\documentclass[a4paper, twocolumn]{article}
\usepackage{polyglossia} \usepackage{authblk}
\usepackage[sfdefault]{inter}
\usepackage[math-style=french]{unicode-math}
\setmathfont{Fira Math}
\usepackage{graphicx}
\usepackage[hmargin=2.1cm, vmargin=2.97cm]{geometry}
\usepackage{hyperref}
\usepackage{siunitx}
\sisetup{
	mode=text,
	reset-text-family=false,
	reset-text-series=false,
	reset-text-shape=false,
	propagate-math-font=true,
}

\setmainlanguage{english}

\usepackage[
	backend=biber,
	style=iso-authoryear,
	sorting=nyt,
]{biblatex}
\bibliography{library}

\hypersetup{
	pdftitle={Analysis of the displacement of a large concrete block under an extreme wave},
	pdfauthor={Edgar P. Burkhart}
}

\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 \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.

% 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 \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
\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 \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.

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 \SI{30}{\s}.

\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 \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.

\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 \SI{400}{\m} long in front of the wave where the water level is below \SI{0}{\m}.

\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 \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
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}
\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 \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}.

\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}
\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
\SI{14.7}{\m}, over twice the significant wave height of \SI{6.3}{\m} 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 $T+\SI{60}{\s}$ for instance, the water level is under \SI{0}{\m} for \SI{400}{\m}, and
then over \SI{0}{\m} 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 $t=\SI{1500}{\s}$) 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 \SI{19.4}{\m\per\s} The results from the Olaflow model yield a maximal wave velocity during
the displacement of the \SI{50}{\tonne} concrete block of \SI{14.5}{\m\per\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{Buoy data analysis}

\subsubsection{Rogue wave identification}

Identifying rogue waves requires two main steps: computing the significant wave height, and computing the height of
individual waves. The first step is straightforward: $H_s=4\sigma$, where $\sigma$ is the standard deviation of the
surface elevation. Computing the height of individual waves is conducted using the zero-crossing method: the time domain
is split in sections where water level is strictly positive or negative, and wave size is computed according to the
maxima and minima in each zone. This method can fail to identify some waves or wrongly identify waves in case of
measurement errors or in the case where a small oscillation around 0 occurs in the middle of a larger wave. In order to
account for those issues, the signal is first fed through a low-pass filter to prevent high frequency oscillations of
over \SI{0.2}{\Hz}.

\subsubsection{Wavelet analysis}

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}.

\subsection{SWASH models}

\subsubsection{Domain}

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.
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 \SI{4}{\hour} in order to generate a fair range of conditions. The wave
transformation study was conducted over a \SI{1}{\hour} 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.

\subsection{Olaflow model}

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

\subsubsection{Domain}

A \SI{150}{\m} long domain is built in order to obtain the hydrodynamic conditions on the Artha breakwater during the
passage of the identified extreme wave. The bathymetry with \SI{50}{\cm} resolution from \textcite{poncet2021} is used.
The domain extends \SI{30}{\m} up in order to be able to capture the largest waves hitting the breakwater. Measurements
are extracted \SI{20}{\m} shorewards from the breakwater crest. The domain is displayed in Figure~\ref{fig:of}.

A mesh in two-vertical dimensions with \SI{20}{\cm} resolution was generated using the interpolated bathymetry. As with
the SWASH model, the porous armour was considered at a macroscopic scale.

\subsubsection{Model}

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.

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}

Initial and boundary conditions were generated using the output from the SWASH wave transformation model. The boundary
condition is generated by a paddle-like wavemaker, using the water level and depth-averaged velocity computed by the
SWASH model.

\printbibliography
\end{document}
