Coastal Services Center

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Cross-shore and Longshore Transport Models for Large Scale Geological Processes

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Introduction

The processes and conditions affecting changes in the cross-section and shoreline of the beach are complex and include storm water levels and waves, sediment characteristics, and, for nourished beaches, initial project characteristics and the project setting. A beach nourishment project area can be located on a long straight beach, within a pocket beach, or adjacent to an inlet. To describe these processes and the response of the shoreline after sand placement, numerical models are employed in the design phase of a beach nourishment project. The capabilities of these models to accurately represent the evolution of the planform and profile of the shoreline are key to successful project performance and therefore of interest to stakeholders and decision-makers. These same models can be applied to natural beaches, but due to the large variability in the natural system, the models are more effective for cases which have been modified significantly, such as by beach nourishment.

Beach nourishment projects represent a planform anomaly or "bulge" in the shoreline. In addition, sand is placed at slopes that are steeper than the "natural" or equilibrium profile. The steeper profile and the "bulge" in the waterline relative to the pre-nourished beach system induce sediment transport in both the longshore and cross-shore directions. Under normal wave conditions, transport in the longshore direction causes sand to be moved from the project area to adjacent beaches where sediment deposition occurs. Similarly, transport of sand in the onshore or offshore direction (cross-shore transport) results in an adjustment toward an equilibrium profile. High waves and water levels during storms result in accelerated and modified longshore and cross-shore sediment (sand) transport processes.

Although the two induced sediment transport components (longshore or cross-shore) occur simultaneously in nature, the need for simplicity in engineering applications has resulted in modeling them separately. Longshore sediment transport models have been used for about 50 years, whereas cross-shore sediment transport models have been of interest for less than two decades. It is possible to develop simplified models for planform evolution that provide substantial insight and quantitative understanding of the processes. The available cross-shore transport models are less developed than for longshore sediment transport, but reasonably reliable predictions are possible for limited situations.

Factors Limiting Performance Predictability

There are several factors which limit the design professional's capability to predict the performance of a beach nourishment project, including: (1) variability of the natural processes, including waves which affect the project, (2) differences between the "actual" sediment characteristics and beach slopes of the constructed project from those considered in design, and (3) approximations inherent in our models of cross-shore and longshore sediment transport processes. The National Research Council (1985) has developed the estimates presented in Table 1 that show the more localized the prediction and the more complex the setting, the less the accuracy of the prediction. All of the factors noted above contribute to uncertainty in design and prediction. This paper focuses on models available for these purposes and a limited assessment of the validity of such models.

Longshore Sediment Transport Models

Longshore sediment transport is the result of waves approaching the shoreline at an angle. Within the area where waves are generated, the waves acquire energy and momentum from the winds. As the waves propagate from the generating area toward shore, energy and momentum are transported to the shore as in Figure 1. Within the wave breaking zone, drastic modifications occur to the wave energy and momentum. The energy dissipation of the waves within the surf zone and the generation of turbulence are evident through the foam and chaotic motion apparent to the observer. Within the surf zone the waves also transfer momentum. If waves are breaking at an angle to the shoreline, the momentum will impart a longshore force to the water within the breaking zone, resulting in a longshore current (which is evident to swimmers being moved along the coastline). These currents are generally small, as shown in Figure 2 which presents a histogram of 5,591 longshore currents measured on the Pacific Coast of the United States. (Note that approximately 86 percent of the currents are less than one foot per second.) In conjunction with the sediment mobilization by the breaking waves, the resulting currents can transport substantial quantities of sediment. The combination of weak currents generated by the oblique waves and the mobilization of sediments due to turbulence result in transport of large quantities of sediment.

A nourishment project represents a planform anomaly as shown in Figure 3a. Following sand placement, waves tend to smooth out this anomaly. A method of representing this evolution is through a numerical longshore sediment transport model. Figure 4 represents a schematic of a control volume as represented in a numerical longshore transport model. From this figure it is seen that if the sediment transport is greater into than out of a control volume, the sediment volume within that element will increase. Conversely, if the volume of sediment transport into a given area (control volume) is less than that moved out of the given area, the volume of sand within the element will decrease (erode). The "bookkeeping" of these sediment transport differences is in the so-called "continuity equation," discussed later in this paper.

In applying longshore sediment transport models, it is necessary to specify boundary conditions that best approximate controls that will govern the project evolution. When nourishment occurs along a long, straight shoreline, areas far from the project area remain unaffected. The boundary conditions of the model must represent the situation. In cases where a complete littoral barrier is present (such as downdrift of a jettied inlet), the transport is set equal to zero.

Pelnard Considère discovered a similarity between the models of heat conduction and diffusion from classical physics and shoreline evolution. His equations provide several useful results that are described as follows. Quite surprisingly, if a beach nourishment project is constructed with sand that is compatible with the native beach sand on a long straight beach, the evolution of the beach planform is relatively insensitive to wave direction. This may seem counterintuitive; however, the explanation is that the same amount of sand flowing into the nourishment area as that moving away from the nourishment area results in a planform centroid whose position is unchanged. If the nourishment sand is finer or coarser than the native sand, the nourishment planform centroid migrates in the downdrift and updrift directions, respectively (Dean and Yoo 1992). Moreover, the planform evolution of a beach nourishment project is approximately proportional to the cumulative wave energy that has been expended on the project since its construction. Thus, the evolution at a particular time is insensitive to storm sequencing, that is, whether a severe storm precedes a milder storm or whether the milder storm occurs first followed by the severe storm.

Longshore transport depends on a number of factors including wave height and direction relative to the shoreline and the sediment size. The role of sediment size in longshore sediment transport has been examined by several investigators and is represented in terms of a sediment transport coefficient. Studies by Dean (1987) show that an inverse relationship exists such that the larger the sediment size, the smaller the transport coefficient.

As noted earlier, the continuity equation is a formalized approach to the bookkeeping of sediment flows and accounts for the differences of transport into and out of selected control volume, as shown in Figure 4. It is possible to interpret this volume change in terms of a shoreline change. For the simplest case in which the nourishment and native sediments have same size characteristics, it is reasonable to assume that the profile translates seaward or landward without change of form in response to an increase or decrease, respectively, of volume change per unit beach length. The shoreline change may then be calculated as the volume change per unit beach length divided by the vertical dimension of the active profile. This is clearly an approximation, and allows interpretation of shoreline changes from volume changes. This procedure is referred to as the "profile translation method" and the vertical depth of the active profile is an essential quantity in this determination of shoreline changes from volume changes. This dimension is the sum of the active portions of the profile above and below the water level. In most cases, the vertical dimension above the water level is taken as the elevation of the berm (see Figure 4), and the dimension of the portion of the active profile below the water level, which is referred to as the "closure depth." It is intuitive that the closure depth is related to the wave climate. In medium-sized bays the closure depths are on the order of four to six feet. On the Florida Atlantic Coast, they range from 12 to 20 feet, and on the Oregon and Washington shorelines, they are on the order of 40 feet. If the total vertical dimension of the active profile is 27 feet, the rule of thumb applies that each cubic yard of compatible sand added as beach nourishment material results in one square foot of increased beach plan area.

With the models described above, it is now possible, given an initial shoreline, wave climate, and conditions at the boundaries of the computations, to calculate the planform evolution of a beach nourishment project. The planform model "GENESIS" (Hansen and Kraus 1989) is such a model used by the Corps of Engineers. Other examples of planform evolution models are provided in Dean and Dalrymple (2001) and Dean and Yoo (1992).

Cross-shore Sediment Transport Models

In addition to the longshore sediment transport induced by the planform anomaly, beach nourishment projects are usually constructed with slopes that are steeper than equilibrium (see Figure 3b) which induces sediment transport in the cross-shore direction. In contrast to longshore sediment transport, which has been studied for more than 50 years, cross-shore sediment transport investigations have been conducted for a much shorter period and thus predictive capabilities are not as well developed. Natural beaches tend to adopt a so-called "equilibrium beach profile" (EBP) which depends primarily on sediment size, although wave conditions also affect the EBP to a lesser degree. EBP concepts are now being used increasingly in the design of beach nourishment projects. Because the EBPs of beaches composed of coarser sand are steeper than those for finer sand, the additional dry beach width depends substantially on the nourishment sand size relative to the native sand size.

In addition to the EBP to which the project profile is considered to evolve, the time scales of this "profile equilibration" are of interest to designers and stakeholders. Numerical models have been developed to represent profile evolution and usually require the profile to ultimately converge to the EBP. These models are also discussed in Dean and Dalrymple (2001). Although these models have been demonstrated to represent beach profile recession during storms reasonably well, they tend to predict too rapid of a recovery rate. Additionally, no models have been demonstrated to predict the generally slower profile adjustment associated with beach nourishment projects. Simpler models have been fitted to measurements of profile change and have documented that, for the projects examined, approximately 50 percent of the profile equilibration occurs within a period of two to three years (Dean 2002). This is important because beach renourishment usually occurs some five to eight years following initial project construction. The model SBEACH (Larson and Kraus 1989) is the cross-shore model used by the US Army Corps of Engineers.

The EBP for an increased water level will be elevated by the amount of the increased water level and displaced landward to achieve a sediment balance. The landward profile displacement (recession) is proportional to the elevated water level. This relationship was developed by Bruun in 1962 and is called the Bruun Rule. The Bruun Rule predicts an equilibrium shoreline recession that is approximately 50 to 100 times the increased water level, although this factor depends inversely on the profile slope. This is of particular significance when accounting for sea level rise, where profile slopes are very mild, as can be found along the Louisiana and Mississippi coastlines. Thus, a one inch increase in sea level can result in 50 to 100 inches of shoreline recession.

Summary

The time scales of beach nourishment profile equilibration are generally considered to be shorter than for beach nourishment planform evolution. This has led to the engineering consideration that the cross-shore equilibration occurs soon after the project is constructed. Much engineering effort is devoted to the estimation of the project's design life, or the time required for the beach width or volume remaining within the project area to erode to a specified "design value." Models are available which allow approximate calculations of the shoreline and profile changes associated with beach nourishment projects.

References

Dean, R. G. 1987. "Coastal Sediment Processes: Toward Engineering Solutions." Proceedings, Coastal Sediments, ASCE. Pages 1 to 24.

Dean, R. G., and C.–H. Yoo. 1992. "Beach–Nourishment Performance Predictions." Journal of Waterway, Port, Coastal, and Ocean Engineering, ASCE. Volume 118, Number 6. Pages 567 to 586.

Dean, R. G. 2002. "Beach Nourishment." Ed. M. Isobe and N. C. Kraus. Chapter 17 in Coastal Engineering, a Book in Tribute to Professor Kiyoshi Korikawa.

Dean, R. G., and R. A. Dalrymple. 2001. Coastal Processes With Engineering Applications. Cambridge University Press, United Kingdom, 475 pages.

National Research Council. 1995. Beach Nourishment and Protection. National Academy Press, Washington, DC.

Pelnard–Considère, R. 1956. "Essai de Théorie de l' Evolution des Formes de Rivage en Plages de Sable et de Galets." 4th Journées de l' Hydraulique, Les Energies de la Mer, Question III, Rapport No. 1.