Two-line Model for Inverse Estimation of Cross-shore and Longshore Transport Rates on Nourished Beaches

7/24/2019 Two-line Model for Inverse Estimation of Cross-shore and Longshore Transport Rates on Nourished Beaches

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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]

2545
mailto:[email protected]:[email protected]


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COASTAL ENGINEERING 2008

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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COASTAL ENGINEERING 2008 2555

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.

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Two-line Model for Inverse Estimation of Cross-shore and Longshore Transport Rates on Nourished Beaches