River deltas are among the most important environments on earth, housing productive ecosystems and a large fraction of the human population. The key process for delta development is the deposition of mouth bars in front of delta distributaries. Predicting mouth bar formation on marine coastlines is complex because of the interactions between waves, tides, water and sediment discharge. Here we study the effects of waves on mouth bar growth with the coupled sediment transport and wave model Delft3D-SWAN. Our results show that wave characteristics (height, period, and direction) play an important role in the formation of mouth bars. In our numerical experiments waves affect bar development in three ways: by modifying the direction of the river jet, by increasing bottom shear stresses at the river mouth, and by changing bottom friction and hence increasing jet spreading. We further show that high waves with long period prevent the formation of mouth bars; in particular, wave angles between 45° and 60° are the least favorable to bar formation, likely producing a deflected river mouth.
 River deltas are the most important depositional systems on earth, providing fertile land for agriculture, housing half a billion people and some of the most productive ecosystems [Syvitski and Saito, 2007]. Despite this critical role for society and life, many processes at the heart of delta development and evolution are still poorly understood [Fagherazzi and Overeem, 2007]. One of them is the deposition of mouth bars in front of fluvial distributaries. Mouth bars constitute much of the delta: once formed the river bifurcates around it, shifting deposition and delta development seaward [Edmonds and Slingerland, 2007]. Predicting mouth bar formation on marine coastlines is complex because of the interactions between buoyancy, waves, and tides [Wright, 1977]. However, most studies have considered mouth bar formation into a lake in the absence of coastal forces [Mikhailov, 1966; Edmonds and Slingerland, 2007].
 Waves might play a crucial role in the formation of mouth bars along coastlines. Studies based on the relationship between the distributary network of wave-influenced deltas showed that wave energy controls the network topology by suppressing mouth-bar development, thereby preferentially eliminating smaller-scale distributaries [Jerolmack and Swenson, 2007].
 Researchers have begun to consider and isolate how waves effect the growth of the entire deltaic system [Ashton et al., 2001; Ashton and Giosan, 2011; Geleynse et al., 2011], but few studies have focused on how waves affect mouth bar growth. Ashton and Giosan  showed that waves approaching from one direction can lead to the asymmetric evolution of a delta, with the possible formation of offshore extending spits and alongshore sand waves. In another modeling study, Geleynse et al.  noted that the presence of waves reduces the number of active distributaries, with the few remaining often flowing in a direction parallel to the shoreline.
 Understanding sediment transport driven by waves, and its relationship to the discharge of river sediments, is crucial for the morphological evolution of bars and therefore of the entire delta. To this end here we study the effects of waves on mouth bar formation with the coupled sediment transport and wave model Delft3D-SWAN [Lesser et al., 2004; Booij et al., 1999].
 Delft3D solves the full Navier-Stokes equations with the shallow water approximation for unsteady, incompressible, turbulent flow. The hydrodynamic and morphodynamic modules are fully coupled so that the flow field adjusts in real time as the bed topography changes. In the model SWAN, wind waves are described with the two-dimensional wave action density spectrum [Booij et al., 1999]. The model simulates wave generation, propagation, and nonlinear wave-wave interactions. Bottom dissipation, whitecapping, and depth induced breaking are fully accounted for in a dissipation term.
 We simulate bar formation in a basin of 72 by 44 computational cells with different sizes, ranging from 15 m to 30 m (Figure 1a). The basin is horizontally planar with an initial uniform depth of 3 m. A rectangular river channel with width, B, of 90 m and depth of 3 m is carved into a sandy shoreline along the eastern boundary of the grid, extending 100 m basinward. Tests show that the beach width does not alter the numerical results. The lower, upper, and left boundaries are open with a steady and uniform water surface elevation equal to mean sea level (Figure 1a). Five meters of non-cohesive sediment are initially available for erosion at the bottom.
 We utilize the van Rijn transport formulation and the erosion and deposition shear stresses are based on the Shields parameter for sediment resuspension. Suspended sediment eddy diffusivities are a function of the fluid eddy diffusivities and are calculated using horizontal large-eddy simulations and grain settling velocity. A time step of 0.2 s is adopted to satisfy all stability criteria. Every numerical experiment is run for 12 model hours before allowing morphological change. We speed up bed adjustments by multiplying the bed sediment flux at each time step by a morphological scale factor of 50.
 The 273 simulations of bar formation and progradation are conducted in which 6 different steady river discharges from 155 m3/s to 390 m3/s, carrying a steady concentration of non cohesive sediment, with D50 of 200 μm and specific density of 2.650 kg/m3. The geometry of the river mouth and the discharge range are similar to those of the distributaries of the Apalachicola River, Florida, USA (see Figure S1a in the auxiliary material).
 The sediment concentration in the river is maintained constant in each simulation and equal to the equilibrium concentration for the initial cross section in absence of waves. Hence in simulations utilizing a higher discharge the sediment concentration is higher, in order to keep the river mouth in equilibrium. As a result the mouth is neither eroding nor silting in time and the amount of sediment introduced in the domain is independent of wave conditions.
 We choose 3 peak periods (Tp = 5, 8, 10 s) and 4 significant wave heights (Hs = 0.3, 0.5, 0.8, 1.0 m). The simulations are further divided into 7 wave angles ranging between 0° and 90°, every 15°; this range of wave characteristics is typical of Apalachicola Bay, where the Apalachicola River debouches (see Figure S1a in the auxiliary material).
3. Effect of Waves on River Mouth Jet
 To study the effect of waves on the water jet exiting from a river mouth we vary wave characteristics, river discharge, and the flux of sediments in our simulations while holding all other factors constant.
 Waves attacking the river mouth at an angle lead to a deflection of the river jet downdrift [Wright, 1977, see Figure 1b]. The deflection increases for high wave angles, and it is maximum when the waves propagate almost parallel to the shore (Figure 1b). Higher waves with long period increase jet deflection until a limit angle, above which waves do not change the jet curvature (Figures 1c and 1d).
 The presence of waves also reinforces the mean bottom shear stress. This increase is caused by the non-linear interaction between the wave and current boundary layers [Grant and Madsen, 1979]. The increase in mean shear stress can be expressed using Soulsby et al.  formulation:
where τcrepresents the bed shear-stress generated by the current alone andτw is the bed shear stress generated by the waves. The maximum bed shear stress is considered to be the driving variable for sediment resuspension, entrainment, and transport, and it is obtained by combining the wave bottom shear stress with the current shear stress:
where α is the angle between the current and the direction of the waves. From equations (1) and (2) we determine that waves increase bottom shear stresses, but this effect is less important when the waves reach the jet at an angle. Larger bottom shear stresses allow the sediment to remain in suspension for longer time, and therefore favor the transport and subsequent deposition of material far from the mouth.
 Waves also have a second effect on the river jet. Yoon and Liu  numerical analysis indicates that the increase in bottom shear stresses due to waves results in a higher lateral spreading of the jet. In fact higher shear stresses increase bottom friction, which has been shown to control jet spreading [Wright, 1977].
 Here we compute jet spreading as the relative difference of velocity at the mouth (Uinlet) and at a fixed distance of 900 m measured along the centerline of the jet (U900):
where S is the dimensionless spreading rate. A series of simulations covering a range of wave conditions and river velocities show that the jet spreading decreases with wave angle (Φ), and it is maximum for waves propagating perpendicularly to the coast (Figure 2a).
 A planimetric comparison of jets affected by waves propagating at different angles shows how deflected jets remain laterally confined, while a jet affected by frontal waves spreads wider with a sharp decrease in velocity (Figure 1b).
 Waves have a threefold effect on the river jet: i) they change the jet direction; ii) when the wave angle decreases, the jet spreading increases (Figures 2b and 2c); an increase in jet spreading produces a sharp decrease of jet velocity, favoring the deposition of sediments close to the river mouth; iii) the maximum shear stress at the river mouth and along the entire centerline increases for a decreasing wave angle (Figures 2b and 2c), maintaining sediment in suspension and transporting it farther away from the river mouth.
4. Morphological Evolution of a Mouth Bar Attacked by Waves
 Numerical simulations of bar formation under the influence of waves result in four possible river mouth morphologies: central bar, side bar, deflected mouth, and wave dominated (Figure 3). For low wave conditions the bar forms at the center, producing a bifurcation of the flow similar to the case without waves [Edmonds and Slingerland, 2007]. When moderate waves propagate onshore at an angle, the bar forms updrift (side bar case, Figure 3 after 50 days), still triggering a bifurcation of the flow and the formation of two channels. In time the channel closer to the shoreline silts up forming a uniform deposit that extends to the shoreline. The river jet is then deviated toward the direction of the waves, forming a second downdrift bar (see side bar case in Figure 3 after 75 days). Strong waves with a large angle deflect the river mouth, leading to a jet that flows parallel to the shoreline. In this case large quantities of sediments are deposited between the jet and the shoreline, producing a swash bar that extends along the coast (deflected mouth case, Figure 3). A fourth case (wave dominated case, see Figure 3) occurs when strong waves reach the mouth perpendicular to the shoreline (Φ = 0°), destabilizing the jet. The jet starts oscillating, spreading sediments on a vast area without forming a distinct bar.
 In analogy with the work of Edmonds and Slingerland , for the cases of central and side bar we consider the bar to be fully developed when it reaches an elevation equal to 60% of the depth at the mouth of the river, because at that height the flow is forced to go around the bar and progradation effectively stops. At this point the position in front of the river mouth determines whether we have a central or side bar. We define a side bar when its top part is located outside of the longitudinal projection of the river mouth. In a deflected mouth a swash bar develops when the sediment is delivered to the shoreline and the flow is unable to fork around the deposit. Finally, the wave dominated case occurs when the bar is continuously destroyed and only lateral levees are built.
 The different final morphologies are a function of wave angle and the relative strength of waves with respect to the river flow. To quantify the relative importance of waves, we introduce a non-dimensional numberW equal to the ratio of the wave shear stress at the right domain boundary (and therefore outside the influence of the river jet), and the current shear stress at the river mouth:
where Ub is the significant wave bottom orbital velocity, fw is the wave friction coefficient, Uinlet is the current velocity at the river mouth, g the gravity acceleration, and C2D is the drag coefficient due to the current.
 For each of the 273 simulations, we determine the final morphology after 60 days, and we report the results as a function of W and wave angle Φ (Figure 4). Remarkably, all simulations group themselves in the four morphological cases reported above.
 For wave angles below 15° and large waves (W > 0.9) the sediment is scattered in the domain and no bar forms (wave dominated case in Figure 4). For intermediate wave heights (0.03 < W < 0.9) and low wave angles (Φ < 15°), a central bar forms in front of the river mouth. A central bar is also the final outcome when the wave bottom shear stress is very small compared to the shear stress of the river flow, irrespectively of wave direction (W < 0.06, see Figure 4). Under these conditions the waves are not strong enough to deflect the jet and move the bar to the side.
 For wave angles above 15° and moderate wave bottom shear stresses a side bar forms (side bar case, Figure 4), whereas for the same wave angles and stronger wave bottom shear stresses the river mouth is completely deflected giving rise to the swash bar between the jet and the shoreline (deflected mouth case in Figure 4). The boundaries between central bar, side bar, and deflected mouth cases in the phase space of Figure 4 are complex and depend on both parameters W and Φ.
 In our experiments, waves affect bar development in three ways: by modifying the jet direction, by increasing bottom friction and therefore jet spreading, and by increasing bottom shear stresses and sediment resuspension. The second and third hydrodynamic processes have the opposite effect on sediment transport. Jet spreading favors deposition of sediments near the mouth and the formation of a bar, whereas high bottom shear stresses maintain sediment in suspension transporting it farther away. However the two processes occur at different wave angles. At low wave angles, large waves reach the river mouth, and jet spreading is the dominant process (Figure 2a). Higher jet spreading favors the deposition of sediments near the mouth, and the formation of a mouth bar. At high wave angles the jet is completely deflected and bottom shear stresses are low since waves are perpendicular to the flow (equation (2) and Figure 2b). As a result sediment deposition occurs very close to the river mouth, likely forming a mouth bar. Therefore an intermediate angle must exist for which: i) jet spreading is limited, thus maintaining a fast and confined jet that carries all the sediments away from the coastline; ii) the wave angle is low enough to trigger high bottom shear stresses that keep the sediment in suspension thus favoring sediment delivery far from the mouth. Figure 4 shows that the most adverse conditions for mouth bar formation are wave angles between 45° and 60°. At these angles small waves are sufficient to trigger a transition from side bar to a fully deflected river mouth (deflected mouth case in Figure 4). In fact the waves deflect the jet, which then carries the sediments away from the mouth to be deposited in the lateral swash bar at the shoreline.
 For the same range of wave angles, between 45° and 60°, it is also easier for waves to create a side bar rather than a central bar, since the sediments are carried laterally for a longer distance. We conclude that waves propagating at angles between 45° and 60° are the least favorable to the formation of bars, creating ideal conditions for a deflected river mouth.
 It is important to point out that in our simulations longshore transport is limited, since we consider waves propagating on a flat bottom. The surf zone is therefore absent, and waves break only near the bar when it emerges. We are therefore unable to simulate deltaic features typical of coastlines with strong longshore currents, like spit formation in front of the river mouth [Bhattacharya and Giosan, 2003]. Our results better describe mouth bars forming in shallow shelves or in bays (for example the Apalachicola delta in Apalachicola bay, see Figure S1a in the auxiliary material).
 Our results show that wave climate (height, period, and direction) plays an important role in the formation of mouth bars. Specifically:
 1. Waves affect the evolution of mouth bars in three ways: by increasing river jet spreading, by augmenting bottom shear stresses, and by deflecting the jet. The first two occur at low wave angles, while the third is driven by high wave angles.
 2. High waves with long period prevent the formation of a mouth bar. In these conditions the river jet is either deflected if the wave angle is high or destabilized when the wave angle is low. Once the jet is destabilized, it oscillates laterally without forming a mouth deposit.
 3. Waves of intermediate height approaching the shore at an angle lead to a side bar; while small waves are unable to deflect the river jet producing a central bar.
 4. Wave angles between 45° and 60° are the least favorable to bar formation, likely producing a deflected river mouth.
 This research was supported by the ACS-PRF program award 51128-ND8, by NSF award OCE-0948213 and through the VCR-LTER program award DEB 0621014, by the Office of Naval Research award N00014-10-1-0269. We would also like to thank Doug Edmonds for the productive discussions on this topic.
 The Editor thanks A. Brad Murray and an anonymous reviewer for assisting in the evaluation of this paper.