Citation |

- Permanent Link:
- http://ufdc.ufl.edu/UF00091080/00001
## Material Information- Title:
- Development of erosional indices and a shoreline change rate equation for application to extreme event impacts
- Series Title:
- UFLCOEL-2001012
- Creator:
- Miller, Jonathan K., 1976-
University of Florida -- Civil and Coastal Engineering Dept - Place of Publication:
- Gainesville Fl
- Publisher:
- Coastal & Oceanographic Engineering Program, Dept. of Civil & Coastal Engineering
- Publication Date:
- 2001
- Language:
- English
- Physical Description:
- xii, 99 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Coast changes -- Mathematical models ( lcsh )
Storm surges -- Environmental aspects ( lcsh ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (M.S.)--University of Florida, 2001.
- Bibliography:
- Includes bibliographical references (leaves 97-98).
- General Note:
- Cover title.
- Statement of Responsibility:
- by Jonathan K. Miller.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 49535169 ( OCLC )
## UFDC Membership |

Full Text |

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. Mfichael 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 ACKNOWLEDGMENTS ......................................................................1i LIST OF TABLES .............................................................................vii LIST OF FIGURES............................................................................. viii ABSTRACT ................................................................................... xii INTRODUCTION ............................................................................... 1 Coastal Erosion Processes..................................................................... 1 Objectives and Scope.......................................................................... 5 LITERATURE REVIEW ........................................................................ 7 Model Hurricane............................................................................... 7 Bathystrophic Storm Surge.................................................................. 10 Erosion Models..............................................................................11I Shoreline Recovery Model................................................................... 14 Seasonal Shoreline Change.................................................................. 14 SHORELINE CHANGE ANALYSIS......................................................... 16 Site Descriptions.............................................................................. 18 Volusia County............................................................................. 18 St. Johns County ........................................................................... 18 Martin County.............................................................................. 19 Sources and Analysis Techniques........................................................... 19 Results and Conclusions ..................................................................... 21 EROSION INDICES............................................................................ 28 Event Longevity Parameter Index (ELP)................................................... 31 Theory .................................................................................... 31 Winter Storm Related Surge-Reconstructed Hydrographs ........................... 33 Hurricane Related Surge-Bathystrophic Storm Surge Model......................... 34 Results .................................................................................... 39 Hurricane Erosion Index (HBI).............................................................. 49 V Theory ....................................................................................................................... 49 Results ....................................................................................................................... 50 M odified Hurricane Erosion Index (M HEI) .................................................................. 53 Theory ....................................................................................................................... 53 Results ....................................................................................................................... 55 SHORELINE RESPONSE RATE .................................................................................... 58 Rate Equation ................................................................................................................ 59 Theory ....................................................................................................................... 59 Results ....................................................................................................................... 60 Determ ination of Equilibrium Shoreline Position ......................................................... 65 Theory ....................................................................................................................... 65 Results ....................................................................................................................... 70 CONCLUSIONS AND RECOM M E ND ATIONS ........................................................... 88 Erosion Indices .............................................................................................................. 88 Summ ary and Conclusions ........................................................................................ 88 Recom m endations for Future Study .......................................................................... 90 Shoreline Response Rate Equation ............................................................................... 92 Summ ary and Conclusions ........................................................................................ 92 Recom m endations for Future Study .......................................................................... 93 LIST OF REFERENCES .................................................................................................. 97 BIOGRAPHICAL 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 mEI ...................................................................................... 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 MHEI .................................................................................. 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 t, from 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 Definition sketch of (x, 0, and On.............................................................. 30 4.2 Definition sketch of Osin-rotwvect .........................................................................30 4.3 Definition sketch of w and 0 ............................................................ 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 storms are not .................................................................. 48 4.11 Plot of ELP Index versus erosion at the Volusia South Site. Setup is included, winter storms 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, 5, and wave setup, 1j.............................. 54 4.15 Plot of MHEI versus erosion at the Volusia South Site................................... 56 4.16 Plot of MHiEI 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=1 and no external forcing................... 62 5.3 Application of Eq. (5.3) for the case of ke=ka=1 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 k,=l, k.=0.2 with external forcing applied at t= O and t= 10 ........................................................................................................ 64 5.7 Site and data station location m ap ................................................................................... 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 w ave 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 d ata ..................................................................................................................... 7 4 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 d ata ..................................................................................................................... 7 5 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 d ata ..................................................................................................................... 7 6 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 I I 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 implicitly 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 im plicitly 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. Longterm 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 Hr .. s 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. OPALSTORM SURGE HYRGRPH Panama Cty, GULF 0r Beach Pier 4 12 ;--4 30 6 r 8 4-. 1 113 1-4-11 11 2 0 a 12 ; : i : ; + . i s t 24t A 42' i 4 d 4' 0 16 i 2l+ 1013 1014 1oll 1016 TIME (hrs.(UTC):sarl1tng 0:00 on 1003195 Figure 5. Slorm surge hydrograph for Opal at Panama City Beach HURRICANE OPAL High Water Marks ... L" . Figure 6. Surveyed high water marks after Opal along 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 19501995, while the second phase dealing with the modifed 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 tol976-1995. 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. Y1Vf 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) = p. + 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- -p. (2.2) in which p. 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 .0 Vf r 6 4 C S2 0 C .0 -2 I(D E -4 * -6 1 0 z -8 -10 I I I I I -.-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, U,, 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 Uc = ApR (2.3) P. 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 R 2R/ u = Pa r (2.4) g = 2oRsin4 where o and ( 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 lVf sin P U + c (2.5) Y=iL u:n u~ c 9 ) where Vf is the event forward speed and 03 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 + 1 Y) (2.6) where the gradient wind vectors are rotated inward by approximately 18'. 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. 100 z 8 \ .: TA , 6: -1 Z , C:C cc ( 2 C 00. -10 .1 a ,! -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 811 1 FT'. 1Ey] 1 8p(27 8x- gD[ p, PWg 8x dqY 1 ( Tb) (2.8) dt pw where x and y represent the cross-shore and long-shore directions respectively, g is gravity, qj is the water surface elevation, D is the total local depth, 'r, and Tb are the surface wind and bottom shear stresses respectively, 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 onedimensional 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 BBqYI +- w (2.9) Ax Fr, in+ + n+I n+I n+I 1 Pi+I1 i+l =li + -EqyI + (2.10) gDi pw Pw8g fat qy BB =1.0+ (2.11) D2 where the boundary conditions include initiation from rest (qy=0 & TI=0) and ri due to barometric pressure alone at the seaward edge of the grid. Ap = (2.12) Pw 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 = 16 (tS2Vgr (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 12 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 tr =0.00175 g (2.14) Vf 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. 1., 6 Storm Tie 4 Pro-StoIm Elevation Abo~vo#ASL 3 (m) 2 Normal Pro.Storm Tidli Cond motions "0 20 Te40 os) 60 SO Time Oours) 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 W. (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 Ay = -S h (2.16) A /2 1+ Bh 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 f 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 ,EE = 0 W1 WA(x, y)cos0dxdy = ]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 1.2 10 2 0.8 0.4 0.2 0.00 1 2 3 4 5 6 7 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) = R_(1-eXT) (2.19) where R,, represents the maximum or equilibrium response, and T, 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 T, and various R(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 Czemniak (1977) supports Dewall's results. CHAPTER 3 SHORELIN4E 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 0, over a several mile stretch were analyzed. Location X is near the intersection of Route AlA and Driftwood Avenue. Locations 0 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, 0, 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 7t1h Street and Route AlA. Location 0 is approximately 1 mile south of Location X, near the intersection of East 24 th Street and Route AlIA. Location is approximately 3000 feet north of Location X, near the intersection of Ocean Avenue and Route Al IA. 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 AlA 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 A IA. 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 AlA. The centerlines of nearby streets were used as control structures at both locations. Martin Count 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 I 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. IL -IL 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 197 1, 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 - _B 0 0 -60 _ _ -90 I -150- -_ _ ______ _ 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 Figure 3.3 Shoreline positions at the St. Augustine Site ST. JOHNS SHORELINE POSITION 300 -- 270 - 4B 240-- IA 210 g 180 -- 0 1500 a-120 ___90 ____ ___60 'y 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 I A .-4.- A_____150 _--.e-- X A -0 120 O 90 -___. 60 0 -30 -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 120 90 -4- Ast 60 -- X 60 _-30 -90 -120 -150 -180 -210 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 Year Figure 3.6 Shoreline positions at the Volusia South Site MARTIN COUNTY SHORELINE POSITION 90 -- 60 _____30 _0 -30 c -60 ___-0 -90 -_ -_0 a. -120 -150 - X -180 A -210 -240 1965 1970 1975 1980 Year 1985 1990 1995 2000 Year Figure 3.7 Shoreline positions at the Martin County Site COMPARISON OF AVERAGE SHORELINE POSITION AT THE FIVE SITES 270 1 i I i I 240 4 Martin -- St. Johns 210 A St. Augustine -Volusia North 180 0 Volusia South __150 1120 - a A,&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 (X is one such exception. The angle oc is defined as the angle between the shoreline and the yaxis and is clockwise positive. Several angles defined in the conventional manner and used throughout include 0, Osin, Osln_rotwvect, and 03. 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 Osn = 0+ aX (4.1) The angle of interest is the angle of the gradient wind vector with respect to the shoreline normal. In terms of 0sin, this angle, Osin_rotwvect, is defined as 0sIn_rotwvect = 0sln 90' + 18* (4.2) The 18' accounts for the inward rotation of the gradient wind vectors due to frictional effects. The angle 13 is defined as the angle between the velocity vector and the radius, and is given in terms of o and 0 as 13 = 180 -0(+0 (4.3) Here ow defines the direction of the velocity vector. Figures 4.1-4.3 provide definition sketches of the problem geometry. (delE.deIN) Shore F 4Normal E Figure 4.1 Definition sketch of (x, 0, and 0,1n rmal Rotated Wind Vector Figure 4.2 Definition sketch of Osln_rotwvect Figure 4.3 Definition sketch of w and P3 Event Longevity Parameter Index (ELP) TheorEy 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= 1-2 (S2t"F-y5 (4.4) 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 onedimensional 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 Hydrographs 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 12 F10 .0 8 -v--b-er-ed- *1 W 6 Surge=2.04' e= .9 ------ Predicted+ l' 4 t -9 6hrs t L 84h rq S urge (O bs-P red) 2 0 .- 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 onedimensional 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 onedimensional 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 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 + BB q' + At (4.5) lwy n+1 n+1 AxF ~ n+1 pf~ n+1 n+ l~l -n+l Pi --Pi+I 46 fli+ =l i + gD -8qy + 1+1 (4.6) BB = 1.0+ D 2 (4.7) 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+1 n+1 P Pi+i (4.8) P. g 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 I- q n+1] (4.9) gDj qy, The Coriolis parameter E is defined as 2osino, where 4 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). Ax "tw_1 (4.10) gDi i pw j where 'wx is the wind shear stress given by ,, = PwkW2 (-COSOsn-rotwvect ) (4.12) The previously undefined factor k is the Van Dom air-sea friction coefficient given by 1.1X10-6 w < w k = 1.1x106 +2.5x106 '-E-J 2 > W (4.14) Wc 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 crossshore 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 Tw, =w kwW 2 (sin OsIn rotwvect ) (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. - _H2 2khb 3y2/8 l6hb sinh(2khb) +3y2 /8 hb 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. T"11 = p_ Pi (4.15) p9g Here p, is the pressure in the absence of the hurricane and pl 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 R 2=1_ Y" IJ -ZJ]_ (4.16) ( y2)_ n y) 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 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 Rsquared 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 1 ELP Index vs Erosion St. Augustine Site XI 1 V u-,. - 15 ___- ---. y =0.3797 -1.9187 0 , e R2=04551 0 10 0 ,0 y. 0.4143x- .4.305 = 0.3567x 4.7668 "R' = 0.70 9 R2 = 0.3 104 A 0 _ U __ __ 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. 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. Setup and winter storms are not included. ELP Index vs Erosion St. Augustine Site * Cu 0 yry =0.4'77x 4.782' 30 3yr X 5yr .. 35 - -Linear(yr) - -Linear(3yr) R 0.7516 25 inear(5yr) Linear (Cu e) 25 r y =.4204x- 1.9293 0 20 00, .0 0 20- 0- 0 C so 0*0 10-___(= 0.3628x 2.8005 0 5R 461x- 14.31 4 R =0.8696 0 10 20 30 40 50 60 70 80 90 Erosion (ft) Figure 4.7 Effect of removing all four points associated with from Figure 4.5. the 1989 erosional event 42 ELP Index vs Erosion St. Johns Site 35 30 25 20 15105 - * Cum U 1 yr A 3yr X 5yr -o - Linear (3yr) Linear (lyr) 00 Linear (5yr) --- Linear (Cum) A0 y =00.5431 x 1.11.9035 1 RR0.5307 00.506 -- P- 1 =0.485x + 1. 13111 0 10 oy= '- y = 0. 537x 10.079 1 y = 0.5677x 3.4; g1o 2= ,4 R2 = 0.9874 V o R 0.7234 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 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 197 1. Here the long interval of application is questioned. Examination of the ELP Index associated with each individual ston-n 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 350 300 250 X 200 a. 150 100 50 0 0 10 20 30 Erosion (ft) 40 50 60 Figure 4.9 Plot of ELP Index versus erosion at Martin County Site. included, setup is not. Winter storms are 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 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 trendiness 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 5r-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. ELP Index vs Erosion Volusia South Site North Site. Setup is * Cum 1 yr A 3 yr S5 yr - - Linear (1 yr) - - Linear (Cum) - - Linear (3 yr) - - Linear (5 yr) y = -4.0669x R2 = 0. +331.43 - n - .4586x + 230.7 R = 0.538 S- I -- __J~KZL___ Ii m. - - - R2 = 0.003 ', --y =2.012x 40.445 12, R2= 0.7367 0A 1 0 3 ( 45 010 20 30 Erosion () 40 50 60 7 Figure 4.11 Plot of ELP Index versus erosion at the Volusia South Site. Setup is included, winter storms are not. 300 250 200 X (0 S150 0. -j 100 50 0 . - " m m m Hurricane Erosion Index (I{EI' 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. IOR HEI= f p,,,kW Y (cos 0,. otwecl )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 crossshore 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, m. The shoreline orientation is taken as -17.80, north of state plane coordinate 833,400.0 (approximately West Palm Beach), and +6.70 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 Rsquared 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 # Cum i 1yr 10- A 3yr X 5yr - Linear (3yr) Linear(1lyr) - -- Linear (5yr) Linear (CUM) 8 6 -R2=0.008 2 2R=0'04620 4 y=-0.016;:2 + "4- 711 3 "0 ", ,, -.19 + .828-A R2 = 0.0669 12 (3462 2 "'" ,== ='.-= 0.0058 ; 0,9696 ................--......-----------0 R =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 0 10 20 30 40 50 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 1960-197 1 from Figure 4.12. Modified Hurricane Erosion Index (MIHEI) 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 HEL 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 HEL, 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. 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 MHEL. In addition to an elevated water level, S, we introduce the wave induced setup, il. 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 f(B-S- (yy+ JA(y-Ay)2/3dy Ay Ay W.+Ay W.+Ay f Ay2/dy+ f S+71(y))dy (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 Ay = -W. 0 Hb (4.19) 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 - I.28Hb (S + 0.068Hb) Ay= J.E / B (4.20) 1 +1.28Hb Now we will define the numerator as the Modified Hurricane Erosion Index MIEI= f(S+0.068Hb 1.28HbdA (4.21) huricane 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 cur a 1 yr A 3 yr X 5 yr y =4. 585x + 230.49 X 500 Linear (5 yr) - Linear (3 yr) R =0.7782 400 * -300 -.. ...- 094.-..269.43" R2=0.1008. 200 y=4.4021X -84.379 A * R2 = 0.9429 * 100 -10 ., y = 7.63 7x 152.31 R2 = 0.8238 010 20 30 Erosion (t) 40 50 60 70 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 600 A 3 yr X 5 yr 6R -0.42 - Linear (1 yr) - Linear (3 yr) XR 2 - Linear (5 yr) - -Linear (CUM) 500-- ...... . X 400 y= -.-02 3x + 218.45W A. 2 i R2 0.0981 300- -u m a y=1.0685x + 220. 200* R= 0.1018 0 100y=- .898x + 6.:7 ",,,' R = 0.2908 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 I 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 I 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-terrn 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 Thepa 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. ay - = ka (Y eq (t) Y(O) (5.1) at Here k, 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 k, resulting in y(t) = y0ek t + kalek"(tt)yeq (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. y n 1lk.At)+ kaAtyeq yl = k At (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, yeq 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 90' 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 Yeq. A quasiequilibrium 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 62 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 k. 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 quasiequilibrium 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=l) 1.25 A i Al i 6 ii IIl i i i I i I I II II II II II II II I I II II II II l I II II I 0o.5 ,!I I ,I 0.25 2 t-Ht 1 4v U LLU- mI I m 1I 1-- 14 1 1111111 11 1 A-i W IA, V/ II I I I 1 0 a. . -0.25 0 -0.75 -1.25 IV~~ ~ IV IVr iriv iv tT rrVII IV IVi wrIrV 1VI W ITI_T _i i i' i Ii ii'i i'ii iI I I I I I I I I I I I I I I II I I I I II I I I I I II il Jfil i i l i1 If IfilI il l ill If il l l Il Ii V Yeq VVVVVVVVVVVVVVVV 0 2 4 6 8 10 12 14 16 18 20 Time (yrs) Figure 5.2 Application of Eq. (5.3) for the case k=ka=1 and no external forcing RATE EQUATION (Ka=Ke=1) 0 2 4 6 8 10 12 14 16 18 20 Time (yrs) 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) 1.25 I I 'I ~IA lII~Ii "'I'll 'Ii I' I I I I 1A! 1 I I I ('fit' I V! I' Ii~~I I II I II I ib'Ii !A!I (i I -- .I -I I 4 I I I * it ii j ,ii ii- ... . :i .. i j .... :i .. .. I ji ii ji I I I'"l l I 1 1 I I I I I l l i l l I I I I I I I I I I I I I I I I I II II II I I I I I I I II Ii II 1 1 I Ij II I II I 1 1 1 1 1 1 1 1I1 11 1 1 1 11 1 11 I I I I I I I I I I, I I I --- Yeq Y 0 2 4 6 8 10 Time (yrs) 12 14 16 18 20 Figure 5.4 Application of Eq. (5.3) for the case of ka=ke=5 and no external forcing 3 2 0o 0 S-1 -2 -3 I I I ! ~iAt 0.75 a) 0.25 0 25 -0.25 0 -0.75 -1.25 RATE EQUATION (Ka=0.2,Ke=1) 1.25 L A II III II I i I ii I I I 'I J LI II JI A II I Ii I II I II iIl II II II III I S jk li I t .I I if I I i I Il/ '[ Ii I II II IV V V V V V -Yeq ---V A tj 27TV~V 27127 ni; j: :1111 ]1II I I V V I I I I V V II II I I1~ V V 0 2 4 6 8 10 Time (yrs) III' ~~II I i~I I V V 'II I V V liii rrl'r I I iT-4F V V III' 1rjr I I il-tv V V 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) -3 V-I I 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 0.75 0.25 0 0 C -0.250 , -0.75 -1.25 i -4 1 i i i i I 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 threehourly 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, H,, 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=.626H,. 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. Wetapo Q5 L A A 8 00 9 0 80 76 *72 F04 25 7 0 Data 0 Locabons 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. Westhampton, New York Jupiter Island, 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 Hb 2.60 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 1/3 0.188 ft 1/3 Subaerial Beach Slope 1 on 10 1 on 10 IShoreline 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 N11LW 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 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) -9 1 J1 L - I -,IJ- 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 Setup Implicit) Westhampton (Setup Implicit) Year Max Min Range Year Max Min Rangie 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 10.000 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 1-3.612 3.612 1992 1.634 -4.589 6.223 1992 -0.634 1-8.171 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 13.076 -6.653 9.730 1995 -0.533 -5.143 4.610 Average 3.840 -4.264 18.104 Average 0.479 -4.754 15.233 Jupiter Island Setup Explicit) Westhampton (Setup Exp lit) 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 -1 2.188 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 8.7 9.669 Average -0.983 -7.737 6.754 ENVELOPE OF AY VALUES JUPITER ISLAND (WIS DATA SETUP IMPLICIT) 10 ___ ___ __4 0 2~ -4 k -6 ___ ___ __-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 ___ ___ __-2 0 1C -6 0 -8 -10 -12 -1A_________ __ _ 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 _____________0 -2 .4 p -6 NAWO -8 -10 __-12 -14 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. ENVELOPE OF AY VALUES JUPITER ISLAND (BUOY DATA SETUP IMPLICIT) 0 -2 0~-4 S-6 -12-___ ___ _ 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) -4 -8 0-10 -12 -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. 77 ENVELOPE OF AY VALUES WESTHAMPTON (BUOY DATA SETUP IMPLICIT) -1 -2 ~--3 o-4 0 -5 -7 1 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) -1A -2 -3 0~-5 0 -6 2-7 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 (SetupImpi cJ 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 1 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 Expicit) 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 -1 0.555 8.522 1994 1.001 -10.548 11.549 1997 -2.409 -7.687 5.278 1995 0.048 -1 6.913 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 5.145.17. 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 60 ..........- Jupiter Island Buoy Data Setup Included Impicitl. . j ................I....................................................................... 40 20 0 0 CD -20 -40 -60 -80 on-"""" lallweerI StrM harch 19 Storm -100 100 urrl no Dea i Gabriel a, and H1 9go -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) L I I I I I I I I *7' urricano Gordon Hall ween Stc rm ___ _. . IFI- Nochl Storm Janu gag 199torm March 1T96 Sto______ 91 92 93 94 95 95 Year 88 89 90 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) 07 5 .......................... ..... .. ........... .... ...... ....... -. ....... .. ....................... .. ..... ..... .......... ........ .................... .. ..................................... 50 25 20 -25 .50 '$_ _*" . -7 5 'r h U- Hurricane Hurd o Floyd ' -100 ,' a. -125 . -150 -Superm -175 ' Decomber'1992 orm -200.0.................... ...........................I....................................................-........................................................................................................................... 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) .. I ... ralCO 41 . Hurric in Bertha $ Hurrica LlHurric a Floyd * Super storm tJ I ~Decorrbar 1992 1t-r 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) s9 -1 1A 1987 1988 1989 1990 1991 Year 1992 1993 1994 1995 1996 Figure 5.22 Potential shoreline change at Jupiter Island calculated using WIS data, assuming the setup is implicitly included. -25 U -50 -75 U) 100 Q. -125 -150 -175 -200 91 80 60 40 20 0 0 -2 -20 3-4 0 -60 -80 -100 _1986 84 Potential Shoreline Change Jupiter Island WIS Data (Setup Calculated Explicitly) 80 60 40 20 0 N -20 0 40 0 -60 -St perstorm -80 rricane ndrew 100 anuary1 Storm urricane Dean, G braille, & ugo - 100 Sh 1987 S 1989 torm Janua 1987 Sto m Dec mber 19 Storm Hurric es Flore ce & Gor on -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 100 75 50 Potential Shoreline Change Westhampton WIS Data (Setup included Implicitly) I I, 1~ 25 0 -25 -50 -75 -100 -125 -150 -175 -200 1986 I. 1.3 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 Is a3 * Potential Shoreline Change Westhampton WIS Data (Setup Calculated Explicitly) 100 . 75 50 __-25 'g 0 0 L -25 ..444 -1 75 .-12 -100 -- -o_ Decemb r 1990t rm-*~~Sprtr -125 ___Oco r 1988 St~r N,.ove o 93tr -1 emer19 Storm 4; -175 December 1996 Storm Storm 120 -2 0 ................. ... .. .. ........... .. ...... ........ 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. 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 1127 166 150 110.75 1108.25 73 74 189 163 123 112 118 106 128 107 93 83 122 121 227 208. 97 87 173 157_ 82 181 156 157 97 91 144.38 130.94 100 82 126 127 85 126.55 120 105 83 114.80 j98j 2 8... |