UFL/COEL96/010
MODULATION OF INLET EBB TIDAL DELTAS BY
SEVERE SEA
by
Paul T. Devine
Thesis
1996
MODULATION OF INLET EBB TIDAL DELTAS BY SEVERE SEA
By
PAUL T. DEVINE
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
1996
ACKNOWLEDGMENTS
I would like to extend special thanks to Dr. A. J. Mehta for his intricate attention to
detail, and constantly keeping me focused on the path of least resistance by challenging me
to resist my impulses. I thank Dr. D. M. Sheppard for opening my eyes to lands west of the
Atlantic. I thank Dr. R. G. Dean for always having an open ear, and for being so
diplomatically delicate in his responses. I appreciate the understanding of all nine of my
office mates who were a part of my regimented lunch routine. I would like to express my
sincere gratitude to everyone who gave me their time and energy; without their assistance the
road would have been additionally windy and infinitely bumpier.
This study was partially funded through contract DACW3994K0043 from the
Coastal Engineering Research Center (CERC) of the U.S. Army Corps of Engineers
Waterways Experiment Station, Vicksburg, MS. Thanks are due to Mr. Gary Howell of
CERC for project support and management.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS .......................................... ....... ii
LIST OFFIGURES .............................................. ...... vi
LIST OF TABLES ....................... ............................... x
LIST OF SYM BOLS ....................... ............................. xi
ABSTRACT ......................................................... xv
CHAPTERS
1 INTRODUCTION ............................................ 1
1.1 Problem Statement ................. ............................ 1
1.2 Delta Mining ...................................................3
1.3 Some Previous Studies on Ebb Deltas .............................. 9
1.4 Outline of Presentation ..................................... 23
2 PARAMETERIZATION OF EBB DELTA VOLUME .................... 24
2.1 Introduction ..................................................24
2.2 Delta Volume ................................................ 24
2.3 NoInlet Geometry Parameterization ............................... 28
2.4 Delta Geometry Parameterization ................................. 29
3 DELTA EROSION MODEL ........................................40
3.1 Introduction ..................................................40
3.2 Profile Erosion ........................................ ........ 42
3.3 Volume Calculation ........................................... 43
3.5 Erosional TimeConstant ........................................43
3.6 Depth of Limiting Motion ...................................... 46
3.7 D eposit Profile ................................................48
4 DELTA VOLUME AND EROSION .............................. 51
4.1 Introduction .................................................. 51
4.2 Dependence of Delta Volume on Profile Parameters .................. 51
4.3 Changes in Delta Volume ................... .................. 55
4.4 Case Studies ......................... ............... ..........67
5 SUMMARY AND CONCLUSIONS ................................. 81
5.1 Summary and Conclusions ................ ..................... 81
5.2 Recommendations for Future Investigations ......................... 83
5.3 Note of Caution ............................................... 84
REFERENCES ...................... ............ ....... ... ..... ...... 85
APPENDICES
A FORTRAN CODE FOR LEAST SQUARES FIT FOR THE
NOINLET PARAMETERS ....................................... 91
B MATLAB CODE FOR LEAST SQUARES FIT
OF VI, A AND MPARAMETERS ..................................93
C MATLAB CODE FOR LEAST SQUARES FIT FOR
PARABOLIC RELATION OF APARAMETER ........................ 96
D MATLAB CODE FOR THE LEAST SQUARES FIT OF POWERLAW
RELATION WITH HYPERBOLIC APARAMATER .................... 98
E MATLAB CODE FOR THE LEAST SQUARES FIT FOR
PARABOLIC RELATION OF MPARAMETER ....................... 99
F MATLAB CODE FOR GAUSSIAN FIT OF VIRTUAL
ORIGIN PARAMETER .......................................... 101
G FORTRAN CODE FOR MODULATION OF DELTA VOLUME ......... 104
BIOGRAPHICAL SKETCH ............................................ 114
LIST OF FIGURES
1.1: A schematic description of an ebb delta, including ebb and flood channels and a
swash bar platform (from Hayes, 1973). ............................... 2
1.2: Bathymetry from Boca Raton Inlet, Florida (from Coastal Planning and
Engineering, 1993). ..............................................2
1.3: Flow chart showing criteria to be evaluated for ascertaining the feasibility of
delta sand mining. Return pathway emanating from a particular set of criteria
implies that these criteria were not satisfied, thus requiring selection of a new
site (from Mehta et al., 1996). ....................................... 4
1.4: Wave spectra from stations near Norderney, Germany (adapted from Niemeyer,
1985). ......................................................... 4
1.5: Representation of the grid system for calculating delta volume, and the no inlet
contours for a typical inlet (from Dean and Walton, 1973). ................ 14
1.6: The length of active profile extends from the shoreline to the depth of closure,
hd. The crossover depth, ho, separates nearshore and offshore zones within
the active length. Dashed curve corresponds to a storm profile, whereas the
monotonic curve represents profile formed under comparatively mild wave
conditions (from Mehta et al., 1996). ................................ 20
1.7: Schematic drawing of an idealized crossshore profile near an inlet and
episodic change in profile geometry due to wave action, where "y" is the
distance offshore, "h" is the depth of the inlet profile, "y," is the distance from
the beginning of the channel to the origin of the profile defining the seaward
slope of the delta, "Uo" is the current emanating from the inlet, "U c" is the
current over the delta crest, "ho" is the depth of the inlet channel, "h d" is the
depth of closure and "ho" is the crossover depth (from Mehta et al., 1996). ... 20
1.8: Diagram depicting the three different delta classifications (from Hoekstra,
1988). ......................................................... 22
2.1: Delta planform schematic. ........................................... 25
2.2: Schematic of delta elevation profile. ................................... 26
2.3: Least squares fit of PBC R3 1978 and 1990 data to Eq. 2.2. ................. 30
2.4: Sample placement of profile origin, and baseline on the ebb delta at New Pass,
Florida. .........................................................30
2.5: Profiles #14 from Jupiter Inlet delta, February, 1980 ....................... 32
2.6: Profiles #58 from Jupiter Inlet delta, February, 1980 ....................... 32
2.7: Profiles #911 from Jupiter Inlet delta, February, 1980....................... 33
2.8: AParameter data from Jupiter Inlet delta, February 1980, and corresponding
parabolic fit. .................................................... 34
2.9: Profiles #14, fit with parabolic relation for A(n) and VI(n) from
Appendix A...................................................... 34
2.10: Profiles #58, fit with parabolic relation for A(n) and VI(n) from
Appendix A...................................................... 35
2.11: Profiles # 911, fit with parabolic relation for A(n) and VI(n) from
Appendix A...................................................... 35
2.12: MParameter fit with A derived from hyperbolic relationship. ............... 37
2.13: Polar plot of virtual origin, VI data, and the corresponding average VI values
for February, 1980 Jupiter Inlet survey. Numbers on the exterior of the plot
are the profile angles with respect to the updrift shoreline. Concentric circles
are lines of equal radii in meters ..................................... 37
2.14: Average VI data normalized by the maximum alongshore and offshore distances
of the virtual origin distribution, and the corresponding Gaussian fit with K =
1.52, February, 1980 Jupiter Inlet survey. ........................... . 38
2.15: Plot of the 37 data points of VI for input into delta modulation model. Numbers
on the exterior of the plot are the profile angles with respect to the updrift
shoreline. Concentric circles are lines of equal radii in meters. ............. 38
3.1: Bruun Rule of erosion, where "s" is the recession, "1" is the active profile length,
"a" is the sea level rise, "d" is the depth of active sediment movement, "e" is
the berm height, and "z" is the point of no net sediment transport (from
Bruun, 1988). ................................................... 41
3.2: Bruun Rule of erosion, with deposit slope. (from Bruun, 1988) ............... 41
3.3: Beachprofile response to waves and surge where R. is the recession at
equilibrium, and R(t) is the recession at a certain time, t. .................. 44
3.4: Differential angular planform area, where 0 is in degrees .................... 44
3.5: Delta radial profile response where R. is at equilibrium, and R(t) is at a specified
time, t. ................... ...................................... 47
3.6: Planform view of the delta illustrating the assumed magnitudes and approach
angles for the breaking waves, HB, on the crest of the delta. .............. 48
3.7: Nondimensionalized deposit slope, 6 = 0.2443461. ....................... 50
3.8: Deposit slope for an arbitrary set of values of 6, S, R(t) and Hm ................ 50
4.1: Isometric (a) and side (b) views of an arbitrary delta volume with parameters:
CW = 100 m, HC = 3 m, Ani = 0.05, Cn, = 0.7, AK = 0.079, Ami = 0.05,
MK = 0.34, Mmx = 0.8, K = 1.5 m, o = 0.35 m, VImx = 1400 m, VImy = 800 m. .53
4.2: Isometric (a) and side (b) views of an arbitrary delta volume with parameters
set to those in Figure 4.1, except with Ci = 0.675. ....................... 53
4.3: Dependence of delta volume on the noinlet shape and scale parameters, Ani and
Mi. The remaining delta parameters are equal to those in Figure 4.1 ......... 54
4.4: Plot of nondimensional timedependent profile recession versus phase of storm,
trt/Td, with P = 0, corresponding to no lag in profile response and, 3 = t/2,
selected for determining delta response. ............................... 56
4.5: Plot of delta volume during storm versus storm surge, S, for variable wave
height Ho and p = x/2. The initial delta condition is depicted in Figure 4.1. .. 58
4.6: Plot of delta volume during storm versus storm surge, S, for varying wave height
Ho and p = 7t/2. The initial delta condition is depicted in Figure 4.2. ........ 60
4.7: Plot of delta volume during storm versus storm surge, S, for variable wave height
Ho and p = 7t/2. The initial delta condition is that of Figure 4.1 with VImx =
1000 and Vimy = 500, effectively producing a smaller delta ............... 60
4.8: Plot of delta volume during storm versus storm surge, S, for variable wave height
Ho and p = 7t/2. The initial delta condition is that of Figure 4.2 with VImx =
1000 and Vimy = 500, effectively producing a smaller delta ............... 60
4.9: End of storm radial profiles for the steep noinlet condition with [(a) and (b)],
and without [(c) and (d)] the accretionary portions of the profile included.
Theta refers to the profile angular orientation with respect to the updrift
shoreline. The equilibrium profile is the solid line, and the maximum
recession profile is the dashed line. .................................. 62
4.10: End of storm profiles for the initial condition in Figure 4.2 with [(a) and (b)],
and without [(c) and (d)] the accretionary portions of the profile included.
Theta refers to the profiles angular orientation with respect to the updrift
shoreline. The profile is the solid line, and the maximum recessed profile is
the dashed line. ............................................... 63
4.11: Planform views of the evolution of the top of delta of Figure 4.1 with increasing
values of storm surge and changes in incident wave height, H,. Solid line is
the initial condition and the dashed lines correspond to increasing values of
storm surge from 0.5 m to 3.5 m in increments of 0.5 m, with the largest
deviation from the initial condition occurring at the largest surge ........... 65
4.12: Planform views of the evolution of the top of delta of Figure 4.2 with increasing
values of storm surge and changes in incident wave height, Ho. Solid line is
the initial condition and the dashed lines correspond to increasing values of
storm surge from 0.5 m to 3.5 m in increments of 0.5 m, with the largest
deviation from the initial condition occurring at the largest surge ........... 65
4.13: Isometric (a) and side (b) views of the maximum erosion of the delta of Figure
4.1 due to incident wave height, Ho = 5.5 m, maximum surge, S = 3.5 m, and
3 = x/2 ............................................... ......... 66
4.14: Isometric (a) and side (b) views of the maximum erosion of the delta of Figure
4.1 due to incident wave height, H, = 5.5 m, maximum surge, S = 3.5 m, and
S= /2. ......................................................... 66
4.15: Bathymetric chart of Jupiter Inlet (from Coastal Planning and
Engineering, 1994) ...............................................69
4.16: Ebb delta volume versus year with modelcalculated growth envelope for
Jupiter Inlet with a = 0.17, H, = 0.54 m, and a = 0.27, Ho = 0.68 m (from
Dombrowski, 1994). ............................................. 69
4.17: Plot of delta volume at Jupiter Inlet and WIS hindcast wave data for the time
period 19561993 (from USACOE, 1996) .............................70
4.18: Plot of the Jupiter Inlet delta volume versus maximum hindcast WIS wave
height during the preceding year. ................................... 70
4.19: Bathymetric chart of South Lake Worth Inlet (from Olsen and Associates, Inc.
1990). ......................................................... .72
4.20: Ebb delta volume versus year with modelcalculated growth envelope for
South Lake Worth Inlet with a = 0.03, Ho = 0.19 m, and a = 0.60, H =
0.28 m (from Dombrowski, 1994). .................................. 73
4.21: Plot of delta volume South Lake Worth Inlet and WIS hindcast wave data for
the time period of 19651993 (from USACOE, 1996) .................... 74
4.22: Plot of the South Lake Worth Inlet delta volume versus maximum hindcast WIS
wave height during the preceding year. ...............................74
4.23: Bathymetric chart of East Pass (from USACOE, 1990) .................... 76
4.24: Plot of delta volume at East Pass and WIS hindcast wave data for the time period
19651976 (from USACOE, 1996). ................................ 76
4.25: Plot of East Pass delta volume versus maximum hindcast WIS data during the
preceding year. ....................................................77
4.26: Bathymetric chart of the area surrounding Katiakati Inlet, New Zealand,
including the ebb delta (from Hume et.al., 1996). ....................... 79
4.27: Ebb delta at Katiakati Inlet (New Zealand) ebb delta protrubance from the
shoreline over a 50 year period (adapted from Hume et.al., 1996) ............ 79
LIST OF TABLES
1.1: Inlets where ebb delta mining has been performed. .......................... 6
1.2: Benefits, adverse impacts and monitoring at inlets given in Table 1.1............ 7
1.3: A summary of previous delta studies..................................... 17
4.1: Dependence of delta volume on the noinlet scale parameter for a given delta
seaward slope. ................................................... 54
LIST OF SYMBOLS
a = Sea level rise in Bruun Rule definition
A = Powerlaw relation scale factor
AK = Scale factor in determining the parabolic relationship for A
AMN = Minimum value of A for eleven fits of digitized data
ANI = Noinlet scale factor
AREA(r) = Projected angular differential area between two given radial
distances
B = Berm height in convolution method
C = Constant in convolution method
CW = Flood channel width
d = Depth of active sediment movement in Bruun Rule definition
DH(n,r) = Difference in elevation between the delta height and the noinlet
contour
DVOL(r) = Angular differential volume between two radial distances
e = Berm height in Bruun Rule definition
f (t) = Time dependent storm surge forcing
g = Acceleration due to gravity
GAUS = Modified Gaussian distribution fit to VIm(n)
h = Depth of profile in Figure 1.7
hco = Crossover depth in Figure 1.7
hd = Depth of closure in Figure 1.7
ho = Depth of the inlet channel in Figure 1.7
H = Profile Depth
Hb(n) = Breaking wave height
HNI = Noinlet profile depth
Hm(n) = Depth of limiting motion
Ho = Unrefracted wave height in shore normal direction
HS = Depth of the crest of delta
1 = Active length of profile in Bruun Rule definition
M = Powerlaw shape factor
MK = Scale factor in determining the parabolic relationship for M
MMA = Maximum value of M for eleven fits of digitized data
M = Noinlet shape factor
n = Index referring to the value of the variable of particular profile
number
OF = Offset from virtual origin to delta crest
P = Spring tidal prism
Po = Wave power
r = Radial distance from origin: index referring to the value of the
variable at particular radial location
r2, r, = Specific radii between which the differential area is calculated
rm(n) = Radius at the depth of limiting motion
R = Radial distance from origin
R(t) = Time dependent virtual origin recession
R. = Equilibrium recession of the radial virtual origin
s = Profile recession in Bruun Rule definition
S = Storm surge
T = Wave period
Td = Storm duration
T, = Tidal period
Ts = Characteristic erosion timeconstant
Uo = Current emanating from inlet
Uc = Current over the delta crest
VI(n) = Virtual origin for shoal seaward delta profile
VIT = Temporary virtual origin during least square fitting of A, VI
parameters
VIy(n) = Alongshore component of VI for use in determining Gaussian
distribution
VIN(n) = Normalized alongshore component of VI
VIyM = Maximum value of the alongshore component of VI
VIxM = Maximum value of the offshore component of VI
X(n,r) = Normal distance from shoreline to a point on delta
Xb = Surf zone width in convolution method
y = Distance offshore in Figure 1.7
yo = Distance from the beginning of the channel to the origin of the
profile in Figure 1.7
z = Point of no net sediment transport in Bruun Rule definition
2ao = Spring tidal range
a = Ratio of normal incident wave energy to tidal energy at inlet
3 = Ratio of erosion time scale to the storm duration
6 = Angular offset for determining deposit slope
60 = Angular separation in between radial profiles
Y, = Unit weight of seawater
K = Scale factor for fitting Gaussian distribution of VI, data
a = Standard deviation of VIy
0 = Angular orientation of present profile relative to updrift shoreline
I: = Time differential
5, 6 = Coefficients for modification of the deposit profile
Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science
MODULATION OF INLET EBB TIDAL DELTAS BY SEVERE SEA
By
Paul T. Devine
August, 1996
Chairman: Dr. Ashish J. Mehta
Major Department: Coastal and Oceanographic Engineering
Sandy ebb deltas in the microtidal sea environment subject to storm waves and surges
are represented in terms of an idealized geometry guided by measured offshore profiles in
the vicinity of inlets, and the method of Dean and Walton for calculating delta volumes.
Characteristic delta features include an arcuate planform shape, ebb channel, marginal flood
channel boundaries, a swash platform and offshore slopes that are defined by a generalized
powerlaw akin to that commonly used for delineating beach profiles. Delta volumes
calculated by using realistic values of the geometric constants are shown to result in values
that are comparable to those measured at inlets.
A simplified model for assessing the response of the delta to severe waves and
associated storm surges is developed by using the convolution method of Kriebel and Dean.
In this model, erosion of the delta occurs due to a recession of the offshore portion of the
delta. It is found that the characteristic erosion timeconstant for the delta to attain
equilibrium is an order of magnitude greater than that corresponding erosion timeconstant
for beach erosion. This finding, which is a reflection of the relatively slower response of the
delta than beaches to eroding forces, is seemingly in agreement with natural events, and
requires further exploration through detailed pre and post storm monitoring of inlet deltas.
The model is used to show that reductions in delta volume due to wave forcing are
commensurate with volumetric changes recorded at several inlets.
CHAPTER 1
INTRODUCTION
1.1 Problem Statement
When planning a beach nourishment project it is often found that an economical,
beach compatible offshore sediment source is not available (Walther, 1995). In recent years,
therefore, the trend has been to look to the ebb deltas at sandy inlets for a comparatively
inexpensive source of beach quality sand. At present, the potential effects of mining the
delta for sand on the littoral system cannot be accurately predicted. Some possible effects
of mining are alterations in the inlet hydraulics and sediment transport pathways, focusing
of wave energy on the nearby shoreline causing erosional "hot spots," and deprivation of the
downdrift shoreline of its sediment supply due to a reduction in the natural bypassing ability
of the inlet system.
According to Bruun and Gerritsen (1959, p. 82), "A submerged bar in front ofan inlet
or harbor entrance on a littoral drift coast will often function as a 'bridge' upon which sand
material is carried across the inlet or entrance. Every channel dredged through the bar will,
therefore be subject to deposits. Both bar and tidal flow bypassing include cases with
irregular transfer of large amounts of materials in migrating sand humps or by change in
the location of channels." Thus, since the delta could be the primary sediment transport
pathway to the downdrift shoreline, it is necessary to have a better understanding of how the
inlet ebb delta responds to oceanic forcing. Figure 1.1 depicts a schematic of an arcuate ebb
Figure 1.1: A schematic description of an ebb delta, including ebb and flood channels and
a swash bar platform (from Hayes, 1973).
Figure 1.2: Bathymetry from Boca Raton Inlet, Florida (from Coastal Planning and
Engineering, 1993).
3
delta, including ebb and flood channels and a swash bar platform. Figure 1.2 is the
bathymetry of a typical sandy microtidal inlet ebb delta system. This study is concerned with
the response of such deltas to forcing by episodic waves and the associated storm surges. As
a basis for this work, we will first consider criteria for delta mining and illustrate examples
of mining from available sources.
1.2 Delta Mining
Using knowledge of the beach fill template and length of beach nourishment project,
a volume requirement of sand can be determined. In order to make an educated decision
whether to mine this volume of sediment from the delta, three criteria must be satisfied. The
first criterion is the availability of this volume of sediment, and the corresponding cost for
transportation and placement on the beach. Too small a volume, or too large a pumping
distance for the dredge could exclude the sediment source. The second criterion is the
composition and size of sediment available for mining. A sediment size smaller than native
will provide a smaller dry beach width than the same fill volume of native sediment (Dean
and Dalrymple, 1996). The third criterion is based on assessments of physical and ecological
impacts of the proposed operation. Acceptable changes to the hydrodynamic and sediment
transport processes surrounding the mining area must be predicted. Some probable
environmental impacts to the benthic habitats due to inlet modifications are briefly
mentioned by Mehta and Montague (1991). Figure 1.3 is a flow chart showing the three sets
of criteria that must be satisfied when considering delta mining.
Excluding impacts to the ecological system, the main issues concerning influences
to the physical system due to delta mining are rate of infilling of the sand borrow pit and
Figure 1.3: Flow chart showing criteria to be evaluated for ascertaining the feasibility of
delta sand mining. Return pathway emanating from a particular set of criteria implies that
these criteria were not satisfied, thus requiring selection of a new site (from Mehta et al.,
1996).
1500
Offshore (I)
l 1 000 '
soo
"Z Inshore (III)
0.00 0.06 0.12 0.1 D.24 0.30 0.36 0..2 0.48
Frequency (Hz)
Figure 1.4: Wave spectra from stations near Norderney, Germany (adapted from Niemeyer,
1985).
5
changes to the nearby shorelines. One cause of shoreline change would be the reduction in
sediment supply to the downdrift shoreline caused by impoundment of littoral sediment in
the dredge pit. Bruun and Gerritsen (1959) describes the transfer of sand from the updrift
to downdrift sides of the inlet as occurring over the, "shoal or offshore bar." A dredge pit
could possibly interrupt the main sediment pathway across the delta in a bar bypassing type
of inlet system, and thereby change the shorelines in the vicinity. Another cause of shoreline
change would be modification of the wave climate due to the reduction of the wave
sheltering capability of the ebb delta due to mining. Figure 1.4 shows measured wave spectra
at three stations near Norderney in Germany (Niemeyer, 1985). Note that there is a
substantial reduction in wave energy from the offshore station I to the ebb delta sheltered
station II.
Examples of delta mining operations are summarized in Table 1.1. Table 1.2
provides a summary of information on benefits, adverse impacts and monitoring of mining
at inlets in Table 1.1. These include seven inlets in Florida: John's Pass (Walther and
Douglas, 1993; Army Corps of Engineers, Jacksonville District, personal communication),
Longboat Pass and New Pass (Applied Technology and Management, 1992; 1993a; 1993b),
Redfish Pass (Tackney and Associates, 1983; Coastal Planning and Engineering, 1992), Boca
Raton Inlet (Coastal Planning and Engineering, 1993), Jupiter Inlet (Coastal Planning and
Engineering, 1994) and Nassau Sound Entrance (Olsen and Associates, personal
communication). Five examples are from South Carolina: Port Royal Sound Entrance
(Bodge et al., 1993), Fripp Inlet (Coastal Science and Engineering, 1990a;1990c), Port Isidro
Entrance (Coastal Science and Engineering, 1989; 1993),
Table 1.1: Inlets where ebb delta mining has been performed.
Inlet Mining Site Year Volume Placement Method of
(m3) Location Dredging
John's Pass, FL Seaward side of 1988 405,000 Updrift beach 61 cm
northern delta lobe ___ Cutterhead
Longboat Pass, FL Seaward side of 1993 1,020,000 Downdrift Dustpan dredge
delta __beach
New Pass, FL Seaward side of 1993 720,000 Updrift beach Dustpan dredge
delta
Redfish Pass, FL Center of delta from 1981 501,000 Downdrift 53 cm
channel to seaward 1988 1,220,000 beach Cutterhead
extent of delta
Boca Raton, FL Top of delta 1985 169,000 Downdrift 61 cm
beach Cutterhead
Jupiter, FL Seaward side of 1995 392,000 Downdrift 76 cm
delta beach Cutterhead
Nassau Sound, FL Outer, updrift and 1994 2,140,00C Updrift beach 76 cm
relict portion of cutterhead
delta
Port Royal Sound, SC Seaward edge of 1990 596,000 Downdrift 76 cm
delta beach Cutterhead
Fripp, SC North delta of 1974 469,000 Updrift beach 5161 cm
entrance 1980 1,080,000 pipeline
Port Isidro, SC Landward edge of 1990 524,000 Beach eroded 7684 cm
delta by entrance pipeline
flood channel
Captain Sam's, SC Closed off migrating 1983 134,000 Operation Earthmovers and
old entrance, (by tides meant to bulldozers,
creating a new and nourish 99,000 m3 to
entrance updrift of waves) downdrift beach close old
the old one ___ channel
Hog, SC Shoreattached delta 1990 288,000 Updrift beach Hydraulic hoe at
low tide
Townsend's, NJ Updrift entrance 1978 483,000 Updrift beach 76 or 91 cm
swash bar complex 1983 626,000__ Cutterhead
Townsend's, NJ Downdrift entrance 1987 1,030,000 Surrounding 76 or 91 cm
swash bar complex beaches Cutterhead
Great Egg, NJ Undetermined 1992 to 4,900,000 Downdrift Dustpan and
portion of delta 1994 beach Hopper dredges
Absecon, NJ Spit attached to 1986 765,000 Downdrift 76 or 91 cm
north ietty __ beach Cutterhead
Captain Sam's Inlet (Kana and Mason, 1988) and Hog Inlet (Coastal Science and
Engineering, 1990b;1992). Finally, three examples from the New Jersey coast are:
Townsend's Inlet (Ippolito and Sorensen, 1990), Great Egg Inlet (New Jersey Department of
Table 1.2: Benefits, adverse impacts and monitoring at inlets given in Table 1.1.
Entrance Benefits Adverse Impacts Monitoring
John's Pass, FL Beach was restored; Increased beach erosion Borrow area shoaled at the
reduced shoaling in in the vicinity due to rate of 24,000 m3 from
entrance channel due to reduced sand by 1988 to 1992
sand trapping in borrow passing
area
Longboat Pass/ New Calculated wave None were predicted to Between Dec. 91 and Dec.
Pass, FL refraction patterns occur; no impacts were 92 Longboat Pass borrow
showed a reduction of monitored area volume increased by
sediment "trapped" by 150,000 m3; however,
deltas, and a more even between Dec.'92 and
spreading of wave Apr.'93 (due to March 13
energy to the south of 14 storm) borrow area
each entrance volume decreased by
41,000 m3
Redfish Pass, FL Erosion protection for No specific studies Approximately 80,000 m3
downdrift shoreline; were performed of finegrained material
accretion to the north of was carried into the borrow
the entrance, and south area within 18 months after
of the project 1981 project; between
1989 and 1991 35,000 m3
filled both (1981 and 1988
borrow areas
Boca Raton, FL Beach erosion Feeder beach within Beach 1 km south of
contained, improved 600 m of entrance entrance grew by an
navigational conditions eroded critically; beach average of 12 m; the entire
over delta and nourishment project delta, including borrow
maintenance of water planned for 1995 using area exceeded pre project
quality in Lake Boca delta sediment volumes
Raton
Jupiter, FL Erosion protection for None were predicted to Pre and postproject
downdrift shoreline occur, including surveys of fill area to be
focusing of wave carried out
energy on jetties,
changing of littoral
pattern or increased
salinity in Loxahatchee
River
Nassau Sound, FL Refraction analysis for None were predicted to No monitoring of the relict
borrow area showed a occur delta region
reduction of wave
energy on shoreline, and
an increase of energy in
the sound, resulting in
decreased sediment
transport
Table 1.2 Continued
Port Royal Sound, SC Mitigated chronic None were predicted to No monitoring of delta
erosion problem with a occur
predicted 8 year project
life
Fripp, SC Temporary beach No studies were Rapid recovery of borrow
nourishment performed; no impacts area, approximately
were monitored 153,000 m3 accumulated in
delta since 1980
Port Isidro, SC Dredging the delta None were predicted to Sediment filled the borrow
moved the channel 125 occur; no impacts were area, and the channel
m offshore, removing monitored slowly migrated landward
the source of scour, and towards its equilibrium
renourishing the beach position
Captain Sam's, SC Relict delta "pushed" None were predicted to Entrance began migrating
ashore by wave action occur; no impacts were to the south at it's previous
nourished the beach at monitored rate
the rate of 130,000 m3/y
between Mar.'83 to May
'85; totally 1,150,000
m3 by 1993
Hog, SC Emergency nourishment Delta filled with Position of Hog Inlet
for heavily armored sediment at the expense channel did not shift
sections of shoreline of deltas further toward Myrtle Beach
following hurricane offshore and the shoreline; six months after
Hugo downdrift shoreline; the project, 95% of the dry
7,100 m3 of remedial beach was recovered due tc
nourishment became nourishment and seasonal
necessary to guard effects
downdrift shoreline
Townsend's, NJ Beach nourishment Mining redirected the Critical beach erosion
ebb channel northward along Avalon shoreline,
through the delta, and growth of a large spit
resulting in changed in the interior of the
channel hydraulics entrance where the channel
once occurred
Townsend's, NJ Emergency nourishment, No studies were done; Monitoring to verify
redirection of channel no impacts were channel position
monitored
Great Egg, NJ Beach nourishment No studies were Monitoring of delta system
preformed planned
Absecon, NJ Beach nourishment No studies were 43% of fill remained in
preformed 1991, some sediment was
lost offshore; no
monitoring of delta
Environmental Protection, personal communication) and Absecon Inlet (Weggel and
Sorensen, 1991). The minimum volume of material transported was 1.34 x 105 m3 at Captain
9
Sam's Inlet to a maximum of 4.9 x 106 m3 at Great Egg Inlet, and in the majority of cases the
material was placed on a segment of the beach downdrift of the inlet. The method of
dredging included the use of cutterhead dredges, dustpan dredges and a hydraulic hoe at Hog
Inlet. Artificial closure of Captain Sam's Inlet was a novel method making use of natural
forces to transport relict delta sediment to the downdrift beach. Closure of the old entrance
was carried out by earthmovers and bulldozers. It should be emphasized that the information
in Tables 1.1 and 1.2 is based on the given citations only, and therefore does not include any
subsequent studies, e.g., monitoring, that may have been carried out at the inlets considered.
Although as noted, each of the beaches in Tables 1.2 received the benefit of
additional sediment within their system, in some cases there was no preproject study to
predict possible adverse effects on the nearby shoreline due to mining of the delta. Also,
postproject monitoring was not always performed. Where it was performed, adverse effects
were recorded in all cases. If a standardized procedure were developed for delta mining site
selection, prediction of impacts and postproject monitoring, the processes involved could
be better understood in the future, and mining planned better.
1.3 Some Previous Studies on Ebb Deltas
In order to better understand the interaction of inlets with the downdrift shoreline,
Bruun and Gerritsen (1959) discuss the two different types of sediment bypassing by natural
action that occur at an inlet. Bypassing over an offshore bar occurs when the ratio of the
rate of littoral drift to the volume of flow through the inlet in one tidal cycle is high. On the
other hand, bypassing by tidal flow action occurs when this ratio is low. Inlets exposed to
tides and waves generally exhibit both types of bypassing. Also, a relationship is obtained
10
between the intensity of the wave forcing and the magnitude of the bypassing since,
according to Bruun and Gerritsen (1959, p.98) "experience demonstrates that such by
passing cannot exist without considerable wave action."
In response to the need for prediction of the navigability and stability of an inlet,
Bruun and Gerritsen (1960) defined an inlet stability ratio. The ratio of the spring tidal prism
passing through the inlet over onehalf tidal cycle to the annual average rate of littoral drift
reaching the inlet was considered as the relevant stability ratio. Large values of this ratio
(>150) correspond to inlets which have good flushing characteristics, and large clearances
over the delta. As the value of this ratio decreases the ebb delta becomes more pronounced,
and the controlling depth of water above the delta decreases. When values of the stability
ratio are small enough (<50) the natural channels are very shallow and unreliable becoming
dangerous for even shallow draft vessels.
To further quantify the stability of an inlet, O'Brien (1971) defined a ratio, a, of the
normally incident wave energy over a tidal period to the tidal energy through the inlet over
the tidal period. Wave energy over one tidal period is a function of the wave power, Po,
width of the inlet throat, w, and the tidal period, Tr. The tidal energy through the inlet over
the tidal period is a function of the spring tidal range, 2ao, the spring tidal prism, P, and the
unit weight of seawater, y,. Thus,
PwT,
a = (1.1)
2aosYwP
Considering a representative wave height, Ho, and period, T, wave power is defined as:
P, Y 2T (1.2)
S 32rT
where g is the acceleration due to gravity. Mehta and Hou (1974) plotted the values of a
versus tidal prism for many inlets of known stability. A line was drawn that separated the
unstable high values of the coefficient from the stable low values, thus confirming the
stability relationship (Eq. 1.1).
A method was needed to predict the quantity of sediment to be removed from the
inlet channel during maintenance dredging. For that purpose, Gole et al. (1971) proposed
an empirical formula for sedimentation in dredged channels within inlets by determining the
percentage of the total littoral drift that will be entrapped by the increased depths and the
corresponding decreased current velocities across the channel.
Realizing that inlet ebb deltas are comparatively large deposits of littoral material that
is diverted from the shorelines downdrift of the inlets, Dean and Walton (1973)
qualitatively described the sedimentary processes associated with inlet flood and ebb
currents. The ebb currents were compared to a turbulent jet escaping a nozzle, while flood
currents were considered to be analogous to a concentric sink flow pattern into the inlet.
Since the turbulent ebb jet entrains water, and thus sediment from the adjacent shorelines,
both flood and ebb currents have the effect of transporting sediment into the inlet from both
the updrift and downdrift shorelines. Since the turbulent jet displaces the same volume of
water as the sink flow, but is concentrated within a narrower area, higher ebb velocities exist,
and these transport sediment offshore forming the ebb delta. Dean and Walton (1973) stated
12
that wave action will limit the volume of the delta to some equilibrium volume by
transporting sediment deposited by the ebb currents back toward the inlet.
Dean and Walton (1973) also developed a method for determining the sediment
volume in the ebb delta. In this method, a grid, as well as the elevation contours that would
exist in the absence of the inlet, or the "noinlet" contour lines, is superimposed on the plot.
The difference between the noinlet and delta elevation at each node of the grid box is
calculated, averaged over the four corer values for the grid, and multiplied by the surface
area of grid to find the delta volume within the grid box. The total delta volume is then the
sum of all of the grid box volumes. This method was applied to twentythree Florida inlets.
Figure 1.5 is a representation of the grid system and the noinlet contours for a typical inlet.
Continuing the work of Dean and Walton (1973), Walton and Adams (1976) analyzed
the delta volumes of fortyfour tidal inlets around the United States. The inlets were divided
into three different incident wave energy groups, and the delta volumes were plotted versus
tidal prism for each group. A powerlaw relation was fit to all three data sets. The
differences in the powerlaw coefficients implied the dependence of delta volume on wave
energy.
In order to better predict the lifespan of, and processes involved in, channel dredging
at tidal inlets, Galvin (1982) presented a method to determine the shoaling rates, and times
for overdredged channels to infill to the design depth. Galvin (1982) observed that the
maximum rate of shoaling occurs immediately after dredging, shoaling rate is increased
when dredging reduces the currents in the channel, shoaling rate increases when zero by
13
passing occurs, and shallow dredged channels tend to approach the natural depth at a slower
rate than a deep dredged channels.
Marino (1986) applied Dean and Walton's (1973) method to Florida inlets. A
dimensional analysis was preformed to determine relationships between relevant inlet
parameters. From this analysis and its application to Florida inlet data, Marino (1986)
concluded that the delta volume not only depends on the tidal prism, but also on the ratio of
throat width to channel depth.
In order to understand sediment transport processes in the vicinity of the tidal
inlet, Ozsoy (1986) analytically modeled the sediment transport resulting from the turbulent
jet. Ozsoy's (1986) results were: 1) Increasing bottom friction caused jet flow sediment
concentrations to decrease and, in general, the resulting sediment deposition to become fan
shaped and relatively close to the inlet mouth. 2) Decreasing the bottom slope caused
sediment to be transported further offshore in elongated shoals flanking the channel. 3)
Material with small settling velocities was jetted far offshore, while grains with large settling
velocities were deposited closer to the inlet mouth. 4) Currents less than the critical value
for the initiation of sediment movement caused sediment to deposit close to the mouth, while
currents with a value much greater than critical caused sediment to be jetted farther from the
mouth.
To improve the understanding of inlet delta dynamics, Oertel (1988) used field data
to relate the relative strengths of ebb and flood currents over different portions of the delta.
It was shown that the resultant of the two flows having differing strengths and directions
leads to flood and ebb dominated zones on the delta. Also, it was shown that ebb tidal deltas
14
Idealized NoInlet
Contour Lines
Existing Contour Lines Oceon
Figure 1.5: Representation of the grid system for calculating delta volume, and the no inlet
contours for a typical inlet (from Dean and Walton, 1973).
derive their shape from the irregularity in the shear stresses produced from the summation
of the two flows. Oertel (1988) further examined the combined effects of wave
inducedalongshore current and the ebb and flood currents. Depending on the relative
strengths of the alongshore and tidal currents, littoral material may be deposited as a spit
originating from the end of the updrift barrier island, or transported into the channel to either
remain there, or be jetted into either the flood or ebb tidal deltas. Oertel (1988) concluded
that the downdrift biased flow field could change the position and angular orientation of the
inlet channel, and the resultant shape of the ebb tidal delta. The extra sediment supply to the
15
delta system from the addition of littoral drift material could remain in the delta system or
be bypassed to the downdrift shoreline.
Hayter et al. (1988) performed a laboratory study using a moveable bed model in
order to examine the effects of both waves and tidal flows on inlet delta formation. One
result of the study was that the delta volumes were larger in the experiments run with waves
and tidal currents than in experiments using only tidal currents. An explanation for this
phenomenon is that the presence of waves increases the bottom shear stress over the current
only condition, suspending additional littoral material which is ultimately deposited in the
enlarged ebb delta. Ultimately, Hayter et al. (1988) concluded that the delta volume was
proportional to tidal prism, and inversely proportional to wave energy, agreeing with
O'Brien.
In response to the recent trend of mining the ebb deltas for beach nourishment
sediment, Mehta and Montague (1991) discussed the need for development of strategies to
artificially bypass sediment, while preserving the natural environment of the inlet. The inlet
management plan includes: 1) maintenance of the inlet channel for navigation purposes. 2)
restoration of the sediment bypassing pathway to the downdrift side of the inlet interrupted
by dredging the inlet. 3) maintaining the premodification ecological balance in the vicinity
of the inlet. An "activepassive boundary" was identified as the depth of closure defined by
a certain storm return period. Sediment bypassing due to waves will occur at depths less
than the active depth. Mining of delta sediment for beach nourishment purposes should
therefore take place beyond the activepassive boundary, so that there are no significant
16
impacts to the natural state of inlet bypassing. Several possible impacts to the benthic
habitats in the vicinity of the tidal inlets due to inlet modifications were discussed.
To quantify reduced bypassing across the dredge pit at an ebb delta, Walter and
Douglas (1993) developed a method to predict the shoaling rate within the borrow area. The
method was similar to the work on sedimentation in entrance channels by Gole et al. (1971).
Walther and Douglas (1993) considered two borrow pits of the same volume but differing
dredged depths. For the particular input conditions of littoral drift, predredged depth, and
volume removed, the shallow pit was found to have twice the initial bypassing rate as the
deep pit. The rate of increase in bypassing was larger for the deep pit, which returned to the
predredged condition in about five years, while the shallow pit required about fourteen years
to equilibrate, with both pits eventually bypassing equal after equilibrium was achieved.
To determine the equilibrium volumes of ebb deltas, Dombrowski (1994) developed
a diagnostic model to simulate the rate of delta growth starting from a newly opened inlet
until the combined shear stress due to currents and waves equals the critical stress for
deposition, when further deposition ceases. Delta growth rate was shown to be a function
of wave and tidal energy, suspended sediment concentration, and sediment grain size.
Increasing the sediment concentration increased the rate of approach to equilibrium, but not
the equilibrium volume. Increasing the sediment diameter increased both the rate of growth
and the ultimate volume due to increasing shear stress needed to initiate sediment
movement. Increasing the wave height decreased the rate of growth due to increases in the
near bed orbital velocities, and associated shear stresses. Increasing the a parameter defined
by O'Brien (1971) according to Eq. 1.1 had the effect of decreasing the delta volume. This
relationship between the delta volume and a was shown to be in agreement with the observed
variation in delta volume at four microtidal inlets.
In order to evaluate the depositional characteristics of inlet tidal flows, Merz (1995)
developed a computer code that combined previous analytical expressions for flood and ebb
flows over sloping bottoms with bedload and suspended sediment transport formulas to
predict sediment deposition over a tidal cycle. The model was applied to Jupiter Inlet and
was shown to match the actual depositional characteristics of the delta.
Table 1.3 is a short summary of the above mentioned studies on ebb delta related
topics.
Table 1.3: A summary of previous delta studies.
Source Significant Findings
Bruun and Described the two modes of sand transfer at inlets, bar bypassing and tidalflow bypassing.
Gerritsen Defined the ratio of the rate of littoral drift to the tidal discharge through the inlet. A large ratio
(1959) corresponds to bar bypassing and a low ratio corresponds to tidal flow bypassing. Waves and
currents of an episodic nature are forces behind bar and tidalflow bypassing. The result is
complexities in bypassing.
Bruun and Defined inlet stability ratio as the ratio of spring tidal prism passing through inlet during onehalf
Gerritsen tidal cycle to the average annual littoral drift reaching the inlet. Large values of the stability ratio
(1960) correspond to stable and deep deltas, while small values correspond to unstable and shallow deltas.
O'Brien Noted that an inlet will be in equilibrium when the normally incident wave energy that tends to close
(1971) the inlet is balanced by the tidal energy which keeps the inlet open. Determined that inlet stability
is a function of wave power, width at throat, tidal period, spring tidal prism and the spring tidal
range.
Gole et al. Developed a formula to calculate the percentage of total littoral drift which will accumulate in a
(1971) dredged channel due to increased depths, and the resulting decreased current velocities across
channel.
Dean and Described the nozzle and sink characteristics of tidal flows through inlets, both of which result in
Walton net flow toward the inlet along the shoreline close to the inlet. Developed a method for measuring
(1973) delta volume.
Walton and Grouped 44 deltas into three categories by wave energy. Related ebb delta volume to tidal prism
Adams for each category, and determined that delta volume is dependent on both prism and wave energy.
(1976)
Table 1.3: Continued
Galvin Developed a method to determine the time for channel shoaling to reduce project depth from the
(1982) initial overdredged depth to the design depth by calculating the channel shoaling rate.
Marino Described problems inherent in applying Dean and Walton's (1973) method to calculate delta
(1986) volume at inlets. Found that the delta volume not only depends on the tidal prism, but on the ratio
of throat width to channel depth.
Ozsoy (1986) Analytical model of a turbulent ebb jet produced by tidal flows. Related sediment fall velocity,
bottom friction bathymetry, and initial velocity at the throat to sediment depositional patterns in
the delta area.
Hayter et al. Moveable bed laboratory model study to determine the effects of wave action and tidal flow on the
(1988) delta formation. Concluded that an increase in tidal prism, or a decrease in wave energy will
increase the volume of the ebb delta.
Mehta and Discussed the need for development of strategies to artificially bypass sediment, while preserving
Montague the natural environment inlets. Discussed elements of inlet management plans, and the "active
(1991) passive delta" concept. Sediment bypassing due to waves occurs in depths less than the passive
delta, and mining of delta sediment for beach nourishment purposes should take place beyond the
activepassive boundary so that there are no significant impacts to the natural state of inlet by
passing. Several possible impacts to the benthic habitats in the vicinity of the tidal inlets due to inlet
modifications were discussed.
Walther and Applied Gole's (1971) theory on bypassing/sedimentation rates around a dredged channel to two
Douglas hypothetical borrow sites at an ebb tidal delta. Initially the deeper site had half the bypassing rate
(1993) of the shallow site, but returned to the predredged condition in 9 years, where the shallow site took
14 years to recover.
Dombrowski Diagnostic model considering the effects of wave energy, suspended sediment concentration, and
(1994) grain size on delta growth, and ultimate equilibrium volume. It was shown that increasing sediment
concentration increased the rate at which the delta achieved equilibrium. Increasing wave energy
decreased the equilibrium volume, but increased the time to achieve equilibrium. A larger sediment
size increased the equilibrium volume due to the larger shear stresses needed to mobilize the
sediment. The relationship between wave and tidal effects was found to agree with previous
observations.
Merz (1995) Continuation of Ozsoy's (1986) work. Wrote a computer code which determined bed and suspended
sediment loads due to both ebb and flood tidal current forcing. If offshore slope, or the friction
factor are increased, the deposition patterns are held closer to shore and become narrower in the
longshore direction, while the deposited depth increases. Depositional patterns for the Jupiter Inlet
simulation agreed with the actual depositional patterns surrounding the inlet.
If ebb deltas are to be repeatedly mined for beach nourishment, prediction of the
shoaling rates within the borrow area is extremely important. The decision concerning if and
19
where to borrow sediment from the delta must be made considering the possible effects to
the surrounding shorelines due to mining, and requirements for refilling of the pit for
renourishment. As noted, in response to this issue Mehta and Montague (1991) defined
"active" and "passive" zones, which correspond to the frequency and severity of sediment
transport due to episodic wave conditions. The active zone represents the area of the most
frequent wave action, and where sediment bypassing customarily occurs. It is in the active
zone that delta mining should be avoided, so that any potentially adverse effects to the
surrounding shorelines can be prevented.
Mehta et al. (1996) suggested that the limit to the active zone of sediment transport
can be found by applying, with minor modifications, concepts derived from beach profile
dynamics. As shown in Figure 1.6, the seaward limit to the active zone of sediment transport
due to severe wave conditions is defined by, hd, the depth of closure. The depth at which
there is no net sediment transport is defined by the crossover depth, hco. The profile from the
shoreline to the socalled crossover depth will experience erosion due to severe storms, while
the profile from the crossover depth to the depth of closure will experience accretion in the
form of a bar. It should be noted that the depth of closure, the length of active profile and
the crossover depth are directly related to the return period of the wave episode, or increasing
bottom response time (Dean et al., 1993). Thus, for example, the depth of closure for the
Atlantic coast of Florida has been estimated to be from 4 m to 7 m (Dean and Grant, 1989),
while on the Genkai Sea in Japan the depth of closure has been estimated to be around 35 m
(Kojima et al., 1986).
Figure 1.6: The length of active profile extends from the shoreline to the depth of closure,
hd. The crossover depth, h,, separates nearshore and offshore zones within the active length.
Dashed curve corresponds to a storm profile, whereas the monotonic curve represents profile
formed under comparatively mild wave conditions (from Mehta et al., 1996).
I yo 
DELTA CREST
EROSION,
BAR
Figure 1.7: Schematic drawing of an idealized crossshore profile near an inlet and episodic
change in profile geometry due to wave action, where "y" is the distance offshore, "h" is the
depth of the inlet profile, "yo" is the distance from the beginning of the channel to the origin
of the profile defining the seaward slope of the delta, "Uo" is the current emanating from the
inlet, "Uc" is the current over the delta crest, "h," is the depth of the inlet channel, "hd" is
the depth of closure and "ho0" is the crossover depth (from Mehta et al., 1996).
In Figure 1.7, a shorenormal bottom profile, which extends from the inlet channel
to the depth of closure, is shown. Since the ebb current (Uo) decreases to almost a negligible
value (Uc) at point A, the profile, h(y), between point A and B can be considered to be acted
on by waves only. One can then assume that the prestorm seaward slope of the ebb delta
can be described by a function h = f,(y), and the storm seaward slope of the ebb delta as h
= f2(y). The poststorm delta profile exhibits erosion in the active zone of transport, and
accretion in the passive zone. If the recession of the seaward profile were to exceed the
landward extent of the delta crest at point A, then wave action would penetrate the delta, thus
possibly affecting the nearby shorelines significantly. The sediment which rebuilds the
eroded delta profile after the storm may be transported into the eroded area either by the inlet
currents, by littoral current, or by waves from the bar area. Additional modifications to the
delta profile will depend on how much accretion occurred inbetween wave events, and the
severity of the next event. The active zone landward of the crossover depth can only be
determined by the instantaneous configuration. Thus, the interpretation of the active zone
of delta for mining purposes is dependent on the episodic return period of storm forcing.
It has been noted by Dombowski (1994) that a delta will approach an equilibrium
volume if it is acted on by a steady state wave environment. However, Dombrowski (1994)
also noted large fluctuations in the volume of sediment which defined the delta at Jupiter
Inlet. It is hypothesized here that these fluctuations are a result of episodic storm events.
Thus, it is the main objective of this study to develop a method to simulate the change in
volume of the delta due to such episodic events. In general, the processes which occur at
22
inlets are very complex, and the problem as a whole is best attacked in parts. The results
from this study should provide a new perspective into one of these processes, namely the
erosion of deltas due to storm waves and associated surges.
This study will focus on microtidal ebb deltas, because deltas in the mesotidal and
macrotidal environment tend to behave differently. This is because the tidal energy in the
latter environments is larger compared to the available wave energy, so that the relative roles
of waves and tides are different from those at microtidal systems. Figure 1.8 is a diagram
depicting three different delta classifications and some examples. This study deals with the
environment of the bottom left third of that figure.
ELONGATE
(birdsfoot)
RIVER
ELONGATE DOMINATED
(birdsfoot)
/Solo wet season
/ e\  \
MISSISSIPPI
L E LOBATE
LOBATE PO
Figure 1.8: Diagram depicting the three different delta classifications (from Hoekstra, 1988).
R__________ ___ ESTUARINE/
STRAIGHT IRREGULAR EMBAYED
1.4 Outline of Presentation
Chapter 2 deals with the construction of a generalized delta based on relevant delta
characterizing parameters. A method is developed to generate the required "noinlet"
bathymetry for defining delta volume. The delta surface is constructed from equations
representing the channel section, the seaward slope of the delta. The method by which the
delta parameters are calibrated is described. Chapter 3 describes the delta erosion process
and the methods on which the process is based. Chapter 4 investigates the sensitivity of the
delta volume to the choice of the noinlet profile. The results of the delta erosion model
applied to a generalized ebb delta are discussed, and qualitatively related to case studies of
delta evolution at four different inlets. Finally, in Chapter 5 the study is summarized and
general conclusions are presented. This chapter concludes with some recommendations for
future studies related to the subject.
CHAPTER 2
PARAMETERIZATION OF EBB DELTA VOLUME
2.1 Introduction
For the purpose of this study, it is necessary to develop a geometric method for
construction of the ebb delta volume. The volume is represented by a series of generalized
mathematical expressions characterized by seven delta related parameters. A specific delta
can be simulated by varying these parameters until the surface generated by the mathematical
expressions mimics the actual shape of the delta in question.
2.2 Delta Volume
The standard procedure to determine delta volume involves constructing parallel
contour lines which would exist if no inlet were present. The depth differences between the
noinlet bathymetry and the actual bathymetry of the delta determine the volume of the ebb
delta in question. This method is described in detail in Dean and Walton (1973), and was
briefly mentioned in Chapter 1.
A schematic planform diagram of the delta is shown in Figure 2.1. The delta baseline
is defined as zero distance offshore, and as a result the flood channel width, CW, is defined
as having a negative value. The shorenormal distance, X, is defined as:
X(n,r) = r SINOCW (2.1)
n2
CW X
IY I Baseline I
SShoreline r RMAX
Alongshore Distance
Figure 2.1: Delta planform schematic.
where n is the integer for delta radial profile number, r is the radius along the profile and 0
is the angle of the profile with respect to the updrift shoreline. In constructing the delta
surface, n = 37 profiles are developed for an adequate representation of the delta surface.
Thus, each profile represents a 60 = 50 differential slice of delta. A general elevation profile
is shown in Figure 2.2. The assumed prestorm erosion delta profile is shown as a solid line,
and the differential delta volume is identified by the hatched area.
The first step in developing the delta volume is to define the "noinlet" profile. This
profile is assumed to take the powerlaw form of Eq. 2.2, (Dean, 1977), where H is depth,
X is normal distance offshore, A is a "scale factor," and M is a "shape factor."
H = AX M (2.2)
I I I I I I I I 1
VI(n) OF(n)
Theoretical
seaward slope HS
Intersection of channel
HC profile and seaward slope
 H(n,r) = A(n)*[rVI(n)]M(n)
f Intersection of seaward slope
CW 1and no inlet profile
HN,(n,r) = AN, X(n,r)CNI
H(n,r) = HS + (HCHS) TANH(rt)/2 (HCHS)/2*
TANH(it + Tt/2*r/[VI(n)+OF(n)])
Radius, r
Figure 2.2: Schematic of delta elevation profile.
The form of Eq 2.2 selected for representing the noinlet contours, where subscript NI means
"noinlet" is:
HNI = AN XMNI
The determination of the variable scale and shape parameters AM and MN, is described in
Section 2.3. The volume of sediment above the noinlet elevation is considered part of the
prestorm delta.
The maximum depth in the channel from the actual delta bathymetry is used to set
the depth of the ebb channel, HC (Figure 2.2). The width of the flood channel, CW, is set
equal to the average distance from the shoreline to the delta baseline. Depth of delta, HS is
27
assumed constant for every radial profile, and is taken to be equal to the depth of the noinlet
profile at a distance offshore equal to the flood channel width, i.e.,
HS = AI CWMN (2.4)
Delta depth, HS, is defined thus to avoid any discontinuities when considering delta profiles
that are parallel to the shoreline, i.e., corresponding to 0 = 0.
In order to generate the seaward slope of the radial profiles of the delta, Eq.2.2 is
modified by changing the A and M parameters, so that they exhibit a dependance on their
angular position on the delta with respect to the updrift shoreline. The distance offshore is
changed to distance from the radial origin of the delta, r, and a virtual origin, VI(n) is added
to the profile, i.e.,
H(n,r) = A(n) [rVI(n)]M(n) (2.5)
Determinations of VI(n), A(n), and M(n) are given in Section 2.4. The virtual origin defines
a reference point for the beginning of the seaward profile, and sets the depth of the
theoretical seaward slope to zero at the origin. The seaward slope is the portion of the delta
slope above the intersection with the channel profile. When the depths over the seaward
slope exceed the corresponding depth of the no inlet profile, the value of the seaward slope
is replaced by the noinlet depth.
Prior to developing the flood channel portion of the delta, the intersection of the
channel profile and the seaward slope must be determined. This intersection occurs at a
28
radial distance equal to the virtual origin of the seaward slope plus a radial offset, OF(n)
(Figure 2.2) defined as,
r
OF(n)= HS ]Mn (2.6)
A(n)]
where OF(n) is the distance from the virtual origin to where the seaward slope equals the
delta depth. Next, a hyperbolic tangent relation is assumed for the channel profile, because
of its zero slope characteristics at the endpoints. Then an equation is developed so that the
channel depth decreases from HC at r = 0 to HS at r = VI(n) + OF(n), i.e.,
H(n,r) = HS + (HCHS)tanh(Tc/2) HCHStanh 7c+T/2 [V) (2.7)
2 [VI(n)+OF(n)]
2.3 NoInlet Geometry Parameterization
Historic surveys were used as a source of the shorenormal profiles at locations far
enough updrift from the delta being studied, and far enough downdrift from the closest
updrift inlet, so as not to be influenced by the delta. As an example, the profile chosen for
evaluating the noinlet condition for Jupiter Inlet, a typical microtidal entrance on the east
coast of Florida, was Florida DNR Palm Beach County survey monument R3, which is
approximately 2,440 m updrift of that inlet. Two surveys taken of Palm Beach County
monument R3, in 1978 and 1990, were combined to obtain a representative noinlet profile.
Eq. 2.4 was fitted to this profile by the least squares method to obtain the AN, and MN,
29
parameters. Appendix A gives the code used for calculating these two parameters. The
result of the averaging and fitting of Eq. 2.2 is given in Figure 2.3.
2.4 Delta Geometry Parameterization
Bathymetric plots were used for determining the parameters HC, CW, A(n), M(n) and
VI(n) for constructing the delta. A location that was "central" on the delta was designated
as the origin. Placement of this origin was somewhat arbitrary, and it is difficult to draw a
distinct criterion for this placement. However, in general, it should be noted that the origin
should be in the ebb channel, so that a line drawn through it is just seaward and parallel to
both the updrift and downdrift flood channels. In case there is no discernable flood channel,
the line should be drawn over, and parallel to, the delta crest, or at the inflection in the shore
normal profile. In case there is a considerable delta offset, i.e., asymmetry with respect to
the updrift and downdrift shorelines, the line should cross just seaward of the recessed
shorelines flood channel, and traverse the seaward shoreline. If there is a significant angular
difference between the shorelines, the line should be directed over the crest of the most
discernable ebb/flood channel, and traverse the other shoreline. The average depth of the
channel region should be selected as HC, and the average distance from the shoreparallel
line to the shoreline as CW. Figure 2.4 is an example of placement of the profile origin, with
shoreparallel and shorenormal lines.
For convenience of delta representation, eleven radial profiles were digitized from
the delta surveys, spacing them every 18', from 18 to 162. Two additional profiles were
digitized at 9" and 171 to better represent the behavior of the three remaining parameters,
A(n), C(n) and VI(n), at very small and very large angles with respect to the updrift shoreline,
0.534
2 H=0.211' X
o
0 PBR31978 o
4 o PBR31990
i6 O0 "....
6
O0 0
10
0 200 400 600 800 1000 1200 1400 16
Offshore Distance, X (m)
Figure 2.3: Least squares fit of PBC R3 1978 and 1990 data to Eq. 2.2.
LONGBOAT KEY
0 R26
0 R27
OR28
0OR31
OR33.5
Figure 2.4: Sample placement of profile origin, and baseline on the ebb delta at New Pass,
Florida.
00
respectively. These profiles were loaded into a program that: 1) interactively determined
which digitized data point was the first on the seaward slope of the delta, 2) set the
temporarily selected value of the virtual origin, Vr, as a fraction of the first data point, 3)
interpolated the distances and depths over a selected 10 m interval from Vrr to the last data
point, 4) shifted the data by a distance equal to the Vrr so the first distance and depth values
would be zero, 5) converted the data to a loglog format for input into the function "Polyfit"
which found, M(n) = slope and A(n) = exp(yintercept), 6) calculated the least squares error,
7) incremented Vrr by the selected value of 1 m and, 8) executed the process untir V
equaled the distance of first depth point on the seaward slope of the delta. The value of Vrr
that had the smallest error in the least squares fit of A(n) and M(n) was selected as VI(n), the
final value of the virtual origin for that particular profile. Appendix B gives the code for
fitting a digitized radial profile to Eq.2.5. Plots of the eleven profile fits for the selected
February, 1980 Jupiter Inlet survey are shown in Figures 2.5, 2.6 and 2.7. It was necessary
to describe the variation of the three variable parameters, A(n), C(n), and VI(n) in Eq.2.5 as
functions of angular orientation on the delta planform. For that purpose, the values of all
eleven A(n) were obtained through a code which fitted a parabolic equation to the data, i.e.,
A(n)=[AK (6x/2)]2+AMN (2.8)
The value of AMN was set equal to the smallest A(n) value in the data set, and AK was varied
until the least squares error was the smallest. Appendix C gives this code which fits Eq.2.8
0
0.426
H = 0.305' (R139.5)
2
4
6
8
10 9 deg. from updrift
0 500 1000 15(
Radius, R (meters)
0
0.614
H = 0.127 (R95.51)
2
4
6
8
36 deg. from updrift
0 500 1000
Radius, R (meters)
U
0.519
H =0.189' (R104.7)
2
4
6
8
18 deg. from updrift
0 500 1000 15(
Radius, R (meters)
0
0.64
H = 0.13' (R118.3)
2
4
6
8
10 4 deg. from updrift
0 500 1000 15(
Radius. R (meters)
1500
Figure 2.5: Profiles #14 from Jupiter Inlet delta, February, 1980.
0
0.775
H = 0.062' (R123.3)
2
4
6
8
10 72 deg. from updrft
0 500 1000 151
Radius, R (meters)
0
0.81
H = 0.058' (R246.9)
2
4
6
8
108 deg. from updrift
0 500 1000 151
Radius, R (meters)
30
0.899
H = 0.031' (R155.5)
2
4
6
8
0 0 deg. from updrift
0 500 1000 15'
Radius, R (meters)
0
0.611
H = 0.185' (R337.9)
2
4
6
8
10 126 deg. from updrift
0 500 1000 15
Radius, R (meters)
Figure 2.6: Profiles #58 from Jupiter Inlet delta, February, 1980.
6
0.514
H = 0.266* (R385.8)
10 44 deg. from updrift
0 500 1000 15(
Radius, R (meters)
2
4
6
0.317
8 H = 0.487 (R795.9)
171 deg.from updrift
0 500 1000
Radius, R (meters)
4
6
0.301
8 H = 0.742' (R579.9)
S162 deg. from updrift
0 500 1000 15(
Radius, R (meters)
1500
Figure 2.7: Profiles #911 from Jupiter Inlet delta, February, 1980.
to the data of A(n). Figure 2.8 shows the fit of the parabolic relation to the A(n) data for the
February, 1980 Jupiter Inlet survey. Using the virtual origin data for the eleven profiles, and
the parabolic relationship for A(n), values of M(n) were then solved for using the least
squares method. Appendix D gives the code for this calculation. Plots of the eleven profile
fits to find M(n) for the February, 1980 Jupiter Inlet survey are given in Figures 2.9, 2.10,
and 2.11. The values of all eleven M(n) were then inputted to a code which fitted a parabolic
equation to the data, i.e.,
M(n)=MMA[MK (O6i/2)]2 (2.9)
~1,
< 0.5
0.3 )
0 0.5 1 1.5
Radians
2 2.5 3
Figure 2.8: AParameter data from Jupiter Inlet delta, February 1980, and corresponding
parabolic fit.
U V
0.355 0.409
H = 0.465* (R139.5) H = 0.374' (R104.7)
2 2
T 4. 4
Q6 Q 6
8 8
19 deg. from updrift 1 8 deg. from updift
0 500 1000 1500 0 500 1000 1500
Distance, R (meters) Distance, R (meters)
0 0
0.526 0.657
H = 0.224" (R95.51) H= 0.117" (R118.3)
2 2
I 4 \ 4
6 6
8 8
136 deg. from updrift 10 54 deg. from updrift
0 500 1000 1500 0 500 1000 1500
Distance, R (meters) Distance, R (meters)
Figure 2.9: Profiles #14, fit with parabolic relation for A(n) and VI(n) from Appendix A.
AMIN = 0.031
Least squares AK value = 0.466
W
/
/
/
/
t"
"% ( J'
0
0.804
H = 0.052 (R123.3)
2
4
6
8
10 72 deg. from updrift
0 500 1000 15(
Distance, R (meters)
0
0.823
H = 0.052* (R246.9)
2 ,
4
6
8
10 108 deg. from updrift
0 500 1000 151
Distance. R (meters)
0.9
H= 0.031' (R155.5)
2
4
6
8
10 90 deg. from updrift
0 500 1000 15(
Distance, R (meters)
0
0.69
H= 0.117' (R337.9)
2
4,
6
8
1026 deg. from updrift
0 500 1000 15(
Distance. R (meters)
Figure 2.10: Profiles #58, fit with parabolic relation for A(n) and VI(n) from Appendix A.
0 500 1000
Distance, R (meters)
1500
0 500 1000
Distance, R (meters)
15001
2
r 4
a) 6
S6 0.325
H = 0.465' (R795.9)
8
171 deg. from updrift
0 500 1000 1500
Distance, R (meters)
Figure 2.11: Profiles # 911, fit with parabolic relation for A(n) and VI(n) from Appendix A.
2
4
6
0.543
8 H = 0.224' (R385.8)
S144 deg. from updrift
U
2
2 *
4
6
0.419
H = 0.374' (R579.9)
8
162 deg. from updrift
36
The Value of MMA was set equal to the largest M(n) value in the data set, and M Kwas varied
until the least squares error was the smallest. Appendix E gives the code, which fits the
parabolic relation (Eq. 2.9) to the data of M(n). Figure 2.12 shows the fit of the parabolic
relation to the M(n) data for the February, 1980 Jupiter Inlet survey.
A method was developed to take the eleven virtual origin values and interpolate them
so that thirtyseven data points were generated, as required for the delta modulation method
described in Chapter 3. A listing of the following method is given in Appendix F.
The VI(n) values obtained from the code in Appendix A were asymmetric because
the Jupiter Inlet delta is skewed in the downdrift direction. This skewness is a result of the
relative strength of large northeast storm wave energy causing a significant southward
longshore drift, versus the relatively small tidal energy. To simplify subsequent treatment,
the raw VI values are converted to a symmetrical form by averaging. Thus, Profile #1 was
averaged with profile #11, profile #2 with profile #10, etc. This averaging was done in order
to obtain a symmetric delta. Figure 2.13 is the polar representation of the fitted VI values
and the corresponding average VI values for the February, 1980 Jupiter Inlet survey. The
eleven average VI values were converted from the polar to the rectangular coordinate system,
VIx(n), and normalized, VIxN(n), by the maximum value, VIxm, according to:
VIx(n) =VI(n) COSO (2.10)
VIx(n)
VIXN(n) = (2.11)
VIXM
 \
I
/ \
I
/ r \
I
I \
\
MMAX = 0.9
Least squares MK value = 0.557
Ins
0.3F /
0 0.5 1
1.5
Radians
2 2.5 3
Figure 2.12: MParameter fit with A derived from hyperbolic relationship.
Figure 2.13: Polar plot of virtual origin, VI data, and the corresponding average VI values
for February, 1980 Jupiter Inlet survey. Numbers on the exterior of the plot are the profile
angles with respect to the updrift shoreline. Concentric circles are lines of equal radii in
meters.
0.81
0.4 
i. . I
Figure 2.14: Average VI data normalized by the maximum alongshore and offshore distances
of the virtual origin distribution, and the corresponding Gaussian fit with K = 1.52, February,
1980 Jupiter Inlet survey.
270
Figure 2.15: Plot of the 37 data points of VI for input into delta modulation model. Numbers
on the exterior of the plot are the profile angles with respect to the updrift shoreline.
Concentric circles are lines of equal radii in meters.
The standard deviations, o, of the VI, data were then evaluated from:
C = VI,(n)2 (2.12)
11 n=i
It should be noted that there is no mean term in the o expression since a symmetrical delta
has zero mean VI,(n). A modified Gaussian distribution was next fitted to the VI xn) data
by the least squares method, varying the K value in Eq.2.13 until a best fit is achieved. The
Gaussian distribution is:
GAUS(n)= exp [VxN(n)]2 (2.13)
O2( (22 2302
Figure 2.14 shows a plot of the Gaussian fit in the normalized coordinate system, which was
necessary for this representation. The values were scaled up by their maximum values, and
the relationship was used to solve for VI(n) in selected 5" increments for input into the delta
modulation code (Chapter 3). Figure 2.15 shows the thirtyseven data points input into that
code.
CHAPTER 3
DELTA EROSION MODEL
3.1 Introduction
After developing a means for geometrically describing the delta volume, a method
for modulating the volume is derived. For that purpose, and approach based on Kriebel and
Dean's (1993) convolution method for timedependent beach profile response to forcing by
waves and associated storm surge was used to generate poststorm deltas.
The basis for the convolution method is the Bruun Rule of erosion (Bruun, 1982).
Bruun Rule is simply a response of beach profile to sealevel rise. This response is
determined by shifting the profile (assumed to be in equilibrium with the prevalent sea
conditions) upward by sealevel rise, and then landward until there exists a conservation of
sediment mass eroded from the nearshore zone and deposited in the offshore zone, out to the
limiting depth for sediment movement. A graphical representation of the Bruun Rule is
shown in Figure 3.1. The limit to the zone of sediment movement in the profile was defined
by Bruun (1982) as a depth equal to twice the breaking wave height. The profile
discontinuity at the limiting depth (vertically equal to "a", and horizontally equal to "s" in
Figure 3.1), which is created by shifting the active profile up and back, is replaced with a
deposit slope (Bruun, 1982). A diagram depicting the sealevel rise, landward shift of
equilibrium profile and resulting deposit slope is given in Figure 3.2.
ii
Figure 3.1: Bruun Rule of erosion, where "s" is the recession, "1" is the active profile length,
"a" is the sea level rise, "d" is the depth of active sediment movement, "e" is the berm height,
and "z" is the point of no net sediment transport (from Bruun, 1988).
AFTER RISE
S\ BEFORE RISE
DEPOSIT SLOPE
Figure 3.2: Bruun Rule of erosion, with deposit slope. (from Bruun, 1988).
3.2 Profile Erosion
The convolution method assumes that the profile responds to a storm surge, S,
analagous to sea level rise in the Bruun rule of erosion. The resulting profile recession is
given by:
t
R(t)=R_ (le T) (3.1)
where R. is the equilibrium response of the profile to the steady state storm surge, S, as
determined by the Bruun Rule, R(t) is the erosion at a specified time t, and T, is a
characteristic erosion time constant. If a time dependent storm surge forcing, f(t), is used
then the R(t) is found by solving the equation:
R t (
T,
R(t) = Jf(z) e Ts dc (3.2)
s o
Solving this integral with a function which closely mimics most storm surges, f(t) =
sin2(ct/Td), where TD is the storm yeilds:
R(t) 1( 2 2._ t 1
Rt) 1 e2 exp( T27D) ( ) [cos27rt/TD+sin27Tt/TD (3.3)
R. 2 (1+p2) TD (I+p2)
where p = 27iT/Td. The erosion timeconstant, T, was determined by Dean and Kriebel
(1993) by running a numerical erosion model for sandy beaches. During these numerical
43
tests, the water level was elevated, the profile was allowed to equilibrate, and the time record
of recession data fitted to Eq. 3.1. The resulting equation for the timeconstant is:
Hbl / Hb 0.78 m Xb, 1
Ts = C 11+ + (3.4)
g 1/2 A 1/3 0.78 B Hb
where Hb is the breaking wave height, A, is the scale factor for the profile, g, is the
acceleration due to gravity, hb is the depth of breaking, B, is the berm height, m is the beach
face slope, Xb, is the surf zone width, and, C, is a constant. A graphical representation of the
results of the convolution method is shown in Figure 3.3. It should be noted that this method
is not dependent on the wave period, and assumes the wave heights are constant for the entire
storm duration.
3.3 Volume Calculation
In order to calculate the volume of sediment above the noinlet contour, the
differential area between each radial profile was calculated according to,
AREA(rl)=3 T (r2r2) (3.5)
360
where 60 is in degrees and r, and r2 are the specific radii between which the area is being
calculated. Figure 3.4 shows a sketch of the differential area. The corresponding differential
volume is then calculated by taking the difference in elevation, DH(n,r,), between the delta
envelope and the noinlet contour at each of the four points at the covers, and then
multiplying by the projected planform area according to:
BREAKING WAVES
ELEVATED WATER LEVEL N
EQUILIBRIUM RESPONSE
Figure 3.3: Beachprofile response to waves and surge where R. is the recession at
equilibrium, and R(t) is the recession at a certain time, t.
UpdriftBaslne
Figure 3.4: Differential angular planform area, where 0 is in degrees.
DH(n,rl) = H N(n,rl)H(n,rt) (3.6)
DVOL(r)=A A DH(n,rl) + DH(n,r2) + DH(n+l,r,) + DH(n+l,r2) (37)
DVOL(rl)=AREA (3.7)
4
The total volume is the sum of all of the differential volumes, DVOL, for all of the selected
radii between each of the angular sections.
3.4 Profile Erosion Model
The convolution method was applied to modulate the idealized ebb delta volume
under wave and surge conditions. The equilibrium recession of the virtual origin and the
seaward slope of the delta to the depth of limiting motion, Hm, for a constant surge level, S,
was calculated for each selected 50 differential slice of the delta. Waves acting on each
profile were assumed to be refracting in such a way that the wave energy was considered to
propagate radially toward the profile origin. The recession for each profile was then found
by shifting the active profile up by the selected maximum storm surge, and towards the
profile origin by R., which was incremented by 1 m intervals until the volumes eroded in the
nearshore zone, and accreted in the offshore zone were equal. The deposit slope connecting
the seaward end of the shifted profile with the initial depth of limiting motion was
lengthened with every increment in recession and, as explained later, was idealized as a
modified sinesquare curve, with the initial slope of the deposit slope matching the slope of
the delta profile at the depth of limiting motion. The timedependent recession, R(t), was
46
then calculated from Eq.3.3 with an assumed value of P, the dimensionless timeconstant.
Finally, the change in volume of the delta was calculated as the sum of volumes eroded
between the pre and poststorm profiles. A graphical representation of the convolution
method applied to an single ebb delta radial profile is shown qualitatively in Figure 3.5. The
code for modulation of the ebb delta volume is given in Appendix G.
3.5 Erosional TimeConstant
The calculation of equilibrium recession, R., of a delta radial profile is simply a
geometrical exercise, but to calculate the timedependent recession, R(t), a timeconstant, T,,
is needed. As noted further in Section 4.3, because of a lack of poststorm surveys of deltas,
and no preceding analytical or numerical methods for determining the change in delta volume
due to storm effects, the Ts relation Kriebel and Dean (1993) proposed for application to
beach profiles could not be applied to the delta profiles. As a result, in order to apply Eq.3.4
a value was assigned to P that gave reasonable recessions of the seaward slope of the delta
for extreme storm conditions.
3.6 Depth of Limiting Motion
In order to apply the Bruun Rule, the limiting depth of sediment movement, Hm,
must be
found for each radial profile. A sine function relation was used to simulate refraction of
the breaking wave heights, Hb around the delta.
Hb(n)=HosinO (3.8)
Where Ho is defined as the unrefracted wave height in shore normal direction. As noted
NoInlet Profile f'. Depth of Limiting Motion
Equilibrium Response
Area of Accretion
Radius
Figure 3.5: Delta radial profile response where R. is at equilibrium, and R(t) is at a
specified time, t.
earlier, the depth of limiting motion is taken to be twice the breaking wave height
representative of that particular profile (Bruun, 1982), i.e.,
Hm(n) = 2 Hb(n) (3.9)
Figure 3.6 is a planform view of the delta, which illustrates the relative magnitudes and
approach angles for the waves breaking on the crest of the delta.
Figure 3.6: Planform view of the delta illustrating the assumed magnitudes and approach
angles for the breaking waves, HB, on the crest of the delta.
3.7 Deposit Profile
The convolution method assumes a constant deposit slope connecting the seaward
end of the shifted profile to the initial depth of limiting motion. When considering large
surges over the delta profile, this unrealistic constant deposit slope tends to become a
significant portion of the poststorm profile. Thus, to introduce a better approximation, the
deposit slope was modified in order to better resemble a natural profile and to eliminate the
abrupt change in slope at the endpoints. A sinesquare relation was chosen for the deposit
profile, since it would be possible to match the slope at the endpoints to the slope of the
delta profile at the depth of limiting motion, and still achieve the necessary change in
elevation between the endpoints. Thus, the equation of the deposit slope is described by:
H(n,r)=Hm + S (ii + 1) (3.10)
2[rm(n) r]
R(t)
6[ ( [rm(n) r] )
= sin2 6 + (286t/2) tr( 1
R(t)
Where 6 is the angular offset and Rm(n) is the radius to initial sediment mobilization depth,
Hm. The delta volume modulation code in Appendix G calculates a value for the slope of the
delta profile at the depth of limiting motion, and solves for 6, which sets the initial slope of
the deposit profile equal to the slope of the delta profile. Figure 3.6 is a nondimensional
plot of the deposit profile with an arbitrary 6. Figure 3.7 is a plot depicting the deposit
profile, for arbitrary values of 6, S, R(t) and Hm. The solid lines correspond to the delta
profile, while the dashed line is the deposit profile.
50
0
0.1
0.2
0.3
0.4
0.5
0 0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
[Dm(n) D(n,r)]/R(t)
Figure 3.7: Nondimensionalized deposit slope, 6 = 0.2443461.
4
+
4
Radius, R (m)
Figure 3.8: Deposit slope for an arbitrary set of values of 6, S, R(t) and Hm.
CHAPTER 4
DELTA VOLUME AND EROSION
4.1 Introduction
In this chapter geometric deltas are constructed, and the effects of differing noinlet
profiles on the resulting delta volume are studied. Values of the noinlet shape and scale
factors are varied, producing both steep and flat profiles, and the corresponding changes in
the delta volume are documented. Then, realistic ranges of wave and storm surge forcing are
used to erode the delta, and the reduction in delta volume as well as changes in the location
of the top of the delta are examined. Finally, several case studies related to delta volume
change are presented and discussed.
4.2 Dependence of Delta Volume on Profile Parameters
In order to define the delta volume, the intersection of the delta profiles with the no
inlet profile must be found. Under certain combinations of the parameters A(n), M(n), VI(n),
Ani and Mi, the powerlaw, i.e., Eq.2.5, for delta profiles and the equation for the noinlet
profile, i.e., Eq.2.3, do not intersect. Therefore, the parameters for input into Eq.2.5 had to
be chosen so that this non intersecting situation did not occur.
Using the relationships for the coefficients A(n), M(n) and VI(n) derived from the
least squares fitting of the data from the Jupiter Inlet delta, a characteristic delta was
developed. The input parameters for this delta were: CW = 100 m, HC = 3 m, AK = 0.079,
Amin = 0.05, M~ = 0.34, Mvax = 0.8, K = 1.5, a = 0.35 m, Vj = 1400 m, yj = 800 m.
52
These particular values were chosen so that the resulting delta would have a maximum
volume of approximately 107 n3 above the noinlet profile. It should be noted that the
relatively small selected value of AK produced a much flatter parabolic curve shown in
Figure 2.8.
For a given set of A(n), M(n), and VI(n) in the seaward slope equation, i.e., Eq.2.5,
the delta volume was found to be sensitive to the slope of the noinlet profile, i.e., Eq.2.2.
When a steep noinlet profile was chosen, the seaward slope of the delta intersected the no
inlet profile in deeper water and at a larger radial distance from the profile origin, resulting
in a larger delta volume than for the same seaward slope parameters intersecting a flatter no
inlet profile. Figures 4.1 and 4.2 show isometric and side views of the initial delta seaward
slope specified above, intersecting a steep and flat noinlet profile, respectively. Figure 4.3
is a plot of the delta volume versus the Ani scale parameter for different values of the N'i
shape parameter. This plot illustrates the significant dependence of delta volume on the
choice of the noinlet profile. Table 4.1 attempts to quantify this sensitivity of the delta
volume to the definition of the noinlet profile for a specific delta seaward slope, in terms of
delta volumes for two values of the noinlet profile scale parameter, and 10% deviations from
these values. As observed, the larger scale parameter, corresponding to a steeper profile, had
larger differences in volume when varying the scale parameter by 10%, than for the flatter
profile.
Thus, in conclusion, it is noted that the geometric parameterization of the delta
volume has quantified the significance of the choice of the noinlet profile in the
determination of the volume of sediment contained in the delta. Generally, the profiles
500 *inn .. .. :,.. .......
~~Z 71
. . ... ..... ;.. .
0 (
01 0 ..............
CD 10
0 10 .. ... .. ... .. .. .. .. ... .. ..
0 500 1000 1500 2000
Offshore Distance (m)
2500 3000
Figure 4.1: Isometric (a) and side (b) views of an arbitrary delta volume with parameters:
CW = 100 m, HC = 3 m, An = 0.05, Ci = 0.7, AK = 0.079, Amin = 0.05, MK = 0.34, Mmax =
0.8, K = 1.5 m, o = 0.35 m, VImx = 1400 m, VIy = 800 m.
....... .... ...................
.  ,,~,;,'~".'~': ........ .
5 . :......... ......
. ..... 2000
0 500 1000 1500 2000
Offshore Distance (m)
2500 3000
Figure 4.2: Isometric (a) and side (b) views of an arbitrary delta volume with parameters set
to those in Figure 4.1, except with C,i = 0.675.
(b)
.. . . . . . . .. . . . .. . .
. .. .. .. ..
Figure 4.3: Dependence of delta volume on the noinlet shape and scale parameters, Ani and
Mi. The remaining delta parameters are equal to those in Figure 4.1.
Table 4.1: Dependence of delta volume on the noinlet scale parameter for a given delta
seaward slope.
Mni Ani +/ 10% An, Vol (m3) % Change in Volume
0.6 0.070  3.66 x 106 
0.063 2.87 x 106 22
0.077 + 4.66 x 106 +27
0.6 0.020  4.86 x 106 
0.018 4.22 x 106 13
0.022 + 5.53 x 106 +14
downdrift of inlets tend to be much steeper than the profiles updrift of the inlet. Thus, for
example, if the downdrift noinlet profile were used instead of the updrift profile, the result
would be a significantly larger delta. Therefore, when for instance planning a beach
M i 8 :0.:...................................
Mi : i=0.8 : OT: : : . . ..
M:ni 0.6
S: Mi = 0.S
.I.....Il i 1.. ..
55
nourishment project using delta sediment, it is important to accurately represent the noinlet
profile for delta volume calculation.
4.3 Changes in Delta Volume
In order to calculate the decrease in the delta volume due to severe seas, the method
of Kriebel and Dean (1993), described in Chapter 3.2 was applied to each of the radial
profiles. In summary, this method assumes a timelag between storm forcing and profile
response. Timedependent profile recession is defined as a function of the ultimate profile
recession and the erosional time constant for that particular profile. The ultimate recession
is defined as the maximum recession if the storm forcing conditions were allowed to become
the new ambient conditions. Ultimate recession, as described in Chapter 3.1, was
investigated by Bruun (1982). The Bruun Rule of shoreline erosion due to a rising sea level
states that the ultimate profile recession is derived by shifting the profile (out to the depth of
limiting sediment motion defined by the ambient wave conditions) upward by the elevated
water level and then landward until the volume of sediment per unit beach width eroded in
the nearshore zone equals the corresponding volume accreted in the offshore zone.
In order to apply the method of Kriebel and Dean (1993), it was necessary to define
both the specific portion of the radial profile to be shifted, and the erosional timeconstant.
The profile to be shifted is the entire seaward slope of the delta from the virtual origin out
to the depth of limiting sediment motion. A value was set for the erosional timeconstant
which resulted in a maximum profile recession under storm conditions of approximately one
half of the ultimate recession. The maximum timedependent profile recession is shown in
Figure 4.4. It is the intersection of the line of the nondimensional timedependent profile
Figure 4.4: Plot of nondimensional timedependent profile recession versus phase of storm,
7rt/Td, with p = 0, corresponding to no lag in profile response and, P = 7/2, selected for
determining delta response.
response, R/R(t), derived with no lag between the sin2(itt/T ) storm surge forcing and profile
recession (corresponding to P = 0 in Eq. 3.3) with the line of nondimensional profile
response with a time lag between forcing and response corresponding to the selected value
of p = 7t/2. It should be noted that the maximum profile recession will always occur prior
to the end of the storm event. Profile response after that time is recovery, which has not been
considered in this study.
With reference to the choice of P = 7/2, note that since at a beach both the berm and
the swash zone are eroded, and are nonexistent at a delta profile, values of the erosional
timeconstants chosen by Kriebel and Dean (1993) for beach profile response cannot be
applied to delta erosion. Thus, in order to investigate the characteristic maximum storm
57
induced changes in the delta volume, a value for Ts in Eq.3.3 was chosen to be equal to one
quarter of the storm duration, i.e. Ts = T /4, which yielded P = Tc/2 from Eq.3.3. The
reasoning behind choosing this value was that the resulting maximum delta erosion was
seemingly reasonable relative to the ultimate delta erosion that would occur if the delta were
allowed to equilibrate with respect to the increased maximum water level and characteristic
wave height.
The Bruun Rule of erosion assumes profile recession in such a way that there exists
a conservation of sediment eroded in the nearshore zone, and accreted in the offshore zone,
as shown in Figure 3.1. In the present analysis, the delta volume during the storm is defined
as initial volume above the noinlet profile minus the eroded volume, as shown in Figure 3.5.
Defining the delta volume in this manner does not include the accretional portion of the
delta profile. However, this definition seems to be qualitatively consistent with the usual
way in which delta volumes at inlets are measured from surveys.
The method of Kriebel and Dean (1993) for crossshore profile change simulation has
the greatest accuracy when applied to a shoreline with straight and parallel depth contours.
If waves break at a nonnormal angle to the shoreline, the resulting longshore sediment
transport will be constant along the shoreline, and only the crossshore component of
transport will change the beach profile. When anomalies in the shoreline exist, the shore
normal and tangential components of wave height do not remain constant, due to the
changing angle of breaking waves with respect to the shoreline. As a result, both crossshore
and the longshore sediment transport vary along the shoreline in the vicinity of the anomaly.
Ho = 5 (m)
8.9
91 ......... .......... ......... ......... ......... ......... ......... . ... .. ..... ....
8.8 9 Ho 5.5 (m). .
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Surge (m)
Figure 4.5: Plot of delta volume during storm versus storm surge, S, for variable wave height
Ho and P = 7t/2. The initial delta condition is depicted in Figure 4.1.
In this case the crossshore profile evolution model results will have larger errors than in the
case of straight and parallel contours.
An ebb tidal delta is a large shoreline anomaly, and the present attempt at modulating
this volume incorporates a modified crossshore relationship, as applied to the delta radial
profiles. For simulating wave refraction, the wave heights acting along each radial profile
have been modified so that the height is maximum on the shorenormal profile, and zero
along the shoreparallel profile, as shown in Figure 3.6. The effects of the component of
wave height tangential to the radial profile have not been considered, and result in increasing
error in the poststorm characteristics of the radial profiles as the angular orientation with the
closest shoreline decreases. It is possible that the deletion of the accretionary portion of the
59
x106
5 . .... ......... ......... ......... ..... .... ............................. ......... .
5.2 .... ....... .......... .m
5
S 4 .8 .........4(m)
S4.4 Ho r'5' (m) ... !......
0 4.2
4 .. ....... ...... ..H o .5 5.(m .... .........
3 .8 ...... .. ................. ... ... ................. ..... ..... .................. .......... .........
.4 Ho =.55 (m) .
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Surge (m)
Figure 4.6: Plot of delta volume during storm versus storm surge, S, for varying wave height
Ho and p = Tc/2. The initial delta condition is depicted in Figure 4.2.
profile can be justified on the basis that the tangential component of the wave energy at the
profile, which is not considered here, could be responsible for transporting this sediment
away from the delta region, toward the adjacent shorelines.
The delta recession model was executed for the initial delta configurations in Figures
4.1 and 4.2, forced by different combinations of wave heights and storm surges. Figures 4.5
and 4.6 are plots of the minimum delta volume during the storm, versus storm surge, for
increasing wave heights at deltas with steep and flat noinlet profiles. Figures 4.7 and 4.8 are
similar to Figures 4.5 and 4.6 for an initial bathymetry with a more compact distribution of
the virtual origin. Reducing the radial extension of the virtual origin effectively simulates
the reduction in the magnitude of the tidal currents jetting the sediment offshore and the
corresponding reduction in the delta volume, as predicted by O'Brien (1971). It was found
3 .1 . ... .. . ...... .. ...... .. .. ..... .. ... . .. . ... ..... H ... : .. ....
j 3 .. .............. .. :.... ..... ...... ........ ......... ... .......... ....... ......... ... .......
Ho= 45 (m)
2 .6 ......... ... .... ........... .......... ... ....... ........................ ......... ......... .........
S 2.8 ........S.(m).. .....
0
2 Ho 55 (m)
2.7
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Surge (m)
Figure 4.7: Plot of delta volume during storm versus storm surge, S, for variable wave height
Ho and p = t/2. The initial delta condition is that of Figure 4.1 with VImx = 1000 and Vimy
= 500, effectively producing a smaller delta.
i1.6
o
1.5
 1.4
E 1.3
E
S1.2
1.1
1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Surge (m)
Figure 4.8: Plot of delta volume during storm versus storm surge, S, for variable wave height
Ho and p = ir/2. The initial delta condition is that of Figure 4.2 with VImx = 1000 and Vimy
= 500, effectively producing a smaller delta.
1 1 i 1 1 1 1
......... ......... ............ .......... .......... o ....... ....
SH ...... .......... .............. o = 4 (m )......
Ho 4(M)
Ho 45 (m)
SHo = 5(m)
Ho = 5 5 (m)
61
that for the given initial depth of the delta crest, small waves did not mobilize sediment to
a great enough depth for any considerable profile adjustment to occur. As a result, the delta
with the flatter noinlet profile had both a larger percentage of its initial volume eroded, as
well as a larger total volume of sediment eroded for the same wave and surge conditions.
It should also be noted that delta erosion was a function of both wave height and storm surge.
The dependence of profile recession on storm surge is inherent to the method of Kriebel and
Dean (1993). As a result, there would be no profile erosion in the case of of increased wave
action without an accompanying storm surge. This effect is evidenced by the zero change
in delta volume when forced by zero surge and any wave height in figures 4.5, 4.6, 4.7 and
4.8. Increased wave action from typical tropical or "Northeaster" extratropical storm systems
will always have an associated storm surge and setup; therefore the above mentioned effect
is very unlikely to occur in nature.
Plots of the delta profile recession for the steep and flat noinlet conditions acted on
by the maximum wave and surge conditions considered are given in Figures 4.9 and 4.10,
respectively. Views (a) and (b) are of profiles at 90 and 450 degrees with respect to the
updrift shoreline, whereas views (c) and (d) are of the same profiles with the accretionary
portions of the profiles removed. The ultimate recession is depicted by the solid line, and
the maximum recession during the storm is depicted by the dashed line. For the flat noinlet
profile case, sediment was mobilized over a much larger portion of the radial profile, and
much larger recessions of the virtual origin occurred than in the steep noinlet profile case.
It should also be noted that in the extreme erosional case depicted in Figure 4.10, there was
Ho = 5.5 (m)
0
2
E 4
6
8
0 1000 2000
Radial Distance (m)
0 1000 2000
Radial Distance (m)
3000
3000
0 1000 2000 3000
Radial Distance (m)
0 1000 2000
Radial Distance (m)
3000
Figure 4.9: End of storm radial profiles for the steep noinlet condition with [(a) and (b)], and
without [(c) and (d)] the accretionary portions of the profile included. Theta refers to the
profile angular orientation with respect to the updrift shoreline. The equilibrium profile is
the solid line, and the maximum recession profile is the dashed line.
a large recession of the delta seaward slope, which caused an increase in the depth of the
delta crest. Such a reduction in the height of the delta can cause a larger percentage of wave
energy to be transmitted across the delta, which could possibly have an adverse effect on the
stretch of coastline no longer partially sheltered by the delta.
63
Ho = 5.5 (m)
Ssurge = 3.5 (m): surge = 3.5 (m)"
2 ." .. . ...... ...ta ... ...... .. 2' "..^ t ..... "
E 4 .. .. ... .... . . . . .. . . . . .. 4 .. . . . . . . . ..
Stheta:= 90 2 theta:= 45
Surge = 3.5 (m) .surg = 3.5 (m).
Q 2 ^ Q r
8 8
(a) s (b))
10 ................ .. i .... 10 
0 1000 2000 3000 0 1000 2000 3000
Radial Distance (m) Radial Distance (m)
Sue surge = 3.5 (m) surge = 3.5 (mi)
to the total thickness of the delta above the noinlet profile, the accr'''etionary offshore portion
(c) (d)
1 0 ............ ........ ... . .. ...... ... ......................10... .
0 1000 2000 3000 0 1000 2000 3000
Radial Distance (m) Radial Distance (mn)
Figure 4.10: End of storm profiles for the initial condition in Figure 4.2 with [(a) and (b)],
and without [(c) and (d)] the accretionary portions of the profile included. Theta refers to the
profiles angular orientation with respect to the updrift shoreline. The profile is the solid line,
and the maximum recessed profile is the dashed line.
The plots of the extreme erosional case in Figure 4.10 can explain the slight upturn
in the minimum volume versus surge plot occurring for storm surges greater than 3 m in
Figures 4.6 and 4.7, and greater than 2 m in Figure 4.8. As the surge became large relative
to the total thickness of the delta above the noinlet profile, the accretionary offshore portion
of the profile was dominated by the deposit slope. Because delta volume increases when
64
storm surges became large relative to the delta thickness, the deposit slope determined by
Eq. 3.9 incorrectly simulated a large portion of the accretionary region of the radial profile.
However, it should be noted that despite this problem, that assumed form of the deposit slope
is an improvement over the linear deposit slope used in the Bruun rule of erosion, and
provides a better qualitative description of the eroded profile.
Planform views of the evolution of the top of delta for four different wave heights and
storm surge increasing from 0.5 to 3.5 m in increments of 0.5 m for the steep and flat noinlet
profiles are shown in Figures 4.11 and 4.12, respectively. The recession of delta nose was
larger in the case of the flat noinlet contours than for the steep noinlet contours, as also
evident from Figures 4.9 and 4.10. The plots in Figures 4.11 and 4.12 have apparent "hinge
points" about which the top of delta recession swung with increasing storm surge. This
hinge point is a result of the assumed mode of wave refraction around the delta, as described
in Section 3.6. As the depth of sediment mobilization due to wave action decreased along
with the angle between the profile and the nearest shoreline, the surf zone width on the delta
became so small that little or no profile recession occurred. Also, the profiles in the range
of 0 to 30' with respect to the nearest shoreline for the extreme normally incident wave
height of Ho = 5.5 m were not acted on by large enough waves to cause any significant profile
adjustment.
The three isometric and side views of the delta volume due to the maximum storm
forcing conditions for the steep and flat noinlet profiles are in Figures 4.13 and 4.14,
respectively. The decrease in profile erosion for small angles is evident in the isometric
views. Since both the steep and flat noinlet profiles had the same delta seaward profile
65
1000 1000
(a) (b)
800 /800
800 Ho.4(m) Ho.4.5(m
s 600 600
2 400 400
S / Delta Nose
S200 200
0
0 0
2000 1000 0 1000 2000 2000 1000 0 1000 2000
1000 1000
(c) (d)
2800 800
Hinge Ho.S(m) Ho=5.5(m
600 600
2 400 400
S200 200
0
0 0
2000 1000 0 1000 2000 2000 1000 0 1000 2000
Alongshore Distance (m)
Figure 4.11: Planform views of the evolution of the top of delta of Figure 4.1 with increasing
values of storm surge and changes in incident wave height, Ho. Solid line is the initial
condition and the dashed lines correspond to increasing values of storm surge from 0.5 m to
3.5 m in increments of 0.5 m, with the largest deviation from the initial condition occurring
at the largest surge.
1000 1000
(a) (b)
S800 800
800Hinge Ho.4(m) Ho=4.5m
S600 600
S400 400
200 200
0
2000 1000 0 1000 2000 2000 1000 0 1000 2000
1000 1000
(c) (d)
800 / 800 HS.S(
5 600 600
S 400 400
= 200 200
0 0
2000 1000 0 1000 2000 2000 1000 0 1000 2000
Figure 4.12: Planform views of the evolution of the top of delta of Figure 4.2 with increasing
values of storm surge and changes in incident wave height, Ho. Solid line is the initial
condition and the dashed lines correspond to increasing values of storm surge from 0.5 m to
3.5 m in increments of 0.5 m, with the largest deviation from the initial condition occurring
at the largest surge.
.. ..a .. ...... .. ...... . 
0 500 100 500 2000 2500 3000
. . .. . . .
2000
 (b)
5 ^
1500 2000
Offshore Distance (m)
15'
0 500 1000 1500 2000 2500 3000
Offshore Distance (m)
Figure 4. 14: Isometric (a) and side (b) views of the maximum erosion of the delta of Figure
4.1 due to incident wave height, Ho = 5.5 m, maximum surge, S = 3.5 m, and P = 7c/2.
(b)
...................
Figure 4.13: Isometric (a) and side (b) views of the maximum erosion of the delta of Figure
4.1 due to incident wave height, H, 5.5 m, maximum surge, S 3.5 m, and It /2.
..........~.ia
... . . ...
,0 .....
55
.......... 2000
0 500 1000.......0
1500 2000 200 300 200
67
shape and the same initial nose position, a larger shorenormal recession of the flat noinlet
profile delta is evident in the two side views.
4.4 Case Studies
In connection with the above analysis, the following case studies illustrate the role
of episodic sea wave conditions in modulating ebb delta volumes. For relating historical
delta volumes with episodic extreme sea conditions necessary for the case studies, the
hindcast Wave Information Studies, or WIS, data were acquired from the US Army Corps
of Engineers (USACOE, 1996). WIS data used are wave heights calculated from historical
wind speed and direction in variable depths of water at approximately every 0.50 longitude
along the Florida panhandle, and approximately every 0.250 latitude along the Atlantic coast
of Florida. The calculated delta volumes at the inlets chosen, i.e., Jupiter Inlet, South Lake
Worth Inlet, and East Pass, were plotted versus the corresponding extreme WIS wave height
during the preceding year. Note that similar plots were also made using, for example, the
mean wave height of the preceding year. However, the maximum wave height, being an
indicator of the extreme storm condition, yielded the best correlative results.
4.3.1 Jupiter Inlet, Florida
Jupiter Inlet in Florida is a natural entrance connecting the Loxahatchee River to the
Atlantic Ocean. The spring tidal range in the vacinity of this Inlet is 1.0 m (Dombrowski,
1994). This inlet was unstable, and has had a history of numerous openings and closures as
well as updrift migration. In order to stabilize the inlet, rock jetties were constructed in 1922,
extended in 1929 (Marino, 1986), and again in 1968, when the updrift fillet of sand had
shifted the shoreline to the end of the updrift jetty (DCOE, 1969). Figure 4.15 is a
68
bathymetric chart of the area surrounding Jupiter Inlet, including the ebb delta (Coastal
Planning and Engineering, 1994). Figure 4.16 is a plot from Dombrowski (1994) showing
delta volumes versus year with two modelcalculated growth curves with a = 0.17 (from Eq.
1.1), Ho = 0.54 m, and a = 0.27, Ho = 0.68 m. The significance of this plot is that is shows
how the changing wave climate changes the rate of delta development and the ultimate value
of delta volume for a given value of tidal prism at the inlet.
Figure 4.17 is a plot of delta volume at Jupiter Inlet and WIS wave heights from
Station Number A2013, located in 45 m of water, versus year for the period 1956 to 1993.
In order to relate the change in delta volume to the episodic wave events, the measured delta
volume is plotted versus the highest WIS wave height during the preceding year in Figure
4.18. The spike in the wave record, and the drastic decrease in delta volume between the
1979 and 1980 surveys, can be attributed to the influence of Hurricane David, which made
landfall in the vacinity of Jupiter inlet on September 3, 1979. Although only the relationship
between maximum wave height during the proceeding year and delta volume is being
investigated, it is assumed that the large wave events have an accompanying storm surge.
It should be pointed out that both the north and south jetties were extended in 1968, in order
to stabilize the mouth of the inlet and improve navigation. The two lines in Figure 4.18
correspond to the delta in the pre and postjetty extension conditions. It appears that there
is an inverse relationship between the wave height during the preceding year and the delta
volume for both the pre and post jetty extension conditions. Therefore, the extension of the
jetty system, and the corresponding increase in delta volume would explain the much smaller
delta volume in 1957 than in 1993, even though in the preceding years extreme wave heights
ooo~o~~oa
L.C SW a PU
Primary Borrow Source
For 1995 Nourishment
Figure 4.15: Bathymetric chart of Jupiter Inlet (from Coastal Planning
1994).
0.0
1
)40
1950
1960
and Engineering,
1970 1980 1990 2000
Year
Figure 4.16: Ebb delta volume versus year with modelcalculated growth envelope for Jupiter
Inlet with a = 0.17, Ho = 0.54 m, and a = 0.27, Ho = 0.68 m (from Dombrowski, 1994).
   
O .17:
................... .................. . ........ .................. ..... ......... ........... .......................
............ A  .... .
=0.27a 102
S.. .......... ................................... .....................
.............. .. ....... ............. .. ...................... ........................ ........................ .......................
I ft I
^'"* 

x 10s
S .......... ............. .. ........ ............ ............... ............ ') ..................
E
S6 ....................... ....................... ...................... ...... .. ............ ........
.5
06
> 5 .......... ......................... ............ ......... ... ............ .. .......................
3    i  i  K 
5
1955 1960 1965 1970 1975 1980 1985 1990 1995
Year
8
4
2
0
1955 1960 1965 1970 1975 1980 1985 1990 1995
Year
Figure 4.17: Plot of delta volume at Jupiter Inlet and WIS hindcast wave data for the time
period 19561993 (from USACOE, 1996).
8x 10sc
1967
7.5 ..... ....... ... ....... 993 .. .................... ...........
7 .... .......... ..1986 ... ........
1979
6 .5 . . . . ........... ...... ... ... ..... . ........ ... ............ ...........
S ........ ... ....... ... .. ..... .. ..... ... ... .. .......... ............ ............ ............
6.5
E
5 ..... ..tty Extent.or .... Post Jetty Ext ....
3 45 ......... I ..... ......... ..... ....... ....... .. ........
3.5
1980
4 4.5 5 5.5 6 6.5 7 7.5 8
Maximum Ho (m)
Figure 4.18: Plot of the Jupiter Inlet delta volume versus maximum hindcast WIS wave
height during the preceding year.
were almost the same. It should be noted that the volumes for Jupiter Inlet in 1978 and 1981
included in Figure 4.16 were not included in Figure 4.17, because the small scale of the
National Ocean Service nautical charts used to calculate those volumes greatly increased the
probability of error in volume.
Customarily, the addition of jetties will cause the littoral sediment to be jetted further
out into deeper water, causing the resulting delta to attain a much larger volume than in the
nojetty configuration (Marino and Mehta, 1986). Tomlinson (1991), in a study of the
Tweed River Entrance in Australia whose jetties had been extended by approximately 400
m in 1960, calculated an accretion in the delta region of approximately 2x106m3. He stated
that a new equilibrium will occur when the controlling depth over the delta decreases to the
previous controlling depth, and the sediment bypassing pathway is reestablished by the
delta crest completely connecting with the downdrift sand bar system. Thus, from the effects
of jetties apparent in Figure 4.18, it can be considered that fluctuations in delta volumes at
microtidal inlets must be examined carefully for likely causative factors, in this case both
waves and structural modifications.
4.3.2 South Lake Worth Inlet, Florida
South Lake Worth Inlet was artificially opened and stabilized in 1927. The spring
tidal range near this inlet is 1.24 m (Dombrowski, 1994). A sand transfer plant was installed
near the end of the north jetty in 1937. In 1967 the plant was moved seaward by 36 m and
its pumping capacity was enlarged in conjunction with the north jetty extension of 125 m.
Figure 4.19 is a bathymetric chart of the area surrounding the inlet, including the ebb delta.
0BORROW, /4C 85
Figure 4.19 Bathymetric chart of South Lake Worth Inlet (from Olsen and Associates, Inc.
1990).
Significant shoreline offset between the updrift and downdrift shorelines of approximately
75 m exists at this inlet, as visible in Figure 4.19. The ebb delta accreted at an average net
rate of 9,850 m3/yr from 1955 to 1979, and 9,550 m3/yr from 1979 to 1990.
Figure 4.20 is a plot of delta volume versus year from Dombrowski (1994) with a
model calculated growth envelope with a = 0.03, H, = 0.19 m, and a = 0.60, Ho = 0.28 m.
As in the case of Jupiter Inlet, this plot shows that the changing wave climate will modify
the rate of delta development, and also the ultimate delta volume. Dombrowski (1994) did
not include the 1990 volume because a larger planform delta area than that given by the
method of Davis and Gibeaut (1990), which was used to obtain the delta base area from the
2.4
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
1920 1930 1940 1950 1960 1970 1980 1990
Year
Figure 4.20: Ebb delta volume versus year with modelcalculated growth envelope for South
Lake Worth Inlet with a = 0.03, Ho = 0.19 m, and a = 0.60, Ho= 0.28 m (from Dombrowski,
1994).
tidal prism. In that connection it should be pointed out that if the timedependent growth
curves are correctly redrawn, they would abruptly change slope at the 1967 data point,
corresponding to the extension of the jetties, and achieve an equilibrium volume greater than
that in the 1990 survey sometime beyond the turn of the century.
Figure 4.21 is a plot of delta volume at South Lake Worth Inlet and WIS wave height
from Station A2011, located in 90 m of water, versus year for the period 1965 to 1993. It
should be noted that the first data point in Figure 4.21 is the only value prior to the major
jetty extension. An inverse relationship between wave height and delta volume in Figure
4.18 is not apparent in Figure 4.22. The fact that the changes in wave height seemingly did
not affect the delta volume is because that during the time considered, the delta was still in
the process of accumulating sediment, not having achieved its new equilibrium volume due
to changes in the jetty system. Therefore, if we isolate the 1967, 1968 and 1969 cluster
............ ;... .................. ................. .................. .................... ................... .....

 
a=0.03: ."" ,
........... ....... . ..... .............. .. .... ............ .. ...................................................
i  
I III
2.5
E 2
0
E
.5 1.5
0
0.5
19(
x 10
1980
Year
1990
8
2
0
1965 1970 1975 1980 1985 1990 1995
Year
Figure 4.21: Plot of delta volume South Lake Worth Inlet and WIS hindcast wave data for
the time period of 19651993 (from USACOE, 1996).
2.4
c(990
2'^
2 .................. .: = ... ......... ............. .......................... ...................
1 .8 ................. ................ ........... .. ................... ............ I ......
1.6 ... ................. .... ... ...... ......... ........ ...................
2 ............................................ ............. .................................. .
1.4 .....,,.
1.2
1 .2 ................... ". ........... '. .................. ....... .. .. .. ..................
01969 61979
1 \
0.6....... .................
0 . .'......... ...... .... .. ... . .. . ...... ...... ............. .........................
d 01967 0
0.4 .
3.5 4 4.5 5 5.5
Maximum Ho (m)
Figure 4.22: Plot of the South Lake Worth Inlet delta volume versus maximum hindcast WIS
wave height during the preceding year.
Jetties Extended
. ..... .. ............ .. ...... ... . ................... ................
X/ .. . .. . .

65
75
corresponding to the years close to the jetty extension, we find that the 19901979 (dashed)
line does indicate the anticipated inverse relationship between delta volume and wave height.
It should be noted that the volume for 1978, which was included in Figure 4.20, was not
included in Figure 4.21 because of the small scale of the NOS nautical chart used to calculate
that volume.
4.3.3 East Pass, Florida
A navigation channel was dredged at East Pass in 1930. The spring tidal range at this
inlet is 0.43 m (Dombrowski, 1994). The channel was deepened and widened in 1945, and
required subsequent maintenance dredging to sustain the 3.7 m project depth. In 1967, the
U.S. Army Corps of Engineers began the construction of converging rubblemound jetties
with a weir section in the west jetty to stabilize the mouth of the pass and improve
navigation. In 1977, a spur jetty was added to the landward end of the east jetty in order to
divert the ebb channel toward the centerline of the jetties. In 1985, the weir section in the
west jetty was closed. The inlet channel is continuously dredged across the ebb delta in order
to avoid hazardous navigation situations that tend to occur due to the rapid channel migration
and shoaling (Morang and Irish, 1996).
Figure 4.23 is a bathymetric chart of the area surrounding East Pass, including the
ebb delta. Figure 4.24 is a plot of the measured delta volumes at East Pass (from Morang
and Irish, 1996) and hindcast WIS wave heights from Station G1028, located in 24 m of
water, versus year for the period 1965 to 1975. Figure 4.25 is a plot of delta volumes versus
the highest WIS wave heights during the preceding year. It appears that there is an inverse
relationship between the wave height during the preceding year and delta volume. As noted,
Figure 4.23: Bathymetric chart of East Pass (from USACOE, 1990).
1970 1975 1980
Year
1985 1990
< NoData Available
02
0
1965 1970 1975 1980 1985 1990 1995
Year
Figure 4.24: Plot of delta volume at East Pass and WIS hindcast wave data for the time
period 19651976 (from USACOE, 1996).
x 10
3
E
o 2.8
> 2.6
02.4
19
 .... ....
. ................. ... .. ................ ..... ..... ..... ............... ...............
. ........ ... .. ................ . ................ ................ ,................ ................
x .

65
2.9x 106
1967
2.8
2.7
E
.966
2 .4 ........ ......... ......... ...... ...... ...... ... ..... .. ... ..... .. ....... .
2.4
2 .3 ......... ......... ....... . .. ....... ......... .........97 ... ....... ..
S 1974
2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4
Maximum Ho (m)
Figure 4.25: Plot of East Pass delta volume versus maximum hindcast WIS data during the
preceding year.
construction of the converging jetty system was begun in 1967. Customarily, the addition
of jetties causes the littoral sediment to be jetted further out into deeper water causing the
resulting ebb delta to attain a much larger volume than for the nojetty configuration.
Therefore, the addition of a jetty system would not be responsible for the reduction in the
delta volume as seen in Figure 4.25. Apparently, the relative reduction in delta volume due
to increased wave heights was much more significant than the likely increase in delta volume
due the equilibration of the delta system under the jettied configuration. Although WIS data
were not available for the period extending to 1990, delta volumes did increase by
approximately 0.7x106 m3 above the prejetty volume in 1967.
78
On October 4, 1995, Hurricane Opal occurred with an estimated 4.45 m storm surge
and a maximum wind speed of 67 m/sec. The elevated water level overtopped the barrier
island, and in conjunction with the increased flood tide through the inlet and fresh water run
off, caused an extreme elevation in the Choctawhatchee River Basin. As a result,
considerable water exited through East Pass. There was an extreme reorientation and erosion
of the inlet channel, but no appreciable change in the planform of the inlet ebb delta crest
(from Morang and Irish, 1996). As found in the delta erosion model, the maximum change
in delta volume occurred some time during the storm. A poststorm survey measured the
accretion on the delta that was actually derived from channel sediment when the surge
receded, and the high ebb currents deposited channel sediment onto the delta.
4.3.4 Katiakati Inlet, New Zealand
The Katikati Inlet in New Zealand is bounded by a rock headland acting as a jetty
stabilizing a sandy coastline on the north and a sand barrier island on the south. The spring
tidal range is 1.64 m (Hume et.al., 1996). Figure 4.26 is a bathymetric chart of the area
surrounding Katiakati Inlet, including the ebb delta. The delta has been classified as a
storage cell for sediment, being a poor tidal bypasser under normal circumstances, but a bar
bypassing delta under north storm conditions. By examining 12 aerial photographs and
relating them to historical storms, Hume et.al. (1996) determined that large storms had a
significant effect on the morphology of the delta, and the time period of recovery to a similar
prestorm geometry took years.
In March 1975 Tropical Cyclone Alyson, which was classified as a 30 year storm,
produced large waves for a three day period. The storm was responsible for changing the
0 1000 2000 m
5' 5,
Figure 4.26: Bathymetric chart of the area surrounding Katiakati
including the ebb delta (from Hume et.al., 1996).
2300
2200
2100
2000
1900
1800
1700
1600 ..
1500
1940
1950 1960 1970
Date
Inlet, New Zealand,
1980 1990 2000
Figure 4.27: Ebb delta at Katiakati Inlet (New Zealand) ebb delta protrubance from the
shoreline over a 50 year period (adapted from Hume et.al., 1996).
iv
o
. *
,,
80
delta from a horseshoelike tidedominated geometry to a wavedominated geometry that was
flattened against the coastline. This storm event changed the morphodynamics of the delta
for approximately a decade. Smaller storms, which did not cause the magnitude of change
experienced during the storm of 1975, have correspondingly had smaller timescales for
readjustment to the prestorm condition. Figure 4.27 is a plot of the extent of the ebb tidal
delta (ETD) in the shorenormal (SN) direction, or protrubance of the ebb delta at Katiakati
Inlet over the years. Note that the excursion of the proturbance, or the nose, is qualitatively
consistent with the simulation given in Figure 4.12.
CHAPTER 5
SUMMARY AND CONCLUSIONS
5.1 Summary and Conclusions
In recent years at least sixteen inlet ebb deltas on the United States East and Gulf
Coasts have been mined for beach nourishment sediment. Presently, the effects of these delta
minings on the processes of the inlet/beach system are not well understood. However, it is
known that the mining of sediment can change the wave refraction pattern, which in turn can
result in changes in the magnitude and possibly the direction of sediment transport in the
vicinity of the inlet.
The main objective of this study was to determine generalized responses of the delta
volume to storm forcing, and from that to develop an understanding of how real deltas
respond to storms. A geometrical delta was developed by fitting a powerlaw equation to
radial delta profiles digitized from a typical microtidal ebb delta. Then, a previously
developed method of beach profile response was modified for application to the delta
profiles. Finally, the response of the delta was compared qualitatively to relevant case
studies. The following is a brief summary of this work.
A generalized delta geometry was first developed. This delta was idealized as a set
of radial profiles originating from a central point on the delta, with a hyperbolic tangent term
describing the inlet channel and swash platform, and a powerlaw equation for description
of the seaward slope of the delta. The scale, shape and virtual origin parameters
82
characterizing the seaward slope equation were variable with respect to the radial profile
angle relative to the updrift shoreline.
In order to determine the quantity of material in the delta, the elevations of delta
profiles were referenced to the corresponding elevations of the profile that would exist if the
inlet were not present, i.e., the noinlet profile. The calculation of the delta volume thus
obtained was found to be extremely sensitive to the choice of the noinlet profile. Since there
often exist large shoreline offsets or changes in profile slope between the updrift and
downdrift sides of an inlet, considerable care must be taken when choosing the location of
the origin and slope of the noinlet profile when calculating the volume of an actual delta.
To simulate the reduction of ebb delta volume due to waves and elevated surge
conditions, a method derived by Kriebel and Dean (1993) to predict timedependent beach
profile response was modified and applied to the seaward slope portion of the radial profiles
around the delta. This method also attempts to predict the recovery process for beach
profiles. However, since the recovery process of ebb deltas characteristically differs from
that of beach profiles, the application of the model was limited to the duration of the storm
over which erosion occurs. Because insufficient poststorm survey data exist to calibrate the
erosional timeconstant for delta profiles, a value was assumed which gave reasonable results
of profile recession.
The results of the model tests, which ran a range of waves and storm surges over a
selected ebb delta geometry, indicate that delta erosion is dependent on both waves as well
as surge. Also, the relative magnitude of ebb delta erosion was found to be a strong function
of the steepness of the noinlet profile. Deltas with flatter noinlet profiles are found to have
83
a much larger percentage of volume lost than deltas with steeper noinlet profiles under the
same wave and surge conditions. It was also found that the nose of the delta, or the region
of the delta furthest away from the shoreline, showed considerable recession, while the delta
region closer to the shoreline did not experience significant adjustment. The study by Hume
(1996) on the ebb delta at the Katiakati Inlet in New Zealand, where the delta morphology
was determined from aerial photographs for a period of approximately 50 years, seems to
agree with the model results of the overall flattening of the delta against the shoreline during
extreme storm events.
Comparisons of historical delta volumes with WIS wave data showed an inverse
relationship between delta volume and the maximum wave height during the preceding year
at Jupiter Inlet and East Pass, confirming the laboratory observations of Hayter et at. (1988)
and the analyses of Walton and Adams (1976) and Dombrowski (1994). On the other hand,
there was no distinct relationship between the delta volume and maximum wave height at
South Lake Worth Inlet, presumably because the ebb delta there is still growing, and has yet
to achieved a quasiequilibrium with respect to the average wave conditions.
5.2 Recommendations for Future Investigations
It is concluded that not enough surveyed delta bathymetries exist for a better
understand of the processes and timescales involved in the storminduced delta erosion and
recovery cycle. In order to make a more accurate prediction of the influence of increased
wave height and surge on the delta volume, or the effects of delta mining and borrow pit
recovery on the surrounding shorelines, a data base of periodic as well as poststorm
bathymetric surveys must be accumulated.
