Barrier islands are narrow strips of land formed by waves through a variety of (hypothesized) mechanisms (e.g., Gilbert, 1885; McGee, 1890; Penland et al., 1985) associated with relatively slow SLR rates and primarily passive margins (FitzGerald et al., 2008; Stutz and Pilkey, 2011, McBride et al., 2013). The emergence of many barrier islands can be traced back to about 6000 years before present, when Holocene SLR slowed down (McBride et al., 2013).

However, the relationship between barrier islands and SLR is complex. Under no SLR, barrier islands are generally not observed as their associated back-barrier environments would fill completely (e.g., Beets and van der Spek, 2000). In contrast, under moderate SLR rates marshes and tidal flats generally occupy back-barrier environments. In this case, to maintain their elevation with respect to sea level, barriers migrate towards land as storm overwash and flood–tidal flows deposit sediment. Under higher SLR rates, however, it is more difficult for barriers to maintain their subaerial portion above sea level. Consequently, when onshore-directed sediment fluxes are insufficient, barrier islands drown in place and are left offshore (Rodriguez et al., 2001; Mellet and Plater, 2018). Additionally, when onshore-directed sediment flux events are very intense and frequent, barrier islands are unable to maintain their geometry as they rapidly migrate towards land, which also results in drowning (Lorenzo-Trueba and Ashton, 2014). This potentially delicate balance between SLR and barrier response, together with the current projections of future acceleration in SLR, highlight the need to better constrain onshore-directed sediment fluxes in different barriers island systems (Carruthers et al., 2013; Lazarus, 2016; e.g., McCall et al., 2010; Rogers et al., 2015).

One way for sediment to be transported across the barrier is through storm overwash. Differences in water-level setup between the ocean and the lagoon during a storm can force the flow of water and sediment through and above the subaerial portion of the barrier. Most frequently this flow is directed landward, resulting in the transport of sediment from the ocean to the bayside where it deposits as the flow spreads laterally into the lagoon (Carruthers et al., 2013; Donnelly et al., 2006). Although this process is complex and highly intermittent, individual storm events integrated over time result in a net landward sediment flux, which allows barriers to keep pace with SLR over geological timescales (Leatherman, 1983). Despite its importance in terms of future barrier island morphodynamic response and vulnerability to flooding (Miselis and Lorenzo-Trueba, 2017), this long-term landward sediment flux is generally poorly constrained, and its relationship with modern overwash fluxes is not straightforward (Carruthers et al., 2013; Donnelly et al., 2006; Lazarus, 2016; Rogers et al., 2015). This lack of constraints on long-term overwash fluxes has resulted in a suite of barrier island models that do not compute overwash processes as a function of single storm events. Instead, such models parameterize overwash volume fluxes as a function of barrier geometry and observations of barrier island migration. For example, Leatherman (1979) observed that narrow barrier islands tend to be more susceptible to overwash events than wide barrier islands. They defined a “critical barrier width” below which overwash is frequent and the barrier migrates rapidly and above which overwash and barrier migration tend to be slow. Based on these findings, overwash is often parameterized by assuming the volume is inversely proportional to island width (e.g., Jiménez and Sánchez-Arcilla, 2004; Lorenzo-Trueba and Ashton, 2014) and additionally adjusted based on local factors such as land use (e.g., Rogers et al., 2015).

Aside from storm overwash, tidal inlets have also been found to be a major contributor to barrier transgression (Inman and Dolan, 1989; Moslow and Heron, 1978; Pierce, 1969). Tidal inlets derive their transgressive potential through the deposition of flood–tidal deltas. The volume of flood–tidal delta deposits correlates with the size of the associated inlet (Powell et al., 2006). Simple equilibrium models (e.g., Stive et al., 1998) suggest that initially flood–tidal deltas grow fast but that their growth slows down as they approach an equilibrium volume and the bay fills up near the inlet. Inlet migration can therefore add to transgressive transport by exposing new bay to flood–tidal delta deposition (Nienhuis and Ashton, 2016). For these two reasons, it has been hypothesized that short-lived and rapidly migrating inlets are most efficient for barrier transgression (Pierce, 1970). However, if the migration rate and life span of tidal inlets correlate with sediment import, their potential for transgression should then depend on factors such as basin size, ocean waves, and tidal conditions.

Along extended barrier coastlines, barrier morphodynamics are complicated by the existence of multiple tidal inlets. Tidal inlets interact through their control on water surface elevation in the tidal basin. This interaction can cause inlets to close or change size (van de Kreeke et al., 2008; Roos et al., 2013). Observations of tidal inlet spacing (Davis and Hayes, 1984), corroborated by a recent modeling study (Roos et al., 2013), found that increasing tidal range and basin size can allow inlets to exist closer together. For a barrier coast, a greater number of inlets likely enhances their contribution to barrier transgression.

The joint long-term effect of storm overwash and tidal inlets on barrier island evolution remains difficult to quantify. On the one hand, engineering models typically assess barrier island changes over annual to decadal timescales, which includes overwash fluxes and tidal inlet formation during storm events. For example, models such as XBeach (McCall et al., 2010; Roelvink et al., 2009) resolve wave dynamics coupled with sediment transport during storm events and are able to capture barrier morphological changes, including breaching. On decadal timescales, models like Delft3D (Deltares, 2014) have been applied to study inlets, but these typically do not include the effect of storms or SLR (e.g., Tung et al., 2009; NA16). On longer timescales, models no longer use laboratory-validated sediment transport relationships but rather use various degrees of conceptual relationships between barrier geometry and barrier island movement (Cowell et al. 1995; Storms et al., 2002; Stolper et al., 2005; Masetti et al., 2008; Wolinsky and Murray, 2009; LTA14). Some of these models are morphokinematic, i.e., based upon the conservation of mass and the maintenance of barrier geometry (Cowell et al., 1995, Wolinsky and Murray, 2009; Stolper, 2005). The models developed by Storms et al. (2002), Masetti et al. (2008), and LTA14 are morphodynamic as they account for sediment fluxes along the shoreface and across the barrier island. LTA14 represents a significant simplification compared to other morphodynamic models, making it suitable for model extensions and model coupling (such as the one presented here).

Coming from a different angle, the ASMITA (Aggregated Scale Morphological Interaction between Tidal inlets and the Adjacent coast) model couples coasts to their back-barrier environment via sediment exchanges determined by the deviation of a morphological element (ebb delta, tidal flat, etc.) from an assumed equilibrium volume. ASMITA has been developed in part to understand the effects of SLR on inlets and their back-barrier environments (van Goor et al., 2003; Stive et al., 1998; Townend et al., 2016). Inlets cannot close or migrate, and the model does not account for overwash processes (Stive et al., 1998). In ASMITA, as well as other back-barrier models (van Maanen et al., 2013; Mariotti and Canestrelli, 2017), the maximum potential sediment import through tidal inlets exerts a first-order control on the ability of back-barrier environments to sustain themselves during SLR.

Here, we describe a new model (BRIE) that accounts for tidal inlet dynamics, including opening, closing, and lateral migration, combined with barrier overwash processes as described by LTA14. Within the realm of coastal geomorphological models, BRIE can be considered a large-scale coastal behavioral (LSCB) model (de Vriend et al., 1993). It seeks to represent only the main governing mechanisms of the coast at appropriate timescales, without fully resolving the mechanics of fluid and sediment transport. There is a rich body of literature concerning LSCB models, ranging from rocky coasts (Walkden and Hall, 2011), barrier islands (Stolper et al., 2005; LTA14), tidal basins (Townend et al., 2016), tidal inlets (Kraus, 2000), and sandy coastlines (Ashton et al., 2001), to aggregates of LSCB models that couple these elements (Ashton et al., 2013; Payo et al., 2017). To our knowledge, BRIE would be the first to explicitly couple barrier islands and tidal inlet morphodynamics. Despite its simplicity, BRIE provides a novel approach to study the evolution of barrier islands under decadal to millennial timescales. Moreover, it allows us to explore complex barrier dynamics across a wide range of parameter values.

After initializing the environment (typically ∼100 km long barrier island with periodic boundaries) and determining wave climate and shoreface parameters (Table 1), we run each time step in a for loop. In each iteration, we first raise sea level (Fig. 2). SLR affects subaerial barrier volume and shoreface slope, which in turn drives overwash and shoreface fluxes (Sect. 3.2).

Next, we determine if new inlets should be formed (Sect. 3.3.1), in which case we analyze their hydrodynamics and calculate their equilibrium dimensions. For each inlet, we distribute sediments into the flood–tidal delta, the barrier island, and the shoreface (Sect. 3.3.3). Flood–tidal delta deposition changes the back-barrier location (Fig. 1). After each time step, we add the different sources and sinks to the coastal zone, including diffusive wave-driven alongshore sediment transport, and implicitly determine a new shoreline, back-barrier, and shoreface toe position. See Table 1 for an overview of all model parameters and units.

Inlets can form along barrier island chains if there is sufficient potential for tidal flow between the lagoon and the open ocean (Escoffier, 1940). In turn, the potential for tidal flow is determined by factors influencing the potential tidal prism (e.g., the proximity of other tidal inlets nearby, the width and depth of the basin and the barrier, the marsh cover) and factors reducing tidal flow (e.g., tidal inlet friction, wave-driven transport into tidal inlets). Once inlets exist, they alter barrier morphodynamics by distributing sediments and enhancing storm overwash potential.

After we have determined the various sources and sinks of sediment to the nearshore environment, we distribute sediment alongshore between the different cells based on alongshore sediment transport. We use an implicit Crank–Nicolson scheme (Crank and Nicolson, 1947) to solve for shoreline change, governed by the following nonlinear diffusion equation: 36Δxs,jΔt=DjΔ2xΔy2+Δxs,j,extΔt, which includes the effect of wave refraction and shoaling and is therefore suitable to apply based on offshore (deepwater) wave conditions (AM06). We have added a source–sink term xs,j,ext (m) to account for cross-shore sediment movement. Dj is a nonlinear term and accounts for the fact that diffusivity depends on the wave approach angle (AM06), 37D(θ)=k(Hb+DT)H012/5T1/5Eϕ0⋅Ψϕ0-θ, where k is ∼0.06 m3/5 s-6/5 (Nienhuis et al., 2015) and θj=(xs,j+1-xs,j)/Δy. Ψ is the angle dependence of the diffusivity (AM06) that we compute as follows: 38Ψϕ0-θ=cos⁡1/5ϕ0-θcos⁡ϕ0-θ2-65sin⁡2ϕ0-θ, which we convolve with the normalized angular distribution of wave energy E(ϕ), 39E(ϕ0)=a⋅hif-12π<ϕ0<-14πa+(1-h)if-14π<ϕ0<0(1-h)(1-a)if0<ϕ0<14π(1-a)hif14π<ϕ0<12π, to generate a long-term, wave-climate-averaged shoreline diffusivity for every alongshore location j.

We rewrite the shoreline diffusion Eq. (28) into 40xsjn+1-xsjnΔt=Dj2xsj+1n+1-2xsjn+1+xsj-1n+1+xsj+1n-2xsjn+xsj-1nΔy2+Δxs,j,extΔt, where n and j denote the specific time and space locations. We solve this equation by inverting this nearly tri-diagonal matrix: 41⋱⋱00-βj⋱⋱⋱000-βj1+2βj-βj000⋱⋱⋱-βj00⋱⋱⋮xsj-1n+1xsjn+1xsj+1n+1⋮=⋮⋮xsjn+βjxsj+1n-2xsjn+xsj-1n+Δxs,j,ext⋮⋮, where βj=DjnΔt/2/Δy2. Because we use D at n instead of n+1 this is simply a linear diffusion equation. Indices in the lower right and upper left corner indicate periodic boundary conditions. The source xs,j,ext (m) can be described by 42Δxs,j,ext=Δxs,j,ow,sf+Δxs,j,inlet, representing offshore and onshore sediment fluxes that can erode and accrete the shoreline and flood–tidal delta deposition that acts as a littoral sink.

The shoreline model is unconditionally stable and second-order accurate in space and time. We discretize the coastline into cells with width Δy (typically 100 m). We use a time step Δt (typically 0.05 years) to ensure smooth inlet migration and reasonably accurate shoreline change. However, we note that the overwash and inlet elements of this model are not solved by Eq. (41) and are therefore not necessarily second-order accurate nor unconditionally stable. Section 5 presents the grid and time resolution tests.

At the end of each time step, we update the shoreface toe position xt, 43xtn+1=xtn+Δxt,ow, back-barrier location (xb), 44xbn+1=xbn+Δxb,ow+Δxb,inlet+Δxb,breach, and barrier height (H), 45Hn+1=Hn+ΔHow+ΔHinlet, independently for all alongshore locations j and run another time step.

After a model simulation (typically 10 kyr) we obtain shoreline, back-barrier, and shoreface morphodynamics for different scenarios given by, for example, SLR rates, wave climates, and tidal conditions. One aspect of particular interest, and the primary motivation for this model, is the transgressive flux due to inlet activity. We define a ratio F, 46F=QinletQoverwash+Qinlet, where Qoverwash=Δy⋅∑yQow,b (m3 yr-1), and Qinlet is the along-coast average transgressive sediment flux by inlet formation and flood–tidal delta deposition for all inlets, 47Qinlet=Δxb,breachΔtH+DinletWb+∑i=1..ninlet1-β-βrQs,in,i. F quantifies the fraction of the total transgressive flux due to inlets and can range from 0 to 1.

Aside from the usual output, such as transgressive fluxes, inlet morphodynamics, and barrier island change, the model can also compute the synthetic stratigraphy of a barrier at a certain location xstrat for all grid cells j (Fig. 6). When xb exceeds xstrat, the model saves the location j, lagoon depth Dlagoon, the sediment deposit thickness (i.e., Dlagoon-z), and the responsible process, either flood–tidal delta deposition or storm overwash. While xs<xstrat<xb, we record the height of the barrier H as dune construction or erosion bounded vertically by z and H. If an inlet is present, it erodes the deposit up to a depth dinlet. Inlet migration forms sedimentary facies between dinlet and z. If an inlet is closed, it forms inlet fill facies. These barrier island facies allow us to compare model output to geological reconstructions of barrier islands (e.g., Mallinson et al., 2010).

We first investigated a simulation without inlets, focusing on the effect of alongshore transport gradients and barrier overwash on barrier evolution. As we might expect, in the case of no inlets and uniform initial conditions alongshore, the barrier retreats uniformly and alongshore sediment fluxes do not affect barrier response. We also performed a model experiment with an initially variable barrier width driven by spatial changes in the bay shoreline location (Fig. 5). In this scenario, the initially narrower barrier stretches overwash more than the wider stretches and therefore transgress faster. As shoreline curvatures increases, the magnitude of the alongshore sediment fluxes directed to the narrow stretches also increases, which reduces the width of the initially wider stretches. Interestingly, we find that time lags in shoreline interconnectivity can cause the initially rapidly transgressing stretch to stay in place and eventually become landward of other portions of the coast, a phenomenon also reported by Ashton and Lorenzo-Trueba (2018). Eventually, after a few oscillations that can last for hundreds of years, the barrier approaches a spatially uniform migration rate (Fig. 5).

Including tidal inlets, we see a richer set of model dynamics. We investigated barrier change, including inlets, for an SLR rate of 2 mm yr-1, a wave height of 1 m, and a tidal range of 1 m. After an initial spin-up phase associated with large overwash fluxes, barrier island response stays highly dynamic and does not converge to an equilibrium response, despite the imposition of constant boundary conditions (Fig. 6). Inlets open, close, interact, and migrate preferentially with the direction of the littoral drift. Inlet migration rates vary gradually, and inlet sediment distribution is initially dominated by alongshore sediment bypassing and gradually becomes more flood–tidal delta dominated (Fig. 6e). The inlet transgressive sediment flux is highest when the flood–tidal delta deposition and alongshore sediment bypassing are roughly equal (Fig. 6f). Barrier stratigraphy at that time shows that inlet migration facies make up most of the barrier, even though not all of the transgression is due to the inlet (Fig. 6b).

To investigate model mass conservation, we summed the volume of the barrier and offshore deposits (Fig. 7). Comparison to an identical model without inlets shows that slight losses and gains can be attributed to inlet morphodynamics, likely inlet migration and closure (Fig. 7b). For example, we do not track the sediment lost or gained as inlets change their cross-sectional area from an initial breach width. We also assume that increases in the back-barrier location can be considered small enough so that there is one depth Dlagoon, whereas in reality these deposits exist on a surface with slope sbackground. Regardless of these assumptions, model volume (offshore deposits and the barrier island itself) does not depend on the time step Δt and the grid length Δy; these values only deviate a few percent around their mean, with no obvious trend in time (Fig. 7b).

For model verification, we compared model results to the original cross-shore model of barrier change that only includes overwash (LTA14) (Fig. 8). Our model without inlets produces the same dynamics as the original cross-shore model (LTA14), resulting in the same overwash flux (Fig. 8a). Comparing the cross-shore model to the BRIE model with inlets (forced nonuniformity) we see some clear differences. Even though the average shoreline location along the 100 km barrier follows roughly the same trajectory (Fig. 8c) and therefore has a similar transgression (erosion) rate (Fig. 8b), the individual locations vary significantly. The straight barrier reproduced by the BRIE model without inlets is now variable alongshore. Transgression rates vary from -2 m yr-1 (progradation) to +10 m yr-1 (erosion). Even though the overall trajectory is a result of the sea level history and the passive inundation of the main (non-barrier) coast (Wolinsky and Murray, 2009) (Fig. 8c), significant deviations from this trend appear and are reflected in the overwash rates (Fig. 8a). In particular, the inlet transgressive sediment flux rates are variable.

We investigated the sensitivity of the model output (Qoverwash, Qinlet, and F) by varying the grid resolution and time resolution and holding all other parameters constant. In general, we find that these fluxes vary approximately ∼20 % between different settings (Fig. 9). These deviations appear only in simulations that include inlets and are likely caused by a sensitivity to small perturbations such as random wave angles. For example, comparing multiple simulations with equal settings, including grid and time, we obtain a variability in F (Fig. 10), with a standard deviation of 0.025. Sensitivity to grid spacing and time steps can also be caused by the discretization of inlet migration rates and distances (Eq. 35).