Evaluating Transport Formulations for Application to Nearshore Berms

Introduction

Methods

Model Overview

Throughout this paper, berm deflation is defined as the removal of sediment from a control volume delimited by the original placement footprint (Fig. 1). As in previous studies, it is assumed that transport removes sediment from the control volume but cannot add new sediment (Johnson and Work 2005), such that a reversal in transport direction cannot generate berm “reinflation.” It is clear that this assumption simplifies the dynamics of sediment transport in the nearshore region. For example, Andrassy (1991) and Barnard et al. (2009) both report brief periods of sediment accumulation within a longer trend of berm deflation. Nevertheless, the simplification is reasonable considering that the nearshore berm is a significant perturbation to the beach profile that will deflate as the profile equilibrates.

Considering the experimental results that identified longshore transport as the dominant deflation direction (Smith et al. 2015; Bryant and McFall 2016; Smith et al. 2017), this study evaluates the utility of several longshore transport formulations for predicting nearshore berm deflation. However, because various field studies note advection and/or diffusion in the cross-shore direction (e.g., Andrassy 1991; Bodge 1994; Otay 1995; Ojeda et al. 2008; Marinho et al. 2018), a cross-shore transport component is also included in the calculations. The longshore and cross-shore components are treated as orthogonal vectors which are calculated independently and then superimposed to produce a total deflation rate. To distinguish between longshore and cross-shore motion, the subscript y indicates transport parallel to the y-axis (the longshore direction), and the subscript x indicates transport parallel to the x-axis (the cross-shore direction; see Fig. 1).

All methods investigated focus on transport from waves and wave-induced current. Tidal currents are excluded from this analysis to evaluate the feasibility of applying transport equations with minimal input to provide a first-order estimate of a nearshore berm’s deflation rate. Application of these techniques adjacent to tidal inlets is cautioned. In addition, although wave refraction around the edge of nearshore berms can cause a focusing of wave energy and induce localized shoreline erosion, most nearshore berms are constructed as shore-parallel linear features to mitigate wave focusing and maximize wave attenuation benefits for a larger stretch of shoreline (McLellan et al. 1990; Priestas et al. 2019). Setup currents generated by breaking waves have been observed to transport water shoreward to the lee side of the nearshore berm, inducing a flow alongshore on the lee side, and then return offshore at the ends of the berm (van Duin et al. 2004). In some examples, the nearshore placements are segmented with gaps to allow for return flow (e.g., Fort Myers Beach, FL; Brutsché et al. 2014). Such circulation patterns are not explicitly quantified in the present methodology, but are implicitly incorporated because increased wave breaking at the berm crest increases the circulation current and the longshore transport.

Longshore Transport Calculations

Calculation of berm deflation via the longshore transport equations in Table 1 requires two sets of inputs. First, it is necessary to define several time-invariant parameters, including the depth and cross-shore position of the berm at mean water level (h_crest and x_crest, respectively), the landward and seaward positions of initial sediment placement (x_L and x_S), the diameter d₅₀ of the placed sediment, and an estimate of the beach slope m at the site. Second, the model is driven by time series values of the offshore significant wave height H₀(t), wave direction θ₀(t), and peak period T_p(t). For this study, these were obtained from the closest Wave Information Study (WIS; Hubertz 1992) station, which supplies a hindcast of all required values at 1-hourly intervals for a user-specified interval between 1980 and 2014. All calculations maintain the WIS timestep of Δt = 1 h.

The offshore WIS data are transformed into nearshore values using the following procedure. At each timestep, the breaker height H_b(t) is calculated as

$$H_b=0.39g^{1/5}{\left(T_p{H}_0^2\right)}^{2/5}$$

(1)

and the breaking depth h_b(t) is calculated from the breaker index γ_b = H_b/h_b = 0.78 (Komar and Gaughan 1972). Wavelengths at h_b and the WIS station depth h₀ are then calculated from the dispersion equation as

$$L_0=\frac{g}{2\pi }{T}_p^2\tanh \left(\frac{2\pi h_0}{L_0}\right)\mathrm{and}{L}_b=\frac{g}{2\pi }{T}_p^2\tanh \left(\frac{2\pi h_b}{L_b}\right)$$

(2)

using the numerical approximation of Hunt (1979). The offshore wave direction is rotated into a shore-normal coordinate system following the convention of McFall et al. (2016). The shore-normal breaker angle θ_b(t) is calculated using Snell’s law:

$$\sin {\theta }_b=\frac{L_b}{L_0}\sin {\theta }_0$$

(3)

Examples of the raw WIS time series and its transformation to nearshore values are shown in Figs. 2(a–c). The nearshore characteristics are then applied to the longshore transport rate equations shown in Table 1. When performing these calculations, the density of water (ρ_w = 1,025 kg/m³), density of sediment (ρ_s = 2,650 kg/m³), porosity (a = 0.4), friction coefficient (c_f = 0.005), and CERC coefficient (K = 0.4) are treated as universal constants rather than varying between sites.

The volumetric longshore transport rate Q_y(t) represents the total longshore transport over the entire nearshore region; i.e., $Q_y(t)={\int }_{\mathrm{\forall }x}{q}_y(x,t)dx$, where q_y represents the cross-shore variation in longshore transport. The magnitude of total longshore transport Q_y will exceed the berm’s volumetric longshore deflation rate because the nearshore berm only fills a portion of the cross-shore littoral zone. The raw values of Q_y calculated from Table 1 must therefore be scaled by some fraction between 0 and 1 to avoid producing an overestimate of the deflation rate. To estimate the fraction of Q_y which transports sediment out of the control volume, the proposed method utilizes a compilation of experimental cross-shore transport profiles from Bodge and Dean (1987) and Smith et al. (2009). These datasets were selected because they offer complete spatial coverage between the still-water shoreline at x = 0 and the experimental breaker position x_b. To allow for direct comparison of data at varying spatial scales, all experimental profiles were nondimensionalized as:

$$\hat{x}=x{x}_b^{-1}\mathrm{and}{\hat{q}}_y=q_y{x}_b{Q}_y^{-1}$$

(4)

A representative longshore transport profile ${\hat{q}}_y(\hat{x})$ was then produced by averaging the individual datasets [Fig. 3(a)]. Because ${\int }_{\mathrm{\forall }\hat{x}}{\hat{q}}_{y}d\hat{x}=1$, integrating ${\hat{q}}_y$ between the landward and seaward control volume boundaries will produce the desired fraction for scaling Q_y.

Before using ${\hat{q}}_y(\hat{x})$ at any particular field site, it is necessary to determine the location of the placed sediment in nondimensional space. As described in the previous section, the depth at breaking h_b(t) varies at 1-h intervals; consequently, the breaker position x_b (and, therefore, the nearshore berm’s nondimensional location) is also time-dependent. To accommodate scenarios in which a full bathymetric profile is unavailable, the beach profile landward and seaward of the placed sediment is represented by the line

$$h(x)=\left(\frac{h_{\mathrm{crest}}^{\mathrm{pre}}}{x_{\mathrm{crest}}}\right)\cdot x,x\le x_L\mathrm{or}x\ge x_S$$

(5)

where $h_{\mathrm{crest}}^{\mathrm{pre}}$ is depth at the crest position prior to nearshore berm placement. For x_L < x < x_S, the berm is assumed to be triangular with its peak at the point (x_crest, h_crest). Then, at each WIS timestep, the breaker position x_b(t) is assigned to the seaward-most occurrence of h_b(t), which may on the seaward side of the berm [Fig. 3(b)] or at some other location in the nearshore region [Figs. 3(c and d)] depending on the berm’s geometric parameters. The nondimensional landward and seaward limits of the control volume are then calculated as

$${\hat{x}}_L(t)=\frac{x_L}{x_b(t)}\mathrm{and}{\hat{x}}_S(t)=\frac{x_S}{x_b(t)}$$

(6)

where x_L and x_S are the dimensional distances between the mean-water-level shoreline and the landward and seaward boundaries of sediment placement.

Having determined ${\hat{x}}_L$ and ${\hat{x}}_S$, the berm’s instantaneous deflation rate is given by

$$Q_y^{\mathrm{berm}}(t)=|Q_y(t){\int }_{{\hat{x}}_L}^{{\hat{x}}_S}{\hat{q}}_y(\hat{x})d\hat{x}|$$

(7)

that is, the product of the raw Q_y-value and the fraction of longshore transport within the control volume. Taking the absolute value prevents reversals in the longshore transport direction from reinflating the nearshore berm. An example of using Eq. (7) is shown in Figs. 2(d–f).

Cross-shore Transport Calculations

The cross-shore deflation component is calculated using a modified method from Dronkers (2016), which was derived from the earlier results of Bailard (1981). This energetics-based transport model was selected based on the work of Thornton et al. (1996) and Gallagher et al. (1998), who used the Bailard (1981) model to explain the cross-shore migration of the offshore bar at Duck, NC. In unmodified form, the per-unit-width net bedload transport at a distance x from the shoreline is calculated as

$$⟨q(x,t)⟩=\alpha [\lambda m⟨|u_w{|}^3⟩-⟨|u_w{|}^2{u}_w⟩]$$

(8)

Following the sign convention in Fig. 1, the first term represents the gravity-driven offshore transport, and the second term represents the wave-driven onshore transport. In Eq. (8), α and λ are empirical parameters, m = Δh/Δx is the beach slope, u_cr is the critical velocity for sediment motion, and u_w(x, t) is the near-bed horizontal orbital velocity, which has maximum magnitude $u_w^{\max }$. Angular brackets 〈 〉 indicate averaging over a wave period. The transport direction of 〈q〉 is assumed parallel to the local wave angle θ and consequently contains both a longshore and a cross-shore component if |θ| > 0. To avoid double-counting the longshore transport when the results are superimposed, the present study modifies Eq. (8) into

$$⟨q_x(x,t)⟩=\alpha [\lambda m⟨|u_w{|}^3⟩-⟨|u_w{|}^2{u}_w⟩(1-\kappa )]\cdot \cos {\theta }_{\mathrm{crest}}$$

(9)

where θ_crest is the wave direction at the crest of the nearshore berm. As an additional modification, a critical velocity scaling term (1 − κ) with $\kappa =\min [u_{\mathrm{cr}}/u_w^{\max },1]$ has been introduced in Eq. (9) to prevent overprediction of transport during low-energy wave conditions. Hudson et al. (in prep.) recently validated Eq. (9) for a series of placements at the Columbia River mouth and found that α = 3 × 10⁻⁵ and λ = 1.7 were optimal parameter values. These values are within the range of values cited in Dronkers (2016) and are retained as constants in the present study.

To implement Eq. (9) in the model, the offshore WIS wave height H₀(t) is transformed to H_crest(t) at the berm crest using the conservation of energy flux and Snell’s law (e.g., Komar 1998). Stream function wave theory is then used to determine the near-bed velocity u_w(x_crest, t). Although expressions for the maximum and minimum values of u_w are provided by Ahrens and Hands (1998), a closed-form equation for u_w is not available. Instead, tabulated values of the stream function near-bed velocity from Dean (1974) were digitized for use in the transport model. For each 1-h model timestep, the phase-resolved velocity was selected from among the 40 cases in the table based on values of h_crest/L₀(t) and H_crest(t)/L₀(t) (see Fig. 4 for examples). The wave-averaged values 〈|u_w|² u_w〉 and 〈|u_w|³〉 were then calculated from u_w. Meanwhile, the critical velocity for sediment motion is calculated as

$$u_{\mathrm{cr}}=\{\begin{array}{cc}{\displaystyle {\left(\frac{8g{d}_{50}({\rho }_s-{\rho }_w)}{{\rho }_w}\right)}^{1/2},}& d_{50}\le 2.0\mathrm{mm}\\ {\displaystyle {\left(\frac{0.46g{T}_p^{1/4}(\pi d_{50}{)}^{3/4}({\rho }_s-{\rho }_w)}{{\rho }_w}\right)}^{4/7},}& d_{50}>2.0\mathrm{mm}\end{array}$$

(10)

following the method of Ahrens and Hands (1998).

The volumetric cross-shore deflation rate for the nearshore berm is expressed as

$$Q_x^{\mathrm{berm}}(t)=\left|⟨q_x(x_{\mathrm{crest}},t)⟩\right|\cdot \mathrm{\Delta }y$$

(11)

where Δy is the length of the nearshore berm in the shore-parallel direction. Eq. (11) is labeled the “D16” method in the results. As in Eq. (7), the absolute value reflects the assumption that the nearshore berm is exclusively a sediment source, and a reversal in transport cannot reinflate the berm. Example output of Eq. (11) is shown in Fig. 2(g). Finally, the total predicted volume loss from the berm over some duration of interest is obtained by time-integrating the total volumetric transport rate:

$$V_{\mathrm{loss}}={\int }_{t_{\mathrm{start}}}^{t_{\mathrm{end}}}(Q_y^{\mathrm{berm}}+Q_x^{\mathrm{berm}})dt$$

(12)

Historical Projects

The deflation estimation techniques are evaluated using monitoring results from 11 historical nearshore berms along the Pacific, Atlantic, and Gulf coasts of the United States (Fig. 5). These sites span a range of wave conditions and placement geometries to permit testing across a variety of nearshore berm scenarios. Measurements from several additional nearshore placements in the United States were omitted from this study because the placement locations were several kilometers offshore (e.g., Dam Neck, VA, Mobile, AL, and Brunswick, GA; see Priestas et al. 2019), making wave-induced longshore transport inapplicable, or because the placed sediment did not form an identifiable mound on the seabed (e.g., 2018 placement at South Padre Island, TX; Figlus et al. 2021).

A summary of geometric parameters from each location is displayed in Table 2. Whenever possible, Table 2 directly records the values reported in the original publication. If an author reported a range of values, the average is used to produce a single representative measurement for the entire berm. At several sites, depth was reported relative to a datum other than local mean sea level (MSL). Water depth h_crest above the berm was therefore corrected to its MSL value using the nearest National Oceanic and Atmospheric Administration (NOAA) benchmark. All descriptions of depth in this manuscript are relative to the local MSL datum unless otherwise noted. Beach slope was rarely reported and is ambiguously defined for a nonplanar bathymetric profile. Beach slope was therefore defined as $h_{\mathrm{crest}}^{\mathrm{pre}}/x_{\mathrm{crest}}$, where $h_{\mathrm{crest}}^{\mathrm{pre}}$ is the depth at the future crest position prior to placement. A brief description of the sites evaluated in this study is given chronologically in the following paragraphs. The sites are further summarized by Fig. 6, which displays the distribution of calculated breaker positions relative to the landward boundary of each historical placement.

	Fire	New	Port	Ft. Myers	Perdido	South	Silver	Newport	Ocean Beach			Coos	North
Parameter	Island	River	Canaveral	Beach	Key	Padre	Strand	Beach	First	Second	Third	Bay	Head
Parameters used for deflation rate prediction
Monitoring start	15-Jun-87	13-Aug-76	28-Jul-92	1-Oct-09	1-Oct-91	4-Jan-89	29-Dec-88	1-Nov-92	8-Jun-05	2-Jun-06	11-Jul-07	1-Jan-88	21-Sep-2018
Monitoring end	15-Dec-87	16-Sep-76	1-Jul-93	1-May-13	1-Nov-93	14-May-90	1-Nov-90	1-May-95	14-May-06	25-May-07	11-Dec-07	1-Jan-89	20-Nov-2018
WIS station #	63120	63284	63439	73295	73164	73020	83107	83102	83066	83066	83066	83031	NOAA 46248
WIS depth (m)	25	18	17	5	20	36	848	467	39	39	39	131	181
Shoreline angle	255${}^{\circ }$	248${}^{\circ }$	198${}^{\circ }$	298${}^{\circ }$	259${}^{\circ }$	173${}^{\circ }$	342${}^{\circ }$	303${}^{\circ }$	6${}^{\circ }$	6${}^{\circ }$	6${}^{\circ }$	31${}^{\circ }$	8${}^{\circ }$
d50 (mm)	0.40	0.49	0.40	0.16	0.30	0.13	0.18	0.27	0.18	0.18	0.18	0.28	0.25
xL (m)	250	110	530	80	700	910	200	150	500	500	500	2,400	1,870
xcrest (m)	300	170	560	130	825	1,070	245	400	600	510	510	2,950	1,880
xS (m)	390	240	730	200	1,150	1,220	400	590	1,200	1,200	1,300	3,500	3,300
hcrest (m)	3.5	1.1	5.1	0.5	4.5	6.5	3.9	4.5	10	9.9	9.8	16.4	12.9
$h_{\mathrm{crest}}^{\mathrm{pre}}$ (m)	5.6	3.6	6.7	1.7	6.1	8.3	5.9	9.1	11.5	10.3	10.3	24.1	13.5
Beach slope	0.019	0.021	0.012	0.013	0.007	0.008	0.024	0.023	0.019	0.020	0.020	0.008	0.007
Δy (m)	2,300	215	910	1,600	3,750	1,220	360	900	800	800	800	480	120
Measurements used for validation only
Vinit (m3)	3.1 × 105	2.7 × 104	1.2 × 105	1.7 × 105	3.0 × 106	1.3 × 105	1.2 × 105	9.8 × 105	3.4 × 105	2.6 × 105	4.1 × 105	6.9 × 105	3.8 × 104
Vend (m3)	9.9 × 104	6.7 × 103	5.2 × 104	1.0 × 105	3.0 × 106	2.6 × 104	2.1 × 104	7.0 × 105	1.6 × 105	2.6 × 105	4.0 × 105		1.3 × 104
Qberm (m3/day)	1180	600	200	50	0	200	150	300	490	0	90	360	410

New River Inlet, North Carolina: In August 1976, a nearshore berm was constructed between the 2.5 and 4.5 m depth contours southwest of New River Inlet (Schwartz and Musialowski 1977). The sediment was placed in mounds which coalesced to form a linear feature spanning a 215-m length of shoreline. The crest was approximately 170 m offshore with an initial depth at crest of h_crest = 1.1 m. Schwartz and Musialowski (1977) report that 75% of placed sediment was removed from the initial berm footprint within 34 days of placement. Because the New River Inlet project predates the earliest WIS hindcast values, longshore transport at this location was calculated using August and September WIS data for each year between 1980 and 2014. The results were then averaged to estimate the transport rate during the 1976 monitoring period. During the 35 years of WIS hindcast data, the mean value of H₀ in August and September was 0.8 m, and the maximum was 6.1 m. The mean value of T_p during these months was 7 s with a maximum of 20 s. As shown in Fig. 6, a large fraction of waves are predicted to break seaward of the berm crest at this site.

Fire Island Inlet, New York: A 2.2-km long linear nearshore berm was constructed near the 5 m depth contour adjacent to Fire Island Inlet on the southeast coast of Long Island, New York, in June 1987 (McLellan et al. 1988, 1990). The crest was approximately 300 m offshore, with an initial depth at crest of h_crest = 3.5 m relative to MSL. A bathymetric survey in December 1987 indicated that 68% of the original placement volume had been removed from the site. Over the 7-month period of June through December 1987, the mean value of H₀ was 0.8 m with a maximum of 2.6 m, and the mean value of T_p was 7 s with a maximum of 13 s. Fewer than 1% of the hourly wave events are predicted to break within the placement footprint (Fig. 6).

Coos Bay, Oregon: An offshore site between the 21 and 27 m depth contours near Coos Bay has been used for dredged sediment disposal since 1979, with an average annual placement of 4.6 × 10⁵ m³ between 1979 and 1988 (Hartman et al. 1991). The mound dimensions increased over this period due to the continued placement of sediment. In 1988, the landward and seaward boundaries were approximately 2.4 and 3.5 km offshore, respectively, and the depth at crest was 16.4 m. Although transport measurements for each individual year were not provided, Hartman et al. (1991) report an average deflation rate of 1.3 × 10⁵ m³/year over this period. Because the mound’s cross-shore boundaries have changed through time and the site report only displays geometry at the end of the monitoring period, the present study calculates transport conditions using WIS data from January through December 1988. During calendar year 1988, the nearest WIS station reports average values of H₀ = 2.5 m and T_p = 12 s. The maximum significant wave height and period were 8.4 m and 24 s, respectively. All waves are predicted to break shoreward of the berm crest.

Silver Strand, California: The Silver Strand nearshore berm was constructed in December 1988 using fine sand dredged from the San Diego Bay entrance channel (Juhnke et al. 1990; Andrassy 1991). The placement was configured as a linear bar along a 360-m span of beach, with its crest approximately 245 m offshore and h_crest = 3.9 m. Monitoring results indicated that 82% of the original placement volume was removed from the footprint after 22 months (Andrassy 1991). The nearest WIS station reports average values of H₀ = 0.9 m, with a maximum value of 3.9 m. The WIS data have a bimodal distribution of T_p, with peaks at 6 and 14 s and a maximum value of T_p = 24 s. However, a temporary gauge near the placement site recorded T_p between 7 and 9 s for 53% of measurements (Andrassy 1991), likely due to sheltering by Point Loma to the northwest. To account for this wave sheltering, waves with an offshore angle ${\theta }_0>{280}^{\circ }$ relative to north are assumed to produce zero transport in the model. The majority of waves are predicted to break landward of the placement boundary.

South Padre Island, Texas: In January 1989, a linear berm was constructed near the 8 m depth contour approximately 1 km offshore of South Padre Island (Aidala et al. 1992; Ahrens and Hands 1998). The initial depth at crest was h_crest = 6.5 m. After 17 months of monitoring, the placement volume had been reduced by 79%. WIS data from the monitoring period indicate average values of H₀ = 1.2 m and T_p = 6 s, with maximum values of 5.4 m and 19 s, respectively. Fewer than 1% of waves are predicted to break within the placement footprint (Fig. 6).

Perdido Key, Florida: In September 1991, a nearshore berm was constructed along a 4 km length of the Gulf Coast’s Perdido Key near the 6 m depth contour. The berm crest was 825 m offshore with h_crest = 4.5 m (Otay 1995; Work and Otay 1996). The placed sediment at this location displayed no significant movement over the 13-month period examined by Otay (1995). Waves at this location are smaller in amplitude and higher in frequency compared with many of the other locations, with average values of H₀ = 0.6 m and T_p = 4.5 s at the nearest WIS station during the project monitoring period. The maximum value of H₀ during this time was 3.7 m, and the maximum peak period was 11 s. Breaking is almost exclusively landward of the nearshore berm.

Port Canaveral, Florida: The Port Canaveral nearshore berm was constructed in July 1992 and spanned a 900 m length of coastline between the 5.8 and 7.5 m depth contours (Bodge 1994). The placement was configured as a linear bar with x_crest = 560 m and h_crest = 5.1 m. Bodge (1994) reports that 57% of the original sediment volume was removed from the initial footprint after two years. Because this site is located on the Atlantic coast of Florida, wave energy tends to be higher than sites along Florida’s Gulf Coast. During the monitoring period, average values of H₀ and T_p at the nearest WIS station were 1.3 m and 9 s, respectively. The maximum value of H₀ was 4.5 m, and the maximum value of T_p was 16 s. Although most waves break landward of the berm, 6% of the hourly waves are predicted to break within the control volume.

Newport Beach, California: The nearshore berm at Newport Beach was constructed in October 1992 as a series of irregular mounds spanning a 900-m length of beach. The sediment was placed between the 4 and 9 m depth contours (relative to MSL), with its crest approximately 400 m offshore and h_crest = 4.5 m at placement (Mesa 1996). Because the survey area from the original site report extends outside the initial berm footprint, the deflation rate for Newport Beach was recalculated for this study to ensure that the value was specific to the control volume of interest. Using the time series of bathymetric profiles in Mesa’s (1996) Newport Beach report, the deflation rate was calculated from the initial placement volume of 9.8 × 10⁵ m³ and an apparent 28% reduction in the berm's vertical cross-sectional area over the 2.5-year monitoring duration. The local WIS hindcast records a bimodal distribution of T_p with peaks at 7 and 15 s and a maximum period of 24 s. The average of H₀ was 0.8 m with a maximum of 3.9 m. Few waves are predicted to break within the placement boundaries.

Ocean Beach, California: At Ocean Beach, three nearshore berms were constructed at around 1-year intervals between 2005 and 2007 using dredged sediment from the San Francisco Bay shipping channel (Barnard and Hanes 2006; Barnard et al. 2009). The berms were constructed as irregular mounds occupying the same footprint, with the crest 500 to 600 m offshore and h_crest = 10 m. Because the nearshore berms did not fully deflate before subsequent placement at the same location, the initial volume of the second and third berms is defined as the sum of the reported placed volume plus any volume remaining in the study area from prior placement(s). For the second berm, tabulated values from Barnard et al. (2009) indicate an initial volume of 2.6 × 10⁵ m³ (i.e., 9.3 × 10⁴ m³ of newly placed sediment plus 1.6 × 10⁵ m³ remaining from the berm that was placed a year earlier); after a year, the volume is reported to have increased to 3.7 × 10⁵ m³. It is not clear whether the increase in volume is due to sedimentation shoreward of the berm or whether sediment was actually deposited within the berm control volume. As the present conceptual model does not accommodate berm “inflation,” in this instance it is assumed that the entire placed volume of 2.6 × 10⁵ m³ was remaining at monitoring end for a deflation rate of ∼0 m³/year. The third berm then has an initial volume of 4.1 × 10⁵ m³ (i.e., 1.5 × 10⁵ m³ of new sediment plus 2.6 × 10⁵ m³ remaining from the earlier placement). WIS data from 2005 through 2007 have average values of H₀ = 1.7 m and T_p = 12 s, with maxima of 6.4 m and 27 s. All waves are predicted to break landward of the placement.

Fort Myers Beach, Florida: The Fort Myers Beach (Gulf Coast) nearshore berm was constructed in September 2009 and is the shallowest placement in this study, with h_crest = 0.5 m at a distance x_crest = 130 m from the shoreline. The placement spanned a 1.6-km length of beach and was configured as an irregular linear bar with several gaps to allow recreational vessel access to the beach (Brutsché et al. 2014). Because the control volume from the original site report extended shoreward of the placement footprint, the deflation rate for Fort Myers Beach was recalculated for the present study using digital elevation models provided by Brutsché et al. (2014). At this location, the initial placement volume was found to be 1.7 × 10⁵ m³, with 1.0 × 10⁵ m³ remaining in the footprint after 3.5 years (i.e., 41% volume loss). During the study period, the nearest WIS station recorded a bimodal distribution of H₀ with peaks at 0.1 and 0.5 m; the average value of T_p was 4 s. However, Brutsché et al. (2014) reported that two major storm systems during the third year of monitoring caused significant transport relative to typical conditions. Both events appear in the WIS hindcast, and their effect on transport should therefore be represented by the proposed model. Within the 3.5-year monitoring period, approximately half of the waves are predicted to break seaward of the nearshore berm crest.

North Head, Washington: In September 2018, an offshore deposit of fine to medium sand was constructed approximately 3 km north of the Columbia River mouth between the 13.5 and 17.1 m depth contours (USACE Portland District, personal communication, 2020). Unlike the sites listed previously, the North Head placement had a shore-perpendicular orientation (width Δx = 1,500 m in the cross-shore direction and length Δy = 120 m in the longshore direction). The deposit was designed to have a constant thickness of 0.6 m relative to the preplacement seabed, positioning the berm crest near the landward placement boundary of x_L = 1870 m with h_crest = 12.9 m. Survey results indicated that the initial placement of 3.8 × 10⁴ m³ decreased to 1.3 × 10⁴ m³ (65% volume loss) over a 61-day period. Because the WIS hindcast terminates in 2014, longshore transport at this location was calculated using time series of H₀, θ₀, and T_p collected by the National Data Buoy Center (NDBC) during the project monitoring period. The buoy (NDBC station #46248) was located 40 km offshore of the Columbia River mouth and recorded at 30-minute increments; these measurements were then downsampled to Δt = 1 h to mimic the format of the WIS hindcast used at the other locations. Over the duration of site monitoring, the average values of H₀ and T_p were 2.1 m and 11 s with maxima of 4.5 m and 20 s. No breaking is predicted within the control volume (Fig. 6).

Results

Discussion

Recommendations and Conclusions

Data Availability Statement

Acknowledgments

Notation

A

Dean’s equilibrium beach profile shape parameter;

a

porosity;

c_f

friction coefficient;

d₅₀

median grain size;

F_b(t)

wave energy flux;

g

gravitational acceleration;

H_b(t)

significant wave height at breaking;

H_crest(t)

significant wave height at crest of nearshore berm;

H₀(t)

deep-water significant wave height;

h(x)

depth profile;

h_b(t)

depth at breaking;

h_crest

minimum water depth above the berm;

$h_{\mathrm{crest}}^{\mathrm{pre}}$

water depth at x_crest before sediment placement;

K

CERC coefficient;

K_sw

coefficient related to the percentage of low-period swell waves (see van Rijn 2014);

L₀(t)

wavelength at offshore WIS station;

MTR

local mean tide range;

m

beach slope (Δh/Δx);

$Q_x^{\mathrm{berm}}(t)$

nearshore berm deflation rate in the cross-shore direction;

$Q_y^{\mathrm{berm}}(t)$

nearshore berm deflation rate in the longshore direction;

$Q_y(t)={\int }_x{q}_{y}dx$

total volumetric longshore transport rate for the entire nearshore zone;

q_x(x, t)

volumetric cross-shore transport profile;

q_y(x, t)

volumetric longshore transport profile;

${\hat{q}}_y(\hat{x})$

normalized volumetric longshore transport profile;

T_p(t)

peak wave period; and

t

time;

u_cr

critical velocity for sediment motion;

u_w(x, t)

near-bed wave orbital velocity;

$\overline{V}(t)$

longshore current velocity, averaged over x;

w_s

sediment fall speed;

x

shore-perpendicular coordinate;

$\hat{x}$

normalized shore-perpendicular coordinate;

x_b(t)

breaker location relative to the still-water shoreline;

x_crest

x-coordinate corresponding to h_crest;

x_L

landward boundary of placed sediment;

x_S

seaward boundary of placed sediment;

y

alongshore coordinate;

α

empirical transport parameter from Dronkers (2016);

γ_b

breaker index H_b/h_b;

ɛ

Bayram et al. (2007) transport coefficient;

θ_b(t)

wave angle at breaking relative to shore-normal;

θ_crest(t)

wave angle at x_crest;

θ₀(t)

offshore wave angle;

$\kappa =min[u_{\mathrm{cr}}/u_w^{\max },1]$

critical velocity scaling term;

λ

empirical transport parameter from Dronkers (2016);

ξ

Iribarren number;

ρ_s

density of sediment; and

ρ_w

density of water.

References