Division of Earth and Ocean Sciences, Nicholas School of the Environment, Center for Nonlinear and Complex Systems, Duke University, Durham, North Carolina, USA
Corresponding author: K. Ells, Division of Earth and Ocean Sciences, Nicholas School of the Environment, Center for Nonlinear and Complex Systems, Duke University, 103 Old Chemistry Bldg., Box 90227, Durham, NC 27708, USA. (email@example.com)
 The future of large-scale coastline evolution will be strongly coupled to human manipulations designed to prevent erosion. We explore the consequences of this coupling using a numerical model for large-scale coastline evolution to compare the long-term, non-local effects of two generalized classes of shoreline stabilization: 1) beach nourishment (the addition of dredged sand to an eroding beach), and 2) hard-structures (e.g., seawalls, groynes, etc.) which fix the position of the shoreline without adding sand. In centurial model experiments where localized stabilization is maintained in the context of changing climate forcing, both forms of stabilization are found to significantly alter patterns of erosion and accretion at distances up to tens of kilometers. On a cuspate-cape coastline similar to the North and South Carolina coast, USA, with stabilization applied to the eroding updrift flank of a single cape, perturbations to coastline evolution are qualitatively similar within ∼20 km for each stabilization scenario, though they differ in magnitude both updrift and downdrift of the stabilized shoreline. The “human” signal in coastline change can extend as far as a neighboring cape (approximately 100 km away), but these long-range effects differ for each scenario. Nourishment resulted in seaward growth of the stabilized cape, increasing the extent that it blocked sediment flux in downdrift regions of the coast through wave shadowing. When stabilized with a hard structure the cape's initial position remain fixed, decreasing wave shadowing.
 Local shoreline change rates are frequently manipulated through various forms of shoreline stabilization to protect property and infrastructure from the threat of erosion. The short term (years to decades) and local (kms) effects of various types of shoreline stabilization have attracted considerable study. Typical analyses of beach nourishment (the emplacement of sand from a non-local source on an eroding beach) focus on the transient effects of single perturbations to an equilibrium shoreface profile or the diffusion of plan-view shoreline shape that results from alongshore redistribution of sand by waves approaching either normal or at slightly oblique angles to the local shoreline orientation [Dean, 2002; Dean and Yoo, 1992]. In contrast to beach nourishment, hard structures are often built to maintain a desired beach width or fix the position of the shoreline without adding sediment nearshore. Shore-perpendicular structures such as groynes and jetties act as artificial littoral boundaries, trapping sand on their updrift side with consequent erosion in their lee, leading to the construction of “groyne fields” that often extend several kilometers alongshore (e.g., Westhampton Beach, New York [Nersesian et al., 1992]). Kraus and McDougal reviewed the effects of shore-parallel structures (e.g., seawalls and revetments), noting that transient erosional effects in their vicinity may be more likely to result from gradients in alongshore sediment flux than from wave reflection or long-term profile changes.Basco  reinforced this conclusion, indicating that the effects of seawalls on adjacent shorelines are most likely analogous to groynes or headland/bay systems.
 Exploring the possibility for local shoreline stabilization to alter patterns of coastline evolution far beyond the relatively local vicinity of individual engineering projects has only recently been considered [Slott et al., 2010; van den Berg et al., 2011], in light of recent insight into the instability in plan-view shoreline shape resulting from highly oblique offshore wave incidence [Ashton et al., 2001; Ashton and Murray, 2006a, 2006b; Falqués and Calvete, 2005; van den Berg et al., 2012]. Previous numerical modeling studies showed that large-scale (10–100 km) shoreline shapes (e.g., sand waves, cuspate capes) can self-organize on decadal to millennial time scales when the offshore wave climate (distribution of wave-approach angles before refracting and shoaling over nearshore bathymetry) is dominated by waves approaching at angles greater than ∼45° with respect to the general shoreline orientation [Ashton et al., 2001; Ashton and Murray, 2006a, 2006b]. Subsequent work, starting with a complex large-scale coastline shape similar to the North and South Carolina capes, USA (Figure 1a), showed that even slight shifts in offshore wave energy distribution (as may be expected from global warming related changes in storm patterns [Knutson et al., 2010]) can induce rapid coastline shape change (Figures 1c and 1e) and accelerated erosion, especially near the seaward extended capes (Figures 1d and 1f) [Slott et al., 2006]. Extending this analysis, Slott et al. included beach nourishment at different locations along the simulated cuspate-cape system. They found that effectively fixing the position of a local stretch of shoreline in the long term could alter rates of change on spatial scales commensurate with the scale of the coastline pattern (up to 100 km, in the case-study cuspate coastline), and that these effects can be intensified by a shifting wave climate. In long-term numerical experiments (decades to centuries), the long-range effects of local stabilization propagate both progressively alongshore and through a non-local mechanism, wave shadowing, where large seaward promontories (e.g., capes) reduce the wave energy reaching distant shorelines.
 As the concentration of human population in coastal zones increases [Small and Nichols, 2003], the prevalence of shoreline stabilization is not likely to abate, and the long-term sustainability of beach nourishment has come into question when considering feedbacks between coastline morphodynamics and economic decisions for stabilization driven by coastline change [McNamara et al., 2011]. If the paradigm of shoreline stabilization were to shift from beach nourishment to hard structures (e.g., due to increased costs or depletion of the sand resource [McNamara et al., 2011]) the human influences on coastline change will be quite different.
 In this work we use a numerical model to compare the way a cuspate-cape coastline similar to the Carolina capes responds to sustained hard-structured stabilization and beach nourishment in the context of a wave-climate shift based on observations of increased influence from hurricane generated waves [Komar and Allan, 2008]. We present results for three model scenarios: one involving localized hard structures, one involving localized beach nourishment, and a control run representing no stabilization. Although the unique morphology and long history of stabilization on the Carolina capes [Pilkey et al., 1998; Valverde et al., 1999] provides an illustrative case study, our intent is not to simulate or make quantitative predictions of the details of a particular coastline, but rather to gain insight into general behaviors that may apply on large-scale, complex-shaped coastlines where human forcing is present. To our knowledge this is the first study to address the effects of hard structures on these spatial and temporal scales, especially in the context of complex coastline shapes where the non-local effect of wave shadowing is a relevant physical mechanism.
2.1. Numerical Model
 We use a numerical model that is described in detail by Ashton and Murray [2006a]. The model is based on the assumption that on large spatial and temporal scales changes in the cross-shore positionηof a sandy shoreline are driven by gradients in wave-driven alongshore sediment fluxQs, and that erosion and accretion extends to a depth D, where the offshore extent of the equilibrium shoreface profile intersects with the continental shelf. Assuming no sediment losses or gains this can be cast in a continuity equation,
where x is alongshore distance. Alongshore sediment flux is calculated with a common formula [Komar, 1971] that relates breaking wave height Hbto breaking wave-crest angleϕb and local shoreline orientation θ (with respect to the general shoreline trend),
where K is an empirical constant that requires calibration to field measurements. Here we set K = 0.4 m1/2 s−1 based on previous model calibration with fifty years of observed shoreline change along the Outer Banks of North Carolina [Slott et al., 2006, 2010]. (Historical shoreline change rates can be found at http://dcm2.enr.state.nc.us/index.htm, NC50hereafter.) The model domain is discretized into a global two-dimensional grid of cells with 1 km × 1 km resolution. Here we setD equal to 10 m, approximating observed conditions off of the Carolina coast (Figure 1a). Wave inputs—alongshore-uniform height and angle relative to the global coordinate system—are best interpreted as representing wave conditions at the seaward extent of approximately shore-parallel contours (i.e., the base of the shoreface). For each shoreline cell, the input wave is iteratively shoaled and refracted over contours assumed to parallel the local shoreline (no spatially explicit wave propagation is involved), until the height/depth ratio reaches a value deemed to cause depth-limited breaking. The shoreline then evolves according to (1) and (2). Wave shadowing occurs when protruding shoreline features block waves from reaching adjacent shorelines. Whenever a shoreline is in a shadowed region for the current incident wave angle sediment flux is set equal to zero.
2.2. Wave Climates
 For each daily model iteration, a new input wave is chosen randomly from a probability distribution of wave approach angles described by two parameters: Ais the proportion of left-approaching waves (looking offshore, relative to the global coordinate system), andU is the proportion of waves approaching from high angles (>45°). Wave climates dominated by high angle waves (i.e., U > 0.5) tend to cause undulations in the shoreline to grow [Ashton et al., 2001], while predominantly low angle wave climates smooth plan view shoreline shapes.
 We use values for U and A based loosely on the recent wave climate along the Carolina coast, determined from 20 years of wave data from Wave Information Study hindcast station 509, located off of Cape Fear (Figure 1a; hindcast data can be retrieved at http:/wis.usace.army.mil/, hereafter WIS). These wave data represent conditions in water depths considerably greater than the base of the shoreface, which would be a source of inaccuracy if we intended to reproduce the evolution of the actual Carolina coastline. Our intent, however, is merely to generate a model coastline that shares some of the main characteristics of the prototype. We hold offshore significant wave height H0 = 1.7 m, based on the effective average wave height 〈H012/5〉5/12 [Ashton and Murray, 2006b] from the WISdata. Calculating the relative influences on alongshore transport from the different wave-approach directions (discretized into four bins in this case) [Ashton and Murray, 2006b] yields wave climate parameters A = 0.55 and U = 0.60 for recent conditions off the Carolina coast [Slott et al., 2006].
 Recent work documented an approximate 0.054 m/y increase in hurricane-season related offshore significant wave heights at NOAA buoy 41002 [Komar and Allan, 2008]. McNamara et al.  applied this linear increase to wave heights in the WIS509 record, finding that hurricane-related waves approached dominantly from the east and northeast (from the left, looking offshore as inFigure 1b). We simplify this wave climate shift by adjusting the asymmetry parameter by 10%, so that A = 0.65 and U = 0.60. This magnitude of wave-climate shift simplifies comparison with the results ofSlott et al. , although we note that they interpreted this parameter change as a scenario of increasing influence from extra-tropical storms. However, this wave climate shift is not intended to be a robust prediction of future storm behaviors, but rather one possible scenario. In addition, for simplicity we model an instantaneous shift in wave climate, and hold the offshore wave heightH0 = 1.7 m. This approach implicitly averages over seasonal shifts in wave climate parameters in order to explore the coastline response to changes in the longer-term wave climate.
2.3. Shoreline Stabilization
 To represent the long-term effects of nourishment, select shoreline cells are designated as nourishing coastal communities. In a given time step, if a divergence of alongshore sediment flux causes one of these locations to erode past its initial position, sand is added at a rate that will counteract the flux divergence. We assume that nourishment sand is retrieved from outside of the modeled nearshore system, and timescales of intermittent shoreface-adjustment processes (months to years) [List and Farris, 1999] are ignored because of our focus on the longer timescales of decades to centuries. We also do not consider economic factors affecting communities' decisions to nourish their beach, including optimal nourishment intervals based on cost-benefit analyses [Smith et al., 2009] or the scarcity of the nourishment sand resource [McNamara et al., 2011].
 Hard structures are represented by holding a designated section of the initial shoreline fixed throughout the model run (no erosion landward of the initial position is allowed) with no addition of sand. Accretion can occur in front of a hard structure if sediment flux converges there, and each landward cell behind the stabilized shoreline cell is also held fixed, so adjacent shorelines may not erode into areas landward of the structure. We neglect the relatively small scale effects that hard structures have on nearshore sediment transport processes, such as scour or wave reflection at a seawall [Kraus and McDougal, 1996].
3.1. Initial Cuspate-Cape Coastline and Responses to Changing Wave Climate
 In the following experiments we use an initial model coastline that was generated by subjecting a straight coastline with random cross-shore perturbations to theWIS based wave climate (A = 0.55, U = 0.60) until capes with an alongshore scale of approximately 100 km evolve. These capes exhibit cross-shore shore amplitudes and aspect ratios similar to the Carolina capes (Figure 2a). When the initial cape coastline is forced with 200 simulated years of the same wave climate, with no influence from stabilization, the capes maintain their general shape (Figure 1c), with patterns of erosion updrift of the cape tip (∼1 m/y) and accretion downdrift (∼1.2 m/y). Increasing A, the coastline adjusts by attaining a less symmetrical shape (Figure 1e), accompanied by shifts in the distribution and magnitude of erosion (∼2.3 m/y) and accretion (∼6.3 m/y) [Slott et al., 2006].
3.2. Coastline Responses to Stabilization and Increased Asymmetry
Slott et al. conducted a series of six independent model experiments, each with nourishment added to a different 10 km section of a single cape that were then compared to a control run with no stabilization. With increased wave climate asymmetry the long-term human signal in shoreline change was most pronounced when nourishment was added to the updrift flank of the cape's seaward extent. This stabilization-induced shoreline perturbation (the difference between the final shoreline positions in the stabilized and non-stabilized model runs,η′ = ηhuman − ηnatural) is reproduced in Figure 2b (blue line) and compared to the case of hard structured stabilization (red line), where the initial model coastline is shown in Figure 2a for reference. Positive and negative perturbations represent sections of coastline with final positions more seaward or more landward than in the control run, respectively. In this experiment we used a wave climate representing an increase in storm generated wave influences (A = 0.65), again for 200 simulated years (experiments without a wave climate shift show the same qualitative behavior but with lower magnitude, similar to the findings of Slott et al. ).
 Within roughly 20 km for both forms of stabilization shoreline perturbations show similar trends; erosion was prevented updrift (η′ > 0) and accretion was prevented downdrift (η′ < 0). However, their relative magnitudes differ, with the hard structure less positive than nourishment updrift and more negative downdrift. Within the stabilized section (vertical dashed lines) the perturbation is positive everywhere for nourishment but crosses from positive to negative for the hard structure. This occurs because nourishment maintains a saturated (transport-limited) sediment flux locally (flux is not limited by sediment availability), while the hard structure limits flux, which prevents some accretion (flux convergence) near the downdrift extent of this section. (This effect is represented explicitly in similar experiments on an initially straight coastline; seeauxiliary material.)
 Further downdrift, nourishment induced a slightly positive perturbation within the bay, followed by a small negative perturbation to the downdrift cape tip. The negative perturbation adjacent to stabilization extends farther for the hard-structure, followed by a sharper increase to a positive perturbation that extends as far as the downdrift cape tip, where the nourishment perturbation was negative.
Figure 3 compares the final shorelines for the control run (black), beach nourishment (blue), and the hard structure (red). Figure 3a shows the stabilized and downdrift capes, and Figure 3bshows a magnified view of the downdrift, non-stabilized cape. Local nourishment builds the proximal cape outward in both the cross-shore and alongshore directions, partly negating the tendency of the cape's shape to adjust to the shifting wave climate. The hard structure, on the other hand, prevents any extension of the cape.
3.3. Physical Mechanisms
 Changes in the plan view shoreline position result from gradients in alongshore sediment flux that are created by two physical mechanisms: 1) local shoreline curvature and 2) alongshore gradients in the amount of wave shadowing. For each model scenario we calculate time averaged net flux Qnet at each shoreline location, where Qnet is positive for rightward and negative for leftward fluxes. We then express the “flux perturbation” as the difference between the net flux for the stabilization scenarios and that of the control run, or Q′net = Qnet, human − Qnet, natural (Figure 2d). Due to the sign convention for calculating alongshore sediment flux (positive for rightward and negative for leftward directed fluxes) Q′net > 0 may be interpreted as either an increase in rightward or a decrease in leftward-directed fluxes due to stabilization [Slott et al., 2010]. Figure 2c compares alongshore gradients in the flux perturbations, to the cross-shore perturbations fromFigure 2b. Sections of shoreline resulting in a positive (negative) cross-shore perturbation correspond to negative (positive) gradients in the flux perturbation, such that
Because stabilization occurs near the seaward extent of the capes in this experiment, wave shadowing plays a substantial role in how the large-scale signal is transmitted. In each model run we also tracked the net influence that wave shadowing had on each shoreline cell as the amount of sediment flux prevented because of shadowing,Qshad. Hence, Qshad > 0 for rightward and Qshad < 0 for leftward fluxes prevented by shadowing. Then, similar to the flux perturbation, we calculate the wave shadow perturbation Q′shad as the difference between the flux prevented by shadowing in the stabilization and control runs. Figure 2e shows the wave shadowing perturbation, compared to the flux perturbation in Figure 2d. From this figure we can isolate sections of the coastline where stabilization affected the amount of wave shadowing as opposed to local-curvature-related gradients in sediment flux. For example, the sections adjacent to the stabilized shoreline whereQ′net is negative correspond to negligible Q′shad, showing that perturbations there primarily result from altering shoreline orientations adjacent to stabilization and therefore the relative angle that approaching waves make with them according to (2).
 Alterations to wave shadowing have large influence on the far-field responses to stabilization. Within the bay between the stabilized cape and its updrift neighbor we note a small positive flux perturbation where there is a negative shadow perturbation of comparable magnitude for both beach nourishment and the hard structure. In both cases this can be attributed to a decrease in leftward fluxes that results from an increase in the amount that waves approaching from the right are shadowed by the stabilized cape.
 Downdrift, beach nourishment leads to a positive flux perturbation within the bay yet a negative one closer to the downdrift cape tip, both attributable to wave shadowing. The positive flux perturbation within the bay (near ∼400 km alongshore position; Figure 2d) occurs because the nourished cape initially shadows this region less than the control run. The alongshore shift of the maximum cross-shore extent of the cape is prevented by nourishment, exposing this updrift section of the bay to more waves from the left (i.e.,Q′shad < 0). As the nourished cape extends further seaward than the control run (Figure 3), shadowing is increased near the downdrift cape (Figure 2e, ∼425–445 km). At the downdrift cape tip, the alongshore gradient in the shadowing perturbation tends to cause increased flux divergence (Figure 2c), and hence increased erosion relative to the control run (i.e., η′ < 0; Figure 2b). Without the addition of nourishment sand, the cape stabilized by hard structures remains fixed (no seaward or alongshore translation; Figure 3), decreasing shadowing of rightward fluxes throughout the downdrift bay (Figure 2e). This ultimately increases flux convergence at the downdrift cape tip, decreasing erosion relative to the control run (i.e., η′ > 0; Figure 2b).
 The modeling approach used here involves several simplifications of processes occurring on time and space scales that are small relative to our scales of interest (decades to centuries and tens of kilometers). The sediment transport equation assumes that wave transformation to breaking wave quantities occurs over shore parallel contours. However, linear stability analysis of the high-angle wave instability considering variations in bathymetric contours, wave heights, and wave periods [Falqués and Calvete, 2005] show this to be a reasonable approximation for coastline features with large wavelengths (∼10 km or greater). Wave shadowing, a crucial driver for the far-field responses to stabilization, is treated with a rule-based approach, ignoring sediment fluxes driven by wave energy from local wind waves or waves diffracted (or refracted over non-shore-parallel contours) around capes, therefore neglecting the effects of shoals near the cape tips [McNinch and Luettich, 2000]. In addition, we model instantaneous shifts in wave-climates rather than those that occur more gradually over decades to centuries. Although sea-level rise (and consequent cross-shore sediment redistribution) is expected to produce shoreline erosion [Moore et al., 2010], we neglect this approximately alongshore-uniform component of shoreline change, since our goal here is to investigate the alongshore-transport driven, spatially heterogeneous responses of a complex-coastline to shoreline stabilization in the context of changing wave climate.
 The model simplifications accounting for these discrepancies would be a significant drawback if our intent were to make detailed predictions about the evolution of a particular coastline. They are, however, appropriate for the exploratory modeling approach we have taken, where we seek insight into essential interactions and mechanisms rather than detailed predictions [Murray, 2007]. Nonetheless, we can make some comparisons with the Carolina coastline. Modeled nourishment sand volumes were found to agree with those documented on the North Carolina coast, particularly at Wrightsville Beach and Carolina Beach [Slott et al., 2010; Valverde et al., 1999], located on the updrift flank of Cape Fear, the southernmost and most stabilized cape in Figure 1a. There is some disagreement with the behaviors found in our results and observed shoreline change rates on Cape Fear; the downdrift flank of this cape is found to be highly erosive in spite of a long history of high-volume nourishment on the updrift flank (NC50). There are, however, other modifications to this region that confound direct comparison to the field, such as the jetties at Masonboro Inlet (the downdrift boundary of Wrightsville Beach) and the heavily dredged Cape Fear River inlet on the cape's southern flank. These human influences would need to be added to model the behavior of this specific coastline.
 More generally, as Slott et al. found for the case of localized beach nourishment in the context of shifts in storm behaviors and associated shifts in wave climates, our results show that stabilization through hard structures can have long-range effects in the long term. Our results also show that the long-range effects of hard structures differ significantly from those of nourishment, and can be much more pronounced in some locations.
 The two forms of stabilization in this study are implemented without regard to the economic decision making processes of humans. Recent work has revealed emergent behaviors arising from coupling between human and coastline dynamics when beach nourishment decisions depend on an increasing cost of sand and the distribution of property values along a complex-shaped coast [McNamara et al., 2011]. This coupling is likely to be different if hard structures are introduced as an alternative stabilization strategy. A town located downdrift of a community protecting their shore with a seawall, for example, may cease to benefit from a convergence of sediment flux and thus become forced to alter their own stabilization behaviors. This work has demonstrated the differences between the large-scale, long-term effects of two fundamentally different forms of localized human manipulation of the coastline system. Future studies including a combination of stabilization methods at different locations, and dynamic stabilization decisions, are likely to yield new insights into the nature of this morpho-economic coupling.
 The National Science Foundation grant EAR-095182 supported this work. Thanks to Jordan Slott and Jennifer “JJ” Johnson for technical assistance, and to Déborah Idier and an anonymous reviewer for helpful comments.
 The Editor thanks Déborah Idier and Peter Adams for assisting in the evaluation of this paper.