An analytical model to predict dune erosion due to wave impact

Coastal Engineering 51 (2004) 675 – 696 www.elsevier.com/locate/coastaleng

An analytical model to predict dune erosion due to wave impact Magnus Larson*, Li Erikson, Hans Hanson Department of Water Resources Engineering, Lund Institute of Technology, Lund University, Box 118, S-221 00, Lund, Sweden Available online 23 August 2004

Abstract An analytical model is developed to calculate recession distance and eroded volume for coastal dunes during severe storms. The transport relationship used in the model is based on wave impact theory, where individual swash waves hitting the dune face induce the erosion. Combining this relationship with the sediment volume conservation equation describes the response of the dune to high waves and water levels. Four different data sets on dune erosion, originating from the laboratory and the field, were employed to validate the model and to determine the value of an empirical transport coefficient appearing in the analytical solutions. The time evolution of dune recession observed in the different data sets was well described by the model, but the empirical coefficient showed some variation between cases, especially for the field data. In practical applications of the model, it is recommended to use a range of coefficient values to include an uncertainty estimate of calculated quantities, such as recession distance and eroded volume. D 2004 Elsevier B.V. All rights reserved. Keywords: Dune erosion; Analytical model; Wave impact; Large wave tank data; Storm erosion; Runup height; Swash bore

1. Introduction Coastal dunes often constitute the final defense line against high waves and water levels during severe storms. If they are overtopped or breached, serious damage due to flooding and direct wave attack could occur, resulting in loss of life and property. Thus, it is of significant value to be able to predict the impact of a storm on a dune in terms of recession distance, eroded volume, and probability of breaching. Several analytical and numerical models have been developed * Corresponding author. Fax: +46 46 222 4435. E-mail address: [email protected] (M. Larson). 0378-3839/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.coastaleng.2004.07.003

for this purpose (see discussion in the following section), but many of them lack a clear physical basis and have only been tested against limited data. Analytical models typically require marked simplifications in the description of the governing processes, forcing, and initial and boundary conditions, whereas numerical models can deal with these aspects with less restrictions. However, analytical models still have their use since the simplicity makes them easy to apply, which is valuable in the initial stage of a project when approximate estimates are required. In addition, the limited work needed to apply them, as well as the required amount of input data, makes the analytical models attractive to employ. Another benefit is that

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the main governing parameters are easily identified and asymptotic behavior can be explored. Two classical approaches exist to estimate the effect of high waves and water levels during a storm on coastal dunes, namely the equilibrium profile approach (Vellinga, 1986; Kriebel and Dean, 1993) and the wave impact approach (Overton et al., 1987, 1994a). Equilibrium profile theory assumes that the beach profile strives toward an equilibrium state defined by the wave and water level conditions, which geometrically determines the response of the dune. The wave impact approach estimates the sediment transport from the dune and associated profile change by the waves directly hitting it. Existing numerical models to predict the response of dunes tend to be based on the equilibrium theory with limited description of the physical processes (Kriebel and Dean, 1985; Larson and Kraus, 1989a), although some attempts have also been made to use the wave impact approach (Overton et al., 1994b; Nishi and Kraus, 1996). These models typically focus on local areas since they describe the evolution of single profile lines. However, when simulations are to be used in decisions concerning design, planning, and management in the coastal zone, it is often the response at the regional scale that is of interest. Presently, numerical models have difficulties to address this scale in time and space, and analytical models may be an alternative when assessing the effects over large areas, and for long time series of data. The main purpose of this study was to develop a simple, yet physically based model of dune erosion for approximate estimates of eroded volume and dune retreat during severe storms. Such a model could be applied to large input data sets, in time and space, which would provide a basis for statistical estimates of exceedance frequencies for specific events. For these types of applications, it was decided that a model, which allowed for analytical solutions, would be advantageous to develop. By combining transport relationships based on impact theory with the sediment volume conservation equation for the dune, a model of the dune response was obtained for which analytical solutions were available, if the geometrical, forcing, and initial conditions were simple enough. The needed schematization of these conditions implied that an empirical coefficient appeared in the solution with a value estimated using a number of high-quality data sets on dune erosion from the field

and laboratory. The basic model formulation is easily solved numerically for arbitrary conditions, but this method is not discussed in the present paper. In the following section, previous efforts to model dune erosion are first reviewed with emphasis on analytical approaches. Then, an overview of the data sets employed is given, encompassing four different sets from both the laboratory and the field. The theoretical development of the dune erosion model is described, and several analytical solutions are presented for varying forcing conditions. Finally, the data sets are used to validate the general model formulation and to determine the value of the main empirical transport coefficient. The paper ends with some conclusions.

2. Previous studies Several analytical models describing dune erosion have been developed (e.g., Bruun, 1962; Edelman, 1972; Dean and Maurmeyer, 1983; Vellinga, 1986; Kobayashi, 1987; Kriebel et al., 1991; Kriebel and Dean, 1993; Komar et al., 1999). Bruun (1962) presented a method to estimate shoreline retreat as a function of sea-level rise based on his equation describing the equilibrium shape of a beach (Bruun, 1954). The main assumption was that over a long time period, the beach profile would adjust to an increase in sea level through a shoreward retreat, where the magnitude of the retreat would be determined by translating the equilibrium profile upward and landward in such a manner that sediment was conserved. Basically, such an approach implies that the horizontal recession is directly proportional to the increase in sea level. Edelman (1972) developed an equation to estimate dune erosion due to storm surge employing the same basic assumption as Bruun (1962). The resulting equation involves the active width of the profile, the breaker depth (before and after storm surge), and dune height referenced to mean sea level (before and after storm surge). Dean and Maurmeyer (1983) presented an implicit equation for the nondimensional recession distance as a function of storm surge and breaker depth. The primary difference between Dean’s model and Edelman’s model is that the former considers an abrupt rise in sea level with the equilibrium occurring over a long time period after

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the sea-level rise. The resulting offshore end of the profile is vertical, whereas it is wedge-shaped in the profile described by Edelman (1972). Vellinga (1986) refined and improved the Edelman approach and Van de Graaff (1986) developed a probabilistic design method for dunes based on the Dutch predictive methods. Kobayashi’s (1987) analytical model is based on the continuity equation of bottom sediments combined with an empirical sediment transport relationship, relating the transport rate to the local bottom slope using a depth-dependent diffusion equation. The model resulted in a partial differential equation that could be solved for simple initial and boundary conditions. Kobayashi applied his model to Hurricane Eloise data (Chiu, 1977) with a rectangular storm surge hydrograph and found reasonable agreement with the field data, showing the characteristic erosion time scale was much greater than the equivalent storm surge duration. Kriebel et al. (1991) presented an analytical model relating the recession distance (or eroded volume) to the storm surge, dune height, and foreshore length. The equation was based on a convolution formulation, where the water level was the prime driving force for the erosion. They provided solutions for several different beach and dune system configurations, including those with a wide horizontal backshore. A common characteristic of most of these models is that the sea level is assumed to rise suddenly at t=0 and to remain constant during the storm (not assumed by Kriebel et al., 1991 though), and sediments eroded from the dune are deposited in the vicinity of the initial breaker point, causing the breaker point to move offshore. Most of the models also assume that the dune face recedes as a vertical front. These efforts at predicting the rate of dune erosion depend upon the development of a post-storm equilibrium profile, where the volume of sand lost from the dune is determined by the quantity of sand required to establish the new profile (Hughes and Chiu, 1981). In reality, morphologic response of the profile is slow in comparison to the faster variation of the hydrodynamic forcing. Storm surge and wave conditions are not constant, and storm durations seldom exceed the time necessary for a profile to reach a post-storm equilibrium profile. As such, many of these timeindependent analytical models tend to over-predict the

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amount of dune retreat. Dette (1986) and Dette and Uliczka (1987) observed from large wave tank (LWT) experiments that the recession of beach contours, as well as the eroded volume above specific contours, decreased exponentially in time approaching an equilibrium state. Larson and Kraus (1989a) accounted for the decay of the transport rate q with time by using an empirically based equation q(t)=q mo/ (1+at), where q mo is the transport at time t=0 and a an empirical rate coefficient. Kriebel et al. (1991) found an empirical time decay response described by a similar equation with the decay term being an exponential function. Such relationships could theoretically be used with the maximum erosion potential derived by different authors to make the relationships sensitive to the storm duration. An alternative method to predict dune erosion was proposed by Fisher and Overton (1984) and Nishi and Kraus (1996), where erosion from the impact of each individual swash bore is used to determine dune recession. The total storm erosion is a function of the frequency and intensity of the impacts. Wave impacts can be described mathematically by a change in momentum flux of the waves as they hit the dune face. The wave impact theory has been empirically validated using small- and prototype-scale wave tank data. Overton et al. (1987, 1994a) and Overton and Fisher (1988) conducted a series of small-scale wave tank experiments whereby they generated single bores of varying height that impacted a vertical dune face at the opposite side of the tank. Following each bore impact, they measured the amount of sand eroded and found a linear correlation between measured eroded volume and swash force. Using a small wave tank, Overton et al. (1994a) showed that sand grain size and dune density are significant parameters in the swash force and dune erosion relationship. They found that in general, decreasing the sand grain size and increasing density of the dune improved the strength of the dune. Furthermore, increasing the dune density of sediments with D 50=0.22 mm only marginally improved the strength of the dunes, while the same increase in dune density with D 50=0.78 mm improved the strength of the dunes significantly. Although they showed that dune density was statistically significant, the effect of dune density in the force–erosion relationship was much less significant than sand size.

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An investigation with the same aim of assessing the difference in volume eroded by changing the grain size was conducted using the large wave tank facility at Oregon State University (Overton et al., 1990). Once again, a linear relationship between calculated swash force (based on measured wave heights and bore velocities in front of the dune) and measured eroded volume was observed. However, there was no significant correlation with respect to a change in grain size. They suggested that this was due to the relatively small difference in grain size (D 50=0.23 and D 50=0.33 mm) compared to the prototype-scale hydrodynamic conditions. Data from the SUPERTANK experiment (test runs ST50 and ST60; Kraus and Smith, 1994) have also been used to verify wave impact theory. Overton et al. (1994b) found a strong relationship between measured swash force (derived from wave height and bore velocity) and eroded volume. The measurements were based on the summation of the force in a given interval of time versus the volume eroded in the same time interval. The SUPERTANK data have also been used to verify two numerical models employing wave impact forcing terms at the dune face (Overton et al., 1994b; Nishi and Kraus, 1996). Both models showed that a linear relationship between the summation of calculated swash forces over selected time intervals and eroded volumes could predict the measured evolution of the dune.

3. Data employed Four different data sets on dune erosion originating from both the laboratory and the field were employed in this study to validate the analytical model and to determine applicable values of the main empirical coefficient in the model. The laboratory data included both small-scale and prototype-scale experiments carried out for monochromatic and random waves with constant or varying water level. Some data encompassed time series of profile measurements, whereas other data only yielded the total eroded volume during an experimental case. Most of the field cases studied were obtained from a database describing the impact of major storms at several locations along the United States North Atlantic coast. For the field data, only the total eroded volume during an

event was available to quantify impact of the storms on the dune. In the following, brief descriptions of the four data sets are given, whereas a summary of the main characteristics for the experimental cases or storm events in each data set are given in the section where the analytical model is applied. The first data set was derived from four experiments carried out in three LWTs, namely the Delta Flume (Vellinga, 1986), the bGrosser Wellen KanalQ (GWK; Dette and Uliczka, 1987; Dette et al., 1998), and the O.H. Hinsdale Wave Research Laboratory at Oregon State University (OSU; Kraus and Smith, 1994). These experiments were all designed to simulate dune erosion under storm conditions at prototype scale. Both monochromatic and random waves were employed, and the water level was kept constant during an experimental case. Several profile surveys, including the dune region, were performed during such a case yielding a series of data points, which were used for comparison purposes for each case when employing the analytical model. In total, 10 experimental cases were available for comparison with the analytical model, namely three from the Delta Flume, four from the GWK, and three from OSU. This data set also yielded general insights to the process of dune erosion forming the basis for some of the key assumptions in the analytical model. The second data set involved a small-scale laboratory experiment on dune erosion with a time-varying mean water level. Hughes and Chiu (1981) conducted a small-scale experiment on dune erosion with the objectives to derive new laws for hydraulic similitude and to study the effects of wave height, wave period, surge level, surge duration, and dune height on dune erosion. Various combinations between the experimental parameters gave 41 different cases for which tests were performed. Each case started from an equilibrium profile backed by a dune and the water level was increased linearly over a typical time span to the peak surge level. The wave conditions were monochromatic and a median grain size of about 0.15 mm was used in the experiment. The third data set originated from Hasaki Beach on the Pacific Coast of Japan, where Kubota et al. (1999) conducted several field experiments on sediment transport and profile evolution in the swash zone. During two of the experiments (carried out in 1994 and 1997), the profile shape that evolved on the

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foreshore had the characteristics of a dune with waves causing transport and erosion by direct impact. Profile surveying was done regularly during an experimental case, but because these measurements were made at poles that were fixed in space, the measurement resolution of the dune face retreat only permitted the estimate of the total eroded volume during a particular case. The experiments were made during rising tide, but for the data used in the model comparison the change in the water level was small and it was neglected. At Hasaki Beach, the median grain size is approximately 0.18 mm. The fourth data set was derived from the storm erosion field data presented by Birkemeier et al. (1988). In that report, data from 13 storm events impacting the United States North Atlantic coast were compiled summarizing the effects on seven different beaches. These beaches were Nauset Beach, Massachusetts; Misquamicut Beach, Rhode Island; Westhampton and Jones Beaches, New York; Long Beach Island, Atlantic City, and Ludlam Beach, New Jersey. At each site, 7–19 profile lines were surveyed before and after the storm providing reliable estimates on the amount of erosion and recession distances resulting from each storm. The time history of hindcasted wave data for shallow water and measured water elevations were also presented. Individual events were selected from the storm database for which the profile shape were similar to the schematized shape assumed in the analytical solution, that is, the profile had a distinct foreshore backed by a clearly identifiable dune (to be discussed in the following).

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Fig. 1. Definition sketch for modeling dune erosion due to the impact of runup waves.

where C E is an empirical coefficient. The weight of the eroded volume DV is given by DW ¼ DV qs ð1  pÞg

ð2Þ

where q s is the density of the sediment, p porosity, and g acceleration of gravity. It is less straightforward to estimate the swash force F, but it is a result of the change in the momentum of the bore hitting the dune according to (for a single bore generating the swash force F o) Fo ¼

d duo ðmo uo Þ ¼ mo dt dt

ð3Þ

where m o is the mass of the bore (conserved), u o the speed of the bore, and t time. The mass of the bore (per unit width) is estimated as (Nishi and Kraus, 1996) mo ¼

1 qho so 2

ð4Þ

in which h o and s o are the height and length of the bore, respectively, and q the density of water. The deceleration of the bore is obtained from (Nishi and Kraus, 1996)

4. Theoretical developments 4.1. Governing equation

duo uo c dt T

ð5Þ

A basic assumption in estimating dune erosion from bwave impact theoryQ is that there is a linear relationship between the impact (force F on the dune due to change in the momentum flux of the bores impacting the dune) and the weight (DW) of the sediment volume eroded from the dune. This may be written (Fisher et al., 1986; Overton et al., 1987; Nishi and Kraus, 1996; see Fig. 1 for a definition sketch)

where T is the period at which waves hit the dune taken to be equal to the incident wave period. Furthermore, it is assumed that the speed of the bore is related to the bore height according to (e.g., Cross, 1967; Miller, 1968) pffiffiffiffiffiffiffi uo ¼ Cu gho ð6Þ

DW ¼ CE F

where C u is an empirical coefficient. If the bore wavelength is taken as the product between the bore

ð1Þ

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speed and the period, the swash force for a single bore is expressed as 1 2 1 u4 quo ho ¼ q o2 2 2 gCu

Fo ¼

ð7Þ

For a number of bores impacting the dune during a time period Dt, the total swash force may be written as F¼

1 u4o Dt q 2 gCu2 T

ð8Þ

where Dt/T represents the number of incoming waves. Equating swash force and weight of eroded volume (Eqs. (2) and (8)) gives DV qs ð1  pÞg ¼

u4o

1 Dt CE q 2 2 gCu T

ð9Þ

Some rearranging yields DV 1 CE q u4o 1 ¼ Dt 2 Cu2 qs g2 T ð1  pÞ

ð10Þ

Although the process of dune erosion is more or less discrete, depending on the failure mechanism (Erikson et al., 2003), an average rate of dune erosion q D might be derived, qD ¼

dV 1 CE q u4o 1 ¼  dt 2 Cu2 qs g2 T ð1  pÞ

ð11Þ

This estimate of u o is obtained by regarding the bore as a slug of water moving along the foreshore (friction neglected; see Waddell, 1973; Hughes, 1992). By substituting this expression into the governing differential equation, we can solve for how z o varies in time, if the volume of sand in the dune V is expressed in terms of z o. Examining laboratory data (primarily from the German GWK), it seems that the foreshore slope is relatively constant during dune retreat (the assumed profile shape during retreat is indicated by the dashed line in Fig. 1), and if this assumption is employed, V is given by V ¼

1 ðDs  zo Þ2 2 tanbf

ð13Þ

where D s is the elevation of the dune crest above the reference level from which z o is taken and b f the foreshore slope. The change in dune volume with time might be expressed as dV dV dzo ðDs  zo Þ dzo ¼ ¼  dt dzo dt tanbf dt

ð14Þ

Substituting this into the governing equation (Eq. (11)) together with the expression for u o (Eq. (12)) yields  2 2 us  2gzo dzo ðDs  zo Þ ¼ Cs tanbf dt g2 T

where a minus sign was introduced since the dune volume must decrease with time (erosion). This equation will form the basis for calculating dune erosion. It can easily be implemented into a numerical model (see Nishi and Kraus, 1996) requiring that u o is provided as well as a scheme for distributing the eroded sediment on the foreshore.

where

4.2. Analytical model

4.3. Analytical solution

In order to continue the derivation and arrive at an analytical model, some simplifying assumptions are introduced. First of all, the bore speed in front of the dune face (u o) is needed, and it is estimated as

The governing differential equation (Eq. (15)) for z o is separable and can be integrated if u s is constant (in the general case, u s would be varying with time). However, the solution is implicit in u o according to

u2o ¼ u2s  2gzo

ð12Þ

where u s is the velocity of the bore as it starts its travel up the foreshore, and z o is the elevation difference between the dune foot and the beginning of the swash.

Cs ¼



1 CE q 1 2 2 Cu qs ð1  pÞ

ð15Þ

ð16Þ

 2gDs  u2s 

2gðzm  zo Þ   u2s  2gzo u2s  2gzm  2  us  2gzo t þ ln 2 ¼  Cs 4tanbf T us  2gzm

ð17Þ

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where the initial condition z o=z m when t=0 was applied (z m is the initial dune foot elevation). Besides the coefficient C s and the geometry of the dune configuration, the speed u s has to be specified in the solution. Although it might be possible to derive this speed by using a number of data sets and trying to correlate u s with the wave properties, another option is to express u s in terms of the runup height (R). At the limit of the runup, the velocity u o should be zero, implying that (Eq. (12) with z o=R and u o=0) u2s ¼ 2gR

ð18Þ

This means that predictive formulas for R may be used to derive u s. Rewriting Eq. (17) in non-dimensional form and using Eq. (18) gives 







Ds ðzm =R  zo =RÞ 1  zo =R 1 þ ln 1  zm =R R ð1  zo =RÞð1  zm =RÞ t ð19Þ ¼  Cs 4tanbf T

This solution is only valid for certain values of z o, that is, z m /Rbz o /Rb1, where 0bz m /Rb1. Also, if D s / Rb1, z o /R should be less than D s /R (after that, the wave would have eroded back to a point where the foreshore crosses the top of the dune, which corresponds to the maximum allowable retreat). For example, Fig. 2 shows the time evolution of the dune foot for several different values of D s/R and z m/R=0.2.

681

The eroded volume after a certain time (DV E) when the dune foot is located at elevation z o is given by DVE ¼

 1  2Ds ðzo  zm Þ þ zm2  zo2 2tanbf

ð20Þ

In deriving Eq. (17), it was assumed that the dune face is vertical and that the retreat of the dune proceeds along a constant foreshore slope b f. These two assumptions may be relaxed without any major modifications of the solution. If the dune face has a slope of b D and the retreat of the dune occurs with a different foreshore slope b R, Eq. (17) (and other equations) is still valid if the following substitution is made: 1 Ytanbf 1 1  tanbR tanbD

ð21Þ

For the case b D=908 and b R=b f, the original solution is retained. Note that the dune face slope must be larger than the slope in front of the dune during retreat (b DNb R) for this solution to make sense. 4.4. Simplified solutions The LWT data indicated that the foreshore slope (b f) approximately remained constant during dune face retreat, implying that z o varies with time making

Fig. 2. Non-dimensional time evolution of the relative dune foot location for different relative dune heights (z m/R=0.2).

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the analytical solution more complicated. If the variation in z o with time is ignored, which may be appropriate when the retreat of the dune is small, more complicated cases may be treated including a timevarying runup height or mean water level (or both simultaneously). Thus, Eq. (11) may be written

Thus, a simple model for the water level and runup height variation during a storm would be   pt ð27Þ zo ¼ zi  za sin Ts

dV ð R  zo Þ2 ¼  4Cs dt T

R ¼ Ri þ Ra sin ð22Þ

where the definition in Eq. (16) was used together with Eqs. (12) and (18). For the case when R and z o are constants, the following solution to Eq. (22) is obtained V ¼ Vo  4Cs ð R  zo Þ2

t T

ð23Þ

where the initial condition V=Vo at t=0 was employed. From this equation, the time t B needed to erode away an entire dune (i.e., breach the dune) with the volume Vo is given by tB ¼

T Vo 4Cs ð R  zo Þ2

ð24Þ

Also, the eroded volume DV E=VoV after time t may be determined from Eq. (23) to be DVE ¼ 4Cs ð R  zo Þ2

t T

ð25Þ

The effect of changing the mean water level on dune erosion can be modeled by specifying a varying z o. For example, a linear increase in the water level would be described by z o=z iat, where z i is the vertical distance from the beginning of the swash to the dune foot at t=0, and a is a coefficient describing the rate of increase (the negative sign indicates a reduction in the vertical distance with time, which corresponds to a water level increase). Substituting this expression for z o into Eq. (22) and solving yields   Cs 1 ð R  zi Þ2 t þ að R  zi Þt 2 þ a2 t 3 DVE ¼ 4 3 T ð26Þ Another water level variation of interest that might be used to schematize the conditions during a storm, for example, is a sinusoidal change. Furthermore, during a storm, the wave conditions would typically vary with time producing a varying runup height.



pt Ts

 ð28Þ

where z i is the vertical distance from the beginning of the swash to the dune foot at t=0, z a the amplitude of the sinusoidal water level variation, R i the runup height at t=0, R a the amplitude of the sinusoidal runup variation, and Ts the duration of the storm (0btbTs). Substituting Eqs. (27) and (28) into Eq. (22) produces the following equation to solve !2   dV Cs pt ¼ 4 ð29Þ  zD RT sin dt Ts T where RT =R a+z a and z D=z iR i. In order for Eq. (29) to be meaningful, dV/dtb0, that is, the dune is eroding. If z Db0, this condition is always fulfilled, but if z iNR i, the condition is only fulfilled during a portion of the storm when the waves hit the dune causing erosion. The time t L when the waves start hitting the dune is determined by   Ts zD tL ¼ arcsin ð30Þ p RT Thus, the time when the dune is exposed to erosion is t LbtbTst L. The solution to Eq. (29) gives    Cs Ts 1 2 2 R þ zD DVE ¼ 8  tL 2 T T 2     ptL Ts ptL 2 Ts þ RT sin 2  2RT zD cos 4p Ts p Ts ð31Þ Since the function on the right-hand side of Eq. (29) is symmetric, the integration of the equation may be performed between t L and Ts/2 after which the result is multiplied by a factor 2. A sinusoidal variation in water level with time may be suitable for some storms, but in other cases, the storm surge hydrograph would be more peaked with a shorter duration for the high water levels. Larson and Kraus (1989b) simulated the response of beach fills to different types of storms. The hydrograph during a

M. Larson et al. / Coastal Engineering 51 (2004) 675–696

hurricane was described by an inverse hyperbolic cosine square, which yielded a narrower more peaked hydrograph than for a northeaster for which a cosine squared function was employed. Thus, for some storms an alternative schematization to Eq. (27) of the water level variation might be necessary. One possibility is to use the following expressions:  n 2t zo ¼ zi  za ð32Þ Ts  R ¼ Ri þ Ra

2t Ts

n ð33Þ

where n is an empirical power that determines the shape of the hydrograph and 0 bt b Ts /2 (z o and R are assumed to be symmetric around t=Ts /2). The runup was schematized in the same manner as the water level to simplify the analytical solution. For nN1, a concave hydrograph shape is obtained, for nb1, a convex shape, and for n=1, a linear variation. Thus, the smaller n is, the longer the duration will be that exceeds a specific high water level. As for the case of the sinusoidal water level variation, it is only when the waves hit the dune that erosion will occur. Analogous to Eq. (30), the time when waves start hitting the dune is tL ¼

1 Ts 2

zD RT

ð Þ

1=n

ð34Þ

Solving the governing equation (Eq. (22)), using Eqs. (32) and (33) for z o and R, respectively, yields (integrating between t L and Ts/2, and multiplying the result by a factor of 2)  2n    Cs R2T Ts 2tL RT zD DVE ¼ 8  tL  2 T 2n þ 1 2 Ts nþ1  n     Ts 2tL Ts   tL tL þ z2D  2 Ts 2 ð35Þ

5. Model comparison with data The four data sets employed provided different possibilities for validating the analytical model and for determining parameter values. Data from the LWT

683

experiments were first utilized to test the overall capabilities of the model to reproduce the retreat of the dune face. This data set was also used to determine the two main parameters appearing in Eq. (19), namely the transport coefficient C s and the runup height R, through a least-square fit procedure. As described below, an empirical equation similar to the Hunt (1959) formula was developed to predict R from which the optimum value of C s was obtained through comparison with the LWT data. This empirical equation for R was used when the analytical model was employed for the other data sets. The simplified set-up, as well as the quality and extent, of the LWT data made it possible to use the full solution in Eq. (19) when comparing it with the data. However, for the other data sets which mainly provided the total eroded volume during a case or an event, and where the mean water elevation varied with time, the simplified solutions ((Eqs. (25), (26), and (31)) had to be employed. The Hughes and Chiu data set allowed for testing of a linearly increasing water level with time and its effect on dune erosion. For practical applications, validation against field data was necessary to ensure that the model produced reasonable predictions. However, in order to find an analytical solution describing the field data, marked schematization of the wave and mean water level input was necessary, which contributed to the larger scatter in the predictions compared to the laboratory data. The empirical relationship for R derived from the LWT data was used when modeling the three other data sets and only the value on C s was left to determine in the least-square fitting process. 5.1. Large wave tank data As previously indicated, data from three LWTs were used in the first step to develop and validate the analytical model, namely from the Delta Flume (Vellinga, 1986), the GWK (Dette and Uliczka, 1987; Dette et al., 1998), and the OSU flume (Kraus and Smith, 1994). The conditions for each experimental case are listed in Table 1, including the number of surveyed profiles, dune elevation (above still water level), median grain size, foreshore and under-water slopes, deepwater significant (or monochromatic) wave height (unrefracted), and wave period. The studied profiles had a nearly vertical

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Table 1 Overview of experimental cases from the LWT data used for validating the analytical model of dune erosion Experimental case

N

Dune elevation D s (m)

D 50 (mm)

Foreshore slope (m)

Underwater slope (s)

H o (m)

T (s)

MAST ST50 ST602 ST603 Vellinga1 Vellinga2 Vellinga5 Dette1 Dette2 Dette3

10 4 3 3 3 3 3 8 17 12

1.8 1.6 1.1 0.9 2.0 2.1 3.7 2.0 2.0 2.0

0.30 0.22 0.22 0.22 0.23 0.23 0.23 0.33 0.33 0.33

0.21 0.16 0.17 0.18 0.12 0.10 0.06 0.26 0.24 0.20

0.07 0.07 0.07 0.07 0.04 0.05 0.05 0.25 0.25 0.05

1.3 0.8 0.8 0.5 1.6 1.6 2.0 1.6 1.6 1.6

5.0 4.0 4.5 6.0 5.4 5.4 7.6 6.0 6.0 6.0

dune face and sloping foreshore between the dune face and still water level in correspondence with the assumed shape in the analytical solution. Such a vertical dune face represents idealized conditions of dune face retreat during a storm (e.g., Edelman, 1972; Kobayashi, 1987; Kriebel and Dean, 1993). The following notation was used for the experimental cases employed: Vellinga1, Vellinga2, and Vellinga5 are from Vellinga (1986); Dette1, Dette2, and Dette3 are from Dette (1986); ST50, ST602, and ST603 are from Kraus and Smith (1994); and MAST from Dette et al. (1998). The total number of points (N) in each case deemed usable for comparison with the analytical model is the same as the number of profiles given in the second column of Table 1. Dune elevation ranged from about 1 m to nearly 4 m, and the median grain size, D 50, varied between tests ranging from 0.22 to 0.33 mm. The dune was compacted for ST50 while dunes from ST602 and ST603 were not compacted (it is not clear if the dunes in the other experiments were compacted). Foreshore and underwater slope values were obtained from published profile plots if no estimates were given in the literature, and slopes ranged from 0.06 to 0.26 and from 0.05 to 0.25, respectively. The offshore deepwater wave height varied between 0.5 and 2.0 m and the wave periods between 4.0 and 7.6 s. It should be noted that the values listed here for SUPERTANK experiment ST50 are mean values, as the significant wave height and period varied somewhat during the simulation. Wave conditions corresponded to a TMA spectrum for all cases except the Dette data set, where the first and third

case employed monochromatic waves and the second case a Jonswap spectrum. The general solution of the dune face retreat, taking into account the effect of a changing z o on the bore front velocity, is given by Eq. (19). In addition to the geometry of the profile (D s, z m, and b f) and T, there are two parameters (R and C s) that determine how z o varies with t. Initially, the optimum values of these two parameters were determined through a simultaneous least-square fit toward the data on z o from the 10 cases. For all cases, it was possible to obtain a good fit between the model and the data. Figs. 3 and 4 are examples of the agreement between the model and the data for cases Dette2 and MAST, respectively. This agreement lends credibility to the basic model formulation and derived solution, although in order to obtain a model that can be used for reliable estimates of dune erosion R and C s must be accurately predicted. Initially, several different runup height formulas were tested to predict R (e.g., Hunt, 1959; Mase and Iwagaki, 1984; Holman, 1986; Larson and Kraus, 1989a; Mayer and Kriebel, 1994), but the resulting optimum values on C s displayed a large variation between the individual cases making application of the model difficult. Instead, a new empirical relationship to calculate the runup height was derived based on the R estimates obtained in the general least-square fit procedure involving both parameters resulting in R ¼ 0:158

pffiffiffiffiffiffiffiffiffiffi Ho Lo

ð36Þ

where H o is the wave height and L o the wavelength, both taken in deep water, and the root-mean-square

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Fig. 3. Measured time evolution of the relative dune foot location for experimental case Dette2 with both runup height (R) and transport coefficient (C s) optimized through least-square fitting.

(rms) wave height was used for random waves. Fig. 5 displays the agreement between Eq. (36) and the optimum R values obtained through least-square fitting Eq. (19) to LWT data. Eq. (36) is formally identical to the Hunt (1959) formula, if a slope of

about 98 (tan b f =0.158) is employed. The mean foreshore slope for the cases studied was approximately 88 (tan b f =0.14) implying that an overall representative foreshore slope would be suitable for computing R. Using calculated slope values in the

Fig. 4. Measured time evolution of the relative dune foot location for experimental case MAST with both runup height (R) and transport coefficient (C s) optimized through least-square fitting.

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Fig. 5. Optimum runup height for the analytical dune erosion model determined through least-square fitting against large wave tank data and an empirical relationship.

Hunt formula from the individual cases produced less agreement with the data compared to Eq. (36). This may partly be due to the difficulties in defining the foreshore slope. Thus, it is believed that Eq. (36) will provide a good basis for computing R to be used in Eq. (19). It should be noted that although R is the runup height, the simplicity in the present description

implies that R in fact represents an overall measure of the hydrodynamic impact on the dune and as such may not be directly described by standard runup formulas. Using the predictions of R by Eq. (36), optimal C s values were again derived by least-square fitting the solution toward the data in order to arrive at a

Fig. 6. Measured time evolution of the relative dune foot location for experimental case Dette2 with the transport coefficient (C s) optimized through least-square fitting and runup height (R) estimated from a predictive relationship.

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consistent method to estimate C s. This produced a more reasonable range of variation in the value of C s than what was obtained when both R and C s were allowed to vary freely. The mean C s value was 1.8103 with a standard deviation of 0.89 and about 80% of the values being in the range 1.0–2.5103. This still gives a marked variation in C s, but the effect on the dune retreat is typically less dramatic. Figs. 6 and 7 illustrate the fit obtained for Dette2 and MAST, respectively, using Eq. (36) and the optimum value on C s. These two cases implied the smallest and largest change in the fit compared to allowing both parameters to vary (compare with Figs. 3 and 4). The change for Dette2 is marginal, whereas for MAST the total amount of erosion is underestimated mainly because the predicted R by Eq. (36) is not large enough. In principal, R determines the amount of erosion needed before equilibrium is attained, whereas C s gives the rate at which equilibrium is approached. 5.2. Hughes and Chiu small-scale laboratory data The data on dune erosion from Hughes and Chiu (1981) allowed for testing of the analytical model with a linearly varying mean water level in time. Table 2 summarizes the experimental cases in terms of the key parameters. In total, 41 cases were

687

conducted but three of them involved no surge and were omitted in the present application leaving 38 cases for comparison. Monochromatic waves were employed and the height was measured in 0.46 m water depth. The dune configuration was geometrically similar in all cases with an initially constant foreshore slope of tan b f =0.21 seaward of the dune. Three different dune heights (D s) were employed, namely 0.24, 0.30, and 0.35 m. Regarding the hydrodynamic forcing, the following input conditions were used in various combinations (see Table 2): wave height=0.77–0.132 m; wave period=0.97, 1.16, and 1.36 s; peak surge=0.097, 0.134, and 0.171 m; and duration until peak surge=21–68 min. The experimental procedure involved applying a linearly increasing mean water level with time up to the peak surge after which the water level was constant and the dune was allowed to erode toward equilibrium conditions. Only data on eroded volume from the first part of the experiment when the water level was increasing were utilized in the present study. Eq. (26) was used to estimate the volume eroded from the dune during each experimental case. In order to obtain the best possible fit for the equation, the right-hand side of Eq. (26) leaving out C s was plotted against the measured eroded volume for each case

Fig. 7. Measured time evolution of the relative dune foot location for experimental case MAST with the transport coefficient (C s) optimized through least-square fitting and runup height (R) estimated from a predictive relationship.

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Table 2 Overview of experimental cases from the Hughes and Chiu data used for validating the analytical model of dune erosion Case no.

Wave height (m)

Wave period (s)

Surge level (m)

Surge duration (min)

Dune height (m)

Eroded volume (m3/m)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 38 39 40 41

0.077 0.077 0.077 0.128 0.128 0.128 0.102 0.077 0.102 0.128 0.077 0.102 0.128 0.079 0.106 0.132 0.079 0.106 0.132 0.079 0.106 0.132 0.080 0.107 0.080 0.107 0.080 0.107 0.079 0.106 0.079 0.106 0.079 0.106 0.079 0.106 0.079 0.106

0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.36 1.36 1.36 1.36 1.36 1.36 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16

0.098 0.098 0.098 0.098 0.098 0.098 0.098 0.134 0.134 0.134 0.171 0.171 0.171 0.098 0.098 0.098 0.134 0.134 0.134 0.171 0.171 0.171 0.098 0.098 0.134 0.134 0.171 0.171 0.098 0.098 0.134 0.134 0.171 0.171 0.134 0.134 0.171 0.171

30 57 17 59 21 41 38 40 43 48 45 52 51 39 42 49 48 51 55 53 54 55 45 47 49 52 52 58 38 43 47 51 48 54 50 51 52 68

0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.244 0.295 0.295 0.295 0.295 0.295 0.295 0.345 0.345 0.345 0.345

0.0291 0.0232 0.0244 0.0270 0.0182 0.0256 0.0286 0.0399 0.0506 0.0469 0.0470 0.0609 0.0622 0.0317 0.0327 0.0390 0.0498 0.0558 0.0514 0.0615 0.0729 0.0700 0.0423 0.0458 0.0431 0.0578 0.0615 0.0744 0.0331 0.0377 0.0591 0.0594 0.0543 0.0777 0.0591 0.0672 0.0668 0.0852

(see Fig. 8). The parameter a in Eq. (26) was obtained by dividing the peak surge level with the duration until this surge was attained. It should be noted that the basic transport relationship describing dune erosion (Eq. (1)) does not include the dune height implying that the analytical solution is independent of this variable, unless there is a connection between dune volume and horizontal retreat, which is the case when deriving Eq. (19). The runup height R was calculated using Eq. (36), as previously mentioned. In some cases the predicted R was not large enough to

initially reach the dune foot (Rbz i), but the surge had to reach a certain level before this occurred. For those cases the time when the runup waves would start hitting the dune was determined from t L=(z iR)/a and the analytical solution was adjusted to start at this time. Fig. 8 illustrates that the analytical solution describes the data quite well (the solid line is the result of a least-square fit of Eq. (26) to the data). The estimated optimal C s is 0.82103, where the line was forced through the axis origin. This value is

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Fig. 8. Eroded volume as a function of an impact parameter obtained from an analytical solution for the laboratory data by Hughes and Chiu (1981).

approximately within the range of optimal values obtained for the LWT data, although it is somewhat lower than the mean for the LWT data. Several explanations may be found for the difference in C s value such as scale effects, schematization of the hydrodynamics, and neglecting changes in the geometry occurring during dune retreat in Eq. (26) (more discussion on the variation in the C s value will follow later). If the straight line in Fig. 8 was not forced through the origin, a better fit could be obtained but at the expense of the physics since this would imply that a zero impact parameter would cause erosion. A power curve could provide a better fit than the line in the Figure, simultaneously as it would go through the origin. However, a nonlinear equation would be in conflict with the simple analytical formulation used here. Another reason for the deviation might be flooding of the dune. For the highest water level producing the largest erosion, the still water level may reach the dune face. In such cases, it is expected that the estimate of the wave impact through R is less reliable, even if the analytical solution works from a mathematical point of view. The line corresponding to the optimal C s value from the LWT data is also plotted in Fig. 8 (dashed line). It seems like this line approximately provides an upper limit to the amount of erosion and would thus represent a conservative estimate. However, for the

larger values on the impact factor the eroded volume will be significantly overestimated using the C s=1.8103. Another aspect to consider is that for the higher values on DV E, it is more likely that the dune foot elevation increases, which implies that Eq. (26) overestimates the impact factor (the increase in z o is neglected). This overestimation could in turn be balanced by a reduction in the C s value, explaining the lower value on C s obtained for the Hughes and Chiu data compared to the LWT data. 5.3. Kubota et al. field data Two experimental cases from Kubota et al. (1999) were suitable for comparison with the analytical model. In one case carried out in 1994 (HA94), a dune-like feature was built on the foreshore where it was exposed to erosional waves. Another case, originating from 1997 (HA97), encompassed the artificial steepening of the foreshore by adding material causing seaward transport and erosion. During the latter part of this case, a scarp formed that experienced direct wave impact with associated erosion. The experiments were made for a rising tide, but the change in mean water level was small during the period of the measurements used here. Measured profiles at the start and end of the study period for each case were utilized to compute the dune foot

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retreat and eroded volume, which were used in the comparison with the analytical model. The basic hydrodynamic data for the two cases were: HA94: H s,o=0.41 m and T m=11 s; and HA97: H s,o=0.35 m and T m=13 s, where H s,o is the unrefracted deepwater rms wave height and T m is the mean wave period. The distance from the mean water elevation to the dune foot was 0.39 and 0.18 m for HA94 and HA97, respectively. Case HA94 experienced an eroded volume of about 0.64 m3/m during 30 min and HA97 0.37 m3/m during 36 min. Eq. (25) was used to compute C s values for cases HA94 and HA97 yielding 1.6103 and 0.92103, respectively. Both these values are in good agreement with the previous estimates, especially for case HA94. The field experiments by Kubota et al. (1999) were similar to the LWT data in scale, so the agreement with the C s values from the LWT data is expected. 5.4. Birkemeier et al. field data based on storm erosion The storm erosion field database summarized by Birkemeier et al. (1988) provided an excellent opportunity to test the analytical model. However, due to the complexity of the hydrodynamic forcing during a storm, significant schematization was needed when describing the wave and water level input. Such simplifications may affect the C s value somewhat, but hopefully the essence of the forcing is captured by the selected schematization. In addition, data on eroded volumes were only taken from profiles that displayed a shape similar to the assumed configuration shown in Fig. 1, that is, the profile had a distinct foreshore backed by a steep dune. Thus, in total, 45 cases were identified from the 13 storm events impacting seven different locations. Table 3 summarizes the background data for these cases. The different beaches were abbreviated as follows (in the table and the text): NB=Nauset Beach; MB=Misquamicut Beach; WB=Westhampton Beach; JB=Jones Beach; LBI=Long Beach Island: AC=Atlantic City; and LB=Ludlam Beach. Numbering of the events in Table 3 is as they consecutively appeared in Birkemeier et al. (1988). In order to apply the analytical solution given by Eq. (31), a sinusoidal variation in time was assumed for the water level change due to the storm surge as

well as for the runup height. Thus, for each case, the storm was first identified from the wave and water level records (measured or hindcasted) and the duration and peak surge level was determined. Often, a sinusoidal shape was satisfactory to describe the water level changes in time, but in some cases, marked deviations were observed, especially if the tide was completely out of phase with the surge. Such deviations contributed to the scatter in the results. The maximum wave height often occurred approximately at the time of the peak surge, and the duration for which the wave height exceeded normal wave conditions was typically similar to the surge duration. These observations were the basis for selecting Eqs. (27) and (28) to describe how z o and R, respectively, varied with time during a storm. The mean wave period was estimated based on the entire surge duration. For the storm events studied, the maximum surge level was in the range 1.2–2.5 m with duration from 10 to 36 h. The maximum significant wave height (H max) varied between 2.4 and 5.5 m and the mean wave period between 7 and 10.5 s. Two other parameters of importance were the wave height prior to the storm (H i) and the distance from the water level to the dune foot (z i) at t=0 The height H i was based on the annual average wave height reported by Savage and Birkemeier (1987) and z i was obtained from the surveyed profiles prior to the storm attack. As before, the runup height was calculated using Eq. (36), after the waves had been backed out to deep water and transformed to the rms height. Thus, R i was based on H i and R max on H max, and R a was given by R maxR i. The eroded volume was determined as the amount of material removed from the dune above z i as given in Birkemeier et al. (1988). Dune heights (distance from the dune foot to the top) are typically in the range of 4–8 m for the studied beaches, with the exception of NB where bluffs of a height of 15–25 m are present. The median grain size for the beaches is between fine/medium to medium/coarse (Savage and Birkemeier, 1987), approximately corresponding to the interval 0.3–0.5 mm, respectively. Eq. (31) was compared with the storm erosion data by plotting DV E as a function of the right-hand side of the equation leaving out C s. Fig. 9 shows the data plotted in this way where each field site is denoted with a different symbol. The scatter is significant, but if the NB data is ignored, a clear trend is observed. As

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Table 3 Overview of storm erosion events from the Birkemeier et al. data used for validating the analytical model of dune erosion Case no.

Field site

Max wave height (m)

Wave period (s)

Surge level (m)

Surge duration (h)

Eroded volume (m3/m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

LBI LBI AC LB AC LBI AC LB LB LB LB LB LBI LBI LBI LB LB NB NB WB JB JB JB LBI AC NB NB NB NB MB MB WB WB JB JB AC AC LB LB LB NB LBI LB LB LB

2.6 2.6 2.6 2.6 1.8 3.4 2.6 3.4 3.4 3.4 3.4 2.0 1.9 1.9 1.9 2.1 2.1 2.4 2.4 2.8 3.0 3.0 3.0 2.5 2.3 3.6 3.6 3.6 3.6 3.8 3.8 3.9 3.9 3.9 3.9 3.0 3.0 3.0 3.0 3.0 3.0 1.8 1.8 1.8 1.8

9.0 9.0 8.0 8.0 8.0 8.0 9.0 8.0 8.0 8.0 8.0 7.0 8.0 8.0 8.0 7.0 7.0 8.0 8.0 9.0 8.0 8.0 8.0 7.0 8.0 9.5 9.5 9.5 9.5 10.5 10.5 10.0 10.0 9.0 9.0 8.0 8.0 8.0 8.0 8.0 8.0 10.0 10.0 10.0 10.0

1.5 1.5 1.5 1.5 1.2 1.4 1.4 1.4 1.4 1.4 1.4 1.2 1.5 1.5 1.5 1.5 1.5 2.0 2.0 1.5 1.5 1.5 1.5 1.5 1.5 2.5 2.5 2.5 2.5 2.0 2.0 2.0 2.0 2.0 2.0 1.8 1.8 1.8 1.8 1.8 2.0 1.5 1.5 1.5 1.5

14 14 14 14 24 24 24 24 24 24 24 12 36 36 36 36 36 10 10 24 24 24 24 24 24 12 12 12 12 10 10 12 12 12 12 10 10 10 10 10 34 12 12 12 12

7.9 22.1 9.6 2.5 14.4 1.6 5.1 6.2 6.0 6.2 11.3 4.4 28.8 33.0 20.0 6.0 13.3 27.1 23.8 2.4 35.8 27.7 23.9 14.4 20.3 17.4 24.9 44.1 21.0 6.1 7.8 19.4 7.2 8.9 5.2 9.9 4.1 4.7 1.8 3.2 16.1 14.9 1.5 4.3 6.1

previously pointed out, bluffs of large heights prevail at NB and the simple impact model employed here does not seem to be suitable for describing such features. Neglecting the NB point, an optimum C s value of 1.3104 was obtained (straight line), which

is considerably lower than the values derived for the other data sets. Several reasons are possible for this difference, such as overestimation of the forcing through the sinusoidal schematization of the variation in time (which leads to a lower C s value) and

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Fig. 9. Eroded volume during a number of storms from the Birkemeier et al. (1988) database as a function of an impact parameter obtained from an analytical solution.

dependence of C s on some physical parameter not included in the analytical model derivation. More discussion on this follows in the next section. In order to investigate the importance of the schematization of z o and R, the solution given by Eq. (35) was also least-square fitted to the Birkemeier et al. data. The general agreement between model and data was about the same as for Eq. (31) (see Fig. 9), but the optimum value of C s was strongly dependent upon n. For example, n=1 gave C s=2104, n=2 gave C s=3.4104, n=4 gave C s=6.3104, and n=6 gave C s=9.2104. Thus, a bcalibrationQ of the parameter n could produce an optimum C s value that was in good agreement with the C s values derived for the other data sets. However, preferably the selection of an appropriate storm surge hydrograph should be based on analysis of field observations and not on calibration. Since such an analysis was outside the scope of this paper, the possibility of adjusting the C s value by selecting the value of n was not further pursued. 5.5. Non-dimensional transport coefficient Optimum values of the non-dimensional transport coefficient (C s) were derived based on the four studied data sets by least-square fitting the different analytical

solutions to the available data points. For the LWT and Kubota et al. data sets, one value was obtained for each experimental case, whereas for the other two data sets a single optimum value of C s was estimated for the entire set. The result of this analysis showed marked scatter in the C s values, although three of the data sets had similar mean values and less standard deviation (LWT, Hughes and Chiu; Kubota et al.). However, the Birkemeier et al. data set on storm erosion displayed significantly larger scatter and the mean value was much lower (for the sinusoidal water level variation). As briefly discussed above, this discrepancy might depend on the parameterization of the storm (i.e., variation in wave runup and mean water level with time) as well as the selection of the characteristic value in this parameterization. Another cause for the observed variation in C s may be the dependence of this coefficient on some other physical quantity not included in the simple analytical model. For example, the geotechnical properties of the dune are expected to affect the strength and thus the amount of erosion during wave impact (Erikson et al., 2003). In the present study, limited amount of information was available about the properties of the dune besides its geometry and the grain size. One nondimensional quantity that may possibly affect C s is the

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wave height over the grain size (H rms,o/D 50). In order to investigate this dependence C s was plotted as a function of H rms,o/D 50 for all cases studied (see Fig. 10). Values of C s were estimated for each individual case and no general fitting was employed as was previously the case for the Hughes and Chiu data and the Birkemeier et al. data. Fig. 10 shows a clear trend in the data where C s decreases with increasing H rms,o/ D 50 values. This inverse dependence may be related to matrix suction (Overton et al., 1994a; Erikson et al., 2003) which produces higher dune strength for smaller grain size, implying a smaller C s value because less material is eroded for specific wave and water level conditions. In Fig. 10, two empirical equations are also plotted where an exponential function was least-square fitted to the entire data set, and a power function to the data encompassing prototype-scale cases (these two types of functions were best suited for the least-square fitting). Neglecting the small-scale laboratory data does not markedly affect the predictions of C s. Thus, the following equation, valid over the entire range of H rms,o /D 50 values studied, seems to capture the trend: Cs ¼ Ae

b

Hrms;o D50

ð37Þ

where the coefficient values were found to be A=1.34103 and b=3.19104. If matrix suction

693

is the explanation for the inverse dependence of C s on D 50, it is expected that Eq. (37) is only valid over the range of grain sizes investigated here, that is, 0.15bD 50b0.50 mm. In the LWT, Hughes and Chiu, and Kubota et al. data sets, the forcing was well known through the measurements and could be accurately described in the analytical solution due to its simplicity. This was not the case for the Birkemeier et al. data, but both the knowledge of the forcing and its description in the analytical solution were schematic. Thus, as an alternative to Eq. (37) for estimating C s in practical applications, different, but constant, values on C s may be used for conditions corresponding to the three former data sets or to the latter data set. The overall mean for the LWT, Hughes and Chiu, and Kubota et al. data sets was C s=1.4103 with a standard deviation of 0.74103, whereas corresponding values for the Birkemeier et al. data set were 1.7104 and 2.5104, respectively.

6. Conclusions In this study, an analytical model was developed to describe the erosion and recession of coastal dunes impacted by high waves and water levels during severe storms. Analytical models are useful because

Fig. 10. Dimensionless transport coefficient as a function of wave height over grain size for all individual cases studied.

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their simplicity and limited data requirements make them easy to apply. In addition, the main governing parameters are readily identified and the asymptotic behavior can be derived. In engineering projects, analytical models are useful in the initial stage of a project when approximate estimates are needed or in projects where the response of large areas to long time series of data is investigated. By combining a transport relationship for the dune based on wave impact theory with the sediment volume conservation equation, analytical solutions were developed for different geometrical configurations and forcing characteristics. The general solution takes into account the changes in elevation of the dune foot during retreat, whereas simplified solutions neglect this variation and assumes the elevation to be a constant. Complex forcing conditions, involving time-varying waves and water level, require that this latter assumption is invoked in order to arrive at analytical solutions. The analytical solutions presented in this paper encompassed cases where the water level increased with time linearly, sinusoidally, and according to a power function. In addition, in some cases, a time-varying runup height was simulated to describe a variation in the incident wave conditions. The analytical model contains two parameters which determine the solutions, namely the runup height (R) and an empirical transport coefficient (C s). A Hunt-type formula was developed to predict R after which the optimum value of C s was derived by leastsquare fitting the solutions to four different data sets. These data sets originated from the laboratory and the field, covering both monochromatic and random waves, as well as small-scale and prototype-scale conditions. Overall, the agreement between the data and the analytical model was satisfactory, both for more detailed comparisons involving time series of dune foot recession and for integrated quantities such as the total eroded volume. For three of the data sets, C s attained similar values in the fitting procedure, but for the data set consisting of eroded volumes observed during a number of storms on the North Atlantic coast of the United States, a lower optimum C s value was obtained. Several different explanations are possible for this discrepancy, for example, the schematization of the geometry and forcing for the storm cases and the dependence of C s on some

physical quantity not described by the model. Analysis showed that C s displayed a negative correlation with the ratio between the rms wave height and the median grain size, although marked scatter was observed. This inverse dependence on grain size may possibly be attributed to matrix suction, which implies that finer material gives the dune a greater strength than coarser material during wave attack. It is believed that the present analytical model will produce reliable quantitative estimates of dune recession and erosion during storms, provided that the forcing conditions are known and that the geometry of the dune configuration is similar to the one assumed here (plane-sloping foreshore backed by a vertical dune). In practical applications, it is recommended to use a range of values on C s to obtain an estimate of the variability in dune response. To reconcile the difference in C s values between some of the data sets employed in this study, storm events from the field should be analyzed more carefully to establish the characteristics of the forcing and how to schematize it. In addition, the geotechnical properties of the dune and their influence on the strength should be investigated further. List of Symbols a rate coefficient for water level increase (s1) A empirical coefficient b empirical coefficient Cs empirical coefficient CE empirical coefficient C U empirical coefficient Ds dune crest elevation (m) D 50 median grain size (m) F swash force (N/m) Fo swash force for a single bore (N/m) g acceleration of gravity (m/s2) ho height of bore (m) Hi significant wave height during period prior to storm (m) H max maximum significant wave height (m) Ho deepwater wave height (monochromatic or root-mean-square) (m) H rms,o deepwater root-mean-square wave height (m) H s,o significant deepwater wave height (m) Lo deepwater wavelength (m) mo mass of bore (kg/m)

M. Larson et al. / Coastal Engineering 51 (2004) 675–696

n N p q qD R Ra Ri R max RT so t tB tL T Tm Ts uo us V Vo za zD zi zm zo a bD bf q qs Dt DV DV E DW

empirical power determining shape of surge hydrograph number of experimental cases porosity cross-shore transport rate (m3/m s) cross-shore transport rate from dune face (m3/m s) runup height (m) runup height amplitude during surge (m) runup height at t=0 (m) runup height for maximum offshore wave (based on H max) (m) =R a+z a (m) length of bore (m) time (s) time required for dune breaching (s) time when bores start impacting the dune during surge (s) wave period (s) mean wave period (s) duration of surge (s) speed of bore (m/s) speed of bore where uprush starts (m/s) sand volume in dune (m3/m) sand volume in dune at time t=0 (m3/m) amplitude of water level variation during surge (m) =z iR i (m) initial elevation of dune foot for varying water level (m) initial elevation of dune foot (m) elevation above location where uprush starts (m) transport decay coefficient (s1) dune-face slope foreshore slope water density (kg/m3) sediment particle density (kg/m3) duration of bore impact on dune face (s) eroded volume (m3/m) eroded volume at a specific time (m3/m) weight of eroded volume (N/m)

Acknowledgements The support of the Swedish Natural Science Research Council (NFR) is gratefully acknowledged

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by ML. Financial support for LE and HH was provided by VINNOVA (Swedish Agency for Innovation Systems, contract No. 1998-0188). This work was also partly funded by the project bStudy on erosion control of national landQ, through the Interdisciplinary Global Joint Research Grant for Nihon University, Japan, in 2001. The assistance from Dr. Hans Dette and Mr. Jqrgen Newe, Technical University of Braunschweig, regarding the transfer and interpretation of the MAST data is highly appreciated. Dr. Susumu Kubota from Nihon University provided the field data from Hasaki Beach. Dr. Steve Hughes from the Coastal and Hydraulics Laboratory, U.S. Army Engineer Research and Development Center provided a careful and stimulating review of the manuscript.

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