End report

nature/library.bib

@@ -119,3 +119,19 @@
 	year={2020},
 	publisher={Elsevier}
 }
+@book{violeau2012,
+  title={Fluid mechanics and the SPH method: theory and applications},
+  author={Violeau, Damien},
+  year={2012},
+  publisher={Oxford University Press}
+}
+@article{violeau2007,
+  title={Numerical modelling of complex turbulent free-surface flows with the SPH method: an overview},
+  author={Violeau, Damien and Issa, Reza},
+  journal={International Journal for Numerical Methods in Fluids},
+  volume={53},
+  number={2},
+  pages={277--304},
+  year={2007},
+  publisher={Wiley Online Library}
+}

nature/main.tex

@@ -56,47 +56,59 @@ initiation. A parametrisation of waves depending on their source is also used to
 on the type of scenario --- wave or tsunami. Those equations were later revised by \textcite{nandasena2011}, as they
 were found to be partially incorrect. A revised formulation based on the same considerations was provided.
 
-The assumptions on which \citeauthor{nott2003, nandasena2011} are based were then critisized by \textcite{weiss2015}. In
+The assumptions on which \textcite{nott2003, nandasena2011} are based were then critisized by \textcite{weiss2015}. In
 fact, according to them, the initiation of movement is not sufficient to guarantee block displacement.
 \textcite{weiss2015} highlights the importance of the time dependency on block displacement. A method is proposed that
-allows to find the wave amplitude that lead to block displacement.
+allows to find the wave amplitude that lead to block displacement. Additionally, more recent research by
+\textcite{lodhi2020} has shown that the equations proposed by \textcite{nott2003, nandasena2011} tend to overestimate
+the minimum flow velocity needed to displace a block.
 
 % Lack of observations -> observation
 Whether it is \textcite{nott2003}, \textcite{nandasena2011} or \textcite{weiss2015}, all the proposed analytical
-equations suffer from a major flaw; they are all based on simplified analytical models and statistical analysis.
-Unfortunately, no block displacement event seems to have been observed directly in the past.
+equations suffer from a major flaw: they are all based on very simplified analytical models and statistical analysis.
+Unfortunately, no block displacement event seems to have been observed directly in the past, and those events are
+difficult to predict.
 
 In this paper, we study such an event. On February 28, 2017, a \SI{50}{\tonne} concrete block was dropped by a wave on
-the crest of the Artha breakwater. Luckily, the event was captured by a photographer, and a wave buoy located
-\SI{1.2}{\km} offshore captured the seastate. Information from the photographer allowed to establish the approximate
-time at which the block displacement occured. The goal of this paper is to model the hydrodynamic conditions near the
-breakwater that lead to the displacement of the \SI{50}{\tonne} concrete block.
+the crest of the Artha breakwater (Figure~\ref{fig:photo}). Luckily, the event was captured by a photographer, and a
+wave buoy located \SI{1.2}{\km} offshore captured the seastate. Information from the photographer allowed to establish
+the approximate time at which the block displacement occured. The goal of this paper is to model the hydrodynamic
+conditions near the breakwater that lead to the displacement of the \SI{50}{\tonne} concrete block.
 
 % Modeling flow accounting for porous media
 Several approaches can be used when modelling flow near a breakwater. Depth-averaged models can be used to study the
-transformation of waves on complex bottoms. Studying the hydrodynamic conditions under the surface can be achieved using
-smoothed-particles hydrodynamics (SPH) or volume of fluid (VOF) models. SPH models rely on a Lagrangian representation
-of the fluid, while VOF models rely on an Eulerian representation. VOF models are generally more mature for the study of
-multiphase incompressible flows.
+transformation of waves on complex bottoms. Studying the hydrodynamic conditions under the surface can be achieved
+using smoothed-particles hydrodynamics (SPH) or volume of fluid (VOF) models. SPH models rely on a Lagrangian
+representation of the fluid \parencite{violeau2012}, while VOF models rely on an Eulerian representation. VOF models
+are generally more mature for the study of multiphase incompressible flows, while SPH models generally require more
+processing power for similar results \parencite{violeau2007}.
 
-In this paper, we first use a one-dimensionnal depth-averaged non-linear non-hydrostatic model to verify that the signal
-measured by the wave buoy can be used as an incident wave input for the determination of hydrodynamic conditions near
-the breakwater. For this model, we use a SWASH model \parencite{zijlema2011} already calibrated by \textcite{poncet2021}
-on a domain reaching \SI{1450}{\m} offshore of the breakwater.
+In this paper, we first use a one-dimensionnal depth-averaged non-linear non-hydrostatic model to verify that the
+signal measured by the wave buoy can be used as an incident wave input for the determination of hydrodynamic conditions
+near the breakwater. For this model, we use a SWASH model \parencite{zijlema2011} already calibrated by
+\textcite{poncet2021} on a domain reaching \SI{1450}{\m} offshore of the breakwater.
 
 Then, we use a nested VOF model in two vertical dimensions that uses the output from the larger scale SWASH model as
-initial and boundary conditions to obtain the hydrodynamic conditions on the breakwater. The models uses olaFlow
+initial and boundary conditions to obtain the hydrodynamic conditions on the breakwater. The model uses olaFlow
 \parencite{higuera2015}, a VOF model based on volume averaged Reynolds averaged Navier-Stokes (VARANS) equations, and
 which relies on a macroscopic representation of the porous armour of the breakwater. The model is qualitatively
-calibrated using photographs from the storm of February 28, 2017. Results from the nested models are finally compared to
-the analytical equations provided by \textcite{nandasena2011}.
+calibrated using photographs from the storm of February 28, 2017. Results from the nested models are finally compared
+to the analytical equations provided by \textcite{nandasena2011}.
+
+\begin{figure*}
+	\centering
+	\includegraphics[height=.4\textwidth]{fig/pic1.jpg}
+	\includegraphics[height=.4\textwidth]{fig/pic2.jpg}
+	\caption{Photographs taken during and after the wave that displaced a \SI{50}{\tonne} concrete block onto the Artha
+	breakwater.}\label{fig:photo}
+\end{figure*}
 
 \section{Results}
 \subsection{Identified wave}
 
 Preliminary work with the photographer allowed to identify the time at which the block displacement event happened.
 Using the data from the wave buoy located \SI{1250}{\m} offshore of the Artha breakwater, a seamingly abnormally large
-wave of \SI{14}{\m} amplitude was identified that is supposed to have lead to the block displacement.
+wave of \SI{14}{\m} amplitude was identified that is supposed to have led to the block displacement.
 
 Initial analysis of the buoy data plotted in Figure~\ref{fig:wave} shows that the movement of the buoy follows two
 orbitals that correspond to an incident wave direction. These results would indicate that the identified wave is
@@ -115,14 +127,10 @@ with a period of over \SI{30}{\s}.
 
 \subsection{Reflection analysis}
 
-The results from the large scale SWASH model using two configurations --- one of them being the real bathymetry, and the
-other being a simplified bathymetry without the breakwater --- are compared in Figure~\ref{fig:swash}. The results
-obtained with both simulations show a maximum wave amplitude of \SI{13.9}{\m} for the real bathymetry, and \SI{12.1}{\m}
-in the case where the breakwater is removed.
-
-The 13\% difference between those values highlights the existence of a notable amount of reflection at the buoy.
-Nonetheless, the gap between the values is still fairly small and the extreme wave identified on February 28, 2017 at
-17:23:08 could still be considered as an incident wave.
+The results from the large scale SWASH model using two configurations --- one of them being the real bathymetry, and
+the other being a simplified bathymetry without the breakwater --- are compared in Figure~\ref{fig:swash}. The results
+obtained with both simulations show a maximum wave amplitude of \SI{13.9}{\m} for the real bathymetry, and
+\SI{12.1}{\m} in the case where the breakwater is removed.
 
 \begin{figure*}
 	\centering
@@ -146,12 +154,10 @@ crest increases, with a zone reaching \SI{400}{\m} long in front of the wave whe
 	qualitatively estimated position of the wave front.}\label{fig:swash_trans}
 \end{figure*}
 
-\subsection{Wavelet analysis}
-
 In an attempt to understand the identified wave, a wavelet analysis is conducted on raw buoy data as well as at
 different points along the SWASH model using the method proposed by \textcite{torrence1998}. The results are displayed
 in Figure~\ref{fig:wavelet} and Figure~\ref{fig:wavelet_sw}. The wavelet power spectrum shows that the major component
-in the identified wave is a high energy infragravity wave, with a period of around \SI{60}{\s}.
+in identified rogue waves is a high energy infragravity wave, with a period of around \SI{60}{\s}. 
 
 The SWASH model seems to indicate that the observed transformation of the wave can be characterized by a transfer of
 energy from the infragravity band to shorter waves from around \SI{600}{\m} to \SI{300}{\m}, and returning to the
@@ -159,8 +165,9 @@ infragravity band at \SI{200}{\m}.
 
 \begin{figure*}
 	\centering
-	\includegraphics{fig/wavelet9312.pdf}
-	\caption{Normalized wavelet power spectrum from the raw buoy timeseries.}\label{fig:wavelet}
+	\includegraphics{fig/wavelet.pdf}
+	\caption{Normalized wavelet power spectrum from the raw buoy timeseries for identified rogue waves on february 28,
+	2017.}\label{fig:wavelet}
 \end{figure*}
 \begin{figure*}
 	\centering
@@ -172,16 +179,17 @@ infragravity band at \SI{200}{\m}.
 
 The two-dimensionnal olaFlow model near the breakwater allowed to compute the flow velocity near and on the breakwater
 during the passage of the identified wave. The results displayed in Figure~\ref{fig:U} show that the flow velocity
-reaches a maximum of \SI{14.5}{\m\per\s} towards the breakwater during the identified extreme wave. Although the maximum
-reached velocity is slightly lower than earlier shorter waves (at $t=\SI{100}{\s}$ and $t=\SI{120}{\s}$, with a maximum
-velocity of \SI{17.3}{\m\per\s}), the flow velocity remains high for twice as long as during those earlier waves. The
-tail of the identified wave also exhibits a water level over \SI{5}{\m} for over \SI{40}{\s}.
+reaches a maximum of \SI{14.5}{\m\per\s} towards the breakwater during the identified extreme wave. Although the
+maximum reached velocity is similar to earlier shorter waves, the flow velocity remains high for twice as long as
+during those earlier waves. The tail of the identified wave also exhibits a water level over \SI{5}{\m} for over
+\SI{40}{\s}.
 
 \begin{figure*}
 	\centering
 	\includegraphics{fig/U.pdf}
-	\caption{Horizontal flow velocity computed with the olaFlow model at $x=\SI{-20}{\m}$ on the breakwater armor. The
-	identified wave reaches this point around $t=\SI{175}{\s}$.}\label{fig:U}
+	\caption{Horizontal flow velocity computed with the olaFlow model at $x=\SI{-20}{\m}$ on the breakwater armor.
+	Bottom: horizontal flow velocity at $z=\SI{5}{\m}$. The identified wave reaches this point around
+	$t=\SI{175}{\s}$.}\label{fig:U}
 \end{figure*}
 
 \section{Discussion}
@@ -194,12 +202,19 @@ twice the significant wave height over a given period. The identified wave fits
 rogue waves often occur from non-linear superposition of smaller waves. This seems to be what we observe on
 Figure~\ref{fig:wave}.
 
-The wavelet power spectrum shows that a very prominent infragravity component is present, which usually corresponds to
-non-linear interactions of smaller waves. \textcite{dysthe2008} mentions that such waves in coastal waters are often the
-result of refractive focusing. On February 28, 2017, the frequency of rogue waves was found to be of 1 wave per 1627,
-which is considerably more than the excedance probability of 1 over 10\textsuperscript4 calculated by
-\textcite{dysthe2008}. Additionnal studies should be conducted to understand focusing and the formation of rogue waves
-in front of the Saint-Jean-de-Luz bay.
+As displayed in Figure~\ref{fig:wavelet}, a total of 4 rogue waves were identified on february 28, 2017 in the raw buoy
+timeseries using the wave height criteria proposed by \textcite{dysthe2008}. The wavelet power spectrum shows that a
+very prominent infragravity component is present, which usually corresponds to non-linear interactions of smaller
+waves. \textcite{dysthe2008} mentions that such waves in coastal waters are often the result of refractive focusing. On
+February 28, 2017, the frequency of rogue waves was found to be of 1 wave per 1627, which is considerably more than the
+excedance probability of 1 over 10\textsuperscript4 calculated by \textcite{dysthe2008}. Additionnal studies should be
+conducted to understand focusing and the formation of rogue waves in front of the Saint-Jean-de-Luz bay.
+
+An important point to note is that rogue waves are often short-lived: their nature means that they often separate into
+shorter waves shortly after appearing. A reason for which such rogue waves can be maintained over longer distances can
+be a change from a dispersive environment such as deep water to a non-dispersive environment. The bathymetry near the
+wave buoy (Figure~\ref{fig:bathy}) shows that this might be what we observe here, as the buoy is located near a step in
+the bathymetry, from around \SI{40}{\m} to \SI{20}{\m} depth.
 
 \subsection{Reflection analysis}
 
@@ -207,10 +222,11 @@ The 13\% difference between those values highlights the existence of a notable a
 Nonetheless, the gap between the values is still fairly small and the extreme wave identified on February 28, 2017 at
 17:23:08 could still be considered as an incident wave.
 
-Unfortunately, the spectrum wave generation method used by SWASH could not reproduce simlar waves to the one observed at
-the buoy. As mentionned by \textcite{dysthe2008}, such rogue waves cannot be deterministicly from the wave spectrum. For
-this reason, this study only allows us to observe the influence of reflection on short waves, while mostly ignoring
-infragravity waves.
+Unfortunately, the spectrum wave generation method used by SWASH could not reproduce simlar waves to the one observed
+at the buoy. As mentionned by \textcite{dysthe2008}, such rogue waves cannot be deterministicly from the wave spectrum.
+For this reason, this study only allows us to observe the influence of reflection on short waves, while mostly ignoring
+infragravity waves. Those results are only useful if we consider that infragravity waves behave similarly to shorter
+waves regarding reflection.
 
 \subsection{Wave transformation}
 
@@ -236,11 +252,16 @@ Those results tend to confirm recent research by \textcite{lodhi2020}, where it
 threshold tend to overestimate the minimal flow velocity needed for block movement, although further validation of the
 model that is used would be needed to confirm those findings.
 
-Additionally, the flow velocity that is reached during the identified wave is not the highest that is reached in the
-model. Other shorter waves yield similar flow velocities on the breakwater, but in a smaller timeframe. The importance
-of time dependency in studying block displacement would be in accordance with research from \textcite{weiss2015}, who
-suggested that the use of time-dependent equations for block displacement would lead to a better understanding of the
-phenomenon.
+Additionally, similar flow velocities are reached in the model. Other shorter waves yield similar flow velocities on
+the breakwater, but in a smaller timeframe. The importance of time dependency in studying block displacement would be
+in accordance with research from \textcite{weiss2015}, who suggested that the use of time-dependent equations for block
+displacement would lead to a better understanding of the phenomenon.
+
+Although those results are a major step in a better understanding of block displacement in coastal regions, further
+work is needed to understand in more depth the formation and propagation of infragravity waves in the near-shore
+region. Furthermore, this study was limited to a single block displacement event, and further work should be done to
+obtain more measurements and observations of such events, although their rarity and unpredictability makes this task
+difficult.
 
 \section{Methods}
 
@@ -261,14 +282,23 @@ over \SI{0.2}{\Hz}.
 
 All wavelet analysis in this study is conducted using a continuous wavelet transform over a Morlet window. The wavelet
 power spectrum is normalized by the variance of the timeseries, following the method proposed by
-\textcite{torrence1998}.
+\textcite{torrence1998}. This analysis extracts a time-dependent power spectrum and allows to identify the composition
+of waves in a time-series.
 
 \subsection{SWASH models}
 
 \subsubsection{Domain}
 
+\begin{figure}
+	\centering
+	\includegraphics{fig/bathy2d.pdf}
+	\caption{Bathymetry in front of the Artha breakwater. The extremities of the line are the buoy and the
+	breakwater.}\label{fig:bathy}
+\end{figure}
+
 A \SI{1750}{\m} long domain is constructed in order to study wave reflection and wave transformation over the bottom
-from the wave buoy to the breakwater. Bathymetry with a resolution of around \SI{1}{\m} was used for most of the domain.
+from the wave buoy to the breakwater. Bathymetry with a resolution of around \SI{1}{\m} was used for most of the domain
+(Figure~\ref{fig:bathy}).
 The breakwater model used in the study is taken from \textcite{poncet2021}. A smoothed section is created and considered
 as a porous media in the model.
 
@@ -322,12 +352,12 @@ the SWASH model, the porous armour was considered at a macroscopic scale.
 
 A volume-of-fluid (VOF) model in two-vertical dimensions based on volume-averaged Reynolds-averaged Navier-Stokes
 (VARANS) equations is used (olaFlow, \cite{higuera2015}). The model was initially setup using generic values for porous
-breakwater studies. A sensibility study conducted on the porosity parameters found a minor influence of these values on
-the final results.
+breakwater studies. A sensibility study conducted on the porosity parameters found the influence of these values on
+the final results to be very minor.
 
-The k-ω SST turbulence model was used, as it produced much more realistic results than the default k-ε model, especially
-compared to the photographs from the storm of February 28, 2017. The k-ε model yielded very high viscosity and thus
-strong dissipation in the entire domain, preventing an accurate wave breaking representation.
+The k-ω SST turbulence model was used, as it produced much more realistic results than the default k-ε model,
+especially compared to the photographs from the storm of February 28, 2017. The k-ε model yielded very high viscosity
+and thus strong dissipation in the entire domain, preventing an accurate wave breaking representation.
 
 \subsubsection{Boundary conditions}