<%namespace name="tools" file="tools.tex"/>

In this section, a detailed comparison between simulated physics over an evolving bathymetry and the measurements obtained during the Deltaflume experiment in 2006 \citep{VanGent2008} is made. For brevity this comparison is performed only for test T01 (this test corresponds best to the Dutch normative conditions). The simulations are performed on a regular grid with dx = 1m and input to the model are time series of short wave varying energy (low pass filtered on the wave group time scale) and incoming (bound) long waves. The time series are constructed from pressure and flow measurements at x = 41m from the wave board. The short wave group velocity (associated with advection of wave action) is based on the $T_{m-1,0}$ wave period. Other model settings can be found in \citet{VanThielDeVries2009}.

Wave height transformation and wave setup (\autoref{fig:deltaflume2006_t01_hydro}) are favourably reproduced with the model. The long wave height is slightly underestimated whereas the wave setup is slightly overestimated. The correlation between measured short wave variance and long wave water surface elevations (\autoref{fig:deltaflume2006_t01_corr}) corresponds reasonably well with the measurements. Towards the shoreline this correlation increases \citep{Abdelrahman1987,Roelvink1989} meaning the highest short waves travel on top of long waves, which likely causes that more short wave energy gets closer to the dune face. 
Short wave skewness and asymmetry are reasonably predicted with the extended Rienecker Fenton model (\autoref{fig:deltaflume2006_t01_waveshape}, panel 1). However, in the inner surf zone both wave skewness and asymmetry are overestimated. Possible explanations are wave breaking, which limits the steepness and height of waves and the presence of free harmonics in the flume. Both these effects are not included in the wave shape model but indeed are present in the flume test \citep{VanThielDeVries2009}. From simulated skewness and asymmetry it follows that the total nonlinearity of a short wave is overestimated close to the dune face (\autoref{fig:deltaflume2006_t01_waveshape}, panel 2). The phase $\beta$ is favourably simulated with the model but is underestimated further offshore.

${tools.image('Deltaflume2006/T01/', 'hydro.png', 'Wave height transformations and flow velocities', 'deltaflume2006_t01_hydro')}
${tools.image('Deltaflume2006/T01/', 'corr.png', 'Correlation between short wave variance and long wave surface elevations', 'deltaflume2006_t01_corr')}
${tools.image('Deltaflume2006/T01/', 'waveshape.png', 'Wave shape and nonlinearity', 'deltaflume2006_t01_waveshape')}

The simulated test and depth averaged flow velocity shows the same trend as in the measurements and increases towards the shoreline (\autoref{fig:deltaflume2006_t01_undertow}). However, in the simulation the cross-shore range with a high offshore mean flow is smaller and extends less far seaward than in the measurements. This is possibly explained by differences in profile development (\autoref{fig:deltaflume2006_t01_final_profiles}) or inaccurate measurements. In addition, another explanation may be found in the incorrect modeling of the roller energy dissipation. Simulations (not shown) with a smaller roller dissipation rate revealed that roller energy in the inner surf increases, leading to higher return flow over a broader cross-shore range. 
Long waves contribute to the time and depth averaged flow close to the shoreline. The contribution of long waves to the mean flow is explained by on average larger water depths during the interval associated with shoreward flow velocities in relation to the interval with offshore flow velocities. Considering continuity and a uniform vertical structure of the long wave flow this means a time and depth averaged offshore directed flow should be present. 
Nonlinear waves may cause onshore sediment transport presuming non-uniform sediment stirring over the wave cycle and a positive correlation between sediment suspension and the intra wave flow. In order to include the wave averaged effect of nonlinear waves on the sediment transport a mean flow $u_A$ is computed, which is added to the mean (Eulerian) flow $U_m$ \citep{VanThielDeVries2009}. The simulated time averaged flow associated with nonlinear waves shows a comparable evolution as in the measurements but is overestimated especially closer to the dune face. Near the shoreline the wave skewness related sediment transport vanishes (\autoref{fig:deltaflume2006_t01_waveshape}, panel 1) since waves develop towards fully saw tooth shaped bores that have negligible skewness.
The orbital flow velocity (\autoref{fig:deltaflume2006_t01_hydro}, panel 2) is favourably predicted by the model. The short wave orbital flow velocity is slightly overestimated whereas the long wave orbital flow is underestimated. The underestimation of the simulated long wave orbital flow corresponds well to the slight underestimation of the observed long wave water surface variance.

${tools.image('Deltaflume2006/T01/', 'undertow.png', 'Wave motion velocities and undertow', 'deltaflume2006_t01_undertow')}

The simulated test and depth averaged sediment concentration increases towards the shoreline but is underestimated, especially in deeper water where the modeled sediment concentration is smaller (\autoref{fig:deltaflume2006_t01_concentration}). In the proximity of the dune face the simulated mean sediment concentration is within a factor two with the measurements. Further offshore the discrepancy between simulations and measurements is larger. The sharp rise in the near dune sediment concentration compares well with the bore averaged near-bed turbulence intensity (\autoref{fig:deltaflume2006_t01_turbulence}) that also increases towards the shoreline. This increase in turbulence intensity through the inner surf is explained by more intensive wave breaking (turbulence pro-duction at the water surface increases) and by decreasing water depth (generated turbulence at the water surface is more effective in reaching the bed). 
The simulated time averaged sediment transport compares well with the measured sediment transport computed from profile changes (\autoref{fig:deltaflume2006_t01_transports}, panel 1). Sediment is eroded from the dune face via avalanching and as a result the sediment transport associated with avalanching is dominant over the dune face and in the swash zone. From the swash zone seaward, the flow based sediment transport becomes more important. At 205m from the wave board, in a water depth that varies between 0.1m and 0.2m, the flow related sediment transport is dominant.
The simulated flow related sediment transport is separated in sediment transports as-ociated with nonlinear waves ($S_W$), long waves ($S_L$) and the short wave driven under-tow ($S_R$) (\autoref{fig:deltaflume2006_t01_transports}, panel 2):

$S_W = \overline{u_A ch}$
$S_L = \overline{u^L ch}$
$S_R = \overline{(u^E-u^L) ch}$

The offshore sediment transport results from the short wave and roller driven under-tow ($S_R$) combined with the transport associated with the long waves ($S_L$). The transport that follows from the short wave undertow is dominant in the present simu-lation but the long wave related sediment transport cannot be neglected (about 30\% at the location of the maximum offshore transport). The wave related sediment transport ($S_W$) is onshore and suppresses the offshore sediment transport with some 30\%.
Profile evolution and dune erosion volumes are favourably predicted with the model during test T01 (\autoref{fig:deltaflume2006_t01_profile} and \autoref{fig:deltaflume2006_t01_morpho}, panel 2). Between t = 2.04 and 6.0 hours (interval E) the dune erosion rate is slightly underestimated. At the offshore edge of the developing foreshore, the model seems not capable to reproduce the steep transition from the original (unaffected) profile towards the newly developed foreshore. A bar type feature is observed at this transition that is hypothesized to be related to (partly) plunging breakers that generate a water jet, which penetrates in the water column and causes additional sediment stirring when it reaches the bed. Though the effect of wave breaking induced turbulence on sediment suspension is included in the simulation, the applied model only considers spilling breakers, which are expected to be less efficient than plunging breakers in stirring up sand.

${tools.image('Deltaflume2006/T01/', 'concentration.png', 'Sediment concentrations', 'deltaflume2006_t01_concentration')}
${tools.image('Deltaflume2006/T01/', 'turbulence.png', 'Wave induced turbulence', 'deltaflume2006_t01_turbulence')}
${tools.image('Deltaflume2006/T01/', 'transports.png', 'Sediment transports by transport type and forcing', 'deltaflume2006_t01_transports')}
${tools.image('Deltaflume2006/T01/', 'final_profiles.png', 'Profile development', 'deltaflume2006_t01_final_profiles')}
${tools.image('Deltaflume2006/T01/', 'morpho.png', 'Bed level changes and erosion volumes', 'deltaflume2006_t01_morpho')}

It is concluded that profile evolution and dune erosion during test T01 are favourably simulated. Also simulated wave heights, flows, sediment concentrations and sediment transports compare reasonably well with measurements. However,looking at the results in more detail some discrepancies are found:
\begin{enumerate}
  \item The long wave height and especially associated long wave orbital flows are un-derestimated.
  \item The test and depth averaged flow between x = 170 m and x = 200 m is underestimated. Close to the shoreline no reliable measurements are available to verify the model results. 
  \item The simulated sediment concentration compares well with measurements close to the dune face. However, for smaller sediment concentrations in deeper water the simulated concentration is underestimated.
  \item The offshore sediment transport is mainly driven by the short wave and roller induced undertow (O(70\%) at the location of the maximum offshore transport) whereas the offshore directed long wave related sediment transport cancels out with the onshore sediment transport due to nonlinear short waves.
\end{enumerate}
It is remarked that shoreward of the maximum offshore sediment transport, the importance of the long wave related transport increases and eventually becomes dominant in relation to the transport associated with short wave and roller driven undertow. Considering the mainly long wave associated sediment transport in proximity of the dune face and the importance of long wave run-up for avalanching it is expected that long waves are mainly responsible for the swash zone sediment transport.

An overview of the Model Performance Statistics for test T01 with respect to the measurements is given in \autoref{tab:deltaflume2006_t01_skills}.

${tools.table('Deltaflume2006/T01/', 'skills.tex', 'Model Performance Statistics for test T01', 'deltaflume2006_t01_skills')}