.51
-5220.00 368.52 383.63 368.39 380.61 351.40 368.52 347.33
-S510.00 369.84 373.26 368.39 392.23 349.59 378.03 350.96
-5800,00 380.42 373.17 343.83 382.84 363.62 396.80 369.70
6090.00 399.05 355.69 376.58 391.54 366.98 395.78 374.80
6380.00 406.77 3609.40 368.39 394.23 366.55 374.02 370.99
-6670.00 397.14 371.28 343.83 403.96 406.03 387.31 396.64
6960.00 380.26 377.04 352.02 399.30 359.02 398.27 372.55
-7250.00 378.92 377.80 376.58 398.33 339.69 398.57 360.83
~7540.00 368.26 378.€7 352.02 372.55 334.20 387.91 48.66
7830.00 333.79 377.18 327.46 368.23 345.21 369.82 336.77
-8120.00 320.83 366.00 297.99 330.77 265.82 337.21 278.77
~8410.00 323.07 347.67 294.71 313.28 269.54 323.07 277.82
$80.00 1462.02 1508.24 1555.44 1454.79 1876.63 1529.14 1556.18
870.00 1331.98 1409.17 1391.71 1364.26 1391.30 148.35 1363.12
«| 1160.00 1270.90 1322.42 1318.03 1310.48 1330.39 1275.81 1300.10
S| 1450.00 1217.10 1282.84 1219.79 1245.69 1250.28 1205.64 1231.38
$5 1740.00 1192.73 1180.02 1187.04 1200.87 1227.33 1197.64 1220.21
3+} 2030.00 1165.84 1141.13 1154.30 1201.74 1170.91 1183.85 1180.72
™ &! 2320.00 1121.49 1111.99 1105.18 1174.79 1167.40 1170.64 1167.08
3 & 2610.00 1092.75 1115.03 1088.80 1148.23 1163.55 1135.32 1147.05
2900.00 1053.97 1110.21 1090.44 1112.73 1098.91 1131.01 1099.02
3190.00 1018.51 1075.61 1064.25 1090.39 1085.86 1084.00 1067.99
3480.00 | _1020.92 1028.68 1023.31 1085.09 1048.59 1016.01 1043.63
\Determined by aerial photos.
47
ee —— NITY INID JDNVYLND YOBEVH
°o ° ° ° o °
°o ° ° ° °
bod oO v e %
(44) JYOHSI4O
48
° wn
S fe)
Py p
O
oS
joe
ce
Es}
4
4
o
3
a
rw
ae)
: :
3 2
= =
s
o
PS)
o
isc)
n
ey
ise)
™~
a
So
fs) SI
ey a
0 n
S
3
60
“cd
= us
Shee.
=
z ~
(S
w
re
a
° g
p
3
yn
“ =
fe)
“4
p
“od
wn
°
a,
®
5
3 oH
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0 4
e)
iS)
wn
Figure 23.
\
ee ears ee cw se es = BNITUI NID JINVYULNS YORYVH
” r) - ° 7
GV 00) MOL 3ED9V wOIsOUR
————— —es
49
22
(247 00)
13,000
10,000
5,000
25,000
(ft)
N
Smoothed average rates of shoreline erosion and accretion
(feet per year) at Holland, Michigan.
Figure 24.
Table 2. Summary of cross-sectional changes and computed
volumetric changes for 1945 and 1973 surveys. !
[ Above LD { ___below LWD lee observed shoreline
Line ||Distance |} Cross section | Avg. rate of }/ Cross section | Avg. rate of || Total mean v | accretion- i
g ' g ol. | erosion
No. || between area change vol. change area change | vol. change change rate change rate
profiles (1945-73) (1950-73)
(ft) (£2), (yd?/yr/ft) || (ft?) | (ya /yr/ ft) (yd3/yr/ft) (ft/yr)
1 2,298 1,528 z
1,150 +1.47 +1.10 +2.57 +2.0
2 -78 134
800 +0.84 +0.65 +1.49 7113)
3 1,347 855
Harbor
4 2,144 928
740 +1.16 +0.30 +1.46 -0.3
§ 391 -479
880 +0.17 -0.53 -0. 36 -0.6
6 642 -324
1,210 -0.24 -0.72 -0.96 -0.9
7 821 -761
605 -0.91 -2.80 -3.71 -1.0
8 -562 -3,480
830 -1.59 -3.39 -4.98 -1.2
8) -1,848 -1,660
815 +0.39 -0.88 -0.49 -0.9
One| +2,431 I 305 |
poet a ee Nee ee ee | | 2 =
lL = accretion; - = erosion
As demonstrated previously in the analysis of the aerial surveys, the har-
bor influence on the north side extends about 4,200 feet. Farther north it is
assumed that the observed erosion rate will be close to the natural erosion
rate for this area of the Lake Michigan coast. At 4,200 to 15,400 feet to the
north of Holland Harbor, the average erosion rate during the 1950-73 period
(from aerial photos) was 1.28 feet per year per foot of shoreline. If the
approximation of 1 cubic yard of volume per square foot of beach is used, the
corresponding sediment volume loss is 1.28 cubic yards per year per foot of
shoreline plus the erosion rate of the dunes. The bluffs which have an average
height of about 40 feet in this area would contribute an additional sediment
loss of 40 x 1.28/27 = 1.90 cubic yards per year per foot. Hence, the total
sediment loss to the offshore in this area is estimated to be averaging 1.90
+ 1.28 = 3.18 cubic yards per year per foot of shoreline during the past 23
years. This rate of loss is considered to represent the natural rate of loss
for littoral sediments in the Holland area due to all factors except the navi-
gation structure. It must be remembered that the erosion rate of 3.18 cubic
yards per year per foot of shoreline is an average rate for the past 23 years
(1950-73). Since lake levels have been much higher during the past few years,
the present erosion rate must be expected to be higher than the average and
therefore will probably exceed 3.18 cubic yards per year per foot of beach.
9. Sediment Budget.
a. North Side. An independent evaluation of the net littoral drift has
been obtained from various surveys as summarized below in which a sand budget
analysis is made for the north side of Holland Harbor. The shoreline changes
shown in Figure 25 provide information for an area measurement of accretion.
An approximation for volumetric accretion can subsequently be made from the
relationship, wherein 1 square foot of change in beach surface area equals 1
cubic yard of beach material. This relationship has previously been shown to
be reasonably valid for the north side. Subsequent corrections may be neces-
sary because of the dune-building phenomenon.
50
HOLLAND
BUILT 187-73
DREOGING LIMITS —@
;
Fur
fe iG 1? 9: fol) D0 1) ee Eh bd (ZU ins en
- 50 Co oo re
aS ae S71, Le on ec
ea ee co < Ost
ocr ou os
t er
al 921
Historical shorelines north of Holland, Michigan, between 1849
and 1945 (from U.S. Army Engineer District, Detroit, 1975).
Figure 25.
S|
The area changes corresponding to the waterline shown in Figure 21 were
determined for a length of shoreline extending 2,550 feet north of the north
breakwater. These areas are summarized in Table 3. A correction is applied
for the variation in lake level by the relationship
Correction in 1,000 square feet = sp ~ take Tevet ot rference ~ 2,580
which follows the assumption of an average beach slope of 1 on 45 for this area
as determined from the 1945 surveys.
Table 3. Summary of shoreline accretion rates north
of Holland Harbor (1849-1945).
Measured area from | Lake level Lake level Area Net area Rate
breakwater to (N.Y. datum) difference | correction
2,550 feet north
(1,000 ft? (ft) (ft) (1,000 £t2) | (1,000 ft?) | (1,000 £t2/yr)
The results of the computations (Table 3) indicate rates of accretion vary-
ing from 3,400 to 51,900 cubic yards per year if the general rule of 1 square
foot of beach area corresponds to 1 cubic yard of beach material is applied.
These accretion rates should be interpreted according to the history of the
construction. The first breakwaters were constructed in 1856-60 and trapped
most, if not all, of the littoral drift. Since the accretion of 51,900 square
feet per year during the period 1856-71 is an average, then the corresponding
littoral drift rate must have exceeded 51,900 cubic yards per year (it must be
assumed that by 1871 some of the littoral transport was passing around the break-
water heads into the navigation channel).
The accretion was slowed to 3,400 square feet per year from 1871 until the
harbor breakwater extensions were started in 1906. Therefore, the littoral
transport passing around the north breakwater must have been quite high from
1871 to 1906; also, the building of new dunes by wind action probably occurred
during this period.
From 1906 to 1933 the accretion averaged 13,700 square feet per year; after
1933 this rate dropped slightly to 13,400 square feet per year. However, some
inland dune building could still be taking place. Measurements from the 1945
survey show an accumulated volume of 137,400 cubic yards above elevation 585
(1945 datum) on the north side of Holland Harbor. This volume would only
account ror about 1,900 cubic yards per year from 1871 to 1945 but windblown
sand losses farther inland would increase this.
The detailed analysis of aerial photos taken from 1950 to 1973 indicates
an average accretion rate of about 1.65 feet per year per foot of shoreline for
the first 4,000 feet of shoreline north of the harbor, a total of 6,600 square
feet per year. The accretion rate based on a comparison of the 1945 and 1973
hydrographic surveys for profiles 1, 2, and 3 in Figure 10 was about 4,800
52
cubic yards per year for the first 2,550 feet north of the breakwater. The
incoming littoral drift supply must be providing the observed accretion since
there are expected offshore losses of 3.18 cubic yards per year per foot of
shoreline for the first 4,000 feet of shoreline north of the harbor, and
material is apparently being accumulated in the dunes in Holland State Park.
To summarize the present situation, it is believed that about 61,000 cubic
yards of littoral material is being transported toward Holland Harbor from the
north. The disposition of this material is summarized in Figure 26. About
12,700 cubic yards per year is lost offshore (3.18 x 4,000), 6,600 cubic yards
per year is accumulating on the beach face on the north side, and about 1,900
cubic yards per year is being accumulated in the dunes of Holland State Park.
Although the exact amount is unknown, an estimated 1,300 cubic yards per year
of sand is lost inland. Hence, it is calculated that about 38,500 cubic yards
per year of sand arrives at the harbor entrance but at the most 25,000 cubic
yards is trapped. It is likely that some of the 25,000 cubic yards dredged in
the harbor entrance comes from the south side; therefore, at least 13,500 cubic
yards per year is lost offshore. Consequently, it is concluded that the harbor
structures cause a loss of materials from the littoral region of at least
13,500 cubic yards per year. In addition, the 25,000 cubic yards per year
that is dredged is dumped offshore. The overall result is that the south side
of the harbor is being starved of 61,000 cubic yards per year.
3
13,500 yd~ lost offshore
a
25, 000:
deposited ins RagGoU>
entronce
channel
yd 37 fos? offshore
4
Total 12,700 yas lost offshore
ae wan
Total 9,800 yd > to State Beach Park
61,000 yd 2
Incoming littoral delft
ee
Total 1,900 yd 3 to dunes
Y
HOLLAND STATE
PARK
3.18 yd 374 eroded from shoreline
Wet bio taay
3
Total 1,300 yd to Inland
4,000ft
Figure 26. Summary of sand budget north of Holland, Michigan.
b. South Side. Figures 27 and 28 provide information for an area measure-
ment of shoreline accretion and erosion. The beach area changes are summarized
in Table 4 by areas north and south of a section located 1,200 feet south of the
south breakwater (an arbitrarily selected station which appears to correspond to
a stationary shoreline point for the period 1856-1945). The table summarizes
the corrections in lake level variation about an average offshore beach slope
of 1 on 24.5 for the first 1,200 feet south, and an average beach slope of 1
53
MACATAWA
2G rs IAT ITS IDS 02
—
Historical shorelines immediately south of Holland,
Michigan, between 1849 and 1945 (from U.
Engineer District, Detroit, 1975).
Figure 27.
Army
.
S
54
Figure 28. Historical shorelines farther south of Holland, Michigan, between
1871 and 1945 (from U.S. Army Engineer District, Detroit, 1975).
55
Table 4. Summary of beach area changes south of Holland Harbor.
Measured area Lake level Lake level Area Net area Rate
(N.Y. datum) difference | correction
(1,000 ft?) (ft) (£t) (1,000 ft?) | (1,000 ft?) | (1,000 £t2/yr)
Breakwater to 1,200 feet south
on 12.1 for a length from 1,200 feet to 3,600 feet south of the breakwater.
The values shown in Table 4 can be added for a representative length of shore-
line of 3,600 feet to yield the result that an accretion rate as high as 48,500
Square feet per year in the period 1856 to 1871 decreased to a rate of about
6,000 square feet per year from 1871 to 1906, then increased again to 20,500
square feet per year following construction of the breakwater extensions after
1906, and finally showed an average erosion rate of 8,000 square feet per year
from 1933 to 1945. This last value corresponds to an erosion rate of 0.75 foot
per year as determined from analysis of the aerial photos. However, it should
be emphasized that the recently observed erosion rates reflect two effects:
(a) The length of shoreline near the harbor has been protected by rubble and
groins, and (b) the unprotected shoreline farther south is backed by dunes with
an average height of about 120 feet with peaks more than 200 feet above the
lake level.
The rate of accretion at times on the south side may seem somewhat surpris-
ing, but it must be recalled that the littoral drift often reverses (see Fig.
22) and in some years may be completely reversed. The variable direction of
littoral drift is also evidenced by the two surveys in Figure 12, which shows
the harbor entrance shoal from the south side as being apparently larger than
that from the north in 1973.
Using the approximation of 1 cubic yard of sediment per square foot of beach
the 1950-73 observed average erosion rate for about 9,000 feet south of Holland
Harbor is 0.75 cubic yard per year per foot of shoreline, plus the bluff contri-
bution of 0.75 x 120/27 = 3.33 cubic yards per year per foot for a total loss
rate for unprotected shoreline of 4.08 cubic yards per year per foot of shore-
line. When compared with the erosion rates observed for the shoreline about 1
mile or more north of Holland Harbor of 3.18 cubic yards per year per foot, the
difference of 0.9 cubic yard per year per foot of 22 percent can be attributed
to the navigation structure. The effect is very small and should not be ex-
pected to be readily apparent. The rate of loss of 0.9 cubic yard per year per
foot of bluff material would correspond to a shoreline erosion rate of less than
0.2 foot per year for a bluff height of 120 feet.
56
The actual loss of land to the south of Holland (attributed to the naviga-
tion structure) is relatively small, at the most 22 percent. This figure has
been derived as a 23-year average, but with current high lake levels the pres-
ent natural erosion rate is certainly higher than the average.
The evolution of the shoreline at Holland was studied using the present
model. The relevant physical data and the estimates of offshore sediment losses
were used in the analysis. Interpolation of values shown in Table 1 gives the
shoreline every 100 feet along the base line. The channel at the harbor en-
trance was collapsed to zero so that the south reach ended at O- and the north
at 0+. These shorelines indicate that the shore is stable at the breakwater;
therefore, when the direction of transport is toward the breakwater it is as-
sumed that the sand transported to the breakwater is entirely lost offshore.
Monthly lake levels were taken from Figure 21. The results of these computa-
tions are not given since they are essentially previously described results.
The average beach slope at the waterline was determined to be 1:10; the beach
profiles in Figure 17 near the breakwater show that this is a reasonable esti-
mate although the slopes are not constant from profile to profile. The height
of the berm is assumed to be 10 feet. The depth to no sediment motion was esti-
mated at 30 feet, based on visual consideration of the offshore bathymetry and
the use of Weggel's method (J.R. Weggel, personal communication). An offshore
loss of 3.2 cubic yards per year per foot of beach is also included (see Fig. 26).
The transport equation for this situation can be written (see eq. 53)
A een 3 2 eh
Be Os (KpQ) m dt dt
where
Q | COS G, Sin hy and 2 — (K5Q)
m = beach slope at waterline
D = dimensionless lake level
ea = dimensionless offshore line loss
This equation will be applied on each side of the breakwater. When the in-
coming wave direction gives drift toward the breakwater, the boundary condi-
tions expressed in Q are
Q | b
= COS a sin ap
xX = wo
Conversely, when the incoming wave direction gives drift away from the break-
water, the conditions become
Le)
i]
[e)
and
cos a sin On,
57
Choice of wave climate is the remaining input parameter to be determined, and
is the most controversial. The wave climate most desirable for the study of
shoreline evolution is a time series giving wave height, H, period, T, and
direction, D. Unfortunately, this is seldom available and hence monthly sta-
tistical summaries must be used, such as those by the SSMO for southern Lake
Michigan. One possible use of the summaries is to construct monthly times t
for each possible (H,T,D) triple, i.e., t(H,T,D), and then calculate the evo-
lution of the shoreline as the (H,T,D) triples are chosen in some determin-
istic order or at random. This method would be computationally very expensive
and is not used. The most simple approach is to assume that these are but two
(H,D,T) triples representing the gross transport north and south, each occur-
ring for some length of time per month. The entire shoreline is alternately
calculated for an incremental time, assuming the direction of the incoming
wave iS positive, then negative. The period, T, used is taken to be the
average T; i.e.,
ei Osis) 1b
2p (H,T)
where the p(H,T) are the (H,T) probabilities given in Table 19 of the SSMO
(National Oceanic and Atmospheric Administration, 1963-73). The choice of (H,D)
for north and south, denoted (Hjy,Djy) and (Ho,Dg), respectively, must now be made.
This choice is subject to the condition that the actual northerly transport, as
calculated from the statistics and given a straight shoreline for the reach of
interest, be preserved; 1.e.,
QT oa : a = D2) .
tH cos a, sin a, = t, » p(T) "p.(H, D) HE cos a, Sin o%
Eel Dis gaivemnne:
north transport
where
t; = number of hours in a given month
p(T|H) = conditional probability T occurs given H
p(1,D) = probability of (H,D) pair (using Table 18 in SSMO as a data base)
A, = Dy - shoreline orientation
Ci a(S)
ty = number of hours the average wave condition exists
Hy, = average wave height
ae =) Dy = shorelanel orientation
Dy = average direction
and similarly for the directions giving south transport.
The average directions of the shoreline at the breakwater are calculated
from the historical records. The directions are chosen for the incoming wave
angles since the complex geometry of the harbor breakwater shields the nearby
shoreline from waves arriving from most directions. At present, the time dura-
tion of waves arriving from the north is assumed to be the same as from the
58
south; therefore, ty = tg = 0.5 tz. The conservation of transport equation is
then used to calculate the average wave heights Hy and Hs. The results of
these calculations are given in Table 5.
Table 5. SSMO averages for Holland Harbor. ?
Month
ANDMNNMUNNKHUNNUNAAD
er Oa 6 noe Oitenieiie °° °
° °
68)
-0
.5
6
oS)
ol
8
-8
-0
-6
-0
of
3
NILWNWWDNODNDDNPNON BG
0. Wea oOo Of Gasol 0 Oa GS-6
OLD DADUNFPWAHEHLODANW
CMIWVUOUADWWHEUWOFH
Annual
+
ho
“I
°
tEor Da 885 Ohne
The computer program described in the Appendix was used to calculate the
evolution of the shoreline from September 1967 to May 1968. The historical
1967 and 1968 shorelines, as well as the computed 1968 shoreline, are shown
in Figure 29. The calculation assumes that Ax = 100 feet with > = 1 which
gives a value for At which varies from 8 to 20 hours depending on the monthly
wave characteristics. The principal discrepancy between the predicted and
actual 1968 shoreline occurs near the breakwater. Although the shapes agree,
there is an erosion in the calculated shoreline which is probably due to the
approximations used in the calculation of the diffraction coefficients, and
incoming wave angles which are functions of x in the shadow region of the
diffraction zone. The unaltered theory of Penny and Price (1944) was incor-
porated into the numerical scheme since most breakwaters can be represented
as line barriers, and hence is almost always useful. However, for the case
of Holland Harbor a universally valid prediction of the shoreline would re-
quire the detailed calculation of the diffraction effects due to the geometry
of the breakwaters. Also, the convenient choice of incoming wave direction
obscures the fundamental problem of how to properly use the statistical wave
summaries.
IV. CONCLUSIONS AND RECOMMENDATIONS
The basic idea of Pelnard-Considere (1956), i.e., to investigate shoreline
evolution by concentrating on conservation of mass as a special one-dimensional
problem, has been generalized to essentially its limits of applicability. These
physical processes of refraction and diffraction (where applicable) have been
incorporated, including deterministic variations in lake level, bluff height,
and beach slope. The inclusion of refraction makes possible the proper use of
the known physical relationships between wave energy and littoral drift on a
priori basis without necessarily determining these as results from the past
recorded shorelines at a given location. Accurate determination of shore be-
havior in the lee of a breakwater requires inclusion of diffraction in some
39
1967
SSS eS S> 1968 ACTUAL
—— -—— 1968) /PREDICTED
BREAKWATER
PLANFORMS
CALCULATED
—40 —30 —20 -10 0 10 20 20 40 50 60 70 80 90
x100 FT
Figure 29. Computed data versus actual data.
form. This could be done in a heuristic manner, either in the global approxi-
mation described previously or in the use of the constant depth theory of Penny
and Price (1952). It could also be done in a more rigorous manner which would
include the effects of a sloping beach. Thus, quantitative predictions of the
shoreline can, in theory, be attempted in situations where onshore-offshore
transport of sand is either negligible or is known from other sources.
The resulting theory is presented in two equivalent forms, one in terms of
the behavior of the shoreline y(x,t) alone, the other expressed explicitly in
the longshore transport Q(x,t) and implicitly in y(x,t). The former has the
advantage that numerical schemes, such as that of Crank-Nicolson can qualita-
tively indicate the behavior of the shoreline in regions of rapid change. How-
ever, the conservation of mass is difficult, if not initially impossible to
achieve since any approximation of a transport-derived term (i.e., a term
arising from 3Q/3x) will alter the transport balance. The later form allows
the use of analytical or numerical approximations in the transport equation
which will not disturb the total sand content of the system, but only its
local distribution.
The most severe and unavoidable limitations to the engineering application
of these methods are the use of the statistical wave summaries. One possible
use of these statistics was shown; however, many others are possible. Efficient
and accurate use of the offshore wave statistics is endemic to the problem of
large-scale shoreline prediction, and must be achieved before any theory (whether
one line, multiple lines, or grid) can successfully produce accurate results.
The problem of shoreline evolution sensitivity to time step in the input
wave climatology would require further research. Despite this limitation, if
the effects of wave refraction, wave diffraction, and change of lake level are
taken into account (as in this report), and if the method is generalized, then
a mathematical model with multiple bottom contour lines could be formulated
which would (if the problem of wave statistics input is solved) permit a cal-
culation of the evolution of the complete bottom topography.
60
It is important to point out that wave refraction effect on shoreline evo-
lution has been found particularly important. It is particularly necessary in
order to determine a planform stability criteria which can be established from
the present formulation.
The problem of shoreline stability needs to be investigated, both physi-
cally and numerically, as for some deepwater wave angle, shoreline perturbance
may increase by instability instead of being flattened out.
Perhaps one of the most significant results of this mathematical approach
to shoreline evolution is to point out the need for more research to quantify
the phenomenology relevant to shoreline evolution. The insertion of empirical
parameters has allowed the investigator to fit (to a large extent) a form of
shoreline evolution; however, the processes involved in each of these param-
eters are not well known, and, what fits at Holland, may not necessarily apply
elsewhere. For example, the research topics which will improve the model are
(a) onshore-offshore movement; (b) quantity of sand (silt) lost by rip currents
or density currents; (c) percentage of sand in suspension; (d) distribution of
sand discharge as a function of the distance from shore; and (e) more impor-
tantly, how to treat the wave climatology in finite time intervals to obtain
an equivalent result. The last topic may be the most difficult since the noise,
(i.e., daily variation and effect of storm) may exceed the signal (i.e., the
long-term trend). Shoreline evolution is due to a succession of extreme events
separated by long-period time effects of equal importance. Therefore, the
treatment of long-term evolution from an average wave climatology is question-
able, since the complete time history may have to be considered on a daily basis.
This topic can be investigated by a sensitivity analysis to wave definition;
€.g., the present model could be used, as well as a research guideline for fill-
ing many knowledge gaps in shoreline processes.
6|
LITERATURE CITED
COLE, A.L., and HILIKER, R.C., "Wave Statistics for Lake Michigan, Huron, and
Superior," Department of Meteorology and Oceanology, University of Michigan,
Ann Arbor, Mich., 1970.
LE MEHAUTE, B., "A Theoretical Study of Waves Breaking at an Angle with a Shore-
line,"' Journal of Geophystcal Research, Vol. 66, No. 2, Feb. 1961, pp. 495-499.
LE MEHAUTE, B., and KOH, R.C., "On the Breaking of Waves Arriving at an Angle to
the Shore," Journal of Hydraulte Research, Vol. 5, No. 1, 1967, pp. 67-88.
LE MEHAUTE B., and SOLDATE, M., "Mathematical Modeling of Shoreline Evolution,"
MR 77-10, U.S. Army, Corps of Engineers, Coastal Engineering Research Center,
Fort Belvoir, Va., Oct. 1977.
MOBAREK, I.E., and WIEGEL, R.L:, "Diffraction of Wind Generated Waves,'' Proceed-
tings of the 10th Coastal Engineering Conference, American Society of Civil
Engineers, Ch. 13, Vol. I, 1966, pp. 185-206.
NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION, ''Summary of Synoptic Meteoro-
logical Observations (SSMO) for Great Lakes Areas," National Climatic Center,
Asheville, North Carolina, 1963-73.
PELNARD-CONSIDERE, R., ''ESsai de Theorie de 1'Evolution des Formes de Rivage en
Plages de Sable et de Galets," Fourth Journees de l'Hydraultque, Les Energies
de la Mer, Question III, Rapport No. 1, 1956.
PENNY, W.G., and PRICE, A.T., "The Diffraction Theory of Sea Waves and the
Shelter Afforded by Breakwaters," Phtlosophtcal Transactions, Royal Society
of London, Serial A, Vol. 244, 1952, pp. 236-253.
SAVILLE, T., Jr., "Wave and Lake Level Statistics for Lake Michigan," TM-36,
U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, D.C., Mar.
1953).
US. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, Shore Protec-
tion Manual, 3d ed., Vols. I, II, and III, Stock No. 008-022-00113-1, U.S.
Government Printing Office, Washington, D.C., 1977, 1,262 pp.
U.S. ARMY ENGINEER DISTRICT, DETROIT, "Detailed Project on Shore Damage for
Holland Harbor, Michigan,"' Tetra Tech Report, Detroit, Mich., Apr. 1975.
62
APPENDIX
COMPUTER PROGRAM
This appendix presents the listing and brief explanation of the computer
program written to investigate the behavior of a shoreline which contains a
complete littoral barrier at x = 0. The numerical scheme is based on the
finite-difference method of Crank-Nicolson, used to solve the cyclic nonlinear
transport equations in Q and y. Although the program is written expressly
for Holland Harbor, Michigan, hopefully, enough explanation is given to modify
the program, if necessary, to suit a particular application.
The program consists of a main program, which is simply a calling routine
to the controlling subroutine, and eight subroutines. The interrelationship
of these programs is shown in the figure below; a brief discussion of each
subroutine follows.
INPUTQ |
f : 3 ; Sica
Hy f 5 a i :
i OREGION { smmems:| CALCKDQ fem ; SREGION |
be . H & ts 2
Venera f | e sea ee i hismesz es
Figure. Program structure.
MAINQ Calls subroutine EVOLVEQ
EVOLVEQ The controlling subroutine which organizes the numerical method in a
global sense and calls various other subroutines as required. The
first call is to INPUTO, which reads the controlling parameters, his-
torical shorelines and lake levels, and statistical wave information.
For each month in the time period of interest, subroutine CALCKDQ is
called. (It is assumed that during a month the diffraction coeffi-
cients change negligibly.) For alternating equal incremental times
the angle of the incoming waves is changed and subroutine SOLVEQ is
called. This is repeated until a month's time has been completed.
Then this is repeated until another historical shoreline is reached.
63
INPUTO
This routine reads the information required to define a particular
problem.
Below are the cards required with an explanation of the
variables required. Title cards are user information devices which
are printed but never used. The card(s) are numbered by their order
as defined by READ statements.
CARD il:
CARD 2:
CARD 3:
CARD 4:
CARD 5:
CARD 6:
CARD 7:
CARD 8:
CARD 9:
CARD 10:
CARD 11:
title card
LAMBDA = A = At/Ax2
B = bluff height (feet)
DO = depth to no sediment motion (feet)
DX = Ax (feet)
BERM = berm height (feet)
BWL = length of breakwater (feet)
BWD = depth of breakwater tip (feet)
SLOPE = slope of shore profile at waterline
XLINE = offshore line loss (square feet per hour)
title card
N1 = number of grid points to left of breakwater
N2 = total number of grid points on both sides of the
breakwater
ND = number of historical shorelines
NY = number of years lake levels are to read
INYR = year of the first historical shoreline (must be
year of first lake level as well)
for all years with historical shorelines read
year (I); I = 1,ND
month (I); I = 1,ND
hour (I); I = 1,ND where hour is the day x 24
for each grid point I read historical shorelines (feet)
title card
for each year read lake level for each month
title card ()
for each month (including annual which is not at present
used) input (in the notation used for Holland) Hg, Hy,
Deep Dino Wp ie
The input data above are printed without explanation along with
several assays used internally. Note that in this routine i is
redefined slightly so that a month's time is exactly 2nAt.
64
SOLVEQ
CALCKDQ
QREG ION
SREG ION
CS
CETC
Note I.
Note II.
This subroutine calculates the shoreline for one time step assuming a
previous shoreline (and shoreline angle = tan7! (dy/ax)) and diffrac-
tion coefficients as calculated in subroutine CALCKDQ. The scheme is
essentially as is described in Section 2. The matrix inversion is
achieved using a standard algorithm for tridiagonal matrices.
This routine calculates the diffraction coefficient and incoming wave
angle a = a(x) in the diffraction zone using the theory of Penny and
Price (1952) for both incoming wave directions (the angle convention
is shown in subroutine INPUTQ).
Calculates diffraction in the ''Q'" region.
Calculates diffraction in the "'S'" region.
Calculates complex valued Fresnel integral.
Calculates a given depth and period.
In subroutine EVOLVEQ
L1 = Ist year index
L2 = last year index
Calculation
begins at date: year (Ll), month (L1), hour (L1)
ends at date: year (L2+1), month (L2+1), hour (L2+1)
In a general program
L1 =
L2 = ND-1 where ND is defined in subroutine INPUTQ ND must be
passed to subroutine EVOLVEQ.
Program as given in this report is set up to allow complete bypassing
(lost offshore) when waves are not diffracted, i.e.,
To change program for no bypassing, four statements must be changed
in subroutine SOLVEQ.
2 statements after 1001
change QOLD (N1P1)
to QOLD (N1PT)
(1-ID) x QOLD (N1P2)
0
1 statement after 1001
change QOLD (N1) = (ID-1) x QOLD (N1M1)
to QOLD (N1) = 0
1 statement after 3000
change P(N1) = 1. - (1D-1) x Q(N1M1)
to P(N1) = 1
3 statements after 3000
change Q(N1P1) = 0 + (2-ID)/P(N1P1)
to Q(N1P1) = 0
65
aaAgD
nan
aan
1001
1oo2
1100
diol
1104
PROGRAM MALTNQCINPUT® OUTPUT eo TAPES@YNPUT 9 TAPEGSOUTPUT)
CALL EVULYEQ
CALL EXIT
EMD
SUSRUUTINE EVOLVEW
COMYONs BUATAZ LAMBUAV Re DQ eDX > HERMyNI pN@oBWLOBWOVDXHATsSLUPE OAL INE
CUOMHONSOTFRES KKNeC!2]0115) 9 AAC 20115)
COMMONS SSMUS DIR(130¢) oPERC 1302) OWH( 1502) oXB( S306) oAMPC I B02)
COMMON SLAKELVLS KL( 24942) j
COMMON /SFRLINEZ VATE (398) oSL(195,A)
COMMUNSZ3S/23013,2)
OIMENSTUN HOLES) oVCRISIOICCLIS) oVxCSL5) 0VV(4S5) oRGLC115) oMD(13)
REAL LAMBDA
DATA MO/31 6289315309319 3500310319 300310300319 305/
LIBIMNnEX UF FIRST BHOKELINE
L136
Lato
FREAD tPUT DATA
CALL phPUTa
NiMpsniel
HTP LENle)
NLPeauie?2
NaMy ence]
TXde( fol) eOxneDK
OU 1001 T3t0NJ
Txz] aoa
ICiT)sIx
Tapely
NIPLEN1 064
DUO {0ne LT3N{PyoNa
Tut]la,0x
IC(L) 1k
DEFL%¢ ORIGINAL SHORELINE AND SHORELINE ANGLE
Gai ,4(8+090)
OU $1n0 1£4—N2
YCL)85bCT0L1) oF
E970 e5/4( R00)
ZE(yCadev(1))sOxHat
vyy¥(y)aATaN(Z)
OO inl Te2aeNtM)
ZelCy(yeloev(Tel)y%e9
V¥( 1) 24TAN(Z)
ZeCv(N LOT ONIME))/DAMAT
VV ONL) FATANCZ)
ZaCv(nNiPe)@eV(NIPy))/OKHMAT
VY(NIPL)BATAN(Z)
DO ine LENyPagnamt
Zz(V( pol) oV¥( Tol), #69
VV¥(T)@ATAN(Z)
Z3(Y(N2Z)°Y(N2M1) )/DKHAT
VY¥(N@)2ATAN(Z)
MaIn aNALYSIS LOOP
IVE AR eh
50 9009 L®LicLa@
Toisk 0009 OD
LTy1s0aTe (lol)
IMieOsTE Ceol)
Ly2eDaTE (lolol)
T#7sNaTe (eet od)
NMONaCTY2° 7 v1) *1arlh2ermMiol
MUNaITMi ef
QO 2379 LeLeteN¥4On
MUNEMNG |
IF (MON gNE, 13) GUTO 3
MOMNe]
1VEARSIYE ARS)
TTLIMaXS (ONS)
ZL3Z3( "UNG )
2P205("UNO2)
CALL CAL CKNUC Ye MON)
DTHAT=xXS("0ONG3)
DIRLEN Ik CHONG)
OLR eanIk (MONG?)
IF(LL o£%, 1) GOTO 11
IF(LL EG, NMON) GOTU 42
N1TaaS¢(*UNeu)
O1HaT=*£S(M0N93)
66
oan
aan
200
3000
too0
lool
MLGEXS (HONG 2)
GOTU 44
X¥EMQe“UN) S24
FaCTS(KMN@SLATE( SSL) ISXM
GUTG 45
MPEMI(PON) O24
FACTSnATE(S9L)/x™
XLOSFACTEXS( HON) 2)
CTBRACTEXS(MONGUY
OTMATSFACTOXS(4ONG 3)
DLLOTEXLCIVEAROMAN) sed,
NOLLHATELLEOTeDTy(b+b0)
PRINT Jug
FURMAT(C IHL» /)
PRINT 150eLeoTVYEAR MONO TILIMeD HAT ,OLLHAT ODT
FURMAT(® FOR PARAMETEKS®o STO0® TTL IMPDTHATODLLHATeDIa¢eJO0 3EI1,3)
PRINT 151 0MON¢ (25 (MUNOK) oKE1 96)
FURMAT(/e® LISTING OF ARRAY KS POR MUNTH# oI 5S060F10,3)
OxeeNULLHATsSL OPE
OO 2991 TetelTlLI™
TALL SULVEQC VO VY oXoVXoZLoXLVeDOLLHaToOTHAT es te AMP( MONG 1) oDT)
CALL SULYEQCMoVKeVoVVYoZ2oXLVoDLLHaToDTHATe 20 AMP( MONO?) 0 DT)
CONTINUE
PRINT SHOKELINE atl END UF MONTH
JVEARSTYI®{2eLLeaeIMi
JMONZ VE AR@L2¢(JyEAR/I2) 64
JVEARZJYEAR/I2
WRITE (60200) JYEARe JMON
FURMAT(C//0% FOR YEARS eT90% MONTHS, FSe% LISTING UF FINAL SHORELINE
1%)
DO 3090 I34oN2
RELCT)SVCI)/E
OO 30ot 13198
JLS(Teol) 91561
J2zsiel4
IF(T (60, 8) JesqiS
WRITE( 60201) (ICU) o JO eJ2)
FORMAT(///98 X 991518)
WRITE( 60202) (RSLC J) odadioJ2)
FORMAT(s»® ¥ So15F B01)
CONTINUE
CONTINUE
CONTINUE
RETURN
ENO
SUBROUTINE SOLVEQ(VVeVeXXo SLANG o LyeXLGoOLLHAT oDT AT 2100 AMP_DT)
SULVE FOR ARRAY yX USING IMPLICIT SCHEME FUR TRANSPORT QNEW
CUNMONS BDATAS LAMBDAH DOD oRERMeNI oN2e BML eBWDODXHAT SLUPE OC ALINE
CUKMONSUTERES KeMeCPo1IS) eAaCeoLI4)
CIMENSTUN MXCLISDOVYCALD) oVC115) oSLANGCIIS) eVYOLUCI15) ovV6119)
DIME SIUM POLLS) oWCLIS)oL (115) 910415) eNC113) oGNEW( 11S) pQOLN( 115)
Re AL
BGA SCOANG
Eat} o/b 8
NIM] Bry leo]
NLP Bylo.
NIPQ2an1¢2
N24 sr eol
D) $000 T8yeN2
VILOCPIBYYCT)
CaLCULATE VLD & (VOLO) BASED UN OLD SHORELINE AND FRESENT DIRECTION
DU 1001 ITFt—eN2
V¥CTIsVCI)
AUGaaAal(IOcl)ev( Ty
BaAnGeahGeZy
QILOCTISAMPETOS( ANG) SSINC BANG) XK H2CT Del)
GILDONIDECTOSL) #QULDONI MY)
CILUCNIP1) 3 £2810) *8OLD(NIP2)
auGebal lot)
BANGTanGeZy
QLT4isa“P COS (ang) *SIN(C BANG) #XKDA(TD01)
67
amor
ana
aon
aan
2000
2001
2002
eos
3000
5001
40090
4001
4902
4003
5000
5901
5002
AIGZAQ(TDON2)
BaAnGesNnGely
Q.1M4 «PF ANP3COS (ANG) ® SINC BANG) ®&XKD2(IDONG@)
TTLasy33
9) 9999 ITsyolTLaST
CALCULATE akRAYS LoD
D) 2090 TeaeniMj
M<zaAKMe(100])
E=(vy¢(lelevvy( ley )) #9
AvGahal(lOol)ovv(y)
BavGtanGely
CTAMFa(LS®CUS( ANG) FCOB( BANG) CSIN( ANG) SSIN(BANG))/(1,%E%E)
LCL) axtoeCoxn
O02 etal TSN)PQeNnaMt
WKeRaHe(10eT)
Es(vv¥ (le1eovv( ley) )#b9
ANGzsallColT)eovv(7)
BanGsanoely
C3a4P 9 (239CUS( ANG) *COBC BANG) CSIN( ANG) *SIN(BANG))/(1 ,¢6%E)
LCL) 8 xl 99Caxk
DO 20n2 JEreNj™]
e
OCTISQHULO(TIOLOCT)®(COLD( 141) 02,90NL0(1)*@0LD(1e}))
00 20n3 IBN{P2eN>MI
OCT) SQOLO(CT) LOCI) ® (CULD( 11 )e2,%00LU( 3) *UOLD(Jol))
O01) 80LI41
OCLC (NIP1) 20,
O02) sGLIKNP
CALCULATE aRRAYS Pol
P(1) 840
Q(1)8ne
OU $059 IBaeNy™1
PCT) aL (CII (e( Tey) ee2odol,
MC1)e_Cl)sO¢(7)
PCy )el pe(1Dey)FQ( NIM)
PCy Py )Z16
GC 41P 1) B00e(P@ID)/P(NIPI)
DU 39g ITENLPQ29NDM]
FCT)TL C1) 8(eS( lay )teedoly
91) FL CLIP (7)
Pl tedsle
SO-vé FUR THANSPORY GNEW
UCL) 3nNC1)7P 04)
DO 49nO TB ayNI™]
UCT) S(OCTI oC (CL enCTel))7PC1)
Onna (NL) EUCNI)EDEINI SPOON)
DO dnl Tet oNtM]
Jevyjeory
AN aly) BOC S)FQNEwW(JO1) 6UCJ)
UCYIPy)E0CNIPL) s/PCNIPI)
DO 49n2 T=N{P20N—MI
UCT ECOCTIoL C1) enClel)) PCT)
Q\2a(s2)2U0N2)3D(N2) /P(N2)
DU GOn3d TEnNyPyenant
Jzv2en te]
GNiw(JIEQCJ)OONEW( IOI I OUCJ)
IPUINTEQ
TE CIPRINT FQ, 0) GUTU 5O
PRINT 400
FO«mar(/o*® LISTING UF QNEwS)
PRINT GWOLe(UNEN(T) 0181 9N2)
FOIMATC {Xe t OF 10,4)
CALCULATE New SHORELINE
M009) 20,98(QNEN( 2) PUNERW(1)¢QOLD(2)eGUL0(1))
DU $970 T82N1 4}
e072 29 254 (QNEW( LOL CQNENW( Tel )OQHLOCIT¢1)eQVULO(I@1))
MMONL)F0,S8(QNEw( NI @UNEW(NI@] CQQLD( NI) PGOLD(NI@1))
AKENEP1) 20,59 (GONE MC NIO2) CQNEW(NI 61) *UOLDINIO2) CUOLDINIOS))
00 5091 13NIP20ND™1
MKCL BO PS*(QVEMC TOL SQNen(CTel)¢QQLO0( 191) eGULD(I"4))
KH0N2) 50,599 (QNEW(N2) PUNEM(N204)¢QOLD(N2) eQ0LD( Nem) )
FaytaTs/Oxmay
FL2C0LLMAT/S5LOPE
F2AALTNESLT/(HODG) BF2
DU 5092 1419Ne
MRC LT) 2F xn LyoVv¢( look leh
ZECKA(2)OKK CL) sPAHAT
68
oan
aan
SLANG( 1) S47 AN(Z)
00 $03 1z2N1™1
Ze( UK Lop youn( ley) )9e9
S003 SLANG(I)FaTaN(Z)
ZECKAOND oxn(Niey)) /DAHAT
SLANG(NL)SATAN(Z)
ZECHACNLPSVAKKCNG OLD) SDKHAT
SLANG(N1¢})FaTANCL)
DU 5094 TEN{P20NQM1
Use( anc Topeoxn( Loy) )Fb9
$008 SLAnG(J)saTAN(Z)
Za(xtcN2)enx(n20y))/0RHAT
SLANG (N2)BATAN(Z)
TFCIT 4&9, ITLAST) GUTO 9999
00 6000 134 on2
VV (1) sSLANG(1)
6000 VOL y)akacT)
9999 CGT ue
ME TURD
END
SUBRUWTINE CALCKHUCY Ye MUN)
COMMUNS BOATA/S LAMBDA eRe DQ eDK yHERM NI pN@o BML eo BWDeDXHAT » SLOPE OAL INE
COMMONSUTFROEs KKN2L20115) 9 AAC 20115)
COMHURS SSMUZ DIR( 1502) PERC 1302) NPH( 1302) oXSC1S,6) pAMPC1 302)
OIMENSIUN VY(S15)
REAL «3
OxK230 S/UKMAT
NYP Landed
CALC OJFFRACTION FOR LH®
A=DIR¢MON4)
CALL GETC( RAD pPER( MUNG 1) 943)
Ageo)k
DU 3090 JE1oN]h
Tanyeyed
WekedDy
Vatal.eVY¥(J) @(8+n0)
Rypzytyovey
R1sSUPTCRL) say
Zeasy
Oysatan(2)
TFC AL olLT,. A) GOTO 4
CALL QHEGION( RS pao ALOR)
AaCyoy) sa
GOTU 2a
1 CALL SREGION(R 1p Ao AL OBS)
AAC Joy) 344
2 xKD2CyoT) aK sony
$000 CONTINUE
OO 1091 JENIPLeND
mKO2C e121,
JOOt AAChoy) BA
CalC mIFFRACTION FOR RRB
AceDIR(MCN, 2)
CALL GETC(AwDyPER( MUNG 2) on3)
Aszeo/e
DU 2000 JENIPLOND
uaneDy
veRoleoYY(1)*(B+00)
Aysrty+yey
RLSSURTCRL)/H3
Zzasy
ayzatTan(lZ)
TF(A1 eLT. A) GOTU 31
CALL nREGION( Rig AVAL ORY)
Ad(Qoy)BeA
GuUTO y2
11 CALL SREGION( RI odo AL oKS)
AAC2e7) 2044
12 XKD2(aeT) BK 5eKS
2000 CONTINUE
NC PO91 TB 0eN4
me O2C201) 51,
2001 Abl(2ey)zoa
ae TUN
b.0
SueBROLTINE JINPUTQ
CLMMONS BUATAS LaMSL4e Re DOsDK eo BERMoNS gN2oBWL eo BwDoDXHAT eo SLOPE SALINE
COMMU SSARLINES VATE ($08) oSL(115,A)
69
nan
aon
onan
NMNAMNRANDQAANAAGAAIE
CiMMON ZLAKELVLZ KL (C24—9}2)
COMMUN SSHUO/ DIR( 1592) ePER( 13902) DWH( 1502) 0X90 1506) oAMP(I S02)
COMMON/Z3S/7230130e)
DIMENSION 72LCA4e12) 9 4D(E3) eo ALIST(40)
REAL LAMBUAGLAMBNAM
MATA MD/3L Abe Se S00SL odo SL odio SH yp S10 3003105057
PRINT 999
999 FORMAT(IH19% DATA A5 READ BY SUBROUTINE INPUT%9/)
READ SPACITAL PARAMETERS
READ(G¥L01) CALISTCI) eo f31010)
JO1 FLRMNA? (1008)
READ( 5190) LAMBNACBeDOODKe BERMe Bwl eo BWDe SLOPE
$90 FORMAY (8510.1)
READ( 5100) MLINE
PEINT 1OLeCALIST¢1) 0181910)
PFINT 100°LAMHDAs beDOODXeHERMO AML »ANDe SLOPE o XL INE
READ HISTURICAL SHUORFLINE DATA
READ(Ge101) CALISTCI) 0181510)
READ(50102) NtoNaeNDoNVoINYR
1o2 FORMAT(5I10)
READ( 50100) C(DATECJ oT) e131 9ND) oJato3)
DL 1050 TBq_—he
BEAD (50100) (SLC Ted) odsteND)
$000 CONTINUE
PFINT LOL CALIST(I) 9181010)
PRINT 1000((DATE( Je) 0181910); Jai,3)
OU 2000 1349N2
2000 PRINT 1000(SL( Ted) eJ840ND)
READ HISTORICAL LAKE LEVELS AND COMPUTE DLLJOT
Fet,/(8%00)
READ(C S101) CALISTCI) 0191010)
OU 1001 IateNy
READ( 50100) C2LC Ted) odatosa)
$001 CONTINUE
PRINT JOLOCALIST (I) 0381910)
Du 20m! 1349NV
IFBIOTNYR@{
2003 PRINT 99B01P o(ZLCled) pJeiol2)
998 FURMAT(IL 7oSu012F10—e2)
K20
OC 10n2 13,eNV
IvelNyReley
LyYslvo48(lysu)
Kr ko]
ME CLoy I BCLLCT 2) edb (101) /MD(1)
Mzts(y*Ly)
MP seOre)em
MLCLersCLLCTod elk (102). Km
99 10ne J3$e11
MLCLeseCdeCrodey edb Cred) )/MDCJ)
IFC] ,tU,. NY) GOTU 1002
MOCTeyeeCZUCLerodrece (tote sMb(42)
1002 CONTINUE
READ HIRECTIONSy PSXIUD8, WAVE HETGHTS FUR EACH MONTH
DIRECTION CONVENTIONS
CA & 1 6 2 OLEp WATER ANGLE LESS THAN 0
. x + 1 DEEP WATER ANGLE GREATER THAN O
° M 4
e a +
% x o
oh +
one
MOCO ye OOK OK
€9=364415926/180,
RrAD(5e101) CALISTCL) 0181510)
0) 1993 “B1913
R-61(60100) DeH( Mol) oDWHE Moe) oDIR( MOL) eDIR(Me2) oPER(My1) OPER (Moe)
DIF(M,1)Z01TR(Me 1) *C9
70
1003 DiP(M,2)3UTR(Me2)9C9
PAINT LO1eCALISTCL) o 184010)
D2 2003 “81013
DIRyerIR(h.1)/C9
DLAeIHITR(Me2)/C9
2003 PRINT 1L00eDRH(My 1) pOMH( Me 2) gDIR1 DIRS sPER( M91) PER( M2)
aoaonan
annan
CALCULATE PARAMETEHS FON SCALING
PINT 200
200 FIRMAT(/0% LISTING OF SCOLE PARAMETERS AS CALCULATED JN SUBROUTINE
1 [NPUTUe)
OY LOn4 M3193
2180 025¢5eo5#DMH (Mel) / (Sel QUBBPER( My) 92)
225002585 eS8DWH (He 2)/ (Sef 2UBSPER(My2) 942)
DLS Ko BSINKH( Mey) P2HPER( MGS)
AATUT HE oBSENRH( MD) OR2EPER(MQ)
ArmA,
IrF( Op ,GT, As) azde
K3(M04) B0XHAT BOK (8400)
OTHATSLAMBDASDXHAT#E2
OTSDIHATS (RDO) Se37A
W)TsMn(*)F12
NaxDT,/OTe]
13N
Drandy/Nn
13 (Moa) SL AMBNAMa¢OT/0K 092) 8A/(BO09)
W3(“oz)BOTHATSLAMBDAMPNKHATSOD
x5(Meu)207
X5(™065)3N
15(M0q)3(B900)F(RODGISA
MOBS 12UBRPER(M,1)PPER( M1)
Z3CM9 4 E0255 eS MVAM (MeL ISNLO
Z$0% 02) 800255 eS @VNK (Moe) SKLO
A*PCM,L)BALZA
wePOm PVEAIS“A
POINT 2010Me(XS(MOK) oKSh 0G) oZ3( Moy) 02S( Moe) oAMP( M91) po AMP( M2)
ZOL FORMAT (® MONTHB®,T30% AHRAY SAFO ZFOoSePOgL OFS ,O0F Oo 30% ARRAY 250%
Le2booye® ARKAY AMPS eek O,3)
1004 CuNTINve
RETURN
END
SJBNUNTINE SREGION( Ro Ag Al KS)
CALCULATES KD IN S REGION
(ASSUMES 1 8 DISTANCE/WAVE LENGTH)
REAL wh
P133,4415920535
C$BA2,n2R4271 25
STESING(CALSA) S24)
SazSIu( (Alea) se.)
C2zcIs(aiea)
tCrzcus(alea)
RZSSUPT(H)
ULEC Sah S457
U2zoC yeh 305g
MTP TEULFUL 2,
CALL cS(%o360C6)
ViF(1,755°C6) 72,5
w1z(3,°CO) 72,
MEPL HV 2e2z2,
Call CS(XS6sC6)
V23(1,"S6°Co)/2,5
W2=(S,°CO) s2,
XuBQesPlOKeC]
MFZEQoePLORSCD
CssCUs(x4)
C52CU8(K5)
SusSIn (ug)
S325], 045)
DasvielCUsveeCSowysSuUonoess
Dozen} eC UenreCSayy*SUevo9ss
KsSD4eDUeO5eN5
K52SGQT(KS)
RETURW
ED
SIBROYTINE QREGION( Re Ao Alo K3)
CALCULATES KM IN @ HKEGTON
(ASSUMES R @ DISTANCE/WAVE LENGTH)
REAL «3
7|
mae:
10090
100
PT2304 415920535
CBBPeRAKILTIAS
SIZGBIN (CALA) S24)
SBPSIN((ALOA) S25)
C1eCOs( Alea)
CetCUsl Aiea)
K32Supl(R)
ULsC bak3es7
U2zelyek3esy
yePTS 18UlLs2,
CALL CS(1X9S6eC6)
YLECL*S6°CO)/2,
b1z(SeeCo) 2,
VEIT) 2%UCs2,
CALL C30 X9S6eC6)
v2z(t ,°Sb°Co)s2,
#22(94°CO) s2,
wusesplexecy
rS=etpl]eReco
CuzCus(xu)
C5=C IS (x5)
Su=SIN(x4)
SS=STN(KS)
CUTC4eV IP CumHt FS ys V28C5on2esgy
NSFeSurvitSuew{@sueVve®SSow2ecs
HE2DUSUUCUSEOS
KY=SSaRTCKS)
RETURR
END
SUBROUTINE GETC (DoT ow 55
GIVEN OEPTH(O)e PERIODCT) CALCULATE WAVE LENGTH(W3)
G3232,2
P12 301415926535
Pleas? 9P]
W3EG3eT/F 12
ZyzPLoayst
U$sutoI2¢0/(G3FTeT)
VY$3E 4003)
TFCKS ok Te 16,0) GOTO 3
Ceeb3eT/Ple
Gulu 2a
V3tEXP (KS)
T3ECVyPL)/(V3e1)
Ch2zGSeT@BUQT( TS) sPl2
0) 1000 124050
M$32%73/C5
YS3ZEXP( KS)
TSE(VEPLI/(Y3O1)
OssndeT3
M12489(C%9N3)
M250 0en001%AK8(03)
I-( xl okbe K2) GOTO 2
C3e(CyOOS) 2,
CONTINUE
w$eC3eT
MITE (S0100) wdeC30D0T
FuRMAT(/)® EAROR IN COMPUTATION OF WAVE LENGTHS04F10930% CALL EXIT
1%)
CALl EXIT
asectely
RETURN
(aD)
SIHROYUTINE C8(KeG60C6)
Z5R4AHS(K)
Tt (24 oG1, 4,0) GUTO 4
Corsuet(Z3)
So32Z3eCo
Z$=(4 eZ3)%(4,¢7%)
CAFC ((Se1IOTASE M1 182305 e2UU297ER9) O¢ 345045114267) 82343027 3300E 05)
CorCom((COCS#L3¢4 D204 KE aS) #Z564 1 CASU4Em2)9Z23)¢1 6U0905E@1)
S5B(CeoOTTHBIESI NALS 9S HASISKEOH)eZSOS COLIUI ESO) #23
S5ES6a0(0(5442,441616E 4) ¥7366,121 32EC3) 92548, 02049Fe2)
GOTO a
C5zCUS(23)
$$291N0Z3)
Thue/ly
ASSCC (RB, THAC5HERUPZ50U,169289E 93) 82547, 97094 SE OS) *2106,792R01LeS)
ASC CC A342 = 3, COC SUEM4) 92365, 972151 E93) *Z571 pb00U2BEe5) *l30
12,4595422F aD
A3=4b02394 uuu o9Ee9
BS2(( (eo eb3S92bt me UL 5% 5 UOLYNGE DS) HL 307,271 69E3) ¥Z347,UCB2bbt as)
BSFC(C (BRL yeu, O27LUSES4) PZ IA, BIUGIESS) ¥Z3m) o20799B8EoG) #23
BSEHSe Le IIUTI ILE wy
7 $=5GaT( 235)
C520 65*Z3*(C3¥SAS4 53963)
S$o=005%Z259(S3h3aC 38H)
RETURN
ED
ee
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