The effect of asymmetric dune roughness on tidal asymmetry in the Weser estuary

The bed of estuaries is often characterized by ripples and dunes of varying size. Whereas smaller bedforms adapt their morphological shape to the oscillating tidal currents, large compound dunes (here: asymmetric tidal dunes) remain stable for periods longer than a tidal cycle. Bedforms constitute a form roughness, that is, hydraulic flow resistance, which has a large‐scale effect on tidal asymmetry and, hence, on hydrodynamics, sediment transport, and morphodynamics of estuaries and coastal seas. Flow separation behind the dune crest and recirculation on the steep downstream side result in turbulence and energy loss. Since the energy dissipation can be related to the dune lee slope angle, asymmetric dune shapes induce variable flow resistance during ebb and flood phases. Here, a noncalibrated numerical model has been applied to analyze the large‐scale effect of symmetric and asymmetric dune shapes on estuarine tidal asymmetry evaluated by residual bed load sediment transport at the Weser estuary, Germany. Scenario simulations were performed with parameterized bed roughness of symmetric and asymmetric dune shapes and without dune roughness. The spatiotemporal interaction of distinct dune shapes with the main drivers of estuarine sediment and morphodynamics, that is, river discharge and tidal energy, is shown to be complex but substantial. The contrasting effects of flood‐ and ebb‐oriented asymmetric dunes on residual bed load transport rates and directions are estimated to be of a similar importance as the controls of seasonal changes of discharge on these net sediment fluxes at the Lower Weser estuary. This corroborates the need to consider dune‐induced directional bed roughness in numerical models of estuarine and tidal environments.


| INTRODUCTION
Estuarine morphology is largely determined by residual sediment transport patterns that depend to a large extent on tidal asymmetry (Dronkers, 1986). Tidal asymmetry refers to the inequality between flood and ebb flow velocities and associated tidal phase durations, or the difference between slack water periods before ebb and flood (Dronkers, 1986). In estuaries, tidal asymmetry is governed by interacting mechanisms such as tidal wave dampening and distortion or tidal straining and estuarine circulation, to name a few, and has been extensively discussed in literature (e.g., Burchard et al., 2011;De Swart & Zimmerman, 2009;Dronkers, 1986;Friedrichs & Aubrey, 1988). The effect of directional hydraulic drag induced by large asymmetric tidal dunes has yet been given little attention in this discussion.
Bedforms, such as ripples, megaripples, or large dunes, cause a local hydraulic flow resistance, which has a large-scale effect on hydrodynamics and sediment dynamics of rivers, estuaries, and coastal seas (e.g., Blondeaux, 2012). Active tidal bedforms are defined as subaqueous flow transverse features, which significantly influence the flow field (Winter et al., 2015). The resistance to the flow induced by bedforms, that is, hydraulic bedform roughness, is associated with the flow expansion on their lee side resulting in kinetic energy loss, as exemplified for river dunes by Engelund and Fredsoe (1982). Additional turbulence generated in the case of flow separation behind the bedform crest and recirculation on the downstream side greatly enhances this effect (e.g., Vanoni & Hwang, 1967). The expansion loss and the rate of velocity decrease downstream of the bedform crest can be related to the lee slope angle (Best & Kostaschuk, 2002;Kwoll et al., 2016;Motamedi et al., 2013;Paarlberg et al., 2007). While small bedforms such as ripples and megaripples are assumed to change their asymmetric orientation through morphological development with the oscillating tidal flow, large tidal dunes retain their morphological shape and orientation on time scales much longer than a tidal cycle. Consequently, lee slope angles of asymmetric dunes are different for reversing tidal flow and effective dune roughness changes between ebb and flood tidal phases. Field measurements of flow, turbulence, and transport patterns over large asymmetric dunes have revealed differences between tidal phases. For a tidal channel in the Danish Wadden Sea, Kwoll et al. (2014) showed that, when flow and primary bedform orientation are aligned (gentle stoss side facing the flow), water-depth-scale macroturbulent structures develop in the bedform lee-side, which are coupled to increased amounts of fine sediment in suspension. When flow and bedform orientation are opposed, no evidence of flow separation associated with primary bedforms was found. Lefebvre et al. (2011Lefebvre et al. ( , 2013 found that asymmetric large primary bedforms are mainly contributing to form roughness, when the tidal flow is in alignment with dune asymmetry, that is, coming up the gentle stoss side and flow expansion and recirculation on the dune at the steep lee side. The effect of these findings on the estuarine scale, however, has not been shown yet. Energy loss above bedforms must be considered in numerical hydro-and morphodynamic model simulations through bed friction coefficients that are associated with grain (i.e., skin friction) and bedform (i.e., form drag) roughness. In particular, the effect of form roughness on the flow has received little attention in large-scale model applications. Although computer power increases steadily, numerical coastal domain models are typically still restricted in horizontal and vertical grid resolutions to properly represent all topographical and morphological features; common grid cell sizes of process-based models are between 20 and 200 m. This implies the parameterization of the effect of bedform roughness elements that are of the dimension (dune lengths of 10-100 m) of a model grid cell or less. Thus, bedforms are usually considered as of "sub-grid-scale" (Sandbach et al., 2012). Even at high spatial resolution and adequate three-dimensional discretization of bedforms (and flow), their resistance to the flow is not necessarily accounted for. Flow separation and turbulence generation over bedforms require simulations with a fully nonhydrostatic model configuration (Lefebvre et al., 2014;. The use of nonhydrostatic models is computationally very intensive and still not applicable for large-scale coastal domains, which hence requires appropriate methods to parameterize the hydraulic effect of bedforms in, for example, numerical process-based models. Even more than for capturing flow properties, the correct parameterization of the effective bed roughness is crucial for the prediction of sediment transport. Although common in fluvial studies (e.g., Paarlberg et al., 2010), there are only few numerical model studies that apply bedform roughness predictors and deal with the effect of bedform drag on hydro-and morphodynamics at the estuarine and coastal system scales (e.g., Brakenhoff et al., 2020b;Davies & Robins, 2017;Herrling et al., 2017;Villaret et al., 2011Villaret et al., , 2013Wang et al., 2016).
The form resistance exerted by bedforms, caused by local flow separation and recirculation, depends on bedform dimension, flow, and sediment characteristics (Karim, 1999). Hydraulic bedform roughness predictors are usually based on the ratio of height and length (i.e., steepness) of a bedform (Julien & Klaassen, 1995;Karim, 1999; van Rijn, 1984;Yalin, 1964) and thus require field data or a prediction of bedform geometry. Yet the predictor of van Rijn (2007a) directly expresses the bedform roughness height depending on flow velocity, sediment grain size, and water depth. Thus, in tidal environments, variations of current speed and water depth may result in variations of bedform roughness over a tidal cycle. However, since primarily developed for unidirectional flow, van Rijn (2007a) disregarded the asymmetrical geometry of tidal dunes that results in exposures of different lee slope angles during reversing tidal flows. A modified application of van Rijn's (2007a) dune roughness predictor here considers the directionality of bedform roughness induced by asymmetric tidal dunes.
The objectives of the present study are twofold. The first is to determine the effect of asymmetric dune shapes parameterized through tidal-phase-dependent bed roughness on spatiotemporal flow and sediment transport at the Weser estuary. The second is to discuss the combined effects of freshwater discharge, tidal energy conditions, and dune-induced flow resistance on the estuarine tidal asymmetry.
Several different harmonic and statistical methods exist to characterize tidal asymmetry (Guo et al., 2019). Here, the tidal asymmetry is shown by the direction and magnitude of residual bed load sediment transport, because of the specific role of bed load controlling both local bedform-related (Naqshband et al., 2014) and large-scale changes of estuarine morphology.

| Weser estuary
The Weser estuary is located at the southeastern coast of the North Sea, Germany, (Figure 1a) and can be classified as partially mixed (Grabemann & Krause, 1989). The distance from the tidal barrier at Bremen-Hemelingen (km À5) to the mouth of the estuary (km 112) where the main channel opens into North Sea waters is 117 km ( Figure 1b). The official kilometrage of the navigation channel starts at Bremen Weserbrücke (km 0). In the following, the distance to the upstream tidal barrier at Hemelingen is marked by À5 km. The estuarine reach can be subdivided into the Lower Weser (Hemelingen to Bremerhaven, km À 5 to 67) and the Outer Weser downstream from Bremerhaven to the open North Sea. At the Outer Weser, several channel-shoal systems with large intertidal areas drain into the estuarine channel. The estuary is characterized as mesotidal with tidal range increasing upstream from 2.9 m at the mouth (km 112) to 4.2 m at Bremen (km 0). The mean water depth along the navigational channel increases from approximately 6 m at Bremen to up to 22 m at the Outer Weser ( Figure 2b). The channelized fairway, with an almost constant width of 200 m between Bremen (km 0) and the port of Brake (km 39), there abruptly expands to a width of approximately 300 m.
The sedimentology of the channel bed is characterized by medium to coarse grained sands with locally high amounts of cohesive sediments ( Figure 2a). Grain sizes are variable along the navigational channel with lowest values to be found in the estuarine turbidity maximum zone (ETM) from km 56 to 73. The cohesive sediment content in the main channel is largely below 25%, except for local peaks of 50-75% along the port of Bremen (km 5-8) and at the estuarine turbidity maximum.
Intertidal flats flanking the channel at the mouth of the estuary are mainly composed of fine to very fine grained sands. Statistics of freshwater discharge for the years 1941-2017, that is, mean of the observed low discharges (MLQ), annual mean (MQ), and mean of the observed high discharges (MHQ), measured at Intschede at the riverine part upstream of the tidal barrier are 123, 319, and 1,220 m 3 /s, respectively.

| Evaluation of tidal dune characteristics
Regular monitoring of the navigational channel and maintenance dredging are carried out by the local authorities (Waterways and Shipping Authorities of Bremen and Bremerhaven). High-resolution (2 Â 2 m) multibeam bathymetric data (see example Figure 1c) was processed by filtering and zero-upcrossing methods . Dune parameters are calculated for spatial dune data segmented in along channel beds with lengths of 500 m and crosschannel widths of approximately 200 m depending on spatial coverage of the measurements. Dune dimensions, asymmetry, and lee-side angles were determined for two characteristic morphological states in The dune length is here defined as the distance between two consecutive troughs. The dune height was calculated as the distance of the crest from a straight line connecting two troughs, determined as a vertical line through the crest. Bedform asymmetry is determined as the length of the ebb stoss side (facing the ebb stream) to the total length of the dune (A d > 0.5 ebb-directed asymmetry and A d < 0.5 flood-directed asymmetry). The hydraulically relevant steep segment of the dune lee slope, that is, the slip face , is here determined as the maximum dune lee-side angle for flow directions during ebb and flood, respectively. Slip-face angles of 11 are considered an onset for initiation of flow separation, while fully developed and persistent flow separation and maximum hydraulic flow resistance is expected for lee-side angles steeper than 24 (Kwoll et al., 2016;. In view of the diversity of natural dune shapes, characteristic values for F I G U R E 1 The Weser estuary in the southeastern North Sea, Germany (a). The model from the tidal barrier at Bremen-Hemelingen to the mouth of the estuary (b). Visualization of typical tidal dunes (c) measured by high-resolution (2 Â 2 m) bathymetrical surveys of the waterways and shipping authorities of Bremen and Bremerhaven. Water level gauges are depicted by dots with names and Weser km [Color figure can be viewed at wileyonlinelibrary.com] these maximum angles are determined as follows. First, the maximum angle is obtained from along channel bed elevation profiles, as it occurs locally along the lee side of each dune. Then, for all dunes found in along-channel segments of 500 m, the 50th (median) and the 90th percentile of these local maximum angles are calculated, that is, per segment 50% (or 90%, respectively) of all dunes obtain a smaller local maximum angle.
Dune dimensions vary along the navigational channel with median dune heights ranging between 0.5 and 2.5 m and median dune lengths between 20 and 100 m (Figure 2c,d). Dune crest orientation was found to be perpendicular to the main flow direction at the channelized cross-section . In the Lower Weser from km 20 to 55, large tidal dunes are identified to be mostly ebb-oriented (asymmetry > 0.5) with gentle stoss sides facing towards the ebb current and steep lee sides particularly after high discharge events as, for example, seen for spring 2009 (Figure 2e). Although many dunes at this reach are ebb-oriented, also symmetrical (asymmetry $0.5) and even flood-oriented (asymmetry < 0.5) dunes are observed between km 10 and 33 in fall after times of low discharge ranging between 100 and 200 m 3 /s in summer. Downstream of km 55, tidal dunes are found to be symmetrical to partially flood-oriented irrespective of the discharge regime. Their maximum lee-side angles are gentle and exceed median values of 11 or 90th percentile ranks of 15 only sporadically, and thus, dune-induced form roughness is assumed to be hydraulically not relevant in this study (Figure 2f,g). At the upper estuary between km 20 and 55, however, these thresholds are exceeded, in particular during ebb tidal phases. Maximum lee-side angles mostly between 11 and 15 suppose flow separation to be initiated for 50% of the dunes (50th percentile). For 10% of the dunes (90th percentile), maximum lee-side angles exceed 15 and are locally up to 25 . Here, flow separation, turbulence, and, thus, form drag are supposed to be more frequent and well developed. Furthermore, these tidal dunes are found to change their morphological shape and asymmetry with seasonal variations of discharge regime (km 10-55).
The variability and extent of these bedforms motivated us to this idealized study that aims to evaluate the effect of different dune shapes and roughness parameterizations on tidal asymmetry.
Findings, here derived from data limited to two morphological states in spring and fall 2009, are supported by the analysis and interpretation of extensive spatiotemporal data of tidal dunes in the Weser estuary .

| Model setup
The modeling system Delft3D (Deltares, 2014) has been applied to set up and run process-based models in a baroclinic, threedimensional (3D) configuration to simulate hydrodynamics and F I G U R E 2 Observed surface sediment grain-size characteristics (a) and channel water depth (b) along the navigational channel from Bremen (km 0) to the mouth of the estuary (km 112). Measured dune dimension and geometry: Median dune height H d (c), median dune length L d (d), median dune asymmetry A d (e), 50th-percentile maximum dune lee slope (f) and 90th-percentile maximum dune lee slope (g) in spring 2009 after high discharge and fall 2009 after months of low-discharge regime. Data were not available for sections depicted in gray [Color figure can be viewed at wileyonlinelibrary.com] sediment transport in the Weser estuary. The modeling system solves the horizontal momentum equations, the continuity equation, and the transport equation on a staggered model grid by use of an implicit finite-difference scheme. The evolution of turbulent flow is simulated by the application of the k-ϵ turbulence closure model. For a detailed description of the equations and implementation into Delft3D, the reader is referred to Lesser et al. (2004) or the documentation distributed with the modeling system (Deltares, 2014).
The Weser model covers the estuary from the tidal barrier at Bremen-Hemelingen and extends seawards to coastal water depths of 25 m (Figure 1b). Here, water level boundary conditions were prescribed by nesting into the larger process-based EasyGSH-model covering the North Sea (Hagen et al., 2019). The Weser model is structured by a curvilinear grid that was set up with an increased reso-  Samples of sediment grainsize composition (Milbradt et al., 2015;Valerius et al., 2015) available in 12 sediment grain-size classes from cohesive sediments to very coarse sand of 2,000 μm were used to determine an initial sediment distribution of the Weser model. Samples were collected at channel cross-sections that each were distanced by 250 m downstream. The sampling density was lower on intertidal flats at the estuarine mouth. The original sediment distributions of 12 sediment classes were clustered to five sand classes and one cohesive sediment class. Considering the area of the Lower Weser estuarine channel (km 0-70), mass-weighted mean grain sizes were determined for individual sand classes, that is, 91, 182, 302, 501, and 1,232 μm. The sand fraction with a mean grain size determined to be 501 μm, for example, was merged from two original sand classes: (1) 354-500 μm with mean grain size of 427 μm and relative total mass of 9.75% and (2) 501-707 μm with mean grain size of 604 μm and relative total mass of 7.06% (calculated as: 0.0975/ (0.0975 + 0.0706) Â 427 μm + 0.0706/(0.0975 + 0.0706) Â 604 μm = 501.3 μm). Its mass is thereby determined as the total mass of the original classes merged into this new fraction. The distribution of each sediment fraction is accounted for as mass percentages at bed cells in the model. Van Rijn's (2007b) transport formula, as implemented in the modeling system, is applicable only for sand fractions ≥100 μm. For this reason, the smallest determined sand fraction with a mean grain size of 91 μm is prescribed in the model as 100 μm. Percentiles from grain size distribution, that is, D50 and D90 (Figure 2a), were calculated from composition and spatial distribution of five sand fractions incorporated to the model. The availability of a specific grain size fraction, that is, mass percentage at a grid cell, is accounted for by a factor that controls the sediment transport rate. Although the transport formula applied ( Van Rijn, 2007b) computes suspended and bed load sediment transport, only the sediment part that is transported by bed load processes is considered for the subsequent analysis. Our study is on time-averaged residual bed load sediment fluxes that were shown to especially contribute to dune morphodynamics and migration (Naqshband et al., 2014). The feedback between high suspended sediment concentrations and the effective hydraulic drag in estuaries (Winterwerp, Lely, & He, 2009) and the effect of dune morphology on dynamics of very fine sediments (e.g., Kwoll et al., 2013) or vice versa, that is, the contribution of suspended sand to dune morphology (Hendershot et al., 2016;Kostaschuk & Best, 2005), are neglected in this study.
For the definition of hotstart conditions, preceding simulations were performed allowing for a sufficiently long spin-up to establish realistic horizontal salinity gradients along the estuarine reach and to smooth the initially prescribed sediment distribution. These salinity gradients and spatial sediment distributions were used to initiate subsequent simulations to be evaluated.
A plausibility test was conducted to verify whether the uncalibrated model meets expectations. Water levels and salinity concentrations were simulated from April 2 to May 2, 2009, and compared with observations at 13 water level gauges and at 10 salinity gauges maintained by the local authorities (see Figure 1 for the position of water level gauges). Salinity concentrations are surveyed at water depths of approximately 1 m below mean low water level. Root mean square errors (RMSEs) between measured and modeled water levels are 0.1 m at the mouth of the estuary (km 115), <0.25 m at the outer, and <0.15 m at the inner estuary. Simulated salinity concentrations are compared with observations by RMSEs <2.2 psu at the outer estuary, <0.5 psu at the spatial range of the estuarine turbidity maximum, and <0.1 psu upstream from there. For this plausibility test, measured freshwater discharge was imposed, and bed roughness coefficients parameterizing the frictional drag of symmetric dunes (Sections 3.3 and 3.4) were prescribed. The spatiotemporal variability of water levels and salinity gradients was assumed to be sufficiently represented by the model. Model quality was thus accepted to be good enough for the subsequent parameter study. The focus here is on along-channel residual bed load sediment transport (RBLT). Fluxes of RBLT (m 3 /s/m) integrate the mass of all sand fractions (specific and dry bed density of 2,650 and 1,600 kg/ m 3 ) being transported during 14 tidal cycles and were determined along grid cells between five curvilinear grid lines following the largely channelized bed of the fairway. These slightly diverging grid lines were purposely aligned to cover the deeper cross-sectional part of the channel bed that itself keeps widening in downstream direction.

| Scenario simulations
Note that the residual sediment transport (m 3 /s/m) integrates the flux of the sediment mass per unit width, irrespective of the crosssectional area that was taken into account for spatiotemporal integration. Sediment fluxes perpendicular to the channel orientation, for example, due to secondary currents in river bends, were not taken into consideration when computing along-channel RBLT.

| Different dune shape and roughness parameterizations
Tidal dunes in the Weser estuary, with typical heights ranging between 1 and 2 m and lengths between 40 and 70 m, are those primary bedforms that retain their asymmetric shape and orientation throughout tidal cycles and even on time scales of weeks and months (e.g., Lefebvre et al., 2020). For reversing tidal flow, form drag is thus assumed to be only effective when the flow is in line with the dune asymmetry, that is, converging on the gentle stoss side and diverging on the dune at the steep lee side. Ripple and megaripples, which are often superimposed to large tidal dunes, are assumed here to be hydraulically active independent of the tidal flow direction (Section 3.6).
In nature, different dune shapes are observed in the main channel, predominantly symmetric dunes in the outer part and ebb-oriented, symmetric, or flood-oriented in the inner part of the estuary ( Figure 2e). For the sake of simplicity and to better distinguish the effects of individual dune shapes on tidal asymmetry and bed load sediment fluxes along the estuary, we refrain from schematizing mixed dune shapes for one and the same scenario simulation. Each simulation represents one distinct dune shape, that is, dune roughness parameterization.
T A B L E 1 Twelve scenario simulations were conducted with four different dune roughness parameterizations (Sections 3.3 to 3.4) and three different discharges. This overview indicates where outputs of these 12 simulations are presented No dunes, that is, dune roughness deactivated Ebb-oriented asymmetrical dunes, that is, dune roughness activated for ebb phases Symmetrical dunes, that is, dune roughness activated for ebb and flood phases Flood-oriented asymmetrical dunes, that is, dune roughness activated for flood phases

| Dune roughness height prediction
Dune morphology, and how it varies in time and space, has strong effects on bedform roughness height. After Warmink et al. (2013), bedform roughness predictors can generally be classified into analytical, semianalytical, and empirical roughness approaches: analytical predictors (e.g., Yalin, 1964) are directly based on the mass and momentum conservation laws; semianalytical roughness predictors (e.g., Karim, 1999) on the conservation laws, but are calibrated to fit data from flume experiments; and empirical roughness predictors (van Rijn, 1984(van Rijn, , 2007aVanoni & Hwang, 1967) rely on empirical relations between bedform and flow characteristics and measured bed roughness. van Rijn (2007a) bedform roughness predictor as applied here directly determines the ripple-, megaripple-and dune-induced roughness coefficients governed by dynamically simulated water depth, flow velocity, and median grain size. Van Rijn's bedform roughness predictor (2007) and particularly the dune roughness predictor have primarily been developed for steady, riverine conditions. The scheme has already been applied in large-scale coastal area models and appears to be robust (Davies & Robins, 2017;Herrling et al., 2017;Villaret et al., 2011Villaret et al., , 2013Wang et al., 2016). A recent study  explored the parameterization of bedform hydraulic roughness in a model (Delft3D) of the Weser estuary. Wellestablished bedform roughness predictors of van Rijn (2007avan Rijn ( , 1984 were related to observations of dune dimension evaluated from highresolution bathymetrical surveys of a dune field at the upper Weser estuary. The dune roughness prediction after van Rijn (2007a) gave a satisfactory representation of the bed friction, while its predecessor (van Rijn, 1984) underestimated dune-induced roughness. Both predictors had been evaluated following van Rijn's expectation that the dune roughness height (Nikuradse, k s ) should be "on the order of half the dune height." In the present study, van Rijn's (2007a)  To account for the effect of different dune roughness parameterizations between km 10 and 55, the dynamic dune roughness prediction is either activated or deactivated for particular tidal phases (Sections 3.3, 3.4 and Figure 4). The computation of depth-averaged flow velocity and instantaneous bed load sediment transport is shown to be dependent on this inter-tidal-phase variable dune roughness ( Figures A3 and A4, see appendix).

| Stationary bedform roughness induced by ripples
The primary tidal dunes in the Weser estuary retain their shape and asymmetry for periods longer than a tidal cycle. Secondary bedforms, that is, ripples and megaripples partially superimposed to the large compound (tidal) dunes, however, reverse in response to the changing flow direction (e.g., Ernstsen et al., 2011;Lefebvre et al., 2011). These ripples morphologically adapt by redistributing sediment during each tidal phase. We thus assume that ripples and megaripples exert a frictional influence on the flow most of the time during the tidal cycle, since flow energy may be dissipated both during (i) the morphological adaptation when the ripples reverse and (ii) through turbulence as the flow recirculates at the bedform lee side once the ripple is fully reversed and hydraulically effective. Based on this assumption, a simulation with ripple and megaripple bedform roughness prediction (van Rijn, 2007a) was performed from neap to spring tide during 14 tidal cycles to compute the spatiotemporal variation of ripple roughness coefficients considering no bedform hysteresis but equilibrium conditions. By temporal averaging of predicted ripple and megaripple roughness coefficients, a stationary but spatially variable bed roughness was generated. It was then applied as a "stationary ripple roughness" for subsequent simulations in the entire model domain, except for the channelized bed between km 10 and 55 where dune roughness was imposed ( Figure 4). Here, ripple-, megaripple-, and dune-induced roughness were predicted dynamically by applying van Rijn's (2007a) empirical formulae dependent on simulated hydrodynamic parameters and sediment characteristics (Section 3.5).
It is noted that Nikuradse roughness heights (k s ) of ripples and megaripples are in the order of a few centimeters and up to a decimeter, respectively. The value of dune-induced roughness is higher than that of ripples and megaripples, in the order of several decimeters, up to the scale of a meter ( van Rijn, 2007a).

| RESULTS
Dune shape and roughness significantly influence estuarine tidal asymmetry. Small-and large-scale estuarine hydrodynamics and sediment transport are here shown as a proxy for tidal asymmetry. The application of either flood-or ebb-oriented dune roughness parameterizations (FAD or EAD) reveals higher tidal ranges in comparison with symmetrical dunes (SD) (Figure 5a). Overall, the tidal range is highest without the application of dune roughness ("no dunes," ND), that is, bed friction composed of grain, ripple, and megaripple roughness only. Tidal low water is approximately at the same level for ND and FAD, but lower in relation to the level of SD and EAD.
For tidal high water, however, SD and FAD have a similar level and are relatively lower than the level of ND and EAD.
Compared with symmetrical dunes, the tidal wave propagates faster up-estuary during flood when imposing ebb-oriented dunes, that is, dune roughness only active during ebb (Figure 5a). Hence, peak flood velocity exceeds peak ebb velocity with a shorter flood than ebb phase (Figure 5b). Likewise, instantaneous bed load F I G U R E 4 Combined bed roughness (Nikuradse, k s ) applied for scenario simulations considering "no dunes" (a), "ebb-oriented asymmetrical" (b), "symmetrical" (c), or "flood-oriented asymmetrical dunes" (d) along the estuarine channel exemplarily shown for two tidal cycles at mid neapspring period and discharge of 450 m 3 /s. Horizontal white lines indicate instants of slack tide when dune roughness prediction (km 10-55) was activated or deactivated in between tidal phases for asymmetrical dunes (cf. b and d) [Color figure can be viewed at wileyonlinelibrary.com] sediment transport shows higher peak values during flood than during ebb ( Figure 5c).
The opposite results from flood-oriented dunes. The water level rises as fast as for the roughness parameterization of symmetrical dunes, but decreases much faster during ebb (Figure 5a). While symmetrical dunes cause peak flow velocity and peak bed load transport to be similar during ebb and flood, flood-oriented dune roughness causes ebb-dominant peak values. This is because roughness is not active during the ebb phase (Figure 5b,c).
The model run with dune roughness excluded (ND) reveals maximal peak velocity and maximal peak sediment transport of bed material among all scenarios, in particular during the flood.
The residual bed load transport (RBLT) is calculated from the instantaneous bed load flux at intervals of 10 minutes between consecutive low water slack tides of the aforementioned tidal cycle (compare Figure 5c). RBLTs of 5.4 Â 10 À5 , 1.9 Â 10 À5 , 5.7 Â 10 À5 , and

| Along-estuary tidal asymmetry in response to discharge, tidal range, and dune roughness
It can be shown that tidal asymmetry is not uniform along the estuary.
Its orientation and strength depend on variable boundary conditions such as tidal energy and freshwater discharge and interact with the frictional effect of bedforms. Normalized residual bed load sediment transport (RBLT) integrated over 14 tidal cycles between neap and spring tides qualitatively shows spatiotemporal effects on tidal asymmetry ( Figure 6). A strong variation in RBLT between neap and spring tide (cf. shaded colors spanning an area in Figures 6 and 7) indicates an important influence of tidal range. Likewise, the effect of discharge is evaluated to be important when the variability of RBLT between different discharge regimes is high (Figure 6a Tidal asymmetry along section IV (km 41-55) is relatively symmetrical at low discharge, and gets moderately ebb-dominant with F I G U R E 6 Tidal asymmetry quantified by residual bed load sediment transport (RBLT) in response to discharge regimes of (a) 150, (b) 450, and (c) 750 m 3 /s, tidal range, and the interaction with distinct dune roughness. Note that positive values reveal ebb dominance.
Values are normalized to the spring tide RBLT computed for flood-oriented dune parameterization at a discharge of 750 m 3 /s at km 39 (cf. Figure 6c); values > 1 are not presented in the figure. Shaded color patches show the range of RBLT between neap and spring tide conditions. The spatial extent of sections I-VI are depicted by gray rectangles (cf. Figure 6b) [Color figure can be viewed at wileyonlinelibrary.com] F I G U R E 7 This is zoomed in on section III and IV (cf. Figure 6b) to highlight the effect of mean, neap, and spring tidal energy conditions on RBLT for discharge of 450 m 3 /s [Color figure can be viewed at wileyonlinelibrary.com] increasing discharge. In analogy to section III, tidally asymmetric dune roughness schematizations EAD and FAD represent the smallest and largest RBLT, respectively.
Section V (km 56-72) covers the estuarine turbidity zone where dune roughness prediction was prevented and only bedform roughness induced by ripples is effective. Tidal asymmetry is generally well balanced: RBLT is small for all simulations and shows the tendency to change from flood to ebb dominance with increasing discharges.
Like in section V, dune roughness was not prescribed at the most downstream section VI (km 73-110). RBLT shows flood dominance; however, there is a tendency to ebb dominance from km 105 to 110 for smaller discharge rates. The effect of tidal range on tidal asymmetry is shown to be strong in section VI.

| Inter-tidal-phase variation of dune roughness
Large tidal bedforms, that is, dunes and sand waves, are shaped by persistent flow conditions, such that their asymmetry is usually assumed to represent the dominant and residual sediment transport direction (e.g., Barnard et al., 2013;Knaapen, 2005). Large tidal dunes in the Weser estuary are quasi morphostatic in view of their hydraulic flow resistance; their overall asymmetric shape and orientation do not reverse with tidal flow direction but are expected to migrate in the dominant flow direction (Nasner, 1977). On timescales of a hydraulic year, however, reversal of tidal dune asymmetry is governed by discharge seasonality, in particular in the upper reach of the Weser estuary ( Figure 2e) . Tidal dunes in the upper reach of the flood dominant Elbe estuary also reverse dune asymmetry and migrate seawards during high-discharge conditions (Zorndt, Wurpts, & Schlurmann, 2011). At the mouth of the Gironde estuary, an opposite dune asymmetry to discharge behavior was observed and explained as a combination of tidal and fluvial changes (Berne et al., 1993).
The present study reveals the overall significance and spatiotemporal effect of dune shape asymmetry on tidal asymmetry. Simulations with bed roughness coefficients parameterizing the hydraulic flow resistance of symmetric and asymmetric dune shapes show substantial small-and large-scale effects on water levels, current velocity, and bed load sediment transport along the estuarine channel. A new aspect of this study is to consider the dependence of asymmetric dune-induced drag. We tested the effect of dune-induced roughness that is hydraulically effective solely in one tidal flow direction for asymmetrical dune shapes (Kwoll et al., 2014;Lefebvre et al., 2013).
A number of studies explored flow resistance of angle-of-repose bedforms, that is, asymmetrical geometries with steep lee-side angles of at least 30 (Best & Kostaschuk, 2002;Kwoll et al., 2016;Lefebvre et al., 2014;Motamedi et al., 2013;Paarlberg et al., 2007). The geometry of natural bedforms and dunes For this scenario of low discharge (150 m 3 /s) and bed roughness parameterization of ebb-oriented dunes (Figure 6 a, blue line)-both being representative for the situation in early summer after high discharges in spring-our idealized model predicts flood-directed net bed load fluxes at the upper estuary, which seems counterintuitive at first given the dunes' ebb-orientation. The residual sediment transport as a response to the tidal asymmetry is hence interpreted as showing the imbalance of the prescribed boundary conditions, that is, lowdischarge regime and ebb-oriented dune shape. The computation of flood-directed residual sediment transport caused by a flooddominant flow regime may thus be interpreted as the system seeking to change from prescribed ebb-oriented dunes to symmetrical or even flood-oriented dune shapes. The here modeled RBLT and particularly its directionality show the response of the estuarine system, notably the dune field, to changes in river discharge aiming to reach a new morphodynamic equilibrium.

| Tidal asymmetry influenced by dune-induced friction
Tidal asymmetry and associated effects on residual sediment transport and morphological evolution of estuaries have been extensively discussed (e.g., Dalrymple et al., 2012;De Swart & Zimmerman, 2009;Friedrichs & Aubrey, 1988). For estuarine systems without a tidal weir, tidal energy tends to increase into the estuary but then decreases toward the upper reaches, following a gradual transition to river-dominated flow and sediment transport at the head of the estuary (Dalrymple et al., 2012). Factors for a possible distortion of the tidal wave contributing to the spatial variation of tidal asymmetry may be the interplay with the estuarine basin geometry including channel convergence by changes in width (e.g., Dronkers, 1986), the ratio of tidal wavelength to basin length determining propagating or standing tidal wave characteristics (e.g., Hunt, 1964), areas of intertidal storage influencing tidal propagation speed (e.g., Friedrichs & Aubrey, 1988), density currents (e.g., Burchard et al., 2018), or bed friction owing to the spatially and temporally varying character of sediments, vegetation, or bedforms. In estuaries, many of these factors independently cause flood dominance, that is, shorter rising than falling tides associated with higher flood than ebb velocities, often attributed to the generation of higher-frequency overtides, such as M4/M2 interaction.
This flood dominance does not necessarily imply net flood-directed sediment transport, since the dominant direction of RBLT is not only determined by flow intensity but also by the effective duration of transport critical bed shear stress. The peak velocity and the peak bed load transport are higher during the flood than during the ebb for EAD, ND, and SD, respectively, which commonly implies flood dominance. However, the tidal asymmetry determined on the basis of residual bed load fluxes is clearly ebb-dominant for all dune roughness scenarios and discharge of 450 m 3 /s (Section 4).
Moreover, other interacting mechanisms can enhance ebb dominance such as Stokes drift return flow (e.g., Dronkers, 1986Dronkers, , 2005Van der Wegen et al., 2008) or river flow (e.g., Guo et al., 2015b), both promoting ebb-directed net sediment fluxes. Variations in river discharge affect estuarine morphodynamics by supplying sediment, enhancing ebb currents, and dampening flood currents, but foremost through the nonlinear interaction with the propagating tidal wave (Guo et al., 2016;Zhang et al., 2016). Zhang et al. (2016) proposed that the tidal asymmetry is one important "degree of freedom" to allow the estuarine system to seek a state of minimum work by adjusting tidal wave distortion under the influence of varying freshwater discharge. The Weser tidal dunes change shape and asymmetry with varying discharge regime and contribute to the distortion of the tidal wave by inter-tidal-phase varying hydraulic drag. Guo et al. (2016) decomposed the overall tidal asymmetry to identify distinct contributions of riverine discharge, tidal energy, and river-tide interaction, that is, the nonlinear modulation effects of fluvial discharge on tides but without considering the changing patterns of density-driven circulation. The authors further concluded that the flow asymmetry induced by the complex river-tide interaction could be more efficient in favoring ebb-directed residual sediment transport in comparison with the solely river-reinforced residual currents (Guo et al., 2015a). We suspect that the contribution of river-tide interaction decomposed by the authors may comprise the frictional effect of dunes on tidal flow. Our

| Spatiotemporally varying bedform roughness in estuarine and coastal area models
The frictional influence of the seabed on the flow in tidal environments and, hence, on sediment transport rates and morphological development is usually schematized by a bed roughness coefficient.
Nowadays, it is still common practice to use bed roughness as a general parameter for fine-tuning of numerical models. Adjusting bed roughness coefficients spatially along estuarine channels by calibrating computed water levels and gradients utilizing observations at tidal gauges has been widely used, but neglects inherent physical processes and causes. In practical applications, bed roughness coefficients have often been assigned to vary with water depth to achieve good calibration results (e.g., Cheng et al., 1993;Elias & Hansen, 2013). Common bedform roughness predictors (e.g. van Rijn, 1984van Rijn, , 2007a may allow for a more realistic, spatially and temporally varying description of the seabed roughness, but still do not account for the bedform alignment with respect to the tidally varying flow direction. There are only a few studies in which the validity of these predictors has been verified by means of in situ measurements. Davies and Robins (2017)  Additional studies on contrasting tidal environments are necessary before arguing a general recommendation to incorporate time-and space-dependent aspects into the calculation of bedform roughness in coastal-scale numerical models (Brakenhoff et al., 2020a;Brakenhoff et al., 2020b). It is shown to be crucial, however, for coastal settings with large tidal dunes, which preserve their geometry and shape because of an extended morphological response time that is much longer than the semidiurnal tidal cycle. Our investigation denotes an important step in quantifying the influence of asymmetrical dune shapes, which create unidirectional friction, on tidal asymmetry and sediment dynamics at the coastal scale. Further studies need to incorporate the spatiotemporal interaction of bedform roughness directionality with the primary forcing conditions, that is, tides and river discharge, to be fully coupled, allowing investigation of the sensitivity of these factors to long-term morphodynamics. Any such analysis will have to incorporate the spatiotemporal distribution of tidal dune characteristics, obtained from sufficiently long time series of bathymetrical field data (e.g., Krämer et al., 2019;Lefebvre et al., 2020).

| CONCLUSION
The present study discusses the effect of different dune shapes and associated dune roughness on tidal asymmetry, quantified by the direction and magnitude of residual bed load sediment transport.
The frictional effect of asymmetric and symmetric dune shapes was parameterized through the modified application of a dune roughness predictor ( van Rijn, 2007a) in a process-based sediment transport model of the Weser estuary. This new approach focused on tidalphase-dependent bedform roughness induced by dune asymmetry. It was shown how tidal asymmetry depends on dune flow resistance interacting on spatiotemporal scales with the combined influence of discharge and tidal energy. Our idealized model study revealed that shape and alignment of tidal dunes and associated tide-variable hydraulic resistance substantially affects large-scale estuarine hydrodynamics and bed load fluxes in particular at the inner estuary, upstream of the estuarine turbidity maximum.
In the case of ebb-oriented asymmetric dunes, which exert flow resistance during ebb only, the tidal wave propagates faster upestuary compared with the case of symmetric dunes, that is, flow resistance during both ebb and flood. Therefore, at very low river discharge, the presence of ebb-oriented dunes can promote flood dominance at the upper estuary, with higher velocities, shorter tidal phase, and enhanced bed load sediment transport during the flood than during the ebb. For higher-discharge regimes, however, the tidal asymmetry becomes ebb-dominant for ebb-oriented dunes.
At the outer Weser estuary, spatiotemporal observations reveal dunes to have low lee-side angles that exert negligible flow resistance.
There is little response to discharge, and residual sediment transport is largely flood-directed irrespective of the dune roughness that was prescribed only at the upper estuary. The reduction of dune-induced drag on the flow, for example, as a possible effect of outer estuarine channel and dune crest dredging, is supposed to increase flooddirected net bed load fluxes. This may promote the convergence of large-scale net bed load transport patterns at the center of the estuary.
The interaction of dune shape, discharge, and tidal energy is shown to be complex. Contributions of the interacting processes have been discussed. The effect of different dune asymmetries on tidal asymmetry is of a similar importance as influences of seasonal discharge variability on tidal asymmetry. Our study suggests that the nonequilibrium nature of large asymmetric dunes in tidal flow is critical to tidal asymmetry and large-scale residual bed load transport and needs to be addressed through inter-tidal-phase varying bedform roughness in numerical models covering estuarine and coastal environments. Finally, the authors are grateful to both reviewers, the special issue editor, and the chief editor for their careful reading of the manuscript, their valuable comments, and appreciated suggestions. F I G U R E A 4 Instantaneous bed load sediment transport for scenario simulations considering "no dunes" (a), "ebb-oriented asymmetrical" (b), "symmetrical" (c), or "flood-oriented asymmetrical dunes" (d) along the estuarine channel exemplarily shown for two tidal cycles at mid neapspring period and discharge of 450 m 3 /s [Color figure can be viewed at wileyonlinelibrary.com]