Development of erosional indices and a shoreline change rate equation for application to extreme event impacts

UFL/COEL-2001/012

DEVELOPMENT OF EROSIONAL INDICES AND A SHORELINE
CHANGE RATE EQUATION FOR APPLICATION TO EXTREME
EVENT IMPACTS

by

Jonathan K. Miller

Thesis

2001

DEVELOPMENT OF EROSIONAL INDICES AND A SHORELINE CHANGE RATE
EQUATION FOR APPLICATION TO EXTREME EVENT IMPACTS

By

JONATHAN K. MILLER

A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2001

This thesis is dedicated to all of those who have believed in me along the way as well as
those who have not. Without the influence of both I would not have possessed the desire
and dedication required to get to this point.

ACKNOWLEDGMENTS

First of all, I would like to thank my advisor, Dr. Robert Dean, for his inspiration

and guidance. His knowledge and enthusiasm truly are contagious. Our Saturday and

Sunday morning "breakfast" meetings have been invaluable. Many were the times I

came to him frustrated and dejected, only to be reinvigorated by an enlightening Saturday

morning marathon meeting.

I would like to thank the other members of my supervisory committee, Dr. Daniel

Hanes and Dr. Robert Thieke, for their insightful comments and thought provoking

lectures. I would also like to thank the rest of the faculty and staff, particularly Becky

Hudson, for making these past two years as rewarding as possible. I also feel it is

necessary to thank several of the students who made my transition to Gainesville and

graduate school much easier. The help of Jamie MacMahan, Sean Mulcahy, David

Altman, Cris Weber and especially Al Browder was much appreciated.

I would be remiss in not thanking both the Florida Sea Grant and the American

Society for Engineering Education for providing financial support for this project. Their

contributions have been greatly appreciated.

Several teachers I have met along the way have played an integral role in my

development as a student and as a person. First of all, I would like to thank Br. Paul

Joseph who first instilled in me a desire to become a teacher. His enthusiasm and love

for his students shined through every day. Next, I need to thank Dr. K.Y. Billah and Dr.

Michael Bruno. Dr. Billah believed in me always and encouraged me to learn as much as

possible. He made me want to learn for the sake of learning, not for a grade. He is the

true epitome of a teacher. I need to thank Dr. Bruno for encouraging me to follow my

heart and pursue the field of coastal engineering when others told me I would be better

off doing something more "practical."

Without the continued support and assistance of my family this would not have

been possible. They have always been there to back me and encourage me in whatever I

have chosen to do.

Last and most importantly, I need to thank my wonderful wife Diana. Without

her emotional support and encouragement over the past 2 years this would not have been

possible. She has always been there for me even when I was not there for her. I can

never thank her enough for her infinite love and patience.

TABLE OF CONTENTS

page

ACKNOW LEDGM ENTS..................................................................... ... ..................... .. iii

LIST OF TABLES .......................................................................................................... vii

LIST OF FIGURES ...................................... ......................... viii

ABSTRACT ..................................................................................................................... xi

INTRODUCTION.................................................................................................. 1

Coastal Erosion Processes ..................... ...... ........................................................... 1
Objectives and Scope .................................................................. .............................. 5

LITERATURE REVIEW ........................... ................................. .................................. 7

M odel Hurricane ........................................................................................ .. .......... 7
Bathystrophic Storm Surge ............................... ........................... ......................... 10
Erosion M models ................................................................................. ...................... 11
Shoreline Recovery M odel................................................. ............ .......... 14
Seasonal Shoreline Change ........................................................................................... 14

SHORELINE CHANGE ANALYSIS .................................................... .................... 16

Site Descriptions ................................................................................. ............ .............. 18
Volusia County............................................................................ ............ ........... 18
St. Johns County................................................................. .............................. 18
M martin County .............................................. ....................................................... 19
Sources and Analysis Techniques .................................................... 19
Results and Conclusions.................................................... .................................... 21

EROSION INDICES ........................................................ ........................................... 28

Event Longevity Parameter Index (ELP) ....... ... ................................... ................... 31
Theory ....................................................................................................................... 31
W inter Storm Related Surge-Reconstructed Hydrographs ..................... ............ 33
Hurricane Related Surge-Bathystrophic Storm Surge Model............................. 34
Results.......................................................................... .....................................39
Hurricane Erosion Index (HEI) ................................................... .......................... 49

v

Theory ................................................................................................................. 49
Results ....................................................................................................................... 50
M odified Hurricane Erosion Index (M HEI).......................................... ........ ..... 53
Theory ................................................................................................................. 53
Results ................................................................................................................. 55

SH ORELINE RESPON SE RATE .................................................. .......................... 58

Rate Equation .......................................................................................................... 59
Theory ................................................................................................................. 59
Results....................................................................................................................... 60
D eterm nation of Equilibrium Shoreline Position................................... ........... ... 65
Theory ................................................................................................................. 65
Results ....................................................................................................................... 70

CONCLUSIONS AND RECOMMENDATIONS ....................................... .......... .. 88

Erosion Indices.............................................................................................................. 88
Sum m ary and Conclusions................................................ ................................ 88
Recom m endations for Future Study.............................. .................................. 90
Shoreline Response Rate Equation ............................................... ........................ 92
Sum m ary and Conclusions................................................ ................................ 92
Recom m endations for Future Study.............................. ................................. 93

LIST OF REFEREN CES ........................................................................................... 97

BIOGRAPH ICAL SKETCH ....................................................... ............................. 99

LIST OF TABLES

Table Page

4.1 R-squared values indicating the correlation between the observed erosion and the
ELP Index for the case of no setup and no winter storms............................... 40

4.2 R-squared values resulting from the removal of the 1989 data set and the cumulative
data point where the interval of application is greater than 10 years ................ 44

4.3 R-squared values indicating the correlation between the observed erosion and the
ELP Index for the case of no setup, including winter storms............................ 45

4.4 R-squared values indicating the correlation between the observed erosion and the
ELP Index for the case of including setup and no winter storms.................... 47

4.5 R-squared values indicating the correlation between the observed erosion and the
H E I. ..................................................................................................................... 50

4.6 R-squared values resulting from the removal of the 1989 data set and the cumulative
data point where the interval of application is greater than 10 years .............. 52

4.7 R-squared values indicating the correlation between the observed erosion and the
M H E I.................................................................................................................. 56

5.1 Relevant site characteristics ........................................ ................ .............................. 68

5.2 Maximum and minimum average monthly potential shoreline change values
calculated from Eq. (4.19) using the WIS data set. Values are reported in feet. 73

5.3 Maximum and minimum average monthly potential shoreline change values
calculated from Eq. (4.19) using buoy data. Values are reported in feet............ 78

5.4 Range of potential shoreline change values based on individual extreme values.
Values are reported in feet...................................................... .................... 87

LIST OF FIGURES

Figure Page

1.1 Storm surge and high water marks for Hurricane Opal. Calculation of the setup
utilizing associated wave conditions indicates a large portion of the increase in
the mean high water marks with respect to the measured surge can be
attributed to setup................................................................................................ 4

2.1 Definition sketch of the model hurricane................................................ ................ 7

2.2 Plan view of the pressure field associated with the model hurricane............................ 8

2.3 Plan view of the gradient wind field associated with the model hurricane................... 9

2.4 Illustration of the determination of trfrom a typical storm surge hydrograph................ 12

2.5 Variation of hei with non-dimensional distance from the hurricane center ................ 14

3.1 Sites selected for aerial photograph analysis............................................ ............ ... 17

3.2 Example of the aerial photograph analysis technique .............................................. 20

3.3 Shoreline positions at the St. Augustine Site ........................................... ............ ... 22

3.4 Shoreline positions at the St. Johns Site................................................................ 23

3.5 Shoreline positions at the Volusia North Site .................................... ............ 23

3.6 Shoreline positions at the Volusia South Site .................................... ................ 24

3.7 Shoreline positions at the Martin County Site ..................................... ............... 24

3.8 Comparison of the average shoreline positions at each of the five Sites ........................ 25

4.1 D definition sketch of a, 0, and ............................................ .............................. 30

4.2 D definition sketch of sin_rotwvect ................................ .................................................. 30

4.3 D definition sketch of o and ................................................................ ..................... 31

4.4 Example of a recreated storm surge hydrograph using the Mayport, Florida data. In
this case two distinct surge events are observed. ............................................ 34

4.5 Plot of ELP Index versus erosion at the St. Augustine Site. Setup and winter storms
are not included. ............................................................................................. 40

4.6 Plot of ELP Index versus erosion at St. Johns Site. Setup and winter storms are not
included. ........... ...................................................... ..................................... 41

4.7 Effect of removing all four points associated with the 1989 erosional event from
Figure 4.5. ............................................................... .................................... 41

4.8 Effect of removing all four points associated with the 1989 erosional event and the
cumulative point associated with the period 1960-1971 from Figure 4.6........... 42

4.9 Plot of ELP Index versus erosion at Martin County Site. Winter storms are included,
setup is not..................................................................................................... 45

4.10 Plot of ELP Index versus erosion at the Volusia North Site. Setup is included,
winter storm s are not......................................... ................ .......................... 48

4.11 Plot of ELP Index versus erosion at the Volusia South Site. Setup is included,
winter storm s are not. ...................................................... ....................... 48

4.12 Plot of HEI versus erosion at the St. Johns Site....................................... ............ 51

4.13 Effect of removing all four points associated with the 1989 erosional event and the
cumulative point associated with the period 1960-1971 from Figure 4.12......... 53

4.14 Definition sketch for the calculation of the change in shoreline position resulting
from an increased water level, S, and wave setup, ....................................... .. 54

4.15 Plot of MHEI versus erosion at the Volusia South Site .............................................. 56

4.16 Plot of MHEI versus erosion at the Volusia North Site .............................................. 57

5.1 Graphical depiction of the effect of storm sequencing on the perceived effectiveness
of the erosion indices. In both cases the value of the erosion index will be the
same however the final position of the shoreline will be much different. ......... 59

5.2 Application of Eq. (5.3) for the case ke=ka=l and no external forcing ....................... 62

5.3 Application of Eq. (5.3) for the case of ke=ka=l with external forcing applied at t=0
and t= 10 ............................................................................................. ............ 63

5.4 Application of Eq. (5.3) for the case of ka=ke=5 and no external forcing................... 63

5.5 Application of Eq. (5.3) for the case ke=1, ka=0.2 and no external forcing ................ 64

5.6 Application of Eq. (5.3) for the case of ke=1, ka=0.2 with external forcing applied at
t=0 and t= 10 ...................................................................................................... 64

5.7 Site and data station location map................................................ .......................... 67

5.8 Average shoreline positions at Jupiter and Westhampton as reported in Dewall (1977)
and D ew all (1979).............................................................. ....................... 72

5.9 Average monthly potential shoreline change values. S represents the measured tide.
Eta is the wave induced setup effect. ......................................... ............ ... 72

5.10 Envelope of average monthly potential shoreline change values at Jupiter Island
calculated from Eq. (4.19) using the WIS data set, assuming the setup is
included implicitly. Each line represents a single year of the 20 years with
available data....................................................................................................... 74

5.11 Envelope of average monthly potential shoreline change values at Jupiter Island
calculated from Eq. (4.19) using the WIS data set, including the setup
explicitly. Each line represents a single year of the 20 years with available
data. ................. ................................................. ... .. ................................ 74

5.12 Envelope of average monthly potential shoreline change values at Westhampton
calculated from Eq. (4.19) using the WIS data set, assuming the setup is
included implicitly. Each line represents a single year of the 20 years with
available data ................................................................................................ 75

5.13 Envelope of average monthly potential shoreline change values at Westhampton
calculated from Eq. (4.19) using the WIS data set, including the setup
explicitly. Each line represents a single year of the 20 years with available
data. ..................................................................................................................... 75

5.14 Envelope of average monthly potential shoreline change values at Jupiter Island
calculated from Eq. (4.19) using buoy data, assuming the setup is included
implicitly. Each line represents a single year of the 11 years with available
data............................................................................................................... 76

5.15 Envelope of average monthly potential shoreline change values at Jupiter Island
calculated from Eq. (4.19) using buoy data, including the setup explicitly.
Each line represents a single year of the 11 years with available data .............. 76

5.16 Envelope of average monthly potential shoreline change values at Westhampton
calculated from Eq. (4.19) using buoy data, assuming the setup is included
implicitly. Each line represents a single year of the 9 years with available data. 77

5.17 Envelope of average monthly potential shoreline change values at Westhampton
calculated from Eq. (4.19) using buoy data, including the setup explicitly.
Each line represents a single year of the 9 years with available data........... 77

5.18 Potential shoreline change at Jupiter Island calculated using buoy data, assuming the
setup is im plicitly included ................................... ......................................... 81

5.19 Potential shoreline change at Jupiter Island calculated using buoy data, explicitly
calculating the setup............................................................. ...................... 82

5.20 Potential shoreline change at Westhampton calculated using buoy data assuming the
setup is implicitly included ................................................... ................. 82

5.21 Potential shoreline change at Westhampton calculated using buoy data, explicitly
calculating the setup. ........................................................................................... 83

5.22 Potential shoreline change at Jupiter Island calculated using WIS data, assuming the
setup is implicitly included. ...................................................................... 83

5.23 Potential shoreline change at Jupiter Island calculated using WIS data, explicitly
calculating the setup...................................................... .... .................. 84

5.24 Potential shoreline change at Westhampton calculated using WIS data assuming the
setup is implicitly included ................................... .................................. 84

5.25 Potential shoreline change at Westhampton calculated using WIS data, explicitly
calculating the setup. .............................................................. .................. .. 85

6.1 Illustration of the potential application of the shoreline change rate equation for
determination of the appropriate rate constants. In this case ka=0.2 and
ke= 1.0. .................................................................... ............................... 94

6.2 Determination of error at a single point ............................................... .............. 95

6.3 Potential results of trial and error solution for determining ka and ke............................. 96

Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

DEVELOPMENT OF EROSIONAL INDICES AND A SHORELINE CHANGE RATE
EQUATION FOR APPLICATION TO EXTREME EVENT IMPACTS

By

Jonathan K. Miller

August 2001

Chairman: Robert G. Dean
Major Department: Civil and Coastal Engineering

Several indices are developed for the prediction of the erosive potential associated

with extreme events. The indices utilize readily available storm parameters. The

proposed indices are compared with historical shoreline changes as determined from

aerial photographs. Analysis of the results obtained from applying the indices to predict

the historical shoreline changes leads to the proposal of a shoreline change rate equation.

The shoreline change rate equation is a function of the displacement from the equilibrium

shoreline position and rate constants with different values for erosion and accretion.

Seasonal shoreline changes are investigated in an effort to determine the unknown

parameters of the rate equation.

CHAPTER 1
INTRODUCTION

The population of America, and indeed the world, is in love with the beach.

Although true throughout the world, nowhere is this more true than in the United States.

The thirty coastal states (including those bordering the Great Lakes) contain 62 percent of

the U.S. population and twelve of the thirteen largest cities. In all, 53 percent of the U.S.

population lives within 50 miles of the shore (Edwards, 1989). It has been reported that

the United States largest trade surplus is the tourism industry with beaches accounting for

a large percentage of this surplus. Several states, Florida, California, and New Jersey

among them, list tourism among their fastest growing industries (Douglass, Scott L.,

University of South Alabama, personal communication, 2001). More people visit Miami

Beach per year than the three largest national parks combined (Douglass, Scott L.,

University of South Alabama, personal communication, 2001). With such a large social

and economic stake in the health of our nations shorelines, beach protection and

preservation are a national concern.

Coastal Erosion Processes

Coastal erosion is the result of numerous processes, some of them yet to be fully

understood. In the most general sense, erosion is the result of sand being removed from

the beach either by man or nature. Although many natural processes tend to erode the

beaches, they usually restore them as well. In many cases, the single largest cause of

beach erosion is man made structures that interrupt the dynamic natural processes. By

removing sand from the natural system whether directly through sand mining/dredging,

or indirectly through inlet stabilization, breakwater construction or any number of other

construction activities, the result is often chronic erosion. As coastal engineers, we are

too often faced with the task of attempting to reintroduce sediment into a system in which

we are the primary cause of the sediment deficit.

Aside from anthropogenic effects, there are many natural processes that induce

erosion. Numerous studies have been undertaken on the shoreline changes resulting from

various natural processes. Everything from wind speed, ground water table, and sea

temperature, to more intuitive processes such as wave action and sea level rise has been

studied. The most damaging erosion however is caused by extreme storm events. Long-

term erosional trends can be addressed by a number of coastal engineering remedies. The

sudden, catastrophic erosion associated with hurricanes and winter storms however, is

difficult to predict and even more difficult to prepare for. Extreme event induced erosion,

and methods to quantify it through the development of a predictive erosion index,

comprise the main focus of this thesis.

Much of the coastal damage associated with hurricanes can be attributed to the

massive storm surges they produce. In many cases, surges of over 15 feet have been

reported with the maximum being 22.4 feet at Pass Christian, Mississippi caused by

Hurricane Camille in 1969 (Dean and Dalrymple 2001). In addition to causing beach

erosion, storm surge often leads to substantial inland flooding which is the primary cause

of associated with hurricanes. This deadly surge can be decomposed into four separate

components that combine during a hurricane to form a massive wall of water. Extreme

low barometric pressure at the hurricane's center acts to "suck up" the water surface with

respect to water around it. The enormous winds associated with hurricanes create a shear

stress on the water surface, in effect pushing the water ahead of it until it is forced to pile

up at a boundary, the beach. The strong winds also create large longshore currents that,

combined with the Coriolis force, either reinforce (in the case of southerly currents on the

east coast in the Northern hemisphere) or negate the previous two effects. The fourth

component is an increase in the mean water level caused by increased wave action and

termed the wave setup. In general the wind shear stress is the dominant term, although in

some cases such as Hurricane Opal in 1995 and in the case of winter storms, the wave

setup can actually dominate. Figure 1.1 illustrates the important contribution of wave

setup during Hurricane Opal. Nielsen (1988) has shown that wave induced setup values

of 40 percent of the deepwater Hn value for waves between 2 and 8.5 feet is not

unusual.

When investigating extreme event induced erosion and the subsequent shoreline

recovery, the different time scales of the processes involved must be considered. In

general, erosion occurs at a much faster rate than accretion, hence it can take a beach a

substantial amount of time to recover from a relatively short duration storm event.

Several models exist that predict erosion rates under different conditions with various

degrees of success, but to the author's knowledge, an adequate shoreline recovery model

has yet to be developed. There are vast complexities governing the dynamics of the

fluid-particle interactions, however as with most dynamic processes, a quasi-equilibrium

state is presumed to exist. How quickly a shoreline reaches this equilibrium state

depends on both the time scale of the processes involved, and the magnitude of its

displacement from the equilibrium position. An equation expressing the shoreline

4

response as a function of the equilibrium position and the time scales of the governing

processes will be proposed later in this thesis.

OPAL Ir---- --
STORM SURGE
HYDROGRAPH "*' '9 mm,

Peak Storm
i .. ...; .r ; srg .hy fl:fr ;a : .... .

HURR OPA L Hgh at rMr
0 W 2 24 h .2' ,
I-I

10a3 1014 1015 1016
TIME (hr#,UTC);ala flng 0:00 on 10/03/95
Figure 5. Storm surge hydrograph for Opal at Panama City Beach

HURRICANE OPAL- High Water Marks

Cr.

4 0* 16 20 oMNLES
S OJ----------------- ^n
Figure 6. Surveyed high water marks after Opal long the Panhandle Coast

Figure 1.1 Storm surge and high water marks for Hurricane Opal. Calculation of the
setup utilizing associated wave conditions indicates a large portion of the increase in the
mean high water marks with respect to the measured surge can be attributed to setup
(Leadon et al. 1998).

As if this problem were not complex enough, the equilibrium state itself is

unsteady. Any attempt to sufficiently define shoreline response rates must first

5

sufficiently define the equilibrium position. Even the casual beachgoer knows that the

beach tends to be narrowest in the winter and widest during the summer. This is part of

the beach's natural defense mechanism to help dissipate the increased winter wave

energy more efficiently. The second objective of this thesis is to examine these seasonal

shoreline fluctuations with the goal of developing an expression for the equilibrium

shoreline position for input into the proposed rate equation.

Objectives and Scope

The main objective of this project is to develop an index capable of correlating

storm characteristics to potential shoreline changes. To achieve this, several previously

defined indices have been modified and applied to historical events in an effort to predict

the observed shoreline changes. The results obtained during the investigation of the

primary objective led to the development of a secondary objective. The secondary

objective is to develop an equation that accurately predicts the rate of shoreline response

to non-equilibrium conditions, while accounting for the different time scales of the

processes involved. Seasonal shoreline changes were investigated for potential

application to the rate equation model.

The scope of the investigation varies. The spatial scope of the erosion index

portion of the project is limited to five sites located in three counties within Florida. The

initial phase of the hurricane index investigation addresses the time period from 1950-

1995, while the second phase dealing with the modified erosion indices is limited by the

available data to 1976-1995.

The spatial scope of the secondary objective is limited to two sites: Westhampton,

New York and Jupiter Island, Florida. In both cases the maximum time period of

analysis is limited by the available wave data to1976-1995.

The scope of the primary objective is limited to attempts at correlating historical

events to the observed shoreline changes though the application of several existing

hurricane erosion indices. Based on the data available and the results achieved,

expansion to statistical parameters and predicted events is not warranted at this time.

The scope of the secondary objective is limited to the suggestion of an equation to

describe the varying rates of erosion and accretion. An attempt is made to define a

seasonally varying equilibrium shoreline position for use in this rate equation. Further

analysis and more data are needed to determine the appropriate constants for application

in the shoreline change rate equation.

CHAPTER 2
LITERATURE REVIEW

Model Hurricane

Of intrinsic importance to this study is the characterization of a model hurricane.

Wilson (1957) provides the characterization of the model hurricane utilized throughout

this thesis. Figure 2.1 provides a definition sketch.

yvt

Figure 2.1 Definition sketch of the model hurricane

The model hurricane has both a characteristic pressure field and wind field associated

with it. The pressure field is composed of concentric isobars and is defined by the

equation

p(r) = po + Ape-R/r (2.1)

where po is the central pressure, R is the radius to maximum winds, r is the distance from

the location of interest to the hurricane center, and Ap is the central pressure deficit

defined by

Ap = p. -po (2.2)

in which po is the ambient pressure at a location sufficiently far as to be free from the

influence of the hurricane. Figure 2.2 provides a plan view of the pressure field

associated with the model hurricane.

10"
8 .-

4 "
2
S0

.Z -2 I-
-4 '-

-10
-10 -8 -6 -4 -2 0 2 4 6 8 10
Non-Dimensional Distance, x/R
Figure 2.2 Plan view of the pressure field associated with the model hurricane

The wind field associated with the model hurricane is more complex. Three

related wind speeds are defined: cyclostrophic, geostrophic, and gradient. The

cyclostrophic wind speed, Uc, is a balance between the pressure field gradients and the

centripetal force, neglecting the effects of friction and the Coriolis force. The

cyclostrophic wind field is composed of concentric isovels and is given by

U /AP Re (2.3)
Pa r

where pa is the mass density of air. The geostrophic wind speed, Ug, is the speed that

would occur if the pressure field were in balance with the Coriolis force and is defined as

Ap R2 -
U = Par (2.4)
g= 2o)R sin <

where to and 0 are the rotational speed of the earth in radians per second and the latitude

of the location of interest respectively. The gradient wind speed, W, is the speed of

interest in most calculations, and is the wind speed measured approximately 30 feet

above the water surface. The gradient wind speed is related to the cyclostrophic and

geostrophic wind speed through the parameter y

1 Vf sin p U
Y= + c (2.5)
2 U U
S c g)

where Vf is the event forward speed and 1 is the angle defined in Figure 2.1. The

gradient wind speed is then defined in terms of y and Uc as

W= 0.83U,( y+ ) (2.6)

where the gradient wind vectors are rotated inward by approximately 180. The gradient

wind speed is the speed utilized in all of the models. Figure 2.3 represents a plan view of

the gradient wind speed associated with the model hurricane.

10. '

-10 -8 -4 -4 -2 0 2 4 6 8 10
I r)-- -f
8u

0 -2
Z -8 I

-10 -8 -6 -4 -2 0 2 4 6 8 10
Non-Dimensional Distance, x/R
Figure 2.3 Plan view of the gradient wind field associated with the model hurricane

Bathystrophic Storm Surge

Freeman, Baer, and Jung (1957) developed a simple method for calculating the

one-dimensional storm surge due to a translating atmospheric disturbance. They made

use of the following assumptions to reduce the problem to one-dimension.

1. Minimal cross-shore transport takes place.

2. Divergence of the velocity field does not significantly affect the height of
the water surface.

3. Change in height in the alongshore direction is assumed negligible.

4. Space derivatives of the current are assumed negligible when compared to
the Coriolis force.

Assumption one necessitates the existence of a compensating alongshore current termed

the bathystrophic current. Assumption two is a result of the fact that any "bumps"

created by a diverging velocity field spread out at a faster rate than the velocity field can

build them up. Assumption three is a common assumption in many models. The final

assumption is justified by the small magnitude of the currents involved relative to the vast

spatial scales.

Applying the above assumptions to the depth integrated equations of motion

results in the following differential equations

81 1 [Tw x 1 1 8p
x- gD p, Pg x(2.7)

dq- 1 ( _by) (2.8)
dt p,

where x and y represent the cross-shore and long-shore directions respectively, g is

gravity, ri is the water surface elevation, D is the total local depth, Tw and Tb are the

surface wind and bottom shear stresses respectively, e is the Coriolis parameter, pw is the

density of water, p is the atmospheric pressure, and q is the volumetric flow per unit

width. Chiu and Dean (1984) used the bathystrophic surge model as the basis for a one-

dimensional model against which a larger two-dimensional model was calibrated.

Rewriting Eqs. (2.7) and (2.8) in finite difference form yields

1 At 1
q n+I q + -Pw (2.9)
qy, BB. p, I

axI n+I Pi -pi+l
Ix+lI Ax Ii 1 p (2.10)
gDi L Pw Pwg1

fAt qy
BB = 1.0+ (2.11)
Di2

where the boundary conditions include initiation from rest (qy=0 & T=0) and Ir due to

barometric pressure alone at the seaward edge of the grid.

A= (2.12)
Pwg

Erosion Models

Balsillie (1985, 1986, 1999) utilizes empirical methods to determine the

volumetric erosion due to 14 erosional events associated with 11 major hurricanes, and

22 erosional events associated with other severe storms, with good success. Balsillie's

equation incorporates a measure of the event longevity, tr, and gives the volume per unit

length eroded above MSL as

Qe = 1 6 t,S2 (2.13)
1622

where tr is the storm tide rise time, and S is the combined peak storm tide elevation

including both the astronomical tide and the dynamic wave setup. The storm tide rise

time is defined as "the final continuous surge of the storm tide representing impact of the

event at landfall" (Balsillie 1999, p.8), and does not include pre-storm setup. Figure 2.4

illustrates the definition of rise time as determined from a storm surge hydrograph.

Balsillie also parameterized the storm tide rise time in terms of a more readily available

quantity, the event forward speed as

t, = 0.00175 g (2.14)
V,

where g has units of length per hour squared, the coefficient has units of hours squared,

and Vf has units of length per hour and is measured at the time the radius of maximum

winds makes landfall.

7*3y1-- --' -- --i-- *--i-- :~
Storm Tide

4 Pre*-Storm
Elevation Lo S tup
Above MSL 3
(m)
2 Normal Pro-Storm
Tidal Conditions

o a------ -----~- ,-~ ,-----

20 4 0 60 0s
Time cours)

Figure 2.4 Illustration of the determination of tr from a typical storm surge hydrograph
(Balsillie, 1999)

Dean (1999) presents a different method for calculating the erosive potential of a

storm. He calculates a Hurricane Erosion Index (HEI) based primarily upon the Bruun

Rule given by Eq. (2.15).

Ay = -S (2.15)
(h. +B)

Here the shoreline change Ay is dependent on the change in water level, S, the berm

height, B, and the width and depth of the active profile, W* and h* respectively. Relating

h* and W* through equilibrium beach profile concepts, Eq. (2.14) can be rewritten

h^
Ay = -S- (2.16)
AY(I1+ h

where A is the sediment scale parameter and has units of length to the one-third power.

Dean argues that the storm surge is dominated by three primary factors: onshore

wind stress, pressure reduction, and wave set-up. Recognizing that two of the three

effects are proportional to the wind speed squared (wind stress and pressure reduction),

and that the other is directly proportional to the wind speed, Dean proposes

S(x) o W2 cosOdy (2.17)
Y1

where 0 is defined as the angle between the rotated wind vector and the shoreline normal.

Recognizing that the quantity h* is also proportional to the wind speed, Dean defined the

Hurricane Erosion Index (HEI) as

T1 T1 W (xy)cos~dxdy To ,
HEI=' T W (xy)cosdxdy= hei(x)dx (2.18)
0 0 Vf 0

where R is the radius to maximum winds, and Vf is included as a measure of the time a

given storm acts upon the adjacent shoreline. Lower case hei is a local erosion index,

which can be integrated alongshore to obtain the global hurricane erosion index, HEI.

Figure 2.4 depicts the variation of hei with distance from the storm's center.

1.6
1.4

1.0
:1 0.8
o.08
0.4
0.2
0.0, 24 ,i
0 1 2 3 4 5 6 7 8 9 10
Non-Dimensional Distance, x/R
Figure 2.5 Variation of hei with non-dimensional distance from the hurricane center

Shoreline Recovery Model

Kriebel and Dean (1993) present a method of calculating the time varying beach

profile response to a given forcing function. Based upon laboratory observations, an

approximate equation for the time dependent beach response to a steady state forcing

function can be written as

R(t) = Rj 1-e-/t (2.19)

where Ro represents the maximum or equilibrium response, and Ts is the representative

time scale of the response. It then follows from the differential equation that the rate of

recovery is proportional to the difference between the instantaneous and equilibrium

profile responses

dR(t) 1 (R.(t)-R(t)) (2.20)
dt T,

which can be integrated analytically for constant Ts and various Ro(t).

Seasonal Shoreline Change

Dewall (1977, 1979) studied seasonal shoreline fluctuations at Jupiter Island,

Florida and Westhampton, Long Island. Dewall observed the beach at both locations

tended to advance seaward during the summer, and retreat shoreward during the winter in

response to the fluctuations in local wave energy. Unfortunately, Dewall's study may

have been biased by several factors:

1. The construction of groins and beach nourishment projects during the study.

2. The insufficient spatial density of the surveys (2 Jupiter profile lines separated
by only 250 feet).

3. The insufficient temporal density of the surveys (3 data points in 10 years
defines July average position at Westhampton).

4. An admitted biasing of the surveys towards winter post-storm events
(particularly Westhampton).

In spite of this less than perfect data set, a significant correlation was observed

between the breaking wave conditions and the observed shoreline fluctuations.

Shoreline advance tended to be associated with the mild wave conditions more

prevalent during the summer months, and retreat with more energetic winter

conditions. Dewall's study encompassed both long and short-term behavior. In

general, an inverse relationship between the magnitude of the fluctuations and their

associated time scales was observed. The largest shoreline changes occurred on the

smallest time scales as a result of individual storms, while the smallest changes were

observed over the longest time scales, and were associated with changes in the yearly

mean shoreline position. Within these extremes, Dewall also observed large cyclical

fluctuations of up to 90 feet in one year at Westhampton and 70 feet in one year at

Jupiter. A later study of several New Jersey beaches by Everts and Czeriak (1977)

supports Dewall's results.

CHAPTER 3
SHORELINE CHANGE ANALYSIS

Dean, Cheng, and Malakar (1998) have shown that on average, Florida's east

coast beaches are accreting at a long-term, pre-nourishment rate of approximately 4

inches per year. This conclusion is based on the analysis of shoreline data obtained from

a database maintained by the Florida Department of Environmental Protection, Office of

Beaches and Coastal Systems (OBCS). The OBCS data consists of surveys referenced to

fixed monuments located approximately every 1000 feet along Florida's sandy

shorelines. Because the OBCS data set is too sparse temporally to capture the short-term

dynamic behavior of the beaches needed for this study, aerial photographs from several

sources are analyzed to obtain a more temporally dense coverage over the study period.

Considerable photographic data exists; however, the use of many of these photographs

was precluded by their cost.

Five sites within three counties were selected as locations representative of typical

Florida East coast conditions. Site selection was based on three primary criteria. First

and foremost, locations representative of the entire east coast were desired. In order to

achieve this, an attempt was made to include widely spaced locations. Secondly, the sites

should be as free from anthropogenic effects as possible. This criterion eliminates areas

within the limits of nourishment projects or within the influence of inlets, and

consequently makes the fulfillment of the first criteria much more difficult. This is

particularly true in the case of south Florida. Photograph availability and suitability was

the final criterion. Suitability incorporates, photo quality, scale, and the location of a

sufficient number of control structures within the available photographs. Figure 3.1

depicts the five selected sites based on fulfillment of the above criteria. Within each site

several locations were selected for analysis to ensure accuracy and to eliminate any

anomalous trends. In order to make the analysis easier to follow, an attempt has been

made to consistently refer to the locations within each county as Sites, and to refer to the

individual locations within each Site as Locations.

Figure 3.1 Sites selected for aerial photograph analysis

Site Descriptions

Volusia County

As indicated in Figure 3.1, two sites in Volusia County were selected. The

northernmost site is located in Daytona Beach and is centered about DNRBS Monument

R-76. Three locations, A, X, and O, over a several mile stretch were analyzed. Location

X is near the intersection of Route A1A and Driftwood Avenue. Locations O and A are

approximately one mile north and one mile south of Location X respectively. At all three

locations, buildings were utilized as control structures. This site is referred to as the

Volusia North Site.

The second Volusia County site is located south of Ponce De Leon Inlet in New

Smyrna Beach, and consists of three locations, X, O, and *. This site is centered about

Monument R-172 and is referred to as the Volusia South Site. Location X is at the

intersection of East 7h Street and Route A1A. Location O is approximately 1 mile south

of Location X, near the intersection of East 24h Street and Route A1A. Location is

approximately 3000 feet north of Location X, near the intersection of Ocean Avenue and

Route A1A. The centerlines of nearby roads were utilized as reference points at all three

locations.

St. Johns County

Two sites were also selected in St. Johns County. The northernmost site, located

north of St. Augustine Inlet and centered about Monument R-117, is referred to as the St.

Augustine Site. Three locations, A, B, and C were analyzed at this site. Location C is

the southernmost location, and is situated near the intersection of Meadow Avenue and

Route A1A in St. Augustine. Locations B and A are located 2000 and 6000 feet north of

Location A respectively. The control structure at all three locations is the centerline of

Route A1A.

The second St. Johns County site is located approximately five miles north of

Mantanzas Inlet near Monument R-170. Two locations, A and B, are analyzed at this site

which is referred to as the St. Johns Site. Location B is situated at the intersection of

Mantanzas Avenue and Atlantic Boulevard in Crescent Beach. Location A is also located

in Crescent Beach, approximately 2 miles south of Location B near the intersection of

Route 206 and Route A1A. The centerlines of nearby streets were used as control

structures at both locations.

Martin County

Martin County is the least well documented of the five selected sites. In Martin

County only one site was selected due to the scarcity of control structures and the limited

availability of the photographs. The Martin Site is located one mile north of St. Lucie

Inlet on Hutchinson Island and is centered about Monument R-26. The Site spans

approximately 3.5 miles encompassing locations, A, X, and Z. Location X is adjacent to

the entrance to the Elliot Museum on Hutchinson Island. Location Z is approximately 1

mile south of Location X and less than 1 mile north of St. Lucie Inlet. Location A is

approximately 3000 feet north of Location X near the entrance to the Little Ocean Club

on Hutchinson Island. Buildings, roadway centerlines, and an inland marina were

utilized as control structures at the Martin Site.

Sources and Analysis Techniques

Aerial photographs were obtained from several sources. Most of the photographs

were obtained from the Florida Department of Transportation and the United States

Department of Agriculture, with several being obtained from other sources. Photographs

range in scale from 1:2,400 to 1:40,000. The extent of photographic documentation

varies from county to county. Photographs for Volusia County and St. Johns County are

available for the periods 1943-1992 and 1942-1998 respectively. In many cases the older

photographs for both counties were not utilized due to poor photograph quality and a lack

of suitable control structures. In Martin County, photographs were only available for the

thirty year period from 1966 to 1996. In general there is a marked increase in the

frequency of the aerial surveys in all three counties between 1980 and the present.

The photographic analysis is performed utilizing a Hewlett Packard Scanjet to

digitally capture the images. Standard photo editing software is utilized to pinpoint and

enlarge areas of interest. Shoreline location is measured from an established reference

point to the wetted sand line using the initial shoreline position as a reference. Permanent

structures such as roadway centerlines, and buildings are utilized as reference points, and

are considered fixed from photograph to photograph. In rare cases, the photo-editing

software is utilized to clarify the distinction between wet and dry sand. This is avoided

wherever possible. Due to the accuracy requirements for the current study, no attempt is

made to rectify the images.

Figure 3.2 Example of the aerial photograph analysis technique

Results and Conclusions

Figures 3.3-3.7 depict the shoreline positions at each location within each site.

Shoreline position is with respect to the shoreline location in the earliest photograph. In

examining Figures 3.3-3.7, it should be noted that the period from 1940-1950 was the

most active for Category 3 hurricanes in Florida's history (Hebert and Case 1990). Data

points obtained immediately after this decade reflect an initially eroded state, therefore

care should be taken when comparing two sets of shoreline position data. For example,

the St. Johns Site data indicates a long-term accretional trend while the St. Augustine Site

data indicates an erosive trend. A significant portion of this difference can be attributed

to the fact that the first St. Augustine data point in 1942 is prior to or near the very

beginning of this active period and may reflect a more normal beach width, whereas the

first St. Johns data point is immediately after this period and will reflect an initially

eroded state. As previously indicated, all future shoreline positions are recorded with

respect to this initial condition, leading to the difference in the long-term trends observed

in Figures 3.3 and 3.4.

In general the major shoreline trends are similar at all 5 sites, however the

magnitude of the major trends varies from site to site in response to the different local

conditions. Superimposed on these major trends are smaller more localized trends.

Examining the two sites within each county together yields some insight into the

major shoreline trends. At the St. Augustine Site and the St. Johns Site, the observed

shoreline trends are very similar. As previously mentioned, the effects of the active

hurricane period from 1941 to 1950 are reflected in the erosion indicated by the first two

St. Augustine data points. Both sites exhibit an accretional trend during the 1950's,

indicating the gradual recovery from the previous decade's erosion. The effects of two

major storm events, the Ash Wednesday Storm in 1962 and Hurricane Dora in 1964, are

reflected in the erosion indicated at both sites by the first post storm data points.

Comparison of the magnitudes of the erosional events is made difficult because the first

post storm data point at the St. Johns Site is in 1971, and reflects 7 years of recovery from

the storms. Another erosional event occurs between 1971 and 1974, and is followed

immediately by an accretional trend between 1974 and 1975. The erosion indicated by

the 1975 and 1980 data points is most likely the result of Hurricane David in 1979. The

most recent data points indicate a more naturally dynamic shoreline. This is reflective of

both the increased temporal density of the photographs and the lack of a singular severe

hurricane impact during the period. The erosion caused by two major events, the 1984

Thanksgiving Storm and the passage of Hurricanes Dean, Gabrielle, and Hugo offshore

in 1989, is indicated at both sites by the data points immediately following their

occurrence.

ST. AUGUSTINE
SHORELINE POSITION
60

30-
+ -

30
0.

-120 __

-150
1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000
Year

Figure 3.3 Shoreline positions at the St. Augustine Site

ST. JOHNS
SHORELINE POSITION
300 -

270 B B

240 -- --A

210

S180---
o 150-
U)
0
a. 120 1

90- -

60-____ ___

30

0
1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000
Year

Figure 3.4 Shoreline positions at the St. Johns Site

VOLUSIA NORTH
SHORELINE POSITION
180
--- A
150- -
-r- 0
120
120- --- --O--

90 ______

.' 60

0-

-60
-60 *----- ----- ------- --- ----- ---- ----- ---------------
1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995
Year

Figure 3.5 Shoreline positions at the Volusia North Site

VOLUSIA SOUTH
SHORELINE POSITION

-A-/ _____,______ ___
,AAst

- -~ -X~
-Z-0

-__

-_

120
90
60
30
0
-30
-60
-90
-120
-150
-180
-210
1!

1990 1995

Figure 3.6 Shoreline positions at the Volusia South Site

MARTIN COUNTY
SHORELINE POSITION

-e
-+.'

---,
....... ............ -

^.^\N. N l\__ ___ __ __

1980 Year
Year

1990

1995

2000

Figure 3.7 Shoreline positions at the Martin County Site

1945 1950 1955 1960 1965 1970 1975 1980 1985

940

90
60
30
0
-30
-60
-90
-120
-150
-180
-210
-240
1

965

COMPARISON OF AVERAGE SHORELINE POSITION
AT THE FIVE SITES
270
240 Martin -U- St. Johns
210 --St. Augustine -0-Volusia North -
180- -Volusia South

120 --

-30

-60 --

-90 30
-120-
-150
1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000
Year

Figure 3.8 Comparison of the average shoreline positions at each of the five Sites

Both sites within Volusia County also exhibit similar trends. The erosion caused

by the Ash Wednesday Storm and Hurricane Dora is reflected in the first post 1964 data

point at both sites. This erosional trend is followed by an accretional trend as both sites

recover from the Ash Wednesday and Hurricane Dora induced erosion, and accrete to

maximum seaward positions by 1978. The effects of Hurricane David are reflected in the

erosion indicated by the 1979 data points at each site. Again the most recent data points

indicate a more naturally dynamic shoreline position. The major trend indicated by the

most recent 20 years of photographs is the erosion related to the passage of Hurricanes

Dean, Gabrielle, and Hugo in 1989, and the subsequent recovery. In this case,

comparison between the two sites is made more difficult due to the different time periods

between photographs.

In Martin County only one site was analyzed due to the scarcity of photographs

containing suitable control structures. The shoreline at Locations A and X exhibit similar

behavior. The deviation of the trends observed at Location Z from those observed at

Locations A and X is likely due to the fact that Location Z is too close to St. Lucie Inlet.

It appears likely that Location Z is more affected by the inlet dynamics than the local

shoreline change trends, particularly in the case of small short-term trends. Locations A

and X experience an erosional trend between 1971 and 1974, but recover and accrete to

maximum seaward positions in 1976. Once again, the effects of Hurricane David are

observed in the erosion indicated by the 1980 data points. Between 1980 and 1996 the

shoreline changes are much milder than in the preceding decade with all three locations

exhibiting an erosional trend between 1983 and 1986 followed by an accretional trend

between 1986 and 1992. It should be noted that the 1992 data points are based upon

photographs taken in January and thus do not reflect the impact of Hurricane Andrew.

Averaging the several locations at each site and plotting the average shoreline

positions at each site as in Figure 3.8 yields some interesting results. Several major

trends are be observed. The erosion indicated by the data points in the early 1940's and

the early 1950's reflects the impact of the abnormally large number of Category 3

hurricanes making landfall during the 1940's. The erosion caused by Hurricane Dora and

the Ash Wednesday Storm is reflected in most of the post 1964 data points. The

relatively calm period between 1964 (Dora) and 1979 (David) is indicated by the

accretional trends exhibited at all five sites. Most sites accrete to a maximum beach

width between 1975-1978. The influence of Hurricane David at all five sites is exhibited

in the erosion indicated by the first post 1979 data points. In all cases, the most recent

data appears to indicate a more dynamic shoreline due to closer spacing of the

photographs and the lack of a large number of extreme storm events. The only major

trend exhibited at all 5 sites in the last 20 years is the erosion resulting from the passage

offshore of Hurricanes Dean, Gabrielle and Hugo in 1989.

Based on Figures 3.3-3.7 shoreline change rates may be calculated. Individual

shoreline change rates range from -6.5 feet per year at Location Z of the Martin Site to

+4.6 feet per year at Location B of the St. Johns Site. Based on the individual trends, an

average shoreline change rate of -0.05 feet per year is calculated. Removing Location Z

of the Martin Site (due to the expected influence of St. Lucie Inlet) from the average

calculation results in an average shoreline change rate of +0.41 feet per year, or

approximately 5 inches per year. This number agrees well with what Dean has reported.

In analyzing these results, it is interesting to note that shoreline trends are largely

dependent on the initial position chosen for comparison. Choosing an initially eroded

state will most likely yield an accretional trend while choosing an initially accreted state

will most likely result in the calculation of an erosional trend. In general the layperson

does not appreciate the magnitude of these naturally dynamic conditions. The shoreline

change data presented here indicates natural shoreline fluctuations of over 200 feet in the

past 50 years at several locations. It is interesting to note that the beginning of the coastal

construction boom in Florida, during the mid to late 1970's, coincided with the period of

maximum beach width.

CHAPTER 4
EROSION INDICES

As discussed in the introduction, the primary objective of this thesis is to correlate

observed shoreline fluctuations with readily available storm parameters. The goal is to

develop an index capable of accurately predicting the erosive potential associated with

extreme storms. Dean's hurricane erosion index and Balsillie's equation for calculating

volumetric erosion are modified and applied in an attempt to develop an index capable of

predicting the observed shoreline changes caused by historical storms.

Input data common to both models includes shoreline position data obtained from

the OBCS database and hurricane data obtained from the Colorado State University

hurricane database. Additional storm parameters are obtained from the National

Hurricane Center website and NOAA Technical Report NWS 15 (Ho, Schwerdt, and

Goodyear, 1975). Hurricane track information is first translated from latitude-longitude

coordinates to Florida State Plane East coordinates using the US Army Corps of

Engineers' Corpscon program. Once the hurricane tracks are converted, all hurricane

positions with negative coordinates, indicating locations well south and west of the study

area, are eliminated. This procedure limits the hurricane data set to 337 hurricanes and

major tropical storms occurring during the study period from 1950 to 1997. Of the

required hurricane parameters including: location, forward velocity, maximum winds,

radius to maximum winds, and central pressure, only the radius to maximum winds and

central pressure are not consistently recorded. In the case of central pressure, a sufficient

number of records exist, such that linear interpolation to determine the missing values is

valid. The missing radius to maximum wind data is more problematic. A large number

of records are missing this vital parameter. The statistical analysis presented in Ho,

Schwerdt, and Goodyear is utilized to assign all hurricanes lacking a recorded radius to

maximum winds a value of 23.5 miles, corresponding to the approximate median radius

of landfalling hurricanes along Florida's East coast. It is recognized that this assumption

limits the index accuracy; however, lacking more complete data, it is required.

A brief review of the geometry utilized in the development and application of the

indices will facilitate future discussions. The base coordinate system is the Florida State

Plane East coordinate system with the East axis corresponding to the x-axis and positive

offshore, and the North axis corresponding to the y-axis and positive north. Most angles

are defined with respect to the x-axis and are positive counterclockwise, although

exceptions exist and will be noted. The shoreline orientation defined by the angle a is

one such exception. The angle a is defined as the angle between the shoreline and the y-

axis and is clockwise positive. Several angles defined in the conventional manner and

used throughout include 0, sin, Osrotwvect, and P. The angle 0 defines the hurricane

location with respect to the base coordinate system. Oin is the angle between the radius

and shoreline normal and is defined in terms of 0 and a as

0,, =0+a (4.1)

The angle of interest is the angle of the gradient wind vector with respect to the shoreline

normal. In terms of Osin, this angle, Osin_rotwvect, is defined as

rotwv = Os, 90 +18" (4.2)

The 18 accounts for the inward rotation of the gradient wind vectors due to frictional

effects. The angle 3 is defined as the angle between the velocity vector and the radius,

and is given in terms of o and 0 as

3 = 180 -o0+ (4.3)

Here o defines the direction of the velocity vector. Figures 4.1-4.3 provide definition

sketches of the problem geometry.

.N (delE.delN)
Shore
R Normal

E

Figure 4.1 Definition sketch of a, 0, and %in

irmal

Rotated
Wind
Vector

Figure 4.2 Definition sketch of 0sin_rotwvect

Figure 4.3 Definition sketch of ( and 13

Event Longevity Parameter Index (ELP)

Theory

The Event Longevity Parameter (ELP) Index is an application of Eq. (2.13). The

index name refers to the inclusion of a measure of the event longevity, the storm tide rise

time, which led Balsillie to originally refer to the parameter raised to the four-fifths

power in Eq. (2.13) as the "event longevity parameter". Required input parameters for

the ELP Index are the maximum storm surge and the associated storm surge rise time.

Ideally the surge and rise time can be determined from storm surge hydrographs as

indicated in Figure 2.4. Due to the large scope of the project (337 hurricanes), and the

scarcity of available storm surge hydrographs, alternative methods of determining the

required parameters are employed. The maximum surge associated with each hurricane

is determined from a one-dimensional storm surge model. The associated rise time is

calculated from the event forward speed according to Eq. (2.14).

Because most winter storm related surge is caused by wave induced setup, an

alternate method of determining the surge related to winter storms is required. The local

wind conditions utilized in the storm surge model often give little indication of winter

storm related surge. Artificial hydrographs are created for the winter storms based on the

difference between the observed and predicted water levels. Both methods of

determining the surge are described in detail in the next section.

Once the storm surge and rise time have been determined, the ELP Index is

defined as

ELP= 2 (S jtg'5 (4.4)
1622

where S is in feet, tr is in hours, and g is in feet per hour squared. In order to more

accurately represent the dynamic conditions associated with a hurricane, the tracks are

interpolated to hourly intervals, prior to the application of the ELP Index.

There are several minor differences between the ELP Index and Balsillie's

original equation. Balsillie defines S as the maximum storm tide including astronomical

effects. Here, S is taken as the maximum storm surge in absence of astronomical effects.

The storm surge is used in place of the storm tide because most of the surge values are

calculated from a storm surge model. Since Eq. (4.4) is intended to be an index,

obtaining the astronomical tides and synchronizing the results of the surge model to the

astronomical tides is thought to be unnecessary at this stage. Another difference between

the index and the original equation is in the determination of the event rise time. Balsillie

defines the forward velocity, Vf, in Eq. (2.14) as that corresponding to the time at which

the radius to maximum winds makes landfall. Since in the ELP Index the surge data are

calculated from a model, it is possible that the radius to maximum winds never makes

landfall. Here, the instantaneous velocity corresponding to the time at which the

maximum surge occurs is used in the calculation of the storm surge rise time.

The ELP Index is applied for a total of three cases. In the first case a one-

dimensional storm surge model is applied without including the effects of setup. In the

second case, the same model is applied but several winter storms are introduced. The

third and final case considers only hurricanes and utilizes the same surge model but

includes wave setup.

Winter Storm Related Surge-Reconstructed Hvdrographs

The main component of the surge generated by winter storms is the wave induced

setup. A storm surge model based upon local conditions, as applied to the hurricane data,

cannot capture the effects of remotely generated swell. Reconstructed hydrographs are

developed for all 25 included winter storms, utilizing observed water level data and

predicted tidal elevations. The reconstructed surge hydrograph is simply the observed

minus the predicted water level. Figure 4.4 is an example of a hydrograph created using

this technique. Maximum surge elevations and rise times are obtained directly from the

reconstructed hydrographs as indicated in Figure 4.4.

Tide predictions are obtained from National Oceanic and Atmospheric

Administration (NOAA) tide tables and the Mote Marine Laboratory website. (The

website data are comparable to the NOAA tables but are electronically formatted, making

them easier to analyze.) Observed water levels are obtained from the NOAA Center for

Operational Oceanographic Products and Services (CO-OPS) database. Unfortunately,

the CO-OPS data are incomplete for many of the Florida stations. Mayport is the station

nearest to all three sites that contains a complete data set; therefore the artificial

hydrographs are created based upon conditions at Mayport. It is recognized this will

introduce some error into the indices particularly at the Martin County Site, however

because the indices are only intended to compare the effects of various storms at a single

location and are not intended to compare the effects of individual storms between

locations, it is hoped that any error will be reasonably small.

Ash Wednesday Storm
March 1962

18
16
14
12

t-
F 10
I 8

m 6
4
2
0
-2

1 4 7 10 13 16 19
Day

Figure 4.4 Example of a recreated storm surge hydrograph using the Mayport, Florida
data. In this case two distinct surge events are observed.

Hurricane Related Surge-Bathystrophic Storm Surge Model

In order to calculate the storm surge associated with passing hurricanes, a one-

dimensional storm surge model is employed. The storm surge created by a hurricane can

be considered as the sum of four individual components: wind stress, barometric pressure

reduction, Coriolis contribution, and wave setup. Generally the Coriolis and pressure

effects are small compared to the wind stress and wave setup components. The one-

dimensional model employed here does not include the effects of wave setup, therefore it

is calculated explicitly utilizing WIS hindcast data. Unfortunately, the WIS data prior to

Observed+10'
Surge=2.04' Surge=1.9' ------ Predicted+10'
-t-6hrs 111 84hr_----Surge (Obs-Pred)

1976 do not include the effects of hurricanes and tropical storms. Because of this

limitation, the storm surge model was applied for the full study period from 1950-1997

excluding the setup, and also for the reduced period from 1976-1995 including the setup.

The one-dimensional storm surge model is based upon the bathystrophic surge

model originally proposed by Freeman, Baer, and Jung (1957), and later applied by Chiu

and Dean (1984). Applying the assumptions of Freeman, Baer, and Jung, as described in

Chapter 2, and writing the simplified equations of motion in finite difference form results

in

q n+ = 1 F +At y, (4.5)
qi_ BB y + tw" (4.5)

n+1 n+1 Ax w1 n+1 n++
4T =f + -- q I + P P (4.6)
gDi I p, y p0g

BB = 1.0+ f (4.7)
Di2

where the previously undefined f is a bottom friction factor taken as 0.01. The subscripts

i and n refer to the grid location and time step respectively. Reviewing the coordinate

system, x represents the cross-shore direction and is positive onshore, and y represents

the longshore direction and is positive North. At this point it is convenient to break the

surge into its constituent parts and describe each component individually.

The surge generated by the atmospheric pressure gradient associated with a

hurricane is given by

n+I n+I
Pi Pi+i (4.8)
Pwg

The pressure at each location along the grid is calculated from Eq. (2.1) and is based

upon the exponential pressure distribution associated with the model hurricane. As

evidenced by Figure 2.2, the magnitude of the pressure reduction associated with a

hurricane dies out rapidly. Close examination of Figure 2.2 and Eq. (2.1) reveals that

unless the hurricane passes within several radii of the coastline, the barometric induced

surge is not expected to be a dominant contributor to the total surge.

The contribution of the Coriolis force to the total surge is given by the second

term inside the brackets of Eq. (4.6).

Ax -Eq +1] (4.9)
gDi

The Coriolis parameter E is defined as 2osino, where 0 is the latitude of the location of

interest, and o is the angular frequency of the earth's rotation. Due to the large length

scales, and significant long-shore currents, a hydrostatic gradient (surge) is developed to

compensate for the Coriolis effect. The negative sign indicates that a negative longshore

current, directed south in this case, results in a positive contribution to the surge. As was

the case with the barometric contribution, the contribution of the Coriolis effect is small

when compared to the contributions of wave setup and wind stress.

The wind stresses in both the cross-shore and longshore directions are important

components of the storm surge even in this one-dimensional model. The cross-shore or

x-component of the shear stress is included directly in the calculation of the surge

through the first bracketed term in Eq. (4.6).

Axr'i p, (4.10)
gDL Pw J

where Twx is the wind shear stress given by

T = pkW2 (-cos 0 Iotwvet ) (4.12)

The previously undefined factor k is the Van Dor air-sea friction coefficient given by

1.1x10-6 W k = Wc 2 (4.14)
S1.1x10-6+2.5x10-6 1- W >WC

We is a critical velocity equal to 16.09 miles per hour. The negative cosine term in Eq.

(4.12) is the result of two different coordinate systems. Eqs. (4.5-4.7) consider the cross-

shore coordinate to be positive onshore, however the Florida State Plane system used as

the base system for calculating the indices, defines the cross-shore coordinate as positive

offshore. The negative sign simply reverses the direction of the wind stress term

rendering the systems compatible. The longshore or y-component of the wind stress is

included in the surge calculation indirectly through its influence on the volumetric

longshore transport. The longshore component of the wind stress is given by

T, = P kW2 (sins_rotwvc) (4.13)

Eq. (4.5) gives the volumetric longshore transport per unit width as a function of the

longshore wind stress. The effect of the longshore component of the wind stress on the

surge is eventually incorporated into the contribution of the Coriolis term discussed

previously. The direct and indirect contributions of the wind stress, along with the wave

induced setup, are the largest contributors to the total storm surge.

The contribution of the wave setup must be calculated and included explicitly.

Prior to determining the effectiveness of either index, it was determined that calculation

of the wave fields associated with each hurricane to obtain local wave conditions was

beyond the scope of this project. The largest sets of wave data available are the WIS

hindcast data, which are available for the period 1950-1995. The WIS data are composed

of two sets of hindcasts, the first of which does not include the effects of hurricanes and

therefore is excluded. This reduces the usable range of WIS data to the period from 1976

to 1995. The WIS data report the wave parameters in variable water depths, at 0.25

spacing, on three-hour intervals. The WIS data are time synchronized to each individual

hurricane. A filter is applied based upon the wave angle with respect to the shoreline

orientation to eliminate any offshore propagating waves. Linear theory and the

assumption of straight and parallel contours are utilized to refract and shoal the waves to

the breakpoint. Depth limited breaking assuming a breaking index of y=0.78 and the

familiar Longuet-Higgins (1964) result ares utilized to calculate the setup at the shoreline

from the breaking conditions.

H 2kh, 3y/8 (4.4
b + hb
16hb sinh(2khb) 1+3 /8

Here k is the wave number and Hb and hb are the breaking wave height and depth

respectively. Recognizing that waves are always present and that a background setup

exists, the average setup over the 25-year data set is calculated and subtracted from the

result, isolating the hurricane induced setup. The contribution of the setup is included

only at the shoreline. In order to simplify the calculations, the setup is not assumed to

modify the depth at each location along the profile.

The one-dimensional storm surge model, encompassing Eqs. (4.5-4.7), is solved

over the full time period neglecting the setup, and over the shortened time period

including the setup. The initial and boundary conditions for the model include initiation

from rest and surge due to barometric effects alone at the seaward edge of the grid.

S1=P Pi (4.15)
P g

Here poo is the pressure in the absence of the hurricane and pi is the local pressure at the

seaward edge of the profile.

Results

The ELP Index is calculated for each storm at the monument corresponding to the

center point of the three locations analyzed at each of the 5 sites. In an effort to

determine an optimal interval of application, the index is summed over one, three, and

five years prior to each photograph as well as over the period between photographs. The

ELP Index is then plotted versus the average observed shoreline change at each site.

When the accretion is plotted versus the ELP Index, no trend is observed. This should be

expected, as the index is not intended to predict accretion. When erosion is plotted

versus the ELP Index a minor trend is observed. As expected, larger values of the ELP

Index tend to indicate more erosion, although the correlation is less than expected. A

best-fit line is added based upon the R-squared value. The R-squared value is a measure

of the goodness of fit and is defined as

R2 = 1 __ -j (4.16)

n

Both exponential and linear trends were fit to the data. In the case of a linear trend, an

attempt was made to force the line through the origin, as ideally an index value of zero

indicates no erosion. In most cases, the best-fit line takes a linear form and is not forced

through zero. The parameters associated with the best-fit lines vary substantially from

location to location, in some cases even indicating a decreasing trend.

Table 4.1 gives the R-squared values for each case at each site. Underlined values

correspond to trendlines with negative slopes, indicating poor agreement. Table 4.1

40

indicates the best correlation as determined by the R-squared values occurs when the ELP

Index over the year preceding each photograph is utilized. In general the correlation

between the observed shoreline changes and the ELP Index is not good, with only one R-

squared value greater than 0.5. Examining several of the plots allows us to see where and

why the index failed.

Table 4.1 R-squared values indicating the correlation between the observed erosion and
the ELP Index for the case of no setup and no winter storms.
R SQUARED VALUES
Site 1 year prior 3 years prior 5 years prior Between Photos Data Points
St. Johns 0.039 0.005 0.002 0.044 6
St. Augustine 0.710 0.433 0.455 0.380 6
Volusia North 0.055 0.138 0.014 0.000 4
Volusia South 0.307 0.148 0.265 0.130 6
Martin 0.498 0.285 0.201 0.304 4

Average 0.322 0.202 0.187 0.172

ELP Index vs Erosion
St. Augustine Site

I i I I I I I

25

20
x
O
- 15
Lu
-J
10

5

0

Cum
3yr
- Linear (lyr)
- Linear (5yr)

S___ 2.7
K9
c-

S,

y=0.3797: -1.9187 .
R2 = 0 4551 /
*------- x 7--

0" y =0.4143x 14.305
= 0.3567x 4.7668 R2 = 0.709
R2 = 0.304 of
--- -M 4-

0 10 20 30 40 50 60 70 80 90
Erosion (ft)

Figure 4.5 Plot of ELP Index versus erosion at the St. Augustine Site. Setup and winter
storms are not included.

1yr
5yr
- Linear (3yr)
- Linear (Cum)

2 = 0.4327

I

-

-

ELP Index vs Erosion
St. Johns Site

0 10 20 30 40 50 60 70 80 90
Erosion (ft)

Figure 4.6 Plot of ELP Index versus erosion at St. Johns Site.
are not included.

ELP Index vs Erosion
St. Augustine Site
35
Cum lyr
30- A 3yr X 5yr
- Linear (lyr) - Linear (yr) X
- -Linear (5yr) - Linear (Cum)
25
y = 0.4204x 19293

S15-- ^-- --

10 ----to--- A -
= 0.3628x 2.8005 0 0
'02_n ,- - =O. 46
2 5 .y=0.

0 i

Setup and winter storms

0 10 20 30 40 50 60 70 80 90
Erosion (ft)

Figure 4.7 Effect of removing all four points associated with the 1989 erosional event
from Figure 4.5.

42

ELP Index vs Erosion
St. Johns Site

35

30

25
x
c 20

Si15

10

5

n

0 10 20 30 40 50 60
Erosion (ft)

Figure 4.8 Effect of removing all four points associated with the 1989 erosional event and
the cumulative point associated with the period 1960-1971 from Figure 4.6.

The St. Augustine Site, represented by Figure 4.5, is the case where the index

appears to work the best. The trend is clear and the R-squared values for the best-fit lines

are reasonably high. Figure 4.6, representing the St. Johns Site, is an example of a case

where the index appears to fail. The St. Johns data appear to indicate larger values of the

ELP Index are associated with less erosion. Not only are the R-squared values low, the

best-fit lines indicate a trend completely opposite of what was expected. Closer

examination reveals one or two data sets responsible for the poor agreement at the St.

Johns Site. Examination of these data sets indicates why the index fails. The data points

in Figure 4.6 associated with over 80 feet of erosion and low values of the ELP Index

cause the best-fit lines to slope in the wrong direction. This data set corresponds to

shoreline change over the period 1988-1989. Extensive analysis of the local conditions

during this period revealed extreme wave conditions associated with the three offshore

Cum N lyr
A 3yr X 5yr
S- -Linear (3yr) Linear (1yr)
- Linear (5yr) - Linear (Cum)

y = 0.5431x+ 1.935
R = 0.5307 01 a 0 __ "0
O

= 0.485x+ 1.131 _______
R2=0.5046 o O

R2 = 0.9874

Hurricanes Dean, Gabrielle, and Hugo. The surge model does not predict large surges

associated with these storms due to their distance offshore. These storms behave much

like winter storms in that a large proportion of the storm surge will be due to wave

induced setup. As is the case with winter storms, an alternate method of analysis is

required to capture the setup effect.

The data point corresponding to an index value of over 150 feet in Figure 4.6 also

appears to be distorting the results. This data point corresponds to the cumulative ELP

Index between a photograph in 1960 and the next available photograph in 1971. Here the

long interval of application is questioned. Examination of the ELP Index associated with

each individual storm occurring between the available photographs indicates that nearly

75% of the index value is associated with Hurricane Donna in 1960, and Hurricane Dora

in 1964. In this case, the interval of application is believed to be too long. The 28 feet of

erosion indicated by the two available photographs reflects both the erosion caused by the

storms, and 8 years of subsequent recovery. Very few storms impacted St. Johns County

between 1964 and 1971, therefore it is likely that the shoreline recovered substantially

prior to the second photograph. Here the index fails because the photographs taken over

10 years apart do not capture the true erosional effect of the storm. Figure 4.8 illustrates

the improvement observed in the correlation between the ELP Index and the erosion at

the St. Johns Site when the cumulative data point and all four data points corresponding

to 1989 are removed. Although the correlation at the St. Augustine Site was good to

begin with, Figure 4.7 shows additional improvement resulting from the removal of the

1989 data set. No cumulative data point is removed at St. Augustine due to the higher

density of photographs, including one in 1963.

The substantial increases in R-squared values at the St. Johns and St. Augustine

Sites are observed at all locations when the 1989 data set and any cumulative data points

calculated by summing the ELP Index over 10 or more years are eliminated. Table 4.2

gives the R-squared values resulting from the removal of these data points. It should be

noted that the value of 1.0 reported for the cumulative data point at the Volusia North

Site reflects the fact that after removing the 1989 data set and the cumulative point for the

period 1958-1969 only two data points remain.

Table 4.2 R-squared values resulting from the removal of the 1989 data set and the
cumulative data point where the interval of application is greater than 10 years.
R SQUARED VALUES
Site 1 year prior 3 years prior 5 years prior Between Photos Data Points
St. Johns 0.723 0.504 0.531 0.987 5 (4 cum)
St. Augustine 0.870 0.716 0.621 0.752 5
Volusia North 0.453 0.000 0.405 1.000 3 (2 cum)
Volusia South 0.640 0.586 0.821 0.712 5
Martin 0.498 0.285 0.201 0.304 4

Average 0.637 0.418 0.516 0.751

A second set of plots is generated by adding 25 severe winter storms to the

analysis. The procedure for analyzing the results is the same as in the first case. Again

the best-fit line, as determined from the R-squared value, is linear in all cases. Table 4.3

presents the R-squared values from these plots. Comparing the values in Table 4.3 with

those in Table 4.1, it is observed that inclusion of the winter storms drastically reduces

any correlation between the ELP Index and the observed erosion. Table 4.3 indicates that

nearly half of the best-fit lines have negative slopes as indicated by the underlined values.

Only the St. Augustine Site produces results that follow the expected trend. Although all

four trendlines at the St. Augustine Site have positive slopes, the small R-squared values

are indicative of the large amount of deviation from the trendline. Excluding cumulative

data points calculated over 10 years or more and the 1988-1989 data set, as done in the

previous case, has a negligible effect on these results. An example of the results obtained

by including winter storms is given in Figure 4.9.

ELP Index vs Erosion
_ Martin Site (Winter Storms Included)
t-3yr, ,

300

250

e 200

1 150

100

50

0

Cum
t-5yr
- Linear (t-5yr)
- Linear (t-1vrt

t-3yr
t-1yr
- Linear (Cum)
- Linear ft-3vr

y = -.4384x + 494.39
--2 0.200721
4* A

y=02079x+130.2 -
2 = 0.0012

y = 0.6687x 27.764
R2=0.4 84
I ..XV
10 20 30 Erosion (ft) 40 50 60
0 10 20 30 Erosion (ft)40 50 60

Figure 4.9 Plot of ELP Index versus erosion at Martin County Site. Winter storms are
included, setup is not.

Table 4.3 R-squared values indicating the correlation between the observed erosion and
the ELP Index for the case of no setup, including winter storms.
R SQUARED VALUES
Site 1 year prior 3 years prior 5 years prior Between Photos Data Points
St. Johns 0.312 0.055 0.001 0.263 6
St. Augustine 0.001 0.008 0.000 0.002 6
Volusia North 0.203 0.464 0.500 0.018 4
Volusia South 0.065 0.382 0.202 0.121 6
Martin 0.498 0.607 0.499 0.001 4

Average 0.216 0.303 0.240 0.081

y = -6.2481 x
2 ,

+ 559.74
0n

Several possible explanations are given for the reduction in the index's

effectiveness. Although in general the magnitudes of the winter storm related surge is on

the order of the hurricane related surge, the rise times are five to ten times larger.

Although not surprising physically, this results in ELP Index values for winter storms that

are four to eight times larger than those associated with even the strongest hurricane. The

analysis of seasonal shoreline changes presented in Chapter 5 indicates that in selecting

only 25 storms, numerous other important winter storms are excluded. The winter season

is characterized by a large number of severe storms, both named and unnamed, that cause

substantial erosion. Without representing the background effect of all of these storms,

the index will not work. The summer season on the other hand, is characterized by more

predictable, mild conditions. In 90% of the cases, extreme summer erosion is associated

with the 337 analyzed hurricanes. The background effect is much less important in the

summer, allowing for a better application of the ELP Index.

As mentioned in the discussion of the results from Case 1, excluding the effect of

wave setup appears to be limiting the index's effectiveness. Although this is particularly

true in the case of the 1988-1989 data, it is assumed the exclusion of the wave setup

affects other data sets as well. A third case was examined explicitly including the wave

setup in the storm surge model, however, as previously mentioned, this has the

consequence of limiting the scope of analysis. The duration of the analysis is limited to

1976-1995 by the available wave data and produces inconclusive results. In the previous

analyses, the best-fit lines were based upon a maximum of six data points. Reducing the

study period reduces the number of data points on which the trendlines are calculated,

thereby reducing the amount of confidence that can be placed in the results.

The method of analysis is the same as in the previous two cases. The results are

plotted and the linear best-fit trendline is added. Table 4.4 gives the R-squared values

associated with the best- fit lines. As in the previous two cases, underlined values

indicate negatively sloped trendlines. The large variability in both the R-squared values

and the slope directions within each site is a direct consequence of calculating the

trendlines based on only three data points. The best agreement is observed at the Volusia

North Site, which is presented as Figure 4.10. Here, three of the four best-fit lines

indicate increasing trends, and the R-squared value for both the three-year and the

cumulative trendlines is nearly one. Figure 4.11 is an example of a case where little

correlation between the ELP Index and the historical erosion is observed. In this case

two of the four trendlines are sloping in the wrong direction and of the two positively

sloping trendlines only one has an R-squared value greater than 0.5. Theoretically,

including the setup should improve the correlation between the ELP Index and the

observed erosion, however with only three data points on which to base the results no

definitive conclusion can be made.

Table 4.4 R-squared values indicating the correlation between the observed erosion and
the ELP Index for the case of including setup and no winter storms.
R SQUARED VALUES
Site 1 year prior 3 years prior 5 years prior Between Photos Data Points
St. Johns 0.976 0.773 1.000 0.995 3 (5yr-2)
St. Augustine 0.248 0.437 1.000 0.083 3 (5yr-2)
Volusia North 0.846 0.949 0.134 0.960 3
Volusia South 0.737 0.004 0.538 0.992 3
Martin 0.177 0.144 1.000 0.047 3 (5yr-2)

Average 0.597 0.461 0.734 0.615

48

ELP Index vs Erosion
Volusia North Site

15 25 35 45 55 65 75 85 95 105
Erosion (ft)

Figure 4.10 Plot of ELP Index versus erosion at the Volusia
included, winter storms are not.

North Site. Setup is

ELP Index vs Erosion
Volusia South Site

Cum y = -4.0669x +331.43
1yr R = 0.92 y = -C.4586x+230.7
RA = 0.538
A 3yr R - ,
5 yr .....
S-- Linear (1 yr)
- Linear (Cum) _
Linear (3 yr)
Linear (5yr) - - --
'=0.0761 +118.8----l-
R2 = 0.003'

Sy = 2.012x-40.45
S- S R2=0.7367

0 10 20 30 Ersion () 40 50 60 71
Erosion (ft)

Figure 4.11 Plot of ELP Index versus erosion at the Volusia South Site. Setup is
included, winter storms are not.

300

250

200

_ 150
a.
-J

100

50

0

Hurricane Erosion Index (HEI)

Theory

The HEI is very similar to Dean's original index discussed in Chapter 2, with

several minor modifications made. In the new index, the forward velocity in the

denominator of the original HEI, is eliminated. Here, the forward speed is accounted for

directly when integrating over a hurricane track recorded at a constant time interval (6

hours). Density and friction factor terms are also added to the original index, resulting in

an equation more closely resembling the dominant wind stress component of the storm

surge.

10R
HEI= J pkW (-cosO rtwvt)dr (4.17)
0

As described previously, the negative sign preceding the cosine term is the result of the

two different coordinate systems. The negative sign reverses the direction of the cross-

shore coordinate, making the onshore direction positive. Reversing the direction of the

cross-shore coordinate results in positive index values corresponding to an onshore wind

stress. Based on Figure 2.5, the limits of integration are kept the same.

In applying Eq. (4.17), two geometrical approximations are employed. The first

assumption is that the shoreline is located at the monument location. In reality, the

monuments are not located at the shoreline, however the large spatial scales involved in

calculating the HEI justify this simplification. An approximation is also employed for the

shoreline orientation angle, a. The shoreline orientation is taken as -17.80, north of state

plane coordinate 833,400.0 (approximately West Palm Beach), and +6.7 to the south.

Both of these angles are approximate and are determined graphically.

Results

The results obtained from the application of the HEI are very similar to those

obtained with the ELP Index. The methodology applied in analyzing the results is the

same. The HEI associated with each hurricane is calculated at the center of each site and

then summed between each photograph, and one, three, and five years prior to each

photograph. The results are plotted versus the average observed shoreline change over

each interval at each site. As expected, periods of accretion are characterized by small

values of HEI. Trendlines are added to the plots of HEI versus erosion, with the best-fit

lines determined from the R-squared values. In all cases, the best-fit lines are linear,

although the slopes and intercepts exhibit significant scatter. Table 4.5 presents the R-

squared values for each case at each site, where underlined values indicate trendlines with

a negative slope. Again the correlation between the observed erosion and the HEI is less

than expected. Several of the trendlines slope in the wrong direction, and only three of

the twenty R-squared values are greater than 0.5. Examination of the results obtained at

the St. Johns Site as presented in Figures 4.12 and 4.13, yields some insight into the

apparent failure of the index.

Table 4.5 R-squared values indicating the correlation between the observed erosion and
the HEI.
R SQUARED VALUES
Site 1 year prior 3 years prior 5 years prior Between Photos Data Points
St. Johns 0.006 0.067 0.046 0.081 6
St. Augustine 0.514 0.180 0.197 0.173 6
Volusia North 0.027 0.717 0.367 0.012 4
Volusia South 0.311 0.003 0.293 0.270 6
Martin 0.556 0.454 0.066 0.140 4

Average 0.283 0.284 0.194 0.135

HEI vs Erosion
St. Johns Site
12
1 Cum lyr
10- A 3yr X 5yr
- Linear (3yr) Linear (1yr)
S- Linear (5yr) Linear (Cum)
8 ______-

W 6 -00445X4 5474 -
R= 0.0E 08

R2= (.0669 " R2= 00462
"'" ,, ,'.-'.--'-'--- = =.

= 0.0058 0.696.6 ... .- -- - -- -- -- ----- -
R2 = 0. 055
0 10 20 30 40 50 60 70 80 90
Erosion (ft)

Figure 4.12 Plot of HEI versus erosion at the St. Johns Site.

Upon examination of Figure 4.12, it is apparent that problems similar to those

affecting the application of the ELP Index affect the application of the HEI. Contrary to

what was expected, Figure 4.12 indicates larger HEI values correlated with smaller

amounts of erosion. Similar to the problem with ELP Index, this can be attributed to the

data set indicating over 80 feet of erosion, corresponding to the shoreline changes

between 1988 and 1989. As discussed previously, three hurricanes producing substantial

storm surges, passed offshore of Florida during 1989. The effects of these hurricanes are

not captured because they fall outside the limit of integration placed on the HEI.

Increasing the limits of integration has little effect due to the rapid decay of the wind field

with distance. Similar to winter storms, the erosion caused by these offshore hurricanes

is likely the result of an increased water level due to the setup resulting from hurricane

waves generated well offshore.

The cumulative data point corresponding to a HEI of approximately 11 also

appears to be distorting the results. This data point corresponds to the cumulative HEI

between a photograph in 1960 and the next available photograph in 1971. As discussed

in the ELP Index section, the interval of application is too long. Analysis of the HEI

values attributed to individual storms during this period reveals over half of the index

value can be attributed to Hurricane Donna in 1960. It is likely that the shoreline changes

indicated by the 1971 photograph reflect not only the erosion caused by Hurricane

Donna, but also a substantial amount of post-storm recovery.

Figure 4.13 illustrates the effects of removing the 1989 data points, and the

cumulative data point from the ST. Johns data. In this case, as in all cases, the R-squared

values increase substantially. The R-squared values resulting from the removal of

cumulative data points calculated over periods of ten or more years, and the entire 1989

data set, are illustrated in Table 4.6. Upon removal of the questionable data points, more

than half of the R-squared values increase to greater than 0.5, a substantial improvement.

Summing the HEI between each photograph and over the 5 years prior to each

photograph results in the best correlation.

Table 4.6 R-squared values resulting from the removal of the 1989 data set and the
cumulative data point where the interval of application is greater than 10 years.
R SQUARED VALUES
Site 1 year prior 3 years prior 5 years prior Between Photos Data Points
St. Johns 0.428 0.825 0.881 0.896 5 (4 cum)
St. Augustine 0.647 0.425 0.436 0.434 5
Volusia North 0.216 0.592 0.518 1.000 3 (2 cum)
Volusia South 0.643 0.019 0.791 0.793 5
Martin 0.556 0.454 0.066 0.140 4

Average 0.498 0.463 0.538 0.653

HEI vs Erosion
St. Johns Site

5

4 -

y = 0.0802x + (.7099 A -
3 0.811 y 0931x-1.2557
2 x + F 2 = 0.4275
Sy = 0.0784x + 0.6466 R=0.8254 Cum I lyr
S0 A 3yr X 5yr
1 y = U.U084x 4 .4614 0 - Linear (3yr) - Linear (lyr)
R2=0.8 58 Linear (5yr) - Linear(Cum)

010 20 30 40 50 60
Erosion (ft)

Figure 4.13 Effect of removing all four points associated with the 1989 erosional event
and the cumulative point associated with the period 1960-1971 from Figure 4.12.

Modified Hurricane Erosion Index (MHEI)

Theory

As discussed in the previous section, two major factors limit the HEI's

effectiveness. The first factor is the excessive time period between applications of the

index, particularly during the early period of the study. Errors resulting from attempts at

applying the HEI over intervals greater than the limits of its effectiveness can only be

fixed by obtaining more, better data, eliminating data points where the HEI is pushed

beyond its limits, or developing a method to account for the amount of recovery between

storms. The second factor hindering the effectiveness of the HEI, the exclusion of wave

effects, is somewhat easier to correct. A more detailed index for calculating the erosive

potential of a given storm including the effects of increased wave heights is developed

and termed the Modified Hurricane Erosion Index or MHEI.

w.

Ay<0

Figure 4.14 Definition sketch for the calculation of the change in shoreline position
resulting from an increased water level, S, and wave setup, Tj (Dean and Dalrymple
2001).

Once again, the Bruun Rule provides the basis for the MHEI. In addition to an

elevated water level, S, we introduce the wave induced setup, rl. Figure 4.14 graphically

depicts the problem setup. Equating the volume eroded from the foreshore and deposited

offshore utilizing equilibrium beach profile methodology results in

0 W.+Ay
I(B-S-1I(y))y+ A(y-Ay)2/3dy=
Ay Ay

W.+Ay W.+Ay
J Ay2/3dy + J+ i(y))y (4.18)
0 0

where A is the sediment scale parameter in units of length to the one third. All other

variables are defined in Figure 4.13. Integrating Eq. (4.18) and simplifying for the case

of small relative shoreline displacement results in

0.068Hb +S" .
Ay = -W. .68Hb j (4.19)
(y BB+1.28Hb

where W* is the distance to the breakpoint and is calculated from the breaking depth

assuming an equilibrium beach profile. Writing W* in terms of Hb and dividing both the

numerator and denominator by 1.28 Hb results in

1.28Hb (S + 0.068Hb)
Ay =- A B(4.20)
Y A3/2 B +12H
1 1.28Hb

Now we will define the numerator as the Modified Hurricane Erosion Index

MHEI= (S + 0.068Hb )1.28HbdA (4.21)
hurricane

Here the surge, S, is calculated from the previously described one-dimensional storm

surge model. The integral is taken over the sea surface area affected by the hurricane.

The contribution of the wave setup is included through the breaking wave height, which

is calculated from the WIS data. As with the setup term in the surge model, a

background breaking wave height is subtracted from the storm related breaking wave

heights. The background wave height is obtained by averaging the long-term breaking

wave heights at each site, which are calculated by applying the shoaling routine to the

WIS data. Again the dependence on WIS data limits the period of analysis to 1976-1995.

Results

The shorter period of analysis imposed by the availability of the wave data allows

only a limited analysis. The same procedure as described previously is utilized to

compare the observed shoreline changes with the MHEI. Unfortunately, the reduced

study period leaves only three erosional trends for analysis. It is difficult to draw

conclusions from trendlines fit to only three points. The R-squared values for each case

at each site are presented in Table 4.7, where the underlined values indicate decreasing

trends. It is obvious from Table 4.7 that the MHEI does not appear to improve the

results. Aside from the Volusia South Site, only one trendline has an R-squared value

greater than 0.5 and a positive slope. Figures 4.15 and 4.16 represent cases where the

index appears to succeed and fail respectively. In viewing the results, the lack of

sufficient data must be kept in mind. A more complete data set is required before a

definitive judgment on the effectiveness of the MHEI can be made.

Table 4.7 R-squared values indicating the correlation between the observed erosion and
the MHEI.
R SQUARED VALUES
Site 1 year prior 3 years prior 5 years prior Between Photos Data Points
St. Johns 0.731 0.017 1.000 0.046 3 (5yr-2)
St. Augustine 0.095 0.865 1.000 0.799 3 (5yr-2)
Volusia North 0.291 0.102 0.103 0.098 3
Volusia South 0.824 0.943 0.778 0.101 3
Martin 0.015 0.270 1.000 0.015 3 (5yr-2)

Average 0.391 0.439 0.776 0.212

MHEI vs Erosion
Volusia South Site
600
Cum U 1 yr
A 3 yr X 5 yr y = 4. 585x + 230.49 X
500 - Linear (5yr) - Linear (3 yr) R = 0.7782
- Linear (lyr) - Linear(Cum) ,
400 .--
- .- x J.
300 y-=-8094x- 26943 - -
R2=0.1008 0 4 e "
200 ,
y =4.4021x-- 84.379 A *
R2 = 0.9429 I0
100
S y=7.63 7x 152.31
0 R2 = 0.8238

0 10 20 30 Erosion (ft) 40 50 60 70

Figure 4.15 Plot of MHEI versus erosion at the Volusia South Site

MHEI vs Erosion
Volusia North Site
700
CUM 1 yr
A 3 yr X 5 yr
-600 y- -.342x+ -. I- -Linear (1 yr) -Linear (3 yr)

SR2= 0.0981
300- ..... -
300 x y= 1.0685x+220.

200

100- y = -1.983x + 260.87 -
R = 0.2908
0
0 20 40 60 80 100 120
Erosion (ft)

Figure 4.16 Plot of MHEI versus erosion at the Volusia North Site

CHAPTER 5
SHORELINE RESPONSE RATE

The results presented in Chapter 4 suggest that at least a portion of the failure of

the erosion indices is due to the different time scales governing the erosion and accretion

processes. Ideally the indices should be compared with changes observed immediately

after each storm event, however in many cases they are applied over periods of a decade

or longer, as dictated by the photograph spacing. In this case, storm sequencing plays an

important role in determining the index's effectiveness as illustrated in Figure 5.1. If we

assume the index is perfectly correlated with the observed shoreline changes, the

following two scenarios are proposed. In Case 1, a large erosional event and

corresponding large index value occur at the beginning of a fixed time period. In Case 2,

the same event occurs at the end of an equal time period. The value of the erosion index

will be the same for both cases. In Case 1 the shoreline will gradually recover over the

remainder of the period. In Case 2, because the erosional event occurs immediately

before the end of the period, the beach will not have time to recover. Depending upon

the length of the period, and the rates of the erosion and accretion processes, the

perceived effectiveness of the index can vary substantially. In Case 2 the erosion index

performs admirably; in Case 1 it fails.

There are many models able to predict erosion adequately, but few if any, that

predict the recovery accurately. A satisfactory equation representing the rate of shoreline

response is needed to help describe the long-term evolution of the beach, particularly the

recovery phase. Potential applications of such an equation are numerous.

Effect of Storm Sequencing
on Perceived Erosion Index Effectiveness

0 1 2 3 4 5 6 7 8 9 10
Time

Figure 5.1 Graphical depiction of the effect of storm sequencing on the perceived
effectiveness of the erosion indices. In both cases the value of the erosion index will be
the same however the final position of the shoreline will be much different.

Rate Equation

Theory

An effective shoreline change rate equation should incorporate two key concepts.

First, the shoreline response should be related to its displacement from an equilibrium

position; the farther the shoreline is displaced from equilibrium, the faster it should

respond. Secondly, the equation should allow for the different time scales governing the

erosion and accretion processes to be represented. An equation similar to Eq. (2.20) is

proposed based upon these criteria.

Sk (t)- y(t)) (5.1)

Here ka is a rate constant with different values for erosion and accretion, and yeq is the

equilibrium shoreline position, which is also a function of time. Eq. (5.1) can be solved

analytically for constant ka resulting in

y(t)= yoe-kat + kJe-k,(t-t y(t)dt (5.2)
0

where yo represents the shoreline location at some initial time to. The contribution due to

yo approaches zero for steady state conditions and long time intervals. Analytic solutions

to Eq. (5.1) exist for a limited number of functions. A more direct application of Eq.

(5.1) is made by numerically solving the finite difference form of the equation.

Syln- k1 At2 ++kAty.q
y =+ kAt (5.3)

The difficulty in solving Eq. (5.3) lies in the fact that both rate constants (erosion and

accretion) are unknown, as well as yeq. As discussed in Chapter 2, yq is a also a function

of time, varying in response to the different seasonal conditions throughout the year.

Results

Shoreline changes predicted by Eq. (5.3) are presented for several idealized cases

to illustrate the effect of changing the rate constants. In all cases, the seasonal

equilibrium shoreline position is approximated by a sinusoidal function. Artificial

erosional and accretional events are applied to help illustrate the effect of different rate

constants on the shoreline response.

Figure 5.2 illustrates the shoreline response assuming both rate constants are

equal to one, although in reality the erosion constant is larger than the accretion constant.

The shoreline response is a sinusoidal function, damped and phase lagged by nearly 900

with respect to the equilibrium shoreline position. Figure 5.3 illustrates the shoreline

response under the same conditions to both an erosional and accretional forcing. Because

the rate constants are equal, the shoreline requires the same amount of time to "recover"

from both events. Figure 5.4 illustrates the effect of increasing the rate constants. Notice

the shoreline fluctuations approach those of the equilibrium shoreline, and the phase lag

exhibited in Figure 5.2 is reduced as the shoreline responds at a faster rate.

Figures 5.5 and 5.6 illustrate the effect of reducing the accretional rate constant to

one-fifth the erosional constant. Reducing the accretion constant has the effect of

reducing the amount of recovery that is achieved in a given time. When compared with

Figure 5.2, Figure 5.5 exhibits two critical differences. First a new quasi-equilibrium

position is established for the given rate constants. Initially, the erosion and accretion

cycles occur over the same time period, but because the erosion proceeds at a faster rate,

the entire response curve shifts downwards. As the curve shifts down, the shoreline

spends a longer period of time in an accreted state with respect to ye. A quasi-

equilibrium state is reached when the accretion time is sufficient for the amount of

accretion to equal the amount of erosion over one period of the equilibrium function. The

second difference is that the shape is no longer sinusoidal. The larger value of the

erosion constant results in a steeper slope during the erosional phase, which contributes to

the response curve shifting to the quasi-equilibrium position.

Figure 5.6 examines the effects of introducing both an erosional and an

accretional forcing event while keeping the accretional constant at one-fifth the value of

the erosional constant. As described in the previous paragraph, a new quasi-equilibrium

state develops for the chosen values of the rate constants. From Figure 5.6 it is observed

that for events of the same magnitude, the recovery from the erosional event takes much

longer. This is a direct result of reducing the magnitude of ka with respect to ke. Ten

years after the erosional event, the shoreline has not reached a quasi-equilibrium state.

Due to the larger value of the erosional constant, the shoreline reaches a quasi-

equilibrium state within approximately seven years after the accretional event. This

behavior is similar to what is observed in nature. For events of the same magnitude, a

shoreline will erode to its equilibrium state from an artificially accreted state much faster

than it will accrete to its equilibrium state from an initially eroded state. The exact time

scales of the processes will be determined by the relative magnitudes of the rate

constants.

RATE EQUATION (Ka=Ke=1)
1.25 -

0.75

t--
S-0.25

-1.25
0
0-

-0.75

1.25

0 2 4 6 8 10 12 14 16 18 20
Time (yrs)

Figure 5.2 Application of Eq. (5.3) for the case ke=ka=1 and no external forcing

AI 01 ii iI 1A I l I, I, I, I, It I' 11 _ II_ I I

IC IA I I A I A I Al IA I'A IA I, IA I A IA I I

IVIVIV VI I---YV IV IV V ' 9 1 I I IV IV
H Hy \ 1 IH

RATE EQUATION (KaK.=1)

-Yeq

2-

A 0

a. I \ L I I l II &ly \ I I

CO

-2

-3 ____ ____ ____ _____________

0 2 4 6 8 10
Time (yrs)

12 14 16 18 20

Figure 5.3 Application of Eq. (5.3) for the case of ke=ka=1 with external forcing applied
at t=0 and t=10

RATE EQUATION (Ka=Ke=5)

AI IA I 11 I

A I

1111
I
In Il
111 11
i IIR II

A l

'I Ii

fII l 1 '~ II III II

SIIRi AI

i'BII

I I
IAI

~IA Ii
' I Ill II

R IIUI
f18i~i!

Iill IIll liii 11111i 111(i111 1 1111
I Il t il il I I I

I 1 I

I I I I I I I I I-1I I

- - Yeq Y

0 2 4 6 8 10
Time (yrs)

ii 'I
1IA

12 14 16 18 20

Figure 5.4 Application of Eq. (5.3) for the case of ka=ke=5 and no external forcing

1.25

0.75

-, 0.25
0

-0.25
0
0 -0.25
C

-0.75

-1.25

RATE EQUATION (Ka--Ke-5)

I

ml

A ;, A
I It I 111

64

RATE EQUATION (Ka=0.2,Ke=1)
-I

-!
[ l I A I II AI A I I A I A !I A A

Il I 1 11 11 II II 11 I1_I I i I i i II I I i I I

Tl I, I i i11, i1i1 i 1i i i11i i i i i i ii 1 i ii i i II 1 1 ii

- - Yeq -- Y

0 2 4 6 8 10
Time (yrs)

12 14 16 18 20

Figure 5.5 Application of Eq. (5.3) for the case ke=1, ka=0.2 and no external forcing

RATE EQUATION (Ka=0.2,Ke=1)

0 2 4 6 8 10 12 14 16 18 20
Time (yrs)

Figure 5.6 Application of Eq. (5.3) for the case of ke=1, ka=0.2 with external forcing
applied at t=0 and t=10

1.25

0.75

t-

0.

' -1.25
S-0.25

0
CO
-0.75

-1.25

Potential applications of Eq. (5.3) are numerous. The key is describing both the

equilibrium shoreline position function and the appropriate rate constants accurately.

Unfortunately, these parameters are themselves functions of numerous complex

variables, among them sediment characteristics, beach conditions, and local wave

environment. For use as an aid in assessing the accuracy of a hurricane erosion index, a

shoreline response equation could provide guidance in determining a maximum time

period of applicability, as well as a method for calculating shoreline recovery between

individual storm events.

Determination of Equilibrium Shoreline Position

Theory

As discussed in Chapter 2, Dewall found substantial seasonal shoreline

fluctuations in his studies at Westhampton and Jupiter Island. A clear relationship was

found between the average beach width and the average breaking wave conditions. In

general the more energetic winter wave conditions were correlated with the narrowest

beach width. Associated with these more energetic wave conditions is an increase in the

mean water elevation as a result of wave setup. A simple equation based upon the Bruun

Rule is applied in an attempt to reproduce Dewall's results, and develop an expression for

the equilibrium shoreline position as a function of time

For purposes of this study, the shoreline recession or advancement is assumed to

be in response to an increase in local water elevation. The equation utilized to define the

MHEI, Eq. (4.19), is reapplied to calculate potential shoreline changes. Figure 4.14

illustrates the problem setup, and Eq. (4.19) defines the potential shoreline change Ay.

Again W* is taken as the width of the surfzone and is calculated from the breaking depth,

assuming an equilibrium beach profile. Here S is the observed water level taken from the

CO-OPS database and the wave setup effect is represented by the breaking wave height

Hb. The potential shoreline changes are calculated and compared to the seasonal changes

observed by Dewall.

The shoreline changes calculated by Eq. (4.19) are referred to as potential changes

because the conditions causing the change may not persist long enough for the shoreline

to reach an equilibrium state. Values calculated from Eq. (4.19) assume that a new

equilibrium state is reached, however in nature the conditions are so dynamic that often

the shoreline is not able to respond completely to the local conditions before they change

and a new equilibrium results. Here, Eq. (4.19) is applied on either an hourly or three-

hourly basis as dictated by the wave data set. Due to the frequency with which Eq. (4.19)

is applied, the shoreline will rarely reach an equilibrium state; therefore, the results

obtained represent only the potential shoreline response, assuming the beach has

sufficient time to evolve to a new equilibrium state.

The required inputs for Eq. (4.19) are the local wave conditions and water surface

elevations. Offshore wave data were obtained from two sources. Both WIS (Stations 13

& 78) and NDBC (Buoys 41009 & 44025) data were utilized. Each data set has some

advantages as well as disadvantages. Advantages of the WIS data set include: proximity

to the sites of interest, length of the data set, and inclusion of wave direction. The main

disadvantage of the WIS data set is the fact that it is a statistical hindcast based on wind

data, and often excludes important local effects. Although the NDBC data consist of

actual wave measurements, this data set also has some disadvantages. These include:

distance from the site, lack of wave direction data, and the frequent malfunctioning of the

equipment. Both data sources provide significant wave heights, Hs, and the associated

wave periods. For application to Eq. (4.19), the significant wave heights are converted to

average wave heights using the relation Havg=0.626Hs. A filter based upon the wave

direction (assumed to be the same as the wind direction in the case of buoy data) is

applied to eliminate offshore propagating waves.

17 4BP0
: 17 00A
S15 Locations 7 80
s 1 *1**** 1
11 7 72 25
0 .Data
/9 70 Locations
/ 7 .

Figure 5.7 Site and data station location map

Water level data are obtained from the NOAA CO-OPS database. Due to the

length of the study and the incomplete nature of the NOAA CO-OPS data sets, conditions

at representative stations are utilized and converted to approximate equivalent local

conditions. The representative stations for Westhampton and Jupiter Island are Montauk

and Mayport respectively. The time offset of the high and low tides is under one hour

between the representative and local stations allowing for a direct correlation with the

hourly and three hourly wave data once the tidal amplitudes have been adjusted.

Amplitude modification factors as reported in the NOAA tide tables are applied to

account for the difference in the magnitude of the tidal fluctuations between the sites and

the representative locations. The amplitude modification factors for low and high tide are

applied to all water level observations falling below and above the uncorrected average

tidal elevations respectively. Table 5.1 lists these factors and other pertinent site

characteristics. The astronomical tides were not removed from the water level data,

which results in the astronomical tides being considered as erosional and accretional

events. Since we are most interested in obtaining the maximum range of shoreline

positions and are calculating the potential shoreline changes, inclusion of the

astronomical tide in the water level observations is allowable.

Table 5.1 Relevant site characteristics
'.V S ? Westhampton, Jupiter Island,
S- New York Florida
Surge Data .. .
Mean Tidal Range 2.90 ft 2.46 ft
Spring Tidal Range 3.50 ft 2.95 ft
Representative Station Montauk Mayport
Maximum time difference 51 min 17 min
High Tide Modification Factor 1.22 0.58
Low Tide Modification Factor 1.09 0.75
Wave Data
WIS Station 78 13
Recording Depth 89 ft 148 ft
Average Hb 3.21 ft 3.02 ft
Buoy Number 44025 41009
Recording Depth 131 ft 138 ft
Average H2.60 ft 2.97 ft
Site Conditions
Berm Height 9.5 ft 6.9 ft
Sediment Diameter 0.45 mm 0.30 mm
Sediment Scale Parameter 0.230 ft/3 0.188 ft1/3
Subaerial Beach Slope 1 on 10 1 on 10
Shoreline Azimuth 67 -15

The site conditions recorded in Table 5.1 are from either the aforementioned

sources, or directly from Dewall's reports, with the exception of sediment size. Dewall

briefly mentions the sediment size at Westhampton as being in the range of 0.3 to 1.2

millimeters. A sediment size of 0.45 millimeters was assumed for the calculations at

Westhampton based upon previous experience. Sediment sample data is included in

Dewall's report for the Jupiter Island site, however the maximum sample depth is only 6

feet. The average sediment size of the samples collected between the MLW line and the

-6 foot contour was approximately 0.35 millimeters. Previous experience indicates this

value is too large. Dean and Dalrymple (2001) and Charles, Malakar, and Dean (1994)

show the sediment size at Jupiter Island drops sharply at approximately the -6 foot

contour, due in part to a nearby Coquina rock outcropping. Since we need a median

sediment size applicable across the entire active profile, a more representative sediment

size of 0.3 millimeters was chosen. A sensitivity test was performed on the results to

examine the effects of changing the sediment scale parameter. Decreasing the sediment

size at Westhampton to 0.3 millimeters results in increasing the average range of

shoreline change values by approximately 2 feet. Increasing the sediment size at Jupiter

Island to 0.45 millimeters has the opposite result, decreasing the average range this time

by approximately 2 feet. This result is not surprising as the sediment scale parameter

only enters into Eq. (4.19) through the calculation of W*, which simply acts as a

multiplier, modifying the magnitude of any trends, but not changing them.

A question that arises in the analysis of the data is whether the setup is already

captured in the water level observations recorded at the gauges. Both representative

stations are somewhat sheltered compared to the open coast; therefore it is assumed these

stations capture only a small fraction of the setup occurring on the open coast.

Nonetheless, calculations are performed for both cases; assuming the setup is implicit in

the water level observations, as well as calculating and including the set up explicitly.

Two sets of calculations are performed at each site for each data set resulting in a total of

eight distinct cases.

Results

Dewall's observations of the average shoreline position at each location during

each month are reproduced in Figure 5.8. Clear seasonal trends are observed at both

sites, with maximum ranges of approximately 35 feet in both cases. Dewall's results are

compared with the potential shoreline changes predicted by Eq (4.19). Figure 5.9

displays the results of averaging the potential shoreline change values for each month

throughout the study period, and plotting the average values. As can be expected from

inspection of Eq. (4.19), including the setup explicitly shifts the curves towards a more

erosional value and increases the range of values. Surprisingly, Figure 5.9 indicates a

greater range of average potential shoreline change for Jupiter Island. This result is

contrary to what was found by Dewall. Although both sites appear to exhibit some

seasonal fluctuations, the trends do not coincide with those observed by Dewall. The

Jupiter Island curves mimic the observed trend from August to December, but predict a

maximum accreted or only slightly eroded state (depending on the method of including

the setup), from January to April, the most eroded months according to Dewall's

observations. The Westhampton curves predict only slight variations in the average

potential monthly shoreline position, and are slightly out of phase with Dewall's results.

Figure 5.9 also reveals that the average potential shoreline changes for each

month at Westhampton indicate erosion, while the Jupiter Island data indicate both

erosion and accretion. This is primarily due to the relative magnitudes of the predicted

potential shoreline changes. As discussed later in the chapter, the magnitudes of the

erosional changes are generally much larger than the magnitudes of the accretional

changes at Westhampton, skewing the average potential shoreline changes towards

negative values.

A somewhat surprising feature of Figure 5.9 is the minimal effect explicitly

including the wave setup has during the summer months, at Jupiter Island. This indicates

very mild wave conditions even though July may be considered as the beginning of

hurricane season. At both Westhampton and Jupiter Island, the effect of including the

setup explicitly is more substantial during the winter months which is expected due to the

larger wave heights at both locations during the winter months.

The maximum range in the monthly average potential shoreline changes at

Westhampton and Jupiter Island are 3.32 and 7.92 feet respectively. In comparing these

results to Dewall's, it should be noted that the averaging technique employed here does

not incorporate any measure of the time scales of the erosion and accretion processes as

will be discussed in Chapter 6. Also, Dewall's averages are admittedly biased towards

post storm surveys.

Another useful method of analyzing the data is to examine the average potential

shoreline positions on a monthly basis within each year. Table 5.2 presents the maximum

and minimum average potential monthly shoreline change values calculated using Eq.

(4.19) and the WIS data. The range column is simply the maximum monthly average

potential shoreline change minus the minimum monthly average potential shoreline

change as presented in the table. The average range over the entire period of the study is

calculated as the average of the yearly range values, excluding years containing

incomplete data sets. In Figures 5.10-5.13, the average monthly potential shoreline

change for each month during the study period is plotted. Each line represents a single

72

year of the study. Table 5.3 and Figures 5.14-5.17 present the same information

calculated using buoy data in place of the WIS data.

AVERAGE SHORELINE POSITION AT JUPITER AND WESTHAMPTON
AS OBSERVED BY DEWALL
9_ ,---,

--* Jupiter Island
- I Westhampton

__II~~-2

0-- ^- 1

10 I I I I
15 I
20
35m

1 2 3 4 5 6 7 8 9 10 11 12
Month

Figure 5.8 Average shoreline positions at Jupiter and Westhampton as reported in Dewall
(1977) and Dewall (1979)

AVERAGE CALCULATED AY VALUES
(BOTH LOCATIONS, BOTH DATA SOURCES)

1 2 3 4 5 6 7 8 9 10 11 12
Month

Figure 5.9 Average monthly potential shoreline change values. S represents the
measured tide. Eta is the wave induced setup effect.

(

73

Table 5.2 Maximum and minimum average monthly potential shoreline change values
calculated from Eq. (4.19) using the WIS data set. Values are reported in feet.
Jupiter Island ISetup Implicit) Westhampton (Setup Implicit)
Year Max Min Range Year Max Min Range
1976 5.684 -2.761 8.445 1976 2.377 -2.140 4.517
1977 6.685 -2.903 9.588 1977 1.842 -5.159 7.001
1978 4.506 -6.097 10.603 1978 0.103 -8.642 8.745
1979 8.454 -2.221 10.675 1979 0.000 -5.265 5.265
1980 3.564 0.280 3.284 1980 0.000 0.000 0.000
1981 6.831 -2.375 9.207 1981 0.699 -2.873 3.572
1982 5.252 -4.910 10.162 1982 1.793 -2.299 4.092
1983 1.599 -5.108 6.707 1983 -0.885 -6.349 5.465
1984 3.710 -6.507 10.217 1984 -0.295 -8.191 7.896
1985 5.308 -7.344 12.651 1985 0.160 -6.018 6.178
1986 3.005 -3.119 6.125 1986 2.263 -3.679 5.941
1987 2.531 -3.482 6.013 1987 -0.330 -4.351 4.021
1988 2.980 -4.360 7.340 1988 0.987 -2.368 3.354
1989 1.795 -3.791 5.586 1989 0.640 -2.864 3.504
1990 3.197 -1.632 4.829 1990 1.248 -2.555 3.803
1991 1.998 -6.784 8.782 1991 0.000 -3.612 3.612
1992 1.634 -4.589 6.223 1992 -0.634 -8.171 7.537
1993 2.253 -3.681 5.934 1993 -0.560 -4.447 3.888
1994 2.737 -7.247 9.985 1994 -0.026 -4.825 4.799
1995 3.076 -6.653 9.730 1995 -0.533 -5.143 4.610
Average 3.840 -4.264 8.104 Average 0.479 -4.754 5.233
Jupiter Island Setup Explicit) Westhampton (Setup Explicit)
Year Max Min Range Year Max Min Range
1976 3.039 -7.849 10.888 1976 -0.053 -4.900 4.847
1977 2.426 -7.111 9.537 1977 0.591 -8.905 9.496
1978 1.858 -11.802 13.660 1978 -1.398 -10.956 9.558
1979 2.247 -5.423 7.670 1979 -0.468 -11.285 10.817
1980 0.817 -6.215 7.032 1980 -0.738 -5.679 4.941
1981 1.562 -6.332 7.894 1981 -0.616 -7.990 7.374
1982 2.268 -9.755 12.023 1982 -0.210 -3.412 3.201
1983 0.101 -9.553 9.653 1983 -1.332 -11.992 10.660
1984 0.874 -12.722 13.596 1984 -1.387 -13.193 11.806
1985 0.877 -12.188 13.065 1985 -1.322 -8.716 7.394
1986 0.324 -7.008 7.331 1986 -1.237 -7.909 6.673
1987 0.510 -10.723 11.233 1987 -1.242 -6.646 5.404
1988 0.342 -7.167 7.509 1988 -0.685 -3.529 2.844
1989 1.146 -7.134 8.280 1989 -1.172 -6.688 5.516
1990 1.758 -3.449 5.207 1990 -0.746 -4.728 3.982
1991 0.155 -10.518 10.672 1991 -1.020 -6.534 5.514
1992 0.409 -7.288 7.696 1992 -1.374 -11.064 9.691
1993 0.736 -6.991 7.728 1993 -1.209 -7.421 6.213
1994 0.802 -11.992 12.794 1994 -1.206 -6.781 5.575
1995 1.561 -8.346 9.907 1995 -1.717 -8.158 6.441
Average 1.191 -8.478 9.669 Average -0.983 -7.737 6.754

ENVELOPE OF AY VALUES JUPITER ISLAND
(WIS DATA SETUP IMPLICIT)
10

2 2

-4
o -2

-6
-8
-10
1 2 3 4 5 6 7 8 9 10 11 12
Month

Figure 5.10 Envelope of average monthly potential shoreline change values at Jupiter
Island calculated from Eq. (4.19) using the WIS data set, assuming the setup is included
implicitly. Each line represents a single year of the 20 years with available data.

ENVELOPE OF AY VALUES JUPITER ISLAND
(WIS DATA SETUP EXPLICIT)
4

0

-2

-4

-6

-8

*10

-12 '
.AA

2 3 4 5 6

7 8 9 10 11 12

Month

Figure 5.11 Envelope of average monthly potential shoreline change values at Jupiter
Island calculated from Eq. (4.19) using the WIS data set, including the setup explicitly.
Each line represents a single year of the 20 years with available data.

ENVELOPE OF AY VALUES WESTHAMPTON
(WIS DATA SETUP IMPLICIT)

1 2 3 4 5 6 7 8 9 10 11 12
Month

Figure 5.12 Envelope of average monthly potential shoreline change values at
Westhampton calculated from Eq. (4.19) using the WIS data set, assuming the setup is
included implicitly. Each line represents a single year of the 20 years with available data.

ENVELOPE OF AY VALUES WESTHAMPTON
(WIS DATA SETUP EXPLICIT)

.2
-4

-10
-12
.4

-14
-1 4 -------------------------------- I ---- I ---- I ---
1 2 3 4 5 6 7 8 9 10 11 12
Month

Figure 5.13 Envelope of average monthly potential shoreline change values at
Westhampton calculated from Eq. (4.19) using the WIS data set, including the setup
explicitly. Each line represents a single year of the 20 years with available data.

76

ENVELOPE OF AY VALUES JUPITER ISLAND
(BUOY DATA SETUP IMPLICIT)

-12
-8 ___ ___

1 2 3 4 5 6 7 8 9 10 11 12
Month

Figure 5.14 Envelope of average monthly potential shoreline change values at Jupiter
Island calculated from Eq. (4.19) using buoy data, assuming the setup is included
implicitly. Each line represents a single year of the 11 years with available data.

ENVELOPE OF AY VALUES JUPITER ISLAND
(BUOY DATA SETUP EXPLICIT)

-2
-4

-6

-18
-14
-16 -------
-18 --- --- ---
1 2 3 4 5 6 7 8 9 10 11 12
Month

Figure 5.15 Envelope of average monthly potential shoreline change values at Jupiter
Island calculated from Eq. (4.19) using buoy data, including the setup explicitly. Each
line represents a single year of the 11 years with available data.

1
0
-1

-2

-7

-6
-7
-8

-9

77

ENVELOPE OF AY VALUES WESTHAMPTON
(BUOY DATA SETUP IMPLICIT)

I 2 3 4 5 6 7 8 9 10 11 12

Month

Figure 5.16 Envelope of average monthly potential shoreline change values at
Westhampton calculated from Eq. (4.19) using buoy data, assuming the setup is included
implicitly. Each line represents a single year of the 9 years with available data.

ENVELOPE OF AY VALUES WESTHAMPTON
(BUOY DATA SETUP EXPLICIT)

-1
-2
-3
44
-5
0 -6
-7
-8
-9
-10
-11
1 2 3 4 5 6 7 8 9 10 11 12
Month

Figure 5.17 Envelope of average monthly potential shoreline change values at
Westhampton calculated from Eq. (4.19) using buoy data, including the setup explicitly.
Each line represents a single year of the 9 years with available data.

Table 5.3 Maximum and minimum average monthly potential shoreline change values
calculated from Eq. (4.19) using buoy data. Values are reported in feet.
Jupiter Island (Setup Implicit) Westhampton (Setup Implicit)
Year Max Min Range Year Max Min Range
1988 1.809 -4.817 6.626 1991 0.000 -2.896 2.896
1989 1.930 -5.184 7.114 1992 -0.924 -7.654 6.730
1990 3.661 -1.800 5.461 1993 -0.537 -3.201 2.664
1991 2.787 -7.723 10.510 1994 -0.082 -2.821 2.739
1992 1.949 -6.014 7.963 1995 -0.743 -3.876 3.132
1993 0.690 -4.214 4.904 1996 -1.221 -8.359 7.138
1994 1.790 -7.128 8.918 1997 -1.152 -5.361 4.209
1995 1.634 -10.969 12.603 1998 -0.660 -4.330 3.670
1996 1.609 -6.073 7.681 1999 0.236 -4.689 4.926
1997 0.558 -3.330 3.888 Average -0.635 -5.036 4.401
1998 1.214 -7.425 8.639
Average 1.782 -5.986 7.768_

Jupiter Island (Setup Explicit) Westhampton (Setup Explicit)
Year Max Min Range Year Max Min Range
1988 0.394 -8.168 8.563 1991 0.000 -4.215 4.215
1989 1.354 -9.331 10.685 1992 -1.472 -10.683 9.211
1990 1.746 -4.330 6.076 1993 -1.129 -5.099 3.970
1991 -0.617 -12.573 11.956 1994 -1.034 -4.554 3.520
1992 -0.160 -11.124 10.964 1995 -1.479 -4.908 3.430
1993 0.355 -7.696 8.051 1996 -2.033 -10.555 8.522
1994 1.001 -10.548 11.549 1997 -2.409 -7.687 5.278
1995 0.048 -16.913 16.961 1998 -1.456 -6.246 4.790
1996 0.548 -10.991 11.539 1999 0.000 -6.463 6.463
1997 -1.335 -5.591 4.256 Average -1.377 -7.024 5.648
1998 0.590 -12.290 12.880
Average 0.353 -10.139 10.492

Tables 5.2 and 5.3, illustrate that slightly "tweaking" the way the average range is

calculated, produces different results. Calculating the average range based on the range

of potential shoreline change values within each year results in an increase of

approximately 3 feet versus the previously defined range. Including the setup explicitly

increases the yearly range values by approximately 2 to 4 feet. It should be noted, that in

no case does the maximum range obtained by including the setup implicitly occur in the

same year as when calculating it explicitly. This supports the previous assumption that

the water level observations do not capture the entire setup.

In general it is observed that the maximum monthly shoreline changes do not vary

substantially, especially when the setup is calculated explicitly. Most of the variation in

the yearly range values results from a decrease in the minimum average monthly

potential shoreline change values. In other words, there is a greater tendency for a large

range of potential shoreline changes to be caused by an extremely erosional month than

an extremely accretional one.

A maximum range of 11.81 feet is calculated at Westhampton in 1984 using the

WIS data. From Figure 5.13, it is observed that the minimum monthly potential shoreline

change for 1984 occurs in March. As would be expected in Long Island, this is most

likely the result of a spring Nor'easter. Using the buoy data, a maximum range of 9.21

feet is calculated at Westhampton in 1992. From Figure 5.17, it is observed that this

event corresponds to a minimum average potential shoreline change occurring in

December of that year. Data presented later in the chapter indicate that a majority of this

minimum average shoreline change can be linked to a single severe storm resulting in

several erosional shoreline change values in excess of 100 feet, with a maximum value of

over 175 feet.

In general the average range of potential shoreline positions at Jupiter Island is 3

to 5 feet more than those at Westhampton. The largest range of average monthly

potential shoreline change values at Jupiter Island is 16.96 feet, and occurs in 1995.

Figure 5.15 indicates that this maximum range is associated with a minimum average

potential shoreline position occurring in August of 1995. Further research reveals that an

incredible 7 hurricanes, including two category 4 hurricanes, passed offshore of Florida

during this month. Comparing the average range calculated from the WIS data with

those calculated from the buoy data for 1995 illustrates one of the disadvantages of the

WIS data set. In this case, significant wave events created by the 7 offshore hurricanes

are not captured in the WIS data set and result in a substantially different range

prediction.

In general the results obtained from the WIS data, Figures 5.10-5.13, exhibit a

slightly greater range of values than the figures based on the buoy data, Figures 5.14-

5.17. Although this is true at both sites, it is most pronounced at Westhampton during the

winter months. This increase in the range of values is due to the difference in breaking

wave height predicted by the two data sets. The average breaking wave height calculated

using the WIS data is larger than that calculated using the buoy data, with a more

pronounced difference at the Westhampton site. The breaking wave height affects the

calculation of W*, which acts as a multiplier in Eq. (4.19). Increasing W* increases the

range of potential shoreline changes predicted regardless of the method of including the

setup.

The variability observed in the average potential shoreline change for each month

is surprising. Due to the method and period of averaging employed, smaller fluctuations

were expected. Figure 5.13 shows the average monthly values fluctuating through more

than 10 feet during 4 of the 12 months. Not only are the ranges of the average potential

monthly shoreline changes surprising, but also the spread within the range. In most

cases, the large range of values in a given month is the result of many values spread

evenly between the two extremes rather than several closely spaced averages with a

single outlier responsible for increasing the range. The envelope plots help illustrate the

substantial variability in the potential shoreline change values predicted during the winter

months as compared to the summer months.

Each individual potential shoreline change value is plotted in Figures 5.18

through 5.25. All of the figures presented previously have depicted the results of various

averaging techniques. Although averaging is oftentimes useful in facilitating

comparisons between large data sets, here it actually obscures the dynamic behavior we

are most interested in. As the period of averaging is decreased, the results become more

interesting. Figure 5.9 indicates very little variability in the average potential shoreline

change values between January and April, however examination of the corresponding

envelope plots reveals a large range of values that is obscured by the averaging

technique. The previously discussed erosional event of December 1992 provides a

convenient platform for discussion.

Potential Shoreline Change
Jupiter island Buoy Data Setu Included mpcity.

40

20 -

-40

S-60

-80allow Storm
-100 Hunlc nto Dee Gabriel., and H go

-120
88 89 90 91 92 93 94 95 96 97 98 99
Year

Figure 5.18 Potential shoreline change at Jupiter Island calculated using buoy data,
assuming the setup is implicitly included

82

Potential Shoreline Change
Jupiter Island Buoy Data (Setup Calculated Explicitly)

1 .. .... I .. .. ... .

S Supersto

4* urrican Gordon
SHal ween St rm I
SLHurricar s Dean, Gabrielle and H lu j arch 199 Storm
___ Janury 1989 storm March 1 96 Stor ______

88 89 90

91 92 93 94 95 96
Year

97 98 99

Figure 5.19 Potential shoreline change at Jupiter Island calculated using buoy data,
explicitly calculating the setup

Potential Shoreline Change
Westhampton Buoy Data (Setup Included Implicitly)
75 ................-. -.. ..-.-.................. ...-----------

50
25

0
-25

-20 H.-u- ---- - -
0 -12
-75 D b-o

91 92 93 94 95 96 97 98 99 100

Year
0 -125 "

-150 ; S

-17 5 ......................

-200 ............................... .............................
91 92 93 94 95 96 97 98 99 100
Year

Figure 5.20 Potential shoreline change at Westhampton calculated using buoy data
assuming the setup is implicitly included

0 -

-20

-40 -

Potential Shoreline Change
Westhampton Buoy Data (Setup Calculated Explicitly)

91 92 93 94 95 96 97 98 99 100
Year

Figure 5.21 Potential shoreline change at Westhampton calculated using buoy data,
explicitly calculating the setup.

Potential Shoreline Change
Jupiter Island WIS Data (Setup Included Implicitly)

R L
It*

40 - -
B. A.^.J^ T

0

( 20
0
* 0

- -20

0 -40
a
-60

-80

-100
1986

1987 1988 1989 1990 1991
Year

1992 1993 1994 1995 1996

60

Figure 5.22 Potential shoreline change at Jupiter Island calculated using WIS data,
assuming the setup is implicitly included.

84

Potential Shoreline Change
Jupiter Island WIS Data (Setup Calculated Explicitly)

"; l ^t

____ 3~r L -I- L Z1 I .4 4i U

-BO rricane indrew
-100anuary Storm hurricane Dean, G brile, & go
V10 Mai :h 1987 S orm M >rch 1989 Storm
-120
1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996
Year

Figure 5.23 Potential shoreline change at Jupiter Island calculated using WIS data,
explicitly calculating the setup

Potential Shoreline Change
Westhampton WIS Data (Setup Included Implicitly)

100
75
50

1987 1988 1989 1990 1991 1992 1993 1994 1995 1996
Year

Figure 5.24 Potential shoreline change at Westhampton calculated using WIS data
assuming the setup is implicitly included

S'-Speretorm \

25
0 |

6 -25
| -50
0
, -75 -

| -100
0-
-125
-150
-175
-200
1986

~f~'r

I I~

I $

Potential Shoreline Change
Westhampton WIS Data (Setup Calculated Explicitly)

100---- ----
75

00 / Decembr1990 Strm Supe"rst ; -
-125

Octol r 1988 St rm Nov1 r 1993 torm -- ___ __
-150---- De ember 19 4 Storm
-500

-7500 ------------
-125
-150 Do( ember 199 Storm

7D comber 1! 6 Storm Dec ber 1br992 Storm
1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996
Year

Figure 5.25 Potential shoreline change at Westhampton calculated using WIS data,
explicitly calculating the setup.

Close examination of this event depicted in Figure 5.21, reveals 13 potential

shoreline change values greater than -100 feet, occurring over a time period of only 13

hours out of a total of 744 hours in the month of December. The true impact of this large

scale event is masked by the 731 "normal" values throughout the month. The erosion

caused by an event of this magnitude would likely persist throughout the month, a fact

that is lost when the different time scales of the erosion and accretion processes are not

accounted for. A more correct method of averaging the potential shoreline change values

is discussed in the following chapter.

After closely examining the December 1992 storm at Westhampton and the effect

averaging has on its perceived magnitude, it is useful to examine the August 1995 storms

at Jupiter Island. The December 1992 storm and the August 1995 storms contribute to

the maximum range of average potential shoreline changes calculated at each site

utilizing the buoy data. As discussed previously, these maximum ranges are due

primarily to the minimum potential shoreline changes predicted in December of 1992 and

August of 1995. Figure 5.21 indicates that the large average erosional change predicted

for December of 1992 is due primarily to the effects of one severe short-lived storm. In

contrast, examination of Figure 5.19 indicates the large average potential erosional

change predicted in August of 1995, appears to be due to the longer duration of a number

of smaller events. Without including a measure of the time scales of the erosional and

accretional processes, no distinction exists between a singular massive event, and several

smaller, longer duration events.

Several other conclusions may be drawn from Figures 5.18 to 5.25. In general the

magnitudes and the durations of the large erosional events are much greater than those of

the accretional events. This manifests itself in the disproportionate number of potential

erosional changes predicted, particularly at Westhampton, when the results are averaged.

Table 5.4 presents the maximum range of the individual potential shoreline change values

for each year as well as the average of these ranges. The data calculated using both wave

data sets indicate a slightly larger range of values at Westhampton. This is in contrast to

the larger ranges predicted at Jupiter Island when the data are averaged, but in agreement

with what Dewall observed. The averages in Table 5.3 are based upon the potential

shoreline change values and are slightly larger, but of the same order of magnitude as the

90 feet at Westhampton and 70 feet at Jupiter Island reported by Dewall. Considering

that the calculated ranges are based on potential ranges representing the maximum

change, the agreement of the results is considered promising.

87

Figures 5.18-5.25 also help to distinguish the major causes of erosion at each site.

Keeping in mind the scale difference between the Jupiter Island and Westhampton plots,

it is clear that the Jupiter Island site is more influenced by hurricanes than the

Westhampton site. Although the effects of several hurricanes are observable at the

Westhampton site, most notably, Hurricanes Lilli and Floyd, the magnitude of their

effects and the frequency of their occurrence are small compared to that of the major

winter storms. At Jupiter Island, although the effects of several winter storms are

observed, their magnitude and frequency are less than that of the major hurricanes.

Table 5.4 Range of potential shoreline change values based on individual extreme values.
Values are reported in feet.

WIS DATA BUOY DATA
Westhampton Jupiter Island Westhampton Jupiter Island
Explicit Implicit Explicit Implicit Explicit Implicit Explicit Implicit
186 164 95 100 201 177 121 100
110 88 120 116 142 146 105 94
119 117 160 122 117 117 110 104
91 81 175 153 69 69 95 93
119 114 135 134 112 112 90 83
102 86 170 164 98 98 120 105
205 183 138 125 58 58 96 82
163 164 127 106 89 89 142 107
123 127 166 150 110.75 108.25 73 74

189 163 123 112
118 106 128 107
93 83 122 121
227 208 97 87
173 157 82 81
156 157 97 91
144.38 130.94 100 82

126
127
85
126.55

120
105
83
114.80

98 86
105.00 92.80

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