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record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd leader nam Ka controlfield tag 001 001681063 003 fts 005 20060215071203.0 006 med 007 cr mnuuuuuu 008 051222s2005 flu sbm s000 0 eng d datafield ind1 8 ind2 024 subfield code a E14SFE0001055 035 (OCoLC)62734905 SFE0001055 040 FHM c FHM 049 FHMM 090 GC11.2 (Online) 1 100 Fauver, Laura A. 0 245 Toward predicting barrier island vulnerability h [electronic resource] : b simple models for dune erosion / by Laura A. Fauver. 260 [Tampa, Fla.] : University of South Florida, 2005. 502 Thesis (M.S.)University of South Florida, 2005. 504 Includes bibliographical references. 516 Text (Electronic thesis) in PDF format. 538 System requirements: World Wide Web browser and PDF reader. Mode of access: World Wide Web. 500 Title from PDF of title page. Document formatted into pages; contains 173 pages. 520 ABSTRACT: The objective of this study is to quantify the accuracy of two engineering models for dune erosion (SBEACH and EDUNE), and to determine which of the two models is best suited for predicting barrier island vulnerability due to extreme storm events. The first model, SBEACH, computes sediment transport using empirically derived equations from two large wave tank experiments. The second model, EDUNE, theoretically relates excess wave energy dissipation in the surf zone to sediment transport. The first mechanism for model comparison is sensitivity testing, which describes the response of the model to empirical, physical, and hydrodynamic variables. Through sensitivity tests, it is possible to determine if responses to physical variables (e.g. grain size) and hydrodynamic variables (e.g. wave height) are consistent with theoretical expectations, and whether the function of each variable is properly specified within the governing equations.With respect to empirical parameters, model calibrations are performed on multiple study sites in order to determine whether or not the empirical parameters are properly constrained. Finally, error statistics are generated on four study sites in order to compare model accuracy. Crossshore profiles of dune elevation are extracted from coastal lidar (light detecting and ranging) surveys flown before and after the impact of major storm events. Three study sites are taken from 1998 lidar surveys of Assateague Island, MD in response to two large northeasters that produced significant erosion along the Assateague shoreline. Two additional study sites are obtained from 2003 lidar surveys of Hatteras Island, NC in response to erosion caused by Hurricane Isabel. Error statistics generated on these study sites suggest that the models are statistically equivalent in their ability to hindcast dune erosion due to extreme storm events. 590 Adviser: Peter A. Howd. 653 Sbeach. Edune. Lidar. Assateague island. Northeaster. Hatteras island. Hurricane Isabel. Breaching. 690 Dissertations, Academic z USF x Marine Science Masters. 773 t USF Electronic Theses and Dissertations. 4 856 u http://digital.lib.usf.edu/?e14.1055 PAGE 1 Toward Predicting Barrier Island Vulnerability: Simple Models for Dune Erosion by Laura A. Fauver A thesis submitted in partial fulfillment Of the requirements for the degree of Master of Science College of Marine Science University of South Florida Major Professor: Peter A. Howd, Ph.D. Asbury H. Sallenger, Ph.D. Gary T. Mitchum, Ph.D. Date of Approval: March 29, 2005 Keywords: SBEACH, EDUNE, lidar, Assateague Island, northeaster, Hatteras Island, Hurricane Isabel, breaching Copyright 2005, Laura A. Fauver PAGE 2 Acknowledgements I would like to express my deepest thanks to my thesis advisor Peter Howd, for his time and dedication to this project and to my pers onal development as a member of the scientific community. Peter has served as an outstanding mentor, and has been supportive of my endeavors not only in the research and academic realms, but also of my desire to pursue a leadership role within the college. My continuous involvement in school projects undoubtedly distracted me from thesis work from time to time, and I ca nnot thank him enough for his unending patience. I would also like to thank Abby Salleng er for serving as a member of my committee, supporting my research by providing data and trav el funds, and asking to ugh questions about the societal implications of dune erosion modeling. Additionall y, Gary Mitchum has provided valuable insight as a committee member, lending his expertise in both physical processes and statistical evaluation of the models. This research could not have been complete d without the support of many people at the USGS, including Hilary Stockdon, Meg Palmsten, Dave Thompson, Karen Morgan, and Kristy Guy. A special thanks is due to my wonderful lab mate Justin Brodersen, whose comic relief and distractions made coming to work enjoyable. Funding for this research came from numerous sources, including a University Graduate Fellowship, the VonRosenstiel Fellowship pr ovided generously by Anne and Werner VonRosenstiel, and the Gulf Charitable Trust O ceanographic Fellowship kindly endowed by Mr. Claude Greene. I have also been privileged to receive funding from the National Defense Science and Engineering Graduate Fellowship, and travel funds from the USGS National Assessment of Coastal Change Hazards program. PAGE 3 i Table of Contents List of Tables iv List of Figures v Abstract xi Chapter One: Introduction 1 Study Objective 1 Background 4 SBEACH Model Description 8 Wave Height Module 10 Sediment Transport Module 13 Profile Change Module 17 EDUNE Model Description 18 Sediment Transport Module 20 Profile Change Module 25 Assateague Island Study Site 26 Hatteras Island Study Site 33 Chapter Two: Methods 41 Description of Lidar Data Sets 41 SBEACH Program Interface 44 EDUNE Program Interface 45 Statistical Analysis 48 Chapter Three: SBEACH Results 53 Introduction 53 Model Sensitivity and Calibration: Empirical Parameters 53 PAGE 4 ii K, the Transport Rate Coefficient 54 the Slope Coefficient 59 the Maximum Local Slope Before Avalanching 61 the Landward Surf Zone Depth 62 1, the Transport Rate Decay Coefficient 65 Model Sensitivity: Physical and Hydrodynamic Variables 66 D50, Median Grain Diameter 66 Water Temperature 70 Wave Height 71 Water Elevation 76 Wave Period 79 Wave Angle 80 Model Results: Assateague Island 82 Calibration Site: Assateague North 82 Berm Profiles: Assateague South 85 Extreme Longshore Variability: Chincoteague 88 Model Results: Hatteras Island 93 Verification Site: Hatteras North 93 Extreme Impacts: Hatteras Breach 96 Chapter Four: EDUNE Results 99 Introduction 99 Simplifying Assumptions 100 Runup Equations 100 Tides 103 Breaking Wave Criterion 104 Model Sensitivity and Calibration: Empirical Parameters 108 K, the Transport Rate Coefficient 109 etanb, the Equilibrium Beach Slope 112 etand, the Equilibrium Dune Slope 114 tanrep, the Active Profile Critical Angle 115 tanoff, the Offshore Critical Angle 116 Model Sensitivity: Physical and Hydrodynamic Variables 117 D50, Median Grain Diameter 117 Storm Surge Elevation 120 Breaking Wave Height 123 Significant Offshore Wave Height 124 Wave Period 125 Model Results: Assateague Island 125 Calibration Site: Assateague North 125 Berm Profiles: Assateague South 128 PAGE 5 iii Extreme Longshore Variability: Chincoteague 131 Model Results: Hatteras Island 134 Verification Site: Hatteras North 134 Extreme Impacts: Hatteras Breach 138 Chapter Five: Discussion 140 Introduction 140 Model Comparison 140 Longshore Variability and Barrier Island Breaching 146 Recommendations 150 Chapter Six: Conclusions 151 References 156 PAGE 6 iv List of Tables Table 2.1 Mean and standard deviations of both gross volume error (GVE) and rootmeansquare (RMS) e rrors between original lidar profiles and modified EDUNE profiles for each of the five study sites. 47 Table 5.1 Compiled error statistics for SBEACH model runs for the four nonbreaching study areas. 142 Table 5.2 Compiled error statistics for EDUNE model runs for the four nonbreaching study areas. 142 PAGE 7 v List of Figures Figure 1.1 Schematic diagram of the four sediment transport regions determined by crossshore wave height profile. 15 Figure 1.2 Illustration of the potential erosion prism method used to calculate sediment transport rat es between the landward edge of the surf zone and the edge of runup. 24 Figure 1.3 Maps showing the location of the three Assateague Island study areas, the location of NDBC Buoy 44009 at Delaware Bay, and NOAA Tide Gauge 8570283 at Ocean City Inlet, MD. 27 Figure 1.4 Storm characteristics of the two northeasters that affected Assateague Island during January and February 1998. 28 Figure 1.5 Dune crest elevations are plotted as a function of latitude along Assateague Island. 30 Figure 1.6 Observed values of gross volume change are plotted as a function of latitude for the Assateague North study site, and three example profiles display the varying impacts of the storm on dune erosion. 31 Figure 1.7 Observed values of gross volume change are plotted as a function of latitude for the Assateague South study site, and three example profiles display the varying impacts of the storm on dune erosion. 32 Figure 1.8 Observed values of gross volume change are plotted as a function of latitude for the Chincoteague study site, and three example profiles display the varying impacts of the storm on dune erosion. 33 Figure 1.9 Maps showing the location of the two Hatteras Island study areas, the location of NDBC Buoy 41025 at Diamond Shoals, and NOAA Tide Gauge 8654400 at the Hatteras Island Fishing Pier. 34 Figure 1.10 A true color image of Hurricane Isabel taken by NASAs Terra satellite at 11:50 EST on September 18th, 2003. 35 PAGE 8 vi Figure 1.11 Storm characteristics for Hurricane Isabel that affected Hatteras Island during September 2003. 37 Figure 1.12 Dune crest elevations as a function of longshore distance along Hatteras Island. 38 Figure 1.13 Observed values of gross volume change are plotted as a function of latitude for the Hatteras North study site, and three example profiles display the varying impacts of the storm on dune erosion. 39 Figure 1.14 Three profiles displaying dune change at the Hatteras Breach study site. 40 Figure 2.1 Illustration of the lidar (light detection and ranging) system developed for coastal surveying applications. 42 Figure 2.2 Original lidar profile and modified profile that conforms to EDUNE program requirements for (a) a profile with a significantly elevated berm, and (b) a profile without a significantly elevated berm. 47 Figure 2.3 Horizontal boundaries on the subaerial profile are chosen at the landward dune base and mean high water. 49 Figure 2.4 An example of a gross volume error (GVE) calculation. 50 Figure 2.5 GVE and RMS error are plotted as a function of dx. 51 Figure 2.6 Net volume error (NVE) calculations were not used as a measure of accuracy, because local positive and negative errors often canceled over the length of the profile. 52 Figure 3.1 Profile 4362 from Assateague North is used as an example of SBEACH model response to changing values of K (m4/N). 56 Figure 3.2 SBEACH sensitivity to (a) K, the transport rate coefficient, (b) the slope coefficient, (c) the maximum local slope before avalanching, (d) the landward surf zone depth, and (e) 1, the transport rate decay coefficient. 57 Figure 3.3 SBEACH calibration curves fo r values of K, the transport rate coefficient on the Assateague North study site. 58 Figure 3.4 Profile 4362 from Assateague North is used as an example of SBEACH model response to changing values of (m2/s). 60 Figure 3.5 SBEACH calibration curves for values of the slope coefficient on the Assateague North study site. 61 PAGE 9 vii Figure 3.6 Profile 4362 from Assateague North is used as an example of SBEACH model response to changing values of (m). 64 Figure 3.7 Profile 4362 from Assateague North is used as an example of SBEACH model response to changing values of D50 (mm). 68 Figure 3.8 SBEACH sensitivity to (a) D50, median grain diameter and (b) water temperature. 69 Figure 3.9 SBEACH calibration curves for values of D50, the median grain diameter on the Assateague North study site. 69 Figure 3.10 SBEACH sensitivity to water temperatures ranging from 0 C to 40 C. 71 Figure 3.11 Example time series of wave heights utilized to test SBEACH model sensitivity. 72 Figure 3.12 SBEACH sensitivity to (a) wave height (H), (b) water elevation ( ), (c) wave period (T), and (d) wave angle ( ). 73 Figure 3.13 SBEACH calibration curves for modified time series of wave height on the Assateague North study site. 74 Figure 3.14 Profile 4362 from Assateague North is used as an example of SBEACH model response to modified time series of wave height. 75 Figure 3.15 Example time series of water elevations utilized to test SBEACH model sensitivity. 77 Figure 3.16 Profile 4362 from Assateague North is used as an example of SBEACH model response to modified time series of water elevation. 78 Figure 3.17 SBEACH calibration curves for modified time series of water elevation on the Assateague North study site. 79 Figure 3.18 Relative frequency histograms of the original wave angle time series ( ) and two of the modified time series used to test SBEACH model sensitivity. 82 Figure 3.19 Ten profiles selected from Assateague North and the predicted SBEACH results. 84 Figure 3.20 RMS error and GVE as a function of latitude for SBEACH model results on the Assateague North study site. 85 Figure 3.21 Ten profiles selected from Assateague South and the predicted PAGE 10 viii SBEACH results. 87 Figure 3.22 RMS error and GVE as a function of latitude for SBEACH model results on the Assateague South study site. 88 Figure 3.23 SBEACH calibration curves for increasing values of K, the transport rate coefficient on the Assateague South study site. 88 Figure 3.24 Ten profiles selected from Chincoteague and the predicted SBEACH results. 90 Figure 3.25 RMS error and GVE as a function of latitude for SBEACH model results on the Chincoteague study site. 92 Figure 3.26 SBEACH calibration curves for increasing values of K, the transport rate coefficient on the Chincoteague study site. 92 Figure 3.27 Ten profiles selected from Hatteras North and the predicted SBEACH results. 94 Figure 3.28 RMS error and GVE as a function of longshore distance for SBEACH model results on the Hatteras North study site. 95 Figure 3.29 SBEACH calibration curves for increasing values of K, the transport rate coefficient on the Hatteras North study site. 96 Figure 3.30 Ten profiles selected from Hatteras Breach and the predicted SBEACH results. 98 Figure 4.1 Three representations of time varying runup elevations specific to Profile 4362 from Assateague North, and corresponding EDUNE model results. 102 Figure 4.2 Two representations of time varying water levels, and corresponding EDUNE model results for Profile 4362 from Assateague North. 104 Figure 4.3 Vertical RMS error is plotted as a function of latitude on the Assateague North study site when b=0.78. 106 Figure 4.4 Sensitivity of the EDUNE model to K, the transport rate coefficient when b=0.78. 107 Figure 4.5 Vertical RMS error is plotted as a function of latitude on the Assateague North study site when b=0.32. 107 Figure 4.6 Sensitivity of the EDUNE model to K, the transport rate coefficient when b=0.32. 108 PAGE 11 ix Figure 4.7 Profile 4362 from Assateague North is used as an example of model response to changing values of K (m4/N). 110 Figure 4.8 EDUNE sensitivity to (a) K, the transport rate coefficient, (b) etanb, the equilibrium beach slope, (c) etand, the equilibrium dune slope, (d) tanrep, the active profile critical angle, and (e) tanoff, the offshore critical angle. 111 Figure 4.9 EDUNE calibration curves for values of K, the transport rate coefficient on the Assateague North study site. 112 Figure 4.10 Profile 4362 from Assateague North is used as an example of model response to changing values of etanb (). 113 Figure 4.11 EDUNE calibration curves for values of etanb, the equilibrium beach slope on the Assateague North study site. 114 Figure 4.12 EDUNE sensitivity to (a) D50, the median grain diameter, (b) S, the storm surge elevation, (c) Hb,rms, the rms breaking wave height, (d) H, the significant offshore wave height, and (e) Tp, the peak wave period. 119 Figure 4.13 EDUNE calibration curves for values of D50, the median grain diameter on the Assateague North study site. 120 Figure 4.14 Profile 4362 from Assateague North is used as an example of model response to changing representations of storm surge elevation. 122 Figure 4.15 EDUNE calibration curves fo r varying representations of the storm surge elevation time series on the Assateague North study site. 123 Figure 4.16 EDUNE calibration curves for varying representations of the rms breaking wave height time series on the Assateague North study site. 124 Figure 4.17 Ten profiles selected from Assateague North and the predicted EDUNE results. 126 Figure 4.18 RMS error and GVE as a function of latitude for EDUNE model results on the Assateague North study site. 128 Figure 4.19 Ten profiles selected from Assateague South and the predicted EDUNE results. 129 Figure 4.20 RMS error and GVE as a function of latitude for EDUNE model results on the Assateague South study site. 130 PAGE 12 x Figure 4.21 EDUNE calibration curves for increasing values of K, the transport rate coefficient, and et anb, the equilibrium beach slope on the Assateague South study site. 131 Figure 4.22 Ten profiles selected from Chincoteague and the predicted EDUNE results. 132 Figure 4.23 RMS error and GVE as a function of latitude for EDUNE model results on the Chincoteague study site. 133 Figure 4.24 EDUNE calibration curves for increasing values of K, the transport rate coefficient, and et anb, the equilibrium beach slope on the Chincoteague study site. 134 Figure 4.25 Ten profiles selected from Hatteras North and the predicted EDUNE results. 136 Figure 4.26 RMS error and GVE as a function of latitude for EDUNE model results on the Hatteras North study site. 137 Figure 4.27 EDUNE calibration curves for increasing values of K, the transport rate coefficient, and et anb, the equilibrium beach slope on the Hatteras North study site. 137 Figure 4.28 Ten profiles selected from Hatteras Breach and the predicted EDUNE results. 139 Figure 5.1 Mean RMS error and mean GVE are plotted as a function of study site for both the SBEA CH and EDUNE models. 142 Figure 5.2 Mean RMS error and mean GVE are plotted for SBEACH results as a function of Kvalue for four study sites. 144 Figure 5.3 Mean RMS error and mean GVE are plotted for EDUNE results as a function of Kvalue for four study sites. 145 PAGE 13 xi Toward Predicting Barrier Island Vulnerability: Simple Models for Dune Erosion Laura A. Fauver Abstract The objective of this study is to quantify the accuracy of two engineering models for dune erosion (SBEACH and EDUNE), and to determ ine which of the two models is best suited for predicting barrier island vulnerability due to extreme storm events. The first model, SBEACH, computes sediment transport using empi rically derived equations from two large wave tank experiments. The second model, EDUNE, theoretically relates excess wave energy dissipation in the surf zone to sediment transport. The first mechanism for model comparison is sensitivity testing, which describes the response of the model to empirical, physical, and hydrodynamic variables. Through sensitivity tests, it is possible to determine if responses to physical variables (e.g. grain size) and hydrodyna mic variables (e.g. wave height) are consistent with theoretical expectations, and whether the function of each variable is properly specified within the governing equations. With respect to empirical parameters, model calibrations are performed on multiple study sites in order to dete rmine whether or not the empirical parameters are properly constrained. Finally, error statisti cs are generated on four study sites in order to compare model accuracy. Crossshore profiles of dune elevation are extr acted from coastal lidar (light detecting and ranging) surveys flown before and after the impact of major storm events. Three study sites are taken from 1998 lidar surveys of Assateague Isla nd, MD in response to two large northeasters that produced significant erosion along the Assatea gue shoreline. Two additional study sites are PAGE 14 xii obtained from 2003 lidar surveys of Hatteras Island, NC in response to erosion caused by Hurricane Isabel. Error statistics generated on these study sites suggest that the models are statistically equivalent in thei r ability to hindcast dune erosion due to extreme storm events. However, observed inconsistencies in the performance of the EDUNE model undermines confidence in the model to correctly predict dune erosion. These inconsistencies include illconstrained empirical parameters, as well as a flaw ed description of onshore transport that leads to anomalous accretionary features on the beach and dune. Therefore, SBEACH is viewed as the preferred model for future studies employing macr oscale approaches to dune erosion modeling. PAGE 15 1 Chapter One Introduction Study Objective Hazards posed by coastal change arise fr om many different environmental causes, including longterm coastal change due to sea le vel rise, and shortterm events such as cliff failure and erosion caused by extreme weather c onditions. This study focuses on coastal change that arises as a consequence of extreme storm even ts that often reshape entire beach environments and cause immense property damage. Tall sand dunes that line many beaches absorb wave energy produced by small storms, thereby protecting adjacent coastal properties. In fact, sand dunes are so effective at protecting coastal prope rty that many state governments in the United States have paid millions of dollars to create, re inforce, and heighten coastal dunes. However, even tall and massive dunes cannot withstand the most severe storms, and when dunes fail, water and sand inundate entire coastal areas, destroying and damaging property. The fallible nature of the dune system has le d researchers to create predictive models to quantitatively describe dune response to extrem e storm events. Two of these models are EDUNE, a first principles model that quantifi es beach change based on the theory of an equilibrium beach profile, and SBEACH, an em pirical model derived from observations of sediment transport in large wave tank studies. In order to predict coastal change in areas of special concern, both models have been used by government agencies including the Federal Emergency Management Agency and the U.S. Army Corps of Engineers. However, both models incorporate multiple empirical parameters to account for unknown physical processes, some of which are capable of drastically changing model results. Because calibration values tend to vary PAGE 16 2 widely between coastal areas, the models will remain questionable as predictive tools until all parameters are properly constrained. The limite d availability of stormrelated beach surveys has hampered this process, with reports of calibrati on values still ranging over an order of magnitude on different beaches (Larson and Kraus, 1989; Larson, Kraus, and Byrnes, 1990). The advent of lidar (light detecting and ra nging) mapping, an airplane based system for producing highresolution elevation maps, now ma kes it possible to better constrain empirical parameters and to verify these models with an ex tensive number of beach profiles. In order to accomplish these goals, the models will first undergo sensitivity testing and calibration with a set of profiles from Assateague Island, MD, which we re eroded as a result of two northeasters that traversed the Atlantic Ocean in early 1998. Se nsitivity tests will first identify parameters that affect the accuracy of model results, and subseque nt calibrations will constr ain the parameters to values that minimize model RMS error. The calibra ted models will then be verified with profiles from Hatteras Island, NC that experienced ex tensive erosion as a result of the passage of Hurricane Isabel in 2003. The verification pro cess questions whether model error is minimized with the calibration values generated with the Assat eague Island profiles, d espite changes in dune morphology and storm characteristics. If the ca librated values are stationary between the two study areas, then the model is considered we ll constrained with respect to its empirical parameters. Three additional study areas are selected from Assateague and Hatter as Islands for further testing, each exhibiting unique characteristics in dune morphology and storminduced erosion. The first additional study area consists of a set of berm profiles on Assateague Island that lack the presence of a dune. As a result of the storms, these profiles experienced severe erosion on the foreshore, and the development of overwash deposits landward of the original berm location. The second additional site consists of tall dunes on Assateague Island that experienced great variability in dune response, from complete destruct ion of the dune, to mild beach erosion. The PAGE 17 3 variation in dune response does not exhibit a c onsistent longshore tre nd, and may be due to gradients in longshore forcing and sediment transport rates. Finally, the third additional site is a section of Hatteras Island coastline that was brea ched by storm surge and waves during Hurricane Isabel. This inlet is the first stormrelated breaching event recorded by lidar, and creates an excellent case study for the models ability to pr edict extreme impacts. Because neither model has been tested with similar data sets in th e past, this analysis will expand the potential applications of both models. Model performance will be evaluated on two levels, including the qualitative ability to correctly predict impact regime, as well as quantitative accuracy in reproducing the measured poststorm profile. First, impact regimes are qualita tive descriptions of dune erosion that can be incorporated into maps that predict the vulnerab ility of the coastline to storm events. The four impact regimes utilized in this study were developed by Sallenger (2000), and include swash, collision, overwash, and inundation. Vulnerabilit y maps have tremendous potential to identify weak sections of the dune system, and therefore, both models will be analyzed for their ability to correctly predict impact regime for vulnera bility mapping applications. Secondly, two quantitative measures of error will be used to eval uate the accuracy of the models in predicting measured poststorm dune shape. Quantitative me asures of accuracy will allow an assessment of model performance between study sites, and w ill facilitate a comparison of the two models. This research is conducted as a part of the USGS National Assessment of Coastal Change Hazards program, a project of the USGS Coastal and Marine Geology program that seeks to create a better understanding of coastal processes of so cietal concern. The societal relevance of this research is threefold in that (1) because it seeks to predict the vulnerability of different coastal areas, it will serve as a planning tool fo r counties and states when creating zoning laws and construction setback limits, (2) this researc h will bring about a better understanding of the fundamental nature of nearshore hydrodynami cs and sediment transport during storm events, PAGE 18 4 knowledge that can be applied to all coastlines a ffected by extreme weather events, and (3) the correct application of numerical models should provide improved real time predictions of beach change, comparable to the National Weather Serv ices hurricane tracking system. Quantification of the vulnerability of U.S. coastlines to storm even ts has multiple scientific and societal impacts, all of which stem from the pressing need for comprehensive knowledge of the response of coastal geomorphology to physical forcing created by storms. Background Beginning in the 1940s, coastal scientis ts performing beach surveys observed a remarkably consistent seasonality in profile shap e. During summer months when wave heights were small, the beach maintained a wide, flat be rm, and in general the nearshore zone was void of sandbars. In winter months when wave height s increased, the berm narrowed and sandbars were formed. Shepard (1950) called these differing shapes summer profiles and winter profiles after documenting five years of this behavior at Scripps Pier in La Jolla, CA. Bascom (1953) also observed this cyclic behavior with two years of observations at Carmel, CA. Both assumed that the volume of sand was conserved between the berm and the sandbars over these seasonal cycles, although neither measured far enough offshore to confirm this hypothesis. Literature now commonly identifies the two profile shapes as berm profiles and bar profiles (Komar, 1998). While these differences have generally been ob served on seasonal cycles, they also apply to changes that result from large storm events. At first, attempts to delineate between th e two cases were based on critical values of offshore wave steepness, H/L, where H is the significant offshore wave height, and L is the associated wave period. Johnson (1949) reported th at based on his data, values of wave steepness less than 0.025 always resulted in berm profiles, and steepness values greater than 0.03 always generated bar profiles. The critical wave steepne ss varied greatly between studies, and was given PAGE 19 5 by Rector (1954) to be 0.016. Rector remarked that differences in critical H/L values could possibly be attributed to differences in grain size between beaches. This observation initiated a large number of studies attempting to incorporate grain size into the critical threshold for the occurrence of bar and berm profiles. Most notably, Dean (1973) developed a criti cal relationship between wave steepness and the dimensionless fall velocity, ws/gT, where ws is the fall speed of a specific grain size, g is acceleration due to gravity, and T is the wave peri od. Dean theorized that sediments are primarily suspended during the crest phase of a wave, wh en onshore horizontal velocities dominate. Therefore, if the time required for a grain to fall out of suspension is small compared to the wave period, the grain will move onshore. However, if the settling time is much greater than the wave period, the sediment is transported offshore into the bar. The equation developed by Dean as a critical boundary between bar and berm profil es based on small scale laboratory data is: gT w L Hs 7 1 (1.1) Several subsequent studies have attempted to revise this model, including Larson and Kraus (1989) who developed the relationship: 300070 0 T w H L Hs (1.2) based on observations from large wave tank studies. This equation is used by Larson and Kraus to predict the direction of sediment transport in the SBEACH model, the implications of which will be discussed in Chapter 3. A more advanced approach to predicting pr ofile evolution is the development of time dependent models that calculate profile change as a function of hydrodynamic forcing. Multiple process based models attempt to quantify profile change by developing governing equations for wavesediment, currentsediment, and sedimen tsediment interactions. These models often PAGE 20 6 incorporate internal modules for computing waves, currents, and sediment transport. Hedegaard, Deigaard, and Freds e (1991) present a deterministic mode l for predicting the morphology of nearshore profiles. The model focuses on the de velopment of the wavecurrent boundary layer and the resultant transport of sediments. The hydrodynamic portion of the model accounts for wave heights across the profile, nonlinear wave interactions in the turbulent wave boundary layer, wave drift, and mean flows. The sedim ent transport module is based on the channel flow approach of Engelund and Freds e (1976), which separates transport into bedload and suspended modes. Nairn and Southgate (1993) and Southgate and Nairn (1993) also developed a processbased model for predicting sediment transport acr oss the nearshore zone. This model also includes internal hydrodynamic modeling, including longshore processes such as the Sxy radiation stress component. The specification of longshore fo rcing allows the model to predict longshore gradients in transport leading to profile cha nge. The Nairn and Southgate sediment transport model is based on the classic energetics approach of Bagnold (1963), which states that a portion of wave and current produced energy is expende d in transporting suspended sediment loads. Several other deterministic profile change m odels have been developed, each employing different variations of several existing processba sed sediment transport models, including those mentioned above, the Grant and Madsen (1979) m odel, and others. These process based models focus on the transport of sand in the nears hore zone, which is exceedingly important for predicting erosion on the beach and dune. Howeve r, no models to date have attempted to incorporate these deterministic models with descr iptions of the swash and avalanching processes that define the basic mechanisms for subaerial profile erosion. Several notable approaches to predictin g dune erosion have focused on empirical considerations of wave runup created by st orms. Fisher, Overton, and Chisholm (1986) constructed fullscale sand dunes in the swash zone at the FRF Pier, and videotaped the gradual PAGE 21 7 erosion process. Vertical stakes and capacitan ce wave gauges were used to characterize the specific force of each swash bore, and the videos were used to calculate the eroded volume of sand created by individual bores. The specific fo rce was found to correlate linearly to the volume of sediment eroded from the dune. Overton, Fish er, and Fenaish (1987) then developed a twodimensional momentum equation to predict the specific force created by each bore, and Overton et al. (1993) investigated the roles of grain size and dune density on the erosion of the dune by the individual bores. Sallenger (2000) proposed a conceptual model that classifies interactions between runup and dune morphology into one of four impact regimes including swash, collision, overwash, and inundation. Lidar data were used to identify th e elevations of the dune crest and dune base, denoted as dhi and dlo respectively. Holman s (1986) equations for calculating runup based on offshore wave characteristics were used to cal culate the high and low elevations of runup, named rhi and rlo. Comparisons of the runup and dune elevations place each dune into one of the four regimes. For example, if the high el evation of runup exceeds the dune crest, overwash is expected. This model does not have the capability to predict the volume of sediment eroded or the shape of the poststorm dune. Additionall y, because the model is not time dependent, it neglects storminduced changes in dune morphology that may affect the prediction of impact regime. Edelman (1968, 1972) developed a simple ge ometric dune erosion model specific to the Dutch coastline, where a series of dikes and ta ll dunes protect the country from inundation by the North Sea. This model predicts the volume of eroded sediment and resulting poststorm profile shape based on the highest elevation of storm surge occurring during the storm. The profile shape is predicted by a single equation that re lates the horizontal position of each contour to the maximum storm surge elevation. This model produces profiles in favorable agreement with PAGE 22 8 observed field data in the Netherlands, but do es not perform satisfactorily on other coastlines because of its specific calibration to Dutch dunes. More recent models have attempted to in corporate time dependence and additional hydrodynamic variables such as wave height and wave period. One such model under investigation in this study is SBEACH (Lar son and Kraus, 1989). This model employs an internal wave model developed by Dally, D ean, and Dalrymple (1984, 1985), and empirical relationships that define sediment transport ra tes based on observations from two large wave tank studies. The large wave tank data revealed the ex istence of four crossshore regions with unique sediment transport rate structures. This model makes use of excess energy dissipation as the main driving force behind sediment transport, a mech anism that is also derived from wave tank observations. The SBEACH model is described in more detail in the following section, and model results are presented in Chapter 3. The second model explored in this study is EDUNE (Kriebel and Dean, 1985). This model is based on equilibrium beach profile theo ry developed by Dean (1976, 1977). Deans theory states that individual beach profiles should come into static equilibrium with constant waves and water levels. The profile maintain s its equilibrium shape because waves generate uniform energy dissipation per unit volume across th e profile. However, when water levels rise, wave shoal further inshore and dissipate excess en ergy over the shallow portions of the profile. Subsequently, the profile evolves toward equilib rium with the new hydrodynamic forcing. The EDUNE model is described later in this chapter, and model results are presented in Chapter 4. SBEACH Model Description In an attempt to quantitatively predict pr ofile change, the SBEACH model was developed by a group of coastal engineers at the U.S. Army Corps of Engineers Waterways Experiment Station. Their primary objective was to develop a tool that could predict beach change in the vicinity of coastal engineering projects. The model takes a macroscale approach to predicting PAGE 23 9 beach change, invoking sediment transport over le ngth scales of meters, and time scales of hours. This approach was preferred to a microscale sed iment transport model, because at the time of development little was known about sedimentcu rrent, sedimentwave, and sedimentsediment interactions on the seafloor. Although these theori es have been advanced in recent years, the computing time required to run these models is s ubstantial, covering spatial scales limited to a few square meters rather than the hundreds of meters that a profile typically encompasses. Rather, the evolution of macroscale features such as berms and sandbars exhibit amazing regularity in time, and therefore the devel opment of a macroscale model was prudent. The numerical scheme makes two main assumptions about the nature of sediment transport in the coastal environment. First, the model assumes that over short time scales and under storm conditions, profile change is dominat ed by crossshore transport, and therefore longshore sediment transport gradients can be excl uded. Secondly, the developers assumed that sediment transport rates observed during two large wave tank experiments could be applied to the field environment. Initial tests of the model w ith field data showed that adjustments were required to correctly scale the model to real wo rld beach environments. After these adjustments were made, favorable agreement was found betw een calculated and measured poststorm profiles for several beaches on the east and west coasts of the United States. A full description of the large wave tank e xperiments, model development, and model tests can be found in a series of technical reports published by the USACE. The first report by Larson and Kraus (1989) describes the large wave ta nk experiments, as well as the data analysis techniques used to derive the fundamental equa tions implemented in the numerical model. A second report by Larson, Kraus, and Byrnes (1990) describes model development and the numerical scheme in greater detail. Wise, Smith and Larson (1996) report verification tests and the development of an optional randomized wave model that can be implemented within the SBEACH program. PAGE 24 10 For each simulation, the user defines the shape of the prestorm profile, which the model interpolates to the predefined grid spacing. The user also supplies time series of offshore significant wave height, period, angle, and water el evation, which are interpolated to the chosen time increment. At each time step, values from th e time series are used to calculate the crossshore wave height profile, by employing the wa ve model of Dally, Dean, and Dalrymple (1984, 1985). The wave heights are then used to calculate sediment transport rates, which are applied to the profile change module. The following paragrap hs describe each step of the process in detail, including the method for calculating wave heights across the profile, the transport rate equations, and computations of profile change. Wave Height Module Beginning at the seawardmost grid cell, wave height is known from the input time series, and energy flux can be calculated with the following equation: gEC F (1.3) where E is the wave energy density, and is defined as: 28 1 gH E, (1.4) and Cg is the wave group speed defined as: nC Cg (1.5) In the group speed equation, C is the wave ph ase speed and is equal to wavelength divided by period, where wavelength is determined from th e full equations for linear wave theory. The quantity n is defined as: ) 2 sinh( 2 1 2 1d d n (1.6) PAGE 25 11 where is the wavenumber, and d is the total depth defined by: h d. (1.7) where is setdown (or setup) caused by excess moment um flux due to waves. In the seaward most calculation cell, setdown is determ ined analytically with the equation: ) 2 sinh( 42d L H (1.8) and is used to calculate total depth in Equati on 1.7. Utilizing Equations 1.4 through 1.8, the energy flux at the seaward boundary of the pr ofile can be calculated with Equation 1.3. Once F is calculated in the seawardmost cell, the model utilizes the conservation of energy flux equation developed by Da lly, Dean, and Dalrymple (1984, 1985): ) ( ) sin ( ) cos (sF F d k F y F x (1.9) where Fs is the stable energy flux, is the wave angle with respect to the bottom contours, and k is the wave decay coefficient. A convenient tw odimensional form of Equation 1.9 that assumes alongshoreuniform wave conditions and straight and parallel bottom contours is written as: ) ( ) cos (sF F d k F dx d (1.10) A finite difference form of Equation 1.10 is u sed to calculate the value of F at the adjacent landward cell. To perform this calculation, the value is calculated from a finite difference form of the twodimensional Snells Law equation that assu mes straight and parallel bottom contours: 0 sin L dx d (1.11) where is given at the seawardmost cell, and L is calculated in both cells with the full linear wave theory equations. Before waves break, k is set at a value of zero, indicating that the only PAGE 26 12 changes in energy flux are due to changes in wave angle as waves shoal over intermediate and shallow depths: 0 ) cos ( F dx d. (1.12) After F is calculated at the adjacent grid cell, wave height is determined with Equations 1.3 through 1.7. Setdown (or setup) is determined from a finite difference form of the radiation stress equation: dx d gd dx dSxx (1.13) where Sxx is the xdirected flux of xdirected momentum given by the equation: 2 1 ) 1 (cos 8 12 2 n gH Sxx. (1.14) This process continues cell by cell, m oving landward towards the shoreline. After wave height is determined at each gr id cell, the model checks for broken waves with the following empirical criterion: 21 0 2 / 1) / ( tan 14 1 L H h Hb b (1.15) where tan is the slope of the local profile. Breaki ng waves occur when the ratio of wave height to water depth meets or exceeds the condition specifi ed by the right hand side of this equation. When the first instance of breaking waves occurs, E quation 1.10 is modified so that k is set at a standard value of 0.15 determined by large and small wave tank experiments. This coefficient value determines the rate at which energy is di ssipated as the waves break. The stable energy flux in Equation 1.10 (Fs) is defined as the energy flux necessar y for stable conditions to occur once the waves break, and is calculated by the equation: g s sC E F (1.16) PAGE 27 13 In this equation, Es is the stable wave energy density determined by: 28 1s sgH E (1.17) where Hs is a function of water dept h and the empirical parameter : d Hs (1.18) The value of is set at 0.40, which was also determin ed from large and smallscale wave tank experiments. Finally, the model attempts to simulate wave reformation by identifying grid cells where F=Fs. When this condition is met, the wave is cons idered to reform, and k is again set at a value of zero. The model then searches for the ne xt break point using a new breaking criterion: 1b bh H. (1.19) This criterion for waves breaking after reformation is used in order to maintain numerical stability in the shallow region of the surf zone. The end r esult of this module is a crossshore profile of wave heights calculated at the given time step. The wave heights are then fed into the sediment transport rate module, which is described in the following section. Sediment Transport Module The large wave tank experiments used to develop the SBEACH model revealed four distinct crossshore regions, each defined by a unique sediment transport rate structure. The four zones are illustrated in Figure 1.1, and are refe rred to as (I) the prebreaking zone, (II) the breaker transition zone, (III) the broken wave zone, and (IV ) the swash zone. The location of each region shifts landward and seaward in time with the br eak point, while Zones II and III often occur multiple times in the presence of wave reforma tion. Wave heights calculated by the methods described in the previous section are used to divide the profile into these four regions. PAGE 28 14 First, the boundary between Zones I and II is defined at the location of the break point. The breaking wave criterion establishing this location is defined in Equation 1.15, and is dependent on the ratio of wave height to water depth. The boundary between Zones II and III is located at the plunge point, a distance 3Hb landward of the break point. The distinct separation of the break point and the plunge poi nt is based on observations that a certain distance is required after wave breaking before turbulent conditions become uniform. Finally, the boundary between Zones III and IV is depende nt on an arbitrary depth assigned by the user, which generally ranges between 0.1 and 0.5 meters. The effects of assigning different values of on model results are explored in Chapter 3. The transport rate equations derived from the large wave tank experiments are as follows: Zone I: ) (1 bx x b se Q Q x xb (1.20) Zone II: ) (2 px x p xe Q Q b px x x (1.21) Zone III: sQ dx dh K D D Keq dx dh K D Deq 0 dx dh K D Deq p zx x x (1.22) Zone IV: r z r z sx x x x Q Q, z rx x x (1.23) In these equations, each subscript refers to a speci fic location on the crossshore profile, where b, p, z, and r refer to the location of the break poi nt, the plunge point, the landward edge of the surf zone, and the runup limit respectively. PAGE 29 15 Figure 1.1 Schematic diagram of the four sediment transport regions determined by crossshore wave height profile. Figure taken from Larson, Kraus, and Byrnes (1990). Equation 1.22 defines the sediment transport rate Qs in the broken wave zone. The DDeq term accounts for excess energy dissipation acro ss the profile, which is the main mechanism driving sediment transport in this zone. This equation was derived from the large wave tank observation that excess wave energy dissipation accounted for 50 to 70% of the variability in transport rate, while the local profile slope account ed for an additional 10% of the variability. The term D in this equation is the wave energy dissipation per unit volume, and is defined by the equation: ) (2sF F d k D (1.24) Equilibrium energy dissipation Deq is the energy density required to maintain the instantaneous profile form, and the equation for computing Deq is: 24 52 2 / 3 2 / 3 g A Deq (1.25) PAGE 30 16 where is the ratio of wave height to water de pth at breaking, and A is the profile shape parameter. The value of A is determined by the equation: x A g K h2 / 3 2 2 / 3 2 / 35 24 (1.26) which is based on the equilibrium profile shape developed by Dean (1977). Additional terms in Equation 1.22 are K, the transport rate coefficient, an empirical coefficient that directly controls the ma gnitude of the sediment transport rate, and an empirical coefficient governing the slope dependent term. Values of K generally range between 2.5x107 and 2.5x106 m4/N, and calibration values from previous studies have varied widely across this approximate range. Values of range between 5.0x104 and 5.0x103 m2/s, although little calibration work has been done with this para meter. Chapter 3 discusses the effects of both empirical parameters on model results, and provides a calibration value for each. Sediment transport rates in Zones I and II decrease exponentially moving away from the plunge point, and the rate decay coefficients are defined by: 47 0 50 1H D 0.4 b, and (1.27) 1 22 0 (1.28) Subsequent versions since the orig inal model release have turned 1 into an empirical coefficient defined by the user, which genera lly ranges between 0.1 and 0.5 m1. Chapter 3 will also discuss the effect of 1 on model accuracy. Once 1 and 2 are defined, sediment transport rates offshore of the breakpoint are computed with Equations 1.20 and 1.21. Finally, sediment transport rates in Zone IV decrease linearly from the landward edge of the surf zone to the limit of runup. The limit of runup is defined as the active subaerial profile height (zr) developed from the large wave tank data: PAGE 31 17 79 0 2 / 1) / ( tan 47 1 L H zr (1.29) Once zr is calculated, sediment transport rates in the swash zone are computed with Equation 1.23. Once the transport rate magnitudes have b een determined for the entire length of the profile, the model employs a final equation to determine the direction of transport: 300070 0 T w H L Hs. (1.30) Both the large wave tank studies and field data show that this equation has good predictive skill in delineating between episodes of onshore and offshore transport. When the left hand side of the equation is larger (smaller) than the right ha nd side, onshore (offshore) transport takes place over the length of the profile. Profile Change Model Once sediment transport rates are calculated acr oss the profile, a finite difference form of the conservation of mass equation is used to evolve the profile to its new form: .x Q t hs (1.31) The model solves for profile change by adjusti ng the vertical elevation at fixed horizontal locations along the profile. The landward boundary for the application of this equation is the horizontal limit of runup, and the offshore bounda ry occurs where the sediment transport rate decreases to zero. The profile change model also employs the c oncept of avalanching, which occurs when the slope between adjacent grid cells becomes overly steep. When the slope of the profile exceeds the angle of initial yield avalanching occurs between neighboring grid cells, until the PAGE 32 18 residual angle 10 is achieved. This mechanism safeguards the model from numerical instabilities that would occur with nearly vertical slopes. The large wave tank data suggests that the angle should be set at 28 with a residual angle of 18 The effects of assigning different values of on model results are discussed in Chapter 3. EDUNE Model Description In an attempt to predict the Nyear frequency distribution of extreme coastal erosion events, coastal engineers at the University of Delawares Center for Applied Coastal Research developed the EDUNE model to quantitatively fo recast the volume of erosion produced by a randomly generated storm. Existing Monte Carlo storm surge models were used to calculate storm surge from randomly selected meteorological parameters, and breaking wave heights were also estimated based on the parameters of the storm. A theoretical erosion model named EDUNE was then developed to incorporate storm surge a nd breaking wave height values. The model was used to calculate eroded volumes on simulated b each profiles, and these volumes were used to calculate the Nyear frequency distribution of erosional events. The development of the EDUNE model is doc umented in multiple papers, including the original report in the Proceedings of the 19th Coastal Engineering Conference which details its development for calculating erosional frequenc y distributions (Kriebel and Dean, 1984). Improvements to the model are documented in a second report by Kriebel and Dean (1985), and two users manuals published as technical memos for the Beaches and Shores Resource Center at Florida State University (Kriebel, 1984(a); Kriebe l, 1984(b)). Final improvements to the model are documented in the Proceedings of the 22nd Coastal Engineering Conference (Kriebel, 1990), and a users manual published by the University of Delawares Center for Applied Coastal Research (Kriebel, 1995). PAGE 33 19 Although EDUNE was developed as a first prin ciples model with the goal of calculating eroded sediment volumes for frequency distributions, it was quickly popularized for a variety of coastal engineering tasks because it was the fi rst quantitative model de veloped to predict sediment transport on the subaerial profile. Most notably, the U.S. Army Corps of Engineers and the Federal Emergency Management Agency used the model to predict the impact of storms on United States beaches. Several years later, EDUNE became outdated when the SBEACH model was developed and adopted by the USACE and FEMA. However, EDUNE remains a good case study of a predictive erosion model, because of its theoretical approach to beach erosion. The guiding principle of the equilibrium beach profile is incorporated into many models, and therefore it is important to understand the EDUNE mode l implicitly before attempting to create and understand more advanced models. The first simplifying assumption made by EDUNE is that sediment transport gradients in the crossshore dominate over short time scales and under storm conditions, and therefore longshore processes can be excluded from the mode l. As a consequence of this assumption, crossshore forcing generates all change in th e beach profile. The second assumption alleges that the energy dissipation per unit volume generated by constant wave heights and water levels produce a profile in static equilibrium with th e forcing. The amount of energy dissipation required to maintain the shape of the profile is called the equilibrium energy dissipation, and is a function of wave height, water level, and shape of the profile. The profile produced by constant waves and water levels is calle d the equilibrium profile. Once a profile comes into equilibrium with hydrodynamic forcing, a rise in water level allows waves to shoal further inshore, increasi ng the energy dissipation per unit volume over the shallow profile. Because the actual energy dissi pation is greater than the equilibrium value for the given profile shape, the profile must evolve in order to attain equilibrium with the new forcing. The evolution of the profile proceeds with the erosion of sediment from the beach and PAGE 34 20 surf zone, which translates the profile upward and landward. As water levels and wave heights fluctuate over time, a continual evolution of the profile takes place as the energy dissipation fluctuates at levels greater than equilibrium values. The profile will continue to evolve until wave heights and water levels are c onstant and the profile once again reaches equilibrium with the forcing. For each simulation, the user defines the prestorm profile shape, which must conform to the requirement that elevations decrease monotonically on either side of the dune crest. Because actual profiles seldom meet this requirement, a r outine was developed to transform lidar profiles into this form, and is documente d in detail in Chapter 2. The user also supplies time series of storm surge elevations, wave period, breaking wave heights, and runup elevations. Because breaking wave height is not recorded by an y traditional observational methods, the SWAN model (Booj et al., 1999) is used to generate break ing wave heights from measured offshore wave parameters. At each time step, values from the breaking wave height time series are used to calculate the location of the surf zone, and sub sequently, sediment transport rates are computed across the width of the active profile. Calculat ed sediment transport rates are then used to compute the evolution of the profile based on profile change equa tions. The following paragraphs describe each step of the process in de tail, including calculations of sediment transport rates and the profile change module. Sediment Transport Module The first step in calculating sediment trans port rates is defining horizontal boundaries on the surf zone. The seaward boundary is define d as the location of wave breaking, and is calculated from empirical evidence that suggests that the wave height is a fixed percentage of the water depth when it breaks (McCowan, 1894; Sverdrup and Munk, 1946): 78 0 b b bh H (1.32) PAGE 35 21 At each time step, the breaking wave height Hb is known from the input time series, and the depth at breaking hb can be calculated. The horizontal location of this depth defines the seaward boundary of the surf zone. The landward boundary is defined at the depth where the concave underwater beach profile is tangent to the equilib rium beach slope. This slope, named etanb, is an empirical parameter that is defined by the u ser for each model run, and the horizontal location of this depth defines the landward boundary of th e surf zone. The effects of assigning different values of etanb on model results are explored in Chapter 4. The EDUNE model does not employ an inte rnal wave module, and therefore, wave heights are not calculated at each grid cell within the surf zone. Rather, the authors developed a method to calculate sediment transport rates at each grid cell without knowledge of the local wave height. As stated in preceding paragr aphs, sediment transport occurs when energy dissipation per unit volume exceeds the equilibrium energy dissipation specific to that profile. Therefore, it is only necessary to calculate equilibrium dissipation (Deq) and actual dissipation (D) in order to compute sediment transport rates. To calculate Deq, the model utilizes Deans equilibrium profile theory, which states that equilibrium energy dissipation per unit volume exis ts for each profile, and is a function of profile shape, wave height, and water level. Equilibri um energy dissipation is calculated from Deans equation that relates Deq to the shape parameter A, wave height, and water depth: 2 2 2 / 3 2 / 324 5 h H g A Deq (1.33) An equation for determining A was derived by Moore (1982), and suggests that the shape parameter is a function of grain size: A = 924 0 237 0 log5010D 262 050 D (1.34) 30 3 264 2 log5010D 263 050D PAGE 36 22 where D50 is the median grain diameter in millimeters. Because H is not calculated at each grid cell, it must be removed from Equation 1.33 through theoretical principle. Assuming that the waves are spilling breakers, the value of is set equal to b from Equation 1.32, indicating that wave height decreases proportionally as a function of water depth as waves break across the surf zone: 78 0 h H (1.35) This assumption allows Equation 1.33 to become independent of wave height: 24 52 2 / 3 2 / 3 g A Deq (1.36) In order to calculate sediment transport rates, it is then necessary to compare the value of Deq to the actual dissipation of energy per unit volume across the width of the surf zone. The actual energy dissipation per unit volume (D) is calculated at each grid cell using the equation: dx dF h D1 (1.37) where F is the energy flux. The quantity F is determined from the linear theory equation for waves propagating in shallow water: gEC F (1.38) where E is the energy density, and Cg is the wave group speed. Energy density is defined as: 28 1gH E (1.39) and the equation for wave group speed in shallow water is: 2 / 1) ( gh Cg (1.40) PAGE 37 23 Combining Equations 1.39 and 1.40, Equation 1.38 becomes: 2 / 1 2 2 / 38 1h H g F. (1.41) Spilling breaker theory from Equation 1.35 is agai n used to eliminate the dependence of F on wave height, and the equation becomes: 2 / 5 2 2 / 38 1h g F (1.42) This value of F is used to compute D at each grid cell with Equation 1.37. Once D and Deq are calculated across the width of the surf zone, sediment transport rates are calculated as a function of the excess wave energy dissipation: ) (eq sD D K Q (1.43) where K is the transport rate coefficient that gove rns the magnitude of the sediment transport rate. The transport rate coefficient is a free parame ter in the model formulation, and was found by Moore (1982) to be 2.2x106 m4/N. Later calibration by Kriebel (1986) with a largescale laboratory experiment performed by Saville (1957) increased the best estimate of K to 8.7x106 m4/N. The effects of assigning different values of K on model results are discussed in Chapter 4. The transport rate calculated at the landward edge of the surf zone is used as a boundary condition for calculating sediment transport rates in the swash zone. Sediment transport rates are computed according to geometric arguments rather than a physically based swash zone transport model. This process is described as the potentia l erosion prism method, and the numerical scheme aims to evolve the profile slope to match the e quilibrium beach slope, etanb. Figure 1.2 provides a schematic diagram of this method, which is descr ibed in detail by Kriebel (1990, 1995). If the equilibrium slope is steeper than the existing slope, the situation is defined as Case I, and the contours at the bottom of the prism have a greater erosi onal potential than those at the top. In Case II situations, the existing slope is steeper than the equilibrium slope, and correspondingly the top of the PAGE 38 24 prism has the greatest erosion potential. When th e two slopes are identical, as shown by Case III, all contours erode uniformly. Figure 1.2 Illustration of the potential erosion prism method used to calculate sediment transport rates between the landward edge of the surf zone and the edge of runup. This figure is taken from Kriebel (1990). One serious limitation found in the models nu merical scheme is the inability to simulate overwash processes when water levels overtop the existing dune crest. The imposed continuity requirement states that sediment must be conser ved between the active dune crest and the most seaward grid cell, and therefore sediment cannot be transported outside of this zone (i.e. landward). A second limitation of the model is th at onshore transport is simulated only when water levels fall and the quantity DDeq becomes negative. This condition for onshore transport is not physically based, but rather a byproduct of th e numerical scheme. No attempts have been made to determine the validity of this representa tion of the recovery process, although it has been PAGE 39 25 suggested that it produces much quicker recovery than is experienced in nature (Larson and Kraus, 1989). Larson and Kraus (1989) also noted that maximum onshore transport rates are simulated when energy dissipation reaches a minimum at zero. Previous studies have shown that a cutoff energy dissipation exists under which no tran sport occurs, suggesting that this process is an inaccurate description of system dynamics. Profile Change Module Once sediment transport rates are calculated for each grid cell between wave breaking and the landward edge of runup, the profile cha nge module employs a finite difference form of the continuity equation: h Q t xs (1.44) The EDUNE model solves this equation for profile change by calculating the horizontal location of fixed vertical contours. The equation is solv ed in the form of a tridiagonal matrix relating three adjacent contours, and a recursion formula th at is employed in a doublesweep procedure to determine the change in the position of each contour over time. The specifics of this process are described in detail by Kriebel and Dean (1985). After Equation 1.44 is utilized to compute the resulting profile, the numerical scheme checks the slope between adjacent grid cells to assu re that the profile has not become overly steep at any location. The maximum allowable slope between any two adjacent contours depends on their position across the profile. The empirical para meters tanoff, tanrep, and etand are defined as the maximum allowable slopes seaward of the break point, along the active profile, and between the edge of runup and the dune crest, respectiv ely. If the slope between adjacent contours exceeds the corresponding critical value, the hor izontal distance between the fixed vertical contours is adjusted to meet the critical value, and eroded sediment is redistributed to neighboring contours. The slope above the runup limit is a sp ecial case, in which sand that is eroded from the PAGE 40 26 dune as a result of oversteepening is redistributed on the beach face until the slope there becomes uniform. Additional sand is distributed in a nonuniform manner across the active profile, decreasing in thickness moving offshore. Each of these three slope values is defined by the user, and the effects of each value on model re sults will be explored in Chapter 4. Assateague Island Study Site Assateague Island is an unpopulated barrier island encompassing approximately 57 kilometers of Maryland and Virginia coastline. The island hosts two national wildlife refuges, Assateague NWR in the north and Chincoteague NWR in the south, and is undeveloped except for a state highway that runs along the coastline. A map detailing the location of Assateague Island and the three local study areas is provided in Figure 1.3. The first study area, Assateague North, is used for sensitivity testing and calib ration with respect to empirical, physical, and hydrological parameters. The second and third s ites, Assateague South and Chincoteague, are used to investigate model accuracy, and to confirm empirical parameters calibrated with Assateague North data. In October 1997, a cooperative program between the USGS, NASA, and NOAA produced detailed elevation maps of the island using lidar mapping techniques. The data gathered by this system are spatially dense, rendering approximately one elevation measurement per square meter, and providing an excellent snap shot of the instantaneous topography of this transient system. These data were gathered as pa rt of the lower 48 mapping project aiming to characterize the barrier island system. Soon after the data were collected, a large storm system moved north along the Atlantic Coast, causing ex tensive beach and dune erosion. This event provided an excellent opportunity for mapping th e poststorm island topography, creating one of the first highly dense sets of pre and poststor m beach profiles. The following paragraphs describe these storms in more detail. PAGE 41 27 Figure 1.3 Maps showing the location of the three Assateague Island study areas, the location of NDBC Buoy 44009 at Delaware Bay, and NOAA Tide Gauge 8570283 at Ocean City Inlet, MD. On January 26, 1998, a low pressure cell origin ating in Texas migrated quickly across the southern United States, turning northeast and cro ssing into the Atlantic Ocean over the mouth of the Chesapeake Bay around 19:00 EST on January 28th. The storm system moved north along the Atlantic Coast, buffeting the shoreline with wi nds exceeding 80 km/hour, and generating offshore significant wave heights exceeding 7 m. The storm system moved north, with wind speeds decreasing by 19:00 on January 29th. Several days later on February 3rd, a second low pressure cell originating in the Gulf of Mexico migrat ed north across the coastal states, entering the Atlantic Ocean over the North CarolinaVi rginia border at 19:00 on February 4th. This storm was stronger than the previous system, producing wind speeds exceeding 95 km/hr, and offshore significant wave heights again in excess of 7 m. This storm lasted approximately 24 hours before fading offshore of the Maine coastline. A more detailed description of the storm system is provided by Ramsey et al. (1998). Figure 1.4 displays storm data local to Assateague Island beginning at 08:00 on January 27th, and ending at 10:00 on February 13th. Water level elevations ( ) were obtained from NOAA PAGE 42 28 Tide Gauge 8570283 at Ocean City Inlet, MD, and ar e referenced to mean sea level. Wave data were gathered from NDBC Buoy 44009 in Dela ware Bay, including significant offshore wave height (H), peak period (Tp), and wave angle ( ). The wave angle measurements in this figure have been transformed from the true north orient ation recorded by the buoy into the convention used by the SBEACH model, where 0 indicates waves approaching the beach shore normal, positive angles indicate waves approaching from the northeast, and negative angles denote waves approaching from the southeast. This transforma tion was made with the first order approximation that the Assateague Island coastline runs directly nor th to south. These data are used to simulate the storm system in both models, for a ll locations along Assateague Island. Figure 1.4 Storm characteristics of the two northeasters that affected Assateague Island during January and February 1998. These time series were used as input data into both models. Time 0 corresponds to 08:00 EST on January 27th, 1998. PAGE 43 29 For both models, the absolute timing of the events is not as important as an accurate representation of the water levels and wave he ights, with particular importance assigned to keeping the relative timing of the waves and wate r levels as accurate as possible. The relative proximity of the tide gauge and buoy to one anot her provides data that are well synchronized in time, when compared to other possible combina tions of offshore buoys and coastal tide gauges that were operational during this time period. Ad ditionally, the large extent of the storm system compensates for the distance between the instrume nts and the study sites. Unlike the compact nature of a hurricane, northeasters tend to affect large areas of coastline in a relatively uniform manner. Figure 1.5 plots dune elevation as a function of latitude for the length of Assateague Island. The top plot shows prestorm dune elev ations gathered in 1997, and the bottom plot displays poststorm dune elevations collected in 1998. As this figure indicates, significant losses in dune crest elevation occurred around a few hots pots, while the majority of dunes maintained their prestorm dune elevations. The locations of the three study sites, including two from Assateague Island NWR in Maryland, and one from Chincoteague NWR in Virginia, are highlighted in yellow. Each site is approxima tely three kilometers long, encompassing 300 crossshore profiles. Because of limitations on computing time, 30 evenly spaced profiles from each area were chosen for the study. PAGE 44 30 Figure 1.5 Dune crest elevations are plotted as a function of latitude along Assateague Island. The top graph displays prestorm elevations, and the bottom graph presents poststorm elevations. The Assateague North study area is situated at the north end of the island, and is composed of high dunes averaging 6.5 m in elevati on at the crest. Over the course of the storm, these dunes experienced erosion on the beach and du ne face. Figure 1.6 plots the observed gross volume change for each of the 30 profiles as a functi on of latitude, indicating that erosion in this area was relatively constant, increasing slightly movi ng to the north. This figure also plots three profiles from this study area in order to demonstr ate several examples of observed profile change. Towards the south, the profiles experienced little change, with small volumes of sediment eroded from the beach. Along the northern end of the st udy area, the dune face was consistently eroded back from its original location, although never beyond the location of the dune crest. PAGE 45 31 Figure 1.6 Observed values of gross volume change are plotted as a function of latitude for the Assateague North study site, and three example profiles display the varying impacts of the storm on dune erosion. In each figure the blue profile repr esents the prestorm profile, and red represents the poststorm profile. The Assateague South study area is composed mainly of berm type profiles averaging 2.8 meters in elevation at the berm crest. Fi gure 1.7 plots observed gross volume change on this section of island as a function of latitude, along w ith three profiles demonstrating the typical response of the beach and berm to the storms. On most profiles, the storm eroded sediment from the top of the berm, and transported a portion of the sand landward to form thin overwash deposits. Profiles of this nature are ubiquitous along the U.S. Atlantic and Gulf Coasts, and therefore, it is essential to understand the beha vior of both models on this study site. PAGE 46 32 Figure 1.7 Observed values of gross volume change are plotted as a function of latitude for the Assateague South study site, and three example profiles display the varying impacts of the storm on dune erosion. In each figure the blue profile repr esents the prestorm profile, and red represents the poststorm profile. Finally, the Chincoteague study area is situ ated on the south end of the island, and is composed of high dunes with a mean elevation of 5.8 meters at the dune crest. From Figure 1.5, it is apparent that this is an area of high longshore variability in dune response, with some dune crest heights changing by more th an three meters, while others experience no change at all. Figure 1.8 displays the ob served gross volume change for each of the 30 profiles as a function of latitude, along with three corresponding profiles. The longshore variability is even more expressed in this figur e, with some dunes experiencing total erosion and overwash, while others expe rienced only mild beach erosion. There is no discernable pattern to the erosion, suggesting that longs hore gradients in local hydrodynamic forcing may have caused these variations in dune response. PAGE 47 33 Figure 1.8 Observed values of gross volume change are plotted as a function of latitude for the Chincoteague study site, and three example profiles display the varying impacts of the storm on dune erosion. In each figure the blue profile represents the prestorm profile, and red represents the poststorm profile. Hatteras Island Study Site Hatteras Island is a narrow barrier island that encompasses more than 80 kilometers of North Carolina coastline. The northern portion ex tends from Oregon Inlet in the north to Cape Hatteras in the south, and includes Pea Isla nd Wildlife Refuge and several small village communities. At Cape Hatteras the island turns to the southwest, ending approximately 20 km away at Hatteras Inlet. This section of the is land encompasses the small to wn of Hatteras Village at its southwestern edge. The entire island is in cluded in the Cape Hatteras National Seashore, and therefore developments are small and growth is highly restricted. The two Hatteras Island study areas both lie on the southwest to northeast trending portion of the island, displayed in Figure 1.9. The first, Hatteras North, is used for model verification after the models are PAGE 48 34 calibrated to Assateague Island profiles. The sec ond site, Hatteras Breach, is the site of island breaching that took place during Hurricane Isabel. Figure 1.9 Maps showing the location of the two Hatteras Island study areas, the location of NDBC Buoy 41025 at Diamond Shoals, and NOAA Tide Gauge 8654400 at the Hatteras Island Fishing Pier. Hurricane Isabel developed as a tropical wave off the western coast of Africa on September 1, 2003. The system moved slowly westward, and was given hurricane status on September 7th. The storm intensified over the following days, and was classified as a Category 5 hurricane on September 11th, with maximum sustained winds in excess of 260 km/hr. On September 15th, the system turned northnorthwest and encountered increased vertical wind shear, which weakened the storm to a Category 3 hurricane. On September 18th, Hurricane Isabel made landfall at Drum Inlet, North Carolina as a Category 2 storm, with maximum sustained surface winds of 128 km/hr recorded near Cape Hatteras. The location of landfall was approximate ly 45 kilometers to the southwest of the study sites on Hatteras Island. As the storm passed over the Outer Banks, significant wave heights at the location of the Diamond Shoals buoy reached 15 m, as predicted by the WaveWatch3 model. Storm surge along the coast was consistently greater than 2 meters for locations lying to the PAGE 49 35 northeast of landfall. A NASA true color satellite image of the storm is presented in Figure 1.10. Additional details and descriptions of Hurricane Isabel are available in the National Hurricane Centers Tropical Cyclone Report (Beven and Cobb, 2004). Figure 1.10 A true color image of Hurricane Isabel take n by NASAs Terra satellite at 11:50 EST on September 18th, 2003. As Isabel slowly advanced on the east coast of the United States, lidar surveys were flown over much of the coastline on September 16th, just as waves from the storm began to approach the shore. This is the first lidar data set flown immediately before the strike of the storm, eliminating questions of profile change between the first flight and storm impact. Lidar surveys were gathered postimpact on September 21st, providing an ideal snapshot of beach change caused by the hurricane. Although the da ta coverage extends the entire length of the Outer Banks, the quality of the prestorm profil es in some areas is compromised because of the hurried nature of the flights prior to the arrival of the storm. Because it was difficult to identify a PAGE 50 36 long stretch of coastline without gaps in high qua lity coverage, the study areas were narrowed to 300 meters, with profiles spaced approximately 10 meters apart in the longshore. The intensity of the hurricane caused many observational instruments to fail near the peak of the storm. For this reason, the data used for model runs combine observed and modeled storm characteristics from a variety of sources. Figure 1.11 plots the storm characteristics as a function of time, beginning at 00:00 EST on September 16th. Water level information is derived from two NOAA tide gauges, 8651370 at the Duck FRF Pier, and 8654400 at the Cape Hatteras Fishing Pier. Although the Cape Hatteras tide gauge is perfectly positioned in relationship to the study areas (Figure 1.9), it failed prior to the peak of the storm on September 18th. Data recorded prior to gauge failure were compared to several other coastal tide gauges in the area, and were found to best match the record at the Duck Pier, more than 100 km to the north. Because of a lack of alternatives, the Duck r ecord was used as a best estimate of conditions at the study site. Because the Duck gauge was significantly furt her from landfall than the study site, the two records begin to diverge immediately before the Ha tteras tide gauge failed. For this reason, data at the peak of the storm is replaced with wa ter level predictions from the National Hurricane Centers SLOSH model, which estimates that water levels reached 2.2 meters at the height of the storm. Wave heights and periods were recorded offshore at NDBC Buoy 41025 at Diamond Shoals until several hours before the peak of the storm. For model runs, predictions from NOAAs WaveWatch3 model extracted at the lo cation of the Diamond Shoals buoy were substituted for observed data. The model also pr ovides peak periods and wave angles, which are illustrated in Figure 1.11. The wave angles in this figure have been transformed to the conventions used by SBEACH, where 0 indicates a wave approaching shore normal, positive angles indicate waves arriving from the east, and negative angles indicate waves arriving from the PAGE 51 37 south. This transformation relies on the appr oximation that the southern portion of Hatteras Island trends exactly southwest to northeast. Figure 1.11 Storm characteristics for Hurricane Isabel that affected Hatteras Island during September 2003. These time series were used as input data into the models. Time 0 corresponds to 00:00 EST on September 18th, 2003. In Figure 1.12, dune crest elevations from Ha tteras Inlet to Cape Hatteras are plotted as a function of longshore distance, and the two study areas are highlighted in yellow. As mentioned previously, the two study areas identified as Ha tteras North and Hatteras Breach both encompass a longshore extent of 300 m, consisting of 30 profiles spaced approximately 10 m apart in the longshore. The average prestorm dune crest elevation on this stretch of beach was 4.5 meters, with a standard deviation of 1.8 meters. As this figure indicates, significant losses in dune crest elevation were recorded along the southwestern section of the island that was closest to landfall. PAGE 52 38 Figure 1.12 Dune crest elevations as a function of long shore distance along Hatteras Island. The top graph displays prestorm elevations, and the bottom graph presents poststorm elevations. The Hatteras North study area is composed of tall dunes averaging 6.5 m in elevation at the dune crest. The storm caused extensive dama ge on this study site, lowering the mean dune crest elevation to 5.3 m. Observed erosion on th is site is confined mostly to the dune face and crest. These observations are captured in Figure 1.13, which plots observed gross volume change as a function of latitude, and three preand poststorm profiles from this area. On these profiles, erosion is observed at the top of the dune, wh ile the beach appears to have accreted through deposition of the eroded sand. PAGE 53 39 Figure 1.13 Observed values of gross volume change are plotted as a function of latitude for the Hatteras North study site, and three example profiles display the varying impacts of the storm on dune erosion. In each figure the blue profile represents the prestorm profile, and red represents the poststorm profile. The Hatteras Breach study site consists of shorter dunes averaging 4.5 m in elevation at the dune crest. As seen in Figure 1.14, the poststorm profiles clearly show that the dune and beach were completely destroyed as a consequence of the storm. However, because lidar cannot penetrate water, the poststorm profiles are only reflections off the waters surface, and do not measure the depth of the channel scoured by currents moving over the island. Because lidar cannot capture the true poststorm bathymetry, cal culations of eroded volumes are not practical. The extreme nature of the impact along this section of coastline may be due to a number of factors, including extreme local water levels and wave heights, short and narrow dunes, the narrowing of the island at this location, and tr ansport by currents flowing over the island. PAGE 54 40 Figure 1.14 Three profiles displaying dune change at th e Hatteras Breach study site. In each figure the blue profile represents the prestorm prof ile, and red represents the poststorm profile. PAGE 55 41 Chapter Two Methods Description of Lidar Data Sets Topographic data sets were obtained from ai rborne topographic lidar (Light Detection and Ranging) surveys. Lidar technology is a laser mapping technique with numerous remotesensing applications, which in recent years has been adapted for coastal applications including the rapid surveying of narrow strips of beach and ba rrier island environments. The rapid nature of the surveying method makes lidar ideal for collecting storm related beach change data sets. Descriptions of coastal lidar theory and met hods are presented in Brock et al. (2002), and are briefly summarized here. Analogous to radar theory, lidar systems em it laser pulses that reflect off the earths surface, allowing the computation of elevation from the twoway transmission time. These high frequency laser pulses are emitted from NASAs air borne topographic mapper (ATM), a lighttransmitting device that is mounted on a small twine ngine aircraft that flies at an altitude of approximately 700 meters. The ATM emits lase r pulses at a frequency of 20,000 Hz in an elliptical scanning pattern as the aircraft flies overl apping flight lines over the survey site. The pulses are emitted with a near step function shape, and return time measurements are referenced to a predefined amplitude on the leading edge of the pulse. The return time, or twoway transmission time, is used to calculate the distan ce between the airplane and the earths surface, a measurement that is also known as range. An illust ration of the lidar system is provided in Figure 2.1. PAGE 56 42 Figure 2.1 Illustration of the lidar (light detection and ranging) system developed for coastal surveying applications. Pict ure courtesy of the USGS. Additionally, the aircraft is fitted with an Inertial Navigation System (INS), which records pitch, roll, and heading of the plane at a frequency of 64 Hz and an accuracy of 0.1 degrees. Measurements from the INS are used to correct range measurements from simple return time calculations. The aircraft is also equippe d with Global Positioning Systems (GPS), which records the location of the plane with an accuracy of 5 centimeters when operated in conjunction with a GPS ground station. The ground station is usually located at the airport where the local aircraft operations are based. The raw data in cluding range measurements and GPS locations are converted to latitudes and longitudes in NAD83 a nd elevations in NAVD88. Postprocessed data sets consist of approximately one elev ation measurement per square meter. In order to investigate storminduced beach change, the postprocessed data are utilized to construct a series of crossshore elevation prof iles spaced approximately ten meters apart in the longshore direction. Elevation points lying with in +/1 meter of a predefined profile line are incorporated into each crossshore profile. Each profile encompasses several hundred meters of beach and dune morphology, from landward of th e frontal dune, to seaward of the beach/water PAGE 57 43 interface. Lidar data gathered either from th e lower 48 mapping project conducted in the mid to late 1990s (Assateague Island study areas), or from immediately before the storm event (Hatteras Island study areas) are utilized in constr ucting prestorm profile lines. Lidar surveys are flown within several days of the passage of ma jor storm events, and crossshore profiles are extracted at longshore locations matching those of the prestorm profiles. The vertical accuracy of ATM survey data was investigated as part of the 1997 nearshore field experiment Sandy Duck held at the U.S. Army Corps of Engineers Field Research Facility (FRF) in Duck, North Carolina. Field methods and an indepth statistical analysis from this study are presented in Sallenger et al. (2003), a nd are summarized here. On September 26th, seven overlapping ATM surveys were conducted over a 70km stretch of beach encompassing Bodie Island from Corolla to Oregon Inlet. On the same day, three distinct ground surveying methods were used to conduct surveys of three regions of Bodie Island encompassed by the ATM surveys. The ground surveys were carried out to allow a statistical determination of the vertical accuracy of the ATM data. The first ground surveying method, the l ong transect buggy, consisted of a single transect line along the beach from Corolla to Ore gon Inlet, obtained from an Ashtec GPS receiver mounted on an ATV. The second method, the l ocal transect buggy, gathered multiple crossshore survey lines from the region of runup to th e dune base over a longshore distance of 3 km in the vicinity of the FRF, and consisted of a Trimble GPS mounted on an ATV. The final ground surveying method, the GPS rod survey, consiste d of a series of beach transects from the region of swash to the dune base over a 100meter longshor e stretch near Corolla. This survey was conducted manually with an Ashtec GPS mounted on a stadia rod. It is important to note that each of the ground surveys encompassed only the b each region, so that error estimates obtained from this study can only be applied to the beach, a nd not to the steeply sloping face of the frontal dune. PAGE 58 44 Statistical analysis of the data included in tercomparisons of the seven overlapping ATM surveys, as well as comparisons between the ATM and groundbased surveys. The study shows that the vertical accuracy of individual eleva tion measurements obtained by ATM surveys is 15 cm. While this error estimate may not be acceptable for some applications, 15 cm is a reasonable instrument error relative to vertical beach change recorded during storm events. Sallenger et al. cite the 1998 Assat eague Island storms, and note that for the profiles investigated in their study, maximum vertical beach cha nge was approximately 2 meters, and maximum vertical dune change was approximately 3 meters. SBEACH Program Interface The SBEACH program (Version 3.01G, released 2003) was purchased through VeriTech Inc., a coastal science and engineering soft ware company. Publications documenting model theory consist of five Army Corps of Engin eers technical reports, including Larson and Kraus (1989), which discusses the models empirical foundation, and Larson et al. (1990), which introduces the numerical scheme. The three additional manuals include Rosati et al. (1993) which is a users manual for an early DOS version of SBEACH, Wise et al. (1996) which discusses a random wave model that the user can ch oose to incorporate in model runs, and Larson and Kraus (1998) which discusses modifications to the model that allow nonerodible hard bottoms. SBEACH runs on a Microsoft Windows inte rface, and the code is not available to the user for modification. Therefore, all work done with the SBEACH model was performed within the user interface developed by VeriTech Inc. Profiles are compiled in Matlab, and transf erred individually into the SBEACH user interface in tabular format. The subaerial profile consists of the exact lidar profile bounded on the seaward edge by the horizontal location of mean sea level. The subaqueous profile is calculated with Deans h=Ax2/3 equilibrium equation and M oores equation for A (Equation 1.34), and is fitted to the subaerial profile at the location of mean sea level. Time series of water PAGE 59 45 level, offshore significant wave height, wave angl e, and peak period are also transferred into SBEACH in tabular format. Various windows w ithin the program allow the user to define physical variables and empirical parameters. Fo llowing each model run, final profile data is saved in tab delimited data files with the help of an automated keyboard software program. A Matlab program then converts the ta bdelimited data into Matlab .mat files in a format suitable for statistical analysis programs. EDUNE Program Interface The EDUNE model is available online through the University of Delawares Center for Applied Coastal Research (CACR, 1994). The CACR website provides code for EDUNE in the Fortran programming language, as well as exampl e data files for two profiles and their corresponding storms, profile R41 from a Hurricane Eloise data set, and profile F9 from a Hurricane Frederic data set. A users manual written by Kriebel (1995) describes program variables, as well as the individual algorithms that complete the framework of the program. The following paragraphs describe how this informati on was utilized and implemented in the current study. Before EDUNE was utilized for the current study, the model was translated from its original Fortran to the Matlab programming langua ge, in order to allow faster computing times and convenient storage of output data. The Matla b version was verified with the example data files provided on the CACR website. When running EDUNE in Matlab, the user must first create an input file defining physical variables and empirical parameters. The program prompts the user for profile numbers, which are identical to USGS profile numbers extracted from lidar data sets. The program also prompts the user for a number assigned to the storm event of interest. The program extracts profile data and time series data of storm surge, peak period, offshore significant wave height, and breaking wave heights from co rresponding data folders. Following each model run, output data including the final profile is automatically saved in a separate file folder. PAGE 60 46 One significant limitation of EDUNE is that input profiles must monotonically decrease in elevation both seaward and landward of the dune crest. Profiles of this form are seldom observed in lidar surveys, and cannot encomp ass backbarrier dunes, nor accurately represent elevated berms that occur seaward of the dune crest. In order to transform each lidar profile to meet this requirement, a Matlab program was deve loped to convert measured elevations into the correct monotonic form while maintaining as much of the original profile shape as possible. Within this program, if the original lidar prof ile does not decrease monotonically in elevation on both sides of the dune crest, the program performs a three point moving average across the profile and then assesses the averaged profile for the same monotonic criterion. The program continues this process using progressively larger moving average windows until the criterion is met. The subaqueous profile is calculated from Deans h=Ax2/3 equilibrium equation and Moores equation for A (Equation 1.34), and is fitted to the subaerial profile at the location of mean sea level. Modified profiles show only small departur es from the original lidar measurements, except in cases where elevated be rms are present. Figure 2.3 illustrates the differences between the lidar and modified profiles for two cases on As sateague North, one with and one without a significantly elevated berm. Gross volume erro rs (GVE) and rootmeansquare (RMS) errors (both statistical measures are described in more de tail later in this chapter) between the original lidar profiles and the modified profiles are presented in Table 2.1 for each of the five study areas. The mean RMS error for all 150 profiles is 15 cm, w ith a standard deviation of 5 cm. This error is the same magnitude as the rms error of the lidar measurements themselves. PAGE 61 47 Figure 2.2 Original lidar profile and modified profile that conforms to EDUNE program requirements for (a) a profile with a signifi cantly elevated berm, and (b) a profile without a significantly elevated berm. EDUNE profiles must decrease monotonically in elevation both landward and seaward of the dune crest. Both profiles are from the Assateague North study area. Study Area # of Profiles Mean GVE Standard Deviation GVE Mean RMS Error Standard Deviation RMS Error Assateague North 30 0.09 m3/m 0.02 m3/m 0.13 m 0.04 m Assateague South 30 0.12 m3/m 0.03 m3/m 0.18 m 0.05 m Chincoteague 30 0.08 m3/m 0.01 m3/m 0.12 m 0.02 m Hatteras North 30 0.10 m3/m 0.05 m3/m 0.15 m 0.07 m Hatteras Breach 30 0.10 m3/m 0.04 m3/m 0.15 m 0.05 m All Study Sites 150 0.10 m3/m 0.04 m3/m 0.15 m 0.05 m Table 2.1 Mean and standard deviations of both gross volume error (GVE) and rootmeansquare (RMS) errors between original lid ar profiles and modified EDUNE pr ofiles for each of the five study sites. PAGE 62 48 Statistical Analysis Statistical tests performed in this study ser ve two purposes: (1) defining the sensitivity of the models to changing input variables, and (2) determining the accuracy of the models for both model tests and calibration purposes. Three sta tistical measures will be discussed, including gross volume change (GVC), gross volume error (GVE), and rootmeansquare (RMS) error. First, calculations of GVC are used to measure th e sensitivity of the models to changing values of input parameters. GVC is a quantitative measur e of the normalized gross change in volume between the measured prestorm profile and the model predicted poststorm profile. Secondly, both RMS error and GVE are used as quantitative me asures of model accuracy. Both calculations measure differences between the surveyed poststorm profile and the model prediction of the poststorm profile. GVE quantifies the normalized gross difference in volume between the two profiles, while RMS error is the rootmeansquare error in elevation measurements. A model prediction that matches the measured profile exactly will have both a GVE and an RMS error equal to zero. The following paragraphs discuss the numerical methods u sed to calculate these three statistical measurements. The area of interest for the present study is the subaerial profile including the dune and the beach. Therefore, the horizontal boundaries for statistical calculations are defined to encompass only on the subaerial profile. The la ndward boundary is chosen at the landward edge of the dune and is user defined. The seaward boundary is chosen at the location of mean high water on the poststorm profile, and is determined with a Matlab program developed specifically for this application. MHW is defined on the poststorm profile because it eliminates contamination of the profile by waves. The sel ection of horizontal boundaries is illustrated in Figure 2.3. PAGE 63 49 Figure 2.3 Horizontal boundaries on the subaerial profile are chosen at the landward dune base and mean high water. Example profile is 4342 from Assateague North. (a) The shaded area indicates the portion of the profile included in statistical analysis. The landward boundary is chosen manually on the prestorm profile, and the seaw ard boundary is chosen by a mean high water crossing method on the poststorm profile. (b) An enlarged view of the subaerial profile. GVC is calculated as: dx z z L GVCpre i N i i) ( 1, 1 (2.1) where L is the length of the profile, z is the predicted elevation, zpre is the measured prestorm elevation, the subscript i refers to the series of interpolated elevations points, N is the total number of elevation points, and dx is the horizontal distance between elevation points. Measurements of GVC are used in sensitivity test s to describe the response of model results to empirical, physical, and hydrodynamic variables. The calculation of GVE is similarly specified by: dx z z L GVEpost i N i i) ( 1, 1 (2.2) PAGE 64 50 where zpost is the poststorm dune elevation. Figure 2.4 illustrates the method for calculating GVE with an example profile from Assateague North. Additionally, RMS error calculations make use of the interpolated elevations along the horizontal extent of the subaerial profile: 2 / 1 1 2 ,) ( 1 N i post i iz z N RMS (2.3) Both GVE and RMS error are used in calibration tests to confirm calibration values of empirical coefficients and to describe the accu racy of the calculated profile. Figure 2.5 demonstrates how GVE and RMS error change as a function of horizontal spacing between elevation points for five different profiles from Assateague North. A value of dx=0.1m is chosen because while smaller intervals allow error statistics to converge to a limiting value, they utilize excessive computation tim e for the corresponding increase in accuracy. Conversely, greater values of dx cause both error st atistics to diverge from the limiting value. Figure 2.4 An example of a gross volume error (GVE) calculation. Each individual box has a volume of dx z zi i) ( with height ) (i iz z width dx and unit thickness. GVE is the normalized sum of the individual volumes. For illustrative purposes, dx = 1.0 m in this graphic. Example data is from an SBEACH model run for profile 4342 from Assateague North. PAGE 65 51 Figure 2.5 GVE and RMS error are plotted as a function of dx As dx decreases, both error statistics converge to a limiting value. For the current study, dx was set at 0.1 m for both calculations. Data is compiled from an SBEACH run on five profiles on Assateague North. Two additional statistical measures were inves tigated, but their values were determined to be of limited use for the current study. First, net volume error (NVE) was investigated because it calculates the net positive or negative error asso ciated with the model fit, whereas this information is lost in the calculation of GVE. The NVE calculation is similar to GVE: N i post i idx z z L NVE1 ,) ( 1. (2.3) This statistical measure had no value as a measure of accuracy, due to local positive a nd negative errors canceling out over the length of the subaer ial profile. NVE calculations often resulted in error estimates near zero when the true mode l fit was poor, such as the example provided in Figure 2.6. Additionally, contour retreat errors were measured at the mean sea level contour, the 3meter contour, and the 5mete r contour. These measurements proved problematic, as the mean PAGE 66 52 sea level contour was often obscured by waves, a nd as the poststorm dune often did not exceed 3 meters. In some cases of low berm profiles, the 3 and 5 meter contours did not exist on the prestorm profile. Although these statistical measures ha ve been used in previous studies, their value for this study was limited. Figure 2.6 Net volume error (NVE) calculations were not used as a measure of accuracy, because local positive and negative errors often canceled over the length of the profile. In this example, NVE is near zero, while the true model fit is poor. In this case, GVE and RMS error provide a better measure of accuracy. For illustrative purposes, dx = 1.0 m in this graphic. Example data is from an EDUNE model run for profile 4352 from Assateague North. PAGE 67 53 Chapter Three SBEACH Results Introduction The objective of this chapter is to quantif y the accuracy of calculated SBEACH profiles for a variety of model tests, and to identify pa rameters that influence the magnitude of model error. Sensitivity tests are performed for a wide variety of variables, in order to determine the relative effect of each variable on model results. The first section in this chapter examines the sensitivity and calibration of the model to empiri cal parameters (e.g. K, the transport rate coefficient), while the second section examines the sensitivity of the model to physical and hydrodynamic variables (e.g. grain size, time series of water levels). All sensitivity and calibration runs are calculated with lidar data from the Assateague North study area. In the third section, model results and erro r calculations are presented for the three Assateague Island study sites. Analysis presented in this section confirms the robust nature of the calibrated transport rate coefficient between stud y sites on the same island. The fourth and final section discusses results from the Hatteras North and Hatteras Breach study areas. Hatteras North data are used to produce a sitespecific calibration value for the transport rate coefficient, which is determined to be the same as the calibrated Assat eague value. This section also briefly proposes two transport mechanisms that may account for the models inability to reproduce the breach. Model Sensitivity and Calibrati on: Empirical Parameters The results of sensitivity tests performed on the SBEACH model are important for two reasons. First, all previous publications about SBEACH have commented on model parameters PAGE 68 54 in terms of their effects on the development and movement of the sandbar. Because this study seeks to use SBEACH to hindcast the poststorm subaerial profile, it is important to examine sensitivity relative to the dune rather than the bar system. It is imperative to understand these differences in light of the fact that many e ngineering firms and governmental agencies now use SBEACH to predict dune erosion. Secondly, sev eral recent changes to the SBEACH model have gone unnoted in the literature, most recognizably, the addition of several empirical and physical parameters. The user now defines variables su ch as the landward edge of the surf zone, the transport rate decay coefficient, and water temper ature. The effect of these parameters on model results must be fully understood before SBEACH can be considered a viable predictive tool. K, the Transport Rate Coefficient The first empirical parameter inherent to the SBEACH model is K, the transport rate coefficient, which enters the SBEACH model formulation through Equation 1.22: sQ dx dh K D D Keq dx dh K D Deq (3.1) 0 dx dh K D Deq K governs the magnitude of the sediment transport rate Qs when excess wave energy dissipation is present. As values of K increase, gross volume change is expected to increase proportionally, as large values of K should increase transport ra tes across the entire profile. Previous studies have found bestfit Kvalues ranging from 7.0x107 to 2.0x106 m4/N. A range of 2.5x107 to 2.5x106 m4/N is chosen for sensitivity analysis in or der explore model results over an order of magnitude, while incorporating the previous calibration values. Model results for each value of K are presente d in Figure 3.1 for an example profile from Assateague North. As predicted, the shape of the calculated profile varies widely, simulating too little erosion for small values of K, and producing excessive erosion for large values of K. PAGE 69 55 Results of the sensitivity tests for K are shown in Fi gure 3.2(a). In this figure, the predicted gross volume change is reported as a mean value for th e 30 Assateague North profiles, with each error bar representing the 95% confidence interval on the mean value. The apparent trend reveals a statistically significant increase in mean GVC with increasing K. The change in the mean value is 369% across this one order of magnitude of Kvalues, and is significant at the 95% confidence level. At the smallest value of K tested, mean gross volume change is 0.47 m3/m, which is evident as a deficiency in erosion on both the beach and dune. Conversely, the entire dune is erroneously eroded at the greatest value of K, with mean gross volume change equaling 2.2 m3/m. Figure 3.2 indicates that among the five empirical parameters, model results are most sensitive to K. For this reason, it is essential to establish an accepted Kvalue that can be transferred between study areas and storms, if SBEACH is to be used in the future as a predictive tool. Calibration tests were performed to determin e the bestfit Kvalue through minimization of both RMS error and GVE. Calibration curves are presented in Figure 3.3, indicating that error statistics are minimized at values of 5.0x107 and 7.5x107 m4/N. Tests of the equality of means and variances indicate that there is no statistical difference between the errors produced at these two values. The value of K=5.0x107 m4/N is chosen as the bestfit K value for Assateague Island, minimizing mean RMS error at 39 cm, and mean GVE at 0.27 m3/m. The sizeable error bars that are observed at large Kvalues are due to numerical instabilities that occur when transport rates become large. In these cases, ex cessive sediment is transported in and out of single grid cells, which results in calculated profiles containing numerous jagged peaks and valleys often ranging several meters in amplitude. PAGE 70 56 Figure 3.1 Profile 4362 from Assateague North is used as an example of SBEACH model response to changing values of K (m4/N). In each figure, the blue profile represents the prestorm profile, red represents the poststorm profile, and black indicates the calculated model result. PAGE 71 57 Figure 3.2 SBEACH sensitivity to (a) K, the transport rate coefficient, (b) the slope coefficient, (c) the maximum local slope before avalanching, (d) the landward surf zone depth, and (e) 1, the transport rate decay coefficient. Sensitivity is evaluated through a m ean calculation of predicted gross volume change (m3/m) for the 30 Assateague North profiles. Each error bar represents the 95% confidence interval on the mean value. PAGE 72 58 This calibration value is appreciably lower than Kvalues chosen in previous SBEACH studies, including K=1.6x106 m4/N from the original large wave tank studies, K=7.0x107 m4/N from the Duck, NC field study, K=1.4x106 m4/N from the Torrey Pines Beach, CA field site, and K=2.0x106 m4/N from the New Jersey beach field sites (Larson and Kraus, 1989; Larson et al., 1990). Figure 3.3 indicates that any of these previously determined values would produce large errors on the Assateague North study site. Howeve r, each of the four previous calibration values was generated by minimizing an error statistic that considered the entire length of the profile, of which the subaqueous profile was the dominant por tion. For this reason, the new Assateague Island calibration value is important for model tests that focus on the accuracy of the model on the subaerial portion of the profile. Further tes ting will investigate whether this calibration value remains constant between different field sites. Figure 3.3 SBEACH calibration curves for values of K, the transport rate coefficient on the Assateague North study site. Each error bar represents the 95% confidence interv al on the mean value. The large error bars seen at large K values are a result of model instabilities that occur with high sediment transport rates. PAGE 73 59 the Slope Coefficient The empirical parameter is found in the sediment trans port rate calculation in Equation 3.1, where it acts as a coefficient for the local slope term. Large values of are expected to smooth locally steep slopes occurring in the broken wave zone, enhancing o ffshore transport and sandbar formation in the offshore region. In th eory then, because waves break over much of the beach during storms, increasing values of should increase GVC on the subaerial profile by smoothing beach slopes and transporting sand seaward. However, because transport is affected only through the slope term, the c onsequences of changing values of are expected to be small. A calibration value of 2.0x103 m2/s is reported for Torrey Pines Beach, CA, and therefore, the limits for sensitivity tests are set at 5.0x104 and 5.0x103 m2/s in order to encompass one order of magnitude surrounding the previously reported value. Model results for each value of are presented in Figure 3.4 for an example profile from Assateague North. The observed differences in m odel results are much more subtle than those observed for changing values of the empirical coe fficient K. However, careful inspection shows that as the value of increases, subaerial erosion increases through the flattening of the beach. Figure 3.2(b) indicates that SBEACH is most sensitive to changes in the slope coefficient in the lower range of values. The observed trend suggests that as hypothesized, total erosion on the subaerial profile increases with increasing values of The change in the mean value of predicted gross profile change across one order of magnitude of values is 102%, and is statistically significant with 95% confidence. Calibration plots are shown in Figure 3.5, i ndicating that neither error statistic changes appreciably for values greater than 2.0x103 m2/s. The mean error statistics reported for values in the range of 1.5x103 to 4.5x103 m2/s are not statistically different from one another at the 95% confidence level. As a result, any value between 1.5x103 and 4.5x103 m2/s can be PAGE 74 60 Figure 3.4 Profile 4362 from Assateague North is used as an example of SBEACH model response to changing values of (m2/s). In each figure, the blue profile represents the prestorm profile, red represents the poststorm profile, and black indicates the calculated model result. PAGE 75 61 viewed as a valid for calibration purposes. Lars on et al. (1990) calibrate SBEACH to Torrey Pines Beach, CA field data, and report =2.0x103 m2/s as the bestfit value. The error statistic used to generate this calibration value is dom inated by the subaqueous portion of the profile including multiple sandbars, and therefore =2.0x103 m2/s can be used as a robust value that minimizes error on both the subaerial and subaqueous portions of the profile. Figure 3.5 SBEACH calibration curves for values of the slope coefficient on the Assateague North study site. Each error bar represents the 95% confidence interval on the mean value. the Maximum Local Slope Before Avalanching The third empirical parameter inherent to the SBEACH model is the maximum slope that can exist between adjacent grid cells. When the model detects a slope that exceeds the angle avalanching occurs until the slope stabilizes at a residual angle of 10 Small values of cause frequent small avalanches on the dune face as the slope of the dune base steepens due to transport by swash and breaking waves. The sediment that is eroded during avalanching will ultimately be carried into the surf zone, a nd will migrate further offshore under high wave conditions. Large values of reduce the sediment transport on the dune by allowing the dune PAGE 76 62 base to achieve steeper angles before avalanchi ng occurs. Therefore, gross profile change calculated by SBEACH should exhibit an inverse dependence on with erosion on the subaerial profile decreasing with increasing A preliminary value used by Larson and Krau s (1989) and Larson et al. (1990) of 28 is based on observations from the large wave tank st udies. For the purpose of sensitivity tests, is varied between 24 and 42 in order to accommodate this value and initial observations that the optimal value of for Assateague Island may be larger than 28 Contrary to the original hypothesis, Figure 3.2(c) indicates that the model is insensitive to with virtually no change occurring in predicted gross volume change across the test range of values. Tests of the equality of means and variances confirm that none of the mean values of GVC are statistically different from one another at the 95% confidence le vel, signifying that erosion on the subaerial extent of the model is insensitive to values in the given range. Despite this result, a visual inspection of the 30 profiles from Assateague North indicates that an optimum value of does exist for this study area. A value of 38 allows the model to most accurately reproduce the slope of the poststorm dune. These optimal slopes are achieved through the avalanching of small volumes of sand off the dune face and stabilizing at the residual angle of 28 This observation suggests that the optimal value of should be chosen based upon the slope of the poststorm dune and the residual angle after avalanching. Knowledge of poststorm dune shape from prior storms will facilitate the user in choosing the correct value of for any study area. the Landward Surf Zone Depth One fundamental principle of the SBEACH mode l is the existence of distinct crossshore transport regions, between which the structure of the sediment transport rate equations vary significantly. The rate equations for each of the four regions were devel oped with data from two PAGE 77 63 large wave tank experiments, and are given in Eq uations 1.20 to 1.23. Naturally, the successful use of these equations is in part dependent on the correct placement of horizontal boundaries between the zones. Unfortunately, the exact criterion for the placement of boundaries is unknown, and the task is left to the best scientific judgment of the user. The horizontal boundary between transport ra tes in the broken wave zone (Zone III) and the swash zone (Zone IV) is marked at an arbitrary depth generally ranging from 0.1 to 0.5 meters. When the location of the boundary occu rs at shallow depths, the width of the broken wave zone increases, and waves expend energy over a greater width of the profile. In theory, as increases, predicted gross volume change s hould decrease as waves dissipate energy over a shorter distance. Model results for each value of are presented in Figure 3.6 for an example profile from Assateague North. At values of greater than 0.15 m, the profiles often exhibit sharp peaks and valleys on the beach. These instabilities occu r when Zones III and Zones IV exist on the subaerial profile during the storm, and a portion of this profile is excluded from the smoothing effects of changing wave heights due to large values. When waves are allowed to expend energy closer to shore with smaller values of local spikes in the profile are smoothed as energy dissipation over that portion of the profile fluctuates with time. Figure 3.2(d) indicates that the mean value of GVC remains constant as values increase over one order of magnitude, from 0.05 to 0.5 m. The means and variances of predic ted GVC values are statistically equivalent at the 95% confidence level, indicating that th e subaerial model results are insensitive to Therefore, a fixed value of =0.1 m is chosen for all SBEACH runs, in order to avoid the jagged profiles that occurs for greater values of PAGE 78 64 Figure 3.6 Profile 4362 from Assateague North is used as an example of SBEACH model response to changing values of (meters). In each figure, the blue prof ile represents the prestorm profile, red represents the poststorm profile, and black indicates the calculated model result. PAGE 79 65 1 the Transport Rate Decay Coefficient The sediment transport rate decays exponentia lly offshore of the br oken wave zone. The rate of decay is controlled by 1, the transport rate decay coeffici ent, which is found in Equation 1.20: ) (1bx x b se Q Q (3.2) Large values of 1 create sharp gradients in the transport rate, and facilitate the development of slowly moving sandbars in the offshore region. Smaller values of 1 do not encourage the accumulation of sand in any single area, and allow uniform sheets of sediment to move slowly offshore. Because 1 controls the shape of the offshore profile, it should have little effect on model results on the subaerial profile. Initial versions of SBEACH calculated the value of 1 at every time step with the equation: 47 0 50 1H D 0.4 b (3.3) where Hb is the breaking wave height. Subsequent versions abandoned this timedependent version of 1 for a constant value defined by the user. For sensitivity testing, 1 was varied between values of 0.1 and 0.5 m1, which correspond to breaking wave heights of 6.2 and 0.2 meters respectively from Equation 3.3. Figure 3.2( e) displays mean values of GVC for the test range of 1 values. The means and variances for the GVC values are not statistically different from one another at the 95% confidence level, confirming that the subaerial model results are insensitive to 1. A visual inspection of the 30 profiles from Assateague North displays no bias towards any specific value, and therefore, an arbitrary fixed value of 1=0.2 m1 is chosen for all SBEACH model runs. PAGE 80 66 Model Sensitivity: Physical and Hydrodynamic Variables D 50 Median Grain Diameter Median grain diameter, given the notation D50, plays two distinct roles in the SBEACH model. First, D50 influences the shape of the offshore profile calculated with Deans h=Ax2/3 equilibrium profile equation, where A is a function of grain size. It is important to note that the representation of the offshore profile by Deans equation is not standard to SBEACH, but was chosen for lidar data sets that do not include subaqueous data. As grain size increases the profile steepens, allowing waves to shoal further inshore before breaking. When the waves break further inshore, the energy dissipated over the shallo w depths increases and causes an increase in total transport. Therefore, this causeandeffect relationship implies that an increase in grain size should increase total sediment transport. However, wellknown sediment transport theory states that greater energy is required to entrain larger grains in the flow, and that these la rge grains settle out of the water column quickly after initial suspension. This concept is inco rporated into the SBEACH model through the sediment fall speed, ws. Grain size directly influences the sed iment fall speed, with larger grains settling at greater velocities than their smaller counterparts. Fall speed enters the numerical formulation in Equation 1.30: 300070 0 T w H L Hs. (3.4) The exact equations for fall speed have not been published, making this an important test for users who wish to know how grain size effects mode l results. This equation is calculated at each time step, and acts as the defining condition between onshore and offshoredirected transport. Onshore transport occurs when the left hand side of the equation is greater than the right hand side, so an increase in grain size (and hence fall sp eed) allows the right hand side of this equation to decrease rapidly with the cubed term. Therefor e, as grain size increases, total onshoredirected PAGE 81 67 transport will increase over the duration of the simulated storm. Paired with the effect of increased total transport due to shoaling over a steep er profile, a large amount of sand is expected to move onshore, causing a significant decrease in erosion with increasing grain size. The true value of D50 is 0.33 mm, which is the average median grain size from multiple sediment samples taken during a 1995 U.S. Army Corps of Engineers study along Assateague Island (USACE, 1998). For sensitivity tests, D50 is varied about this value at intervals of 0.1 mm, ranging from 0.13 to 0.93 mm, corresponding roughly to the lower boundary of fine sand (0.125 mm) and to the upper boundary of coarse sand (1 .0 mm). Model results for each value of D50 are presented in Figure 3.7 for an example profile from Assateague North. Additionally, Figure 3.8(a) presents results of the sensitivity tests, displaying mean values of predicted GVC for all grain sizes. An interesting trend is illustrated in these two figures, as mean GVC does not strictly increase or decrease as grain size increases, but rather reaches a maximum value at D50 = 0.23 mm. Most likely, transport on the upper profile is suppressed when D50 = 0.13 mm, because the corresponding offshore profile is exceedingly shallow, confining the dissipation of energy to the offshore region. When D50 increases to 0.23 mm, the profile steepens slightly, allowing waves shoal further inshore. This increases overall trans port, and therefore, predicted GVC increases. However, as grain size increases beyond this poin t, an increasing proportion of the total transport is directed onshore, decreasing GVC via Equation 3.4. In simpler terms, even as wave energy increases, large grain sizes do not erode as easily as small grain sizes, and when they do, they settle quickly before being transporte d away from their original location. Encouragingly, Figure 3.9 indicates that both error statistics are minimized at the true median grain diameter of 0.33 mm. Error measurements increase rapidly for values of D50 greater and less than 0.33 mm, suggesting that a small measurement error or bad judgment in choosing this value could cause significant errors in model results. Background literature does not provide PAGE 82 68 Figure 3.7 Profile 4362 from Assateague North is used as an example of SBEACH model response to changing values of D50 (millimeters). In each figure, the blue profile represents the prestorm profile, red represents the poststorm profile, and black indicates the calculated model result. PAGE 83 69 Figure 3.8 SBEACH sensitivity to (a) D50, median grain diameter and (b) water temperature. Sensitivity is evaluated through a mean calculation of predicted gross volume change (m3/m) for the 30 Assateague North profiles. Each error bar represen ts the 95% confidence interval on the mean value. Figure 3.9 SBEACH calibration curves for values of D50, the median grain diameter on the Assateague North study site. Each error bar represen ts the 95% confidence inte rval on the mean value. PAGE 84 70 recommendations for choosing the correct value of D50, and therefore these results validate the choice of median grain diameter taken from m ean high water to mean low water and averaged over multiple profiles (USACE, 1998). Because recent sediment surveys will not always be available for each study site, the user must use their best scientific judgment when choosing an appropriate value of D50. Water Temperature Like median grain diameter, water temperature also enters the model formulation through sediment fall speed in Equation 3.4. As temp eratures increase, the viscosity of the water decreases and sediments to settle with a greater velocity. Increasing fall speeds allow the right hand side of Equation 3.4 to decrease rapidly, therefore increasing total onshoredirected transport over the duration of the storm. Theref ore, in theory, increasing water temperatures will impede profile change because of faster settling velocities. The true water temperature of 6.5 C was obtained from NDBC Buoy 44009, as a mean of hourly sea surface temperatures. Because the surf zone is a wellmixed environment, sea surface temperature should accurately approximate the temperature of the entire water column. However, because the measurement is taken offshor e of the surf zone, an error of several degrees is quite possible. Therefore, sensitivity tests are performed with temperatures ranging from 1.5 C to 11.5 C. Greater differences in temperature that could potentially occur only on seasonal scales can be excluded from the possible range of measureme nt error. Figure 3.8(b) presents sensitivity test results for temperatures varying about the true temperature of 6.5 C. Changes in the mean value of predicted GVC are not discernable, and none of the means or variances are statistically different from one another at the 95% confiden ce level. The model results are therefore insensitive to changes in water te mperature within a reasonable range of the measured value. Although the original hypothesis states that GVC should decrease with increasing temperatures, PAGE 85 71 this result is not surprising. The influence of water temperature on the settling velocity is very small over a 10 C range, and consequently is un detectable in the results. In Figure 3.10, model sensitivity is explored over a wider range of water temperatures. This graph shows that the model results ch ange slightly as temperatures rise above 15 C. As temperatures increase, the mean value of pred icted GVC decreases, confirming the original hypothesis that increased temperatures will decr ease total erosion. A statistically significant change in mean GVC of 31% is recorded between 15 C and 40 C. This result indicates that the accuracy of the temperature measurement becomes increasingly important at higher temperatures. Therefore, it is more important to have an accu rate measure of temperature for a hurricane that occurs in August, than for a northeaster that strikes during the winter months. Figure 3.10 SBEACH sensitivity to water temperatures ranging from 0 C to 40 C. Sensitivity is evaluated through a mean calculation of predicted gross volume change (m3/m) for the 30 Assateague North profiles. Each error bar represents th e 95% confidence interv al on the mean value. Wave Height In order to investigate the effect of wave height on model results, the original time series of significant offshore wave heights (H) from NDBC Buoy 44009 was modified to vary between 0.5H and 1.5H These values correspond roughly to mean and maximum wave height (0.60H and 1.57H respectively) from a Rayleigh distribution of wave heights for broadspectrum hurricane waves (Goodknight and Russell, 1963). The original time series, as well as the extreme time series reco rds used in the sensitivity tests are shown in PAGE 86 72 Figure 3.11. In theory, larger wave hei ghts should increase energy dissipation across the length of the surf zone, and therefore, greater erosion across the entire profile. However, a greater wave height also causes the waves to br eak further offshore, dissipating some of the excess energy at greater depths, and therefore damping the effect of increasing energy dissipation on the subaerial profile. Figure 3.12(a) shows the sensitivity test results for the modified wave height time series. The observed trend exhibits a slight increase in mean GVC with increasing wave height. The total change in the mean value of predicted GVC over the entire range of wave heights is 148%, which is statistically signifi cant with 95% confidence. The small signal in model sensitivity is not only due to the dissipati on of energy at greater depths, but also to the nature of the sensitivity measurement. GVC me asurements are calculated on the subaerial profile, while wave heights dissipate energy that transports sediment in the surf zone. Therefore, if GVC were calculated over the entire profile length, sensitivity would increase dramatically. Because the study is focused on the dune and beach, sensitivity to wave height is less what is intuitively expected. Figure 3.11 Example time series of wave heights utilized to test SBEACH model sensitivity. The H record was taken from NDBC Buoy 44009 in Delaware Bay. Time 0 corresponds to 08:00 EST on January 27, 1998. PAGE 87 73 Figure 3.12 SBEACH sensitivity to (a) wave height (H) (b) water elevation ( ), (c) wave period (T), and (d) wave angle ( ). Each error bar represents the 95% confidence interval on the mean value. PAGE 88 74 Figure 3.13 presents error statistics over the range of modified wave height time series. Interestingly, both mean RMS error and mean GVE are minimized at 1.3H, a value that roughly corresponds to the calculation of H1/10=1.25H given by Goodknight and Russell (1963). However, the means and variances of the GVE and RMS error statistics for wave height time series ranging from H to 1.3H are not statistically different from one another at the 95% confidence level. Therefore, this result does not justify running the model with wave heights of 1.3H without additional confirmation. M odel results for each value wave height time series are presented in Figure 3.14 fo r an example profile from Assateague North. Figure 3.13 SBEACH calibration curves for modified time series of wave height on the Assateague North study site. Each error bar represents the 95% confidence interv al on the mean value. PAGE 89 75 Figure 3.14 Profile 4362 from Assateague North is used as an example of SBEACH model response to modified time series of wave height. In each figure, the blue profile represents the prestorm profile, red represents the poststorm profile, and black indicates the calculated model result. PAGE 90 76 Water Elevation The original time series of water elevations ( ) was obtained from NOAA Tide Gauge 8570283 at Ocean City Inlet, MD. In order to investigate the sensitivity of the model to variations in the entire time series is shifted vertically by a fixed amount ranging from 1 m to 1 m. Example time series from the sensitivity test s are given in Figure 3.15. In theory, elevated water levels allows waves to break closer to the beach, which should cause increased erosion on the subaerial profile. Figure 3.12(b) confirms this expectation, with the mean value of predicted GVC increasing consistently over this range of with a statistically significant increase of 265%. Figure 3.16 presents model results for an example profile for Assateague North, indicating that the model performs most accurately with a water elevation time series that is elevated above the original tide gauge data. This suggestion is supported by Figure 3.17, which plots mean error statistics for the different time series. Mean error is minimized at a +0.2, which indicates that the original time series does not allow waves to shoal close enough to shore to produce observed levels of erosion. The variances of GVE and RMS error calculated at and +0.2 are statistically different from one another at the 95% confidence level, suggesting that a greater level of accuracy is achieved with the modified time series. However, there is no reasonable justification in recommending that th e water level time series should always be elevated above the observed values. Water levels are known to fluctuate along the shoreline, and may be smaller or larger than the measured valu es at a local tide gauge depending on the location of the shoreline relative to the storm. The significance of this finding lies in the na ture of storm surge along the coast. Water levels vary significantly in the longshore direction, depending on the location of the storm relative to the coastline. As a result, the use of a single water level time series for a long stretch of coast may cause varying degrees of model error at each location. This effect will become PAGE 91 77 increasingly important when modeling hurricane even ts, as water levels are elevated on one side of the eye, and depressed on the other. For exam ple, for a hurricane on the east coast of the U.S., utilizing a tide gauge to the north of landfall for a section of coastline that lies south of landfall could potentially lead to an error in water levels of several meters. However, an error of several meters in water level could also occur in less ex treme situations, making it important for future research to attempt to incorporate longshore variab le time series of water levels from models such as NOAAs SLOSH or DELFT3D. Figure 3.15 Example time series of water elevations utilized to test SBEACH model sensitivity. The record was taken from NDBC Tide Gauge 8570283 at Ocean City Inlet, MD Time 0 corresponds to 08:00 EST on January 27, 1998. PAGE 92 78 Figure 3.16 Profile 4362 from Assateague North is used as an example of SBEACH model response to modified time series of water elevation. In each figure, the blue profile represents the prestorm profile, red represents the poststorm profile, and black indicates the calculated model result. PAGE 93 79 Figure 3.17 SBEACH calibration curves for modified time series of water elevation on the Assateague North study site. Each error bar represen ts the 95% confidence inte rval on the mean value. Wave Period Wave period (T) plays two distinct and opposing roles in the SBEACH model formulation. First, period appears in Equation 3.4, which acts as the defining condition between onshore and offshore transport. Increases in wa ve period cause the right hand side of the equation to decrease rapidly, increasing total onshor e transport over the duration of the simulated storm. However, increasing wave period also leads to an increase in wavelength, which according to SBEACH equations, allows waves to s hoal closer to shore before breaking. This theoretical statement is represented numerically in Equation 1.15, which defines the depth at breaking with the breaking wave criterion: 21 0 2 / 1) / ( tan 14 1 L H h Hb b (3.5) As waves break closer to shore, greater amount s of energy are dissipated over shallow depths, increasing erosion on the upper extent of the profile. These two opposing roles of wave period should act to offset any trend in profile change that either might produce individually. PAGE 94 80 To investigate the effect of wave period on model results, the original time series from NDBC Buoy 44009 was modified by shifting the entire series by a fixed amount, ranging from a decrease of four seconds to an increase of six seconds. The variation in period for this test was asymmetric in order to avoid negative periods, but to allow testing over a ten second range. Figure 3.12(c) shows evidence that the model is re latively insensitive to changes in wave period, with the mean value of GVC increasing by only 28% over the entire range of wave period records. Because the increase in mean GVC is statistically significant with 95% confidence, we can conclude that period is most influential th rough its effect on wavelength, allowing waves to shoal further inshore with increasing period. When compared to the sensitivity of the model to both wave height and water leve l, variations in model results with period are relatively insignificant. Wave Angle Wave angle is incorporated into the mo del through Equation 1.11, a onedimensional form of Snells Law that assumes straight and parallel bottom contours: 0 sin L dx d (3.6) Snells Law allows the computation of refraction as waves travel across the surf zone. The wave angles calculated from Equation 3.6 are incorporat ed into the conservation of energy flux equation (1.10): ) ( ) cos (sF F d k F dx d (3.7) where cos determines the proportion of total energy flux moving in the crossshore direction. When waves arrive in the surf zone at low angles, th e majority of total energy flux is available in the crossshore over the entire length of the surf zone. However, when waves arrive at high angles, the available crossshore energy flux is onl y a small percentage of the total. Crossshore PAGE 95 81 flux then increases as waves refract and changes with decreasing depth, but the total energy expended in the crossshore is appreciably less than for waves approaching at low angles. Therefore, the expectation arises that waves w ith high offshore angles will cause less erosion because of the decreased amount of energy di ssipated across the width of the profile. In order to test the sensitivity of the model to wave angle, the original time series from NDBC Buoy 44009 was modified to vary between 50 to +50 An angle of 0 is shore normal, positive angles indicate waves arriving from the northeast, and negative angles denote waves arriving from the southeast. Relative freque ncy histograms of the original time series and the two extremes are shown in Figure 3.18. The original time series is skewed towards waves arriving at low angles from the southeast, but is essentially centered at shore normal. The extreme cases cause waves to approach at higher angles from both the northeast and the southeast. Therefore, erosion should be highest with the original time series, and should decrease as wave angles increase in both directions. Figure 3.12(d) indicates that the model is essentially insensitive to changes in wave angle, with no discernable trend in calculated GVC as changes across the given range. A statistically significant decrease in GVC of 20% is recorded between the original time series, and 50 but no significant changes occur be tween the original time series and +50 Because of the relative insensitivity of the model to wave a ngle, a greater level of measurement error is acceptable than would be the case for wave heights and water levels. Add itionally, if wave angle information is unavailable, SBEACH assumes a constant shore normal approach over the duration of the simulated storm. This method do es not significantly change the results at the 95% confidence level, and therefore, SBEACH can be operated without wave angle input without losing accuracy in model results. PAGE 96 82 Figure 3.18 Relative frequency histograms of th e original wave angle time series ( ) and two of the modified time series used to test SBEACH model sensitivity. An angle of 0 is shore normal, positive angles indicate waves arriving from the northeast, and negative angles denote waves arriving from the southeast. The record was taken from NDBC Buoy 44 009 in Delaware Bay. Model Results: Assateague Island Calibration Site: Assateague North The Assateague North study area is comprised of 30 profiles characterized by high dunes, with elevations at the dune crest ranging between 5 and 8 meters. Dune scarping is observed on the northern end of the site, while only beach erosi on is observed in the sout h. Figure 3.19 plots SBEACH model results for ten Assateague North pr ofiles spaced approximately 300 meters apart in the longshore. These results were calculated using calibrated values of empirical parameters from the previous section, including a Kvalue of 5.0x107 m4/N. The model displays an exceptional ability to reproduce th e basic shape of the beach, a lthough the vertical placement often varies slightly from the measured postst orm location. Additionally, the model reproduces the slope of the dune well, even though the exact location of the sloping dune face is not always correct. These results indicate that the tran sport mechanisms (wave energy dissipation on the beach, and avalanching on the dune face) are empiri cally correct, despite e rrors in the magnitude of their effects. Figure 3.20 presents RMS error and GVE as a f unction of latitude. From this plot it is apparent that RMS error is small and relatively constant for profiles that experience only beach erosion, and that the error increases significantly for those profiles that experience dune scarping. PAGE 97 83 Mean RMS error for the entire study site was 0.39 m, with a standard deviation of 0.19 m. In comparison, the eighteen beach erosion profiles had a mean RMS error of 0.30 m, while the twelve dune scarping profiles had a mean RMS error of 0.53 m. The two means are statistically different from one another at the 95% confiden ce level, suggesting an inverse relationship between erosion and model accuracy on the subaer ial profile. The most important observation from this site is that the model correctly de lineates between beach erosion and dune scarping in all 30 cases. Because the accurate prediction of impact regime has been a recent subject of study, the ability of SBEACH to differentiate between beach and dune erosion is a promising result. PAGE 98 84 Figure 3.19 Ten profiles selected from Assateague North and the predicted SBEACH results. The first profile (4122) is the southernmost profile in th e study area, and the last profile (4392) is the third northernmost profile. The profiles are spaced approximately 300 meters apart in the longshore direction. In each figure, the blue profile represen ts the prestorm profile, red represents the poststorm profile, and black indicates the calculated model result. PAGE 99 85 Figure 3.20 RMS error and GVE as a function of latitude for SBEACH model results on the Assateague North study site. The north end of the island is characterized by dune scarping, and the south end of the island is ch aracterized by beach erosion. Berm Profiles: Assateague South The Assateague South study area is characterized by berm profiles whose elevations at the berm crest seldom exceed three meters. The storm system caused extensive erosion on these low lying profiles, resulting in thin deposits of sand occurring landward of the original berm location. These deposits are akin to overwash that occurs for profiles with high dunes, and therefore, they will be referred to as overwash de posits from this point on. Figure 3.21 displays model results for ten profiles from Assateague Sout h, moving from south to north with a constant spacing of 300 meters between profiles. The cal culated profiles exhibit a noticeable trend from south to north, simulating too much erosion on the southern profiles, and insufficient erosion on the northern profiles. Any unaccounted for longshore gradient in either wave height or water level could produce this type of trend. Additi onally, this figure reveals that in general, the calculated results do not accurately reproduce the el evation or location of the overwash deposits. Figure 3.22 plots GVE and RMS error measurements on Assateague South as a function of latitude. The south to north trend observed in the profiles themselves is absent from these PAGE 100 86 plots, because the net profile change signal is not present in either error statistic. Both error statistics are relatively low and constant across the study area, with a mean RMS error of 0.37 m, and a mean GVE of 0.30 m3/m. The mean RMS error is sta tistically lower than the value calculated on Assateague North at the 95% confidence level, while mean GVE is not statistically different. Additionally, because the model is highly sens itive to the empirical parameter K, it is important to confirm that the calibration value of 5.0x107 m4/N calculated with Assateague North data is appropriate for the given study area. Encouragingly, Figures 3.23 shows that calibration curves for the transport rate coefficient ar e comparable between Assateague South and Assateague North, and that both error statistics are minimized in the range of 5.0x107 to 7.5x107 m4/N. Larger values of K create large instabiliti es in the profile, with occurrences of jagged peaks and valleys, and severe erosion on the beach and berm portions of the profile. PAGE 101 87 Figure 3.21 Ten profiles selected from Assateague South and the predicted SBEACH results. The first profile (3692) is the southernmost profile in th e study area, and the last profile (3962) is the third northernmost profile. The profiles are spaced approximately 300 meters apart in the longshore direction. In each figure, the blue profile represen ts the prestorm profile, red represents the poststorm profile, and black indicates the calculated model result. PAGE 102 88 Figure 3.22 RMS error and GVE as a function of latitude for SBEACH model results on the Assateague South study site. Figure 3.23 SBEACH calibration curves for increasing values of K, the transport rate coefficient on the Assateague South study site. Each error bar re presents the 95% confiden ce interval on the mean value. Extreme Longshore Variability: Chincoteague The Chincoteague study area profiles are ch aracterized by high dunes that display immense longshore variability in profile response, ranging from beach erosion to a complete loss of the dune and the developm ent of thin overwash deposits. Figure 3.24 presents SBEACH PAGE 103 89 results for ten profiles from this site, spaced evenly in the longshore by a distance of approximately 300 meters. As th is figure shows, the model results were highly variable, with a general trend towards overpredicting erosion in cases of beach erosion and dune scarping, and underpredicting erosion for more severe cases when the dune lost elevation or overwash occurred. Figure 3.25 plots the mean error statistics as a function of latitude, with a mean RMS error of 0.77 m and mean GVE of 0.59 m3/m. Both error statistics are significantly greater than those recorded on both Assateague North and Assateague South with 95% confidence. Interestingly, SBEACH results are accurate in predicting the general cases of beach erosion and dune scarping, with no instances of the model erroneously flattening the dune. Conversely, and more problematic, is the fact that in several instances the model predicts that the dune remains standing after the storm, when in r eality it suffered severe erosion and was either lowered or overwashed. This observation has serious consequences in the engineering and emergency management aspects of this subject, where it is highly undesirable for a model to underpredict the storms true impact. However, if the goal of the modeling effort is not to predict impact regime on individual profiles but rather over longer spatial scales, the SBEACH model can be seen as an appropriate modeling tool. For example, vulnerability mapping seeks to predict the most extreme impact over a binned section of coastline, which would be overwash in the case of the Chincoteague study area. The SBEACH model does correctly predict overwash on several profiles (not pictured in Figure 3.24), and therefore would produce accurate results for vulnerability mapping purposes. PAGE 104 90 Figure 3.24 Ten profiles selected from Chincoteague and the predicted SBEACH results. The first profile (550) is the southernmost profile in the study area, and the last profile (840) is the third northernmost profile. The profiles are spaced approximately 300 meters apart in the longshore direction. In each figure, the blue profile represen ts the prestorm profile, red represents the poststorm profile, and black indicates the calculated model result. PAGE 105 91 Because the prestorm dunes were fairly uniform in shape and elevation along the study area, longshore gradients in forcing must have existed in order to produce the extreme variability observed in profile response. The buoy record of wave height and the tide gauge record of water level utilized to produce results for the length of Assateague Island cannot account for potential longshore variability in these processes. Additiona lly, many processes that act in the longshore direction are not included in model formulation, including wave refraction, circulation cells and rip currents, and currents induced by longshore pressure gradients. Consequently, the models inability to hindcast extreme local variability in prof ile response is reflected in the model error. The implications of gradients in longshore forcing are discussed in more detail in Chapter 5. Figure 3.25 presents error statistics as a functi on of K, the transport rate coefficient, for this length of coastline. Both RMS error a nd GVE are minimized between Kvalues of 5.0x107 and 7.5x107 m4/N, the same as the Assateague North ca libration. Once again, large values of K result in instabilities in the calcu lated profiles, with occurrences of jagged peaks and valleys, which are denoted by the larger error bars about the mean as K increases. This observation has been constant across all Assateague study sites, suggesting that the utility of setting K at a low value may be more closely related to avoiding prof ile instability than to finding a coefficient to predict the correct sediment transport rate. PAGE 106 92 Figure 3.25 RMS error and GVE as a function of latitude for SBEACH model results on the Chincoteague study site. Figure 3.26 SBEACH calibration curves for increasing values of K, the transport rate coefficient on the Chincoteague study site. Each error bar represen ts the 95% confidence inte rval on the mean value. PAGE 107 93 Model Results: Hatteras Island Verification Site: Hatteras North Verification tests of the calibrated SBEACH model are performed with lidar data from the Hatteras North study area taken before and afte r Hurricane Isabel. This section of coastline is characterized by high dunes that are eroded beyond the dune crest, with a small portion of the dune remaining intact and preventing overwash. The model is run with the Assateague North calibration value for K, and with storm data from Hurricane Isabel, which is described in Chapter 1. Figure 3.27 presents model results for ten profiles from the st udy area, spaced evenly in the longshore direction by approximately 45 meters. This series of plots indicates that the mode l produces profile shapes that are significantly different from the observed poststorm profiles. The model produces rounded, convex dunes, while lidar data indicates that the poststorm profiles are narrow and peaked, with a concave profile leading from the dune crest to the beach. The rounded profile form predicted by SBEACH is the result of the stormspecific wate r level and wave height time series that were observed during Hurricane Isabel, and their combin ed interaction in the SBEACH model. The consequence of oscillating water levels paired w ith extreme wave heights is to rapidly move shallow sandbars offshore, allowing waves to shoa l further inshore before breaking. The excess energy dissipated on the profile lowers the dune so that runup overtops the profile and creates the calculated rounded shape. However, despite the models inability to predict the correct profile shape, it consistently predicts the lowering of the dune crest for all profiles. PAGE 108 94 Figure 3.27 Ten profiles selected from Hatteras North and the predicted SBEACH results. The first profile (17790) is situated on the southwest end of th e study site, and the last profile (17817) is positioned at the northeast end of the study ar ea. The profiles are spaced approximately 45 meters apart in the longshore direction. In each figure, the blue profile represents the prestorm profile, red represents the poststorm profile, and black indicates the calculated model result. PAGE 109 95 Figure 3.28 presents RMS error and gross volume error as a function of longshore distance. A slight trend suggests that model error decreases moving to the northeast away from the location of the breach. Quite obviously, extr eme forcing was present at the breach, and the effect of this forcing most likely reaches hundreds of meters in either direction. These effects are unaccounted for in the model runs, and may explain the trend in error moving away from the breach. Mean RMS error for the 30 profiles is 0.54 m, and GVE is 0.41 m3/m, both of which are significantly greater than the Assateague No rth value at the 95% confidence level. Model error is plotted as a function of K, the transport rate coefficient, in Figure 3.29. Once again, model results indicate that for higher values of K, the model become unstable. Both error statistics are minimized in the range of 5.0x107 to 1.0x106 m4/N, which encompasses the previous calibration value of 5.0x107 m4/N. This test confirms that the calibration value is transferable between locations and storms, and a llows the calibration value of K to be utilized when testing the extreme impact that is observed in the Hatteras Breach site. Figure 3.28 RMS error and GVE as a function of longshore distance for SBEACH model results on the Hatteras North study site. PAGE 110 96 Figure 3.29 SBEACH calibration curves for increasing valu es of K, the transport rate coefficient on the Hatteras North study site. Each error bar repr esents the 95% confidence interval on the mean value. Extreme Impacts: Hatteras Breach Although the occurrence of a breach is rare, th e ability to predict this type of event is highly desirable from a coastal planning perspectiv e. Figure 3.30 presents model results for ten profiles evenly spaced along this section of coas tline by a distance of approximately 45 meters. The first profile presented in this figure is situ ated immediately to the southwest of the breach, and the last profile is situated to the northeast of the breach. Inspection of these results shows that the model calculated a consistent response am ong profiles, with the dune fully flattened and a rounded volume of sand approximately two meters high remaining in its place. Qualitatively, it is obvious that the model does not have the capab ility to reproduce breaching scenarios. Because the true profile response (channel formation) is not captured in the lidar profiles, it is impractical to calculate error statistics fo r this section of coastline. Transport mechanisms not included in the m odel formulation may be responsible for the lack of skill in predicting breaches. First, the breach occurred at the narrowest section of the island, indicating that transport due to longs hore funneling of water may have initiated an PAGE 111 97 erosional hotspot and inundation of the island by storm surge and waves. Once inundation occurs and water flows freely over the island, the pressure gradient formed between the ocean and the estuary allows currents to scour the island. In this situation, transport by currents becomes the most important mechanism for moving sand in to the estuary. The assumption of longshore homogeneity in hydrodynamics does not allow for the funneling of water to the narrowest sections of the island, nor does the model incl ude any calculations of current velocities or transport by currents. Therefore, it is possible to identify at least two transport mechanisms that may be incorporated into future versions of the model in order to more accurately hindcast breaches. An indepth discussion of breaching processes and model implications is presented in Chapter 5. PAGE 112 98 Figure 3.30 Ten profiles selected from Hatteras Breach and the predicted SBEACH results. The first profile (17736) is situated on the southwest edge of the breach, and the last profile (17763) is positioned at the northeast edge of the breach. The profiles are spaced approximately 45 meters apart in the longshore direction. In each figure, the blue profile represen ts the prestorm profile, red represents the poststorm profile, and black indicates the calculated model result. PAGE 113 99 Chapter Four EDUNE Results Introduction The three main objectives of this chapter ar e to explore several simplifying assumptions inherent to the EDUNE model, to test the sensitiv ity of model results to a variety of parameters, and to quantify and discuss model accuracy. Th e first section explores three processes that are considerably simplified by model assumptions, including runup, tides, and the breaking wave criterion. First among these process is runup, which the original model treats as a constant input value that is to be estimated based on previ ous knowledge of the fi eld site. However, measurements of runup during storms are rare, a nd measurements from one storm are not directly applicable to another storm because of their de pendence on wave height characteristics and the slope of the beach. Multiple field studies have concentrated on developi ng equations describing runup as a function of waves and beach slope (Hunt, 1959; Guza and Thornton, 1982; Holman and Sallenger, 1985; Holman, 1986), and therefore, several timedependent runup equations are tested as an alternative to a constant value. The second process explored in this section is the treatment of storm surge as a proxy for observed water levels. While this substitution allows the m odel to avoid onshore transport produced in shallow water, it also assumes that storm surge alone can account for erosion due to elevated water levels. Erosion of the profile should be implicitly linked to timevarying water levels produced by tidal cycles, and therefore it is important to quantify differences in model outcome that result from this substitution. Theref ore, the model is tested with both time series, PAGE 114 100 and error statistics are examined in order to dete rmine if storm surge is indeed an appropriate proxy. The third process explored in this section is the simplified breaking criterion Hb/hb=0.78, which defines the seaward boundary of the surf zone. Breaking wave heights are input from the SWAN model, and are compared with water depths until the specified ratio of 0.78 is met or exceeded. The input wave height st atistic for this process is the rms breaking wave height, as recommended by Kriebel (1995). Howeve r, several field studies have established that rms breaking wave heights correspond to much lower values of than 0.78 (Thornton and Guza, 1983; Sallenger and Holman, 1985) Therefore, values of b resulting from these field studies are tested as alternatives. The second section of this chapter explores th e sensitivity of model results to various several empirical parameters, and develops calibration values for K, the transport rate coefficient, and etanb, the equilibrium beach slope. In the third section, model results are found to be sensitive to each of the four hydrodynamic variab les, although the magnitude of the sensitivity varies widely. EDUNE is most sensitive to cha nges in the storm surge elevation time series, a result that is expected based on the ability of st orm surge to shift the surf zone landward. The final two sections test the accuracy of the model on the five study sites, using calibration values developed on Assateague North. Calibration curves for K and etanb are calculated at each site, to determine if the values remain constant between study sites. Simplifying Assumptions Runup Equations Maximum runup elevations generated by stor ms can exceed several meters, and depend mainly on wave heights and beach slope. The la ndward extent of wave runup defines the edge of the swash zone, and hence the boundary for the application of swash zone sediment transport PAGE 115 101 processes. The original EDUNE model is formul ated to operate with a constant runup value, which is not a realistic representation of a process that is highly dependent on wave characteristics and beach slope, both of which change constantly over the duration of the storm. The three primary components of runup include setup, which is the change in mean water level due to momentum flux gradients associated with wave breaking, fluctuations about mean setup due to incident waves (incident swash), and fluctuations due to infagravity waves (infagravity swash). The resulting runup eleva tions can be calculated from equations based on field observations. Holman (1986) determined that the maximum 2% of all runup elevations exceed the value predicted by the formula: ) 2 0 83 0 (% 2 H R (4.1) where is the deepwater Iribarren num ber, which is defined as: 2 / 1) / ( tan L H (4.2) where tan is the beach slope. The 2% exceedence va lue of runup will be tested as a possible representation of runup in the EDUNE model. Because R2% is an extreme value, it is also prudent to test two moderate represen tations of runup as well. Assuming a Rayleigh distribution of runup elevations, these R2% values can be transformed to rms runup and significant runup with the formulas: 9779 1 /% 2R Rrms and (4.3) rms sR R 4142 1 (4.4) These three representations of runup, R2%, Rs, and Rrms are tested with the 30 Assateague North profiles and EDUNE calibration values from publishe d literature, including a Kvalue of 8.7x106 m4/N. PAGE 116 102 In order to implement these equations in the m odel, the slope of the beach is calculated at each time step, at the location of mean high wate r +/0.5 vertical meters. Additionally, wave heights and periods are taken from buoy measuremen ts, and wavelengths are calculated with the deepwater linear wave equations. Figure 4.1 di splays the three time series of runup specific to Profile 4362 on Assateague North, and the corresponding model results. The R2% record produces excessive erosion on the dune, as does the Rs time series. The model results corresponding to Rrms provide the best fit for all 30 Assateague North profiles, with a mean GVC value of 0.48 m3/m, and mean RMS error of 0.63 m. Both error statistics are statistically smaller than those calculated with R2% and Rs representations of runup at the 95% confidence level. Based on these test results, Rrms is chosen as the preferred repr esentation of runup elevations for all EDUNE model runs. Figure 4.1 Three representations of time varying runup elevations specific to Profile 4362 from Assateague North, and corresponding EDUNE model results. Blue profiles represent the prestorm profile, red profiles indicate the poststorm profile, and black represents the model results. PAGE 117 103 Tides Water levels affect energy dissipation per unit volume in the surf zone, via Equations 1.42: 2 / 5 2 2 / 38 1 h g F (4.5) Kriebel (1995) notes that when water levels are extremely low, actual energy dissipation can fall below equilibrium energy dissipation, generati ng onshore transport over the shallow portions of the profile: ) (eq sD D K Q (4.6) This process is a byproduct of the numerical schem e rather than theory or observation. In order to avoid extremely low water levels due to the fluctuations of tides, and therefore minimize the occurrences of onshore transport, Kriebel recomme nds using storm surge elevations as a proxy for observed water levels. As me ntioned previously, water elevati on is important in determining the crossshore location of the surf zone, a nd therefore this substitution will impede the movement of the surf zone back and forth across th e profile. Therefore, the model is also tested with the observed water level time series (tides pl us storm surge), in order to determine if significant differences are observed in model accuracy. Figure 4.2 displays the two water level time series, and corresponding model results for Profile 4632 on Assateague North. Little differen ce is observed in the results produced by the two methods, and the mean error statistics for all 30 profiles are not significantly different from one another at the 95% confidence level. This observation suggests that storm surge is an appropriate proxy for observed water level. The l ack of change in model results may be due to a high dependence on water levels at the height of the storm, which both time series represent accurately. Additionally, when results are gene rated with the observed water level time series, the models interpolation scheme shortens the time increment significantly due to the rapidly PAGE 118 104 fluctuating water levels, which increases comput ation time by 400%. Based on these results, all EDUNE runs are performed with storm surge r ecords rather than observed water level time series. Figure 4.2 Two representations of time varying water levels, and corresponding EDUNE model results for Profile 4362 from Assateague North. Blue profiles represent the prestorm profile, red profiles indicate the poststorm profile, and black represents the model results. Breaking Wave Criterion The width of the surf zone is primarily dete rmined by the breaking wave criterion, which defines the depth at breaking through the wave hei ght to water depth ratio. The seaward edge of the surf zone is defined by the criterion for spilling breaking waves found in Equation 1.32: 78 0 b b bh H. (4.7) PAGE 119 105 The value of 0.78 is also used in the definiti ons of equilibrium and actual energy dissipation in Equations 1.36 and 1.42, in order to eliminat e the dependence of the model on the crossshore wave height profile. Initial EDUNE tests showed that the combination of b=0.78 and rms breaking wave height input from the SWAN model allowed small waves to break landward of the swash zone boundary. The model compensated for this instability by setting the landward and seaward boundaries of the surf zone at the same location, eliminating the presence of the surf zone altogether. Figure 4.3 plots three prof iles and corresponding EDUNE model results, along with pdfs depicting the width of the surf zone over the duration of the storm. These pdfs show that the surf zone went undefined for a majority of time steps. Because the surf zone did not exist in th e presence of small waves, the entire profile seaward of the swash zone was defined as a pa rt of the offshore region. This region was governed by the critical slope tanoff, and evolved in a linear fashion towards this empirical slope value. These linear slopes can be seen in the pr edicted profiles in Figure 4.3, which depart significantly from the observed concave shape of the poststorm profile. Additionally, model results were only moderately sensitive to changing values of K, the transport rate coefficient, which governs sediment transport rates in the surf zone. Figure 4.4 shows that calculated mean GVC increased by only 68% over one order of ma gnitude of Kvalues, which is small compared to the SBEACH sensitivity value of 369%. Both Thornton and Guza (1983) and Sallenger and Holman (1985) found that the value of rms ( based on rms breaking waves) is much less th an the 0.78 value used by EDUNE. Thornton and Guza found a value of rms=0.42 based on field data from Torrey Pines Beach, CA, and Sallenger and Holman found rms=0.32 based on data from Duck, NC. The model was tested with b values of 0.32 and 0.42, and as predicted these smaller values allowed the rms breaking wave heights to break further offshore, creating a wide surf zone over which the transport rate equations were employed. Figure 4.5 displays model results for three profiles from Assateague PAGE 120 106 North and corresponding relative frequency histograms of surf zone width when b is set at 0.32. This figure shows that while the surf zone is of ten narrow in the presence of small waves, the effective width is never zero. The beach profiles now display a concave shape similar to that of the observed profiles. Figure 4.3 Vertical RMS error is plotted as a function of latitude on the Assateague North study site when b=0.78. Also plotted are three EDUNE model results, and corresponding relative frequency histograms of surf zone width. PAGE 121 107 Figure 4.4 Sensitivity of the EDUNE model to K, the transport rate coefficient when b=0.78. Sensitivity is evaluated through a mean calculation of predicted gross volume change (m3/m) for the 30 Assateague North profiles. E ach error bar represents the 95% co nfidence interval on the mean GVC value. Figure 4.5 Vertical RMS error is plotted as a function of latitude on the Assateague North study site when b=0.32. Also plotted are three EDUNE model results, and corresponding relative frequency histograms of surf zone width. PAGE 122 108 With b=0.32, sensitivity to K increases to an e xpected level, with mean GVC changing by 207% over one order of magnitude of Kvalues as shown in Figure 4.6. It is important to note that by decreasing the value of b, the value of K must increase in order to produce the correct volume of sediment transport in the surf z one. This implies that the calibration value of K has a dependence on the value of b. The model was calibrated to a bestfit Kvalue with both b=0.32 (K=2.0x105 m4/N) and b=0.42 (K=1.5x105 m4/N). Error statistics for model runs with b=0.32 were significantly smalle r than errors produced with b=0.42 at the 95% confidence level. For this reason, b is set at a value of 0.32 for all model runs. Figure 4.6 Sensitivity of the EDUNE model to K, the transport rate coefficient when b=0.32. Sensitivity is evaluated through a mean calculation of predicted gross volume change (m3/m) for the 30 Assateague North profiles. Each error bar represen ts the 95% confidence interval on the mean value. Model Sensitivity and Calibration: Empirical Parameters The EDUNE model employs five empirical parameters, among which only the transport rate coefficient appears in the models governing equations. The remaining empirical parameters each describe the critical slope assigned to one of f our distinct regions of the profile, including the dune face, swash zone, surf zone, and the offshore region. Because the location of each region fluctuates with breaking wave height and wa ter level, each grid cell may be subject to two or three different critical angles over the duration of the storm. This confounds the true effect of PAGE 123 109 each parameter on model results, and limits interpre tation of the sensitivity of the model to each slope parameter. The following subsections expl ore model sensitivity to K, as well as the four critical slope values. K, the Transport Rate Coefficient The function of K in the EDUNE model is th e same as in SBEACH, directly influencing the magnitude of the sediment transport rate in the surf zone. Sediment transport rates should increase linearly as the value of K increases, as d escribed in Equation 4.6. Therefore, eroded sediment volumes should increase as the value of K increases. Moore (1982) calibrated the EDUNE model and found a bestfit Kvalue of 2.2x106 m4/N. Later, Kriebel (1986) performed a second calibration with laboratory data, and revised the best estimate of K to 8.7x106 m4/N. However, both of these calibration values were determined with a b value of 0.78. Using the new b value of 0.32, neither calibration value produces sufficient erosion in the surf zone to minimize model error. Values of K used for sens itivity testing are therefore larger than the previously determined values, ranging over an order of magnitude from 5.0x106 to 5.0x105 m/N. Figure 4.7 plots the results of sensitivity test s for a single profile from Assateague North. As hypothesized, erosion on the dune increases as the value of K increases. Differences in erosion are seen mostly on the dune, while the sh ape and elevation of the beach exhibit little change with increasing values of K. Figure 4.8( a) shows that mean calculated GVC increases by 207% over one order of magnitude of Kvalues, which is statistically significant at the 95% confidence level. This figure also indicates that among the five empirical parameters, the model is most sensitive to K. Calibration curves are pr esented in Figure 4.6, indicating that the bestfit value of K falls in the range of 2.0x105 to 3.0x105 m4/N. A value of K=2.0x105 m4/N is chosen for all subsequent model runs, minimizing mean RMS error at 0.51 m, and mean GVE at 0.38 m3/m. As discussed in the previous section, this calibration value of K is specific to b=0.32. PAGE 124 110 Figure 4.7 Profile 4362 from Assateague North is used as an example of model response to changing values of K (m4/N). In each figure, the blue profile represents the prestorm profile, red represents the poststorm profile, and black indicates the calculated model result. PAGE 125 111 Figure 4.8 EDUNE sensitivity to (a) K, the transport rate coefficient, (b) etanb, the equilibrium beach slope, (c) etand, the equilibrium dune slope, (d) tanrep, the active prof ile critical angle, and (e) tanoff, the offshore critical angle. Sensitivity is eval uated through a mean calculation of predicted gross volume change (m3/m) for the 30 Assateague North profiles. Each error bar represents the 95% confidence interval on the mean value. PAGE 126 112 Figure 4.9 EDUNE calibration curves for values of K, the transport rate coeffi cient on the Assateague North study site. Each error bar represents the 95% confidence interv al on the mean value. etanb, the Equilibrium Beach Slope The equilibrium beach slope, etanb, perfor ms two roles within the models numerical scheme. First and foremost, the landward boundary of the surf zone is defined at the location where the concave underwater profile is tangent to the slope value defined by etanb. As the value of etanb increases, the boundary moves landward, increasing the width of the surf zone. This landward shift of the surf zone boundary shoul d increase erosion on the subaerial profile, and based on this particular role of etanb in the numerical scheme, GVC should increase with increasing values of etanb. Secondly, the potentia l erosion prism method illustrated in Figure 1.2 generates sediment transport rates that evolve the swash zone towards the slope defined by etanb. Because the volume of the erosional pris m does not change as a function of the target slope, this role of etanb should have little e ffect on total eroded volume. Therefore, eroded volumes should increase as etanb increases, due to the widening of the surf zone. For sensitivity testing, the value of etanb is varied from 1.5 to 6 encompassing slope values for a variety of beach types. Figure 4.10 presents the results of sensitivity tests for an individual profile from Assateague North. Contrary to the previously stated hypothesis, erosion on the dune and beach decreases significantly as va lues of etanb increase. This observation is PAGE 127 113 Figure 4.10 Profile 4362 from Assateague North is used as an example of model response to changing values of etanb (). In each figure, the blue profile represents the prestorm profile, red represents the poststorm profile, and black indicates the calculated model result. PAGE 128 114 confirmed in Figure 4.8(b), which shows that m ean calculated GVC decreases for values of etanb greater than 2 The change in mean GVC between etanb values of 2 and 6 is 65%, which is statistically significant at the 95% confidence leve l. Calibration plots are given in Figure 4.11, which indicate that both error statistics are minimized and statistically equivalent at etanb values of 2 and 2.5 A bestfit value of 2.5 is chosen for all subsequent model runs. One possible explanation for the unexpected inverse relationship between etanb and GVC lies within the undercutting and avalanching mechan isms that shape the face of the dune. At the height of the storm, the swash zone impinges on the steep dune face, and the volume of sediment defined by the erosional prism is eroded from the dune base. When etanb is small, the dune is undercut at the dune base, which initiates a signi ficant redistribution of sand onto the active profile due to avalanching of the dune face. Conve rsely, if etanb is large, the undercutting of the dune will not be as dramatic and will result in less erosion. Figure 4.11 EDUNE calibration curves for values of etanb, the equilibrium beach slope on the Assateague North study site. Each error bar represen ts the 95% confidence inte rval on the mean value. etand, the Equilibrium Dune Slope The equilibrium dune slope, etand, defines the critical angle assigned to the portion of the profile lying between the dune crest and the landw ard edge of the swash zone. When the slope PAGE 129 115 between any two grid cells in this region ex ceeds etand, avalanching occurs until the residual slope matches the empirical value of etand. Sediment eroded by this process is redistributed across the swash and surf zones. Large values of etand impede the avalanching process and hinder erosion on the subaerial profile. Therefore, GVC should be inversely dependent on etand, decreasing with increasing values of the critical slope. For sensitivity testing purposes, values of etand are varied over a wide range from 25 to 70 Figure 4.8(c) shows that mean GVC is in sensitive to changing values of etand, as opposed to the hypothesized inverse dependence between th e two values. None of the means or variances are statistically different from one another at th e 95% confidence level, confirming that the model is insensitive to etand. However, like the SBEA CH model, a visual inspection of model results shows a slight variation in profile shape with cha nging values of etand. While this change is not large enough to make a significant impact on the mean GVC value, an etand value of 60 best allows the model to reproduce the poststorm dune sl ope, and therefore is used for all model runs. tanrep, the Active Profile Critical Angle The empirical parameter tanrep is the critical angle assigned to the surf and swash zones. Within both regions, if the slope between grid cells exceeds tanrep, spacing between contours is adjusted until the critical angle is established. Se diment that erodes as a result of this adjustment is redistributed to neighboring cells. Because the e volution of the swash zone is first governed by the parameter etanb, tanrep only serves as a second check on slopes in order to prevent instabilities in this region of the profile. Conver sely, tanrep is the governing value that prevents oversteepening of the surf zone profile. As w ith etand, large values of tanrep impede the redistribution of sediment in the surf zone, and th erefore reduce erosion on the subaerial profile. This implies that an inverse relationship betw een tanrep and erosion should exist, with GVC decreasing as the value of tanrep increases. PAGE 130 116 For sensitivity testing purposes, tanrep was varied from 2 to 20 Figure 4.8(d) shows that GVC is insensitive to values of tanrep ranging between 4 and 20 and are not statistically different from one another at the 95% confiden ce level. An arbitrary tanrep value of 10 is therefore chosen for all EDUNE model runs. Anot her noticeable trend in Figure 4.8(d), is that the mean GVC value calculated with a critical angle of 2 is substantially higher than for the rest of the range. The higher GVC value occurs because the value of tanrep is less than the calibrated etanb value of 2.5 At each time step, the potential erosion prism method attempts to establish a slope of 2.5 and subsequently the tanrep mechan ism redistributes sediment in order to establish a flatter slope of 2 As a result, the competing processes generate excessive erosion on the subaerial profile. Therefore, it is recommende d that the value of tanrep should always exceed the value of etanb. tanoff, the Offshore Critical Angle The offshore critical angle, tanoff, is the critical angle established for grid cells seaward of the surf zone. If any slope value in this region exceeds tanoff, spacing between contours is adjusted until the critical angle is reached, a nd eroded sand is redistributed among neighboring cells. Because this readjustment does not aff ect the subaerial profile in any way, eroded volume on the dune and beach should not be affected by differing values of tanoff. Proving the hypothesis, Figure 4.8(e) shows that model resu lts are insensitive to values of tanoff ranging from 2.5 to 8.5 The means and variances reported for each value of tanoff are not statistically different from one another at the 95% confidence level. An arbitrary tanoff value of 4.5 is used for all model runs. PAGE 131 117 Model Sensitivity: Physical and Hydrodynamic Variables The EDUNE model inherently relies on one ph ysical and two hydrodynamic variables. The physical parameter found in the models numerical formulation is D50, the median grain diameter. This value affects the shape parame ter A, which influences the equilibrium energy dissipation per unit volume and the slope of the initial offshore profile. The two hydrodynamic variables are S, the storm surge elevation, and Hb,rms the rms breaking wave height. The storm surge time series directly influences the cro ssshore location of the surf zone, while the breaking wave height time series affects surf z one width. The addition of time dependent runup equations introduced earlier in the chapter produces a model dependence on two additional hydrodynamic variables, H, the offshore significant wave height, and L, the corresponding offshore wavelength. Both of these values are utilized to calculate the elevation of runup, which in turn determines the width of the swash zone. D 50 the Median Grain Diameter The physical setting of the dune environmen t is described by a single parameter, D50. Median grain diameter plays two opposing rol es within the model formulation, both through calculations of the shape parameter A. The first role of D50 is defined in Equation 1.34: A = 924 0 237 0 log5010D 262 050 D (4.8) 30 3 264 2 log5010D 263 050D Through this equation, A increases nonlinearly as D50 increases. Subsequently, equilibrium energy dissipation defined Equation 1.36 increases: 24 52 2 / 3 2 / 3 g A Deq. (4.9) PAGE 132 118 Increased values of Deq decrease sediment transport rates in the surf zone (Equation 4.6) and limit erosion of the subaerial profile. Therefore, an increase in D50 should bring about a corresponding decrease in GVC. This relationshi p is conceptually co rrect, as research has shown that energy required for transport increases as grain size increases. The second effect of D50 on model results is due to the dependence of the initial offshore profile slope on the shape parameter. Because lidar does not penetrate water to record bathymetric data, the shape of the initial offs hore profile is calculated with Deans h=Ax2/3 equilibrium profile. As D50 increases the profile steepens, which allows waves to shoal further inshore before breaking. This effect should cause GVC to increase as D50 increases. Therefore, any observed trend in GVC will be the net result of the two opposing processes. Figure 4.12(a) shows that the mean calculated GVC value decreases by 40% as D50 increases from 0.13 to 0.93 mm. This change is statistically significant at the 95% confidence level. This observation implies that the increase in Deq with increasing D50 is the dominant process, the magnitude of which cannot be full y discerned due to the simultaneous steepening of the initial profile. Figure 4.13 presents calibration curves for D50, implying that error is not minimized at the true value of 0.33 mm. Inst ead, both error statistics are smallest at 0.13 mm, and are statistically different from the other mean error statistics at the 95% confidence level. This observation suggests that model accuracy can be improved by decreasing the values of Deq calculated in Equation 4.9, and that the function of D50 is not correctly defined within the numerical scheme. PAGE 133 119 Figure 4.12 EDUNE sensitivity to (a) D50, the median grain diameter, (b) S, the storm surge elevation, (c) Hb,rms, the rms breaking wave height, (d) H, the significant offshore wave height, and (e) Tp, the peak wave period. Sensitivity is evaluated through a mean calculation of predicted gross volume change (m3/m) for the 30 Assateague North profiles. Each error bar represents the 95% confidence interval on the mean value. PAGE 134 120 Figure 4.13 EDUNE calibration curves for values of D50, the median grain diameter on the Assateague North study site. Each error bar represen ts the 95% confidence inte rval on the mean value. Storm Surge Elevation Storm surge elevation effects erosion of the subaerial profile by altering the crossshore location of the surf zone. As water levels in crease, the location of breaking moves landward onto the beach and the dune. As the surf z one moves landward across the beach, the swash zone also moves landward, and swashinduced tr ansport impinges on the dune face. For this reason, erosion on the subaerial profile should incr ease with increasing storm surge elevation. In order to investigate the sensitivity of the model to storm surge, the original record from NOAA tide gauge 8570283 is offset vertically by fixed values ranging from 1 m to 1 m. Figure 4.14 presents plots of a single profile from Assateague North and corresponding model results for each representative storm surge time serie s. The trend in profile change is clear, in that erosion increases on the dune as storm surge el evations increase. Only mild beach erosion occurs with the smallest surge values, while th e dune is completely flattened when surge values increase. It is important to note that when the dune erodes completely, overwash deposits do not form due to the models continuity require ment that sediment is conserved between the dune crest and the seawardmost contour. PAGE 135 121 Figure 4.12(b) presents sensitivity results fo r storm surge, concluding that mean GVC increases rapidly with increasing storm surge eleva tion. The change GVC across the test range is 455%, which is statistically significant at the 95% confidence level. Therefore, it is important to obtain the most accurate storm surge time series possible when running the EDUNE model, as slight deviations from the tr ue time series can significantly alter model results. Figure 4.15 presents both error sta tistics as a function of storm surge, and encouragingly shows that both error statistics are minimized with the observed storm surge time series. PAGE 136 122 Figure 4.14 Profile 4362 from Assateague North is used as an example of model response to changing representations of storm surge elev ation. In each figure, the blue profile represents the prestorm profile, red represents the poststorm profile, and black indicates the calculated model result. PAGE 137 123 Figure 4.15 EDUNE calibration curves for varying representations of the storm surge elevation time series on the Assateague North study site. Each er ror bar represents the 95% confidence interval on the mean value. Breaking Wave Height Breaking wave height has a single function in the EDUNE model, which is to determine the depth at breaking as defined by the criterion b=0.32. Large waves break further offshore than their smaller counterparts, increasi ng the width of the surf zone. When the surf zone width increases, waveinduced transport acts over a greater length of the profile. This effect is extremely important at the height of the storm when the surf zone impinges on the beach and dune. Therefore, increases in Hb should increase GVC values. In order to test model sensitivity to Hb, the entire time series is multiplied by constant coefficients ranging from 0.5 to 1.5. Figure 4.12(c) showing that mean GVC increases by 54% over the range of Hb time series, confirming the hypothesis that larger wave heights should generate greater erosion. Sensitivity to break ing wave height is not as large as expected, because the GVC measurement only captures change on the subaerial profile as opposed to the entire profile. Figure 4.16 presents calibration curves for th e breaking wave height time series. The means and variances of the error statistics for all representations of Hb are not significantly different from the error statistics produced by the t rue time series at the 95% confidence level. PAGE 138 124 Because error does not change over this range of variation in the input time series, it is possible to conclude that the accuracy of the breaking wave height time series is not as critical as the accuracy of the storm surge time series. This is an important observation, as breaking wave heights must be predicted by an external wave model. Figure 4.16 EDUNE calibration curves for varying representations of the rms breaking wave height time series on the Assateague North study site. Ea ch error bar represents th e 95% confidence interval on the mean value. Significant Offshore Wave Height Although EDUNEs original numerical scheme does not include significant offshore wave height, the runup calculations introduced in Equations 4.1 through 4.3 are all dependent on H. As values of H increase, runup elevations increase, which expands the width of the swash zone and increases erosion across the subaeria l profile. Therefore, a direct relationship between H and eroded volume is hypothesized, with mean GVC increasing as H increases. In order to explore the sensitivity of model results to this variable, the H time series is multiplied by constant coefficients ranging from 0.5 to 1.5. Figure 4.12(d) displays sensitivity results for the various H time series, and shows that mean calculated GVC increases slightly (35%) over the range of modified time series. This observation confirms the hypothesis that increased offshore wave heights enhance subaeria l erosion by increasing the width of the swash PAGE 139 125 zone. However, the relatively small sensitivit y suggests that small inaccuracies in the H time series will not significantly alter model results. Wave Period Similarly, wave period (Tp) enters the models numerical scheme through calculations of runup elevations in Equations 4.1 through 4.3. Increasing values of wave period decrease wave steepness, therefore increasing the value of the Iribarren number in Equation 4.2. This increase in Iribarren number subsequently incr eases runup elevations, widening the swash zone and increasing the total erosion on the subaerial profile. Therefore, a direct relationship between period and erosion is expected, with GVC increasing as Tp increases. For sensitivity tests, the peak period record was modified by shifting the entire time series by several seconds, ranging from 4 to 6 s. The modifications are asymmetric in order to avoid negative wave periods, but also to test the sensitivity of the model to a 10 second range. Sensitivity results are presented in Figure 4.12( e), which shows that as hypothesized, mean calculated GVC values increase by 24% over the range of modified Tp time series, which is statistically significant at the 95% confidence level. As with H, small inaccuracies (<4s) in the peak period time series should not significantly alter model results. Model Results: Assateague Island Calibration Site: Assateague North The Assateague North study area is comprised of 30 profiles with tall dunes averaging 6.5 meters in elevation at the dune crest. Figure 4.17 displays ten profiles from the study area with accompanying EDUNE model results. Th ese results were calculated using calibration values of empirical parameters from the prev ious section, including a Kvalue of 2.0x105 m4/N. These profiles suggest that the model correctly delineates between cases of beach erosion and dune scarping. With respect to profiles that erode only on the beach, the EDUNE model PAGE 140 126 Figure 4.17 Ten profiles selected from Assateague Nort h and the predicted EDUN E results. The first profile (4122) is the southernmost profile in the study area, and the last profile (4392) is the third northernmost profile. The profiles are spaced approximately 300 meters apart in the longshore direction. In each figure, the blue profile represents the prestorm profile, red represents the poststorm profile, and black indicates the calculated model result. PAGE 141 127 correctly predicts this response but does not rep licate the true concave nature of the observed profiles. In instances of dune scarping, the dune b ase occurs consistently between elevations of 2.8 and 2.9 m, which correspond favorably with the elevations of maximum storm surge plus runup. This observation indicates that the shape of the poststorm profile predicted by EDUNE is highly dependent on the processes that act at th e height of the storm. This dependence on conditions at the height of the storm is con ceptually correct, seeing that the measured poststorm profiles similarly maintain a dune base in the same elevation range. The process that allows EDUNE to delineate between beach erosion and dune scarping appears to be the conditioning of the beach prior to the arrival of maximum storm surge. If the beach erodes significantly prior to the height of the storm, runup will impinge on the dune and the dune face will erode through swashinduced tr ansport and avalanching. If a large volume of sand remains on the beach at the peak of the storm, the beach protects the dune from the brunt of swashinduced erosion, and the dune face will not erode. Therefore, it is appropriate to conclude that the prestorm volume of the beach is an important variable in the dune erosion process. Figure 4.18 plots error statistics for the 30 Assateague North profiles as a function of latitude. Errors are particularly low to the sout h, corresponding to profiles that experienced only beach erosion. In the north, errors are larger due to the tendency of the model to either overpredict erosion on the dune face or underpredict erosion on the beach. Mean RMS error is 0.51 m, which is significantly higher than SBEACH model results at the 95% confidence level. Similarly, the mean GVE of 0.38 m3/m is also significantly higher than for SBEACH model results. PAGE 142 128 Figure 4.18 RMS error and GVE as a function of la titude for EDUNE model results on the Assateague North study site. Berm Profiles: Assateague South The Assateague South study area consists of 30 profiles with an average maximum elevation of 2.8 m at the berm crest. Beca use these profiles lack a dune, sediment transported by waves accumulates landward of the original berm crest forming overwash deposits. Figure 4.19 plots ten profiles from Assateague South and corresponding EDUNE model results. These plots show that for a major ity of cases, EDUNE does not predict sufficient erosion on the beach, nor does it reproduce the small overwash deposits. As noted previously, overwash deposits do not form due to the models continuity requirement that sediment is conserved between the dune crest and the seawardmost contour. Additionally, Figure 4.20 presents error st atistics for the 30 Assateague South profiles as a function of latitude. A portion of the error is due to the inability of the model to predict the overwash deposits, while most of the remaining error is due to the lack of erosion on the beach. The mean RMS error statistic of 0.51 meters is significantly greater than SBEACH model results at the 95% confidence level. Additionally, the mean GVE statistic of 0.44 m3/m is also significantly greater than SBEACH results. PAGE 143 129 Figure 4.19 Ten profiles selected from Assateague Sout h and the predicted EDUN E results. The first profile (3692) is the southernmost profile in the study area, and the last profile (3962) is the third northernmost profile. The profiles are spaced approximately 300 meters apart in the longshore direction. In each figure, the blue profile represen ts the prestorm profile, red represents the poststorm profile, and black indicates the calculated model result. PAGE 144 130 Figure 4.20 RMS error and GVE as a function of la titude for EDUNE model results on the Assateague South study site. Previous analysis has shown that the EDUNE model is sensitive to both K, the transport rate coefficient, and etanb, the e quilibrium beach slope. Figure 4.21 presents calibration curves for both empirical paramete rs, in order to compare calibration values between study sites. Mean error statistics appear to decrease with increasing Kvalues, leveling off for values greater than the calibration value of K=2.0x105 m4/N. Because the mean error statistics do not reach a statistica lly significant minimum in this range of Kvalues, it is impossible to confirm the validit y of the Assateague North calibration value on this study site. Calibration curves for the equilib rium beach slope, etanb, are similar to those produced with Assateague North data. Both error statistics are minimized in the range of 2 to 3.5 encompassing the calibration value of 2.5 PAGE 145 131 Figure 4.21 EDUNE calibration curves for increasing values of K, the transport rate coefficient, and etanb, the equilibrium beach slope on the Assateague South study site. Each error bar represents the 95% confidence interval on the mean value. Extreme Longshore Variability: Chincoteague The Chincoteague study area consists of 30 profiles with an average dune crest elevation of 5.8 m. The response of the dunes varied signi ficantly across the length of the site, from mild beach erosion to overwash accompanying the complete erosion of the dune. As noted in previous chapters, because the dunes themselves were similar in prestorm shape and size, the variability in response may be due to gradients in longshore fo rcing. Ten profiles from the Chincoteague study area are plotted in Figure 4.22, along with corresponding EDUNE model results. Similar to SBEACH model results, EDUNE is unable to captu re the variability in dune response observed along this section of coastline. However, unlik e SBEACH, model results are more likely to PAGE 146 132 Figure 4.22 Ten profiles selected from Chincoteague and the predicted EDUNE results. The first profile (550) is the southernmost profile in the study area, and the last profile (820) is the third northernmost profile. The profiles are spaced approximately 300 meters apart in the longshore direction. In each figure, the blue profile represents the prestorm profile, red represents the poststorm profile, and black indicates the calculated model result. PAGE 147 133 predict extreme impacts, correctly predicting th e complete erosion of the dune on 5 of 6 overwashed profiles. Figure 4.23 plots error statistics for the 30 Chincoteague profiles as a function of latitude. Error varies considerably along the study area, with a mean RMS error of 0.66 m, and a mean GVE of 0.50 m3/m. Although both of these means are sm aller than reported for SBEACH model runs, the large variances associated with th e means render SBEACH and EDUNE errors statistically equivalent at the 95% confidence le vel. Figure 4.24 plots calibration curves for K and etanb on the Chincoteague study site. The large variance associated with both error statistics makes it impossible distinguish the mean errors from one another at the 95% confidence level, and therefore no conclusions can be draw n about the suitability of the Assateague North calibration values to this study site. Figure 4.23 RMS error and GVE as a function of la titude for EDUNE model results on the Chincoteague study site. PAGE 148 134 Figure 4.24 EDUNE calibration curves for increasing values of K, the transport rate coefficient, and etanb, the equilibrium beach slope on the Chincoteague study site. E ach error bar represents the 95% confidence interval on the mean value. Model Results: Hatteras Island Verification Site: Hatteras North The Hatteras North study area consists of 30 profiles with an average dune crest elevation of 6.7 m. As a consequence of the conditions generated by Hurricane Isabel, the dunes along this stretch of coastline eroded vertically by several meters. Ten profiles from Hatteras North are plotted in Figure 4.25, along with correspondi ng EDUNE model results. The calculated profiles reproduce both the slope of the beach and the narro w and peaked shape of the dune well, although the placement of the beach and dune is often incorrect. The general shape of the profiles PAGE 149 135 produced by EDUNE is much different than tho se predicted by SBEACH, which exhibited wide, rounded dunes that had been eroded several meters below the dune crest. Another feature present on several of the EDUNE profiles is the accretion of the beach above the initial prestorm elevation. In so me instances the accretion is significant, causing considerable deviations between the predicted and observed profiles. The general mechanism that produces these accretionary features is ons hore sediment transport that is initiated when storm surge elevations slightly exceed the eleva tion of the beach. This situation produces an extremely shallow surf zone, and consequently energy dissipation in this region is extremely low. Therefore, onshore transport is simulated across this portion of the profile via Equation 4.6. Subsequently, the beach grows in elevation as long as accretion can keep pace with water levels. The onshore transport mechanism shuts down only when surge rises significan tly at the peak of the storm. This type of profile response has not been observed previously, because it requires precise pairing of beach morphology and water le vels. As Figure 4.25 shows, these accretional features appear on only a few profiles along this section of coastline, indicating that they occur infrequently even when model conditions are fa vorable. These observations show that this affect can be significant, producing wide, elevated beaches that protect the dune from erosion. Figure 4.26 plots error statistics for the 30 profiles as a function of latitude. RMS error and GVE are both relatively low along most of the coast, with spikes occurring as a result of the accreted beach profiles. The mean RMS error for this section of coastline is 0.50 m, which is not significantly different than the mean RMS e rror for SBEACH model results at the 95% confidence level. Likewise, the mean GVE value of 0.36 m3/m is not statistically different than for SBEACH. Calibration curves for K and etanb are presented in Figure 4.27, all showing little structure and indistinguishable minimums. Ther efore, no conclusions can be drawn about the suitability of the Assateague North calib ration values to this study site. PAGE 150 136 Figure 4.25 Ten profiles selected from Hatteras North and the predicted EDUNE results. The first profile (17790) is on the southwestern edge of the study area, and the last profile (17819) is the third profile from the northeastern e dge. The profiles are spaced approxi mately 30 meters apart in the longshore direction. In each figure, the blue profile represents the prestorm profile, red represents the poststorm profile, and black indicates the calculated model result. PAGE 151 137 Figure 4.26 RMS error and GVE as a function of latitude for EDUNE model results on the Hatteras North study site. Figure 4.27 EDUNE calibration curves for increasing values of K, the transport rate coefficient, and etanb, the equilibrium beach slope on the Hatteras North study site. Each error bar represents the 95% confidence interval on the mean value. PAGE 152 138 Extreme Impacts: Hatteras Breach The Hatteras Breach study area consists of 30 profiles averaging 4.5 meters in elevation at the dune crest. Conditions during Hurricane Isabel caused the island to breach in this location, scouring a channel several meters below sea level. Figure 4.28 plots ten profiles from the Hatteras Breach site, as well as co rresponding EDUNE results. Most notably, EDUNE does not correctly predict the breach or ev en the flattening of the dune. On several profiles, the beach accretes significantly on bo th the beach and dune face. This process is described in detail in the previous section, a nd is due to a flawed description of onshore transport in the models governing equations. On ce accretion is initiated, the profile continues to build as water levels rise slowly, and ceases only when water levels rise significantly above the beach. These observations show that onshor e transport allowed by EDUNE can severely mask the true impact caused by an extreme storm. PAGE 153 139 Figure 4.28 Ten profiles selected from Hatteras Breach and the predicted EDUNE results. The first profile (17736) is on the southwestern edge of the study area, and the last profile (17763) is the third profile from the northeastern e dge. The profiles are spaced approxi mately 45 meters apart in the longshore direction. In each figure, the blue profile represents the prestorm profile, red represents the poststorm profile, and black indicates the calculated model result. PAGE 154 140 Chapter Five Discussion Introduction The objective of this chapter is to provi de a comprehensive comparison of the two models, as well as recommendations for future dune erosion studies involving a macroscale modeling approach. The first section of this chap ter compares the two models in terms of overall accuracy in hindcasting erosional events, sens itivity to empirical parameters, and observed inconsistencies in model performance. The second section comments on both models lack of predictive skill on the Chincoteague and Ha tteras Breach study sites, and discusses the implications of gradients in longshore forcing which most likely contribute to the lack of model skill on Chincoteague. This section also provides a possible explanation for the inability of the models to predict the Hatteras Island breach, a nd includes a discussion of hypothesized transport mechanisms that may lead to breaching ev ents. Finally, the third section presents recommendations for future studies of macrosca le dune modeling, including priorities for modifications to the models sediment transport processes. Model Comparison One desired outcome of this study is to de termine which of the two models is most accurate in its prediction of dune erosion, a dete rmination that will provide direction for future macroscale modeling approaches. The quantitative accuracy of the two models can be measured with mean RMS error and mean GVE statistics from the four nonbreaching study sites. In PAGE 155 141 Figure 5.1, mean error statistics are plotted as a function of study site with each error bar representing the 95% confidence interval about the m ean value. A fifth entry to these graphs is labeled as total, and specifies mean error for all 12 0 profiles included in the four study sites. As these graphs show, mean error produced by SBEA CH is significantly lower than EDUNE at the 95% confidence level on both the Assateague No rth and Assateague South study sites. In comparison, the mean error statistics on Chinco teague and Hatteras North are statistically equivalent between the two models. Total error for the four study sites is statistically equivalent at the 95% confidence level. Tables 5.1 and 5.2 list the mean and standard deviations of RMS error and GVE for both models. The statistical equivalence of total error s uggests that both models are equally suited to the task of predicting dune response to extrem e storms. However, results presented in the previous two chapters suggest that a quantitative comparison of the two models is not sufficient evidence to make this conclusion. In order to determine which model is best suited for predictive purposes, it is necessary to consider additional f actors, including the consistency of calibrated empirical parameters between study sites and r ecognized inconsistencies in model performance. The following paragraphs will address both of th ese aspects in more detail, and will make the case that while the quantitative accuracy of the two m odels is statistically equivalent, SBEACH is the preferred model for predictive applications. PAGE 156 142 Figure 5.1 Mean RMS error and mean GVE are plotted as a function of study site for both the SBEACH and EDUNE models. The final entry labeled total, depicts model error combined for all four study sites. Each error bar represents the 95% confidence interval on the mean value. SBEACH RMS RMSs GVE GVEs Assateague North 0.39 m 0.19 m 0.27 m3/m 0.10 m3/m Assateague South 0.37 m 0.09 m 0.30 m3/m 0.08 m3/m Chincoteague 0.77 m 0.37 m 0.59 m3/m 0.29 m3/m Hatteras North 0.54 m 0.13 m 0.41 m3/m 0.10 m3/m Total 0.52 m 0.27 m 0.39 m3/m 0.21 m3/m Table 5.1 Compiled error statistics for SBEACH model r uns for the four nonbreaching study areas. Total error is based on the 120 profiles included in these four study sites. EDUNE RMS RMSs GVE GVEs Assateague North 0.51 m 0.18 m 0.38 m3/m 0.12 m3/m Assateague South 0.51 m 0.20 m 0.44 m3/m 0.17 m3/m Chincoteague 0.66 m 0.38 m 0.50 m3/m 0.30 m3/m Hatteras North 0.50 m 0.20 m 0.36 m3/m 0.16 m3/m Total 0.55 m 0.26 m 0.42 m3/m 0.21 m3/m Table 5.2 Compiled error statistics for EDUNE model ru ns for the four nonbreaching study areas. Total error is based on the 120 profiles included in these four study sites. PAGE 157 143 A primary consideration for determining whic h model is best suited for predicting dune erosion is overall confidence in the calibration values of empirical parameters. Because empirical parameters enter the models to account for unk nown processes, it is important that they are constrained to account for the same processes at ea ch study site. It is necessary for empirical parameters to be robust between coastal locations in order that sitespecific calibrations are not required to minimize model error. The followi ng paragraphs will review and discuss the ability of the calibrated empirical parameters to minimi ze error across the four nonbreaching study sites. This discussion suggests that empirical parameters are well constrained by the SBEACH model, while it is difficult to have confidence in the ca libration values of empirical parameters inherent to the EDUNE model. SBEACH exhibits sensitivity to the empirical coe fficient K, the transport rate coefficient. Model results are highly sensitive to K, displa ying a 369% increase in mean GVC over one order of magnitude of Kvalues. As shown in Figure 5.2, the calibration curves for each of the four nonbreaching study sites is similar in shape to th e original curve produced on Assateague North, and error is consistently minimized in a na rrow range encompassing the calibration value of K=5.0x107 m4/N. This observation suggests that K accounts for the same unknown processes at each study site, which is a desired characteristic of all empirical parameters. Therefore, despite the high level of sensitivity to the Kvalue, the calibration is well constrained across the four nonbreaching study sites. Therefore, sitespecific calibration is not a requirement for the SBEACH model. PAGE 158 144 Figure 5.2 Mean RMS error and mean GVE are plotted fo r SBEACH results as a function of Kvalue for four study sites. Each error bar represents the 95% confidence interv al on the mean value. The EDUNE model also exhibits sensitivity to the empirical coefficient K. When values of K are varied over one order of magnitude on the Assateague North study site, mean GVC increases by 207%. While EDUNE is less sensitiv e to K than SBEACH, the calibration is not well constrained between study sites, as seen in Figure 5.3. Each calibration curve takes on a different shape, with the range of error mi nimizing Kvalues increasing considerably on the Assateague South, Chincoteague, and Hatteras Nort h study areas. This figure suggests that the empirical Kvalue absorbs different processes at each study site, and that the role of K is not well constrained within the models governing equations. This observation indicates that the bestfit value of K is dependent on study site, whic h implies that the model should be calibrated separately at each distinct location along the coast The necessity of sitespecific calibration with PAGE 159 145 respect to K prevents the use of EDUNE for pr edictive applications. The sensitivity of the EDUNE model to an empirical parameter that is not well constrained argues against its use as a tool for widespread application for the prediction of dune erosion. Figure 5.3 Mean RMS error and mean GVE are plotted fo r EDUNE results as a function of Kvalue for four study sites. Each error bar represents the 95% confidence interv al on the mean value. An additional fundamental concern about th e ability of EDUNE to predict erosional events is the inconsistency in model performan ce that results from the incorrect description of onshore transport in response to shallow water dept hs. The result of the rapid onshore transport mechanism is the formation of accretionary feat ures on the beach, which are observed on both the Hatteras North and Hatteras Breach study sites. The negative consequence of this causeandeffect relationship is most clearly seen in Figure 4.28, where instead of predicting the flattening of the dune and breaching, the model predicts th at the dune grows over the course of the storm. PAGE 160 146 The occurrence of these anomalous accretionary features weakens confidence in the model to correctly predict dune erosion. Therefore, despite the good performance of the model on the other study sites, EDUNE is rendered ineffective due this inconsistency in model performance. Based on total error statistics compiled for th e four study sites, the two models are statistically equivalent in their ability to predic t dune erosion. However, several concerns about inconsistencies in EDUNE model results create a preference for the use of SBEACH for the prediction of dune erosion. Inconsistencies in calibration curves for the empirical parameter K necessitate sitespecific calibration, which is not conducive to predicting erosion at new study sites. Additionally, the onshore transport m echanism that leads to anomalous accretionary features on the beach is of serious concern, as it may mask extreme impacts similar to model results on the Hatteras Breach study site. Conver sely, the SBEACH model exhibits consistency in calibration value for the empirical parameter K, and no inconsistencies were noted in model results during the testing phase of this study. Th erefore, the conclusion of this study is that SBEACH is most suitable for macroscale modeli ng of dune erosion during extreme storm events. Longshore Variability and Barrier Island Breaching One conclusion that can be drawn from this study is that model error increases significantly in the presence of extreme longshore variability in dune response and barrier island breaching. As shown in Figure 5.1, mean erro r statistics for both models are highest on the Chincoteague study site, which exhibits the gr eatest variation in dune response among the five study areas. Dune response on this section of coastline varies from mild beach erosion to complete destruction of the dune, with no observe d longshore trend (Figure 1.8). This variability in dune response does not appear to be a f unction of prestorm dune morphology, which is relatively uniform in the longshore direction. Th e dune crest elevation along this stretch of coastline varies only slightly, with a mean of 5.6 m and a standard deviation of 0.35 m. PAGE 161 147 In the absence of extreme variability in prest orm profile shape, the observed differences in dune response are most likely a function of pr ocesses that act in the longshore direction, and that may exhibit significant gradients on spatia l scales smaller than 3 kilometers. Potential processes that may cause extreme longshore variatio ns in dune response include the dissipation of wave energy over longshorevariable sandbars, wave refraction over arbitrary bathymetry, rip currents and associated circulation cells, and lo ngshore currents that flow along the beach in response to gradients in dune elevation. Because the SBEACH model excludes longshore sediment transport processes by assuming that cro ssshore gradients in transport dominate under storm conditions, the presence of any of these four processes may cause significant deviations between observed and pred icted dune response. While the pr ocesses described in the following paragraphs are not exha ustive, the magnitude of any one of these processes may be large enough to account for extreme variability in dune erosion on spatial scales on the order of several kilometers. The dissipation of energy by sandbars can significantly reduce the magnitude of wave energy available to act on the subaerial profile. Sandbars can be considerable in longshore extent, stretching along the coast for many kilometers. Br eaks in sandbars occur where rip currents have scoured the surface of the seafloor, making wave energy dissipation over the sandbar a longshore variable process. Profiles that align with an offshore sandbar will experience significantly less wave energy dissipation, as sandbars initiate the breaking of large waves well offshore. Profiles that align with breaks in the sandbar system will experience greater dissipation of energy closer to shore, leading to greater erosion and leaving the dunes more vulnerable to overtopping by waves. In addition to single sandbar systems, a r ecent study by Kannan, Lippmann, and List (2003) has shown that the existence of multiple o ffshore bars may also effect the response of the shoreline to storm events. These multiple bar sy stems were shown to correspond to areas of the coast that experienced little to no shoreline cha nge during large storm events. The existence of PAGE 162 148 multiple bars significantly decreases shoreline erosi on in comparison to beaches that are protected by a single longshore bar. If bathymetric data were available for model runs, simulating this process would not require that the model be th reedimensional, because the longshore variability in offshore sandbars would be captured across each profile line. A second mechanism that may cause longshore variability in dune response is the refraction of waves over arbitrary shallowwater bathymetry. While the east coast does not boast as many complex hardrock bathymetric features as the west coast, small shoals and stationary transverse bar systems can significantly alter the wave ray path, focusing energy on small sections of the coast, and defocusing energy in other areas. The focusing of energy on a particular location will significantly intensify er osion, increasing the vulnerability of the dune to overtopping and overwash. If specific offshore bathymetry was added to these simple dune erosion models, a threedimensional approach to wave refraction would be preferred to the simpler twodimensional SBEACH approach that assumes plane and parallel bottom contours. Additionally, differences in wave height al ong the coast create longshore gradients in setup elevation at the shoreline. These gradients in water elevation can lead to the generation of circulation cells and rip currents that extend we ll past the breaker zone. Komar (1971) observed that longshore currents created by variations in setup elevation produce erosional hotspots where two longshore currents converge at the location of the rip channel. Erosion decreases moving away from the location of the rip channel, to the point of divergence in longshore currents that occurs where two circulation cells meet. The sp acing between rip currents can range from tens to hundreds of meters and tends to increase with increasing wave height (McKenzie, 1958). Rip currents and associated longshore currents are ther efore another potential mechanism for creating longshore variability in dune response that is not explained by dune morphology and crossshore transport processes alone. PAGE 163 149 One process that is essential to the prediction of both longshore variability and barrier island breaching is the funneling of water along the beach in response to gradients in dune elevation. Once waveinduced erosion creates a break in the continuous dune, a pressure gradient is established, causing water to flow along the beach. Water flowing from both directions will converge at the opening in the dune and create an erosional hotspot. This mechanism increases erosion closest to the original break, and decr eases erosional pressure in neighboring locations. The initial break may be established where the dune is short and narrow, or where refraction concentrates wave energy on a small section of co astline. The creation of an erosional hotspot may initiate inundation of the island leading to breaching, or simply intensify longshore variability in dune response. Because the SBEA CH model does not include a threedimensional component linking the profiles in the longshore direction, this effect cannot be simulated at the present time. This discussion of erosional hotspots naturally leads into the question of barrier island breaching. Model results from the Hatteras Breach study area reveal that the SBEACH model is not capable of predicting barrier island breaching (Figure 3.29). While SBEACH readily predicts the overtopping of dunes on the Hatteras North site there is a clear divide between the models ability to predict overtopping and breaching events The dunes at the Hatteras Breach study site are short and narrow, and are protected by a ve ry narrow beach. The SBEACH model predicts that the dunes erode quickly as water levels rise, allowing water to inundate the island and flow into the bay. What is not simulated in the m odel is that differences in water elevation between the seaward and landward sides of the island estab lish a pressure gradient that produces currents moving across the island at significant speeds. Once water flows freely across the island, th e main mechanism for erosion shifts from waves to currents, which are not included in th e SBEACH governing equations. In reality, these pressureinduced currents scour sediment from the top of the island and deposit it in large PAGE 164 150 estuarine deltas. In the case of the Hatteras Is land breach, current velocities were great enough and were sustained for a period of time sufficient to scour a channel several meters deep and approximately 600 meters wide. The inability of the models to pr edict barrier island breaching is not unexpected, based on the lack of currenti nduced sediment transpor t mechanisms in the governing equations. In order to create a dis tinction between the overtopping and breaching, pressuregradient induced currents and associated tr ansport must be added to the models. These corrections would increase the complexity of th e model considerably, as the number of input variables and processes increases. Recommendations As discussed earlier in this chapter, SBEACH distinguishes itself as the model that is best suited for predicting dune erosion. For future applications of SBEACH to dune erosion studies, the main priority for model development is the inclusion of mean flows generated by pressure gradients, both in the crossshore and longshore directions. The specification of crossshore pressure gradients and associated currents will enhance the ability of SBEACH to delineate between overwash and breaching occurrences. Add itionally, a threedimensional form of the model that links profiles in the longshore will a llow for the funneling of water to breaks in the continuous dune, improving the ability of the m odel to predict longshore variable dune response and erosional hotspots. A threedimensional form of the model will also create the possibility of adding processes such as wave refraction and circul ation cells. However, the implementation of pressureinduced currents significant increases in model complexity, through requirements for additional input variables and the tracking of ma ny new processes. One potential approach to these improvements is to link SBEACH to another model such as DELFT3D that already account for pressureinduced currents and associated sediment transport. It is important to note that increasing the complexity of the model in or der to predict breaching and longshore variable dune response reduces the capacity to produce vulne rability maps for large extents of coastline. PAGE 165 151 Chapter Six Conclusions Macroscale approaches to earth systems mode ling are appealing for predictions of system behavior on large spatial scales. These models allow the description of landform features with lengths scales of meters, without requiring the specification of smallscale processes that necessitate excessive computation time. SBEACH and EDUNE are two such models that allow the prediction of coastal change over many kilome ters of shoreline using only the simple inputs of dune morphology and time series of storm charact eristics. The objective of this study was to determine the potential of each model to accura tely and consistently predict dune erosion on spatial scales of several kilometers. SBEACH is the more sophisticated of the two simple models, employing a selfcontained wave module that calculates the crossshore wave height profile. This capability allows the specification of the breaking wave location as a f unction of offshore wave height and wavelength, water depth, and beach slope. Sediment transport rates are calculated by empirically determined equations in each of four crossshore regions. The direction of sediment transport is established with an empirical equation that is governed by the offshore wave height and wavelength and the sediment fall speed. Profile change is calculated at each time step by adjusting the vertical elevation at fixed horizontal locations. This study revealed that the SBEACH mode l is significantly sensitive to the empirical parameter, K, the transport rate coe fficient. A calibration value of K=5.0x107 m4/N was established with Assateague North profiles, and remained stationary over the four nonbreaching PAGE 166 152 study sites. The model is well constrained with respect to K, and therefore can be operated without sitespecific calibration. The m odel is also sensitive to the value of D50, the median grain diameter. An increase in grain diameter decreases erosion on the subaerial profile, which is consistent with the theory that increasing ener gy is necessary to suspend and transport larger particles. Model error is minimized at the true D50 value of 0.33 mm on Assateague Island, suggesting that the function of grain size is pr operly specified within the governing equations. The model is only moderately sensitive to the offshor e wave height, which is due to the fact that waves generate profile change in the surf zone while sensitivity is as a function of subaerial profile change. The model is also sensitive to water elevations, which dictate the location of the broken wave zone. Model error is appropriately minimized with the time series of water elevations obtained from the local NOAA tide gauge. The accuracy of the model is ascertained from tests on five separate study areas, three from Assateague Island, MD, and two from Hatteras Island, NC. The model performs well with respect to prediction of impact regime, correc tly delineating between the swash and collision regimes on Assateague North, and accurately repr oducing dune overtopping on Hatteras North. On the Chincoteague study site, the model is less skillful at reproducing the exact poststorm profile, because of the extreme longshore variability in dune response. On the Hatteras Breach study site, the SBEACH model predicts the i nundation and flattening of the dune, but no additional erosion indicating the formation of a br each. With respect to quantitative measures of model accuracy, the mean RMS error for the four nonbreaching study sites is 0.52 m, and mean GVE is 0.39 m3/m. Comparatively, EDUNE does not support an internal wave module and relies on calculations of breaking wave height from an extern al source, which for the purposes of this study is the SWAN model. A simple criterion determines the location of wave breaking based on the ratio of breaking wave height to water depth. Once the location of the surf zone is established, PAGE 167 153 sediment transport rates are calculated with a th eoretical formula that relies on calculations of excess wave energy dissipation per unit volume. As a byproduct of this formula, onshore transport is simulated when water depths are extremely shallow and wave energy dissipation is very low. Profile change is calculated at each ti me step by solving for the horizontal location of a fixed vertical contour. The resulting profile is highly controlled by critical angles that are operational along different horizont al sections of the profile. EDUNE employs several simplifying assumptions that were analyzed prior to model testing. First, the model was previously reliant on a constant value of runup estimated by the user. Three representations of runup were consid ered for inclusion in the governing equations, and the rms runup equation described in Chapte r 4 was chosen based on the minimization of model error. Additionally, the breaking criterion of b=0.78 was found to be inconsistent with the rms breaking wave height statistic, allowing waves to break in the swash zone and eliminating the presence of the surf zone. Sallenger and Holmans (1985) rms value of 0.32 was found to minimize model error, by allowing waves to break offshore and defining an appropriate surf zone width. The EDUNE model is sensitive to two empi rical parameters, K, the transport rate coefficient, and etanb, the equilibrium beach slope The calibration value of K was determined to be 2.0x105 m4/N with the Assateague North profiles, but was not well constrained across the four nonbreaching study sites. Additionally, the calibration value of etanb=2.5 was not well constrained between sites. EDUNEs illdefined empirical parameters establish a preference for the SBEACH model for future predictions of dune erosion. The EDUNE model also displays sensitivity to D50, the median grain diameter. As with the SBEACH model, EDUNE predicts that s ubaerial dune erosion decreases as grain size increases. However, model error is not minimized at the true value of 0.33 mm, but continues to decrease as the value of D50 decreases. This suggests that the function of D50 is not properly PAGE 168 154 specified within the governing equations. A dditionally, subaerial model results are only moderately dependent on the breaking wave height due to the fact that increases in wave height mainly increase erosion in the surf zone. The model is highly sensitive to the storm surge elevation, which determines the location of th e surf zone across the profile. Model error is appropriately minimized for the storm surge time series obtained from the local NOAA tide gauge. With respect to the prediction of impact regime, EDUNE is capable of correctly delineating between swash and collision on the A ssateague North study site, and the occurrence of dune overtopping on the Hatteras North site. Like SBEACH, model results are less reliable on Chincoteague where dune response is extremely va riable in the longshore direction. On the Hatteras Breach site, the model does not predict th e flattening of the dune, but rather simulates the growth of the dune on many profiles. These anomalous accretionary features are a byproduct of the numerical scheme that allows onshore tr ansport when extremely shallow water depths exist. Because these features are the result of an incorrect description of system dynamics and their occurrence cannot be predicted, the EDUNE mo del is not well suited for predictions of dune erosion. With respect to quantitative measures of EDUNE accuracy, the mean RMS error for the four nonbreaching study sites is 0. 55 m, and the mean GVE is 0.42 m3/m. The mean values of RMS error and GVE on th e four nonbreaching sites are statistically equivalent between the two models at the 95% c onfidence level. This result suggests that both models are equally suited for predictions of du ne erosion. However, previously mentioned inconsistencies in the EDUNE model make it in appropriate for both vulnerability mapping and predictions of poststorm profiles. The inconsiste ncies include empirical parameters that are not well constrained, and the illdefined function of D50 within the governing equations, and the formation of accretionary features when sha llow water depths exist. Each of these PAGE 169 155 inconsistencies prevents the use of EDUNE as a predictive tool for vulnerability mapping applications. The failure of the models to accurately reproduce dune erosion on the Chincoteague study site is likely the result of longshore vari able processes that are un accounted for in the model governing equations. Several of these processes we re highlighted in the previous chapter, including the dissipation of wave energy over lo ngshore variable sandbars, refraction of waves over arbitrary bathymetry, the generation of ri p currents and circulation cells, and longshore currents that respond to gradients in dune elev ation. The failure to predict barrier island breaching can be attributed to crossisland pressure gradients and associated currents that are not accounted for by either model. The future use of the SBEACH model for vul nerability mapping is promising, due to the ability of the model to correctly delineate betw een the swash, collision, and overwash regimes. With respect to the prediction of poststorm dune shape, it is important to include crossisland pressure gradients and descriptions of currentinduc ed sediment transport in order to correctly predict breaching occurrences. Additionally, it is necessary to specify the longshore pressure gradients and associated sediment transport that results from gradients in the height of the continuous dune. Both of these corrections w ill necessitate a considerable increase in model complexity, requiring definition of numerous additional hydrodynamic variables, and the specification of multiple additional macroscale pr ocesses. 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