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Group Title: UFL/COEL (University of Florida. Coastal and Oceanographic Engineering Laboratory) ; 90/001
Title: Equilibrium beach profiles
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 Material Information
Title: Equilibrium beach profiles characteristics and applications
Series Title: UFLCOEL
Physical Description: v, 70 leaves : ill. ; 28 cm.
Language: English
Creator: Dean, Robert G ( Robert George ), 1930-
United States -- National Oceanic and Atmospheric Administration
National Sea Grant College Program (U.S.)
University of Florida -- Coastal and Oceanographic Engineering Dept
Publisher: Coastal & Oceanographic Engineering Dept., University of Florida
Place of Publication: Gainesville Fla
Publication Date: 1990
 Subjects
Subject: Coast changes -- Mathematics   ( lcsh )
Beach erosion -- Mathematics   ( lcsh )
Shorelines -- Mathematics   ( lcsh )
Coastal and Oceanographic Engineering thesis M.S
Coastal and Oceanographic Engineering -- Dissertations, Academic -- UF
Genre: government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )
 Notes
Bibliography: Includes bibliographical references (p. 66-70).
Funding: Sponsored by the Sea Grant College Program, National Oceanic and Atmospheric Administration.
Statement of Responsibility: by Robert G. Dean.
General Note: "January 15, 1990."
General Note: Final.
 Record Information
Bibliographic ID: UF00076132
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved, Board of Trustees of the University of Florida
Resource Identifier: oclc - 22468527

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Table of Contents
    Report documentation page
        Unnumbered ( 1 )
    Title Page
        Title Page
    Table of Contents
        Table of Contents
    List of Figures
        List of Figures 1
        List of Figures 2
        List of Figures 3
    Abstract
        Page 1
    Introduction
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
    Modified equilibrium beach profile
        Page 9
        Page 10
        Page 11
        Page 8
    Application of equilibrium beach profiles of the form: h=Ay2/3
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
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        Page 36
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        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
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        Page 47
        Page 48
        Page 49
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        Page 54
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        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
    Summary and conclusions
        Page 64
        Page 65
    Acknowledgements
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
Full Text


___REPORT DOCUMENTATION PAGE
1. Report No. 2. 3. Recipient'# Accession No.


4. Title and Subtitle 5. Report Date
EQUILIBRIUM BEACH PROFILES: January 15, 1990
CHARACTERISTICS AND APPLICATIONS 6.

7. Author(s) 8. Performing Organization aRport mo.
Robert G. Dean UFL/COEL-90/001

9. Performing Organization iNae and Address 10. Project/Task/Work Unit no.
Coastal and Oceanographic Engineering Department
University of Florida 11. Ctrt or rant No.
336 Weil Hall R/C-S-22
Gainesville, FL 32611 13. Typ of report
12. Sponsoring Organization Name and Address
Sea Grant Program Final
National Oceanic and Atmospheric Administration

14.
15. Suppleentary Notes


16. Abstract
An understanding of equilibrium beach profiles can be useful in a number of types of coastal engineering projects. Empirical
correlations between a scale parameter and the sediment size or fall velocity allow computation of equilibrium beach
profiles. The most often used form is h(y) = Ay2/3 in which h is the water depth at a distance y from the shoreline
and A is the sediment-dependent scale parameter. Expressions for shoreline position change are presented for arbitrary
water levels and wave heights. Application of equilibrium beach profile concepts to profile changes seaward of a seawall
include effects of sea level change and arbitrary wave heights. For fixed wave heights and increasing water level, the
additional depth adjacent to the seawall first increases, then decreases to zero for a wave height just breaking at the
seawall. Shoreline recession and implications due to increased sea level and wave heights are examined. It is shown, for
the equilibrium profile form examined, that the effect of wave set-up on recession is small compared to expected storm
tides during storms. Profile evolution from a uniform slope is shown to result in five different profile types, depending
on initial slope, sediment characteristics, berm height and depth of active sediment redistribution. The reduction in
required sand volumes through perching of a nourished beach by an offshore sill is examined for arbitrary sediment and
sill combinations. When beaches are nourished with a sediment of arbitrary but uniform size, it is found that three types
of profiles can result: (1) submerged profiles in which the placed sediment is of smaller diameter than the native and
all of the sediment equilibrates underwater with no widening of the dry beach, (2) non-intersecting profiles in which the
seaward portion of the placed material lies above the original profile at that location, and (3) intersecting profiles with
the placed sand coarser than the native and resulting in the placed profile intersecting with the original profile. Equations
and graphs are presented portraying the additional dry beach width for differing volumes of sand of varying sizes relative
to the native. The offshore volumetric redistribution of material due to sea level rise as a function of water depth is of
interest in interpreting the cause of shoreline recession. If only offshore transport occurs and the surveys extend over
the active profile, the net volumetric change is zero. It is shown that the maximum volume change due to cross-shore
sediment redistribution is only a fraction of the product of the active vertical profile dimension and shoreline recession.
The paper presents several other applications of equilibrium beach profiles to problems of coastal engineering interest.
17. Oritgnator's Key Words 18. Availability Stament
Beach erosion
Equilibrium beach profiles
Nourishment
Sea level rise
Seawalls
1*9. U. S. Security Classif. of the Report 20. U. S. Security Classl. of This Page 21. No. of Pafes 22. Price

Unclassified Unclassified 1 75

















EQUILIBRIUM BEACH PROFILES:
CHARACTERISTICS AND APPLICATIONS









Robert G. Dean

January 15, 1990









Sponsored by:

Sea Grant College Program
National Oceanic and Atmospheric Administration









Coastal and Oceanographic Engineering Department
University of Florida
Gainesville, FL 32611


I








TABLE OF CONTENTS

ABSTRACT 1

INTRODUCTION 2

Modified Equilibrium Beach Profile 8

Applications of Equilibrium Beach Profiles of the Form: h = Ay2/3 12
Effect of Sediment Size on Beach Profile ................. .... 12
Beach Response to Altered Water Level and Waves .............. 12
Equilibrium Profile and Recession Including the Effect of Wave Set-Up 18
Profile Adjustment Adjacent to a Seawall Due to Altered Water Level and
W aves . . . . . . . . .. .. 22
Response from Initial Uniform Slope ................. .... .. 29
Type 1 Profile .............................. 32
Type 2 and Type 4 Profiles .................. ...... 34
Type 3 and Type 5 Profiles ........................ 35
Limits of Profiles Types ........................ 35
No Terrace Present ........................... 36
Terrace Present .............................. 36
Perched Beach ................... .................. 36
Profile Response to Beach Nourishment . . . ..... 45
Intersecting, Non- Intersecting and Submerged Profiles . ... 45
Variation in Sediment Size Across the Surf Zone . . . ... 49
Continuous Arbitrary Distribution of Sand Sizes Across the Surf Zone 49
Piecewise Uniform Sand Size Across Surf Zone . . ... 53
Comparison with Empirical Orthogonal Functions . . ... 53
Effects of Sea Level Rise on Beach Nourishment Quantities . ... 55
Case I Nourishment Quantities for the Case of No Onshore Sediment
Transport ... ... ... ... ..... .. ... .. ... ..... 55
Case II Nourishment Quantities for the Case of Onshore Sediment
Transport ... ... ... ... .. .. .. ... ... ... ..... 56
Cross-Profile Volumetric Redistribution Due to Sea Level Rise ...... .. 59
Trailing Beach Profile Signature Due to Sea Level Rise . . ... 64

SUMMARY AND CONCLUSIONS 64

ACKNOWLEDGEMENTS 65

REFERENCES 66









LIST OF FIGURES


1 Beach Profile Factor, A, vs Sediment Diameter, D, in Relationship h = Ay2/3
(modified from Moore, 1982) ................... ......... 5
2 Profile P4 from Zenkovich (1967). A Boulder Coast in Eastern Kamchatka.
Sand Diameter: 150 mm 300 mm. Least Squares Value of A = 0.82 m1/3
(from M oore, 1982). ................... ............. 6
3 Profile P10 from Zenkovich (1967). Near the End of a Spit in Western Black
Sea. Whole and Broken Shells. A = 0.25 m1/3 (from Moore, 1982) . 6
4 Profile from Zenkovich (1967). Eastern Kamchatka. Mean Sand Diameter:
0.25 mm. Least Squares Value of A = 0.07 m1/3 (from Moore, 1982). . 7
5 Variation of Sediment Scale Parameter, A, With Sediment Size and Fall
Velocity (Dean, 1987a). ................... ......... 9
6 Comparison of Calibrated Profile Response Model With Large Wave Tank
Data by Saville (1957), From Kriebel (1986). . . . ... 10
7 Comparison of Calibrated Profile Response Model With Field Profile Affected
By Hurricane Eloise as Reported by Chiu (1977). From Kriebel (1986). 11
8 Comparison of Equilibrium Beach Profile With and Without Gravitational
Effects Included. A = 0.1 m1/3 Corresponding to a Sand Size of 0.2 mm. 13
9 Definition Sketch for Profile Response Due to Sea Level Rise. . ... 14
10 Equilibrium Beach Profiles for Sand Sizes of 0.3 mm and 0.6 mm A(D = 0.3
mm) = 0.12 m1/3, A(D = 0.6 mm) = 0.20 m/3. . . . ... 15
11 Isolines of Dimensionless Shoreline Change, Ay', vs Dimensionless Storm
Breaking Depth, h./B, and Dimensionless Storm Tide, S'. . . 17
12 Profile Geometry and Notation For Shoreline Advancement Due to Lowering
of W ater Level ................... .............. 19
13 Non-Dimensional Wave Set-Up and Equilibrium Beach Profile. ...... .. 21
14 Beach Recession Due to Waves and Increased Water Level, Including the
Effect of Wave Set-Up ...................... .......... 21
15 Definition Sketch. Profile Erosion Due to Sea Level Increase and Influence
of Seawall .. .. . . . . . . . . 24
16 Isolines of Dimensionless Seawall Toe Scour, Ah'~ vs Dimensionless Storm
Tide, S' and Dimensionless Breaking Depth h. . . . ... 26
17 Non-Dimensional Change in Water Depth at Wall, Ah',, as a Function of
Non-Dimensional Storm Surge. Example for h' = 6.0. . . .... 27
18 Definition Sketch, Profile Response Adjacent to a Seawall for Case of Lowered
Sea Level . . . . . . . . . 28









19 Non-Dimensional Shallowing Adjacent to a Seawall Due to Lowering of Wa-
ter Level. Variation With Non-Dimensional Breaking Depth, h., and Non-
Dimensional Water Level, S'. ........................... 30
20 Illustration of Five Equilibrium Profile Types Commencing From an Initially
Uniform Slope. ................... .................. 31
21 Regimes of Equilibrium Profile Types Commencing From An Initially Planar
Profile Showing Five Types of Equilibrium Profiles and Non-Dimensional
Profile Advancement, Ay'.............................. 33
22 Example of Type 1 (recession) Profile Response From An Initially Uniform
Slope and Associated Volumetric Transport. Note Only Positive (seaward)
Transport. ....................................... 37
23 Example of Type 2 (Recession) Profile Response From An Initially Uniform
Slope and Associated Volumetric Transport. Note Both Positive (Seaward)
and Negative (landward) Transport. . . . . .. 38
24 Example of Type 3 (Recession) Profile Response From An Initially Uniform
Slope and Associated Volumetric Transport. Note Only Negative (Landward)
Transport .. . . .. .. . . . . . 39
25 Example of Type 4 (Advancement) Profile Response From An Initially Uni-
form Slope and Associated Volumetric Transport. Note Both Positive (Sea-
ward) and Negative (Landward) Transport. . . . . ... 40
26 Example of Type 5 (Advancement) Profile Response From An Initially Uni-
form Slope and Associated Volumetric Transport. Note Only Negative (Land-
ward) Transport ................... .............. 41
27 Perched Beach, Demonstration of Nourishment Volumes Saved. ...... .. 42
28 Three Generic Types of Nourished Profiles. . . . ... 43
29 Effect of Nourishment Material Scale Parameter, Ap, on Width of Resulting
Dry Beach. Four Examples of Decreasing AF With Same Added Volume Per
Unit Beach Length. ............ ................... 46
30 Effect of Increasing Volume of Sand Added on Resulting Beach Profile. Ap
= 0.1m1/3, AN = 0.2 m1/3,h = 6 m, B = 1.5m................ 47
31 (1) Volumetric Requirement for Finite Shoreline Advancement (Eq. 54);
(2) Volumetric Criterion for Intersecting Profiles (Eq. 53). Variation with
AF/AN. Results Presented for h./B = 4.0. . . . .... 50
32 Variation of Non-Dimensional Shoreline Advancement Ay/W*, With A' and
V. Results Shown for h,/B = 2.0. ............ ........... 51









33 Variation of Non-Dimensional Shoreline Advancement Ay/W,, With A' and
V. Results Shown for h,/B = 4.0. ........................ 52
34 Comparison of Beach Profile Elevation Changes By Equilibrium Profile Con-
cepts With Results From Field Measurements. Example in Panel b) for h,/B
= 4.0 and S/B = 0.5. ................... ............ 54
35 Possible Mechanism of Sedimentary Equilibrium (After Dean, 1987b). 58
36 Definition Sketch and Non-Dimensional Volume Redistributed as a Function
of Non-Dimensional Depth. Due to Sea Level Rise, S. Case of B/h. = 0.25,
S/B = 0.5 ................... ................ 60













EQUILIBRIUM BEACH PROFILES:

CHARACTERISTICS AND APPLICATIONS

R. G. Dean


Abstract
An understanding of equilibrium beach profiles can be useful in a number of types of
coastal engineering projects. Empirical correlations between a scale parameter and the
sediment size or fall velocity allow computation of equilibrium beach profiles. The most
often used form is h(y) = Ay2/3 in which h is the water depth at a distance y from the
shoreline and A is the sediment-dependent scale parameter. Expressions for shoreline
position change are presented for arbitrary water levels and wave heights. Application of
equilibrium beach profile concepts to profile changes seaward of a seawall include effects
of sea level change and arbitrary wave heights. For fixed wave heights and increasing
water level, the additional depth adjacent to the seawall first increases, then decreases to
zero for a wave height just breaking at the seawall. Shoreline recession and implications
due to increased sea level and wave heights are examined. It is shown, for the equilibrium
profile form examined, that the effect of wave set-up on recession is small compared to
expected storm tides during storms. Profile evolution from a uniform slope is shown to
result in five different profile types, depending on initial slope, sediment characteristics,
berm height and depth of active sediment redistribution. The reduction in required
sand volumes through perching of a nourished beach by an offshore sill is examined for
arbitrary sediment and sill combinations. When beaches are nourished with a sediment
of arbitrary but uniform size, it is found that three types of profiles can result: (1)
submerged profiles in which the placed sediment is of smaller diameter than the native
and all of the sediment equilibrates underwater with no widening of the dry beach,
(2) non-intersecting profiles in which the seaward portion of the placed material lies
above the original profile at that location, and (3) intersecting profiles with the placed
sand coarser than the native and resulting in the placed profile intersecting with the
original profile. Equations and graphs are presented portraying the additional dry beach
width for differing volumes of sand of varying sizes relative to the native. The offshore
volumetric redistribution of material due to sea level rise as a function of water depth
is of interest in interpreting the cause of shoreline recession. If only offshore transport
occurs and the surveys extend over the active profile, the net volumetric change is zero.
It is shown that the maximum volume change due to cross-shore sediment redistribution
is only a fraction of the product of the active vertical profile dimension and shoreline
recession. The paper presents several other applications of equilibrium beach profiles
to problems of coastal engineering interest.









INTRODUCTION


A quantitative understanding of the characteristics of equilibrium beach profiles is central

to rational design of many coastal engineering projects and to the interpretation of nearshore

processes. Several features of equilibrium beach profiles are well-known: (1) they tend to

be concave upwards, (2) smaller and larger sand diameters are associated with milder and

steeper slopes, respectively, (3) the beach face is approximately planar, and (4) steep waves

result in milder slopes and a tendency for bar formation.

In a broad sense, it is obvious that sand particles are acted upon by a complex sys-

tem of constructive and destructive generic "forces" with the constructive forces acting to

displace the sediment particle landward and vice versa. Constructive forces include land-

ward directed bottom shear stresses due to the nonlinear character of shallow water waves,

landward directed "streaming" velocities in the bottom boundary layer (Bagnold, 1946;

Phillips, 1966), the phasing associated with intermittent suspended sediment motion, etc.

The most obvious destructive force is that of gravity coupled with the destabilizing effects of

turbulence induced by wave-breaking; others include the effect of seaward directed bottom

undertow currents and forces due to wave set-up within the surf zone (e.g. Svendsen (1984),

Stive and Wind (1986)). Indeed, the above represents only a partial listing of the complex

force system acting on sediment particles and serves to illustrate the difficulty of a rational

physics-based prediction of equilibrium beach profiles.

Several approaches have been pursued in an attempt to characterize equilibrium beach

profiles. Keulegan and Krumbein (1919) investigated the characteristics of a mild bottom

slope such that the waves never break but rather are continually dissipated by energy losses

due to bottom friction. Bruun (1954) analyzed beach profiles from the Danish North Sea

coast and Mission Bay, CA and found that they followed the simple relationship


h(y) = Ay2/3 (1)









in which h is the water depth at a seaward distance, y, and A is a scale parameter which

depends primarily on sediment characteristics. Eagleson, Glenne and Dracup (1963) de-

veloped a complex characterization of the wave and gravity forces acting on a particle

located outside the zone of "appreciable breaker influence" and developed expressions for

the seaward limit of motion and for the beach slope for which a sand particle would be in

equilibrium.

Swart (1974) carried out a series of wave tank tests and developed empirical expressions

relating profile geometry and transport characteristics to the wave and sediment conditions.

The active beach profile was considered as four zones and empirical expressions were de-

veloped for each zone. Vellinga (1983) investigated dune erosion using wave tank tests and

developed the following "erosion profile" which included the effect of deep water significant

wave height, Ho, and sediment fall velocity, w,
(7.6\.) h 7.6 18 w 0.56 0.5
0.47 68 y+ 18 -2.0 (2)
Ho., Ho, .0268
in which the values of all variables are in the metric system. It can be shown that Eq. (2)

is in reasonably good agreement with Eq. (1). Sunamura and Horikawa (1974) examined

and characterized beach profiles for two sizes of sediments, and ranges of wave heights and

periods and initial slopes of planar beaches. Three beach profile types were established

by laboratory experiments including one erosional and two accretional types. Suh and

Dalrymple (1988) applied concepts of equilibrium beach profiles to address the same problem

as Sunamura and Horikawa and identified one erosional profile type and one accretional

type. Comparison of laboratory data demonstrated good agreement with their criteria and

predictions of profile changes.

Numerous investigations have been carried out to develop appropriate scale modeling

criteria including Dalrymple and Thompson (1976), Noda (1972), Hughes (1983) and Van

Hijum (1975). Hayden, et al. (1975) apparently were the first to apply the concept of

empirical orthogonal functions (EOF) to extract the principal modes of change from a set

of beach profile data. Numerous later investigations have used this approach to investigate









the character of the dominant modes of profile change, e.g. Winant et al. (1975), Weishar

and Wood (1983), Aubrey et al. (1977), and Aubrey (1979). The EOF is a purely descriptive

method and does not address the causes or processes of profile change.

Hayden, et al. (1975) assembled a data set comprising 504 beach profiles along the

Atlantic and Gulf coasts of the United States. Dean (1977) analyzed these profiles and used

a least squares procedure to fit an equation of the form

h = Ay" (3)

to the data and found a central value of n = 2/3 as Bruun had earlier. It was shown that

Eq. (3) with n = 2/3 is consistent with uniform wave energy dissipation per unit volume,

D.. It can be shown that P, and A are related by

A 24 D. (D) /3(4)
A 5 pg3/22 (4)

in which p is the water mass density, g is gravity, D is sediment particle diameter and C

is a constant relating wave height to water depth within the surf zone. The interpretation

of Eq. (3) was that a particle of given size is characterized by an associated stability and

that the wave breaking process results in the transformation of organized wave motion

into chaotic turbulence fluctuations; these fluctuations are destructive forces and, if too

great, cause mobilization of the sediment particle with resulting offshore displacement and

a milder beach slope, which reduces the wave energy dissipation per unit volume eventually

resulting in an equilibrium profile. Later Moore (1982) collected and analyzed a number

of published beach profiles and developed the relationship between A and D as shown in

Figure 1. As expected, the larger the sediment size, the greater the A parameter and the

steeper the beach slope. Figures 2, 3 and 4 present several profiles employed by Moore in

establishing the relationship presented in Figure 1. The profile in Figure 2 is of particular

interest as the sediment particle size ranges from 15 cm to 30 cm, approximately the size

of bowling balls! Dean (1987a) has shown that when the relationship presented in Figure 1

is transformed to A(w) rather that A(D), where w is the fall velocity, the relationship is
















1.0
wI Suggested Empirical -
1H Relationship -
w
< From Hughes'
(L Where a Range of Sand Si;
SWas Given




_Z -From Swart's
IW Laboratory Results


L 0.01
0.01 0.1 1.0 10.0 100.0

SEDIMENT SIZE, D (mm)


Figure 1. Beach Profile Factor, A, vs. Sediment Diameter, D, In Relationship
h = Ay23 (Modified from Moore, 1982).











DISTANCE OFFSHORE (m)


Figure 2. Profile P4 From Zenkovich (1967). A Boulder Coast in Eastern
Kamchatka. Sand Diameter: 150 mm 300 mm. Least Squares
Value of A = 0.82 m113(from Moore, 1982).



DISTANCE OFFSHORE (m)
0.00 60.00 120
0.00 I I
E -- Least Squares Fit
Actual Profile
I-
LU
S 3.00

UI


6.00



Figure 3. Profile P10 From Zenkovich (1967). Near the End of a Spit In
Western Black Sea. Whole and Broken Shells. A = 0.25 m/3
(from Moore, 1982).





















DISTANCE OFFSHORE (m)


0.00
0.00 1N


1.00




2.00


Figure 4. Profile From Zenkovich (1967). Eastern Kamchatka. Mean Sand
Diameter: 0.25 mm. Least Squares Value of A = 0.07 m'l3(from
Moore, 1982).









surprisingly linear (on a log- log plot) as presented in Figure 5.

Kriebel (1982), Kriebel and Dean (1984, 1985) and Kriebel (1986) have considered

profiles out of equilibrium by hypothesizing that the offshore transport is proportional to

the difference between the actual and equilibrium wave energy dissipation per unit volume,

i.e.

Q = K(D PD) (5)

Eq. (5) and a sand conservation relationship have been incorporated into a numerical

model of sediment transport with generally good confirmation between laboratory profiles

(Figure 6) and field results (Figure 7). Larson (1988) and Larson and Kraus (1989) have

considered the active region of sediment transport in four zones and have developed em-

pirical counterparts to Eq. (5) for each zone, thus allowing solution of the transient beach

profile problem including a capability for generating longshore bars.

Modified Equilibrium Beach Profile


An unrealistic property of the form of the equilibrium beach profile represented by Eq. (1)

is the predicted infinite slope at the shoreline. Large slopes induce correspondingly large

gravity forces which are not represented in Eq. (1). A slight modification to include gravi-

tational effects is
D, ah 1
+ (ECG) = D (6)
m ay hay
Gravity Effect Turbulence Effect


in which the two terms on the left hand side represent destabilizing forces due to gravity

and turbulent fluctuations due to wave energy dissipation, m is the beach face slope, and

as before D. represents the stability characteristics of the sediment particle but now the

interpretation of ,D is broadened beyond equilibrium energy dissipation per unit volume to

include gravity as an additional destabilizing force. Eq. (6) can be integrated to:

y = A3/2h (7)















SEDIMENT FALL VELOCITY, w (cm/s)
1.0 10.0


0.0
1.0







0.10







0.01
0.0


100.0


100.0


SEDIMENT SIZE, D (mm)

Figure 5. Variation of Sediment Scale Parameter, A, With Sediment Size and
Fall Velocity. (Dean 1987a)


1I


10.0


Suggested Empirical
Relationship A vs. D
(Moore)
From Hughes'
Field Results 0.44
From Individual Field 0
Profiles where a Range
of Sand Sizes was Given _
Based on Transforming
A vs D Curve using
Fall Velocity Relationship

/ From Swart's
Laboratory Results


1












10.00


0.00
z
Predicted
Sa40hat 40 hrs :
J -5.00 Observed
Si at40hrs



-10.00-




-15.00 I I I I I II
0.00 40.00 80.00 120.00 160.00 200.00 240.00 280.00 320.00
DISTANCE (ft)


Figure 6. Comparison of Calibrated Profile Response Model With Large Wave
Tank Data by Saville (1957), From Kriebel (1986).






































-150


-100


.. I.... I...... ...I .*.....*...... ...
S. i.... ....;.... .... .... .. ....I... .... .. .. .... I ... i.... I ..i... I...I. .... I ..i .... .... .. I I .... I.... I.... ..i I I... ..* I .. ...
L i i- I i- I 1 i .

i T !" !"! T" T!.... ....!.." P i l ..:". e T i I '!"!
...... ......... ...... Pre-storm Pro .e: :....



.... .. ... observe ... ii ... ....

Predicted ......
II** *
... .... ... .. .... .. ... .... .... I....... ,....... .. .. ... .. .. ... re ic t. .... ... ......... .. .... .... .....I .. ... ...... ........ ...
:.. : : : i : :: i:: 1:: :,, : 1::: ., : i: i;: i :: i: :: :: i... : :: .. : .

. .j. ....:.: I: ;i..l..l .....I.... .j...v...e d I...j..... ......... ..........


50 100


150


200


MONUMENT (ft)




Figure 7. Comparison of Calibrated Profile Response Model With Field Profile Affected
By Hurricane Eloise as Reported By Chlu (1977). From Kriebel (1986).


-10


-250


-200


250









surprisingly linear (on a log- log plot) as presented in Figure 5.

Kriebel (1982), Kriebel and Dean (1984, 1985) and Kriebel (1986) have considered

profiles out of equilibrium by hypothesizing that the offshore transport is proportional to

the difference between the actual and equilibrium wave energy dissipation per unit volume,

i.e.

Q = K(D PD) (5)

Eq. (5) and a sand conservation relationship have been incorporated into a numerical

model of sediment transport with generally good confirmation between laboratory profiles

(Figure 6) and field results (Figure 7). Larson (1988) and Larson and Kraus (1989) have

considered the active region of sediment transport in four zones and have developed em-

pirical counterparts to Eq. (5) for each zone, thus allowing solution of the transient beach

profile problem including a capability for generating longshore bars.

Modified Equilibrium Beach Profile


An unrealistic property of the form of the equilibrium beach profile represented by Eq. (1)

is the predicted infinite slope at the shoreline. Large slopes induce correspondingly large

gravity forces which are not represented in Eq. (1). A slight modification to include gravi-

tational effects is
D, ah 1
+ (ECG) = D (6)
m ay hay
Gravity Effect Turbulence Effect


in which the two terms on the left hand side represent destabilizing forces due to gravity

and turbulent fluctuations due to wave energy dissipation, m is the beach face slope, and

as before D. represents the stability characteristics of the sediment particle but now the

interpretation of ,D is broadened beyond equilibrium energy dissipation per unit volume to

include gravity as an additional destabilizing force. Eq. (6) can be integrated to:

y = A3/2h (7)









where, as before, A is related to D, by Eq. (4). In shallow water, the first term in Eq. (7)

dominates, simplifying to

h = my (8)

i.e. the beach face is of uniform slope, m, consistent with measurements in nature. In

deeper water, the second term in Eq. (7) dominates with the following simplification

h = Ay2/s (9)

as presented earlier.

Figure 8 presents a comparison of Eq. (7), which includes the planar portion near the

water line and Eq. (1) which has an infinite slope at the water line. A form similar to Eq.

(7) was adopted by Larson (1988) and Larson and Kraus (1989).

Applications of Equilibrium Beach Profiles of the Form:
h = Ay2/3

The utility of Eq. (1) will be illustrated by several examples. A definition sketch of the

system of interest is presented as Figure 9. In results presented here, it will be assumed

that within the surf zone the wave height is proportional to the local water depth with the

proportionality factor, i, i.e. H = ich(. w 0.78). In particular, the breaking wave height,

Hb, and breaking depth, h,, are related by Hb = Kh,.

Effect of Sediment Size on Beach Profile

Figure 10 presents two examples of the effect of sediment size on beach profile. The

scale parameter A is determined for various sediment sizes from Figure 1 and the profiles

computed from Eq. (1).

Beach Response to Altered Water Level and Waves

The effects of elevated and lowered water levels will be treated separately.

An elevated water level, S, with wave and sediment conditions such that the profile

is reconfigured out to a depth, h*, is assumed to result in equilibration with the final state



















DISTANCE OFFSHORE (m)


100


200


*I I I I I

-


SGravitational Effects Included








Only Wave Dissipation Included
~~ ~ -. .


300


Figure 8. Comparison of Equilibrium Beach Profiles With and Without Gravitational
Effects Included. A = 0.1 m1/3 Corresponding to a Sand Size of 0.2 mm.


1


3
















































Figure 9. Definition Sketch For Profile Response Due to Sea Level Rise.


B--~- -------------- ---------


FjS




q~17.
















DISTANCE OFFSHORE (m)
100


200


Figure 10. Equilibrium Beach Profiles for Sand Sizes of 0.3 mm and 0.6 mm
A(D = 0.3 mm) = 0.12 ml'/A(D = 0.6 mm) = 0.20 m1/3.









being the same profile form as before, but relative to the elevated water level. This situation

could pertain to a storm tide of long duration or to sea level rise.

Referring to Figure 9, the sand volume eroded, VE, is equal to the volume deposited,

VD

VE =VD (10)

When Eqs. (1) and (10) are combined the following implicit equation for the shoreline

change, Ay, is obtained

3 h,W, Ay\ 5/3 =3 hiWf /S S
5B W 5 B B
^ -Ay 5. -(1-" s}; (-W (11)

in which W* is the seaward limit of the active profile (* = (H)3/2) and for this case of

shoreline recession Ay < 0. Eq. (11) can be expressed in non-dimensional form as


Ay'- [ (+ Ay')/] S' =0 (12)

in which the non-dimensional variables are

Ayy
W*
BAy =



B' =
h*


S


Eq. (12) is plotted in Figure 11. For small values of Ay', Eq. (12) can be approximated by

W
Ay = -S (14)
(h, + B)

first proposed by Bruun (1962) and now referred to as the "Bruun Rule".

For the case of lowered water levels, there will be an excess of sand in the active system

and a resulting advancement of the shoreline. The profile equilibration depth, h,, will occur

at a distance, W2, from the original shoreline. The equilibrium depth is considered to extend



















1.0

,.. o.8-- --- ^-- 7--T-7-- -
cn / /

0.8 -


0 0.6 . ..... ....

I-
S 0.4 "" ,,-
z
. 0.2 I
o .2 0.0
5 0.0 : II I I,, I I I I II
0.1 0.2 0.5 1.0 2.0 5.0 10.0

DIMENSIONLESS STORM BREAKING DEPTH, -=
,BB'



Figure 11. Isolines of Dimensionless Shoreline Change, Ay', vs.
Dimensionless Storm Breaking Depth, h,/B, and Dimen-
sionless Storm Tide, S'.










17


; I









as a horizontal terrace from the distance W2 noted above to a landward location consistent

with the equilibrium profile and the shoreline advancement, Ay. For this case, shown in

Figure 12, the non-dimensional shoreline advancement can be shown to be

S2 (1 S'B')6/2 1 (15)
5 (B' SB' + 1)

where it is emphasized that for this case, S' < 0. Additionally,


W= = (1- S'B')/2 (16)

and the landward location of the terrace, W1, is


W' = W l1 + Ay' (17)
W.

The other non-dimensional parameters are as defined by Eqs. (13).

Equilibrium Profile and Recession Including the Effect of Wave Set-Up

The equilibrium beach profile, h = Ay2/3, interpreted as resulting from uniform wave

energy dissipation per unit water volume does not include the effect of wave set-up. As a

precursor to developing recession predictions due to increased water levels and wave set-up,

it is useful to first establish the equilibrium beach profile, including the effect of wave set-up,

q].

We start with the well-known solution for wave set-up across the surf zone (Bowen et

al., 1968)

q(y) = b + J[hb h(y)] (18)

in which hb is the breaking depth (hb = h. qb S), qlb is the set-up (actually negative) at

breaking, and

3m'/8
J 2/ (19)
1 + 3i2/8


1 kH2 (20)
8 sinh 2kh.





















I I JIIIM ,UI. T, Y


H L------------------------------------------------------ -
SL---------- S<

hi *1 h

\ Original h
Profile





Horizontal
Terrace




Figure 12. Profile Geometry and Notation For Shoreline Advancement Due to Lowering
of Water Level.









where k is the wave number. Since i is positive over most of the surf zone, it is reasonable

that it contributes much like a tide in causing recession, especially for the larger breaking

wave heights. The equilibrium profile based on uniform wave energy dissipation per unit

volume commences from

((h)) (ECG)= -D (21)
(h + S +q 4) aye
in which ye is directed landward, E is the wave energy density, and CG is the group velocity.

Assuming shallow water,

ECG = g'(h+S+ q g)/ [(h+S+ 4)

If the algebra is carried through, the not-too-surprising result is obtained

h + S + f = Ay2/3 (22)

where the y-origin is now the location where h + S + 4 = 0. Figure 13 presents a plot of

the non-dimensional wave set-up and the associated equilibrium beach profile.

In the following development, the effects of a uniform storm tide, S, and the set-up

which varies across the surf zone (Eq. (22)) will be considered as presented in Figure 14.

Following procedures similar to those used for determining recession with a water level

which is uniform across the surf zone, volumes are equated as
f 0 W .+Ay &2W.+Ay /W.+Ay
[B-S-q(y)]dy+ A(y-Ay)23dy = Ay 2/3+f [S+(y)]dy (23)
y Ay J0 0
which after considerable algebra yields

3 5/3 1 (3/5 J)
Ay' + [1+ Ay'] I -S -176 (24)
5B' B' (1-J) (

in which Eqs. (13) have been used for non-dimensionalization and 14 = 7b/B.

The question of the relative roles of breaking waves and storm surges can be addressed

by simplifying Eq. (24) for the case of relatively small non-dimensional beach recessions,

|Ay'l << 1), which upon adoption of n = 0.78 and expressing in dimensional form yields

Ay 0.068( + (
1+ (25)
W. 1 + 1.28H
B










































Figure 13. Non-Dimensional Wave Set-Up and Equilibrium Beach Profile.


U"


Ay .4-


Figure 14. Beach Recession Due to Waves and Increased Water level,
Including the Effect of Wave Set-Up.









It is necessary to exercise care in interpreting Eq. (25), as the surf zone width, W. includes

the effect of the breaking wave height,


W. = (26)

As shown by Eq. (25), the dimensionless beach change (Ay/W,) is much more strongly

related to storm surge than wave height with the storm surge being approximately 16

times as effective in causing the dimensionless beach recession. However, during storms the

breaking wave height may be two to three times as great as the storm tide and the larger

breaking waves may persist much longer than the peak storm tides. The reason that the

storm tide plays a much greater role than that due to breaking wave set-up is evident from

Figure 13 where it is seen that the wave set-up (actually the set-down) acts to reduce the

mean water level over a substantial portion of the surf zone.

For the case in which storm surge is not important and the ratio of breaking wave height

to berm height is large, Eq. (25) can be simplified to represent only the effects of waves

and wave-induced set-up
= -0.053
W.
or
( /b 3/2 (27)
Ay = -0.053 ( ) (27)

Thus for a doubled wave height, the recession induced by wave set- up increases by a factor

of 2.8.

Profile Adjustment Adjacent to a Seawall Due to Altered Water Level
and Waves

We first consider the case of profile lowering adjacent to a seawall due to an elevated

water level.

It is well-known that during storms a scour trough will occur adjacent to a sea wall.

For purposes here it is appropriate to consider this scour as a profile lowering due to two

components: (1) the localized and probably dominant effect due to the interaction of the









seawall, waves, and tides, and (2) the effect due to sediment transport offshore to form a

profile in equilibrium with the elevated water level. Applying equilibrium profile concepts,

it is possible to calculate only the second component.

The system of interest is presented in Figure 15. The profile is considered to be in

equilibrium with virtual origin yl = 0. For a water level elevated by an amount, S, the

equilibrium profile will now be different and will have a virtual origin at y2 = 0. We denote

the distances from these virtual origins to the wall as yw, and yw, for the original and

elevated water levels, respectively. As in previous cases, the approach is to establish the

origin, yw2, (now virtual) such that the sand volumes seaward of the seawall and associated

with the equilibrium profiles are equal before and after the increase in water level. In the

following, all depths (h values) are referenced to the original water level except h. which as

described previously is a reference depth related to the breaking wave height.

Equating volumes as before

/W.-Yw,+Ywi rW
f .-Y2+ hl(y)dyl = W h2(y2)dy2
"W1 fYw2

which can be integrated using hi(y) = Ay2/3, and h2(y) + S = Ay/3 and simplified to yield

h- [h + S'] 5 h2 +S' 1 3/2 5/3 5/2
m- 1 "


[S' (h',2+ S,)32 =0 (28)


in which the primes represent non-dimensional quantities defined as

S hw,



h2 = hw
hW
hw1



S' = (29)
hw,


















Yw2

Ey--
m&


// / / // ///////

Virtual Origin, K,
Increased
Sea Level '

Virtual Origin,
Original
Sea Level


//


/
/r


W*



Seawall


hw
< -Ahw h2 h*
h,
.


_ Increased Sea Level
---------------- Y2

,,-Original Sea Level


Figure 15. Definition Sketch. Profile Erosion Due to Sea Level Increase and Influence of Seawall.









Eq. (28) is implicit in h'y2 and must be solved by iteration. Defining the change in depth

at the wall, Ahw, as

Ahw = hw, hw1

and in non-dimensional form
A Ahw
-h, = --
hw,
The quantity AhM, is now a function of the following two non-dimensional variables:

S' and h'. The relationship Ah',(h',S') is presented in Figure 16 where it is seen that

for a fixed h' and increasing S', the non- dimensional scour, Ah'y, first increases and then

decreases to zero. Figure 17 presents a specific example for h, = 6. The interpretation

of this form is that as S' increases, the profile is no longer in equilibrium and sand is

transported seaward to develop the equilibrium profile and the water depth adjacent to the

seawall increases. However, as sea level rises further, with the same total breaking depth,

the active surf zone width decreases, such that less sand must be transported seaward to

satisfy the equilibrium profile. With increasing storm tide, the surf zone width approaches

zero at the limit

S + hw, = h,

or

S' = h' 1

which corresponds to the upper line in Figure 16. It is emphasized that the increased depth

at the seawall predicted here does not include the scour interaction effect of the seawall and

waves.

We next consider the case of lowered water level (S < 0) adjacent to a seawall as shown

in Figure 18. In this case sediment will move landward due to the disequilibrium caused

by the lowered water level. The notation is the same as in the previous case. Equating

volumes eroded and deposited is expressed as

/WA-W, +Y hi(yi)dyi = h2 (2)dy2 + h*(WA W*) (30)
vW1 f1W2






















En 1-0 h .-=
LF 8.0 hw







1.0.8 --- \ K --
S 6.80 -2-
_____ -___ \ 5.0


e 0.64 .0 -



0.4 0.8
z 1.5


0 0.4 0.1
0.4
0.2 I I I I I
1 2 4 6 810 20 40 6080100

DIMENSIONLESS STORM BREAKING DEPTH, h',


Figure 16. Isolines of Dimensionless Seawall Toe Scour, Ah'w vs Dimensionless
Storm Tide, S' and Dimensionless Breaking Depth, h,.
























ZLU
00
z<


Sw






ZO
OZ 0


.5


.0


Figure 17.


1 2 3 4 5

NON-DIMENSIONAL STORM SURGE, S'


Non-Dimensional Change in Water Depth at Wall Ah'w as a Function
of Non-Dimensional Storm Surge. Example for h' = 6.0.























Original Water Level


Virtual Origins- -









Hori

Figure 18. Definition Sketch, Profile Response Adjacent to a Seawall
for Case of Lowered Sea Level.


Terrace









which can be integrated and simplified to yield

(h' + S')3/2 [ (h + S') h 2(h S')5/2 + h S' + h = 0 (31)

The non-dimensional distance WI(- WA/W*) is given by

WA 3/2 t t (32)
A= (1~ T1) + yW2 yWI (32)

The non-dimensional depth change (decrease) at the wall is

Ah', = h' 1 (33)

which can be established by solving Eq. (31). Figure 19 presents a plot of Ahty (h', S').

In Eq. (31), the term h'y + S' represents the non-dimensional total depth at the

seawall on the equilibrium profile for the lowered water level. A limiting case for the above

formulation is for this water level to be zero, i.e. h'~ + S' = 0, yielding

2(h S)5/2 + (h' S' + h' /= 0 (34)


which is plotted as the upper dashed line in Figure 19.

Response from Initial Uniform Slope

For simplicity, many wave tank tests commence with an initially planar beach slope,

mi. It is of interest to examine the relationship of the equilibrium and initial profiles. As

presented in Figure 20, there are five types of equilibrium profiles that can form depending

on the initial slope, mi, and the sediment and wave characteristics. Wave tank tests by

Sunamura and Horikawa (1974) identified profile Types 1, 2 and 5. Shoreline responses

for Types 1 and 3 have been investigated by Suh and Dalrymple (1988) using equilibrium

profile concepts and good agreement with wave tank data was demonstrated.

Referring to Figure 20, for the Type 1 profile the initial slope is much steeper than

that for the equilibrium profile and only seaward sediment transport occurs. An additional

characteristic is that a scarp is formed at the shoreline and no berm is deposited. The

wave tank profile presented in Figure 6 is an example of a Type 1 profile. Type 2 profile,





























--Limiting Case
1%% h'2+ S' = 0


-0.4 = A h'w
-0.3
-0.2


-0.1




S I i I I IlI I I I I i 1111


5 10 20


50 100


NON-DIMENSIONAL BREAKING DEPTH, h' = h,/hw




Figure 19. Non-Dimensional Shallowing Adjacent to a Seawall Due
to Lowering of Water Level. Variation With Non-Dimensional
Breaking Depth, h and Non-Dimensional Water Level, S'.


-1.01


-0.5


-0.2-


-0.1



-0.05


-0.02 F-


' '"~













Type 1 Profile
(Recession, No Terrace)











Type 2 Profile
(Recession, No Terrace)


Ay< 0


Profile


Ay.* 0


Type 4 Profile
(Advancement, No Terrace)


- *Initial Profile
h,*


Ay> 0


Type 5 Profile


Figure 20. Illustration of Five Equilibrium Profile Types Commencing
From an Initially Uniform Slope.


Initial
Shoreline









also a case of shoreline recession, occurs for a somewhat milder relative slope (initial to

equilibrium), sediment transport occurs in both the onshore and offshore directions and a

berm is formed at the shoreline. With still milder relative slopes, sediment transport occurs

only shoreward resulting in a Type 3 profile characterized by shoreline recession with all of

the sediment transported deposited as a berm feature. A terrace or "bench" (here assumed

horizontal) is formed at the seaward end of the equilibrium profile. This type profile is

probably the least likely to occur due to the unrealistically high berm elevation required.

Type 4 profile is one of shoreline advancement, occurring for still milder initial slopes and is

characterized by sediment transport in both the landward and seaward directions. Finally,

Type 5 profile is one of shoreline advancement with only landward sediment transport

and leaving a horizontal terrace or bench at the seaward end of the equilibrium profile.

The following paragraphs quantify the profile characteristics and shoreline changes for each

of these five types. As described previously, shoreline recession and advancement will be

denoted by negative and positive Ay, respectively. It will be shown that the non-dimensional

shoreline change, Ay'(= Ay/W,) is a function of the non- dimensional depth of limiting

profile change, h',( h,/W.mi) and non-dimensional berm height, B'(= B/W.mi). The

developments associated with these profile types will not be presented in detail. Methods

are similar to those applied earlier in this report for example for the case of shoreline

recession due to an elevated water level. Figure 21 presents Ay'(h',, B') and the associated

regions of occurrence for the five profiles types.


Type 1 Profile

The non-dimensional advancement, Ay', can be expressed in terms of the non-dimensional

depth of limiting motion, h', as

Ay' = 3h 1 (35)
5 2










2.0
.4'0 / ,,S-0.2
.. ,, ,
1 5,y ./ /3A.
1. 5 0 .8 -0.1 /
Sy1.5 / / / T //
S/ /
. / y/ / +0.1 0 /
ri/ +0.3

S10 / / .5 -0.4
, / +o.6 /
/ /+0.7
/ / / pe /
U 0+5 0e
Sype / / / / //+0.

- /
o ^ T ype 4 / / / / 1.0
0 1.0 2.0 3.0
h,
NON-DIMENSIONAL SEAWARD DEPTH OF ACTIVE MOTION, h',= m,

Figure 21. Regimes Of Equilibrium Profile Types Commencing From An initially
Planar Profile Showing Five Types Of Equilibrium Profiles And Non-
Dimensional Profile Advancement, Ay'.









where for this profile type, Ay' < 0. Because no berm is formed, the berm characteristics

do not appear in this expression and B is simply a reference quantity when plotted in

Figure 21.

It can be shown that the non-dimensional volume, V, transported seaward past any

location, y', is

= 1 [ 12 3 h2r 3 ,
V-BW- 2B' ( 2] + AY')5/3 Ay' < y' < 1 + Ay' (36)

in which

y' (37)

For this type profile, Suh and Dalrymple (1988) have denoted h,/W. as the equilibrium

slope, me, and have shown a correlation between the ratio of initial to equilibrium slopes

and the resulting profile changes.


Type 2 and Type 4 Profiles

For these profile types, the non-dimensional change, Ay', depends on h', and non- di-

mensional berm height, B',

Ay'= -(B' + 1)+ 2B' + h, (38)

For Type 2 and Type 4 profiles, the values of Ay' will be negative (recession) and positive

(advancement), respectively.

The equations for non-dimensional volumes transported seaward past any location, y',

for Type 2 and Type 4 profiles are


y = ([B'2- y2] (y'+ B') -B' < y' < Ay' (39a)

-(Ay'B') 2B'(y'A Ay'< y'< 1+ Ay (39b)

S= -(Ay' + B') -(-/y1 B'-)-+ 3 y Ay')5/s Ay' < y' < 1 + Ay' (39b)









Type 3 and Type 5 Profiles


Both of these equilibrium profile types include a horizontal terrace at their seaward

limits. The expressions for Ay' are identical:

1 h B2 4 h'
Ay' = (40)
2 h'. + B'
where again Ay' is negative and positive for profile Types 3 and 5, respectively.

The equations for non-dimensional volumes transported seaward past any location, y',

for Type 3 and Type 5 profiles are


V = --1(B y'2)-(y'+ B') ,-B' < y' < Ay' (41a)
2B'



v = '( B ) + 3 '(y Ay)5/3 (Ay'+ B') Ay' < y' < 1+ Ay' (41b)

2Bh' h B' 12 h'

4"V= Ay' (+ y-z l+ Ay < y' < h' (41c)
5 B' 2 B' 2 2 B' B



Limits of Profiles Types

The criterion for berm formation is

B' + 3h' 1 < 0, No Berm Formed (42)
5 2 > 0, Berm Formed

where here B' is interpreted as the non-dimensional berm height that would be formed if

the berm height exceeds the scarp height cut by the shoreline recession into the uniform

slope profile. For berm formation, Ay > -B/mi.

The limit for terrace formation is

62 < 0 No Terrace
(B' + h')2 2B'- Terrace (43)
5 {> 0, Terrace

The limit for no shoreline change, Ay' 0, depends on whether or not a terrace is

present.









No Terrace Present


-- f= 0, No Shoreline Change
B'2- h, + 1 > 0, Shoreline Recession (44)
< 0, Shoreline Advancement

Terrace Present


= 0, No Shoreline Change
B'2 h + h, > 0, Shoreline Recession (45)
< 0, Shoreline Advancement
Figures 22-26 present examples of profile response and associated volumes transported

for profile Types 1-5.

Perched Beach

The offshore extent of a perched beach is terminated by a shore parallel structure which

prevents the sand from moving seaward. In conjunction with a beach nourishment project, it

is possible in principle to obtain a much wider beach for the same volume of added sediment.

Denoting the "native" and "fill" sediment scale parameters as AN and Ap respectively and

referring to Figure 27 for terminology, the required volume, V, for the case of the sand just

even with the top of the submerged breakwater is

-V = B[( h, )"'3 h2 )3/2


3 r h, \s/2 h ( _2 5/2
+ AN ) -A([A (46)
5 \AN AF

and the added beach width, Ay, is

&Y ( h, 3/2 2 3/2 (47)
AN Ap

Referring to Figure 27, it is possible to calculate the reduction in required volume

through a perched beach design. The results can be developed for intersecting and non-

intersecting type profiles (c.f. Figure 28a and b). For simplicity, only the results for non-












0.3.

/0.2


-0.2


h 0.4-
h* 0.6


BWi.. ~


y
W*


0.6


a) Volumetric Transport Past Any Location


b) Profile Response


Figure 22. Example of Type 1 (Recession) Profile Response From An
Initially Uniform Slope and Associated Volumetric Transport.
Note Only Positive (Seaward) Transport.


F


I \ I

















i I


-
-


-0.2 -- 0.2 0.4 0.6 0.8 1.0

a) Volumetric Transport Past Any Location


Equilibrated Profile


b) Profile Response


Figure 23. Example of Type 2 (Recession) Profile Response From An Initially Uniform
Slope and Associated Volumetric Transport. Note Both Positive (Seaward)
Transport and Negative (Landward) Transport.


BW-
BW*-


-0.2


















SI I I I I I


y


.0.8 1.-


a) Volumetric Transport Past Any Location


.6 0.8 1.0 Y
--- W*

Inital Profile

Equilibrated Profile
< b) Profile Response


Figure 24. Example of Type 3 (Recession) Profile Response From An
Initially Uniform Slope and Associated Volumetric Transport.
Note Only Negative (Landward) Transport.


BV
BW*


-0.8


-0.6 -- -0.4


-0.2


-0.1


\ -0.2
I ,-


S.-


w*


















S -- I'


0.4 -0.6


1.0 w*


a) Volumetric Transport Past Any Location


0.8 1.0 Y
II -I



:ial Profile
Equilibrated Profile


b) Profile Response


Figure 25.


Example of Type 4 (Advancement) Profile Response From An
Initially Uniform Slope and Associated Volumetric Transport.
Note Both Positive (Seaward) and Negative (Landward) Transport.


BW-
Bw,


0.1 -


-0.2 --


,i N


0.2


- ""


-0.2 I-


y


ill













V
BW*
I


^^-0.2
%% -0.1 -


-0.3
-0.4-


-0.5 1-


y


1.6 W*


._ ... a)Volumetric Transport Past Any Location


1.6. Y


I aI I I I I I

SInitial Profile
Equilibrated Profile


Figure 26.


Example of Type 5 (Advancement) Profile Response From An
Initially Uniform Slope and Associated Volumetric Transport.
Note Only Negative (Landward) Transport.


I I


-0.6


-0.6


-0.


BW 012


S


0.6
0.8
1.0-


I I I


-- ~I1


I


1.2,,714


w,


m

























Yi


<-Ay-I- y2


Perched Beach



h2





Toe Structure




Figure 27. Perched Beach, Demonstration of Nourishment Volumes Saved.


m~







A:--yj---

--.. .-F

B


w,-
-<-Ay--


Added Sand --:

b) Non-Intersecting Profile


Ay -1 L


Virtual Origin of
Nourished Profile


Added Sand


c) Submerged Profile AF

Figure 28. Three Generic Types of Nourished Profiles.









intersecting profiles (without the sill) will be presented here. The fractional reduction in

volume, AV /-V, is given by


AV Ay' + [(h)5/2- (A )32 )
A- 5B, [ (h )/2]5/3 N (h i)/2
A V = 1-" I I A 2 ( 4 8 )


in which VI = the volume that would be required to advance the shoreline seaward by an

amount, Ay, without the sill and

y' = y2/w.



h' = hi/h,



h' = h2/h.



Ay' = Ay/W*



B1 B
h,
As an example of the application of Eq. (48), consider the following parameters


AN = AF = 0.15 m/3
h, = 6m
hi= 4m
h2 = 3m
B = 2.0 m
Ay = 48.3 m (From Eq. (47))
The width of the surf zone without the sill, W., is

W, = )3/2 = 253.0m
AN/
and
/(2 3/2
1Y2 3/2 = 89.4m
\AF/









The fractional reduction in volume is


AV
V 0.342

i.e., there is a 34% reduction in sand volumes with the perched beach design.

Profile Response to Beach Nourishment

Intersecting, Non-Intersecting and Submerged Profiles Beach nourishment is con-

sidered with sediment of arbitrary but uniform diameter. As indicated in Figure 28,

nourished beach profiles can be designated as "intersecting", "non-intersecting" and "sub-

merged" profiles. A necessary but insufficient requirement for profiles to intersect is that

the placed material be coarser than the native. Similarly, non-intersecting or submerged

profiles will always occur if the placed sediment is the same size as or finer than the na-

tive. However, non-intersecting profiles can occur if the placed sediment is coarser than

the native. For "submerged" profiles to occur, the placed material must be finer than the

native. Figure 29 illustrates the effect of placing the same volume of four different sized

sands. In Figure 29a, sand coarser than the native is used, intersecting profiles result and a

relatively wide beach Ay is obtained. In Figure 29b, the same volume of sand of the same

size as the native is used, non-intersecting profiles result and the dry beach width gained is

less. More of the same volume is required to fill out the milder sloped underwater profile.

In Figure 29c, the placed sand is finer than the native and much of the sand is utilized in

satisfying the milder sloped underwater profile requirements. In a limiting case, shown in

Figure 29d, no dry beach is yielded with all the sand being used to satisfy the underwater

requirements. Figures 30a through 30d illustrate the effects of nourishing with greater and

greater quantities of a sand which is considerably finer than the native. Figure 30d is the

case of formation of an incipient dry beach, i.e. the same as in Figure 29d. With increasing

volumes, the landward intersection of the native and placed profiles occurs closer to shore

and the seaward limit of the placed profile moves seaward.










92.4m
7 '_-- ~B = 1.5m


a) Intersecting Profiles, -
AN= 0.1mi/AF = 0.14m1/3


45.3m


b) Non-intersecting Profiles
AN= AF= 0.1m1/3


c) Non-Intersecting Profiles'
AN= O.lm 3,AF = 0.09m1/3


d) Limiting Case of Nourishment Advancement, 1 3
Non-Intersecting Profiles, AN= 0.1ml/3,AF = 0.085m1

I I i I! I


100


200


300


400


500


h,= 6m


600


OFFSHORE DISTANCE (m)

Figure 29. Effect of Nourishment Material Scale Parameter, A F,on Width of
Resulting Dry Beach. Four Examples of Decreasing A F With Same
Added Volume Per Unit Beach Length.


I


S115.9m










OFFSHORE


200


DISTANCE (m)
300 400


b) Added Volume = 490 m3 Im


Added Volume = 1660 m3/m
Case of Incipient Dry Beach


Figure 30. Effect of Increasing Volume of Sand Added on Resulting
Beach Profile. A F= 0.1 ml/3,AN= 0.2 m1/3,h, = 6.0 m, B = 1.5 m.


100


-,+4

I 0
0
O


" 10
.J


500


I 1 1 I








We can quantify the results presented in Figures 28, 29 and 30 by utilizing equilibrium
profile concepts. It is necessary to distinguish the three cases noted in Figure 28. The first is
with intersecting profiles such as indicated in Fig. 28a and requires Ap > AN. For this case,
the volume placed per unit shoreline length, V1 associated with a shoreline advancement,
Ay, is presented in non-dimensional form as

J/ j = A + A)5[/ ( 3/212/3 (49)

in which V1 (- V /BW,) is the non-dimensional volume, B is the berm height, W. is a refer-
ence offshore distance associated with the breaking depth, h., on the original (unnourished)
profile, i.e.

W= ( 32 (50)
(hN
and the breaking depth, h, and breaking wave height, Hb are related by

h, = Hble/

with ni(s 0.78), the spilling breaking wave proportionality factor.
For non-intersecting but emergent profiles (Figure 28b), the corresponding volume V2
in non-dimensional form is

V, = Ay' + 3/2 5/3 3/2 (51)

It can be shown that the critical value of (Ay') for intersection/non-intersection of
profiles is given by

A' + (AN )3/2 1 < 0, Intersecting Profiles
y' + -F 1 (52)
Ap > 0, Non-Intersecting Profiles
The critical volume associated with intersecting/non- intersecting profiles is

'^ = + 3 ) 1_-(A) (53)
and applies only for (AF/AN) > 1. Also of interest, the critical volume of sand that will
just yield a finite shoreline displacement for non-intersecting profiles (AF/AN < 1), is

( )2= AN5 32 i -1 (54)









Figure 31 presents these two critical volumes versus the scale parameter ratio AF/AN for

the special case h,/B = 4.0, i.e. B' = 0.25.

The results from Eqs. (49), (51) and (52) are presented in graphical form in Figures 32

and 33 for cases of (h./B) = 2 and 4. Plotted is the non-dimensional shoreline advancement

(Ay) versus the ratio of fill to native sediment scale parameters, AF/AN, for various isolines
of dimensionless fill volume V' (= W- ) per unit length of beach. It is interesting that the

shoreline advancement increases only slightly for AF/AN > 1.2; for smaller values the

additional shoreline width, Ay, decreases rapidly. For AF/AN values slightly smaller than

plotted, there is no shoreline advancement, i.e. as in Figure 29d.

Referring to Figure 28c for submerged profiles, it can be shown that

A 1- AN \3/2
=i (1 (55)
yi AF

where Ay < 0 and the non-dimensional volume of added sediment can be expressed as


S(--3 (-AN //2(
5B' A 32 23 A


Variation in Sediment Size Across the Surf Zone

All cases presented earlier have considered the sand size to be of uniform size across the

surf zone. In most cases, there is some sorting with the sand grading to finer sizes in the

seaward direction. With the relation of A(D), and thus PD(D), known (c.f. Eq. (4)) it is

possible to calculate equilibrium profiles for cases of a continuum of sand sizes across the

surf zone and a distribution composed of piecewise uniform diameter segments.

Continuous Arbitrary Distribution of Sand Sizes Across the Surf Zone The

differential equation for an equilibrium beach profile is given by

=2 A3/2(D) (57)
























00
Fn >

z


0
oe





Figure 31.


1 2
AF/AN


3

LL





z
200
1
o
z O
0
z


(1) Volumetric Requirement for Finite Shoreline Advancement
(Eq. 54); (2) Volumetric Criterion for Intersecting Profiles
(Eq. 53). Variation with AF/AN. Results Presented for h,/B = 4.0.














10.


1.0





1 . .
B %/ V' = 0.5



0.10 V' = 0.1

V' = 0.05




0.01' = 0.01


-y V' = 0.005


:; V = V/BW, = 0.002
hA .. .. ..

0.001 -Definition Sketch- ---

0 1.0 2.0 2.8
A'= AF/AN


Figure 32. Variation of Non-Dimensional Shoreline Advancement Ay /W,, With
A' and : Results Shown for h.B = 2.0.









1.0


W Asymptotes "I-A M
for Ay = 0
0'= 0.02
^ ^npt^^E --;-F----^


0.01







0.001 AV'= VIBB = 0.001


Definition Sketch
I _




0.0 001 00

-AY f AF V
I B N__ I W*

0 1.0 2.0 2.8
A' = A/AN
Figure 33. Variation of Non-Dimensional Shoreline Advancement Ay I/W, With
A' and V: Results Shown for hi/B = 4.0.









from which Eq. (1) is obtained readily. Integrating across the surf zone yields the equilib-

rium profile

h3/2(y) = j AS/2(D)dy (58)

Piecewise Uniform Sand Sizes Across the Surf Zone Denoting the sand size as,

D,, over the segment y, < y < yn+l, the water depth in this region is obtained from a

slight variation of Eq. (58)

h3/2(y) = h3/2(yn) + A /(Dn)[y yn] (59)

which applies for yn < y < yn+i.

Comparison with Empirical Orthogonal Functions

It is instructive to compare results of profiles obtained from a simple application of the

equilibrium beach profile with those developed by various researchers (e.g. Winant, Inman

and Nordstrom (1975) and Weishar and Wood (1983)) in their application of Empirical

Orthogonal Function (EOF) methods to time series of natural beach profiles. This method

has been described by Hayden, et al. (1975). For purposes here, we note that the first EOF

is analogous to the equilibrium beach profile and the second EOF is termed the "berm-bar"

function.

We will consider the change in profile elevation resulting from a single elevated water

level and wave and sediment conditions that would mobilize sediment out to a depth h..

Consideration of Figure 9 and utilizing Eq. (12), the first EOF is the average profile, and

the second EOF can be shown to be approximately
Ah = Ah' B'(1 S') + (/' Ay')2/3 Ay < y' < 0
h. (y' Ay')2/3 y2/ S'B 0 < y' < 1 + Ay (60)
and where the primed (non-dimensional) quantities are as defined by Eqs. (13). Figure 34

presents a comparison between Eq. (60) and the second EOF as determined by Winant et

al. (1975) based on field measurements at Torrey Pines, CA. The similarities between the

EOF obtained by these investigators and those developed by equilibrium profile synthesis

are quite evident.



























SEAWARD DISTANCE, y(m)


a) Most Significant Eigenfunction of Profile Change
(From Winant, Inman and Nordstrom, 1975)


Y


b) Non-dimensional Elevation Changes Based
On Equilibrium Beach Profiles


Figure 34.


Comparison of Beach Profile Elevation Changes by Equilibrium
Profile Concepts With Results From Field Measurements. Example
In Panel b) for hJB = 4.0 and S/B = 0.5.


o c

> C
-i-
-J
w~


z
uJ*






wo



0
z









Effects of Sea Level Rise on Beach Nourishment Quantities

Recently developed future sea level scenarios (Hoffman, et al., 1983) have been developed

based on assumed fossil fuel consumption and other relevant factors and have led to concern

over the viability of the beach nourishment option for erosion control. First, in the interest

of objectivity, it must be stated that the most extreme of the scenarios published by the

Environmental Protection Agency (EPA) amounting to sea level increases exceeding 11 ft

by the year 2100 are extremely unlikely. While it is clear that worldwide sea level has

been rising over the past century and that the rate is likely to increase, the future rate

is very poorly known. Moreover, probably at least 20 to 40 years will be required before

our confidence level of future sea level rise rates will improve substantially. Within this

period, it will be necessary to assess the viability of beach restoration on a project-by-

project basis in recognition of possible future sea level scenarios. Presented below is a basis

for estimating nourishment needs for the scenarios in which there is no landward sediment

transport across the continental shelf and there is a more-or-less well-defined seaward limit

of sediment motion; in the second case the possibility of onshore sediment transport will be

discussed.

Case I Nourishment Quantities for the Case of No Onshore Sediment Transport

Bruun's Rule (1962) is based on the consideration that there is a well-defined depth limit,

h,, of sediment transport. With this assumption, the only response possible to sea level

rise is seaward sediment transport. Considering the total shoreline change Ay, to be the

superposition of recession due to sea level rise Ays and the advancement due to beach

nourishment, AyN,

Ay = Ays + AlN (61)

and, from Bruun's Rule (Eq. 14)

W,
s = -Sh, (62)

in which S is the sea level rise, W, is the distance from the shoreline to the depth, h,,









associated with the seaward limit of sediment motion and B is the berm height. Assuming

that compatible sand is used for nourishment (i.e. Ap = AN)


AY N = (63)
h, + B

and V is the beach nourishment volume per unit length of beach. Therefore

1
Ay = [V -SW.] (64)
(h. + B)

The above equation can be expressed in rates by,

dy 1 [dV dS 1
1 B) W. (65)
dt (h- + B) dtW dt

where S- now represents the rate of sea level rise and H is the rate at which nourishment

material is provided. It is seen from Eq. (65) that in order to maintain the shoreline stable

due to the effect of sea level rise the nourishment rate is related to the rate of sea level

rise dS by
dv dS
t= W. t (66)

Of course, this equation applies only for cross-shore mechanisms and therefore does not

recognize any other causes of background erosion or longshore transport losses from the

project area. It is seen that W, behaves as an amplifier of material required. Therefore, it

is instructive to examine the nature of W, and it will be useful for this purpose to consider

the equilibrium profile given by Eq. (1),


W*- Hb )3/2(67)

i.e. W, increases with breaking wave height and with decreasing A (or sediment size).

Case II Nourishment Quantities for the Case of Onshore Sediment Transport

Evidence is accumulating that in some locations there is a substantial amount of onshore

sediment transport across the continental shelf. Dean (1987b) has noted the consequences

of the assumption of a "depth of limiting motion" in allowing only offshore transport as a

response to sea level rise and proposed instead that if this assumption is relaxed, onshore









transport can occur leading to a significantly different profile response to sea level rise.

Consider that there is a range of sediment sizes in the active profile with the hypothesis

that a sediment particle of given hydraulic characteristics is in equilibrium under certain

wave conditions and at a particular water depth. Thus, if sea level rises our reference

particle will seek equilibrium which requires landward rather than seaward transport as

required by the Bruun Rule. Figure 35 summarizes some of the elements of this hypothesis.

Turning now to nourishment requirements in the presence of onshore sediment transport,

the conservation of cross-shore sediment yields

= + sources sinks (68)
ay at
in which h is the water depth referenced to a fixed vertical datum and the sources could

include natural contributions such as hydrogenous or biogenous components, and suspended

deposition or human related contributions, i.e. beach nourishment. Sinks could include

removal of sediment through suspension processes. Eq. (68) can be integrated seaward

from a landward limit of no transport to any location, y

(y) o (sources sinks)dy = Y -dy (69)

If only natural processes are involved and there are no gradients of longshore sediment

transport, the terms on the left hand side of Eq. (69) represent the net rate of increase of

sediment deficit as a function of offshore distance, y. For y values greater than the normal

width, W., of the zone of active motion, the left hand side can be considered as representing

the "ambient" deficit rate due to cross-shore sediment transport resulting from long-term

disequilibrium of the profile and source and sink terms.

In attempting to apply Eq. (69) to the prediction of profile change and/or nourishment

needs under a scenario of increased sea level rise, it is reasonable to assume that over the

next several decades the ambient deficit rate (or surplus) of sediment within the active zone

will remain constant. However, an increased rate of sea level rise will cause an augmented

demand which can be quantified as W, [(-s) ()-o] in which (!) is the reference











POSSIBLE MECHANISM OF SEDIMENTARY EQUILIBRIUM


Increased Sea Level
SO al Sea Level___
StS &- Orialnlal Sea Level


Sediment
Particle


Subjected to a Given Statistical Wave Climate, A Sediment
Particle of a Particular Diameter is in Statistical Equilibrium
When in a Given Water Depth

Thus When Sea Level Increases, Particle Moves Landward


Figure 35. Possible Mechanism of Sedimentary Equilibrium (After Dean, 1987b).









sea level change rate during which time the ambient demand rate is established. Thus the
active zone sediment deficit rate will be

New Deficit Rate = [ dy + W, ( ] (70)
[f ol at 10 dt dt dt

in which t represents the nourishment rate and the subscript "0" on the bracket represents

the reference period before increased sea level rise. In order to decrease the deficit rate to

zero, the required nourishment rate is

dV I* \fdS\ fdS\
dV [W ahd]W [(dS) (dS)] (71)
dt at dt dt

These models may assist in evaluating the vulnerability of various shoreline systems to

increased rates of sea level rise.

Cross-Profile Volumetric Redistribution Due to Sea Level Rise

Eq. (12) was developed earlier to describe the shoreline change, Ay, due to a sea level

rise, S. Associated with this recession is a cross-shore transfer of sediment from the upper

to the lower portions of the profile. However there will be no net change of volume across

the entire active profile. This statement would not hold, of course, if a portion of the eroded

profile were peat or lagoonal muds that would be transported by suspension well beyond

the normal limits of the active profile.

It is instructive to consider the change in volume that would be measured due to surveys

extending out to a depth, h,, which is less than the active profile depth, h,. Referring to

Figure 36b, the volumetric change per unit length due to profile equilibration as a result of

sea level rise, S, must be considered for four regions.

_h.
VA = Aydh -B < h, <-S
-B


VB = VA (-S) Ay+ (h 3 dh -S < h, < 0
J-S L A




















0.050 -. 10 S
0 --.%, 1 BW,*




0.5




1.0

a) Non-Dimensional Volume b) Definition Sketch.
Redistributed vs. Non-Dimensional
Depth.




Figure 36. Definition Sketch and Non-Dimensional Volume Redistributed as a Function
Of Non-Dimensional Depth. Due to Sea Level Rise, S. Case of B/h, = 0.25,
S/B = 0.5.









-VC = VB(O) + -Ay+ ()3/2 ( )3/2 dh O O


yD = c(h-)+ -Ay+ d h, -S A.h,-S [_s A;l
(72)

The depth, he, at which the profiles cross will represent a maximum volume, Vma, and

is given by
h, = h, -Ay +h' 3/2 2/3 (7)
h =h -S (73)

and must be solved by iteration. The maximum volume, Vma, which occurs at he is


ma= -Ay (B + h ) _S W* S 3/(74)

The height A can be shown to be


A = h I + 2 1+ S- (75)

The above integration results can be cast into non- dimensional form with the following

non-dimensional parameters


Ay A = y







M h-
h'e = h
h*

A
h*



St S
B









Bi= B



S/ B (76)
WB

The non-dimensional volumes are


A = -Ay'(l+ -B'< h<--S'B' (77)



S= -Ay'(1+ (h' S 'B') SB < hB < 0 (78)



v = -A (l+2+- [(h)/2 -(, B)5/2 < h' < S'B' (79)




B'] 5B'E 5/ 5B
4 = -Ay'(1+ 5 \Bh)5/2' 5 1- <.^<1-I'B'+A'

(80)

It can be shown by substituting the value of A' from Eq. (75) and Ay from Eq. (12)

into Eq. (80), that V', (h' = 1 S'B' + A') = 0 as would be expected.

To illustrate the volume changes that would be determined by surveying to various

depths, h., consider the following example,


B' = B/h = 0.25

S' = S/B=0.5

For this case, the following results are obtained


Ay' A -0.10285
W,.


h', = he = 0.23947
h,









A
A' = =0.05520
h,



Jmax = VmaZ /(WB) = 0.11795

The variation of the cross-shore volumetric transport versus depth below the elevated

water level, S, is shown in Figure 36a for this example. It is seen as expected that the total

volumetric change at h = h, + A is zero.

It is of interest to compare the maximum volumetric change, Vmaz, with the volume

deficit associated with a sea level rise, S. The latter quantity is simply WS. Thus this

ratio, r, is
V ,,Vmax W* "Bmax
WS S'
and for our example
(0.11795)
r (0.11795) = 0.2359
0.5
Thus, the maximum volumetric survey error that could occur due to the surveys not ex-

tending to a sufficient depth is approximately 24% of the volume deficit associated with the

sea level rise.

It is also possible to develop approximate expressions describing the volumetric redis-

tribution. First, we approximate the non- dimensional recession (Eq. (12)) by

S S'
(h, + B)- + ( -)
which for our example yields Ay' = 0.10 vs the complete equation result of- 0.10285. The

approximate expression for h' is
4 2Ayr2
h = (81)
9S'B'
which for our case yields ht = 0.30089 versus the complete equation result of 0.23947.

The value of A' obtained from the approximate value of Ay' is A' = 0.05717 vs 0.05520

obtained from the complete equation. Finally, the maximum non-dimensional volume, Vmaz,

is 0.092036 versus 0.093497 obtained from the full equation. In general, it is seen that for

this example the approximations to the full equations are quite reasonable.









Trailing Beach Profile Signature Due to Sea Level Rise


If, as discussed by Bruun (1962), and as implied by Eq. (14), the beach profile moves

landward and upward in response to sea level rise, it is possible to infer a simple trailing

beach profile signature, which can then be compared with measured cross- continental shelf

profiles. The processes are complex due primarily to the landward boundary of the profile

and the implicit assumption in Eq. (14) that the sand is transported only offshore. These

potential shortcomings aside, Eq. (14) predicts that for each unit of vertical rise, the

landward retreat of every element on the equilibrium beach profile will be, W./(h. + B),

a value usually considered to be in the range of 50 to 100. Thus the trailing profile slope

should be the inverse of this ratio. A comprehensive investigation of profiles may provide

insight into conditions under which Eq. (14) is most valid and/or of the ratio W,/(h. + B).


SUMMARY AND CONCLUSIONS


Equilibrium beach profile concepts provide a useful basis for application to a number

of coastal engineering projects. In addition to addressing conditions at equilibrium, these

concepts establish a foundation for considering the response of profiles out of equilibrium.

Based on analysis of numerous profiles representing laboratory and field scales, a reason-

able approximate and useful form of a monotonic beach profile appears to be h(y) = Ay2/3

in which h is the water depth at a distance, y, offshore and A is a scale parameter depending

on sediment characteristics. A representation of the sediment scale parameter is presented

and provides a rational basis for assessing the relative merits of utilizing beach sand of

different sizes.

Methods are presented for quantifying the shoreline response due to elevated water levels

and wave heights on natural and seawalled shorelines. Additionally, results are presented for

calculating nourishment quantities for sand of uniform but arbitrary diameter. Depending

on volumes and sizes of sediment added, three types of profiles can occur: intersecting,

non-intersecting and submerged. The advantages of using coarser sand are quantified and









equations are presented expressing the volume of a particular sand size required to yield

a desired additional beach width. Many laboratory studies of beach profiles commence

with a planar slope which could be much steeper or coarser than the overall equilibrium

slope consistent with the sand size in the experiments. Applying equilibrium beach profile

concepts, it is shown that five profile types relative to the initial profile can occur. Three

of these types are erosional and two are accretional. Three of these profile types have

been identified in laboratory studies. Conditions under which profile type will occur are

quantified and all results including shoreline change are incorporated into a single graphical

representation.

The volumetric redistribution of sediment across the profile due to sea level rise is

examined in detail and compared with the total sediment "demand" as a result of the sea

level rise. An application is the possible error if the survey does not extend over the full

depth of effective motion. It is shown that the maximum error is only a fraction of the

sediment "demand".

The effects of sea level rise on nourishment needs are evaluated for cases with and without

onshore sediment transport across the continental shelf. It is shown that the sediment

volumes required to maintain a shoreline position vary directly with wave height and sea

level rise rates and inversely with profile slopes.

It is hoped that the results presented herein will provide guidance for coastal engineering

projects and serve as a framework for interpretation of project performance and behavior

of natural beach systems.


ACKNOWLEDGEMENTS


Support provided by the Sea Grant College Program of the National Oceanic and Atmo-

spheric Administration under Project R/C- S-22 is hereby gratefully acknowledged. This

support has enabled the author and students to pursue a range of problems concerned with

rational usage of the shoreline. Ms. Cynthia Vey and Lillean Pieter provided their usual









equations are presented expressing the volume of a particular sand size required to yield

a desired additional beach width. Many laboratory studies of beach profiles commence

with a planar slope which could be much steeper or coarser than the overall equilibrium

slope consistent with the sand size in the experiments. Applying equilibrium beach profile

concepts, it is shown that five profile types relative to the initial profile can occur. Three

of these types are erosional and two are accretional. Three of these profile types have

been identified in laboratory studies. Conditions under which profile type will occur are

quantified and all results including shoreline change are incorporated into a single graphical

representation.

The volumetric redistribution of sediment across the profile due to sea level rise is

examined in detail and compared with the total sediment "demand" as a result of the sea

level rise. An application is the possible error if the survey does not extend over the full

depth of effective motion. It is shown that the maximum error is only a fraction of the

sediment "demand".

The effects of sea level rise on nourishment needs are evaluated for cases with and without

onshore sediment transport across the continental shelf. It is shown that the sediment

volumes required to maintain a shoreline position vary directly with wave height and sea

level rise rates and inversely with profile slopes.

It is hoped that the results presented herein will provide guidance for coastal engineering

projects and serve as a framework for interpretation of project performance and behavior

of natural beach systems.


ACKNOWLEDGEMENTS


Support provided by the Sea Grant College Program of the National Oceanic and Atmo-

spheric Administration under Project R/C- S-22 is hereby gratefully acknowledged. This

support has enabled the author and students to pursue a range of problems concerned with

rational usage of the shoreline. Ms. Cynthia Vey and Lillean Pieter provided their usual









flawless typing and illustrative services, respectively. I appreciate the many discussions and

collegial support provided by present and former students.


REFERENCES


Aubrey, D.G. (1979) "Seasonal Patterns of Onshore/Offshore Sediment Movement", Jour-

nal of Geophysical Research, Vol. 84, No. C10, p. 6347-6354.

Aubrey, D.G., D.L. Inman and C.E. Nordstrom (1977) "Beach Profiles at Torrey Pines,

CA", Proceedings of the Fifteenth International Conference on Coastal Engineering,

American Society of Civil Engineers, p. 1297-1311.

Bagnold, R.A. (1946) "Motion of Waves in Shallow Water Interaction Between Waves

and Sand Bottoms", Proceedings Royal Society, Series A, Vol. 187, p. 1-15.

Bowen, A.J., D.L. Inman and V.P. Simmons (1968) "Wave Set-Down and Wave Set-Up",

Journal of Geophysical Research, Vol. 73, No. 8, p. 2569-2577.

Bruun, P. (1954) "Coast Erosion and the Development of Beach Profiles", Beach Erosion

Board, Technical Memorandum No. 44.

Bruun, P. (1962) "Sea Level Rise as a Cause of Shore Erosion", Journal of Waterway, Port,

Coastal and Ocean Engineering, American Society of Civil Engineers, Vol. 88, No.

117,

Chiu, T.Y. (1977) "Beach and Dune Response to Hurricane Eloise of September 1975",

Proceedings, American Society of Civil Engineers, Specialty Conference on Coastal

Sediments, '77, p. 116-134.

Dalrymple, R.A. and W.W. Thompson (1976) "Study of Equilibrium Profiles", Proceed-

ings, Fifteenth International Conference on Coastal Engineering, American Society of

Civil Engineers, p. 1277-1296.









Dean, R.G. (1973) "Heuristic Models of Sand Transport in the Surf Zone", Proceedings of

Conference on Engineering Dynamics in the Surf Zone, Institution of Civil Engineers,

Australia, Sydney, p. 208-214.

Dean, R.G. (1977) "Equilibrium Beach Profiles: U.S. Atlantic and Gulf Coasts", Depart-

ment of Civil Engineering, Ocean Engineering Report No. 12, University of Delaware,

Newark DE.

Dean, R.G. (1987a) "Coastal Sediment Processes: Toward Engineering Solutions", Pro-

ceedings, American Society of Civil Engineers, Specialty Conference on Coastal Sedi-

ment '87, p. 1-24.

Dean, R.G. (1987b) "Additional Sediment Input to the Nearshore Region", Shore and

Beach, Vol. 55, Nos. 3-4, p. 76-81.

Dean, R.G. and E.M. Maurmeyer (1983) "Models for Beach Profile Response", Chapter 7

in CRC Press, Inc., P.D. Komar, Editor, Boca Raton, FL, p. 151-166.

Eagleson, P.S., B. Glenne and J.A. Dracup (1963) "Equilibrium Characteristics of Sand

Beaches", Journal of Hydraulics Division, American Society of Civil Engineers, Vol.

89, No. HY1, p. 35-57.

Hayden, B., W. Felder, J. Fisher, D. Resio, L. Vincent and R. Dolan (1975) "Systematic

Variations in Inshore Bathymetry", Department of Environmental Sciences, Technical

Report. No. 10, University of Virginia, Charlottesville, VA.

Hoffman, J.S., D. Keyes and J.G. Titus (1983) "Projecting Sea Level Rise; Methodology,

Estimates to the Year 2100 and Research Needs", Washington, D.C., U.S. Environ-

mental Protection Agency.

Hughes, S.A. (1983) "Movable Bed Modeling Law for Coastal Dune Erosion", Journal of

Waterway, Port, Coastal and Ocean Engineering, Vol. 109, No. 2, p. 164-179.









Keulegan, G.H. and W.C. Krumbein (1919) "Stable Configuration of Bottom Slope in

a Shallow Sea and Its Bearing on Geological Processes", Transactions of American

Geophysical Union, Vol. 30, No. 6, p. 855-861.


Kriebel, D.L. (1982) "Beach and Dune Response to Hurricanes", M.S. Thesis, Department

of Civil Engineering, University of Delaware, Newark, DE.


Kriebel, D.L. (1986) "Verification Study of a Dune Erosion Model", Shore and Beach, Vol.

54, No. 3, p. 13-21.

Kriebel, D.L. and R.G. Dean (1984) "Beach and Dune Response to Severe Storms", Pro-

ceedings, Nineteenth International Conference on Coastal Engineering, American So-

ciety of Civil Engineers, p. 1584-1599.


Kriebel, D.L. and R.G. Dean (1985) "Numerical Simulation of Time-Dependent Beach and

Dune Erosion", Coastal Engineering, Vol. 9, No. 3, p. 221-245.


Larson, M. (1988) "Quantification of Beach Profile Change", Report No. 1008, Depart-

ment of Water Resources Engineering, Lund University, Lund, Sweden.

Larson, M. and N.C. Kraus (1989) "SBEACH: Numerical Model for Simulating Storm-

Induced Beach Change Report 1: Empirical Foundation and Model Development",

Technical Report CERC-89-9, Coastal Engineering Research Center, Waterways Ex-

periment Station, Vicksburg, Mississippi.

Moore, B.D. (1982) Beach Profile Evolution in Response to Changes in Water Level

and Wave Height", Masters Thesis, Department of Civil Engineering, University of

Delaware.


Noda, E.K. (1972) "Equilibrium Beach Profile Scale Model Relationship", Journal Water-

ways, Harbors and Coastal Engineering Division, American Society of Civil Engineers,

p. 511-528.


I









Phillips, O.M. (1966) "The Dynamics of the Upper Ocean", Cambridge University Press,

Cambridge, England.

Saville, T. ('957) "Scale Effects in Two-Dimensional Beach Studies", Transactions of the

Seventh General Meeting of the International Association of Hydraulic Research, Vol.

1, p. A3-1 to A3-10.

Stive, M.J.F. and H.G. Wind (1986) "Cross-Shore Mean Flow in the Surf Zone", Coastal

Engineering, Vol. 10, No. 4, p. 325- 340.

Suh, K. and R.A. Dalrymple (1988) "Expression for Shoreline Advancement of Initially

Plane Beach", Journal of Waterway, Port, Coastal and Ocean Engineering, American

Society of Civil Engineers, Vol. 114, No. 6, p. 770-777.

Sunamura, T. and K. Horikawa (1974) "Two-Dimensional Beach Transformation Due to

Waves", Proceedings, Fourteenth International Conference on Coastal Engineering,

American Society of Civil Engineers, p. 920-938.

Swart, D.H. (1974) "A Schematization of Onshore-Offshore Transport", Proceedings, Four-

teenth International Conference on Coastal Engineering, American Society of Civil

Engineers, p. 884-900.

Svendsen, I.A. (1984) "Mass Flow and Undertow in a Surf Zone", Coastal Engineering,

Vol. 8, p. 347-365.

Van Hijum, E. (1975) "Equilibrium Profiles of Coarse Material Under Wave Attack", Pro-

ceedings of the Fourteenth International Conference on Coastal Engineering, American

Society of Civil Engineers, p. 939-957.

Vellinga, P. (1983) "Predictive Computational Model for Beach and Dune Erosion During

Storm Surges", Proceedings American Society of Civil Engineers, Specialty Conference

on Coastal Structures '83, P. 806-819.










Weishar, L.L. and W.L. Wood (1983) "An Evaluation of Offshore and Beach Changes on

a Tideless Coast", Journal of Sedimentary Petrology, Vol. 53, No. 3, p. 847-858.


Winant, C.D., D.L. Inman and C.E. Nordstrom (1975) "Description of Seasonal Beach

Changes Using Empirical Eigenfunctions", Journal of Geophysical Research, Vol. 80,

No. 15, p. 1979-1986.


Zenkovich, V.P. (1967) "Processes of Coastal Development", Editor: J. A. Steers, Oliver

and Boyd, Edinburgh.


L




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