5a. TOPIC 1 – Tidal amplification
In this topic we study the factors that influence tidal amplification/tidal damping in an estuary. Moreover, we look at how tidal amplification in an estuary can be stopped or even be reduced. To study this topic, we used tidal data and topobathymetry data (see §3.1.2 and §3.1.1). Secondly, we calculated the tidal damping scale (§3.2.2), a parameter which describes tidal amplification/damping in an estuary.
Tidal range
The tidal range characteristics are unique for each estuary (Figure 29). The most important differences between the four estuaries are:
 The tidal range at the mouth
 The maximum tidal range
 The position of the maximum in tidal range
 The number of kilometers with increase/decrease in tidal range
 The strength of the increase/decrease in tidal range (i.e. the tidal range gradient)
The smallest and highest tidal ranges at the mouth are respectively observed for the Elbe and Humber. Scheldt and Weser have intermediate tidal ranges and are comparable (Figure 29). The tidal range in the Scheldt, Elbe and Weser estuaries is almost everywhere higher than the tidal range at the mouth (Figure 30, Figure 31: >80% estuary length). For the Humber, this is only the case for about 50% of the estuary length (Figure 31). Although the tidal range in the Weser estuary is for the entire estuary higher than the tidal range at the mouth, it is featured by the lowest TR_{x}/TR_{0} values (Figure 30 and Figure 31). Two factors are important in determining TR_{x}/TR_{0} values : (1) the value of the tidal range gradient, and (2) the distance over which increase/decrease in tidal range occurs (see Figure 32 and Figure 33). As a consequence, the rather low maximum in tidal range increase (2 cm/km), and the limited distance over which increase in tidal range occurs results in rather low TR_{x}/TR_{0} values for the Weser. This is in contrast to the Scheldt estuary, where larger maximum values occur (up to 3 cm/km), and where the increase in tidal range occurs over a large distance (Figure 32 and Figure 33). In general we may conclude that tidal amplification is the highest in the Scheldt (TR_{x}/TR_{0} ~ 1.4), followed by the Elbe (TR_{x}/TR_{0} ~ 1.3) and Humber (TR_{x}/TR_{0} ~ 1.15), and that tidal amplification in the Weser is the smallest (TR_{x}/TR_{0} ~ 1.1). Similar values are observed by van Rijn (2011)
5a. Damping or amplification?
Tidal damping and tidal amplification was in this study evaluated in two ways:
 By the theoretical derived tidal damping scale parameter (Savenije, 2001) (see §3.2.2): if 1/β > 0 amplification prevails, if 1/β < 0 damping prevails
 By the observed tidal range gradient: if 𝛁 TR > 0 amplification prevails, if 𝛁 TR < 0 damping prevails
We observe that there is a good agreement between the theoretical tidal damping (1/β) and the observed changes in tidal range gradient. Data points with 1/β > 0 (amplification) correspond with an increase in tidal range (𝛁 TR > 0), data points with 1/β < 0 (damping) correspond with a decrease in tidal range (𝛁 TR < 0) (Figure 34). These observations are valid for all estuaries. Note that for the Weser, the tidal damping is very limited (see also Figure 32 and Figure 33).
5a. Critical threshold values for depth and estuary convergence
It is assumed that the convergence of the estuary is an important driver for the tidal amplification, while limited water depth will cause friction and lead to a damping of the tidal range. In the next paragraphs we present threshold values in estuary depth, for which we can expect tidal amplification/damping, taking into account the estuary convergence.
Based on the theoretical tidal damping (1/β)
The relationship between the cross section averaged depth at LW (measure for the friction) and the theoretical tidal damping (1/β) shows that for the most convergent estuary (Humber, 1/b = 4.18*10E5, see Table 4 and Figure A 4) tidal amplification (1/β > 0) occurs at DLW > 4.2 m and that tidal damping (1/β < 0) occurs at DLW < 4.2 m (i.e. based on a logarithmic trendline) (Figure 35 and Table 7). For the least convergent estuary (Elbe, 1/b = 2.75*10E5, see Table 4 and Figure A 2) the critical threshold value for DLW occurs at 6.4 m, and for the intermediate converging estuary (Scheldt, 1/b = 3.39*10E5, see Table 4 and Figure A 1) at 5.6 m (Figure 35). For the Weser it was not possible to establish a proper trendline due to the lack of data points representing tidal damping (Figure 35).
Based on the found threshold values for DLW (Figure 35) and the estuary convergence, tidal amplification and tidal damping in an estuary can be described as a function of the estuary convergence and the estuary depth (Figure 36). We observe that for more convergent estuaries, amplification occurs at more shallow conditions compared to less convergent estuaries. The found threshold values for tidal amplification/damping range between 4.2 m (for the most convergent estuary, i.e. the Humber) and 6.4 m for the least convergent estuary (i.e. the Elbe).
Based on the observed tidal range gradient (𝛁 TR)
The relationship between the cross section averaged depth at LW (measure for the friction) and the observed tidal range gradient (𝛁 TR) shows that for the most convergent estuary (Humber, 1/b = 4.18*10E5, see Table 4 and Figure A 4) tidal amplification (𝛁 TR > 0) occurs at DLW > 5.3 m and that tidal damping (𝛁 TR < 0) occurs at DLW < 5.3 m (i.e. based on a logarithmic trendline) (Figure 37 and Table 7). For the least convergent estuary (Elbe, 1/b = 2.75*10E5, see Table 4 and Figure A 1) the critical threshold value for DLW occurs at 7.7 m, and for the intermediate converging estuary (Scheldt, 1/b = 3.39*10E5) at 7.4 m (Figure 37).
Based on the found threshold values for DLW (Figure 37) and the estuary convergence, tidal amplification and tidal damping in an estuary can be described in function of the estuary convergence and the estuary depth (Figure 38), similarly as for the theoretical tidal damping parameter 1/β (Figure 36). Again, we observe that for more convergent estuaries, amplification occurs at more shallow conditions compared to less convergent estuaries. The found threshold values for tidal amplification/damping range between 5.3 (for the most convergent estuary, i.e. the Humber) and 7.7 for the least convergent estuary (i.e. the Elbe).
Table 7 – R² and DLWcr for the logarithmic regression lines in Figure 35 and Figure 37 . All found trendlines are significant. DLWcr is the critical DLW value for tidal damping/amplification. Below this value tidal damping occurs, above this value tidal amplification occurs

Humber 
Scheldt 
Elbe 

R² 
DLWcr 
R² 
DLWcr 
R² 
DLWcr 
DLW ~ 1/β 
0.91 
4.2 
0.88 
5.6 
0.9 
6.4 
DLW ~ 𝛁 TR 
0.89 
5.3 
0.67 
7.4 
0.69 
7.7 
Conclusions
The two most important factors that influence tidal amplification and tidal damping in an estuary are: (1) the funneling of the estuary (i.e. estuary convergence) leading to tidal amplification, and (2) the friction in the estuary (controlled by the estuary depth) which leads to tidal damping. Based on the theoretical tidal damping parameter 1/β, we found that tidal amplification for several converging estuaries occurs at an estuary depth larger than 4.2  6.4 m (crosssection averaged at LW) and that tidal damping occurs at an estuary depth smaller than 4.2  6.4 m. Based on the observed tidal range gradients (𝛁 TR) this critical threshold value is somewhat higher, ranging from 5.3  7.7 m. The range in these threshold values (both for 1/β and 𝛁 TR) is influenced by the estuary convergence: the more convergent an estuary, the lower the critical threshold depth.
We recommend for the observed estuaries an estuary depth smaller than 4.2  7.7 m (i.e. crosssection averaged depth at LW) to have important tidal damping. As analysis were performed over 5 km blocks, this critical estuary depth should be present for at least 5 km along the estuary. The range in critical estuary depth (4.2 – 7.7 m) is a consequence of the estuary convergence: the more convergent the estuary, the smaller the critical estuary depth. However, we are convinced that it is necessary to include more estuaries in the analysis to improve the accuracy of the found threshold values in estuary depth.
Shallowing of the subtidal channels is thus a possible measure to lower the crosssection averaged depth at LW, and in this way increase the tidal damping. However, also other measures could be taken to introduce more friction in an estuary, for example by creating more intertidal area. This is further elaborated in the next topic (§5.2) where we relate tidal damping/amplification to habitat occurrence.
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