Nature v1
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nature/main.tex
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nature/main.tex
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@ -1,9 +1,19 @@
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\documentclass[a4paper, twocolumn, draft]{article}
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\documentclass[a4paper, twocolumn]{article}
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\usepackage{polyglossia} \usepackage{authblk}
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\usepackage[sfdefault]{inter}
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\usepackage[math-style=french]{unicode-math}
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\setmathfont{Fira Math}
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\usepackage{graphicx}
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\usepackage[hmargin=2.1cm, vmargin=2.97cm]{geometry}
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\usepackage{hyperref}
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\usepackage{siunitx}
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\sisetup{
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mode=text,
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reset-text-family=false,
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reset-text-series=false,
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reset-text-shape=false,
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propagate-math-font=true,
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}
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\setmainlanguage{english}
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@ -33,21 +43,21 @@
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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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or tsunamis. For instance, \textcite{cox2018} studied coastal deposits on the coast of Ireland in relation to the storms
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from winter 2013--2014, and extracted criteria for analysing such deposits. Similarly, \textcite{shah2013} found boulder
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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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forces depending on the initial setting of the block, allowing to extract a minimal flow velocity necessary for movement
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initiation. A parametrisation of waves depending on their source is also used to provide minimal wave heights depending
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on the type of scenario --- wave or tsunami. Those equations were later revised by \textcite{nandasena2011}, as they
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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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The assumptions on which \citeauthor{nott2003, nandasena2011} are based were then critisized by \textcite{weiss2015}. In
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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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@ -56,44 +66,44 @@ Whether it is \textcite{nott2003}, \textcite{nandasena2011} or \textcite{weiss20
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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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In this paper, we study such an event. On February 28, 2017, a \SI{50}{\tonne} concrete block was dropped by a wave on
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the crest of the Artha breakwater. Luckily, the event was captured by a photographer, and a wave buoy located
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\SI{1.2}{\km} offshore captured the seastate. Information from the photographer allowed to establish the approximate
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time at which the block displacement occured. The goal of this paper is to model the hydrodynamic conditions near the
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breakwater that lead to the displacement of the \SI{50}{\tonne} 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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transformation of waves on complex bottoms. Studying the hydrodynamic conditions under the surface can be achieved using
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smoothed-particles hydrodynamics (SPH) or volume of fluid (VOF) models. SPH models rely on a Lagrangian representation
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of the fluid, while VOF models rely on an Eulerian representation. VOF models are generally more mature for the study of
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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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In this paper, we first use a one-dimensionnal depth-averaged non-linear non-hydrostatic model to verify that the signal
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measured by the wave buoy can be used as an incident wave input for the determination of hydrodynamic conditions near
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the breakwater. For this model, we use a SWASH model \parencite{zijlema2011} already calibrated by \textcite{poncet2021}
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on a domain reaching \SI{1450}{\m} 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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calibrated using photographs from the storm of February 28, 2017. Results from the nested models are finally compared to
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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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Using the data from the wave buoy located \SI{1250}{\m} offshore of the Artha breakwater, a seamingly abnormally large
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wave of \SI{14}{\m} 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{fig:wavelet} highlights a primary infragravity wave in the signal, with
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a period of over 30s.
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The wavelet power spectrum displayed in Figure~\ref{fig:wavelet} highlights a primary infragravity wave in the signal,
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with a period of over \SI{30}{\s}.
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\begin{figure*}
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\centering
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@ -105,10 +115,10 @@ a period of over 30s.
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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 results from the large scale SWASH model using two configurations --- one of them being the real bathymetry, and the
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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 \SI{13.9}{\m} for the real bathymetry, and \SI{12.1}{\m}
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in the case 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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@ -123,11 +133,11 @@ Nonetheless, the gap between the values is still fairly small and the extreme wa
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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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The free surface obtained with the SWASH model using raw buoy measurements as an elevation boundary condition is plotted
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in Figure~\ref{fig:swash_trans}. Those results display a strong transformation of the wave between the buoy and the
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breakwater. Not only the amplitude, but also the shape of the wave are strongly impacted by the propagation over the
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domain. While the amplitude of the wave is reduced as the wave propagates shorewards, the length of the trough and the
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crest increases, with a zone reaching \SI{400}{\m} long in front of the wave where the water level is below \SI{0}{\m}.
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\begin{figure*}
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\centering
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@ -141,11 +151,11 @@ the crest increases, with a zone reaching 400m long in front of the wave where t
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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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in the identified wave is a high energy infragravity wave, with a period of around \SI{60}{\s}.
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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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energy from the infragravity band to shorter waves from around \SI{600}{\m} to \SI{300}{\m}, and returning to the
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infragravity band at \SI{200}{\m}.
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\begin{figure*}
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\centering
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@ -162,16 +172,16 @@ energy from the infragravity band to shorter waves from around 600m to 300m, and
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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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reaches a maximum of \SI{14.5}{\m\per\s} towards the breakwater during the identified extreme wave. Although the maximum
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reached velocity is slightly lower than earlier shorter waves (at $t=\SI{100}{\s}$ and $t=\SI{120}{\s}$, with a maximum
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velocity of \SI{17.3}{\m\per\s}), the flow velocity remains high for twice as long as during those earlier waves. The
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tail of the identified wave also exhibits a water level over \SI{5}{\m} for over \SI{40}{\s}.
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\begin{figure*}
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\centering
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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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\caption{Horizontal flow velocity computed with the olaFlow model at $x=\SI{-20}{\m}$ on the breakwater armor. The
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identified wave reaches this point around $t=\SI{175}{\s}$.}\label{fig:U}
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\end{figure*}
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\section{Discussion}
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@ -180,13 +190,14 @@ exhibits a water level over 5m for over 40s.
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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. The identified wave fits this definition, as its amplitude is
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14.7m, over twice the significant wave height of 6.3m on that day. According to \textcite{dysthe2008}, rogue waves
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often occur from non-linear superposition of smaller waves. This seems to be what we observe on Figure~\ref{fig:wave}.
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\SI{14.7}{\m}, over twice the significant wave height of \SI{6.3}{\m} on that day. According to \textcite{dysthe2008},
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rogue waves often occur from non-linear superposition of smaller waves. This seems to be what we observe on
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Figure~\ref{fig:wave}.
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The wavelet power spectrum shows that a very prominent infragravity component is present, which usually corresponds to
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non-linear interactions of smaller waves. \textcite{dysthe2008} mentions that such waves in coastal waters are often
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the result of refractive focusing. On February 28, 2017, the frequency of rogue waves was found to be of 1 wave per
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1627, which is considerably more than the excedance probability of 1 over 10\textsuperscript4 calculated by
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non-linear interactions of smaller waves. \textcite{dysthe2008} mentions that such waves in coastal waters are often the
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result of refractive focusing. On February 28, 2017, the frequency of rogue waves was found to be of 1 wave per 1627,
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which is considerably more than the excedance probability of 1 over 10\textsuperscript4 calculated by
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\textcite{dysthe2008}. Additionnal studies should be conducted to understand focusing and the formation of rogue waves
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in front of the Saint-Jean-de-Luz bay.
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@ -196,20 +207,20 @@ The 13\% difference between those values highlights the existence of a notable a
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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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Unfortunately, the spectrum wave generation method used by SWASH could not reproduce simlar waves to the one observed
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at the buoy. As mentionned by \textcite{dysthe2008}, such rogue waves cannot be deterministicly from the wave spectrum.
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For this reason, this study only allows us to observe the influence of reflection on short waves, while mostly ignoring
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Unfortunately, the spectrum wave generation method used by SWASH could not reproduce simlar waves to the one observed at
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the buoy. As mentionned by \textcite{dysthe2008}, such rogue waves cannot be deterministicly from the wave spectrum. For
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this reason, this study only allows us to observe the influence of reflection on short waves, while mostly ignoring
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infragravity waves.
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\subsection{Wave transformation}
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The SWASH model yields a strongly changing wave over the domain, highlighting the highly complex composition of this
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wave. Although the peak of the amplitude of the wave is reduced as the wave propagates, the length of the wave is
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highlighted by the results. At T+60s for instance, the water level is under 0m for 400m, and then over 0m for around
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the same length, showing the main infragavity component of the studied wave.
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highlighted by the results. At $T+\SI{60}{\s}$ for instance, the water level is under \SI{0}{\m} for \SI{400}{\m}, and
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then over \SI{0}{\m} for around the same length, showing the main infragavity component of the studied wave.
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The wavelet analysis conducted at several points along the domain (Figure~\ref{fig:wavelet_sw}) show that the energy of
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the studied wave (slightly before t=1500s) initially displays a strong infragravity component. Energy is then
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the studied wave (slightly before $t=\SI{1500}{\s}$) initially displays a strong infragravity component. Energy is then
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transfered from the infragravity band towards shorter waves, and back to the infragravity band. This behavior is quite
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unexpected, and further investigations should be conducted to understand and validate those results.
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@ -217,8 +228,9 @@ unexpected, and further investigations should be conducted to understand and val
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The hydrodynamic conditions on the breakwater are the main focus of this study. Considering an initially submerged
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block, analytical equations proposed by \textcite{nandasena2011} yield a minimal flow velocity that would lead to block
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displacement by saltation of 19.4m/s. The results from the Olaflow model yield a maximal wave velocity during the
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displacement of the 50T concrete block of 14.5m/s. The results from the model are 25\% lower than the analytical value.
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displacement by saltation of \SI{19.4}{\m\per\s} The results from the Olaflow model yield a maximal wave velocity during
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the displacement of the \SI{50}{\tonne} concrete block of \SI{14.5}{\m\per\s}. The results from the model are 25\% lower
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than the analytical value.
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Those results tend to confirm recent research by \textcite{lodhi2020}, where it was found that the block displacement
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threshold tend to overestimate the minimal flow velocity needed for block movement, although further validation of the
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@ -232,28 +244,49 @@ phenomenon.
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\section{Methods}
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\subsection{Buoy data analysis}
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\subsubsection{Rogue wave identification}
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Identifying rogue waves requires two main steps: computing the significant wave height, and computing the height of
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individual waves. The first step is straightforward: $H_s=4\sigma$, where $\sigma$ is the standard deviation of the
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surface elevation. Computing the height of individual waves is conducted using the zero-crossing method: the time domain
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is split in sections where water level is strictly positive or negative, and wave size is computed according to the
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maxima and minima in each zone. This method can fail to identify some waves or wrongly identify waves in case of
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measurement errors or in the case where a small oscillation around 0 occurs in the middle of a larger wave. In order to
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account for those issues, the signal is first fed through a low-pass filter to prevent high frequency oscillations of
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over \SI{0.2}{\Hz}.
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\subsubsection{Wavelet analysis}
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All wavelet analysis in this study is conducted using a continuous wavelet transform over a Morlet window. The wavelet
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power spectrum is normalized by the variance of the timeseries, following the method proposed by
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\textcite{torrence1998}.
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\subsection{SWASH models}
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\subsubsection{Domain}
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A 1750m long domain is constructed in order to study wave reflection and wave transformation over the bottom from the
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wave buoy to the breakwater. Bathymetry with a resolution of around 1m was used for most of the domain. The breakwater
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model used in the study is taken from \textcite{poncet2021}. A smoothed section is created and considered as a porous
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media in the model.
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A \SI{1750}{\m} long domain is constructed in order to study wave reflection and wave transformation over the bottom
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from the wave buoy to the breakwater. Bathymetry with a resolution of around \SI{1}{\m} was used for most of the domain.
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The breakwater model used in the study is taken from \textcite{poncet2021}. A smoothed section is created and considered
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as a porous media in the model.
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A second domain is constructed for reflection analysis. The second model is the same as the first, excepted that the
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breakwater is replaced by a smooth slope in order to remove the reflection generated by the structure.
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The reflection analysis is conducted over 4h in order to generate a fair range of conditions. The wave transformation
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study was conducted over a 1h timeframe in order to allow the model to reach steady-state before the studied wave was
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generated.
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The reflection analysis is conducted over \SI{4}{\hour} in order to generate a fair range of conditions. The wave
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transformation study was conducted over a \SI{1}{\hour} timeframe in order to allow the model to reach steady-state
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before the studied wave was generated.
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\subsubsection{Model}
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A non-linear non-hydrostatic shallow water model (SWASH, \cite{zijlema2011}) is used to model wave reflection and
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transformation on the studied domain. The study is conducted using a layered one-dimensional model, that allows to
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consider porous media in the domain.
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The reflection analysis was conducted with 2 layers as to prevent model instability in overtopping conditions. The
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study of wave transformation and the generation of boundary conditions for the Olaflow model is done with 4 layers.
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The reflection analysis was conducted with 2 layers as to prevent model instability in overtopping conditions. The study
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of wave transformation and the generation of boundary conditions for the Olaflow model is done with 4 layers.
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\subsubsection{Porosity}
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@ -264,37 +297,37 @@ calibrated in \textcite{poncet2021}.
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Two different sets of boundary conditions were used for both studies. In all cases, a sponge layer was added to the
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shorewards boundary to prevent wave reflection on the boundary. In the reflection analysis, offshore conditions were
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generated using the wave spectrum extracted from buoy data during the storm. The raw vertical surface elevation
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measured by the wave buoy was used in a second part.
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generated using the wave spectrum extracted from buoy data during the storm. The raw vertical surface elevation measured
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by the wave buoy was used in a second part.
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\subsection{Olaflow model}
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\subsubsection{Domain}
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A 150m long domain is built in order to obtain the hydrodynamic conditions on the Artha breakwater during the passage
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of the identified extreme wave. The bathymetry with 50cm resolution from \textcite{poncet2021} is used. The domain
|
||||
extends 30m up in order to be able to capture the largest waves hitting the breakwater. Measurements are extracted 20m
|
||||
shorewards from the breakwater crest. The domain is displayed in Figure~\ref{fig:of}.
|
||||
|
||||
A mesh in two-vertical dimensions with 20cm resolution was generated using the interpolated bathymetry. As with the
|
||||
SWASH model, the porous armour was considered at a macroscopic scale.
|
||||
|
||||
\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.
|
||||
(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.
|
||||
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}
|
||||
|
||||
|
|
|
|||
Loading…
Reference in a new issue