UFL/COEL94/016
APPLICATION OF EQUILIBRIUM BEACH PROFILE
CONCEPTS TO FLORIDA'S EAST COAST
by
Lynda L. Charles
Thesis
1994
APPLICATION OF
EQUILIBRIUM BEACH PROFILE CONCEPTS
TO FLORIDA'S EAST COAST
By
LYNDA L. CHARLES
A THESIS PRESENTED TO THE GRADUATE SCHOOL
,' OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
1994
ACKNOWLEDGEMENTS
First and foremost, I wish to express a great deal of thanks to my advisor and supervisory
committee chairman, Dr. Robert G. Dean, who provided much more than support and guidance.
Thanks also go to Florida Sea Grant for providing the financial support which made this project
possible and to the Department of Coastal and Oceanographic Engineering for providing matching
funds. I would also like to thank Dr. Daniel M. Hanes and Dr. Robert J. Thieke for their
participation as supervisory committee members.
The contribution of many people made the completion of this project possible. I am
especially indebted to those who at various times allowed me to drag them up and down the state
in quest of the ultimate data set. They include Rajesh Srinivas, Emre Otay Paul Work, Mark
Pirello, Victoria Jones, Chris Jette, David St. John, Suna Birsen, Mark and Scott who
participated in the onshore work and Mark Sutherland, Vic Adams, Don Mueller, Lee Townsend,
and Jon Grant for the offshore work.
Another group I must thank, are those who participated in the design, construction,
calibration, and operation of the rapid sediment analyzer (RSA). They include Sidney Schofield;
Chuck Broward, Vernon Sparkman, Danny Brown, Jim Joiner, and George Chappell.
Finally, I would like to thank Subarna Malakar for his assistance in the graphical
reduction of the, at times, almost unmanageable data set. As well, I would like to express
appreciation for the help of the office staff: Becky Hudson, Sandra Bivens, Lucy Hamm, Sonya
Brooks, HIlen Twedell, and John Davis.
TABLE OF CONTENTS
page
ACKNOWLEDGEMENTS ........................................ ii
LIST OF TABLES ............................................. v
LIST OF FIGURES ............................................ vi
ABSTRACT ................................................. xi
CHAPTERS
1 INTRODUCTION .......................................... 1
1.1 The Power Law Form of the Equation for an Equilibrium Beach Profile (EBP) 1
1.2 Scope of This Thesis ..................................... 4
2 BACKGROUND AND REVIEW ................................. 6
2.1 Curve fitting h =Ay" and h =Ay2 for invariant A ....... ........... 6
2.2 Varying A Along the Profile .................................. 9
2.3 The Gravity Term ...................................... 12
2.4 Application Beyond the Surf Zone ............ ................ 13
2.5 Motivation for Different m Values ............................ 13
2.6 Fitting Two Curves to the Nonmonotonic Beach Profile ............. 14
2.7 Tim e Varying A ....................................... 15
3 DATA COLLECTION AND STUDY AREA ......................... 21
3.1 Study Area .................................. ....... 21
3.2 Field Data Collection: Beach Profiles and Sediment Samples ........... 22
3.3 Lab Methods for Sediment Size Analysis ....................... 25
4 PROCEDURES FOLLOWED IN DATA ANALYSIS .................... 28
5 RESULTS AND DISCUSSION .................... ............... 29
5.1 Sediment Size Data ............. ......................... 29
5.2 Results from the "Blindfolded" Test on Beach Profile Predictability
using Sediment Size Data ............................... 30
5.3 Comparison Between EBP Calculations Performed With and Without the
Gravity Term ....................................... 34
5.4 Results from the Curve Fit Analysis of h=Ay" and h=Ay' for Invariant A 35
5.5 Motivation for Different m Values ............................ 37
5.6 Relationship Between A and Sediment Size ...................... 37
5.7 Application Beyond the Surf Zone and Dependence on Profile Depth ...... 38
5.8 Nonmonotonic Beach Profiles .............................. 40
5.9 Time Varying Beach Profiles ...... ......................... 40
CHAPTER 6 SUMMARY AND CONCLUSIONS ......................... 90
6.1 Sum m ary ............................................... 90
6.2 Conclusions ............................................. 91
APPENDIX DATA COLLECTION DATES FOR EACH PROFILE LOCATION ..... 92
REFERENCES .............................................. 100
BIOGRAPHICAL SKETCH ....................................... 103
LIST OF TABLES
Table page
31 Data Summary by County .................................... 23
32 Data Collection Dates for each Field Location Identified by the FDEP Range
Number  Nassau County, Florida .............................. 24
51 Median diameter of sediment samples collected at depths of 0.0, 0.9, 1.8,
3.7, 5.5, 7.3, and 9.1 meters averaged over the entire east coast of Florida
and averaged for each of the 12 eastcoast counties .................... 30
52 Root Mean Square (RMS) deviations of predicted equilibrium beach profiles
(EBP) from the measured beach profile for each measured profile extending out
to depths of 1.8, 3.7, 5.5, 7.3, and 9.1 meters; these individual RMS values
for each profile are averaged over the entire east coast of Florida and
averaged for each of the 12 eastcoast counties; EBP predictions based on
calculations (1) without the gravity term and (2) with the gravity term . ... 31
53 Root Mean Square (RMS) deviations of the average predicted equilibrium beach
profile (EBP) from the average measured beach profile extending out to depths
of 1.8, 3.7, 5.5, 7.3, and 9.1 meters; EBP predictions based on calculations
(1) without the gravity term and (2) with the gravity term. ............... 33
54 Averages, for the entire east coast of Florida and countybycounty, of bestfit
A and m parameters from curve fit analysis of h =Ay" to measured beach profiles
out to depths of 1.8, 3.7, 5.5, 7.3, and 9.1 meters ................... 36
55 Averages, for the entire east coast of Florida and countybycounty, of bestfit
A parameter for m=2/3 from curve fit analysis of h=Ay" to measured beach
profiles out to depths of 1.8, 3.7, 5.5, 7.3, and 9.1 meters . . . .. 36
LIST OF FIGURES
Figure paa
11 Map of Florida showing the 12 coastal counties on the east coast shaded ....... 5
21 Moore's Curve: scale parameter, A, versus sediment diameter, D, and fall
velocity, w, in EBP relationship h=Ay3 .......................... .17
22 Definition sketch showing Inman's compound curve fitting scheme where crosses
denote the origin for the barberm profile (dotted) and for the shorerise profile
(dashed) .. ... . . ... . . .. . . ... .. . 18
23 Seasonal beach changes as observed with Inman's compound curve approach comparing
mean summer (solid) with mean winter (dotted) profiles ....... .,....... 18
24 Fitted A and m parameters for Inman's barberm profiles plotted against available
beach face sediment size ...................................... 19
25 Schematic illustrating the decomposition of Pruszak's timevarying A parameter in
h=Ay23 ............................................... 20
26 Long term variation in A parameter for Pruszak's Baltic Sea data ........... 20
27 Short term variation of A parameter for Pruszak's Black Sea data ........... 20
31 Median sediment diameter from sieve analysis versus sediment median diameter
calculated from settling velocities using Gibbs' equation, all in phi units,
solid straight line is bestfit line defining calibration equation also shown at 
the top of the graph ........................................ 27
51 Variation in median sediment diameter, in millimeters, along the east coast of
Florida for all of the sediment samples analyzed ................... .. 42
52 Variation in median sediment diameter, in millimeters, along the east coast of
Florida for all of the sediment samples collected at the waterline ............ 43
53 Variation in median sediment diameter, in millimeters, along the east coast of
Florida for all of the sediment samples collected at 0.9 meters depth ......... 44
54 Variation in median sediment diameter, in millimeters, along the east coast of
Florida for all of the sediment samples collected at 1.8 meters depth . ... 45
55 Variation in median sediment diameter, in millimeters, along the east coast of
Florida for all of the sediment samples collected at 3.7 meters depth . ... 46
56 Variation in median sediment diameter, in millimeters, along the east coast of
Florida for all of the sediment samples collected at 5.5 meters depth . ... 47
57 Variation in median sediment diameter, in millimeters, along the east coast of
Florida for all of the sediment samples collected at 7.3 meters depth ......... 48
58 Variation in median sediment diameter, in millimeters, along the east coast of
Florida for all of the sediment samples collected at 9.1 meters depth ......... 49
59 Best performance of predicted profiles: (A) crossshore variation in sediment
median diameter, in mm; and (B) measured beach profile (solid line),
calculated equilibrium beach profile: (1) without the gravity term (dotted),
and (2) with the gravity term (dashed) for Range number 27 in Duval County,
Florida . . . . . . . . . . . . 50
510 Worst performance of predicted profiles: (A) crossshore variation in sediment
median diameter, in mm; and (B) measured beach profile (solid line),
calculated equilibrium beach profile: (1) without the gravity term (dotted),
and (2) with the gravity term (dashed) for Range number 1 in Broward
County, Florida .......................................... 51
511 Longshore distribution of RMS values based on cumulative deviations of the
calculated equilibrium beach profile: (1) without the gravity term (solid)
and (2) with the gravity term (dotted) from the measured profile out to 1.8.
meters depth ...................................... .. .... 52
512 Longshore distribution of RMS values based on cumulative deviations of the
calculated equilibrium beach profile: (1) without the gravity term (solid)
and (2) with the gravity term (dotted) from the measured profile out to 3.7
m eters depth ............................................ 53
513 Longshore distribution of RMS values based on cumulative deviations of the
calculated equilibrium beach profile: (1) without the gravity term (solid)
and (2) with the gravity term (dotted) from the measured profile out to
5.5 m eters depth .......................................... 54
514 Longshore distribution of RMS values based on cumulative deviations of the
calculated equilibrium beach profile: (1) without the gravity term (solid)
and (2) with the gravity term (dotted) from the measured profile out to
7.3 m eters depth .......................................... 55
515 Longshore distribution of RMS values based on cumulative deviations of the
calculated equilibrium beach profile: (1) without the gravity term (solid)
and (2) with the gravity term (dotted) from the measured profile out to
9.1 meters depth ............................... ....... ... 56
516 Average measured profile (solid line), and average calculated equilibrium
beach profile: (1) without the gravity term (dotted) and (2) with the gravity
term (dashed) for the entire east coast of Florida . . ... . .. 57
517 Average measured profile (solid line), and average calculated equilibrium
beach profile: (1) without the gravity term (dotted) and (2) with the gravity
term (dashed), and maximum and minimum measured depths (thick solid line)
for Nassau County, Florida .................................... 58
518 Average measured profile (solid line), and average calculated equilibrium
beach profile: (1) without the gravity term (dotted) and (2) with the gravity
term (dashed), and maximum and minimum measured depths (thick solid line)
for Duval County, Florida .................................... 59
519 Average measured profile (solid line), and average calculated equilibrium
beach profile: (1) without the gravity term (dotted) and (2) with the gravity
term (dashed), and maximum and minimum measured depths (thick solid line)
for St. Johns County, Florida .................................. 60
520 Average measured profile (solid line), and average calculated equilibrium
beach profile: (1) without the gravity term (dotted) and (2) with the gravity
term (dashed), and maximum and minimum measured depths (thick solid line)
for Flagler County, Florida ................................... 61
521 Average measured profile (solid line), and average calculated equilibrium beach
profile: (1) without the gravity term (dotted) and (2) with the gravity term
(dashed), and maximum and minimum measured depths (thick solid line) for
Volusia County, Florida ............................ ......... .. 62
522 Average measured profile (solid line), and average calculated equilibrium
beach profile: (1) without the gravity term (dotted) and (2) with the gravity
term (dashed), and maximum and minimum measured depths (thick solid line)
for Brevard County, Florida .................................. 63
523 Average measured profile (solid line), and average calculated equilibrium
beach profile: (1) without the gravity term (dotted) and (2) with the gravity
term (dashed), and maximum and minimum measured depths (thick solid line)
for Indian River County, Florida ................................. 64
524 Average measured profile (solid line), and average calculated equilibrium
beach profile: (1) without the gravity term (dotted) and (2) with the gravity
term (dashed), and maximum and minimum measured depths (thick solid line)
for St. Lucie County, Florida ................................... 65
525 A.erage measured profile (solid line), and average calculated equilibrium
beach profile: (1) without the gravity term (dotted) and (2) with the gravity
term (dashed), and maximum and minimum measured depths (thick solid line)
for M artin County, Florida ................................... 66
526 Average measured profile (solid line), and average calculated equilibrium
beach profile: (1) without the gravity term (dotted) and (2) with the gravity
term (dashed), and maximum and minimum measured depths (thick solid line)
for Palm Beach County, Florida ................................ 67
527 Average measured profile (solid line), and average calculated equilibrium
beach profile: (1) without the gravity term (dotted) and (2) with the gravity
term (dashed), and maximum and minimum measured depths (thick solid line)
for Broward County, Florida .................................. 68
528 Average measured profile (solid line), and average calculated equilibrium
beach profile: (1) without the gravity term (dotted) and (2) with the gravity
term (dashed), and maximum and minimum measured depths (thick solid line)
for Dade County, Florida .................................... 69
529 Generalized representation of the relationship between the EBP calculation:
(1) without the gravity term (dotted), and (2) with the gravity term (dashed);
the gravity term alone is also shown (solid) ................... ... .70
530 Longshore variation in bestfit A (solid line) and m (solid line with symbols)
parameters as well as A (dotted line) for m=2/3 from curve fit analysis of h =Ayf
to measured beach profiles out to a depth of 1.8 meters ................. 71
531 Longshore variation in best fit A (solid line) and m (solid line with symbols)
parameters as well as A (dotted line) for m =2/3 from curve fit analysis of h =Ayf
to measured beach profiles out to a depth of 3.7 meters ................. 72
532 Longshore variation in bestfit A (solid line) and m (solid line with symbols)
parameters as well as A (dotted line) for m=2/3 from curve fit analysis of h=Ay"
to measured beach profiles out to a depth of 5.5 meters ........... ......... 73
533 Longshore variation in bestfit A (solid line) and m (solid line with symbols)
parameters as well as A (dotted line) for m=2/3 from curve fit analysis of h=Af
to measured beach profiles out to a depth of 7.3 meters ................. 74
534 Longshore variation in bestfit A (solid line) and m (solid line with symbols)
parameters as well as A (dotted line) for m= 2/3 from curve fit analysis of h =Ay"
to measured beach profiles out to a depth of 9.1 meters ................. 75
535 Variation of A parameter with median sediment size. A parameter is bestfit
value to depths of 1.8,3.7, 5.5, 7.3, and 9.1 meters. Sediment size is average
of all samples which extend from the waterline to each of the designated depths 76
536 Scatter plot of average county A values out to 1.8 meters depth versus sediment
site averaged over all profiles in county and over all landward samples in each
profile . . . . . . . . . . . . 77
537 Scatter plot of average county A values out to 3.7 meters depth versus sediment
size averaged over all profiles in county and over all landward samples in each
profile . . . .. . . . . . . . . 78
538 Scatter plot of average county A values out to 5.5 meters depth versus sediment
size averaged over all profiles in county and over all landward samples in
each profile ............................................. 79
539 Scatter plot of average county A values out to 7.3 meters depth versus sediment
size averaged over all profiles in county and over all landward samples in
each profile ............................................. 80
540 Scatter plot of average county A values out to 9.1 meters depth versus sediment
size averaged over all profiles in county and over all landward samples in
each profile ......... ....... .. ......... ... ........ .. .. 81
541 Plots of average county A values out to 1.8 meters depth and sediment sizes
averaged out to that depth. County code denotes the county sequence
commencing from the north on Florida's east coast . . . ........ 82
542 Plots of average county A values out to 3.7 meters depth and sediment sizes
averaged out to that depth. County code denotes the county sequence
commencing from the north on Florida's east coast ................. .. .83
543 Plots of average county A values out to 5.5 meters depth and sediment sizes
averaged out to that depth. County code denotes the county sequence
commencing from the north on Florida's east coast .................. 84
544 Plots of average county A values out to 7.3 meters depth and sediment sizes
averaged out to that depth. County code denotes the county sequence
commencing from the north on Florida's east coast ............... 85
545 Plots of average county A values out to 9.1 meters depth and sediment sizes
averaged out to that depth. County code denotes the county sequence
commencing from the north on Florida's east coast ... .. . . . 86
546 Illustration of effect of a seaward displacement of the equilibrium profile,
possibly by an offshore reef ................................... 87
547 Comparison of measured and predicted profiles for Range R15 in Indian
River County. Note that the measured profile appears to be displaced seaward
by a distance of approximately 200 meters .................. .. 88
548 Comparison of measured and predicted profiles for Range R15 in St. Lucie
County. Note that the measured profile appears to be displaced seaward by
a distance of approximately 600 meters .......................... 89
Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science
APPLICATION OF
EQUILIBRIUM BEACH PROFILE CONCEPTS
TO FLORIDA'S EAST COAST
By
Lynda L. Charles
December 1994
Chairman: Dr. Robert G. Dean
Major Department: Coastal and Oceanographic Engineering
This thesis aims to test and evaluate the latest modifications to Dean's' equlibrium beach
profile concept and to evaluate Moore's empirical relationship between grain size and A. Toward
this end the data set collected, analyzed, compiled, and examined in this report is considerably
more extensive than those employed in previous works.
This report examines data from Florida's sandy east coast which is comprised of 12
counties and encompasses 531 kilometers of coastline. A total of 1834 sediment samples was
collected for this study. Where possible, 9 surfacesediment samples were collected along each
of 207 profiles at depths of 0.0, 0.9, 1.8, 3.7, 5.5, 7.3, and 9.1 meters as well as a dune and
berm saniple. The spacing between these 207 profiles averages 2.7 kilometers. The profile
locations extend from the St. Marys River on the GeorgiaFlorida border (Nassau County) to
Government Cut in Miami (Dade County).
Results indicate that, on average, Dean's equilibrium beach profile equation of the form
h=Ay23, which is computed solely from sediment size information, adequately predicts the
measured beach profile from the east coast of Florida out to a depth of approximately 4 meters.
The depths of the computed profile between 4 and 9 meters depths, however, are under predicted
by this method. As this methodology was based on Moore's empirical relationship between the
A parameter and sediment size, no change was recommended for this relationship in the sand size
range. Also, the results indicate a regional dependence in the relationship between sediment size
and A.
CHAPTER 1
INTRODUCTION
1.1 The Power Law Form of the Equation for an Equilibrium Beach Profile (EBP)
Modelling of nearshore processes, and profile or shoreline response to these processes,
generally requires some quantitative depiction of beach profile shape. Specifically, shoreline
position and sand volume calculations for beach nourishment purposes have been found to be
rather sensitive to the profile shape selected for these computations. Although various equations
have been proposed, the most widely used in numerous aspects of coastal engineering research
and project design is a power law form of the equation for an equilibrium beach profile (EBP)
empirically developed and then theoretically motivated by Bruun (1954), Dean (1977), and
Kaihatu (1990). This equation, in its simplest and most familiar form, is
h=Aym (1.1)
where h is water depth at a seaward distance y from the shoreline, and m and A are coefficients
related to wave climate and sediment stability characteristics', respectively. Both of these coeffi7
clients have been determined by empirical, as well as by various theoretical, means. The
empirical method (Dean, 1977) involves application of least squares, to find the A and m values
of the EBP which provide the bestfit to the actual measured profile. The theoretical development
which has been the most popular was proposed by Dean (1977) and is based on the consideration
of uniform wave energy dissipation per unit water volume across the surf zone for a beach profile
in equilibrium.
1. Sediment stability characteristics include grain size, density, shape, and surface texture, or fall
velocity which is a measure which incorporates all of these properties.
1
2
In this particular development by Dean, the governing equation in terms of energy
conservation is
hDeq (1.2)
where, Deq, is the wave energy dissipation per unit water volume for the equilibrium case and the
energy flux, F, is obtained from shallow water linear wave theory as
F= pgI2gf (1.3)
Assuming a spilling breaker characterized by wave height, H= h, Dean (1977) obtained a value
of 2/3 for m, so that (1.1) becomes
h=Ay < (1.4)
and A is given by
3/2
[24 Dj (1.5)
[5pg3/2 K 2
where g is gravitational acceleration, p is the density of water, K is the breaking index2, and D,
can be expressed as
Deq=[. K 2 h2 (1.6)
From (1.5) note that A is related to wave energy dissipation, D., which in turn is related
to profile'slope as shown in (1.6).
Dean develops this concept further, introducing sediment size into these relationships by
postulating that Deq represents the wave energy dissipation rate per unit water volume under
2. The breaking index is taken as a constant, usually 0.78 (see McCowan, 1891).
3
which a sediment particle of a certain size3 is stable. The utility associated with this particular
relationship is that it provides this equation with a predictive capability if grain size data are
available. Later, using data compiled from the existing literature, Moore (1982) developed an
empirical relationship between A and grain size, which allows profile shape to be determined
simply by grain size, thereby avoiding the complexities of wave energy dissipation.
Originally, the relationship between grainsize and the A parameter allowed for only a
single grain size for the entire profile. Most recently, however, attempts have been made to
account for the natural crossshore variation in sediment size along the profile (Work and Dean,
1991; Larson, 1991; Dean et al., 1993; Duncan, 1993).
In response to these recent modifications to the EBP concept, some studies have indicated
a need to revise Moore's empirical relationship (Stockberger, 1989; Madalon, 1990; Kriebel et
al., 1991). Others have contended that a relationship between the A parameter and grain size
does not exist (Pilkey et al., 1993). Whatever the case, evaluating the relationship between the
A parameter and grainsize is not a trivial task. This is primarily due to the fact that the data
collection process necessary to evaluate such a relationship (including beach profile surveying,
sediment sample collecting, and grainsize analysis) is extremely labor intensive.
Although published profile and sediment size data are available, they are rarely both
available for the same location and time. When they do occur together, there is generally no
standard method of representing sediment data, thus, making the compiled data sets difficult to
compare and therefore unsuitable for testing, evaluating, and if necessary, revising Moore's
empirical relationship.
3. Or other stability characteristic.
1.2 Scope of This Thesis
This thesis aims to test and evaluate the latest modifications to the EBP concept, and to
evaluate Moore's empirical relationship between grain size and A. Toward this end the data set
collected, analyzed, compiled, and examined in this report is considerably more extensive than
those employed in previous works.
This report examines data from Florida's sandy east coast which is comprised of 12
counties and encompasses 531 kilometers of coastline, see Figure 11. The Florida Department
of Natural Resources (DNR) maintains 1753 monuments in this area for which extensive profile
data are available. At each of these monument locations, both long and shortterm shoreline
change rates have been established by Grant (1992). Shoreline stability information is useful to
any field study involving equilibrium beach concepts. In addition to the DNR survey data, which
extends out to a depth of at least 10 meters, 155 profiles were surveyed to deaths of approximate
ly 4 meters specifically for this study to correspond with the sediment sampling efforts.
A total of 1834 sediment samples was collected for this study. Where possible, 9 surface
sediment samples were collected along each of 207 profiles at depths of 0.0, 0.9, 1.8, 3.7, 5.5,
7.3, and 9.1 meters as well as a dune and berm sample. The spacing between these 207 profiles
averages 2.7 kilometers. The profile locations extend from the St. Marys River on the Georgfa
Florida border (Nassau County) to Government Cut in Miami (Dade County). The particular
DNR monuments studied in this thesis are presented by county in the Appendix along with tie
dates the profiles were surveyed and the dates each sediment sample was collected. DNR
provides maps of all their monument locations for all 12 eastcoast counties (Penquite et a.,
1983). Sediment size analyses were performed using both the sieving and settling techniques.
*>,
I I
Duval
St Johns
S Flagler
Volusia
. Brv8ard
SLLudc
Palm Beach
lBroward
Dade
Figure 11. Map of Florida showing the 12 coastal counties on the east coast shaded.
CHAPTER 2
BACKGROUND AND REVIEW
2.1 Curve fitting h =Av and h =Av/3 for invariant A
The original empirical studies determined the m and A parameters which provided the best
fit to the measured beach profile data. The earliest of these studies include a study by Bruun'
(1954) and a study by Dean2 (1977). Bruun found that the best fit to the data was obtained with
m=2/3 for profiles extending out to at least a 30 foot depth, but that for profiles extending to
depths greater than 40 or 50 feet, a better approximation of the data occurs if m=1/2. Dean
obtained as an average value, m=0.66, for all of the 502 profiles he examined; his group
averages, however, ranged from 0.52 to 0.82.
Currently, the most widely used value is m=2/3. Boone and Green (1988), however;
proposed m =0.5 as a more appropriate choice for the typical carbonate beaches of the Caribbean
as it provides a better fit to that particular data.
A straightforward least squares method can be applied to determine the m and A
parameters which provide the best approximation to the measured data. Equation (1.1) is first
linearized by taking the log of each side of the equation
logh= logA +mlogy (2. 1)
and then substituting H = log h
C = logA
Y= log Y
1. Bruun (1954) examined 25 profiles from Mission Bay, CA, and 30 profiles from the Danish
North Sea Coast.
2. Dean (1977) examined 502 profiles along the U.S. Atlantic and Gulf coasts from Long Island,
NY, to Galveston, TX.
to obtain
H=C+mY
which is linear in H and Y. For (2.2) the normal equations are
N N
NC+mE (Y,)=E (H,)
i=1 i1l
N N N
CE (Yc) +m (Y2) i(YH)
i=l i=1 i=1
where
(2.2)
(2.3)
(2.4)
N = number of data points
Solving (2.3) and (2.4) for C and m and replacing C with log A yields the following expressions
for A and m
N N N N
EH,E (Y2)E(YYi) (YH,)
A=exp il t= i=i i=1
Nii
NE ( Y2) E(Yi)
i= 1 i= 1
N N N
NE (YiHi) (Y.) (Hi)
i=1 i=1 i 2
N
m= 
NE (Y}) (Y,)
iil i*
(2.5)
(2.6)
For the case with, m= 0.67, the calculation simplifies and A becomes
N N
EH,0.67E Yi (2.7)
A= i=2 i1
N
Following the above procedure for m=0.67, Moore (1982) compiled and analyzed data
from the available literature. He located data from 38 profiles3 with known sediment sizes
ranging from 0.1 to 300 millimeters. He plotted the resulting A value for each profile against
its corresponding sediment size and drew a smooth curve through the data points. Moore's
empirical relationship between A and sediment size is commonly referred to as Moore's curve
which is presented as the solid line in Figure 21.
Moore's curve has been used extensively as it provides a predictive capability to the EBP
calculation. Dean (1987) converted the grainsize values in Moore's curve to fall velocities and
found the data could be well described by the loglinear relationship
A=0.067w0. (2.8)
where w is the fall velocity in centimeters per second and the dimensions of A are meters to the
onethird power. This relationship is shown as the dashed line in Figure 21. Since sediment
data are usually presented as grain size4, Dean's relationship is not as commonly used as
Moore's unless the data are converted to settling velocities.
3. Of the, 38 profiles Moore studied, 10 came from lab experiments and 28 from field
investigations. The size data were given usually as a:median diameter but occasionally as a size
range for each profile.
4. Even if sediment is analyzed using a settling tube or rapid sediment analyzer (RSA) which
directly provides the distribution of sediment settling velocities, as a matter of convention, the
data is almost always converted to size data which is representative of the data obtained directly
from the standard sieving technique.
9
Thus, for a beach of known sediment size, the relationships presented in Figure 2.1
provide an A value which can then be used in equation (1.2) to calculate a predicted EBP. Then,
if profile data are available, this calculated EBP can be compared to the actual measured
profile. Interpretations of these comparisons involve discussions on the state of equilibrium (or
disequilibrium) of the actual profile and whether there may be an excess or deficit of sand in the
profile (Dean et al., 1993). In the case of beach nourishment, this method is applied in
volumetric calculations and is especially useful when the fill material has a grainsize distribution
(GSD) which differs from that of the native beach which is to be filled. Also, in the case of
modeling dynamic profile response to various nearshore conditions, this EBP concept has been
found useful in providing either a target profile for which to aspire or a control profile with
which to base comparisons.
At this point, the methodology thus explained allows only one vatie of grainsize to
represent the sediment characteristics for the entire profile. Usually the median diameter (D50)
is used however the mean diameter has also been used as well as an equivalent value (either mean
or median) of settling velocity. It is well known, however, that the GSD varies in the crossshore
direction along the profile on natural beaches. If grainsize data are available from several
sample locations along the profile, an averaging technique (Hughes, 1978) may be used to obtain
a single composite GSD for the entire profile from which the composite D50 may be obtained.
This procedure, however, has not been widely used.
2.2 Varving A Along the Profile
Recently, there have been attempts to account for the variations in sediment characteris
tics which naturally occur along the profile by allowing A also to vary along the profile.
Referring to the original derivation by Dean (1977), the governing equation is
d( pgK2h gf) (2
=hD
dy = eq
Taking the derivative of (2.9) yields the common differential form of (1.2) as follows
Sh32 _24 Deq =A32 (2.
a 5 pgo2
Work and Dean (1991) examined a linear and exponential form for A given respectively as
A(y)=A + ny
A(y)=Aeky
.9)
10)
(2.11)
(2.12)
By separately substituting (2.11) and (2.12) into (2.10) and integrating, respectively, the
following EBP equations are obtained:
h(y)= [ [A(Ao+nY)s]n (2.13)
h(y) =A. 2 1
(2.14)
where n and k are the coefficients which provide the bestfit to the actual measured profile using
an iterative scheme (Work and Dean, 1991). This technique is not predictive, however, unless
n and k are known quantities.
11
The relationship between A and D50 (or w50) here again provides the method with
predictive capability. With D50 (or w50) data from samples collected at various locations along
the profile, A values along the profile can be determined using the relationships presented in
Figure 21.
By considering A to be locally constant, say between y, < y < y,+;, we can integrate
(2.10) to obtain
h32 = A3y + C (2.15)
for which the constant of integration can be found by using the condition that at y,, h =h,. This
results in the final equation
h(y)=[h.2 +A n(y _y.)]2 (2.16)
where A, is the constant A value between y, and y,+,.
This particular method, which allows A to vary with grain size, was introduced by Dean
(1991) and is applied to data from Florida and the Caribbean by Work and Dean (1991)5. A
modified version of this method was also used by Dean et al. (1993)6 and Duncan (19937) t6
examine data from New Zealand. Another modified approach, which also incorporates a varying
sediment size along the profile, was introduced by Larson (1991) using the same data as Work
and Dean (1991).
5. Work et al. (1991) examined data from four profiles in Florida (two from Escambia County
on the westernmost Panhandle; one from Pinellas County on the peninsular Gulf coast; and one
from Martin County on the Atlantic Coast) and two profiles in the Caribbean (one from Mullet
Bay, Sint Maarten, and one from Baie Rouge, St. Martin).
6. Dean et al. (1993) examined 10 profiles from New Zealand.
7. Duncan (1993) studied one profile from New Zealand for which sediment data from 22
locations along the profile were available.
2.3 The Gravity Term
This most recent modification to the EBP concept, introduced by Dean et al. (1993), is
basically the addition of a second term into the equation. It corrects a problem inherent in the
original equation which is the prediction of a vertical slope at the shoreline where y= 0. This
added term is referred to as the gravity term.
Dean hypothesizes that gravity acts through the beach slope as a destructive force in
addition to wave energy dissipation. Thus, the governing equation in terms of energy
conservation is expressed as
1d pg dh (2.17)
D q = dy +Bg'dh
S dy dy
where B is an unknown constant.
Taking the derivative of (2.16) yields the following differential form
p5 gJ h3+Bg a=D (2.18)
24 dy ay
integrating and substituting for A yields
hBg ha3 (2.19)
Deq A32
In their application, Dean et al. (1993) substituted the measured beach face slope, BSL, for
DW/Bg in (2.18) and obtained the following differential form
dh=[ 1 3 hI2 (2.20)
dy BSL f2A 
13
which was then applied using the following simple Euler forward difference scheme to calculate
the predicted beach profile:
h(y,,) =h(y,) + Ay (2.21)
2.4 Application Beyond the Surf Zone
In most engineering design applications, information regarding the total profile extending
from the shoreline to the limiting depth of motion is required. Although the original theoretical
developments are applicable only within the surf zone, the EBP equation in various modified
forms as discussed above has been applied out to these depths when profile shape information is
necessary. Hughes (1978) applied equation (1.2) to both long and shortened profiles from
Florida's northeastern coast and the southeastern shoreline of Lake Michigan and found
significant differences in the A values obtained. One exception in Hughes study was in Brevard
County, Florida; he suggests the closure depth may extend out further for this particular county.
2.5 Motivation for Different m Values
The theoretical basis by Dean (1977) for m=2/3 in (1.1) was briefly reviewed in Chapter
1 of this thesis. As well, it was pointed out that empirical fits to the data yield a similar value
for m on average; although group averages may range from 0.52 to 0.82 (Dean, 1977). Dean
(1977) also discusses two additional theoretical motivations for a value of 2/5 for m. In these two
theoretical developments, Dean considers the primary destructive force involved in molding the
beach profile to be due to (1) uniform alongshore shear stress, and (2) uniform wave energy
14
dissipation per unit surface area. Another theoretical development proposed by Kaihatu (1990)
yields a value of 14/17 (or 0.82) for m.
Kaihatu (1990) follows Deans' theoretical development of uniform wave energy
dissipation per unit water volume, but incorporates a spectral breaking model instead of Dean's
fixed break point assumption. Kaihatu employed the simple random wave dissipation model of
Thornton and Guza (1983, equation 27).
Based on field evidence in Australia8, Kotvojs and Fried (1991) divided beaches into 3
groups: (1) reflective, (2) dissipative, and (3) intermediate. They obtained the following average
values for m from a best fit analysis: (1) m=0.49 for reflective, (2) m=0.83 for dissipative, and
(3) m=0.65 for intermediate.
2.6 Fitting Two Curves to the Nonmonotonic Beach Profile
Using an iterative technique, Inman et al. (1993) fit field data9 with two curves of the
form h =Aym. One curve, the barberm profile extends from the berm crest to the breakpointbar.
The other profile, the shorerise profile extends from the breakpointbar to at least a 12 meter
depth. Figure 22 shows both curves fitted to a typical profile; notice that the origin of the
shorerise profile is at mean sea level (MSL) whereas the origin of the barberm profile averages
1.4 meters above MSL. This particular compoundcurve fitting approach removes the monotonic
restriction of the single curve fit as well as removing the vertical slope at the water line. In the
case of multiple offshore bars, the shorerise profile begins at the outer bar.
8. Kotvojs and Fried (1991) studied 25 profiles from reflective beaches in New South Wales, 957
profiles (9 transects over a nine year period) from intermediate beaches in the Narrabeen, and
12 profiles from dissipative beaches around Goolwa in South Australia.
9. Inman et al. (1993) examined 51 profiles obtained from 8 range lines in San Diego, CA,
collected over a 40year period. This data was also supplemented with a number of profiles from
Torrey Pines, CA; Duck, NC; and the Nile Delta, Egypt.
15
In this study by Inman et al. (1993), the best fit to the data yields an m value generally
around 2/5 for both profiles. Distinct seasonal variations in the profiles were observed as simple
displacements of the two curves as the offshore bar moved farther offshore in the winter (Figure
23); the A value also tended to increase in winter.
One of the most interesting findings in this study was the relationship with A in the
shorerise profile and average beachface sand size (Figure 24). Unfortunately, sediment size
information was limited for this particular data set. Figure 24 clearly shows, however, that a
relationship exists at least for this particular data set using Inman's approach.
2.7 Time Varving A
An interesting field study by Pruszak'1 (1993) proposed a timevarying A expressed as
A(t)= A+A+A+A' ... (2.22)
where
A = 5(sediment size, shoreline geomorphology)
A, = .(longterm cyclic changes, e.g., in sea level, sediment supply, etc.)
A2 = F(mesoterm cyclic changes, e.g., seasonal changes in wave conditions, etc.)
A' = 9(shortterm random changes, e.g., single storm events, etc.)
10. Pruszak studied profile data collected from Lubiatowo, Poland, on the Baltic Sea and data
from Gold Beach, Bulgaria, on the Black Sea. The Baltic Sea data includes 20 range lines spaced
150 meters apart for which data was collected periodically from 1964 to 1991. The Black Sea
data consists of monthly surveys of one range line from September of 1972 through June of 1978.
Pruszak specified A(t) more specifically as
A(t)=A+ acos 27r l +0 +az cos 27rt +42 +A (2.23)
where a,, T,, 4, a2, 2, 02 in the Fourier series correspond to the amplitude, period, and phase
of the longterm, A1, and mesoterm, A2, cyclic changes, respectively. Figure 25 illustrates this
decomposition of A(t).
Pruszak fit h=A(t)y2 to data from the Baltic Sea after simplifying A(t) as
A(t)= A+A= A+ a cos (wt+ ) (2.24)
where t is time in years. For this data set, which ranged from 1964 to 1991, Pruszak obtained
the following values for the parameters in (2.24)
A = 0.075 [m"3]
a, = 0.022 [mi3]
wl = 0.230 [yr']
Thus, the longterm profile changes have a period of approximately 25 to 30 years. The
associated long term variation in A is presented in Figure 26.
Figure 27 shows the time variation in A which Pruszak obtained after fitting h =Ay/ to
profile data from the Black Sea. The Black Sea data were collected at a higher frequency
(monthly) and over a smaller duration (5 years) than the Baltic Sea data. Figure 27 shows'that
for this data, the A2 component can be identified. For this Black Sea data, Pruszak does not give
any calculated values for each component.
I
SEDIMENT FALL VELOCITY, w (cm/s)
SEDIMENT SIZE, D (mm)
Figure 2,1. Moore's Curve: scale parameter, A, versus sediment diameter, D, and fall velocity,
w, in EBP relationship h =AyZ (Dean, 1987, modified from Moore, 1982).
'
Figure 22. Definition sketch showing Inman's compound curve fitting scheme where crosses
denote the origin for the barberm profile (dotted) and for the shorerise profile (dashed) (Inman
et al., 1993).
4 2W 7
Basic Data Set *2XS' I 5
A 1W
MSL o
1 I
Distance, meters
Figure 23. Seasonal beach changes as observed with Inman's compound curve approach
comparing mean summer (solid) with mean winter (dotted) profiles (Inman et al., 1993).
3
2
D
I0**
S 
SN
TPeD
0
TP
.5
.4
*N
.3
1.2
0.5 
0.3 
100
1
Diameter, upm
Figure 24. Fitted A and m parameters for Inman's barberm profiles plotted against available
beach face sediment size (Inman et al., 1993).
TP *.
*N
u.I
A(t) 20
A, A
time
t
Figure 25. Schematic illustrating the decomposition of Pruszak's timevarying A parameter in
h=Ay23 (Pruszak, 1993).
Al(1)
A(tl A A, = o acos tt
0.12 y A().x =0.075.0022 cos(0.23t)
0.10
0 0.075
960 1964 1968 1972 1976 1980 1984 1988 1992 t
Figure 26. Long term variation in A parameter for Pruszak's Baltic Sea data (Pruszak, 1993).
025 A)t
0.15 ~
0,10
005* I I I '* I x a r X
.Pr72 1973 1974 1975 1976 1977 1978
Figure 27. Short term variation of A parameter for Pruszak's Black Sea data ruszak, 1993).
Figure 27. Short term variation of A parameter for Pruszak's Black Sea data (Pruszak, 1993).
CHAPTER 3
DATA COLLECTION AND STUDY AREA
3.1 Study Area
The study area includes all 12 coastal counties on the east coast of Florida (Table 31),
spanning some 531 kilometers from the St. Marys River entrance at the GeorgiaFlorida border
to Government Cut in Miami, Florida (Figure 11). This segment of the Atlantic Coastline is
exposed to moderate to high wave energy levels and is characterized by numerous sandy barrier
islands separated by 17 tidal inlets. Fringing coral reefs occur in shallow water along the
coastline in various areas primarily south of Cape Canaveral. In the northeast, linear shore
parallel shoals have been documented as well as arcuate capeassociated shoals just offshore froni
Cape Canaveral (Meisburger and Field, 1975).
For the most part, the sediment forming the eastcoast beaches is merely a thin veneer
(geologically speaking) spread over a Pleistocene limestone core (the Anastasia Formation). This
type of barrier island is called a "perched" barrier island by some coastal geologists. The
limestone core which parallels the Florida coastline is believed to have been a carbonate ridge
at one time during the Pleistocene (Tanner, 1964). This Pleistocene core acts to stabilize the
shoreline with a smooth rather than irregular configuration. It is exposed in various places along
the coast, especially in Flagler and Palm Beach Counties.
The northeastern beaches are made up of relatively fine siliceous sand and are primarily
rather wide and gently sloping. Flagler county is an exception, however. In many areas of this
county, the beach is relatively narrow and steep while the sediment exhibits an increase in
coquina shell material derived from the outcropping Anastasia Formation. Similar to the Flagler
22
County beaches, the southeastern beaches are generally narrow and steep. In this region, as well,
the beach sediment is dominated by highly permeable shell mixtures.
In the Southeast, tidal inlets are more numerous than in the northeast. Stabilization of
these inlets for navigational purposes has resulted in severe erosion at some downdrift locations.
3.2 Field Data Collection: Beach Profiles and Sediment Samples
Along Florida's entire coastline there exists a series of permanently located baseline
monuments which serve as benchmarks for beach profile surveys. These monuments are spaced
305 meters apart on average and are referenced with the state's grid system'. The monuments
were installed and are maintained by the Florida Department of Natural Resources (DNR) as an
integral part of their effort to establish and regulate a Coastal Construction Control Line (CCCL)
for the sandy beaches in Florida. Since the 1970's, when the first CCCL line was established,
the DNR has periodically surveyed the beaches and offshore regions of Florida on a countyby
county basis. The DNR records survey data and benchmark elevations in feet relative to
NGVD2 and profile azimuths in degrees relative to magnetic North. From each benchmark,
beach profiles are surveyed using a standard rodandlevel procedure out to "wading" depths
averaging 1.5 meters. On every third monument, the DNR extends the surveys out to at least
1000 meters offshore which corresponds todepths greater than 10 meters on the east coast.
These offshore surveys are performed using a survey vessel equipped with a fathometer and
electronic positioning system. Currently, this is the most widely used method of surveying the
beach profile. This type of surveying yields beach profile data with a vertical accuracy of 2 to
1. This grid system is the Florida State Plane Coordinate System.
2. NGVD represents the National Geodetic Vertical Datum which was formerly called the 1929
Mean Sea Level (MSL) Datum and currently corresponds to a level somewhere between MSL
and MLW (mean low water) due to the modern trend of rising sea level.
23
3 centimeters for the rodandlevel procedure and a vertical accuracy in the range of 15 tb 20
centimeters for the offshore portion (Otay et. al, 1994).
As mentioned previously, this report examines data from Florida's east coast which is
comprised of 12 counties and encompasses 531 kilometers of coastline. DNR maintains 1753
monuments in this area for which extensive profile data are available. In addition to DNR survey
data, 155 profiles were surveyed to "swimming" depths of approximately 4 meters specifically
for this study. Sediment samples were also collected from 207 monument locations correspond
ing to approximately every ninth DNR survey. This information is summarized in Table 31.
Table 31. Data Summary by County.
kilometers total number of number of number of profiles
County of coastline FDEP profiles UF profiles with sediment data
Nassau 21 82 0 7
Duval 26 80 0 9
St. Johns 66 209 23 23
Flagler 29 100 11 11
Volusia 74 234 0 26
Brevard 64 219 14 24
Indian River 35 119 12 12
St Lucie 35 115 13 13
Martin 35 127 34 34
Palm Beach 72 227 26 26
SBroward 39 128 13 13
Dade 34 113 9 9
total: 12 530 1753 155 207
24
A total of 1834 samples were collected for this study. Nine surfacesediment samples
were collected along each profile at depths of 0.0, 0.9, 1.8, 3.7, 5.5, 7.3, and 9.1 meters as
well as a dune and berm sample. The dune, berm, waterline (or 0.0 meter), 0.91, and usually
the 1.8 meter samples were collected using a coffee can as a scoop during the "swimming"
profile surveys when possible. The offshore samples were collected using a drag sampler from
a boat equipped with a fathometer for depth determination and a loran for profile positioning.
Depths were not corrected for tidal fluctuations at the time of sampling. On average, these
sampling techniques produced 500 gram samples which were placed in labelled cloth sample
bags. Table 32 presents sampling and survey dates for the Nassau County data; this table is
reproduced in the Appendix along with similar tables for each of the remaining 11 counties.
Table 32. Data Collection Dates for each Field Location Identified by the FDEP Range Number
 Nassau County, Florida.
NASSAU COUNTY:
beach proi survey date sediment sampling dates
FDEP UF PDEP FDEP Om .91m 1.83m 3.66m S.49m 7.32m 9.14m
Benchmark Survey Onhore Offsbore dune berm WL 3ft 6ft 12ft 18ft 24ft 30ft
R15 N/A (1028/81) (12/16/81) (8/20/92) (80) (2 (8/202) ) ) (82092) (8/20/92) (8M/92) (80/92) (8/0/92)
R27 N/A (10/27/81) (12/16/81) (8/20/92) (8/2092) (8/2092) (8/2092) N/A (8/20/92) (8/20/92) (8092) (8(092)
R36 N/A (10/27/81) (121681) /292) (820/92) (8/2092) (8/2/92) (8/2092 92)8202) (8202) (820/9) (82092)
R45 N/A (1W22/81) (12/16/81) (8/2/92) (820/92) (8/20/92) (8/2/92) N/A (820/92) (8/2092) (8/2092) (892)
R57 N/A (1w14/81) (12/17/81) (8/20/92) (8/2092) (8/20/92) (8/2092) N/A (82092) (8/20/) ( (82/92) (82092)
R66 N/A (1007/81) (12/17/81) (820/92) (8/2092) (8/20/92 V92)20/92) (8092) (8/2092) (82092) (802) (82092)
R72 N/A (10/07/81) (12/17/81) (8/20/92) (8/20/92) (8/20/92) (8292) N/A (8/20/92) (8/2/92) (8/20/92) (8/20/92)
TOTAL7:p(ofiles subtotali: 7 7 7 7 3 7 7 7 7
SNI toL spi: 59
NOTE: N/A = not available or not attempted.
3.3 Lab Methods for Sediment Size Analysis
Initially a somewhat random selection of 500 samples was prepared for grainsize
analysis. These 500 samples were first dried in an oven at 500 centigrade afterwhich a subsample
weighing approximately 100 grams was obtained using a Jonestype splitter. These subsamples
were then placed in the top of a stack of sieves which was in turn secured to a Rotap' mechanical
shaking device with a runtime set at 15 minutes. Each subsample was sieved through at least 12
different sieves with different screen sizes which varied depending on the bulk characteristics of
the sample. The median diameter of each sediment sample was calculated from the resulting
grainsize distribution.
The standard sieving technique proved infeasible for the number of samples requiring
analysis, so an alternative method was devised. A settling tube apparatus was constructed and
the 500 sieved samples were used specifically for calibration of the samples collected in this
study. The remainder of the samples were analyzed using this settling tube apparatus, also called
a rapid sediment analyzer (RSA).
For the RSA, subsamples weighing approximately 40 grams were obtained again using
the Jonestype splitter. These subsamples were placed in a plastic cup containing enough water
to saturate the sample. The saturated sample was then placed into the release mechanism at the
top of the RSA by first scooping with a spoon and then rinsing the remaining sample out of the
cup using a wash bottle. The electronically triggered samplerelease mechanism at the top of the
RSA is controlled through an IBM_AT computer. After the sample is released, it falls 246
centimeters to an aluminum pan which is suspended from a load cell above the plexiglass settling
tube. This load cell continuously measures the total pan weight as the sediment particles fall
while the computer accesses this information at quartersecond intervals. Thus, the raw data file
consists of a time series of cumulative weights sampled at a frequency of 4 hertz. The raw data
26
are then reduced to an effective grainsize distribution using the following empirical formula
(Gibbs et al., 1971):
3v + 9p'2+gd2(s1)(0.003869+0.02480d) (3.1)
W,= 0.011607+ 0.07440d
where settling velocity, w., is in cm/s, grain diameter, d, in centimeters, and the fluid viscosity,
v, in cm2/s. Median diameters were obtained from these calculated distributions and corrected
with the following calibration formula:
sieve size[O] = 0.1252 + 1.232135 (RSA size[o]) (3.2)
where the size is given in phiunits.
By expressing the sediment size in phiunits, the calibration procedure was reduced to a
simple linear regression of the size data from both measurements. Figure.31 is a plot of the
sieve size versus the RSA size, both in phi units, as well as the bestfit calibration line defined
by (3.2). The conversion from phi units to millimeters is written
size[mm] = 2""'e (3.3)
SIEVE [phi] = 0.125 + 1.232 RSA[phi]
0.5 1 1.5
RSA median diameter [phiunits]
Figure 31. Sediment median diameter from sieve analysis versus sediment median diameter
calculated from settling velocities using Gibbs equation, all in phi units, solid straight line is best
fit line defining calibration equation also shown at the top of the graph.
CHAPTER 4
PROCEDURES FOLLOWED IN DATA ANALYSIS
This study applies (1.4) and (2.18) in a predictive or "blindfolded" manner using Moore's
relationship between sediment median diameter and the A parameter. The A parameter was
allowed to vary along the profile in accordance with the corresponding variation in sediment size.
Sediment size, and therefore the associated A value, was considered to vary linearly between each
sample location. As mentioned in the previous chapter, Work and Dean (1990) applied (1.4) in
this way and Dean et al. (1993) applied (2.18) in the same "blindfolded" manner.
After calculating predicted profiles using both of the abovementioned techniques, the root
mean square difference (RMS) was calculated for each profile in an attempt to quantify the
quality of the "fit" to the actual measured profiles. The RMS error was calculated by
RMS= (hmhc,)2 (31)
where N is the number of data points, hm, and hci are the measured and calculated depths at the
i' location, respectively.
To further investigate the validity of the A versus D50 relationship, least squares was
applied in the manner discussed in Section 2.1 in order to determine the A and m values which
provided the best approximation to the measured profile. This was done for m=2/3 using (2.7)
as well as m= best fit using (2.5) and (2.6).
InWlight of the discussion on the applicability of the EBP equation beyond the surf zone,
all of the above analyses were performed out to 5 specified depths for each profile. These depths
correspond to the sediment sample locations, i.e., depths of 1.8, 3.7, 5.5, 7.3, and 9.1 meters.
CHAPTER 5
RESULTS AND DISCUSSION
5.1 Sediment Size Data
The variation of the sediment median diameter along the east coast of Florida is plotted,
in Figure 51 for all of the sediment samples analyzed in this study. Figures 52 through 58
present similar plots of the sediment size variation at each of the following depths used in the
EBP calculations: 0.0, 0.9, 1.8, 3.7, 5.5, 7.3, and 9.1 meters.
Much of the scatter in the data presented in Figures 51 through 58 is due to the shell
content of the sediment samples. Increased amounts of shell material in the sediment yields a
larger sediment size for those sample locations which occurred around reefs or shell pockets. I.3
general, the 0.0 and 0.9 meter samples contained the greatest amount of shell material. As well,
south of Brevard County, all of the sediment samples tended to contain more shell material than
the samples from the counties to the north. Specifically, Indian River and Broward counties may
be identified in Figures 54 through 56 by the scatter in their respective size data. These
counties were especially characterized by offshore reef systems.
Table 51 lists the average median diameters for the sediment samples collected at each
of the 7 sampling depths mentioned above. These averages were done over the entire east coast
as well as for each individual county. The average median diameter for all of the sediment
samples is 0.24 millimeters. Two notable trends in this data are that generally the sediment size
becomes finer in the offshore direction and that the southernmost counties, i.e., Dade and
Broward counties, comprise the coarsest sediment.
Table 51. Median diameter of sediment samples collected at depths of 0.0, 0.9, 1.8, 3.7, 5.5,
7.3, and 9.1 meters averaged over the entire east coast of Florida and averaged for each of the
12 eastcoast counties.
averaging average sediment sample median diameters [mm]
spatial
domain depth of sediment samples [m]
0.0 0.9 1.8 3.7 5.5 7.3 9.1
east coast 0.3792 0.3153 0.1972 0.1563 0.2018 0.1826 0.2561
Nassau 0.3456 0.1985 0.2903 0.1678 0.3379 0.1032 NA
Duval 0.1723 0.1770 0.1400 0.1301 0.1196 0.1119 NA
St. Johns 0.2142 0.1912 0.1603 0.1218 0.1590 0.1093 0.1564
Flagler 0.5858 0.2016 0.1719 0.1396 0.1259 0.1281 0.1301
Volusia 0.2279 0.2527 0.1454 0.1382 0.1264 0.1378 0.1573
Brevard 0.3303 0.4894 0.3727 0.0898 0.0849 0.1384 0.3518
Indian River 0.4929 0.3910 0.2493 0.3061 0.3630 0.3128 0.6078
St. Lucie 0.3872 0.1314 0.1412 0.2500 0.3867 0.3917 0.5291
Martin 0.3030 0.3094 0.1495 0.1123 0.1010 0.1019 0.8464
Palm Beach 0.5154 0.3680 0.1924 0.1598 0.1692 0.1786 0.1933
Broward 0.4644 0.3611 0.2225 0.3067 0.5027 0.4902 0.2782
Dade 0.6663 0.6586 0.2305 0.1429 0.4491 0.3440 NA
NOTE: NA = not available
5.2 Results from the "Blindfolded" Test on Beach Profile Predictability using Sediment Size Data
Initially, for each profile the measured profile (solid line) was plotted with both calculated
profiles; one without the gravity term (dotted line) and the other with the gravity term (dashed
line). Above these graphs, the cross shore variation in sediment median diameter for the
measured profile was plotted. These figures have been omitted from this thesis to reduce its
bulk, however, they are presented in Charles (1994). Figure 59 and 510 present two examples
of these figures showing the best and worst fit to the measured profiles, respectively.
31
The RMS deviation of the calculated from the measured profile depths was computed for
each profile location in order to quantify the quality of the fit to the measured profile and to
simplify comparisons, see Equation (3.1). For each profile location, RMS values were computed
for the profiles truncated at the following 5 depths: 1.8, 3.7, 5.5, 7.3, and 9.1.meters. The
variation of these RMS values, along the east coast of Florida, is plotted in Figures 511 through
515 (one plot for each of the 5 depths mentioned above); the solid line is for the calculation
without the gravity term and the dotted line is for the calculation with the gravity term.
Presented in Table 52 are the averages of the individual RMS values for each of the two
calculations and for each of the 5 depths. These averages were done over the entire east coast,
as well as for each individual county.
Table 52. Root Mean Square (RMS) deviations of predicted equilibrium beach profiles (EBP)
from the measured beach profile for each measured profile extending out to depths of 1.8, 3.7,
5.5, 7.3, and 9.1 meters; these individual RMS values for each profile are averaged over the
entire east coast of Florida and averaged for each of the 12 eastcoast counties; EBP predictions
based on calculations (1) without the gravity term and (2) with the gravity term.
averaging average RMS deviations of each predicted profile from measured
spatial profile truncation depth [meters]
domain 1.8 3.7 5.5 7.3 9.1
(1) (2) (1) (2) (1) (2) (1) (2) (1) (2)
east coast 0.42 0.39 0.65 0.77 0.88 1.08 1.21 1.54 1.58 2.02
Nassau 0.44 0.51 0.48 6.66 0.81 139 1.85 2.68 NA NA
Duval 0.33 0.46 0.49 0.98 1.01 1.58 0.39 1.46 NA NA
St. Johns 0.30 0.33 0.39 0.69 0.57 0.96 1.08 1.59 1.74 2.29
Flagler 0.36 0.32 0.78 0.99 0.82 1.08 1.16 1.43 1.73 2.01
Volusia 0.44 0.40 0.43 0.67 0.48 0.98 0.83 1.54 1.24 2.19
Brevard 0.48 0.40 0.70 0.65 0.99 1.04 1.46 1.70 2.07 2.33
Indian River 0.51 0.38 1.41 1.26 1.61 1.35 1.65 1.30 0.62 0.75
St. Lucie 0.41 0.44 0.75 0.78 1.84 1.67 2.26 2.16 1.95 2.16
Martin 032 0.32 0.50 0.52 0.64 0.77 1.09 1.31 NA NA
Palm Beach 0.49 0.44 0.74 0.73 0.77 0.87 1.02 1.24 135 1.71
Broward 0.42 0.43 0.68 0.82 0.97 0.97 1.43 134 NA NA
Dade 0.41 0.26 0.69 0.77 0.75 1.11 NA NA NA NA
NOTE: NA=not available
32
The first set of RMS values, the first two columns in Table 52, were calculated for
profiles truncated at a 1.8 meter depth. These RMS values range from 0.26 to 0.51 meters
indicating that both methods of EBP calculation provide a reasonably good fit to the measured
data. For this set of values, the calculation including the gravity term gives a slightly better fit,
0.39 meters on average, than does the other EBP calculation which has an overall average of 0.42
meters. This would be expected since the additional term, the gravity term, includes a measured
value for the beach face slope (BSL).
Overall, there are two obvious trends in the data of Table 52. The first is that as we
include more of the profile in our calculations, the RMS values increase and thus the quality of
the EBP fit to the measured profile decreases. The second trend is that except for the profile
truncated at 1.8 meters, which was already discussed, the EBP calculation without the gravity
term consistently outperforms the EBP calculation including the gravity termT This second trend
is reasonable since Moore's curve, which was used to calculate the A parameter from the
sediment size in both EBP calculations, was derived using the EBP calculation without the gravity
term. If the EBP calculation including the gravity term term is, to be judged fairly, a new
relationship between A and sediment size should be determined specifically for this calculation.
To provide a reference for discussion, an average measured profile was computed for the
entire east coast. As well, an average calculated profile for both techniques was also computed
for the entire east coast. These three average profiles are plotted together in Figure 516.
Examination of this figure shows that the EBP calculation without the gravity term (dotted line)
obviously outperforms the EBP calculation including the gravity term (dashed line). For profiles
truncated 4t 1.8 meters depth, the RMS values for the EBP calculation (based on profile averages
out to a particular depth) with and without the gravity term are 0.30 and 0.08 meters,
respectively. For profiles extending out to 3.7 meters depth, the RMS error increases slightly
to 0.59 and 0.11, respectively.
33
To further facilitate analysis of the results, average measured profiles and calculated
profiles for both techniques were computed for each of the 12 eastcoast counties. An average
calculated profile for both techniques was also determined for each of the counties. These group
averaged profiles are presented in figures 517 through 528. Along with the average measured
profile and the averaged calculated EBP by two techniques, the maximum and minimum
measured depths for all profiles in the group are also plotted versus distance to give an indication
of profile variation within the group.
Table 53 summarizes the RMS values calculated from these groupaveraged profiles.
There is great improvement in the quality of the fit of the average measured profiles to the
average EBP calculated profiles than was indicated by the individual profiles discussed earlier in
this section.
Table 53. Root Mean Square (RMS) deviations of the average predicted equilibrium beach
profile (EBP) from the average measured beach profile extending out to depths
of 1.8, 3.7, 5.5, 7.3, and 9.1 meters; EBP predictions based on calculations (1) without the
gravity term and (2) with the gravity term.
averaging RMS deviations of average predicted profiles from average measured
spatial profile truncation depth [m]
domain 1.8 3.7 .5 7.3 9.1
(1) (2) (1) (2) (1) (2) (1) (2) (1) (2)
east coast 0.08 030 0.11 0.59 NA NA NA NA NA NA
Nassau 0.22 032 0.22 0.62 NA NA NA NA NA NA
Duval 0.29 032 0.20 0.81 NA NA NA NA NA NA
St. Johns 0.09 030 0.26 0.68 0.74 1.04 1.11 1.40 NA NA
Flagler 0.20 0.19 0.75 1.00 034 1.03 0.76 1.56 NA NA
Volusia 0.23 0.44 0.24 0.68 0.63 0.99 NA NA NA NA
Brevard 0.30 0.16 0.28 0.40 NA NA NA NA NA NA
Indian River 0.18 0.16 NA NA NA NA NA NA NA NA
St. Lucie 0.17 0.24 034 0.66 0.23 0.63 NA NA NA NA
Martin 0.14 0.47 0.25 0.64 0.40 0.66 0.99 132 NA NA
Palm Beach 0.05 0.27 0.18 0.40 0.46 0.81 NA NA NA NA
Broward 0.22 0.45 033 0.63 NA NA NA NA NA NA
Dade 0.25 0.14 NA NA NA NA NA NA NA NA
NOTE: NA=not avaiable
34
5.3 Comparison Between EBP Calculations Performed With and Without the Gravity Term.
Examination of Figures 59, 510, and 517 through 537 in Section 5.2 shows that there
is a consistent relationship between the two EBP calculations, with and without the gravity term.
Perusal of these plots shows that the EBP calculation including the gravity term always predicts
a shallower depth than the other EBP calculation for any given location, y, along the profile.
This relationship between the 2 calculations can be simply explained by expressing both
BBP equations in the same form as follows
yh (5.1)
y= [h 32. h (5.2)
A BSL
Inspection of Figure 529, makes this effect of the additional term, h/BSL, in (5.2) more obvious.
On average, the EBP calculation without the gravity term outperforms the EBP
calculation including the gravity term. As was mentioned in Section 5.2, the A versus sediment
size relationship given by Moore's curve was determined using the EBP calculation without the
gravity term and for an invariant A along the profile. Thus, Moore's relationship is slightly more
appropriate for the EBP equation (5.1) although both applications in this study are for a variable
A along the profile. Perhaps Moore's relationship could be reevaluated for (5.2) at least for the
case of an invariant A and then reapplied to (5.2).
The reevaluation of the A versus sediment size relationship for (5.2) was not performed
in this study. It was felt that the predictive capability of (5.1) using Moore's relationship was
quite good and that further study of (5.2) was not justifiable. Thus, the remainder of this chapter
considers (5.1) and any additional discussion of (5.2) is omitted
35
5.4 Results from the Curve Fit Analysis of h=Avm and h=Av23' for Invariant A
In order to investigate further the h=Ay" relationship, a curve fit analysis was performed
using least squares as was discussed in section 2.1. The fitting of h =AY" to the measured profile
data involved using (2.5) and (2.6) to determine m and A, respectively. The alongshore variation
of these A (solid line) and m (solid line with empty square symbols) values are plotted in Figures
529 through 533 for profiles truncated at each of the following 5 depths: 1.8, 3.7, 5.5, 7.3, and
9.1 meters. Fitting h=Ay'/ to the measured data required application of (2.7) to determine the
A value. This A value is plotted as the dotted line with asterisk symbols in Figures 530 through
534.
The most notable feature of the data plotted in Figures 530 through 534 is the inverse
correlation between A and m; as A increases, there is a corresponding decrease in m. This
inverse relationship is inherent in the relationship: h=Ay" and may be explained further by
returning to the original analysis, that of linear regression. When fitting a straight line (Equation
2.2) through a set of data points, as the slope (the m value) is increased, naturally the yintercept
(the logarithm of A) will decrease and vice versa. As Dean (1977) points out, it is doubtful that
this feature in the data has any physical significance.
Table 54 presents the average A and m values from (2.5) and (2.6) discussed above,
whereas Table 55 presents the average A values from (2.7) for m=2/3. These averages were
performed over the entire east coast as well as for each individual county.
It must be mentioned that due to the nature of the logscaling, the linearized fit to the data
using (2.2) is slightly biased toward the portion of the profile closer to the waterline than a direct
linear fit of (1.1) to the data. The added difficulty in curvefitting (1.1) is not outweighed,
however, by the increased accuracy of the fit.
Table 54. Averages, for the entire east coast of Florida and countybycounty, of bestfit A and
m parameters from curve fit analysis of h=Af to measured beach profiles out to depths of 1.8,
3.7, 5.5, 7.3, and 9.1 meters.
averaging curvefit analysis results averaged overall and by county
spatial profile truncation depth [meters]
domain 1.8 3.7 1 5 73 9.1
A m A m A m A m A m
east coast 0.12 0.66 0.13 0.65 0.13 0.65 0.12 0.68 0.11 0.69
Nassau 0.07 0.75 0.07 0.76 0.03 0.89 0.02 0.97 NA NA
Duval 0.05 0.74 0.05 0.79 0.05 0.79 0.04 0.76 NA NA
St. Johns 0.07 0.72 0.08 0.72 0.08 0.73 0.06 0.78 0.06 0.81
Flagler 0.11 0.65 0.11 0.71 0.13 0.66 0.13 0.67 0.12 0.70
Volusia 0.07 0.72 0.07 0.72 0.07 0.74 0.05 0.78 0.05 0.79
Brevard 0.13 0.62 0.13 0.63 0.12 0.65 0.11 0.67 0.10 0.68
Indian River 0.19 0.55 0.26 0.49 0.27 0.48 0.23 0.52 0.19 0.55
St. Lucie 0.15 0.59 0.17 0.57 0.18 0.58 0.17 0.60 0.12 0.67
Martin 0.15 0.59 0.19 0.54 0.17 0.56 0.10 0.66 NA NA
Palm Beach 0.17 0.60 0.22 0.53 0.20 0.56 0.18 0.60 0.16 0.61
Broward 0.19 0.63 0.18 0.65 0.21 0.60 0.26 0.55,. NA NA
Dade 0.11 0.71 0.10 0.76 0.13 0.67 NA NA NA NA
NOTE: NA=not available
Table 55. Averages, for the entire east coast of Florida and countybycounty, of best fit A
parameter for m =2/3 from curve fit analysis of h =Ay to measured beach profiles out to depths
of 1.8, 3.7, 5.5, 7.3, and 9.1 meters.
averaging averaged results of curvefit analysis m=2/3
spatial profile truncation depth [meters]
domain 1.8 3.7 5.5 73 9.1
east coast 0.10 0.10 0.10 0.10 0.11
Nassau 0.08 0.08 0.09 0.11 NA
Duval 0.06 0.08 0.09 0.07 NA
St. Johns 0.08 0.09 0.09 0.09 0.09
Flagler 0.10 0.12 0.11 0.12 0.12
Volusia 0.07 0.08 0.08 0.08 0.09
Brevard 0.11 0.10 0.10 0.11 0.11
Indian River 0.13 0.11 0.10 0.10 0.10
St. Lucie 0.11 0.10 0.10 0.10 0.12
Martin 0.12 0.10 0.10 0.10 NA
Palm Beach 0.13 0.11 0.11 0.11 0.11
Broward 0.16 0.15 0.13 0.13 NA
Dade 0.12 0.13 0.14 NA NA
NOTE: NA=not available
5.5 Motivation for Different m Values
Inspection of the results from the curve fit analysis indicates that for the east coast of
Florida on average, m=0.67is not an unreasonable value. While the group averages range from
a low of 0.48 and a high of 0.97, the average m value for the entire east coast is 0.67 for profiles
extending out to all 5 depths examined in this study.
As well, there are no major groupings of m values around any of the other values for
which theoretical justifications have been made (i.e., m=0.4 and m=0.8).
5.6 Relationship Between A and Sediment Size
The EBP calculation, without the gravity term and allowing A to vary along the profile,
proved onaverage to be quite a useful approximation of the actual profile (Figure 517), at least
out to a depth of 4 meters. Since this calculation was performed using sedimentsize data and
Moore's empirical relationship between sediment size and A, it would seem that there should be
some obvious relationship between A and sediment size in the data set analyzed in this thesis.
To investigate the nature of this relationship in the data set at hand, various graphical
representations were constructed.
The data selected for this investigation include the average A values calculated from the
curve fit analysis of Section 5.4 (Table 54) and the average sediment median diameters presented
in Table 51. The A values calculated from h =Ay were selected since this equation is that which
was tested in this study. For each of the 5 depths, for which average A values are available, the
sediment data represents the average of all of the average sediment size data which are available
over the lengths of each of the 5 truncated profiles. So, for instance, the A value calculated from
the curve fit to a profile extending to 3.7 meters depth would correspond to the average sediment
size of the 0.0, 1.8, and 3.7 meter sediment sizes listed in Table 54 as averages for each county.
38
Figure 535 is a graph of average A versus average sediment median diameter for the 12
counties and for each of the depths: 1.8, 3.7, 5.5, 7.3, and 9.1 meters; so there are basically 12
data points, one for each county, for each of the 5 depths for which data are available.
Inspection of this figure shows a widely scattered but positive relationship between A and average
sediment size.
The next step is to examine the same data plotted separately for each individual depth.
Thus Figures 536 through 540 are plots of the same data in Figure 535 except they correspond
to the data at each depth. There is an obvious positive relationship between A and average
sediment size for all of these figures (Figures 536 through 539) except perhaps for the profiles
extending out to 9.1 meters depth (Figure 540).
Finally, the average A values and average sediment median diameters are plotted, by
county, north to south; this is plotted for profiles extending out to depths ofl.8, 3.7, 5.5, 7.3,
and 9.1 meters corresponding to Figures 541 through 545, respectively. Plotted in this relative
spatial scale, the relationship between the average best fit A values and average sediment size
becomes more apparent. This would seem to indicate that perhapsA is also a function of other
regional characteristics of the beach profile as Pruszak (1993) indicated for his A.
5.7 Application Beyond the Surf Zone and Dependence on Profile Depth
The theoretical explanations, applied to the h=Ay" formulation for an EBP, consider
breakingwave and associated dynamics inside the surf zone. Most engineering applications,
however, require information regarding the total profile, which extends outside the surf zone and
out to the limiting depth of motion. In the absence of a better method, the EBP equation has
been applied out to these closure depths when profile shape information is necessary.
39
A 10 meter closure depth is generally recommended for the east coast of Florida. In an
attempt to investigate the depth dependence for the EBP, all calculations in this study were
repeated for profiles truncated at depths of 1.8, 3.7, 5.5, 7.3, and 9.1 meters, which correspond
to the depths of the sediment samples. All of the data presented so far exhibit marked variation
depending on this limiting depth applied to the profile.
The most striking depth dependence is the predictability of the measured profile using
h =Ay/3 where A varies with the crossshore distribution of sediment median diameter following
Moore's empirical relationship. On average, this predictability is quite good out to a depth of
around 4.0 meters, see Figure 516. In fact, for this east coast average profile, the RMS
deviation of the average calculated EBP from the average measured profile out to a depth of 3.7
meters is 0.11 meters; Otay and Dean (1994) estimate the profile survey error out to this depth
is between 0.15 and 0.20 meters.
Figures 517 through 527 show the average calculated and the average measured profiles
for averages taken over each of the 12 counties; there is generally a marked decrease in quality
of the EBP approximation to the measured profile beyond 4 meters' depth. Thus, application of
this EBP calculation beyond the 4 meter depth and out to the closure depth is not an ideal
method. To remedy this problem, Inman et al. (1993) proposed using 2 profiles of the form
h=Ay.
Inman pointed out that by combining 2 profiles, one inside and one outside the surf zone,
the dynamics associated with shoaling waves outside the surf zone could be handled separately
with the second profile. Inman achieved a much better fit to the data using his compoundcurve
approach as opposed to the single EBP curve. Interestingly enough, similar values for A, as well
as m, for both the shorerise and the barberm profiles were obtained from the compoundcurve
fitting scheme employed by Inman. Inman's method was not applied to the data in this study.
5.8 Nonmonotonic Beach Profiles
Unfortunately, the plots of the measured and calculated individual profiles could not be
included in this report due to spatial constraints but they are presented in Charles (1994).
Nevertheless, after perusal of these graphs, an unusual trend becomes apparent inmany of the
measured profiles. It seems as though there is a consistently sloping profile for most of the
length of the profile, but that at the locations of offshore bars and offshore reefs, it is horizontally
displaced in the offshore direction; after this displacement, the same sloping profile continues.
Figure 546 presents this observation in an exaggerated form. Again going back to the original
theoretical development of the EBP concept, that of uniform wave energy dissipation per unit
volume across the surf zone, we know that Dq is related to depth and slope as h'ddh/dy.
Conceptually we might expect a plot of D, across the profile to look something like that which
is also plotted for the exaggerated profile in Figure 546.
Referring to the graph of the trend in RMS values along the east coast, we see that there
are a few noticeably large values. If we examine more carefully the profile plots of the largest
of the two, we see something quite interesting. Figures 547 and 548 show these particular
profiles from Indian River and St.Lucie counties, respectively. In both measured profiles,
offshore reefs are evident. With a little bit of imagination, one might see the consistent slope of
the profile and the horizontal displacement of this sloping profile at the locations of these reefs.
5.9 Time Varving Beach Profiles
It is well known that beach profiles are constantly evolving, both under field and lab
conditions. For this reason, the EBP concept has always been associated with a sort of dynamic
equilibrium. The time varying A parameter proposed by Pruszak (1993), is quite interesting.
Unfortunately at the present time, the data set for the east coast of Florida is unsuitable for a test
41
of this approach. With the sediment size data now available, perhaps in the future, profile
measurements will be performed at a higher frequency so that the timevarying nature of A can
be examined fully. Pruszak's idea of decomposing A does however seem plausible given the
results of the data presented in this report.
First, the application of h =Ay23 for A =A where A is a function of sediment size (here
following Moore's empirical relationship) results in a reasonably good fit to the data only on a
group average basis. Perhaps some of the variations in the individual profiles which are averaged
out could be better explained with a timevarying A as Pruszak proposed. All of the beach
profiles were not measured during the same season of the year or during the same general time
frame, some profiles were measured 10 years apart, e.g., Dade 1980 and Broward 1979 versus
Palm Beach 1990 and Duval 1990. Seasonal shoreline changes in Florida are relatively small
however.
Average Sediment Size in mm
River
Figure 51. Variation in sediment median diameter, in millimeters, along the east coast of
Florida for all of the sediment samples analyzed.
Average Sediment Size in mm
Nassau
Duval
St Johns
Flagler
Volusia
Brevard
Indian River
St Lucie
Martin
Palm Beach
Broward
Dade
Figure 52. Variation in sediment median diameter, in millimeters, along the east coast of
Florida for all of the sediment samples collected at the waterline.
Average Sediment Size in mm
Indian River
Figure 53. Variation in sediment median diameter, in millimeters, along the east coast of
Florida for all of the sediment samples collected at 0.9 meters depth.
Figure 54. Variation in sediment median diameter, in millimeters, along the east coast of
Florida for all of the sediment samples collected at .1.8 meters depth.
Average Sediment Size in mm
Figure 55. Variation in sediment median diameter, in millimeters, along the east coast of
Florida for all of the sediment samples collected at 3.7 meters depth.
Average Sediment Size in mm
Johns
DM Indian River
S' St Lucie
Martin
SP Palm Beach
RO Broward
DAD Dade
Figure 56. Variation in sediment median diameter, in millimeters, along the east coast of
Florida for all of the sediment samples collected at 5.5 meters depth.
Average Sediment Size in mm
Johns
River
Figure 57. Variation in sediment median diameter, in millimeters, along the east coast of
Florida for all of the sediment samples collected at 7.3 meters depth.
Average Sediment Size in mm
Figure 58. Variation in sediment median diameter, in millimeters, along the east coast of
Florida for all of the sediment samples collected at 9.1 meters depth.
0.155
0.15
E
0.145
a
E
0.14
c
10.135
g 0.13
0.125
U.1
1
_2
a
Z .3
24
.5
7
0
0 100 200 300 400 500 600 700 800 90
Distance from Waterline [m]
B
DUVAL County: Range027
t
100 200 300 400 500 600 700
Distance from Waterline [m
800 000
Figure 59. Best performance of predicted profiles: CA) crossshore variation in sediment median
diameter, in mm; and (B) measured beach profile (solid line), calculated equilibrium beach
profile: (1) without the gravity term (dotted), and (2) with the gravity term (dashed) for Range
number 27 in Duval County, Florida.
* .
II I
......... .............. ........... ........... ........... ......... .. .................................
.......... .......... ........... ........... ........... ........... i.......... ".. ........... !...........
........... ....... ... ......................................
...... ... .. *............................. .. .... ....... .. .... ... ...........
i
:"""""\"'"
;I''"i"'' i'""'
e t t .ii~;
. ... . . . . .. . . . . . .. .
i........... .. .. .. .. .. .. .. .
... .... ... ... ... ... I.. ... ... ... ... ... ... ... ......
)
,


0.5
0.7
E
S0.4
*0.6
0.2
0
0.1
70 80 00 100
BROWARD County: Range051
10 20 30 40 50 60
Distance from Waterline [m]
70 80 90 100
Figure 510. Worst performance of predicted profiles: (A) crossshore variation in sediment
median diameter, in mm; and (B) measured beach profile (solid line), calculated equilibrium
beach profile: (1) without the gravity term (dotted), and (2) with the gravity term (dashed) for
Range number 1 in Broward County, Florida.
I I ~ I I I
. .. .. .................... ....... ........ .......... ......... .........
... ... ....................... ................... .......... ......... .......... ......... .
 *** ** ** ..... ............ ............
L
10 20 30 40 50 60
Distance from Waterline [m]
0.5
*1
0
 *1.5
z
L 2
S2.5
.3
3.5
.4
0
. .......... ..... ..... .. ..........
. : : :
. .......... ... ..................... ........ ...... .........
.... ... ... .' ... ..... ...... ....... .... .... .................... ,
. ....... ....................... .................... ;....... ...* .. .. ...... .... ......... ..........
 ......... ... ... .
t s t
I
1.8m
4.
0i
0 20 40 60 80 100 120 140 160 18(
relative distance from North to South
6  i    _________
relative distance from North to South
Figure 511. Longshore distribution of RMS values based on cumulative deviations of the
calculated equilibrium beach profile: (1) without the gravity term (solid) and (2) with the gravity
term (dotted) from the measured profile out to 1.8 meters depth.
3.7m
4
3
2
0
0 20 40 60 80 100 120 140 160 18(
relative distance from North to South
Figure 512. Longshore distribution of RMS values based on cumulative deviations of the
calculated equilibrium beach profile: (1) without the gravity term (solid) and (2) with the gravity
term (dotted) from the measured profile out to 3.7 meters depth.
7.3m
4
1  i      
5.   ~c V   1 1 
3.
1.
0
S 20 40 60 80 100 120 140 160 18i
relative distance from North to South
Figure 514. Longshore distribution of RMS values based on cumulative deviations of the
calculated equilibrium beach profile: (1) without the gravity term (solid) and (2) with the gravity
term (dotted) from the measured profile out to 7.3 meters depth.
55
5.5m
6
4
0 20 40 60 80 100 120 140 160 180
relative distance from North to South
Figure 513. Longshore distribution of RMS values based on cumulative deviations of the
calculated equilibrium beach profile: (1) without the gravity term (solid) and (2) with the gravity
term (dotted) from the measured profile out to 5.5 meters depth.
56
9.1m
6,
S.
1Co
2 A  
0
0 20 40 60 80 100 10 140 1 1
distance from North to South
1' _    __ __ __
.*
Figure 515. Longshore distribution of RMS values based on cumulative deviations of the
calculated equilibrium beach profile: (1) without the gravity term (solid) and (2) with the gravity
term (dotted) from the measured profile out to 9.1 meters depth.
S100 150 0 250 300
distance from shoreline [m]
Figure 516. Average measured profile (solid line), and average calculated equilibrium beach
profile: (1) without the gravity term (dotted) and (2) with the gravity term (dashed) for the entire
east coast of Florida.
east coast of Florida.
.4
0 ASO 1o 200 2 300
distance from shoreline [m]
Figure 517. Average measured profile (solid line), and average calculated equilibrium beach
profile: (1) without the gravity term (dotted) and (2) with the gravity term (dashed), and
maximum and minimum measured depths (thick solid line) for Nassau County, Florida.
50 100 UO 200 0 0 300 350 400 450 50
distance from shoreline [m]
Figure 518. Average measured profile (solid line), and average calculated equilibrium beach
profile: (1) without the gravity term (dotted) and (2) with the gravity term (dashed), and
maximum and minimum measured depths (thick solid line) for Duval County, Florida.
T4
1 S 100 O 200 250 300
distance from shoreline [m]
Figure 519. Average measured profile (solid line), and average calculated equilibrium beach
profile: (1) without the gravity term (dotted) and (2) with the gravity term (dashed), and
maximum and minimum measured depths (thick solid line) for St. Johns County, Florida.
4
79
10 lillllli
50 100 so 200 250 30 350 400 450 50
distance from shoreline [m]
Figure 520. Average measured profile (solid line), and average calculated equilibrium beach
profile: (1) without the gravity term (dotted) and (2) with the gravity term (dashed), and
maximum and minimum measured depths (thick solid line) for Flagler County, Florida.
.7
So10 200 30 400 s5 00
distance from shoreline [m]
Figure 521. Average measured profile (solid line), and average calculated equilibrium beach
profile: (1) without the gravity term (dotted) and (2) with the gravity term (dashed), and
maximum and minimum measured depths (thick solid line) for Volusia County, Florida.
63
3.
z 4
4
0 50 100 SO 200 20 3;0 310 460
distance from shoreline Im]
Figure 522. Average measured profile (solid line), and average calculated equilibrium beach
profile: (1) without the gravity term (dotted) and (2) with the gravity term (dashed), and
maximum and minimum measured depths (thick solid line) for Brevard County, Florida.
U
0 10 20 30 40 0 60 70 so
distance from shoreline [m]
Figure 523. Average measured profile (solid line), and average calculated equilibrium beach
profile: (1) without the gravity term (dotted) and (2) with the gravity term (dashed), and
maximum and minimum measured depths (thick solid line) for Indian River County, Florida.
65
z
4
.O0
ditance from shoreline [m]
Figure 524. Average measured profile (solid line), and average calculated equilibrium beach
profile: (1) without the gravity term (dotted) and (2) with the gravity term (dashed), and
maximum and minimum measured depths (thick solid line) for St. Lucie County, Florida.
0 1 200 .3$ 40 o00 sdo 700
distance from shoreline [m]
Figure 525. Average measured profile (solid line), and average calculated equilibrium beach
profile: (1) without the gravity term (dotted) and (2) with the gravity term (dashed), and
maximum and minimum measured depths (thick solid line) for Martin County, Florida.
67
4
ID   :5  r'
L7.
40
S10 2000 400 s5o 600
distance from shoreline [m]
Figure 526. Average measured profile (solid line), and average calculated equilibrium beach
profile: (1) without the gravity term (dotted) and (2) with the gravity term (dashed), and
maximum and minimum measured depths (thick solid line) for Palm Beach County, Florida.
0 o100 o 20 230o 3o 310 400 "
distance from shoreline [m]
Figure 527. Average measured profile (solid line), and average calculated equilibrium beach
profile: (1) without the gravity term (dotted) and (2) with the gravity term (dashed), and
maximum and minimum measured depths (thick solid line) for Broward County, Florida.
4,
0 10 2 0 40 so 60 10 s
distance from shoreine [m]
Figure 528. Average measured profile (solid line), and average calculated equilibrium beach
profile: (1) without the gravity term (dotted) and (2) with the gravity term (dashed), and
maximum and minimum measured depths (thick solid line) for Dade County, Florida.
70
equivalent
'\ displacements
\y = (h/A)^(3/2)
\ y= (h/A)^(3/2)+(h/BSL)
. 1 " y=(h/BSL)
relative distance from shoreline
Figure 529. Generalized representation of the relationship between the EBP calculation: (1)
without the gravity term (dotted), and (2) with the gravity term (dashed); the gravity term alone
is also shown (solid).
71
1.8m
1'
o.  i  1 i 
.,7.
Us
0 20 40 60 80 100 120 140 160 180
relative distance from North to South
Figure 530. Longshore variation in best fit A (solid line) and m (solid line with symbols)
parameters as well as A (dotted line) for m=2/3 from curve fit analysis of hAy to measured
beach profiles out to a depth of 1.8 meters.
3.7m
*ii  
1
1.7
0 20 40 60 80 100 120 140 160 180
relative distance from North to South
Figure 531. Longshore variation in best fit A (solid line) and m (solid line with symbols)
parameters as well as A (dotted line) for m=2/3 from curve fit analysis of h=Ay to measured
beach profiles out to a depth of 3.7 meters.
73 .
5.5m
.9
II
.4
0 20 40 60 80 100 120 140 160 180
relative distance from North to South
Figure 532. Longshore variation in best fit A (solid line) and m (solid line with symbols)
parameters as well as A (dotted line) for m=2/3 from curve fit analysis of h=Ay" to measured
beach profiles out to a depth of 5.5 meters.
7.3m
.4
W, M
0 20 40 60 80 100 120 140 160 180
relative distance from North to South
Figure 533. Longshore variation in best fit A (solid line) and m (solid line with symbols)
parameters as well as A (dotted line) for m=2/3 from curve fit analysis of h=Ay" to measured
beach profiles out to a depth of 7.3 meters.
9.1m
1
C333
0 20 40 60 80 100 110 140 160 180
relative distance from orth to South
Figure 534. Longshore variation in best fit A (solid line) and m (solid line with symbols)
parameters as well as A (dotted line) for m=2/3 from curve fit analysis of h=Ay" to measured
beach profiles out to a depth of 9.1 meters.
I 0
parameters as well as A (dotted line) for m==2/3 from curve fit analysis of =4y" to measured
0.07 0.08 0.09 0.1 0.11 0.12
A parameter [m^ 113]
0.13 0.14 0.1S
Figure 535. Variation of A parameter with median sediment size. A parameter is bestfit value
to depths of 1.8,3.7, 5.5, 7.3, and 9.1 meters. Sediment size is average of all samples which
extend from the waterline to each of the designated depths.
0 1
03
0.45
O
0.4
035
03
0.25
02
0.15
n S'
0.06
U
.
"m
U m
n
U U 
U)
U
U
0.07 0.08 0.09 0. .1 1 0.12
A parameter [m 1/3]
Figure 536. Scatter plot of average county A values out to 1.8 meters depth versus sediment size
averaged over all profiles in county and over all landward samples in each profile.
. 0.4
S0.35
03
So
0.2
0n.
n
U
U
U
UI
U
U
U
U
rU
U
rU
0.06
0.13 0.14 .15
9
U.43
0.4
0.35
03
0.25
0.2
U
nA t ,
0.07
0.08 009 0.1 0.11 0.12
A parameter [m^ 1/3]
0.3
0.13
0.14 0.15
Figure 537. Scatter plot of average county A values out to 3.7 meters depth versus sediment size
averaged over all profiles in county and over all landward samples in each profile.
" "
0.11
A parameter 1/3
A parameter [m ^ 1/3]
Figure 538. Scatter plot of average county A values out to 5.5 meters depth versus sediment size
averaged over all profiles in county and over all landward samples in each profile.
79
0.45 1
0.4
033
03
025
02
U
IU
a
U
rU
UI
UI
IU
U
U
(1~1*I
0.08
U *
0.08 0.09 0.1 0.11 0.12 0.!
A parameter [m ^ 1/3]
Figure 539. Scatter plot of average county A values out to 7.3 meters depth versus sediment size
averaged over all profiles in county and over all landward samples in each profile.
045
0.4
035
0.3
I
I
0.2
0.15
0.1
0.07
0.4 1
0.35
035
0.2
0.15
0.0
9
0.095
005o
Aparameter [m^ 1/3]
Figure 540. Scatter plot of average county A values out to 9.1 meters depth versus sediment size
averaged over all profiles in county and over all landward samples in each profile.
UI
U
U
U
U
Ur
0.115
1.8 meters depth
county code
 A parameter 9 cumulative size
Figure 541. Plots of average county A values out to 1.8 meters depth and sediment sizes
averaged out to that depth. County code denotes the county sequence commencing from the
north on Florida's east coast.
83
3.7 meters depth
county code
 A parameter  cumulative size
Figure 542. Plots of average county A values out to 3.7 meters depth and sediment sizes
averaged out to that depth. County code denotes the county sequence commencing from the
north on Florida's east coast.
5.5 meters depth
county code
4 A parameter S .cumulative size
Figure 543. Plots of average county A values out to 5.5 meters depth and sediment sizes
averaged out to that depth. County code denotes the county sequence commencing from the
north on Florida's east coast.
85
7.3 meters depth
county code
1 A parameter  cumulative size
Figure 544. Plots of average county A values out to 7.3 meters depth and sediment sizes
averaged out to that depth. County code denotes the county sequence commencing from the
north on Florida's east coast.
CI
9.1 meters depth
county code
+ A parameter E cumulative size
Figure 545. Plots of average county A values out to 9.1 meters depth and sediment sizes
averaged out to that depth. County code denotes the county sequence commencing from the
north on Florida's east coast.
87
I I
1h=AyA(273) 
profile
displacemi t] 
relative distance from shoreline
relative distance from shoreline
Figure 546. Illustration of effect of a seaward displacement of the equilibrium profile, possibly
by an offshore reef.
