Tide schematization

For the schematization of the tide, a so called ‘morphological tide’ will be derived. This morphological tide is a harmonic tide of two consecutive tidal cycles that should result in the same morphological development as a daily average of the astronomical tide for approximately a full spring-neap tidal cycle. The morphological tide will be chosen such to match the change in the morphology for an entire spring- neap cycle in reality as closely as possible. Because of the daily inequality of the tidal water levels in the study area, a double morphological tide (or two consecutive tides) will be derived.

In this study the method of Roelvink & Reniers (2012) is used which is in principle the same as Latteux (1995). Roelvink & Reniers (2012), however, recommend to only calculate one spring-neap tidal cycle instead of a full year of morphological behaviour to derive the morphological tide. The method is easy to apply and uses a scaling factor directly derived from linear regression instead of a calibration of parameters such as in the method of Lesser (2009). The following procedure should be executed to derive the morphological tide according to Roelvink & Reniers (2012):

 First, a simulation of flow and sediment transport including bottom changes over approximately a full spring-neap tidal cycle should be performed as a reference situation (simulation without waves).

 The next steps has to be performed for each consecutive tide (double tide in this study):

1. Calculate the correlation between the tide-averaged transport rates or bottom changes in all grid points for the full spring-neap tidal cycle and all consecutive double tides separately. The linear correlation coefficient is defined as:

𝑟 = 𝑐𝑜𝑣(𝑥, 𝑦)

𝜎(𝑥) ∗ 𝜎(𝑦) 2

Where a value r = 1 indicates a perfect match of the shapes of both datasets. 𝑐𝑜𝑣(𝑥, 𝑦) is the covariance of datasets x and y. 𝜎(𝑥) and 𝜎(𝑦) are the standard deviations of both datasets. The linear correlation coefficient indicates the correctness of the shape of the bottom changes; the correlation represents the rate of linearity between these bottom changes (McClave, Benson, & Sincich, 2010). A correlation of 1 means that the bottom changes of both periods (double consecutive tide and full spring-neap tidal cycle) is fully linear and positive. A correlation of zero means no linearity at all between the two datasets.

2. Together with the correlation coefficient, the slope of the linear regression line between the bottom changes of the full spring-neap tidal cycle and each double tide has to be calculated. This slope parameter indicates the correctness of the magnitude of the bottom changes; a slope equal to the number of double tides simulated is a perfect match. This slope parameter is used as a time-scale factor. The computed bottom changes should be multiplied with this factor to obtain the actual bottom changes for a full spring-neap tidal cycle.

Both parameters, the correlation coefficient and the slope of the linear regression line combined provide quantitative information of the shape and magnitude of all double consecutive tides in which they represent the shape and magnitude of bottom changes of the simulated spring-neap tidal cycle. Ideally, a double tide should be chosen with a correlation coefficient closest to one and a slope parameter closest to the number of double tides simulated.

For this study, the following parameters are calculated for the approximately simulated spring-neap tidal cycle (2/4/2013 to 30/4/2013, 28.45 days), see Figure 14 (here only the best fit of all tides is shown, the results of the other double tides can be found in Appendix B.1). From the full series, the tide with the highest correlation and the most adequate slope parameter is no. 20 (Figure 14). This tide has a correlation coefficient of 0.9993 and a slope parameter of 20.342. This tide is visualized in Figure 15. The tide period is from 11 April 2013 06.50 h to 12 April 2013 07.40 h and is chosen as the morphological tide for this study. The morphological development (cumulative sedimentation and erosion pattern) of this full spring-neap tidal cycle can be seen in Appendix B.2.

Figure 14 - Correlation and slope parameter of morphological tide

To make the selected tide a perfectly periodic boundary condition with a base frequency equal to the M2

tidal constituent frequency, a harmonic analysis of the selected time series is performed. For this harmonic analysis, the selected morphological tide is repeated several times to be able to extract various harmonic constituents from this signal. The main tidal constituents derived from this signal are the M2 and higher

harmonics M4, M6 and M8 tidal constituents. Amplitudes and phases values of the harmonic tidal

constituents for the grid corner points northwest and southwest are listed in Table 3. The choice for only incorporating these four constituents is supported by the tidal form factor F which is defined as (Pugh, 2004):

𝐹 = 𝜁̂𝐾1 + 𝜁̂𝑂1

𝜁̂𝑀2 + 𝜁̂𝑆2= 0.14 3

This value of 0.14 means that the tide is semidiurnal and thus mainly determined by the M2 constituent

distorted by the higher harmonics. The derived harmonic morphological tide is shown in Figure 16. The derived harmonic morphological tide shows the characteristics of the increasing tidal amplitude towards the South and towards the coast as expected by the explanation of the characteristics of the tide in the North Sea (section 2.3.1).

Table 3 - Harmonic tide conditions

Location Constituent Amplitude (m) Phase (°)

Northwest M2 0.6624 142.90

M4 0.2213 178.27

M6 0.1115 239.92

M8 0.0613 303.66

Maximum total amplitude 1.0565

Southwest M2 0.7138 99.53

M4 0.2804 151.84

M6 0.0477 214.41

M8 0.0546 212.15

Maximum total amplitude 1.0965