On the sensitivity of the sedimentary system in the East Frisian Wadden Sea to sea-level rise and wave-induced bed shear stress

Abstract

The paper addresses the individual and collective contribution of different forcing factors (tides, wind waves, and sea-level rise) to the dynamics of sediment in coastal areas. The results are obtained from simulations with the General Estuarine Transport Model coupled with a sediment transport model. The wave-induced bed shear stress is formulated using a simple model based on the concept that the turbulent kinetic energy (TKE) associated with wind waves is a function of orbital velocity, the latter depending on the wave height and water depth. A theory is presented explaining the controls of sediment dynamics by the TKE produced by tides and wind waves. Several scenarios were developed aiming at revealing possible trends resulting from realistic (observed or expected) changes in sea level and wave magnitude. The simulations demonstrate that these changes not only influence the concentration of sediment, which is very sensitive to the magnitude of the external forcing, but also the temporal variability patterns. The joint effect of tides and wave-induced bed shear stress revealed by the comparison between theoretical results and simulations is well pronounced. The intercomparison between different scenarios demonstrates that the spatial patterns of erosion and deposition are very sensitive to the magnitude of wind waves and sea-level rise. Under a changing climate, forcing the horizontal distribution of sediments adjusts mainly through a change in the balance of export and import of sediment from the intertidal basins. The strongest signal associated with this adjustment is simulated North of the barrier islands where the evolution of sedimentation gives an integrated picture of the processes in tidal basins.

Appendix 1

Here, we will use for the bed shear stress Eq. 10 and, as done in many theoretical studies of the bottom layer, we parameterize $\tau_b^t$ as

$$ \tau_b^t =\rho c_D u^2, $$

(22)

where $u$ is the mean velocity.

We will apply the above equations to an idealized topography with a constant slope $a=\frac{dh}{dx}$ (Fig. 15). For simplicity, we will assume that $\zeta=\frac{r}{2}cos\omega t$ where $r$ is the tidal range (Fig. 16a), that is we neglect the spatial dependence. As demonstrated by SFW, this assumption is approximately valid over the tidal flats. The volume of water between the location where the depth is $H_{lw}$ (see Fig. 15) and the movable boundary is

$$ V=[x_{lw}(\frac{H_{lw}}{2}+\zeta)+\Delta x \frac{\zeta}{2}]y_0,$$

(23)

where $y_0$ is the extension along the coast of the area, $x_{lw}=\frac{H_{lw}}{a}$ is the distance between location where the depth is $H_{lw}$ and the coast, $\Delta x=\frac{\zeta}{a}$ The total transport is $Tr=\frac{dV}{dt}=x\frac{d\zeta}{dt}y_0$ where $x=\frac{(H_{lw}+\zeta)}{a}$ is the extension of the flooded area. Velocity is computed as $v=\frac{1}{y_0(H_{lw}+\zeta)}\frac{dV}{dt}$ and for the given $\zeta$ is $v=-\frac{r\omega}{2a}sin\omega t$ The corresponding bed shear stress computed from Eq. 22

$$ \tau_b^t=\rho c_D (\frac{r\omega}{2a})^2\sin^2\omega t=\rho c_D (\frac{r\omega}{2a})^2\frac{(1-cos2 \omega t)}{2} $$

(24)

is shown in Fig. 16b. This is a bimodal oscillation shaped by the maxima of current speed.

Fig. 15

The simplified topography

Fig. 16

Temporal variability during one tidal period of sea level (a), bed shear stress due to tides (b), and bed shear stress due to wind waves (c and d). In this case $f_w=0.015$ and the limiting factor $(H/h)_{lim}$=0.15. The horizontal lines at 0.2$Nm^{-2}$ indicates critical shear stress

The importance of the limiting factor $(H/h)_{lim}$ in Eq. 12 is revealed below from the comparison between Figs. 16, 17, and 18 in which $(H/h)_{lim}$ is 0.15, 0.5, and 0.8, correspondingly. The friction factor in the first and third case is changed compared to the basic case (Fig. 17) in a way to ensure comparable magnitudes of bed shear stress. Obviously, beyond some values of $(H/h)_{lim}$ the temporal variability follows a similar pattern (compare Figs. 17 and 18), however, for small limiting factors such as those reported by Le Hir et al. (2000), the course of curves change substantially (some maxima disappear).

Fig. 17

Temporal variability during one tidal period of bed shear stress due to wind waves. In this case $f_w=0.005$ and the limiting factor $(H/h)_{lim}$=0.5. The horizontal lines at 0.2$Nm^{-2}$ indicates critical shear stress

Fig. 18

Temporal variability during one tidal period of bed shear stress due to wind waves. In this case $f_w=0.0031$ and the limiting factor $(H/h)_{lim}$=0.8. The horizontal lines at 0.2$Nm^{-2}$ indicates critical shear stress

Following the simple model of Eq. 12 with $(H/h)_{lim}=0.5$ and $f_w=0.005$ the bed shear stress at $x=0$ (local depth $H_{LW}$) due to wind waves follows different patterns depending on the wave height (Fig. 17a,b). For small waves (Fig. 17a), the minimum stress occurs during low and high water. Both minima are lower than the critical value of 0.2N$\rm {m^{-2}}$ thus enabling a deposition during these times. The temporal variability of bed shear stress due to high waves (Fig. 17b) reveals that only during low water are the values lower than the critical ones. Although there is a secondary minimum during high water, the corresponding values are too high to allow sedimentation. Thus, the total shear stress (Fig. 19b) demonstrates that depending on the wave height, favorable conditions for deposition (blue color) could occur two times per tidal period (for small waves) and one time for high waves. As seen in Fig. 19a, for waves with a constant height, the dependence of bed shear stress on the bottom slope is also very pronounced. For small slopes velocity is large and the bimodal oscillations are better pronounced. For relatively steeper bottom, the control of wind waves becomes dominant (the time of erosion decreases and that of deposition increases).

Fig. 19

Bed shear stress ($Nm^{-2}$) due to tides and wind waves. Time vs bottom slope diagram (a) and time vs wave height diagram (b). In this case, $f_w=0.005$ and the limiting factor $(H/h)_{lim}$=0.5

The above behavior of the bed shear stress due to breaking wind waves becomes clear if we express Eq. 11 as:

$$ \tau_b^w = \frac{1}{8}\rho g f_w (H/h)_{lim}(h_0+\zeta),$$

(25)

for small depths and

$$ \tau_b^w = \frac{1}{8}\rho g f_w H_i^2/(h_0+\zeta),$$

(26)

for large depths where $h_0$ is the mean depth. Thus, depending on the ratio between wind-wave magnitude and local depth, the control on bed shear stress is taken either by Eq. 25 or 26. The first case is clearly represented in Fig. 16d.

Because in the near-coastal zone $\zeta$ reaches values that are comparable with the mean depth, the bed shear stress is expected to show a strong sensitivity to oscillations in sea level. Critical conditions (maximum in Fig. 3) are reached two times per tidal period. This is the reason for the bimodal oscillations of bed shear stress (Fig. 17). It is noteworthy that depending on the wave height, the “spikes” in the bed shear occur earlier for high waves and with some delay for the low wave heights. Therefore, there is an asymmetry of the response of bed shear stress to tidal forcing characterized with a pronounced spatial pattern.