Development of tidal watersheds in the Wadden Sea

River, Coastal and Estuarine Morphodynamics: RCEM2011 © 2011 Tsinghua University Press, Beijing

Development of tidal watersheds in the Wadden Sea

WANG Zheng Bing

Dealtares & Delft University of Technology, Faculty of Civil Engineering and Geosciences P.O. Box 177, 2600 MH Delft, the Netherlands

Julia VROOM, Bram C. VAN PROOIJEN, Robert J. LABEUR, Marcel J.F. STIVE

Delft University of Technology, Faculty of Civil Engineering and Geosciences Stevinweg 1, 2628 CN Delft, the Netherlands

Maarten H.P. JANSEN

Witteveen+Bos

P.O. Box 2397, 3000 CJ Rotterdam, the Netherlands

ABSTRACT: The Wadden Sea consists of a series of tidal lagoons which are connected to the North Sea by tidal inlets. Boundaries to each lagoon are the mainland coast, the barrier islands on both sides of the tidal inlet, and the tidal watersheds behind the two barrier islands. Behind each Wadden Island there is a tidal watershed separating two adjacent tidal lagoons. The locations of the tidal watersheds in the Wadden Sea are not fixed. Especially after a human interference in one of the tidal lagoons, a tidal watershed can move and thereby influence the distribution of area between the lagoons. This appears to be important for the morphological development in not only the basin in which the interference takes place, but also in the adjacent basins. This paper describes theoretical analyses and numerical modelling aimed at improving the insights into the location of the tidal watersheds, their movements, and the impact of the movements of tidal watersheds on the morphological development of a multi-basin system like the Wadden Sea.

1 INTRODUCTION

The Wadden Sea is separated from the North Sea by a series of barrier islands (Figure 1). It contains the largest coherent tidal flat area in the world, connected to the North Sea via tidal inlets between the barrier islands. Because of its exceptional size, great beauty and richness in unique natural assets this coastal wetland became a World Heritage Site in 2009.

One of the unique characters of the geomorphology of the Wadden Sea is that all the tidal inlets have a well-developed flood-tidal delta (Stive and Wang, 2003). Behind each of the barrier islands a morphological high is present, which are the tidal watersheds. These tidal watersheds divide the Wadden Sea into a series of tidal lagoons. The presence of these relatively high tidal watersheds is essential for the stability of the multi-inlet system of the Wadden Sea according to Van de Kreeke et al. (2008).

In previous studies (Oost, 1995, Elias, 2006) it has been shown that the locations of the tidal watersheds in the Wadden Sea cannot be considered as fixed. Especially after a human interference in one of the tidal basins a tidal watershed can move and thereby influence the distribution of area between the basins. Due to the closure of the Lauwerszee, which was a part of the tidal basin of the Frisian Inlet, the tidal watershed behind the Schiermonikoog Island has moved eastwards (Oost, 1995), and the tidal watersheds behind the Engelsmanplaat have moved westwards (Wang and Oost, 2011). After the construction of the Afsluitdijk closing the Zuiderzee from the Texel Inlet and the Vlie Inlet the tidal watershed separating Marsdiep and Vlie has been moving such the basin Marsdiep becomes larger at the cost of the Vlie basin.

The movement of a watershed after an interference appears to be important for the morphological development in not only the basin in which the interference takes place, but also in the adjacent basins. As a consequence the morphological equilibrium in a basin changes in time. Understanding of the movement of the tidal watersheds is thus important for the prediction of the morphological development and the sediment demand in the Wadden Sea on the long-term, and thus relevant for the practice of coastal maintenance and the sustainable management of the Wadden Sea. The tidal basins in the Wadden Sea can thus not be considered as isolated from each other when studying their morphological development responding to environmental changes and human interferences.

Figure 1 The Wadden Sea (www.waddensea-secretariat.org)

How the movement of a tidal watershed depends on the morphology of the adjacent basins is still insufficiently understood. In the past the positions of the tidal inlet and thereby also the positions of the tidal watersheds were more or less anchored by the large basins behind them. Because of the closures of (parts of) these basins these anchors are removed. Nowadays, the defence of the heads of the barrier islands (on the west side of the islands) plays a role.

watershed. A morphological tidal watershed is defined as the line between the two basins with the highest bed level. The corresponding hydraulic tidal watershed is the division line between the two basins on the basis of tidal filling. Whether a particular area behind a barrier island belongs to one or the other tidal basin depends on via which tidal inlet the area is filled during flood and emptied during ebb. A practical definition can be the division line between two basins with the smallest variance (standard deviation) of the flow velocity. A hypothesis is that a system is in equilibrium concerning the locations of the tidal watershed when the two kinds of watershed are at the same location and when out of equilibrium the morphological tidal watershed moves in the direction of the hydraulic tidal watershed.

In this paper we first analyse the influence of the movement of a tidal watershed on the morphological development by considering a two-basin system. Then we attempt to obtain insights into which factors and how these factors influence the location of the hydraulic tidal watershed by theoretical analysis and numerical modelling.

2. INFLUNCE OF LOCATION OF TIDAL WATERSHED ON MORPHOLOGICAL EQUILIBRIUM 2.1 Morphological equilibrium

For tidal basins with well-developed flood-tidal delta empirical relations exist for the morphological equilibrium (Stive and Wang, 2003). The tidal basin can be schematised into large morphological elements of which the equilibrium sizes are related to hydrodynamic parameters like the tidal range and the tidal prism. The tidal prism is strongly dependent on the size of the basin, especially for the short basins like most of the Wadden Sea basins, in which the tidal range is more or less constant and weakly dependent on the morphology. This means that movement of tidal watersheds can have substantial influence on the morphological equilibrium of the Wadden Sea as it influences the size of the basins. The morphological equilibrium of the basins has significant influence on the sediment exchange between the basin and the North Sea coasts of the barrier islands. This sediment exchange, in most cases implying a sediment import to the Wadden Sea basin, is important for the coastal maintenance as well as for the sustainable management of the Wadden Sea. It is important for the coastal maintenance because import to the Wadden Sea basins causes erosion of the North Sea coasts of the Wadden Islands. It is important for the management of the Wadden Sea because the import determines to which extent the Wadden Sea bottom can keep pace with the relative sea-level rise and in how far the Wadden Sea can restore its morphological equilibrium after a human interference. The movements of the tidal watersheds introduce extra uncertainties concerning the morphological equilibrium and the sediment demand of the basins. Here this is illustrated by considering a two-basin system like the Marsdiep-Vlie system after first considering a single basin showing the influence of the size of the basin.

2.2 Single basin

Within a tidal basin two morphological elements are distinguished, viz. the aggregated inter-tidal flat and channel elements. For both elements empirical relations are available defining their morphological equilibrium value.

For the inter-tidal flat there are two empirical relations, one for its area and one for its height (Renger and Partenscky, 1974, Eysink and Biegel, 1992).

5 0.5 1 2.5 10 fe b b A A A      (1) fe fe h  H ₍₂₎

Herein Afe [m2] is equilibrium tidal flat surface area; Ab [m2] is basin surface area; H is tidal range and according to Eysink (1991)

9

0.24 10

fe f Ab

with f=0.41. The equilibrium volume of the inter-tidal flat, i.e. the sediment volume between low water (LW) and high water (HW), is thus per definition:

fe fe fe

V A h (4)

The channel volume is defined as the water volume under LW in the basin. Its equilibrium value is related to the tidal prism as follows:

1.55

ce c

V  P (5)

The tidal prism P is the wet volume in the basin between LW and HW, thus

b f

PA H V (6)

Using these equations the morphological equilibrium of a tidal basin can be determined from two parameters, the total basin area Ab and the tidal range H. As an indication for the sediment demand in a basin one can use the total wet volume of the basin under HW:

b c

V V  P (7)

The difference between the actual value of Vb and its value at equilibrium is the amount of sediment a basin needs to achieve equilibrium, i.e. the sediment demand of the basin.

2.3 Two-basin system

Consider two adjacent and inter-linked tidal basins with a fixed total basin area:

1 2 constant

b b b

A A A  (8)

with subscript 1 and 2 referring to basin 1 and basin 2. Consider first the situation that the tidal ranges in the two basins are the same:

1 2

H H H (9)

As the boundary between the basins is movable the area of each basin can vary between zero and Ab,

thus 1

0

1

A

A



 



For each division between the two basins, i.e. for each value of  between 0 and 1, the equilibrium volume of the inter-tidal flat, the corresponding tidal prisms and the equilibrium volume of the channel can be calculated for each basin, using the equations in the previous subsection. As an example for the tidal prisms we have:





















Similar relations can be derived for the wet volumes under HW in the two basins and for the two basins together we have:





From Equation (12) it becomes clear that the total tidal prism is maximum when the two-basin system becomes a single basin, and it is minimum when the two basins are equal in size. The same behaviour applies for the total wet volume under HW. This is illustrated in Figure 2, which shows the results calculated using the values of the Marsdiep-Vlie system concerning the total basin area and the tidal range. Note that the difference between the two extreme cases (=0 and =0.5) concerning the sediment demand can be very large as shown by the figure.

Figure 2 The total tidal prism and the total (wet) volume under HW of a two-basin system. Used values: Ab=1370 km2, H=1.8 m, c=10-5 m-0.55

In reality the tidal ranges in the two basins are not always the same, as the Marsdiep-Vlie case clearly illustrates. Also for such a case the same calculations can be carried out, see Figure 3 in which the tidal range in basin 1 (representing Marsdiep) is taken as 1.52 m and for basin 2 (representing Vlie) is taken as 1.89 m, following Van Geer (2007). The minimum value of the total tidal prism as well as of the total wet volume under HW now occurs at a larger value of  (>0.5), i.e. when the Marsdiep basin is larger than the Vlie basin. For making both parameters dimensionless (see Eq. 12 & 13) the average value of the two tidal ranges is used.

Figure 3 The total tidal prism and the total (wet) volume under HW of a two-basin system. Used values: Ab=1370 km2, H1=1.52 m, H2=1.89 m, c=10-5 m-0.55

As mentioned above, the total wet volume is an indication of the sediment demand. As an example, the sediment demand with respect to the initial condition in 1970 concerning the volumes of the inter-tidal flats and the channels as reported by Steetzel and Wang (2003), is shown in the following figure. The sediment demand is maximal when the total equilibrium wet volume in the two basins is minimal. For the system under consideration this will occur when the Marsdiep basin is slightly larger than the Vlie basin. At present the Marsdiep basin is still smaller than the Vlie basin but the Marsdiep basin is increasing in size at the cost of the Vlie basin. The system is thus developing towards a situation with larger sediment demand.

Total sand demand

-2.00E+09 0.00E+00 2.00E+09

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

relative area Marsdiep

cub

ic

m

et

er

Figure 4 The total sediment demand. Used values: Ab=1370 km2, H1=1.52 m, H2=1.89 m, c=10-5 m-0.55 The total sediment demand in such a two-basin system depends thus on the location of the tidal watershed between them, or the area distribution between the two basins. The question arises what is the most likely end situation when morphological equilibrium is established. As an attempt to answer this question the „shortest way‟ hypothesis is put forward here. Consider the sediment demands in the two basins as the two components of a vector in a two-dimensional space, then this hypothesis says that the system will end at such an area division between the two basins that the length of this vector is minimal. It is called „shortest way‟ because this is the state to which the system probably needs the least time to

achieve. This reasoning implies the assumption that this state represents a stable morphological equilibrium. In Figure 5 the sediment demands in the two basins are plotted separately, as well as the length of the defined vector. The „shortest way‟ hypothesis suggests that the system will end at a situation in which the Marsdiep has an area which is 60% of the total area of the system. This is near the state that the total sediment demand is maximal (see Figure 4). The present development seems in the direction to this state, as can be seen in Figure 6, which shows the movements of the tidal watersheds since the closure of Zuiderzee and changes of the sizes of the various tidal basins in the Dutch Wadden Sea. The basins Marsdiep and Eierlandse Gat have been expanding at the cost of the Vlie. A tidal flow simulation with the same open sea boundary conditions has been carried out for each year with a measured bathymetry. The tidal watersheds have been determined as the places at which the variance of the flow velocity magnitude is minimal.

Sediment demand in the two basins

-5.00E+09 -4.00E+09 -3.00E+09 -2.00E+09 -1.00E+09 0.00E+00 1.00E+09 2.00E+09 3.00E+09 4.00E+09 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Ab1 / A_tot cub ic m et er basin 1 basin 2

length of sediment demand vector

0.00E+00 1.00E+09 2.00E+09 3.00E+09 4.00E+09 5.00E+09 6.00E+09 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Ab1 / A_tot C ub ic m et er

Figure 5 Sediment demand of the two-basin system. Left: sediment demand in the two basins. Right: length of the vector with the sediment demands of the two basins as components

1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 -10 -8 -6 -4 -2 0 2 4 6 8x 10 7 Time [years] A b( t) -A b( fir st ) [ m 2 ] Marsdiep Eierlandse gat Vlie Amelander Zeegat

Figure 6 Development of the positions of tidal divides (top) and the corresponding changes of the areas of the various basins (bottom), (After Van Geer, 2007)

3 FACTORS INFLUENCING THE LOCATION OF A HYDRAULIC TIDAL WATERSHED 3.1 Theoretical analysis for a simple case

In order to obtain more understanding about the behaviour of the tidal watershed we analyse a simple case: The region between two adjacent tidal inlets, i.e. the area behind a barrier island, is considered as a prismatic channel. The problem is then simplified to the tidal propagation through a channel with constant water depth. We consider the propagation of a single tidal component. At one side of the channel the water level is then described by a cosine function and at the other side of the channel the amplitude as well as the phase may be different.

 

 







 



 



Figure 7 Sketch of the simplified problem

We now ask our self the question will there be a hydraulic tidal watershed, i.e. a place in the channel where the amplitude of the flow velocity is minimal? Which factors determine whether or not a tidal watershed is present? What is the position of the tidal watershed? Which factors determine the position of the tidal watershed?

First we try to answer the questions using an analytical approach. To do this we solve the linear equation for tidal propagation with the assumption that the effect of bottom friction can be neglected:

2 2 2 2 c 2 t x    _    (14) Herein  = Water level t = Time

c = Propagation velocity of tidal wave x = Coordinate along the channel (island)

The solution of this equation satisfying the boundary conditions as shown in Figure 1 is

 



_{ }



 

_{ }

 





sin sin k L x kx t x t a t kL kL          (15) Herein

 = frequency of the tidal wave

L = the length of channel / island k = Wave number = /c

a = Amplitude ratio between the two ends of the channel

 = Phase lag between the two ends of the channel

The flow velocity can be determined from the following equation:

0 u g t x   _  _   (16)

The solution for the flow velocity is:

 



_{ }



 

_{ }

 





sin sin k L x kx gk gk u t x t a t kL  kL             (17)

The amplitude of the flow velocity is then

 





_{ }

 

_{ }

 

2 2

cos cos cos cos

ˆ sin sin sin k L x a kx kx gk u x a kL kL     _  _     _ _ _ _     (18) Solving the equation that the derivative of the amplitude of the flow velocity is zero yields:

 

_



_



 

_{ }

sin 2 2 cos sin

tan 2

cos 2 2 cos cos

kL a kL kx kL a a kL       (19)

This is the equation determining the location of the tidal watershed. Figure 8 shows how the position of the hydraulic tidal watershed is influenced by the ratio between the tidal amplitude at the two ends a and the phase-lag  for various values of kL.

The following observations are made from the analytical solution:

 If the tidal amplitudes at the two ends are the same, a tidal watershed exists and is located in the middle for the case that there is no phase difference between the two ends. This is the trivial case that the tides at the two ends are exactly the same.

 It is remarkable that the phase difference has no influence on the position of the tidal watershed in case the incoming waves have equal amplitudes. But the phase difference does have an influence on the absolute value of the minimum of the flow velocity amplitude in this case. The phase difference has to be smaller than kL, if it is equal to kL a purely propagating wave will be present.

 Even more remarkable is the influence of the amplitude difference between the two ends. The tidal watershed is closer to the side with larger tidal amplitude than the side with smaller amplitude. According to the linear solution the amplitude ratio has even more influence on the existence and the position of tidal watersheds than the relative phase difference. This means that it is not the propagation of the tidal wave but the spatial variation of the tidal range that has the most influence on the location of the tidal watershed behind a barrier island.

 There is not always a tidal watershed. If the tidal amplitudes at the two ends are not equal there is a limit to the phase difference between the two ends for the existence of a tidal watershed. When the relative phase difference between the two ends becomes larger than the limit, no tidal watershed can be found (no solution for x0 in the realistic domain 0<x<L). The limit for the phase difference becomes smaller if the ratio between the amplitudes at the two ends differs more from unit. Interpreted for the Wadden Sea case this means that a tidal watershed only can exists if the propagation of the tidal wave on the sea side of a barrier island is much faster than the propagation in the basin behind the island. In other words, the tidal watershed can only exist if the Wadden Sea is relatively shallow.

 The phase difference limit for the existence of the tidal watershed becomes larger if the island is longer and/or the basin behind the island is shallower. It seems thus that the barrier islands in should have lengths above a certain limit depending on the characteristics of the tide and the tidal propagation.

3.2 Numerical modelling simulation for the simple case

In order to verify the conclusions drawn from the analytical solution a series of numerical simulations have been carried out for the same simple case as considered by the analytical solution in the previous section. The simulations are carried out with Delft3D flow. The set up of the simulations are given in the following table.

Table 1 Set up of the Delft3D simulations Run Common Boundary conditions

000

L=10000m H=5 m C=80 m1/2_s-1

Water level, single component with the same amplitude at both ends. 001 Same amplitude & 0.5 degree phase difference between the two ends 002 Same amplitude & 2 degree phase difference between the two ends 003 no phase difference, but at the end factor 1.005 increase of amplitude 004 no phase difference, but at the end factor 1.01 increase of amplitude 005 0.5 degree phase difference, at the end factor 1.01 increase of amplitude 006 -2 degree phase difference，factor 1.01 increase of amplitude

KL=0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.85 0.9 0.95 1 1.05 1.1 1.15 a X o /L phi/KL=0 phi/KL=0.1 phi/KL=0.2 phi/KL=0.3 phi/Kl=0.5

Figure 8 Location of the tidal watershed according to the analytical solution KL=0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.85 0.9 0.95 1 1.05 1.1 1.15 a X o /L phi/KL=0 phi/KL=0.1 phi/KL=0.2 phi/KL=0.3 phi/Kl=0.5 KL=0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.75 0.85 0.95 1.05 1.15 1.25 a X o /L phi/KL=0 phi/KL=0.1 phi/KL=0.3 phi/KL=0.5 phi/KL=0.7

The tidal period is set at 12 hour and tidal amplitude (for run 000) is set to 0.1 m in order to be close to the conditions for which the linear model for tidal wave propagation is applicable. The results of the simulations for the amplitude of the flow velocity are show in Figure 9. It can be seen that all the conclusions from the analytical solution are confirmed by the numerical results.

Amplitude of flow velocity

0.00 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0 2000 4000 6000 8000 10000 x (m) U _a m p (m /s ) run 000 run 001 run 002 run 003 ron 004 run 005 run 006

Figure 9 Simulated amplitude of the flow velocity along the channel for the various runs, tidal amplitude is 0.1 m

Figure 10 shows the results of the same series of simulations but now with the tidal amplitude equal to 1 m instead of 0.1 m. This means that the tidal amplitude to water depth ratio is no more small, which means that the non-linear effects become more important. The figure shows that the phase lag between the two ends does have influence on the position of the minimum of the amplitude of flow velocity (tidal watershed), even for the case that the tidal amplitudes at the two ends are the same. The tidal watershed moves to the side with larger phase, i.e. where the tidal wave arrives later.

Amplitude of flow velocity

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0 2000 4000 6000 8000 10000 x (m) U _a m p (m /s ) run 000 run 001 run 002 run 003 ron 004 run 005 run 006

Figure 10 Simulated amplitude of the flow velocity along the channel for the various runs, tidal amplitude is 1 m

In summary the following conclusions are drawn from the numerical modelling results:

 All conclusions from the analytical solution are confirmed by the numerical simulations with relatively small amplitude to water depth ratio, i.e. for conditions at which the linear solution is applicable.

much smaller than the water depth, the relative phase lag between the tides at the two ends become more important for the existence and the position of the tidal watershed. The tidal watershed moves to the side with the largest phase, i.e. the end where the tidal wave arrives later.

4 CONCLUSIONS

The back barrier basins of the tidal inlets of the Wadden Sea are separated from each other by tidal watersheds. The tidal watersheds do not have fixed locations but can move, especially after human interventions like closures of a part of the Wadden Sea. The positions of the tidal watersheds have direct influence on the sizes of the corresponding tidal basins. Using the empirical relations for the morphological equilibrium it is demonstrated that the movement of the tidal watersheds after e.g. the closure of the Zuiderzee in the Dutch Wadden Sea has substantial influence for the total amount of sediment import needed to restore morphological equilibrium for the remaining basins.

In order to obtain more insights into the locations of the tidal watersheds a theoretical analysis has been carried out for a simplified case. The tidal propagation behind an island is schematised into a simple channel flow with two open boundaries representing the two inlets where the tidal variation of water level is represented by a single tidal component with different amplitudes and phases at the two boundaries. The hydraulic watershed is defined at the location where the amplitude of the flow velocity is minimal. The analytical solution of the linear tidal propagation equation shows that a tidal watershed does not always exist. It can only exist when the island is relatively long and / or the back barrier basin is relatively shallow. The analytical solution shows further surprisingly that the difference in tidal amplitude between the two ends of the channel (island) has more influence on the location of the tidal watershed than the phase difference. When the tidal amplitudes at the two ends are the same the tidal watershed is always in the middle, independent of the phase difference as long as the phase difference is smaller than for the case of a progressive wave which means that the propagation velocity on the sea side is faster than that in the back barrier basin. It is further surprising that the tidal watershed moves to the end with the larger tidal amplitude when the amplitudes at the two ends are not equal. The limit for the phase difference between the two ends for existence of a tidal watershed becomes smaller if the tidal amplitude increases in the direction of tidal propagation. If the difference between the amplitudes at the two ends is too large and / or the phase difference between the two ends is too large no tidal watershed exists because the amplitude of the flow velocity will be monotonously changing along the channel.

The conclusions from the analytical solutions have been verified with numerical simulations using Delft3D for the same simplified case as considered by the theoretical analysis. The numerical results confirm all the conclusions from the analytical solution if the tidal amplitude to water depth ratio is small. When the amplitude to depth ratio is larger, the non-linear effects become more important. This has the consequence that the influence of the phase difference between the two ends becomes more important than according to the linear analytical solution. In contrast to what the analytical solution indicates the position of the tidal watershed does depend on the phase difference if the tidal amplitudes at the two ends are the same. The tidal watershed moves to the end where the tidal wave arrives later, which is more in agreement with the classical view. Our final conclusion is that the variation of the tidal amplitude as well as the direction of the tidal wave propagation on the sea side of the islands has influence on the location of the hydraulic tidal watersheds in the Wadden Sea, but which parameter is more dominant is depending on the water depth - wave height ratio.

5 ACKNOWLEDGEMENTS

The work described in this paper is financed by various projects at Deltares and at TUDelft: The coastal research project finance by Rijkswaterstaat, The various Building with Nature projects financed by Ecoshape, the project “Effect of human activities on eco-morphological evolution of rivers and estuaries” (project nr. 08-PSA-E-001) within the framework of Programme of Strategic Scientific Alliance between China and The Netherlands (PSA), financed by Royal Dutch Academy of Sciences and Arts (KNAW).

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