UFL/COEL91/013
BEACH NOURISHMENT PERFORMANCE
PREDICTIONS
by
R. G. Dean
and
ChulHee Yoo
October 1991
REPORT DOCUMENTATION PAGE
1. Report No. 2. 3. Recipient's Accessioon o.
4. Title oad Subtitle 5. Report Date
October, 1991
BEACH NOURISHMENT PERFORMANCE PREDICTIONS 6.
7. Author(s) S. Performing Organizatioo Report No.
R. G. Dean
ChulHee Yoo UFL/COEL91/013
9. Performing Organization Name and Address 10. project/lask/Work Unit No.
Coastal and Oceanographic Engineering Department
University of Florida 11. contract or crant No.
336 Weil Hall
Gainesville, FL 32611 13. Type of ~port
12. Sponsoring Organizaton Name and Address
Miscellaneous
14.
15. Supplementary Notes
16. Abstract
A simple method is developed for representing wave refraction and shoaling in the vicinity of a
beach nourishment project. The method applies for the case of a oneline model of shoreline evolution in
which the active profile is displaced seaward or landward without change of form. The model can include
the presence of shoreperpendicular structures and background erosion. Background erosion rates are
formulated in terms of crossshore and longshore transport. An underlying concept of the method is that
in cases where large perturbations, such as nourishment projects, are placed in the natural system, the
system erodes on two time scales with the shorter time scale associated with the planform perturbation. It
is recommended that shoreline modeling be carried out by conducting an ad hoc transformation in which
the preproject contours are represented as straight and parallel.
The simple method is compared to a oneline model which includes a more detailed gridbased refrac
tion and shoaling algorithm. For all cases tested, the simple method of representing refraction and shoaling
results in shoreline evolution in good correspondence with the detailed method. The models are used to
illustrate the effects of several features of beach nourishment projects that are of engineering interest. For
a long uninterrupted shoreline, which has been nourished with the same material as the native, and in
the absence of structures and background erosion, initially symmetric nourished planforms remain nearly
symmetric as they evolve, even under oblique wave attack. This is interpreted as due to the small aspect
ratio (additional beach width to length) of the nourishment project. For no background erosion or retention
17. Originator's Key Words 18. Availability Statement
Beach nourishment
Sediment transport
Shoreline evolution
Shoreline models
19. U. S. Security Classif. of the Report 20. U. S. Security Classif. of This Pale 21. No. of Page 22. Price
Unclassified Unclassified 47
structures, the proportion of sand remaining in the project area over a twenty year period is illustrated
for projects of various lengths and various effective wave heights. The effects of shoreperpendicular struc
tures placed at the ends of the project with and without background erosion are illustrated. For normally
incident waves and no background erosion, the project with or without retention structures causes only
accretion to the adjacent shorelines. However, in the presence of background erosion or oblique waves,
retention structures can cause localized erosion to the adjacent shorelines.
Finally, the effects of nourishing with material more and less transportable than the native are illus
trated. If less transportable, the nourishment tends to "armor" the project area and oblique waves can
cause localized downdrift erosion. It is concluded that the simple method can provide reliable predictions
of the evolution of nourishment projects and is thus useful in conducting benefit/cost analyses.
UFL/COEL91/013
BEACH NOURISHMENT PERFORMANCE PREDICTIONS
by
R. G. Dean and ChulHee Yoo
October, 1991
TABLE OF CONTENTS
LIST OF FIGURES
LIST OF TABLES
Abstract
INTRODUCTION
BACKGROUND
Governing Equations ................................
METHODOLOGY
Procedures Common to Both Methods ......................
Grid System and Transformation of Initial Geometry ...........
Background Erosion .............................
Numerical Solution of Governing Equations ...............
Procedures Which Differ for the Two Methods Wave Refraction and Shoaling
Simplified Wave Refraction and Shoaling .................
Detailed Refraction and Shoaling ......................
Effective Wave Height and Period .........................
RESULTS
SUMMARY AND CONCLUSIONS
Appendices
I. REFERENCES ...........
II. NOTATION ............
LIST OF FIGURES
FIGURE PAGE
1 Definition Sketch .................... ........... 11
2 Definition Sketch for Numerical Model ..................... 14
3 Recommended Ad Hoc Transformation for Modelling Coastal Systems in
which Large Perturbations are to be Introduced . . ... 16
4 Definition Sketch for Effect of Beach Nourishment on Contours . 18
5 Example 1. Comparison of Beach Nourishment Evolution for Simple and
Detailed Methods of Wave Refraction and Shoaling, Normal Wave Inci
dence, Ho = 0.6 m, T = 6.0 sec, co = 900. No Background Erosion 24
6 Example 2. Comparison of Planform Evolution Obtained by Two Methods
for 200 Oblique Waves, Ho = 0.6 m, T = 6.0 sec, ao = 700. No Background
Erosion . . . . . . . . .. 25
7 Example 3a. Planform Evolution by Detailed Method for Deep Water
Wave Directions, oa = 700, 800, 900. Ho = 0.6 m, T = 6.0 sec. No
Background Erosion ................... .......... 27
8 Example 3b. Planform Evolution by Simple Method for Deep Water Wave
Directions, ao = 700, 800, 900. Ho = 0.6 m, T = 6.0 sec. No Background
Erosion ................. ................. 28
9 Example 4. Even and Odd Components of Shoreline Position After 10
Years for Deep Water Wave Directions, ao = 700 and 800. Results Ob
tained by Detailed Method ....................... 29
10 Example 5. Illustration of Wave Height Effect on Rate of Planform Evo
lution. Results Based on Simple Method, ao = 900, T = 6.0 sec. Results
Shown for 1, 3, 5 and 10 Years. No Background Erosion . ... 31
11 Example 6. Effects of Various Project Lengths and Wave Heights on
Project Longevity ................... .. .......... 32
12 Example 7a. Effects on Planform Evolution of Two ShoreNormal Reten
tion Structures of Length Equal to OneHalf the Initial Project Width.
Normal Wave Incidence, No Background Erosion. Ho = 0.6 m, T = 6.0
sec . . . . . . . . . 34
13 Example 7b. Effects on Planform Evolution of Two ShoreNormal Reten
tion Structures of Length Equal to OneHalf the Initial Project Width.
Normal Wave Incidence. Uniform Background Erosion Rate at 0.5 m/yr,
Zero Background Transport at Project Centerline. Ho = 0.6 m, T = 6.0
sec. . . . . . . . . . 35
14 Example 7c. Effects on Planform Evolution of Two ShoreNormal Reten
tion Structures of Length Equal to OneHalf the Initial Project Width.
Normal Wave Incidence. Uniform Background Erosion Rate at 0.5 m/yr,
Zero Background Transport Located 4,500 m to Left of Left Structure. Ho
= 0.6 m, T = 6.0 sec. ........................... .. 36
15 Plot of K vs D (Modified from Dean, et al., 1982) . . ... 38
16 Example 8a. Planform Evolution for Nourishment Sand Less Transportable
than the Native (KF = 0.693, KN = 0.77). Note Centroid of Planform
Migrates Updrift Variation of Surface Layer K Values with Time at
Locations A, B, C, D and E are presented in Figure 18. Wave and other
Project Conditions Presented in Table I. . . . ... 39
17 Example 8b. Planform Evolution for Nourishment Sand More Trans
portable than the Native (KF = 0.847, KN = 0.77). Note Centroid of
Planform Migrates Downdrift Variation of Surface Layer K Values with
Time at Locations A, B, C, D and E are presented in Figure 19. Wave
and other Project Conditions Presented in Table I. . .... 40
18 Example 8c. Variation of Surface Layer Longshore Transport Coefficient
K with Time at the Five Locations Shown in Figure 16. Case of KF =
0.693, KN = 0.77, Ymi = 2.0 m. ...................... 42
19 Example 8d. Variation of Surface Layer Longshore Transport Coefficient
K with Time at the Five Locations Shown in Figure 17. Case of KF =
0.847, KN = 0.77, Ymi = 2.0 m. ...................... 43
LIST OF TABLES
1 Characteristics of Examples Presented ................. 23
TABLE
PAGE
BEACH NOURISHMENT PERFORMANCE PREDICTIONS
R. G. Dean and ChulHee Yoo
Abstract
A simple method is developed for representing wave refraction and shoaling in the vicinity of
a beach nourishment project. The method applies for the case of a oneline model of shoreline
evolution in which the active profile is displaced seaward or landward without change of form.
The model can include the presence of shoreperpendicular structures and background erosion.
Background erosion rates are formulated in terms of crossshore and longshore transport. An
underlying concept of the method is that in cases where large perturbations, such as nourishment
projects, are placed in the natural system, the system erodes on two time scales with the shorter
time scale associated with the planform perturbation. It is recommended that shoreline modeling
be carried out by conducting an ad hoc transformation in which the preproject contours are
represented as straight and parallel.
The simple method is compared to a oneline model which includes a more detailed gridbased
refraction and shoaling algorithm. For all cases tested, the simple method of representing refraction
and shoaling results in shoreline evolution in good correspondence with the detailed method. The
models are used to illustrate the effects of several features of beach nourishment projects that are
of engineering interest. For a long uninterrupted shoreline, which has been nourished with the
same material as the native, and in the absence of structures and background erosion, initially
symmetric nourished planforms remain nearly symmetric as they evolve, even under oblique wave
attack. This is interpreted as due to the small aspect ratio (additional beach width to length)
of the nourishment project. For no background erosion or retention structures, the proportion of
sand remaining in the project area over a twenty year period is illustrated for projects of various
lengths and various effective wave heights. The effects of shoreperpendicular structures placed at
the ends of the project with and without background erosion are illustrated. For normally incident
waves and no background erosion, the project with or without retention structures causes only
accretion to the adjacent shorelines. However, in the presence of background erosion or oblique
waves, retention structures can cause localized erosion to the adjacent shorelines.
Finally, the effects of nourishing with material more and less transportable than the native are
illustrated. If less transportable, the nourishment tends to "armor" the project area and oblique
waves can cause localized downdrift erosion. It is concluded that the simple method can provide
reliable predictions of the evolution of nourishment projects and is thus useful in conducting
benefit/cost analyses.
INTRODUCTION
Beach nourishment, the placement of large quantities of sand in the nearshore region to advance
the shoreline seaward, is being applied increasingly as a method of erosion control. Advantages
over other methods include maintaining a nearnatural wide beach for storm protection and recre
ational purposes. In some localities the wide beach also provides suitable nesting areas for several
endangered species of sea turtles. Sand placed in this manner along a long, uninterrupted shore
line represents a perturbation which, over time, tends to be smoothed out by longshore sediment
transport. Additionally, if placed at an initial profile that is steeper than "equilibrium", offshore
transport will occur with an associated narrowing of the dry beach. Realistic prediction of the
performance of beach nourishment projects includes analysis of the crossshore and longshore
transport processes and is of considerable significance to a rational evaluation of the economic
benefits of the project. Unfortunately, at present there are few beach nourishment projects that
have been documented adequately to allow detailed and quantitative evaluation of calculation
procedures.
Methods have been proposed by a number of investigators to predict the performance of nour
ishment projects. A beach nourishment manual has been developed based on experience in the
Netherlands (Pilarczyk and Overeem, 1987) which, in addition to performance, addresses dredge
equipment, manner of placement, environmental effects, etc. Krumbein and James (1965), James
waves and no background erosion, the project with or without retention structures causes only
accretion to the adjacent shorelines. However, in the presence of background erosion or oblique
waves, retention structures can cause localized erosion to the adjacent shorelines.
Finally, the effects of nourishing with material more and less transportable than the native are
illustrated. If less transportable, the nourishment tends to "armor" the project area and oblique
waves can cause localized downdrift erosion. It is concluded that the simple method can provide
reliable predictions of the evolution of nourishment projects and is thus useful in conducting
benefit/cost analyses.
INTRODUCTION
Beach nourishment, the placement of large quantities of sand in the nearshore region to advance
the shoreline seaward, is being applied increasingly as a method of erosion control. Advantages
over other methods include maintaining a nearnatural wide beach for storm protection and recre
ational purposes. In some localities the wide beach also provides suitable nesting areas for several
endangered species of sea turtles. Sand placed in this manner along a long, uninterrupted shore
line represents a perturbation which, over time, tends to be smoothed out by longshore sediment
transport. Additionally, if placed at an initial profile that is steeper than "equilibrium", offshore
transport will occur with an associated narrowing of the dry beach. Realistic prediction of the
performance of beach nourishment projects includes analysis of the crossshore and longshore
transport processes and is of considerable significance to a rational evaluation of the economic
benefits of the project. Unfortunately, at present there are few beach nourishment projects that
have been documented adequately to allow detailed and quantitative evaluation of calculation
procedures.
Methods have been proposed by a number of investigators to predict the performance of nour
ishment projects. A beach nourishment manual has been developed based on experience in the
Netherlands (Pilarczyk and Overeem, 1987) which, in addition to performance, addresses dredge
equipment, manner of placement, environmental effects, etc. Krumbein and James (1965), James
(1974), and Dean (1974) have proposed ad hoc but quantitative methods of assessing the quality
of material. Application of the James method is described in the Shore Protection Manual (1984).
In general, these methods attempt to establish the required volume of borrow material which is
equivalent to one unit volume of native material. However, other parameters critical to project
performance are not addressed by these procedures.
Pilkey and his coworkers (Pilkey and Clayton, 1989; Leonard, Clayton and Pilkey, 1990;
Leonard, Dixon and Pilkey, 1990) have conducted "broad brush" analyses of the performances of
many projects along the East and Gulf coasts of the United States. Attempts have been made
to define project performance in terms of the actual versus design lifetimes of the projects. Also
performance was plotted versus project length, sediment grain size, etc. In general, they found
little or no correlation between expected and actual performances. Rather, the following equation
was proposed for the East Coast of the United States to represent the required renourishment
volumes, VR, conducted at intervals of n (in years) for an initial restoration volume, V, and a
project life, N (in years).
R = N (1)
n
It was recommended that n = 9 for Florida, n = 3 for New Jersey, and n = 5 for the remaining
portions of the East coast. Eq. (1) simply states that it will be necessary to renourish with an
amount equal to the initial restoration volume every n years. The rationale for Eq. (1) is not
apparent from sediment transport considerations. The studies by Pilkey and his coworkers do
not recognize explicitly the effects of background erosion, sediment quality or project length on
project performance.
Dean (1983) reviewed available methods of predicting beach nourishment performance and
showed that based on the Pelnard Considere (1956) solution and in the absence of background
erosion, the time, tp, required for a project to lose p percent of the material placed is
tp a 2 (2)
in which i is the project length and Hb is the height of the breaking waves which mobilize the
sediment causing the spreading out beyond the project limits. Recently Dean (1988b) has shown
that the proportionality factors for 50% "loss" are approximately, for the units as shown
t50% = K 2 (3)
H5/2
in which t50% is in years and K" = 0.172 for i in km and Hb in meters, and K" = 8.7 for I in
miles and Hb in feet. It is emphasized that the material "lost" from the project area is transported
alongshore to project adjacent areas and continues to provide benefits there (Dean, 1988a).
BACKGROUND
The bases for predicting the performance of beach nourishment projects are the equations of
continuity and transport. In general, these may be used to develop a oneline model in which
only one contour (usually the mean water line) is used to represent shoreline changes (e.g. Bakker,
1968; LeMehaute and Soldate, 1977; Perlin, 1978; Walton and Chiu, 1979; Perlin and Dean, 1985;
Hanson, 1989; Hanson and Kraus, 1989). Bakker (1968) has developed a twoline model which,
for example, allows for profile steepening and flattening updrift and downdrift of a structure,
respectively. Hanson and Kraus (1987) have compared the results of oneline numerical models to
analytical solutions applicable for several beach nourishment initial planforms. An nline model
has been developed by Perlin and Dean (1985) in which an arbitrary number (n) of contour lines
is used to represent the beach profile. A restriction of the multiline models developed to date
is that the profiles represented must be monotonic. An alternative formulation method would be
to represent the topography by a grid system as is commonly done in hydrodynamic modeling
thereby eliminating the monotonic requirement. For models which represent the profile by more
than one contour, it is necessary to specify a relationship for crossshore sediment transport.
Such models have been developed by Kriebel (1982) and Kriebel and Dean (1985) for profiles
which must vary monotonically and Larson (1988) and Larson and Kraus (1989) for models not
requiring monotonocity.
Governing Equations
The equation of sediment conservation can be expressed for three dimensions as
Oh Oq Oy (4)
t= V = x + y (4)
+
81 Ox ay
in which h is the water depth relative to a fixed datum, t is time, V is the horizontal vec
tor differential operator (V () = + j q is the horizontal sediment transport vector
(q=i q,+ j q,) as presented in Figure 1, and i and j are the unit vectors in the x and y
directions, respectively. Integrating Eq. (4) in the crossshore direction from a landward location,
yi, where the crossshore transport (qy) is zero to a seaward location, y2, where qy is similarly
zero, yields
a v2 9 vY2
I j hdy I2 qdy = 0 (5)
The first term is recognized as minus the time rate of change of volume of sand,V and the second
integral is the total longshore sediment transport, Q. Making these substitutions
&V aQ
+ = 0 (6)
Ot Ox
In onedimensional model formulations, it is usually assumed that accretion or erosion of a profile
is associated with a seaward or landward displacement respectively of the profile without change
of form. The vertical extent of this change is from some depth, h., of limiting sediment motion
up to the berm elevation, B. Thus the change, Ay, in any contour associated with a change in
volume, AV, is
Ay =AN (7)
h. + B
which when substituted in Eq. (6) yields the onedimensional equation for conservation of sand,
ay 1 8Q
y+ 1 0 (8)
Ot + (h + B) Ox
The onedimensional equation of sediment transport can be expressed as (Komar and Inman,
1970)
I = KPt, (9)
Depth
Contours
Crest
. h*( (x,y)
Depth of Limiting Motion
Figure 1. Definition Sketch.
in which I is the immersed weight sediment transport rate, Pt, is the longshore energy flux factor
and K is a nondimensional sediment transport proportionality factor. These two quantities can
be expanded
I = Qpg(s 1)(1p) (10)
Pt, = EbCG sin Ob cos b (11)
in which s is the ratio of mass densities of sediment to water (r 2.60), g = gravitational acceleration,
p = inplace sediment porosity (taken here as 0.35), E = wave energy density, CG = wave group
velocity, 0 = angle between the wave crests and the bottom contours, and the subscript "b" denotes
that the subscripted variable is to be evaluated at the breaker position. Based on smallamplitude
shallow water wave theory and assuming that the breaking wave height Hb is proportional to the
breaking water depth, hb (Hb = Khb)
H2
Eb = pg9 (12)
CGb = Cb = = = gHb/ (13)
in which K is a proportionality constant (; 0.78) and Cb = wave celerity at breaking. Substituting
Eqs. (10), (11), (12) and (13) into Eq. (9),
K Hs5/2gi/2
Q = K ( _1)( sin Ob cos Ob (14)
8 (s 1)(1 p)
Eqs. (8) and (14) form the basis for a onedimensional numerical model. PelnardConsidere
(1956) has shown that by linearizing Eq. (14) with respect to perturbations in the predominant
shoreline alignment and combining the result with Eq. (8), the oneline model is transformed into
the heat conduction equation
ay y
S= G (15)
Tt Ox2
in which G is the longshoree diffusivity", defined as
K H 5/2)(1 )
G 8()(p)(= (16)
8(s 1)(1 p)(h* + B)
In the finite difference solution of Eq. (15), it can be shown that a critical time increment,
Act, exists which, if exceeded, will cause the numerical solution to become unstable,
1 A2(17)
At 2 G (
in which Ax is the alongshore grid spacing. By inspecting Eqs. (16) and (17) it can be seen
that the smaller the grid spacing and the larger the wave height, the smaller the allowable time
increment.
For multiline models, it is necessary to specify relationships for the distributed longshore
(q.) and crossshore (qy) sediment transport. Kriebel (1982) and Kriebel and Dean (1985) have
proposed the following for the crossshore transport rate
qy = K'(V ?D.) (18)
in which D and 9. are the actual and equilibrium values of wave energy dissipation per unit water
volume and K' is a proportionality factor.
METHODOLOGY
Only oneline models are discussed here. Two models using quite different methodologies for
representing wave refraction and shoaling will be presented, applied and the results compared. In
the first model, refraction and shoaling will be represented by a very simple onestep procedure
whereas in the other a detailed gridbased solution is used.
Procedures Common to Both Methods
Grid System and Transformation of Initial Geometry The continuity and transport
equations (Eqs. (8) and (14)) were solved using an explicit method with the grid system shown
in Figure 2. The shoreline displacements are maintained fixed while the transport is computed
and in the second part of the same time step the transport is held constant while the shoreline
displacements are computed. Thus although the primitive equations are employed, the limiting
time step specified by Eq. (17) is still valid. In areas of special interest where greater resolution is
required, smaller grid elements, (Ax), can be used in the grid system presented in Figure 2.
and the effective value of C02/C. (or equivalently, wave period) to be used in Eq. (29) is the
denominator of Eq. (35) raised to the 2.4 power.
RESULTS
Several examples were investigated to demonstrate the effects of various design options and to
establish, by comparison with the detailed procedure, the relative validity of the simple procedure.
In the examples the background erosion will be taken as zero unless otherwise stated. For those
examples including effects of background erosion, only longshore transport contributions will be
represented. Unless noted otherwise, the grid spacing, (Ax), and project length for all examples
were 150 m and 6,000 m, respectively. At the lateral boundary conditions, the shoreline position
changed according to the background erosion rate and the domain was selected to be sufficiently
large that the perturbation effects at the ends of the domain were insignificant. Other character
istics of these examples are presented in Table I.
Example 1 Initially Rectangular Planform, i = 6000 m, Y = 30 m, Ho = 0.60 m,
T = 6.0s, ao = Po = 90, h*, = 5.5 m, B = 2.5 m.
The results of this example comparing the two methods is presented in Figure 5 for 1, 3, 5
and 10 years after nourishment. The simple and detailed procedures yield very similar results
which is somewhat surprising in view of the extremely simple algorithm employed to represent
wave refraction and shoaling. Again the interpretation is that the aspect ratio of the nourishment
project (Y/i) is quite small.
Example 2 Initially Rectangular Planform, = 6000 m, Y = 30 m, Ho = 0.6 m,
T = 6s, o = 700, o = 900, h. = 5.5 m, B = 2.5 m.
Conditions for this example are the same as for Example 1, except the deep water wave di
rection is 200 oblique to the shoreline. Reference to Figure 6 demonstrates that the two methods
are in quite good agreement with surprisingly little planform asymmetry due to the oblique wave
direction. This asymmetry is examined in greater detail in the next two examples.
Q.
\ ii;
Note:
Ob=1P,ab
Yi
Ax+ y i+
x
Figure 2. Definition Sketch for Numerical Model.
In conducting numerical modeling of generally straight shorelines where a large perturba
tion has been introduced, it is recommended, in general, that the original shoreline and offshore
bathymetry be represented by straight and parallel contours. This is equivalent to the ad hoc
transformation shown in Figure 3. The rationale for the transformation is that, with the exception
of the longterm background erosion that will be discussed subsequently, the nearshore system has
approached a nearequilibrium, the details of which present modeling techniques cannot represent
adequately. The equilibrium may depend on rather subtle and perhaps nonlinear threedimensional
wave transformation (refraction, diffraction and shoaling) over minor bathymetric features. Usu
ally, if a model is applied to a shoreline situation such as Figure 3a, it is found that the shoreline
and offshore contours will approach unrealistically straight alignments and/or the changes will
occur over time scales that are much shorter than actual. Thus numerical modeling is much more
effective in cases where substantial perturbations (humaninduced or natural) have placed the sys
tem out of balance. The transformation presented in Figure 3 recognizes that the changes caused
by the perturbation introduced will be of a much shorter time scale than those associated with
the preexisting nonequilibrium causative features. Subsequent to the modeling, the results can
be transformed back to the actual system.
Background Erosion Longterm background shoreline changes must be accounted for ap
propriately if the nourishment evolution is to be predicted with good accuracy. This is especially
the case if stabilizing structures are to be installed. Longterm background changes could be
caused by divergences in longshore or crossshore transport (cf Eq. (4)). Thus, the general treat
ment considers the longterm background changes composed of a longshore transport component,
YB,L, and a crossshore transport component, YB,C
8yB 9YB,L 9YB,C (19)
at at+ at19)
and the background change, y, is expressed as background longshore transport, according to
Eq. (8),
,() YBL at
Shoreline
/r Contours
Ad Hoc Transformation
b
Shoreline
/Contours
a) Initial Actual Shoreline b) Initial Shoreline and
and Contours Contours to be Modeled.
Figure 3. Recommended Ad Hoc Transformation for Modelling Coastal Systems in which Large
Perturbations are to be Introduced.
In the finite difference solution of Eq. (15), it can be shown that a critical time increment,
Act, exists which, if exceeded, will cause the numerical solution to become unstable,
1 A2(17)
At 2 G (
in which Ax is the alongshore grid spacing. By inspecting Eqs. (16) and (17) it can be seen
that the smaller the grid spacing and the larger the wave height, the smaller the allowable time
increment.
For multiline models, it is necessary to specify relationships for the distributed longshore
(q.) and crossshore (qy) sediment transport. Kriebel (1982) and Kriebel and Dean (1985) have
proposed the following for the crossshore transport rate
qy = K'(V ?D.) (18)
in which D and 9. are the actual and equilibrium values of wave energy dissipation per unit water
volume and K' is a proportionality factor.
METHODOLOGY
Only oneline models are discussed here. Two models using quite different methodologies for
representing wave refraction and shoaling will be presented, applied and the results compared. In
the first model, refraction and shoaling will be represented by a very simple onestep procedure
whereas in the other a detailed gridbased solution is used.
Procedures Common to Both Methods
Grid System and Transformation of Initial Geometry The continuity and transport
equations (Eqs. (8) and (14)) were solved using an explicit method with the grid system shown
in Figure 2. The shoreline displacements are maintained fixed while the transport is computed
and in the second part of the same time step the transport is held constant while the shoreline
displacements are computed. Thus although the primitive equations are employed, the limiting
time step specified by Eq. (17) is still valid. In areas of special interest where greater resolution is
required, smaller grid elements, (Ax), can be used in the grid system presented in Figure 2.
in which QB,L(XR) denotes the background transport at some reference location, XR, to be based
on local knowledge or other calculation procedures not discussed here.
Numerical Solution of Governing Equations The azimuth, Pi, of the shoreline normal
at the nth time level, is established to represent the value at the grid line associated with Qg (see
Figure 2)
i" = ~ + tan1 ( 1 (21)
2 Zi+l Xi/
The background transport, QB,L, (Eq. (20)), is added to the longshore transport resulting from
the planform anomaly to yield the total transport, Qn,
Q = Q1 + QB,L, (22)
Finally, the shoreline position is updated from the nth to the (n + 1)th time level
y?"1 = y" + z(h + B)( T QT+) + t (23)
Procedures Which Differ for the Two Methods Wave Refraction and Shoaling
Simplified Wave Refraction and Shoaling We start by showing several results from lin
ear wave theory using the wave and contour directions shown in Figure 4. The difference Ap,
between the nourished contour orientation, s,(h < h.), and the deep water contour orientation,
0o, (h > h.), can be considered small since the ratio of nourished beach width, Y, to length, ,
is generally on the order of 0.02 at most. Conservation of wave energy flux from deep water to a
water depth, h,+, is
EoCGo cos(flo ao) = E+ CG.+ cos(/3o a.+) = E.CG. cos(Po a.) (24)
in which the subscripts "+" and "*" indicate conditions just seaward and landward of the depth
transition respectively (cf. Fig. 4). The energy flux just landward of the transition may be equated
to that at breaking by
E.CG. cos(po a,) = EbCGb cos(3o ab) (25)
Since 3,(x) = 3o + Ap(x)
EC cos(, ) = EoCGo cos(13o Qo) EbCGb sin(o, ab) sin() (26)
EbCGb cosi b) (26)
cos(Ap)
North
h=h*
Y
Shoreline
i.
**.
*
AX
Contours 
x)
Waves
Deep Water
Contour
N
ao
y ep
V 0
A
_t
a) Planform Showing Perturbed Contours to Depth h,
b) Profile Through AA
Figure 4. Definition Sketch for Effect of Beach Nourishment on Contours.
18
Utilizing shallow water linear wave theory and the wave height breaking proportionality factor,
neglecting terms modified by sin(A3) sin(3, ab) and approximating cos(A/) by unity (At is
small),
S2H CG0, coS(PO 0) 0.2
cb = [ ^cos(f3,a,) J (27)
Cb 2 COS(p, b)
which will be useful later. Applying Snell's Law in a similar manner across the transition and to
breaking,
sin(, a.+) sin(p, a.) sin(s, ab)
(28)
C*+ C. Cb
and a,+ can be determined by applying Snell's law from deep water to h,.+ Eqs. (26), (27) and
(28) will be used to express the transport relationship (Eq. (14)) primarily in terms of deep water
wave conditions. With a limited amount of algebra, we obtain
KH2.4C 2 0.4 COS1.2(PO )
Q = 0 0sin(P, a, ) (29)
8(s 1)(1 p)C.o.4 a29
in which terms modified by sin(Af) sin( ab) have been neglected and cos(Af) has been ap
proximated by unity. It is interesting to note that if Eq. (29) is linearized in the form of Eq. (15),
the appropriate value of longshore diffusivity, G, in terms of deep water wave conditions can be
expressed as
G" KHI 2Co.4 C.20.4 cos1.2(po a0) cos 2(o a.) (
8(s 1)(1 p)C.no.4(h, + B) cos(o a) a.(30)
Eq. (29) is the relationship for the longshore sediment transport rate that completes the simple
model development.
Detailed Refraction and Shoaling The detailed method of solution differs from the sim
ple method as refraction and shoaling are carried out by the detailed procedure. This method as
first presented by Noda (1972), and employed by Perlin and Dean (1983) and Dalrymple (1988),
is based on the irrotationality of wave number and conservation of wave energy. The reader is
referred to the papers noted above for further detail.
V x = 0 (31)
in which k is vector wave number. Eq. (31) can be expanded to
(k sin 0) = (k cos 0) (32)
In the present model, a twodimensional grid is established and Eq. (32) is expressed in finite
difference form as
S[r(k cos O)_l,_ + (1 2r)(k cos 0)i,j+1
1 = cos1 +r(k cos O)i+1,+x (33)
2_ "((k sin O)ij+l (k sin 0), j)
in which r is a smoothing factor taken as 0.25 in application here and all k and 0 values on the
right hand side are understood to be at the (n + 1) time step.
The above equation is solved by considering the wave direction, 0, at the offshore boundary as
known and the wave directions along the lateral boundaries given by Snell's Law. These conditions
along the lateral boundaries should not affect the interior solution if the lateral boundaries are
sufficiently distant from the area of interest.
With the refraction solution for a given time step, the conservation of wave energy equation is
solved to determine the wave height field, Hi,. The equation for conservation of wave energy (Eq.
(24)), is expressed in finite difference form as
r(H2CGsin 8)i_,j+l + (1 2r)(H2CG sin )i,j+ 1/2
S (C sin +r(H2CG sin O)i+,j+1 (34)
i (CG sin 0)jj
+A ((H2CG cos 0)i+,,j (H2CG cos 0)i,_,)
As with wave refraction, in solving Eq. (34), the wave height along the outer boundary is considered
as known and along the lateral boundaries, it is assumed that a = 0. Eq. (34) is iterated until
convergence of the solution is obtained to within an arbitrary predefined limit.
The wave height calculations are carried landward to a location where the inequality Hf[, >
Khj occurs. The transport, Q', is then calculated by transforming the wave conditions at i,j' 1
to the breaking location much as was done for the simple method in transforming wave conditions
from deep water to the breaking point.
This concludes description of the simple and detailed procedures for calculating shoreline evo
lution following a nourishment project. While the required computer times are not large for either
procedure, the times for the simple procedure were approximately 1/250 of those for the more
detailed procedure.
Effective Wave Height and Period
Although the wave height and period change continuously with time, it is evident that in the
absence of littoral barriers, there is an effective constant wave height and period that will produce
the same spreading out of the beach nourishment material as the actual timevarying values.
In the following development, a Rayleigh distribution f(H) for wave heights will be assumed
2f(H) = (H/Hm (35)
f() H2rms
in which Hrm, is the rootmeansquare wave height. Referring to the sediment transport equation
(Eq. (29)) expressed in terms of deep water conditions, for a given sea state characterized by the
significant deep water wave height H,, the effective deep water wave height, Ho.,,, is given by
Hoeff = [m H2.4p(H)dH] (36)
Evaluating Eq. (36) numerically and using the approximate Rayleigh distribution relationship
between significant and rootmeansquare wave heights (H, VI2Hms), it can be shown that
Ho,,f = KrmsHrms = KHs (37)
where Krms = 1.04 and K, = 0.735. Thus the longterm effective wave height Ho.ff at a particular
location is
Ho.f = E (K, HsW)24 (38)
in which H,, is the significant wave height of the nth record in a series of N records encompassing
the time period of interest. Examining Eq. (29), it is clear that a somewhat more appropriate but
U1.
more cumbersome value of effective wave height, H'1 ,is given by
eff = (39)
[1 N GoE n 2.
N n=1 C*n
TABLE I. CHARACTERISTICS OF EXAMPLES PRESENTED*
Wave Characteristics Results Special
Example Ho T ao in Characteristics Purpose of Example
(m) (sec) (0) Figure(s)
0.6
0.6
0.6
0.6
0.4,0.8
Variable
0.6
0.6
900
700
700,800,900
700,800
900
900
900
700
* Unless Stated Otherwise, the Nourished Length, of Shoreline is 6,000 m.
5
6
7,8
9
10
11
12,13,14
16,17
None
None
None
None
None
Various Wave
Heights and
Project Lengths
Two Project
Retention Structures
Present, With and
Without Background
Erosion
Nourishment with
Sands of Different
Transport
Characteristics
Compare Two
(Simple and Detailed)
Methods for Normal
Wave Incidence
Compare Two Methods
for Oblique Waves
Compare Two Methods
for Range of Wave
Directions
Detailed Examination
of Antisymmetric
Component of
Response
Effect of Wave Height
Effects of Wave Heights
and Project Lengths on
Project Longevity
Illustrate Effects of
Retention Structures
for Various Background
Transport Conditions
Effect on Evolution
Patterns
50.0 I1 I I
... SIMPLE METHOD
DETAILED METHOD
Initial Planform
30.0 
S 1 year
S//:. ...... 3 years
y m) 5 years
10 years
10.0 
 10 .0 I I I I I I I I i I
0. 1800. 3600. 5400. 7200. 9000. 10800. 12600. 14400. 16200. 18000.
x(m)
Figure 5. Example 1. Comparison of Beach Nourishment Evolution for Simple and Detailed
Methods of Wave Refraction and Shoaling, Normal Wave Incidence, Ho = 0.6 m, T = 6.0 sec, co
= 900. No Background Erosion.
I
50.0
30.0 ,
1 year
3 years
y (m) 5 years
S10 years
10.0
10.0 I I I i I I I I
0. 1800. 3600. 5400. 7200. 9000. 10800. 12600. 14400. 16200. 18000.
x(m)
Figure 6. Example 2. Comparison of Planform Evolution Obtained by Two Methods for 20
Oblique Waves, Ho = 0.6 m, T = 6.0 sec, ao = 70. No Background Erosion.
Example 3 Same As Example 2, Except Response to Different Wave Directions
Figures 7 and 8 present shoreline planforms at 1, 3, 5 and 10 years after nourishment for wave
directions of 70, 800 and 900, i.e. 200, 100 and 0 obliquity to the general shoreline alignment.
It is seen that the results obtained by the two methods are quite similar and that the effects of
wave direction are relatively small. Although slight, the major effect seems to be that for the
more oblique angles, there is less wave energy flux toward the shoreline resulting in less longshore
sediment transport. A small asymmetry is evident.
Example 4 Same as Example 2, Except Detailed Examination of Planform Asym
metry for Oblique Wave Directions
The purpose of this example is to examine the asymmetry resulting from an oblique wave acting
on a nourished planform which is initially symmetric. Figure 9 presents the results ten years after
nourishment for deep water wave directions, ao, of 700 and 800, i.e. 200 and 100 obliquity to the
prenourished shoreline alignment. In order to quantify the antisymmetry, the planforms were
separated into even and odd components, ye(x') and yo(x') where x' is the longshore coordinate
with origin at the project centerline. Denoting yT(x') as the total shoreline displacement, it can
be shown that the even and odd components are determined as
1
ye(x') = [yr(x') + YT(x')] (40)
o(x') = [YT(') yr(x')] (41)
The maximum displacement associated with the odd component is 1 m. The reason that the odd
components are so small is that the aspect ration (Y/g) of the nourished planform is so small, in
this case 30 m/6000 m = 1:200. Thus the asymmetries are small and are primarily due to the
nonlinearities in the sin 20 term of the transport equation.
and the effective value of C02/C. (or equivalently, wave period) to be used in Eq. (29) is the
denominator of Eq. (35) raised to the 2.4 power.
RESULTS
Several examples were investigated to demonstrate the effects of various design options and to
establish, by comparison with the detailed procedure, the relative validity of the simple procedure.
In the examples the background erosion will be taken as zero unless otherwise stated. For those
examples including effects of background erosion, only longshore transport contributions will be
represented. Unless noted otherwise, the grid spacing, (Ax), and project length for all examples
were 150 m and 6,000 m, respectively. At the lateral boundary conditions, the shoreline position
changed according to the background erosion rate and the domain was selected to be sufficiently
large that the perturbation effects at the ends of the domain were insignificant. Other character
istics of these examples are presented in Table I.
Example 1 Initially Rectangular Planform, i = 6000 m, Y = 30 m, Ho = 0.60 m,
T = 6.0s, ao = Po = 90, h*, = 5.5 m, B = 2.5 m.
The results of this example comparing the two methods is presented in Figure 5 for 1, 3, 5
and 10 years after nourishment. The simple and detailed procedures yield very similar results
which is somewhat surprising in view of the extremely simple algorithm employed to represent
wave refraction and shoaling. Again the interpretation is that the aspect ratio of the nourishment
project (Y/i) is quite small.
Example 2 Initially Rectangular Planform, = 6000 m, Y = 30 m, Ho = 0.6 m,
T = 6s, o = 700, o = 900, h. = 5.5 m, B = 2.5 m.
Conditions for this example are the same as for Example 1, except the deep water wave di
rection is 200 oblique to the shoreline. Reference to Figure 6 demonstrates that the two methods
are in quite good agreement with surprisingly little planform asymmetry due to the oblique wave
direction. This asymmetry is examined in greater detail in the next two examples.
50.0
30.0 .......  
1 year
S// 3 years
y m) /( m 5 years
1. 10 years
10.0
10.0 ' ''
10.0
0. 1800. 3600. 5400. 7200. 9000. 10800. 12600. 14400. 16200. 18000.
x(m)
Figure 7. Example 3a. Planform Evolution by Detailed Method for Deep Water Wave Direc
tions, ao = 70, 800, 900. Ho = 0.6 m, T = 6.0 sec. No Background Erosion.
0. 1800. 3600. 5400. 7200. 9000. 10800. 12600. 14400. 16200. 18000.
x(m)
Figure 8. Example 3b. Planform Evolution by Simple Method for Deep Water Wave Direc
tions, ao = 70, 800, 90. Ho = 0.6 m, T = 6.0 sec. No Background Erosion.
50.0
30.0
y(m)
10.0
10.0
50.0
30.0   .
ym ) Even Components
Y( i
/.0 \
10.0 / Odd Components
10.0
_ __K_ ' .
10 .0 I I I i I I I i I i I I i I
0. 1800. 3600. 5400. 7200. 9000. 10800. 12600. 14400. 16200. 18000.
x(m)
Figure 9. Example 4. Even and Odd Components of Shoreline Position After 10 Years for
Deep Water Wave Directions, ao = 70 and 80. Results Obtained by Detailed Method.
1
Example 5 Effect of Wave Height
Figure 10 presents the planforms calculated by the simple method at 1, 3, 5 and 10 years after
nourishment and for 0.4, and 0.8 m wave heights and normal wave incidence. Of particular interest
is the major role of wave height in causing spreading out of the nourished planform. This effect is
also evident from Eq. (2) which indicates that the spreading out is proportional to H'/2.
Example 6 Effects of Various Wave Heights and Project Lengths
Earlier examples have demonstrated (for homogeneous sediment conditions) the relative in
significance of wave direction. The simple model was exercised to demonstrate the effect on the
fraction of sand remaining in the project area for a wide range of wave heights and project lengths.
These results are presented in Figure 11 where the proportion remaining is shown over a 20 year
time period for nine combinations of wave heights and project lengths. For a small wave height
and long project (Ho = 0.30 m, i = 24,000 m), it is seen that at the end of a 20 year period,
over 95% of the material remains. By contrast, for a fourfold larger wave height and a length
onesixteenth of the former, the sand remaining after one year is less than 40% of that placed.
This illustrates the significance of project length and wave height. It has been noted earlier that
the longevity of a project varies with the square of the project length and inversely with the wave
height to the 2.5 power. It is emphasized that the results presented in Figure 11 do not include
effects of background erosion which can be of considerable magnitude, especially over long time
periods.
Example 7 Effects of Retention Structures
Several examples will be presented illustrating the effects of retention structures, with and
without background erosion. The algorithm for the transport boundary condition at a structure
is fairly complex and will not be described fully here. In general the algorithm is consistent with
Eqs. (21), (22), (23) and (29) with transport determination based on relative positions of structure
tip and the shorelines on the two adjacent grid cells and the background erosion transport rate
components. Diffraction was not included in the examples presented here.
50.0
30.0
10.0 
S /// \
10.0 
0. 3000. 6000. 9000. 12000. 15000. 18000. 21000. 24000. 27000. 30000.
x(m)
Figure 10. Example 5. Illustration of Wave Height Effect on Rate of Planform Evolution.
Results Based on Simple Method, so = 900, T = 6.0 sec. Results Shown for 1, 3, 5 and 10 Years.
No Background Erosion.
1.0 .._  
S"  Ho=0.3 m, 2= 24000 m
ft 0 8... H,=0.6 m, 24000 m
Z 0.8 \!  H=1.2 m, = 24000 m
Sl Ho=0.3 m, j= 6000 m
LI > %
DLu 0.6 \  .
S, Ho=0.6 m, = 6000 m
^O \ru .. "". ^ 
0.4 ". Ho=1.2 m, .= 6000 m
o Ho=0.9m, 9= 1500
0Q Ho=0.6m, A=1500
0.
a
CL 1Ho=1.2m, J= 1500m
0 2 4 6 8 10 12 14 16 18 20
YEARS AFTER NOURISHMENT
Figure 11. Example 6. Effects of Various Project Lengths and Wave Heights on Project Longevity.
Example 7a and Figure 12 present results for no background erosion and normal wave incidence.
Structures onehalf the length of the initial project width (30 m) are present at the two ends of
the project. Here, the structure length should be interpreted as the "effective" structure length.
Consistent with intuition, it is seen that the shorelines immediately adjacent to the project advance
as sand from the project area "spills" around the structures; however as the project recedes
approaching the structure length, less and less sand is transported to the project adjacent areas and
the planform evolves toward one of shoreline segments straight and parallel to the incoming waves
with the beach width within the project area equal to the lengths of the stabilizing structures.
Example 7b and Figure 13 present the results for the same conditions as Example 7a, except
there is a uniform background erosion of 0.5 m/year and the reference background erosion is taken
as zero at the project centerline. The transient results are qualitatively similar to those of Exam
ple 7a. However, because of the background erosion, an equilibrium planform exists only within
the confines of the stabilization structures; within the project area, the equilibrium planform is
concave outward and symmetric about the project centerline. For equilibrium conditions within
the project area, the background erosion transport is exactly balanced locally by the planform
orientation transport due to the waves. This accounts for the character of the planform in Fig
ure 13. At a great distance from the project area, the shoreline retreats at the background rate
of 0.5 m/year; however, immediately adjacent to the retention structures, the erosion rate would
be greater to compensate for the effect of the reduced erosion (at later times) within the project
area.
The final example (7c) that will be presented illustrating the effects of structures is the same as
example 7b, except now the zero reference background transport is located 4,500 m to the left of
the left structure in Figure 14. It is seen that updrift of the left structure, transport is toward the
structure and the shoreline accretes there. Inside the two structures, sand is transported initially
in both directions, but somewhat later the positive transport prevails and sand is carried past
only the right hand structure. Eventually, as before, the planform within the structures will be
aligned for equilibrium and both the downdrift and updrift shorelines will erode. For zero reference
50 .0 I I I I I ii
Initial Planform
30.0 .......
1 year
S3 years
ym 5 years
10 years
100 Structure Structure
// /
10.0~~~~  
10.0 I I i I I I
0. 3000. 6000. 9000. 12000. 15000. 18000. 21000. 24000. 27000. 30000.
x(m)
Figure 12. Example 7a. Effects on Planform Evolution of Two ShoreNormal Retention
Structures of Length Equal to OneHalf the Initial Project Width. Normal Wave Incidence, No
Background Erosion. Ho = 0.6 m, T = 6.0 sec.
50.0
30.0 
1 year
/ . \ 3 years
y(m),/
Uy( 5 years
  10 years
10.0 Structure Structure
10.0
0. 3000. 6000. 9000. 12000. 15000. 18000. 21000. 24000. 27000. 30000.
x(m)
Figure 13. Example 7b. Effects on Planform Evolution of Two ShoreNormal Retention Struc
tures of Length Equal to OneHalf the Initial Project Width. Normal Wave Incidence. Uniform
Background Erosion Rate at 0.5 m/yr, Zero Background Transport at Project Centerline. Ho =
0.6 m, T = 6.0 sec.
50.0
30.0 ... ....
1 year
// 3 years
m5 years
10 years
10.0 
Structure Structure
 ructur
 1 0 0 I I I I I I I g I
0. 3000. 6000. 9000. 12000. 15000. 18000. 21000. 24000. 27000. 30000.
x(m)
Figure 14. Example 7c. Effects on Planform Evolution of Two ShoreNormal Retention Struc
tures of Length Equal to OneHalf the Initial Project Width. Normal Wave Incidence. Uniform
Background Erosion Rate at 0.5 m/yr, Zero Background Transport Located 4,500 m to Left of
Left Structure. Ho = 0.6 m, T = 6.0 sec.
transport located much farther to the left than for the example shown here, sand would bypass the
updrift structure, flow past the infrastructure segment and onto the downdrift shoreline. However,
both the updrift and downdrift shorelines at great distances from the project would continue to
erode at the background rate and only the updrift shoreline immediately adjacent to the project
area would experience a net accretion.
Example 8 Effect of Different Transport Characteristics for Native and Nourishment
Sands
Dean, et al. (1982) have examined the sediment transport factors, K (Eq. (9)), determined
from various field programs and have proposed the dependency of K on sand diameter, D, as
shown in Figure 15. Intuition suggests that the effects of nourishing with KF 0 KN could have a
significant effect on the adjacent shorelines. The subscripts "F" and "N" denote fill and native,
respectively. This effect will be greater for the case of oblique waves.
Figures 16 and 17 present the case of waves at a 200 obliquity acting on nourishment projects
with KFIKN = 0.9 and 1.1, respectively. It is seen from Figure 16 that for the case in which
the nourishment material is less transportable than the native, the nourishment planform acts as
an erodible barrier with an associated accretion and erosion on the updrift and downdrift sides of
the barrier, respectively. For the case in which the sand is more transportable than the native,
Figure 17, the pattern is qualitatively a mirror image of that noted.
As the project evolves, the calculation procedure requires determination of the degree to which
the sand exposed to the waves is of nourishment and native character. This was accomplished by
the following algorithm. A mixed layer of minimum thickness, Ymix, was assumed. If, during a
time increment, erosion occurred such that the mixed layer was less than Ymix thick, the remaining
material within the mixed layer was mixed with the underlying sand to reestablish a thickness
of Ymix. The character (i.e. K) of the mixed layer was calculated. If deposition occurred, the
mixed layer thickness and character were determined based on the thickness and character at
the previous time step and the character of the material in the littoral stream. Obviously, it was
2.0 1 1 1 1
2.0
I
LL
ZLU \
1.o \ 
0 0.5 1.0
DIAMETER, D (mm)
Figure 15. Plot of K vs D (Modified from Dean, et al., 1982).
50.0
30. 0
Initial Planform
S1 year
/^ \\ 3 years
/ _/_ \ l 5 years
/0/ / 10 years
10.0 I I I
0. 3000. 6000. 9000. 12000. 15000. 18000. 21000. 24000. 27000. 30000.
x(m)
Figure 16. Example 8a. Planform Evolution for Nourishment Sand Less Transportable than
the Native (KF = 0.693, KN = 0.77). Note Centroid of Planform Migrates Updrift. Variation
of Surface Layer K Values with Time at Locations A, B, C, D and E are presented in Figure 18.
Wave and other Project Conditions Presented in Table I.
50.0
30. 0 ..
 Initial Planform
I II
// /  5 years
/i / '\\ \
10.0 / \\\ 10 years
10.0
 10 .0 I I I I I I I I I I I
0. 3000. 6000. 9000. 12000. 15000. 18000. 21000. 24000. 27000. 30000.
x(m)
Figure 17. Example 8b. Planform Evolution for Nourishment Sand More Transportable than
the Native (Kp = 0.847, KN = 0:77). Note Centroid of Planform Migrates Downdrift. Variation
of Surface Layer K Values with Time at Locations A, B, C, D and E are presented in Figure 19.
Wave and other Project Conditions Presented in Table I.
necessary to calculate the character of the sand flowing into and out of a cell; this was accomplished
by starting with the updrift grid line as a boundary condition and, on cells where erosion occurred,
modifying the magnitude and character of the material in the littoral stream. In the nourished
area, the K value was set equal to KN when the nourished thickness reached zero. Figures 18
and 19 show the composite surface layer K values at the five locations indicated in Figures 16
and 17, respectively. It is noted that the centroids of the planform anomalies migrate updrift and
downdrift for KF < KN and KF > KN, respectively and that this profile migrational signature
could possibly be used in conjunction with a field monitoring program to establish the relative
sediment transport coefficients, KF and KN.
SUMMARY AND CONCLUSIONS
Two numerical methods have been presented for calculating shoreline evolution subsequent to
a beach nourishment project. These methods fall within the class of oneline shoreline models
in which the active vertical portion of the profile is represented by only one contour line. Both
models are capable of representing shoreperpendicular structures such as groins that might be
used to stabilize beach nourishment projects. One model is quite simple, with refraction and
shoaling from deep water to breaking conditions occurring in a closed form. In the second method,
refraction and wave shoaling are carried out on a two dimensional grid using conditions of wave
number irrotationality and conservation of wave energy, respectively. The simple model requires
approximately 1/250 of the computer time of the more detailed model.
In applying the two methods, it is recommended that the actual preproject bathymetry be
replaced by contour lines which are straight and parallel with an alignment of the general shoreline.
This recognizes implicitly the shorter time scales associated with the project evolution as compared
to that of the original bathymetric and shoreline disequilibrium. Preproject background erosion is
interpreted as due to crossshore and longshore transport components and is assumed to continue
unchanged after project construction.
0.80 I i I I
..T
0.8  i ii  ] 
Co
0 /  AT
9/ I
c 0.70 /
bO ( '    
I
 T E
0.60
S T C
 AlT E
0.60 II I I i I I I I I I
0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Years After Nourishment
Figure 18. Example 8c. Variation of Surface Layer Longshore Transport Coefficient K with
Time at the Five Locations Shown in Figure 16. Case of KF = 0.693, KN = 0.77, Ymix = 2.0 m.
0n n
I I I I I I I I
U)
c.
0
o 8
C I
0  
, 0.8 
0)
O
. A T C
AT E
0)
0.70 I T I
0 .7 0  ^ } 
0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Years After Nourishment
Figure 19. Example 8d. Variation of Surface Layer Longshore Transport Coefficient K with
Time at the Five Locations Shown in Figure 17. Case of KF = 0.847, KN = 0.77, Ymi, = 2.0 m.
1 I I 1 _
Comparison of results of applying the two models has established that they yield essentially
the same evolution. In the absence of littoral barriers, the planform asymmetry due to oblique
waves is shown to be small and is interpreted to be a result of the small aspect ratio (width to
length) of the nourished project.
The interaction of nourishment retention structures with background transport is examined.
Such structures can cause a wide range of effects within and adjacent to the nourished area
depending on the characteristics of the background transport.
The effects of nourishing with sediment of transport characteristics different than the native
are investigated. If the nourishment material is transported less readily, the nourishment project
acts as an erodible barrier causing accretion and erosion on the updrift and downdrift sides of
the project, respectively. For cases in which the nourishment material is transported more read
ily than the native, the shorelines updrift and downdrift of the project both accrete with the
greater accretion occurring on the downdrift side. For nourishment materials which are less and
more transportable than the native, the centroids of the planform anomalies migrate updrift and
downdrift respectively with time.
The results presented herein predict much greater longevity than those by Pilkey and co
workers although it is cautioned that most of examples do not include background erosion which
can be a significant sitespecific factor.
It is concluded that for the range of cases and conditions examined, the simple model yields
results as valid as the more detailed model which requires much greater computer time. Finally,
models such as those developed herein can be effective in rational benefit/cost analyses of potential
beach nourishment projects.
necessary to calculate the character of the sand flowing into and out of a cell; this was accomplished
by starting with the updrift grid line as a boundary condition and, on cells where erosion occurred,
modifying the magnitude and character of the material in the littoral stream. In the nourished
area, the K value was set equal to KN when the nourished thickness reached zero. Figures 18
and 19 show the composite surface layer K values at the five locations indicated in Figures 16
and 17, respectively. It is noted that the centroids of the planform anomalies migrate updrift and
downdrift for KF < KN and KF > KN, respectively and that this profile migrational signature
could possibly be used in conjunction with a field monitoring program to establish the relative
sediment transport coefficients, KF and KN.
SUMMARY AND CONCLUSIONS
Two numerical methods have been presented for calculating shoreline evolution subsequent to
a beach nourishment project. These methods fall within the class of oneline shoreline models
in which the active vertical portion of the profile is represented by only one contour line. Both
models are capable of representing shoreperpendicular structures such as groins that might be
used to stabilize beach nourishment projects. One model is quite simple, with refraction and
shoaling from deep water to breaking conditions occurring in a closed form. In the second method,
refraction and wave shoaling are carried out on a two dimensional grid using conditions of wave
number irrotationality and conservation of wave energy, respectively. The simple model requires
approximately 1/250 of the computer time of the more detailed model.
In applying the two methods, it is recommended that the actual preproject bathymetry be
replaced by contour lines which are straight and parallel with an alignment of the general shoreline.
This recognizes implicitly the shorter time scales associated with the project evolution as compared
to that of the original bathymetric and shoreline disequilibrium. Preproject background erosion is
interpreted as due to crossshore and longshore transport components and is assumed to continue
unchanged after project construction.
Appendix I
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APPENDIX II. NOTATION
The following symbols are used in this paper:
b = subscript denoting breaking;
B = beach berm height;
C = wave celerity;
CG = group velocity;
D = characteristic sediment size;
V = wave energy dissipation per unit volume;
E = wave energy density;
f = probability distribution;
g = gravitational constant;
G = longshore diffusivity;
h = water depth;
h. = depth of closure;
h.+ = depth of original profile, immediately adjacent to filled profile, see Figure 4b;
H = wave height;
I = immersed weight sediment transport rate;
k = wave number;
K = sediment transport proportionality factor;
K' = proportionality factor for crossshore sediment transport;
K" = proportionality factor for project longevity;
L = nourishment project length;
n = renourishment interval;
N = project life;
p = in place sediment porosity;
Pes = longshore energy flux factor;
s = relative sediment density, ps/p;
t = time;
T = wave period;
qx,y = sediment transport distribution in x, y directions;
Q = total longshore sediment transport;
V = volume;
x = longshore coordinate;
y = crossshore coordinate, position seaward;
Y = initial nourished beach width;
a = wave direction, relative to north;
P = shoreline or contour orientation, relative to north;
0 = wave crest orientation, relative to bottom contours;
K = breaking wave height proportionality factor;
pI = measure of ambient shoreline orientation;
p = water mass density;
p, = sediment mass density;
r = smoothing factor; and
V = horizontal vector differential operator.
