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\title{Analysis of the displacement of a large concrete block under an extreme wave}
\author[1]{Edgar P. Burkhart}
\author[*,1]{Stéphane Abadie}

\affil[1]{Université de Pau et des Pays de l’Adour, E2S-UPPA,  SIAME, France}
\affil[*]{Corresponding Author, stephane.abadie@univ-pau.fr}

\begin{document}
\maketitle

\section{Introduction}
% Displacement of blocks studies
Displacement of large blocks or boulders by waves is an interesting phenomenon in the study of extreme historical
coastal events. The existence of block deposits at unusual heights can be a clue to past events such as extreme storms
or tsunamis. For instance, \textcite{cox2018} studied coastal deposits on the coast of Ireland in relation to the
storms from winter 2013--2014, and extracted criteria for analysing such deposits. Similarly, \textcite{shah2013}
found boulder deposits on the mediterranean coast to be evidence of extreme storms in the Little Ice Age.

% Need for analytical equations
In order for those studies to be possible, analytical criterias are needed in order to ascertain the cause of the
displacement of a block. \textcite{nott1997,nott2003} proposed a set of equations that have been widely used for that
purpose. Those equations rely on an equilibrium relation between the lift force produced by a wave and restraining
forces depending on the initial setting of the block, allowing to extract a minimal flow velocity necessary for
movement initiation. A parametrisation of waves depending on their source is also used to provide minimal wave heights
depending 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 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.

% 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.

In this paper, we study such an event. On February 28, 2017, a 50T 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 1.2km 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 50T 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.

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 1450m 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
\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}.

\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 1250m offshore of the Artha breakwater, a seamingly abnormally large wave of
14m amplitude was identified that is supposed to have lead 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
essentially an incident wave, with a minor reflected component.

The wavelet power spectrum displayed in Figure~\ref{fig:wavelet} highlights a primary infragravity wave in the signal, with
a period of over 30s.

\begin{figure*}
	\centering
	\includegraphics{fig/ts.pdf}
	\includegraphics{fig/out_orbitals.pdf}
	\caption{\textit{Left}: Free surface measured during the extreme wave measured on February 28, 2017 at 17:23UTC.
	\textit{Right}: Trajectory of the wave buoy during the passage of this particular wave.}\label{fig:wave}
\end{figure*}

\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 13.9m for the real bathymetry, and 12.1m 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.

\begin{figure*}
	\centering
	\includegraphics{fig/maxw.pdf}
	\caption{Free surface elevation obtained with the SWASH model in two configurations. \textit{Case 1}: With breakwater;
	\textit{Case 2}: Without breakwater.}\label{fig:swash}
\end{figure*}

\subsection{Wave transformation}

The free surface obtained with the SWASH model using raw buoy measurements as an elevation boundary condition is
plotted in Figure~\ref{fig:swash_trans}. Those results display a strong transformation of the wave between the buoy and
the breakwater. Not only the amplitude, but also the shape of the wave are strongly impacted by the propagation over
the domain. While the amplitude of the wave is reduced as the wave propagates shorewards, the length of the trough and
the crest increases, with a zone reaching 400m long in front of the wave where the water level is below 0m.

\begin{figure*}
	\centering
	\includegraphics{fig/x.pdf}
	\caption{Propagation of the wave supposed to be responsible for the block displacement; highlighted zone:
	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 60s.

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 600m to 300m, and returning to the infragravity band at
200m.

\begin{figure*}
	\centering
	\includegraphics{fig/wavelet9312.pdf}
	\caption{Normalized wavelet power spectrum from the raw buoy timeseries.}\label{fig:wavelet}
\end{figure*}
\begin{figure*}
	\centering
	\includegraphics{fig/wavelet_sw.pdf}
	\caption{Normalized wavelet power spectrum along the SWASH domain.}\label{fig:wavelet_sw}
\end{figure*}

\subsection{Hydrodynamic conditions on the breakwater}

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 14.5m/s towards the breakwater during the identified extreme wave. Although the maximum reached
velocity is slightly lower than earlier shorter waves (at t=100s and t=120s, with a maximum velocity of 17.3s), 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 5m for over 40s.

\begin{figure*}
	\centering
	\includegraphics{fig/U.pdf}
	\caption{Horizontal flow velocity computed with the olaFlow model at x=-20m on the breakwater armor. The identified
	wave reaches this point around t=175s.}\label{fig:U}
\end{figure*}

\section{Discussion}

\subsection{Incident wave}

According to the criteria proposed by \textcite{dysthe2008}, rogue waves can be defined as waves with an amplitude over
twice the significant wave height over a given period. The identified wave fits this definition, as its amplitude is
14.7m, over twice the significant wave height of 6.3m on that day. According to \textcite{dysthe2008}, 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.

\subsection{Reflection analysis}

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.

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.

\subsection{Wave transformation}

The SWASH model yields a strongly changing wave over the domain, highlighting the highly complex composition of this
wave. Although the peak of the amplitude of the wave is reduced as the wave propagates, the length of the wave is
highlighted by the results. At T+60s for instance, the water level is under 0m for 400m, and then over 0m for around
the same length, showing the main infragavity component of the studied wave.

The wavelet analysis conducted at several points along the domain (Figure~\ref{fig:wavelet_sw}) show that the energy of
the studied wave (slightly before t=1500s) initially displays a strong infragravity component. Energy is then
transfered from the infragravity band towards shorter waves, and back to the infragravity band. This behavior is quite
unexpected, and further investigations should be conducted to understand and validate those results.

\subsection{Hydrodynamic conditions on the breakwater}

The hydrodynamic conditions on the breakwater are the main focus of this study. Considering an initially submerged
block, analytical equations proposed by \textcite{nandasena2011} yield a minimal flow velocity that would lead to block
displacement by saltation of 19.4m/s. The results from the Olaflow model yield a maximal wave velocity during the
displacement of the 50T concrete block of 14.5m/s. The results from the model are 25\% lower than the analytical value.

Those results tend to confirm recent research by \textcite{lodhi2020}, where it was found that the block displacement
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.

\section{Methods}

\subsection{SWASH models}
\subsubsection{Domain}

A 1750m 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 1m was used for most of the domain. 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.

A second domain is constructed for reflection analysis. The second model is the same as the first, excepted that the
breakwater is replaced by a smooth slope in order to remove the reflection generated by the structure.

The reflection analysis is conducted over 4h in order to generate a fair range of conditions. The wave transformation
study was conducted over a 1h timeframe in order to allow the model to reach steady-state before the studied wave was
generated.

\subsubsection{Model}
A non-linear non-hydrostatic shallow water model (SWASH, \cite{zijlema2011}) is used to model wave reflection and
transformation on the studied domain. The study is conducted using a layered one-dimensional model, that allows to
consider porous media in the domain.

The reflection analysis was conducted with 2 layers as to prevent model instability in overtopping conditions. The
study of wave transformation and the generation of boundary conditions for the Olaflow model is done with 4 layers.

\subsubsection{Porosity}

In the SWASH model, the porous breakwater armour is represented using macroscale porosity. The porosity parameters were
calibrated in \textcite{poncet2021}.

\subsubsection{Boundary conditions}

Two different sets of boundary conditions were used for both studies. In all cases, a sponge layer was added to the
shorewards boundary to prevent wave reflection on the boundary. In the reflection analysis, offshore conditions were
generated using the wave spectrum extracted from buoy data during the storm. The raw vertical surface elevation
measured by the wave buoy was used in a second part.

\begin{figure*}
	\centering
	\includegraphics{fig/aw_t0.pdf}
	\caption{Domain studied with Olaflow. Initial configuration.}\label{fig:of}
\end{figure*}

\printbibliography
\end{document}