AGU2018Poster.tex

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%Title text and graphics
\title{\parbox{\linewidth}{\centering Modelling Convection, Phase Change and Salt Fluxes from Mushy Sea Ice}} %with Adaptive Mesh Refinement
\titlegraphic{ \includegraphics[height=4cm]{figures/AGU2018.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{0.21}

\block{Introduction}{
\begin{itemize}[leftmargin=0.4cm]
    %\item 22 million square kilometres of sea ice form every year \cite{fetterer-02}
    \item Sea ice is a mushy layer of ice crystals and brine.
    \item Dense brine drains during ice formation (Fig. 1), affecting ocean circulation. \\
    \textbf{Our goal: constrain brine fluxes} using numerical simulations.
    %\item We use a \textbf{low concentration ratio}, \\ $\ConcRatio \approx S_\text{ocean}/(S_\text{eutectic} - S_\text{ocean}) \ll 1$, \\ as is appropriate for sea ice.
\end{itemize}

\vspace{-0.0cm}

\begin{tikzfigure}
%[Mushy layer grown in a lab, $\sim$10cm deep. \\ (Schulze \& Worster, 1998) %\cite{Schulze1998}]
\label{fig:lab-mushy-layer}
%\includegraphics[width=\linewidth]{../Images/SchulzeWorsterMushyLayer.png}  \vspace{-2em}
\begin{tikzpicture}
\node[anchor=north west] (fig1) at (0,0) {\includegraphics[width=10.5em, trim={0.0cm, 8.0cm, 0.0cm, 0.0cm}, clip]{figures/MiddletonEtAlBrineChannels.png} }; %0.47 \linewidth

%\node[anchor=north west] (fig2) at (fig1.north east) {\includegraphics[width=0.55\linewidth, trim={0.0cm, 0cm, 0cm, 0cm}, clip]{figures/Wettlaufer1997BrineChannel.png} };
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\end{tikzfigure}
\figureTitle{Fig. 1:} \figureCaption{Solidification of salt water viewed from (a) side; Middleton et. al. (2016) and (b) below; Wettlaufer et. al. (1997). Arrows indicate where salty plumes leave the ice. } 

}


\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} };

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\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 $T_l$ decreases, more ice forms and the residual brine salinity $S_l$ increases making the fluid denser, which can drive convection. 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:
    
%     \hspace{-\baselineskip}
%     \begin{minipage}[t]{8cm}
%     \textit{Advective terms:}  \\
%     explicit, 2\textsuperscript{nd} order unsplit Godunov.
%     \end{minipage} \hfill \vrule{} \hfill
%     \begin{minipage}[t]{12cm}
%     \textit{Nonlinear diffusive terms: } \\
%     semi implicit, geometric multigrid.
%     \end{minipage}

%    \item Timestepping: Backward Euler.

    %\nohyphens{
    {
    \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}
}




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This work was funded by NERC and a travel grant from the Royal Society.
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\vspace{-0.75cm}
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%\column{0.55}
%\column{0.42}
\column{0.79}


%\block{Scaling laws for confined convection}{
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%\LARGE \textbf{Conclusions:}
\vspace{-1.0em}
\begin{minipage}[c]{9cm}

\vspace{-1.5em}
\LARGE \textbf{Summary:}

\end{minipage}\hfill
\begin{minipage}[c]{\linewidth - 9cm}
\large 
\begin{itemize}
\item \textbf{During transient growth, channel spacing increases over time.} 
\item \textbf{Scaling predictions for steady state growth are consistent with experimental observations}
\end{itemize}
\end{minipage}
\vspace{-0.5em}
\end{tcolorbox}



{\LARGE \textcolor{oxfordblue}{\textbf{Simulated transient growth}}}

Water of initial salinity $S_0=35\,\mathrm{g\,kg}^{-1}$ and temperature $-2^\circ$C is frozen from above in a Hele-Shaw cell (plate separation $d=1$mm). We assume $K_0=10^{-9}\,\mathrm{m}^2$, and vary the atmospheric temperature $T_a$.
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\begin{tikzfigure}
\begin{tikzpicture} 
\node[anchor=north west, inner sep=0pt, outer sep=0pt] (timeseries) at (0,0) {\includegraphics[width=28cm, trim={0.5cm, 0.5cm, 0.5cm, 0.5cm}, clip]{figures/timeSeries-t.png} };
\node[anchor=north west, inner sep=0pt, outer sep=0pt] (timeseries) at ([xshift=0cm,yshift=-0.3cm]timeseries.south west) 
{\figureTitle{Fig. 4:}\figureCaption{Ice grown from a fixed boundary.  https://goo.gl/4n9STV}};
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\end{tikzfigure} 
 



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\vspace{-0.3\baselineskip}
\figureTitle{Fig. 3 ($\uparrow$).}\figureCaption{Domain and colorbars for transient simulations ($\leftarrow$).}
%(b) Solution for $T_a=-15^\circ$C, before the onset of convection.
\vspace{0.3em}
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\vspace{-0.3em}
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\figureTitle{Fig. 5 ($\uparrow$).}\figureCaption{(a) Total salt rejected from ice. (b) Ice depth increases roughly $\propto \sqrt{t}$. (c) Average solid fraction $\bar{\phi}$ increases following the onset of convection. (d) Average channel spacing $\bar{L}$ increases over time.  }

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%\LARGE \textbf{Conclusions:} 
\begin{minipage}[c]{9cm}
\vspace{-0.5em}
\LARGE \textbf{Summary:}

\end{minipage}\hfill
\begin{minipage}[c]{\linewidth - 9cm}
\large 
\begin{itemize}
\item \textbf{During transient growth, channel spacing increases over time.} 
\item \textbf{Scaling predictions for steady state growth are consistent with experimental observations}
\end{itemize}
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\vspace{-1.8em}
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{\LARGE \textcolor{oxfordblue}{\textbf{Steady state solutions}} }

\begin{minipage}[t]{0.54\linewidth}
%\vspace{10pt}
For ice grown at a constant rate $V$ from water of initial salinity $S_0$ we investigate the sensitivity to the Rayleigh number $Rm$ and concentration ratio $\mathcal{C}$,
\begin{align}
   \textcolor{dukeblue}{ \mathcal{C} = \frac{S_0}{S_e - S_0}.} \nonumber
\end{align}


In fig. 6 (right $\rightarrow$), we plot the porosity $\chi$, streamlines (white) and salinity contours (red) for steady state solutions, where the domain width is chosen to maximise the solute flux. \highlightText{Decreasing $\mathcal{C}$ reduces the porosity, confining convection to a narrow porous layer at the mush-liquid interface.} \\

Below, we construct a scaling argument for $\mathcal{C} \ll 1$, then compare to numerical simulations (fig. 7) and experimental results (fig. 8).

\end{minipage} \hfill
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\vspace{-3em} 
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%\node[anchor=south, inner sep=0pt, outer sep=0pt] (timeseries) at ([xshift=2cm,yshift=-1.2cm]timeseries.south) 
%{Go to https://goo.gl/4n9STV or scan QR code for movie};
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\end{minipage} 



 

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\vspace{-2em}

 
%% Validation of scaling laws
\begin{minipage}[t]{0.42\linewidth}
\begin{tikzfigure}
\label{fig:fluxScaling}
\begin{tikzpicture}

\node[anchor=north west] (fig1) at (0,0) {\includegraphics[height=11cm, trim={0.0cm, 0.0cm, 0.0cm, 0.0cm}, clip]{figures/fluxScalingPoster.eps} };
 
\end{tikzpicture}
\end{tikzfigure}
%Fig. x: Numerically calculated porosity (color scale and black contours), liquid salinity (magenta), and streamlines (gray, clockwise) for ice grown from 30 g/kg salt water with $\RmS=400$. 
\figureTitle{Fig. 7:}\figureCaption{Numerically calculated \highlightText{salt flux increases nearly linearly with $(\Rm - \Rm_{crit})^{3/4}$} when $\mathcal{C} \ll 1$, where $\Rm_{crit}$ is the critical Rayleigh number for convection.}
\end{minipage} \hfill
\begin{minipage}[t]{0.55\linewidth}
\begin{tikzfigure}
\label{fig:WWH}
\begin{tikzpicture}

\node[anchor=north west] (fig1) at (0,0) {\includegraphics[height=11cm, trim={0.0cm, 0.0cm, 0.0cm, 0.0cm}, clip]{{figures/WWHPosterCollapse-LiquidusSlope-0.1}.pdf} };    
 
\end{tikzpicture}
\end{tikzfigure}


\figureTitle{Fig. 8:}\figureCaption{(a) Change in salinity of the underlying liquid  during transient growth measured by \cite{wettlaufer-et-al-97}. (b) Collapse of the data predicted by $F\sim \mathcal{C} Rm^{3/4}$, where $t_c$ is the time when convection starts and $\gamma = l \; \kappa_l^{-3/8} \left(g K_0/\nu \right)^{-3/4}$.}
\end{minipage}

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

\begin{subcolumns}  
\subcolumn{0.77} 



\block{Governing Equations for Flow in Porous Mushy Sea Ice}
{
%Continuous equations for conservation of momentum, mass, energy, and salt are found by averaging over lengths greater than the pore scale of sea ice \cite{Worster1991} and nondimensionalised. For flow in a narrow Hele-Shaw cell, the momentum equation is well approximated by Darcy's law everywhere. \\
Solve for flow and solidification in reactive porous media \cite{Worster1991} with Darcy's law applied in a narrow Hele-Shaw cell with variable ice porosity. \\
Scales: length $\sim l$, time $\sim l^2/\kappa_l$, velocity $\sim \kappa_l/l$, temperature $\Delta T \sim T_e-T_0$, salinity $\Delta S \sim S_e-S_0$, permeability $\sim K_0$, pressure $\sim \eta \kappa_l / K_0$.



\begin{minipage}[t]{0.55\linewidth}
 \vspace{-0.1em}
\begin{minipage}[t]{6cm}
\eqnLabel{Momentum}
\end{minipage}
\begin{minipage}[t]{0.4\linewidth}
\vspace{-1.5em}
\begin{align} 
    \mathbf{U} = Rm \Pi \left(-\nabla p - S_l \right), \nonumber
  \end{align}
\end{minipage} \hfill \\
\begin{minipage}[t]{6cm}
\vspace{0.25em}
\eqnLabel{Energy}
\end{minipage}
\begin{minipage}[t]{0.4\linewidth}
\vspace{-0.5em}
\begin{align} 
    \frac{D H}{Dt} + \mathbf{U} \cdot \nabla T = \nabla \cdot \left[\chi + (1-\chi) k \right] \nabla T , \nonumber \label{eq:energy-cons} 
  \end{align}
\end{minipage} \hfill \\
\begin{minipage}[t]{6cm}
\vspace{1em}
\eqnLabel{Mass, salt}
\end{minipage}
%\hfill
\begin{minipage}[t]{0.48\linewidth}
\vspace{-0.0em}
\begin{align} 
    \nabla \cdot \mathbf{U} = 0, \hspace{10ex} \frac{D S}{Dt} + \mathbf{U} \cdot \nabla S = \Le^{-1} \nabla \cdot \chi \nabla S. \nonumber
     \end{align}
\end{minipage} \hfill \\
%\vspace{0.3em}
%\textcolor{dividingLines}{\hrule}
%\vspace{0.3em}
\begin{minipage}[t]{6cm}
\vspace{1.2em}
\eqnLabel{Parameters}
\end{minipage}
%\hspace{-5em} 
\begin{minipage}[t]{0.78\linewidth}
\vspace{0.2em}
\begin{align} 
   \textcolor{dukeblue}{ Rm = \frac{ \, \rho_0 \, g \, \beta \, \Delta S \, K_0 \,l}{\kappa_l \, \eta}}, \;\; Le = \frac{\kappa_l}{D_l}, \; \;St = \frac{L}{c_{p,l} \Delta T}, \; \; c_p = \frac{c_{p,s}}{c_{p,l}}, \; \; k = \frac{k_{s}}{k_{l}}. \nonumber
\end{align}
\end{minipage}

\iffalse
\begin{minipage}{\linewidth}
\begin{minipage}[t]{0.2\linewidth}
Length: $\sim l$.  \\
Time: $ \sim l^2/ \kappa_l$.
\end{minipage}
\begin{minipage}[t]{0.35\linewidth}
Permeability: $\sim K_0$.  \\
Pressure: $ \sim (\rho_0 K_0 / \eta^2)$.
\end{minipage}
\begin{minipage}[t]{0.45\linewidth}
%Temperature, salinity: $(T,S)_\text{eutectic} - (T,S)_0$. \\
Temperature: $\Delta T = \sim T_e - T_0$. \\
Salinity: $\Delta S \sim S_e - S_0$. 
\end{minipage}
\end{minipage}
\begin{minipage}{0.9\linewidth} % 0.9 here forces minipage to the left
\begin{align} 
   & \mathbf{U} = Rm \Pi \left(-\nabla p - S_l \right), \quad \frac{D H}{Dt} + \mathbf{U} \cdot \nabla T = \nabla \cdot \left[\chi + (1-\chi) k \right] \nabla T , \nonumber  \\
   & \nabla \cdot \mathbf{U} = 0, \hspace{15ex} \frac{D S}{Dt} + \mathbf{U} \cdot \nabla S = \Le^{-1} \nabla \cdot \chi \nabla S. \nonumber \\
   & Rm = \frac{\beta \, \rho_0 \, g \, \Delta S \, K_0 \,l}{\kappa_l \, \eta}, \;\; Le = \frac{\kappa_l}{D_l}, \; \;St = \frac{L}{c_{p,l} \Delta T}, \; \; c_p = \frac{c_{p,s}}{c_{p,l}}, \; \; k = \frac{k_{s}}{k_{l}}. \nonumber
\end{align}
\end{minipage}
\fi
\end{minipage} \hfill
\begin{minipage}[t]{0.4\linewidth}
\vspace{-1.2\baselineskip}
\begin{align*}
&\varColor{\mathbf{U}}\; \varLabel{(Darcy velocity)}, \quad \varColor{\chi} \; \varLabel{(porosity)}, \quad \varColor{p} \; \varLabel{(pressure)}, \quad \varColor{T} \; \varLabel{(temperature)},\\
&\varColor{S_l} \; \varLabel{(liquid salinity)}, \quad \varColor{S = \chi S_l} \; \varLabel{(bulk salinity)},  \\
&\eqnColor{H = St \chi + \left[\chi + (1-\chi) c_p \right] T} \; \varLabel{(enthalpy)}, \\
%&\eqnColor{H = \rho_0 \left\{ L \chi + \left[\chi c_{p,l} + (1-\chi) c_{p,s}\right] T \right\}} \; \varLabel{(enthalpy)}, \\
&\eqnColor{\Pi(\chi)^{-1} = \Pi_\text{Hele-Shaw}^{-1} + \chi^{-3}} \; \varLabel{(permeability)}, \\
&\paramColor{\eta} \; \varLabel{(viscosity)}; \paramColor{D_l} \; \varLabel{(salt diffusivity)};\paramColor{\beta} \; \varLabel{(haline expansion)}; \\
&\paramColor{c_{p,l}}, \paramColor{c_{p,s}} \; \varLabel{(liquid/solid specific heat)}; \paramColor{k_l}, \paramColor{k_s} \; \varLabel{(liquid/solid heat conductivity)}; \\
&\paramColor{d} \; \varLabel{(Hele-Shaw cell thickness)}; \paramColor{K_0} \; \varLabel{(reference permeability)}; \\
&\paramColor{l} \; \varLabel{(box height)}; \; \paramColor{\kappa_l} \; \varLabel{(liquid heat diffusivity)}; \; \paramColor{\rho_0} \; \varLabel{(reference density)}.
\end{align*}
\vspace{-1.9em}
\end{minipage}

%\begin{minipage}[t][][b]{0.48\linewidth}
%\begin{align*}
%&\mathbf{U}\; \text{(Darcy velocity)}, \quad \chi \; \text{(porosity)}, \quad \Pi \; \text{(permeability)}, \quad p \; \text{(pressure)}, \\
%&H = \chi S_t + T \; \text{(enthalpy)},\quad T \; \text{(temperature)}, \quad S \; \text{(salt concentration)}, \\
%&Ra - \text{Rayleigh number, determining the importance of buoyancy}, \\
%&Da - \text{Darcy number, determining the importance of viscosity}, \\
%&Le - \text{Lewis number, ratio of heat diffusion to salt diffusion}, \\
%&S_t - \text{Stefan number, quantifies energy release during freezing}.
%\end{align*}
%\end{minipage}

% \iffalse
% Continuous equations for conservation of momentum~\eqref{eq:mom-cons}, mass~\eqref{eq:mass-cons}, salt~\eqref{eq:energy-cons} and energy~\eqref{eq:salt-cons} are found by averaging over lengths greater than the pore scale of sea ice \cite{Worster1991,LeBars2006}. \\
% \begin{minipage}[t]{0.49\linewidth}
% \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},  \\
%   \nabla \cdot \mathbf{U} &= 0, \label{eq:mass-cons}  \\
%     \frac{\partial S}{\partial t} + \mathbf{U} \cdot \nabla S_l &= \nabla \cdot \chi D_l \nabla S_l, \label{eq:salt-cons} \\
%     \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}
% \end{align} 
% \end{minipage}
% \hfill
% \begin{minipage}[t][][b]{0.48\linewidth}
% \vspace{-1.75\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}
% \fi

}
 

\subcolumn{0.23}
 
 \block{Future Work}
{
\begin{itemize}
\item Consider growth over longer time periods (days to weeks), with time varying atmospheric forcing.
\item Simulations with the liquid region governed by the Navier-Stokes equation, rather than flow in a Hele-Shaw cell.
\item Three dimensional simulations, utilising the Adaptive Mesh Refinement capabilities of our code.
%\item Investigation of salt fluxes from warming sea ice in the spring/summer.
%\item Investigate salt fluxes over longer time periods, and with time dependent atmospheric forcing.
%\item Simulations with a liquid region governed by eq (1), rather than flow in a Hele-Shaw cell.
%\item Simulations in three dimensions, including Adaptive Mesh Refinement.
\end{itemize}
}

 
\end{subcolumns}

\end{columns}

\end{document}