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The effect of geometry and tidal forcing on hydrodynamics and net sediment transport in semi-enclosed tidal basins

A 2D exploratory model

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Abstract

A new depth-averaged exploratory model has been developed to investigate the hydrodynamics and the tidally averaged sediment transport in a semi-enclosed tidal basin. This model comprises the two-dimensional (2DH) dynamics in a tidal basin that consists of a channel of arbitrary length, flanked by tidal flats, in which the water motion is being driven by an asymmetric tidal forcing at the seaward side. The equations are discretized in space by means of the finite element method and solved in the frequency domain. In this study, the lateral variations of the tidal asymmetry and the tidally averaged sediment transport are analyzed, as well as their sensitivity to changes in basin geometry and external overtides. The Coriolis force is taken into account. It is found that the length of the tidal basin and, to a lesser extent, the tidal flat area and the convergence length determine the behaviour of the tidally averaged velocity and the overtides and consequently control the strength and the direction of the tidally averaged sediment transport. Furthermore, the externally prescribed overtides can have a major influence on tidal asymmetry in the basin, depending on their amplitude and phase. Finally, for sufficiently wide tidal basins, the Coriolis force generates significant lateral dynamics.

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Funding

The first author is a doctoral research fellow of IWT-Vlaanderen (project IWT 141275). The computational resources (Stevin Supercomputer Infrastructure) and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by Ghent University, the Hercules Foundation, and the Flemish Government, department EWI.

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Correspondence to Thomas Boelens.

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Responsible Editor: Emil Vassilev Stanev

Appendices

Appendix A: Model verification—error analysis

The model will be verified in two steps. Firstly, the water motion will be simulated in a tidal basin with a flat bottom, ignoring Coriolis force. The numerical results of the leading order system of equations (2.13a) and boundary conditions (2.14a) will be compared with an exact analytical solution (see, e.g. Dronkers 1986). Secondly, the influence of the domain extension is investigated when a cross-sectional profile, as described in Eq. 1, is used and the Coriolis force is included.

Regarding the first verification step, a straight channel of length L = 110 km and width B = 1.75 km is considered, with a flat bottom profile (H = 8 m). Denoting the exact solution as χ and the numerical solution as \(\chi _{\tilde {h}}\), where \(\tilde {h}\) is the maximum cell diameter of the triangular grid, the error function \(E_{\tilde {h}}\) is defined as \(E_{\tilde {h}} = \chi -\chi _{\tilde {h}}\). The numerical solution \(\chi _{\tilde {h}}\) converges to the exact solution χ if \(\|E_{\tilde {h}}\|_{L_{2}}\rightarrow 0\) as \({\tilde {h}}\rightarrow 0\), with \(\|\cdot \|_{L_{2}}\) the L2 norm. To make the error measure independent of the size of the domain and the range of the solution, the relative error is defined as

$$ E_{\text{rel}}(\tilde{h}) = \frac{\|E_{\tilde{h}}\|_{L_{2}}}{\|\chi\|_{L_{2}}}. $$
(A.1)

The order of convergence p is the rate at which the numerical solution \(\chi _{\tilde {h}}\) converges to the exact solution χ and is given by

$$ p = \frac{\log\left( \|E_{\tilde{h}_{1}}\|_{L_{2}}/\|E_{\tilde{h}_{2}}\|_{L_{2}}\right)}{\log\left( \tilde{h}_{1}/\tilde{h}_{2}\right)}. $$
(A.2)

For linear (quadratic) basis functions, we expect second (third) order convergence of the numerical solution, provided numerical integrals are computed sufficiently accurately (Kumar et al. 2016).

To investigate the convergence properties of the numerical solution, the number of nodes is systematically increased, with both linear and quadratic basis functions. Figure 22a illustrates that the relative error defined in Eq. A.1 decreases for an increasing number of nodes. Note that for the same number of nodes, the relative error using quadratic basis functions is at least 100 times smaller than the relative error found with linear basis functions. Figure 22b demonstrates that the order of convergence for linear basis functions converges to 2 and for quadratic basis functions, the order of convergence approaches 3. To conclude, the numerical solution for the amplitude of the surface elevation converges with the expected order of convergence for both linear and quadratic basis functions. Similar results are obtained for the velocity amplitudes. As mentioned earlier, quadratic basis functions will be used for the leading order system of equations (2.13a), since the solutions (and their derivatives) are used in the first order system (2.15a) and therefore, the errors should be as small as possible.

Fig. 22
figure 22

a Relative error and b order of convergence for the amplitude of the surface elevation. The blue line shows the results for linear basis functions (P1 elements) and the red line for quadratic basis functions (P2 elements)

As a second verification step, the influence of the length of the domain extension on the water motion is considered. For both a short (L = 25 km, = 0.063) and a long (L = 110 km, = 0.277) channel, the domain has been extended with different fractions (0.001, 0.002, 0.005, 0.01, 0.015, 0.02, 0.025) of the frictionless tidal wavelength \(L_{g}^{*}\). The top row of Fig. 23 shows the relative error for the tidally averaged longitudinal velocity 〈u〉 between the solution calculated on the extended grid and the solution calculated on the grid with the largest extension (\(0.025L_{g}^{*}\)). It can be seen that both for a short and long channel, the solution starts to converge for a channel extension of approximately \(0.005L_{g}^{*}\). The bottom row of Fig. 23 depicts the tidally averaged velocity vectors 〈u〉 at the open boundary x = 0. If the extension of the domain is too small, the tidally averaged velocity vectors point in all directions and the solution is obviously not realistic. For grid extensions with a length exceeding about \(0.005L_{g}^{*}\)\(0.01L_{g}^{*}\), the vectors align well at the open boundary and do not change appreciably compared to larger extensions. Therefore, an extension of \(0.015L_{g}^{*}\) will be used (Table 1).

Fig. 23
figure 23

Error analysis for a short (a and c) and a long (b and d) channel, consisting of a cross-sectional profile as given in Eq. 1. a and b The L2 error norm of the difference between the numerical and the analytical results, derived in Li and O’Donnell (2005), i.e. \(\frac {\|\left \langle u \right \rangle _{\text {Li}}-\left \langle u \right \rangle \|_{L_{2}}}{\|\left \langle u \right \rangle \|_{L_{2}}}\). c and du〉 Along the open boundary of the physical domain, x = 0.

Appendix B: Model verification—comparison to different model results

In the previous appendix, the model was verified in leading order against analytical solutions. In this appendix, full model results will be compared to different models, found in the literature: an analytical model, describing the tidally averaged dynamics in a rectangular channel (Li and O’Donnell 2005) and a Delft2DH simulation model, considering a converging channel (van Rijn 2011).

To verify the robustness of the present exploratory model, tidally averaged results are compared to the analytical solutions from Li and O’Donnell (2005). Figure 24a shows the numerical solution of the tidally averaged water level along the channel and Fig. 24b presents the L2 error norm, compared to the analytical solutions, for different channel lengths, ranging from 5 to 150 km. The other parameters are chosen as in Table 1, but no Coriolis force is taken into account. The results show a relative error between the numerical solution and the analytical solution from Li and O’Donnell (2005), which is about 0.4% for the shortest channel and less than 0.1% for the other channel lengths. The relative error is higher for the shortest channels, because \(\|{\left \langle {\zeta } \right \rangle } \|_{L_{2}}\) is also very small for the shortest channels.

Fig. 24
figure 24

The mean sea level \(\left \langle \zeta \right \rangle (=\left \langle \zeta ^{1} \right \rangle )\) along the channel for different channel lengths. a The numerical results from the present exploratory model. b The relative L2 error norm of the difference between the numerical and the analytical results, derived in Li and O’Donnell (2005), i.e. \(\frac {\|\left \langle \zeta \right \rangle _{\text {Li}}-\left \langle \zeta \right \rangle \|_{L_{2}}}{\|\left \langle \zeta \right \rangle \|_{L_{2}}}\)

As a final test, the model results are compared to a simulation model. Figure 25 depicts the tidal range along a long converging channel without tidal flats. The present model is able to reproduce the results of the Delft2DH model from van Rijn (2011) quite accurately, contrary to the analytical solution that was proposed in van Rijn (2011), which is also shown.

Fig. 25
figure 25

Tidal range along a converging channel with L = 180 km, \(B_{0}^{*}= 25\) km, \(L_{b}^{*}= 25\) km, H = 10 m, and r = 0.0025 ms− 1. The full line corresponds to results generated with the present model; the dashed and dotted lines correspond respectively to a Delft2DH and an analytical model, presented in van Rijn (2011)

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Boelens, T., Schuttelaars, H., Schramkowski, G. et al. The effect of geometry and tidal forcing on hydrodynamics and net sediment transport in semi-enclosed tidal basins. Ocean Dynamics 68, 1285–1309 (2018). https://doi.org/10.1007/s10236-018-1198-9

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