Influence of geometrical variations on morphodynamic equilibria in short tidal basins
Abstract
The existence of cross-sectionally averaged morphodynamic equilibria of tidal inlets is investigated, using a cross-sectionally averaged model, and their sensitivity to variations of geometry, deposition parameter, frictional effects and advective sediment transport is analysed. Different geometries, from exponentially converging to exponentially diverging, are considered for inlets with lengths typical for the Dutch Wadden Sea. Standard continuation techniques are employed to numerically obtain morphodynamic equilibrium solutions, i.e. solutions for which the tidally averaged bed level does not change anymore. It is known that when the water motion at the entrance of the inlet is only forced by a M_{2} tidal constituent assuming the water level to be spatially uniform and only diffusive sediment transport is considered, the morphodynamic bed equilibrium has a constantly sloping profile for a rectangular inlet. We find that the bed profile in equilibrium becomes convex (concave) when we change the frictionless embayment geometry to a diverging (converging) geometry. Upon letting the deposition parameter depend on the depth, a more convex bed profile for all geometries considered is found. Including frictional effects in the momentum equation has a minor effect when only diffusion is considered, but the bed profile changes significantly when advection is included. When the tidal forcing of the sea surface elevation depends on a M_{4} tidal constituent as well, the morphodynamic equilibrium bed varies from very deep to shallow, depending on the relative phase. For a diverging inlet geometry, there are combinations of the relative phase and tidal basin length for which we show the existence of multiple equilibria. This implies that for these geometries, the cross-sectionally averaged bed profile in morphodynamic equilibrium can change significantly when the relative phase or the embayment length is changed. The magnitude of the perturbation necessary to actually evolve towards the other equilibrium and the time scale associated with this change can not be inferred from the analysis presented in this paper.
Keywords
Tidal basin Idealised model Morphodynamic equilibrium Sediment transport Multiple equilibria1 Introduction
A barrier coast consists of several barrier islands with a tidal inlet between these islands connecting one or more back barrier basins to the sea or ocean. This type of coastal feature occurs at approximately 10% of world’s coastline, Glaeser (1978). Large parts of these basins fall dry during a part of the tidal cycle, which makes them important for ecological, economical and recreational purposes. To manage these different interests, it is essential to obtain a better understanding of these systems and their sensitivity to natural and human interference.
Already quite some research has been conducted on this topic. Both the water motion and the morphodynamic evolution in shallow tidal inlet systems have been studied extensively. The linear dynamics of the tidal motion was first studied by Green (1837). It was shown in Parker (1991) and Zimmerman (1981) that nonlinear interactions result in the generation of overtides and residual currents. In the last decades, many analytic solutions have been presented for the tidal motion in a wide range of estuarine geometries, see Friedrichs (2010) for a review. In the studies mentioned above, no feedback to morphology was considered. When taking the interaction of the currents with the erodible bed into account, complex patterns can develop, see De Swart and Zimmerman (2009). A lot of research has been conducted on the sensitivity of these bed patterns to various physical parameters. The influence of sediment supply on morphodynamic bed equilibria was studied by Van der Wegen et al. (2017), Maan et al. (2015) and Robert et al. (2000), the influence of frictional effects by Lanzoni and Seminara (1998, 2002), and Schuttelaars and de Swart (2000), the influence of including mud in the sediment transport by Van Ledden et al. (2004) and the influence of geometry by Lanzoni and Seminara (1998, 2002) and Van Leeuwen et al. (2000).
Although many studies have investigated a similar topic, one has to take care in directly comparing the results, because the studies often use different models and solution methods. For a classification of these different types of morphodynamic models, see De Vriend and Ribberink (1996) and Murray (2013). Here, we restrict ourselves to cross-sectionally averaged idealised process-based models, which are mathematical models based on first physical principles. The equations are simplified such that only those processes are taken into account which, according to a detailed scaling analysis, are important. Note that by averaging the models over the width, observed channel-shoal patterns cannot be reproduced, only cross-sectionally averaged quantities are found. However, a good understanding of cross-sectionally averaged equilibria is essential as the first step in a depth-averaged (linear) stability approach that can be used to unravel the initial formation of channel-shoal patterns and the resulting finite-amplitude patterns (see Dijkstra et al. (2014)). This approach implies that the influence of tidal flats which are shown to be important by Ridderinkhof et al. (2016) and Van Prooijen and Wang (2013) is not parametrically included in the width-averaged model, but only starts to play an important role as tidal flats are formed in a 2DH analysis.
In Van Leeuwen et al. (2000), the authors used such an idealised model to analyse geometric variations of the embayment on the morphodynamic bed profiles. The sea surface elevation was assumed to be spatially uniform and the sediment concentration was given by an advection-diffusion equation. When the sea surface elevation was only forced by a M_{2} tidal constituent and diffusive sediment transport was assumed to be dominant, the authors found that the morphodynamic bed profile becomes more convex as the width convergence increases. When the sea surface elevation was forced by both a M_{2} and a M_{4} tidal constituent and a converging embayment, there was a maximum length for the embayment for which a morphodynamic equilibrium could be found, for a relative phase difference ϕ between M_{2} and M_{4} of ϕ ∈ [0^{∘},180^{∘}]. Letting the deposition parameter depend on the depth, the authors found that the equilibrium bed profiles became more convex.
Instead of fixing the bed at the landward boundary, Lanzoni and Seminara (2002) created an inner boundary condition at the landward side to allow for wetting and drying. Using this approach the authors defined the length of an embayment as the maximum length for which a morphodynamic equilibrium still exists. These results were confirmed by Todeschini et al. (2008). From these studies, it was concluded that the maximum length of the embayment was mainly governed by the convergence length, although frictional effects also influenced the maximum length for weakly convergent embayments.
In Van Leeuwen et al. (2000), and Lanzoni and Seminara (1998, 2002), the morphodynamic equilibria were found by time-integration. A different way of solving a morphodynamic model is to make use of a fixed point seeker. This method was first used in Schuttelaars and de Swart (1996). In their model, the authors neglected effects of waves, density currents, inertia and friction and the sea surface elevation was assumed to be spatially uniform. The authors considered a rectangular embayment with a fixed bed at the seaward and landward boundaries and assumed the deposition parameter to be spatially constant. The authors performed a systematic analysis of the different types of sediment transport. Considering only a prescribed M_{2} tidal forcing at the entrance, a unique morphodynamic equilibrium was found with a constantly sloping bed. When the sea surface elevation at the entrance was forced by both a M_{2} and M_{4} tidal constituent, the bed profile became either convex (0^{∘} < ϕ < 180^{∘}) or concave (180^{∘} < ϕ < 360^{∘}). Schuttelaars and de Swart (2000) extended this analysis to embayments of arbitrary length, including bottom friction and inertia. The existence of multiple stable equilibria was shown for long enough embayments, when the water motion was forced by both a M_{2} tidal constituent and a strong enough M_{4} constituent. These results were confirmed in Hibma et al. (2003) using a numerical simulation model.
In Ter Brake and Schuttelaars (2010), the authors extended the idealised model of Schuttelaars and de Swart (1996) by including a topographically induced transport term in the bed evolution equation. The authors also analysed the influence of different boundary conditions on the bed. The sea surface elevation was again assumed to be uniform and the geometry of the embayment was considered to be rectangular.
The research presented in this paper extends the model of Ter Brake and Schuttelaars (2010) by including inertial and frictional effects in the momentum equation and allowing for geometrical variations in the embayment. The channel-shoal structure of observed patterns develop as instabilities on the width-averaged morphodynamic equilibria. In this article, these width-averaged morphodynamic equilibria are identified. The goal of this article is to analyse the sensitivity of width-averaged morphodynamic equilibria in a rectangular, exponentially converging and exponentially diverging embayment for the following physical parameters: inertia, bed shear stress, depth-dependence of the deposition parameter, inclusion of advective processes and inclusion of an externally prescribed overtide. To validate the model, results are qualitatively compared to profiles obtained from observations.
The outline of this article is as follows. In Section 2, the model geometry and equations are presented. In Section 3 the full model equations are scaled and analysed using an asymptotic expansion. The effect of choosing various geometries for the embayment when only considering diffusive processes is analysed in Section 4. In this section, the influence of the formulation of the deposition parameter, the friction and the inclusion of advective sediment transport for different geometries is also studied. Furthermore, the existence of multiple equilibria is discussed. Conclusions are given in Section 5.
2 The model
2.1 The geometry of the embayment
2.2 Modelling approach
In order to find morphodynamic equilibria, we construct a model that describes the complex interaction between the water motion, sediment transport and bed evolution. The water motion is described by the shallow water equations and transport of suspended sediment is described by an advection-diffusion equation. The bed evolves due to convergences and divergences of suspended load transport, resulting from erosion of the bed and deposition of suspended material and bed load transport. When the bed is steady over the long morphodynamic timescale, a morphodynamic equilibrium is said to be obtained.
2.3 The water motion
2.4 Suspended sediment transport
2.5 The bed evolution equation
3 Solution method
3.1 Scaling the model
Channel | Sediment |
---|---|
L = 20 ⋅ 10^{3} m | c_{d} = 0.001 |
H = 12 m | g = 9.81 ms^{− 2} |
B = 2 ⋅ 10^{3} m | \(\kappa _{h} = 10^{2}~\mathrm {m}^{2}\ \mathrm {s}^{-1}\) |
\(\omega _{s} = 0.015~\text {m s}^{-1}\) | |
Tide | \(\kappa _{v} = 0.1~\mathrm {m}^{2} \mathrm {s}^{-1}\) |
\(A_{M_{2}} = 0.84~\mathrm {m}\) | α = 0.02 kg s m^{− 4} |
\(A_{M_{4}} = 0.08~\mathrm {m}\) | \(\rho _{s} = 2650~\text {kg m}^{-3} \) |
σ = 1.4 ⋅ 10^{− 4} s^{− 1} | p = 0.4 |
ϕ = 195^{∘} | Γ = 7.8 ⋅ 10^{− 5} |
T = 44.9 ⋅ 10^{3}s | \(u_{c} = 0.3 \text {m s}^{-1}\) |
μ = 1.4 ⋅ 10^{− 4}m^{2}s^{− 1} |
Dimensionless parameters for the Ameland Inlet
\(\epsilon = \frac {U}{\sigma L} = \frac {A_{M}2}{H} \sim 0.07\) | \(\frac {\text {tidal excursion length}}{\text {embayment length}}\) |
\(\tilde {r} = \frac {8 c_{d} A_{M_{2}} L}{3 \pi H^{2}} \sim 0.099\) | bottom friction parameter |
\({\Delta }^2 = \frac {g H}{\sigma ^2 L^2} \sim 15.015\) | \(\left (\frac {\text {tidal wave length}}{\text {embayment length}}\right )^2\) |
\(\nu = \frac {\sigma \kappa _{v}}{{\omega _{s}^{2}}} \sim 0.0622\) | \(\frac {\text {deposition timescale}}{\text {tidal period}}\) |
\(\kappa = \frac {\kappa _{h}}{\sigma L^{2}} \sim 1.79 \cdot 10^{-3}\) | \(\frac {\text {tidal period}}{\text {diffusive timescale}}\) |
\(\lambda = \frac {H \omega _{s}}{\kappa _{v}} \sim 1.8\) | \(\frac {\text {vertical diffusion timescale}}{\text {deposition timescale}}\) |
\(\delta = \frac {\alpha U^{2}}{\sigma H \rho _{s}(1 - p)} \sim 7.19 \cdot 10^{-5}\) | \(\frac {\text {tidal period}}{\text {morphodynamic timescale}}\) |
\(\gamma = \frac {A_{M_{4}}}{A_{M_{2}}} \sim 0.095\) | \(\frac {M_{4} \text {amplitude}}{M_{2} \text {amplitude}}\) |
\(\tilde {\mu } = \frac {\mu }{\sigma L^{2}} \sim 2.5 \cdot 10^{-9}\) |
3.2 The analysis of the model
For some simplified situations, the model can be analytically solved, see Schuttelaars and de Swart (1996). However, in general the model has to be solved numerically. As indicated above, the bed evolves due to convergences of different types of sediment transport. In this article, we study the influence of the contribution of these different processes. To systematically assess the importance of the various transport contributions, the model output is analysed in terms of these transport contributions. Leaving out the slope contribution of the bedload transport is not allowed as this would make it impossible to impose boundary conditions for the bed evolution equation at the seaward and landward side. Since it is essential to impose these boundary conditions, at least the slope term of the bedload component has to be retained, although its contribution is (at least in the main part of the embayment) negligible.
The tidal constituents which contribute to the velocity, the sea surface elevation and the sediment concentration for the leading (order one), 𝜖 and γ orders
\(\mathcal {O}(1)\) | \(\mathcal {O}(\epsilon )\) | \(\mathcal {O}(\gamma )\) | |
---|---|---|---|
u,ζ | M _{2} | M_{0},M_{4} | M _{4} |
C | M_{0},M_{4} | M _{2} | M _{2} |
3.3 Morphodynamic equilibria
3.4 Numerical method
To obtain morphodynamic equilibria, the model equations at leading order, order 𝜖 and γ are solved numerically using a finite element method. As a first step, we convert the model equations into their weak formulation and we discretise the model equations using piecewise linear functions. We apply a so-called continuation method: starting with a known equilibrium profile for a specific set of parameters as an initial guess, one can obtain equilibria by slowly varying parameters or the geometry using a Newton-Raphson procedure. The morphodynamic equilibrium that we use as a starting point in the continuation method is the morphodynamic equilibrium of a short, rectangular embayment where sediment transport is dominated by diffusive transport and inertia and friction are neglected. This equilibrium bed profile is constantly sloping, see Schuttelaars and de Swart (1996, 2000) and Ter Brake and Schuttelaars (2010).
4 Results
4.1 Diffusively dominated transport
In this section, we assume that the suspended load transport is dominated by diffusive processes.
Geometry
For the rectangular inlet (blue line in Fig. 2b), we observe that the water depth constantly decreases. This result is similar to what was already found in Schuttelaars and de Swart (1996) and Ter Brake and Schuttelaars (2010), using a model formulation in which inertia was assumed to be negligible in the momentum equation. Our results show that inertia does not influence the bed equilibrium profile. For a tidal embayment with width variations in the along-channel direction, it is found that for a more diverging (converging) inlet, a more convex (concave) equilibrium bed profile is obtained.
Deposition parameter
We observe that for a rectangular inlet, the equilibrium bed profile becomes convex when the depth-dependency of the deposition parameter is incorporated. This result is in agreement with earlier findings in Ter Brake and Schuttelaars (2010) and Van Leeuwen et al. (2000). We see that the same change also occurs when the geometry of the embayment is varied in the along-channel direction. The location where the solution is most convex is found in the middle of the embayment for a diverging geometry and lies more towards the landward boundary as the embayment becomes more converging.
These results can be explained by considering the morphodynamic equilibrium condition, again neglecting the bedload transport. We find that in order for \(F^{0,0}_{\text {diff}} = -\kappa _{h}<WC^{0,0}_{x}>= 0\) to be satisfied, the leading order residual sediment concentration, C^{0,0} cannot depend on x. Using this in Eq. 3.12, it follows that \(<\beta ^{0,0} C^{0,0}> \approx <\left .u^{0,0}\right .^{2}>\). Since C^{0,0} has to be spatially uniform, this implies that \(\frac {(u^{0,0})^{2}}{\beta ^{0,0}}\) must also be spatially uniform, where \( \beta ^{0,0} = \left [1 - h_{\delta } e^{-\frac {\omega _{s}}{\kappa _{v}}(H-h)}\right ]^{-1}\). From Fig. 3, we deduce that the bed profiles are non-decreasing functions of the longitudinal coordinate x. Since the exponent of the leading order deposition parameter depends on the bed profile, this implies that the leading order deposition parameter increases towards the landward boundary when h_{δ} is non-zero. Therefore, for \(\frac {(u^{0,0})^{2}}{\beta ^{0,0}}\) to remain spatially uniform, the amplitude u^{0,0} of the velocity has to increase towards the landward boundary as well. Using expression (4.3), we deduce that for the velocity to increase, H − h has to decrease, i.e. the water depth has to decrease.
Friction
The above results can be explained as follows. The bottom friction parameter depends linearly on the embayment length L. This implies that as the length of the embayment increases, the bottom friction term becomes more important. Furthermore, the bottom friction term is divided by the local water depth in the leading order momentum equation, (3.11). This suggests that this term becomes more significant when the local water depth decreases. As we have seen in the previous section when friction was neglected, the bed is much shallower for a diverging embayment than for a converging one. Therefore, if friction effects are added, their influence is largest for a diverging inlet.
4.2 Advective and diffusive transport without an externally prescribed overtide
For a rectangular inlet, we find that the morphodynamic bed equilibrium is less convex compared to the case where only diffusion is considered. This result is similar to the findings of Ter Brake and Schuttelaars (2010), although the effect of adding internal advection is stronger in our study which is due to frictional effects. When only diffusion is considered, friction only resulted in minor adjustments to the equilibrium bed profile, but the influence of friction is not negligible anymore when internal advection is considered. From Fig. 5, we conclude that for a larger embayment length, the internal advective processes become more important. Also, the importance of internally generated advective sediment transport becomes larger for more strongly divergent tidal inlets widths, for which the equilibrium bed profile is very shallow towards the landward side of the basin. Furthermore, when including internal advection the velocities decrease (increase) into the landward direction when the system is diverging (converging).
4.3 Advective and diffusive transport with an externally prescribed overtide
In this section, the sea surface elevation is forced by both a M_{2} tide and a M_{4} tide. The deposition parameter β depends on the depth and frictional effects are taken into account. In discussing the results, we focus on the sensitivity of morphodynamic equilibria to the relative phase of the overtide. We discuss the sensitivity of the maximal tidal embayment length to the relative phase, see for example Schuttelaars and de Swart (2000), Ter Brake and Schuttelaars (2010), Todeschini et al. (2008), and Seminara et al. (2010). To determine the maximum embayment length, morphodynamic equilibria are obtained for each relative phase by increasing the length from L = 8 km. The maximum embayment length is the largest L for which a morphodynamic equilibrium can still be found numerically for the parameters under consideration. If the embayment length L = 22 km is reached, a maximum length might still exist but it is not in the range of lengths we are focusing on.
4.3.1 Converging tidal embayment
These results can be explained by analysing the different sediment transport contributions. Decreasing the amplitude of the M_{4} tide results in a smaller contribution of the advective transport due to externally generated overtides. Therefore, it is easier to balance this contribution by the internally generated and diffusive sediment transport, resulting in an increase of the maximum length for all values of the relative phase. Increasing the drag coefficient leads to an increase of the internal advection, resulting in a balance for longer tidal inlet systems.
4.3.2 Rectangular tidal embayment
We performed a similar analysis for a rectangular geometry of the embayment and found that the bed profiles are, apart from being less concave, qualitative the same as those for the converging geometry embayment. The dependency of the maximum length on the relative phase is similar to the relation found for the converging embayment. The maximum length decreases as the convergence length increases for all values of the relative phase which agrees to findings of Seminara et al. (2010). The maximum length for a rectangular embayment forced by an overtide with a relative phase ϕ around 270^{∘} is still approximately \(L_{\max } = 8~\text {km}\) which was also found by Ter Brake and Schuttelaars (2010).
4.3.3 Diverging tidal embayment
When we choose the geometry embayment to be diverging, the values of the relative phase for which there is a maximum length start to differ significantly: there are less parameter combinations for which no morphodynamic equilibrium can be found. The region around ϕ = 270^{∘} where no equilibrium is found for larger lengths still exists, although it is much smaller than for a converging and rectangular embayment.
To study this in more detail, the mean water depth is plotted in Fig. 8b. Considering the embayment lengths L = 14 km and L = 18 km, the mean depth of the morphodynamic equilibrium bed profile is large for 0^{∘} < ϕ < 180^{∘} and changes to a shallow profile for a relative phase between 180^{∘} < ϕ < 360^{∘}. For an embayment with L = 18 km, this transition between a deep and a shallow equilibrium profile is more sudden than for L = 14 km. For L = 19.6 km a saddle-node bifurcation takes place at approximately ϕ = 70^{∘} and ϕ = 170^{∘}, respectively. In between these values of ϕ, there exist multiple stable solutions, one of them corresponding to a deep profile and the other one to a more shallow character. The deeper equilibrium still exists when increasing the relative phase ϕ and the more shallow one when decreasing the relative phase. There is also a third unstable equilibrium with a shallow character.
4.4 Comparison with observed bed profiles
In the previous subsections, the geometry of the embayment was taken to be either exponentially converging, exponentially diverging or rectangular. Since it is possible to choose arbitrary width variations in the model, we take a realistic width profile.
We have used data from observations of the Ameland Inlet and the Friesche Zeegat, both systems in the Dutch part of The Wadden Sea, with the water level measured at Nes and Schiermonnikoog.
We have then smoothed the obtained width profile and used this profile in our model to determine the corresponding morphodynamic equilibrium bottom. For the Ameland Inlet, we have used the characteristic values listed in Table 1 and for the Friesche Zeegat, the values used in Ter Brake and Schuttelaars (2010) have been employed. Furthermore, we have performed a harmonic analysis on the measured values of the water level at Nes and Schiermonnikoog resulting in the amplitude of the M2-tide and the M4-tide. A comparison between the data and the model results is presented in Fig. 10, showing a reasonable agreement with the main trends rather well-captured.
Concerning the amplitudes of the sea surface elevation, the difference between the observed and the modelled amplitudes varies from 3 to 10 cm.
The difference between the data and the model results can have multiple reasons. One, it is unknown whether the characteristic values used in this article were the best choices for the physical parameters when the measurements were carried out. We have shown in the previous section that varying physical parameters can have a significant influence on the bed level and, hence, on the amplitudes of the tidal constituents. Second, the data amplitudes have been determined using measurements performed over a whole year, whereas the width and depth values have been obtained at one moment. Third, the measuring locations for the sea surface elevation are close to the coast. It is therefore not necessarily a representative value when comparing to the modelled width-averaged sea surface elevation. Furthermore, lateral processes are only parametrically taken into account, and many possibly important processes were not accounted for.
5 Conclusion
We have analysed the existence of morphodynamic equilibria and their sensitivity to geometrical variations, inertia, bottom friction, depth-dependency of the deposition parameter and the importance of advective processes both with and without an externally prescribed overtide. We considered tidally dominated embayments with a length between L = 8 km and L = 22 km, representative for basins in the Wadden Sea. The geometry of the embayment was chosen to be rectangular, exponentially converging or exponentially diverging. We used the cross-sectionally averaged shallow water equations to describe the water motion and an advection-diffusion equation for the sediment concentration equation. We assumed that the bed evolves due to convergences and divergences of diffusive and advective sediment transport.
We started with only considering diffusive processes and prescribing the sea surface elevation by only a M_{2}-tide. For the frictionless case, the obtained morphodynamic bed equilibrium was highly influenced by varying the geometry. For a rectangular embayment, a constantly sloping bed profile was found. When the geometry was changed, the bed profile became more convex (concave) for a diverging (converging) embayment. Letting the deposition parameter depend on the depth resulted in more concave bed profiles. If we increased the embayment length, the frictional influence became stronger as well. The effect of bottom friction for a converging embayment was negligible, but becomes more significant for a more strongly diverging embayment. When also advective processes were taken into account, the influence of frictional effects increased significantly. When we considered diffusive and advective sediment transport without an externally prescribed overtide, we found that the bed profiles became less convex compared to those with only diffusive processes considered. Although this occurred for all geometries considered, the influence of adding advective sediment transport was much stronger when the geometry of the embayment was diverging.
When the prescribed sea surface elevation consisted of both a M_{2} and a M_{4} tide, we found a maximum length of the embayment for which there still existed morphodynamic equilibria. The maximum embayment length decreased when the geometry became more converging. The smallest embayment length was found for a relative phase of ϕ = 270^{∘} for converging embayments. Making the geometry less converging, increasing the bottom friction or decreasing the amplitude of the M_{4}-tide, resulted in increasing maximum embayment lengths. We obtained two types of bed profiles, depending on the values of the relative phase: a shallow one where the external sediment transport was exporting and a deep one for importing external sediment transport.
When considering a diverging embayment geometry, there is a parameter range of relative phases and embayment lengths for which two stable bed profiles co-exist, along with a third unstable equilibrium. These results indicate that for systems with the characteristics of the Wadden Sea systems, there are parameter values for which both a relatively shallow and a much deeper width-averaged morphodynamic equilibrium can exist. For a proper management, it is important to be aware of the existence of multiple equilibria in tidal inlet systems. The analysis given in this paper does not indicate the magnitude of the perturbation necessary to go from one equilibrium to the other, nor the time scale at which a change would take place, this is topic of further research. However, we would like to stress that the existence of multiple equilibria has been observed in Schuttelaars and de Swart (2000) as well; they found multiple equilibria for a rectangular long embayment with a length of approximately L = 120 km, these model results were confirmed by simulations done with a complex numerical model (Hibma et al. 2003). It would be interesting to investigate the presence of multiple equilibria for longer systems with width variations. For a rectangular system, this has already been done in Schuttelaars and de Swart (2000), where indeed it was found that multiple equilibria can exist as well (see Fig. 6 in that paper), and these results were qualitatively reproduced in the paper of Hibma et al. (2003) using a complex model.
The morphodynamic model derived in this paper can be improved upon by explicitly considering lateral processes what will result in observed complex channel-shoal patterns. Furthermore, the accuracy of the water motion close to the landward boundary can be improved by considering more tidal harmonics, which might be necessary as the parameter 𝜖 is not small near this boundary. This might shed some light on the fact that we obtain equilibria which exist on a long morphodynamic timescale while many simulation models, like Van Ledden et al. (2004), and Maan et al. (2015) result in an infilling of embayments which occur on an even longer timescale. Another interesting extension to the existing model would be the inclusion of wind, waves and density flows, which have been shown to be important (Green and Coco 2014; Gatto et al. 2017; Burchard et al. 2008) and the inclusion of flooding and drying processes.
Notes
Funding information
This research was supported by a grant of the NDNS+ cluster of the Dutch Science Organisation (NWO).
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