EGU2019Poster (another copy).tex

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%Title text and graphics
\title{\parbox{\linewidth}{\centering Convection, phase change, and solute transport in mushy sea ice}} %with Adaptive Mesh Refinement
\titlegraphic{ \includegraphics[height=4cm]{figures/EGU2019Website.eps} %TODO: update QR code
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% Authors
\author[1]{James Parkinson (james.parkinson@physics.ox.ac.uk)}
\author[1]{Andrew Wells}
\author[2]{Dan Martin}
\author[1]{Richard Katz}

\affil[1]{University of Oxford}
\affil[2]{Lawrence Berkeley National Laboratory}



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\begin{document}

\maketitle

\begin{columns} 

\column{0.22}

\block{Motivation}{
\begin{itemize}[leftmargin=0.4cm]
    \item Sea ice is a mushy layer of ice crystals and brine.
    \item Dense brine drains during ice formation, whilst some brine is trapped within sea ice
    \item Observations (Fig. 1) and 1-D simulations suggest that warming sea ice may release some of this brine
    \item \textbf{Our goal: investigate this mechanism} using \mbox{2-D} numerical simulations
\end{itemize}
\vspace{-1em}
\noindent


\begin{adjustwidth}{-1em}{0em}

\begin{tikzfigure}
\label{fig:widellObs}
\begin{tikzpicture}

\node[anchor=north west] (fig2) at (0,0) {\includegraphics[width=0.98\linewidth, trim={0.0cm, 0.0cm, 0.0cm, 0.0cm}, clip]{figures/widdellFig1TaToFsTimeCropped.png} };

\end{tikzpicture}
\end{tikzfigure}
\end{adjustwidth}
\figureTitle{Fig. 1:} \figureCaption{\cite{Widell2006} observed intense salt fluxes $F_s$ below sea ice coincident with changes to the temperature in the atmosphere ($T_a$) and ocean ($T$).}




}


\block{What is a mushy layer?}
{
\vspace{-0cm}

\begin{tikzfigure}
\label{fig:phaseDiagram}
\begin{tikzpicture}

\node[anchor=north west] (fig2) at (0,0) {\includegraphics[width=0.4\linewidth, trim={0.0cm, 0.0cm, 0.0cm, 0.0cm}, clip]{figures/Eicken2000Edited.jpg} };

\node[anchor=north west] (phaseDiagram) at ([xshift=1.0cm]fig2.north east) {\includegraphics[width=0.47\linewidth, trim={0.0cm, 0.0cm, 0.0cm, 0.0cm}, clip]{figures/phaseDiagramVFinal.eps} };

\node[anchor=north west] (fig2label) at (fig2.north west) {(a)};

\node[anchor=north west] (fig1label) at ([yshift=0.0cm]phaseDiagram.north west) {(b)};

\end{tikzpicture}
\end{tikzfigure}
\figureTitle{Fig. 2:} \figureCaption{(a) Sea ice is a porous mixture of solid ice crystals (white) and liquid brine (dark) \cite{Eicken2000}. \\
(b) Trajectory ($\textcolor{red}{\rightarrow}$) of a solidifying salt water parcel through the phase diagram. As the temperature $T$ decreases, more ice forms and the residual brine salinity $S_l$ increases making the fluid denser, which can drive convection. Using a linear approximation for the liquidus curve, the freezing point is $T_f(S_l) = -0.1 S_l$. }
%Below the eutectic ($T_e, S_e$) = (-21.1$^\circ$, 233) the system is entirely solid.} %making the fluid denser which can lead to convection
%the resulting dense fluid can drive convection
%Eicken et. al. (2000).
}
\block{Numerical Method}
{
\vspace{0.5em}
Solve (1)-(4) using Chombo finite volume toolkit:
\begin{itemize}
    \item Momentum and mass: projection method \cite{Martin2008}.
    \item Energy and solute:
   
 
    {
    \raggedright % left align - prevent splitting words over multiple lines as we have space
   \begin{itemize}    
   \item Advective terms: explicit, 2\textsuperscript{nd} order unsplit Godunov method. %\cite{Colella1990}, 
   \item Nonlinear diffusive terms: semi implicit, geometric multigrid. %\cite{Martin1996}
   \item Timestepping: Backward Euler. %\cite{Twizell1996} 
   \end{itemize}
   }
  % }
\end{itemize}
\vspace{-1ex}
}






\column{0.78}


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\vspace{-2.6em}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Start experiment introduction
% sharp corners = uphill
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\vspace{-1.0em}
%\begin{minipage}[t]{0.5\linewidth}
{\LARGE \textbf{The experiment:}} \\
We consider 2-D simulations in a Hele-Shaw cell of width $0.5$m, height $1$m, and plate separation $d$. Water of initial salinity $S_0=30\,\mathrm{g}/ \mathrm{kg}$ and temperature $T_f(S_o) + 0.2^\circ$C is initially frozen from above by applying a fixed atmospheric temperature $T_a = -10^\circ$C. We assume $K_0=10^{-9}\,\mathrm{m}^2$, and initially set $d=0.5$mm to restrict thermal convection in the underlying liquid during growth. After $\sim0.5$m of ice growth, we increase $d$ to $2$mm and spin up simulations for 2 days before considering different warming scenarios.
%\end{minipage} \hfill \textcolor{dividingLines}{\vline  width 1pt} \hfill
%\begin{minipage}[t]{0.16\linewidth}
%\vspace{-2.0em}
%\begin{tikzfigure}
%\begin{tikzpicture}
%\node[anchor=north west] (2d) at (0,0) {\includegraphics[ trim={0.0cm, 0.0cm, 0.0cm, 0.0cm}, clip]{figures/prePerturbationSmall.png}};
%\end{tikzpicture}
%\end{tikzfigure}
%\end{minipage} \hfill
%\begin{minipage}[t]{0.32\linewidth}
%\vspace{0.0em}

%\end{minipage} 

\end{tcolorbox} % \hfill
%% Summary
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        height=9em, left skip=0.5em]

\vspace{-1.0em}
\LARGE \textbf{Summary}
\large 
\begin{itemize}
\item \textbf{We observe significant desalination events from warming sea ice in a 2-D model, with similar magnitude and timing to field observations.}
\item \textbf{Following basal melting, desalination can be caused by fresh melt water rising through brine channels.} 
%\item \textbf{...}
\end{itemize}
\end{tcolorbox}
% End summary
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Nudge content up a bit
\vspace{-0.1em}


\begin{minipage}[t]{0.32\linewidth}
\vspace{1.0em}
{\LARGE \textcolor{oxfordblue}{\hspace{0.0em}\textbf{Full depth desalination}} }
\vspace{1em} \\
\noindent \figureTitle{Fig. 7 ($\rightarrow$):} Porosity $\chi$ in the ice, salinity $S_l$ in the underlying  liquid, and streamfunction (solid black logarithmically spaced contours: CW, dashed: ACW) during basal melting. Remnant brine channels are visible throughout the ice during initial growth. Subsequently the ocean temperature is increased to $T_o=T_f(S_o) + 1.5^{\circ}\mathrm{C}$ and a layer of buoyant water forms due to basal melting, whilst internal melting within the ice gradually increases the porosity. After 32 days a period of strong convection is initiated throughout the ice. Melt water rises through previously active brine channels and sinks through the rest of the ice.

%Our domain is horizontally periodic, with an open bottom boundary and an impermeable, isothermal, top boundary.
\end{minipage} \hfill
\begin{minipage}[t]{0.7\linewidth}
\begin{tikzfigure}
\begin{tikzpicture}
\node[anchor=north west] (diags) at (0,0) {\includegraphics[ trim={1.0cm, 0.0cm, 0.0cm, 0.0cm}, clip]{figures/desalinationEvent-small.png} };
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\end{tikzfigure}
\end{minipage}

\vspace{-2pt}
\begin{minipage}{\linewidth + 1.5em}
\hspace{-0.75em}\textcolor{dividingLines}{\rule{\linewidth}{3pt}}
\end{minipage}
\vspace{-3.0em} 

 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Case 1
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bottomrule=0pt, 
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left skip=-1.0em,  left=0.0em, top=0.5em, right=0.5em,
bottom=0.4em,
 standard jigsaw, opacityback=0]

{\LARGE \textcolor{oxfordblue}{\hspace{1em}\textbf{Case 1: Atmospheric warming}}} 
\vspace{-1.5em}

\begin{tikzfigure}
\begin{tikzpicture}

\node[anchor=north west] (diags) at (0,0) {\includegraphics[trim={0.0cm, 0.0cm, 0.0cm, 0.0cm}, clip]{figures/SurfaceWarmingDiagnosticsNew.pdf} };

\node[anchor=north west] (timeseries2) at ([xshift=-1em,yshift=0]diags.north east) {\includegraphics[ trim={0.0cm, 0.0cm, 0.0cm, 0.0cm}, clip]{figures/SurfaceWarming-3TimeSeries-small.jpg}};

\node[anchor=north](b) at (timeseries2.north) {$T_a = -3^\circ$C};

\node[anchor=north](desalination) at ([xshift=-9.0em, yshift=5.0em]diags.south east) {\normalsize Desalination events};

\draw[->, line width=1mm](desalination.west) -- ([xshift=5.9em,yshift=3.7em]diags.south west);
\draw[->, line width=1mm](desalination.west) -- ([xshift=7.1em,yshift=5.6em]diags.south west);
\draw[->, line width=1mm](desalination.west) -- ([xshift=10.0em,yshift=5.4em]diags.south west);
\end{tikzpicture}
\end{tikzfigure}

\vspace{-2.0em}

\begin{minipage}[t]{1.0em} \end{minipage} \hfill % add some padding
\begin{minipage}[t]{0.39\linewidth}
\figureTitle{Fig. 3 ($\uparrow$):} Vertical fluxes of heat $F_h$ and salt $F_s$ at $z=0.3$m in response to increased atmospheric temperature (see legend).  %Intense desalination events can be seen for $T_a=-3^\circ$C (at $t=$2 days) and $-2^\circ$C (at $t=4$ and $8$ days). After $\sim 15$ days the warming ice stops rejecting significant quantities of salt. 
\keyPoint{Short intense desalination events occur after warming.} After $\sim 15$ days the warming ice stops rejecting significant quantities of salt. 
\end{minipage} \hfill
\begin{minipage}[t]{0.48\linewidth}
\figureTitle{Fig. 4 ($\uparrow$):} Vertical profiles for $T_a = -3^\circ$C. Top panel: minimum vertical velocity, middle panel: change in horizontally-averaged porosity, bottom panel: change in horizontally-averaged bulk salinity. The sea ice interface is indicated by a white line (top panel) or black line (lower panels). \keyPoint{Atmospheric warming increases the ice porosity, which allows salt to drain through the entire depth of ice.}
\end{minipage} 
%\hfill
%\begin{minipage}[t]{0.0em} \end{minipage}


\end{tcolorbox} \hfill \textcolor{dividingLines}{\vline  width 3pt} \hfill
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Case 2
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top=0.5em,  left=-0.15em, right=0.5em,
boxrule=0mm, colback=white, colframe=white,  standard jigsaw, opacityback=0]


{\LARGE \textcolor{oxfordblue}{\hspace{1em}\textbf{Case 2: Basal warming}} }
\vspace{-1.5em}

%\begin{minipage}[t]{0.48\linewidth}
\begin{tikzfigure}
\begin{tikzpicture}

\node[anchor=north west] (diags) at (0,0) {\includegraphics[ trim={0.0cm, 0.0cm, 0.0cm, 0.0cm}, clip]{figures/BasalWarmingDiagnosticsNew.pdf} };

\node[anchor=north west] (timeseries2) at ([xshift=-1.5em,yshift=0]diags.north east) {\includegraphics[ trim={0.2cm, 0.0cm, 0.5cm, 0.0cm}, clip]{figures/BasalWarming-1p5TimeSeries-small.jpg}};

\node[anchor=north](b) at (timeseries2.north) {$T_o = T_f(S_o) + 1.5^\circ$C};

\node[anchor=north](desalination) at ([xshift=10.0em, yshift=5.0em]diags.south west) {\normalsize Desalination events};

\draw[->, line width=1mm]([xshift=-2.0em]desalination.north) -- ([xshift=7.3em,yshift=6.6em]diags.south west);
\draw[->, line width=1mm](desalination.east) -- ([xshift=16.5em,yshift=5.5em]diags.south west);
\draw[->, line width=1mm](desalination.east) -- ([xshift=19.0em,yshift=3.7em]diags.south west);

\node[anchor=north](desalination2) at ([xshift=8.0em, yshift=5.0em]timeseries2.south west) {\normalsize \textcolor{red}{Desalination event}};
% \draw ([xshift=-8.5em,yshift=1em]timeseries2.east) ellipse (1cm and 8cm);
\draw [rounded corners=0.5cm, red, line width=1mm] ([xshift=2.5em,yshift=4em]timeseries2.south) rectangle ++(1.8,15.5);
\draw[->, line width=1mm, red](desalination2.east) -- ([xshift=2em,yshift=4.5em]timeseries2.south);



\end{tikzpicture}
\end{tikzfigure}
\vspace{-2.0em}
\begin{minipage}[t]{1.0em} \end{minipage} \hfill % add some padding
\begin{minipage}[t]{0.39\linewidth}
\figureTitle{Fig. 5 ($\uparrow$):}  Vertical heat $F_h$ and salt $F_s$ fluxes as in Fig. 4 but for simulations with an increased oceanic temperature $T_o$. Salt fluxes are typically positive due to the downward transport of fresh melt water. However, for $T_o>T_f(S_o) + 0.5^\circ$C we observe short intense bursts of negative salt fluxes.
\end{minipage}\hfill
\begin{minipage}[t]{0.47\linewidth}
\figureTitle{Fig. 6 ($\uparrow$):}  Vertical profiles as in Fig. 5, but for oceanic warming with $T_o=T_f(S_o) + 1.5^\circ$C.  An increase in porosity permits stronger convection through a deeper layer at the base of the ice. However \keyPoint{a convectively stable layer of buoyant melt water forms below the ice, preventing rejected salt from mixing with the underlying ocean except in the case of intense desalination events.}
\end{minipage}

\end{tcolorbox}




%\vspace{-2em}

%%%%%%%%%%%%%%%%
% Summary
%\begin{tcolorbox}[boxsep=5mm, width=\linewidth+1.85em,  nobeforeafter, 
%sharp corners, rounded corners=south,
  %      colback=red!5, box align=top, 
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       % before skip=-1em,
       % height=3em]

%\vspace{0.1em}
%\LARGE \textbf{Summary:} {\Large We observe significant desalination events from warming sea ice in a 2-D model, with similar magnitude and timing to field observations.}
%\end{tcolorbox}

\vspace{-0.9em} } 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% end big main block
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{subcolumns}  
\subcolumn{0.75} 



\block{Governing Equations for Flow in Porous Mushy Sea Ice}
{

\begin{minipage}[t]{0.49\linewidth}
Continuous equations for conservation of momentum (1), mass (2), salt~\eqref{eq:salt-cons} and energy~\eqref{eq:energy-cons} are found by averaging over lengths greater than the pore scale of sea ice \cite{Worster1991,LeBars2006}. In a narrow Hele-Shaw cell, the momentum equation is well approximated by Darcy's law everywhere. 
\vspace{0.2em}
\begin{align}
%\rho_0  \frac{\partial \mathbf{U}}{\partial \tau} + \rho_0 \mathbf{U} \cdot \nabla \left( \frac{\mathbf{U}}{\chi} \right) &= - \chi \nabla p + \eta \nabla^2 \mathbf{U} + \chi \rho_l \mathbf{g} - \frac{\eta \chi}{K(\chi)} \mathbf{U},
     \mathbf{U} = -\frac{K(\chi)}{\eta} \left(\nabla p - \rho_l \mathbf{g} \right), & \quad \quad   \nabla \cdot \mathbf{U} = 0, \tag{1, 2}  \\
    \frac{\partial S}{\partial t} + \mathbf{U} \cdot \nabla S_l &= \nabla \cdot \chi D_l \nabla S_l, \label{eq:salt-cons} \tag{3} \\
    \frac{\partial H}{\partial t} + \rho_0 \, c_{p,l} \, \mathbf{U} \cdot \nabla  T &= \nabla \cdot \left[ k_l \chi + (1-\chi) k_s \right] \nabla T . \label{eq:energy-cons} \tag{4}
\end{align} 
\end{minipage}
\hfill
\begin{minipage}[t][][b]{0.48\linewidth}
\vspace{-1.0\baselineskip}
\begin{align*}
&\varColor{\mathbf{U}}\; \text{(Darcy velocity)}, \quad \varColor{\chi} \; \text{(porosity)}, \quad \varColor{p} \; \text{(pressure)}, \quad \varColor{T} \; \text{(temperature)},\\
&\varColor{S_l} \; \text{(liquid salinity)}, \quad \varColor{S = \chi S_l} \; \text{(bulk salinity)},  \\
&\eqnColor{H = \rho_0 \left\{ L \chi + \left[\chi c_{p,l} + (1-\chi) c_{p,s}\right] T \right\}} \; \text{(enthalpy)}, \\
&\eqnColor{\rho_l = \rho_0 \left[1 - \alpha T + \beta S_l \right]} \; \text{(liquid density)}, \\
&\eqnColor{K(\chi)^{-1} = \left(d^2/12\right)^{-1} + \left[K_0 \chi^3 / (1-\chi)^2 \right]^{-1}} \; \text{(permeability)}, \\
&\paramColor{\eta} \; \text{(viscosity)}; \paramColor{D_l} \; \text{(salt diffusivity)}; \paramColor{\alpha}, \paramColor{\beta} \; \text{(thermal/haline expansion)}; \\
&\paramColor{c_{p,l}}, \paramColor{c_{p,s}} \; \text{(liquid/solid specific heat)}; \paramColor{k_l}, \paramColor{k_s} \; \text{(liquid/solid heat conductivity)}; \\
&\paramColor{d} \; \text{(Hele-Shaw cell thickness)}; \paramColor{K_0} \; \text{(Reference permeability)}.
\end{align*}
\end{minipage}

}
 

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 \block{Future Work}
{
\begin{itemize}
\item Consider forcing from reanalysis data, rather than idealised scenarios.
\item Investigate sensitivity of results to the Hele-Shaw cell gap width.
\end{itemize}
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This work was funded by NERC and a travel grant from the Royal Society.
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