\section{Shoal propagation with summer dike\label{case:shoal01}}
\newrefsegment

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\section*{Quality Assurance}
{{\footnotesize
\begin{tabular}{p{22mm-2pt}@{}p{27mm-2pt}@{}p{18mm}@{}p{27mm-2pt}@{}p{18mm}@{}p{27mm-2pt}@{}p{\textwidth-139mm-1pt}} \hline
    \rowcolor{dgrey4}
\textbf{Date} & \textbf{Author} & \textbf{Initials} & \textbf{Review} & \textbf{Initials} & \textbf{Approval} & \textbf{Initials}  \\ \hline
13 Dec 2018 & Andries Paarlberg & & Bert Jagers & & Aukje Spruyt &   \\
            & Lieke Lokin & &  & &  &   \\ \hline
  \end{tabular}
}}

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\section*{Version information}
\begin{tabular}{@{}p{23mm}@{}p{0mm}p{\textwidth-23mm-30pt}}
\textbf{Date of study}& \textbf{:} & 13 Dec 2018 \\
\textbf{Executable}   & \textbf{:} & Deltares, D-Flow FM Version 1.1.294.57940M, Sep 09 2018, 10:46:28 \\
\textbf{Location}     & \textbf{:} & \url{https://repos.deltares.nl/repos/DSCTestbench/trunk/cases/e02_dflowfm/f27_mor1d_tabulated_crosssections/c33_shoal_ds_IBedCond0_us_IBedCond1_toe_sd_low} \\ 
\textbf{SVN revision} & \textbf{:} & -
\end{tabular}

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\section*{Purpose}

\DFLOWFM-1D (mor) can use so-called tabulated cross-sections (cf SOBEK3), where the flow width is specified as a function of the water level. The bed level is updated in the main channel only. \\
%
The purpose of this validation case is to prove that the propagation speed of bed disturbances in \DFLOWFM-1D is correct for different cross-section shapes. In this case a cross section with an increasing width and summer dikes.

%---------------------------------------------------------------------------------
\section*{Linked claims}
\begin{itemize}
\item{\DFLOWFM-1D can simulate the propagation of bed disturbances in cross-sections with summer dikes properly.}
\end{itemize}

\section*{Approach}
We start from the test case with a rectangular cross-section ([NOTE: Check url]), and adjust the cross-section. In this test case the bottom of the cross-section is made smaller and a summer dike is added. Testcase c33 has a 'low' summer dike and testcase c34 has a 'high' summer dike. The low dike is below the equilibrium water level and the high dike well above.

To prove that the computed propagation of bed disturbances non-rectangular cross-sections in \DFLOWFM is correct, the simulated propagation of a shoal is compared with the test case with the rectangular cross-section (c31), for which the propagation has been validated with the analytical approximation and SOBEK3 simulations.

A local shoal is included in the initial bed level of the test case. This local shoal is on top of the equilibrium bed level.

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\section*{Model description}
The model domain, consists of a straight channel with a length of 10 km. The (equilibrium) bed level slope is \num{1d-4}. The shoal is initially located halfway the channel. The initial maximum height of the shoal is 0.4 m above the equilibrium bed level. The cross-section is a semi rectangular profile with a bottom width of 100 m which increases to 200 m at 1 m above the bottom level. For this case, the width of the main channel is equal to the total width of the cross-section. The cross-section of this test case and the reference test case are shown in \autoref{fig:cross_section_c32}.  

\begin{figure}%[H]
	\includegraphics[width=\textwidth]{figures/summerdyke_initial.png}
	\caption{Initial cross-section for the test cases with summer dikes.}
	\label{fig:cross_section_c32}
\end{figure}

The characteristics of the summer dike are shown in \autoref{tab:summerdyke_char}. The height of the summer dike is defined as difference between the bottom level at equilibrium depth and the top of the summer dike.

\begin{table}[H]
  \centering
\caption{Characteristics of the summer dikes.}\label{tab:summerdyke_char}
\begin{center}
\begin{tabular}{| l | l | l | l | l | l | l |} \hline
\textbf{test case} & {\textbf{c33}}   & {\textbf{c34}} \\ \hline
\textbf{name}               &  low             & high     \\ \hline
\textbf{height} [m]         &  3        &  5        \\ \hline
\textbf{flow area} [m$^3$]  &  100      &  100      \\ \hline
\textbf{total area} [m$^3$] &  100      &  100      \\ \hline
\end{tabular}
\end{center}
\end{table}

The model is forced with a constant upstream total discharge of 1000 m$^3$/s. The downstream water level has a fixed value of 4.8775 m AD such that the water depth equal is to the equilibrium water depth. The water level at the downstream end is derived under the assumption that the depth average velocity in the adjusted cross-section is equal to the depth average velocity in the rectangular cross-section. This results in a water level that is 0.25 m higher than for the rectangular cross-section. Any overflow over the summer dike is not taken into account in the derivation of the equilibrium water depth.
The hydraulic roughness of the channel (Ch\'ezy) is 50 m$^{1/2}$/s.

The median sediment diameter is 0.2 mm. The sediment transport is computed with the formula of Engelund-Hansen. As upstream boundary condition, a fixed bed level is applied. At the downstream boundary, no bed level constraints are applied (i.e.\ Neumann boundary condition for sediment transport).

The model has a uniform computational grid size of 250 m. The computational grid of the test case is visualized in \autoref{fig:morphology_shoal_grid}. The time step has a value of 2 minutes.


\begin{figure}%[H]
	\includegraphics[width=\textwidth]{figures/morphology_shoal_network_test011d_test011e.jpg}
	\caption{Computational grid for the testcase.}
	\label{fig:morphology_shoal_grid}
\end{figure}

%
The basis model we use is set-up in SOBEK3. The geometry (tabulated cross-sections, roughness and some other properties), are directly imported into \DFLOWFM using a specific keyword in the MDU-file:
\begin{verbatim} OneDNetworkFile = dflow1d\water flow 1d.md1d \end{verbatim}
Some settings have to be set manually though:
\begin{itemize}
\item{A SOBEK3 import only works if a (dummy) Network file is present. We advise to use a 2D network, to ensure that it doesn't conflict with the 1D network (as imported from SOBEK3).}
\item{Timings such as refdate, start/end time and map-output intervals need to be converted manually to the MDU-file.}
\item{The boundary conditions (locations and values) are not yet imported from SOBEK3. They have to be manually created. We defined one polyline at the upstream boundary for imposing the discharge, and one polyline at the downstream boundary to impose the water level. Note that the boundary lines must be just \emph{outside} the model.}
\item{Recall that in SOBEK3 the water level at the downstream boundary holds at the last pressure point in the grid, while in \DFLOWFM it gets mirrored half a grid cell outside the control volume around each grid point. Effectively, this means that the water level has to be specified as if it is $\Delta x$ further downstream, considering the bed slope. For this case, at the downstream boundary, $\Delta x = 250 m$. For a bed level slope of \num{1d-4} this implies a 2.5 cm lower water level compared to SOBEK3.}
\item{In the MOR input file, the boundaries need to be imposed at the polylines specified in one of the steps above.}
\item{For the upstream (us) and downstream (ds) bed boundary condition we specify: 
\begin{verbatim} IBedCond_us=1 \end{verbatim} 
\begin{verbatim} IBedCond_ds=0 \end{verbatim}
fixed bed and free condition respectively.}

\end{itemize}

\section*{Results}
%
\autoref{fig:shoal_02_initialbedlevel} shows the initial bed level for this test case. \\

\begin{figure}%[h!]
	\includegraphics[width=\textwidth]{figures/c33_fig02_bedlevel_T0.png}
	\caption{Initial bed level.}
	\label{fig:shoal_02_initialbedlevel}
\end{figure}

\autoref{fig:shoal_01_waterlevel_us_ds_Ft} shows the water levels at the upstream an downstream boundaries of the model. The water levels of the summer dike cases are compared with the original rectangular profile (c31). The model with the high summer dike (c34) shows similar behaviour to the case without a summer dike and the same profile (c33).The water level difference between c31 and c34 is almost equal to the imposed water level difference for for these two cases.
In the case with the low summer dike has the same boundary conditions as the case with the high summer dike. At the downstream boundary this results in water levels which are almost equal. The difference is due to the difference in location of the downstream boundary and the grid point. At the upstream boundary the difference between the water levels is larger. In test case c33 the initial water level starts at $\approx 5.7$ m AD and slightly increases in time. In the other two test cases the water level decreases in time.

In the low summer dike case the water depth is not at equilibrium due to overflow. Both, the water level and the bed level start to adjust towards a new equilibrium during.

\begin{figure}%[h!]
	\includegraphics[width=\textwidth]{figures/c33_fig01_waterlevel_us_ds_Ft.png}
	\caption{Water level and water level difference as function of time at the upstream and downstream boundaries}
	\label{fig:shoal_01_waterlevel_us_ds_Ft}
\end{figure}

\autoref{fig:shoal_04_sobek_vs_dflowfm} to \autoref{fig:shoal_06_sobek_vs_dflowfm2} visualize the propagation of the shoal in different ways for a total period of 120 days. \autoref{fig:shoal_04_sobek_vs_dflowfm} shows the longitudinal profile of the bed level with an interval of 10 days. \autoref{fig:shoal_05_sobek_vs_dflowfm} and \autoref{fig:shoal_05_sobek_vs_dflowfm2} compare c31 with c33 and c34 by plotting the bed level in one figure, with an interval of 20 days. \autoref{fig:shoal_06_sobek_vs_dflowfm} and \autoref{fig:shoal_06_sobek_vs_dflowfm2} just give another representation, with an interval of 30 days.

The propagation speed (and evolution of bed level shape) of the shoal in c34 is equal to the reference case. 
These figures shows that the shoal leaves the model domain after about 3 months.
Test case c33 shows different behaviour of the shoal. In this case the shoal does not leave the model domain during the simulation.

\begin{figure}%[h!]
	\includegraphics[width=\textwidth]{figures/c33_fig04_bedlevel_Fx_T0_Tend.png}
	\caption{Shoal propagation in c31, c33 and c34.}
	\label{fig:shoal_04_sobek_vs_dflowfm}
\end{figure}

\begin{figure}%[h!]
	\includegraphics[width=\textwidth]{figures/c33_fig05_bedlevel_Fxt_compare.png}
	\caption{Shoal propagation in c31 and c33.}
	\label{fig:shoal_05_sobek_vs_dflowfm}
\end{figure}

\begin{figure}%[h!]
	\includegraphics[width=\textwidth]{../../c34_shoal_ds_IBedCond0_us_IBedCond1_toe_sd_high/doc/figures/c34_fig05_bedlevel_Fxt_compare.png}
	\caption{Shoal propagation in c31 and c34.}
	\label{fig:shoal_05_sobek_vs_dflowfm2}
\end{figure}

\begin{figure}%[h!]
	\includegraphics[width=\textwidth]{figures/c33_fig06_bedlevel_Fxt_compare.png}
	\caption{Shoal propagation in c31 and c33.}
	\label{fig:shoal_06_sobek_vs_dflowfm}
\end{figure}

\begin{figure}%[h!]
	\includegraphics[width=\textwidth]{../../c34_shoal_ds_IBedCond0_us_IBedCond1_toe_sd_high/doc/figures/c34_fig06_bedlevel_Fxt_compare.png}
	\caption{Shoal propagation in c31 and c34.}
	\label{fig:shoal_06_sobek_vs_dflowfm2}
\end{figure}


The simulated bed level changes for the high summer dike (c34) are equal to the bed level changes found for the same profile without the summer dike. In the test case with the low summer dike (c33) the shoal propagates more slowly through the system. Also the resulting bed slope is milder than the original slope.

The initial and final cross-sections at the middle of the branch, at $x = 1.45 \times 10^5$, are shown in \autoref{fig:cross_section_c32_diff} and \autoref{fig:cross_section_c34_diff}. The bedlevel changes at the corner points at the same location are shown in \autoref{tab:morphology_shoal_cornerpoints}. The difference in bed level change at the corner points in c33 and c35 is due to the assumptions made for bed level chang: Bed level changes are relative to the highest point of the cross-section. 

\begin{figure}%[!tbp]
  \centering
  \begin{minipage}[b]{0.45\textwidth}
    \includegraphics[width=\textwidth]{figures/low_summerdyke_start_end.png}
    \caption{Initial and final cross-section for the testcase c33 at $x = 1.45 \times 10^5$.}
		\label{fig:cross_section_c32_diff}
  \end{minipage}
  \hfill
  \begin{minipage}[b]{0.45\textwidth}
    \includegraphics[width=\textwidth]{../../c34_shoal_ds_IBedCond0_us_IBedCond1_toe_sd_high/doc/figures/high_summerdyke_start_end.png}
    \caption{Initial and final cross-section for the testcase c34 at $x = 1.45 \times 10^5$.}
		\label{fig:cross_section_c34_diff}
  \end{minipage}
\end{figure}



\begin{table}[H]
  \centering
\caption{Comparison of the bed level changes in the corner points of the cross-section at $x = 1.45 \times 10^5$ m. The rectangular cross-section has one corner point while the adjusted has two.}\label{tab:morphology_shoal_cornerpoints}
\begin{center}
\begin{tabular}{| l | l | l | l | l | l | l | l | l | l |} \hline
 & \multicolumn{3}{c|}{\textbf{c31}} & \multicolumn{3}{c|}{\textbf{c33}} & \multicolumn{3}{c|}{\textbf{c34}} \\
 &  \emph{initial}  & \emph{end}       & \emph{diff}      & \emph{initial}   & \emph{end}       & \emph{diff}   & \emph{initial}   & \emph{end}       & \emph{diff}   \\ \hline
1 &  0.9250   &  0.5164   & 0.4086    & 0.9250    & 0.5164    &  0.4009 & 0.9250    & 0.5164    &  0.4009  \\ \hline
2 &           &           &           & 1.8000    & 1.4998    &  0.3007 & 1.8000    & 1.4998    &  0.3007  \\ \hline
\end{tabular}
\end{center}
\end{table}

\subsection*{Low summer dike }
With a high summer dike, eg. the summer dike is always higher than the maximum water level in each grid point, the model behaves equal to a similar case without the summerdike.
When the summer dike is lower than the maximum water level part of the discharge flows over the summer dike, so there is less water in the channel. This lower discharge results in a different equilibrium situation. In the initial situation the flow velocity is lower than in the reference case. This results in less transport. The equilibrium slope will be milder. This results in accretion over the whole domain. In \autoref{fig:shoal_07_U} the flow velocity in the two cases with high and low summer dikes are compared. The difference in flow velocity at time step 1 is 0.06 m/s at the upstream boundary and 0.10 m/s at the downstream boundary. The difference in flow velocity is due to the larger flow area.


\begin{figure}%[h!]
	\includegraphics[width=\textwidth]{figures/c33_fig07_flowvel_Fxt_compare.png}
	\caption{Comparison of flow velocity for c33 and c34.}
	\label{fig:shoal_07_U}
\end{figure}

%
\section*{Conclusion}

\DFLOWFM-1D simulates the propagation of a shoal properly when summer dikes are used in the model.
Using high a high summer dike the propagation of the shoal is similar to the propagation of a shoal in the same situation where no summer dike is implemented.
Using a low summer dike the bed level starts to adjust to a new equilibrium where not all the discharge flows through the channel.

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