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Universidade de Cantabria}, + year={2015} +} +@phdthesis{poncet2021, + title={Characterization of wave impact loading on structures at full scale: field experiment, statistical analysis and 3D advanced numerical modeling}, + author={Poncet, Pierre-Antoine}, + year={2021}, + school={Université de Pau et des Pays de l'Adour}, + chapter={4}, +} +@article{torrence1998, + title={A practical guide to wavelet analysis}, + author={Torrence, Christopher and Compo, Gilbert P}, + journal={Bulletin of the American Meteorological society}, + volume={79}, + number={1}, + pages={61--78}, + year={1998}, + publisher={American Meteorological Society} +} +@article{dysthe2008, + title={Oceanic rogue waves}, + author={Dysthe, Kristian and Krogstad, Harald E and M{\"u}ller, Peter}, + journal={Annu. Rev. Fluid Mech.}, + volume={40}, + pages={287--310}, + year={2008}, + publisher={Annual Reviews} +} +@article{lodhi2020, + title={The role of hydrodynamic impact force in subaerial boulder transport by tsunami—Experimental evidence and revision of boulder transport equation}, + author={Lodhi, Hira A and Hasan, Haider and Nandasena, NAK}, + journal={Sedimentary Geology}, + volume={408}, + pages={105745}, + 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} +} diff --git a/nature/main.tex b/nature/main.tex new file mode 100644 index 0000000..7114d49 --- /dev/null +++ b/nature/main.tex @@ -0,0 +1,369 @@ +\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 \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. 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 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 (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 \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. + +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 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}. + +\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 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 +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. + +\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*} + +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 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 +infragravity band at \SI{200}{\m}. + +\begin{figure*} + \centering + \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 + \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 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. + 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} + +\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}. + +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} + +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. Those results are only useful if we consider that infragravity waves behave similarly to shorter +waves regarding reflection. + +\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, 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} + +\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}. 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 +(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. + +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 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. + +\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} diff --git a/tasks.md b/tasks.md index 072cedf..f5b8ceb 100644 --- a/tasks.md +++ b/tasks.md @@ -18,3 +18,18 @@ Olaflow comparison with photos Comparison of olaflow output with block displacement theories Format rapport: Journal of Geophysical Research + + + +Ajouter Figures: photos: vague & bloc sur la digue, wavelet analysis bouée autres vagues, +Snapshot vagues sortie olaflow + +Étoffer un peu contenu + +Figure 6: Line plot en 1 point de la vitesse du courant + +Faire lien entre photos et splashs dans swash + +Génération vague scélérate : combinaison + dispersion -> zone non dispersive : ajouter profil bathymétrie + +Tester bathy plane avec swash pour voir si transfert énergie IG -> G