An Alternative Analytic Model of Beach Nourishment Planform Behavior Free

Beach nourishment has been used as an erosion-control tool for nearly a century, with frequent usage over the last 50 years. While beach nourishment has been successful in application, the tools to implement the technology are limited. The tools to quantify the evolution of the beach planform include, but are not limited to, zero-dimensional analytic models, one-dimensional shoreline models such as the GENESIS model created by the U.S. Army Corps of Engineers, N-line models, which couple cross-shore sediment transport with alongshore transport, and three-dimensional morphodynamic models, which rely on computationally complex expressions of sediment transport to resolve sediment movements.

The application of beach nourishment tools and models has often been hampered by the poor replication of existing conditions by the tools or models. If a model fails to explain the variable prenourishment erosional stresses along a shoreline, the inclusion of a beach nourishment project in the model will not result in a quantitatively reliable estimate of performance. This poor performance can be overcome by linearly summing the existing or background erosion rates to the results of a modelled beach nourishment.

The zero-dimensional analytic model has been explored by many authors and summarized by Dean (2002). This analytic treatment relates the shoreline evolution processes of beach nourishment to the diffusion of heat and provides symmetric shoreline evolutions about the initial centroid. Dean and Yoo (1991) rederived the combined wave and sediment parameter (beach diffusivity) in terms of offshore wave climate and thus introduced weak influences of the wave period on beach nourishment performance. Since then, minimal efforts have been undertaken to expand the analytic formulation of the evolution of a rectangular sand placement along an initially straight shoreline.

This paper explores the suggestions of two authors (Inman, 1987; Larson and Kraus, 1991) that the inclusion of an advective term in the governing equation will yield more realistic beach nourishment planform evolutions than diffusion-only solutions. It will be shown that the inclusion of the advective term in the governing equation will result in beach nourishment centroid migration and nonsymmetric beach nourishment planform evolution.

While it is recognized that all shoreline evolution models are valuable, the analytic models are attractive because of the following features.

1. Analytic models can be used for verification of numerical models of similar derivation.

2. Analytic models describe the coastal processes through applicable assumptions, physical properties, and relationships. Therefore, the model provides a good instructional tool to understand beach nourishment processes.

3. Analytic models are appropriate for preliminary design evaluations of beach nourishment performance.

4. Analytic models are also appropriate when there is insufficient time or money to establish and calibrate a one-dimensional shoreline model or a process-based numerical model.

5. Analytic models are computationally efficient.

The development of analytic beach planform evolution models is based on a series of assumptions. Here, the derivation of an analytic shoreline model (after Dean, 2002) based on the conservation of sand is reviewed. The shoreline model is then extended by considering the conservation of sediment flux as suggested by Inman (1987), which includes a sand advection term.

The beach diffusivity, G, has units of length²/time.

Equations (7) and (8) are derived under the assumptions that the wave height is uniform across the alongshore model domain and that the wave height is constant in time. The alongshore wave height uniformity assumes that the offshore bathymetry consists of straight and parallel contours such that the wave refraction and diffraction effects of the bathymetry on the offshore wave climate are uniform. The assumption that the wave height is constant in time implies that an average wave height can describe the long-term wave energy forcing.

Equation (8) provides an approximation of the behavior of a beach nourishment planform on an open coast under wave forcing described simply by the wave height. The breaking angle of the waves, α_b, does not appear in the solution; thus, the diffusion of the beach nourishment is principally a smoothing of the beach nourishment planform caused by the wave height as if it were normal to the shoreline. As a result, Equation (8) indicates that the evolution of the shoreline will be symmetric about the center of the initial nourishment, y(x, t) = y(−x, t). For some beach nourishment projects, Equation (8) deviates from the measured data, and one of the principal deviations is that the measured data are not symmetric about the center of the initial beach nourishment. In the following section, a new planform evolution model is proposed that provides for nonsymmetric beach nourishment planforms.

The inclusion of an advective velocity term in Equation (6) to model the behavior of beach nourishments and sand waves has been discussed by several authors.

Larson and Kraus (1991) discussed the inclusion of both advective and diffusion-based processes, provided an approach to solving Equation (9) without the source/sink term, and provided an analytic solution to an initial triangular nourishment. Huisman et al. (2013) described the use of an advective diffusion model to describe the shoreline evolution of the Sand Motor project in The Netherlands. Due to the complexity of the initial shoreline condition (quasi-semicircular), the authors utilized a numerical solution to Equation (9).

Hanson, Thevenot, and Kraus (1996) investigated the alongshore movement of alongshore sand waves along Southampton, New York, utilizing a modified GENESIS model, which included an advective velocity term (Equation [9] without the source/sink term). Due to the complexity of the initial shoreline conditions, a numerical solution was required. They simulated the alongshore movement of the Southampton, New York, sand waves using an average U value of 350 m/y.

While numerical solutions are advantageous when initial conditions are complex and wave climates do not reflect averages, analytic solutions can yield insightful information regarding the primary coastal processes. In this paper, the following assumptions are made:

1. Solutions of Equation (9) are sought for q = 0, and U = constant.

2. The initial planform is a rectangular beach nourishment that extends from the shoreline to a distance Y over a length L, where Y/L is small. The planform is centered on x = 0.

3. Cross-shore processes are ignored, with the overall behavior of the beach nourishment from the landward edge of the berm to the depth of closure being reasonably described by a correlated movement of the shoreline (Equation [1]).

Three calculation examples are provided to identify the effects of sand advective velocities on beach nourishment planforms. First, a literature review is provided to identify a range of advective velocities. From this range of published values, a beach nourishment planform evolution simulation is provided for a range of advective velocities to provide an example of the sensitivity of the planform evolution to the advective velocity. Second, a comparison of the analytic model (Equation [15]) to the data of Grove, Sonu, and Dykstra (1987) demonstrates the applicability of the model to a sand release on the California coast. Lastly, the model is compared to the 1996 nourishment of a beach in Naples, Florida, which demonstrates the model’s strengths and weaknesses in the prediction of beach nourishment planforms.

To develop reasonable estimates of beach nourishment performance of rectangular beach fills, an estimate of the advective velocity must be developed. The following authors provided estimates of the advective velocity from various measurements and analyses.

Grove, Sonu, and Dykstra (1987) presented the results of field measurements of a 153,000 m³ release of sand from a 300-m-long planform onto the beaches of San Onofre, California. Grove, Sonu, and Dykstra (1987) tracked the sand planform over 3 years using beach profiles. The survey data demonstrated that the sand release included both diffusional and advective processes. They concluded that the movement of the centroid of the beach planform varied, and it ranged from U = 730 m/y to 820 m/y.

Inman (1987) provided a detailed discussion of the measurements presented by Grove, Sonu, and Dykstra (1987) from San Onofre, California, and other estimates of U for other California locations. Inman (1987) suggested that U ranged from 500 to 4000 m/y for select locations in California. Inman hypothesized that U (m/y) is equal to the ratio of net alongshore sediment transport (m³/y) to the interannual cross-shore sediment transport volume (m³/m).

Sonu’s data ranged from approximately U = 200 m/y for L = 10,000 m to U = 11,000 m/y for L = 30 m. Sonu’s conclusion has implications for the prediction of beach nourishment planforms as the speed of migration of the nourishment centroid will be reduced with increased initial nourishment length. Conversely, the performance of small-length beach nourishments will suffer from both higher rates of diffusion (Dean, 2002) and increased advective velocity (after Sonu, 1968).

While Equations (19) through (21) provide a more detailed localized advective velocity, U is a function of y(x), which deviates from the assumption that U is a constant as assumed in the derivation of Equation (9). For the numerical simulation conducted by Hanson, Thevenot, and Kraus (1996) of Southampton, New York, sand waves, an average value of U = 350 m/y was determined.

To investigate the effects of advective velocity on the beach nourishment planform, a large beach nourishment design was considered with the following parameters (Table 1): Length, L = 6000 m; diffusivity, G = 0.0354 m²/s; and initial beach width, Y = 30 m. Simulations are presented for t = 8 years. The beach nourishment was then subjected to advective velocities of U = 0 m/y, 500 m/y, and 1000 m/y, respectively. The planform evolution results, at t = 8 years, are shown in Figure 2. Figure 2 shows that the inclusion of advective velocity causes a lateral shift of the beach nourishment planform in the U direction. The planforms demonstrate that the inclusion of advective velocity results in a nonsymmetric planform about the initial center of the beach nourishment planform. For large values of U, most of the beach planforms, relative to the case of U = 0, are transported outside of the initial fill area.

The data of Grove, Sonu, and Dykstra (1987) provide a measured response to an initial release of sand on the California coast. The data were collected through beach profile surveys at select locations and specific times. Solving Equation (15) for U = 820 m/y, a beach diffusivity of G = 0.0042 m²/s, and a beach nourishment of L = 300 m yields shoreline positions for specific time frames corresponding to the survey data (Figure 3). This parameter combination corresponds to a Pe value of 1.9 (Table 1).

Equation (15) demonstrates that the advection of the sand release is substantive for the initial short sand fill placement (300 m). While Equation (15) was calibrated by varying the beach diffusivity, G, until the root mean square error (RMSE) was minimized at 2.41 m, the model represents the shoreline width well, with slight underprediction at t = 0.69 years and slight overprediction at t = 1.69 years and t = 1.91 years.

A comprehensive beach nourishment design may include considerations of the preexisting erosion rate, the variation in the preexisting shoreline orientation or position, the storm protection width, and sediment compatibility and overfill, as well as planform evolution (National Research Council, 1995). Therefore, the beach planform evolution is only one component of a comprehensive beach nourishment design. In practice, the inclusion of all of these design considerations will frequently result in a nonuniform initial nourished shoreline width. Therefore, comparing the predicted planform evolution (Equation [15]) to any prototype data is comparing the results of one design consideration versus a complete design.

The 1996 beach nourishment of Vanderbilt Beach in Naples, Florida, was selected for comparison to the beach nourishment planform predicted by Equation (15). Vanderbilt Beach is located on Florida’s southwest coast and is subject to a relatively mild wave climate and low sediment transport. The beach is not located near a tidal inlet, not directly influenced by limestone rock outcrops, and does not contain any significant coastal structures (groins, seawalls, breakwaters, etc.) within the nourishment area. Thus, the project is a reasonable candidate for evaluation of Equation (15).

The Florida Department of Environmental Protection archives mean high water position data for beach nourishment projects. The mean high water data are collected as part of the series of beach profiles that are collected on a variable spacing that typically ranges between 300 m to 400 m. Preconstruction mean high water positions in 1995 (prior to nourishment) were subtracted from mean high water positions in 1996 (postnourishment) to determine the added shoreline widths (Figures 4 and 5). Figures 4 and 5 show that the added width is not uniform along the shoreline. Due to the nonuniformity of the placed sand, three modelled nourishments were calculated.

The first nourishment simulation held the actual length of the nourishment at L = 2200 m and actual volume = 246,800 m³ and determined the equivalent added shoreline width, Y, which resulted in an average added width of approximately 22 m. Modeling was undertaken by varying the advective velocity, U, and the diffusivity, G, until the RMSE was minimized at 21.2 m. The results are presented in Figure 4 for U = −19 m/y, G = 0.0006 m²/s, and t = 8.8 years. Most of the error was derived from one data point at x = 2000 m. Otherwise, it was noted that the RMSE was generally insensitive to the advective velocity.

The second nourishment simulation held the maximum added width Y = 31 m and the actual volume = 246,800 m³ and determined the equivalent project length, L = 1560 m. Modeling was undertaken by varying the advective velocity, U, and the diffusivity, G, until the RMSE was minimized at 22.1 m. The results are presented in Figure 5 for U = −25 m/y, G = 0.0012 m²/s, and t = 8.8 years. Most of the error was derived from one data point at x = 2000 m. Otherwise, it was noted that the RMSE was generally insensitive to the advective velocity.

The third nourishment simulation held the maximum added width Y = 31 m and the actual volume = 246,800 m³ and determined the equivalent project length, L = 1560 m. Modeling was undertaken by varying the advective velocity, U, and the diffusivity, G, until the predicted shoreline and the measured shoreline were in close visual agreement with emphasis on the region x > 0. The RMSE was not minimized over the data limits. The results are presented in Figure 6 for U = 12 m/y, G = 0.0014 m²/s, and t = 8.8 years. The use of a positive value of U resulted in better fitting of the measured data for x > 0 than in Figures 4 and 5.

All three predicted planform evolutions provide general compliance with the measured data from Vanderbilt Beach. The planform evolutions in Figures 4 and 5 show that the incorporation of a small negative value of U results in reduced RMSE when compared to the measured data. The planform evolution in Figure 6 shows that the incorporation of a small positive value improved the comparison with measured data (visually) for one side of the nourishment (x > 0).

Neither of the first two planform evolutions replicated the shoreline closely, as evidenced by RMSE values that are the same order of magnitude as the initial nourishment width, Y. Even if the data at x = 2000 m are omitted, the RMSE error is high relative to the initial shoreline width.

While the inclusion of an advective velocity term in the formulation appears to improve the planform evolution of an initially uniform nourishment on a straight shoreline (Figures 2 and 3), the model does not appear to forecast future shoreline positions well (Figures 4, 5, and 6). Equation (15) is simulating only the planform evolution of an equivalent rectangular nourishment on a straight shoreline. The model is not addressing variations in initial shoreline orientation, initial shoreline position, background erosion rate, or sediment differences between native and fill sands. Therefore, Equation (15) should best be used in conjunction with other evaluations of the effects of shoreline orientations, sediment differences, and background erosion rates. Alternatively, Equation (15) can be utilized to determine the relative performance of a range of beach nourishment designs for a given project. This is similar to the recommendation of Roelvink and Reniers (2012), who discussed the appropriate application of three-dimensional morphologic modeling to practical coastal problems.

The advective diffusion beach nourishment planform evolution model (Equation [15]), provided herein is an improved model of beach nourishment planform evolution. This model generates nonsymmetric beach planforms and may yield more realistic beach nourishment planform behavior than diffusion-only models.

Three non-complementary methods were provided for the estimation of the advective velocity. Future research into the prediction of the advective velocity is recommended. The data of Sonu support higher advective velocities for shorter beach nourishment fill lengths. Therefore, rapid evolution of the beach nourishment planform is expected as a result of advective processes and diffusive processes.

The authors cited herein are acknowledged for their contributions to the beach nourishment planform solution described here. The constructive criticisms of the anonymous peer reviewers improved the manuscript.