BEACH AND DUNE EROSION DURING SEVERE STORMS
STEVEN ALLEN HUGHES
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
Steven Allen Hughes
Dedicated to my wife, Patty,
and my daughter, Kelsey .
friends of mine who make life worth living.
Many people have provided inspiration and guidance throughout the
course of this research. Above all, the author wishes to express his
gratitude to his supervisory committee co-chairmen, Dr. B.A. Christensen
and Dr. T.Y. Chiu, and to the other committee members, Dr. A.J. Mehta,
Dr. E.R. Lindgren, and Dr. D.P. Spangler. Their valuable advice and
suggestions are deeply appreciated.
A special indebtedness is owed to the personnel of the Coastal
and Oceanographic Engineering Laboratory at the University of Florida,
where the experimental portion of the research was conducted. These
people provided cheerful and enthusiastic help at all times. More
specifically, the author expresses his gratitude to Messrs. G. Howell,
V. Sparkman, S. Schofield, M. Skelton, R. Booze, D. Brown, G. Jones,
and especially to Mr. Jim Joiner. Mr. Joiner served as project co-
ordinator and his skill, organization, and ability to overcome problems
with the wave facility most certainly were the primary forces behind
the successful completion of this project.
Special thanks go to Ms. Rena Herb for typing of the draft, Mrs.
Adele Koehler for typing of the final manuscript, and Ms. Lillean
Pieter for the fine graphics. In addition, the assistance received
from Ms. Lucille Lehmann and Ms. Helen Twedell of the Coastal Engineer-
ing Archives is appreciated.
This work was supported by a grant from the University of
Florida's Sea Grant Program under the National Oceanic and Atmospheric
Administration, Department of Commerce, entitled "Beach and Dune
Erosion Caused by Storm Tides and Waves," and by funds made available
by the Florida Department of Natural Resources.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS . . . . . . . .. . . iv
LIST OF TABLES . . . . . . . . ... . . . ix
LIST OF FIGURES. . . . . . . . . . . . .
ABSTRACT . . . . . . . .. . .... . . .xii
I INTRODUCTION. . . . . . . . . ... . .. 1
Important Factors and Assumptions . . . . . . 4
Objectives. . . . . . . . . ... . .. 6
II PREVIOUS RESEARCH AND EXPERIENCE. . . . . . . 7
Previous Simplified Physical Solutions. . . . ... 8
Previous Experimental Research. . . . . . ... 12
Discussion . . . .. . . ... . . 18
Proposed Method of Problem Solution . . . . .. 19
III WAVE TANK FACILITY. . . . . . . . . ... 21
Air-Sea Wave Tank . . . . . . . .... 21
Computer. . . . . . . . . ... ...... .24
Wave Gauge. . . . . . . . ... .... .25
Rail Mounted Movable Carriage . . . . . .... .26
Profiling Instrument. . . . . . . . . ... 29
IV MODEL LAW . . . . . . . . ... ... .34
Previous Scale-Model Relationships. . . . . . 36
Proposed Model Law for Dune Erosion . . . . . 38
Discussion of the Model Law . . . . . .... .49
V MODEL VERIFICATION. . . . . . . . . ... 54
Selection of Prototype Conditions . . . . .... .56
Selection of Model Scales . . . . . . . . 66
Requirements for Verification . . . . . .... .69
Early Attempts at Model Verification. . . . . ... 70
Irregular Versus Regular Waves. .
Verification Runs . . . . .
Discussion . . . . . .
VI EXPERIMENTAL TEST SERIES. . . . . . . . ... 82
Profile Selection . . . .
Storm Parameters . . . .
General Observations . . .
Typical Test Procedures . . .
. . . 82
. . . 84
. . . 89
. . . 91
. . . 96
VII QUALITATIVE ANALYSIS OF EQUILIBRIUM PROFILES..
Surge Level Rise Duration
The Effect of Dune Height
The Effect of Storm Surge
The Effect of Wave Height
The Effect of Wave Period
Nearshore Beach Profile .
Dune Erosion in General .
Summary . . . . .
VIII EQUILIBRIUM BARRED STORM PROFILE . . .
Breaker Depth Versus Breaker Height
Location of Bar Crest . . . .
Location of Bar Trough . . .
Plunge Distance Hypothesis . .
Nearshore Curve Fit . . . .
Offshore Bar Crest Depth Versus Bar
Wave Runup . . . . . .
Offshore Curve Fit . . . .
Summary . . . . . . .
Sample Prototype Calculation. . .
IX TRANSIENT EROSION EFFECTS . . .
. . . 116
. . . 117
. . . 121
. . . 129
. . . 133
. . . 139
. . . 144
. . . 147
. . . 154
. . 159
. . . 160
. . . 164
Theoretical Development . . . .
Experimental Results . . .
Discussion . . . . . .
Application . . . . . . .
X SUMMARY AND CONCLUSIONS . . . .
Model Law . . . . . . .
Model Verification . . . .
Experimental Test Series . . .
Qualitative Profile Analysis . .
Equilibrium Barred Storm Profile. ..
Transient Dune Erosion . . . .
Applications and Assumptions . .
1 j I j I
I r I I I
Summary . . . . . . . .
Recommendations for the Future. . .
A COMPUTER PROGRAM LISTINGS . . .
B INSTRUMENT CALIBRATIONS . . . .
C TIME-DEPENDENT EXPERIMENTAL PROFILES.
D PROFILE COMPARISONS . . . . .
E DERIVATIONS . . . . . .
BIBLIOGRAPHY . . . . . . . .
BIOGRAPHICAL SKETCH. . . . . . .
. . . . 190
. . . . 200
. . . . 203
. . . . 246
. . . . 272
. . . . 285
. . . . 291
LIST OF TABLES
WALTON COUNTY SAND ANALYSES. . . . . . . ... 60
MODEL SCALE RATIOS . . . . . . . .... 67
FIRST VERIFICATION RUN . . . . . . . .... .75
SECOND VERIFICATION RUN. . . . . . . . ... 77
EXPERIMENTAL PARAMETERS. . . . . . . . ... 87
BREAKING WAVE HEIGHTS. . . . . . . . . ... 120
EROSION VOLUMES. . . . . . . . . ... .. 174
FRACTION OF EQUILIBRIUM EROSION FOR WAVE PERIODS .... .177
FRACTION OF EQUILIBRIUM EROSION FOR SURGE LEVELS .... .177
LIST OF FIGURES
1: EDELMAN'S METHOD APPLIED TO ACTUAL AND IDEALIZED
BEACH PROFILES. . . . . . . . . . . 9
2: FACILITY LAYOUT AND CROSS SECTION. . . . . . 22
3: WAVE GAUGE. . . . . . . . . . . . .. 27
4: INSTRUMENT CARRIAGE . . . . . . . . .. . 28
5: BOTTOM PROFILING INSTRUMENT . . . . . . ... 31
6: PROFILE R-41 LOCATION MAP . . . . . . .... .58
7: PROFILE R-41 AS SCALED IN MODEL . . . . . .... 59
8: USE OF GRAIN-SIZE DISTRIBUTION TO FIND d .. .. . .. 62
9: MODEL AND PROTOTYPE GRAIN-SIZE DISTRIBUTIONS. . . .. 68
10: ATTEMPTED MODEL VERIFICATION USING WAVE SPECTRUM. .... .71
11: VERIFICATION RUN 1, FINAL PROFILE ON CENTERLINE . . . 76
12: VERIFICATION RUN 2, FINAL PROFILE ON CENTERLINE . . . 79
13: ERODED BEACH-DUNE, RUN 2. . . . . . . . .. 80
14: EXPERIMENTAL TEST SERIES PROFILE. . . . . . .. 85
15: RUN 41 SEQUENTIAL EROSION . . . . . . .... .95
16: INSTANTANEOUS VERSUS TIME-DEPENDENT SURGE LEVEL RISE. . 98
17: INSTANTANEOUS SURGE COMPARISONS . . . . . .... 99
18: FIRST COMPARISON VARYING SURGE RISE DURATION. . . ... 100
19: SECOND COMPARISON VARYING SURGE RISE DURATION . . .. 102
20: DUNE HEIGHT COMPARISONS . . . . . . . . .. 103
21: SURGE LEVEL COMPARISONS . . . . . . . . . 106
22: WAVE HEIGHT COMPARISONS. . . . . . . . .
23: WAVE PERIOD COMPARISONS . . . . . . .
24: NEARSHORE PROFILE COMPARISONS . . . . . .
25: BREAKER HEIGHT VERSUS BAR DEPTH . . . . .
26: BAR CREST DISTANCE . . . . . . . . .
27: BAR TROUGH DISTANCE . . . . . . . .
28: RATIO OF BAR CREST DISTANCE TO BAR TROUGH DISTANCE .
29: GALVIN'S BREAKING WAVE PLUNGE DISTANCE DATA. . . .
30: PLUNGE DISTANCE DEFINITION SKETCH. . . . . .
31: VALUES OF "A" FOR NEARSHORE CURVE FIT. . . . .
32: BAR TROUGH DEPTH VERSUS BAR CREST DEPTH . . . .
33: CALCULATED RUNUP VERSUS MEASURED RUNUP . . . .
34: VERTICAL WAVE RUNUP PARAMETER. . . . . . .
35: OFFSHORE CURVE FIT . . . . . . . . .
36: CALCULATED BARRED PROFILE COMPARISONS. . . . .
37: TIME-DEPENDENT EROSION TRENDS . . . . . .
38: TIME-DEPENDENT EROSION FROM PEAK SURGE TO EQUILIBRIUM.
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
BEACH AND DUNE EROSION DURING SEVERE STORMS
Steven Allen Hughes
Chairman: Bent A. Christensen
Co-Chairman: Tsao-Yi Chiu
Major Department: Civil Engineering
With the increasing development of our coastal areas, it becomes
necessary to understand the long and short term behavior of the
dynamic beach-dune system in order to arrive at sound engineering
decisions regarding future use of these regions, and to protect exist-
ing development threatened by beach erosion. One aspect which would
greatly assist these efforts is the ability to accurately predict the
amount of dune recession which would occur during extreme storm events.
This research has been directed toward a semi-empirical solution
through the use of a small-scale movable-bed physical model. Based on
the principles of hydraulic similitude, a new model law has been derived
and verified. The verification involved the reproduction of an actual
storm event to a degree of realism never before attempted, and the
resulting erosion in the model almost exactly reproduced that which
occurred in the prototype.
In order to determine the effects of wave period, wave height,
surge level, surge duration, and dune height, these parameters were
systematically varied in 41 experimental model tests. Beginning with
an equilibrium profile, the surge level was increased over a typical
time span and then maintained at the peak level until near-equilibrium
conditions prevailed. Qualitative comparisons between the resulting
profiles indicated that it is necessary to employ a time-dependent
surge level increase in order to more accurately simulate an actual
storm event. In addition, it was found that each set of conditions
appeared to form an equilibrium barred storm profile which could be
shifted relative to the prestorm profile to obtain a sediment balance
between erosion and deposition, and thus, determine dune recession.
Analysis of the experimental data provided a means of calculating
the equilibrium barred storm profile in terms of the wave period,
breaking wave height, surge level, and profile sediment grain-size.
In most instances, the empirical expressions used in determining this
profile have physical interpretations in terms of the incoming wave
energy flux per unit width.
Finally, a method was developed which corrects the maximum dune
recession in cases when equilibrium conditions are not reached.
Nature has provided delicate measures at the land and sea boundary
on a sandy coast to maintain a balance of forces under ever changing
wave conditions. While the sloping beach and beach berm provide the
first "line of defense" for the absorption of normal wave energy, the
coastal dunes are the last zone of defense in the advent of storm
(hurricane) waves and the accompanying storm surge which succeeds in
overtopping the berm. In essence, the beach-dune system constitutes
the natural coastal structure which effectively adjusts its shape in
the form of erosion or accretion to cope with any weather condition.
Left in its natural state, this system functions in a most efficient
"give and take" fashion to provide a coastal defense mechanism.
However, in recent years, increasing concentration of population and
industry in coastal areas has resulted in developments which encroach
upon the dynamic beach-dune zone and therefore interfere with the
natural shore processes described above. The recent disastrous
destruction of water front, man-made structures and the magnitude
of sediment required by nature (in the form of beach-dune erosion)
in adjusting its defense posture against Hurricane Eloise in Bay and
Walton Counties on the northwest Florida coast serves as a good ex-
ample of what could and did happen to structures placed within this
dynamic zone (Chiu, 1977; Frank, 1976).
Realizing that the ever increasing development of Florida's beaches
was interfering with natural shoreline processes and that the Florida
coastline is in a high risk hurricane area (Neumann et al., 1978), the
legislature established a Coastal Construction Control Line Law in Sep-
tember 1971 (Purpura, 1972). The purpose of the law was to prevent
construction or excavation seaward of a control line for the protection
of the beach-dune system and for the protection of upland structures
from storm activity. The establishment of such an important line de-
pends on a number of factors, however. One of the most important fac-
tors to be considered is the landward erosion distances of the beach-
dune system that could be expected during a severe storm. A capability
of predicting beach-dune erosion distance, caused by storm activity,
with a reasonable degree of accuracy would not only be an enormous
benefit to the State of Florida, but would also address itself to
this great problem on all the world's sandy coasts.
The investigation of any specific problem in the field of applied
engineering usually proceeds in the following manner:
1) Identification of the problem and its important parameters,
2) Development of the physics of the problem and formulation of
suitable assumptions and simplifications based on observations
and common sense,
3) Development of a mathematical theory from the simplified
physics of the system including all the relevant parameters
(predictive in nature and quite often highly empirical), and
4) Theory verification using known prototype data or by small-
scale model testing.
A review of the currently available literature on the subject of
surf zone dynamics quickly leads to the conclusion that a purely
theoretical approach to the problem of dune erosion based on the
Navier-Stokes equations is presently beyond our grasp. Among other
things, this approach would require an understanding of wave trans-
formation within the surf zone and its associated time-dependent ver-
tical water particle velocity distribution, the turbulent fluctuations
and energy losses beneath a breaking wave, bottom friction effects,
sediment transport (both suspended and bed-load) throughout the surf
zone, and consideration of random seas. At present, no suitable solu-
tion exists for any of these problems. However, experience tells us
that nature does indeed have "order," and that many natural phenomena
can be accurately analyzed through empirically derived expressions
relating the important parameters.
Faced with this insurmountable theoretical "wall" the investiga-
tion of the dune erosion problem necessarily becomes one of a parametric
study involving empirically verified relationships with the inclusion
of known physics when possible. A successful solution of this type
would not only provide a suitable engineering solution, but would
also help further our understanding of the basic, underlying physics
which control this process.
There have been two basic approaches to the problem to date, the
gross simplification of the governing physics involved in dune erosion
and the use of small-scale model testing to aid in the parameteriza-
tion of the process. These methods are discussed in Chapter II, but
it should be noted here that none provide results in agreement with
field measurements. This lack of verification is a result of over-
simplification of the process to the point that several important
observed aspects have been omitted, such as offshore bar formation
and time-dependent surges.
Because current methods of predicting dune recession during
severe storms are unsuitable, there remains an immediate need for the
development of an accurate, predictive engineering solution of the
problem to aid coastal zone planners in the orderly development of
the coastal communities without hindering the natural "storm defense"
provided by the beach-dune system. This is the primary goal toward
which this research was directed.
Important Factors and Assumptions
Our present knowledge of the dune erosion process, derived mainly
from observations combined with common sense, make it possible to list
the important parameters involved:
1) Primary dynamic factors:
a) the storm surge water level,
b) the storm surge duration, and
c) the incoming wave period and wave height and their
2) Primary geometrical factors:
a) the configuration of the prestorm dune, beach, nearshore
and offshore zones, and
b) the grain-size distribution on the beach.
No predictive model at present incorporates all of these parameters.
In fact, most models are of a steady-state nature, which is a condi-
tion seldom reached during the short duration of a storm event. This
indicates that any predictive model must give transient beach-dune
Secondary factors of importance are given as:
1) Strong onshore-directed winds,
2) Oblique wave attack,
3) Heavy rains,
4) Beach and dune vegetation, and
5) Interaction with structures.
While the secondary factors listed certainly have some effect
upon the amount of erosion which could occur, it is felt that the
small refinements resulting from inclusion of these factors into the
analysis are not warranted due to the difficulties involved.
In pursuing a solution to the dune erosion process, the following,
almost universally applied, assumptions have been invoked in this
1) Natural beaches with sand-sized sediment distributions;
2) Fairly uniform offshore depth contours and straight beaches;
3) No interactions with coastal structures or tidal inlets;
4) A two-dimensional onshore-offshore sediment transport situ-
ation involving no net sediment losses or gains to the system,
implying an absence of longshore currents (which field data
indicate is a reasonable assumption for severe storm condi-
5) Sufficiently high coastal dune as to prevent wave overtopping
at the peak surge level.
The overall objective of this study has been to develop the
capability of predicting the magnitude of beach-dune erosion as a
result of given storm conditions. More specifically the objectives
include the following:
1) Develop and verify the necessary small-scale movable-bed
modeling relationships for use in an experimental test
2) Determine the characteristics of the poststorm barred profile,
3) Evaluate the effects of storm surge rise duration,
4) Compare and quantify the relative importance of the main
factors listed above, and
5) Evaluate the effect of storm duration on the transient
PREVIOUS RESEARCH AND EXPERIENCE
The specific problem of being able to predict the amount of beach-
dune erosion during severe storm conditions has drawn the interest of
investigators from the fields of coastal engineering, geology, and
land use planning as well as local, state, and federal agencies. And
while the destructive potential resulting from dune erosion is well
recognized, descriptions of the phenomenon, even in the simplest quali-
tative evaluation, provide little insight into the actual physical
At present there are no verified methods for the prediction of
dune erosion during severe storms, even under the limiting assumptions
of a two-dimensional process without bar formation, net erosion due to
longshore currents, or dune overwash.
There are three basic approaches to finding a solution for a
1) A complete physical analysis based on the equations of fluid
flow and sediment transport when suitable simplifying assump-
tions, based upon observation and common sense, are utilized;
2) A numerical representation of the governing equations and
boundary conditions which are solved in an iterative fashion
and verified with extensive field data;
3) The verification and operation of a small-scale model of the
process carried out with the hopes that the dominating forces
have been successfully scaled down.
The complete physical analysis of the problem is presently out of
the question due to the almost total lack of understanding of the
fluid flow details within the surf zone. Likewise, numerical methods
are limited by the lack of detailed prototype measurements before,
during, and immediately following a major storm event. Modeling in a
small-scale wave facility appears to be the most practical, but this
too has its drawbacks, which will be discussed in detail in Chapter IV.
Previous Simplified Physical Solutions
It appears that the first attempt at quantifying the amount of
dune erosion and the resulting recession of the dune field was given
by Edelman (1968). He assumed that the poststorm nearshore profile
could be approximated by a straight line with a slope of about 1:50
for beaches typical to those found in the Netherlands. Under the
assumption of onshore-offshore sediment transport where the eroded
volume equals the deposited volume, Edelman's method could be used to
predict erosion for an actual beach profile as illustrated in Figure
la, or for "ideal conditions" as shown in Figure lb.
For the actual beach profile, recession, R, is obtained by shift-
ing the equilibrium storm profile until erosion area equals deposition
area. For the "ideal condition" the dune recession is calculated by
Storm Tide Level
Equilibrium Storm Profile
la: ACTUAL PROFILE APPLICATION.
Storm Tide Level L
hb= .3H1 Mean Sea Level -
S O I _-b -
Equilibrium Storm Profile
Ib: IDEALIZED PROFILE APPLICATION.
FIGURE 1: EDELilAN'S METHOD APPLIED TO ACTUAL AND IDEALIZED
R = -8 + /82 + 26
m = beach slope,
= D S +,
1 = 1.3Hb S + F,
6 = 1.3Hbu P ,
F = vertical distance from mean sea level to the base of
the dune, and
S = storm surge level.
The other assumptions are given as: (a) the width of the surf
zone is defined by breaking wave height Hb = 1.5(S) and breaking depth
hb = 1.3Hb, where S is the storm surge level; (b) the deposition is
evenly spread over the surf zone; (c) the slope of the poststorm pro-
file is the same as the original slope; and (d) the eroded dune toe
is slightly lower than the highest surge level reached. Both (c) and
(d) above are based upon field observations.
The main weaknesses of Edelman's method lie in the assumption
that the nearshore beach profile is a straight line with the same
slope as the prestorm profile, and in the assumptions for the limiting
depth hb = 1.3Hb and the surf zone width. In addition, the wave
period and the sediment grain-size have not been taken into account,
and the assumption is made that an equilibrium is reached.
Edelman (1972) later conceded that the beach profiles have a
more intricate shape than he assumed, and that storms never lasted
long enough to produce the equilibrium condition.
The Edelman method was modified by Vallianos (1974) and used to
design dune cross sections. One modification included the recognition
of a flatter beach slope after the storm than that of the prestorm
A modified version of Edelman's method has also been used by the
University of Florida's Department of Coastal and Oceanographic
Engineering to estimate erosion distance for use in establishing
coastal construction setback lines for the State of Florida. Al-
though Edelman's method was used successfully by the Department to
test an eroded profile in St. Johns County, Florida, during a three
day northeast storm in early February, 1973, it predicts too much
erosion for Hurricane Eloise. This points to the influence of storm
duration since Eloise was a fast moving storm with a peak surge last-
ing only about half an hour to an hour.
Besides Edelman's assumptions, Dean (1976) considered uniform
energy dissipation and uniform shear stress across the surf zone to
establish an equation for the equilibrium beach profile given as
h (1 ) (2)
x = the horizontal coordinate directed onshore from the
w = the width of the surf zone,
h = the water depth,
hb = the breaking depth,
and a depends upon the assumed energy dissipation mechanism.
By assuming different profile shapes before and after the storm,
and considering the limiting depth and wave assumptions of Edelman,
Dean gave the following dimensionless relationship for beach recession:
R' = 1 [1 (1 R')1 + a] (3)
Wl(hb /hb )
S' = S/B where B is dune height above prestorm mean sea level,
S = surge level,
hb = hb /B,
and the subscripts 1 and 2 refer to prestorm and storm conditions,
respectively. Here again a balance between erosion volume and deposi-
tion volume is assumed. Dean did a comparison between his method and
Edelman's, and while similar trends were observed, the two methods
differed significantly. He also pointed out that a more valid and
verified method is needed which would include the presence of an off-
shore bar, the transient effects, and the fact that surges will never
last long enough to produce equilibrium conditions.
Previous Experimental Research
While much experimental research has been done regarding the
general aspects of beach response to wave conditions, those which have
addressed the specific problem of dune erosion during storm conditions
are relatively few.
Qualitative Model Tests
Van der Meulen and Gourlay (1968) performed two-dimensional small-
scale model tests to examine the influence of dune height, wave steep-
ness, and wave period using a sediment grain-size of d50 = 0.22 mm.
Due to their uncertainty regarding the correctness of the model law
implemented for these tests, the conclusions drawn were given as quali-
tative observations which should be considered for the selection of
future scale-model studies.
These observations include the following:
1) The recession of a sand dune increases
2) For given wave conditions, storm surge
ment, the equilibrium profiles seem to
3) The dune erosion is a function of wave
sediment size, storm surge level, dune
4) The dune foot recession is greater for
regular waves with the same energy and
as the height of the
water level, and sedi-
be independent of the
Bar formations are less predominant for wind wave experiments;
Dune erosion is very rapid in the early stages of the tests
(a direct result of using an instantaneous surge level rise).
All of these observations are basically correct, and later chap-
ters will attempt to physically explain and quantify each point given
Model Tests to Determine Scaling Relationships
Van de Graaff (1977) conducted a series of dune erosion experi-
ments with the twofold purpose of developing a design criterion for
the Dutch coastline and developing relationships for scaling between
model and prototype as well as between models of different dimensions.
All the tests were performed using an instantaneous surge level rise
and a single prototype storm condition represented by a Pierson-Moskowitz
energy density spectrum.
An empirical correlation of the data resulted in the scaling
NT = 1
N = N
NT = morphological time scale,
N = horizontal length scale, and
N = vertical length scale,
valid between the ranges of 26 < N < 150 when the same bottom material
in the prototype is used in the scale model.
More extensive tests by Vellinga (1978) using the same prototype
design conditions as Van de Graaff concluded that the best scale-model
relationship was achieved when the dimensionless fall velocity parameter
H/jT was held constant between prototype and model. When this could
not be met, the profile could be distorted using the relationship
N 1 +a
N -2a (4)
x N 2a
where a = 0.5 for fine sand and N = grain fall-velocity scale. The
morphological time scaling in this case was given as
N = N 1/2 (5)
which is the same as the hydrodynamic time scale.
Unfortunately, both of these experimental series concentrated on
the development of the scaling relationships for one prototype condi-
tion, and then relied upon correlations to determine the distortion
without any physical arguments to support them. Hence, it is quite
possible that the given relationships are valid only for the tested
prototype condition. Furthermore, the use of a fairly wide-banded
spectrum, such as the Pierson-Moskowitz in shallow water, could be
criticized since shallow water spectra are typically more narrow-banded.
This subject will be touched upon again in Chapter IV. Finally, the
testing of only one prototype condition gives no information as to
the effects of varying the storm parameters. In addition, the in-
stantaneous surge level rise negates any type of time-dependency
analysis which could be carried out upon their data. It will be shown
in the present study that the storm surge must rise over a finite
period in order to more accurately predict prototype response from
Ma (1979) carried out a series of dune erosion experiments at the
University of Florida as a preliminary to the present research. The
main thrust of the work was directed towards identifying the scaling
effects present in small-scale models. The tests were designed holding
the dimensionless fall velocity parameter constant between three dif-
ferent vertical scales. The prestorm profile used in the model was
selected to represent an average beach-dune profile existing on the
coast of the Florida Panhandle; and the model series included two dif-
ferent wave heights, wave periods, and storm surge levels. The
scale-model law developed by Van de Graaff was utilized, and three
different sediment grain sizes were tested.
Upon scaling the results for both fine and coarse grains up to
prototype, Ma found good agreement between the differently scaled
models for the same prototype storm conditions. Unfortunately the
comparison between model and actual prototype erosion caused by Hur-
ricane Eloise was not too successful since the test program did not
include a time-dependent surge level increase. However, this research
did lend some credibility to the idea of preserving the parameter
H/wT, and it pointed out some of the difficulties to be expected in
future modeling efforts. In addition, some qualitative insight was
gained with regard to the dune erosion process, and some of the prob-
lems with the wave flume were uncovered and rectified.
Predictive Techniques Based on Model Tests
An exhaustive effort by Swart (1974a, 1974b) examined the results
of many model tests carried out at the Delft Hydraulics Laboratory
over the years. He proposed that the beach profile be characterized
by three distinct zones, each with its own transport mechanism. These
zones are the backshore above the wave run-up limit in which "dry"
transport takes place, the developing profile where a combination of
bed-load and suspended-load transport occurs, and the transition area
where only bed-load transport takes place.
Swart wisely avoided the internal mechanisms of sediment concen-
trations and bottom velocities by making a schematization of the
external properties of the profile development in order to predict
rates of offshore sediment transport. The final empirical relation-
ships given by Swart provided a means of determining both the equili-
brium profile characteristics and the time-dependent sediment transport
Swart subsequently realized that the computational method he de-
scribed was too complicated for normal use, prompting him to modify
the technique in order to ease the computations without significantly
affecting the results (Swart, 1976).
However, even with these simplifications, the method remains
fairly complex and has several drawbacks which preclude it from use
in determining dune erosion during storms:
1) The model law used in the testing has never been fully veri-
fied as representing actual prototype behavior;
2) The tests used primarily a fixed water level which does not
simulate storm surge rise;
3) The empirically derived expressions offer no physical explana-
tion for the interaction of the parameters such as wave
height and period;
4) The time-dependent rates for erosion, when scaled to the
prototype, are assumed to be correct, but never verified;
5) Many of the initial profiles did not represent near-equilibrium
6) Barred profiles are not predicted by the method but are
usually observed in nature during the offshore sediment trans-
As can be seen in the above literature review, the problem of
coastal dune erosion during storms has been approached by two different
methods. On one hand are the simplified physical models offered by
Edelman and Dean, and on the other are small-scale model testing pro-
grams seeking to provide some insight into the process.
The physical models suffer due to the lack of understanding of
the actual complex physics of surf zone and the resultant effects upon
the coastal dunes. There has not even been sufficient field data of
poststorm erosion to indicate that these methods provide even an order
of magnitude estimate of the dune erosion.
The model testing programs, to present, have been concerned
mainly with the establishment of suitable model laws, to be used in
small-scale testing, which would provide accurate results when scaled
up to prototype. In addition, a time-variable surge level increase
has not been used, which later will be shown to be a pertinent factor.
Thus, it is seen that this is an area requiring immediate examina-
tion in order to both identify the important parameters and their
relationship to the process, and to provide an accurate and reliable
means of calculating the dune erosion as a result of selected storm
Proposed Method of Problem Solution
Drawing upon the previous experience discussed above, the follow-
ing steps are proposed in order to logically examine and formulate a
solution method for determining dune erosion during a storm:
1) Derive a suitable small-scale movable-bed model law based
upon sound physics and past experiments which will provide a
basis for the operation of a laboratory model, simulating
storm conditions and the resulting dune erosion.
2) Verify the model law by the approximate reproduction of a
prototype event for which field data are available.
3) Conduct an extensive series of model experiments in which the
important storm parameters are varied within their expected
ranges. The series will attempt to closely reproduce actual
storm conditions by including such factors as a varying storm
surge level and an initial equilibrium profile.
4) The test results will be initially examined overall in an
effort to identify the effects of the important parameters in
a qualitative sense, and to gain some insight into the entire
process. From this overview it may be possible to recognize
the primary physical mechanisms which influence erosion.
5) From the above step, decisions can be made as to which
aspects of the phenomenon can be parameterized in such a
way that some physical justification for the parameters could
6) Correlations will be sought for the dominant features present
in the erosion process in terms of the chosen parameters, and
physical explanations will be given when possible. While the
physical explanations will, for the most part, be qualitative
in nature, it is expected that a go5d amount of insight will
be gained to further add to the present poor understanding of
7) Finally, a proposed engineering solution will be constructed,
based upon the analysis, with the hope that it will provide
an accurate method until the time the process is fully under-
stood and analyzed in physical terms. Along with the proposed
solution will be a summary of the restrictions imposed by the
assumptions and an outline of the range of applications for
which the solution can be deemed reasonably accurate.
WAVE TANK FACILITY
Air-Sea Wave Tank
The Coastal and Oceanographic Engineering wind and wave facility
was dedicated in 1957 and used primarily as a wave tank. Under a grant
awarded in 1967, the Air-Sea tank was updated and improved, giving it
the capability of being used for advanced research into air-sea inter-
actions. These improvements, along with a detailed description of the
facility, are given by Shemdin (1969).
The Wave Channel
Figure 2 gives a general layout of the wave facility which is
6 feet wide and has a total length of 150 feet. The mechanical wave
generator section occupies 11 feet, while the wave absorbing beaches
occupy 19 feet of the total length. The remaining 120 feet are divided
into two bays, each 34 inches wide and 6'4" deep. The maximum
water depth for wave generation is 3 feet. The dune erosion tests
were conducted in the eastern bay since the entire length of the test
section has full height glass observation windows.
While the tank could be filled from a well at the rate of 500
gallons per minute, the water used for this series of experiments came
through a 2-inch city water line equipped with a calibrated valve.
This provided a controlled means of increasing the surge level during
the experiments, and the water was of greater clarity than the well
water, providing for better observation of the sediment movement.
Mechanical Wave Generator
Wave generation in the Air-Sea tank is achieved through an elec-
tronically controlled, hydraulically driven wave paddle measuring 6
feet wide and 4 feet high. Identical waves are generated in both bays.
The wave generator bulkhead is mounted on a carriage and is
driven by two hydraulic rams governed by hydraulic servo-valves. The
system provides independent control of the top and bottom rams in such
a way that the bulkhead can move either as a piston on the carriage or
as a paddle. Any combination of piston and paddle motion is possible.
For the dune erosion tests, the bulkhead motion consisted of a piston
motion combined with a paddle motion about the bottom of the bulkhead.
This gave the upper water particles a greater displacement than the
lower particles, which more closely represents the vertical velocity
distribution found in water waves.
Sinusoidal wave motion was produced by sending an analog sine-wave
signal from a function generator to the servo-valves with the wave
amplitude being controlled by varying the input amplifier gain.
Prior to the experiments, new servo-valves and solid-state elec-
tronic controls were installed in order to provide better frequency
response and system reliability. Throughout the entire testing pro-
gram, the new system performed without any problems.
As is the case with most modern wave facilities, a digital computer
has been employed to aid in the gathering and processing of experi-
mental data. This not only speeded up the process, it also provided
a more accurate means of data acquisition, with a smaller chance of
error. To guard against errors, methods were developed to immediately
check and confirm the accuracy of the computer-made measurements.
Two further benefits of using the computer are that the speed of
profile measuring and recording allowed for a greater number of experi-
ments to be conducted within the time constraints of the project, and
that data reduction and analysis programs could be run immediately
following each experiment on an instantaneous "turn-around" basis.
In the early stages of the project the computer was used to
generate random time-series to a given power spectrum, to operate the
wave paddle to produce these random waves in the tank, to monitor and
record wave data on command, and finally to analyze the recorded wave
data. During the actual test series, when regular waves were used,
the computer was used only to measure and record the beach profiles
and to analyze the data for erosion volume. Listings of these programs
are given in Appendix A.
The computer used in the experimental program was the DECLAB-11/03
with a CPU memory of 28 K bytes and programmable in the Fortran IV
language, complete with diagnostics. The system includes 1) a real-
time clock, 2) an analog-to-digital converter capable of handling 16
single-ended or 8 differential analog signal inputs, 3) a digital-to-
analog converter capable of outputting 4 different analog signals on
command, and 4) an RX01 dual floppy disk drive for program and data
Two terminals were interfaced to the computer. A Tektronix 4006-1
Graphics video terminal was used for writing and editing programs,
while a TI Silent 700 hard-copy terminal was remotely located nearer
to the wave tank for executing profiler programs and data analysis.
The system software included the standard components of: 1) an
editing program for writing or editing Fortran programs, 2) a Fortran
compiler, 3) a linking program which creates the final machine language
program including any hardware input/output commands, 4) a peripheral
exchange program for transferring programs or data to different de-
vices such as tape or hard disk, 5) an extensive Fortran Scientific
Subroutine package, and 6) support routines for the laboratory peri-
The computer realized only a small percentage of its potential
during the dune erosion experiments, and it is expected that many more
applications will be found in future studies involving movable-bed
The wave gauge used for monitoring the waves in the tank was a
capacitance-type gauge designed and built by the Coastal and
Oceanographic Engineering Laboratory at the University of Florida.
Figure 3 shows the gauge in position.
The capacitance wave gauge uses the water as one plate of a vary-
ing capacitor; hence the capacitance of the sensing portion of the
instrument changes as the water level fluctuates during wave motion.
The signal is sent to the control room in the form of a varying fre-
quency where it is then converted to an analog signal and recorded as
a continuous record on a strip chart recorder. A detailed calibration
at the beginning of the experiment confirmed the linearity of the
gauge (see Appendix B), and the resulting conversion factor for volts
to centimeters was confirmed prior to every experimental run. The
gauge is electronically isolated to eliminate any interaction between
other electrical signals present in the laboratory.
Rail Mounted Movable Carriage
A horizontally moving instrument carriage was installed in the
eastern bay of the wave tank to provide a variable-speed platform on
which sensing instruments could be mounted for the monitoring and re-
cording of experimental data. In this research program the carriage
was used solely for the purpose of transporting the beach profiling
instrument over the length of the profile in a controlled fashion.
Figure 4 shows the carriage with profiler in place from several angles.
The rails for the carriage have been installed with a horizontal
tolerance of 0.001 inches, providing an excellent platform from which
to measure beach profiles. The carriage drive train is powered by a
HP electric motor capable of moving the carriage at speeds between
0 to 20 feet per second.
FIGURE 3: WAVE GAUGE.
FIGURE 4: INSTRUMENT CARRIAGE.
A hand-held remote control unit allows the drive motor to be
started or stopped, activates the automatically operated bottom-
following probe on the profiling instrument, and also allows the
manual vertical positioning of the probe.
The horizontal movement of the carriage is monitored by a "follower
wheel" which has a ring of magnets equally spaced around it. A sensor
is activated each time a magnet passes by, and this produces a voltage
variation similar to a square wave between zero and five volts as the
carriage proceeds down the rail. The computer can "count" the number
of downsteps in the voltage signal, and by applying a conversion
factor, it can determine relative horizontal carriage displacement.
The approximate conversion factor was 0.42 inches of horizontal travel
per voltage downstep.
The horizontal displacement can be referenced to any initial
or final carriage location by inputting the appropriate value read
from a measuring tape mounted on the upper portion of the tank side-
The signals from the horizontal position indicator and from the
profiling instrument were relayed to the computer through a cable
connected between the carriage and a junction box mounted on the side
of the tank.
The MK-V Electronic Profile Indicator was developed at the Delft
Hydraulic Laboratory in Holland for the purpose of continuously
measuring bed levels in hydraulic movable-bed models. The instrument
and its mounting on the carriage is illustrated in Figure 5.
The instrument probe is placed vertically in the water where a
feedback servo-mechanism maintains the tip of the probe at a constant
distance (adjustable between 0.5 mm to 2.6 mm) above the sediment
bedform. When the instrument is placed on the carriage and moving
horizontally, the probe continuously follows the bottom profile con-
Monitoring of a ten-turn potentiometer connected to the vertical
shaft holding the probe produces a continuous record of the bottom
variations in terms of voltage. This voltage is easily converted to
vertical distance through the very nearly linear potentiometer cali-
bration constant determined to be 8.25 inches/volt. This calibration
is shown in Appendix B. The vertical probe stroke is about 36 inches,
and the instrument is placed on the carriage so that it can reach
within 1 inch of the bottom of the wave tank.
A sliding track on the carriage allows the profiler to be placed
in any desired lateral position across the width of the tank. For the
dune erosion experiments the profile was measured down the tank center-
line with the thought that this position gave the most representative
profile and was least affected by the side walls.
PROFILER ON CARRIAGE
BOTTOM PROFILING INSTRUMENT.
mo o- y^ `
r, k m
The profiler is normally operated in conjunction with a computer
program which automatically records the vertical position every time
the horizontal carriage position indicator senses a voltage drop.
This translates to a set of profile coordinates every 0.42 inches.
Since it was felt that this much data were not necessary, the
program was operated in a different mode which allowed the operator
to take profile data only where it was deemed necessary.
The recording of the above water portion of the profile was com-
plicated by the fact that the bottom-seeking servo-mechanism is non-
functioning when the probe is out of the water. However, since the
probe potentiometer still produces a voltage signal, the computer
program was written in a way as to permit the taking of a data point
when an external trigger was pressed.
The procedure followed in recording the above-water profile was
to place the carriage in the desired horizontal position, lower the
probe manually by remote control to the beach surface, press the
trigger, and then move the cart, using the remote control, to a new
position where the procedure was repeated. Once the profile was below
the water level, the bottom-seeking feature was activated, and the
points were recorded where desired by stopping the horizontal move-
ment of the carriage and pressing the trigger of the hand control.
The computer kept track of the horizontal positions where data were
taken and paired them with their respective beach elevations. The data
were then stored on floppy disks along with the necessary initial
references and conversion factors needed to convert the digitized
voltage to linear ranges and elevations, both referenced to the initial
intersection point of beach and water level.
The operator was careful to record all the data necessary to
accurately describe the profile features and variations. The typical
profile measuring consisted on the average of 35 to 40 profile data
pairs and covered a horizontal range of 14 feet and a vertical range
of 12 inches. In all cases the measured profiles extended seaward
past the region of any net sediment transport.
While recording the underwater profile, the carriage moved at
about 8 feet/minute, and, after experience was gained, the operator
found that the measuring and recording of the entire profile could be
completed in under 5 minutes from the time the wave action was halted.
Small-scale physical model testing of both natural phenomena and
man-made structures has long been accepted as a useful engineering aid
for the analysis and prediction of the prototype behavior. The first
known hydraulic scale-model experiments were conducted by an English
engineer during the period 1752-53 to determine the performance of
water wheels and windmills (Hudson et al., 1979). The earliest known
tests of a river model using a movable-bed were conducted in France in
1875 and in England by Reynolds in 1885. Since that time great ad-
vances have been made in the development of the model laws which govern
hydraulic similitude, the instrumentation used in the models, and
modeling techniques in general. Perhaps more importantly, many years
of experience and basic research by many fine hydraulic laboratories
throughout the world have made the simulation of more complex phenomena
possible, with more confidence given to the test results.
Many hydraulic problems are fairly simple in nature and can be
adequately solved by analytical methods to a reasonable degree of
accuracy. However, there are many more cases where the problem being
considered is far too complex to be handled analytically. Often our
understanding of the physics of the problem is very limited or even
nonexistent. In these cases simple approximations are insufficient
and other means of obtaining engineering solutions are necessary. This
is when small-scale modeling of a hydraulic phenomenon can prove
beneficial by aiding in the development of a physical solution or by
helping us to arrive at parametric relationships which can aid in our
understanding of the hydraulic process under consideration. This is
because scale models of hydraulic phenomena are essentially a means of
replacing the analytical integration of the differential equations
governing the process, including initial and boundary conditions. Often
model testing provides enough insight into the physics of the process
so that refinements can be made to simple mathematical models.
Physical models allow the investigator to observe the process in
action and distinguish certain characteristics of the flow patterns
which might have been neglected in the analytical approach. In addi-
tion, the model provides a tight control of the important parameters,
a control which is not available in the prototype. Examples include
extreme events where accurate prototype data will be lacking, such as
hurricanes, dam bursts, and flooding. Using the model to predict the
results of these extreme events can provide invaluable information
which more than justifies the high cost of building and operating the
Of all the hydraulic engineering models which can be performed,
movable-bed scale-model investigations of coastal erosion and coastal
sediment transport phenomena are probably the most difficult. In
fact, so many different model laws have been proposed, modeling of
this type should be considered more of an art than a science! How-
ever, by carefully identifying the major forces involved, it should be
possible to derive a model law which can be verified and which will
provide reasonable, quantitative results.
Previous Scale-Model Relationships
Numerous papers have been written proposing similitude relation-
ships for movable-bed coastal models. For a rather complete bibli-
ography of the subject, see page 308 of "Coastal Hydraulic Models"
(Hudson et al., 1979). These relationships range from those which
were derived solely by theoretical considerations to those which were
established on a strictly empirical foundation. In each case the
relationship is assumed valid for a given set of specific flow charac-
teristics. For example, a model law which attempts to scale the
dominant forces involved in incipient motion of sediment in a tidal
inlet over the tidal cycle will not be valid for sediment transport
in the surf zone, where a completely different set of forces dominate
the regime. For this reason it is necessary to first have an under-
standing of the dominant dynamic forces at work for a given coastal
hydraulic problem, and then to examine the proposed scale-model rela-
tionships to determine if these are the forces being scaled.
Perhaps the most thorough investigation using known relationships
of beach processes to determine the proper scaling law for coastal
movable-bed models was that of Fan and LeMehaute (1969). Their result
was a table containing eight proposed scaling relationships, each
being a combination of three or more derived similitude conditions
based on the known beach process relationships. One important point
to note is that all of the proposed model laws satisfied the condition
of the time scale being equal to the square root of the vertical scale.
Using available data, a tentative model law was chosen to be verified
by extensive experiments.
The experimental program was carried out the following year with
the results reported in 1971 (Noda, 1971). Noda conducted the experi-
ments to determine the validity of the proposed model law, and he then
proceeded to derive a completely empirical model law based on simili-
tude of equilibrium beach profiles in the breaker zone. Unfortunately,
while good reproduction was given in the surf zone, the corresponding
dune and beach erosion was not reproduced. Hence, this model law is
not really applicable to the modeling of dune erosion.
The literature appears to contain no reported studies regarding
model similitude relationships which were physically derived specifi-
cally for the case of dune erosion during storms. However, two recent
studies conducted by the Delft Hydraulic Laboratory in the Netherlands
provided a totally empirical model law for dune erosion (Van de Graaff,
1977; Vellinga, 1978).
Without going into details, these studies derived a model law by
empirical correlations of tests done at different scale dimensions
when compared to a single prototype condition. Several questions can
be raised as to the validity of the derived relationships:
1) Since very little physical reasoning has been applied in the
model law derivation, it is more than likely that a prototype
condition other than that tested could not be reproduced.
2) Tests were conducted with a fixed surge level instead of a
variable surge similar to what occurs in the prototype.
Hence, there is no solid evidence that the morphological time
scaling is correct.
3) The hydraulic time scale was derived by Froude scaling and is
equal to the square root of the vertical scale, the same as
Noda's derivation. It will be shown shortly that this is not
necessarily correct when modeling dune erosion.
4) The Pierson-Moskowitz energy density spectrum was used as the
wave climate for the tests. This spectrum is representative
of deep-water wave fields, while the experiments were carried
out in water depths equivalent to 70 feet in the prototype.
At these water depths, energy density spectra are decidedly
narrower than the P-M spectrum. In addition, prototype spectra
contain wave grouping which is not understood, but definitely is
important. These model tests did not attempt to reproduce
this phenomenon. This point will be discussed in the follow-
ing chapter in further detail.
5) Attempts to apply this model law in the initial verification
stages of the current research failed to even come close to
reproducing prototype erosion. In fact, accretion of the
beach above mean sea level occurred for both regular and
random wave conditions. This also will be discussed in the
These factors, combined with the apparent lack of any other suit-
able model law for dune erosion, have led to the following development
of a new scale-model relationship.
Proposed Model Law for Dune Erosion
The requirements for similarity between hydraulic scale-models
and their prototypes are found by the application of several relation-
ships generally known as the laws of hydraulic similitude. These laws,
which are based on the principles of fluid mechanics, define the
requirements necessary to ensure correspondence between model and
prototype (Hudson et al., 1979).
Complete similarity between model and prototype requires that the
system in question be geometrically, kinematically, and dynamically
similar. Geometric similarity implies that the ratios of all linear
dimensions between model and prototype are equal, kinematic similarity
is similarity of motion, and dynamic similarity between two geometric-
ally and kinematically similar systems requires that the ratios of all
forces in the two systems be the same. From Newton's Second Law the
dynamic similitude is achieved when the ratio of inertial forces
between model and prototype equals the vector sums of the active
forces, which are recognized as gravity, viscous, elastic, surface
tension, and pressure in the coastal regime. An additional requirement
is that the ratios of each and every force must be equal. Or, in
(F. ) (F ) (F ) F t) (F ) (F )
F.. TF Fv t \ (F (6)
Im gm vm tm em pm
where subscripts p and m are for prototype and model, respectively.
Five of these force ratios are taken as independent, with one (usually
pressure) being determined after establishment of the others.
Since it is considered impossible to satisfy equation (6) except
with a full-scale model, it is necessary to examine the flow situation
being modeled to determine which forces contribute little or nothing
to the phenomenon. These forces can then be safely neglected with the
goal of reducing the flow to an interplay of two major forces from
which the pertinent similitude criterion may be theoretically de-
veloped (Rouse, 1950).
For models of wave action and ensuing sediment transport the
elastic forces and the surface tension forces are sufficiently small
that they can be neglected, provided that the water wave length in the
model is greater than about 10 centimeters. Since inertial forces are
always present in fluid flow, the condition for dynamic similitude
reduces to equating the ratio of inertial forces to the ratio of
either gravity forces or viscous forces.
For the particular case of dune erosion the main area of interest
is not the offshore zone, but the surf zone and beachface. Here the
waves rush up the beach and then return down the slope, eroding and/or
depositing sediment in their wake. During this process, particularly
during severe storm conditions, the fluid particle velocities near
the bed are well in excess of the critical velocity for incipient
motion, and sediment is in a state of nearly constant motion. Thus,
over the wave cycle, the fluid process can be idealized as unsteady,
unidirectional, open channel flow up a slope followed by flow down the
slope. Of course this simplification can not be used in an analytical
approach due to the complexities involved. However, through this
visualization, it is easy to recognize that the two major forces act-
ing on a sand grain are the inertia forces, due to the turbulent flow
fluctuations near the bed, and the nearly horizontal component of
gravity acting parallel to the beach slope. The viscous forces are
small compared to the forces due to the turbulent fluctuations and
thus can be neglected in this instance.
Before deriving the scale-model relationships, it is important to
point out that the horizontal and vertical length scales will be dif-
ferent, providing a distortion of the model. This is necessary because
the reduction of size of the sand grains, as required to obtain geo-
metric similarity, would result in sand particles so small that co-
hesive forces not present in the prototype would be present in the
model. When sediment material of nearly the same size and specific
weight as found in the prototype is used in the model, distortion of
the model will occur naturally in the mathematical derivation. Some-
times choosing a lighter material to use as the sediment in the model
will result in an undistorted model, but the expenses and difficulties
which arise are seldom worth the effort.
Derivation of Dynamic Similarity Condition
For convenience it is customary to introduce the notation
N = value of parameter "a" in prototype
a value of parameter "a" in model
to represent the scale ratio of a given parameter between the proto-
type and model. Using the symbols:
L = horizontal length,
B = horizontal width,
D = vertical depth,
T = time, and
F = force;
the fundamental model scale ratios can be defined as:
Horizontal Length Scale: N. = p p (7)
Horizontal Length Scale: N, = Ir = J (7)
Vertical Length Scale: N = (8)
Time Scale: NT = p (9)
Force Scale: K = (10)
where again the subscripts p and m are for prototype and model, respec-
tively. From these four scales, all other model scales can be derived.
Following the development of Christensen and Snyder (1975), the
force due to gravity in the nearly horizontal direction of the prin-
cipal flow may be written as
F = pg(volume)(sinB)
where p is the fluid density, g is the gravitational acceleration, and
B is the beach slope. For small beach slopes, sine b 2 i-, so the
force scale for gravity can be written as
K = pg (LBDp)() () ( )NN N (11)
g 0m m m
g D PM \ m m
Pmgm (LmBmDm)( L
The inertial force is best represented as a horizontal, or nearly
horizontal, area multiplied by the shear stress acting over this
Fnertial = Area x Shear Stress
For the turbulent flow experienced next to the bed, which is in the
rough range, the shear stress depends on the rate of momentum transfer
and can be expressed as the Reynold's shear stress which is proportional
to the fluid density and the time mean value of the product of a
vertical velocity fluctuation and the velocity fluctuation in the
direction of the time mean flow. Consequently, the inertial force is
F. = pu'v'(area)
and the inertial force scale ratio can be written as
pPR( )( L ) B ) 3
1 pm N2
Ki L D =( ) 2 (12)
Pm T m)(LmBm) T
For dynamic similitude requiring K. = K gr and noting
that gp = gm, equations (11) and (12) yield the time scale, i.e.,
N N2 = N
N p 2
NT (N )1/2 (13)
Equation (13) is essentially the same as a similarity of the Froude
number between prototype and model, when the Froude number is based on
a vertical length and a horizontal velocity, i.e.,
F* = hrizontal14)
(g D )1/2
Physically, this can be interpreted as a measure of near-horizontal
displacement of a sand grain being held up just above the bed by
turbulent fluctuations. The grain is moved horizontally by a velocity
as it falls back to the bed vertically.
It should also be noted that in the case of an undistorted model
where N, = N equation (13) reduces to
NT = (N)1/2 (15)
which is the time scale for wave motion used in the previous modeling
attempts. It also arises from scaling of the Froude number. However,
the distortion required due to the use of beach-size sand in the model
means that equation (13) must be used for dynamic similarity.
Sediment Transport Similarity Criterion
Besides having dynamic similarity in the model, it is necessary to
find some method of determining the required distortion of the model
arising from the use of natural-sized beach sand. The lack of physical
understanding with regard to this question has led many investigators
to propose a variety of parameters to be scaled in the model which
result in a distortion relationship (Kemp and Plinston, 1968; Noda,
1971). Others have used an empirical approach (Van de Graaff, 1977;
Vellinga, 1978), as already mentioned.
Currently, the most promising parameter used for the prediction
of equilibrium beach slopes is the dimensionless fall velocity, as
presented by Dean (1973). The physical significance of this parameter,
herein designated by
P = HT (16)
H = wave height,
T = wave period, and
w = fall velocity of the sediment,
is whether a sediment particle thrown into suspension by the passage
of a wave will settle to the bed during the time that the water par-
ticle motion is shoreward or seaward, resulting in onshore or offshore
movement of the particle. This parameter has proven to be a good
predictor of onshore-offshore sediment transport (Dean, 1973).
More recent investigations have shown that the dimensionless fall
velocity parameter is also a good predictor of several other surf zone
features. Dalrymple and Thompson (1976) demonstrated that this
parameter could be used successfully to predict the foreshore slope
when H was given as the deepwater wave height. While significant
scatter was present in the data representation, they reported a much
better correlation than when the same data were plotted versus H /L
and a third parameter pertaining to grain-size. They concluded that
the parameter H /wT should be preserved between the model and the
prototype in order to reproduce the same equilibrium profile.
Noda (1978) investigated both full-scale and small-scale model
results for profile similarity and found that a much closer similarity
could be obtained when the H/wT parameter was conserved than when wave
steepness, H /L was held constant. He also offered an empirical
relationship for the selection of model grain-sizes and concluded that
movable-bed coastal models could be distorted, but the validity still
needed to be confirmed.
Saville* (1980) compared prototype model tests using sand against
small-scale tests employing coal as the sediment. The tests were
designed so that the fall velocity of the sediment would be correctly
scaled. His preliminary findings indicated that profile similarity
was best in the surf zone and on the beachface, where settling velocity
might be expected to be a major parameter affecting the modeling.
The comparison seaward of the surf zone is not as good, possibly be-
cause a shear stress modeling relationship would be more appropriate
in this region.
As pointed out in Chapter II, comparisons by Ma (1979) indicate
that scaling between models of different dimensions and sediment is
best achieved by preservation of the dimensionless fall velocity
parameter. In this way there appears to be a profile similarity, and
consequently, erosion volume similarity.
Based on this growing amount of evidence it becomes increasingly
clear that the parameter H/wT should be the same in the model as in the
prototype for profile similarity. This will also allow the sediment
grain-size and specific weight to be incorporated into the model law
as a single variable, w. This requirement becomes
pT p mT
NT = (18)
*Personal communication with Saville indicates that this research has
not been completed as of June, 1981, and thus was not included in the
17th Coastal Conference Proceedings.
Dune Erosion Model Law
Since the time scale for wave motion is the same as the time scale
for the resulting turbulent velocity fluctuations, equation (18) can
be equated to the dynamic similarity criterion of equation (13),
(N )1/2 N
Rearranging gives the final scale-model relationship for the model
distortion as a function sediment fall velocity, i.e.,
N- =--- (20)
which combined with equation (13), given again below,
NT = 1/2 (13)
provides the complete requirements for scale-model testing of dune
erosion during storms using a movable-bed model. Actually, equations
(13) and (20) should fulfill the modeling requirements for the deter-
mination of any profile alterations due to wave action in the surf
zone, not just those due to storm conditions.
It is interesting to note that equation (20) is identical to the
empirically derived distortion given by Vellinga (1978) for fine sand
(see equation (4), Chapter II), with the only difference between the
model laws being the time scale (equation (5) and compared to equation
Morphological Time Scale
Making the morphological time scale the same as the hydraulic
time scale will conserve the number of incoming waves per unit time,
thus conserving the incoming wave energy per unit time, and this seems
most plausible, in view of the scaling of the particle fall velocity
and the length scale distortion. The time scaling of the surge dura-
tion should also be to the same scale.
Unfortunately not enough is known about beach process reaction
times to determine the morphological time scaling unequivocally. How-
ever, results from Saville (1980) indicate that better time-dependent
profile comparisons are given if the time scale is distorted since the
small-scale model beach in Saville's experiments deformed faster than
would be expected when scaled up to prototype using the time-scale
NT = N 1/2 (15)
It is easily seen that the proposed time scaling given by equa-
tion (13) can be expressed as
NT = N 1/2 (21)
n= = model distortion.
Hence, the time scale has the same distortion as the lengths in the
model. If the model distortion is increased, the beach process will
occur in a shorter time in the model.
While the above reasoning appears to be qualitatively correct, a
further confirmation of the morphological time scale for beach pro-
cesses is needed. However, the results of the model verification,
given in Chapter V, indicate that the assumed time scaling is quite
Discussion of the Model Law
The scale-model relationships just derived have resulted from a
combination of basic physics and an observed parametric relationship
which results in a model distortion as a function of the sediment grain-
size scale ratio. The two equations given, equation (13) and equation
(20), are expressed in terms of four variables, N N NT, and N.
This allows the experimenter the freedom of selecting two of the
scaling ratios to suit the model facility.
The main difference between this model law and previous attempts
is the derived time scaling relationship. The other model laws have
expressed, for the most part, the time scaling as
NT = N 1/2
which usually arises as a result of trying to preserve the wave
steepness parameter H /L However, the wide range of values of wave
steepness found for transition between summer and winter profiles by
numerous investigators (Johnson, 1949; Watts, 1954; Rector, 1954;
Saville, 1957a; etc.) indicates that wave steepness may not be such
a good parameter to preserve.
In contrast, this proposed model law preserves the dimensionless
fall velocity parameter, which has been shown to be a better indicator
of beach processes. This results in a time scale given by
NT = CN 1/2 ,
NT = PN
S= model distortion- .
Likewise, the wave steepness is scaled as
H H NH N H
0 J 0 p o p
S T2 -N 2T22 N 2 L
p 2 7 2ir T T m
Lo N N 2 1
p ~ (22)
H NT2 N2 2 (
One drawback that might arise from this distortion is that the reflec-
tion of the incipient waves may become significantly greater in the
model than in the prototype, and care must be taken to minimize this
effect by selection of scales which give a small distortion. For ex-
ample, reflection is not appreciable for beach slopes milder than
about 1:20, so a prototype beach having a slope of 1:40 could be
modeled with a distortion up to n = 2 without much effect due to wave
It is possible to have an undistorted model, using these proposed
relationships, by the proper selection of sediment for the model. This
represents the ideal condition, as long as the resulting model sediment
size is still outside of the cohesive sediment range. Using equations
(13) and (20) with N = N the requirement becomes
N = N 1/2 = NT (23)
with both the parameters Ho/L0 and H/wT being conserved. However, this
condition is more often than not impossible to satisfy due to the
fairly large length scale ratio, N required to model typical sandy
It is worth noting that Battjes' (1974) surf similarity parameter,
H 1/2 (24)
is preserved by the proposed model law. Representing tanB = as
before, and L T2 then
D g 1/2 T
27 HE) /77 NN 1/2
P= p T_ __
m Dm gm 1/2 T N
when (g p/g) = 1.
Substitution of equation (13) for NT yields
E N N 1/2
S 1/2 ---= 1. (25)
5m N 1/2 N
Thus, similarities which Battjes noted were related to the simi-
larity parameter should be successfully scaled using the proposed
An interesting aspect of the power relationship in the scale-model
law between the horizontal and vertical scales is that it is the same
as was derived by Dean for equilibrium beach profiles (Dean, 1977)
and further examined by Hughes (1978). Dean proposed an equation for
the equilibrium beach profile, given as
h = Ax2/3 (26)
h = depth below mean sea level,
x = horizontal distance seaward from the intersection of the
beach and the water level, and
A = function of sediment characteristics,
developed from the theoretical concept of uniform wave energy dissi-
pation per unit volume of water in the surf zone. An examination of
a large number of profiles by both Dean and Hughes lent credibility
to this proposed equation. Rewriting equation (26) as
h = f(x,h) = 1
and requiring similarity of this parameter between prototype and model
Ax 2/3 A x 2/3
A x 2/3 h
A 2/3 h
m x m
Replacing x /xm with the horizontal length scale N and hp/hm with
the vertical length scale N and rearranging gives
N = (28)
N A 3/2 '
which results in the same distortion power as was found in the model
law equation (20). In essence, the proposed model law is also preserv-
ing the equilibrium beach profile as given by equation (26). It is
quite possible that the similarity is only coincidental, but later
analysis in Chapter VIII seems to indicate that this is not the case.
Finally, as a sampler of what follows in the next chapter, it
seems appropriate to mention that attempts to reproduce a prototype
event in the wave tank resulted in complete failure when previous model
laws based on the time scaling NT = N 1/2 were tried, while a quite
reasonable verification was achieved using the proposed relationships.
This fact, plus the seemingly logical development of the model law,
makes it appear that the proposed model relationships given by equa-
tions (13) and (20) provide a viable means of investigating dune
erosion using a small-scale movable-bed model.
Complete verification of a movable-bed scale-model is perhaps the
most difficult task in the whole realm of modeling. The main principle
behind verification is that of being able to reproduce in the model the
results of a prototype event by the scaling of the known parameters of
the event in the model.
Major difficulties arise in two areas:
1) In most cases, complete data on all the important parametric
values during the prototype event are lacking or unreliable.
2) Small-scale testing can introduce secondary effects caused by
the model facility itself. For example, many erosion experi-
ments are done in two-dimensional wave tanks, whereas, the
process may, in fact, be a three-dimensional phenomenon, com-
plete with rip-currents and longshore sediment transport.
The first of these difficulties, the lack of complete data, will
probably never be resolved for the case of dune erosion during storms,
due simply to the fact that prior knowledge of a storm landfall is
not known in time to install all the necessary instrumentation for
recording wave climate, surge level rise, profile changes, etc. Even
if all instrumentation were in place, the severity of a hurricane-
related storm has the habit of destroying or misplacing the equipment.
The second difficulty, scaling effects, can often be resolved by
the careful examination of these effects and the magnitude of change
they cause in the model. In the case of dune erosion, the assumption
is made that the process is strictly that of onshore-offshore sediment
movement. While perhaps not totally correct, the general feeling among
most investigators is that onshore-offshore motion is the primary
mechanism at work during storms, and thus,'the process can be success-
fully modeled in two dimensions with the hope that the other effects
Model verification proceeds loosely in the following order:
1) Select a prototype event with as much available data as
2) Select the model scales to give the best performance and
range of the facility. For instance, a wave tank must be
able to produce good wave forms over the frequency and wave
height ranges determined to represent the upper and lower
extremes expected for the phenomenon.
3) Attempt to reproduce the prototype results by running the
model with the given input conditions scaled as accurately
4) After reproducing the desired end condition to reasonable
satisfaction, repeat the experiment to show that the same
result can be obtained for the same conditions.
5) Investigate the sensitivity of the model by examining the
effects of perturbing the input parameters.
6) Select a different set of scale ratios, as determined by the
model law, and repeat the verification.
Verification of the scale model is a time-consuming but essential
step in an experimental program. With a verified model the experi-
menter can proceed with a carefully planned series of model tests
designed to shed light on the importance of each independent parameter
with reasonable confidence that the results accurately depict what
would happen in the prototype under the same conditions.
Selection of Prototype Conditions
When Hurricane Eloise struck the Florida Panhandle in September,
1975, its storm surge and accompanying destructive wave action eroded
large sections of the relatively undeveloped natural beach-dune system
of Walton County to the right of the storm's landfall, in addition to
the havoc it played with the highly developed Panama City area
Under the Florida Coastal Construction Setback Line Program
(Purpura, 1972), the coastal areas primarily affected by Hurricane
Eloise were surveyed about two years before, with beach-dune profiles
taken approximately every 1000 feet and offshore soundings every 3,000
feet (Coastal and Oceanographic Engineering Laboratory, 1974; Sensa-
baugh et al., 1977). Immediately after the passage of Eloise, survey
teams from the Florida Department of Natural Resources (DNR) resur-
veyed the profiles in the areas most affected by the storm, and these
195 sets of beach-dune profiles were made available by DNR for selec-
tion of a suitable profile to be modeled.
Since the beach and dunes of Walton County came closest to repre-
senting the natural, undisturbed shoreline being considered in this
study, these profiles were examined, and the decision was made to use
the beach-dune profile designated as R-41. Selection was made based
on a number of factors:
1) A 26 feet high dune, which means no wave overtopping occurred.
2) A fairly simple dune geometry making remolding of the dune in
the wave tank considerably easier.
3) Profile R-41 is located almost exactly on the track of the
hurricane's eye, which allows use of existing methods for
calculation of peak storm surge.
4) The other Walton County profiles to the right of the storm
landfall seem to exhibit pretty much the same erosion charac-
teristics as R-41, thus indicating the decision was fairly
arbitrary with regards to poststorm profile configuration.
5) Sand samples taken in the immediate vicinity of profile R-41
were available from DNR for analysis.
The profile location is shown in Figure 6, and the profile selected
for verification tests is given in Figure 7 with the poststorm profile
also shown. Since there was no offshore profile taken at this range,
the offshore portion to be modeled was obtained by averaging profiles
R-39 and R-42 where such data were recorded.
/ Gulf of Mexico
Scale in Miles
FIGURE 6: PROFILE R-41 LOCATION MAP.
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Profile Sediment Analysis
Sand samples taken in the immediate vicinity of profile R-41 were
obtained from DNR and analyzed, in order to arrive at a representative
grain size for the profile. The sieve analysis used standard sieves
numbered 10, 18, 25, 35, 45, 60, 80, 120, 170, and 230, which gave a
linear progression in p units.* The results of the analyses are shown
in Table 1 below.
WALTON COUNTY SAND ANALYSES
Parameter Dune Mean High Mean Low Profile
in (mm) Water Water
d50 0.226 0.215 0.330 R-39
d 0.2199 0.2092 0.3202
d50 0.260 0.280 0.270 R-42
de 0.2517 0.2705 0.2627
d50 0.260 R-45
In the above table, d50 is given as the mean grain-size diameter
taken from the distribution curve at the point where 50 percent of the
sample is finer than that size. The parameter de has been defined by
* = -log2dmm:definition with diameter, dmm, given in millimeters.
Christensen (1969) to be the effective grain-size of a nonuniform natural
sediment. It represents the grain-size of a uniform spherical sediment
that behaves in the same way as the natural nonuniform sediment from
which it was derived, and as such, provides a convenient means for ex-
pressing well-sorted sediment distributions. Calculation of d is done
using equation (29) when the grain-size distribution is a straight
de 2 Cu In Cu
d 2 (29)
50 Cu 1
In equation (29), Cu = -d-, Hazen's uniformity coefficient, and the
values for d10, d50, and d60 are obtained from the sediment distribu-
tion as illustrated in Figure 8. Since the prototype grain-size dis-
tributions for Walton County were nearly straight lines (as can be
seen in Figure 9), de was calculated using equation (29). As shown
in Table 1, well-sorted, narrow grain-size distributions have an effec-
tive grain-size nearly equal to the mean. The average value of de for
all seven samples is 0.255 mm, while the average of the three samples
of profile R-42 is 0.262 mm. Since R-42 is only 1000 feet from the
selected profile R-41, it was decided to use the value of d obtained
from the R-42 samples. Actually the difference between that value and
the overall average is quite insignificant.
Surge Hydrograph Selection
Since no comprehensive storm surge data exist for Hurricane
Eloise, it was necessary to make a close approximation for the surge
0 Fine d ~o d d6 Cc
GRAIN-SIZE DIAMETER d (Millimeters)
Fine do do do Cc
GRAIN-SIZE DIAMETER-d (Millimeters)
FIGURE 8: USE OF GRAIN-SIZE DISTRIBUTION TO FI;iD d
hydrograph using a combination of existing prediction methods and
A first estimate of the peak surge level can be obtained by look-
ing at the eroded dune profile shown on Figure 7. It is seen that the
dune toe is located at a prototype elevation of about 10 feet above
mean sea level. It is obvious that any surge level above 10 feet
would have eroded the dune more than it did. Allowing for wave run-up
would put the peak surge level at somewhere between 7 and 9 feet.
Using the nomogram method for prediction of peak surge for a storm
moving onshore given in the Shore Protection Manual (Coastal Engineer-
ing Research Center, 1975), it was possible to calculate a first
estimate of the surge on the storm track, which passed over the pro-
file in question. Using the storm values of
VF = 20 knots--forward speed,
P = -1.72 inches H --pressure differential,
Rmax = 15 naut. miles--radius of maximum winds,
obtained from the National Weather Service, the peak surge was found
to be approximately 7.7 feet.
Although the method cannot be considered accurate, it is at least
in the neighborhood predicted by common sense.
Finally, a computer prediction for peak surge using the NOAA SPLASH
data is shown in Figure V-5 of Pidgeon and Pidgeon (1977), as calcu-
lated for Hurricane Eloise. From this, the surge elevation of about
7.5 feet is given at the location of profile R-41.
The above reasoning, combined with observations during the veri-
fication runs as to the effects of varying the peak surge, has led to
the final estimate of slightly over 8 feet for the value of peak surge
experienced at this location during Hurricane Eloise.
The time history of the surge level rise was approximated as a
linear increase from mean sea level to peak surge over a time span of
12 hours, a constant peak surge for one hour, and a linear decrease
over 6 hours. This was determined by examining the recorded surge time
histories associated with storms of similar strengths as Eloise
While the assumed surge hydrograph may not be exactly what occur-
red, it is felt that it is a very reasonable estimate, and hopefully the
beach response in the model will bear this out. In fact, preliminary
tests showed that the majority of erosion occurred around the peak
surge value; thus the assumed duration is not quite as important as
the peak elevation. Later observations from the experimental test
series confirmed this.
Wave Climate Selection
While no nearshore wave data were taken precisely at the location
of R-41, it was possible to estimate the probable wave climate using
data recorded at other locations during Hurricane Eloise.
A wave-rider buoy operated by the National Oceanic and Atmospheric
Administration recorded wave data in the middle of the Gulf of Mexico
as the hurricane eye passed within 10 miles (Withee and Johnson, 1975).
At the peak of the storm at that location, it was found that the
dominant spectral wave period was 11 seconds, and it remained constant
over a five hour period. Peak significant wave height was 29 feet. As
the storm made landfall, wave data were also being recorded on the
Naval Coastal Systems Laboratory tower located 11 miles offshore of
Panama City in 105 feet of water. Here the modal wave period was also
11 seconds for the two hours before power failure which occurred within
an hour of the closest approach of the storm's eye (Pidgeon and
Pidgeon, 1977). At that time the maximum significant wave height
was found to be 14 feet.
From these two sets of data, it can be seen that the dominant wave
period appears to be conserved while the storm moved into shallow
water. Thus a dominant wave period of 11 seconds can confidently be
nominated as a prototype parameter for profile R-41, located about
40 miles to the west of the Panama City tower.
Selection of significant wave height is a little less precise due
to the effects of energy dissipation experienced by the wave field
during shoaling. In addition, the significant wave height should be
greater to the right of the track of the storm with the maximum at
about the radius of maximum winds (20 miles in this case).
Some computer numerical models do exist which attempt to predict
energy losses of spectra due to shoaling (Hsiao, 1978), but the ex-
pense and difficulty in running them make their use, in this case,
unjustified.* For this reason, the significant wave height (SWH) in
shallow water (50 feet) at the R-41 profile site was estimated by
assuming a 14 feet SWH existed 11 miles offshore, the same as Panama
City, and that losses incurred in shoaling reduced the significant wave
height to around 12 feet. This value represents the peak value at the
*Dr. O.H. Shemdin, Department of Coastal and Oceanographic Engineering,
University of Florida, personal communication.
height of the storm and associated surge. At lower surge levels, the
SWH value was, of course, less.
While the chosen value for wave period is probably very close to
what actually occurred during the storm, not as much confidence can be
given to the selected SWH. However, the value of 14 feet represents a
maximum, while values of under 10 feet seem slightly low for a category
3 hurricane at landfall. However, it was shown during the experimental
phase that wave height perturbations do not significantly alter the
amount of dune erosion.
Selection of Model Scales
In Chapter IV the scale-model relationships were derived, result-
ing in two equations in four unknowns: the vertical length scale ratio
(N ), the horizontal length scale ratio (N ), the sediment fall velocity
scale ratio (N ), and the time scale ratio (NT). Having the freedom
of selecting two of the ratios, it was found that choosing N and N
was the most convenient. Originally it was hoped that the vertical
scaling could be 1:16 (or N = 16), but this placed a limitation on
the range of wave heights which could be generated in the tank before
extreme nonlinearities began to dominate the wave motion. As a con-
sequence, the vertical scale ratio was chosen to be N = 25.
The sand used to mold the profile in the wave tank was first
sifted to remove all sizes greater than 0.3 mm. This was done to re-
duce the effective grain-size diameter, which in turn increases N with
the result of decreasing the model distortion. A second reason for
removing the larger grains was to try to prevent armoringg" of the
foreshore beach slope.
Two samples were sieve analyzed, producing nearly identical dis-
tributions. A composite using both sample results was plotted, and
the values of d50 = 0.152 mm and de = 0.147 mm were found. The dis-
tribution of grain-sizes for the model sand compared very favorably
with the prototype distributions, one almost appearing to be a linear
offset of the other. Figure 9 shows both the model sand and the
approximate prototype sand grain-size distributions.
The values of particle fall velocity for both the prototype and
model effective grain-size diameters at 25'C are given in the Shore
Protection Manual as:
a = 4.0 cm/sec for d ) = 0.262 mm
m = 1.65 cm/sec for d )m = 0.147 mm ,
which results in
N = = 2.424
From equation (13) and (20) of Chapter IV, the scale ratios for hori-
zontal length and time were calculated along with the model distortion,
S= N-. These values are listed in Table 2 below, and were used
throughout the model verification and the experimental test series.
MODEL SCALE RATIOS
Vertical Length Scale N 25
Horizontal Length Scale N 51.56
Time Scale NT 10.31
Model Distortion Q 2.06
0.05 = 0.2 0.3 0.40.5 I .0
0.05 0.1 0.2 0.3 0.40.5 1.0
iFiODEL AND PROTOTYPE GRAIN-SIZE DISTRIBUTIONS.
Requirements for Verification
There are three major characteristics of dune erosion which must
be reproduced in the model test in order to satisfactorily obtain
1) For the correctly scaled input parameters determined for the
prototype, the total volume of dune material eroded above mean
sea level in the model must approximate the same volume eroded
in the prototype when the proper scale factors are applied.
2) This erosion must occur over the time span of the surge dura-
tion as scaled in the model.
3) The poststorm foreshore beach slope must duplicate that which
existed in the prototype when the distortion factor is applied.
By doing this, the derived distortion of the model is verified.
Referring to Figure 7, the above three points essentially mean
reproducing the poststorm profile in the region above mean sea level.
It is not expected that the berm feature between ranges -10 inches
to +5 inches will be reproduced since it was probably formed by low-
steepness swell wave conditions which occurred between the end of the
storm and the measurement of the profile several days later. The re-
gion above this elevation was not influenced after the surge elevation
decreased. Reproduction of the profile in the surf zone is not
necessary because the poststorm wave climate also caused changes in
Early Attempts at Model Verification
The first attempts to verify a model law were carried out using
the scaling relationships as proposed by the Delft Hydraulics Laboratory
(Vellinga, 1978). The first run was made using irregular waves with a
vertical scale of N = 16. It was seen that this scale was too large
to give proper waves which necessitated reducing the vertical scale to
N = 25. Several tests were conducted at this scaling range using
both regular and irregular waves. In all cases, sediment was moved
from offshore and deposited on the beach as the surge level was raised.
At this point it became obvious that preserving the wave steepness in
the model, which results in the time scaling of NT = N 1/2 = 5, was
not a proper technique. What perhaps worked for others, when an in-
stantaneous surge level was applied, did not work when the water level
rise occurred like it does in nature. In the latter case, the waves
had time to transport sediment landwards and thus altered the beach
configuration, which in turn affects the waves.
At this point in the verification program, a series of undocu-
mented runs were made for the purpose of developing experience and
intuition into the effects of varying the parameters and of gaining
knowledge of the wave tank characteristics, an essential part of any
experimental endeavor. Concurrently, the scale-model relationships
presented in Chapter IV were developed, and the resulting distorted
profile was remolded in the tank.
Following the establishment of the new model law, over 16 attempts
were made at model verification using an irregular wave field. None
of these runs produced results which satisfactorily fulfilled the re-
quirements for verification. An example is shown in Figure 10.
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(S3H3NI) NOIIVATI]3 -13GOV4
As can be seen, the amount of dune recession is good, and the
resultant beach slope is nearly right; but the recurring problem was
that the erosion was not deep enough, and the sand was not being
transported offshore far enough.
Each of the input parameters was varied within reasonable limits
to see if the estimation of the storm characteristics was inaccurate.
None of these variations produced the desired result, although much
was learned in the process. After looking at possible causes for this
failure, from wave reflection to paddle response to spectral represen-
tations and so forth, it was finally decided that the problem was in
trying to use irregular waves in the model. The next run after this
conclusion was conducted using regular waves which gave an almost per-
fect verification of the model. Since it was only a trial, control
was not very strict, and consequently this run was not documented.
Knowing that verification of the model was possible with regular waves
makes the explanation of the failures using irregular waves much easier.
These arguments appear in the following section.
Irregular Versus Regular Waves
Ideally the operation of a small-scale wave model should include
a random-type wave field as would exist in nature. In deep water it is
possible to invoke the assumption of a Gaussian process in order to
derive a suitable power spectrum representation of the wave heights
and frequencies. However, as these waves propagate into shallow water,
they become altered by such effects as shoaling, refraction, and re-
flection, thus turning the wave field into a very non-Gaussian phenomenon
as individual waves begin to interact with each other in ways that are
still not understood. One of the main characteristics of shallow-water
spectra is the presence of phase grouping, sometimes known as surf
beat. This groupiness is a function of both space and time, with well
grouped waves at one location becoming less well defined as they pro-
pagate in time. This must certainly have an effect upon the erosion
of the beach and dune.
The narrow-band spectra generated for use in the model verifica-
tion were developed under the assumption of a Gaussian process since
research into shallow water waves is still in its infancy. And while
the random time series conformed to the statistical representation given
by the desired wave spectrum, there are an infinite number of different
time series which will also conform, none of which depict the wave
groupiness present in shallow-water random wave fields. This is the
major argument against using spectral wave representations in a small-
scale model until more is known about the resonance effect of nearshore
wave spectra, and it probably was the cause of the failure to verify
the model when spectral wave distributions were employed. This view
is also supported in the reviewer's comments presented in a report by
Jain and Kennedy (1979), where the statement is made that small-scale
movable-bed modeling should be restricted to regular waves until more
is known about nearshore random wave phenomena. Further confirmation
of the importance of wave grouping was given by Johnson et al. (1978)
who found that substantially more damage occurred to rubble mound struc-
tures for grouped waves than for nongrouped waves conforming to the
same power spectrum distribution.
To further complicate matters, it has been shown that the genera-
tion of random waves in model facilities has the tendency of generating
long "parasitic" waves because the boundary conditions for second
order wave effects are not satisfied at the wave paddle (Hansen and
Hebsgaard, 1978). This fact was observed in the verification process
on a strip chart recording of a random wave train, which illustrated
a parasitic long wave with a period between 15 and 20 seconds. These
long waves are reflected and continue to build in amplitude. Methods
exist which will eliminate these long waves (Hansen et al., 1980;
Hudspeth et al., 1978), but are primarily theoretical analyses which
are difficult to apply with the present equipment in the Air-Sea wave
For the above reasons, plus the fact that verification could be
reasonably achieved using regular waves, it was decided that the model
verification and ensuing experimental test series be conducted using
regular wave trains. Perhaps future developments will allow the use
of random waves in small-scale erosion tests.
Two documented verification runs were performed using regular
waves and reproducing nearly the same storm conditions. The second
run was required in order to demonstrate that the results of the first
run could be reproduced, and because a slight wave period variation
was observed near the end of the first run. Both tests are detailed
First Verification Run
The parameters for the first run are given in Table 3 with both
model and prototype values given.
FIRST VERIFICATION RUN
Wave Maximum Peak Time to Time at Time of
period wave surge reach peak peak surge
(sec) height level surge surge decrease
Prototype 11 8.5 ft. 8.3 ft. 12.4 hr. 1.9 hr. 5.2 hr.
Model 1.07 4.1 in. 4.0 in. 72 min. 11 min. 30 min.
The surge increase was approximately linear over the time span,
and the wave period was held constant throughout. The prototype wave
height of 8.5 feet gave the same wave energy density as a narrow-band
spectrum with a significant wave height of about 12 feet; and while
the surge level was on the increase, the wave height was gradually in-
creased until the peak value was reached at 38 minutes into the run,
or about when the surge was half its peak value. This value was main-
tained until the surge level began to decrease at which time the wave
height was again reduced. This process hopefully typifies what happens
in nature over the duration of a storm.
Figure 11 gives the measured profile taken down the wave flume
centerline at the end of the verification run. Profiles measured near
the glass wall and the back wall indicated that there was a slight
lateral variation in the eroded profile. However, the erosion which
occurred compared very favorably with the prototype, with the exception
of the extra dune recession. A standard posttest check of the wave
frequency revealed that the function generator knob had been bumped,
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causing an increased wave period (equivalent to 13 seconds in the
prototype) during the latter portions of the peak surge duration. This
was assumed to be the cause of the extra recession, a fact later con-
firmed by the experimental test series.
Second Verification Run
Table 4 gives the storm parameters for the second verification run
which proceeded in much the same manner as the first run.
SECOND VERIFICATION RUN
SWave Maximum Peak Time to Time at Time of
period wave surge reach peak peak surge
(sec) height level surge surge decrease
Prototype 11 8.5 ft. 8.3 ft. 12 hr. 1.4 hr. 5.2 hr.
Model 1.07 4.1 in. 4.0 in. 70 min. 8 min. 30 min.
As before, the surge rise was approximately linear with the wave
period being held constant throughout. The wave height was gradually
increased to its peak value which occurred at about 48 minutes into
the run, and the wave height was decreased as the surge level dropped,
signifying the passage of the storm.
This time there was little discernible lateral variation in the
eroded profile; and the eroded quantity and dune recession gave an
almost unbelievable reproduction of the prototype event as can be seen
in Figure 12, which gives the posttest profile measured on the flume
centerline. Near the glass, the profile was slightly elevated, but not
by very much. Figure 13a shows the profile at the end of the peak
surge duration, while Figure 13b was taken at the end of the surge
Several observations which provide some basic insight into the
erosion process were made during the verification:
1) As the surge level began to rise, there was little change in
the profile until the latter stages of the increase, when more
of the dune was exposed to the erosive wave action. This
points to the importance of the surge level in the erosion
2) The erosion that had occurred by the time the peak surge had
been reached represented between 80 to 90 percent of the final
erosion quantity. Thus it is seen that the most damage is
done during the surge rise and the time the surge remains at
its peak value.
3) The effect of lowering the surge level with decreased wave
energy had only a minor effect on the profile, the most
noticeable being the smoothing out of the profile as the water
level dropped. This was caused by the deposition of sediment
over the beachface and the transition from plunging breakers
back to spilling breakers during the surge decrease.
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13a: PROFILE AT END OF PEAK SURGE DURATION.
13b: PROFILE AT END OF SURGE LEVEL DECREASE.
FIGURE 13: ERODED BEACH-DUNE, RUN 2.
The verification of the proposed model law seems to have been
successful based upon the above results. The estimation of the storm
parameters was fairly reasonable in view of the limited data avail-
able, and the resultant eroded profiles obtained during the model
testing closely resemble the actual prototype profile in terms of re-
cession, eroded volume, and beachface slope. The offshore portion was
not as good, due to the fact that this area was altered by the mild
wave climate which prevailed between the end of the storm and the time
the prototype profile was surveyed. The subsequent experimental series
demonstrated that the erosion is most effected by surge level which
was probably the most accurate of the estimated parameters. Thus the
model would not be too sensitive to incorrect estimates in wave height,
as long as they were within reason.
Of course there is always a need for further verification since
the modeling laws have only been verified on one prototype event, but
this must wait until more prototype data become available. Until such
time, the derived small-scale movable-bed modeling relations presented
in Chapter IV represent suitable modeling laws for use in small-scale
investigations into dune erosion during severe storms. The relation-
ships are based on physics and observations and have been verified in
the laboratory by the reproduction of storm conditions to a degree of
realism never before attempted, including a variable surge hydrograph
and a varying wave height.
EXPERIMENTAL TEST SERIES
The experimental test series to investigate the role of the primary
factors involved in dune erosion by severe storms was designed in such
a way as to cover the range of the storm parameters expected to occur
during a hurricane making landfall on an undeveloped portion of the
Florida coastline. The order of the test runs was usually arranged so
that each successive run reproduced conditions more severe than the
previous run. In this way the regrading of the beach and dune was
optimized, and progressive patterns could be recognized early in the
test series in order to make changes in the parameters if necessary.
This chapter details the experimental test series, the parameters
tested, and the procedures followed.
The selected profile for the model series was chosen to be repre-
sentative of an average Florida beach. From an analysis by Hughes
(1978), it was found that a curve fit of the profile form
h = Ax2/3
on over 400 actual beach profiles from the Florida each coast and
panhandle indicated a modal value of A = 0.15 when both depth (h) and
range (x) are given in feet. Since these profiles probably exhibited
the characteristics of near-equilibrium beach profiles, it was decided
to adopt this equation for the below mean sea level portion of the
profile. This represents a better choice than a plane-sloped beach
since it is desirable to begin the model tests with at least near-
equilibrium conditions. In this way, the resultant changes would more
closely duplicate nature.
Modeling the profile equation using the previously derived rela-
tionship obtained by comparing equations (20) and (28), i.e.,
w = grain-size fall velocity,
p = prototype value, and
m = model value,
resulted in the equation
h 0.19 x 2/3 (30)
where both hm and xm have units of inches. Hughes also observed that
beyond a certain depth the offshore profile became distinctly steeper,
most probably a relic feature from the rising sea level over the past
4000 years (or from previous severe storms). An inspection of prototype
profiles indicated that this portion could be well represented by a
straight line with a prototype slope of 1:40 extending seaward from
about the 13 foot depth.
The prestorm beachface was represented by a straight line having
a prototype slope of 1:10 with a dune toe elevation of 7 feet above
mean sea level. These values were chosen after discussions with an
individual familiar with Florida beaches.* While slightly steeper than
the average beachface calculated for Bay and Walton Counties in the
Florida panhandle (Chiu, 1977), it was felt that the values used were
a good representation of Florida beaches on the whole.
The dune was chosen with an initial height of 20 feet above mean
sea level in the prototype with a dune face slope of 1:2. This slope
is approximately the angle of repose for seciments in the range of
grain sizes found in coastal dunes, and this slope was retained when
the dune height was increased later in the test series.
The same sediment size as was used during the verification tests
was used for the model series. Hence the prototype profile repre-
sents one with an effective grain-size of de = 0.27 mm. For this reason
all the scaling relationship values given for the model verification
in Table 2 still apply for the test series. Figure 14 gives the ini-
tial profile used for the experimental series along with the model
dimensions and slopes as determined by the scale-model relationships.
The storm parameters which were varied during the test series
were as follows:
1) Wave period (10, 12, and 14 seconds in the prototype);
2) Wave height, when measured before shoaling began in the tank
(6, 8, 10 feet in the prototype);
3) Peak surge level above mean sea level (8, 11, 14 feet in prototype);
*Dr. T.Y. Chiu, Department of Coastal and Oceanographic Engineering,
University of Florida, personal communication.
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4) Instantaneous surge level rise as opposed to finite-time
5) Surge level rise duration (4, 7, 15 hours in the prototype);
6) Prestorm dune height (20, 24, 28 feet in the prototype).
Regular waves were used throughout the entire experimental program
which began November 12, 1980, and ended January 20, 1981. The 38
experimental runs, involving a time-dependent surge level increase,
approximated the surge as a linear increase over time. Table 5 lists
both the model and prototype values for the parameters involved in
the test series.
The experimental test runs can be divided roughly into five dis-
tinct series, as noted on Table 5. Series 1 compared three different
wave heights at three different surge levels for a given wave period
and dune height. In addition, the first 8 runs of the series investi-
gated the effects of different time spans in bringing the surge level
from mean sea level to its peak value. The first run used an in-
stantaneous peak surge level in order to see if a finite surge is
even required to obtain the final equilibrium. Runs 36 and 37 of series
4 repeated this test using different surge levels and initial dune
Series 2 repeated series 1 at a different wave period, the only
difference being that the effect of surge level rise duration had
been established so further tests on this variation were not required.
Test series 3 was an abbreviated version of series 1 and 2 using
yet another wave period, but maintaining the same initial dune height
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