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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 use two nested models: a large scale one-dimensionnal model to study the transformation of the wave
from the wave buoy to the proximity of the breakwater, and a VOF model in two vertical dimensions to study the
hydrodynamic conditions near the breakwater. The large scale model uses SWASH \parencite{zijlema2011} a depth-averaged
non-linear non-hydrostatic model that was already calibrated by \textcite{poncet2022}. The nested model uses olaFlow
\parencite{higuera2015}, a VOF model based on volume averaged Reynolds averaged Navier-Stokes (VARANS) equations 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 compared to the analytical equations provided by \textcite{nandasena2011}.

\section{Results}

\section{Discussion}

\section{Methods}
\printbibliography
\end{document}