Chenier Formation Through Wave Winnowing and Tides

Cheniers are ridges consisting of coarse‐grained sediments, resting on top of the fine sediment that forms the otherwise muddy coast. In this paper, we use Delft3D to explore how cheniers are formed through wave winnowing. We identify three phases of chenier development: (a) a winnowing phase, during which mud is washed out of the seabed initially consisting of a mixture of sand and mud, (b) a sand transport phase, when the sand in the upper layer is transported onshore, and (c) a crest formation phase, during which a chenier crest rapidly develops at the landward limit of onshore sediment transport. The main mechanism driving onshore sand transport is wave asymmetry. During calm conditions, sand transport takes place within a narrow band limiting the volume of sand delivered nearshore, and therefore no chenier develops. In contrast, average storm conditions mobilize sufficient sand for a crest to develop. Our results thus reveal that chenier formation through wave winnowing does not require extreme storm conditions. Furthermore, our study showed that chenier formation through wave winnowing is a relatively slow process, with the largest time scales associated with the winnowing and sand transport. Once sufficient sand is available in the intertidal zone, the crest develops rapidly.

TAS ET AL.

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2 of 17 both elements (wave energy and sediment availability) are reinforced, as the waves may now rework sediments that were not available during normal conditions, for example, the chenier formation in the Firth of Thames (Dougherty & Dickson, 2012;Woodroffe et al., 1983).
The objective of this paper is to understand how cheniers are formed through wave-induced winnowing. Despite the large amount of studies on cheniers (as in the references above) their initial formation is largely described in a qualitative sense, and a systematic study in response to wave and tidal processes is lacking. A first step towards modeling of chenier formation was made by Nardin and Fagherazzi (2018), who studied the development of entire chenier plains using Delft3D. In this paper, we focus on the development of a single chenier, hence looking at much smaller spatial and temporal scales. We therefore deploy a process-based numerical model (Delft3D) to identify the responsible processes and mechanisms during the various phases of chenier genesis, and determine under which conditions a chenier develops.
In Section 2.1 we provide an overview of the relevant sediment transport formulae and how they are implemented in Delft3D. Section 2.2 gives a brief overview of the model set-up, which is based on the model developed by Tas, van Maren, Helmi, and Reniers (2022) through validation against field observations (Tas et al., 2020). We explore chenier development through a number of (idealized) scenarios, representative for the conditions in Demak, Indonesia, described in Section 2.3. These scenarios reveal three phases of chenier formation which are presented in Section 3. The implications of our findings, as well as some limitations of our approach are discussed in Section 4 and the conclusions are presented in Section 5.

Methods
The formation of cheniers through wave-induced winnowing is explored using a Delft3D-FLOW morphodynamic model in combination with a SWAN wave model. Delft3D-FLOW solves the unsteady shallow water equations in two (depth-averaged) or three dimensions (Lesser et al., 2004) and SWAN is a third generation numerical wave model (Booij et al., 1999). Below (and in Appendix A) we summarize the relevant sediment transport formulae that are used to calculate sediment transport in Delft3D, followed by a description of the model set-up. The model scenarios are based on the chenier system of Demak, Indonesia (see also Tas et al. (2020)).

Sediment Transport in Delft3D
We consider two types of sediment: cohesive (mud) and non-cohesive (sand). Sand can be transported as bed load and as suspended load. Sediment transported as bed load is limited to a thin layer above the bed; sand particles higher in the water column are transported as suspended load. Mud is transported as suspended load or as fluid mud (for high concentrations the mud is transported as a viscous layer near the bed). Sediment transport (both sand and mud; both suspended and bed load) requires the exceedance of a critical bed shear stress to initiate movement of the sediment particles.
Bed load transport of sand is calculated using the empirical formula of van Rijn (1993). The suspended sediment transport of both mud and sand by currents are computed with an advection-diffusion equation. Additionally, there is also suspended sand transport due to wave asymmetry, as a result of asymmetric wave orbital velocities, which is approximated following van Rijn et al. (2001). Appendix Appendix A gives an overview of the relevant sediment transport formulae.

Model Set-Up
We follow a quasi 1D approach: the model is set up in depth-averaged mode and is alongshore uniform. In order to avoid shadow effects (where the boundaries affect the wave field), the WAVE-domain is much wider than the FLOW domain. The grid cell size Δx reduces from 25 m at deep water to 1.5 m nearshore. The bed has a constant slope of 1:500 and consists initially of a homogeneous mixture of 2 fractions: mostly mud and a small sand fraction (D 50,sand = 235 μm). The mud in Demak is very soft (with a strength of the mud layer being much too low to carry a human). Little is known about the erosion properties of such thick mud bed properties. The most similar conditions for which erosion properties of such thick fluid mud beds have been described in literature are those of van Maren, Winterwerp, and Vroom (2015), using a critical bed shear stress τ cr,e = 0.5 N/m 2 . Settling velocities in similar environments generally range from 0.2 to 5 mm/s (Manning & Dyer, 2007;Soulsby et al., 2013;van 3 of 17 Leussen & Cornelisse, 1993;van Maren, van Kessel et al., 2015). Measurements in Demak suggested a small settling velocity on the lower end of this range (Deltares & BioManCO, 2019), hence we set w s = 0.5 mm/s. The dry bed density is typically around ρ dry = 300 kg/m 3 . With all other parameters largely based on observations or earlier work, the last remaining parameter (the erodibility parameter M ero ) was calibrated such that the bed was dynamic (spatial and temporal variation in the bed level) but still attained dynamic equilibrium (i.e., not completely eroding or filling in with sediments). This was achieved using a value of M ero = 1.10 −4 kg/m 2 /s. This parameter should be re-calibrated before applying this approach to a different chenier system.
The bed composition is modeled using a layered bed stratigraphy: the bed is composed of multiple layers in order to account for winnowing of mud from the mixed bed. The active upper layer is 2 mm thick and always fully mixed. Upon deposition, sediments settle in the upper layer, leading to migration of sediments from the upper layer to the first bed stratigraphy layer. Only the upper layer is available for erosion, and upon erosion sediment from the first stratigraphy layer migrates to the active layer. The stratigraphy layers have a maximum thickness of 10 cm, except the lowest layer when the maximum of 10 layers is reached (this layer is limited by a non-erodible boundary 1 m below the initial bed level).
At the offshore boundary, a (time-varying) water level and wave height is prescribed (see Section 2.3 hereafter). No wind is applied to the model domain. A delayed transfer of wave energy in the landward direction is computed with a roller model (Nairn et al., 1991;Reniers et al., 2004;Svendsen, 1984). This introduces a time lag (and therefore spatial lag) between the moment of wave breaking and the moment of energy dissipation, which is necessary for realistic modeling of cross-shore profile development of barred beaches (Reniers et al., 2004).
The computational time was reduced by introducing a morphological acceleration factor MorFac (multiplying the bed level change every timestep with MorFac). We have scaled the MorFac with the wave height (MorFac ∼1/H s 2 ), using a smaller factor for larger wave heights (hence larger/faster bed level changes) and verified our choice by comparing the model results with a smaller MorFac (i.e., the results were not affected by introducing or increasing the MorFac).

Model Scenarios
Boundary conditions representing the conditions in Demak, Indonesia are defined in multiple model scenarios, see Table 1 (see also Tas et al. (2020) and Tas (2022) for a description of the coast and chenier system of Demak). Three wave scenarios are defined: (a) sea breeze-induced waves, representing the energetic conditions during the SE monsoon season; (b) average storm conditions, which occur multiple times per NW monsoon season; and (c) extreme storm conditions, representing a storm with a return period of 10 years. The extreme storm conditions are estimated based on the peak-over-threshold analysis of Alferink (2022) performed on Wave Watch III data, which were validated against field observations. A simple sinusoidal tide is prescribed at the offshore boundary, composed of the largest constituent in Demak, K1 (Tas et al., 2020), with three varying amplitudes representing no tide (a K1 = 0 m), neap tide (a K1 = 0.25 m) and spring tide (a K1 = 0.5 m). The sand fraction in the initial homogeneously mixed sediment bed is varied between 5%, 10%, and 20%. Each wave scenario is combined with any of the tidal amplitudes and sand fractions (27 scenarios). These idealized scenarios are supplemented with one scenario forced with a realistic time series for water levels and wave conditions at the offshore boundary (see Section 3.6). Figure 1 shows the formation of a chenier under constant wave conditions representing average storm conditions (H s = 1.3 m and T p = 5.6 s, see Table 1), in the absence of tide (constant water level), starting from a fully mixed bed with 10% sand. The chenier develops in three phases: (a) winnowing, during which mud is washed out and the upper layer of the bed becomes increasingly sandy; (b) the sand in the upper layer is transported landward; and (c) crest formation, essentially the abrupt heightening of the chenier crest when sufficient sediment is available. During phase 1 and 2, when a thin sand layer is formed and transported landward, bed level changes are small; most morphodynamic changes take place during phase 3 when the chenier crest develops. The chenier exemplified in Figure 1 reaches its full height after 120 days, after which the speed of bed level changes slow down.

Results
Chenier development is subsequently explored for other wave conditions as well: lower waves (representing an energetic sea breeze-day during the SE monsoon season) and higher waves (representing an extreme storm). The resulting evolution of the sand fraction in the upper layer of the bed (left column, panels 2a, 2d and 2g), bed level changes (middle column, panels 2b, 2e and 2h) and sand transport (right column, panels 2c, 2f and 2i) are given in Figure 2, for extreme storm conditions (upper row), average storm conditions (middle row) and energetic sea breeze conditions (lower row) respectively.
For all wave scenarios, mud is winnowed from the bed and sand is subsequently transported landward, but only for the higher wave scenarios a chenier crest develops (Figures 2b and 2e). The sediment sorting (left column) and sand transport (right column) exhibit similar trends, which suggests that onshore sand transport depends on sand availability, which is a direct result of winnowing. This landward transported sand culminates close to the shoreline and, for the higher wave conditions, develops into a chenier crest.

Phase 1: Winnowing
For all wave scenarios, mud is eroded over almost the entire domain (see Figure 3, left y-axis, when |τ cw,max | − τ cr,e > 0). However, the onset of sand transport requires higher flow velocities (when |v eff | − v cr > 0, right y-axis in Figure 3). This difference is especially relevant for the sea breeze scenario (Figure 3c), where mud is eroded over a much wider area, while sand is only eroded landward of x = −900 m. This means that seaward of x = −900 m, mud is being eroded from the upper bed layer, but the sand particles remain immobile, which results in a higher sand fraction in the upper bed layer (visible in Figure 2g). Landward of x = −900 m, both sediment fractions are mobilized. For higher waves (see Figures 3a and 3b), the erosion thresholds for both sediment fractions are exceeded at increasingly deeper water, resulting in erosion of both mud and sand over almost the entire domain.

Phase 2: Sand Transport
Sand transport (right column in Figure 2) is directly linked to the availability of sand in the upper layer (left column Figure 2). Sand transport is therefore supply-limited, that is, not only governed by the transport capacity, but also by the sand availability (and thus winnowing). During phase 2, the main mechanism for onshore sand transport is wave asymmetry, as demonstrated with a simple sensitivity analysis. The modeled onshore sand transport by wave asymmetry can be modified with a user-defined parameter f SUSW (calibrated to 0.5 using field observations, see Tas, van Maren, Helmi, and Reniers (2022)). Applying a much lower value (f SUSW = 0.05) results in a much lower onshore sand transport (compare panels 4a-4c with panels 4d-4f), revealing that onshore transport indeed results predominantly from wave asymmetry.
Although there is still some onshore sand transport for the scenario with a 10 times smaller value for f SUSW (Figure 4f), leading to a sandy region near the shoreline (Figure 4d), the total volume of onshore sand transport and resulting bathymetric change ( Figure 4e) is small.

Phase 3: Crest Formation
The last phase of chenier formation is rapid sand accumulation at one location, thereby shaping a chenier crest. This process is very fast and abrupt: once a certain threshold is exceeded, the crest height rapidly increases.
Sedimentation (and erosion) can be directly linked to sediment transport gradients. A negative transport gradient is defined as a landward decrease in sediment transport, resulting in sediment deposition. Figure 5 shows the bed level change (left column), the sand transport (middle column) and the sand transport gradient (right column) for a scenario during average storm conditions. The three rows show a different initial sand content, in this 10.1029/2022JF006792 6 of 17 section we focus on the middle row (sand content of 10%), the other rows (5% and 20% sand) will be discussed in Section 3.4.
Initially, sediment transport is maximal in deeper water (x = 1,000 − 1,500 m with water depth d = 2 − 3 m at t = 0 in Figure 5e). Sand is eroded seaward of this sand transport peak (positive transport gradient in Figure 5f)  and deposited landward (negative transport gradient). The winnowed sand body migrates landward, driven by wave asymmetry and a gradual depletion of sand in the sand-mud mixture. The moment the peak sand transport rate is close to the shore (x = 200 m at t = 40 days in Figure 5e), the decrease in transport in landward direction is concentrated over such a narrow zone that a chenier crest emerges ( Figure 5d).

Effect of the Initial Sand Fraction
The size of the developed cheniers, but also the time required to develop a chenier, depends not only on the waves (as evaluated above) but also on the sand fraction in the sand-mud mixture. Figure 6 shows the chenier formation process for three different initial sand fractions (different shades of brown) at three points in time (panels a-c, corresponding to the time markers in Figure 5).
Interestingly, the lowest initial sand fraction (darkest line in Figure 6) leads to the fastest emerging chenier. This initially counter-intuitive observation can be explained by sand winnowing. Because of the lower sand content in the bed, the upper bed layers are more quickly depleted of sand, the maximum transport peak converges more rapidly with the shoreline, and as a result crest formation (phase 3) starts earlier. The onset of crest formation for the scenario with 10% sand is visible around x = −170 m ( Figure 6a) and for the scenario with 20% sand around x = −230 m (Figure 6b). The chenier crest is located where the thin sand layer (which slowly built out from the water line) converges with the steepest transport gradients (see Figures 5c, 5f and 5i).

Effect of Tides
The model scenarios evaluated up to now assume a constant water level. In the following scenarios a single tidal component (K1, the dominant constituent) is prescribed as boundary condition with an amplitude of 0.5 m (the  This difference is related to the time and spatially varying water depths and bed shear stresses. In the intertidal zone, sand is only transported when the bed is submerged (the migrating water line is indicated with the blue line in Figure 7), and therefore the chenier cannot develop during periods of emergence. This explains the slower development of the chenier under tidal conditions. The height of the chenier crest is limited by two factors: the water surface and the available volume of sand. The maximum water surface is higher for tidal conditions, leading to a higher crest height. But in addition to this, the waves propagate further landward during high tide, thus enlarging the bed surface from which sand can be winnowed (i.e., more sand is available under tidal conditions, favoring a higher chenier).

Realistic Boundary Conditions
As a final step, chenier formation under realistic time series for the wave conditions and water levels is evaluated. Wave conditions were derived from the Wave Watch III hindcast data (The WAVEWATCH III Development Group, 2019) for the NW monsoon season between 1 December 2016 and 1 March 2017. The water levels were calculated using all tidal constituents derived from the water level data measured at Semarang (Tas et al., 2020). A morphological factor of 5 was applied to realize bed level changes within acceptable computational periods (Figure 8).
During the first weeks, the wave height is low (rarely exceeding 0.8 m). As a result, sand is only winnowed from a narrow nearshore zone and transported to the coastline (phase 1 and 2), but the amount of sand is insufficient to develop into a crest (phase 3). A thin sand layer develops relatively high in the profile, above MSL.
Winnowing and sand transport occur over a much wider zone during the first storm (t = 100 days), and more sand is transported landward. This leads to sufficient nearshore sand availability for the formation of a chenier crest. The crest is formed at the seaward edge of the sand layer, which is relatively far landward.
After a brief period with calmer conditions (during which the chenier remains static although sand is transported landward to form a thin sand sheet seaward of the chenier), a new storm initiates a second phase of substantial 10 of 17 onshore sand transport and the formation of a second chenier crest. This crest is again located at the seaward edge of the sand layer, and is thus located more seaward than the first chenier crest. This crest quickly grows in height and shelters the older chenier crest from wave action, until it stabilizes with its crest around the highest tidal levels. Later storm periods transport new volumes of sand nearshore, leading to a slight widening of the chenier crest in the seaward direction.
A notable difference between this model scenario with varying boundary conditions and the previous scenarios with constant boundary conditions, is the role of the calmer periods between storms. The wave energy during these calm periods is too weak to transport sufficient sand nearshore to develop into a chenier crest. However, there is still onshore sand transport. This sand supplied during calm conditions provides the material for rapid chenier formation during stormy periods. Therefore cheniers may more rapidly emerge during storms following a period of relatively calm conditions.

Discussion
We have numerically investigated chenier formation for a range of hydrodynamic boundary conditions representing the environmental conditions near Demak, Indonesia (see Table 1). A chenier developed for 18 of the 27 scenarios: the only conditions for which no chenier crest developed were for smaller, sea breeze-induced waves. These various hydrodynamic conditions influence the location where the chenier emerges (x crest ) and the time it takes for the chenier to develop (t crest ). Figure 9 summarizes x crest and t crest for all scenarios with chenier development. Here, the moment of crest formation was defined as the moment the chenier crest reaches 50% of its final crest height, which coincides with the moment of rapid crest heightening (when the highest transport gradients reach the shoreline -see also Figure 5).
The chenier crest develops further offshore for scenarios with higher waves (filled markers) and/or higher initial sand content (yellow markers). Both contribute to a larger sand volume in the nearshore, which allows the initial nearshore sand layer to build out further offshore. A larger tidal range slows down the speed of chenier formation (high t crest ), as the varying water levels lead to a larger range in water depths and bed shear stresses. The diagram also provides more details on the role of the initial sediment substrate. The results in Section 3.4 suggested that a chenier develops slower for higher sand content but closer inspection of Figure 9 reveals that this is only true Figure 9. Location of the chenier crest (horizontal axis) and duration until crest development (vertical axis) for all scenarios that resulted in a chenier (18 out of 27 scenarios described in Table 1, excluding the sea breeze wave conditions). The shape of the data points relate to the tidal range (circle: no tide, triangle: neap tide, square: spring tide), the colors represent the sand content of the bed (lighter for higher sand content) and the two wave conditions are distinguished by the fill (no fill: average storm conditions, filled: extreme storm conditions).
TAS ET AL. Chenier formation has traditionally been associated with energetic conditions (e.g., Brunier et al., 2019;Dougherty & Dickson, 2012;Woodroffe & Grime, 1999). Such conditions correspond to extreme storm conditions in Demak, but our model results suggest that also milder conditions are important for chenier formation. This can be explained with the three phases which drive the formation of cheniers: winnowing, onshore transport, and crest formation. Milder storms regularly occurring during the NW monsoon season are already energetic enough to activate all three phases of chenier formation. Furthermore, small amounts of sand are still transported nearshore during calmer periods in-between. While the winnowing phase has been described qualitatively in previous work Augustinus, 1980;Nardin & Fagherazzi, 2018;Rhodes, 1982;Woodroffe & Grime, 1999), our findings provide the first quantitative investigation on this winnowing process in relation to the development of individual cheniers.
Our model results further suggest that the largest timescales are associated with phase 1 (winnowing) and 2 (onshore transport). The actual crest formation, however, is very fast (see also Section 3.3). In our model approach the timescales of winnowing are dependent on the state of the initial sediment bed (fully mixed), and therefore it takes a long time for the chenier to develop. It is also not realistic that storm conditions occur uninterrupted for 50 days (as in our moderate and high storm conditions simulations). However, a sequence of storms (and calmer periods and seasons in between) may eventually set up a sufficiently pre-sorted situation, which could allow a regular, short storm to form a chenier.
The scenario with a realistic time series for the wave conditions and water levels reveal the development of a second chenier crest, seaward of the first crest. This may seem similar to bar systems which often have multiple shore-parallel bars occurring simultaneously (Masselink et al., 2006;Ruessink et al., 2003;Short, 1991;Walstra et al., 2012). However, in contrast to submerged bars, chenier crests quickly develop to a height around or above MSL, thus emerging during (part of) each tidal cycle. As a result, the most seaward chenier crest cuts off the more landward crest from tidal and wave influences, blocking any further sand supply and effectively stopping its dynamics. This is in agreement with observations by Woodroffe et al. (1983) on the chenier plain in the Firth of Thames, New Zealand. Another difference between barred beach systems and cheniers is the availability of sand; the sediment transport in phase 2 and 3 is limited by the availability of the coarse sediment fraction, resulting in different dynamics compared to submerged bars and swash bars in sandy systems.
The realistic time series for the water levels (and wave conditions) are based on the conditions in Demak, Indonesia, which has a microtidal and mixed, mainly diurnal tide. Under these conditions, the tide generally played a minor role during chenier formation. However, cheniers can be found in many different tidal regimes, ranging from microtidal to macrotidal (e.g., Mont-Saint-Michel Bay, France (Weill et al., 2012) or Gomso Bay, Korea (Lee et al., 1994)). It is possible that for different locations, chenier formation could be more strongly affected by the tidal conditions, similar to for example, the effect spring/neap cycles have on intertidal bar growth (Masselink et al., 2006). Our modeling approach opens up the possibility to investigate the wide range of environmental conditions encompassing chenier systems worldwide.
By using the stratigraphy schematization in Delft3D to model separate bed layers, we modeled the winnowing process in detail. We showed that even for very low sand content (5%), a chenier can be formed. This agrees with the findings of Nardin and Fagherazzi (2018) that sand availability usually is not the limiting factor in chenier formation (except for extremely low sand contents). This finding explains the presence of cheniers in areas without a nearby source of coarser sediments, for example, in Demak, Indonesia.
Possibly the largest limitation of our modeling approach is the absence of sand-mud interactions. In our model set-up, a cohesive (mud) and a non-cohesive (sand) fraction were considered independently: the sand fraction is mainly transported as suspended load due to wave asymmetry once a critical bed shear stress based on the Shields curve is exceeded; the mud fraction is transported as a suspended load, once the critical bed shear stress for erosion is exceeded. However, a sand-mud mixture will behave as a cohesive mixture when the mud content is higher than 30% ( van Ledden et al., 2004). Such a mixture will typically have a higher critical bed shear stress for erosion (van Rijn, 2020). Furthermore, we assumed a relatively low settling velocity and critical bed shear stress for the mud fraction. Altogether, this means that our model results may overestimate the speed (and extent) at which winnowing takes place.

Conclusions
A numerical model (Delft3D) is used to analyze how cheniers develop through wave-induced winnowing and transport. We identify three phases in chenier genesis: (a) winnowing, when the mud is washed out and the upper layer of the sediment becomes increasingly sandy; (b) sand transport, when the sand in the upper layer is transported landward; and (c) crest formation, when the sand culminates at one point, resulting in a rapid heightening of the chenier crest. Winnowing takes place when mud is eroded from the bed and brought into suspension (when τ cw,max > τ cr, e ), leaving behind a thin layer of sand. Sand transport takes places when the threshold for the initiation of motion is exceeded (when v eff > v cr ). The onshore sand transport is mainly driven by wave asymmetry, and is supply-limited (directly depending on the winnowing). Crest formation occurs when the peak of the sand transport rate converges at the shoreline, locally generating steep negative transport gradients resulting in rapid sedimentation.
The model study suggests that chenier formation does not require extreme storm conditions. Winnowing and onshore sand transport (phases 1 and 2) already take place under calm conditions, but crest formation (phase 3) requires sufficient sand to be transported nearshore. Under sea breeze conditions, the area where sand transport takes place is too small to supply sufficient sand to develop into a crest, only creating a thin nearshore layer of sand. However, average storm conditions which are exceeded several times per year, do generate sand transport over a sufficiently large spatial scale for a chenier to develop. Calm conditions therefore do not directly lead to chenier development but do result in landward transport of sand (as long as the upper layer of the bed is not depleted) which speeds up the formation of a crest during more energetic periods following that period of calm conditions.
Starting from a well-mixed bed, chenier formation is a slow process (requiring at least 8-10 weeks of uninterrupted storm conditions). Especially the time period associated with winnowing and sand transport (phases 1 and 2) is long, followed by a rapid crest formation (phase 3). The model study reveals that cheniers generally develop quicker for a lower sand content in the bed because the upper bed layers are more quickly depleted of sand. Therefore the maximum sand transport converges more rapidly with the shoreline (onset of crest formation, phase 3).
On the other hand, larger tidal amplitudes slow down chenier formation, due to temporal and spatial variability of bed shear stresses and periods of emergence in the region of chenier development.

A1. Bed Load Transport
The bed load transport is calculated using the empirical formula of van Rijn (1993): where S b is the bed load transport (in kg/m/s), ρ s is the sediment density (in kg/m 3 ), w s is the settling velocity for the D 50 of the sediment (in m/s), and D 50 is the median diameter of the sediment fraction (in m). M is the sediment mobility number due to waves and currents (see Equation A2) and M e is the excess sediment mobility number (see Equation A3).
The sediment mobility number, M, is defined as: and the excess sediment mobility number, M e , is defined as: TAS ET AL.

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13 of 17 where v eff is the effective velocity due to waves and currents (see Equation A4, in m/s), v cr is the critical velocity for the initiation of motion (see Equation A6, in m/s), s is the relative density (see Equation A5) and g is the gravitational acceleration (in m/s 2 ).
The effective velocity due to waves and currents, v eff , is: where v R is the magnitude of the depth-averaged current velocity (in m/s) and U on is the onshore-directed, high frequency near-bed orbital velocity (in m/s), calculated using a modification of the method of Isobe and Horikawa (1982) by Grasmeijer and van Rijn (1998).
The relative density s is defined as: The critical velocity for the initiation of motion is determined based on a parametrization of the Shields curve (Soulsby, 1997;van Rijn, 1993): where h is the water depth (in m) and D 90 is the sediment diameter (in m) for which 90% of the sediment has a smaller diameter, and is based on the composition of the local sediment mixture. In this case, D 90 = 1.5D 50 .
The bed load transport S b consists of a current-driven component, S b,c , and a wave-driven component, S b,w : with φ the angle between the current and wave direction. From this equation it follows that the current-driven and wave-driven components, S b,c and S b,w respectively, can be calculated as: with:

A2. Suspended Transport
Suspended transport is described by an advection-diffusion equation. In depth-averaged mode (∂/∂z = 0) and assuming longshore uniformity (∂/∂y = 0) this equation reads: where c is the concentration (in kg/m 3 ), h is the water depth (in m), U is the depth-averaged velocity in x-direction (in m/s), w s is the settling velocity (in m/s), ϵ s,x is the horizontal eddy diffusivity (in m 2 /s), E is the erosion flux (in kg/m 2 /s) and D is the deposition flux (in kg/m 2 /s). In Equation A11, the second term gives the advective transport, the third term the diffusive transport and the right hand side represents the source and sink terms (to and from the bed). For cohesive sediment, the erosion and depositional fluxes are calculated with the Partheniades erosion formulation (Partheniades, 1965) and a permanent deposition flux (Winterwerp, 2007): where M ero is an erosion parameter (in kg/m 2 /s), S(τ cw,max , τ cr,e ) is an erosion function (see Equation A14) and c b is the average sediment concentration near the bed (in kg/m 3 ).
The erosion function is defined as: Here, τ cr,e is the critical bed shear stress for erosion (in N/m 2 ) and τ cw,max is the maximum bed shear stress due to current and waves (in N/m 2 ). This maximum bed shear stress is computed with the parametrization of Soulsby et al. (1993): where τ c is the bed shear stress due to currents (in N/m 2 ), τ w is the bed shear stress due to waves (in N/m 2 ), and Z is a dimensionless parameter.
The bed shear stress due to currents, τ c , is defined as where U is the depth-averaged velocity (in m/s) and C is the Chézy coefficient (in m 1/2 /s).
The magnitude of the bed shear stress due to waves alone is defined as: where f w is the wave friction factor (defined in Equation A19) and u orb is the peak orbital velocity (in m/s) and can be calculated using linear wave theory: Here H rms is the root-mean-square wave height (in m), ω = 2π/T is the wave angular frequency (in s −1 ) and k = 2π/L is the wave number (in m −1 ) which can be derived from the linear wave dispersion relationship.
The wave friction factor under pure oscillatory flow is calculated following Swart (1974): TAS ET AL.

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15 of 17 where k s is the Nikuradse roughness height (in m) (which can be derived from the Chézy coefficient, see Equation A21) and A, the orbital excursion length, equals: The Nikuradse roughness height is derived from the Chézy coefficient using the formulation of White-Colebrook: The dimensionless parameter Z from Equation A15 is defined as: with: The value of the parameters a, m, and n can be determined through the following expression: = 1 + 2| cos | + 3 + 4| cos | log 10 (A24) where φ is the angle between the current direction and the direction of wave propagation and χ represents a, m or n. The fitting coefficients based on Fredsøe (1984) are used, see Table A1.
In addition to the suspended transport via advection and diffusion (which in this case is mostly mud transport), there is also suspended sand transport due to wave asymmetry. This component represents the effect of asymmetric wave orbital velocities on suspended sediment transport within about 0.5 m of the bed and can be approximated following van Rijn et al. (2001) where f SUSW is a user-defined tuning parameter, γ is the phase lag coefficient (γ = 0.2), U A is the velocity asymmetry value (in m/s, see Equation A26) and S S is the suspended sediment load (in kg/m/s, see Equation A27).
The velocity asymmetry value, U A , is defined as: In Delft3D, this transport component is included in the bed load vector, because it does not exhibit the relaxation effects of an advection-diffusion relation (Deltares, 2021).