Coupled sandbar patterns and obliquely incident waves
T. D. Price,
Department of Physical Geography, Institute for Marine and Atmospheric Research Utrecht, Faculty of Geosciences, Utrecht University, Utrecht, Netherlands
Now at Subdepartment of Systems Ecology, Department of Ecological Science, Faculty of Earth and Life Sciences, VU University Amsterdam, Amsterdam, Netherlands
Corresponding author: T. D. Price, Subdepartment of Systems Ecology, Department of Ecological Science, Faculty of Earth and Life Sciences, VU University Amsterdam, De Boelelaan 1085, 1081 HV Amsterdam, Netherlands. (firstname.lastname@example.org)
 In double sandbar systems, the alongshore variability in the inner bar often resembles that of the outer bar, suggesting that the outer bar acts as a morphological template for the inner bar. Earlier observations have indicated that this resemblance, also termed “coupling,” may take several forms. Here we apply a nonlinear 2DH morphodynamic model with time-invariant forcing to show that the angle of wave incidence (θ) is crucial for the alongshore-variable morphodynamic evolution of the inner bar, for a given crescentic outer bar. In contrast to previous modeling efforts of double-barred systems, which mostly used highly idealized boundary conditions, we force our model with realistic hydrodynamics and bathymetrical data derived from video observations at the double-barred Gold Coast, Australia. The results show that for small angles of wave incidence (θ<10°) over a crescentic outer-bar, cell-circulation patterns govern the flow at the inner bar, giving rise to rip channels that incise the inner bar at the locations of the landward perturbations in the outer bar (horns). On the other hand, for obliquely incident waves (θ=10°–20°) over a crescentic outer bar, the circulatory nature of the flow disappears and gives way to a meandering alongshore current. The offshore-directed sections of this meandering alongshore current erode the inner-bar downdrift of the outer-bar horns, leading to landward perturbations of the inner bar that are coupled to the outer-bar horns; an observed coupling type that had not been reproduced as yet. Oblique wave incidence thus proves to be crucial to the development of this type of sandbar coupling, as previously hypothesized from sandbar-coupling observations at the Gold Coast. Additional simulations including tidal water level variations and bar depth variations demonstrate the robustness of our findings.
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 The nearshore bathymetry of many sandy beaches is characterized by the presence of one or more sandbars within a distance of several hundreds of meters from the shoreline. Each sandbar may develop a series of three-dimensional patterns [Wright and Short, 1984; Short and Aagaard, 1993], ranging from an alongshore-uniform ridge of sand to remarkably periodic alongshore undulations in both cross-shore position and depth, known as crescentic sandbars and rips. Crescentic sandbars can be seen as a sequence of shallow horns and deep bays, alternating seaward, respectively, and landward of a line parallel to the coast, with typical length scales ranging from 100 to 1000 m [Van Enckevort et al., 2004]. These horns and bays are formed spontaneously because of the positive feedback between horizontal cell-circulation patterns (including rip currents), sediment transport, and the morphology itself (i.e., self-organization mechanisms [Hom-ma and Sonu, 1962; Hino, 1975; Sonu, 1972; Falqués et al., 2000]). In a double sandbar system, the inner-bar morphology generally also develops through these self-organization mechanisms. However, under certain conditions, the alongshore spacing of seaward and landward perturbations in the inner sandbar may be coupled to the horn spacing in the outer sandbar [Van Enckevort and Wijnberg, 1999; Ruessink et al., 2007; Castelle et al., 2007; Quartel, 2009; Price and Ruessink, 2013]. In this case, the crescentic outer-bar morphology acts as a forcing template for the inshore flow patterns through wave breaking and wave focusing across the outer bar [Garnier et al., 2008a; Castelle et al., 2010a, 2010b]. The morphological evolution of the inner bar, therefore, is driven by a mixture of self-organization and outer-inner bar coupling mechanisms [Castelle et al., 2010a, 2010b]. Although morphological coupling between two sandbars suggests a reciprocal influence, observations [Ruessink et al., 2007] and modeling results [Castelle et al., 2010a, 2010b; Thiébot et al., 2012] indicated that the inner-bar morphology does not affect the outer-bar morphodynamics.
 Coupled sandbar patterns can contain a variety of appearances or morphological states [e.g., Bowman and Goldsmith, 1983; Castelle et al., 2007; Ruessink et al., 2007]. Recently, Price and Ruessink  distinguished five coupling types, based on observed breaker lines over a double sandbar system (i.e., barlines) from over 9 years of time-exposure images from the Gold Coast, Australia. Four of the five observed coupling types coincided with a downstate sequence of the outer bar, a sequence during which the outer bar becomes increasingly alongshore non-uniform and propagates onshore [Wright and Short, 1984]. The bars either coupled in-phase, with an outer-bar horn facing a shoreward perturbation of the inner bar line, or out-of-phase, where the outer-bar horn coincided with a seaward bulge in the inner bar line. The morphology of the coupled inner bar was either terraced, with no trough or channels intersecting the bar, or characterized by the presence of rip channels, where landward or seaward perturbations of the terraced inner bar and rip channels constituted the coupled inner-bar features, respectively. The inner-bar morphology may also remain coupled to the outer-bar morphology during upstate transitions (straightening) of the outer bar; the alongshore nonuniformity of the inner bar may actually increase during these upstate transitions as outer-bar horns detach, migrate onshore, and merge with the inner bar [Wijnberg and Holman, 2007; Almar et al., 2010]. Using abbreviations for the aforementioned properties, Price and Ruessink  named the five different coupling types according to whether the bars were coupled in- or out-of-phase (I or O), the outer bar proceeded downstate or upstate (d or u), and whether the inner-bar morphology was terraced or with rips (t or r). Using cross correlation on the inner and outer barlines, they found the most frequent coupling type at their study site to be the Idt type, an outer crescentic bar with landward perturbations of the inner terrace coupled to the alongshore positions of the outer-bar horns (Figure 1).
 Although video observations provide a high-frequency long-term data set of (coupled) sandbar morphology, they do not provide direct information on the morphodynamic processes leading to changes in sandbar morphology. Instead, numerical models are often used to shed light on the observed sandbar morphodynamics. So far, numerical studies of sandbar morphology have largely focused on single-barred beaches [e.g., Ranasinghe et al., 2004; Reniers et al., 2004; Garnier et al., 2006; Tiessen et al., 2011]. The few existing numerical studies of double sandbar systems have mainly focused on the initial development and subsequent evolution of crescentic patterns, either using linear stability analysis [e.g., Klein and Schuttelaars, 2006; Garnier et al., 2008a; Coco and Calvete, 2009; Brivois et al., 2012], nonlinear depth-averaged models [Klein and Schuttelaars, 2006; Smit et al., 2008, 2012; Thiébot et al., 2012], or quasi-three-dimensional models [Drønen and Deigaard, 2007]. Although some of these studies analyzed the effect of a varying cross-shore profile, the initial bathymetries were always alongshore uniform, albeit with initial depth perturbations of O(10−2) m to evoke bed pattern development. Castelle et al.[2010a, 2010b], on the other hand, forced a nonlinear depth-averaged morphodynamic model with an initially crescentic outer bar with varying geometric parameters. They found that coupled inner-bar features arise from horizontal circulation patterns driven by alongshore variations in wave refraction and wave breaking across the crescentic outer bar. Depending on both the outer-bar geometry and the wave conditions, the barlines coupled either in-phase or out-of-phase. Whereas the simulations of Castelle et al.[2010a, 2010b] were performed for shore-normal wave incidence only, Thiébot et al.  performed numerical simulations for a large range of wave angles over initially alongshore-uniform sandbars. For slightly obliquely incident waves (10° and 15° with respect to shore normal at 8 m water depth), they found that initially the inner bar did not develop any alongshore variability due to the large alongshore current. However, when the outer bar started to develop alongshore variability, the alongshore current and the incoming wave field at the inner bar became perturbed, leading to the development of inner-bar features with an alongshore spacing similar to that of the outer-bar horns. This corresponds to the observations of Ruessink et al., , where inner-bar patterns developed in response to the increasingly three-dimensional, onshore migrating outer bar. None of these simulations, however, resulted in the Idt coupling type, where landward perturbations of the inner terraced bar are coupled to the alongshore positions of the outer-bar horns.
 In our previous work [Price and Ruessink, 2013], we hypothesized that the Idt coupling type was related to the dominant oblique wave incidence. For a given crescentic outer bar, we suggested that the type of bar coupling is not solely a function of the amount and alongshore variation of wave breaking over the outer bar (as concluded by Castelle et al. [2010a]), but that the angle of wave incidence plays an important role. In the case of shore-normally incident waves, the outer bar drives a cell-circulation system over the inner bar, resulting in the development of coupled rip channels in the inner bar, either in-phase or out-of-phase. Obliquely incident waves over a crescentic outer bar, on the other hand, drive a meandering alongshore current at the inner bar, similar to the field observations of, e.g., Sonu , MacMahan et al.  and Houser et al. . We hypothesized that the landward perturbations of the terraced inner bar for the Idt coupling type (Figure 1) resulted from this meandering alongshore current, with increased velocities at the more landward oriented parts of the alongshore current at the locations of the outer-bar horns. Although these observations indicated an important role of the angle of wave incidence for the flow pattern and the corresponding coupling type in double-barred systems, we were unable to quantify our hypothesis from the video images alone.
 In this work, we aim to analyze how the angle of wave incidence affects the coupling processes at the inner bar. As the processes and concepts underlying the coupling types that form under shore-normal wave incidence were studied by Castelle et al. [2010a], we here focus on oblique wave incidence and, hence, the Idt coupling type. We use a nonlinear morphodynamic model to analyze the development of the Idt coupling type, and base our modeling exercise on a representative 4 day period of observations during which the Idt coupling type developed. Subsequently, we demonstrate that the angle of wave incidence is crucial to the flow pattern and emerging coupling type at the inner bar. In contrast to existing modeling efforts of double-barred systems, which use synthetic or highly idealized bathymetries, we force our model with realistic bathymetrical data derived from the video observations. In this way, we use the sandbar geometries (e.g., cross-shore distance, wavelength of the crescentic pattern, alongshore depth variation) and wave conditions for which our model should reproduce an Idt coupling type. Using this approach, we aim to bridge the gap between a purely synthetic modeling approach, and a direct comparison of modeling results with field measurements from a natural double sandbar system.
2.1 Nonlinear Morphodynamic Model
 We used a nonlinear depth-averaged model (detailed in Castelle et al. ; Castelle and Coco ) that couples a spectral wave model, a time- and depth-averaged flow model, an energetics-type sediment transport model, and the bed level continuity equation to compute bed level changes. The model has been successfully applied to simulate the formation, the nonlinear evolution, and the finite-amplitude dynamics of rip channels and crescentic sandbars [e.g., Castelle et al., 2010a, 2012]. Furthermore, the model allows for shoreline evolution (and thus the development of erosive mega-cusps at the base of rip channels), and tidal water level fluctuations, which represents a major difference with other nonlinear morphodynamic models applied to 3-D surf zone sandbar behavior. The wave field and resulting radiation stress components are computed from the spectral wave model SWAN [Booij et al., 1999], which solves the spectral wave action balance, here with the parameter settings detailed in Castelle et al. .
 The flow model is based on the phase-averaged nonlinear shallow water equations [Phillips, 1977], comprising the time-averaged and depth-integrated momentum conservation and water mass conservation equations. Using the Einstein summation convention, where subscript i refers to the two horizontal coordinates (with x and y the alongshore and cross-shore axes, respectively), these read as follows:
where Qi=Uihis the water volume flux per unit width in the i-direction, with Ui the component of the flow velocity ; t is time; h is the mean water depth; g is the gravitational acceleration; η is the mean free surface elevation; ρ the water density; Sij are the components of the radiation stress tensor; Tijare the components of the lateral shear stresses, which are the horizontal momentum exchanges due to the combined action of turbulence and the mean current using the formulation proposed by Battjes ; are the components of the wave-averaged bed shear stresses.
 The combined bed load and suspended load sediment transport are computed using the formulations of Bailard . Both the bed load and suspended load parts contain an efficiency factor, ϵband ϵs, respectively. We used the default values for the efficiency of either transport mechanisms: ϵb=0.1 and ϵs=0.02, respectively. Herein, we use the same approach for the sediment transport as in Castelle et al., [2010a, 2010b], which differs from the approach in, e.g., Castelle and Ruessink , where the sediment transport was computed with respect to a predetermined equilibrium bed level (i.e., basic state).
 The new seabed level Zf was computed using the sediment mass conservation equation:
where p=0.4 is the sediment porosity and Qs,j are the components of the sediment transport flux . In the present work, the morphological time step for the bed update scheme, which is different from the hydrodynamic time step (1 s), was 30 min for all the simulations presented herein.
2.2 Model Setup
 A bathymetry containing a double sandbar system can be described using a number of geometrical parameters, described below and illustrated in Figure 2. First, the mean cross-shore profile, excluding sandbars, exhibits a concavity commonly described by the profile shape function d of Dean :
in which d is the water depth (for a given water level h) at a cross-shore distance y, and A is a sediment-dependent scale parameter. Sandbars constitute perturbations on top of this alongshore-uniform profile, with their crests at a certain cross-shore position y and depth D. In Figure 2, the subscripts i and o denote properties of the inner and outer bar, respectively. The outer sandbar in Figure 2a consists of a bar that is separated from the inner bar by a trough. The inner bar, on the other hand, exhibits a terrace, with width wi, where the depth is more or less constant. The alongshore variations of crescentic sandbars can be seen as a regular variation of depth and cross-shore position, in this case for the outer bar (Figure 2b). Accordingly, their shape can be described by a horizontal amplitude Ayof the cross-shore position around the mean cross-shore bar position yo, an alongshore wavelength Lx, and a vertical amplitude Az of the depth around the mean bar depth Do. This amplitude corresponds to the depth differences between the more onshore-positioned horns and more offshore-positioned bays, with depths Dh and Db, respectively (Figure 2a).
 We directly derived the sandbar geometries from field data to create an idealized initial bathymetry using the geometrical parameters described above. In our study, both the bathymetry and the hydrodynamic forcing were derived from observations from the double-barred sandy (D50=250 μm) beach at Surfers Paradise (Queensland, Australia; Turner et al. ), where morphological coupling is frequently encountered [Price and Ruessink, 2013]. We extracted the boundary conditions for the simulations used herein from a representative 4 day period during which the development of an Idt coupling type was observed in time-exposure video images [Price and Ruessink, 2013]: from 25 to 28 January 2006 (see Figures 3a–3d). During this period, the initially alongshore-uniform inner bar coupled to the crescentic outer bar, while the wave conditions remained fairly constant (Figures 3e–3h).
Table 1. Bathymetrical Parameters Used for All Simulationsa
 First, we obtained the mean beach profile d by fitting equation (4) to the alongshore-averaged cross-shore profile from a bathymetrical survey from June 2002, giving d=5−0.245y2/3 (Table 1; Figure 4a). From the time-exposure images, we directly derived the mean cross-shore positions of the inner bar yi and the outer bar yo, which were 100 m apart, the terrace width of the inner bar wi=40 m (Figure 4a), and the wavelength Lx=500 m and cross-shore amplitude Ay=40 m of the outer crescentic bar (Figure 4b). Because of a lack of suitable bathymetric surveys, we applied the assimilation model XBeachWizard [Van Dongeren et al., 2008] to transform the breaking-induced intensity in the video image into estimates of the depth (see Price and Ruessink  for details). Different to the Beach Wizard system described by Van Dongeren et al., , which used the Delft3D model for the assimilation, XBeachWizard uses the process-based model XBeach [Roelvink et al., 2009]; the assimilation scheme itself is identical to the original Beach Wizard system. The assimilation yielded an alongshore depth variation Az=0.5 m between the depth of the horns Dh and the depth of the bays Db. Based on the assimilation results and available measured bathymetries, the mean depth of the outer bar Do and inner bar Di were set at −2.1 and −0.5 m with respect to mean sea level (MSL), respectively. The effect of different outer and inner-bar depths Do and Di on the model results are discussed in section 4. All simulations were performed with the same initial double sandbar bathymetry (Figure 4c) with a (3-D) crescentic outer bar and an alongshore-uniform terraced inner bar, on a computational grid of 3020×780 m (alongshore × cross-shore), 20×20 m grid cells, and periodic lateral boundary conditions. The water depth at the seaward extent of the model, at y=780 m, was 15 m.
 For the reference simulation, we used constant wave conditions corresponding with the mean values of the 4 day observation period (Table 2; Figures 3e–3h): a mean significant wave height Hs of 1.1 m, a mean angle of wave incidence θof 14° (herein, θ is always with respect to shore normal and in 15 m depth, obtained by refracting the measured data using Snell's law), a mean peak wave period Tp of 9 s, and a constant tidal water level ζ of 0 m. Subsequently, we varied θfrom 0° to 20° with steps of 1°, while keeping all other boundary conditions constant. The wave conditions during the 4 day period (Figures 3e–3 eh) were sufficiently constant (standard deviations: θ=6.9°, Hs=0.2 m, Tp=1.2 s), to support the simplified approach of using the averaged, time-invariant wave forcing for the simulations. As θ varied only slightly, and did not change sign (Figure 3f), we do not expect that usage of the mean value in this case will lead to significantly different dynamics as suggested by Castelle and Ruessink  for larger θvariations. We neglected the observed tidal water level variation ζ and applied a constant water level throughout all simulations. Including tide, however, did not significantly affect the model results, further discussed in section 4. We applied a simulation time of 4 days for all computations, corresponding to 192 morphological model time steps.
2.3 Analysis of Model Results
 To examine the evolution and the processes involved in the morphological coupling, we computed a number of parameters representative of the morphological evolution, the flow patterns, and sediment transport, respectively. As this study focuses on the response of the terraced inner bar to a crescentic outer bar for given wave conditions, we analyzed these parameters at an alongshore profile over the inner terrace at cross-shore distance y=120 m, where we found the processes and morphology to be representative of the inner-bar morphodynamics. Additionally, we analyzed the evolution of the depth variation along the outer bar, at cross-shore distance y=220 m.
 In our model, there is no bed diffusion or bed slope transport [Garnier et al., 2008b] likely to dampen the development of instabilities. Depending on the boundary conditions (both the hydrodynamic forcing as the outer-bar morphology), this eventually led to an unrealistic morphology of the inner or outer bar for some simulations. None of the simulations, however, led to a blow up of the model. The analyses of the double sandbar morphology presented below do not represent steady states of the system. Instead, we analyze each model simulation after a simulation time of 2 days, before the appearance of an unrealistic morphology in any of the simulations.
 To quantify the change from horizontal cell-circulation patterns to a more meandering alongshore current, as the angle of wave incidence increases, we compute the swirling strength [Adrian et al., 2000] as a measure for the rotational nature (vortices) of the flow pattern. The swirling strength is extracted from the velocity fields, using critical-point analysis of the local velocity gradient tensor and its corresponding eigenvalues [Zhou et al., 1999]. First, the velocity gradient tensor of the xy-plane is computed as
where Uyand Uxare the cross-shore and alongshore velocities (which constitute ), respectively. Subsequently, complex eigenvalue pairs of V, each consisting of a real part (λcr) and an imaginary part (λci), indicate the presence of a vortex in the velocity field. In this case, λci provides the measure for the swirling strength, where vortices can be identified as regions where λci>0, and represents the period (seconds) required to spiral around the origin of the vortex. Essential to our analysis of the rotational nature of the flow over the inner bar, the swirling strength λci conveniently isolates the rotational parts of the flow and excludes regions with large shearing motion (and no swirling motion), as may be expected for cross-shore differences of the velocity in a purely alongshore (unidirectional) current.
3.1 Reference Simulation
 Before analyzing any change in processes at the inner bar, we first present the reference simulation, which is based on a representative 4 day period during which the development of an Idt coupling type was observed (section 2.2), with Hs=1.1 m, θ=14°, Tp=9 s, and ζ=0 m. This reference simulation serves two purposes: first, the evaluation of the agreement between the computed and the observed inner-bar Idt morphology, and, second, it permits the comparison of model results obtained for varying angles of wave incidence.
 Figure 5a shows the bathymetry after 2 days of simulation. The obliquely incident waves approach from the right top. The inner bar has developed landward perturbations (along y=120 m, e.g., at x≈400, 900, 1400 m), which are slightly offset with respect to the alongshore position of the outer-bar horns (along y=220 m, e.g., at x≈300, 800, 1300 m). This morphology resembles the Idt coupling type, with landward perturbations of a similar magnitude and position with respect to the outer bar as in the observations in Figure 3. In the depth profile along the inner bar (at y=120 m), the flat, terraced morphology is interrupted by perturbations with a depth of approximately 0.4 m (Figure 5b). In our simulations, depth-induced wave breaking across the entire outer bar resulted in lower wave energy shoreward of the outer-bar horns. The dominant direction of the wave-driven flow over the inner bar, and the entire nearshore area (Figure 5c), is from right to left in the figure. At the inner bar, this alongshore flow meanders in onshore and offshore directions. The meandering nature of the alongshore flow becomes clear from the cross-shore component of the flow (the colors in Figure 5c), which shows a change in flow direction from more onshore-directed over the flat terraced area to more offshore-directed at the landward perturbations of the inner bar line. A weaker meandering current also prevails over the outer-bar horns, with more onshore-directed flow downdrift of the outer-bar horns, and more offshore-directed flow updrift of the outer-bar horns. These fluctuations at both the inner and outer bars, however, do not lead to a rotational flow over the inner bar, and the swirling strength remains zero over the entire domain (Figure 5d). Similar to the flow pattern, the main direction of the resulting sediment transport fluxes is alongshore-directed with fluctuations in onshore and offshore direction (Figure 5e). The accretion/erosion patterns show that the landward perturbation at the inner bar is situated directly between an area of accretion and erosion, indicating that the perturbation migrates alongshore, reminiscent of the relation between sediment transport and alongshore rip channel migration identified by Orzech et al. . This alongshore migration, further discussed in sections 3.2.2 and 4, is a result of the alongshore-migrating outer bar, causing the morphological template, and thus the forcing at the inner bar, to change. Accretionary areas at the inner bar coincide with more onshore-directed sediment transport directly updrift of the inner-bar perturbation, whereas erosional areas coincide with more offshore-directed sediment transport originating at the inner-bar perturbation.
 These results indicate that the model is capable of reproducing the landward perturbations in the inner bar observed during the Idt coupling type. In contrast to other coupling types, with more shore-normal wave incidence, the Idt coupling type, where the landward perturbations of the inner terraced bar are coupled to the alongshore positions of the outer-bar horns, had thus far not been simulated. The robustness of this modeling result is further discussed in section 4. In line with the hypothesis based on our previous work [Price and Ruessink, 2013], the landward perturbations are erosional features, and their locations coincide with the turning point of the meandering alongshore current, where more onshore-directed flow and accretion turn to more offshore-directed flow and erosion. In that sense, the landward perturbation in the inner bar is comparable to the erosive mega-cusps in the shoreline at the base of rip channels, often observed in single-barred systems [e.g., see Komar, 1971; Thornton et al., 2007].
3.2 Influence of Wave Angle
 Below, we characterize the change in processes at the inner bar for angles of wave incidence varying from 0° to 20°. Before doing so, we first present a simulation with a more shore-normal wave incidence than in the reference simulation to aid the interpretation of the results.
3.2.1 Simulation With θ=5°
 Figure 6 shows the model results for more shore-normally incident waves (θ=5°), after 2 days of simulation, similar to Figure 5. In this case, the depth perturbations in the inner bar are more pronounced and correspond to clear rip channels with rip-head bars onshore of the outer-bar horns (Figure 6a). This morphology corresponds to the Odr coupling type defined by Price and Ruessink  and the out-of-phase coupling modeled by Castelle et al. [2010a]. Note the subtle mega-cusps in the shoreline, at the base of the rip channels. The depth variations along the inner bar (Figure 6b) show the rip channels with depths of approximately 0.7 m. A cell-circulation pattern characterizes the flow over the inner bar, with onshore flow over the bar, offshore flow through the rip channels (Figure 6c), and feeder currents on either side of the rips. The nonzero angle of wave incidence induces an asymmetric cell circulation with a downdrift deflection of the rip current. It is also downdrift of the rip channel, near the shoreline, where the rotational nature of the flow is most apparent. The swirling strength (Figure 6d) clearly captures this pronounced rotational flow over the inner bar, both downdrift of the rip channel as in the rip channel itself. The resulting sediment transport fluxes and accretion/erosion patterns (Figure 6e) at the inner bar show onshore sediment transport and accretion over the bar and offshore sediment transport through the rip, leading to the erosion of rip channels and the formation of rip-head bars. In contrast to the reference case, where the accretionary area was updrift of the inner-bar perturbation, here the onshore sediment transport flux is the largest downdrift of the inner-bar rip (and outer-bar horn). Note that the outer-bar horns have become more pronounced in this case with a small angle of wave incidence (θ=5°) than in the reference case, with a larger angle of θ=14° (Figure 5).
3.2.2 Depth Perturbations
 Figures 5b and 6b indicate that inner-bar depth perturbations were more pronounced for a smaller angle of wave incidence (θ=5°) than for the reference case with θ=14°. Figure 7 shows the depth perturbations with respect to the alongshore-averaged bathymetry zp and flow (arrows) along the inner bar, at y=120 m, for the simulations with varying angles of wave incidence after 2 days of simulation. The negative zp values correspond to the landward perturbations and rip channels in Figures 5 and 6, respectively. The most pronounced depth perturbations are found for the simulations with θ around 7° (Figure 7c), where relatively deep and narrow negative depth perturbations are formed, i.e., rip channels. This is somewhat surprising, as previous modeling exercises of single bar systems [e.g., Castelle and Ruessink, 2011] found that rip channels were more pronounced when formed during shore-normal wave incidence. For θ>7°, the negative depth perturbations decrease, but become increasingly wider. Similarly, the positive depth perturbations directly downdrift of the negative depth perturbations decrease, indicating an increasingly subdued inner-bar morphology. Toward θ=20°, almost all depth perturbations have disappeared. From θ=0° to approximately θ=10° the magnitude of the flow increases (the arrows become longer), while the flow pattern remains circulatory. For θ>10°, the flow pattern subsequently changes to a meandering alongshore current, becomes increasingly unidirectional, and increases in magnitude.
 Notice that the depth perturbations are located further to the left (downdrift) for larger angles of wave incidence. This relates to the combination of the increased magnitude of the alongshore current and the concurrent alongshore migration and evolution of the outer bar. Figure 7c shows that for small angles of wave incidence (up to θ=7°), the alongshore variability of the outer bar increases with respect to the initial alongshore variability within the 2 day simulation period, whereas the outer bar becomes more alongshore uniform for larger angles of wave incidence (θ>7°). This straightening of the outer bar under oblique wave incidence corresponds with our previous findings from the same study site used herein [Price and Ruessink, 2011] and recent modeling efforts by Garnier et al. . Note that the value of θ=7° coincides with the maximum depth variation along the inner bar (Figure 7c). The alongshore positions of the outer-bar horns, indicated by the black dots in Figures 7a–7b, show that the outer bar migrates further alongshore within the 2 day simulation period (0 to 100 m) as θ increases from 0° to 20°. As would be expected for the morphological template model, the inner-bar depth perturbations follow this alongshore migration of the outer-bar horns, with increased downdrift offsets for larger angles of wave incidence (Figure 7b). For θ=15°–20°, a rapid increase in alongshore offset coincides with a decrease in alongshore variability of both the inner and outer bars (Figure 7c). Inspections of the morphological evolution during these high-angle simulations show that the inner bar initially develops depth perturbations downdrift of the outer-bar horns. However, the subsequent straightening of the outer bar under oblique wave incidence reduces the effect of the outer-bar morphological template on the inner-bar flow pattern, which inhibits the further development of inner-bar features as the flow pattern becomes alongshore uniform.
3.2.3 Swirling Strength
 Figure 8 shows the flow pattern along the inner bar at y=120 m for all θsimulations after 2 days of simulation, together with the swirling strength, to capture the change in rotational nature of the flow over the inner bar. A swirling motion of the flow over the inner bar can only be detected for angles of wave incidence up to ≈10°. For larger angles, the rotational features disappear and a meandering alongshore current prevails, as in the reference case (Figure 5). In line with the depth variations (Figure 7), the maximum swirling strength is found for θ=5° (Figure 8b). Surprisingly, the swirling strength (i.e., the intensity of the cell circulation) is not largest for shore-normal incidence. Instead, the largest swirling strengths develop for waves with a slight oblique incidence (θ=4–6°); the presence of an alongshore component in the mean flow seems to intensify both the flow through the (obliquely oriented) rip channels and the swirling motion on the downdrift side of the rip channel (Figure 6). As θ approaches 10°, the feeder current directly downdrift of the rip channel weakens and eventually disappears as it becomes overridden by the increasingly intense alongshore current. Similarly, for θ<6°, a more subtle swirling motion is detected updrift of the rip channel, over the bar, coinciding with the alongshore position of the outer-bar bay (e.g., at (x,θ)=(650 m, 5°), see also Figure 6d). Here for θ<3°, the onshore-directed flow splits into two feeder channels toward the rip channels on either side. For θ>2°, the onshore-directed flow at this area becomes increasingly alongshore-directed and the splitting point shifts to the area directly downdrift of the more pronounced rip channel described above (e.g., at (x,θ)=(400 m, 3°). Note that smaller outer-bar wavelengths or larger waves may result in stronger circulation patterns [see Castelle et al., 2010a], which persist through larger angles of wave incidence than shown herein.
3.2.4 Sediment Transport
 Figure 9 shows the accretion/erosion rates (dz/dt) and sediment transport flux along the inner bar, at y=120 m, for all simulations at t=2 days. The erosional areas correspond to the coupled inner-bar perturbations. In agreement with the depth and flow variations, the erosion of the rip channels increases from θ=0° to θ=7°. As the rip channel becomes more pronounced (Figure 7), an area of onshore sediment transport and accretion develops downdrift of the rip current, corresponding to the pattern in Figure 6e. As θ is increased above 10°, both the accretion downdrift of the rip channel and the erosion of the rip channel itself decrease substantially, and an area of onshore sediment transport and accretion starts to develop updrift of the inner-bar perturbation. This accretion/erosion pattern corresponds to Figure 5e, which indicates the development of the inner-bar perturbation under more alongshore-directed sediment transport conditions. For even larger angles of wave incidence (θ>17°), the sediment transport direction becomes increasingly alongshore-uniform and bed level changes decrease. The weak erosional areas updrift of the rip channel for θ<6° (e.g., at (x,θ)=(500 m, 5°)) correspond to the aforementioned swirling strength pattern (Figure 8), and indicate the development of subtle channels at the inner bar (see Figure 7).
 In summary, for θ>10° the Idt coupling type developed. In this case, the offshore-directed sections of the meandering current downdrift of the outer-bar horn eroded the inner terrace, causing the coupled inner-bar features to appear as landward perturbations of the terrace edge. For θ<10° cell-circulation patterns governed the flow at the inner bar. In our simulations, where waves broke across the entire outer bar, this circulatory pattern led to the Odr coupling type, with offshore flow and the development of rip channels in the inner bar at the locations of the outer-bar horns, consistent with Castelle et al. [2010a]. The most pronounced rip channels and circulatory flow patterns were found around θ≈7°. These results confirm our hypothesis that the angle of wave incidence is crucial to the flow pattern and sediment transport at the inner bar, and thus the emerging coupling type.
 Our simulations show the importance of the angle of wave incidence over a crescentic outer bar for the inner-bar morphodynamics. In this section, we substantiate our results by discussing the influence of bar depth, including tidal variation in our reference simulation, and by further discussing model limitations.
4.1 Inner and Outer Bar Depth Diand Do
 Although the assimilation results using XBeachWizard provided a suitable estimate of the alongshore depth variation of the outer sandbar, the absolute depth depended strongly on the model settings used for the assimilation, see Price and Ruessink  for details. Accordingly, we reran a series of simulations where the depths of the inner and outer bars were varied within a realistic range, while keeping both the morphological variables (bar position, wavelength, cross-shore extent and alongshore depth variation of the crescentic outer bar) and the hydrodynamic boundary conditions (Hs, θ, Tp, h) constant; after all, these are known. The range over which to vary the outer-bar depth was obtained through the results of the XBeachWizard simulations, which yielded an average outer-bar depth of −1.7 m MSL, and the scarce amount of bathymetries [Ruessink et al., 2009] over the period 1999–2006, yielding outer-bar depths ranging from −2.5 to −1.5 m MSL. In the measured bathymetries, the inner-bar depths ranged from approximately −1 to −0.2 m MSL, and visual inspection of snapshot video images and the corresponding tidal levels (to determine at what water level the inner bar emerged), yielded inner-bar depths of approximately −0.5 m MSL. In total, we performed 42 simulations to assess the variability of the coupled morphology reproduced by the model, with outer-bar depths ranging from −2.5 to −1.3 m MSL, and inner-bar depths ranging from −1.0 to 0 m MSL.
 All of the 42 simulations resulted in a meandering alongshore current and a landward perturbation in the inner-bar downdrift of the outer-bar horn (as in Figure 5). Both for smaller inner- and outer-bar depths, the (cross-shore) amplitude of the meanders increased. Similarly, the morphological response of the inner bar was more pronounced for smaller inner- and outer-bar depths than for larger depths, and eventually led to the inner bar reaching zero depth for small inner-bar depths. Qualitatively, however, the resulting morphology was the same. For the given depth ranges, the inner- and outer-bar depths thus mainly determined the speed of the morphological evolution of the inner bar and not the characteristics of the evolving bathymetry. For the simulations presented herein, we applied an outer-bar depth Do of −2.1 m MSL and inner-bar depth Di of −0.5 m MSL, where the choice for the outer-bar depth was a compromise between a realistic depth and realistic morphological response times.
4.2 Influence of a Tidal Water Level Variation
 Despite variations in Doand Digiving consistent results, we additionally investigated the model predictions by taking tide into account, further bridging the gap toward model-data integration. For our simulations, we used time-invariant wave forcing, including a constant tidal water level ζ of 0 m. The water level variation in our observations of the Idt coupling type, however, show a tidal amplitude of approximately 0.5 m (Figure 3h). Furthermore, in the video images, the inner bar regularly emerged during low tide. Besides a variation in water depth and the potential introduction of tidal-induced currents, a tidal water level fluctuation results in a variation of processes over a cross-shore profile, as the surf zone sweeps across the profile during a tidal cycle [e.g., Masselink and Short, 1993] and in a temporal change in the magnitude of rip current velocities [e.g., Schmidt et al., 2005; Austin et al., 2010]. Although numerous field studies of nearshore currents and sandbar morphodynamics at sites with a tidal range exist, numerical modeling efforts concerning the development of 3-D sandbar behavior under the influence of tides are scarce. Castelle et al. [2010a] suggested that the tide continuously changes the balance between wave breaking and refraction across the outer bar, which they found to be the two most important mechanisms affecting the coupling types for shore-normally incident waves.
 To investigate the influence of the tide and to test whether the model still reproduces the Idt coupling type when tidal variation is included, we ran a number of simulations with different tidal amplitudes. We varied the tidal amplitude from 0.1 to 1 m (the spring tidal amplitude at our study site), with steps of 0.1 m, while applying a 12 hour tidal period and keeping all other boundary conditions the same as in the reference simulation, with Hs=1.1 m, θ=14° and Tp=9 s. The difference between the mean of the high tides and the mean of the low tides in the observation period (Figure 3h) was 1.0 m, corresponding to a tidal amplitude of 0.5 m. As the inner bar has a depth Diof 0.5 m in our simulations, the bar increasingly emerges during a tidal cycle (thus becoming increasingly intertidal) for tidal amplitudes larger than 0.5 m. In section 3, we showed that the development of the landward perturbations in the inner bar coincided with erosional areas downdrift of the outer-bar horns. Figure 10 shows the 4 day evolution of the accretion and erosion rates (dz/dt), together with flow velocity , along the inner bar at y=120 m for the simulation with a tidal amplitude ζ of 0.5 m. During the first 1.5 days, the landward perturbations at the inner bar only develop just before and after low tide (e.g., just before and after t=0.4 days), when both the magnitude and the meandering nature of the alongshore flow become more pronounced. In between these moments, during high tide, the flow is more alongshore-uniform and erosional areas are lacking. After t=1.5 days, the perturbation has become sufficiently pronounced and continues to develop when submerged (i.e., at a minimum water depth). For larger tidal amplitudes, this minimum depth would be reached less frequently during the tidal cycle. For all tidal amplitudes simulated herein, however, our model reproduced the Idt coupling type, similar to the bathymetry in Figure 5a. Figure 11a shows the sum Σ|dzneg| of the erosional areas along the inner bar, at y=120 m, after 4 days of simulation for each tidal amplitude, where a tidal amplitude of 0 corresponds to the reference simulation. This shows that larger tidal amplitudes indeed decrease the total amount of erosion along the inner bar during a tidal cycle, and thus hinder the growth of the inner-bar features, consistent with the findings of Dubarbier and Castelle . This is further illustrated in Figure 11b, which shows the temporal evolution, for each tidal simulation, of the potential energy density Ez of the morphology with respect to the alongshore-averaged profile zmean over the cross-shore extent of the inner-bar area from ya=100 to yb=140 m, similar to Garnier et al.  and Vis-Star et al. :
where Lx is the alongshore extent of the computational domain. Although the smaller tidal amplitudes (<0.5 m) show a decrease in Ez for t>2 days, larger tidal amplitudes (>0.5 m) generally lead to a more subdued (i.e., smaller Ez) Idt coupling type morphology of the inner bar. Moreover, the stepwise development of Ez indicates the increasingly discontinuous morphological development of the intertidal inner-bar morphology for larger tidal amplitudes. The decrease in Ez for smaller tidal amplitudes is a consequence of the decrease in the alongshore variability of the outer bar for smaller tidal amplitudes (not shown). This decreases the effect of the outer-bar morphological template on the inner-bar flow pattern and inhibits the development of inner-bar features, similar to the effect of oblique wave incidence on the morphological template described in section 3.2.2. Hence, the presence of a tidal water level fluctuation ζ helps to maintain the alongshore variability of the initially crescentic outer bar, for a given set of wave conditions.
4.3 Model Robustness and Limitations
 The nonlinear modeling exercise presented herein relies on a number of simplifying assumptions. We neglected the 3-D structure of wave-driven circulations (we assume depth-averaged flow), and wave group-scale forcing. The latter is thought to influence the free development of rip currents and rip channel morphology [Reniers et al., 2004]. However, our study focuses on the development of inner-bar morphology forced by the morphological template of the already crescentic outer bar (i.e., finite-amplitude behavior), and not on the free development of an initially alongshore-uniform nearshore morphology in response to a given set of offshore wave conditions. If the inner-bar morphology were to develop freely, larger alongshore spacings of the inner-bar features would be expected for increasing angles of wave incidence, as demonstrated by, among others, Smit et al.  and Deigaard et al.  for constant offshore wave forcing. Our results, however, showed no change in inner-bar spacing (see Figure 7b), verifying our assumption that the inner-bar morphology is indeed governed by the presence of an alongshore-variable outer bar.
 Overall, the numerical model was capable of reproducing the observed Idt coupling type. The model slightly overpredicted the alongshore migration of the outer-bar patterns and, therefore, the inner-bar features. Figure 10 shows an alongshore migration of the inner-bar features of ≈100 m over the 4 day modeling period, whereas both the inner-bar and outer-bar patterns in the observations hardly exhibit any alongshore migration (Figures 3a–3d). The systematic overestimation of rip channel migration rates with nonlinear morphodynamic models has already been pointed out by Falqués et al. . Furthermore, an inner bar with an initial degree of alongshore variability, as in the observations, may respond differently to a given forcing than an initial perfectly alongshore-uniform inner bar, as in the model [e.g., Tiessen et al., 2011; Price and Ruessink, 2011; Smit et al., 2012]. Similarly, the use of time-variant wave forcing potentially results in different bar morphodynamics than when using mean values [Castelle and Ruessink, 2011]. To test this, we performed a simulation with the hourly measured wave and tide data from the 4 day observation period (not shown). This simulation also resulted in the development of the observed Idt coupling type. The main difference with the reference simulation is the lower rate of the morphological development, as is the case for the simulations including a tidal water level variation in section 4.2. It is only at the end of the 4 day observation period that the large wave height and oblique incidence (see Figure 3, from 28-01 onward) straightened the outer bar, thus removing the morphological template for the inner-bar morphodynamics, similar to the outer-bar morphodynamics described in section 3.2.2.
 We applied a nonlinear model with data-based, time-invariant forcing to test our hypothesis that, for a given crescentic outer bar, the angle of wave incidence θis crucial for the inner-bar morphodynamics. Obliquely incident waves (θ>10°) over a crescentic outer bar lead to cross-shore undulations of the inner terrace edge that are coupled to the outer-bar morphology, confirming our hypothesis that this type of coupling develops during oblique wave incidence. In this case, a meandering alongshore current and more alongshore-directed sediment transport dominates. The offshore-directed sections of the meandering current, downdrift of the outer-bar horn, erode the inner terrace, resulting in the development of landward perturbations of the terrace edge coupled to the outer-bar horns. For more shore-normal wave incidence (θ<10°), on the other hand, cell-circulation patterns govern the flow at the inner bar, leading to coupled rip channels that are incised through the inner terrace. In agreement with both the hypothesis and our observations from a natural double-barred beach, the modeling exercise shows that the angle of wave incidence is crucial to the flow pattern, sediment transport, and thus the emerging coupling type at the inner bar.
 T.D.P. and B.G.R. acknowledge financial support by the Netherlands Organization for Scientific Research (NWO), under contract 818.01.009. B.C. acknowledges financial support through the BARBEC project (ANR N2010 JCJC 602 01). We thank Ton Markus for preparing the figures in this manuscript and Martijn Henriquez for his help on computing the swirling strength.