UFL/CQEL94/010
EBB TIDAL DELTA EVOLUTION AND NAVIGABILITY
IN THE VICINITY OF COASTAL INLETS
by
Michael Richard Dombrowski
Thesis
1994
EBB TIDAL DELTA EVOLUTION AND NAVIGABILITY
IN THE VICINITY OF COASTAL INLETS
By
MICHAEL RICHARD DOMBROWSKI
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
I would like to express my sincere appreciation and gratitude to my advisor and
supervisory committee chairman, Professor Ashish J. Mehta, for his continuous support, guidance
and friendship throughout my study at the University of Florida. My thanks are also extended
to Drs. Robert G. Dean and Robert J. Thieke for serving as members on my supervisory
committee. Special thanks go to Dr. D. Max Sheppard for serving on my committee on such
short notice. Many thanks go to Yigong "Wally" Li for his computer programming expertise and
to Al Browder for his secretarial skills.
I would also like to thank Brett Moore and Ken Humiston for their inspiration and
continued encouragement throughout this adventure. Finally, I am most grateful to my wife, Pat,
for her patience and support through these times.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ................................................................ ii
LIST OF FIGURES .............................................. ....................... vi
LIST OF TABLES .......................................................................... viii
KEY TO SYMBOLS ............................................. ...................... ix
ABSTRACT .................................................................................. xiv
CHAPTERS
1 INTRODUCTION ....................................... .......... 1
Navigability at Coastal Inlets ........................................ 1
Seafloor Evolution ...................................... ............. 2
Study Objectives and Tasks ........................................... 7
2 LITERATURE REVIEW ............................................ 8
Overview .................................................................. 8
Ebb Delta Studies ........................................................ 8
Dean and Walton (1973) ................................... 8
Walton and Adams (1976) .................................. 10
Marino (1986) .............................................. 11
Oertel (1988) .................................... ........... 12
Hayter, Herandez, Atz and Sill (1988) ................. 13
Sediment Transport Studies ............................................. 15
Gole, Tarapore and Brahme (1971) ....................... 15
Sarikaya (1973) ...................................... ....... 16
Galvin (1982) .................................................. 17
Buckingham (1984) .......................................... 19
Ozsoy (1986) .................................... .......... 20
Walther and Douglas (1993) ................................ 21
Inlet Stability Studies .................................................. 23
Bruun and Gerritsen (1960) ................................ 23
O'Brien (1971) .............................................. 24
General Conclusions .................................... ........... 24
3 METHOD FOR DETERMINING DELTA VOLUME AND
GROWTH ......................................... ................ 26
Database ............................................ ................... 26
Ebb Delta Volume ............................................ 26
Tidal Prism .................................... ............ 27
CrossSectional Area ......................................... 27
Significant Wave Height and Wave Period .............. 27
Spring Sea Tidal Ranges ................................... 28
Ebb Delta Volume Estimates ........................................... 28
Ebb Delta Volume .................................... ..... 28
Sensitivity Analysis .................................... .... 29
Diagnostic Examination of Seafloor Evolution ................... 31
Model Development ........................................ 31
Model Parameters ........................................... 40
Ebb delta area ....................................... 40
Suspended sediment concentration ................ 41
Friction factors ...................................... 41
Sediment grain size ................................ 43
Deep water height and period ................... 43
Tidal inlet characteristics .......................... 43
Effects of Significant Physical Parameters on Delta Growth .... 43
Variation in Suspended Sediment Concentration ......... 44
Variation in Sediment Grain Size ........................... 44
Variation in Deep Water Wave Height .................. 47
4 RESULTS .......................................... ................. 48
Ebb Delta ........................................... .................. 48
Estimated Delta Volumes ................................... 48
Delta Volume Sensitivity Analysis ........................ 49
Delta Volume versus Tidal Prism ......................... 51
Time Evolution of Delta Volumes ......................... 54
Jupiter Inlet .......................................... 56
South Lake Worth Inlet ........................... 56
Boca Raton Inlet .................................... 57
Bakers Haulover Inlet .............................. 58
Influence of a on delta growth .................... 59
5 CONCLUSIONS ...................................... ............. 61
Estimated Ebb Delta Volumes ........................................ 61
Ebb Delta Volume versus Tidal Prism ............................. 61
Effects of Significant Physical Parameters on Delta Growth .... 62
Effects of a on Equilibrium Delta Volume ........................ 62
Influence of a on Navigability ....................................... 63
Future Investigations ..................................... .......... 63
Note of Caution ...................................... .............. 64
APPENDIX A EBB DELTA DATABASE ....................................... 65
APPENDIX B FORTRAN MODEL ALGORITHM AND INPUT FORMAT 80
APPENDIX C DATA USED TO CORRELATE TIDAL PRISMBASED
AREA, Ap, TO MEASURED DELTA DEPOSITION AREA, AD 88
REFERENCES ................................................. ......................... 89
BIOGRAPHICAL SKETCH ......................................... ............... 96
LIST OF FIGURES
I1 General sectional view of an inlet channel through a barrier
island and associated ebb delta ....................................... 3
12 General characteristic of an ebb jet and delta ..................... 5
III1 Schematic of three "noinlet" contour conditions in estimating
ebb delta volumes. Seaward exaggerated contours, A (dashed
lines), "bestfit" contours, B (solid lines) and landward
exaggerated contours, C (thick dotted lines) ..................... 29
1112 Schematic of the initial grid and enlarged grid cell patterns
used in estimating ebb delta volumes ............................... 30
1113 Seafloor growth diagrams illustrating the evolution of an ebb
delta: A) initial condition, B) transient condition and C)
equilibrium condition ..................................... .......... 32
III4 Plan view of A) tidal prismbased area, Ap, and B) delta
deposition, AD ........................................................... 36
III5 Crosssectional view and plan view through the inlet and
idealized ebb delta ...................................... ............ 38
III6 Ebb delta area, AD, versus tidal prism based area, Ap, for
twentyone inlets based on data of Davis and Gibeaut (1990) ... 40
1117 Ebb delta volume versus time comparing the effects of three
suspended sediment concentrations, C,, 0.00005, 0.00010 and
0.00020 kg/m3. Runs 1, 2 and 3 ................................... 45
III8 Ebb delta volume versus time comparing the effects of three
grain size diameters, dso, 0.2, 0.3 and 0.4 mm. Runs 4, 5
and 6 ............................................. .................... 45
III7 Ebb delta volume versus time comparing the effects of three
deep water wave heights, Ho, 0.0, 0.4 and 0.8 m. Runs 7, 8
and 9 ............................................. .................... 47
IV1 Ebb delta volume versus tidal prism for mild energy coasts
(030 m2s2) ......................................... ................ 53
IV2 Ebb delta volume versus tidal prism for moderate energy coasts
(30300 m2s2) ..................................................... 53
IV3 Ebb delta volumes versus year with modelcalculated volume
ranges for Jupiter Inlet .............................................. 56
IV4 Ebb delta volumes versus year with modelcalculated volume
ranges for South Lake Worth Inlet .................................. 57
IV5 Ebb delta volumes versus year with modelcalculated volume
ranges for Boca Raton Inlet.......................................... 58
IV6 Ebb delta volumes versus year with modelcalculated volume
ranges for Bakers Haulover ......................................... 58
IV7 Ebb delta volumes versus wave to tidal power ratio, a .......... 60
LIST OF TABLES
II1 Coefficients of Eqn. 2.1 obtained by linear regression for the
relationship of tidal prism with ebb delta volume ............... 11
112 Bruun's stability criterion of sandy coastal inlets related to
ebb delta size and bypassing ......................................... 23
III1 Initial input parameters for examining the effects of variation
in suspended sediment concentrations, C,, grain size diameter,
dso, and deep water wave height, Ho on ebb delta growth ....... 46
IV1 Ebb delta volume for the "bestfit", "noinlet" contours ......... 48
IV2 Ebb delta volumes and percent differences due to changes in
the "noinlet" contours .............................................. 49
IV3 Ebb delta volumes and percent differences due to changes in
the grid cell size ....................................... .............. 51
IV4 Initial input parameters and resulting wave to tidal energy
ratio, ......................................................... .......55
IV5 Comparison of wave to tidal energy, a, and the corresponding
ebb delta volumes ...................................... ............. 59
Ai Ebb delta related parameters for coastal entrances along the
Florida coast ....................................... ................ 66
A2 Ebb delta related parameters for coastal entrances along the
Georgia coast ........................................ ............... 74
A3 Ebb delta related parameters for coastal entrances along the
South Carolina coast ..................................... .......... 75
A4 Ebb delta related parameters for miscellaneous coastal entrances
along the Atlantic and Pacific Ocean coasts ....................... 76
C1 Data used to correlate tidal prism area, Ap, to measured ebb
delta deposition area, A ............................................... 88
KEY TO SYMBOLS
Ac crosssectional area at the inlet throat
A, ebb delta area based on tidal prism
AD ebb delta area based on regression analysis
a,a, regression coefficients, Eqns. 2.1 and 3.1, respectively
a, spring sea tidal amplitude
b,b1 exponential correlation coefficients, Eqns. 2.1 and 3.1, respectively
bo distance between two adjacent deep water wave rays
b, distance between two adjacent nearshore wave rays
C shallow water wave celerity
C CjC,
C, concentration of suspended sediment entering the settling basin
CD drag coefficient
Co deep water wave celerity
C, depthaveraged suspended sediment concentration
d delta accumulation height
de delta equilibrium height
d incremental change in delta height
dso median grain size
E dimensionless wave energy density, Eqn. 2.7
Et tidal energy parameter
E, wave energy parameter
F, sediment settling flux
fc friction factor due to current
f, friction factor due to the combined current and waves
f, friction factor due to waves
g acceleration due to gravity
G parameter including the effects of dso, Ac, and specific gravity, Eqn. 2.4
h water depth at time, t
ha water depth adjacent to the dredged channel
hd water depth in the dredged channel
he equilibrium water depth
hi incremental change in the water depth
ho initial water depth at the inlet throat and over the seafloor
hp design project depth
hR reference water depth
H shoaled nearshore wave height
Ho deep water wave height
HR referenced wave height
H, significant deep water wave height
i timestep subscript
k1,2,3,4 intermediate RungeKutta method steps
k wave number
k' empirical coefficient, Eqn. 2.15
K coefficient of actual sedimentation to effective sediment load
L, length of dredged channel
L wave length
m suspended mass per unit delta bed area
m",m' empirical coefficients, Eqns. 2.12 and 2.15, respectively
M annual average littoral drift reaching the inlet
n Manning's n
p probability of sediment deposition
P spring tidal prism
Pbed bed stability parameter
Po total wave power
Q littoral transport rate
Qp predredging littoral transport rate
R fraction of Q which reaches the dredged channel
re onehalf the distance from the entrance mouth to the outer edge of Ap
rR sediment removal ratio
Re wave Reynolds number
S,n net sedimentation load
Sr shoaling rate
t time
At incremental time in number of tidal periods
tp time interval for an overdredged channel to shoal to a design project depth
T wave period
TR sediment transport ratio
T, tidal period
Ua alongshore current velocity adjacent to the dredged channel
Uc current velocity
Ucb nearbed current velocity
Ucr critical velocity for erosion
Ud alongshore velocity across the dredged channel
U, flow velocity in the x1 direction
U, average inlet velocity over onehalf tidal cycle
Um, maximum velocity through the inlet entrance for a spring tide
Uo characteristic velocity over the delta deposition area
U, total current velocity due to current and waves
U, orbital velocity due to waves
Uw, maximum nearbed orbital velocity due to waves
V ebb delta volume
VT shoaling rate
w width at the inlet throat
Wd dredged channel width
Ws particle settling velocity
W', effective settling velocity of the particles
W, settling velocity of a particle that reaches the seafloor at a distance from the inlet
y vertical distance from the bottom
Y ylh
xl distance in direction of mean flow
X xjlh
Zb distance above the profile origin
Zo theoretical origin of the logarithmic profile
ca wave to tidal energy ratio
E turbulent diffusion coefficient
Eo crosssectional average of the turbulent diffusion coefficient
v kinematic viscosity of seawater
angle between current direction and wave direction
7, unit weight of sediment
7w unit weight of seawater
Pd dry bed density
p, particle density
Pw density of seawater
Tb shear stress due to waves superimposed on current
Tcr critical shear stress
'b maximum nearbed horizontal excursion
Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirement for the Degree of Master of Science
EBB TIDAL DELTA EVOLUTION AND NAVIGABILITY
IN THE VICINITY OF COASTAL INLETS
By
Michael Richard Dombrowski
December 1994
Chairperson: Ashish J. Mehta, Ph.D
Major Department: Department of Coastal and Oceanographic Engineering
For maintaining safe conditions for navigation in the vicinity of coastal inlets, the ebb
delta is frequently dredged. Previous investigations have established the dependence of the delta
volume on wave energy and tidal energy at sandy inlets. In this study, this dependence was
examined with respect to the rate of delta growth and the final equilibrium delta volume starting
with the opening of a new inlet when no delta is present.
A careful examination of the method to estimate delta volumes showed that the choice
of the "noinlet" bathymetric contours and the grid cell size selected for estimating the delta
volume can measurably influence the volume obtained. In general, the greater the complexity
of the bathymetry, the larger the variation in the estimate.
A diagnostic model was developed for examining the influence of the ratio of wave
energy to tidal energy on delta growth. Model sensitivity tests showed that increasing the
suspended sediment concentration in the littoral zone caused the delta to approach equilibrium
faster, but did not affect the equilibrium volume. Increasing the wave height increased the time
of approach to equilibrium but decreased the volume. Finally, increasing the sand size increased
the growth rate as well as the equilibrium volume.
The model was applied to four Florida inlets. It was shown that the observed variability
in the delta volume can be due to the delta not achieving equilibrium and to the degree of
dominance of the wave energy relative to the tidal energy. This dominance also therefore
characterizes navigational access to the inlet.
CHAPTER I
INTRODUCTION
1.1 Navigability at Coastal Inlets
Navigation issues at coastal inlets for commercial and military purposes dates back to at
least 6,000 B.P. (Bruun and Gerritsen, 1960), and within modern times another consideration for
safe navigation has been added, namely recreation. Since early times naturally protected harbors
have been constructed to moor waterborne vessels and load and unload cargo and passengers.
Safe access is provided by inlet entrance channels from deep water through seaward protective
works such as jetties and breakwaters. The purpose of these structures is not only wave
attenuation for the protection of vessels during passage through the channel, but also to stabilize
the channel and interrupt excessive littoral drift from decreasing the channel depth. This problem
of depth decrease is of course not limited to improved channels, since it is also necessary for
unimproved inlet channels to have depths that are sufficient for the navigability of vessels. Thus,
in general, the problem of littoral sediment accumulation at inlets is linked to that of navigability
of the channel.
At the seafloor in the immediate vicinity of the inlet the interrupted littoral sediment tends
to accumulate and raise the floor, leading to the formation of an ebb delta. Thus, the conditions
for navigation, specifically in terms of water depths and wave action, are contingent upon the size
and shape of the ebb delta. At new inlets, or ones which have been closed for a period of time,
the rate at which the seafloor is modified by deltaic formation depends on the prevailing physical
2
conditions, availability of littoral sediment and geologic setting. Thus, from the standpoint of the
requirements for designing and maintaining the entrance channel, it is necessary to develop an
understanding of the mechanisms and means by which delta evolution occurs. The aim of this
study is therefore to examine the interdependence between significant physical parameters
governing sediment transport and the rate of delta formation at coastal inlets. The focus is
restricted to those cases where the bottom material is in the sand size range.
1.2 Seafloor Evolution
As noted, ebb deltas are accumulations of material at the seaward side of coastal inlets
(Figure Ii); the development of these deltas is typically a product of littoral suspended sediment
acted on by tidal and wave forces. Walton and Adams (1976) found that flood deltas on the bay
side of the inlet tend to reach an equilibrium volume over a period of time, and indicated that ebb
deltas volumes also achieve an equilibrium state. At the same time these investigators recognized
that due to changing wave and tidal energy conditions, the volume of ebb delta may be
episodically influenced, thus resulting in the growth of the delta at a new inlet approaching and
fluctuating about some equilibrium volume. Walton and Adams (1976) found the equilibrium
volume to be dependent on the tidal prism and, furthermore, the empirical coefficients
characterizing this relationship were themselves found to depend on the wave energy. For a
given tidal prism, the equilibrium yolume decreased with increasing wave energy. To examine
this important issue of the influence of wave and tideinduced forces on the rate of growth of
sandy ebb deltas, a semiquantitative approach has been attempted in this study. To begin, the
underlying issues in ebb delta development are described first.
The opening of a tidal inlet channel is primarily a result of the breakthrough of a barrier
island (Brown, 1932). This breach can occur during storm conditions when water mass
3
associated with the storm surge and waves is transported across the barrier island, consequently
eroding the beach, the dune system and providing a low relief where tideinduced currents may
contain enough energy to maintain a permanent channel across the barrier island. There are also
cases where coastal inlets are intentionally relocated for the purpose of erosion control along the
downdrift shoreline (Kana and Mason, 1988).
Barr I er
Is I and
s nd  Ebb De I ta
Inlet
Channe I
Figure I1. General sectional view of an inlet channel through a barrier island and
associated ebb delta.
Although barrier island breakthrough is the primary reason for the natural opening of
tidal inlets, many of the resultant openings tend to be unstable, and lead to closure after a
comparatively short period of time. Escoffier (1940) considered the relationship between the
maximum flow velocity through the inlet throat and the throat crosssectional area to determine
if the inlet is stable or unstable. This criterion was based on the sediment depositional conditions
which would either enlarge or reduce the initial crosssectional area. Bruun and Gerritsen (1960)
proposed a stability criterion in which the ability of the channel current to remove the littoral drift
deposited in the channel is considered for the condition for the inlet to remain open. The ratio
of the spring tidal prism to the littoral drift, P/M, is thus a measure of inlet stability. Bruun
(1977) later quantified this relationship for inlets ranging from unstable "overflow channels" to
stable channels with little ebb delta formation and good flushing characteristics. O'Brien (1971)
4
noted that wavetransported material deposited near a coastal inlet may close the inlet depending
on the relative strengths of wave and tidal forces. A greater wave energy would be required to
close a larger inlet than a small one due to the larger tidal prism of the former.
The essential approach to describing the evolution of a ebb delta is to begin with a new
inlet or one that has been closed for a period of time and therefore has no delta. As noted, the
breakthrough of a barrier island by a storm event or artificial means provides an initial opening
between the back barrier bay or lagoon and open water. Once the inlet breaksthrough, a free
jet forms on the seaward side of the inlet as a result of tidal current (Oertel and Dunstan, 1981).
A free jet may also form on the landward side if it discharges into an unrestricted basin. In any
event, due to the initial opening and the formation of the jets, an initial accumulation of material
both on the landward side (flood delta) and the seaward side (ebb delta) of the inlet will occur.
For this study, only the ebb delta accumulation is investigated.
The accumulation of the ebb delta material can be derived from the initial formation of
the cut, especially in areas where the littoral drift is comparatively small. Thus, for example
Oertel (1988) found that approximately 78 percent of the 31.6 x106 m3 of material in the ebb
delta of Quinby Inlet in Virginia consisted of the sediment derived from the newly opened
channel.
A theoretical approach by Ozsoy (1986) presented a model whereby the mass transport
of material is the result of turbulent free jet issuing from the inlet. Thus, a portion of the
suspended sediment supplied by littoral transport is entrained into the inlet opening during the ebb
(as well as flood) phases of flow is ultimately deposited on to the ebb delta (Figure 12). The
depositional pattern of the ebb delta is dependent on the characteristics of the ebb flow jet and
on the wave energy. A low wave energy sandy coast may be characterized by marginal sand bars
advancing seaward and consist of greater quantity of material as compared to high wave energy
5
coasts where the increased wave forces transport the material toward the inlet and nearshore area
(Walton and Adams, 1976).
Ebb Jet  Ebb De Ita
Boundary
I n I et
Channe
Entra I nment of
IIttoral current
and associated
suspended sediment
Figure 12. General characteristics of an ebb jet and delta.
The sediment deposited in the ebb delta can be considered to be contained within a
seafloor area characterized by the area obtained by dividing the spring tidal prism by the water
depth. In Oertel's (1988) conceptual model, material is deposited over an area just beyond the
distal end of the "nearfield" jet as the flow velocity decreases below a certain critical value.
This deposition area, defined as the "farfield", tends to be fanshaped resulting from the lateral
spreading of the jet in the presence of landward approaching waves.
According to O'Brien (1969), the capacity of tidal current in maintaining the channel is
characterized by the maximum velocity in the inlet throat as determined by the tidal prism and
crosssectional area of the inlet opening, whereas, as noted, the current velocity further seaward
governs the deposition or erosion of material within the delta footprint. This characteristic
6
current velocity can be assumed to occur at a point between the inlet mouth and the outer edge
of the ebb delta. This location thus delineates the shoreward end of the "farfield" defined by
Oertel (1988).
The deposition or erosion of material within the confines of the delta footprint is due to
the bottom stress produced by the tidal current. Superimposing oscillatory currents due to wave
action complicates sediment motion near coastal inlets (O'Brien, 1969). The primary result of
wave action is the suspension of material which is then transported by the tidal current. The
critical shear stress for resuspension and transport is dependent on the diameter of the sediment.
If the critical shear stress is exceeded by the combined hydrodynamic stress due to current and
waves, then the seafloor can be expected to be eroded. However, if the combined stress is less
than that of the critical value then material will deposit within the confines of the delta area.
Thus, starting with a situation in which waves are absent, the rate of sediment deposition, hence
delta growth, decreases as waves appear and increase in intensity, until such a condition when
there is no deposition. Further increase in wave action would then cause the delta volume to
decrease due to bottom scour.
As the seafloor elevation rises, and decreases the water depth, not only does the ebb
current velocity increase over the delta in comparison with that at the inlet mouth by continuity,
but the decreased depth also results in increased wave shoaling, thereby increasing orbital velocity
and the wave shear stress contribution. Thus, under constant tide and wave conditions, the
seafloor can only rise to an elevation where there is no further deposition when the combined
hydrodynamic stress equals the critical shear stress. At this stage the ebb delta will have evolved
to an equilibrium volume and shape.
7
1.3 Study Objectives and Tasks
From the above description it is seen that in a given geologic setting, the rate of growth
of ebb delta and its ultimate equilibrium volume depend both on tidal energy and wave energy.
The ebb jet serves to deposit littoral material at a distance from the inlet where the flow velocity
is small enough to be conducive to deposition. Conversely, waves tend to enhance resuspension
and transport of deposited material away from the ebb delta. Thus, as noted, with increasing
wave action one may expect the rate of ebb delta growth to decrease, and the equilibrium volume
to decrease as well. It is the main objective of this study to examine the dependence of the inlet
ebb delta growth rate and the equilibrium volume on tidal, wave and sedimentary characteristics.
This objective is met by completion of the following tasks:
1. Development of a database consisting of inlet ebb delta volumes, throat crosssectional
areas, spring tidal prisms, significant deep water wave heights and wave periods, and
spring tidal ranges.
2. Reexamination of the procedure for ebb delta volume estimation developed by Dean and
Walton (1973) to determine the variation in the estimated delta volumes arising from the
interpretation of bathymetric conditions.
3. Examination of the influence of wave energy on the ebb delta volumetidal prism
relationship following the work of Walton and Adams (1976).
4. Development of a diagnostic model to examine the role of tidal energy and wave energy
in influencing the ebb delta volumetime curve, starting with the novolume condition.
5. Examination of model sensitivity to important governing physical parameters.
6. Examination of the trends in the timevariation of the delta volume with the help of the
model at selected inlets.
CHAPTER II
LITERATURE REVIEW
2.1 Overview
This chapter provides a literature review of relevant studies on ebb delta volumes, inlet
sediment transport and inlet stability. The review of ebb delta volume studies and the physical
processes effecting the inlets in Section 2.2 include 1) measurements of ebb delta volumes; 2)
capacity of deltas to store sand; 3) coastal physical parameters influencing ebb deltas; and, 4)
potential sources of ebb delta material. The second portion of this review in Section 2.3 focuses
on studies on sediment transport relevant to the deposition or erosion of the ebb delta, including
5) prediction of sedimentation in entrance (approach) channels; 6) sedimentation within inlet
channels; 7) erosion/deposition in entrance channels; 8) determination of the removal ratio of
suspended matter in settling basins; and 9) a procedure to quantify the rate of shoaling after ebb
delta dredging. Section 2.4 examines the stability of coastal inlets. Finally, in Section 2.5,
findings based on this review that are relevant to the modeling effort in Chapter IV are
summarized.
2.2 Ebb Delta Studies
2.2.1 Dean and Walton (1973)
Dean and Walton (1973) provided a qualitative description of the hydraulic and
sedimentary processes in the vicinity of coastal inlets. These authors compared inlets to the
9
idealized form of a surface nozzle, where the ebb jet is analogous to the turbulent jet issuing from
the nozzle. On the other hand, the flood flow conforms to a uniform concentric sink flow
converging toward the inlet. Thus, the laterally spreading ebb jet entrains water from the
adjacent shoreline toward the inlet, while during the flood flow water is transported both toward
and into the inlet. As a result, the combination of the ebb and flood flows causes the current to
transport sediment toward the inlet from the adjacent shoreline during all tidal stages. The
sediment thus transported is ultimately removed by the ebb flow and is deposited offshore. Dean
and Walton (1973) noted that the forces acting to transport sediment are not only dependent upon
the effects of ebb and flood currents, but also on waves forces which tend to drive the material
deposited as an delta towards the inlet. These forces have no net effect when the delta volume
reaches a state of equilibrium, at which point it does not exhibit growth or reduction in its
volume.
Twentythree Florida inlets were investigated to estimate the quantities of sand in the ebb
deltas. A method was developed for estimating the volume of accumulated sand. The first step
was to construct a bathymetric record of the inlet and its adjacent shores. A representative shore
normal profile was then determined from contour lines on either side of the inlet corresponding
to the bathymetry of a "noinlet" condition. Contour lines connecting these two profiles were
overlain on the delta and a grid superimposed on the chart. The depth differences at each corer
of the square were measured and averaged for the square. Finally, the summation of the
averaged grid depth differences multiplied by the grid area yielded the ebb delta volume.
Since the estimated delta volumes may be subject to individual interpretation bias, each
author calculated the same eight ebb delta volumes from their individually interpreted "noinlet"
contours. This exercise resulted in delta volume differences of less than 30 percent, with the
inlets with the large volumes having differences of less than 10 percent.
2.2.2 Walton and Adams (1976)
These authors investigated the capacity of deltas at inlets to store sand. The volumes for
six flood tidal deltas were calculated from comparative surveys ranging over time periods from
seventeen to seventyseven years. Preliminary analysis showed that the flood deltas may reach
an equilibrium size over a period of time. It was considered by the authors that the ebb deltas
also reach an equilibrium state. By knowing the volume of sand deposited in the flood and ebb
deltas would give an estimate of the volume of material that would be removed from the adjacent
beaches, thus providing an approximation of the effects to the adjacent shoreline due to the inlet.
Ebb delta volumes for fortyfour tidal inlets around the United States were calculated
using the method described in Dean and Walton (1973). The volume of each of the deltas was
calculated two or more times to obtain a range of values due to the individual interpretation of
the "noinlet" contour lines. This exercise generally resulted in a volume deviation of less than
10 percent.
Tidal prisms were correlated with ebb delta volumes categorized by three wave energy
levels in an effort to explain any effects on the volumes due to differences in the wave energy.
The square of the product of the deep water wave height and period was used to parameterize the
wave energy, thus providing a quantitative description of the available energy to modify the ebb
delta. The wave energy parameter ranges were arbitrarily chosen from 030 (mildly exposed),
30300 (moderately exposed) and > 300 (highly exposed). The data were plotted on a loglog
scale with a linear regression line equation of
V = aP1 (2.1)
where V = ebb delta volume (m3) and P = spring tidal prism (m3). The coefficients a and b
found for the three wave energy regimes are given in Table II1. The dependence of these
coefficients on the wave energy regime in Table II1 implies that the relationship between the
11
delta volume and tidal prism is not unique in the sense that it depends measurably on the
prevailing wave energy.
Table II1. Coefficients of Eqn. 2.1 obtained by linear regression for the relationship of tidal
prism with ebb delta volume.
Wave energy regime Coefficient, a Coefficient, b
High 5.33 x 103 1.23
Moderate 3.77 x 103 1.08
Mild 8.76 x 103 1.24
Source: Walton and Adams (1976)
2.2.3 Marino (1986)
This investigator described the general development of ebb deltas using a geomorphologic
perspective through inlet case studies on the east coast of Florida. These inlets were chosen due
to the unique morphology of each inlet and the associated differences in the development of the
deltas. This examination revealed some of the problems that are encountered in determining the
ebb delta volumes for those inlets where the bathymetric conditions are less than "ideal". For
example, at Ft. Pierce Inlet the updrift and downdrift profile lines have substantially different
slopes, so that the "noinlet" contour lines had to be interpolated between the updrift and
downdrift shorelines. Also, where inlets had significant shoreline offsets with respect to each
other, the grid size had to be decreased from 305 m2 to sizes ranging between 76 and 152 m2,
so that a greater bathymetric detail could be covered. Another example of the problems
encountered was the presence of offshore reefs at St. Lucie Inlet, Hillsboro Inlet, Port Everglades
Inlet and Government Cut. In these cases, the estimated ebb delta volumes may have been
measurably different than actually present, since the reef areas could not be distinguished from
sandy areas.
12
A dimensional analysis was done to determine the physical parameters that are important
in characterizing the ebb delta volume. Dimensionless parameters were found and the functional
relationship was reduced to
V Pw Ac Ei
f [ p 1 = 0 (2.2)
W W3 hoI a, Et
where V = ebb delta volume, w = width at the inlet throat, ho = depth at the throat, P = spring
tidal prism, Ac = crosssectional area at the inlet throat, aos = spring sea tidal amplitude, E, =
tidal energy parameter, and E, = wave energy parameter. Marino (1986) eliminated E,/E,, the
ratio of wave energy to tidal energy, because the wave energies considered were found to be
within the "moderate" range defined by Walton and Adams (1976). The nondimensional V/P
ratio, when plotted against the aspect ratio, w/ho, for different values of A/al did not produce
any strong trends. However, from the relationship, the investigator asserted that as w/ho
decreases, the ebb delta volume tends to increase, leading to the credence that delta volume
depends on not only of tidal prism but also on the aspect ratio.
The relationship of the ebb delta volume (m3) to the spring tidal prism (m3) was
determined by linear regression:
V = 5.59x104 p.39 (2.3)
2.2.4 Oertel (1988)
This investigator considered the ebb jet to be divided into two zones. The first, the near
field, was described as the zone of flow establishment extending from the inlet mouth to a certain
transition point. Within this zone the maximum ebb flow velocities occur. As the jet extends
past the nearfield flow, its boundaries spread out laterally from the inlet. The decrease in flow
13
velocity at the end of the nearfield results in material being deposited in the farfield. The
associated decrease in the water depth compresses the jet vertically and further increases the
lateral spreading resulting in a continual reduction in the flow velocity and a build up of the delta
material.
During a flood tide, the return flow toward the inlet does not form a reverse flow pattern
of the ebb tide; rather the flow distribution radially converges toward the inlet. The velocity field
is distributed over a greater nearfield area than during the ebb flow, thus resulting in lower near
field flood velocities than ebb velocities. Consequently, the summation of the nearfield flood
and ebb velocity vectors results in flood and ebb dominated zones. Oertel (1988) suggested that
the difference between the axially ebbdominated flow and the flooddominated marginal flow be
termed "net tidal delta flow". It is within this region that a spatial asymmetry in the shear
stresses tends to develop ebb marginal deltas.
Oertel (1988) described how resultant vectors of inlet and littoral currents may effect the
flow jet, thus reorientating the inlet gorge and the marginal delta for different types of inlet
situations. Thus, littoral sediment can be deposited in spits at the end of the barrier island, or
transported into the inlet. In the inlet, the material may accumulate in the inlet gorge or, if the
critical shear stress for scour is exceeded, be transported bayward to the flood delta, or seaward
to the ebb delta. Material that accumulates in the ebb delta may remain there to build the delta
itself or may be bypassed downdrift or updrift via marginal deltas and channels.
2.2.5 Havter. Hernandez. Atz and Sill (1988)
These investigators performed a study using a movable bed model to evaluate the effects
of tidal flow and wave action on ebb delta formation and to estimate the effects of shoal mining
on the delta and the adjacent shoreline. The results of the experiment were normalized to include
14
the effects of sediment size, specific gravity, and inlet crosssectional area via the parameter, G:
G = whd,(y, y) (2.4)
where dso = median grain size, y, = unit weight of sediment, and 7, = unit weight of sea
water. The relationship of the delta volume (m3), to the tidal prism (m3), was determined by
linear regression:
V = 4.8x104 1340 (2.5)
G
and
V = 6.9x104' P 1870 (2.6)
G G
where Eqns. 2.5 and 2.6 correspond to the experiments effected by no wave action and by wave
action, respectively. It was found that the data corresponding to the presence of waves was more
scattered than without waves. The two regression lines (Eqns. 2.5 and 2.6) also indicate that
experiments with waves formed larger deltas as compared to deltas without waves. The
explanation provided by the investigators was that more material moved offshore from the inlet
mouth as a result in the increased shear stress induced by the combination of waves and currents.
Secondly, the littoral transport of material into the inlet from the adjacent shoreline was enhanced
by wave action. This observation was however contradicted when the normalized tidal prism,
VIG, was compared to a normalized tidal prism, P/GE, where E is the dimensionless wave
energy density:
E H (2.7)
8h2
where H, = significant deep water wave height and h = inlet water depth.
The resulting regression line is
V = 4.76x103 (2.8)
G y GE
The data plotted in Hayter et al. (1988) and Eqn. 2.7 indicate that the delta volume will increase
with an increase in the tidal prism or with a decrease in the wave energy.
2.3 Sediment Transport Studies
2.3.1 Gole. Tarapore and Brahme (1971)
A method was developed by these investigators to predict sedimentation in entrance
channels based on the analysis of prototype and model data, and analytical studies. This method
was developed to estimate the quantity of material required to be removed from the channel
during maintenance dredging.
The total suspended load crossing a shorenormal channel in the alongshore direction was
calculated from the prototype and hydraulic model data. The prototype data were obtained during
seasonal conditions to account for the variation in tides, wave climate and seasonal sedimentation.
The subsequent analytic step was to determine if a sediment particle will settle out of the water
column and be deposited in the channel or cross over the channel. This criterion was derived
in terms of particle fall velocity. This procedure ultimately determined the fraction of the total
sediment load which contributes to the sedimentation of the channel. It was assumed that the
suspended sediment carrying capacity of the current is proportional to the square of the flow
velocity. Thus, as a result of dredging and the associated increase in the depth of flow, the
alongshore flow velocity across the channel decreases. This decrease in flow velocity results in
the net sedimentation load of the dredged channel (S,):
S = KL, U C, t hd d (h (2.9)
[Udh hChJ
where K = coefficient of actual sedimentation to effective sediment load derived from dredging
records of existing channels, L, = length of dredged channel, Ua = alongshore current velocity
adjacent to dredged channel, Cs = depthaveraged suspended sediment concentration, t = time
(months), hd = water depth in the dredged channel, wd = dredged channel width, W, = particle
settling velocity, ha = water depth adjacent to the dredged channel, and Ud = alongshore
velocity across the dredged channel.
2.3.2 Sarikaya (1973)
This study investigated the removal ratio of suspended sediment in settling basins. This
ratio, rR, is defined as the settled suspended sediment within the deposition area to the suspended
sediment entering the area:
r = l_C (2.10)
where C, = uniformly distributed suspended sediment concentration and Cb = concentration of
suspended sediment entering the settling basin. The differential equation for suspended sediment
transport in a settling basin was derived from the law of conservation of sediment mass. Written
in dimensionless form:
dC _W d 1 d d (2 )
dX U, dY Uh dY dY
where X = xl/h, xl = distance in direction of mean flow, h = water depth, Y = y/h, y =
vertical distance from the bottom, C = C/Cb, U, = flow velocity in the x1direction, and e =
17
turbulent diffusion coefficient. The removal ratio was obtained by employing a finite difference
method, with the initial conditions assuming a uniformly distributed sediment concentration at the
inlet, no reentrainment of sediment particles from the seafloor, and no sediment is introduced
at the water surface. The removal ratio was determined as a function of Whz/2e, where e,
crosssectional averaged turbulent diffusion coefficient, and WF/WV, where Wo = settling velocity
of a particle that reaches the seafloor at a distance from the inlet, if turbulence is neglected. This
relationship indicates that as the effect of turbulence increases Wzh/2e decreases thereby reducing
the removal ratio resulting in comparatively smaller sediment deposition volumes. Conversely,
higher values of the parameter WF/W increase the removal ratio indicating the dependence of
sedimentation on the settling velocity.
2.3.3 Galvin (1982)
This investigator presented a method to determine the channel shoaling rate and a time
estimate to shoal the entrance channel from an initial overdredged depth to a project design
depth. The shoaling rate of the channel was considered to be dependent on the littoral sediment
deposited in the dredged channel. The shoaling rate (Sr) was given as
w ,= e Wd h, (2.12)
where L, = dredged channel length, wd = dredged channel width, Q = littoral transport rate,
R = fraction of Q which reaches the dredged channel, ho = initial water depth, h = water depth
at time t, and m" = exponent depending on whether the dredging increases the crosssectional
area resulting in a reduction in the channel velocity (m" = 5/2), or whether the velocity is the
same compared to predredging condition (m" = 3/2). The time interval, t,, for an overdredged
18
channel to shoal to a project design depth, hp, is defined as
p w (hdh)+ (InAInB (2.13a)
where
(h,h)
A (hh (2.13b)
(h h)
and
Sm" 1 hdho
1+
2 h
B h(2.13c)
+ 1 [hh]
2 h
Channel depth during shoaling versus time after dredging for three combinations of the
natural controlling depth and dredged channel depth were plotted in order to obtain a quantitative
evaluation of the effect of bypassing on channel shoaling for three initial depths dredged to 4
meters. It was found that: a) the curves of channel depth versus time indicated that the maximum
rate of shoaling occurred immediately after dredging; b) the channel shoaled faster when the post
dredging velocity was reduced (m" = 5/2) as compared to when postdredging velocity was not
reduced (m" = 3/2); c) the channel shoaled faster when no bypassing occurred (m" co); and,
d) a shallow dredged channel approached the natural controlling depth at a slower rate than a
deep dredged channel.
It was shown in this study that this method can be used to compute the duration that a
channel will be maintained at a certain project depth, test if a proposed dredging depth will last
a desired length of time, and estimate the decreased rate of bypassing as a result of the dredged
channel.
2.3.4 Buckingham (1984)
Buckingham (1984) studied the erosion of tidal inlet channels and the associated
sedimentation in the vicinity of the channels. A study was conducted using a fixed bed,
undistorted model that combined penetrating oceanic waves and tidal currents in the channel of
Jupiter Inlet, Florida. A bed stability parameter was used to evaluate the sediment
erosion/deposition potential at selected locations within the inlet entrance and interior waters.
This parameter quantitatively relates the flow velocities of the model data to the
erosion/deposition potentials. The bed stability parameter, Pbed, was defined as
P 1 (2.14)
cr
where Tb = bed shear stress due to waves superimposed on current and Tr = critical shear stress.
When b > Tr, (positive Pbd) erosion will occur and when Tb < cr (negative Pbed) material will be
deposited. Under turbulent conditions, the bed shear stress is proportional to the square of the
velocity. Thus, the velocities due to waves and current can be substituted in Eqn. 2.13 to obtain
the following equation for the stability parameter:
2
S l+k2 H m1 j 2 (2.15)
Pbe = 1k'e1
h H U 
where k' and m' are empirical coefficients found to range between 2 to 8 and 1.3 to 1.7,
respectively, HR = reference wave height, hR = reference water depth, H = wave height at the
site in question, h = water depth, U, = current velocity, and Ucr = critical velocity for erosion.
Two limiting assumptions were made in the development of this relationship. First, the critical
shear stress to erode the seafloor was considered to be equal to the shear stress below which
20
sediment will deposit, and secondly, the critical shear stress under waves and current was
considered to be equal to that obtained from Shields' (1936) relationship for incipient grain
motion under steady flow. Thus, the second assumption neglects the shear stress contribution
from wave action. The bed stability parameter was found to produce reasonable results when
interpreting hydrodynamic data to compare the erosion and deposition potentials using fixed bed
models involving both waves and currents within the inlet.
2.3.5 Ozsoy (1986)
This investigator examined transport and sedimentation processes in the vicinity of a tidal
inlet through analytic modeling of sediment transport associated with the turbulent jet produced
by ebb tidal flow. The jet was divided into a "zone of flow establishment" (ZOFE) and a "zone
of established flow" (ZOEF), where the ZOFE is defined from the inlet mouth to a seaward
location where the flow is not fully influenced by the jet boundary shear, hence has a constant
jet centerline velocity and the ZOEF extends seaward of that point, with decreasing jet centerline
velocity.
The analytic model produced a nondimensional distribution of the settlement rate within
the boundaries of the jet. It was found that two maxima and a minimum at the centerline
occurred. The two maxima correspond to the marginal shoals located along both sides of the jet
centerline. Sediment settling was comparatively small along the centerline of the jet due to the
high current velocity. There was no deposition of material outside of the jet boundaries due to
the small or zero ambient sediment concentration.
It was found that bottom friction, bathymetry, sediment settling velocity and the initial
inlet velocity influence the depositional patterns within the jet boundaries. As bottom friction
increased, jet flow sediment concentrations were more rapidly reduced. This resulted in material
21
deposition relatively close to the inlet mouth in building a fanshaped delta. A change in the
bottom slope did not result in any significant change in the concentration within the jet, but did
transport the material further seaward resulting in elongated marginal shoals. The settling
velocities, on the other hand, played a significant role in determining sediment distribution.
Sediments with larger settling velocities (coarser material) were deposited closer to the inlet and
the marginal shoals, while low settling velocity, finer material was jetted further offshore. Flow
velocities were also shown to be influential on sedimentation patterns. When the initial velocity
was less than the critical velocity for erosion, larger deposits of material were found to occur at
the mouth or within the jet core (in ZOFE). When the initial velocity was equal to the critical
velocity, it was found that material is deposited further seaward in the marginal shoals. Finally,
when the initial velocity was greater than critical, scouring occurred and material deposited
further offshore than the marginal shoals, outside the active zone of sediment transport. Ozsoy
(1986) found that the predicted depositional patterns resulted in reasonable qualitative similarity
to those at prototype coastal inlets under certain hydrodynamic conditions.
2.3.6 Walther and Douglas (1993)
These investigators developed a procedure to quantify the shoaling rate within an offshore
dredged area within the ebb delta and sediment transport rate across the inlet after dredging. The
postdredging of the ebb delta will reduce bypassing rate around the inlet, hence the estimation
of this reduction is a matter of considerable engineering interest. The quantity deposited in the
dredged area is determined by the sediment transport ratio, TR, from the work of Gole et al.
(1971) expressed in terms of pre and postdredging depths:
T = h[ 1 (2.16)
where ho = initial water depth, hd = water depth in the dredged channel, and m" = 5/2. The
bypassing rate can then be determined from an estimate of the predredging littoral transport rate.
The shoaling rate (VT) within the dredged area is estimated as the difference between the pre and
postdredging transport rates:
V = QP,(1T,) (2.17)
where Q, = predredging littoral transport rate. As sediment is deposited in the dredged area,
the depth and the trapping rate will decrease, while the sediment bypassing rate will increase.
A comparison of two hypothetical examples were presented to illustrate the recovery and
bypassing rates for a deep and a shallow sand borrow area within the ebb delta. The rate of
littoral material to the ebb delta, the predredged depth and the dredged volume were considered
constant for both cases, with a variation in the borrow area dimensions. The bypassing rate and
the cumulative volume of sand bypassed versus time were determined. The investigators found
that the initial bypassing rate of the shallow cut was approximately twice the rate for the deep cut.
However, the bypassing rate increased faster for the deep cut with the bypassing ratetime curve
intersecting the shallow cut curve by the fifth year, for the particular set of parameters chosen.
The increased bypassing rate for the deep cut resulted in full recovery to the predredging
condition by the ninth year as compared to the fourteenth year for the shallow cut. The
cumulative volume bypassed at the end of the fourteen years was within one percent of each other
for the two cases. It was concluded that the results using this method were sensitive to the
assumed sediment transport rate to the dredged area. As noted, it was also shown that, initially,
the deeper cut trapped more sediment and reduced the bypassing rate compared to the shallow
cut, but eventually the same cumulative sand volume was bypassed in both cases.
23
2.4 Inlet Stability Studies
2.4.1 Bruun and Gerritsen (1960)
Bruun and Gerritsen (1960) proposed a stability criterion in which the condition for the
inlet to remain open was considered to be dependent on the ability of the channel current to
remove the littoral drift deposited on the ebb delta. The ratio of the tidal prism passing through
the inlet during a onehalf tidal cycle to the annual average littoral drift reaching the inlet, P/M,
was proposed as a measure of inlet stability. Bruun (1977) later quantified this relationship for
sandy coastal inlet (Table 112). As the tidal prism increases relative to the littoral drift, thus
increasing the stability ratio, P/M, the inlet entrance has the tendency of maintaining itself.
Conversely, as the stability ratio continues to decrease, the seafloor elevation rises until the inlet
is subject to closure.
Table 112. Bruun's stability criterion of sandy coastal inlets related to ebb delta size and
bypassing.
Ratio range, P/M
P/M > 150
100 < P/M < 150
50 < P/M < 100
20 < P/M < 50
P/M < 20
Inlet conditions with respect to navigability and stability
Conditions are relatively stable and good, little ebb delta formation
and good flushing.
Conditions become less satisfactory, and ebb delta formation
becomes more pronounced.
The ebb delta may be rather large, but they can usually still be
navigated by shallow draft vessels ....
.... all inlets are typical "deltabypasses" .... For navigation, they
present "wildcases", unreliable and dangerous.
.... entrances appear as unstable "overflow channels" rather than
permanent inlets.
Source: Bruun (1977)
I
2.4.1 O'Brien (1971)
O'Brien (1971) reviewed the report by Saville et al. (1957) on a moveable bed hydraulic
model study of an inlet and noted that an inlet that is in equilibrium is due to a balance between
the wave energy which tends to close an inlet and the tidal energy which maintains the opening.
Thus, the ratio, a, of normal incident wave energy over one tidal period to the tidal energy
through the inlet over a tidal period can be used to evaluate the stability of an inlet:
a = T, (2.18)
2 a., ,P
where Po = total wave power, w = width at the inlet throat, T, = tidal period, y, = unit weight
of seawater, P = spring tidal prism, and 2ao, = spring tidal range. Given a single representative
deep water wave height Ho, Po is defined as:
P = 'Yg H2T (2.19)
32vr
where g = acceleration due gravity and T = wave period. Mehta and Hou (1974) and Sedwick
(1974) found a to be a reasonable indicator of inlet stability. The stability coefficient, a, was
plotted against the tidal prism and a line separating relatively low values of a representing
stability from a region of high values of a representing instability was drawn.
2.5 General Conclusions
With regard to the objective of the present study and the modeling effort described in
Chapter IV, the studies reviewed in this section lead to the following relevant observations:
1. Inlets can be modeled as an idealized form of a surface nozzle, where the ebb jet is
analogous to the turbulent jet issuing from the nozzle, and the flood conforms to a
uniform concentric flow converging toward the inlet (Dean and Walton, 1973; Oertel,
1988; and Ozsoy, 1986).
25
2. The relationship between the delta volume and the tidal prism depends measurably on
prevailing wave energy (Walton and Adams, 1976; Hayter et al., 1988).
3. Suspended sediment concentration, grain size, sediment settling velocity and the initial
inlet velocity influence the depositional patterns with the ebb jet boundaries (Gole et al.,
1971; Sarikara, 1973; Ozsoy, 1986)
4. Flow velocities due to waves and current related through the bed stability parameter,
Pbd, determine the sediment erosion/deposition potential (Buckingham, 1984).
5. Ebb jet spreads out laterally and the flow velocity decreases at the end of the nearfield
resulting in the material being deposited in the farfield (Ozsoy, 1986; Oertel, 1988).
6. Curves plotting the water depth during shoaling versus time after dredging indicate that
the maximum rate of shoaling occurred immediately after dredging (Galvin, 1982).
7. Deeper dredged cuts trap more sediment and reduce the bypassing rate as compared to
a shallow cut, but eventually the same cumulative sand volume is bypassed (Walther and
Douglas, 1993).
8. The ratio of the tidal prism to the annual average littoral drift reaching the inlet is an
indicator of conditions for navigation through the entrance. As the tidal prism decreases
relative to the littoral drift, the ebb delta grows and ultimately hinders navigation (Bruun
and Gerritsen, 1960).
9. The stability coefficient, a, defined as the ratio of longshore wave power to the tidal
power can be a reasonable indicator of stable versus unstable inlets (O'Brien, 1971;
Sedwick, 1974; Mehta and Hou, 1974).
CHAPTER III
METHOD FOR DETERMINING
DELTA VOLUME AND GROWTH
3.1 Database
Data included in this compilation of the relevant inlet parameters were primarily obtained
from existing sources, and secondarily calculated in this study. These parameters include ebb
delta volumes, inlet throat width, average water depth at the throat, throat crosssectional area,
spring tidal prism, annual average deep water significant wave height and wave period, and the
spring sea tidal range. These parameters are tabulated, and when available, by year for each
inlet, in Appendix A. The majority of the data were obtained from the literature in the Coastal
Engineering Archives of the Coastal and Oceanographic Engineering Department (COE) at the
University of Florida. Specifically, reports and studies published by COE and other universities,
private engineering consultants, Florida Department of Natural Resources (FDNR), and the U.S.
Army Corps of Engineers. Information was also obtained from the periodicals and journals
housed in the Archives, as well as nautical charts and bathymetric surveys.
3.1.1 Ebb Delta Volumes
Ebb delta volumes were found for eightyone inlets on the three coasts of the United
States, of which fortyfive are in the State of Florida of the 189 volumes, 182 were found in the
literature and 7 estimated as described in Section 3.2.
3.1.2 Tidal Prism
Spring tidal prisms in Appendix A have either been reported in other documents or have
been obtained from Jarrett (1976) based on the formula of O'Brien (1969), knowing the throat
area. O'Brien (1969) found the following relationship between the throat crosssectional area and
the spring tidal prism:
P = aA b(3.1)
where Ac = throat crosssectional area (m2) and P = tidal prism (m3). The coefficients a, and
b, where found for sandy inlets in equilibrium under a semidiurnal tide. Jarrett (1976) later re
analyzed this relationship based on data from inlets along the Atlantic, Pacific and Gulf of
Mexico coasts. In this study, the tidal prisms for both the Atlantic and Gulf of Mexico inlets
were calculated with the coefficients determined for 0, 1 or 2 jetties given by Jarrett (1976).
Coefficients for the Atlantic inlets, a, = 1.94 x 104 and b1 = 0.95, and for the Gulf of Mexico,
a, = 4.06 x 103 and b1 = 1.19 were used to calculate tidal prisms.
3.1.3 CrossSectional Area
Minimum or throat crosssectional area for the inlets are reported from existing
documents, or calculated from United States Geological Survey (U.S.G.S.) quad sheets or
National Ocean Service (NOS) Nautical Charts. Crosssectional areas were obtained by
measuring across the distance across the inlet throat to obtain the inlet width. The width was
then multiplied by an estimated average depth across that line.
3.1.4 Significant Wave Height and Wave Period
Deep water significant wave heights, H,, were obtained from Hubertz and Brooks (1989)
and Hubertz et al. (1993). The hindcast wave information covers a twenty year period from 1956
to 1975 for 108 stations along the Atlantic coast and 51 stations along the Gulf of Mexico coast.
Yearly average height from the station closest to the inlet of interest was used for the year
corresponding to the volume.
28
Mean wave periods were obtained from U.S. Army Corps of Engineers, Coastal
Engineering Research Center (CERC) (Brooks, 1994a, 1994b unpublished) through a special
request. A twentyyear average characteristic value was calculated from these data for the same
stations for which wave heights were obtained.
3.1.5 Spring Sea Tidal Ranges
Spring tidal ranges, 2as, were calculated from Balsillie (1987a, 1987b, 1987c) for the
coast of Florida as the difference of the mean higher high water and mean lower low water
levels. Tidal ranges at locations outside Florida were obtained from two sources: 1) USGS quad
sheets; and 2) NOS Nautical Charts.
3.2 Ebb Delta Volume Estimates
The purpose of this exercise was to examine the procedure developed by Dean and
Walton (1973) for estimating the ebb delta volumes. The first task was to estimate the necessary
volumes, and the second to evaluate the effect of interpretation of bathymetric conditions on the
estimated volumes.
3.2.1 Ebb Delta Volumes
Seven bathymetric surveys were selected to estimate delta volumes. The next step using
each survey was to draw shoreperpendicular lines on either side of the inlet to obtain a
representative "noinlet" profile. These two lines were located at a distance from the inlet in an
effort to minimize the influence of the inlet on the contours. Equal depths along the two lines
were connected to develop a "bestfit" contour configuration overlain on the survey (Figure III1).
The "bestfit" contours, although drawn somewhat subjectively, represent the bathymetric
conditions as if no inlet was present. A grid pattern was then superimposed on the survey. The
size of the grid cells selected was somewhat subjective; however, the choice in general is
dependent upon the complexity of the bathymetric relief to detail the changes in elevation. If the
29
bathymetry shows depth undulations with irregular elevation changes or small pockets of sand
bars, then the grid cell size should be small. However, if the change in elevation is
comparatively smoother, more gradual, the grid cell size can be larger. The elevation differences
at each of the four intersections were estimated within 0.15 meters and averaged for the cell.
The average elevation of each of the grid cells were then summed and multiplied by the total
surface area of all the grid cells to obtain the ebb delta volume.
SNaturaI Contour
......... ............... ........... ... ...........
......... ...... ...............
... Se......... .......eward Contour CA)
.......  
Landward Contour CC ..'
." "BetrPl Contour CB3
.. .................
 ..
Barrie Brrir Is 1 3n
Figure III. Schematic of three "noinlet" contour conditions in estimating ebb delta
volumes. Seaward exaggerated contours, A (dashed lines), "bestfit"
contours, B (solid lines) and landward exaggerated contours, C (thick
dotted lines).
3.2.2 Sensitivity Analysis
Dean and Walton (1973) had each estimated the volumes of eight ebb deltas to evaluate
the differences in the quantities due to the individual interpretation of the natural "noinlet"
contours. The intent of the following exercise was to examine changes in the volumes due to:
1) varying the position of the "noinlet" contours, and 2) changing the grid cell size. For each
of the seven bathymetric surveys mentioned in Section 3.2.1, three "noinlet" contour positions
(Figure III1) were used to estimate the delta volumes. The first step was to determine the
30
volumes from the "bestfit" contours. Next, two additional volume computations were made for
each of the surveys by slightly exaggerating the contours landward and seaward, with the ends
remaining at approximately the same locations. The distance of the landward and seaward
exaggerated contours were subjective, but an attempt was made to retain a reasonable
representation of the "noinlet" contours. The results of this exercise are given in Section 4.1.1.
The next issue was to examine the variation in the volumes due to a change in the grid
cell size (Figure II2). This was undertaken by doubling the size of the initial grid cells for each
of the volumes computed as described above. The elevation differences of the initial grid cell
located at the intersections of the doubled grid size were used to calculate the average elevation
difference each of the grid cells.
Figure III2. Schematic of the initial grid and enlarged grid cell patterns used in
estimating ebb delta volumes.
31
3.3 Diagnostic Examination of Seafloor Evolution
As noted in Section 2.2.2, with the opening of an inlet the ebb delta volume increases
as the inlet tidal current deposits material derived from the littoral system and ultimately reaches
an equilibrium volume when the condition of no net deposition is attained. This process of
monotonic accumulation is influenced by wave action and its seasonal as well as yeartoyear
variation. To examine the influence of the effects of current and waves on the growth rate of ebb
deltas, a diagnostic approach is developed. The growth process of the delta will have an initial
condition of a new inlet with no delta present (Figure I3.A). Delta accumulation height, d,
will be simulated by modeling tidal currents and superimposed waves to determine the combined
shear stress, 7b. The seafloor will continue to rise on the condition that the combined shear stress
is smaller than the critical value, rcr for scour/deposition (Figure III3.B). The model must then
determine the delta volume when the seafloor reaches an equilibrium elevation, de, due to a
balance of the shear stresses, i.e. ,b = rcr (Figure III3.C), and estimate the time for the
equilibrium condition to occur. Derivation of the model is presented in Section 3.3.1 and the
corresponding Fortran algorithm is found in Appendix B.
An important point to note is that for the model development, the tidal period will be
considered to be the smallest timescale. In another words, processes which actually occur over
flood and ebb phases of flow will be treated in a composite manner over the duration of the tide.
3.3.1 Model Development
The net decrease of suspended sediment mass per unit delta bed area, m, with respect to
time is given by
= F (3.2)
dt '
A:) Barrier Island
UI U "
I
.. L0 'cr Seaf I oor
Inlet Channel
B)
U  U Uc b <%cr
C)
U Uo  c r
ho r ^//
F W MMIde
Incrementa I Deposition
Figure UI3. Seafloor growth diagrams illustrating the evolution of an ebb delta: A) initial
condition, B) transient condition and C) equilibrium condition.
where F, = settling flux is defined as
F, = i'C, (3.3)
where W', = effective settling velocity of the particles and C, = depthaveraged suspended
sediment concentration. The effective settling velocity is defined as
pW, (3.4)
where = probability of sediment deposition and W, = particle settling velocity. Krone (1962)
characterized p according to
p = (3.5)
Tcr
where Tb = combined bed shear stress due to waves and current, and cr = critical bed shear
stress. Deposition can occur only when Tb < Tr (p > 0). The settling velocity (Schiller etal.,
1933) can be expressed as
W, a P P, (3.6)
W 3CD Pw
where p, = particle density, p, = seawater density, dso = median particle size, g = acceleration
due to gravity, and CD = drag coefficient. The value of CD outside the Stokes range (Reynolds
number < 1) decreases rapidly then levels off and becomes nearly constant (e.g. 0.43 for spheres)
in the fully turbulent flow regime considered.
Equation 3.2 can thus be expressed as
= Hf b)W, (3.7)
where Hf [x] = heavyside function such that Hf [x > 0] = x, and Hf [x 0] = 0. Next,
pd = AD ] (3.8)
where pd = dry bed density, AD = ebb delta deposition area, m = mass, and V = delta volume.
Furthermore,
dV = dhA, = d(d)AD (3.9)
where dh = change in water depth and d(d) = change in ebb delta height. Substituting Eqns.
3.8 and 3.9 into Eqn. 3.7 results in:
dh W. CIC, ib (3.10)
dt Pd cr)
Given W,, C,, and Pd, Eqn. 3.10 can be solved provided 7b and Tcr are determined.
Komar and Miller (1974) found that data for sediment threshold under oscillatory flows closely
agreed with Shields' (1936) relationship for incipient grain motion under unidirectional flows.
Thus, the work by Shields (1936) for fully turbulent flow can be used to determine the critical
shear stress, cr, for waves and current combined:
Tr, = 0.058(p,p)gdso (3.11)
where p, = density of seawater. Grant and Madsen (1978) prescribed the following relationship
for shear stress, TI,, due to both current and waves:
4 = 0.5 p, f U (3.12)
where U, = total velocity due to waves and current at the seafloor. The friction factor due to
the combined current and waves, fw, is equal to
SIUc Ic f+ IlUb w (3.13)
where fc = friction factor due to current, f, = friction factor due to waves, Ucb = nearbed
current velocity over the delta, Uwb = nearbed orbital velocity due to waves can be obtained
from linear wave theory (Dean and Dalrymple, 1984a):
UWb H cosh(kh) (3.14)
T sinh(kh)
where k = wave number equal to 2r/L, L = wave length, T = wave period and h = water depth
at the delta. As a wave train propagates from offshore into shallower water, the wave height
35
changes as the depth changes. According to linear wave theory (Dean and Dalrymple, 1984b),
the shoaled wave height, H can be expressed as
H H = 2c b, (3.15)
where bo= distance between two adjacent deep water wave rays, b, = distance between two
adjacent nearshore wave rays, Co = deep water wave celerity equal to gT/2ir, C = shallow water
wave velocity equal to (gh)'2, and H,= deep water wave height. Waves approaching the
shoreline in shallower water begin to decrease in celerity and the wave crests tend to align
parallel with the contours of equal bathymetry. As the delta grows these contours are extended
seaward, thereby causing refracting waves to transport material from the adjacent updrift and
downdrift beaches toward the inlet. However, ignoring the refraction process for simplicity, for
the derivation of this model the contours are assumed to remain straight and parallel. Thus, the
refraction coefficient (bo/b,)u2, is set equal to unity.
Grant and Madsen (1978) define U,, the total current velocity due to waves and current
as
Ut = (U b+U,+2Ucb Uwbcos)2 (3.16)
where 4 = angle between the current and wave direction. During ebb flow, when 4 = 7, the
momentum of the ebbing water mass causes the incoming waves to break seaward of the ebb
delta. Conversely during flood flow, when 0 = 0, the waves are able to penetrate over the delta
and into the inlet channel. It is assumed that the combined effect of waves and current over a
tidal period is represented in this model by selecting 4 = 0.
The material is deposited over a certain area as the ebb flow velocity decreases. By
considering the ebb jet boundaries to remain attached to the shoreline, and the entrainment of
36
ambient water into the ebb jet small compared to the tidal prism, this area (Ap) is notionally
defined as the tidal prism divided by the depth of flow (Figure III4.A). The relationship
between Ap and the ebb delta deposition area, AD, is developed in Section 3.3.2.1.
A) B3
AP
AD
I n let
Channe I
Figure 114. Plan view of A) tidal prism based area (Ap), and B) delta deposition area
(AD).
The first step to determine the current velocity over the ebb delta is to obtain the
maximum velocity through the inlet for a spring tide is found from O'Brien (1969):
U 0.86P (3.17)
tA,
where P = spring tidal prism, T, = tidal period, and Ac = throat crosssectional area of the inlet.
The average inlet velocity over onehalf tidal cycle is then obtain from
2 U. (3.18)
As the flow exits the inlet channel it is considered to spread out from the inlet mouth.
To obtain a characteristic velocity, Uo, at the shoreward end of the deposition area, this velocity
is assumed to occur along an arc, onehalf the distance, re, from the entrance mouth to the outer
37
edge of the area A, (Figure mI4.A), where re is obtained from continuity according to
r 2A= p (3.19)
Thus, Uo is obtained from
Uo (3.20)
7r,
where w = width of the entrance.
As the seafloor rises, the water depth decreases with respect to the initial water depth,
whereby to maintain the continuity of flow, the current velocity over the delta, Uc, must increase
(Figures III3 and 1115). As Uc decreases with distance as the flow spreads out over the delta
from its inner to outer limit. For the present purpose, Uc will be defined as its value at the inner
limit of the delta. It should also be noted that the velocity profile of Uc is vertically uniform, it
is therefore necessary to apply a correction factor to obtain the nearbed velocity, Ucb.
From the logarithmic velocity profile (Mehta, 1978), the ratio of the nearbed velocity,
Ub, to the depth averaged current velocity, Uc, is defined as
Ub In(zbzo) (3.21)
U, Iln(h lz)
where Zo = theoretical origin of the logarithmic profile, and zb = distance above profile origin
and is set here equal to 0.05 m. From the ManningStrickler formula, zo can be obtained by
z, = 107n6
(3.22)
Ebb de ta
outer limit
Figure 1115.
Crosssectional view and plan view through the inlet and idealized ebb delta.
Mehta and Ozsoy (1978) noted that a representative Manning's n value of 0.028 can be used for
sandy inlets, and with an initial water depth 4.0 m used in Chapter IV, the ratio of nearbed
velocity to the mean velocity was determined. Thus, the current velocity obtained by continuity
is multiplied by a correction factor of 0.40:
(3.23)
Ulb O0.40UOh
where ho = initial water depth. Note that when the equilibrium delta volume is attained, U =
Ucr, hence Ub = Ur = 0.40 Uo (holh), where h, = equilibrium water depth.
Inserting Eqns. 3.11, 2.12, 3.13, 3.14, 3.16, 3.17, 3.20 and 3.21, into Eqn. 3.10, results
in the governing equation for water depth variation with time, and is expressed as
dh W,C, p,,f (H)2coshkh U+0.16 h Hacoshkh Uho (3.24)
1 [ +0.16 +
dt pd 2~ 4sinh2kh h 2.5sinhkh h
Eqn. 3.24 can be functionally expressed as
h = F(h) (3.25)
dt
This equation was solved by using the 4th order RungeKutta iteration method:
h,., = h,+ (k1+2k +2k,+k) (3.26)
k,=AtF(h,) (3.27)
k=AtF h,+ (3.28)
k,=AtF h+ (3.29)
k,=AtF(h,+k) (3.30)
where h, = incremental change in water depth, for i= 0,1,2,..., j1, kl,2,3,4 = intermediate
method steps, and At = time increment in number of tidal periods.
The incremental change in delta accumulation, Ad, can then be multiplied by the
depositional area, AD, to obtain the ebb delta volume, V. The cumulative volume change is then
plotted to illustrate the effects of waves and currents on ebb delta growth rate and estimate the
duration to achieve an equilibrium volume.
40
3.3.2 Model Parameters
3.3.2.1 Ebb delta area
It is necessary to identify the ebb delta area, AD, over which deposition occurs. This was
achieved by empirically correlating Ap defined in Figure III4 with AD based on measurements.
Davis and Gibeaut (1990) digitized ebb delta features of Florida's lower Gulf Coast inlets from
aerial photographs, and estimated the delta surface areas. Of these, 21 ebb delta areas were
found to be temporally consistent with the tidal prisms from the database in the present study
(Appendix C). The surface area, Ap, characterized by spring tidal prism, P, was obtained from
A, (3.31)
2a,
The plot of Ap against AD is shown in Figure III6.
45 1 11 1 1 1+111 1 I I III 1 I'' "I I I I III + '1 + 
w 3
S2 
+
1000000 ................ ................. .... ..
S6 +
I 4 
S 100000 ... ........................
Ebb Delta Area, Ap (m2) Tidal Prism Method
Figure III6. Ebb delta area, AD, versus tidal prism based area, Ap, for twentyone inlets based
on data of Davis and Gibeaut (1990).
100000 2 3 4 1000000 2 3 4 le+007 2 3 4
Ebb Delta Area, Ap (m2) Tidal Prism Method
Figure III6. Ebb delta area, AD, versus tidal prism based area, Ap, for twentyone inlets based
on data of Davis and Gibeaut (1990).
41
Regression analysis resulted in r2 = 0.65 indicating a reasonable correlation between the AD and
Ap. The equation of the regression line relating the Ap and AD is
A = 2.34AO.8 (3.32)
Equations 3.31 and 3.32 were used in the model to determine AD from P and ao,,
assuming their applicability to the inlets considered.
3.3.2.2 Suspended sediment concentration
Downing (1984) presented a timeseries of sediment concentrations at three locations
across the surf zone at Twin Harbor Beach, Washington. The investigator found two distinct
types of vertical concentration profiles. The first occurred between resuspension events, when
the sediment concentration had vertical uniformity, while during resuspension events a significant
concentration gradient occurred within approximately 0.10 m height above the bed in a total
water column depth of 0.25 m. The uniform concentration between resuspension events ranged
from 0.0002 to 0.0004 kg/m3 and approximately 0.0015 to 0.0100 kg/m3 during resuspension
events. In the present study, the influence of depthaveraged concentration, C,, ranging from
0.00005 and 0.00020 kg/m3 will be examined.
3.3.2.3 Friction factors
The friction factor due to current (f) is proportional to the square of Manning's n and
inversely proportional to the cubic root of the water depth, h (Mehta, 1978):
S 8gn2 (3.33)
c h
As mentioned, Mehta and Ozsoy (1978) noted that a typical mean Manning's n value of 0.028
42
can be used for sandy inlets since the mean grain size at most inlets range between 0.2 and 0.4
mm. The initial water depth used to model the evolution of the ebb deltas in Chapter IV
averaged 4 m, resulting in a characteristic friction factor due to current of 0.039. It should be
noted that Mehta (1978) determined friction factors for three inlets on the Gulf Coast of Florida
ranging between 0.021 to 0.050.
The friction factor due to waves (f,) was obtained from the wave friction factor diagram
developed by Jonsson (1965) which plots the friction factor against the wave Reynolds number
defined as
R U, b (3.34)
where Ub = maximum nearbed orbital velocity due to waves, = maximum nearbed
horizontal excursion and v = kinematic viscosity of seawater. The maximum orbital velocity,
Uwb, and maximum horizontal excursion, b,, are obtained from the linear wave theory for shallow
water waves (Dean and Dalrymple, 1984b):
SH (3.35)
Uwb 2 h
H (3.36)
2kh
Given the typical initial water depth of 4 m, deep water wave height equal to 0.4 m, and wave
period of 8 seconds, R, = 1.7 x 104. From Figure 6 in Jonsson (1966), this wave Reynolds
number corresponds to the fully turbulent flow range. Given the typical variation of Re in the
present study, a representative value off, = 0.005 in the fully turbulent flow range was chosen.
It should be noted that bothfc andf, are water depth dependent, however, for simplicity,
both friction factors will be held constant (0.039 and 0.005, respectively) in this study.
3.3.2.4 Sediment grain size
Mehta and Ozsoy (1978) noted that for sandy inlets the median grain size at most inlets
range between 0.2 and 0.4 mm. This range will be considered in the present study.
3.3.2.5 Deep water height and period
As described in Section 3.4, the deep water wave height has a significant effect on the
growth rate of the ebb delta and its equilibrium volume. By adjusting the wave height, the delta
volumetime curve can be made to pass through the appropriate smallest and largest measured
delta volumes at a given inlet. Deep water wave heights obtained in this way are given in Table
IV4. A characteristic wave period of 8 seconds will be used for all model runs.
3.3.2.6 Tidal inlet characteristics
The tidal inlet characteristics utilized in the analysis are derived from the database in
Appendix A. Specifically, the characteristics include: inlet throat width, throat depth, tidal
prism, and spring tidal range.
3.4 Effects of Important Parameters on Delta Growth
The effects of important parameters on the rate of delta formation at coastal inlets is
examined. The three selected parameters are 1) suspended sediment concentration, C,; 2) median
sediment grain size, d5o; and 3) deep water wave height, Ho. The influence of varying these
parameters on the volume growth curves are next shown in plots of ebb delta volume versus time,
beginning with a new inlet with no delta. The range of values of these three parameters are given
in Table III1. Also given are other model input parameters including wave period T, spring tidal
prism P, spring tidal range 2a,, inlet throat width w, and water depth ho, current friction factor
f,, and wave friction factorf.
44
3.4.1 Variation in Suspended Sediment Concentration
In the Eqn. 3.22 for the rate of water depth change, dh/dt, is proportional to the
suspended sediment concentration, C,. Figure III7 plots the ebb delta volume, V = AD(hoh),
versus time (years) for three suspended sediment concentrations.
The first characteristic that is evident from the growth curves is that the equilibrium ebb
delta volumes are the equal (1.4 x 106 m3) for the three concentrations. However, it is evident
that as C, increases the rate of deposition becomes more rapid. For a concentration of 0.00005
kg/m3, the equilibrium volume is reached in approximately 60 years from the initial formation
of the inlet. By doubling this concentration to 0.00010 kg/m3, the deposition rate is increased,
achieving equilibrium in 25 years. If the concentration is doubled again, to 0.00020 kg/m3, the
time for the delta to achieve equilibrium is reduced to 15 years.
3.4.2 Variation in Sediment Grain Size
Two physical parameters are dependent on the median grain size diameter, dso, the
settling velocity (Eqn. 3.6) and the critical shear stress for sediment transport (Eqn. 3.11). The
ebb delta volume versus time plot for varying sediment diameters (Figure III8) is characterized
by three different growth rates and equilibrium volumes. For a dso of 0.2 mm, the ebb delta
achieves equilibrium in approximately 50 years as compared to 40 years for a dso equal to 0.3
mm, and 30 years for dso equal to 0.4 mm. The increase in the sediment diameter increases the
rate of deposition, due to the dependence of particle fall velocity on sediment size. An increase
in the sediment size also increases the critical shear stress, allowing the sediment bed to remain
more stable as compared to a bed composed of smaller grain size under the same flow conditions.
This effect results in an increase in the equilibrium volume for increasing grain diameters.
Cs =i0.000 0 kg/m3
. ..... ........ ............. .............. ... ....... ..... .. .. ............. .......... ...
) 10' I 0 0 40 50 00010 0 80 0 1
........ ...... ...... ... .. ......... .............. ............. ............ ...... ... .. .... .. ..... .. .......
.... .. ............. ............... I ............. ............. ............. ............. .............
S/ Cs= 0.00005 kg/Mn3
S........... ............. ........................... ............. ........... .............
IF I /
4 1 ... .... .. ..... ...... .
) 10 20 30 40 50 60 70 80 90 100
Years
Figure III7.
Ebb delta volume versus time comparing the effects of three suspended sediment
concentrations, Cs, 0.00005, 0.00010, and 0.00020 kg/m3. Runs 1, 2 and 3.
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Figure HI8.
0
10 20
40
Years
50 60 70 80
Ebb delta volume versus time comparing the effects of three grain size diameters,
dso, 0.2, 0.3 and 0.4 mm. Runs 4, 5 and 6.
............. ... ... 4 n.....
............... ........... /.';. .............. i............... ..................................................................
Si d5o= 0.3mmr
I :
......... ....... ................ ..... .................. .... ................ .. .............. i ......... .....
" i : :
!, ...... ...... ............ .. ".............. ............... ................ ....... ......... .............. ...................
I ..
Initial input parameters for examining the effects of variations in suspended sediment concentration, C,,
grain size diameter, ds, and deep water wave height, Ho on ebb delta growth.
Run C, dso Ho T P 2a, w h, fc /f
(kg/m3) (mm) (m) (s) (x106 m3) (m) (m) (m)
1 0.0002 0.35 0.4 8.0 6.50 1.00 109 4.0 0.039 0.005
2 0.0001 0.35 0.4 8.0 6.50 1.00 109 4.0 0.039 0.005
3 0.00005 0.35 0.4 8.0 6.50 1.00 109 4.0 0.039 0.005
4 0.0001 0.20 0.4 8.0 6.50 1.00 109 4.0 0.039 0.005
5 0.0001 0.30 0.4 8.0 6.50 1.00 109 4.0 0.039 0.005
6 0.0001 0.40 0.4 8.0 6.50 1.00 109 4.0 0.039 0.005
7 0.0001 0.35 0.0 8.0 6.50 1.00 109 4.0 0.039 0.005
8 0.0001 0.35 0.4 8.0 6.50 1.00 109 4.0 0.039 0.005
9 0.0001 0.35 0.8 8.0 6.50 1.00 109 4.0 0.039 0.005
Table III1.
3.4.3 Variation in Deep Water Wave Height
As the waves approach the shoreline, its height increases as the water depth decreases
(Eqn. 3.15). This increase in wave height in turn increases the nearbed orbital velocity, Uwb,
hence reduces the rate of deposition. Figure III9 plots the ebb delta volume versus time
illustrating delta growth due to current alone, i.e. 0.0 m wave height, and two additional deep
water waves heights of 0.4 and 0.8 m.
During sea conditions when the deep water wave height is equal to 0.0 m, the rate of
deposition is observed to be relatively rapid compared to the other two wave conditions. For a
wave height of 0.0 m the equilibrium volume is reached in 20 years and the time is doubled to
40 years to achieve equilibrium for a 0.4 m wave. Equilibrium requires an additional 10 years
when the wave height increases from 0.4 to 0.8 m. Note also the drastic decrease in the
equilibrium volume with increasing Ho.
2.5H OOm
3.0 . I .
S 1 .5 .............. .. ............................... .............. ...................
0 .o
0.0
0 10 20 30 40 50 60 70
Years
Figure III9. Ebb delta volume versus time comparing the effects of three deep water waves
heights, Ho, 0.0, 0.4, and 0.8 m. Runs 7, 8 and 9.
CHAPTER IV
RESULTS
4.1 Ebb Deltas
4.1.1 Estimated Delta Volumes
Seven surveys from a total of four inlets were found to have adequate bathymetric data
to estimate ebb tidal delta volumes by the Dean and Walton (1973) method noted in Section
2.2.1. Each survey had a "bestfit", "noinlet" contours (contour B in Figure III1) and a grid
system superimposed on the bathymetry (Figure III2). The grid cell size was determined
subjectively to provide a tight coverage of the seafloor undulations in order to obtain an
appropriate estimate of the delta volume, as described in Section 3.2. Table IV1 presents the
size of the grid cells and the estimated delta volumes for each of the seven surveys.
Table IV1. Ebb delta volume for the "bestfit", "noinlet" contours.
Inlet Year Grid size (m) Delta volume (x106 m3)
Ponce de Leon 1944 152 2.91
Jupiter 1957 76 0.38
1979 152 0.68
1993 76 0.74
South Lake Worth 1968 61 0.67
Clearwater 1972 122 2.03
1976 122 1.29
4.1.2 Delta Volume Sensitivity Analysis
The ebb delta volume estimates due to the imposed change in the "noinlet" contour
positions are given in Table IV2, and due to change in the grid cell size in Table IV3. The
contours A, B, and C, are described in Section 3.2.1 (Figure III1), and the change in the grid
cell size in Section 3.2.2 (Figure 1112). The results in Table IV2 include the location of the
survey, the year of the survey, and the grid cell size for which the volume was estimated in
columns one, two and three, respectively. The forth column, with the heading "contour A" is
the delta volume obtained using the smallest grid cell size and the "noinlet" bathymetry for the
seaward exaggerated contour position. The same column contains the percent difference in the
volume estimated from contour A to that of the delta volume resulting from the "bestfit", "no
inlet", contour B. The ebb volume estimated from the position of contour B and is given in the
fifth column. The sixth column, "Contour C", contains the volume resulting from the landward
exaggerated "noinlet" bathymetric condition, and the percent difference between volumes
estimated from contour positions B and C. Finally, the last column contains the percent
difference in estimated ebb delta volumes between contour positions A and C.
Table IV2. Delta volumes and percent differences due to changes in the "noinlet" contours.
Delta Volume (x106 m3)
Inlet Year Grid Size Contours A Contours B Contours C Contours
(m) AC
Ponce de Leon 1944 152 2.60 (11%) 2.91 3.38 (+14%) +23%
Jupiter 1957 76 0.25 (34%) 0.38 0.45 (+16%) +44%
Jupiter 1979 152 0.63 (7%) 0.68 0.78 (+13%) +19%
Jupiter 1993 76 0.71 (4%) 0.74 0.87 (+15%) +18%
South Lake Worth 1968 61 0.55 (17%) 0.67 0.83 (+20%) +34%
Clearwater 1972 122 1.91 (6%) 2.03 2.29 (+11%) +17%
Clearwater 1976 122 1.11 (14%) 1.29 1.43 (+10%) +22%
50
The percent difference of the volume estimates between the three contour positions range
from 34% for Jupiter Inlet, 1957, between contours A and B to +44% for Jupiter, 1957,
between contours A and C. Note that the highest percent difference can be expected to occur
between the two surveys with the greatest horizontal change (landward versus seaward) in the
contour positions. By moving the "noinlet" contour positions landward (C) would result in a
comparatively thicker lens of sand and therefore a larger ebb tidal delta volume. Conversely,
moving the contour positions seaward has the opposite effect by reducing the sand lens thickness,
thus resulting in a comparatively smaller delta volume. The percent differences corresponding
to the "bestfit" delta volume show that the minimum and the maximum percent change between
A and B, and B and C are 34% and +16%, respectively.
Using the same contour configurations described above and doubling the size of the grid
cells from the initial grid pattern, the delta volume were reestimated. The results of this exercise
are given in Table IV3, where the forth, fifth and sixth columns respectively give the estimated
ebb delta volumes using the larger grid cells for the same contours A, B, and C. Also included
in each of the columns containing the delta volumes is the percent difference of volumes resulting
from the change in grid cell size for that particular contour position.
The volume percent difference were found to be dependent on the complexity of the
bathymetry. Comparatively straight, widely spaced contours have a tendency of resulting in
smaller percent differences. By having a constant (or near constant) slope between intersecting
corners, the volume of the enlarged grid is close in value to that of the volume of the four initial
smaller grid cells covering the same surface area. Where if the contours were irregularly spaced,
so that the omitted intersections caused a number of small shoals or depressions to be missed, the
change in volume between the grid cell sizes was relatively large. The percent differences for
changing the grid cell size ranged between 22% and +9%.
51
Table IV3. Ebb delta volumes and percent differences due to changes in the grid cell size.
Delta Volume (x106 m3)
Inlet Year Grid Size (m) Contours A Contours B Contours C
Ponce de Leon 1944 305 2.82 (+8%) 3.28 (+11%) 3.66 (+8%)
Jupiter 1957 152 0.21 (19%) 0.34 (12%) 0.40 (13%)
Jupiter 1979 305 0.56 (13%) 0.58 (17%) 0.64 (22%)
Jupiter 1993 152 0.56 (27%) 0.63 (18%) 0.77 (13%)
South Lake Worth 1968 122 0.57 (+3%) 0.68 (+2%) 0.82 (2%)
Clearwater 1972 224 2.01 (+5%) 2.14 (+5%) 2.35 (+3%)
Clearwater 1976 244 1.22 (+9%) 1.31 (+2%) 1.49 (+4%)
The cumulative effect of the interpretation of the bathymetric conditions in selecting the
"noinlet" contours and grid cell size is varied. For example, the 1979 Jupiter Inlet survey had
a relatively smooth, gradual change in the bathymetry. The estimated ebb volume for the "no
inlet" seaward contours (A) with a small grid size (152m x 152m) was equal to 0.63 x106 m3,
and for a larger grid size (305 m x 305 m) with a "noinlet" landward contours (C) the delta
volume was estimated to be 0.64 x106 m3, thus resulting in a relatively small change.
Conversely, the 1944 Ponce de Leon survey was more complex with large undulations including
small shoals. As a result, the difference in volume for the two contour and grid cell size
conditions was estimated to be 3.66 x106 m3 2.60 x106 m3 = 1.06 x106 m3.
4.1.3 Delta Volumes versus Tidal Prism
The ebb delta volume versus spring tidal prism relationship presented by Walton and
Adams (1976) and Hayter et al. (1988) found a significant relationship between the volume and
the spring tidal prism categorized by three wave energy levels (Section 2.2.2). This relationship
is reexamined here using delta volumes, tidal prisms, and synchronous wave data in an effort
to clarify the issue of the dependence of the volumetidal prism relationship on wave energy.
52
The need for synchronous wave information limits the quantity of data available for
analysis. The tidal prism and ebb delta volume must be measured during the same year, and
these data must fall between 1956 and 1975, so as to be able to use the hindcast wave data
obtained from Hubertz and Brooks (1989), Hubertz, et al. (1993) and Brooks (1993a,b).
Twentyeight sets of data met this criterion, of which twenty fell within the range characterized
by Walton and Adams (1976) as a mild energy coast (030 m2s2) and eight data as moderate
energy coast (30300 m2s2). Note that the wave energy is characterized by the parameter, Ho2T~.
The regression plot of the tidal prism against the ebb delta volume characterized by mild
wave energy shows a fairly good correlation (Figure IV1), yielding a regression line with r2 =
0.80. The equation of the regression line for mild energy coasts was found to be
V = 1.86x103P1.3 (4.1)
The data set for moderate energy coasts has more scatter around the regression line
(Figure IV2) than seen for the case for mild coasts. The r2 = 0.45 indicates a poor correlation
of tidal prism against delta volume which may be due to the small data set used in this analysis.
Note also that the data scatter may also be due to the divergence of the V versus P values from
equilibrium. The regression line equation for moderate energy coasts was found to be
V = 5.49x10P32 (4.2)
Figures IV1 and IV2, show that the ebb delta volume does increase with increasing tidal
prism which is in agreement with the observations made by Walton and Adams (1976).
However, due to the small sizes of the data sets, it is inconclusive as to the precise influence of
wave energy on the tidal prism to delta volume relationship. Additional data within both wave
energy regimes are necessary to clarify this issue.
le+008
+ +
0
+
le+007 "" ...................
+ +
1000000
le+007 le+008
Tidal Prism, P (m3)
Figure IV1. Ebb delta volume versus tidal prism for mild energy coasts (030 m2s2).
S l e + 0 0 8 ....... ............... ...................................................   ...................
V le + 0 0 7 ................................ .... ......... ........... .....................................
le+le+00 7 O
*0 V
Tidal Prism, P (m3)
Figure IV2. Ebb delta volume versus tidal prism for moderate energy coasts (30300
2s ).
mV)2.
4.1.4 TimeEvolution of Delta Volume
The following analysis presents the timeevolution of sand volumes of four selected deltas
along the east coast of Florida including those at 1) Jupiter Inlet; 2) South Lake Worth Inlet; 3)
Boca Raton Inlet; and 4) Bakers Haulover Inlet. These inlets were chosen because 1) the date
when each inlet was opened were available, and 2) four or more data points were available per
inlet to represent the timevariation of ebb delta volumes. For each inlet, a plot of the measured
delta volumes versus date of survey with the corresponding volume ranges obtained from the
model is presented.
The theoretical volume curves were derived from the model using the specific
characteristics of the respective inlet. These data including 1) spring tidal prism, P; 2) inlet
throat width, w; 3) inlet depth, ho; and 4) spring tidal range, 2aos, along with other necessary
data presented in Section 3.3.2 are summarized in Table IV4. By adjusting the deep water wave
height, Ho, and the suspended sediment concentration, Cs, the delta volumetime curves were
made to pass through the appropriate smallest and largest measured delta volumes.
The inlet stability parameter, a, introduced by O'Brien (1971) and defined in Section
2.4.1 (Eqn. 2.18) was later expanded by Mehta and Hou (1974) to provide an indicator of the
stability of inlets on the south shore of Long Island, New York. For the present study, a will
be used to provide an indication of the relative effect of waves and tidal current in governing the
rate of growth of the ebb delta. As the deep water wave height is increased at a given inlet, a
increases and reflects a tendency to drive material toward the inlet and the nearshore area, thus
limiting the delta volume. Conversely, if the deep water wave was set to zero, the corresponding
a would equal zero indicating a currentdetermined delta. This condition results in a larger ebb
delta volume as compared to a higher avalue for the same inlet when waves are present.
Table IV4. Initial input parameters and resulting wave to tidal energy ratio, a.
Inlet Cs dso H, T P 2a, w h, f fw a
(kg/m3) (mm) (m) (s) (x106 m3) (m) (m) (m)
Jupiter 0.00015
Jupiter 0.00015
South Lake Worth 0.00006
South Lake Worth 0.00006
Boca Raton 0.00010
Boca Raton 0.00010
Bakers Haulover 0.00020
Bakers Haulover 0.00012
0.35
0.35
0.35
0.35
0.35
0.35
0.35
0.35
0.54 8.0
0.68 8.0
0.19 8.0
0.28 8.0
0.39 8.0
0.46 8.0
0.62 8.0
0.63 8.0
6.46
6.46
1.59
1.59
5.50
5.50
10.20
10.20
1.00 109 4.2 0.039 0.005
1.00 109 4.2 0.039 0.005
0.94 30 3.3 0.039 0.005
0.94 30 3.3 0.039 0.005
0.90 50 3.6 0.039 0.005
0.90 50 3.6 0.039 0.005
0.82 110 3.7 0.039 0.005
0.82 110 3.7 0.039 0.005
0.17
0.27
0.03
0.06
0.05
0.07
0.17
0.18
4.1.4.1 Jupiter Inlet
Nine delta volumes were available for Jupiter Inlet since this entrance was reopened for
navigation in 1947 (Figure IV3). The nonzero delta volumes range between 0.23 x106 m3 and
0.77 x106 m3, for the years 1981 and 1993, respectively. The model was used to simulate the
growth curves matching this volume range. The avalue of 0.17 resulting from a Ho = 0.54 m
yielded a volume of 0.77 x106 m3 in 1993. The higher avalue of 0.27 was calculated for a Ho
= 0.68 m to modify the growth curve to achieve a volume of 0.23 x106 m3 in 1981. It is
therefore surmised that the relative wide range in the delta volumes between the two curves is
the result of waves relative to current. Larger delta volumes correspond to lower values of a and
vice versa. The maximum range of a being 0.17 to 0.27 for this inlet.
I I
0 .7 ...................... .............. ....... ................... .................... ............. ........ .......................
0 .6 ... ..... .. .. ............ .... ....... ........... ......
0 .5 ....................... ....................... ........................ ........................ ........................ .......................
a = 0.17 [..' i !
0
> 0 .4 ..................... ... ...................................................................................... ......................
0.6
  0 .5   
s ,'
S 0 .42 ................... .... .................... ................. .................................. .............
S0.3 
S0..2" a =0.27
0 ,,....... ... .... '.................. ...............................................................................................
5,,
0 .0 ...... ........................ ................................................ ........................ .......................
1940 1950 1960 1970 1980 1990 2000
Year
Figure IV3. Ebb delta volume versus year with modelcalculated volume ranges for
Jupiter Inlet.
4.1.4.2 South Lake Worth Inlet
Six delta volumes for South Lake Worth Inlet were obtained since the entrance was
artificially opened in 1927 (Figure IV4). A growth curve was passed through the lower ebb
57
volume of 0.54 x106 m3 with a Ho = 0.28 m resulting in an avalue of 0.06. Conversely, a
curve was unable to be passed through the largest volume of 2.27 x106 m3 estimated from a 1990
bathymetric survey because to achieve this volume would have required a larger delta area, AD,
than obtained from Eqn. 3.32. It was decided to consider the next highest volume of 1.07 x106
m3 (1969, 1979) for this analysis. To accomplish this, Ho = 0.19 m was used, which resulted
in an avalue of 0.03. Thus, with the exception of 2.27 x106 m3 the other volumes are contained
within the a range of 0.03 to 0.06.
2.4
S 2 .2 ................... .......................................... .................... ......................................... ........... ....
.o ................. ..................... .................... ...................... .................... ................... ..................... .... .
X 1 ...................................... .................... .................... .................... .................... ......................
1.6 .
a = 0.03
S 0 ...................... ..........................
0.6
aa 01 ..~~~;.
0 1 .................. .............. ............. .:. .............................. ..............
0 10" a = 0.06"
0 .2 ...... 0 .8. .. .... .. .................................. .... .... ..... .. ...................... .........
0.8 *. i
0 .0 ...... .. ............. .......... ..................... ........................................ ..................... .....
1920 1930 1940 1950 1960 1970 1980 1990
Year
Figure IV4. Ebb delta volume versus year with modelcalculated volume ranges for
South Lake Worth Inlet.
4.1.4.3 Boca Raton Inlet
Four ebb delta volumes were obtained for Boca Raton Inlet after the inlet was opened in
1925 (Figure IV5). Two of the surveys resulted in a volume of 0.61 x106 m3 (1978,1983), and
the third was 0.84 x106 m3 (1981). Ho = 0.39 m resulting in an avalue of 0.05 passed through
the upper volume and Ho = 0.46 m (avalue of 0.07) through the lower volume.
0.9
0.9 ...... .......
0 .8 ................... ....... .. .
:a = 0.05 :
0 .7 ................................... ............................................. .................... .............. ... .................
/7
0.5 ............. ....
S0.4 
" 0 .4 . .................... ..................... ................... ..0.0 7
0
0 .2 .............. .......................................... ..................... .................... ..................... ..................
o
S 0 1 ...... .......................... ........................................................................................
0.3 1
t 0.2
P 0.1
P 1:
0 .0 ........ .............................................................................................................................
0.0 *
1920 1930 1940 1950 1960 1970 1980 1990
Year
Figure IV5. Ebb delta volume versus year with modelcalculated volume ranges for
Boca Raton Inlet.
4.1.4.4 Bakers Haulover Inlet
Four delta volumes were available for Bakers Haulover Inlet, including zero volume when
the entrance was artificially created in 1925 (Figure IV6). A curve was passed through a point
S0.17  
S 0.5
> 0 .3 ............... ... ........ ............................. ................................. ............... ..................
0 .............. .................... ...... ........................................ ............................................
a 0.18
> 0 3 . . . .. . .. . .. . ... .. . . . .. . . . . . . . . . . .
I
S 0 ...... ........ ... .................. .............................................................. ..................... ................... .
0.0 "
a :
1920 1930 1940 1950 1960 1970 1980 1990
Year
Figure IV6. Ebb delta volumes versus year with modelcalculated volume ranges for
Bakers Haulover.
59
occurring three years after the inlet was opened (0.23 x106 m3) and the largest available volume
of 0.46 x106 m3. This was accomplished with He = 0.62 m resulting in the avalue equal to
0.17. For the lower curve He = 0.63 m had an avalue equal to 0.18, thus defining an a range
of 0.17 to 0.18.
4.1.4.5 Influence of a on delta growth
Table IV5 presents a comparison of the ebb delta volume ranges and the corresponding
wave to tidal energy ratio, a, for each of the four inlets. As a increases, the ebb delta volume
has a tendency to decrease, and vise versa. Figure IV7 further illustrates this trend. Although
there is data scatter and a rvalue of 0.60, which is low, the regression line does show that there
in an inverse relationship between the delta volume to a. The equation of the regression line
relating a and V is
V = 0.17a050 (4.3)
Note that V in this case may not represent the actual equilibrium value, but may be close to it,
given the manner in which the curve fitting was conducted.
Table IV5. Comparison of wave to tidal energy, a, and the corresponding delta volumes.
Inlet a V (x106 m3)
Jupiter 0.17 0.77
Jupiter 0.27 0.23
South Lake Worth 0.03 1.07
South Lake Worth 0.06 0.54
Boca Raton 0.05 0.84
Boca Raton 0.07 0.61
Bakers Haulover 0.17 0.46
Bakers Haulover 0.18 0.38
60
S1 I : : I; '. I I I I I I '. 'I' 1. I I I !
100 .
0 ...... ,. ... ... .. .  b . ; . : .i ... ... .....i.i.i. .... .... .......... .. 
9
5 ... .. ...... ....... ..... ...... .. .. ........ ..... .... .. ....... ...... .... ..... .... ....
S: /..:. ...... ....... ......
..i ' .. .. ..i ". ..... .. .. .
!'"!"'": ".'": ":": "!'.'! . """ "" ":: ':::'" ...... ........ :  .......
i .;" !'"~i \ ". .. "............ .... ". ..'.....L ..I..+ ...
a a
F E
"4 i i ;4 i A, .
ba
2~~~~~ mi:Ip ii:ii iiii:i iii
2 3 4 .... a 7 8 O 2 3
J '''''"""a
Figure I7. Ebbdelta voume aganst waveto tida energy atio, a
CHAPTER V
CONCLUSIONS
5.1 Estimated Ebb Delta Volumes
Personnel interpretation of the bathymetric condition at an inlet measurably influences the
estimated delta volume based on the method of Dean and Walton (1973). The percent difference
due to a change in the "noinlet" contour positions from the a "bestfit" contour positions ranged
between 34% and +20%, and as much as 44% between the landward and seaward exaggerated
"noinlet" contours. The choice of grid size also effected the estimated volumes, ranging between
22% and +9%. In general, the percent volume difference appears to be dependent on the
complexity of the bathymetry. Straight, widely spaced contours have a tendency of resulting in
smaller percent differences as compared to irregularly spaced contours where small shoals or
depressions may be missed, thus resulting in comparatively larger volume differences.
5.2 Ebb Delta Volume versus Tidal Prism
The ebb delta volume versus tidal prism relationship presented by Walton and Adams
(1976) was reexamined in an effort to clarify the issue of the dependence of this relationship on
the wave energy. It was confirmed that the delta volume does increase with increasing tidal
prism; however, the precise influence of wave energy on this relationship could not be examined
conclusively due to the small number of data points. Note, however, that the exponential
correlation coefficient for Eqns. 4.1 (1.30) and 4.2 (1.25) found in this analysis are close to those
62
given by Walton and Adams (1976) for mildly exposed coasts (1.24) and highly exposed coasts
(1.23).
5.3 Effects of Significant Physical Parameters on Delta Growth
Three parameters, namely the suspended sediment concentration, sediment grain size, and
the deep water wave height, were varied in the diagnostic model developed for delta growth to
determined their effects on the rate of delta formation at coastal inlets. It was found that an
increase in the suspended sediment concentration increases the rate of approach to equilibrium,
but does not result in a change in the equilibrium volume. On the other hand, a change in the
sediment grain size and the deep water wave height effect both the rate of growth and the
equilibrium volume. Thus, an increase in the sediment diameter increases the rate of growth due
to the dependence of the particle fall velocity on sediment size, and increases the critical shear
stress resulting in an increase in the equilibrium volume. An increase in the deep water wave
height increases the nearbed orbital velocity at the site of the delta, hence decreases the rate of
growth. The equilibrium delta volume likewise decreases.
5.4 Effects of a on Equilibrium Delta Volumes
It was shown through the application of the model to four Florida inlets that there is a
dependence between the ebb delta volume and the wave to tidal energy ratio, a. An increasing
avalue decreases the accumulated ebb delta volume, and vice versa. This dependence of delta
volume on a partly explains the observed fluctuations in the delta volume at many inlets, since
a tends to vary seasonally as well as annually. Another cause of variation of the delta volume
at a given inlet is that even under constant sea conditions, the equilibrium volume often occurs
after several decades following the opening of an inlet. Thus the delta volumes measured during
the early years of evolution will be lower than the equilibrium volume.
63
5.5 Influence of a on Navigability
As noted in Section 2.4.1, Bruun and Gerritsen (1960) proposed a stability criterion in
which the condition for the inlet to remain open, and therefore the condition for navigation, are
shown to be dependent on the ability of the tidal current to remove the wavedriven littoral drift
deposited at the entrance. O'Brien (1971) and later Mehta and Hou (1974) noted that inlet
stability is governed by the relative dominance of wave energy and tidal energy defined by the
parameter a. In this study it was found that as avalues increased the corresponding delta
volume decreased, thus confirming the work of Walton and Adams (1976) whereby they found
the delta volume to be dependent on the tidal prism and on wave energy. Decreasing the delta
volume increases the controlling depth, thus providing additional underkeel clearance for a vessel
to navigate the entrance. By providing an understanding of the mechanism by which delta growth
is controlled by the prevailing physical parameters, this investigation also provides a potential
means to design safer navigation channels at coastal inlets.
5.6 Future Investigations
If future studies of the ebb delta rate of growth and equilibrium volume are to be
undertaken using the model developed in this study, it is recommended that one important change
be instituted. In nature, the combination of the ebb and flood flows causes the tidal current to
transport sediment toward the inlet from the adjacent shoreline during all tidal stages. It was also
noted that the forces acting to transport sediment are not only dependent upon the effects of ebb
and flood currents, but also on wave refraction which tends to transport sediment toward the inlet
from the adjacent shoreline. Modeling wave refraction and the resulting sediment transport in
the vicinity of an ebb delta was disregarded in the development of this model due to the
complexity of the process. In future studies, it would be a more realistic model to include
64
sediment transport in evaluating the effects of littoral drift on the rate of deposition and
equilibrium volume.
5.7 Note of Caution
The model presented in this thesis is preliminary and conceptual in nature. Many
simplying assumptions have been made with varing levels of justification. This effort is intended
to be a first attempt at understanding and estimating the dynamic nature of ebb tidal deltas.
Therefore attempts to apply the model to actual inlet conditions should be done with caution.
APPENDIX A
DATABASE
Key To Symbols
V ebb delta volume
w width at the inlet throat
ho water depth at the inlet throat
Ac crosssectional area at the inlet throat
P spring tidal prism
H, significant deep water wave height
T 20yr average wave period
2aos spring tidal range
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References for Appendix A
1) Marino and Mehta (1986)
2) Walton and Adams (1976)
3) Hou (1974)
4) Hubertz et al. (1993)
5) Brooks (1994a)
6) Marino (1986)
7) USGS, Fernandina Beach (1975)
8) Eqn. 3.1, Section 3.1.2
9) Balsillie (1987a)
10) Dean and Walton (1973)
11) Work and Dean (1990)
12) Florida Coastal Engineers (1976)
13) Mehta and Jones (1977)
14) Jones and Mehta (1978)
15) Average of volumes, Section 4.1.2
16) Taylor Engineering (1992)
17) UFCOEL (1970)
18) USGS, New Smyrna Beach (1974)
19) Hunt (1980)
20) Mehta, Adams and Jones (1976)
21) Johnson (1976)
22) Coastal Technology (1988)
23) Walton (1974)
24) U.S. Army Corps of Engineers (1974)
25) Coastal Data & Engineering (1980)
26) Coastal Data & Engineering (1985)
27) Harris (1983)
28) Lee (1992)
29) Coastal Planning & Engineering (1993)
30) Jarrett (1976)
31) Olsen Associates (1990)
32) Strock and Associates (1983)
33) Van de Kreeke (1984)
Hine and Davis (1986)
Hubertz and Brooks (1989)
Brooks (1994b)
Basillie (1987c)
UFCOEL (1970)
Davis and Gibeaut (1990)
Basillie (1994c)
Bruun and Gerritsen (1966)
Coastal Engineering Consultants (1988)
Campbell et al. (1990)
NOS, Nautical Chart 11508 (1991)
Oertel (1988)
USGS, Sea Island (1974)
NOS, Nautical Chart 11511 (1974)
USGS, Sapelo Sound (1974)
NOS, Nautical Chart 11509 (1992)
Kraus et al. (1994, in press)
USGS, Savannah Beach North (1975)
Savannah Beach South (1977)
NOS, Nautical Chart 11512 (1974)
NOS, Nautical Chart 11504 (1991)
NOS, Nautical Chart 11506 (1992)
NOS, Nautical Chart 11513 (1974)
NOS, Nautical Chart 11523 (1992)
NOS, Nautical Chart 11521 (1992)
NOS, Nautical Chart 11532 (1992)
NOS, Nautical Chart 12323 (1993)
NOS, Nautical Chart 12318 (1991)
NOS, Nautical Chart 12216 (1994)
NOS, Nautical Chart 12204 (1994)
NOS, Nautical Chart 11547 (1992)
NOS, Nautical Chart 11376 (1993)
Johnson (1972)
O'Brien (1969)
APPENDIX B
FORTRAN MODEL AND INPUT FORMAT
B. I Input Parameter Format
The following is the format of the input parameter file 'I.INP' used with EBBSHOAL.FOR:
bed porosity/current friction/wave friction/median grain diameter/
+ suspended sediment concentration;
tidal prism/spring tidal range/tidal period (hr.)/inlet throat width
+ average inlet depth;
deepwater wave height/wave period/angle between currents and waves/
+ number of tidal steps;
initial output filename ('in quotes');
data output filename ('in quotes');
run date ('in quotes').
B.2 EBBSHOAL.FOR Diagnostic Model
The following model was used to simulate and evaluate the growth of ebb delta volumes:
$LARGE
PROGRAM EBBSHOAL.FOR
C************** ********* ** *************
C Ebb Shoal Volume Model Program
C written in MS FORTRAN 5.0 by
C Michael R. Dombrowski
C Version 1.0
C July 1, 1994
C
C  Variable Definitions  
C
C Variable Description
C   
C AC Minimum crosssectional area (m**2)
C ALPHA Wave to tidal power ratio
C AP Ebb shoal area calculated by tidal prism method (m**2)
C AD Ebb shoal area calculated from regression equation (m**2)
C CONC Sediment concentration (kg/m**3)
C D50 Median grain size diameter (in millimeters)
C D Depth of water array within deposition area (m)
C DO Initial depth of "nodelta" condition (m)
C F Friction due to both waves and currents
C FC Friction due to currents
C FW Friction due to waves
C HT Shoaled nearshore wave height array (m)
C HO Offshore wave height (m)
C N Bed porosity
C NN Number of tidal cycles
C RHOD Density of dry bed (kg/m**3)
C RHOS Density of water (kg/m**3)
C RHOW Density of sand (kg/m**3)
C TCR Critical shear stress (N/m**2)
C TB Shear stress due to currents and waves (N/m**2)
C TP Tidal prism (m**3)
C TPER Tidal period (s)
C TR Spring tidal range (m)
C TWAVE Wave period (s)
C UI Average current velocity over 1/2 tidal cycle (m/s)
C UO Current velocity over 1/2 radius ebb shoal (m/s)
C UMAX Maximum current velocity (m/s)
C VOL Ebb shoal volume array (m**3)
C W Width of inlet mouth and shoal (m)
C  Subroutine Definitions  
C KUTTA Function for the calculation of change in depth (FF).
C SHEAR Calculates current and wave velocities and returns
C shear stresses due to both waves and currents (TB).
C SHOAL Calculates the nearshore shoaled wave height (HT) due to
C the change in the water depth of the ebb shoal.
C Declarations
PARAMETER(IDIM= 15000)
CHARACTER*60 OUT1,DAT1,DATE
INTEGER J,DVOL,NPTS,VOL(IDIM),Z
REAL AC,AD,ALPHA,AP,CONC,C1,D(IDIM),D50,DO,DT,F,FC
REAL FF,FW,G,HO,HT(IDIM),K1,K2,K3,K4,L,N,NN,PHI,PI
REAL RHOD,RHOS,RHOW,T(IDIM),TB(IDIM),TCR,TP,TPER
REAL TR,TWAVE,UAVE,UI,UO,W,WS,X
C Open input file and read data 
OPEN(UNIT= 1,FILE= 'I.INP',STATUS = 'OLD')
C Parameters and filenames read in from I.INP
READ(1,*)N,FC,FW,D50,CONC
READ(1,*)TP,TR,TPER,W,DO
READ(1,*)HO,TWAVE,PHI,NN
READ(1,*)OUT1
READ(1,*)DAT1
READ(1,*)DATE
CLOSE(UNIT= 1)
OPEN(UNIT=2,FILE=OUT1,STATUS= 'UNKNOWN')
OPEN(UNIT= 3,FILE=DAT1,STATUS= 'UNKNOWN')
D50=D50/1000.0
G=9.81
PI=3.141593
PHI=PHI*(PI/180)
RHOD =2650.0*N
RHOS =2650.0
RHOW= 1030.0
TPER=TPER*3600
DT=TPER*NN
DVOL=1
D(1)=DO
J=1
T(1)=0
VOL(1)=0
Z=1
C Ebb shoal area and equivalent shoal length
AP=TP/TRS
AD=2.34*(AD**(0.81))
C Initial current velocity
AC=W*DO
UMAX= ((TP*PI*0.86)/(TPER*AC))
UI= (2.0*UMAX)/PI
RE= (SQRT(2.0*AREA/PI))
UO= (2.0*UI*W)/(PI*RE)
C Settling velocity and critical shear stress
WS = ((4.0/3.0)*((RHOSRHOW)*G*D50)/(0.43*RHOW))**0.5
TCR=0.058*(RHOSRHOW)*G*D50
C Do loop to calculate change in depth and volume with respect to
C changing conditions. The loop is completed when the absolute change,
C DVOL, between VOL(J+1) and VOL(J), is
DO WHILE (DVOL.GT.0.0)
CALL SHOAL(D(J),G,HO,TWAVE,HT(J))
CALL SHEAR(D(J),DO,F,FC,FW,G,HT(J),PHI,RHOW,TB(J),TWAVE,UO,L)
DO WHILE (Z.EQ.1)
PTIDE= (TR*TP*RHOW*G)
PWAVE= ((RHOW*(G**2)/(32.0*PI))*(HO**2)*TWAVE*W*TPER)
ALPHA =PWAVE/PTIDE
Z=Z+1
END DO
A= (WS*CONC)/RHOD
B = (RHOW*F)/(2.0*TCR)
C Solve dh/dt by 4thorder RungeKutta method
X=D(J)
CALL KUTTA(A,B,DO,HT(J),L,PHI,TWAVE,UO,X,FF)
K1 =DT*FF
X=(D(J)+(K1/2.0))
CALL KUTTA(A,B,DO,HT(J),L,PHI,TWAVE,UO,X,FF)
K2=DT*FF
X= (D(J)+(K2/2.0))
CALL KUTTA(A,B,DO,HT(J),L,PHI,TWAVE,UO,X,FF)
K3= DT*FF
X=(D(J)+K3)
CALL KUTTA(A,B,DO,HT(J),L,PHI,TWAVE,UO,X,FF)
K4=DT*FF
D(J+1)=D(J)+(1.0/6.0)*(K1 +2.0*K2+2.0*K3+K4)
VOL(J + 1)= (AD)*(DOD(J + 1))
DVOL = (ABS(VOL(J + 1)VOL(J)))
IF(VOL(J+ 1).LT.0.O) THEN
DVOL=0.0
ELSE
T(J+ 1)=T(J)+(DT/3.1557E+07)
ENDIF
IF(J.EQ. 15000) THEN
DVOL=0
ELSE
J=J+1
ENDIF
END DO
NPTS=J1
C Write out 
WRITE(2,19)OUT1,DATE
WRITE(2,20)RHOD
WRITE(2,21)D50
WRITE(2,22)WS
WRITE(2,23)CONC
WRITE(2,24)TCR
WRITE(2,25)FC
WRITE(2,26)FW
WRITE(2,27)HO
WRITE(2,28)TWAVE
WRITE(2,29)PHI
WRITE(2,30)TP
WRITE(2,31)TR
WRITE(2,32)AD
WRITE(2,33)W
WRITE(2,34)DO
WRITE(2,35)UMAX
WRITE(2,36)UO
WRITE(2,37)ALPHA
WRITE(2,38)NN
19 FORMAT(2X,'Output filename:',A14,'Run date:',A18)
20 FORMAT(2X,'Dry sediment density (kg/m**3) =', F15.5)
21 FORMAT(2X,'Mean grain size diameter (m) =', F15.5)
22 FORMAT(2X,'Settling velocity
23 FORMAT(2X,'Sediment concentration
24 FORMAT(2X,'Critical shear stress
25 FORMAT(2X,'Friction factor for currents
26 FORMAT(2X,'Friction factor for waves
27 FORMAT(2X,'Deep water wave height
28 FORMAT(2X,'Wave period
29 FORMAT(2X,'Angle between waves/currents
30 FORMAT(2X,'Tidal prism
31 FORMAT(2X,'Spring Tidal Range
32 FORMAT(2X,'Ebb shoal area
33 FORMAT(2X,'Minimum throat width
34 FORMAT(2X,'Initial inlet depth
35 FORMAT(2X,'Max throat velocity
36 FORMAT(2X,'Initial velocity over shoal
37 FORMAT(2X,'Wave/tide power ratio, ALPHA
38 FORMAT(2X,'Number of tidal steps, NN
CLOSE(UNIT=2)
WRITE(3,300)(T(K),VOL(K),K= 1,NPTS)
300 FORMAT(F8.4,I12)
CLOSE (UNIT=3)
STOP
END
(m/s)
(kg/m**3)
(N/m**2)
(m)
(s)
(m**3)
(m)
(m**2)
(m)
(m)
(m/s)
(m/s)
=', F15.5)
=', F15.6)
=', F15.5)
=', F15.5)
=', F15.5)
=',F15.5)
=', F15.5)
=', F15.5)
=', E15.5)
=', F15.5)
=', F15.5)
=', F15.5)
=', F15.5)
=', F15.5)
', F15.5)
=', E15.5)
=', E15.5)

C
C
C
C
This subroutine calculates the shoaled nearshore wave height due to
the change in the water depth of the ebb shoal assuming normal waves
with straight parallel contours resulting in Kr= 1.
   Variable Definitions 
C
C Variable
C 
C C
C CO
C HTWAV
IE
Description
Shallow water wave celerity
Deep water wave celerity
Shoaled nearshore wave height
SUBROUTINE SHOAL(D,G,HO,TWAVE,HTWAVE)
C Declarations
REAL C,CO,D,G,HO,HTWAVE,TWAVE
C Calculate shoaled wave height 
CO= (1.56*TWAVE)
C=(SQRT(G*ABS(D)))
HTWAVE= HO*(SQRT(CO/(2.0*C)))
RETURN
END
