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Roberts, Tiffany M.
Limits of beach and dune erosion in response to wave runup from large-scale laboratory data
h [electronic resource] /
by Tiffany M. Roberts.
[Tampa, Fla] :
b University of South Florida,
Title from PDF of title page.
Document formatted into pages; contains 69 pages.
Thesis (M.S.)--University of South Florida, 2008.
Includes bibliographical references.
Text (Electronic thesis) in PDF format.
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Advisor: Ping Wang, Ph.D.
Nearshore sediment transport
Surf zone processes
t USF Electronic Theses and Dissertations.
Limts Of Beach And Dune Erosion In Re sponse To Wave Runup From Large-Scale Laboratroy Data by Tiffany M. Roberts A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Geology College of Arts and Sciences University of South Florida Major Professor: Ping Wang, Ph.D. Charles Connor, Ph.D. Nicole Elko, Ph.D. Rick Oches, Ph.D. Date of Approval: April 30, 2008 Keywords: beach erosion, nearshore sediment transport, swash excursion, wave breaking, physical modeling, surf zone processes, coastal morphology Copyright 2008, Tiffany M. Roberts
Acknowledgements The support and guidance of those who have contributed to the successful completion of this thesis have been greatly appreciated. Most importantly, I would like to sincerely thank Dr. Ping Wang for his knowle dge, instruction, and guidance during this project. From guidance during my undergradu ate Honors Thesis thr ough the present, Dr. Wang has been the most important and influent ial person in my academic, educational, and personal advancements. I would also like to extend sincere thanks to Dr. Nicole Elko, whose guidance through my thesis lead to a more comprehensiv e understanding of coastal environments and processes. Her time and dedication to my academic advancement played an instrumental role in the development and successful completion of this thesis. I would also like to thank Dr. Nicholas Kraus for the SUPERTANK dataset and significant contributions to the study including insights into the broader implications of swash processes in many different coastal a pplications. Thanks to my other committee members, Drs. Chuck Connor and Rick Oche s, whose contribution guided me towards a more comprehensive anal ysis of this topic. Finally, thanks to my family and frie nds who have supported me during my educational endeavors with constant encouragement to pursue my academic and personal aspirations and dreams. Thank you all.
i Table of Contents List of Figures ii List of Tables iv Abstract v Introduction 1 Literature Review 3 Methods 11 SUPERTANK and LSTF 11 Data Analysis 15 Results 18 Beach Profile Change 20 Cross-shore Distribution of Wave Height 31 Wave Runup 37 Discussion 42 Relationship between Wave Runup, Incident Wave Conditions, and Limit of Beach Profile Change 42 A Conceptual Derivation of the Proposed Wave Runup Model 56 Conclusions 60 References 62 Appendices 68 Appendix I Â– Notation 69
ii List of Figures Figure 1 Power density spectra from th e LSTF experiment under spilling and plunging breakers measured from an offshore gauge (above) and a gauge at the shorel ine (below). 7 Figure 2 Beach-type classification base d on beach morphology and slope. 9 Figure 3 The SUPERTANK experiment (upper) and LSTF (lower) during wave runs. 12 Figure 4 The first SUPERTANK wa ve run, ST_10A erosive case. 21 Figure 5 The SUPERTANK ST_30A accr etionary wave run(upper). 24 Figure 6 The SUPERTANK ST_60A dune erosion wave run. 25 Figure 7 The SUPERTANK ST_I0 accreti onary monochromatic wave run. 26 Figure 8 The LSTF Spilling wave case. 27 Figure 9 The LSTF Plunging wave case. 29 Figure 10 The cross-shore random wave -height distribution measured during SUPERTANK ST_10A; yellow arrow indicates the breaker point. 31 Figure 11 The cross-shore random wave -height distribution measured during SUPERTANK ST_30A. 33 Figure 12 The cross-shore random wave -height distribution measured during SUPERTANK ST_60A. 34 Figure 13 The cross-shore monochromatic wave-height distribution measured during SUPERTANK ST_I0. 35 Figure 14 The cross-shore wa ve-height distribution and for the LSTF Spilling and Plunging wave cases. 36
iii Figure 15 Wave runup measured during SUPERTANK ST_10A. 38 Figure 16 Wave runup measured during SUPERTANK ST_30A. 39 Figure 17 Wave runup measured during SUPERTANK ST_60A. 40 Figure 18 Wave runup measured during SUPERTANK ST_I0. 40 Figure 19 Relationship between breaking wa ve height, upper limit of beach profile change, and wave runup for the thirty SUPERTANK cases examined. 43 Figure 20 Comparison of measured and pr edicted wave runup for Eqs. (10), (4), (6), and (7). 46 Figure 21 Measured versus predicted runup for Eq. (10), including the anomalies. 47 Figure 22 Measured versus predicted runup for Eq. (10), excluding the anomalies. 48 Figure 23 Measured versus predicted runup for Eq. (4), excluding the anomalies. 49 Figure 24 Measured versus predicted runup for Eq. (7), excluding the anomalies. 49 Figure 25 Measured versus predicted runup for Eq. (6), excluding the anomailes. 50 Figure 26 Figure 4c from Holman (1986) comparing swash runup (minus setup) and offshore significant wave height (above); the digitized data from Figure 4 (Holman, 1986) with two linear regression analyses (below). 52 Figure 27 Figure 7c from Holman (1986) comparing swash runup(minus setup) and the surf similarity parameter (abo ve); the digitized data from Figure 4 (Holman, 1986) with a linear and exponential regression analysis (below). 53 Figure 28 Schematic drawing of forces ac ting on a water element in the swash zone. 57
iv List of Tables Table 1 Summary of Selected Wave Runs and Input Wave and Beach Conditions (Notation is explained at the bottom of the table). 19 Table 2 Summary of Breaking Wa ve, Maximum Wave Runup, Upper and Lower limits of B each-Profile Changes, and Presence of a Scarp for Each Analyzed Wave Run. 30 Table 3 Summary of Meas ured and Predicted Wa ve Runup Equations. 46
v Limits of Beach and Dune Erosion in Response to Wave Runup from Large-Scale Laboratory Data Tiffany M. Roberts Abstract The SUPERTANK dataset is analyzed to examine the upper limit of beach change in response to elevated wa ter level induced by wave runup. Thirty SUPERTANK runs are investigated, including both erosiona l and accretionary wave conditions under random and monochromatic waves. Two experiments, one under a spilling and one under a plunging breaker-type, from the Large-Sc ale Sediment Transport Facility (LSTF) are also analyzed. The upper limit of beach change approximately equals the maximum vertical excursion of swash runup. Exceptions to this direct relati onship are those with beach or dune scarps when gravity-driven changes, i.e., avalanching, become significant. The vertical extent of wave runup, Rmax, above mean water level on a beach without a scarp is found to approximately equal the significant breaking wave height, Hbs. Therefore, a simple formula max bs R H is proposed. The linear relationship between maximum runup and breaking wave height is supported by a conceptual derivation. This predictive formula reproduced the measured runup from a large-scale 3-dimensional movable bed physical model. Beach and dune scarps substantially limit the uprush of swash motion, resulting in a much reduced ma ximum runup. Predictions of wave runup are not improved by including a slope-dependent surf-similarity parameter. The limit of
vi wave runup is substantially less for monoc hromatic waves than for random waves, attributed to absence of low-freque ncy motion for monochromatic waves.
Introduction Accurate prediction of the upper limit of beach change is necessary for assessing and predicting dramatic morphological changes accompanying storms. It is particularly important for coastal management practices, such as designing seawalls, nourishment profiles, and assessing potential coastal dama ges. The upper limit of beach change is controlled by wave breaking and the subsequent wave runup. During storms, wave runup is superimposed on elevated water levels (due to storm surge). Wang et al. (2006) found that the highest elevation of beach erosi on induced by Hurricane Ivan in 2004 extended considerably above the measured storm-su rge level, indicating that the wave runup played a significant role in the upper limit of beach erosion. The limit of wave runup is also a key parameter for the application of the storm-impact scale by Sallenger (2000). The Sallenger scale categorizes four levels of morphological impact by storms through comparison of the highest elevation reached by the water (storm surge and wave runup) and a representative elevati on of barrier island morphology (e.g. the crest of the foredune ridge). In addition, accurately assessi ng wave runup has numerous engineering applications, including design waves for overtop ping of seawalls, brea kwaters and jetties, and elevation of berm height for beach nourishment (Komar, 1998). Therefore, quantification of wave runup and its relati onship to the upper limit of beach-profile change are essential for understanding and predicting beach and dune changes. Limit of
2 runup typically serves as a crucial parameter for the m odeling of coastal morphology changes.
3 Literature Review Wave runup is composed of wave set up and swash runup, defined as a superelevation of the mean water level and fluctu ations about that mean, respectively (Guza and Thornton, 1981; Holman and Sallenger, 1985; Nielsen, 1988; Yamamoto et al., 1994; Holland et al., 1995). Wave setup can be furthe r defined as a seaward slope in the water surface that provides a pressure gradient or force balancing for the onshore component of the radiation stress induced by the momentum flux of waves (Komar, 1998). The swash runup component of wave runup is defined as the upper limit of swash uprush. The swash uprush is strongly modulated by low-fre quency oscillations of ten referred to as infragravity waves with periods of at least twice the peak incident wave periods (Komar, 1998). Previous studies on the limits of wave runup were primarily based on field measurements at a few locations resulting in several formulas developed to predict wave setup and runup. An early formula for wave setup slope was based on a theoretical derivation for sinusoidal, or monochroma tic, waves on a uniformly sloping beach (Bowen et al., 1968): h K x x 21(12.67) K (1) where h = still-water depth, = wave setup, x = cross-shore coordinate, = H/( + h) and H = wave height. The term essentially describes th e relationship between the
4 breaking wave height and its proportionality to local water depth. Based on both theory and laboratory measurements, the maxi mum set-up under a monochromatic wave, M was found to occur at the stillwater shoreline (Battjes, 1974): 0.3M bH (2) where Hb = breaking wave height. The above equations concern only the setup component of the entire wave runup, not including the porti on of elevated water level induced by swash runup. Taking the commonly used value of 0.78 for the maximum setup yielded from the above equation is about 23% of the breaking wave height. Based on field measurements conducted on dissipative beaches, Guza and Thornton (1981) found maximum setup at the shoreline, s l to be linearly proportional to the significant deepwater wave height, Ho: 0.17 s loH (3) Guza and Thornton (1982) found th at the significant wave runup, Rs, (including both wave setup and swash runup) is also linearly proportional to the significant deep-water wave height: 3.480.71(units of centimeters)soRH (4) Comparing Eqs. (3) and (4), the entire wave runup is approximately 4 times the wave setup, i.e., swash runup constitutes a signifi cant portion, approximately 75%, of the total elevated water level. According to Huntley et al. (1993), Eq. (4) is the best choice for predicting wave runup on dissipative beaches Based on field measurements on highly dissipative beaches, Ruessink et al. (1998) an d Ruggiero et al. (2004) also found linear relationships, but with differe nt empirical coefficients.
5 Based on field measurements on both dissip ative and reflective beaches, Holman (1986) and several similar studies (Holman and Sallenger, 1985; R uggiero et al., 2004; Stockdon et al., 2006), argued that more accurate predictions can be obtained by including the slope-dependent surf-similarity parameter, : 1/2(/)ooHL (5) where Lo = deepwater wavelength, and = beach slope. The surf similarity parameter has been broadly used to parameterize many nearshore and surf-zone processes including morphodynamic classification of beaches, wave breaking, breaker-type, the number of waves in the surf zone, and runup (Iribarre n and Nogales, 1949; Battjes, 1971; Galvin, 1972; Battjes, 1974; Galvin, 1974; Holman and Sallenger, 1985). Holman (1986) emphasized the application of the surf simila rity for setup and swash runup predictions. Holman found a dependence of the 2% exceede nce of runup, R2, on the deepwater significant wave height and the surf similarity parameter: 2(0.830.2)oo R H (6) The 2% exceedence of runup refers to a sta tistic of runup measurem ents of which only 2% of the data exceeded. Stockdon et al. (2006) expanded upon the Holman (1986) analysis with additional data covering a wider range of beach slopes and developed a more complicated empirical equation: 1 2 2 1 2 2(0.5630.004) 1.10.35() 2oof fooHL RHL (7) where f = foreshore beach slope. Realizing the variability of beach slope in terms of both definition and measurement, Stockdon et al. (2006) defined the foreshore beach
6 slope as the average slope over a region of two times the stan dard deviation of continuous water level record. Nielsen and Hanslow (1991) found a relati onship between the surf similarity parameter and runup on steep beaches. However, for beaches with a slope less than 0.1, they suggested that the surf similarity para meter was not related to runup. A subsequent study by Hanslow and Nielson (1993) conducted on the dissipative beaches of Australia found that maximum setup was not dependent on beach slope. Except for the original derivation by Bowen et al. (1968), most predictive formulas for wave runup have been empiri cally derived based on field measurements over primarily dissipative beaches, with a limited number of measurements over reflective beaches. Based on a study by Guza and Thornton (1982), maximum swash excursion on beaches with wide surf z ones was found to depend primarily on the infragravity modulation of sw ash, or the low-frequency os cillations. Because surf bore heights are depth limited, an in crease in offshore wave height increases the distance the bore must travel further dissipa ting its energy, i.e., wave ener gy at the incident frequency has become saturated. In contrast, the en ergy in the low-fre quency range increases shoreward due to non -linear energy transf er (Guza and Thor nton, 1982). This modulation of infragravity swas h motions has been investigat ed in several other studies (Holman and Guza, 1984; Howd et al., 1991 ; Raubenheimer et al., 1996). Energy transfer from incident frequencies to infrag ravity frequencies is non-linear and not well understood. Figure 1 illustra tes the power density spectr a of spilling and plunging breakers measured during the LSTF experiment s at an offshore gauge and at a gauge near the shoreline. Most of the energy peaks w ithin the high-f requency domain prior to wave
7 breaking. However, as the waves propagate onshore after breaking, much of the energy peaks in the low-frequency domain. Although the significance of infr agravity motions is qualitatively recognized, existing theoretical and the empirical equations do not include infragravity wave parameters. Offshore (before breaking) 0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.000.300.600.901.201.50 Frequency (Hz)Power Density (m^2 s) Spilling Plunging Shoreline (after breaking) 0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.000.300.600.901.201.50 Frequency (Hz)Power Density (m^2 s) Spilling Plunging Figure 1. Power density spectra from the LS TF experiment under spilling and plunging breakers measured from an offshore gauge (abov e) and a gauge at the shoreline (below).
8 The definition of dissipative and reflective beaches was developed as a classification scheme for overall beach morphol ogy (Short, 1979a, b; Wright et al, 1979a, b; Wright and Short, 1984). One of the key factors to the morphologi cal classification is beach slope (Fig. 2). However, the region over which beach slope is taken has an influence on the value obtained, and theref ore the classification of the beach. For example, according to Wright and Short (1984), a dissipative beach is classified based on tan = 0.03 within the swash zone, and tan = 0.01 across the inshore profile, or nearshore region (Fig. 1). A reflective beach is defined as one with tan = 0.1 to 0.2 within the swash zone, and tan = 0.01 to 0.02 across the inshore profile (Fig. 2). However, Nielsen and Hanslow (1991) suggested that the distinction between dissipative or reflective should be tan = 0.10. The method of determin ing slope was not specified. The presence of an offshore bar and trough, a ridge-runnel system, rip channels or beach cusps, etc., would result in an intermediate beach type between dissipative and reflective. Therefore, based on Wright and Short (1984), many beaches fall within the intermediate category. In addition, beach types can ch ange with time, e.g., during calm and storm conditions, and at different tid al stages. Holman and Salle nger (1985) suggested that sometimes a beach can be considered as re flective during high tide and intermediate during low tide due to the different inte nsity of bar influence on wave breaking. In contrast to the importance of beach slope for determining the morphodynamic classification of beaches and its influe nce on hydrodynamics, Douglass (1992) suggests the slope term should not be used for the formul ation of runup. After an analysis of the effectiveness of including slope for estim ating runup on beaches using Holman (1986) data, Douglass found the predicti on of runup was just as accura te with the omission of the
9 slope term. He also suggested that because the slope is a dependent variable and varies significantly over time, determining its va lue a priori is difficult and unnecessary. Figure 2. Beach-type classification based on beach morphology and slope (Wright and Short, 1984).
10 Almost all the above field studies focused mainly on the hydrodynamics of wave runup with little discussion of the correspond ing morphologic response, particularly the upper limit of beach changes, which should be closely related to the wave runup. In contrast to numerous studies on wave runup itself, limite d data are available relating wave runup excursion with the resulting morphol ogic change. In other words, it is not well-documented how the limit of wave runup is related to the limit of beach change. In the present study, data from the pr ototype-scale laboratory experiments, including those conducted at SUPERTANK (K raus et al., 1992; Kraus and Smith, 1994) and the Large-Scale Sediment Transport Facil ity (LSTF) (Hamilton et al., 2001; Wang et al., 2002), are analyzed to quantify the limit of wave runup and corresponding limit of beach and dune erosion. Specifically, this st udy examines 1) the levels of swash run-up and wave setup; 2) time-seri es beach-profile changes unde r erosional and accretionary waves; 3) the relationship between the a bove two phenomena; and 4) the accuracy of existing wave runup prediction methods. A ne w empirical formula predicting the limits of wave runup based on breaking wave hei ght, which also infers upper limit of beach change, is based on the prototype-scale la boratory data. The formula predicting maximum wave runup is Rmax=Hb. This first-order estimate has the advantage of giving zero runup for zero wave height, in a much si mpler form than many existing predictive formulas.
11 Methods SUPERTANK and LSTF Experiments Data from two large-scale movable-be d laboratory studies, SUPERTANK and LSTF experiments (Fig. 3), are examined to quantify the upper limits of beach-profile change and wave runup, as well as to determ ine the relationship betw een the two. Both experiments were specifically designed to measure detailed processes of sediment transport and morphological change under va rying prototype wave conditions. Dense instrumentation in the well-controlled labor atory setting allows for the detailed and accurate measurements of hydrodynamic conditions and time-series morphological changes. In addition, large-scale physic al analog models such as SUPERTANK and LSTF are particularly useful in directly ap plying the same conditions to the real-world, without the need for scaling. SUPERTAN K was conducted in a two-dimensional wave channel with beach changes caused primarily by cross-shore processes. LSTF is a threedimensional wave basin, with both cros s-shore and longshore sediment transport inducing beach change. SUPERTANK is a multi-institutional effort sponsored by the U.S. Army Corps of Engineers, conducted at the O.H. Hinsdale Wave Research Laboratory at Oregon State University from July 29 to September 20, 1991. This facility is the largest wave channel in the United States containing a sandy beach having the capability of running
12 experiments comparable to the magnitude of naturally occurring waves (Kraus et al. 1992). The SUPERTANK experiment meas ured total-channel hydrodynamics and sediment transport along with resulting beach-profile changes. The wave channel is 104 m long, 3.7 m wide, and 4.6 m deep (with th e still water level typically 1.5 m below the top) with a constructed sandy beach extending 76 m offshore (Fig. 3 upper). Figure 3. The SUPERTANK experiment (uppe r) and LSTF (lower) during wave runs. LSTF
13 The beach was composed of 600 m3 of fine, well-sorted quartz sand with a median size of 2.2 x 10-3 m and a fall speed of 3.3 x 10-2 m/s. The wave generator and wave channel were equipped with a sensor to absorb the energy of waves reflected from the beach (Kraus and Smith, 1994). The water-le vel fluctuations were measured with 16 resistance and 10 capacitance gauges. The 16 resistance gauges recorded water levels and were mounted on the channel wall, spaced 3.7 m apart, extending from near the wave generator to a water depth of approximately 0.5 m. The 10 ver tical capacitance wave gauges recording wave transformation and runup were mobile with spacing varying from 0.6 to 1.8 m, and extended from the most shoreward resistance gauge to the maximum runup limit, sometimes submerged or partially buried. The 26 gauges provided detailed wave propagation patterns, especially in the swash zone. The beach profile was surveyed following each 20to 60-min wave simulation. The initial profile was constructed based on the equilibrium beach profile developed by Dean (1977) and Bruun (1954) as: 2 3() hxAx (8) where h = still-water depth, x = horizontal distance fr om the shoreline, and A = a shape parameter corresponding to a mean grain size of 3.0 x 10-3 m. The profile shape parameter Â“AÂ” varies with both sediment grai n size and fall velocity carrying dimensions of length to the 1/3 power. The initial beach was built steeper with a greater A -value to ensure adequate water depth in the o ffshore area (Wang and Kraus, 2005). For efficiency, most SUPERTANK test s were initiated with the final profile of the previous simulation. Approximately 350 beach-profile surveys were ma de with an auto-tracking, infrared Geodimeter targeting prism attached to a survey rod mounted on a carriage
14 pushed by researchers. Although three lines, two along the wave -channel wall and one in the center, were surveyed, only the center line was examined in this study, as the surveys showed good cross-tank uniformity. Wave-pro cessing procedures are discussed in Kraus and Smith (1994). To separate incident-band wave motion from low-frequency motion, a non-recursive, low-pass filter was applied to the total wave record during spectral analysis. The period cutoff for the filter was se t to twice the peak period of the incident waves. The LSTF is a three-dimensional wave ba sin located at the U.S. Army Corps of Engineers Coastal and Hydraulic Laboratory in Vicksburg, Mississippi. Details of the design and test procedures are discussed in Hamilton et al. (2001). The LSTF was designed to study longshore sediment transport (Wang et al., 2002). Similar to the SUPERTANK experiments, the LSTF is capable of generating wave conditions comparable to the naturally occurring wa ve heights and periods found along low-energy coasts. The LSTF has effective beach dime nsions of 30-m cross-shore, 50-m longshore, and walls 1.4 m high (Fig. 3 lower). The b each was typically designed in a trapezoidal plan shape composed of approximately 150 m3 of very well-sorted fine quartz sand with a median grain size of 1.5 x 10-3 m and a fall speed of 1.8 x 10-2 m/s. Initial construction of the beach was also based on the equilibrium shape (Eq. 8). The beach profiles were surveyed using an automated bottom-tracking profiler capable of re solving bed ripples. The profiles surveyed in the center of the test basin are used here. The beach was typically replenished after three to nine hours of wave activity Long-crested and unidirectional irregular waves with a relativel y broad spectral shape were generated at a 10 degree incident angle. The wave height and peak wave period were measured with
15 capacitance wave gauges sampling at 20 Hz, w ith statistical wave properties calculated by spectral analysis. The experimental proced ures in LSTF are described in Wang et al. (2002). Data Analysis After inspection of all 20 SUPERTANK test s, 5 tests with a total of 30 wave simulations, or wave runs, were selected for an alysis in the present study. The selection was based on the particular purpose of the wave run, the trend of net sediment transport, and measured beach change. Time-serie s beach-profile changes and cross-shore distribution of wave height a nd mean water level were analyzed. During the 20 tests run during the SUPERTANK experiment, approxi mately 350 beach-profile surveys were conducted. For each profile, the elevation re lative to mean water depth was plotted on the vertical y-axis in meters and the distan ce offshore on the horizontal x-axis in meters. The upper limits (UL) for the beach profile cases we re identified based on the upper profile convergence point, above which no beach change occurred. The other location of importance along th e profile was the lower limit of beach change, sometimes referred to as the depth of closure. The lower limit, or profile convergence point, was identified at the de pth contour below which no beach-profile change occurred. The depth of closure for a given or characteristic time interval is the most landward depth seaward of which there is no significant change in bottom elevation and no significant net sediment exchange betw een the nearshore and the offshore (Kraus, et al., 1999; Wang and Davis, 1999).
16 For all 20 SUPERTANK cases examined, th e water levels and zero-moment wave heights were analyzed, where the mean water level or wave height (y-axis) was plotted against the horizontal distance offshore (x -axis), respectively. The maximum runup (Rmax) was defined by the location and beach elevat ion of the swash gauge that contained a value larger than zero wave height, i.e., wa ter reached that particular gauge. It is important to note that the value of Rmax is not a statistical value, but rather an actual measurement from the experiment. There may be some differe nces in the runup measured in this study as compared to th e video method (e.g., Holland et al., 1995) and horizontally elevated wires (e.g., Guza and Thornton, 1982); however, because of the finite spacing of the capacitance wave gauges in the swash zone, the differences are not expected to be signif icant. Field measurements of wave runup have typically been conducted with video imagery and/or resistan ce wire generally 5 to 20 cm above and parallel to the beach face. Holman and Guza (1984) and Holland et al. (1995) compared these two data collection methods concluding that they are compar able in producing accurate results. Cross-shore distribution of wave height s, or wave-energy decay, was also examined. The wave breaker point was define d at the location with a sharp decrease in wave height (Wang et al., 2002). Alt hough the entire SUPERTANK dataset was available for this study and were analyzed, 5 cases were select ed from the 20 initial test runs. The selection was based on the particular purpose of the wave run, the trend of net sediment transport, and measured beach change. Two LSTF experiments, one conducted under a spilling brea ker and one under a plunging breaker, are examined in this study. The LSTF analysis focused mainly on the
17 upper limit of beach change. The maximum r unup was not directly measured due to the lack of swash gauges. The main objective of the LSTF analysis was to apply the SUPERTANK result to a three-dimensional beach. The dataset from Holman (1986) was digiti zed. Similar statistical analysis, i.e., data normalization and linear regression an alysis, were conducted to evaluate the Â“goodness-of-fitÂ” between the predicted and m easured runup. In addition, polynomial and exponential regression anal yses were also conducted to examine potential non-linear relationships between the measured wave r unup and various parameters, such as wave height, wave period, and beach slope.
18 Results Overall, 30 SUPERTANK wave runs and two LSTF wave runs were analyzed (Table 1). The thirty SUPERTANK cases we re composed of twelve erosional random wave runs, three erosional monochromatic wave runs, seven accretionary random wave runs, three accretiona ry monochromatic wave runs, a nd five dune erosion random wave runs, summarized in Table 1. The first tw o numbers in the Wave Run ID Â“10A_60ERÂ” indicate the major data collection test, the le tter Â“AÂ” (arbitrary nomen clature) indicates a particular wave condition, and Â“60Â” describes the minutes of wave action. In design of the experiment (Kraus et al., 1992) the eros ional and accretionary cases were designed based on the Dean number N bsH N wT (9) where Hbs = significant breaking wave height; w = fall speed of the sediment, and T = wave period. Because the SUPERTANK experiments we re designed to examine the processesresponse at a general beach-dune environment, ra ther than simulating a particular realistic setting, e.g., a certain storm at a certain beach scaling is not a particular concern. Of course, caution should be taken when applyi ng the SUPERTANK findings to different conditions in the real world. However, th is holds true when applying findings of any
19 field experiment, e.g., from the Pacific coasts to the Gulf of Mexico coasts, from one setting to another. Table 1. Summary of Selected Wave Runs and Input Wave and Beach Conditions (Notation is explained at the bottom of the table). Wave Run ID Ho m Tp s Lo m n N Hbs m s Hb_h m Hb_l m Hsl_h m Hsl_l m SUPERTANK 10A_60ER 0.78 3.0 14.0 20 6.4 0. 68 0.10 0.42 0.66 0.15 0.13 0.24 10A_130ER 0.78 3.0 14.0 20 6.8 0.68 0.09 0.38 0. 67 0.15 0.10 0.23 10A_270ER 0.78 3.0 14.0 20 6.9 0.68 0.10 0.42 0. 65 0.16 0.10 0.24 10B_20ER 0.71 3.0 14.0 3.3 6.6 0.65 0.14 0. 58 0.63 0.17 0.10 0.23 10B_60ER 0.73 3.0 14.0 3.3 6.8 0.67 0.11 0. 44 0.65 0.17 0.11 0.24 10B_130ER 0.72 3.0 14. 0 3.3 7.0 0.69 0.09 0. 36 0.67 0.18 0.12 0.25 10E_130ER 0.69 4.5 31.6 20 4.9 0.72 0.11 0.69 0. 71 0.15 0.15 0.16 10E_200ER 0.69 4.5 31.6 20 5.0 0.74 0.12 0.77 0. 72 0.15 0.15 0.18 10E_270ER 0.69 4.5 31.6 20 5.1 0.76 0.09 0.58 0. 74 0.15 0.16 0.20 10F_110ER 0.66 4.5 31.6 3.3 5.1 0.75 0.09 0. 58 0.72 0.18 0.15 0.26 10F_130ER 0.68 4.5 31.6 3.3 5.1 0.76 0.08 0. 48 0.74 0.18 0.13 0.21 10F_170ER 0.69 4.5 31.6 3.3 5.1 0.76 0.08 0. 50 0.73 0.20 0.12 0.24 G0_60EM 1.05 3.0 14.0 M 10.0 1. 18 0.10 0.43 1.18 0.01 0.11 0.03 G0_140EM 1.04 3.0 14. 0 M 10.5 1.04 0.10 0. 41 1.04 0.04 0.08 0.10 G0_210EM 1.15 3.0 14. 0 M 10.8 1.07 0.09 0. 39 1.07 0.04 0.11 0.02 30A_60AR 0.34 8.0 99.9 3.3 1.6 0.41 0.14 2. 24 0.40 0.06 0.24 0.08 30A_130AR 0.33 8.0 99. 9 3.3 1.6 0.39 0.13 2. 09 0.38 0.06 0.24 0.09 30A_200AR 0.34 8.0 99. 9 3.3 1.6 0.41 0.13 2. 02 0.40 0.06 0.25 0.10 30C_130AR 0.31 9.0 126.4 20 1.4 0. 40 0.13 2.36 0.40 0.04 0.18 0.05 30C_200AR 0.31 9.0 126.4 20 1.4 0. 39 0.15 2.31 0.38 0.04 0.19 0.06 30C_270AR 0.31 9.0 126.4 20 1.4 0. 39 0.15 2.60 0.38 0.04 0.20 0.06 30D_40AR 0.37 9.0 126.4 20 1.4 0. 42 0.13 2.00 0.42 0.05 0.17 0.07 I0_80AM 0.60 8.0 99.9 M 2.9 0. 76 0.20 2.78 0.76 0.01 0.38 0.03 I0_290AM 0.63 8.0 99.9 M 3.1 0. 81 0.17 2.35 0.81 0.01 0.34 0.02 I0_590AM 0.60 8.0 99.9 M 2.7 0. 72 0.12 1.64 0.73 0.01 0.25 0.03 60A_40DE 0.69 3.0 14.0 3.3 6.2 0.61 0.12 0. 55 0.58 0.14 0.16 0.24 60A_60DE 0.69 3.0 14.0 3.3 6.2 0.61 0.10 0. 46 0.60 0.14 0.12 0.24 60B_20DE 0.64 4.5 31.6 3.3 4.4 0.66 0.11 0. 74 0.63 0.15 0.18 0.24 60B_40DE 0.63 4.5 31.6 3.3 4.4 0.66 0.11 0. 76 0.62 0.16 0.18 0.25 60B_60DE 0.65 4.5 31.6 3.3 4.4 0.66 0.12 0. 79 0.63 0.17 0.18 0.30 LSTF Spilling 0.27 1.5 3.5 3.3 10.0 0.26 0.11 0.41 N/ C N/C N/C N/C Plunging 0.24 3.0 14.0 3.3 4.4 0.27 0.13 0.96 N/ C N/C N/C N/C ER = erosional random wave; EM = erosional monochromatic wave; AR = accretionary random wave; AM = accretionary monochromatic wave; DE = dune erosion case; Ho = offshore wave height; Tp = peak wave period; Lo = offshore wavelength; n = spectral peakness; N = Dean Number; Hbs = significant breaking wave height ; s. = beach slope defined as the slope of the section approximat ely 1 m landward and 1 m seaward of the shoreline; = surf similarity parameter; Hb_h = incident band wave height at the breaker line; Hb_l = low frequency band wave height at the breaker line; Hsl_h = incident band wave height at the shoreline; Hsl_l = low-frequency band wave height at the shoreline; N/C = Not calculated.
20Beach Profile Change Typically, erosion is defined as a net offs hore transport of sand resulting in a net loss of beach volume above the mean water line, or shoreline. Accretion is defined as a total net onshore transport of sand, building the beach above mean water level. These definitions are applied in this study. For the SUPERTANK wave runs examined, the Dean number N was typically between 5 and 10 for the erosional cases. For the accretionary cases, the N was typically between 1.5 and 3 (Table 1). For the LSTF experiment, the spilling breaker case had a Dean Number of 10, indicating a highly erosive wave. A smaller Dean Number of 4.4 corresponds to the plunging breaker case, indicative of a slightly more accretionary trend (Table 1). The first SUPERTANK wave run, ST _10A, was conducted over the monotonic initial profile (Eq. 8). Significant beach-p rofile change occurred with substantial shoreline recession, along with the development of an offshor e bar. Figure 4 illustrates four time-series beach-profiles surveyed at initial, 60, 130, and 270 minutes. Initially, the overall foreshore exhibited a convex shape while the end profile was concave. After 60 min of wave action, cons iderable erosion occurred in the vicinity of the shoreline. The sand was transported offshore and deposited in the form of a bar. This trend continued during the subsequent wave runs, with additi onal erosion near the shoreline and further accretion of the bar with offshore directed mi gration. After 270 min, the bar had moved 4 m further offshore, compared to the 60-mi nute bar location. Th e 270-min profile was substantially steeper near the shoreline than the initial prof ile. The maximum beach-face recession occurred at the +0.37 m contour line The morphology of a section of the beach profile located landward of th e trough (from 15 to 18 m) did not experience any changes,
21 indicating that the sediment eroded from near the shoreline was transported past this section of the surf zone and deposited on the bar. The upper limit of beach-profile change can be readily identified; for this case, the upper limit was 0.66 m above mean water level (MWL) for all three time segments. There is also an apparent point of profile convergence in the offshore area, at -1.35 m depth contour, beyond which regular profile elevation change cannot be identified. ST_10A: Equilibrium Erosion (Random)-1.8 -1.2 -0.6 0.0 0.6 1.2 1.8 05101520253035 Distance (m)Elevation relative to MWL (m) initial 60 min 130 min. 270 min. Hmo = 0.8 m Tp = 3 s n = 20 Figure 4. The first SUPERTANK wave run, ST_10A erosive case. Substantial shoreline erosion occurred on the initial m onotonic profile with the development of an offshore bar. The horizontal axis Â“distanceÂ” refers to the SUPERTANK coordinate system and not directly related to morphological features.
22 The subsequent wave runs were conducted over the final profile of the previous wave run, i.e., a barred beach. The beach-p rofile changes are detectable, but subtle, especially for the accretionary wave cases w ith lower wave heights. Figure 5 shows an example of an accretionary wave run, ST_30A. Overall, compar ed to the initial run (Fig. 3), the beach-profile ch anges were much more subtle. The top of the offshore bar was eroded with sediment deposition along the land ward slope of the bar (Fig. 5 upper). The subtle profile change near the shoreline, viewed at a smaller scale (Fig. 5 lower), indicates slight accretion. The accreted sand ap parently came from the erosion just below the MWL. Due to the slight change, the iden tification of the upper limit is more difficult than the first case, and is less certain. This upper limit is determined to be at 0.31 m above MWL (Fig. 5 lower). One of the surv eys (60 min.) exhibits some changes above that convergence level, however these changes have been interpreted as survey error. The offshore-profile convergence point was dete rmined to be located at a depth contour of -1 m. A scarp developed during some of the erosional wave runs (Fig. 6). The development of the scarp is induced by wave s eroding the base of the dune or the drybeach, subsequently causing the weight of the overlying sediment to become unstable and collapse. The resulting beach slope immediat ely seaward of the scar p on the beach face tends to be steeper than on a non-scarped be ach. The upper limit of beach change is apparently at the top of the scarp, controlled by the elevation of the beach berm or dune, and does not necessarily represent the vertical ex tent of wave action. The nearly vertical scarp also induces technical difficulties of measuring the runup using wire gages. The upper limit of beach change in this study was selected at the base of the near-vertical
23 scarp, measured at 0.31 m above MWL during ST_60A (Fig. 6). The collapsed sediment from the scarp was deposited just below MW L (40 and 60 min). Therefore, for the scarped case, the upper limit is controlled by both wave action and gravity-driven sediment collapsing. Few beach-prof ile changes were measured offshore. SUPERTANK experiments also includ ed several cases conducted under monochromatic waves. Overall, beach-prof ile changes induced by monochromatic wave action were substantially differe nt from those under the more realistic random waves (Fig. 7). The monochromatic waves tended to cr eate irregular and undulating profiles. For ST_I0, the upper limit of beach-profile change was estimated at around 0.50 m and varied slightly during different wave runs. A pers istent onshore migrati on of the offshore bar also occurred throughout the wave runs, with several secondary bars developed in the mid-surf zone. The erratic profile evolu tion did not seem to approach a stable equilibrium shape, and does not have an appare nt profile convergence point. In addition, it is important to note that the profile sh ape developed under monochromatic waves does not represent profiles typically measured in the field (Wang and Davis, 1998). This implies that morphological changes measur ed in movable-bed laboratory experiments under monochromatic waves may not be applicab le to real world conditions, despite the easier hydrodynamic parameterization and an alysis of monochromatic waves.
24 ST_30A: Equilibrium Accretion (Random)-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 46810 Distance (m)Elevation relative to MWL (m) initial 60 min. 130 min. 200 min. Hmo = 0.4 m Tp = 8 s n = 3.3 Figure 5. The SUPERTANK ST_30A accreti onary wave run. There was subtle shoreline accretion, with an onshore migration of the offshor e bar (upper). The subtle accretion near the shoreline is identified wh en viewed at a smaller scale (lower). ST_30A: Equilibrium Accretion (Ra ndom)-1.8 -1.2 -0.6 0.0 0.6 1.2 1.8 05101520253035 Distance (m)Elevation relative to MWL (m) initial 60 min. 130 min. 200 min. Hmo = 0.4 m Tp = 8 s n = 3.3
25 ST_60A: Dune Erosion 2/2-1.8 -1.2 -0.6 0.0 0.6 1.2 1.8 05101520253035 Distance (m)Elevation relative to MWL (m) initial 40 min. 60 min. Hmo = 0.7 m Tp = 3 s n = 3.3 Figure 6. The SUPERTANK ST_60A dune erosi on wave run. A nearly vertical scarp was developed after 40 minutes of wave actio n, with the upper limit of beach change identified at the toe of the dune-scarp. Similar analyses as described above we re also conducted to a set of threedimensional laboratory movable-bed experime nts at LSTF. The waves generated in LSTF had smaller wave heights as compar ed to the SUPERTANK waves. Two cases with distinctively different breaker types, one spilling and one plunging, were examined in this study. The beach profiles used in th e following discussion are averaged over the middle section of the wave basin, and therefore represent the average condition of the 3dimensional beach.
26 ST_I0: Equilibrium Accretion (Monochromatic)-1.8 -1.2 -0.6 0.0 0.6 1.2 1.8 05101520253035 Distance (m)Elevation relative to MWL (m) initial 80 min 290 min 590 min Hmo = 0.5 m Tp = 8 s MON Figure 7. The SUPERTANK ST_I0 accretionary monochromatic wave run. The resulting beach-profile under monochroma tic waves is erratic and undulating. The spilling wave run was initiated with the Dean equilibr ium beach profile. Because of the smaller wave heights, the beachprofile changes occurred at a slower rate. Similar to the first SUPERTANK wave run, ST _10A, a subtle bar fo rmed over the initial monotonic beach profile (Fig.8 upper). Th e upper limit of beach change was 0.23 m above MWL under the spilling wave s, as identified from the smaller scale plot (Fig. 8 lower). Sediment from the eroding foreshor e was apparently transported seaward to the offshore bar. As the foreshore eroded, the offshore bar accumulated sediment increasingly with each subse quent wave run. Between 5 a nd 9 m offshore, little to no
27 change occurred which could suggest that the sediment bypassed this region and terminated transport at the offshore bar. The profile converges on the seaward slope of the offshore bar. LSTF Spilling Wave Case-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 -1.5-1.2-0.9-0.6-0.30 Distance (m)Elevation relative to MWL (m) 0 min 510 min 890 min 1330 min Hmo = 0.27 m Tp = 1.5 s n = 3.3 Figure 8. The LSTF Spilling wave case. Erosion occurred in the foreshore and inner surf zone. The eroded sediment was deposited on an offshore bar. LSTF Spilling Wave Case-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 -3-2-1012345678910111213141516 Distance (m)Elevation relative to MWL (m) 0 min 510 min 890 min 1330 min Hmo = 0.27 m Tp = 1.5 s n = 3.3
28 For the plunging wave case at LSTF, shoreline advancement occurred with each wave run along with a persis tent onshore migration of th e bar (Fig. 9 upper). This corresponds to a lower Dean number of 4.4. The accumulation at the shoreline was subtle, but can be identified when viewed at a smaller spatial scale (Fig. 9 lower). The upper limit of beach-profile chan ge is located at 0.26 m above MWL, with the lower limit identified at the profile conve rgence point midway on the seawar d slope of the bar. Also, persistent erosion was measured just belo w the MWL and at a region landward of the initial secondary bar. The eroded sediment landward of the trough may have contributed to the onshore migration of the offshore bar (F ig. 9 upper). Overall, the trends observed in the three-dimensional LSTF experiment were comparable to those in the twodimensional SUPERTANK experiment. Table 2 summarizes the upper and lower limits of change during each wave run, including the breaking wave he ight. Overall, for the 30 SUPERTANK wave runs and 2 LSTF wave runs, the incident breaking wave he ight ranged from 0.26 to 1.18 m (Table 2). The measured upper limit of profile change including the scarped cases, ranged from 0.23 to 0.70 m. The lower limit of beach change ranged from 0.50 to 1.61 m below MWL. Relationships between the profile ch anges and wave conditi ons are discussed in the following sections.
29 LSTF Plunging Wave Case0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 -1.8-1.7-1.6-1.5-1.4-1.3-1.2-1.1-1 Distance (m)Elevation relative to MWL (m) pre-test 40 min 80 130 Hmo = 0.24 m Tp = 3 s n = 3.3 Figure 9. The LSTF Plunging wave case. Slight foreshore accretion and landward migration of the offshore bar occurred during the wave run. LSTF Plunging Wave Case-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 -3-2-1012345678910111213141516 Distance(m)Elevation relative to MWL (m) pre-test 40 min 80 130 Hmo = 0.24 m Tp = 3 s n = 3.3
30Table 2. Summary of Breaking Wave, Maximum Wave Runup, Upper and Lower limits of Beach-Profile Changes, and Presence of a Scarp for Each Analyzed Wave Run. Wave Run Hb Rmax UL LL Scarp ID m m m m SUPERTANK 10A_60ER 0.68 0.6 0.66 1.29 No 10A_130ER 0.68 0.6 0.66 1.29 No 10A_270ER 0.68 0.7 0.66 1.29 No 10B_20ER 0.65 0.33 0.67 1.35 No 10B_60ER 0.67 0.69 0.67 1.35 No 10B_130ER 0.69 0.64 0.67 1.35 No 10E_130ER 0.72 0.3 0.74 1.52 No 10E_200ER 0.74 0.78 0.84 1.52 No 10E_270ER 0.76 0.45 0.84 1.52 No 10F_110ER 0.75 0.38 0.43 1.52 Yes 10F_130ER 0.76 0.38 0.42 1.52 Yes 10F_170ER 0.76 0.38 0.48 1.52 Yes G0_60EM 1.18 0.28 0.38 1.61 No G0_140EM 1.04 0.18 0.25 1.61 Yes G0_210EM 1.07 0.32 0.27 1.61 Yes 30A_60AR 0.41 0.42 0.31 1.36 No 30A_130AR 0.39 0.42 0.31 1.36 No 30A_200AR 0.41 0.42 0.31 1.36 No 30C_130AR 0.4 0.41 0.39 1.01 No 30C_200AR 0.39 0.42 0.42 1.01 No 30C_270AR 0.39 0.42 0.42 1.01 No 30D_40AR 0.42 0.23 0.43 0.65 No I0_80AM 0.76 0.27 0.46 1.82 No I0_290AM 0.81 0.16 0.53 1.82 No I0_590AM 0.72 0.35 0.53 1.82 Yes 60A_40DE 0.61 0.17 0.28 1.16 Yes 60A_60DE 0.61 0.17 0.28 1.16 Yes 60B_20DE 0.66 0.16 0.38 0.99 Yes 60B_40DE 0.66 0.16 0.38 0.99 Yes 60B_60DE 0.66 0.17 0.38 0.99 yes LSTF Spilling 0.26 N/M 0.23 0.62 No Plunging 0.27 N/M 0.26 0.5 No Hb = breaker height measured; Rmax= maximum runup measured (not statistical); UL, LL = upper and lower limit of beach change, respectively; NM = not measured.
31Cross-shore Distribution of Wave Height The wave-height decay is representative of the wave-energy dissipation as a wave propagates onshore. Detailed wave decay pa tterns were measured by the closely spaced wave gauges for both SUPERTANK and LSTF experiments. Figure 10 shows timeseries wave decay patterns measured at th e first SUPERTANK wave run, ST_10A. As discussed earlier, considerable beach profile change, for example the formation of an offshore bar, was measured during this wave run (Fig. 4). The substantial morphology change also influenced the pattern of wave decay. ST_10A 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 01020304050607080 Distance (m)Hmo (m) 60 min. 130 min. 270 min. Figure 10. The cross-shore random wave -height distribution measured during SUPERTANK ST_10A; yellow arrow indicates the breaker point.
32 The inflection point of wave decay migr ated slightly as the beach morphology changed from the initial monotonic profile to a barred-beach profile. This point is defined as the location and height at which the wave breaks (Wang et al., 2002). For ST_10A, the significant breaking wave height is 0.68 m. The rate of wave-height decay tends to be smaller in the mid-surf zone (10 to 20 m) as compared to the breaker zone (20 to 25 m) and the inner surf zone (landward of 10 m). The offshore wave height remains largely constant until reaching the breaker line. The wave decay pattern for the longer pe riod accretionary wave case, ST_30A (Fig. 11), was considerably different than the steep erosive waves. The significant breaking wave height was 0.36 m. The time-series wa ve pattern remained constant for each wave run, apparently not influen ced by the subtle morphology change (Fig.5). The offshore bar formed at 30 m during the previous experime nt with higher wave heights. Instead of wave breaking over the bar, shoaling of the long period wave was measured (at around 30 m). The main breaker line is identified at around 20 m.
33 ST_30A 0.0 0.1 0.2 0.3 0.4 0.5 01020304050607080 Distance (m)Hmo (m) 60 min. 130 min. 200 min. Figure 11. The cross-shore random wave -height distribution measured during SUPERTANK ST_30A. For the dune erosion case, the wave-heigh t remained largely constant offshore, with a slight decay in wave height up to the breaker point (Fig. 12). Dramatic waveheight decay was measured over the offshore bar at approximately 30 m, with a breaker height of 0.61 m. A slight increase in wave height, possibly due to wave shoaling, was measured at around 15 m offshore, followed by a sharp decrease in wave energy.
34 ST_60A 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 01020304050607080 Distance (m)Hmo (m) 40 min. 60 min. Figure 12. The cross-shore random wave -height distribution measured during SUPERTANK ST_60A. The cross-shore distribution of wave heights for the monochromatic wave case ST_I0 was rather erratic with both temporal and spatial inconsistencies (Fig. 13). This corresponds to the irregular beac h-profile change observed duri ng this wave case (Fig. 7). The breaking wave height vari ed considerably, from 0.72 to 0.81 m. The wave-height variation in the offshore region, seaward of the breaker line around 30 m, is probably related to oscillations in the wave tank. However, the irregular breaking wave heights were likely caused by reflection of the regular waves off the beach face.
35 ST_I0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 01020304050607080 Distance (m)Hmo (m) 80 min. 290 min. 590 min. Figure 13. The cross-shore monochromatic wave-height distribut ion measured during SUPERTANK ST_I0. The LSTF experiments were designed to examine the effects of different breaker types on sediment transport and morphology ch ange. Similar offshore wave heights of 0.27 m were generated for both cases (Fig. 14). However, the cross-shore distribution of wave heights varies, especially in the vici nity of the breaker line, where the spilling breaker wave dissipation was much more gradual than the plunging breaker. The breaking wave heights were measured at 0. 26 m and 0.27 m; for th e spilling and plunging waves, respectively.. For the spilling wave case, the offshore bar was low with a wide
36 crest at around 12 m offshore (Fig 9) with the main breaker line just seaward of the bar crest (Fig. 14). The narrow bar during the plunging case migrated onshore from 13 to 11 m (Fig. 9), with the main breaker line occurring directly over the crest of this bar (Fig. 14). Again, the wave height decay at the br eaker line was more abrupt for the plunging case than for the spilling case, as expected. Spilling and Plunging 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 024681012141618 Distance (m)Hmo (m) Spilling Plunging Figure 14. The cross-shore wave-height distribution and for the LSTF Spilling and Plunging wave cases.
37Wave Runup The extent and elevation of wave r unup for the SUPERTANK experiments were measured directly by the closely spaced sw ash gages (Kraus and Smith, 1994). Figure 15 shows the cross-shore distributi on of time-averaged water level for the erosive wave run, ST_10A. As expected, the mean water level in the offshore area remained largely at zero. Elevated water levels were measured in the surf zone. It is important to separate the elevation caused by wave setup and swash r unup, although it is often difficult to do. Currently, there is no widely accepted method fo r separating the setup from the swash in total wave runup. In this st udy, the elevated water levels seaward and landward of the still-water shoreline are regarded as wave setup and swash runup, respectively. This convention coincides well with the inflection po int of the change in slope of the crossshore distribution of the mean water level (F ig. 15). The setup approaches the shoreline asymptotically following the slope or curv ature of the beach face, whereas the swash appears to be nearly vertical controlled by th e gauge elevation. For this case, the setup measured at the still-water shoreline was 0. 1 m, which is about 17 percent of the total wave runup of 0.6 m, consistent with the approximation for setup by Guza and Thornton (1982). The slight difference in wave runup am ong different wave runs was attributed to the considerable beach-profile change during the first SUPERTANK wave case.
38 ST_10A -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 01020304050607080 Distance (m)Mean Water Level (m) 60 130 270 Figure 15. Wave runup measur ed during SUPERTANK ST_10A. For the accretionary wave case, ST_30A, the curvature in th e mean-water level also coincides with the still-water shoreline (Fig. 16). However, considerable variations in the wave setup occurred at the shoreline. The slight decrease in mean water level measured just seaward of 20 m offshore dur ing the 130-min wave run is attributed to setdown. The increase measured just seaw ard of 20 m offshore during the 270-min wave run is likely caused by instru ment error at that particular location. The average setup at the shoreline was approximately 0.07 m, also about 17 percent of the total measured wave runup of 0.42m.
39 ST_30A -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 01020304050607080 Distance (m)Mean Water Level (m) 60 130 200 Figure 16. Wave runup measur ed during SUPERTANK ST_30A. Total wave runup was significantly limited by the vertical scarp as shown in the dune erosion case of ST_60A (Fig. 17). A broa d set down was measured just seaward of the main breaker line. The curvature of th e cross-shore distribution of the mean water level occurred between 10 and 11 m, which was di fferent from the still-water shoreline at 8 m. The setup measured at the water-level curvature at 11 m was approximately 0.03 m. Despite the limited total runup, the wave set up contributes 18 percen t of the total wave runup of 0.17 m, similar to the above two cases. The cross-shore distribution of time-averag ed mean water levels for ST_I0 were somewhat erratic (Fig. 18), similar to the e rratic beach-profile changes and cross-shore distribution of wave heights for the monochromatic wave runs. Different from the irregular wave cases, a zero mean-water leve l was not measured at a several offshore wave gauges.
40 ST_60A -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 020406080 Distance (m)Mean Water Level (m) 40 60 Figure 17. Wave runup measur ed during SUPERTANK ST_60A. ST_I0 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 01020304050607080 Distance (m)Mean Water Level (m) 80 290 590 Figure 18. Wave runup measur ed during SUPERTANK ST_I0.
41 In addition, considerable va riances among different wave ru ns were also measured. The total wave runup varied from 0.16 to 0. 35 m, with an average of 0.26 m. The curvature in the mean-water le vel distribution coincides with the still-water shoreline at 5 m. The maximum setup at the still-water sh oreline was 0.08 m, which is 31 percent of the total wave runup. The less er contribution of the swash runup to the tota l wave runup can be attributed to the lack of low-fr equency motion in the monochromatic waves, exemplified in Table 1, in which the lowfr equency components of the swash (Hsl_l) for monochromatic waves were much smaller th an the contribution for random waves.
42 Discussion Relationship between Wave Runup, Incide nt Wave Conditions, and Limit of Beach Profile Change The measured breaking wave height, uppe r limit of beach-profile change, and wave runup from the SUPERTA NK experiments are compared in Figure 19. The thirty cases examined are divided into three cat egories, including non-scarped random wave runs, scarped random wave runs, and monochr omatic wave runs. The upper limit of beach change was almost equal to the maximum excursion of wave runup for the nonscarped cases. In addition, the maximum wa ve runup was nearly equal to the breaking wave height for the non-scarped random wave cases as well. Th is suggests that the breaking wave height is equal to the maximu m elevation of wave runup which is equal to the upper limit of beach change, emphasized by the 16 non-scarped random wave runs, except the three cases 10B_20ER, 10E_270E R and 30D_40AR. The cause of the discrepancy in the data for these outliers varies. During the SUPERTANK experiment, it was confirmed that the capacitance gauges coul d record the wet sand or water surface if the sand was fully saturated. Some degradati on in the response was found if the sand was not fully saturated, which could explain th e limited swash runup measurement during the first 20 min run of ST10B. However, a di rect cause for the other two questionable measurements is not clear.
43 0 0.2 0.4 0.6 0.8 1 1.2 1.410A_60ER 10A_270ER 10B_60ER 10E_130ER 10E_270ER 30A_130AR 30C_130AR 30C_270AR 10F_130ER 60A_40DE 60B_20DE 60B_60DE G0_60EM G0_210EM I0_290AMWave Run No.Upper Limit (m) Breaking Wave Height Beach Change Wave Runup Non-scarped Random Wave CasesScarped Random Wave Cases Monochromatic Wave CasesFigure 19. Relationship between breaking wa ve height, upper limit of beach profile change, and wave runup for the thirty SUPERTANK cases examined. Based on the criteria of identifying the upper limit of beach change at the toe of the scarp, for the scarped random wave cases the breaking wave he ight was much higher than the elevation of wave runup, which was limited by the vertical scarp. The definition was based on the fact that any change occurr ing above this point was likely due to the weight of the overlying sediment collapsing. The true upper limit of change is thus controlled by the elevation of the beach or dune. No relationship among the breaker height, wave runup, and beach-profile cha nge was found for the scarped random wave
44 runs. However, it is important to note that the upper limits of b each changes for scarped cases are not expected to relate to wave runup. The much lower wave runup under monochr omatic waves as compared to the breaker height was found to be caused by the la ck of low-frequency motion. Baldock and Holmes (1999) suggested that simula ting irregular wave s with overlapping monochromatic swash events could repr oduce both low and high frequency spectral characteristics of the swash zone. No SUPERTANK experiments were designed to investigate this. No relationship could be found among the above three factors for monochromatic waves. This further suggest s that despite the c onvenience associated with studying the hydrodynamics associat ed with regular waves, including monochromatic waves in laboratory experime nts related to morphology changes, they do not have direct real -world applications. Based on the above observations from the SUPERTANK data (Fig. 19), a direct relationship between the measured runup he ight on a non-scarp beach and the breaker height is proposed: max bs R H (10) The average ratio of Rmax over Hbs for the 16 non-scarped wave cases was 0.93 with a standard error on the mean of 0.05. Ex cluding the three outliers, 10B_20ER, 10E_270ER and 30D_40AR, the average Rmax/Hbs was 1.01 with a standard error of 0.02. To be conservative due to the limited data coverage, a value of one is used in Eq. (10). In addition, the maximum runup elevation fo r the non-scarped random wave cases was approximately equal to the upper limit of beach change, LU R max (11)
45 This is important for relating morphological changes to wave runup on beaches, which is a main goal of runup studies. However, this re lationship is most dire ctly applied to wave similar to those generated in the two physical analog models. In addition, a wider range of runup and significant wave heights woul d likely increase the reliability of the correlation coefficient. Therefore, it is suggested that the breaker height is equal to the upper limit of wave runup which is equal to the upper limit of beach changes, L bsU R H max (12) Comparisons of the measured wave r unup with the various existing empirical formulas (Eqs. 4, 6, and 7) and the new model (Eq. 10) are summarized in Figure 20 and Table 3. As shown in Fig. 20, the simp le new model reproduced the measured wave runup much closer than the other formulas For the 16 non-scarped SUPERTANK wave cases, 81% of the predictions from Eq. (10) fa ll within 15% of the measured wave runup. In contrast, for Eqs. (4), (6) and (7), only 25%, 6% and 13% of the predictions, respectively fall within 15% of the measured values. Eqs. (6) and (7) under-predicted the measured wave runup for the erosional cases and over-predicted runup for the accretionary wave cases. The loss of predic tive capability is caused by the substantially greater for the gentle long-period accretionary waves than for the steep short-period erosional waves (Table 1). Agreement be tween measured and predicted values was actually reduced by including the surf similarity parameter, The simpler Eq. (4), using just offshore wave height, more accurately reproduced the measured values of wave runup than Eqs. (6) and (7).
46 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.910A_60ER 10A_130ER 10A_270ER 10B_20ER 10B_60ER 10B_130ER 10E_130ER 10E_200ER 10E_270ER 30A_60AR 30A_130AR 30A_200AR 30C_130AR 30C_200AR 30C_270AR 30D_40ARWave Run No.Wave Runup (m) Measured Eq. 10 Eq. 4 Eq. 6 Eq. 7 Figure 20. Comparison of measured and predic ted wave runup for Eqs. (10), (4), (6), and (7). Table 3. Summary of Measured and Predicted Wave Runup Equations. The bold font indicates predicted values that fall within 15% of the measured runup. Wave Run ID Hbs m Rmax m Eq 4 m Eq 6 M Eq 7 m Eq 10 m 10A_60ER 0.68 0.60 0.59 0.43 0.31 0.68 10A_130ER 0.68 0.60 0.59 0.40 0.29 0.68 10A_270ER 0.68 0.70 0.59 0.43 0.31 0.68 10B_20ER 0.65 0.33 0.54 0.49 0.38 0.65 10B_60ER 0.67 0.70 0.55 0.42 0.31 0.67 10B_130ER 0.69 0.64 0.55 0.36 0.26 0.69 10E_130ER 0.72 0.77 0.52 0.53 0.46 0.72 10E_200ER 0.74 0.78 0.52 0.58 0.51 0.74 10E_270ER 0.76 0.45 0.52 0.47 0.41 0.76 30A_60AR 0.41 0.41 0.28 0.70 0.72 0.41 30A_130AR 0.39 0.43 0.27 0.64 0.66 0.39 30A_200AR 0.41 0.42 0.28 0.64 0.66 0.41 30C_130AR 0.40 0.40 0.25 0.67 0.73 0.40 30C_200AR 0.39 0.42 0.25 0.73 0.79 0.39 30C_270AR 0.39 0.42 0.25 0.73 0.79 0.39 30D_40AR 0.42 0.23 0.30 0.69 0.76 0.42
47 Measured wave runup from the SUPERTANK experiment was compared with the predicted runup. For the predic tive formula proposed in this st udy, the linear relationship between breaking wave height and wave runup was found to be improved with the exclusion of the three questionable measurem ents (Figs. 21 and 22). The correlation coefficient was nearly three times better fo r the dataset without the anomalous values than for the analysis with the anomalies, with a R2 of 0.92 and 0.32 respectively. The proportionality coefficient from the linear regression supports the equation suggested by this study. y = 1.0563x R2 = 0.3117 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 00.10.20.30.18.104.22.168.80.9 Measured Wave Runup (m)Eq. 10: Predicted Runup (including outliers) (m) Figure 21. Measured versus predicted r unup for Eq. (10), including the anomalies.
48 y = 0.99x R2 = 0.92 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 00.10.20.30.22.214.171.124.80.9 Measured Wave Runup (m)Eq. 10: Predicted Runup (m) Figure 22. Measured versus predicted runup for Eq. (10), excluding anomalies. A linear regression analysis for Eq. (4 ), presented by Guza and Thornton (1982), also suggested a linear relationship with wave height (also excluding the three anomalies), with a correlation coefficient of R2 = 0.76 (Fig. 23). The predicted runup from Eq. (7), Stockdon et al., (2006), was analyzed with the measured runup, minus the three anomalies (Fig. 24). The R2 was 0.60, with a large intercept of 1.13. The two clusters in the data illustrate the over and under pr ediction of wave runup by Eq. (7) for the erosional and accretiona l waves, respectively. The pr edicted versus measured runup presented by Holman (1986), Eq. (6), was al so analyzed through a linear regression (Fig. 25). A similar trend as the predicted runup from Eq. (7) wa s observed, with a slightly reduced R2 of 0.52. The two cluste rs in the data again illu strate an over and under prediction of the erosive and accretionary waves.
49 y = 0.76x R2 = 0.76 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 00.10.20.30.126.96.36.199.80.9 Measured Wave Runup (m)Eq. 4: Predicted Runup (m) Figure 23. Measured versus predicted r unup for Eq. (4), excluding the anomalies. y = -1.09x + 1.13 R2 = 0.60 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 00.10.20.30.188.8.131.52.80.9 Measured Wave Runup (m)Eq. 7: Predicted Runup (m) Figure 24. Measured versus predicted r unup for Eq. (7), excluding the anomalies.
50 y = -0.66x + 0.93 R2 = 0.52 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 00.10.20.30.184.108.40.206.80.9 Measured Wave Runup (m)Eq. 6: Predicted Runup (m) Figure 25. Measured versus predicted r unup for Eq. (6), exluding the anomailes. Holman (1986) data were digitized from the original paper. The purpose was to reproduce some of his analyses. The Figure 4c in Holman (1986) is reproduced in Figure 26 below, showing the relations hip between runup and wave he ight, without inclusion of setup. The original dataset from Holman ( 1986) is shown above th e digitized data to ensure the figure was accurately reproduced. Two linear regression analyses were conducted, one with a forced zero-intercept and one with out. The linear regression without a forced zero interc ept resulted in a large in tercept of 1.23 with a R2 of 0.45. However, the forced zero-intercept linear regression resulted in a R2 of 0.36. Both suggest that the direct relati onship between wave runup and wa ve height does not exist. It is worth noting that the swash runup is co mpared to significant offshore wave height
51 measured at 8 m water depth, and not to break er height. As shown in Figure 26, for the same wave height, a large vari ation of wave runup values we re measured, which does not seem to be reasonable. It is not clear how and why a 1 m wave induced a 2.5 m swash runup, while a 4 m wave induced less than 2 m of swash runup. It is questionable as to how a 1 m offshore wave can induce a grea ter swash runup as compared to a 4 m offshore wave. The Figure 7c in Holman (1986) showing th e relationship between surf similarity and swash runup (minus setup) is reproduced in Figure 27. The m easured swash runup was normalized with offshore wave height. A linear regression analysis was conducted, resulting in a R2 of 0.62. An exponential fit was also attempted (Figure 27). The correlation coefficient was improved slightly, 0. 69 versus 0.62, as compared to the linear fit.
52 Holman '86: Figure 4 y = 0.44x + 1.23 R2 = 0.45 y = 0.98x R2 = 0.36 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 0.000.501.001.502.002.503.003.504.00 Hs (m)R2 (m) Series1 Linear (Series1) Linear (Series1) Figure 26. Figure 4c from Ho lman (1986) comparing swas h runup (minus setup) and offshore significant wave height (above); the digitized data from Figure 4 (Holman, 1986) with two linear regression analyses (below).
53 Holman '86 Fig. 7 y = 0.46x 0.04 R2 = 0.62 y = 0.83x + 0.20 y = 0.25e0.58xR2 = 0.69 0 0.5 1 1.5 2 2.5 3 3.5 00.511.522.533.54 Surf SimilarityR2 Series1eq.6 Linear (Series1) Linear (eq.6) Expon. (Series1) Figure 27. Figure 7c from Holman (1986) co mparing swash runup (minus setup) and the surf similarity parameter (above); the digitiz ed data from Figure 4 (Holman, 1986) with a linear and exponential regres sion analysis (below).
54 Recall, Douglass (1992) re-analyzed the Ho lman (1986) data set used in Eq. (6) and found no correlation between runup and beach-face slope. D ouglass further argued that beach slope is a dependent variable free to respond to the incident wave energy and should not be included in a runup prediction. In practice, beach face slope is a difficult parameter to define and determine. Excep t for Stockdon et al. (2006), a clear definition of beach slope is not given in most st udies. However, the Stockdon et al. (2006) definition of beach slope is very specific for video measurements and is difficult to utilize for other methods. In the present study, th e slope was defined over that portion of the beach extending roughly 1 m landward and seawar d from the still-water shoreline. Based on this definition, the Holman (1986) dataset overall had a tan = 0.08 to 0.17 across the swash zone; the SUPERTANK dataset had a tan = 0.09 to 0.15 across the swash zone. The resulting beach-profiles from SUPERT ANK under accretionary waves tended to have slopes of 0.13 to 0.20, whereas the beach -profiles from erosional waves had slopes of 0.08 to 0.14 (Table 1). Nielsen and Hans low (1991) also discussed the difficultly in defining beach slope, whether across the beach face or surf zone, w ith the presence of bars on intermediate beaches further comp licating the measurement of beach slope. Including the beach slope thus adds ambiguity in applying the empirical formulas. In addition, Douglass (1992) omitted the beach sl ope term from the Holman (1986) dataset, arguing that the wave steepness term in the su rf similarity parameter was responsible for Â’s predictive capability, rather than the inclusion of slope. This furthe r suggests that the use of the slope term in runup predictions not only induces more uncertainties, but is also generally unnecessary.
55 However, the inclusion of slope may be required under certain circumstances. For a very gentle beach, a large swash excurs ion is associated with a fixed runup, as compared to a steep beach. It is reasonable to argue that if infiltration, saturation, and bottom friction are significant, beach slope cannot be neglected. However, this is beyond the scope of this study and is also not the case for most of the existing studies reviewed (Guza and Thornton, 1982; Nielsen and Ha nslow, 1991; Hanslow and Nielsen, 1993; Ruggiero et al., 1996; Stockdon et al., 2006). Some experiment data have shown that the friction loss due to wave propa gation is small (Komar, 1998). Determining offshore wave height may al so cause uncertainty. In most field studies, the offshore wave height was taken to be the value m easured at a wave gage in the study area. Similarly, here it is taken as that from the offshore-most gauge. Under extreme storm conditions in which wave gauge s often fail, estimating the offshore wave height may not be straightforward (Wang et al., 2006). The definition of an offshore wave height varies between studies, in whic h it is often taken at whatever depth the instrument is deployed (Guza and Thornt on, 1981; Guza and Thornton, 1982; Holman, 1986; Stockdon et al., 2006). Stockdon et al. (2006) concl uded that runup was better predicted using breaking wave height when it is available; however the often lack of measured breaking wave height was argued as the reason for the use of offshore wave height for predicting runup. Theoretically, th e beach slope and wave characteristics are contributing factors in the breaker height and type (Nielsen and Hanslow, 1991). Therefore, using breaking wave height ma y indirectly incorporate these factors.
56 Eq. (10) was applied to examine the th ree-dimensional LSTF data. The wave runup was not directly measured at LSTF. It is assumed here that the maximum runup is equal to the upper limit of beach-profile chan ge. This is a reasonable assumption, based on the SUPERTANK data. For the spilling wave case, using the upper limit of beach change at 0.23 m as the value for total wave runup, the breaking wave height of 0.26 m resulted in an over-prediction of 13%. Fo r the plunging wave case, the upper limit of beach change was 0.26 m, which is almost equal to the 0.27 m breaking wave height. Based on these limited results, the LSTF result s support the predictive capabilities of the new formula (Eq. 10). It is worth noting that the 1.5 x 10 -3 m (0.15 mm) sediment used in the LSTF experiments is finer than the 2.2 x 10 -2 m (0.22 mm) used in the SUPERTANK experiment. A Conceptual Derivation of the Proposed Wave Runup Model Assuming a normally incident wave and neglecting longshor e variations and infiltration, the major forces acting on a wate r element in the swash zone in the crossshore direction, x, (Fig. 28) can be balanced as: 2sin 8 x xV f gxyzxyVxyz t (11) where, = density of water; g = gravitational acceleration; x, y, and z = length, width, and height, of the water element, respectively; = beach slope; f = a friction coefficient; and Vx = velocity. Eq. (11) can be reduced to 2sin 8x x V f gV tz (12)
57 Figure 28. Schematic drawing of forces acting on a water element in the swash zone. Assuming the friction force is negligible, an assumption supported by experiments discussed in Komar (1998), Eq. ( 12) is further reduced to sinxV g t (13) Integrating Eq. (13) with respect to time, yields sinxoVVgt (14) where, Vo = initial velocity. Integrating Eq. (14) again with respect to time gives the swash excursion, x, as a function of time, t : 2()sin 2ogt xtVt (15) From Eq. (14), the maximum uprush occurs at a time, tmax, when the velocity becomes zero: maxsinoV t g (16) Vx g x y z b
58 with a corresponding value of maximum uprush of 2 max() 2sinoV xt g (17) Assuming a small and planar foreshore sl ope (Shen and Meyer, 1963; Mase, 1988; Baldock and Holmes, 1999), i.e. tan sin the elevation of the maximum swash uprush Rsr_max, 22 _maxmax()tantan 2sin2oo srVV Rxt gg (18) Eq. (18) suggests that the maximum elevati on of swash runup is not a function of beach slope when friction forcing is neglected. The initial velocity Vo can be approximated by the velocity of the wave, Cg. In shallow water, the wave velocity is limited by the local water depth, hl. oglVCgh (19) Assuming a linear relationship between local breaking wave height, Hbl, and the water depth, hl bllHh (20) where is the breaker index. Eq. (19) then becomes 22 bl ogH VCg (21) Substituting Eq. (21) into Eq. (18) 2 _max222oblbl srVgHH R gg (22)
59 It is reasonable to assume that the initial Vo can be taken at the main breaker line, using significant breaker height, Hbs. Eq. (22) becomes _max2bs s rbsH R H (23) where = 1/2 Eq. (23) indicates a linear relati onship between breaking wave height and the maximum swash runup, supportin g the findings from the SUPERTANK experiment. Kaminsky and Kraus (1994) examined a la rge dataset, including both laboratory and field measurements, on breaking wave criter ia. They found that the majority of the values range from 0.6 to 0.8, which yields values from 0.63 to 0.83. Based on the discussion of Figs. 15 through 18, the swash runup constitutes approximately 83% of the total wave runup. Adding the 17% contribution from the wave setup, then the total wave runup, Rmax, is roughly proportional to the breaki ng wave height at unity, further supporting the new model developed from the SUPERTANK dataset.
60 Conclusions The SUPERTANK data indicate that the vertical extent of wave runup above mean water level on a non-scarped beach is approximately equal to the significant breaking wave height. A si mple formula for predicti ng the maximum wave runup, Rmax: max bs R H was developed and proven by a concep tual derivation. The new model was applied to the 3-dimensional LSTF experime nts, and accurately reproduced the measured wave runup. By including the surf similari ty parameter, as with several existing empirical formulas, the accuracy of the calcul ated wave runup decreased as compared to the measured values. In addition, the uppe r limit of beach cha nge was found to be approximately equivalent to the maximum vertical excursion. Because the maximum runup was found to be directly related to breaker height, and the upper limit of beach change was approximately equal to the maxi mum runup, the breaker height can be used to assess morphological changes on beach es due to runup. In other words, UL =Rmax =Hb. For monochromatic waves, the measured wave runup was much smaller than the breaking wave height. The l ack of low-frequency modulation limits the wave runup for monochromatic waves. The hydrodynamics and beach-profiles resulting from regular waves are not comparable to the real-world Therefore, results from studies with monochromatic waves are not useful in pr edicting wave runup and beach changes for natural beaches.
61 An exception to the direct relationship between breaking wave height, runup and upper limit of beach change concerns dune or beach scarping. The steep scarp substantially limits the uprush of swash mo tion, resulting in a much reduced maximum level as compared with the non-scarping cases. The nearly vertical scarp also induces technical difficulties of measuring the runup us ing wire gages. Most predictive equations are not expected to apply to scarped cases. Because the actual upper limit of beach change is controlled by the elevation of the backbeach or dune, the upper limit of wave runup was not found to correlate with the breaker height for these cases. Based on the SUPERTANK and LSTF e xperiments, the upper limit of beachprofile change was found to be approximately equal to the maximum vertical excursion of wave runup for waves between 0.20 and 1.00 m. Therefore, the limit of wave runup can serve as an estimate of the landward limit of beach change. Physical situations that are exceptions to this direct relationship are those with beach or dune scarping. Accurately predicting the maximum elevation of wave r unup was best predicted using the breaker height. The applicability of this relationship will contribute to predictive capabilities of assessing beach change for nour ishment and structure designs, engineering, and many other coastal management practices.
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68 Appendix I Notation The following symbols were used in this paper: A shape parameter relating to grain size and fall velocity Cg wave group velocity f friction coefficient g gravitational acceleration h still-water depth H wave height Hb breaking wave height Hbl local breaking wave height Hbs significant breaking wave height Hb_h high frequency component of wave height at the breaker line Hb_l low frequency component of wave height at the breaker line hl local water depth Ho significant deepwater wave height Hsl_h high frequency component of wave height at the shoreline line Hsl_l low frequency component of wave height at the shoreline line LL lower limit of beach change Lo deepwater wavelength n spectral peakness N Dean number Rmax maximum swash runup Rs significant wave runup Rsr_max elevation of maximum swash uprush R2 2% exceedence of runup T wave period tmax time of maximum swash excursion Tp peak wave period UL upper limit of beach change Vo initial velocity Vx velocity of a water particle in the cross-shore direction x cross-shore coordinate beach slope f foreshore beach slope breaker index x length of a water particle y width of a water particle
69 z height of a water particle wave setup M wave setup under monochromatic waves s l wave setup at the shoreline surf-similarity parameter water density sediment fall velocity