The morphod ynamic adaptation of estuaries to sea level fluctuations has been subject of geological studies based on sediment core analysis and qualitative modeling efforts. Limited attention has been paid to understanding bathymetric evolution based on a detailed process level. The current study aims to explore governing morphodynamic processes and timescales by application of a 2D, process-based modeling approach. The starting point of the analysis is an 80 km long and 2.5 km wide basin. Starting from a sandy flat bed, stable channel-shoal patterns emerge within a century under semidiurnal tidal forcing. We impose a gradual rise in sea level (up to 0.67 m per century) and compare the results with a run excluding sea level rise (SLR). Model results show that SLR drowns the basin so that intertidal area disappears. This process generates tidal asymmetry reflected by an emerging M4 tidal constituent. The basin shifts from exporting to importing sediment reflected by shoal patterns migrating in the landward direction. The landward sediment transport remains too limited to compensate for the loss in intertidal area and to restore equilibrium within a millennial time scale. Further sensitivity tests on initial bathymetry, tidal amplitude forcing, and rate of SLR show that shallow basins with limited tidal forcing are most vulnerable to SLR.
 Estuaries are valuable areas with a unique ecological environment, as the result of tidally varying water levels and a salt-fresh water interface. From an economical point of view, sectors such as fishery, aquaculture, and tourism profit from the estuarine richness. Numerous ports are situated along estuaries and form the logistical link between ocean transport and the hinterland. The morphology and morphodynamics form an essential part of the estuarine system.
 Estuarine morphology has different spatial scales, ranging from bed forms (~ cm to m), like ripples and dunes, to channel-shoal patterns, vegetated salt marshes and full basins (~ 1–100 km) [Perillo, 1995, Hibma et al., 2004]. Morphological evolution (morphodynamics) takes place by complex interaction of these different spatial scales. Flow through small irregularities may provoke development of major channels, whereas the estuarine geometry determines the allocation of channels and their bed forms [Van der Wegen and Roelvink, 2012]. The associated time scales range from ripple formation during 6 h ebb or flood currents to decadal development of channel-shoal patterns and filling of complete tidal basins over several millennia. Furthermore, the interaction between morphology and ecology (such as bio-films on tidal flats or macro fauna benthos) may be (locally) significant as well. Examples of morphodynamic developments are the evolution of salt marshes [Allen, 2000, Temmerman et al. 2004, 2005], the siltation of access channels to ports [Kirby, 2002], erosion exposing legacy contaminants [Van Geen and Luoma, 1999, Higgins et al. 2007] or anthropogenic influence by land reclamation [Wei et al., 2010] or by hydraulic mining-induced excessive sediment supply [Gilbert, 1917].
 Understanding estuarine morphodynamic behavior is important to anticipate developments related to the impact of human interference (such as dredging operations or breakwater construction), restoration of natural values (for example evolution of salt marshes and tidal flats), natural evolution (i.e., deepening or infilling of tidal basins), or the impact of long-term changes in forcing (such as sea level rise (SLR) or a changing river discharge regime). The latter is the scope of the current research.
Van der Wegen and Roelvink  distinguished two major morphological timescales in estuarine morphodynamics. One timescale (~ decades) is related to the evolution of channel-shoal patterns. The other timescale is related to the longitudinal bed profile development along the basin with a typical duration of centuries to millennia. The major driver of the latter process is tidal asymmetry describing difference of water levels and velocities during ebb and flood.
1.2 Tidal Asymmetry and Equilibrium
 Tidal asymmetry leads to small spatial gradients in tide residual sediment transport which cause morphodynamic development [Schuttelaars and De Swart, 1996, 2000, Hibma et al., 2003a, Van der Wegen et al., 2008]. Friedrichs and Aubrey  distinguished four causes for tidal asymmetry, which are related to the ratio of shoal volume (Vs) to channel volume (Vc) and the ratio of tidal amplitude (a) to the tidally averaged water depth (h).
 Difference in tidal wave propagation () between high and low water of a progressive wave, with the crest propagating faster than the trough. This results in a shorter flood and higher flood velocities compared to ebb velocities. The effect is larger for increasing (a/h).
 More frictional damping and slower tidal propagation during lower water levels due to nonlinear friction [Dronkers, 1986]. The effect will be larger for increasing (a/h).
 Hampering of high water propagation during flood by storage of water on intertidal area, which lengthens flood duration and decreases flood velocities [see also Boon and Byrne, 1981]. The effect will be larger for increasing (Vs/Vc).
 Higher ebb velocities due to smaller cross-sectional area during ebb (see also Boon and Byrne, 1981). The effect will be larger for higher (Vs/Vc).
 Equilibrium conditions in which the effects of (Vs/Vc) and (a/h) are balanced could lead to a “tidal symmetry” and morphological equilibrium. Dronkers  assumed that morphological equilibrium is present when flood and ebb duration are equal and derived the following expression for equilibrium based on an analysis of the Saint Venant's equations:
where h is the width-averaged water depth, a is the width-averaged tidal amplitude, SHW is the basin area at high water, and SLW is the basin area at low water. Wang et al.  applied this assumption for equilibrium conditions in the schematized model setup by Friedrichs and Aubrey  and derived the following expression for equilibrium conditions in terms of (a/h) and (Vs/Vc):
 This equation describes a curve separating the ebb and flood dominant domains allocated by Friedrichs and Aubrey .
 The impact of SLR on estuarine morphology has been studied in a managerial, geological, and engineering type of context. Nicholls et al.  addressed the vulnerability of the coastal zone to SLR in general terms. Scavia et al. , Fitzgerald et al. , and Broex et al.  focused on a qualitative description of potential impacts on estuarine systems and implications for management strategies.
 In a more geological context, sedimentary data analysis focuses on hindcasting long-term trends with emphasis on Holocene developments. A period of rapid SLR during the Holocene resulted in flooding of river valleys and further deepening of existing estuaries. When sea level became relatively constant over the last millennia, infilling of some of these tidal basins took place by marine or fluvial sediment [Long et al., 1996, 1998, Kelley et al., 1995, Chappell, 1993, Van der Spek and Beets, 1992, Beets and Van der Spek, 2000]. However, the analysis of sediment cores mainly reveals depositional trends as the result of basin infilling on a limited amount of locations. Model application could clarify underlying processes of erosion and deposition with a high resolution in time and space.
1.4 Modeling Efforts
Stefanon et al.  explored the morphological impact of sea level fluctuations (decrease and rise) on a tidal network pattern by means of laboratory flume experiments and show rapid network adaptation as the result of varying mean water levels and associated tidal prisms (i.e., retreat (incision) and contraction (expansion) for lower (higher) sea levels). Most modeling efforts, however, have a mathematical origin. De Vriend et al.  distinguished behavior-oriented modeling and process-based modeling. The first approach is based on empirical relations between geomorphic development and hydrodynamic forcing. Van Dongeren and de Vriend , Stive and Wang , and Van Goor et al.  showed that this type of models have capacity to predict decadal morphodynamic development including the impact of SLR. However, these models heavily depend on empirical relationships without explaining their origin and assume that these relationships are constant over time. In contrast, process-based modeling is based on a detailed description of the underlying physical processes. Although this approach requires a higher level of input, the output of process based models provides more detailed information on governing processes.
 In the last decades, process-based modeling techniques have improved considerably [Lesser et al., 2004, Roelvink, 2006, Warner et al., 2008]. With a highly schematized model setup, realistic reproduction of morphological patterns and morphodynamic behavior is possible over timescales of centuries to millennia [Hibma et al. 2003b, 2004, Van der Wegen and Roelvink, 2008, Van der Wegen et al., 2008, Gelleynse et al., 2010]. Others described successful modeling efforts of characteristic morphological features such as inlet channel orientation [Dastgheib et al., 2008, Dissanayake et al., 2009], pattern development [Marciano et al., 2005, Van Maanen et al., 2011, Van der Wegen and Roelvink, 2012], and the relationship between tidal prism and cross-sectional area [Van der Wegen et al., 2010a]. Ganju et al.  with ROMS and Van der Wegen et al., [2010b, 2011] with Delft3D showed that a process-based modeling approach can lead to significant skill in predicting decadal morphological development in a complex estuarine environment.
 With respect to SLR in tidal basins, earlier studies focus mainly on a hydrodynamic analysis. For example, by extrapolating the basins hypsometry and simply adding (instantaneously) 40 cm on the existing mean sea level (MSL), Friedrichs et al.  mimicked the impact of SLR on several estuaries along the USA coastline. In this way, they found that the reaction of a basin on SLR strongly depends on the shape of the cross-sectional geometry. Based on this hydrodynamic analysis Friedrichs et al.  indicated whether or not the estuary will shift from importing to exporting (or the other way around) as the result of a higher MSL. Wolanski and Chappell  performed a similar analysis for several Australian estuaries. An acknowledged deficit in these approaches is that morphological adaptation during the period of SLR is not taken into account.
Ganju and Schoellhamer  and Dissanayake et al.  were amongst the first to address the morphodynamic impact of SLR inlet systems by means of a numerical model (ROMS and Delft3D, respectively) that includes a direct feedback between hydrodynamic and morphodynamic developments during a period of SLR (SLR). Ganju and Schoellhamer  focused on a case study with a realistic bathymetry including complex processes like density currents, sand and mud transport, and wind waves. They evaluated 20 years of modeled morphodynamic development and compared three different climate change scenarios with a base case of constant forcing conditions. They stressed that added value of the work was comparing the differences between the three scenarios with the base case. The value of the base case prediction itself remained too uncertain for strong conclusions. Similar to the methodology applied in the current research Dissanayake et al.  used a more schematized model setup to avoid uncertainty as the result of a complex model definition. Dissanayake et al.  focused on an analysis of the 100 years change of morphological tidal network elements (ebb tidal delta, channel volume, and flat volume) under two SLR scenarios mimicking an inlet geometry of the Dutch Waddenzee. Dissanayake et al.  concluded that SLR enhances flood dominance and sediment import into the basin. Both Ganju and Schoellhamer  and Dissanayake et al.  concluded that intertidal areas would drown under realistic SLR scenarios.
1.5 Aim and Methodology
 The aim of the current research is to investigate the impact of SLR on the morphodynamic behavior in an elongated tidal basin by means of a 2D process-based modeling approach. The proposed methodology acts as a virtual laboratory. Once the model created realistic, stable patterns, we impose a gradual rise in sea level at the boundary. The schematized model setup allows for a systematic analysis of the SLR impact, whereas the process-based approach allows for a close analysis of governing processes responsible for the observed developments.
 The current work differs from the study by Dissanayake et al.  in the sense that a longer time span is considered (up to 2400 years) in an elongated embayment instead of a lagoon type of environment and that the sensitivity analysis is extended to the tidal difference and the initial bathymetry. Also, this study includes a closer analysis of processes governing the observed morphodynamic evolution, since the applied elongated geometry is especially suitable for a detailed investigation of tidal asymmetry characteristics along the basin. The analysis focuses on the M2 tidal forcing and the generation of tidal asymmetry in the model domain in terms of the M4 overtide. The diurnal M2 tidal component is a major tidal constituent in coastal environments around the world, whereas the M4 overtide is an important higher harmonic generated in estuarine systems [Friedrichs et al., 1990]. The indicators discussed in the current study may act as a guideline for analyzing SLR impact in more realistic estuarine environments in future studies.
2 Model Setup
 This work builds further on research by Van der Wegen and Roelvink  and Van der Wegen et al. . They showed long-term morphodynamic development in a 80 km long and 2.5 km wide basin and describe in detail the model setup, as well as an analysis of the hydrodynamic and morphodynamic processes taking place during 3200 years of bathymetric evolution. For reasons of clarity, the following paragraphs present a short summary of their model setup and finding. Further reference is made to Lesser et al.  for a more detailed description of Delft3D.
2.1 Hydrodynamic and Morphodynamic Model
 The two-dimensional hydrodynamic model is based on a set of shallow water equations. Neglecting the influence of density differences, vertical acceleration terms, wind, and waves, the continuity and the momentum equations read as follows:
where ζ is the water level with respect to datum, h is the water depth, is the depth averaged velocity in x direction, is the depth averaged velocity in y direction, g is the gravitational acceleration, cf is the friction coefficient, n is Manning's coefficient, and ν is the eddy viscosity.
 The Manning's coefficient and eddy viscosity were set constant at values of 0.026 sm−1/3 and 1 m2/s, respectively.
 The velocity field obtained by solving the equation of continuity and the momentum equations is used to calculate the sediment transport field. Use is made of the instantaneous total sediment transport formula developed by Engelund and Hansen  that relates velocity directly and locally to a sediment transport rate and reads as follows:
where S is the sediment transport rate, Sb is the bed sediment transport rate, Ss is the suspended sediment transport rate, U is the magnitude of the depth averaged flow velocity, Δ is the relative density by (ρs − ρw)/ρw, ρs is the sediment density, ρw is the water density, C is the friction parameter defined by , and D50 is the median grain size (200 µm in this study).
 Longitudinal and transverse bed slope effects are taken from research by, respectively, Bagnold  and Ikeda  and Ikeda and Aseada  as presented by Van Rijn . Longitudinal bed slope effects (in the flow direction) are implemented by adjusting the sediment transport rate in proportion to the bed slope in flow direction. Transverse bed slope effects (perpendicular to the flow direction) are accounted for by adding a sediment transport vector perpendicular to the original sediment transport vector. The transverse transport rate is proportional to the bed slope, the sediment transport rate, and a coefficient αbn, with a value of 5 in the current research. Van der Wegen and Roelvink  provide a detailed description of the bed slope effect formulations.
 The following equation represents a balance between the divergence of the sediment transport rate field and the evolution of the bed level corrected for bed porosity:
where ε is the bed porosity (with a value of 0.4), zb is the bed level, Sx is the sediment transport rate in x-direction, and Sy is the sediment transport rate in y-direction.
 The adaptation time scale of morphodynamic developments is much smaller than typical hydrodynamic time scales. In order to enhance the morphodynamic development, we follow Roelvink  and increase the bed level change calculated every hydrodynamic time step by a morphological factor (MF). Roelvink  suggested that this methodology is justified as long as bed level changes do not exceed 5% of the water depth. Sensitivity tests showed that a MF of 200 is justified for this study. For s similar model setup, Van der Wegen and Roelvink  showed that MF values up to 1000 lead to similar patterns compared to runs with lower MF values, although phasing and amplitude differences increase for higher MFs.
 Figure 1 shows the model setup of an 80 km long and 2.5 km wide basin closed at the head, inspired by the dimensions of the Western Scheldt estuary in the Netherlands. The initial bathymetry linearly decays from MSL at the head towards 15 m below MSL at the mouth. The mouth is attached to a 25 m deep basin. Tidal boundary conditions are prescribed along the boundaries of this basin by harmonic forcing with a 1.75 m water level amplitude and equal phase. Boundary conditions at the head and along the basin do not allow transport of water or sediments. Van der Wegen et al. 2008 presented an analysis where bank flooding and erosion of the banks is allowed.
2.2 Previous Modeling Results
 Figure 2 gives an impression of the morphodynamic development after 25, 400, and 3200 years. The development is characterized by two scales [Van der Wegen and Roelvink, 2008]. The first scale is related to local pattern formation with a spatial scale of the basin width and a timescale of decades. These patterns represent a local balance between hydraulic and sediment transport rate adaptation length scales. The second scale is related to the basin length and the development of the longitudinal profile with a timescale of centuries to millennia. Main driving forcing is the continuous interaction between the tidal wave and the bathymetry. The generated overtides lead to sediment redistribution along the basin.
 The first decade pattern formation takes place near the landward end. During following decades, patterns evolve more seaward until the full basin is covered. Multiple channels and shoals remain present near the head, whereas one major channel and an alternating bar develop more seaward. Initially, the tidal wave is damped landward due to the shallow bathymetry. As the basin continuously exports sediment and deepens accordingly, standing wave conditions and resonant behavior develop. This is due to the basin length being approximately 1/4 of the tidal wave length. Initially, the morphodynamic development is fast with a high dissipation of energy (as a proxy for sediment transports and shear stresses). However, after the major patterns developed (after approximately a century), these parameters decay rapidly. Also, on the longer term, the basin shows a decaying trend in energy dissipation, albeit at a decreasingly smaller rate. Energy dissipation becomes more uniformly spread over the basin and over the tidal cycle. Velocity amplitudes during ebb and flood become almost constant along the basin and tide residual transports decay. This means that the morphodynamic process in the basin aims for a state of smaller and slower morphodynamic development.
2.3 Imposing SLR
 The review described in the previous section shows that morphodynamic adaptation time scales may be large and will depend on the initial conditions, the basin length scale, and the forcing conditions. This suggests that the morphology of tidal basins and estuaries observed in reality may seem to be in equilibrium on a decadal timescale, whereas they will show further development when much longer time scales are considered. This means, for example, that tidal basins may be even in a state of “recovery” from stabilization of SLR over the past millennia, albeit that ongoing morphodynamic development will be minor and decreasing compared to previous developments.
 Since tidal basins are subject to long-term evolution with a time scale comparable to the rate in SLR, the impact of SLR may depend on the age of the tidal basin. In order to address this issue, the current study takes a “basic run” covering 2400 years of morphodynamic evolution as a starting point. Subsequently, we impose an 800 year period of SLR on the 200, 800, and 1600 year old bathymetries that were generated by the basic run (referred to as “basic” bathymetries) (see Figure 3). Figure 4 shows the rise in sea level over 800 years imposed at the seaward boundary and described by
where A = 4 m, T = 3160 year, t = morphological time, and C = 4 m. SLR is not imposed abruptly, but the rate of SLR gradually increases to about 0.67 m/century (6.7 mm/yr), with an average of 0.5 m/century. We refer to this forcing as 0.5 m/century SLR. This lies within the range of recent IPCC updates by Allison et al.  who suggest a current (but increasing) SLR of 3.4 mm/yr with an upper level of 2 m by the year 2100.
 The run that is central in the discussion of the modeling results has a 1.75 m tidal amplitude forcing and an average of 0.5 m/century SLR imposed on an 800 year basic bathymetry. In order to assess sensitivity to a different rate in SLR and tidal forcing amplitude, we performed extra runs with a 0.85 m and 3.5 m water level amplitude defined at the seaward boundary as well as SLR rates of 0.25 m and 1 m per century. In summary, we performed 18 runs, of which 3 of 2400 years without SLR with variable tidal amplitude and 15 with 800 years duration (imposing SLR) with varying initial bathymetry, tidal amplitude, and SLR rate. The computational time of one run of 800 years was about 14 days on a 3.0 Ghz, 3.25 Gb RAM PC. The following sections describe the modeling results. If not stated otherwise, the vertical datum is MSL excluding SLR.
3 Model Results
3.1 Morphological Development
 Figure 5a shows the 1600 years evolution of the width-averaged depth profile along the basin under 1.75 m forcing as well as results from imposing 800 years SLR starting from an 800 year basic bathymetry. The crests of the profile coincide with the relatively shallow part in between two alternating bars (see Figure 2), and the deeper parts represent the cross section in which the channel depth is largest just next to the bank.
 Major development takes place during the first 800 years. In the first century, major channel-shoal pattern formation occurs resulting in stable morphological features. Intertidal area develops along the full basin mainly due to local and lateral displacement of sediments (Figure 5c). On the longer term, longitudinal sediment transport gradients caused by overtides result in an ongoing deepening of the basin. The impact of SLR on the mean depth is relatively small (compare, for example, the solid and dashed red lines). Closer analysis in Figure 5b reveals that the crests slowly damp after 800 years but continuously migrate seaward. By including SLR, the seaward hump migration slows down and finally shifts to the landward direction (compare depth profiles after 1000 and 1600 year to 800+200SLR and 800+800SLR in Figure 5b).
 Figure 5c shows the development of intertidal area. The channel-shoal pattern adapts to the relatively slow process of basin deepening. The amount of intertidal area is highest after 100 years. The intertidal area gradually drowns on the longer term, first near the seaward end, and then along the full basin. SLR enhances drowning of intertidal area due to a rising mean water level. Extra deepening of the bed level itself by SLR (compared to the initial MSL), as shown in Figure 5a, is insignificant. This is probably because this basin is deep enough so that the tidal prism does not increase with SLR.
 Due to pattern formation, all basins show a high rise in intertidal area within the first 200 years, after which the intertidal area drops and almost stabilizes (Figure 6). The peaks are most pronounced for the 1.75 m and 3.5 m forcing (after 100–300 years and 100 years, respectively). The basin with 0.85 m forcing shows the peak only after approximately 1500 years. The larger the forcing, the earlier, shorter, and higher the peak, because larger forcing leads to higher transport rates and shorter morphodynamic adaptation time scales. The low peak for 0.85 m forcing is probably due to limited intertidal area development in the seaward part. The drop in intertidal area after the peak is due to loss in intertdal area in the seaward direction, see also Figure 5c. Remarkably, the amount of intertidal area after 2400 years does not differ much for the three tidal forcing scenarios.
 For all basins, intertidal area decreases due to SLR. This effect is most pronounced for the basins with low (0.85 m) tidal forcing or high SLR rate (1 m/century), in which the intertidal area almost disappears. The larger the ratio of tidal forcing to SLR rate, the less sensitive the basin intertidal area is to SLR. The age of the basin seems less important. For example, for similar forcing, the intertidal area left after 800 years of SLR is almost equal for cases starting from the 200, 800, and 1600 year bathymetries, although the amount of intertidal area that disappeares as the result of SLR can vary considerably.
3.2 Sediment Transport
 Under constant forcing, all basins show seaward transport (export) of sediment (Figure 7). The larger the tidal forcing, the larger the sediment export. The basins with 0.85 m and 1.75 m forcing continue to export after 2400 years. The basin with 3.5 m forcing shows major transport during the first 1000 years, followed by a relatively stable period in which hardly any tide residual transport takes place. SLR leads to less export and, in most cases and locations, a shift from export to import. This effect is most pronounced for young basins, midbasin locations, and high SLR rates. Only for the basin with 0.85 m forcing, SLR initially leads to a larger export before turning into less export or import on the longer term (Figure 7b).
3.3 Hydrodynamic Analysis
 The shift from export to import can be expressed in terms of distortion of the M2 harmonic component by the generation of super harmonics (or: overtides), due to friction, the presence of intertidal area, or other processes influencing tidal propagation in the embayment. Since these are the major harmonic components, we focus on an analysis of the M2 and M4 tidal components (referring to M2a and M2p for M2 water level amplitude and phasing, respectively, with the additions of “wl” and “vel” for water level and velocity). The results were obtained after a Fourier analysis of six tidal cycles excluding morphodynamic development on a particular bathymetry in time.
 Figure 8 shows the M4awl and the bathymetry along the basin for different points in time. Submerged shoals generate weak M4awl whereas pronounced shoals generate a high M4awl (compare Figures 8a and 8b). 800 years of SLR results in considerably more M4awl towards the head (compare Figures 8a and 8c) and leads to landward migration of submerged shoals (Figures 8b and 8d between 0 and 35 km) as well as landward orientation of the tips of the shoals (most pronounced at the shoals around 40 and 50 km, Figure 8d).The analysis focuses on width-averaged (W) and bed level (z) corrected values of the tidal harmonic component (Mi) as follows:
in which the over bar indicates the width-averaged value. This implies that that channel values are favored at the expense of shoal values, which seems a reasonable and necessary assumption given the fact that most conveyance of the tidal wave occurs in the channel.
 The initially linearly sloping bathymetry leads to M2awl damping and a strong tidal asymmetry reflected by a large M4awl (Figures 9a, 9c, and 9e). As the basin deepens, the incoming M2 wave is able to protrude into the embayment until it reflects against the head leading to amplification and standing wave conditions in the embayment. 800 years of SLR shows a small decay in M2awl as the result of friction and energy transfer to higher harmonics (solid red line). M4awl is hardly present at the mouth due to the deep and wide area in front of the mouth. On the initial bed, M4awl increases landward, but, as the basin deepens over time, it decays to become almost absent apart from a small section near the shallow head of the basin. However, after 800 years of SLR, M4awl has gained considerable energy again. This is probably due to the loss of intertidal area as we will discuss in the coming paragraphs. Apart from the initial conditions, the velocity amplitudes show a much less smooth profile (Figures 9b, 9d, and 9f) due to the presence of the channel-shoal patterns and a higher variation over the width. Still, general trends resemble the water level profiles with lower (higher) M2avel (M4avel) values in case of SLR.
Friedrichs and Aubrey  systematically investigated the effect of tidal asymmetry on flood or ebb dominance applying the 1D numerical model by Speer and Aubrey . Friedrichs and Aubrey  modeled a short tidal basin with a uniform channel cross section but also allowing for water storage on shoals and suggested the following categorization of tidal phasing for the flood or ebb dominance:
 Flood dominance is defined as the situation in which flood velocities exceed ebb velocities as the result of a shorter flood period. Because of the higher than linear dependency of sediment transport rate to the velocities (see equation (7)), sediment is transported landward. Figure 10 shows water level and velocity phasing for M2 and M4 (a,b,c,d) as well as the relation of model results to the phasing classification by Friedrichs and Aubrey  (e,f). Model results show that all basins are ebb dominant except for a small portion near the head and the basin after 800 years of SLR (compare Figure 7c and Figures 10e and 10f).
 The 1.75 m and 3.5 m basins show a fast increase in basin-averaged M2awl (Figures 11b and 11c). Similar to the development of intertidal area, the larger the tidal forcing, the earlier a peak is reached, after which slowly decays again (see also Figure 9a). The reasons for the decay are not fully clear, but may be attributed to the disappearance of intertidal flat near the head, so that the basin lengthens and becomes less resonant. The remains fairly constant under the 0.85 m forcing, but initially rapidly decays for the other forcing conditions (Figures 11d, 11e, and 11f). Under 3.5 m forcing remains relatively high with some variability (Figure 11f), which seems related to, respectively, a relatively large amount of intertidal area and some variability of intertidal area over time (Figure 6d). Tidal distortion (indicated by /) is highest for the 0.85 m forcing (Figures 11g, 11h, and 11i).
 SLR leads to increasing under 0.85 m forcing (due to development of resonant conditions), but to smaller for the larger forcing conditions (Figures 11a, 11b, and 11c). The latter is probably due to transfer of energy from to and lengthening of the basin because of decay in intertidal area near the head. For all basins, SLR leads to a higher tidal distortion on the long term (Figures 11g, 11h, and 11i). This effect is most pronounced for the 1.75 m forcing. The 0.85 m forcing has less tidal distortion because both and increase. The 3.5 m forcing has less distortion, because less intertidal area disappears (Figure 6c).
3.4 Causes for Tidal Asymmetry
 The following sections evaluate the model results in terms of the equilibrium conditions suggested by Friedrichs and Aubrey , Dronkers , and Wang et al.  represented by equation (2). This equilibrium curve is shown in Figure 12 (in green). The area above and left of the green line is the ebb dominant domain and the area below and right of the green line is the flood dominant domain (at least, for a short tidal basin with uniform cross section).
Van der Wegen et al.  presented a close analysis of the run with 1.75 m forcing without SLR in terms of (Vs/Vc) and (a/h). They observed that the model results compare well with the analytical solution despite the fact that they considered a relatively long basin with a landward decreasing cross section.
 Ebb velocities were continuously higher than flood velocities during the run resulting in an export of sediment. Van der Wegen et al.  attributed this to relatively high pressure gradients under low water conditions and the Stokes' drift-induced return flow. These effects are very limited in a short basin. We observed a 0.35 m water level setup and a (seaward) return flow velocity of 0.1 m/s with the initial, shallow bathymetry, although both values decayed to almost zero after a century.
 Figure 12 shows the results as well as the impact of SLR in terms of the relationship between the ratio of shoal volume (Vs) and channel volume (Vc) and the ratio of tidal amplitude () and the water depth () for different points in time and along the basin. (Vs) is calculated by integrating the maximum volume of water stored above intertidal area over a distance from location (x) to the head. (Vc) is calculated by integrating the channel volume at a local MSL (including tidally averaged water level setup) over a distance from location (x) to the head. () is the water level amplitude averaged over a distance from location (x) to the head (in which the amplitude is half the difference between the lowest and highest width-averaged water level). () is the width-averaged water depth averaged over a distance from location (x) to the head including the tidally averaged water level setup. The ratios (Vs/Vc) and (/) thus cover conditions from the area between location (x) and the head. This seems logical since tidal wave characteristics (or: ebb or flood dominance) will not be determined by local conditions only, but they will be merely a function of interactions of the tide and the geometry in the area between location (x) and the head.
 Model results show fast development towards the analytical solution during the first century. As mentioned before, the fact that sediment export occurs in the flood dominant domain seems contradicting, but is attributed to Stokes' drift-induced return flow and high pressure gradients during ebb flow [Van der Wegen et al., 2008]. Model results are close to the analytical solution without coinciding which is attributed to the different length of the basin and the different behavior of the tidal wave. Model results along the basin follow the curve of the analytical solution. When, after a century, channel-shoal patterns have evolved enough, the evolution at the mouth (Figure 12b) and midbasin (Figure 12c) is almost parallel to the analytical solution. Model results do not lead to a formal equilibrium (compare for example the results after 100, 400, and 800), although development rates decay considerably over time. SLR leads to a pronounced development. At the mouth, this development is parallel to the analytical solution, whereas midbasin the trend seems slightly more towards the flood dominant domain. This latter observation can be related to the clear shift from export to import at the midbasin location (see Figure 7b).
 Figure 13 shows the evolution of the different components applied in Figure 12 for values at the mouth. The mean water level amplitude hardly differs with SLR. The basin mean water depth increases with a similar amount for all basins with a value of approximately the rise in sea level itself (about 4 m over 800 years). Deepening of the bathymetry itself compared to a fixed datum is insignificant (see also Figure 5a). The basin shoal volume decreases because intertidal area drowns as SLRs. The effect is most dramatic for the basin with 0.85 m forcing in which almost all shoal volume disappears. Channel volume increases under SLR with a similar amount for all basins.
 Table 1 shows the model results from Figure 13 in a slightly different manner, namely as the difference between the runs with and without SLR as percentage of the value at the start of the run (when SLR was imposed). For all basins, the shoal volume (Vs) has highest sensitivity to SLR, followed by channel volume (Vc), basin mean water depth (), and, finally, the basin averaged water level amplitude (). Shoal volume decay has highest impact on the Vs/Vc ratio. Increase in basin mean depth has highest impact on the ratio. The basin with 0.85 forcing shows highest sensitivity to SLR in all parameters, although the basin becomes less sensitive when it gets “older”. This latter effect seems not to be present in the basins with other forcing conditions.
Table 1. Difference in Development After 800 Years With or Without SLR in % of Start Time Value (as in Figure 13)
0.85 m Amplitude
1.75 m Amplitude
3.5 m Amplitude
Basin mean amplitude (, m)
Basin mean depth (, m)
Basin shoal volume (Vs, m3)
Basin channel volume (Vc, m3)
 This analysis does not explicitly clarify the major contributions that lead to the increased tidal asymmetry observed in Figure 9 and Figure 11. A large difference in development with or without SLR of a certain parameter does not automatically mean that the parameter has highest contribution to tidal asymmetry. This will depend on the age of the basin and the location in the basin, in other words, on the location in the () versus (Vs/Vc) graph (Figure 12). Larger distance from the equilibrium line given by equation (1) may imply larger tidal asymmetry. Midbasin, where the shift from export to import is highest, the decrease in (Vs/Vc) is larger than the decrease in () and leads to a subtle but larger deviation from the equilibrium curve. Another explanation for the development of tidal asymmetry in case of SLR is the drowning of the channel-shoal pattern so that less water is conveyed through deep channels and more water (with high velocity) is forced over the shallower parts, which generates overtides as well. This concerns not the increase of channel volume or the deepening of the basin, but rather the two-dimensional redistribution of flow over drowned channel-shoal patterns.
 The numerical experiments in this study show the possible effect of SLR on the morphodynamics of a sandy tidal basin. Intertidal area disappears for realistic SLR rates and the basin shifts from a sediment exporting system towards an importing system. A major advantage of the applied modeling approach is that it allows for a close analysis of the processes governing this shift in transport trend without, for example, depending on empirical relations imposed on the model.
 Constant forcing leads to a balanced tidal propagation in the basin reflected by an equal duration of ebb and flood and very limited tide residual transport and morphodynamic development. SLR drowns the intertidal area so that the tidal propagation balance is disturbed. As a result, tidal asymmetry develops reflected by an emerging M4 overtide and a shift in sediment transport. Sensitivity analysis reveal that this trend holds for a range of SLR rates, tidal forcing conditions and initial bathymetries, with one exception. After imposing SLR on the 200 and 800 year basic bathymetries, the basin with 0.85 m forcing reflects an increasing sediment export at the mouth and midbay before it eventually also starts to import sediment (Figures 7b and 7e). We attribute this to the state of the basin, which shows only little morphodynamic development due to the low tidal forcing and relatively small transports. As a result the basin remains quite shallow so that the landward tidal wave damping is considerable. The water level amplitude hardly increases landward the first 1000 years for the basic run (Figure 11a). In contrast, the basins with 1.75 m and 3.5 m forcing deepen fast enough to develop standing wave conditions (landward increase of the water level amplitude and high velocities and transports at the mouth) already during the first decades (Figure 9 and Figures 11b and 11c). When SLR is imposed on the 0.85 m basin, standing wave conditions develop more rapidly compared to the constant forcing conditions, which leads initially to a landward increase in water level amplitude (Figure 11a), higher velocities and an increase in sediment export (Figures 7a and 7e). On the longer term, when the basins have deepened enough, the export decays and even shifts from an sediment exporting system to import like for the 1.75 m and 3.5 m basins.
 The numerical experiments mimic morphodynamic evolution in a schematized sandy tidal embayment. Van der Wegen and Roelvink  showed that the evolving channel-shoals patterns have a characteristic length scale similar to the patterns in the Western Scheldt estuary, whereas Van der Wegen and Roelvink  suggested that the allocation of the patterns emerging from a flat bed with the Western Scheldt geometry resemble the measured Western Scheldt morphology. The study by Van der Wegen and Roelvink  showed that an extreme water level setup causing a diurnal rise in water level similar to the tidal difference, (limited) river discharge, and 3D density currents do not have a significant effect on the morphological pattern. This suggests that the interaction between major tidal forcing, the relatively coarse sediment class observed in the deepest parts of the channel and the estuarine geometry governs the morphology of a tidal basin, at least for the characteristic decadal timescale in which the major patterns emerge and for Western Scheldt conditions. As argued in previous sections, the longer-term evolution of the schematized, sandy basin is a function of tidal asymmetry and the resulting small spatial gradients of the tide residual sediment transport, but in reality, other processes (such as wind waves, mud, and vegetation growth) may play a significant role as well.
 Swell may enter the tidal basin from the seaward boundary, which, together with locally generated wind waves, will probably erode the edges of shoals and salt marshes [Tonelli et al., 2010] and impact salt marsh growth because enhancement of prevailing shear stresses [Fagherazzi et al., 2006]. Another effect of waves is that littoral drift supplies sediment towards the mouth that may later even be transported into the basin [Van der Wegen et al., 2010a]. However, as long as sediment transported by tidal movement is large compared to wave-induced transport, it is expected that equilibrium conditions will develop.
 The impact of the presence of mud is that it will alter the erodibility of sand [Van Ledden et al., 2004], and accretion of shoals is enhanced since mud tends to settle in low energy areas. Vegetation will decrease velocities on the shoals inducing extra mud trapping eventually transferring mud flats into salt marshes. Kirwan et al. synthesized modeling efforts of estimating salt marsh growth in relation to SLR scenarios where salt marsh accretion rates depend on inundation duration, vegetation growth, sediment transport and concentration, and organic versus mineral content. Although a considerable amount of sandy intertidal area drowns in the current study, Kirwan et al.  suggested that salt marshes may survive moderate SLR scenarios when growth conditions are satisfied.
 Inclusion of mud and vegetation growth in modeling exercises as discussed in the current work would probably lead to more intertidal area. This will hamper propagation of flood and probably increase flood duration stimulating export of sediment (or decreasing sediment import), leading to a different balance and equilibrium profile. However, application of vegetation growth requires revisiting the application of the MF. Vegetation growth depends on the season and parameters such as inundation time during a tidal cycle, so that a time variable MF may be required depending on the varying vegetation growth rate. Furthermore, formulations are required to describe the impact of vegetation on turbulence and velocity profiles and the mud erodibility by the presence of roots and by the supply of organic matter due to starvation.
 A final remark concerns the presence of fresh water river flow and sediment supply. This may lead to enhanced ebb velocities, hampered flood propagation, density currents, and turbidity maxima, each with its own effect on the morphodynamic evolution in the basin. Future research may focus on a thorough sensitivity analysis of realistic ranges of each of the mentioned parameters. Also, more realistic estuarine plan forms may be applied, first generating a stable channel-shoal pattern and second imposing SLR so that the vulnerability of the particular estuary becomes clear.
 The current work presents 2D modeling results mimicking 2400 years of morphodynamic development in an 80 km long tidal basin with and without forcing by SLR. Starting from a flat bathymetry, morphodynamic developments are fast, and stable channel-shoal patterns emerge within a century. After that period, development rates decay, but they remain present, even after 2400 years. Higher tidal amplitude forcing leads to more pronounced channel-shoal patterns, faster development, and larger development decay rates.
 Imposing SLR leads to a significantly different morphodynamic evolution, albeit that adaptation timescales remain low. Basins shift from exporting to importing sediment and shoals start moving landward. Nevertheless, the basins “drown”, and the amount of intertidal area decays considerably. The decay is smaller for higher tidal amplitude forcing. Higher rates of SLR lead to faster decay of intertidal area. Young, relatively shallow basins may experience more tidal resonance after SLR since tidal wave dissipation becomes less for larger water depths. These basins are more sensitive to SLR especially when they are subject to a small tidal forcing amplitude as well.
 Without SLR, basins develop limited tidal asymmetry. However, SLR leads to considerable generation of a M4 tidal constituent within the basin. The decrease in intertidal area and its effect on flood propagation are held responsible for this. M4 generation and its phasing compared to the M2 tide in the model lead to a shift from a net sediment export to a net import, which corresponds to theoretical suggestions by Friedrichs and Aubrey .
 We highly appreciated preliminary discussions with Giovanni Coco (Associate Editor) and Barend van Maanen (NIWA, Un. of Waikato, now Un. of Bordeaux) as well as review comments by Alexander Densmore (Editor), and two anonymous reviewers.