188 lines
10 KiB
TeX
188 lines
10 KiB
TeX
\documentclass[a4paper, twocolumn]{article}
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\usepackage{polyglossia} \usepackage{authblk}
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\usepackage[sfdefault]{inter}
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\usepackage{graphicx}
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\setmainlanguage{english}
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\usepackage[
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backend=biber,
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style=iso-authoryear,
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sorting=nyt,
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]{biblatex}
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\bibliography{library}
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\title{Analysis of the displacement of a large concrete block under an extreme wave}
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\author[1]{Edgar P. Burkhart}
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\author[*,1]{Stéphane Abadie}
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\affil[1]{Université de Pau et des Pays de l’Adour, E2S-UPPA, SIAME, France}
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\affil[*]{Corresponding Author, stephane.abadie@univ-pau.fr}
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\begin{document}
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\maketitle
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\section{Introduction}
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% Displacement of blocks studies
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Displacement of large blocks or boulders by waves is an interesting phenomenon in the study of extreme historical
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coastal events. The existence of block deposits at unusual heights can be a clue to past events such as extreme storms
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or tsunamis. For instance, \textcite{cox2018} studied coastal deposits on the coast of Ireland in relation to the
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storms from winter 2013--2014, and extracted criteria for analysing such deposits. Similarly, \textcite{shah2013}
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found boulder deposits on the mediterranean coast to be evidence of extreme storms in the Little Ice Age.
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% Need for analytical equations
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In order for those studies to be possible, analytical criterias are needed in order to ascertain the cause of the
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displacement of a block. \textcite{nott1997,nott2003} proposed a set of equations that have been widely used for that
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purpose. Those equations rely on an equilibrium relation between the lift force produced by a wave and restraining
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forces depending on the initial setting of the block, allowing to extract a minimal flow velocity necessary for
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movement initiation. A parametrisation of waves depending on their source is also used to provide minimal wave heights
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depending on the type of scenario --- wave or tsunami. Those equations were later revised by \textcite{nandasena2011},
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as they were found to be partially incorrect. A revised formulation based on the same considerations was provided.
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The assumptions on which \citeauthor{nott2003, nandasena2011} are based were then critisized by \textcite{weiss2015}.
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In fact, according to them, the initiation of movement is not sufficient to guarantee block displacement.
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\textcite{weiss2015} highlights the importance of the time dependency on block displacement. A method is proposed that
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allows to find the wave amplitude that lead to block displacement.
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% Lack of observations -> observation
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Whether it is \textcite{nott2003}, \textcite{nandasena2011} or \textcite{weiss2015}, all the proposed analytical
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equations suffer from a major flaw; they are all based on simplified analytical models and statistical analysis.
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Unfortunately, no block displacement event seems to have been observed directly in the past.
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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
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the Artha breakwater. Luckily, the event was captured by a photographer, and a wave buoy located 1.2km offshore
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captured the seastate. Information from the photographer allowed to establish the approximate time at which the block
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displacement occured. The goal of this paper is to model the hydrodynamic conditions near the breakwater that lead to
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the displacement of the 50T concrete block.
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% Modeling flow accounting for porous media
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Several approaches can be used when modelling flow near a breakwater. Depth-averaged models can be used to study the
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transformation of waves on complex bottoms. Studying the hydrodynamic conditions under the surface can be achieved
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using smoothed-particles hydrodynamics (SPH) or volume of fluid (VOF) models. SPH models rely on a Lagrangian
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representation of the fluid, while VOF models rely on an Eulerian representation. VOF models are generally more mature
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for the study of multiphase incompressible flows.
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In this paper, we first use a one-dimensionnal depth-averaged non-linear non-hydrostatic model to verify that the
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signal measured by the wave buoy can be used as an incident wave input for the determination of hydrodynamic conditions
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near the breakwater. For this model, we use a SWASH model \parencite{zijlema2011} already calibrated by
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\textcite{poncet2021} on a domain reaching 1450m offshore of the breakwater.
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Then, we use a nested VOF model in two vertical dimensions that uses the output from the larger scale SWASH model as
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initial and boundary conditions to obtain the hydrodynamic conditions on the breakwater. The models uses olaFlow
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\parencite{higuera2015}, a VOF model based on volume averaged Reynolds averaged Navier-Stokes (VARANS) equations, and
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which relies on a macroscopic representation of the porous armour of the breakwater. The model is qualitatively
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calibrated using photographs from the storm of February 28, 2017. Results from the nested models are finally compared
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to the analytical equations provided by \textcite{nandasena2011}.
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\section{Results}
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\subsection{Identified wave}
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Preliminary work with the photographer allowed to identify the time at which the block displacement event happened.
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Using the data from the wave buoy located 1250m offshore of the Artha breakwater, a seamingly abnormally large wave of
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14m amplitude was identified that is supposed to have lead to the block displacement.
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Initial analysis of the buoy data plotted in Figure~\ref{fig:wave} shows that the movement of the buoy follows two
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orbitals that correspond to an incident wave direction. These results would indicate that the identified wave is
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essentially an incident wave, with a minor reflected component.
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The wavelet power spectrum displayed in Figure~\ref{wavelet} highlights a primary infragravity wave in the signal, with
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a period of over 30s.
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\begin{figure*}
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\centering
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\includegraphics{fig/ts.pdf}
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\includegraphics{fig/out_orbitals.pdf}
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\caption{\textit{Left}: Free surface measured during the extreme wave measured on February 28, 2017 at 17:23UTC.
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\textit{Right}: Trajectory of the wave buoy during the passage of this particular wave.}\label{fig:wave}
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\end{figure*}
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\subsection{Reflection analysis}
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The results from the large scale SWASH model using two configurations --- one of them being the real bathymetry, and
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the other being a simplified bathymetry without the breakwater --- are compared in Figure~\ref{fig:swash}. The results
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obtained with both simulations show a maximum wave amplitude of 13.9m for the real bathymetry, and 12.1m in the case
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where the breakwater is removed.
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The 13\% difference between those values highlights the existence of a notable amount of reflection at the buoy.
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Nonetheless, the gap between the values is still fairly small and the extreme wave identified on February 28, 2017 at
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17:23:08 could still be considered as an incident wave.
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\begin{figure*}
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\centering
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\includegraphics{fig/maxw.pdf}
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\caption{Free surface elevation obtained with the SWASH model in two configurations. \textit{Case 1}: With breakwater;
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\textit{Case 2}: Without breakwater.}\label{fig:swash}
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\end{figure*}
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\subsection{Wave transformation}
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The free surface obtained with the SWASH model using raw buoy measurements as an elevation boundary condition is
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plotted in Figure~\ref{fig:swash_trans}. Those results display a strong transformation of the wave between the buoy and
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the breakwater. Not only the amplitude, but also the shape of the wave are strongly impacted by the propagation over
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the domain. While the amplitude of the wave is reduced as the wave propagates shorewards, the length of the trough and
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the crest increases, with a zone reaching 400m long in front of the wave where the water level is below 0m.
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\begin{figure*}
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\centering
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\includegraphics{fig/x.pdf}
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\caption{Propagation of the wave supposed to be responsible for the block displacement; highlighted zone:
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qualitatively estimated position of the wave crest.}\label{fig:swash_trans}
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\end{figure*}
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\subsection{Wavelet analysis}
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In an attempt to understand the identified wave, a wavelet analysis is conducted on raw buoy data as well as at
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different points along the SWASH model using the method proposed by \textcite{torrence1998}. The results are displayed
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in Figure~\ref{fig:wavelet} and Figure~\ref{fig:wavelet_sw}. The wavelet power spectrum shows that the major component
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in the identified wave is a high energy infragravity wave, with a period of around 60s.
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The SWASH model seems to indicate that the observed transformation of the wave can be characterized by a transfer of
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energy from the infragravity band to shorter waves from around 600m to 300m, and returning to the infragravity band at
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200m.
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\begin{figure*}
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\centering
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\includegraphics{fig/wavelet9312.pdf}
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\caption{Normalized wavelet power spectrum from the raw buoy timeseries.}\label{fig:wavelet}
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\end{figure*}
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\begin{figure*}
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\centering
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\includegraphics{fig/wavelet_sw.pdf}
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\caption{Normalized wavelet power spectrum along the SWASH domain.}\label{fig:wavelet_sw}
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\end{figure*}
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\subsection{Hydrodynamic conditions on the breakwater}
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The two-dimensionnal olaFlow model near the breakwater allowed to compute the flow velocity near and on the breakwater
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during the passage of the identified wave. The results displayed in Figure~\ref{fig:U} show that the flow velocity
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reaches a maximum of 14.5m/s towards the breakwater during the identified extreme wave. Although the maximum reached
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velocity is slightly lower than earlier shorter waves (at t=100s and t=120s, with a maximum velocity of 17.3s), the
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flow velocity remains high for twice as long as during those earlier waves. The tail of the identified wave also
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exhibits a water level over 5m for over 40s.
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\begin{figure*}
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\centering
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\includegraphics{fig/aw_t0.pdf}
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\includegraphics{fig/U.pdf}
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\caption{Horizontal flow velocity computed with the olaFlow model at x=-20m on the breakwater armor. The identified
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wave reaches this point around t=175s.}\label{fig:U}
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\end{figure*}
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\section{Discussion}
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\subsection{Incident wave}
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According to the criteria proposed by \textcite{dysthe2008}, rogue waves can be defined as waves with an amplitude over
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twice the significant wave height over a given period.
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\subsection{Reflection analysis}
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The 13\% difference between those values highlights the existence of a notable amount of reflection at the buoy.
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Nonetheless, the gap between the values is still fairly small and the extreme wave identified on February 28, 2017 at
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17:23:08 could still be considered as an incident wave.
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\subsection{}
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\section{Methods}
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\printbibliography
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\end{document}
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