• TABLE OF CONTENTS
HIDE
 Front Cover
 Report documentation page
 Title Page
 Table of Contents
 List of Figures
 Notation
 Introduction
 Background
 Characteristics of realistic beach...
 Equilibrium beach profiles and...
 Cross-shore transport models
 Application of model to predict...
 Prediciton of beach and dune erosion...
 Other applications and publica...
 Summary and conclusions
 Acknowledgments
 Appendix I. References






Group Title: UFL/COEL (University of Florida. Coastal and Oceanographic Engineering Laboratory) ; 83/007
Title: Shoreline erosion due to extreme storms and sea level rise
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Permanent Link: http://ufdc.ufl.edu/UF00076160/00001
 Material Information
Title: Shoreline erosion due to extreme storms and sea level rise
Series Title: UFLCOEL
Physical Description: viii, 58 p. : ill. ; 28 cm.
Language: English
Creator: Dean, Robert George ( Originator )
University of Florida -- Coastal and Oceanographic Engineering Dept
Publisher: University of Florida, Coastal and Oceanographic Engineering Dept.
Place of Publication: Gainesville
Publication Date: 1983
 Subjects
Subject: Storm surges -- Mathematical models   ( lcsh )
Hurricanes   ( lcsh )
Beach erosion   ( lcsh )
Coastal and Oceanographic Engineering thesis M.S
Coastal and Oceanographic Engineering -- Dissertations, Academic -- UF
Genre: federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )
 Notes
Bibliography: Bibliography: p. 56-58.
Funding: Summary of research results developed under
Statement of Responsibility: by R.G. Dean.
General Note: Cover title.
General Note: "December 1983"
 Record Information
Bibliographic ID: UF00076160
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 - 10598838

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Table of Contents
    Front Cover
        Front Cover
    Report documentation page
        Unnumbered ( 2 )
    Title Page
        Title Page
    Table of Contents
        Table of Contents 1
        Table of Contents 2
    List of Figures
        List of Figures 1
        List of Figures 2
        List of Figures 3
    Notation
        Unnumbered ( 9 )
        Unnumbered ( 10 )
    Introduction
        Page 1
        Page 2
    Background
        Page 2
        Page 3
        Page 4
        Page 5
    Characteristics of realistic beach profiles
        Page 6
        Page 5
    Equilibrium beach profiles and applications
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
    Cross-shore transport models
        Page 13
        Page 14
        Page 15
        Page 16
        Page 12
    Application of model to predict beach prfile response to various forcing functions
        Page 17
        Page 18
        Page 19
        Page 16
        Page 20
        Page 21
    Prediciton of beach and dune erosion due to severe storms
        Page 22
        Page 23
        Page 24
        Page 25
        Page 21
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
    Other applications and publications
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 44
    Summary and conclusions
        Page 54
        Page 55
    Acknowledgments
        Page 55
    Appendix I. References
        Page 56
        Page 57
Full Text




UFL/COEL-83/007






SHORELINE EROSION DUE TO EXTREME STORMS
AND SEA LEVEL RISE










By



R. G. Dean


December 1, 1983








REPORT DOCUMENTATION PAGE
1. Report No. 2. 3. Recipient's Accession No.


-. Title sad Subtitle 5. Report Date
SHORELINE EROSION DUE TO EXTREME STORMS AND SEA December 1, 1983
LEVEL RISE 6.

7. Author(s) 8. Performing Organization Report No.
R. G. Dean UFL/COEL-83/007
9. Performing Organization Name and Address 10. Project/Task/Uork Unit No.
Coastal and Oceanographic Engineering Department
University of Florida 11. Contract or crant No.
336 Well Hall R/T-24
Gainesville, Florida 32611 13. Type of Report
Li. Sponsoring Organization Name and Address

Sea Grant Program Final
University of Delaware
Newark, Delaware 19711

15. Supplementary Notes



16. Abstract
A summary is presented of research conducted on beach erosion associated with
extreme storms and sea level rise. These results were developed by the author and
graduate students under sponsorship of the University of Delaware Sea Grant Program.
Various shoreline response problems of engineering interest are examined. The
basis for the approach is a monotonic equilibrium profile of the form h = Ax2/3 in
which h is water depth at a distance x from the shoreline and A is a scale parameter
depending primarily on sediment characteristics and secondarily on wave
characteristics. This form is shown to be consistent with uniform wave energy
dissipation per unit volume. The dependency of A on sediment size is quantified
through laboratory and field data. Quasi-static beach response is examined to
represent the effect of sea level rise. Cases considered include natural and sea-
walled profiles.
To represent response to storms of realistic durations, a model is proposed in
which the offshore transport is proportional to the "excess" energy dissipation per
unit volume. The single rate constant in this model was evaluated based on large
scale wave tank tests and confirmed with Hurricane Eloise pre- and post-storm
surveys. It is shown that most hurricanes only cause 10% to 25% of the erosion
potential associated with the peak storm tide and wave conditions. Additional
applications include profile response employing a fairly realistic breaking model in
which longshore bars are formed and long-term (500 years) Monte Carlo simulation
including the contributions due to sea level rise and random storm occurrences.

17. Originator's Key Words 18. Availability Statement
Shoreline response
Beach erosion
Sea level rise
Hurricane effects

19. U. S. Security Classif. of the Report 20. U. S. Security Classif. of This Page 21. No. of pages 22. Price
Unclassified Unclassified
















SHORELINE EROSION DUE TO EXTREME STORMS AND SEA LEVEL RISE


By

R. G. Dean





Summary of Research Results Developed Under

University of Delaware Sea Grant Project R/T-24





December 1, 1983





















Coastal and Oceanographic Engineering Department

University of Florida

Gainesville, Florida 32611










TABLE OF CONTENTS

Page

LIST OF FIGURES............................................. iv

NOTATION........................................... .......... vii

I. INTRODUCTION.............................................. 1

II. BACKGROUND.................................................... 2

III. CHARACTERISTICS OF REALISTIC BEACH PROFILES.................. 5

General Geometric Profile Characteristics..................... 5

Shape...................................................... 5

Form....................................................... 5

Scale................................. ....... ... ....... 5


Sorting.................................... ..... .... ..... 5

Effect of Water Level Changes................................ 6

IV. EQUILIBRIUM BEACH PROFILES AND APPLICATIONS................... 6

V. CROSS-SHORE TRANSPORT MODELS .................................. 12

VI. APPLICATION OF MODEL TO PREDICT BEACH PROFILE RESPONSE TO
VARIOUS FORCING FUNCTIONS.................................. 16

VII. PREDICTION OF BEACH AND DUNE EROSION DUE TO SEVERE STORMS..... 21

Profile Schematization...................................... 21

Governing Equations......................................... 29

Method of Solution of Finite Difference Equations............. 29

Application of Method to Computation of Idealized Beach
Response........... ............ .............................. 30

Response to Static Increased Water Level................... 30

Effects of Various Wave Heights............................ 33

Effects of Various Storm Tide Levels................... .... 33

Effect of Sediment Size on Berm Recession.................. 33

Effect of Storm Duration................................... 33










Application of Method to Long-Term Beach and Dune Response
Simulations......................... .............. .......... .... 37

Evaluation of Method by Hurricane Eloise Erosion Data...... 37

Long-Term Simulation..................................... 39

VIII. OTHER APPLICATIONS AND PUBLICATIONS.......................... 44

IX. SUMMARY AND CONCLUSIONS ...................................... 54

X. ACKNOWLEDGMENTS............................................... 55

APPENDIX I REFERENCES..................................... 56










LIST OF FIGURES


Figure Page


1. Location map of the 502 profiles used in the analysis
(from Hayden, et al., 1975)................................... 7

2. Characteristics of dimensionless beach profile h (-
for various m values (from Dean, 1977)............. ... 8

3. Equilibrium beach profiles for sand sizes of 0.2 mm and
0.6 mm A(D = 0.2 mm) = 0.1 m1/3, A(D = 0.6 mm) = 0.20 m1/3.... 9

4. Histogram of exponent m in equation h = Axm for 502 United
States East Coast and Gulf of Mexico profiles (from Dean,
1977).................. ............. .................... 11

5. Beach profile factor A, vs sediment diameter, D, in
relationship h = Ax2/3 (modified from Moore, 1982)............ 13

6. 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 Moore, 1982)............. 14

7. Profile P10 from Zenkovich (1967). Near the end of a spit
in Western Black Sea. Whole and broken shells.
A = 0.25 ml/3 (from Moore, 1982).............................. 14

8. Profile from Zenkovich (1967). Eastern Kamchatka. Mean
sand diameter: 0.25 mm. Least squares value of
A = 0.07 m1/3 (from Moore, 1982).............................. 15

9. Model simulation of a 0.5 meter sea level rise and beach
profile response with a relatively mild sloping beach
(from Moore, 1982)...................................... 17

10. Effect of varying the sediment transport rate coefficient
on cumulative erosion during the simulation of Saville's
(1957) laboratory investigation of beach profile evolution
for a 0.2 mm sand size (from Moore, 1982).................... 18

11. Model simulation of a 0.5 meter sea level rise and beach
profile response with a steep 2 meter berm (from Moore,
1982) ........... ................................ ......... 19

12. Model simulation of beach nourishment for a 0.15 mm sand
size (from Moore, 1982)....................................... 20

13. Comparison of the beach profile from the model and Saville's
laboratory 0.2 mm sand after 25 hours (from Moore, 1982)...... 22










14. Initial (January 21, 1980) and final (December 20, 1980)
measured and predicted beach profiles. Leadbetter Beach
Santa Barbara, California (from Moore, 1982)................. 23

15. Model versus prototype for beach profile envelopes (i.e.,
maximum and minimum water depths at each location along
the beach profile) for the entire year (1980) (from Moore,
1982) ...... ............ ................ ................. .. 24

16. Comparison of Eigenfunction analysis results performed on
the predicted and measured beach profiles: First
Eigenfunction, mean-beach function (from Moore, 1982)......... 25

17. Comparison of Eigenfunction analysis results performed on
the predicted and measured beach profiles: Second
Eigenfunction, bar-berm function (from Moore, 1982)............ 26

18. Comparison of Eigenfunction analysis results performed on
the predicted and measured beach profiles: Third
Eigenfunction, terrace function (from Moore, 1982)............ 27

19. Model representation of beach profile, showing depth and
transport relation to grid definitions (from Kriebel, 1982)... 28

20. Characteristic form of berm recession versus time for
increased static water level (from Kriebel, 1982)............. 31

21. Comparison of asymptotic berm recession from model (--)
and as calculated by Eq. (20) (0* ) ........................... 32

22. Effect of breaking wave height on berm recession (from
Kriebel, 1982)................................................ 34

23. Effect of static storm surge level on berm recession (from
Kriebel, 1982)................................................ 35

24. Effect of sediment size on berm recession (from Kriebel,
1982) ..........................o.............................. 36

25. Comparison of the effects of 12, 24, and 36 hrs. storm
surge on volumetric erosion (from Kriebel, 1982).............. 38

26. Flow diagram of N-year simulation of hurricane storm surge
and resulting beach erosion (from Kriebel, 1982)............ 41

27. Average frequency curve for dune recession, developed by
Monte Carlo simulation, Bay-Walton Counties, Florida
(from Kriebel, 1982)....................................... 42

28. Probability or risk of dune recession of given magnitude
occurring at least once in N-years, Bay-Walton Co.,
Florida (from Kriebel, 1982) ................................. 43










29. Nourishment volumes required versus effective wave height
for various native and filled sediment characteristics and
considerations, shoreline advancement = 300 ft (from
Maurmeyer and Dean, 1980) .................................... 45

30. Relationship between volume of sand stored in bar versus
wave height above that required for incipient bar
formation (from Dean, 1983)................................... 46

31. Suggested variation of K with gHb/w2, prototype and
laboratory data (from Dean, 1983).......................... 48

32. Perched beach, demonstrating nourishment volumes saved
(from Dean, 1983)............................................. 49

33. Isolines of dimensionless berm recession, R', vs
dimensionless storm breaking depth hb and dimensionless
storm tide, S', m = 2/3 (from Dean and Maurmeyer, 1983)....... 50

34. Isolines of dimensionless seawall toe scour, h' vs
dimensionless storm tide, S', and dimensionless breaking
depth h' (from Dean and Maurmeyer, 1983).................... 50

35. Generalized shoreline response model due to sea level rise.
Applicable for a barrier island system which maintains its
form relative to the adjacent ocean and lagoon water
levels (Dean and Maurmeyer, 1983)............................. 52

36. 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.................................... 53










NOTATION


A,A' Scale parameter in equilibrium beach profile expression, see
Eq. (10)
B Berm height
D Sediment particle diameter
V Wave energy dissipation rate per unit volume
V* Equilibrium wave energy dissipation rate per unit volume
F Fall velocity parameter, also "function of"
g Gravitational constant
Hb Breaking wave height
Ho Deep water wave height
h Water depth
hb Breaking depth
hl,h2 Depth dimensions to features of perched beach, see Figure 32
hb2 Breaking depth under storm surge conditions
hw1 Water depth at toe of seawall under normal conditions
K,K* Rate constants for offshore sediment transport
K' Longshore sediment transport proportionality factor
kD Constant defined by Eq. (17)
L, Deep water wave length
Lj,L2 Positions of contours L1 and L2
m Shape parameter in equilibrium beach profile, see Eq. (7)

Ps Longshore component of wave energy flux at breaking
p Barometric pressure
Qs Offshore sediment transport rate per unit width
R Beach recession

Rmax Radius to maximum winds in a hurricane
S Water level increase, such as storm tide
T Wave or storm tide period
t Time

VF Translational speed of a hurricane system
W Equilibrium distance between two beach profile contours, also
barrier island width, see Figure 35

W2 Breaking zone width under storm surge conditions











w Sediment fall velocity

YF Landfall location of a hurricane center
x Offshore coordinate
xn Offshore coordinate to nth contour

x1,x2 Horizontal dimensions to features of perched beach, see Figure 32
a Rate constant
B Translation direction of hurricane system
K Proportionality factor in spilling breaker model
a Angular frequency (= 27/T)
6 Average vertical angle over active portion of beach profile
p Mass density of water
Y Specific weight of water (= pg)
I Numerical constant = 3.14159....
n Storm tide


i4 4 4










I. INTRODUCTION

The purpose of this report is to summarize research results
conducted under University of Delaware Sea Grant sponsorship of Project
R/T-24 "Shoreline Erosion to Extreme Storms and Sea Level Rise".

The motivation for this project arose from the recognition of the
need for difficult future decisions with inadequate data/knowledge
relative to shoreline erosion, shoreline development consequences and/or
remedial erosion measures. Much of the shoreline has been developed
with the construction of expensive and substantial upland structures.
The attraction for the placement of these structures included the beauty
and recreational advantages of the beaches. In an era of gradual sea
level rise and associated inexorable erosional trend, the beaches recede
at an average rate of 30 cm to 1 m per year. Single storm events can
cause dune erosion of 30 to 100 meters, depending on the severity and
the degree of instability of the beaches. This ultimately presents the
shorefront property owner or other responsible individual/agency with
three choices: (a) abandon the shoreline, (b) armor the shoreline in
which case the beaches will gradually disappear, or (3) carry out fairly
expensive beach nourishment programs.

As noted, the capability to provide the engineering and economic
data to develop rational responses to the situations discussed above was
clearly inadequate. It was difficult to partition erosion occurring to
natural causes or human-related activities. Moreover, even if the
characteristics of a storm were known precisely, only rudimentary
approaches were available to predict the resulting erosion and the rate
of recovery following the storm. The potential of this problem has been
exacerbated by the predictions resulting from a recent comprehensive EPA
study in which the rate of sea level rise over the next century is
estimated to be between 10 to 30 times that occurring in the last century.

The strategy followed in the research project has been to develop a
quantitative understanding of the mechanisms governing sediment
transport processes and to formulate the understanding into numerical
schemes that can be applied to realistic situations. The problem is
complex and has resisted attempts of complete understanding. However,
it is believed that substantial progress has been made and the basis has










been developed for numerous applications and also for future effective
research programs.

The author of this report has been fortunate to have had the
interest, insight, and motivation of the graduate students that
contributed most significantly to this project. These included: David
Kriebel, Brett Moore, Bill Dally, Osman Borekci, and Peter Williams.


II. BACKGROUND

A complete review of all substantial previous efforts that have
been directed toward equilibrium profiles and beach profile evolution is
beyond the scope of the present report; however, several closely related
studies will be discussed briefly.

Bruun (1954) analyzed beach profiles along the Danish North Sea

Coast and Mission Bay, California and found the following empirical
relationship for the water depth, h, at an offshore distance, x,


h(x) = Ax2/3 (1)


where A is a scale parameter. Bruun proposed two mechanisms responsible
for the equilibrium beach profile. The first considered the onshore
component of shear stress to be uniform and the onshore component of the
gradient of transported wave energy to be constant. This resulted in an
approximate equation of the form found empirically (Eq. (1)). The
second mechanism was based on the consideration that the loss of wave
energy is due only to bottom friction and that the loss per unit area is
constant. A nonlinear wave theory was used with laboratory determined
friction factors leading to the following

x2/3
h(x) = A' _9 (2)
T

where T is the wave period.

Bruun (1962) considered long-term erosion and proposed the
following simple relationship expressing the beach recession, R, in
terms of the increase in sea level, S, and the average beach slope,
tanO, out to the location of limiting motion










been developed for numerous applications and also for future effective
research programs.

The author of this report has been fortunate to have had the
interest, insight, and motivation of the graduate students that
contributed most significantly to this project. These included: David
Kriebel, Brett Moore, Bill Dally, Osman Borekci, and Peter Williams.


II. BACKGROUND

A complete review of all substantial previous efforts that have
been directed toward equilibrium profiles and beach profile evolution is
beyond the scope of the present report; however, several closely related
studies will be discussed briefly.

Bruun (1954) analyzed beach profiles along the Danish North Sea

Coast and Mission Bay, California and found the following empirical
relationship for the water depth, h, at an offshore distance, x,


h(x) = Ax2/3 (1)


where A is a scale parameter. Bruun proposed two mechanisms responsible
for the equilibrium beach profile. The first considered the onshore
component of shear stress to be uniform and the onshore component of the
gradient of transported wave energy to be constant. This resulted in an
approximate equation of the form found empirically (Eq. (1)). The
second mechanism was based on the consideration that the loss of wave
energy is due only to bottom friction and that the loss per unit area is
constant. A nonlinear wave theory was used with laboratory determined
friction factors leading to the following

x2/3
h(x) = A' _9 (2)
T

where T is the wave period.

Bruun (1962) considered long-term erosion and proposed the
following simple relationship expressing the beach recession, R, in
terms of the increase in sea level, S, and the average beach slope,
tanO, out to the location of limiting motion











R 1
T-tta-- (3)

This relation is based on the beach profile remaining the same in
relation to the rising sea level.

Eagleson, et al. (1963) considered a balance between fluid and
gravitational forces and developed a relationship for the equilibrium
profile seaward of the surf zone. Comparison of the results with
laboratory measurements were encouraging.

Edelman (1970) has developed geometric procedures for calculating
the shoreline response due to storms. Basically, it is assumed that the
response time of the beach is short relative to the storm time scale
such that the beach profile relative to the instantaneous water level is
the same as the initial (equilibrium) profile. Comparison of
predictions with measured storm erosion (Chiu, 1981) shows that this
method seriously overestimates the erosion (by factors of 4-10).

Hayden, et al. (1975) applied an empirical eigenfunction method of
analysis to identify characteristic forms of 504 beach profiles along
the Atlantic and Gulf of Mexico shorelines. This method has been
extended and applied by Aubrey (1979) for California beaches. Moreover,
Aubrey has correlated the various eigenfunctions with wave
characteristics in an attempt to develop a predictive approach to beach
profile response. The eigenfunction approach is purely empirical and
does not address the processes associated with beach profile forms and
mechanics of evolution.

Swart (1974) has conducted numerous laboratory studies of beach
profiles and has analyzed these and other relevant data. The results
for equilibrium beach profiles were presented by complicated empirical
relationships. A method was also presented for beach profile evolution
in which the offshore sediment transport, Qs, was presented as


Qs = K*[(L1- L2)t- W (4)










in which K* is a rate constant, W is the equilibrium spacing for two
contours under consideration and (L2 Li)t is the actual time-varying
spacing of those two contours.

On the basis of a heuristic argument, Dean (1973) has proposed a
fall velocity parameter, F, as significant in beach processes,

Hb
F -- (5)
wT

in which Hb is the breaking wave height, w is the fall velocity of the
sediment and T is the wave period. Consideration of bar formation
further leads to the following two parameters

Hb
Wave Steepness: --
0 (6)

Dimensionless Fall Velocity: 2.


where Lo is deep water wave length and g is the gravitational
constant. Comparison of 189 experiments showed that the parameters in
Eq. (6) successfully identified conditions for which bars were formed
for 89% of the data.

Hughes (1983) has proposed a scale relationship for physical models
for beach and dune erosion. The relationship is based on equivalence of
a fall velocity parameter in model and prototype and ratio of inertia to
gravity forces. The modeling requirements allow for model distortion
and include a geomorphological time scale. These relationships were
evaluated against and compare favorably with dune erosion documented as
a result of Hurricane Eloise in 1975.

Based on a series of small and large scale model tests, Vellinga
(1982, 1983) has proposed a predictive procedure that has been shown to
agree reasonably well with measured post-storm profiles associated with
the 1953 and 1976 events in Holland. A reference profile is established
for particular hydrographic and sediment characteristics. The results
are then extended to "non-reference" conditions developed from the model
studies. The method is completely empirical and strictly applicable
only to a constant surge level over a five hour duration. Approximate










methods are presented accounting for storm durations in excess of five
hours. Through modeling relationships, the effects of sediment size are
taken into account.

van de Graaff (1983) has incorporated the methodology of Vellinga
into a procedure for predicting the probability of dune erosion. The
erosion is considered to be the result of seven independent parameters,
each with a probability distribution of known characteristics. Based on
the known probability characteristics of each of the seven variables
contributing to the dune recession, two methods are presented for
establishing the return period dune recession relationship.


III. CHARACTERISTICS OF REALISTIC BEACH PROFILES

Beach profiles in nature are complex and dynamic, always changing
due to altered conditions of tides, waves, winds, currents, or sediment
supply. However, when considering many beach profiles, patterns emerge
that are indicative of the general relationship to the different
variables. Some of these characteristics and general response features
are discussed below.
General Geometric Profile Characteristics

Shape Beach profiles are generally characterized by a concave
upward geometry. Thus the profile is steeper in the.shallow water
depths with the milder slopes occurring offshore. The beach face formed
by the uprush and backwash of the waves is usually nearly planar.

Form Beach profiles can be monotonic or may include one or more
bars offshore. Usually storm waves will cause a bar to form which
thereafter positions the larger breaking waves. Subsequent milder waves
will cause the bar to move ashore in one or more sand "packets" termed
"ridge and runnel systems".

Scale The scale of beach profiles depends to a great extent on
the sediment comprising the profile. Coarse sediment will form a
steeper profile with a lesser tendency for bar formation than beaches
composed of finer sediment.

Sorting Waves are effective sorting agents tending to transport
and deposit the coarser material in shallow water and depositing the










finer portions offshore. In some cases the beach profile includes the
presence of a "step" feature at the base of the beach face. The
coarsest material in the beach profile tends to collect at the base of
the step which is a high energy environment due to the energy
dissipation associated with the backwash. In cases where shell is
present, the base of the step may be composed almost entirely of shell
hash. Bars tend to contain the finer fraction of material available
with the coarser fraction remaining as a lag product on the beaches.
Winter bars form offshore of many California beaches leaving a cobble
beach surface. During the milder summer months, the bars migrate ashore
and a sand beach is formed again.
Effect of Water Level Changes

Many storms are accompanied by increases in mean water level due to
storm surge and/or wave set-up. Additionally, some historically
damaging storms have coincided with extreme astronomical tides. The
effect of an increased water level is to cause the beach profile to be
out of equilibrium and to increase the erosional potential of the storm
waves. The increased water level affects the beach, on a short-term
basis, in the same manner as sea level rise does on a long-term basis.


IV. EQUILIBRIUM BEACH PROFILES AND APPLICATIONS

A number of theories have been advanced attempting to describe the
properties of and mechanisms associated with equilibrium beach profiles.

In the early phases of this study, a data set was located which
consisted of more than 500 beach profiles ranging from the eastern tip
of Long Island to the Texas-Mexico border, see Figure 1. Three fairly
simple possible mechanisms were investigated relating the depth, h, to
the distance offshore, x. Each of these three models predicted a
profile of the following form


h(x) =-Axm (7)


in which A and m are scale and shape parameters, respectively. Figure 2
presents normalized beach profiles for various m values. It is seen
that for m < 1, the profile is concave upward as commonly found in
nature. Figure 3 demonstrates the effect of the scale parameter, A.










methods are presented accounting for storm durations in excess of five
hours. Through modeling relationships, the effects of sediment size are
taken into account.

van de Graaff (1983) has incorporated the methodology of Vellinga
into a procedure for predicting the probability of dune erosion. The
erosion is considered to be the result of seven independent parameters,
each with a probability distribution of known characteristics. Based on
the known probability characteristics of each of the seven variables
contributing to the dune recession, two methods are presented for
establishing the return period dune recession relationship.


III. CHARACTERISTICS OF REALISTIC BEACH PROFILES

Beach profiles in nature are complex and dynamic, always changing
due to altered conditions of tides, waves, winds, currents, or sediment
supply. However, when considering many beach profiles, patterns emerge
that are indicative of the general relationship to the different
variables. Some of these characteristics and general response features
are discussed below.
General Geometric Profile Characteristics

Shape Beach profiles are generally characterized by a concave
upward geometry. Thus the profile is steeper in the.shallow water
depths with the milder slopes occurring offshore. The beach face formed
by the uprush and backwash of the waves is usually nearly planar.

Form Beach profiles can be monotonic or may include one or more
bars offshore. Usually storm waves will cause a bar to form which
thereafter positions the larger breaking waves. Subsequent milder waves
will cause the bar to move ashore in one or more sand "packets" termed
"ridge and runnel systems".

Scale The scale of beach profiles depends to a great extent on
the sediment comprising the profile. Coarse sediment will form a
steeper profile with a lesser tendency for bar formation than beaches
composed of finer sediment.

Sorting Waves are effective sorting agents tending to transport
and deposit the coarser material in shallow water and depositing the










finer portions offshore. In some cases the beach profile includes the
presence of a "step" feature at the base of the beach face. The
coarsest material in the beach profile tends to collect at the base of
the step which is a high energy environment due to the energy
dissipation associated with the backwash. In cases where shell is
present, the base of the step may be composed almost entirely of shell
hash. Bars tend to contain the finer fraction of material available
with the coarser fraction remaining as a lag product on the beaches.
Winter bars form offshore of many California beaches leaving a cobble
beach surface. During the milder summer months, the bars migrate ashore
and a sand beach is formed again.
Effect of Water Level Changes

Many storms are accompanied by increases in mean water level due to
storm surge and/or wave set-up. Additionally, some historically
damaging storms have coincided with extreme astronomical tides. The
effect of an increased water level is to cause the beach profile to be
out of equilibrium and to increase the erosional potential of the storm
waves. The increased water level affects the beach, on a short-term
basis, in the same manner as sea level rise does on a long-term basis.


IV. EQUILIBRIUM BEACH PROFILES AND APPLICATIONS

A number of theories have been advanced attempting to describe the
properties of and mechanisms associated with equilibrium beach profiles.

In the early phases of this study, a data set was located which
consisted of more than 500 beach profiles ranging from the eastern tip
of Long Island to the Texas-Mexico border, see Figure 1. Three fairly
simple possible mechanisms were investigated relating the depth, h, to
the distance offshore, x. Each of these three models predicted a
profile of the following form


h(x) =-Axm (7)


in which A and m are scale and shape parameters, respectively. Figure 2
presents normalized beach profiles for various m values. It is seen
that for m < 1, the profile is concave upward as commonly found in
nature. Figure 3 demonstrates the effect of the scale parameter, A.









































425


500


Figure 1. Location map of the 502 profiles used in the analysis (from
Hayden, et al., 1975).


















0.2
hb h(x)

DEFINITION SKETCH / \
0.4 -

h/hb //

0.6 -


NOTE:
0.8- (a) FOR m IS CONCAVE UPWARD
(b) FOR m=l, BEACH IS
OF UNIFORM SLOPE
m=0 (c) FOR m>I, BEACH IS
CONVEX UPWARD
1.0 I I I
1.0 0.8 0.6 0.4 0.2 0
x/W


m
Figure 2. Characteristics of dimensionless beach profile =
for various m values (from Dean, 1977)
for various m values (from Dean, 1977).

















DISTANCE OFFSHORE (m)


D=0.6


Equilibrium beach profiles for sand sizes of 0.2 mm and 0.6 mm
A(D = 0.2 ) = 0.1 m/3, A(D = 0.6 mm) = 0.20 1/3
A(D = 0.2 mm) = 0.1 m A(D = 0.6 mm) = 0.20 m


Figure 3.










The three models and the associated values of m are:


Model m
1. Uniform Wave Energy Dissipation Per Unit Surface Area 0.40
2. Uniform Longshore Shear Stress 0.40
3. Uniform Wave Energy Dissipation Per Unit Water Volume 0.667
The derivations of each of the three models considered spilling
breaking conditions, i.e. within the surf zone, the wave height, H, is
proportional to the depth, h, through a proportionality constant, K,
i.e.


H = Kh (8)


where K is usually taken as 0.78.

The data from the 502 wave profiles were evaluated employing a
least squares procedure to determine the A and m values for each of the
profiles. The results of this analysis strongly supported a value of
m = 0.667, (see Figure 4) i.e. the value associated with uniform wave
energy dissipation per unit volume and as found earlier by Bruun
(1954). The physical explanation associated with this mechanism is as
follows. As the wave propagates through the surf zone, coherent wave
energy is converted to turbulent energy by the breaking process. This
turbulent energy is manifested as eddy motions of the water particles,
thus affecting the stability of the bed material. Any model must
acknowledge that a particular sand particle is acted on by constructive
and destructive forces. The model here addresses directly only the
destructive (destabilizing) forces. It was reasoned that the parameter
A depends primarily on sediment properties, and secondarily on wave
characteristics, i.e.


A = F(Sediment Properties, Wave Characteristics) (9)


where "F()" denotes "function of" and it would be desirable to combine
wave and sediment characteristics to form a single dimensionless
parameter.












I I

PREDICTED VALUE (0.4)
BASED ON UNIFORM STRE
OR ENERGY DISSIPATION
ACROSS THE
SURF ZONE--


0.2


0.4


I I I I


3S 7 -4 -PREDICTED VALUE (0.67)
BASED ON UNIFORM
ENERGY DISSIPATION PER
'. .: UNIT VOLUME ACROSS
;. ;. THE SURF ZONE














-. -..?:. :.:.. :: -. .:


0.6


0.8


1.0


1.2


EXPONENT m


Figure 4. Histogram of exponent m in equation h = Axm for 502 United States East Coast and Gulf
of Mexico profiles (from Dean, 1977).


0.2


w


00.


CL0 .
_j~


1.4










A portion of Mr. Brett Moore's M.S. Thesis (1982) was directed
toward an improved definition of the scale parameter, A. Moore combined
available laboratory and field data to obtain the results presented in
Figure 5, thereby extending considerably the previous definition of A.
Some of the individual beach profiles used in the development of
Figure 5 are interesting. For example, Figure 6 presents the actual and
best least squares fit to a beach consisting of "sand particles"
15-30 cm in diameter (approximately the size of a bowling ball).
Figure 7 presents the same information for a beach reported to be
composed almost entirely of whole and broken shells. Figure 8 shows a
profile with a bar present resulting in one of the poorer fits to the
data. It is emphasized that the analytical form (Eq. (7)) describes a
monotonic profile.


V. CROSS-SHORE TRANSPORT MODELS

It has been noted that most equilibrium profiles correspond to
uniform energy dissipation per unit volume with the scale of the profile
represented by the parameter A which depends primarily on sediment
characteristics and secondarily on wave characteristics, i.e.


h(x) = Ax2/3 (10)


The parameter, A, and the uniform energy dissipation per unit
volume, D*, are related for linear spilling waves by

9 2/3
A = [24 (11)
pg K

It can be shown that for the spilling breaker assumption and linear
waves, the energy dissipation per unit volume, 9, is proportional to the
product of the square root of the water depth and the gradient in depth,

5 pg3/2K2 h1/2 3h (12)
16 ax

Thus it is clear that an increase in water level such as due to a storm
surge will cause wave energy dissipation to increase beyond the
equilibrium value. It is also known that the beach responds by erosion



















E




01


w
L)





z
Z

U)
O


Beach profile factor, A, vs sediment diameter, D, in relationship h = Ax2/3(modified from
Moore, 1982).


0.1 1.0 10.0 100.0
SEDIMENT SIZE, D(mm)


0.10-








0.01
0.01


Figure 5.












Distance offshore
S.CO 12.50 25.c00
_ ,------


(meters )
37.50


- Least Squsiee Fit
N. -- Ariuml Prpfile












Profile P4 from Zenkovich (1967). A boulder coast in
Eastern Kamchatka. Sand diameter: 150 mm 300 mm.
Least squares value of A = 0.82 m 1/3 ( from Moore, 1982).


a Distance offshore
, 0 .00 30,00 80.00

-----






we

ft
<


(me ter )
90.00


L.ast Squreen Ft
A66iN2 Pffile


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).


50.00


-7
CR.



j'-'
0






-s


Figure 6.


120.00
1


Figure 7.


I m I


I m


t,

























Distance offshore
gO.oo 25.00 5sp.00


meters5 )
75.00


U
Least Sque Fit
S-- Actual PI- tl-


U.


Profile from Zenkovich (1967). Eastern Kamchatka.
Mean sand diameter: 0.25 mm. Least squares value of
A = 0.07 ml/3 (from Moore, 1982).


-_ *u.


0o.00


0

-U
=r^

o

a
r>
<*e
C




tAv


Figure 8.










of sediment in shallow water and deposition of this sediment in deeper
water (Figure 9). It therefore appears reasonable to propose as a
hypothesis that the offshore sediment transport, Qs, per unit width is

given by


Qs = K(D-,*) (13)

where K is a rate constant that hopefully does not vary too greatly with
scale. Moore (1982) evaluated this relationship using large scale wave
tank data of Saville (1957) and found

K = 2.2x10-6 m4/N (14)

Figure 10 presents comparisons of predicted cumulative erosion for
various values of K with the measured values obtained from Saville's
wave tank tests.


VI. APPLICATION OF MODEL TO PREDICT BEACH PROFILE RESPONSE TO VARIOUS
FORCING FUNCTIONS

In an effort to demonstrate model capabilities and to represent a
broad range of beach profiles response, Moore (1982) explored the
possibility of modeling various features of beach profiles, including
longshore bars. This attempt required an improved description of the
breaking wave process across the surf zone. For this purpose, a model
developed by Dally (1980) was employed along with the sediment transport
model given by Eq. (13) and the continuity equation

-h aQs (15)


Moore showed that the model successfully accounted for sea level
rise effects (Figures 9 and 11) with the associated landward erosion and
offshore deposition and that the model could account for the placement
of a volume of sand on the profile, with the subsequent evolution to an
equilibrium profile, see Figure 12.

Moore had limited success in modeling barred beach profiles.
Depending on the type of initial breaking, a bar will develop with a










A portion of Mr. Brett Moore's M.S. Thesis (1982) was directed
toward an improved definition of the scale parameter, A. Moore combined
available laboratory and field data to obtain the results presented in
Figure 5, thereby extending considerably the previous definition of A.
Some of the individual beach profiles used in the development of
Figure 5 are interesting. For example, Figure 6 presents the actual and
best least squares fit to a beach consisting of "sand particles"
15-30 cm in diameter (approximately the size of a bowling ball).
Figure 7 presents the same information for a beach reported to be
composed almost entirely of whole and broken shells. Figure 8 shows a
profile with a bar present resulting in one of the poorer fits to the
data. It is emphasized that the analytical form (Eq. (7)) describes a
monotonic profile.


V. CROSS-SHORE TRANSPORT MODELS

It has been noted that most equilibrium profiles correspond to
uniform energy dissipation per unit volume with the scale of the profile
represented by the parameter A which depends primarily on sediment
characteristics and secondarily on wave characteristics, i.e.


h(x) = Ax2/3 (10)


The parameter, A, and the uniform energy dissipation per unit
volume, D*, are related for linear spilling waves by

9 2/3
A = [24 (11)
pg K

It can be shown that for the spilling breaker assumption and linear
waves, the energy dissipation per unit volume, 9, is proportional to the
product of the square root of the water depth and the gradient in depth,

5 pg3/2K2 h1/2 3h (12)
16 ax

Thus it is clear that an increase in water level such as due to a storm
surge will cause wave energy dissipation to increase beyond the
equilibrium value. It is also known that the beach responds by erosion
















DISTANCE (M )


100.0


120.0


MILD
\ @EACH


MWL FINAL

MWLI
INIW


FINAL BREAK


INITIAL EQUILIBRIUM BEACH PROFILE

FINAL EQUILIBRIUM BEACH PROFILE
AFTER SEA LEVEL RISE


INITIAL BREAK PT. ----

Hb: 1.91 M


Model simulation of a 0.5 meter sea level rise and beach profile response with a
relatively mild sloping beach (from Moore, 1982).


0.0
- f


'1


20.0


40.0


60.0


B0


140.0


160.0


Figure 9.


-.W




















Ve



(VOLUME tRODED)


s -.-.- SAVILLI
-6

2 ---*- K = 2.2 x t
-3
3 ----- K= 2.4 x o

(units of KI M4/HN)


Be*st F t 2


0 5 to0 20 2S 30


TIME (HOURS)


Figure 10.


Effect of varying the sediment transport rate coefficient on cumulative erosion during
the simulation of Saville's (1957) laboratory investigation of beach profile evolution
for a 0.2 mm sand size (from Moore, 1982).


















120.0


140.0


INIT


FINAL BREAK Pt.


INITIAL EQUILIBRIUM BEACH PROFILE

- FINAL EQUILIBRIUM BEACH PROFILE
AFTER SEA LEVEL RISE


INITIAL BREAK PT.


fib M 1.9 M


Figure 11.


Model simulation of a 0.5 meter sea level rise and beach profile response
with a steep 2 meeter berm (from Moore, 1982).


20.0


40.0


DISTANCE (M)










of sediment in shallow water and deposition of this sediment in deeper
water (Figure 9). It therefore appears reasonable to propose as a
hypothesis that the offshore sediment transport, Qs, per unit width is

given by


Qs = K(D-,*) (13)

where K is a rate constant that hopefully does not vary too greatly with
scale. Moore (1982) evaluated this relationship using large scale wave
tank data of Saville (1957) and found

K = 2.2x10-6 m4/N (14)

Figure 10 presents comparisons of predicted cumulative erosion for
various values of K with the measured values obtained from Saville's
wave tank tests.


VI. APPLICATION OF MODEL TO PREDICT BEACH PROFILE RESPONSE TO VARIOUS
FORCING FUNCTIONS

In an effort to demonstrate model capabilities and to represent a
broad range of beach profiles response, Moore (1982) explored the
possibility of modeling various features of beach profiles, including
longshore bars. This attempt required an improved description of the
breaking wave process across the surf zone. For this purpose, a model
developed by Dally (1980) was employed along with the sediment transport
model given by Eq. (13) and the continuity equation

-h aQs (15)


Moore showed that the model successfully accounted for sea level
rise effects (Figures 9 and 11) with the associated landward erosion and
offshore deposition and that the model could account for the placement
of a volume of sand on the profile, with the subsequent evolution to an
equilibrium profile, see Figure 12.

Moore had limited success in modeling barred beach profiles.
Depending on the type of initial breaking, a bar will develop with a

















DISTANCE
112.5


(M)
135.0


157.5


e18.0


MWL









FINAL S1I.


INITIAL EQUILIBRIUM BEACH PROFILE
WITH AN ADDITION OF SAND

.-.-* RESULT EQUILIBRIUM BEACH PROIrLI
WITH A NOURISHED BEACH


Hba 1.80 M
Sandi 0.15 mm


Figure 12. Model simulation of beach nourishment for a 0.15 mm sand size (from Moore, 1982).


0


22.5


45.0


67.5


90.0


0.I
.1.0 I


- 1.0.
x

IQ
a

2.0


31


a 1 1 a a P


&










form similar to those developed in the laboratory or found in nature.
One difficulty encountered was that with a pronounced bar present, the
localized energy dissipation could be so severe as to cause
instabilities. Moore applied a reasonable smoothing function to the
energy dissipation and improved the stability of the computations.
Figure 13 presents a comparison of a barred beach profile measured by
Saville and that computed.

Moore also evaluated his model by comparison against measured
profiles from the Nearshore Sediment Transport Study at Santa Barbara,
California for the period January 21, 1980 to December 20, 1980. The
initial and final prototype and predicted profiles are presented in
Figure 14. The maximum and minimum (envelope) prototype and model
profiles are presented in Figure 15. An empirical eigenfunction
analysis was performed on the measured and predicted profiles. The
first eigenfunction, the so-called "Mean Beach Function" is presented in
Figure 16 where it is seen that reasonably good agreement occurs. The
second or "Berm-Bar" eigenfunction is shown in Figure 17 where it is
evident that the model results have the same general form, but are more
irregular than the measured. The same general comments apply to the
third eigenfunction, the "Terrace Function" presented in Figure 18.


VII. PREDICTION OF BEACH AND DUNE EROSION DUE TO SEVERE STORMS

Mr. David Kriebel conducted the last component of work on the
project to be reported here as a Master's thesis. Most of the previous
work was incorporated and considerable original contributions were
developed into a two-dimensional predictive model of beach and dune
erosion for single storm events and for long-term scenarios in which
many storms occur.
Profile Schematization

The profile was schematized as a series of depth contours, hn, the
locations of which are specified by coordinates, xn, measured from an
arbitrary baseline, see Figure 19. The profile is thus inherently
monotonic and at each time step, the xn values of each of the active
contours is updated.

















SAVILLE
MODEL, K-2.2X106 M/N
INITIAL BEACH 1s15
TIME t 25 HRS.


N MW


Figure 13.


Comparison of the beach profile from the model and Saville's
laboratory 0.2 mm sand after 25 hours (from Moore, 1982).


2M


20 M












DECEMBER 20,1980


MODEL

- PROTOTYPE


INITIAL PROFILE JAN.21, 190


200 240


DISTANCE FROM BASELINE


280


(M)


Figure i4 .


Initial (January 21, 1980) and final (December 20, 1980) measured and predicted
beach profiles. Leadbetter Beach, Santa Barbara, California (from Moore, 1982).















hpmln

/ mmin


um L


hp PREDICTED WATER DEPTH

hm MEASURED WATER DEPTH


MEASURED PROFILE ENVELOPE
- PREDICTED PROFILE ENVELOPE


120


160 200 240


DISTANCE FROM


Figure 15.


BASELINE (M)


Model versus prototype beach profile envelopes (i.e., maximum and minimum water depths
at each location along the beach profile) for the entire year (1980) (from Moore, 1982).


280






















ElOINFUNCTION8 I


MIAN BIACH FUNCTION


S .. \ 160 24
o-O


N.-\





..--- PODIL
PROVOTYPI


DISTIANCI FROM BASILINI (M)






'5



\*. \
S -


"r.:...
~.5.'^


Figure 16. Comparison of Eigenfunction analysis results performed on the predicted
and measured beach profiles: First Eigenfunction, mean-beach function
(from Moore, 1982).










form similar to those developed in the laboratory or found in nature.
One difficulty encountered was that with a pronounced bar present, the
localized energy dissipation could be so severe as to cause
instabilities. Moore applied a reasonable smoothing function to the
energy dissipation and improved the stability of the computations.
Figure 13 presents a comparison of a barred beach profile measured by
Saville and that computed.

Moore also evaluated his model by comparison against measured
profiles from the Nearshore Sediment Transport Study at Santa Barbara,
California for the period January 21, 1980 to December 20, 1980. The
initial and final prototype and predicted profiles are presented in
Figure 14. The maximum and minimum (envelope) prototype and model
profiles are presented in Figure 15. An empirical eigenfunction
analysis was performed on the measured and predicted profiles. The
first eigenfunction, the so-called "Mean Beach Function" is presented in
Figure 16 where it is seen that reasonably good agreement occurs. The
second or "Berm-Bar" eigenfunction is shown in Figure 17 where it is
evident that the model results have the same general form, but are more
irregular than the measured. The same general comments apply to the
third eigenfunction, the "Terrace Function" presented in Figure 18.


VII. PREDICTION OF BEACH AND DUNE EROSION DUE TO SEVERE STORMS

Mr. David Kriebel conducted the last component of work on the
project to be reported here as a Master's thesis. Most of the previous
work was incorporated and considerable original contributions were
developed into a two-dimensional predictive model of beach and dune
erosion for single storm events and for long-term scenarios in which
many storms occur.
Profile Schematization

The profile was schematized as a series of depth contours, hn, the
locations of which are specified by coordinates, xn, measured from an
arbitrary baseline, see Figure 19. The profile is thus inherently
monotonic and at each time step, the xn values of each of the active
contours is updated.






















I I I

\

\


IEOINFUNCTIONI 2
BAR-BERM FUNCTION


i i
I
\ i






--.- Moodt i' !
I I
---- POOTyPi 'i

Ii
j


DISTANCE FROM BASELINE (M)


Figure 17. Comparison of Eigenfunction analysis results performed on the predicted
and measured beach profiles: Second Eigenfunction, bar-berm function
(from Moore, 1982).


_ i_ ___


-.40





























o i \

*0 '


EIGINFUNCTIONs 3
TERRACE FUNCTION


' DISTANCE FROM BASELINE (M)


MoDEl
-- -- PRToTrPe


Figure 18.


Comparison of Eigenfunction analysis results performed on the predicted
and measured beach profiles: Third Eigenfunction, terrace function
(from Moore, 1982).


*,











Xmsl


v SURGE


I hn

SXn-h
+snI 2/3


Sn+l
n xhn= A(xn-xmsi )



L




Figure 19. Model representation of beach profile, showing depth and transport relation to grid
definitions (from Kriebel, 1982).


I
_t I










Governing Equations

As in most transport problems, there are two governing equations.
One is an equation describing the transport in terms of a gradient or
some other feature. The second is a continuity or conservation equation
which accounts for the net fluxes into a cell.

As discussed previously, the offshore transport is defined by
Eq. (13) in terms of the excess energy dissipation per unit volume.
Specifically, in finite difference form

h5/2 h5/2
hn- h
+ = n+1 n (16)
(h' + h')(x )
n+1 n n+1 n

where


k= K2g (17)


The conservation equation is


Ax Kt (V D ) (18)
n Ah n n+1

Method of Solution of Finite Difference Equations

A number of methods could be employed for solving Eqs. (13) and
(18). For example, explicit methods would be fairly direct and simple
to program; however, the maximum time increment would be relatively
small resulting in a program which is quite expensive to run. Implicit
methods are somewhat more difficult to program, but have the desirable
feature of remaining stable with a much greater time step. Because of
the planned application to long-term simulation in which for a 500 year
time period over three hundred storms would be modeled, each with an
erosional phase of six to twelve hours, an implicit method was
adopted. This method will not be described in detail here except to
note that a double sweep approach is used in which the Qsn values and
the xn values are updated simultaneously at each time step. For Ah
values of 1 ft, and a time step of thirty minutes, the system of
equations was stable.










The boundary conditions used were somewhat intuitive. At the
shoreward end of the system, erosion proceeded with a specified slope
above a particular depth, h*. The depth, h*, is the depth that the
equilibrium slope and the slope corresponding to the beach face are the
same. Thus a unit of recession of the uppermost active contour causes
an erosion of the profile above the active contour that is "swept" by
this specified slope. This material is then placed as a source into the
uppermost active contour. The offshore boundary condition is that the
active contours are those within which wave breaking occurs. If an
active contour extends seaward, thereby encroaching over the contour
below to an extent that the angle of repose is reached, the lower
contour (and additional lower contours if necessary) are displaced
seaward to limit the slope to that of the angle of repose.
Application of Method to Computation of Idealized Beach Response

Kriebel (1982) carried out computations for a number of idealized
cases, some of which are reviewed below.

Response to Static Increased Water Level Figure 20 presents the
beach recession due to a static increase in water. The beach responds
as expected. In the early response stages, the rate of adjustment is
fairly rapid with the latter adjustments approaching the equilibrium
recession in an asymptotic manner. Of special relevance is that the
response time to equilibrium is long compared to the duration of most
severe storm systems, such as hurricanes. The form of the response
presented in Figure 20 is reminiscent of that for a first order process
in which the time rate of change of beach recession, R, is represented as


dR aR (19)
dt

for which the solution is


R(t) (1-e-) (20)
R

Figure 21 presents a comparison of the response from the numerical model
and Eq. (20). This similarity forms the basis for a very simple and
approximate numerical model of beach and dune profile response. Such a
model has been developed but will not be presented here.





























z
0
U)
U)




U

0-
S0.








Figure 20.


100 200 300
TIME (hrs)







Characteristic form of berm recession versus time for increased
static water level (from Kriebel, 1982).






































TIME (hrs)


Figure 21.


Comparison of asymptotic berm recession from model (-)
and as calculated by Eq. (20) (* *).


100


200


300










Effects of Various Wave Heights Considering a common increased
water level, but storms with different wave heights, the larger wave
heights will break farther offshore causing profile adjustments over a
greater distance and thus a greater shoreline recession. Simulations
were carried out to examine evolution of the beach under different wave
heights with the results presented in Figure 22. As expected the
greater shoreline recessions are associated with the larger wave
heights. Surprisingly, however during the early phases of the
evolution, the larger wave heights do not cause proportionally larger
erosions. Thus, for storms of short duration, the sensitivity of the
maximum erosion to breaking wave height may not be large.

Effects of Various Storm Tide Levels The counterpart to the
previous case is that of a fixed wave height and various storm water
levels. The results of these simulations are presented in Figure 23.
In contrast to the previous case, the various storm tide levels cause
recession rates in the early stages of the process which are nearly
proportional to the storm water level.

Effect of Sediment Size on Berm Recession The effect of two
different sediment sizes on amount and rate of berm recession is shown
in Figure 24. The equilibrium recession of a coarser material is much
less; however, the equilibrium is achieved in a much shorter time than
that for 'the finer sediment. The explanation for the lesser equilibrium
erosion for the coarser material is that since the beach is steeper, the
waves break closer to shore and thus less material is required to be
transferred offshore to establish an equilibrium profile out to the
breaking depth (considered to be the limit of motion). Presumably the
explanation for the slower approach to equilibrium for the finer
material is that, as will be shown by consideration of the initial and
equilibrium profile geometries, a much greater volume of sediment must
be moved a greater distance to establish equilibrium.

Effect of Storm Duration The effect of storm duration on
shoreline recession was investigated by considering a fixed wave height
and an idealized storm tide variation, expressed as





































100


TIME (hrs)


Figure 22.


Effect of breaking wave height on berm recession (from Kriebel, 1982).






















SS=l.2m
Z
0 30 -





SS)=0.6m

10-
20


0 50 100 150 200
TIME (hrs)


Effect of static storm surge level on berm recession (from Kriebel, 1982).


Figure 23.






































TIME (hrs)


Figure 24. Effect of sediment size on berm recession. (from Kriebel, 1982)










n = 1.2 cos2( (t-18) t-18[ 2

(21)

= 0 t-181> T/2


in which T (= 2w/a) is the total storm duration in hours. The results
are presented for three storm durations in Figure 25. For the shortest
storm duration (T = 12 hours), the potential volume eroded is
approximately 70 m3/m whereas the computed actual maximum volume eroded
is 10 m3/m. With increasing storm tide duration, the computed actual
maximum volume eroded increases. Tripling the storm tide duration to
36 hours doubles the maximum volume eroded to 20 m3/m. It is noted that
this is only approximately 28% of the potential volume eroded, again
underscoring the likelihood that most storms will only reach a fraction
of their potential erosion limit. This feature also highlights the
significance of cumulative effects of sequential storms and of the need
to better understand the recovery process (especially the rates), a
portion of the cycle not addressed in this project.
Application of Method to Long-Term Beach and Dune Response Simulations

The previous section has described the application of the model to
idealized examples of beach and dune response. The model can also be
applied to more realistic situations in which the initial beach and dune
conditions are specified along with time-varying waves and tides.

Evaluation of Method by Hurricane Eloise Erosion Data Kriebel
carried out an evaluation of the method by comparing erosion
computations for Hurricane Eloise (1975) with measurements reported by
Chiu (1977). Although the wave and tide conditions were not measured
along the beaches of Bay and Walton Counties (Florida) of interest, some
tide data were available and wave heights were estimated. Erosion was
computed for twenty combinations of dune slope, wave height and peak
surge. It was found that the volumetric erosion ranged from 21 to
38 m3/m compared to average measured values of 18 to 20 m3/m for Bay and
Walton Counties, respectively and an average of 25 m3/m near the area of
peak surge. Although the predicted values are somewhat larger than the
observed, Chiu (1977) states that the beaches had started to recover at
the time of the post-storm surveys, with approximately 5 m3/m of sand




























Lo





38 HRS
Lii





24 RS --
-1~

12 HRS Actual Volume Eroded \
S" (Computed) co



%.oo s'.0 .00 .00 .00 1 .00 .00 18.00 21.00 2.00 2 .00 b.oo 98.00 8o o
TIME HRS

Figure 25. Comparison of the effects of 12, 24, and 36 hrs. storm surge on volumetric erosion
(from Kriebel, 1982).










having returned to the beach. Thus the maximum eroded volume would be
30 m3/m compared to a maximum calculated value of 38 m3/m, a difference
of approximately 27%. This reasonably close agreement was considered
adequate recognizing the uncertainty in the storm tide employed in the
computations; therefore no further calibration of the model was considered
warranted. It is of interest that the erosion potential associated with
the peak tide is approximately nine times that predicted for the time-
varying conditions included in the computations. This again reinforces
the fact that most storms in nature cause only a fraction of the
potential erosion associated with the maximum conditions in the storm.

Long-Term Simulation With the model reasonably verified for the
Bay and Walton Counties area of Florida, a long-term simulation of beach
and dune erosion was carried out. The hurricane wind and pressure
fields were idealized in accordance with a representation published by
Wilson (1956). The five idealized hurricane parameters


Ap = Maximum Pressured Deficit

Rmax = Radius to Maximum Winds
VF = Hurricane System Translational Speed
B = Hurricane Translational Direction

YF = Landfall Point


were selected by a Monte Carlo method in accordance with the historical
characteristics of hurricanes in the general area. For each hurricane,
the storm tide was calculated using the Bathystrophic Storm Tide Model
of Freeman, Baer and Jung (1957). With the time-varying storm tide and
wave height calculated, the beach and dune model was applied until
maximum erosion was achieved. As the recovery mechanism is not yet
understood to a degree for realistic modelling and because hurricanes
occur approximately on a biennial basis, the erosion for successive
hurricanes was assumed to commence from a fully recovered condition.
This is clearly an approximation as the recovery process occurs at
several rates of magnitude slower than the erosion process. Study of
some recovery stages from severe storms has shown that up to seven years
may be required to achieve approximately 90% recovery. The duration
required for recovery from milder storms would, of course, be less.










Figure 26 presents a "flow chart" describing the elements of the
long-term simulation. In the Bay-Walton Counties area, hurricanes
making landfall within : 150 n.mi. of these counties were considered
requiring a total of 393 hurricanes to simulate a 500 year record. The
return periods associated with various dune recessions as determined
from the simulations are presented in Figure 27. As examples, the dune
recessions for return periods of 10, 100 and 500 years are 4 m, 12 m and
18 m, respectively. Based on these results, Hurricane Eloise is judged
to represent a 20 to 50 year erosional event; however based on results
from a storm surge analysis, Hurricane Eloise was a 75 to 100 year
coastal flooding event.

It is also possible to present the results of the erosion
simulations in a manner that is of maximum relevance to individuals or
agencies responsible for shoreline management. This type of
presentation is demonstrated for the Bay-Walton County area in
Figure 28. This plot includes the contributions from storms and sea
level rise. As examples, without any erosion mitigation measures within
the next 50 years, the erosion due to sea level rise (regarded as a
certainty or probability of 100%) is expected to be approximately 15
ft. Within 50 years, the probability of erosion occurring to a distance
of 40 ft is 85% and for distances of 60 and 80 ft, the corresponding
probabilities are 32% and 9%, respectively. Through the use of figures
such as these it would be possible to weigh the costs of certain erosion
control measures against the potential of damage if those measures are
not carried out.

These procedures provide, for the first time, a basis for
conducting the necessary technical studies to implement the erosion
component calculations of the Flood Insurance Act of 1973 which provides
for the application of methodology to provide the basis for insurance
rates for flooding and erosion coastal hazards. Although the flooding
component of this act has been implemented, the erosion component has
not.

It is noted that the State of Florida Division of Beaches and

Shores of the Department of Natural Resources presently utilizes the
erosion simulation model of Kriebel and simplifications thereof in the


1
















0
CH
o
0

01





C
4
0a

1to
r-4




I)





CO
0)
r.





0
4-1



0)
Cc
C.
--
4-'










0
0



(0
-H



CO



4-4
o
'-I


-P-4

t0








0-H







Pr4
1-1











RETURN PERIOD IN YEARS


0 'I I 1 1 I 1 L 1 -
.2 .1 .05 .03 .02 .01 .005


500


.002


PROBABILITY OF OCCURRENCE OR EXCEEDANCE


Figure 27. Average frequency curve for dune recession, developed by Monte
Carlo simulation, Bay-Walton Counties, Florida (from Kriebel,
1982).




















































0


Figure 28.


25 50 75 100
YEARS FROM PRESENT






Probability or risk of dune recession of given magnitude
occurring at least once in N-years, Bay-Walton Co., Florida
( from Kriebel, 1982).









establishment of the Coastal Construction Control Line and in the
consideration of various applications for coastal construction permits.


VIII. OTHER APPLICATIONS AND PUBLICATIONS

In addition to the above comprehensive contributions by Moore
(1982) and Kriebel (1982), a number of publications and applications
have resulted from this project.

The design of beach nourishment projects has been discussed by
Maurmeyer and Dean (1980) with specific reference to the placement of
sand to minimize overtopping by waves. In addition, an examination was
carried out of the effect of sand size on the usable width of beach
after reconfiguring of the profile by waves of different heights.
Methods were presented for calculating the wave overtopping as a
function of volumes and types (sizes) of beach sand placed. Figure 29
presents, for various sediment characteristics, the required nourishment
volumes to advance the shoreline seaward a distance of 300 ft.

The effect of wave steepness and fall velocity parameter on volume
of material stored in the offshore bar was examined by Dean (1982). A
series of systematic wave tank experiments by Coxe (1978) was employed
to develop a dimensionless relationship for the bar volume. Figure 30
presents the bar volume as a function of the square of the excess wave
height above that required for incipient bar formation.

A number of laboratory and field experiments had been carried out
by various investigators to quantify the immersed sediment transport
rate, It, in terms of the so-called longshore energy flux at breaking,

Pts, i.e.

It = K'Pis (22)

It was found that the laboratory derived values of K' were significantly
lower than the field values and this was taken as grounds that serious
scale effects were present in the modeling of longshore sediment
transport. Examination of the model versus field conditions
demonstrated a scale ratio of approximately 1:10 for the waves, but a
scale ratio of approximately 1:3 to 1:1 for the model sediment. Thus













1 x 105

8


4k


2 L-


1 x 104

8


6


1 x 101
0


o7


4q..

/


/ 0.165 *
*2 *



*
S /Al = 0.15, A2 = 0.30


&-Type 1 Profile

Note: Volume Contained in Dune
of Each Profile = 1800 ft3


Volume Contained in Berm
of Each Profile = 1800 ft3


I I I II I I I I


Effective Wave Height, H (ft.)


Figure 29.


Nourishment volumes required versus effective wave height for
various native and filled sediment characteristics and
considerations, shoreline advancement = 300 ft. (from
Maurmeyer and Dean, 1980).


I I I I I

























2000








1000








0<


2 4 6 8
WAVE HEIGHT ABOVE THAT REQUIRED
INCIPIENT BAR FORMATION, Ho-Hoc,


Figure 30.


Relationship between volume of sand stored in bar
versus wave height above that required for incipient
bar formation. (from Dean, 1983)


10
FOR
(cm)










the model sediment scaled to the prototype is from three to ten times
larger than the field sediment. Dean (1983) recommended that the
scaling be conducted in accordance with the following parameter which
incorporates both sediment and wave characteristics.

Hb
g- (23)
w

where Hb is the breaking wave height and w is the fall velocity of the
sediment. Without presenting the details, Figure 31 demonstrates that
the use of the parameter gHb/w2 allows unification of the laboratory and
field results.

The principles of beach nourishment were reviewed by Dean (1983).
Included were the effects of sediment size on total volumes required and
computational procedures to determine reduction in sand volumes required
through application of a "perched beach" concept, see Figure 32. Also
described were the merits of stabilization of beach nourishment projects
by structures.

Dean and Maurmeyer (1983) presented models for long-term response
to sea level rise. Models were presented in graphical form for a
natural beach profile and a beach profile limited by a seawall.
Figures 33 and 34 present the results for natural and seawalled beaches,
respectively. It is seen from Figure 33 that, for a fixed breaking wave
height, the berm recession for the natural beach increases with
increasing storm tide. However, for the seawalled case, with increasing
storm tide and fixed wave height, the deepening at the base of the
seawall increases, reaching maximum, then with further increases in
storm tide, decreases. The interpretation is that initially an increase
in storm tide requires substantial erosion to meet the demand of
maintaining the offshore profile the same relative to the fixed water
level. However, increasing storm tides will cause the horizontal extent
of the region requiring deposition to decrease to zero, thereby
resulting in zero scour. At the limit, where no wave breaking occurs,
(all the wave energy is reflected), this approximate method predicts
that no erosion would occur. Of course, this result is not completely
realistic. A model for barrier island response was presented which















I- I I 1 1I 111 i I j 11-1 II I 1 11


Johnson and
Galvin,
Prototype
(D= 0.2mm)






Prototype
Bulk Data

Model Bulk
Datag Moore and Cole
Prototype ( D= 1.0mm)


5 103 2 5 104 2
DIMENSIONLESS WAVE HEIGHT-
FALL VELOCITY PARAMETER, gH/w2


5 105


Figure 31.


Suggested variation of K with gHb/w2, prototype and laboratory data
(from Dean, 1983).


2.0









1.0









0


102
























XI


AX916- A X2


a \ r Perched Beach


h2




Toe Structure



Figure 32. Perched beach, demonstrating nourishment volumes saved (from
Dean, 1983).




































Figure 33.


f _. R
R'-
W2

h, hb

S'-
B
0 hbZ (l/m)

10.0


Isolines of dimensionless berm recession, R', vs
dimensionless storm breaking depth, h2, and
dimensionless storm tide, S', m = 2/3. (from Dean
and Maurmeyer, 1983)
w2
x Seowoll
Seawall


20.0



10.0
8.0
6.0


0.2 I ____ l I '
S 2 4 6 8 10 20 40 6080 100
DIMENSIONLESS STORM BREAKING DEPTH, h'b


Figure 34.


Isolines of dimensionless
dimensionless storm tide,
depth h 2. (from Dean and
b2


seawall toe scour, h, vs
S' and dimensionless breaking
Maurmeyer, 1983)


0.2 0.5 1.0 2.0 5.
DIMENSIONLESS STORM BREAKING DEPTH, hb2









accounts for the offshore transport of sand on the ocean and bay sides
and also the upward growth of the barrier island to maintain elevation
relative to the rising sea level, Figure 35.

The equilibrium beach profile represented by Eq. (10) is extremely
simple and has proven useful in a number of applications. It is
recalled that the basis for this profile is that the sediment particle
can withstand a certain level of destructive forces (energy dissipation
resulting in turbulent fluctuations); if the destructive forces exceed
this level, the profile will be remolded by reducing the local slope
and/or depth to again achieve equilibrium. One unrealistic feature of
this consideration is that gravity is not recognized as a destructive
force. Thus it is implicitly assumed that the slopes are so mild that
the gravitational forces are small compared to those induced by
turbulence. Inspection of Eq. (10) shows that the slope of the profile
is infinite at the mean water line. To account for the effect of non-
mild slopes, the equilibrium energy dissipation per unit volume is
modified to include the effect of gravitational forces

dh
= 9, 1 (-h (24)


where pD is the equilibrium energy dissipation on a flat slope
and (d), is the limiting slope for the sand. Equating the above to the
wave energy dissipation per unit volume and simplifying, the solution
relating water depth, h, to distance from shore, x, is

p9 h
5 p3/2 2h3/2 + =o x (25)
-pg K h + 9 x (25)
24 ahax *

Note that the first term on the left hand side and the term on the
right hand side represent the solution developed earlier. Also, the
second term on the left hand side dominates in very shallow water and
predicts a uniform slope which is in accordance with beach face
descriptions.

Considering a nearshore slope of 1:15 (a reasonable value for a
sand size of 0.2 mm), Figure 36 compares the beach profile with and
without inclusion of the gravitational term.





































LAGOON SIDE


RRIER ISLAND
BARRIER ISLAND


Position After Response to Sea
Level Rise ( Upward end Londward
MiQgroton )
Original Position


Figure 35.


Generalized shoreline response model due to sea level rise.
Applicable for a barrier island system which maintains
its form relative to the adjacent ocean and lagoon water levels
( Dean and Maurmeyer, 1983)


OCEAN SIDE











OFFSHORE (m)


100


200


Only Wave Dissipation


Gravitational Effects Included





".,

... .... :. ""...' ...


Figure 36. Comparison of equilibrium beach profile with and without gravitational
effectsincluded. A = 0.1 ml/3 corresponding to a sand size of 0.2mm.


300


DISTANCE









establishment of the Coastal Construction Control Line and in the
consideration of various applications for coastal construction permits.


VIII. OTHER APPLICATIONS AND PUBLICATIONS

In addition to the above comprehensive contributions by Moore
(1982) and Kriebel (1982), a number of publications and applications
have resulted from this project.

The design of beach nourishment projects has been discussed by
Maurmeyer and Dean (1980) with specific reference to the placement of
sand to minimize overtopping by waves. In addition, an examination was
carried out of the effect of sand size on the usable width of beach
after reconfiguring of the profile by waves of different heights.
Methods were presented for calculating the wave overtopping as a
function of volumes and types (sizes) of beach sand placed. Figure 29
presents, for various sediment characteristics, the required nourishment
volumes to advance the shoreline seaward a distance of 300 ft.

The effect of wave steepness and fall velocity parameter on volume
of material stored in the offshore bar was examined by Dean (1982). A
series of systematic wave tank experiments by Coxe (1978) was employed
to develop a dimensionless relationship for the bar volume. Figure 30
presents the bar volume as a function of the square of the excess wave
height above that required for incipient bar formation.

A number of laboratory and field experiments had been carried out
by various investigators to quantify the immersed sediment transport
rate, It, in terms of the so-called longshore energy flux at breaking,

Pts, i.e.

It = K'Pis (22)

It was found that the laboratory derived values of K' were significantly
lower than the field values and this was taken as grounds that serious
scale effects were present in the modeling of longshore sediment
transport. Examination of the model versus field conditions
demonstrated a scale ratio of approximately 1:10 for the waves, but a
scale ratio of approximately 1:3 to 1:1 for the model sediment. Thus









IX. SUMMARY AND CONCLUSIONS


The characteristics of equilibrium beach profiles in nature have
been found to be of the approximate form


h(x) = Ax2/3 (26)


where A is a scale parameter depending primarily on sediment size, and
secondarily on wave characteristics. Eq. (26) was shown to be
consistent with a spilling breaking model and a uniform wave energy
dissipation per unit water volume.

Eq. (26) has been applied to the quasi-static case of predicting
shoreline recession due to sealevel rise along natural shorelines and
deepening in front of seawalled profiles. Additionally, the profile has
been applied to design problems in beach nourishment projects, including
beach widths associated with volumes and diameters of sand used. The
particular case of a perched beach has been treated.

The equilibrium beach profile results have been extended to the
case of non-equilibrium by proposing the following offshore sediment
transport relationship


Qs = K(D-D,) (27)


in which V is the wave energy dissipation per unit volume and, V*
represents the equilibrium value. Eq. (28) has been combined with the
continuity equation to represent a number of problems of interest,
including: many idealized examples, bar formation (which requires a
more realistic breaking model than the spilling breaker model), the
forms of beach change and dune erosion by severe storms, including
simulation of a 500 year period for one location. All of the results
appear realistic and encouraging.

Several problems on which future research should be focused
include:
(1) The recovery phase following erosion which is known to
proceed at a much slower rate,









(2) Comparison of predicted and measured shoreline response,
including normal seasonal storms and severe events,
(3) The effect of wave characteristics on the parameter, A,
(4) The development of improved wave breaking models,
(5) The mechanisms and causes of bar formation, and
(6) The effects of natural offshore rock structures on
shoreline response.


X. ACKNOWLEDGMENTS

The contributions of the graduate students to the research results
summarized in this report are gratefully acknowledged. The excellent
support services of Ms. Cynthia Vey and Ms. Melissa Michaels in typing
and Ms. Lillean Pieter in illustrating are greatly appreciated.
Finally, my graduate students and I sincerely thank Sea Grant for the
continued support in the general area of shoreline processes and
response.









(2) Comparison of predicted and measured shoreline response,
including normal seasonal storms and severe events,
(3) The effect of wave characteristics on the parameter, A,
(4) The development of improved wave breaking models,
(5) The mechanisms and causes of bar formation, and
(6) The effects of natural offshore rock structures on
shoreline response.


X. ACKNOWLEDGMENTS

The contributions of the graduate students to the research results
summarized in this report are gratefully acknowledged. The excellent
support services of Ms. Cynthia Vey and Ms. Melissa Michaels in typing
and Ms. Lillean Pieter in illustrating are greatly appreciated.
Finally, my graduate students and I sincerely thank Sea Grant for the
continued support in the general area of shoreline processes and
response.









APPENDIX I REFERENCES
Aubrey, D.G., "Seasonal Patterns of Onshore/Offshore Sediment Movement,"
Journal of Geophysical Research, Vol. 84, No. C10, Oct., 1979,
p. 6347-6354.
/ Bruun, P., "Coast Erosion and the Development of Beach Profiles," Beach
Erosion Board, Tech. Memo. No. 44, 1954.
SBruun, P., "Sea-Level Rise as a Cause of Shore Erosion," J. Waterways,
Harbors and Coastal Engrg. Div., ASCE, V. 88, WW1, Feb., 1962,
p. 117-130.
Chiu, T.Y., "Beach and Dune Response to Hurricane Eloise of September
1975," Coastal Sediments '77, ASCE, 1977, p. 116-134.
Chiu, T.Y., Personal Communication, 1981.
Coxe, David, "Beach Profile Measurements in a Wave Tank," Unpublished
University of Delaware Report, 1978.
Dally, W.R., "A Numerical Model for Beach Profile Simulation," M.S.
Thesis, Univ. of Delaware, 1980.
/ Dean, R.G., "Heuristic Models of Sand Transport in the Surf Zone," Conf.
on Engineering Dynamics in the Coastal Zone, Sydney, Australia, May,
1973, 208.
J Dean, R.G., "Beach Erosion: Causes, Processes and Remedial Measures,"
CRC Critical Reviews in Environmental Control, Sept., 1976,
p. 259-296.
SDean, R.G., "Equilibrium Beach Profiles: U.S. Atlantic and Gulf Coasts,"
Ocean Engineering Technical Report No. 12, Department of Civil
Engineering and College of Marine Studies, University of Delaware,
January, 1977.
Dean, R.G. and E.M. Maurmeyer, "Models for Beach Profile Response,"
Chapter 7, CRC Handbook of Coastal Processes and Erosion," 1983,
p. 151-165.
SDean, R.G., "Principles of Beach Nourishment," Chapter II, CRC Handbook
of Coastal Processes and Erosion," 1983, p. 217-231.
SDean, R.G., "Physical Modeling of Littoral Processes," Proceeding of a
Conference on Physical Modeling in the Coastal Environment,
University of Delaware, R.A. Dalrymple, Editor, 1983.
Eagleson, P.S., Glenne, B., and Dracup, J.A., "Equilibrium
Characteristics of Sand Beaches," J. Hydraulics Div., ASCE, V. 89,
No. HY1, Jan., 1963, pp. 35-57.










Edelman, T., "Dune Erosion During Storm Conditions," Proceedings of the
Twelfth Conference on Coastal Engineering, ASCE, New York, 1970,
1305.
Freeman, J.C., Baer, L. and Jung, G., "The Bathystrophic Storm Tide,"
J. of Marine Research, Vol. 16, No. 1, 1957, p. 12-22.
Hayden, B., Felder, W., Fisher, J., Resio, D., Vincent, L. and
Dolan, R., "Systematic Variations in Inshore Bathymetry," Tech.
Rept. No. 10, Dept. of Env. Sciences, Univ. of Virginia, Jan., 1975.
Hughes, S.A., "Movable-Bed Modeling Law for Coastal Dune Erosion,"
J. Waterway Port Coastal and Ocean Engineering, Vol. 109, No. 2,
May, 1983, p. 164-179.
Kriebel, D.L., "Beach and Dune Response to Hurricanes," M.S. Thesis,
University of Delaware, 1982.
Kriebel, D.L. and R.G. Dean, "Beach and Dune Erosion by Storms, Part I:
Methodology and Idealized Examples," Manuscript submitted to Coastal
Engineering for publication consideration, 1983.
Kriebel, D.L. and R.G. Dean, "Beach and Dune Erosion by Storms, Part II:
Evaluation and Application to Long-Term Simulation," Manuscript
submitted to Coastal Engineering for publication consideration,
1983.
Maurmeyer, E.M. and R.G. Dean, "Sediment and Overwash Considerations in
the Design of Beach Nourishment Projects," Proceedings of the Second
Symposium on Coastal and Ocean Management, ASCE, Coastal Zone '80,
Vol. IV, November, 1980, pp. 2978-2991.
Moore, B., "Beach Profile Evolution in Reponse to Changes in Water Level
and Wave Height," M.S. Thesis, University of Delaware, 1982.
Saville, T., "Scale Effects in Two-Dimensional Beach Studies,"
Transactions, 7th Meeting, International Association of Hydraulic
Research, Lisbon, 1957, pp. A3-1 to A3-9.
Swart, D.H., "Offshore Sediment Transport and Equilibrium Beach
Profiles," Publication No. 131, Delft Hydraulics Lab., Delft
University of Technology, 1974.
van de Graaff, J., "Probabilistic Design of Dunes," Proceedings, Coastal
Structures '83, April, 1983, p. 820-831.
Vellinga, P., "Beach and Dune Erosion During Storm Surges," Coastal
Engineering, Vol. 6, No. 4, 1982, p. 361-389.




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