Simulating cross-shore material exchange at decadal scale. Theory and model component validation

Coastal Engineering 116 (2016) 57–66

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Simulating cross-shore material exchange at decadal scale. Theory and model component validation Magnus Larson a, Jaime Palalane a,b,⁎, Caroline Fredriksson a, Hans Hanson a a b

Department of Water Resources Engineering, Lund University, Box 118, 22100 Lund, Sweden Department of Civil Engineering, Eduardo Mondlane University, C.P. 257, Maputo, Mozambique

a r t i c l e

i n f o

Article history: Received 27 January 2016 Received in revised form 2 May 2016 Accepted 22 May 2016 Available online 15 June 2016 Keywords: Cross-shore transport Mathematical model Profile evolution Regional coastal evolution Dune erosion Overwash Wind-blown sand Bar volume

a b s t r a c t A model is developed to simulate the cross-shore (CS) exchange of sand and the resulting profile response at decadal scale to be used in regional coastal evolution models. The CS model consists of modules for calculating dune erosion and overwash, wind-blown sand transport, and bar-berm material exchange. The sand transport equations included in these modules are formulated based on relevant physics in combination with empirical observations and then validated towards laboratory and field data. In order to couple the transport equations to the evolution of key morphological features describing the profile a set of sand volume conservation equations are employed and solved together with the transport equations. The features that are modeled include dune height (from dune foot to crest), the locations of the landward dune foot, seaward dune foot, berm crest, and shoreline as well as the longshore bar volume. The CS model is employed and tested at three different field sites in a companion paper (Palalane et al., 2016), whereas in the present paper the properties of the model are demonstrated by simulating the evolution of the dune, berm, and shoreline in connection with accretion and erosion of sand in a groin compartment. This case derives from a study at Westhampton Beach that focused on shortening the length of a number of groins in order to release accumulated sand in the groin field to eroding downdrift beaches. © 2016 Elsevier B.V. All rights reserved.

1. Introduction 1.1. Background Increasing societal and ecological demands on our coastal areas, together with a growing awareness of the risks associated with the anticipated long-term climatic change, have emphasized the need for morphological modeling capabilities that span decades to centuries. Such predictive technology would encompass regional trends and processes, storms, and multiple interacting coastal engineering projects of various kinds (Larson et al., 2002a). This type of modeling focus on evolution at the regional scale; thus, schematization of the governing processes to reduce computational time is required that employs a proper balance between physical descriptions from theoretical considerations and empirical information based on data and observations. Model reliability and robustness are key properties needed to arrive at useful simulation results. During the latest decade several models have been developed to address such time scales (e.g., Larson et al., 2002b; Jiménez and ⁎ Corresponding author at: Department of Water Resources Engineering, Lund University, Box 118, 22100 Lund, Sweden. E-mail address: [email protected] (J. Palalane).

http://dx.doi.org/10.1016/j.coastaleng.2016.05.009 0378-3839/© 2016 Elsevier B.V. All rights reserved.

Sánchez-Arcilla, 2004; Hanson et al., 2008; Hoan et al., 2011; Ranasinghe et al., 2013), but one weakness of these models has been the simplified representation of the cross-shore (CS) material exchange. Typically, CS exchange is described through sources and sinks with schematized values in time and space. In order to improve the predictive capability of these models, physics-based formulations need to be employed for the CS exchange, although simplifications are required as simulations are performed for large areas over long time periods. Larson et al. (2002b) developed a regional coastal evolution model, known as Cascade, to address time and space scales associated with long-term coastal processes, providing predictions suitable in the planning and preliminary design phase of a project. As the first step towards modeling regional coastal evolution, Cascade was developed to bridge the gap between a sediment budget approach and a shoreline evolution model. The dynamics of selected processes and controls of importance for the regional coastal evolution were modeled based on physical representations, whereas other processes were parameterized through time- and space-varying sources and sinks. Cascade was calibrated and validated against high-quality data sets from three coasts to describe the regional coastal evolution after the creation of multiple inlets on open barrier-island coasts. These successful long-term simulations were made for the opening of Moriches and Shinnecock Inlets on Long Island (Larson et al., 2002b), Ocean City and Indian River Inlets on the

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M. Larson et al. / Coastal Engineering 116 (2016) 57–66

Delmarva Peninsula (Larson and Kraus, 2003), and Brunswick County, FL (Kraus, 2008), which has three natural inlets serving as sources for beach nourishment. In Cascade, CS processes are only represented in a schematized manner through source and sink terms, which may yield the correct average material exchange, but does not provide any information on the variability. Also, modeling the CS exchange has to rely completely on historic data and changes in the forcing conditions are difficult to reproduce. Hanson et al. (2008) made a first attempt to merge the Cascade model with the shoreline evolution model GENESIS to cover scales from project to regional level, introducing physically based description of some CS processes, primarily focusing on the subaerial portion of the beach. These processes included dune erosion during storms and the build-up of the dunes by wind. However, overwash was not described as well as material exchange between the subaerial and subaqueous portion of the profile. In this paper, physically based expressions for different CS processes will be developed and validated against laboratory and field data. In a companion paper (Palalane et al., 2016), the complete CS model is employed at three different field sites located in Sweden, Portugal, and Mozambique, to simulate the evolution of key morphological features describing the profile shape. 1.2. Objectives and procedure The main objectives of the present study are to develop and validate sub-models to describe the CS material exchange to be implemented in long-term coastal evolution models spanning decades to centuries. The main processes to include in these sub-models are dune/cliff erosion during storms, dune build-up by wind-blown sand, overwash, and material exchange between the bar and the berm. Although dune erosion is a short-term process (over days) the effects on the coastal system may be long-term, especially with regard to the recovery of the dune by wind-blown sand. Also, proper simulation of the bar-berm material exchange is necessary for reproducing the seasonal variation in the profile response and for realistically describing the effects of sediment release from the dunes during storms to the subaqueous portion of the profile. The paper starts by discussing the context in which CS material exchange will be modeled as well as reviewing the main processes to be included. The theoretical approach to the modeling of each of these processes is then outlined together with the results obtained from calibrating and validating the models towards available data from the laboratory and field. The main processes taken into account are dune erosion and overwash, bar-berm material exchange, and wind-blown sand. The sub-models developed for the CS transport associated with these processes are subsequently combined with sand conservation equations to simulate the evolution of main morphological elements of the beach profile with focus on the shoreline response. An example taken from Westhampton Beach involving a groin field is presented to illustrate how the CS model may be employed to simulate long-term effects on the coastal evolution from CS material exchange. 2. Cross-shore transport processes in a long-term perspective 2.1. Overview CS material exchange occurs on many scales, where not all are relevant for modeling regional coastal evolution (Larson and Kraus, 1995). Transport processes that act over compatible time and space scales should be included in these models, for example cliff erosion or dune build-up by wind, but also short-term processes, such as erosion during storms, must be described since they may cause abrupt changes to the coastal system that have effects over long time periods. Severe storms erode the dunes, possibly resulting in overwash (Donnelly et al., 2006) and breaching, and the recovery could take substantial time, or could even cause permanent changes to the coastal system.

In order to model CS transport and its effects in regional coastal evolution, the main morphological features of the profile need to be schematized and described through a limited set of parameters (see Fig. 1). Here, the dune (or barrier) is given a trapezoidal shape, defined by three parameters: height (s), location of shoreward end (yL), and location of seaward (yS) end (together with the seaward (βD) and shoreward (βL) slopes of the dune/barrier, which are assumed to be constant). During erosion the dune may become triangular if the erosion is severe and the horizontal top portion of the dune is eroded away (the dune may also disappear completely). Overwash will cause the dune to move shoreward, maintaining its shape (i.e., height), if all material that is eroded from the dune face is transported over the dune crest. The berm width is determined by the distance from the seaward location of the dune (dune foot) to the berm crest (located at yB), and the berm is assumed to be approximately horizontal. The foreshore is sloping uniformly (βf) from the berm crest to the still-water shoreline (located at yG), and seaward of the shoreline is the profile described by an equilibrium beach profile. Material may be deposited in a longshore bar (or several), which is characterized by a specific volume (VB). Transport induced by waves and wind causes changes to the profile. These changes are geometrically prescribed so that the schematized profile shape is maintained, but the key parameters are evolving with time. Seaward of the berm crest, the beach profile is translated horizontally without changing its shape between the berm crest height (DB) and the depth of closure (DC). The sum of these two quantities yields the so called active beach profile, which typically corresponds to the depth where significant alongshore sediment transport occurs (Hanson, 1989). However, in this study, where changes in the berm crest location may occur because of dune erosion/build-up, material exchange between the bar and berm, and gradients in longshore sediment transport, DC constitutes a representative depth that quantifies the effect of these types of transport on the berm. However, the term depth of closure will be retained in this paper. As an example of how the profile may change, the impact of a storm is considered. If the waves and the water level in combination produce a sufficient runup level, the dunes are attacked and dune erosion occurs that supplies the subaqueous portion of the beach with sand. Due to this erosion, the dune foot retreats and yS decreases, assuming that the seaward dune face slope remains constant. If the runup level exceeds the dune crest height, overwash occurs and sediment is deposited on the shoreward side of the dune, causing yL to decrease (move shoreward). The shoreward dune face slope is also taken to be constant. Simultaneously berm erosion typically takes place, moving material to the bar leading to a decrease in yB and an increase in VB. The material eroded from the dune will be deposited from the berm crest and seaward, leading to an increase in yB. Thus, in order to compute the location of changing profile features that define the schematized profile presented in Fig. 1, and described above, the following CS transport rates (also presented in Fig. 1) need to be estimated or computed: the sediment eroded from the dune (qD), which will be further divided into a seaward (qS) and landward transport component, the latter only being observed if overwash occurs (qL); the wind-blown sand that will contribute to the dune growth on its landward (qWL) and seaward (qWS) sides; and the exchange of sediment between the berm and the bar, and vice-versa (qB). In the following, a brief description is provided of CS processes at different scales that all have implications for regional coastal evolution and that should be included in such models. The discussion of the governing physics of these processes constitutes a basis for the subsequent modeling. 2.2. Event-based changes The most significant short-term CS changes are induced by storms, which can erode beaches implying retreat of the berm and the dunes, as well as transport sand landward as overwash. Overwash, in turn, is

M. Larson et al. / Coastal Engineering 116 (2016) 57–66

59

Fig. 1. Definition sketch to calculate cross-shore transport and beach profile evolution.

a major contributor to barrier island migration on decadal scale (Leatherman, 1979). Similarly, storms can cause breaching (Kraus and Wamsley, 2003; Wamsley and Kraus, 2005) which are catastrophic events that also must be simulated within the context of regional and long-term modeling. Furthermore, a storm transports material from the berm to the offshore, where it is often deposited as an offshore bar. After the storm, or even during its waning stage, the incoming waves may bring the sand back to the berm, recovering the damage caused by the storm on the foreshore and seaward portion of the berm. If the bar develops in deeper water, where the wave action is small during normal conditions, the onshore transport from the bar occurs at a low rate, implying a slow recovery of the berm. This material exchange between the berm and the bar is also important to model in a regional coastal evolution model since it includes scales compatible with such evolution. 2.3. Annual changes On many coasts there is a seasonal variation in the wave climate with larger waves in the winter and smaller waves in the summer. This is typically reflected in a corresponding variation of the berm width with an associated response in the subaqueous portion of the profile. When larger waves prevail in the winter, sand from the berm is eroded and deposited under water as one or more longshore bars, whereas the smaller waves in the summer bring back the sand to the berm, increasing its width and correspondingly reducing the bar size (Komar, 1998). The CS exchange of material on an annual scale is important to reproduce in a model since the related variability in beach width, and how it varies between years, may be an interesting parameter in coastal planning and management. In some countries the width of the beach constitutes an important quantity; a beach that is too narrow or wide is less attractive to potential visitors. Beaches and dunes typically gain volume through gradual growth by wind-blown sand transport. On the annual scale, this build-up is important to reproduce so that the dune configuration is accurately represented when storms hit the beach and dune erosion occurs. The dunes are typically built by the onshore winds bringing sand from the berm to the dune face (Davidson-Arnott and Law, 1990; Hanson et al., 2008). Thus, dune build-up implies that sand is supplied from the berm, reducing its width. Over the long-term, if the beach is in equilibrium (no longshore effects), the average berm width will correspond to a situation where dune erosion due to the waves are balanced by dune build-up by wind. On a barrier island, growth can also take place on the landward side of the dune when winds are blowing offshore (Leatherman, 1976). 2.4. Decadal changes Long-term climate variability, as well as man-induced changes to the climate, causes responses in the forcing at the coast over decades and centuries. Examples of parameters representing such forcing that affect CS processes are the mean water level, wind speed and direction, wave

height, period, and direction, and the frequency and intensity of storms. Thus, long-term coastal evolution models should employ input data that include expected patterns of variability together with anticipated trends. The physical description of CS processes must be sensitive to long-term changes in the forcing and reproduce the effects on the coastal evolution. Some coastal systems exist through a delicate balance between erosion and deposition, making them particularly sensitive to shifts in the overall forcing conditions. Engineering activities are often designed from a local perspective with little regard for their regional impact. However, some structures and activities, such as jetties, extensive groin systems, maintenance of inlet navigation channels, large-scale beach nourishment projects, and coastal habitat creation, may influence areas far outside the immediate project area, requiring considerations on decadal scale. Engineering activities in general only indirectly affect the CS response through the gradients in the longshore transport, but, for example, the placement of beach nourishment implies modifications to the profile geometry that cause both short- and long-term changes. If material is mainly added to the berm, adjustment of the profile shape back to natural conditions occurs on the annual scale, whereas placement in deeper water may involve changes over several years (Larson and Hanson, 2015). Ongoing relative rise in sea level, where the rate is expected to increase in the coming decades, must be incorporated in long-term coastal evolution models. At present, simple procedures (e.g., Bruun, 1962) are typically employed to determine the effects on the beach profile with limited considerations of site-specific conditions. This type of simple extrapolation at best yields an overall estimate for a schematized coastal stretch, but with regard to predictions for particular coastal areas the result provides insufficient basis for planning and management. Although not explicitly addressed in this study, the present model includes a physically based approach to simulate the effects of relative sea level rise over decades to centuries. A slow rise in mean sea level implies increased ability for the waves to erode the dunes, shifting material seaward that results in berm build-up and a general vertical shift in the profile. In turn, the dunes may grow back to their equilibrium shape, if the wind conditions and sand availability permit it. 3. Dune erosion and overwash 3.1. Background and theoretical formulations Larson et al. (2004a) developed an analytical model of dune erosion based on the work of Fisher et al. (1986) and Nishi and Kraus (1996). In these studies the eroded dune mass is assumed to be proportional to the impact force from the waves hitting the dune face. Employing ballistics theory to quantify the impact force at the dune face, the following equation can be derived for the cross-shore transport rate (qD; see Fig. 1) from the dune face (Larson et al., 2004a),

qD ¼ 4C s

ðR−zD Þ2 T

ð1Þ

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M. Larson et al. / Coastal Engineering 116 (2016) 57–66

where R is the runup height, zD the distance from the mean water level to the dune foot, T the swash period (typically taken equal to the wave period), and Cs an empirical coefficient. The runup height must exceed the dune foot elevation (R N zD), otherwise there is no dune erosion. The runup height is estimated using the formula (Larson et al., 2004a), R ¼ 0:158

pffiffiffiffiffiffiffiffiffiffi H o Lo

ð2Þ

where Ho is the wave height (root-mean-square value for random waves) and Lo the wavelength (mean value), both in deep water. If the runup height exceeds the dune crest level, a part of qD will be transported over the dune crest as overwash (qL), whereas the rest is moved seaward by the backwash (qS = qD − qL; see Fig. 1). Thus, defining α = qL/qS, then qS = qD/(1 + α) and qL = qDα/(1 + α), yielding how much of the eroded dune material that goes offshore and onshore, respectively. Furthermore, if the dune is overtopped by the waves, the impact force of the dune should be reduced, because of the additional momentum flux over the dune. Larson et al. (2009) modified Eq. (1) for the case of overwash to, qD ¼ 4C s

ðR−zD Þs T

ð3Þ

where s is the dune crest height, implying that R− zD N s for overwash to occur. The ratio α is given by, α¼

  1 R−zD −1 A s

Fig. 2. Measured eroded dune volume as a function of an impact parameter for selected field sites and storm events on the US East Coast.

ð4Þ

where A is an empirical coefficient determined to be about 3 by Larson et al. (2009) through comparison with field data.

model, the linear behavior is satisfactorily validated by the field data. However, the slope of a line, which represents the transport rate coefficient Cs (see Eq. (1)), seems to be somewhat site-specific and it is a potential calibration parameter in applications. The overwash model (Eqs. (3) and (4)) was validated in Larson et al. (2009) using data on overwash volumes from severe storms and hurricanes, also obtained on the US East Coast (Eiser and Birkemeier, 1991; Stone et al., 2004; Larson et al., 2004b). 4. Bar-berm material exchange

3.2. Model calibration and validation 4.1. Background and theoretical formulations The general validity of the impact model for describing dune erosion was shown by Overton et al. (1987, 1994) in small-scale wave tank experiments. In these experiments, individual waves (bores) where allowed to hit the dune face and a linear correlation were found between the eroded volume and the swash force (based on the measured wave height and bore velocity in front of the dune). Similar results were obtained in large wave tank tests (Overton et al., 1990). Nishi and Kraus (1996) introduced the impact model in a numerical model of beach profile change (Larson and Kraus, 1989) and could reproduce the recorded evolution in several dune erosion cases from the SUPERTANK experiment (Kraus and Smith, 1994). Larson et al. (2004a) calibrated and validated the impact model through comparisons with data sets from both the laboratory (smalland large-scale experiments) and the field. A variety of forcing conditions were investigated, including constant and varying waves and water levels, making comparisons between measured and calculated dune face retreat and eroded dune volume. The governing equations for the dune evolution, including the impact model, were solved analytically, which required a certain schematization of the forcing conditions and profile geometry. Overall good agreement was found between the analytical solutions and the measurements, although for the field data the scatter was marked, which to a large degree was due to this schematization. Fig. 2 illustrates for selected cases from the field data set by Larson et al. (2004a) the eroded volume, obtained from measurements at three locations on the US East Coast (see Birkemeier et al., 1988) for different storms and survey lines, as a function of an impact parameter given by an analytical solution. Straight lines have also been leastsquares fitted to the data sets from the different locations to see how well the assumption of a linear dependence between eroded volume and the wave impact can reproduce the observed response during a storm. Considering the simplifications introduced in the analytical

It is assumed that the exchange of material between the bar and the berm takes place under conservation, that is, no material is lost offshore. Such losses can easily be added, but must be quantified in some way. Also, the volume eroded from the berm is stored in one offshore bar (or, representative morphological volume) that will reach a certain equilibrium volume (VBE), if the wave conditions are steady and the grain size does not vary. If the bar volume (VB) at any given time is smaller than VBE, then the bar volume will grow, whereas VBE b VB implies decay in bar volume. Growth in bar volume causes the corresponding decrease in berm volume (and shoreline retreat), and decay in bar volume causes an increase in berm volume (and shoreline advance). Based on empirical observations of bar growth (Larson and Kraus, 1989), the change in bar volume is taken to be proportional to the deviation from its equilibrium value, dV B ¼ λðV BE −V B Þ dt

ð5Þ

where t is time and λ a coefficient quantifying the rate at which equilibrium is approached. Note that Eq. (5) describes both growth and decay in bar volume. If VBE and λ are constants, the solution to Eq. (5) is easily obtained as, V B ¼ V BE −ðV BE −V B0 Þ expð−λtÞ

ð6Þ

where VB0 is the bar volume at t = 0. Thus, if VBE N VB0 the bar will grow and if VBE b VB0 the bar will decay. Equivalently, the change in bar volume (ΔVB) during a time Δt may be expressed as: ΔV B ¼ ðV BE −V B0 Þð1− expð−λΔtÞÞ

ð7Þ

M. Larson et al. / Coastal Engineering 116 (2016) 57–66

In order to use Eqs. (6) and (7), the equilibrium bar volume and rate coefficient must be determined. Larson and Kraus (1989) developed an empirically based expression for VBE employing large wave tank (LWT) data (monochromatic waves). The normalized equilibrium bar volume was shown to depend on the dimensionless fall speed and the deepwater wave steepness according to, V BE L2o

 ¼ CB

Ho wT

4=3

Ho Lo

ð8Þ

where Lo is the deepwater water wavelength, CB a dimensionless coefficient (=0.028 for the LWT data), Ho the deepwater wave height, w the sediment fall speed, and T the wave period. In applying Eq. (8) to field conditions and random waves, it is expected that the coefficient value may change, but the general functional relationship should be valid. The LWT data covered the following range of input conditions: H = 0.29–1.68 m, T = 3.1–16 s, and d50 = 0.22–0.47 mm, where d50 is the median grain size (H was taken in the horizontal part of the tank, at 3.5–4.5 m). The rate coefficient (λ) should also depend on the wave and sediment properties, although Larson and Kraus (1989) only found limited correlation between λ and H, T, and w. In spite of this, their analysis of LWT data indicated that λ increases with grain size (or fall speed) and decreases with wave height. Thus, the following relationship is suggested for λ,  λ ¼ λo

Ho wT

m ð9Þ

61

Eq. (5) was also used to simulate bar growth for a field site (Duck, NC) where long-term measurements of the waves and beach profiles were available, from which the bar volume could be determined. However, at Duck, two bars frequently appear (Howd and Birkemeier, 1987; Larson and Kraus, 1992), so Eq. (5) was employed to describe the combined volume of the two bars. The time series simulated extended from 1981 to 1989, encompassing about 300 surveyed profiles, and the measured waves at least every 6 h were available as input. Fig. 4 shows the simulated and measured total bar volume for the studied time period, where the measured bar volume was interpolated to yield values every 15 days (corresponds to the measurement resolution). The period from 1981 to 1985 was used for calibration, whereas the remaining period (1986–1989) was used for validation. During the calibration period and the first half of the validation period the agreement is good; however, towards the latter part of the validation period some fluctuations in the bar volume are not well predicted. The optimum parameter values in Eqs. (8) and (9) obtained for the LWT data were not suitable for the field data. The parameter m = − 1/2 was kept, but CB = 0.080 and λo = 0.002 h−1, indicating that the bars would grow relatively larger in the field and the response would be slower. Also, a multiplier was introduced to reduce λo when onshore transport and bar volume reduction occurred to improve the on agreement with observed bar response (λoff o = λo , λo = 0.3λo). Larson et al. (2013) attributed the difference in the optimal coefficient values obtained for the LWT and field data to (1) random waves prevailing for the field data; (2) two bars frequently appearing in the field that was treated as one single complex; and (3) both bar growth and decay occurring in the field, whereas only bar growth was studied in the LWT experiments.

where λo and m are coefficients with values to be calibrated against data.

5. Wind-blown sand

4.2. Model calibration and validation

5.1. Background and theoretical formulations

Fig. 3 shows the typical result of applying Eq. (5) for describing the bar volume growth for two cases from an LWT experiment (Kraus and Larson, 1988; Larson and Kraus, 1989). Eqs. (8) and (9) were used to calculate VBE and λ, respectively. In this case the wave conditions were steady and the analytical solution to Eq. (5) could be applied, as given by Eq. (6). Comparisons were carried out for a number of other LWT cases (Larson et al., 2013) and in most cases good agreement was achieved. In total nine cases were used for calibration and 13 cases for validation. In all these comparisons Eq. (8) was employed with CB = 0.022, m = −1/2, and λo = 0.15 h−1 (values used in Fig. 3). The calibration and validation displayed similar agreement between calculated and measured bar volumes.

A number of formulas have been developed to estimate the potential sand transport rate by wind, starting with Bagnold (1936). Most formulas take the transport rate to be proportional to the shear velocity, often including a critical value on the shear velocity that needs to be exceeded in order for transport to occur (Sherman et al., 1998). For example, Lettau and Lettau (1977) proposed the following formula for the _ WE ), potential transport rate (m

_ WE ¼ K W m

sffiffiffiffiffiffiffiffi d50 u2 ρ ðu −uc Þ ref a g d50

Fig. 3. Temporal evolution of measured and calculated bar volume for two different experimental cases during a large wave tank test.

ð10Þ

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M. Larson et al. / Coastal Engineering 116 (2016) 57–66

and qWE may be taken as constants for a particular case, Eq. (12) can be solved analytically to yield, qW ðyÞ ¼

qWE qWo expðδyÞ qWE þ qW0 ð expðδyÞ−1Þ

ð13Þ

where qW = qW0 for y = 0. The form of Eq. (12) does not allow for qW = 0 at y = 0; thus, the very initial spatial growth of qW cannot be described by Eq. (13). However, this length is limited and neglecting or schematically describing it through a linear increase from where the transport starts should be satisfactory. A simpler, more heuristic version of Eq. (12) would be:   dqW q ¼ δqWE 1− W dy qWE

ð14Þ

The solution to this equation is: qW ðyÞ ¼ qWE ð1− expð−δyÞÞ Fig. 4. Temporal evolution of measured and calculated bar volume between 1981 and 1989 at Duck, NC.

where dref 50 is the median reference grain size (0.25 mm), ρa the density of air, u⁎ the shear velocity at the bed, u⁎c the critical shear velocity at the bed, and Kw an empirical coefficient (equal to 4.2). Eq. (10) yields _ WE in kg/m/s and in order to convert calculated values to a volumetric m transport (qWE) the density of the sediment grains (ρs) should be employed (if build-up of the dunes by wind-blown sand is considered the porosity needs to be included as well). The critical shear velocity may be calculated from,

uc ¼ AW

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðρs −ρa Þ gd50 ρa

ð11Þ

where AW is a coefficient (about 0.1). The shear velocity can be estimated from the relationship u⁎ = C sW, where W is the wind speed at 10 m elevation and Cs a coefficient (= 0.053; see Horikawa, 1978). However, the transport rate given by Eq. (10) will not be attained immediately, but it will require a certain distance before equilibrium occurs (Hotta, 1984). As the wind blows from the shoreline (or berm crest) towards the dune, the transport rate increases along the foreshore and reaches its equilibrium (potential) value after some distance. This equilibrium distance depends on the local conditions, such as sand grain size, wind speed, and whether the sand is wet or dry. Hotta (1984) reported based on field measurements that a distance of 5–10 m would be sufficient to reach equilibrium conditions, depending on how wet the surface is. In contrast, measurements by Davidson-Arnott and Law (1990) indicated that 20–30 m (or more) may be required before equilibrium is obtained. Sauermann et al. (2001) developed a model for wind-blown sand transport from which an equation to describe the spatial evolution of the transport rate (qW) towards saturation (= equilibrium) was derived,   dqW 1 q ¼ qW 1− W ls dy qWE

ð12Þ

where y is a cross-shore coordinate originating at the berm crest and pointing shoreward, ls is the saturation length, which is a function of several different parameters including the shear stress, and qWE is the transport rate at saturation. Introducing δ = 1/ls, and assuming that δ

ð15Þ

In this case qW = 0 at y = 0, and there is no ambiguity in specifying the boundary condition at the upwind side, although the shape of the function might be less suitable to describe the spatial growth in qW. 5.2. Model calibration and validation In order to test the equation describing the spatial growth of the wind-blown transport towards saturation (equilibrium), data from Davidson-Arnott and Law (1990) were employed. The data on transport _ W instead of qW, but this is irrelevant rates were given in terms of m when testing Eq. (13). In their paper data from three main experimental cases were presented regarding the spatial growth of the transport along the berm. For each case, 3–4 individual measurements were performed from which the average transport at different locations on the berm was determined. The wind velocities for the three cases were ap_ WE was proximately 6, 8.5, and 13.5 m/s. For the two first cases m attained, whereas in the third case the length of the berm did not allow for this to occur. Eq. (13) was least-squares fitted to the data with δ as the main fitting parameter. For the first two cases, the equilibrium transport rate was estimated based on the data by analyzing the measured transport rates at the downwind end, where equilibrium had been attained (estimated _ WE = 0.0014 and 0.0125 kg/m/s for the first and second values were m _ WE was not observed, case, respectively). For the third case, where m _ WE = 0.095 kg/m/ the equilibrium rate was calculated with Eq. (10) (m s). Since no detailed information about the grain size was available, the critical shear velocity was estimated from the measurements to be u⁎c = 0.27 m/s for the third case. For comparison, the equilibrium transport rates for the two first cases were also calculated with Eq. (10) and _ WE = 0.0016 the agreement with the observed values was quite good (m and 0.014 kg/m/s for the first and second case, respectively). Eq. (13) needs a value at x = 0 and this was set to 0.004 kg/m/s; the measurements show a close to linear increase in the transport rate up to this value for all cases. Fig. 5 shows the agreement between the model and the measurements for the three cases. The optimum value on δ was 0.15, 0.21, and 0.14 m−1 for the first, second, and third case, respectively. In general the agreement is satisfactory, although the very initial growth of the transport rate is not described by Eq. (13). The plots only include the distance encompassing the measured rates; thus, for the third case the plotted curve does not go to equilibrium. Although Eq. (13) yields a good description of the shape of the spatial growth curve, it may be convenient to employ Eq. (15) in modeling wind-blown transport since it simplifies the equation and reduces some of the difficulties in setting the boundary values. The latter equation also captures the behavior of the spatial evolution of qW quite well. However, since the exponential

M. Larson et al. / Coastal Engineering 116 (2016) 57–66

63

transport. The term dQL/dx represents the gradient in the longshore transport, which may cause advance or retreat of the shoreline. In a general coastal evolution model substantial effort will go into determining QL. 6.3. Dune evolution The dune is schematized using a trapezoidal shape with fixed side slopes, and the evolution of the dune may occur both on the landward and seaward side. Thus, two different sand conservation equations are used: one for the seaward side involving dune erosion from wave impact and dune buildup from wind, and one for the landward side with build-up from overwash and wind-blown sand. On the seaward side sand conservation requires that: dyS −qD þ qWS ¼ dt s Fig. 5. Measured and calculated spatial evolution of the wind-blown sand transport on a beach (data from Davidson-Arnott and Law, 1990).

shape of Eq. (15) does not fit the initial part of the measured curve very well, best agreement is obtained if y originates somewhat downwind of the location where the measurements started. The starting point for y partly decides the optimal value obtained on δ in Eq. (15); for the first and second case δ is not so sensitive to the selection of this point and the optimal value is about 0.1 m−1. However, for the third case, where _ WE was not measured, δ is more sensitive to where y originates and m the obtained value is about 0.015 m−1, which is much smaller than for the first and second case.

ð18Þ

Sand conservation on the landward side yields: dyL −qL −qWL ¼ dt s

ð19Þ

where qWL is the wind-blown transport rate on the landward side. This rate may also be calculated with Eq. (10); in most cases it may safely be assumed that the equilibrium transport has been attained since the distance over which the transport develops is typically large. 7. Model of sediment exchange in a groin compartment

6. Sediment conservation equations for morphological features

7.1. Background

If the transport equations developed above are to be used in a CS model of profile response (see Fig. 1), a number of sand volume conservation equations are needed to simulate the evolution of the different morphological elements employed in schematizing the profile (see Fig. 1). In the following these equations are presented for the main elements modeled.

In order to illustrate the interaction between the different submodels developed previously in this paper, a simple example is provided here that attempts to simulate the evolution of the shoreline in a groin compartment taking into account cross-shore material exchange. The longshore transport is described in a schematic manner using the reservoir model applied to the groin compartment following Kraus and Batten (2006). The example is based on a study at Westhampton Beach on Long Island where the effects of the shortening of a number of groins on the shoreline response, including the dune foot evolution, were investigated (Hanson et al., 2010). By reducing the length of the groins, sand will be released from the compartments, as well as from the dunes, and supply downdrift beaches that suffer from erosion with sand. The dunes have advanced because of the widening of the berm, which enhanced dune build-up by wind. A model will be helpful in assessing the time scale of material release from the groin compartments and the dunes.

6.1. Bar evolution The growth or decay in the bar volume is coupled to the berm evolution; if sand is eroded from the berm, the bar grows, and vice versa. Thus, the change in bar volume, as determined from Eq. (5), is associated with a CS transport rate (qB) according to: dV B ¼ qB dt

ð16Þ

In this formulation seaward transport is taken as positive. If VBE and λ are constants, then from the solution given by Eq. (6) qB = λ(VBE − VB0) exp(−λt). 6.2. Berm evolution Changes in the berm location are associated with bar-berm material exchange, given by Eq. (16), wind-blown transport towards the dune (qWS), and seaward transport resulting from erosion of the dune (qS). The sand conservation equation is given by,   dyB 1 dQ ¼ −qWS −qB þ qS − L DB þ DC dt dx

ð17Þ

where DB is the berm height, DC the depth of closure (or more correctly, a representative depth for the impact of the different transport types on the berm evolution, as previously discussed), and QL the longshore sand

7.2. Governing equations Fig. 6 provides a definition sketch for the case under study that includes two groins forming a compartment on an open beach. Sand is transported from left to right and initially when the groins are put in place the shoreline is straight and located at yG = yG0. After some time bypassing of the updrift groin occurs and infilling starts of the compartment. Infilling causes a seaward shift in the shoreline position, where it is assumed that the beach profile is translated in the seaward direction without changing its shape (between the berm crest height, DB, and the depth of closure, DC). Employing the reservoir model by Kraus (2000) for the groin compartment (see Kraus and Batten, 2006), only the shoreline position (yG) is required to describe the evolution in the compartment. The groins have a length of LG from the initial shoreline and are placed with a spacing BG. Before the construction of the groin compartment, assuming that no gradients in longshore sand transport rate that could induce shoreline

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where Qin is the transport in to the compartment and Qout the transport out. The reservoir model gives the outflow of sediment from the compartment as (Kraus and Batten, 2006): Q out ¼ Q in

yG LG

ð24Þ

Combining Eqs. (23) and (24) gives:   dQ L Q in yG ¼ −1 dx BG LG

ð25Þ

Thus, in order to obtain the time evolution of yS and yB (or yG), Eqs. (21) and (22) should be solved simultaneously with Eq. (25), where Eq. (20) defines the geometrical relationship between yB and yG. 7.3. Solution of governing equations

Fig. 6. Definition sketch for mathematical model of the interaction between dune/berm complex and shoreline movement in a groin compartment.

change existed, the movements of the dune foot (yS) and the berm crest (yB) were due to cross-shore transport only where the main components are wind-blown transport and dune erosion due to wave impact. Bar/berm material exchange was not included in the simulations discussed here. Since the beach profile has a fixed shape from the berm crest to the depth of closure, the following relationship holds between the berm crest and shoreline location, if the foreshore is plane with a slope βf, yG ¼ yB þ

DB ¼ yB þ LB tanβ f

ð20Þ

where LB is the horizontal distance from berm crest to the shoreline (i.e., foreshore length). In order to model the evolution of the dune foot and the berm crest (or, equivalently, the shoreline position), two sand volume conservation equations are needed. Changes in the dune foot location is determined by the build-up from wind-blown sand and the erosion by impact from waves (compare Eq. (18), in which Eq. (15) was used): dyS qWS q ¼ ð1− expð−δðyB −yS ÞÞÞ− D dt s s

ð21Þ

There is no analytical solution to the system of equations; thus, a numerical solution was implemented. A simple explicit scheme is employed to discretize Eqs. (21) and (22), where the transports are determined at the preceding time step, ykþ1 ¼ ykS þ S

 Δt  k qWS −qkD D

ð26Þ

¼ ykB þ ykþ1 B

 Δt  k −qWS þ qkD −dQ kL =dx DB þ DC

ð27Þ

where k is the index for the time step and Δt the length of the time step. 7.4. Sample calculations In order to test the model two schematic cases were set up based on typical values from Westhampton Beach (WHB). A net transport in to the groin compartment of Qin = 100,000 m3/yr was assumed (a representative bypassing transport of the updrift groin). The active profile height was assumed to be 8 m with DB = 2 m and DC = 6 m, the dune height s = 2 m, and the foreshore slope tanβf = 0.10 (LB = 20 m). The groins were given a length of LG = 150 m with a compartment width of BG = 400 m. Specifying CS sand transport rates by wind and eroding waves are more difficult, but as a first approximation qWE = 40 m3/m/yr and qD = 20 m3/m/yr. The former value was taken from CEM (2015; Part III, Chapter 4) based on the total annual transport capacity by wind for appropriate compass directions. Dune erosion by wave impact corresponds to a few moderate storms hitting the dune every year on the average. In absence of material exchange due to longshore transport gradients, the equilibrium width (lBE) of the berm may be calculated from (see Hanson et al., 2008): lBE ¼

The berm crest evolution is the result of material being lost from the berm to the dune by wind-blown sand as well as beach profile translation associated with material supplied from the dune by eroding waves (no overwash) and the gradients in the longshore sand transport (compare Eq. (17)): dyB qWS q −dQ L =dx ¼− ð1− expð−δðyB −yS ÞÞÞ þ D DB þ DC dt DB þ DC

ð22Þ

In order to close the system of equations an expression for dQL/dx is required. A sand balance equation for the compartment yields, dQ L Q out −Q in ¼ dx BG

ð23Þ

  1 qWE ln δ qWE −qD

ð28Þ

Based on the profile data from WHB, lBE = 40 m, and with the above values on qWE and qD the rate coefficient becomes δ = 0.017 m−1. This value is similar to what was obtained for the third case from the data by Davidson-Arnott and Law (1990), discussed earlier in the paper. In both test cases a time period of 40 years was simulated using a time step of 1 month (=0.0833 years). Case 1. Infilling of groin compartment. The first test case focused on the infilling of the groin compartment, where the initial shoreline was located at the shoreward end of the groins (that is, at yG0 in Fig. 6). Thus, the berm crest was 20 m shoreward of the shoreline and the dune foot another 40 m shoreward of the berm crest (equilibrium conditions assumed according to Eq. 28). Fig. 7 illustrates the simulated time evolution of yS, yB, and yG (=yB + 20) for Case 1.

M. Larson et al. / Coastal Engineering 116 (2016) 57–66

Fig. 7. Simulated evolution of different contour lines in connection with infilling of a groin compartment.

Case 2. Shortening of groins. The second case focused on the effects of shortening the groins when the compartment is initially filled up. It was assumed that the groins are shortened with 30 m, given them a total length of 120 m after modification. The shoreline at t = 0 was in line with the seaward groin tips, that is, yG0 = 150 m. As in Case 1, initially the berm crest was located 20 m shoreward of yG0 and the dune foot another 40 m in the shoreward direction. Fig. 8 illustrates the simulated time evolution of yS, yB, and yG (=yB + 20) for Case 2. Note that LG b yG0 implying that Qout N Qin, which is no problem mathematically but it remains to be determined how physically accurate the reservoir model can describe this situation. The schematic cases shown in Figs. 7 and 8 indicate that the shoreline in the groin compartment mainly responds over a period of about 10 years, whereas the dune foot evolves at a much slower scale. It takes approximately 3–4 times longer for the dune to come close to its equilibrium position compared to the shoreline. 8. Concluding remarks A model was developed to simulate the cross-shore sand transport for application in regional coastal evolution models that describe

65

processes at the decadal scale. The cross-shore model included modules for calculating dune erosion and overwash, wind-blown sand, and barberm material exchange. All modules were calibrated and validated in standalone mode before put together to simulate the evolution of the schematized profile. In order to simulate this evolution, the transport equations were combined with a set of sand volume conservation equations that encompassed the dune, berm, and bar regions. The model was tested in a case study derived from an investigation at Westhampton Beach that focused on the effects of shortening a number of groins to release sand accumulated in a groin field, which caused downdrift erosion. The cross-shore model was coupled to a simple longshore transport equation, illustrating the interaction between these two transport types and the time scales involved. It was shown that the response of the dunes were significantly slower than the berm and shoreline reaction to changes in the longshore transport. The calibration and validation of the modules implied that certain coefficient values were assigned based on comparison with data. Although the transport formulas were derived from considerations of the governing physics, some of these coefficients are expected to be site-specific and data are needed to apply the modules with confidence at a particular site. Examples of such coefficients are Cs in the dune impact model, A in the overwash model, CB and λo in the longshore bar model, and δ in the wind-blown sand transport model. In the case when sufficient data are lacking, however, the coefficient values discussed here should provide a first rough idea about suitable values. The cross-shore model can easily be included in a regional coastal evolution model that simulates for time and space scales over decades to centuries. Although the model may be operated at time step short enough to resolve the impact of individual storms, which often is necessary even in regional models since the consequences may be longlasting, the schematization of the governing processes allows for short computational times.

Acknowledgements This work was partially funded by the Regional Sediment Management Program under the Inlet Geomorphologic Work Unit of the Coastal Inlets Research Program of the U.S. Army Engineer Research and Development Center, and partly by Sida/SAREC under the grants for continued bilateral research cooperation between Swedish institutions and Eduardo Mondlane University in Mozambique, Program Integrated Water Resources Management – Quantitative and Qualitative Aspects of IWRM for Sustainable Development in Southern Mozambique (2011-002102), and SWE-2010-038.

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Fig. 8. Simulated evolution of different contour lines in connection with sand release due to shortening of the groin lengths in a groin compartment.

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