# Future Response of the Wadden Sea Tidal Basins to Relative Sea-Level rise—An Aggregated Modelling Approach

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## Abstract

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## 1. Introduction

#### 1.1. Study Area

#### 1.2. Influence of Sea-Level Rise

#### 1.3. Modelling the Response to Sea-Level Rise

## 2. Modelling Approach—Aggregated Model ASMITA

- The ebb-tidal delta, with its state variable V
_{d}= total excess sediment volume relative to an undisturbed coastal bed profile [L^{3}]; - The inter-tidal flat area in the tidal basin, with its state variable V
_{f}= total sediment volume between mean low water (MLW) and mean highwater (MHW) [L^{3}]; - The channel area in the tidal basin, with its state variable V
_{c}= total water volume below MLW [L^{3}].

_{b}is the basin area and a is the tidal amplitude. As the equilibrium values of V

_{c}and V

_{f}can be calculated from a and P (see [69]), the equilibrium value of V is thus a function of a and A

_{b}:

- t = time [T];
- w = vertical exchange coefficient [LT
^{−1}]; - δ = horizontal exchange coefficient [L
^{3}T^{−1}]; - n = power in the formulation for the local equilibrium concentration [-];
- c
_{E =}overall equilibrium concentration [-]; - R = relative sea-level rise rate [LT
^{−1}].

_{c}is the critical flow velocity for incipient movement of sediment particles. It is larger for coarser sediment than for finer sediment. This means that for the same hydrodynamic condition n is larger for coarser sediment than for finer sediment.

## 3. Analysis and Modelling Results

#### 3.1. Dynamic Equilibrium and Critical SLR Rate

_{c}of sea-level rise rate beyond which the tidal basin will drown:

_{m}as defined by linearizing Equation (4) or Equation (9) for the case R = 0 [73]. The morphological time scale is the time needed for a small deviation from the morphological equilibrium to decrease by a factor e. The relation between the two timescales is:

- Equation (15) for R
_{c}revealed the importance of T. In this relation H_{e}is the equilibrium depth. Empirical relations [69] were available from which its value can be evaluated if the tidal amplitude a and the size of the tidal basin is known, H_{e}= F(A_{b}, a). Moreover, it was also connected to direct observations. For basins which are approximately in equilibrium, as, for example, when the basin has not been impacted by human interference for a long time, the equilibrium depth can be evaluated from the measured bathymetry. Note that a correction is necessary when it is in a dynamic equilibrium as it has been forced by sea-level rise with a constant rate for a long time (See Figure 4). An example within the Dutch Wadden Sea is the Ameland Inlet. - Equation (15) can also be used for estimating the timescale T if R
_{c}can be derived from observations. This is the case for the Texel Inlet, for example, in which a large sediment deficit arose after the closure of the Zuiderzee in 1932 [20,21]. The large sediment deficit (depth of the basin much larger than equilibrium depth, or h much larger than 1) in the basin has practically the same effect on sediment import as a ‘drowned’ system (see Figure 8), implying that the observed sedimentation rate is close to the critical sea-level rise rate. - The power n influences the morphological timescale, but not the critical sea-level rise rate. It does influence the dynamic equilibrium state.

_{e}value for the same dimensionless sea-level rise rate r is smaller. This means that for larger n values, the deviation of the dynamic equilibrium from the morphological equilibrium as defined by the empirical relations is smaller. Figure 4 also shows that for larger n values, the transition from gradual increase to rapid increase of h

_{e}occurred at a larger value of r. However, the magnitude of n also has an influence on the critical SLR rate according to Equations (15) and (16): the larger the n value, the lower the critical rate for the same morphological timescale.

#### 3.2. Transient Development

_{e}(the dynamic morphological equilibrium state):

_{a}is the dimensionless time (value of τ) after which a deviation from the dynamic equilibrium would be decreased with a factor e according to the linearized model. For r = 0, τ

_{a}= 1/n which is the same relation between the morphological timescale and the timescale T as given by Equation (16), implying that a larger n results in a smaller morphological timescale, for the same value of T. The morphological timescale depends on the SLR rate r, the larger the r value, the larger the morphological timescale. This means that for a higher accelerated SLR rate it takes longer for the new dynamic morphological equilibrium to be reached.

_{e}= 1.4). However, if the SLR rate is equal to 90% of the critical value (r = 0.9, Figure 6b) then both are larger. The equilibrium depth increases to more than doubled to 3.2, while the dimensionless timescale increases by an order of magnitude to 15.8. When the SLR rate approaches the critical value, the morphological timescale increases to a very large or even infinitely large value (Figure 7).

_{e}, or h > 1, it is negative and its magnitude is the sediment import rate to the basin. With the help of Equations (5) and (8) the following relation can be derived for the sediment import rate S (depicted in Figure 8):

#### 3.3. Application to the Dutch Wadden Sea

## 4. Concluding Discussions

_{c}= H

_{e}/nT

_{m}. The equilibrium depth H

_{e}can be determined from the field observations, directly or indirectly via empirical relations. The morphological timescale T

_{m}can also be derived from observed development if the considered system is disturbed from its equilibrium. A system can be disturbed not just due to human interventions but also due to, e.g., nodal tidal cycle [74,75], or major storm events [76]. This relation thus makes it possible to determine the critical rate of SLR from field observations of limited time period.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Wadden Sea, the light-colored areas between the barrier islands and the mainland coast (based on picture from www.waddensea-secretariat.org).

**Figure 3.**The three-elements schematisation (

**a**) for a tidal inlet in the aggregated scale morphological interaction between tidal basin and adjacent coast (ASMITA) and the definitions of the hydrodynamic and morphological parameters tidal prism P (

**b**), area A

_{f}and volume V

_{f}of tidal flats (

**c**), area A

_{c}and volume V

_{c}of channels (

**d**).

**Figure 5.**Transient development for various sea-level rise (SLR) rates starting from equilibrium state (h(0) = 1). Influence of the power n can be seen by comparing the two panels, (

**a**) for n = 2 and (

**b**) for n = 5.

**Figure 6.**Transient development starting from various initial conditions (h(0)) for four SLR rates: (

**a**) r = 0.5 (SLR far below critical rate), (

**b**) r = 0.9 (SLR just below critical rate), (

**c**) r = 1.1 (SLR just above critical rate), (

**d**) r = 2 (SLR far above critical rate), in all cases n = 2. The dynamic equilibrium h

_{e}and the morphological timescale τ

_{a}are given in the title of the panels.

**Figure 8.**Relation between sediment import rate and the morphological state. h = 1 represents the morphological equilibrium without SLR. The import rate approaches the maximum value for large h.

**Figure 9.**Dimensionless SLR rate r, i.e., normalized with the critical rate (given between brackets in the legend) as calculated in [21] for the tidal inlets in the Dutch Wadden Sea, increases linearly with the SLR rate R. The solid red line (r = 1) indicates the initiation of drowning and the dashed orange line (r = 0.8) indicates the level above which significant impact of SLR is expected.

**Table 1.**Critical SLR rate for drowning of the various tidal inlet systems in the Dutch Wadden Sea from [21] and the dimensionless SLR rate r for four different SLR rates (2, 4, 6 and 8 mm/y). The equilibrium depth H

_{e}calculated using the empirical relations and the parameters for basin area A

_{b}and tidal range H are also given. The listed time scale T is calculated using the relation with R

_{c}and H

_{e}.

Inlet | A_{b} (km^{2}) | H (m) | H_{e} (m) | R_{c} (mm/y) | T (Year) | r for SLR Rate = | |||
---|---|---|---|---|---|---|---|---|---|

2 mm/y | 4 mm/y | 6 mm/y | 8 mm/y | ||||||

Texel | 655 | 1.65 | 2.8 | 7.00 | 400 | 0.29 | 0.57 | 0.86 | 1.14 |

ELGT | 157.7 | 1.65 | 1.7 | 18.0 | 90 | 0.11 | 0.22 | 0.33 | 0.44 |

Vlie | 715 | 1.9 | 3.5 | 6.30 | 560 | 0.32 | 0.63 | 0.95 | 1.27 |

Amel | 276.3 | 2.15 | 2.7 | 10.4 | 260 | 0.19 | 0.38 | 0.58 | 0.77 |

PinkeG | 49.6 | 2.15 | 1.7 | 32.7 | 55 | 0.06 | 0.12 | 0.18 | 0.24 |

ZoutK | 105 | 2.25 | 2.1 | 17.1 | 125 | 0.12 | 0.23 | 0.35 | 0.47 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lodder, Q.J.; Wang, Z.B.; Elias, E.P.L.; van der Spek, A.J.F.; de Looff, H.; Townend, I.H. Future Response of the Wadden Sea Tidal Basins to Relative Sea-Level rise—An Aggregated Modelling Approach. *Water* **2019**, *11*, 2198.
https://doi.org/10.3390/w11102198

**AMA Style**

Lodder QJ, Wang ZB, Elias EPL, van der Spek AJF, de Looff H, Townend IH. Future Response of the Wadden Sea Tidal Basins to Relative Sea-Level rise—An Aggregated Modelling Approach. *Water*. 2019; 11(10):2198.
https://doi.org/10.3390/w11102198

**Chicago/Turabian Style**

Lodder, Quirijn J., Zheng B. Wang, Edwin P.L. Elias, Ad J.F. van der Spek, Harry de Looff, and Ian H. Townend. 2019. "Future Response of the Wadden Sea Tidal Basins to Relative Sea-Level rise—An Aggregated Modelling Approach" *Water* 11, no. 10: 2198.
https://doi.org/10.3390/w11102198