Water transport
The DEMO model has a very simple formulation for the hydrodynamics (exchange between the bordering compartments and the North Sea). It is assumed that the compartments are homogeneously mixed. For two neighboring compartments (i
and j
), water is exchanged between both compartments with a constant rate of Ex_{i,j}
(m3 d-1). The rate is bidirectional, so the water transport rate from compartment i
to compartment j
is the same as the water transport rate from compartment j
to i
. As a result, there is no net transport of water between the compartments and the volume remains constant.
flowchart LR;
Comp_i <==>|Ex_i,j| Comp_j;
The same formulation is used for the exchange between the North Sea and the compartment neighboring the North Sea (Ex_{i,NS}
). The exchange parameters Ex_{i,j}
and Ex_{i,NS}
are estimated by fitting a tracer model to data, artificially generated by a tracer simulation with the 3-D hydrodynamic model of Jiang et al. (2019)1.
3D Hydrodynamic model
A 3D hydrodynamic model has been developed by Jiang et al. (2019) 1. The model was developed in the open source General Estuarine Transport Model GETM in combination with the General Open Turbulence Model GOTM. The model has a 300x300 meter rectangular grid in the Oosterschelde and a part of the Voordelta in the North Sea.
With the 3D model a tracer simulation with an inert tracer (r
= 0) was done. At the start of the simulation the fraction of North Sea water in the North Sea was 1 (100%) and in he Oosterschelde the fraction of North Sea water was 0 (0%). The model was run for one year from 1 January 2009 to 31 December 2009. For each hour, the depth-averaged tracer concentration and the water level in each (300x300 m) grid-cell was exported. The intrusion of North Sea water into the Oosterschelde with the tidal currents that was calculated by the model is shown in the movie below .
Calibration exchange rates
The exchange rates between the neighboring compartments are estimated by fitting the output of the DEMO tracer model with an inert tracer (r
= 0) to "artificial" data generated by the 3D GetM/GotM model. These artificial data were the result of the tracer simulation with the 3D hydrodynamic model from Jiang et al. (2019) 1. For each DEMO model compartment, the depth-weighted average fraction North Sea water was calculated for each time step (hour). Initial conditions for each compartment were calculated from the artificial data as the average fraction of North Sea water in each compartment during the first four tidal cycles. At the North Sea boundary the tracer concentration was 1. The models are fitted using the modFit function in R (Soetaert and Petzold, 2010)2. The 0D model and the models with the 4 and 28 compartments are fitted separately. The parameters were estimated using the Levenberg-Marquardt algorithm where the objective was to minimize the sum of squared residuals (SSR) between model and "observations".
0D model
For the 0D model only 1 parameter (exchange rate between the North Sea and the Oosterschelde) had to be fitted. In the figure below, the fitted model is compared with the artificial data. The fitted model is used to calculate the average residence time (RT
, d) and turnover time (
TT
, d) of the water within the compartment.
Results of the DEMO tracer model (blue line), together with the artificial data from the 3D the GetM/GotM model (orange dots).
The fitted exchange rate between the North Sea and the Oosterschelde is implemented in the function “Transp_1C()”.
4 Compartments
For the model with 4 compartments, four exchange parameters had to be estimated:
- Exchange rate between North Sea and compartment West (m3 d-1);
- Exchange rate between compartment West and Central (m3 d-1);
- Exchange rate between compartment Central and East (m3 d-1);
- Exchange rate between compartment Central and North (m3 d-1)
Results of the DEMO tracer model (blue lines), together with the artificial data from the 3D the GetM/GotM model (orange dots) for the compartments West, Central, East and North.
The fitted exchange rates between the neighboring compartments are implemented in the function “Transp_4C()”.
28 Compartments
For the model with 28 compartments, a total of 42 exchange rates had to be estimated.
Results of the DEMO tracer model (blue lines), together with the artificial data from the 3D the GetM/GotM model (orange dots) for the compartments 1-15.
Results of the DEMO tracer model (blue lines), together with the artificial data from the 3D the GetM/GotM model (orange dots) for the compartments 16-28.
The fitted exchange rates between the neighboring compartments are implemented in the function “Transp_28C()”.
References
- Jiang, L., T. Gerkema, J. W. M. Wijsman en K. Soetaert (2019) Comparing physical and biological impacts on seston renewal in a tidal bay with extensive shellfish culture. Journal of Marine Systems 194: link
↩️ - Soetaert, K. and T. Petzoldt (2010) Inverse Modelling, Sensitivity and Monte Carlo Analysis in R Using Package FME. Journal of Statistical Software 33(3): 1:28.link
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