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TWO-LINE MODEL FOR INVERSE ESTIMATION OF CROSS-SHORE
AND LONGSHORE TRANSPORT RATES ON NOURISHED B EACHES
Jens Figlus
1
and Nobuhisa K obayashi
Continual erosion necessitates frequent replenishment of the nourished beaches
protecting communities along the Delaware Atlantic coast. Since conventional one-line
models have failed to adequately predict the beach fill performance and the evolution of
the beach profiles on four Delaware beaches, longshore and cross-shore sediment
transport rates are inferred from detailed profile surveys using a two-line model. The
model captures the dominant seasonal changes inherent to all 65 fixed profile lines which
have been surveyed almost semiannually up to 11 times between 1998 and 2005. Cross-
shore and longshore transport rates are shown to be of equal importance on these
relatively steep beaches in a fairly energetic wave environment.
I N T R O D U C T I O N
The gradual loss of sand along the developed beaches of the Delaware
Atlantic coast is combated by strategic placement of sand from offshore sources.
While nourishment costs are certainly justifiable in light of economic and
environmental impacts, there are still large uncertainties associated with the
sediment transport rates dictating the evolution of beach fills over time. Garriga
and Dalrymple (2002) concluded that the application of standard one-line
models to the nourished Delaware beaches reveals considerable difficulties in
predicting shore line and profile evolution.
In the present paper, a method for inverse determination of cross-shore and
longshore sediment transport rates from measured profile data by means of a
two-line model is presented. The model is applied to survey data recorded on
four Delaware beaches with a frequency of about 6 months over the course of 5
years. Since seasonal profile transformations are the main characteristic on these
beaches any modeling effort must accommodate for these cross-shore shape
changes.
Prior to a detailed description of the two-line model the available data is
presented along with the derivation of the relevant profile change parameters
used in the model. Analysis results demonstrate the capability of the two-line
model to capture the dominant seasonality observed from the field
measurements. Furthermore, transport rates in both the cross-shore and
longshore direction are shown to have the same order of magnitude.
Below, the available survey data and analysis are shown only for Dewey
Beach (DE) representative for the entire data set. The focus is on the
1
Center for Applied Coastal Resea rch, Department of Civil Environmental Engineering,
University of D elaware, Newark, DE, 19716, USA, E-mail:[email protected]
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mailto:[email protected]:[email protected] -
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formulation of the two-line model since a more detailed explanation of the full
data set is included in Figlus and Kobayashi 2008).
FIELD DATA ANALYSIS
The State of Delaware located on the US Atlantic coast maintains its
shoreline through frequent nourishment projects. The beach sand is fairly well
sorted and its med ian diameter is approximately 0.4 mm Ram sey 1999). Of
special interest are the beaches at North Shore NS ), Rehoboth RE ), Dew ey
DE ) and Bethany BE ) since they serve developed municipal areas as storm
protection and g enerate tourist reven ues. Figure 1 gives an overview of the
Delaware coastline w here the easting and northing grid corresponds to D elaware
State Plane coordinates in km. Bathymetry is shown using gray scales with
darker shades representing deeper areas. In addition, a reference contour line at
mean sea level M SL) is drawn. At the northern end of
th
map lies the entrance
to the Delaware Bay. NS, RE and DE are located about 12 km north of the
Indian River Inlet IRI) which is bypassed by a discontinuously operated
pumping system transporting sand from south to north at an average rate of
77,000 m
3
per year Garriga and Dalrymple 2002). BE is located abou t 10 km
south of IRI. M ann and Dalrym ple 1983) found the location of a nodal po int to
be situated just south of
BE.
North of the nodal point the net longshore sediment
transport rate is towards the north whereas south of it the direction reverses.
225 230 235 240
x easting) [km]
Figure 1. Bath ym etry m ap of the Delaware At lant ic c oas t sh ow ing the loca t ion of the
fou r nour ished beac hes: Nor th Shore NS), Rehoboth RE), Dewey DE), and B ethany
BE).
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Water surface elevation and hourly wave data are available from the local
tide gauge at Lewes and offshore NOAA buoy 44009 for the entire length of the
surveys but hindcast directional wave information is only available through
1999 at the locations of WIS 154 and WIS 156. Figlus and Kobayashi (2007)
provide a detailed description of measured water levels and wave conditions for
the respective period of
time.
The average offshore significant wave height was
1.3 m and the average spectral peak period was 7.5 s.
As part of the nourishment project monitoring effort dense profile surveys
were initiated along a total beach length of 6 km at NS, RE, DE and BE after
placement of approximately 1,100,000 m
3
of sand in 1998. An inventory for the
available survey data at DE including dates and time steps is given in Table 1.
Table 1. Dates and Time Step of Available
Profile Survey Data for Dewey Beach.
Survey No.
1
2
3
4
5
6
7
8
9
10
Year | Month
1998 11
1999 4
1999 10
2000 4
2000 11
2001 4
2002 5
2002 10
2003 5
2003 10
At (Months)
-
5
6
6
7
5
13
5
7
5
65 survey lines cover the entire region of interest. Survey lines are spaced
about 150 m apart and cover a cross-shore distance of up to 800 m extending
from the dune line to a water depth of 11 m below MSL. The measured data
points on 18 profile lines for DE recorded during the October 1999 survey are
shown in Figure 2 along with the shoreline at MSL. Note that the numbering
convention for the survey lines is from south to north.
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79 r
229 . 5 230 . 0 230 . 5 231 . 0
x easting) [km]
Figure 2. DE survey points recorded in October 1999. 18 fixed profile lines and the
shoreline at M SL are shown .
Seasonal cross-shore variations are dominant in the evolution of the
measured profile shapes. In Figure 3 ten DE profiles from survey line 18 are
superposed including the cross-shore variation of the standard deviation o
z
among the ten profiles. 4o
z
is plotted for clarity in the figure. In limiting the
cross-shore extend of
th
active profile where most of
th
sediment movement
is expected to take place, landward and seaward boundaries are imposed. The
landward limit x
L
corresponds to the dune crest and the seaward limit x
s
is
chosen at the location beyond which survey errors appear to be random and not
associated with actual profile changes. The beach slope between x
L
and x
s
is
about 0.05.
1 2 3 4 5 6
x[m]
Figure 3. Evolution and standard deviation o
z
of the beach profile shape along DE
survey line 18 meas ured over ten surveys.
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For each survey line profile changes are parameterized between points
P
x
L
, z
L
) and P
s
x
s
, z
s
) as depicted in the top panel of Figure 4 where the cross-
shore and vertical coordinates are x and z with z = 0 at MSL. Investigation of
successive surveys yields clear intersection points P
3
x
3
,z
3
) between typical
summer and winter profiles for almost all surveys separating the landward and
seaward areas of profile change, denoted as A
L
and A
s
with
A
L
=r[z{t
2
-z{t, \dx 1)
A
s
=P[z t
2
-z t
x
]dx 2)
where [z t
2
) - z tj)] is the difference between the profile elevation of two
consecutive surveys as indicated in the bottom panel of Figure 4. The important
profile change parameters calculated for each set of consecutive profile surveys
on every survey line are the above m entioned A L and A
s
, the total profile
change A
L
+ A
s
) and the shoreline change Ax at MSL .
_
2
L_ L 1 1 1 L_
50 100 150 200 250 300
x [m]
Figure 4 Parameters obtained from profile change between two successive surveys
Figure 5 shows the evolution of these parameters for all survey lines at DE
including their spatial averages. Survey dates are marked by vertical dotted lines
as listed in Table 1. Additionally, cumulative values of averages over the entire
time frame are given to the right of the respective plots where the overbar and
prime denote averaging among the survey intervals and lines, respectively.
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- - eac h survey line average o f DE beach
EAa/ = -45,?m
4 = 1.5m
SA
7
= - 1 2 1 , 1 m
2
E A ^ 36.0m
2
( A
L
+ A s ) = - 85 .1 m
2
1999 2000 2001 2002 2003 2004
year
F igure 5 . Va r i a t i on o f DE p ro f i l e change pa ramete r s f o r 18 su rvey l i nes ove r t ime
dashe d l i nes ). Ave rag e va lues ind ica ted by the p rime a r e r ep resen ted b y so l id li nes
an d cumu la ti v e v a lue s a r e l i s ted to the r i gh t o f the r espec t ive pane l .
A regression analysis between combinations of Ax, A
L
, A
s
and (A
L
+ A
s
)
shows similar trends for all four beaches, indicating positive correlation between
foreshore quantities Ax and A
L
and clear negative correlation between A
s
and
A
L
associated with the seasonal profile change. A
s
and Ax are correlated
negatively because the shoreline advances when the offshore area erodes. The
correlation betw een Ax and the total profile change (A
L
+ A
s
) is poor and a one-
line model approach based on the shift of an equilibrium profile does not give
realistic results (Figlus and Kobayashi 2007). Figure 6 shows a graphical
representation of the correlation between the above mentioned combinations of
the profile change parameters for the data from DE. Correlation coefficients R
and the slope b of the best fit straight line through the origin are given in each
panel.
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o At 0.5yr; At 1.0 yr
Figure 6. Linear regression analysis for DE between A
s
and A
L
a), A
L
and Ax b), A
s
and Ax c), and A
L
+A
S
) and Ax d) including the correlation coefficient R and the
slope b of the best fit straight line through the origin.
TWO-LINE MODEL
The time-averaged gradient of the longshore sediment transport rate q
c
(m
3
/m/month) and the time-averaged cross-shore sediment transport rate per unit
longshore length q
c
(m
3
/m/month) between successive surveys at times t] and t
2
are computed using a two-line model to invert the measured profile data.
Kobayashi and Han (1988) utilized a similar approach to predict erosion at the
bend of a gravel causeway during a storm. Here, a control volume of unit
alongshore length encompassing the "active"' profile is divided into two
separate zones at the location x
3
, allowing for cross-shore exchange of sediment
between the landward and seaward zones. Figure 7 depicts the conceptual setup
of this model with V and V
2
being the landward and seaward control volumes,
respectively. The cross-shore direction is shown on the x-axis. Alongshore loss
or gain of sediment from either zone is allowed through the associated
longshore transport rates Qi and Q
2
(mVmonth) where total longshore transport
Q = (Qi + Q
2
).
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y ,
A
T
0
}
s
1
y
1
2
J
*; ~
>
X
Figure 7. Schematic of two-line model control volume and geom etry.
After es tabl ishing the conceptual p ic ture of the model , the sediment
continui ty equat ions for both zones can now be wri t ten as
dV
x
_
d t
dV
2
_
3 0
dy
J_Q
dt dy
-1c
+ 1c
3)
4)
which imply th ree unknown var iab les Q
1 ;
Q
2
and q
c
wi th the two equat ions . As
a resul t we assume
Qx=*Q
5)
fi
2
( - i ) e
(6)
w here a is a para m eter betw een 0 and 1 and assum ed constant during the t ime
betw een tw o consecu tive profile surveys . W e wil l presen t a m ore deta i led
explanat ion for th is parameter short ly .
Substituting (5) and (6) into (3) and (4) and expressing the gradient of the
total lo ng sh or e se di m en t tran spo rt rate -f - as q
f
y ie lds
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dV
x
=
-a q
t
~ q
c
(7)
ot
-T~ = -{\-a q
e
+q
c
(8)
Ot
If we time-average (7) and (8) over the interval (t
2
- ti) between two consecutive
profile surveys and take the average volume changes in zone and 2 to be equal
to A
L
and A
s
, respectively, the equations become
A
- ~ - = -aq
e
-q
c
(9)
t
2
t
]
A
~- }-a)q
t
+q
e
(10)
where the overbar denotes time averaging and will be omitted in the following.
The gradient of the longshore transport rate and the cross-shore transport rate
per unit alongshore length are thus expressed as
A, + A
v
q
=
_ ID
aA
s
- l-a) A
L
t
2
f,
(12)
The parameter
a
appears only in the expression for q
c
and
is
related to the
cross-shore distribution of the longshore sediment transport rate, which is still
under investigation (e.g. Kobayashi
et al.
2007).
A
simple geometric
relationship based on the profile width ratio w
r
has been found to be most robust
for the present data set. The width ratio is defined as
w
r
= ^ ^ -
(13)
x
s
x
L
representing the fraction of the landward portion of the profile over the entire
active profile. Note that x
3
is the mean cross-shore location of the intersection
point between successive profiles for the entire period of observation and has
been found to be ofthe order of the average significant offshore wave height.
Figure 8 shows the estimated gradient of the longshore sediment transport
rate for the available profile survey data at DE. The evolution of q
t
for all
18
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profile lines at this site is presented as dashed lines and the spatial average is
shown as the solid line.
20
10
qt
o
-10
each p rofile line
average
qt
> 0 Longshore loss
< 0 Longshore gain
1999 2000 2001 2002
year
2003 2004
Figure 8. Estimated gradient of the net longshore sediment transport rate on 18 DE
profile lines dashed lines) and the spatial average solid line).
The estimated values for q
t
exhibit large spatial and temporal variability in
longshore sediment loss (q
t
> 0) and gain (q
c
< 0) among individual profile lines
with an averag e longshore sediment loss on the order of 1 m
3
/m per month
com paring favorably with the transport rate estimated from the bypassing data at
Indian River Inlet. Larger than average longshore loss between the first two
surveys is associated with initial erosion immediately after beach fill placement.
Elevated values within the last two survey intervals correlate well with
increased storm activity.
In Figure 9 the estimated cross-shore transport rate q
c
for DE is shown
where the clear seasonal variation of the profile changes becomes evident. The
asymmetric variations between onshore transport during summer (q
c
< 0) and
offshore transport during winter (q
c
> 0) leads to gradual beach erosion which
explains the required frequent re-nourishment. Interestingly, the average values
of q
c
and q
c
turn out to be of equal magnitude and importance and neither of
these processes can be neglected in the planning of erosion mitigation strategies.
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each profile line average
q
c
0 Onshore transport
I : I : I i I i
1999 2 2 1 2 2 2 3 2 4
year
Figure 9. Est imated cross-shore sediment t ransport ra te per un i t longshore length on
18 DE pro fi le lines dash ed l ines) and the spatial average sol id l ine).
ON LUSIONS
A practical method for determining the cross-shore sediment transport rate
and the gradient of the longshore sediment transport rate from measured cross-
shore profile data using a two-line model is presented. The model is applied to
survey data obtained semiannually over several years on four nourished
Delaware beaches. Profile changes between successive surveys are used to
obtain the important parameters describing the evolution of the profile shape
between a landward and a seaward cross-shore limit where a clear intersection
point separating landward and seaward area changes is present on almost every
survey line. Traditionally one-line models have been used to compute shoreline
changes and transport rates but a linear regression analysis of several
combinations of profile change parameters shows that for the case of these
relatively steep Delaware beaches in a fairly energetic wave environment they
may not suffice.
Results from the two-line model show a large variability of the estimated
gradient of the longshore transport rate q
t
among individual profile lines due to
the longshore and temporal variations in the measured bathymetry. On average
how ever a fairly constant longshore loss on the order of m
2
per month is
computed. The cross-shore transport rate q
c
is found to have the same order of
magnitude underlining the importance of both processes for beach morphology
modeling applications. Individual profile lines behave more uniformly in a
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cross-shore direction and display the strong seasonality inherent to these
beaches.
ACKNOWLEDGMENTS
The authors would like to thank the Delaware Department of Natural
Resources and Environmental Control (DNREC) for providing the beach
nourishment and survey data used in this study. This study was supported partly
by the National Oceanic and Atmospheric Administration (NOAA), Office of
Sea Grant, Department of Commerce, under Grant No. NA85AA-D-SG033
(Project R/ETE-4) and partly by the U.S. Army Engineer Research and
Development Center in conjunction with the MORPHOS project.
REFEREN ES
Figlus, J., and N. Kobayashi. 2007. Seasonal and yearly profile changes of
Delaware beaches, Res. Rep. No. C AC R-07-01, Center for Applied Coastal
Research, Univ. of Delaware, Newark, D el.
Figlus, J., and N. Kobayashi. 2008. Inverse estimation of sand transport rates on
nourished Delaware beaches, Journal of Waterway, Port, Coastal, and
Ocean Engineering,
134(4), 218-22 5.
Garriga, C M ., and R.A. D alrymple. 2002. Development of a long-term coastal
managem ent plan for the D elaware A tlantic coast, Res. Rep. N o. CA CR -02-
04,
Center for Applied Coastal Research, Univ. of Delaware, Newark, Del.
Kobayashi, N., and K.-S. Han. 1988. Erosion at bend of gravel causeway due to
waves, Journal of Waterway, Port, Coastal, and O cean E ngineering,
114(3), 297-314.
Kobayashi, N., A. Agarwal, and B.D. Johnson. 2007. Longshore current and
sediment transport on beaches, Journal of Waterway, Port, Coastal, and
Ocean Engineering, 133(4), 296-304.
Mann, D.W ., and R.A. D alrymple. 1986. A quantitative approach to Delaw are s
nodal point,Shore and
Beach,
54(2), 13-16.
Ramsey, K.W. 1999. Beach sand textures from the Atlantic coast of Delaware,
Open File R ep. No. 41 , Delaware Geological Survey, New ark, D el.