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%\authorrunninghead{GOLDEN ET AL.}

%\titlerunninghead{Permeability of Sea Ice}

%\authoraddr{K. M. Golden,
%University of Utah, Department of Mathematics, 
%155 S 1400 E RM 233, Salt Lake City, UT 84112-0090, USA.
%(golden@math.utah.edu)}


\begin{document}


%\title{Thermal evolution of permeability 
%        and microstructure in sea ice}


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~\vspace{20ex}





\begin{center}


{\LARGE {\bf Climate Change and the \\ 
\vspace{1.5ex}
Mathematics of Transport in Sea Ice} }


%\author


\vspace{5ex}



{\large Kenneth M. Golden 


\vspace{4ex}


Department of Mathematics \\

\vspace{1ex}

University of Utah} \\

%\vspace{1ex}

%Salt Lake City, Utah, USA}



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%\begin{center}
%{\bf Abstract}
%\end{center} 


%\noindent
%Polar sea ice is a critical component of our 
%climate system, and an acute indicator of global warming.
%Fluid and thermal transport in sea ice mediates a broad range of processes
%which are fundamental to earth's climate and to the polar ecosystems.
%Electromagnetic transport in sea ice is central to remote sensing
%of the polar regions, and in monitoring the impact of climate change. 
%Here we discuss recent developments in
%mathematical modeling of transport in sea ice, and related
%measurements we made on an Antarctic expedition in 2007.








\vspace{1ex}





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%\section{Introduction}


\noindent
As the boundary layer between the ocean and atmosphere in 
the polar regions, sea ice is a critical component of the global climate
system. As temperatures on earth have warmed, the Arctic sea ice
pack in particular has exhibited a dramatic decline in its
summer extent. Indeed, the polar sea ice packs are harbingers
of climate change. Predicting what may happen over the next
10, 50 or 100 years requires extensive modeling of critical sea ice
processes, and the role that sea ice plays in the global climate. 
Currently, large-scale climate models oversimplify
many sea ice processes which can significantly affect predictions.
Moreover, monitoring the state of earth's sea ice packs, in 
particular their thickness distributions, provides key data
on the impact of global warming. Mathematics is currently 
playing an important role in addressing these fundamental
issues, and will likely play an even greater role in the future. 
Here we give a brief, bullet by bullet summary of the contents of
this article.

\vspace{2ex}

\noindent
{\bf 1. What is the role of polar sea ice in global climate?}


\begin{enumerate}
  \item[$\bullet$] Sea ice forms the boundary between the 
                   polar oceans and the atmosphere, and 
                   mediates the exchange of heat and
                   momentum between them. The polar sea ice packs
                   help regulate earth's climate, and are acute
                   indicators of climate change. 
                   
                   

  \item[$\bullet$] While sea water absorbs most incident solar radiation, 
                   sea ice reflects most of it. Earth's sea ice packs                   
                   act as solar heat reflectors, as well as an ocean insulator,
                   keeping significant heat from escaping to the atmosphere.

 \item[$\bullet$] Sea ice is a porous composite of pure ice with liquid
                   brine inclusions. The flow of fluids
                   through sea ice mediates processes
                   important to climate, from melt pond evolution
                   which controls the reflectance of the sea ice packs, 
                   to upward percolation of sea water which floods
                   the surface and then freezes, forming {\it snow-ice},
                   an increasingly important component in the sea ice system. 
                   Brine flow though sea ice can also enhance heat exchange.

 \item[$\bullet$]  Sea ice hosts algal and bacterial communities 
                   which support life in the polar oceans. Nutrient
                   replenishment processes are controlled by fluid transport 
                   through the microstructure.
 

  \item[$\bullet$]  The state of the polar sea ice packs $-$ their extent and 
                    thickness $-$ provide important information
                    in monitoring climate change. 
\end{enumerate}





\vspace{2ex}

\noindent
{\bf 2. What is the role of mathematics in modeling tranpsort in sea ice?}


\begin{enumerate}
   \item[$\bullet$] Various techniques within the field of {\bf homogenization}
                    are used to derive macroscopic information about transport in sea
                    ice from partial information about its microstructure. 
  
  \item[$\bullet$] Variational formulations for the trapping constant and
                   mean survival time for {\bf diffusion processes} which interact
                   with the pore boundaries are used to obtain rigorous
                   bounds on the {\bf fluid permeability tensor} for sea ice,
                   based on general microstructural information.                    

  \item[$\bullet$] Fluid transport in sea ice exhibits a permeable/impermeable
                   transition at a critical brine volume fraction of about 5\%,
                   which controls geophysical and biological processes.
                   {\bf X-ray computed tomography} and mapping of the pore microstructure 
                   onto {\bf random graphs} is
                   used to demonstrate that the brine phase of sea
                   undergoes a transition in connectedness at this brine
                   volume fraction. {\bf Percolation theory} is used to 
                   theoretically predict the transition, and to mathematically
                   characterize the thermal evolution of the fluid pores and 
                   their connectedness.

 \item[$\bullet$] {\bf Lattice and continuum percolation models} are used to 
                  predict critical behavior of the fluid permeability in sea ice,
                  with a universal exponent describing the behavior
                  above the percolation threshold, and {\bf critical path analysis} 
                  yields the scaling factor.


 \item[$\bullet$]  {\bf Hierarchical models} developed for porous rocks are
                   used to predict the fluid permebaility of sea ice over the 
                   entire range of brine porosities.
                    
 

  \item[$\bullet$]  A {\bf random pipe network}, which is equivalent to a random
                    resistor network and is solved using fast {\bf multigrid methods},
                    is used to numerically simulate fluid flow through sea ice.


  \item[$\bullet$] Methods of {\bf complex analysis} and {\bf functional analysis}
                   are used to obtain rigorous bounds on the 
                  {\bf effective complex permittivity} of sea ice, the key 
                 parameter characterizing its electromagnetic
                 behavior and the response of sea ice in remote sensing
                 applications, such as monitoring sea ice thickness.

 \item[$\bullet$]  {\bf Inverse methods} yield microstructural
                   information from complex permittivity data, paving the way for
                   electromagnetic monitoring of fluid and thermal transport in sea ice.

  \item[$\bullet$] We have made {\bf measurments of fluid and electrical transport} in 
                   sea ice on a 2007 Antarctic expedition, in order to validate
                   our models and investigate new phenomena.

  \item[$\bullet$] Large-scale sea ice and climate models currently do not generally
                   incorporate the key processes our work on transport describes.
                   These global models have grid sizes on the order
                   of many kilometers.  
                   {\bf Future mathematical challenges} include quantification of how
                   local transport properies on the scale of individual floes
                    influence pack behavior on much larger scales relevant to 
                   global models.  


\end{enumerate}










This paper is divided into two main parts. In Section 2, we discuss the role
of sea ice in the climate system, and how transport mediates key processes
important to issues of climate change. In the remaining sections we outline
how mathematics is being used to model fluid and electromagnetic transport 
in sea ice. Only a small number of %more general
key references, and starting points for further reading, 
are given here. Many of the references for this work can be
found in \cite{Golden:2007:GRL,Golden:MM-801,Gully:2007:Physica},
and a web link to an extensive bibliography is given at the end.











%\setlength{\baselineskip}{13pt}

%\noindent 
%{\bf \large{Sea Ice and its Role in the Global Climate System}}

%\vspace{1ex}





\eject




\noindent
{\bf \Large{Sea Ice and its Role in the Global Climate System}}

%\section{Sea Ice and its Role in the Global Climate System}

\vspace{1ex}

\noindent
{\bf \large{Decline of the summer Arctic sea ice pack}}



%\subsection{Decline of the summer Arctic sea ice pack}






 
Up until about ten years ago, each summer most of the 
Arctic Ocean was covered with sea ice of areal extent
roughly the same as the continental United States. By 2005, 
this thick, perennial ice pack had lost the areal equivalent of 
the US east of the Mississippi River. 
By the end of the 2007 summer melt season, an area comparable to 
most of the Midwestern US had been lost as well,  
and the Arctic ice pack had shrunk by nearly 40 percent 
from its 1979-2000 average extent. The fabled Northwest Passage 
was finally open to seafarers for the first time in human memory.
These stunning losses in the extent of summer Arctic sea ice
represent perhaps the most dramatic, large-scale indicator 
of global warming on the surface of the earth $-$ so far.







\begin{figure}                \centering
\includegraphics{arctic_ice_extent_bill_version.eps}
   \caption{While winter Arctic sea ice extent (blue) has decreased 
modestly over the past thirty years, summer pack ice (yellow) has 
experienced a dramatic decline, particularly in the last couple years.
The dotted base line represents the average extent 
during the period 1979$-$2000.
Some climate scientists wonder if this critical behavior of the system
means we have passed through a so-called {\it tipping point} into
a new regime. (Figure courtesy of Donald Perovich.)}
         
    \label{arctic_seaice_extent}
        
\end{figure}





The sea ice pack surrounding the continent of Antarctica is
mostly seasonal, meaning it melts in the summer before returning
in the cold, dark winter.
Will the Arctic become mostly ice-free in summer like 
the Antarctic? If so, how long will it take? The answers
to such questions are critical for global climate.
The summer Arctic sea ice pack serves as a major part 
of earth's polar refrigerator, cooling it and protecting 
the Arctic Ocean from absorbing too much solar heat.
One of the central questions of climate science today is to 
understand why the Arctic ice pack has declined so precipitously, 
and to predict how this complex system will evolve in the
coming years.
In order to see how mathematics can play a role in 
addressing such fundamental issues, let's first discuss how
sea ice forms, explore its material structure, and then outline
its role in global climate and polar ecosystems.









\begin{figure}                \centering
\includegraphics{arctic_ice_extent_comparison.eps}
   \caption{In 1980, the areal extent of the summer Arctic sea ice pack
corresponded roughly to the continental US.
By 2005 the area shown in red was gone, and by 2007 the additional area
in purple was gone as well. (Figure courtesy of Donald Perovich.)}
         
    \label{arctic_seaice_extent_comparison}
        
\end{figure}


\vspace{2ex}

%\subsection{Formation and structure of sea ice}

\noindent
{\bf \large{Formation and structure of sea ice}}




Sea ice is frozen ocean water, which freezes at a temperature of about $-1.8^{\circ}$C, or $28.8^{\circ}$F. Initially, tiny plate-like crystals called {\it frazil} form near the sea surface. Often they aggregate into slick patches called {\it grease ice}. When sea ice forms in calm conditions, thin sheets of translucent ice called {\it nilas} are seen first, which eventually thicken through congelation into gray then white ice in large, undeformed floes. Currents or light winds often push the nilas around so that they slide over each other, a process known as {\it rafting}. 
Rafting nilas is shown in Figure \ref{forming_ice} on top.
In turbulent conditions typical of the open ocean, the ``greasy'' frazil suspension undergoes cyclic compression in the wave field, and during compression the crystals can freeze together to form small cakes of slush. They grow larger by accretion and more solid through continued freezing between the crystals, and form plates with raised rims known as {\it pancake ice} \cite{Wadhams:2000}. 
Patches of new grease ice are shown on the bottom in Figure \ref{forming_ice}. Pancake ice is shown in the bottom row of Figure \ref{multiscale_structure_A}. The pancakes in the lower right of Figure \ref{multiscale_structure_A} are ``glued'' together in the process of forming larger floes. Sea ice that melts after one season is called {\it seasonal ice}, and sea ice that survives a summer season is called {\it multiyear ice}. 






\begin{figure}                \centering
\includegraphics{forming_ice.eps}
   \caption{Sea ice forming in Antarctica, with nilas 
       on top and grease ice on the bottom.}         
    \label{forming_ice}
        
\end{figure}








% http://news.yahoo.com/s/nm/20081222/sc_nm/us_climate_seawater_1

% http://www.arctic.noaa.gov/reportcard/seaice.html
















During the austral winter the continent of Antarctica, sitting atop the South Pole, is ringed mostly by seasonal ice on the Southern Ocean, with maximal ice extent in September. The Arctic is almost the geographical opposite, with an ocean at the North Pole, surrounded by continents, and covered with both seasonal and multiyear sea ice, of maximal extent in March. Sea ice should be contrasted with icebergs, glaciers, ice sheets, and ice shelves, which all originate on land. The great ice sheets covering Antarctica and Greenland are up to two miles thick, and are composed of glacial ice formed from snow which has been compressed over thousands or millions of years into ice which flows under great pressure.






\begin{figure}                \centering
\includegraphics{rule_of_fives.eps}
   \caption{Sea ice was observed in 1975 by Cox and Weeks 
(see \cite{Golden:S-2238}) to be 
effectively impermeable to brine drainage
for $\phi$ below about 5\%, and increasingly
permeable with $\phi$ above 5\%.}         
    \label{rule_of_fives}
        
\end{figure} 




As a material, sea ice is quite different from glacial ice. 
When salt water freezes, the result is a polycrystalline 
composite of pure ice with inclusions of liquid brine, air 
pockets, and solid salts. As the temperature of sea ice 
increases, the porosity or relative volume fraction $\phi$ 
of brine increases. Brine inclusions on the submillimeter 
scale are shown in the upper left panel of 
Figure \ref{multiscale_structure_A}, and its 
centimeter scale polycrystalline structure is shown in the upper right panel.






Sea ice exhibits a very interesting and 
important critical phenomenon \cite{Golden:S-2238}. 
For brine volume fractions $\phi$ below about $5\%$, 
sea ice is effectively impermeable to fluid flow, while for 
$\phi$ above $5\%$, it is increasingly permeable.
This critical brine volume fraction $\phi_c \approx 5\%$ corresponds
to a critical temperature $T_c \approx -5^{\circ}$ for a typical
bulk sea ice salinity of 5 parts per thousand, termed the 
{\it rule of fives}. When fluid
flows through sea ice, transport is facilitated by
{\it brine channels} $-$ connected brine structures
ranging in scale from a few centimeters for horizontal
slices, to a meter or more in the vertical direction, 
as shown in the middle panel of Figure \ref{multiscale_structure_A}. 





\vspace{2ex}

\noindent
{\bf \large{Ice$-$ocean$-$atmosphere interactions and global climate}}



%\subsection{Ice$-$ocean$-$atmosphere interactions and global climate}





\begin{figure}                \centering
\includegraphics{boundary_layer.eps}
   \caption{The sea ice pack forms the boundary between the 
ocean and atmosphere in the polar regions. Here sea smoke forms
in the track of the icebreaker {\it Aurora Australis} as heat flows
from the warmer ocean to the colder atmosphere. Rougher ice on the right 
has a larger drag coefficient than smooth ice, and transfers
more momentum from the winds to the ocean below. }         
    \label{boundary_layer}
        
\end{figure}




As the boundary layer between the ocean and atmosphere in the 
polar regions, sea ice is a key player in global 
climate and the world's ocean system, 
and is a leading indicator of climate change 
\cite{Weeks:1986:GSP,Jeffries:1998:ASI,Thomas:2003:SI,Serreze:S-1533}. 
The sea ice pack mediates the exchange of heat, moisture and 
momentum between the ocean and atmosphere, the two prinicpal 
geophysical fluids on earth. For example, as winds push sea 
ice around on the ocean surface, momentum is transferred
to the water below; the rougher the ice, the more efficient
the transfer. This process can cause large-scale overturning
in the upper ocean, bringing warmer water to the surface.
As another example, when sea ice freezes, it rejects cold, dense
brine, forming descending plumes in the polar oceans.
In the Southern Ocean, this process leads to the formation
of Antarctic {\it bottom water}, which then flows like a complex
river through the world's oceans. Another important process in
atmosphere$-$ocean interactions is the transfer of heat between
them. In winter when the ocean is warmer, heat flows
to the atmsophere through the sea ice itself, 
which forms an insulating 
blanket over the ocean, as well as through {\it leads}
or openings in the pack.
The thermal conductivity of sea ice 
is thus an important transport coefficient helping to 
quantify atmosphere$-$ocean interactions in coupled 
climate models. 









Roughly speaking, most of the solar radiation which is 
incident on snow-covered sea ice is reflected, while most of the solar radiation 
which is incident on darker sea water is absorbed. Sea ice is both 
ocean sunscreen and blanket, preventing solar rays from warming 
the waters beneath and thwarting ocean heat from escaping to warm 
the air above. The ratio of reflected sunlight to incident sunlight 
is called {\it albedo}. While the albedo of snow-covered ice is 
close to 1 (larger than 0.8), the albedo of sea water is close 
to zero (less than 0.1). If warming temperatures melt sea ice 
over time, fewer bright surfaces are available to reflect 
sunlight, more heat escapes from the ocean to warm the atmosphere, 
and the ice melts further. As more ice is melted, the 
albedo of the polar oceans decreases, leading to more solar 
aborption and warming, which in turn leads to more melting, 
in a positive feedback loop. It is believed that this so-called 
{\it ice-albedo feedback} has played an important role in the 
marked decrease in Arctic sea ice extent in summer 
\cite{Perovich:GRL-2008}.  
Thus, even a small increase in temperature can lead to greater 
warming over time, making the polar regions the most sensitive 
areas to climate change on earth. Global warming is {\it amplified} 
in the polar regions. 
Indeed, global climate models consistently show
amplified warming in the high latitude Arctic, although
the magnitude varies considerably across different models.
For example, the average surface air temperature at the North Pole
by the end of the 21st century 
is predicted to rise by a factor of about 1.5 to 4 times the 
predicted increase in global average surface air temperature.






\begin{figure}                \centering
\includegraphics{albedo.eps}
   \caption{White, ice$-$covered polar regions are important for 
reflecting much of the incoming solar radiation during summer, 
whereas sea water absorbs most of the incident radiation. The
relectance, or albedo of the sea ice pack is largely determined
by melt ponds on the surface, like those shown on the right
in the Arctic in June 2007. Melt pond evolution itself is largely
determined by the fluid permeability of the sea ice below, as well
as the sun's rays. }         
    \label{albedo}
        
\end{figure}







%\begin{figure}                \centering
%\includegraphics{polar_amplification.eps}
%   \caption{The zonally averaged change in surface air temperature 
% (SAT) normalized by the global average surface air temperature change 
%  from a number of climate models for 2080-2099 minus 1980-1999. The
%  normalized air temperature change provides a measure of amplification. 
%  For example, a value of 3 indicates 3 times the global average surface
% air temperature change at that latitude. Climate models consistently show
% amplified warming in the high latitude Arctic, although the magnitude
% varies considerably across different models (courtesy of Marika Holland).}         
%    \label{polar_amplification}
        
%\end{figure}





In the summer of 2007, the extent of the summer Arctic ice pack
reached a record minimum.
Ice mass balance observations show
that there was an extraordinarily large amount of melting on
the bottom of the ice in the Beaufort Sea during that summer
\cite{Perovich:GRL-2008}. 
Calculations indicate that solar heating of the upper
ocean was the primary source for the observed
melting. An increase in the
open water fraction resulted in a 500\% positive anomaly in
solar heat input to the upper ocean, triggering an ice$-$albedo
feedback, accelerating the pack ice retreat. Also of concern is 
that there has apparently been a significant reduction in the 
relative amount of thicker, multiyear ice in the Arctic basin.
Many authors have recently acknowledged that a relatively 
younger, thinner ice cover is more susceptible to the 
effects of atmospheric and oceanic forcing 
\cite{Serreze:S-1533}.
In the face of predictions for continued warming, 
the persistence of recent atmospheric and oceanic circulation patterns, 
and the amplification of these effects through the ice$-$albedo feedback 
mechanism, it is becoming increasingly likely that the Arctic Ocean
will change from perennially ice-covered to ice-free in the summer. 


As evidenced from the previous discussion, a key determining 
factor in predicting the future trajectory of Arctic sea ice
is understanding how ice pack albedo evolves. Melt ponds which 
form on the surface of the ice, as shown on the right in 
Figure \ref{albedo}, determine its albedo. 
Whether or not these ponds spread, deepen, or drain, 
is controlled largely by the fluid permeability of the sea ice, 
which is one of the main motivations
for our studies of fluid transport in sea ice discussed
below.











Another motivation for studying fluid flow in sea ice 
is the importance of sea ice production from freezing of flooded ice
surfaces, particularly in the 
Antarctic, where it is ubiquitous throughout the pack. 
As snow loading increases during a storm,
a pressure head develops. 
Sea water is forced upward 
through the sea ice, 
as constrained by its permeability. Net flux to the surface can occur
only if the temperature throughout the entire sea ice layer 
has exceeded the critical temperature $T_c$ for fluid flow.
Once the surface of the ice is flooded, the slushy mix
of sea water and snow freezes, which a dominant form 
of ice production in the Antarctic. In fact, witnessing this 
process during a powerful winter storm in the Eastern
Weddell Sea while on the Antarctic Zone Flux Experiment
(ANZFLUX) in 1994, 
and watching sea water flood the ice surface
in a rather dramatic fashion, was what motivated the 
author to begin studying fluid transport in sea ice.
Recent models have even suggested that with increased polar
precipitation accompanying global warming, the Arctic winter
sea ice pack could thicken, and that snow-ice production 
could become more important in the Arctic. 
Fluid flow through sea ice also facilitates 
convection-enhanced thermal transport,
and the input of brine and fresh water into the upper
ocean from freezing, melting, and drainage processes
\cite{Weeks:1986:GSP,Thomas:2003:SI}. 






\begin{figure}                \centering
\includegraphics{sea_ice_ecosystem_photos.eps}
   \caption{The rich Antarctic marine ecosystem in supported
by photosynthetic algae (d), which give sea ice a brownish or yellowish
color (e). Nutrient replenishment for these microbes is controlled by
fluid flow through the porous brine microstructure. Krill (c) feed
on algae, Emperor penguins (a) feed on krill, and killer
whales (b) spy-hop looking for penguins to eat.  }         
    \label{sea_ice_ecosystem_photos}
        
\end{figure}





\vspace{2ex}

\noindent
{\bf \large{Sea ice ecology and fluid transport processes}}



%\subsection{Sea ice ecology and fluid transport processes}















The brine inclusions in sea ice host extensive algal and bacterial communities 
\cite{Lizotte:1998:ASI,Thomas:2003:SI}
which are essential for supporting life in the polar oceans, such as krill, which themselves support fishes, penguins, seals, 
and Minke whales, and on up the food chain to top predators like killer whales, 
leopard seals, and polar bears. 
Nutrient replenishment processes for sea ice microbes are facilitated
by fluid flow through the porous microstrucure. For example, an algal
bloom during the Antarctic fall in the Weddell Sea was fueled by nutrient-laden sea water replacing cold dense brine draining from 
freezing slush at the surface, in a convective overturning process. 
The bloom effectively shut down,
however, once the critical isotherm $-$
above which the ice was impermeable and below which the ice 
was permeable $-$ passed through 
the algae layer during the fall freeze-up \cite{Golden:S-2238}.






\vspace{2ex}

\noindent
{\bf \large{Multiscale structure of sea ice over ten orders of magnitude}}





%\subsection{Multiscale structure of sea ice over ten orders of magnitude}







\begin{figure}                \centering
\includegraphics{multiscale_A.eps}
   \caption{Multiscale nature of sea ice. Sea ice is structured on
      many length scales, ranging from tenths of a millimeter to 
         hundreds of kilometers.}         
    \label{multiscale_structure_A}
        
\end{figure}



\begin{figure}                \centering
\includegraphics{multiscale_B.eps}
   \caption{Multiscale nature of sea ice, continued. Pack ice viewed from 
             ice level (top), from a helicopter (middle), and from a satellite
            (bottom).}         
    \label{multiscale_structure_B}
        
\end{figure}







One of the fascinating, yet challenging aspects of modeling 
sea ice and its role in global climate is the sheer range
of relevant length scales of structure, over ten orders
of magnitude from the submillimeter 
scale to hundreds of kilometers. In Figures
\ref{multiscale_structure_A} and \ref{multiscale_structure_B},
we show principal examples of sea ice structure illustrating
such a range of scales. 
Modeling sea ice on a large scale 
depends on understanding the physical properties of 
sea ice at the scale of individual floes and smaller.
Today's climate models challenge even the most powerful 
super computers to their fullest capacity. However, 
even the largest computers still limit the horizontal 
resolution to tens of kilometers and require clever 
approximations to model the basic physics of sea ice.
One of the central themes is how to use information
at a smaller scale to predict behavior at a larger scale,
such as in classical statistical mechanics. In the sections
to follow, we focus on the microphysics of sea ice.
That is, we address the fundamental issues of characterizing
the brine microstructure and its thermal evolution,
and how to use that knowledge to derive information about
the bulk transport properties of sea ice relevant to 
climate models, as well as monitoring climate change. 
%We first consider the fluid transport problem in sea ice, and 
%bounds on the fluid permeability.



\vspace{8ex}

\eject



\noindent
{\bf \Large{Bounds on the Fluid Permeability of Sea Ice}}

\vspace{2ex}


%\section{Bounds on the Fluid Permeability of Sea Ice}


%\noindent 
%{\bf \large{Bounds on the Fluid Permeability of Sea Ice}}

%\vspace{1ex}

\noindent 
Despite the importance of fluid flow through sea ice 
in considerations of climate change and polar ecosystems,
little attention had been paid to this problem, 
particularly from the theoretical standpoint,
prior to our work 
\cite{Golden:S-2238,Golden:MM-801,Golden:2007:GRL,Zhu:AG-129}
(with \cite{Freitag:1999:HPA} a notable exception).
As a first step toward analyzing the 
vertical fluid permeability $k$ and its microstructural dependence,
we obtained rigorous upper bounds on $k(\phi)$ 
\cite{Golden:MM-801,Golden:2007:GRL}, which we 
briefly review.


Consider low Reynolds number flow 
of a fluid %with viscosity $\nu$
through a porous random medium, occupying 
a region $\Omega \subset {\mathbb{R}}^3$. 
Brine of viscosity $\mu$
occupies the brine pore space 
$\Omega_b \subset \Omega$, 
having a relative volume fraction $\phi$, which
is determined by the 
sea ice temperature $T$ and bulk salinity $S$
\cite{Weeks:1986:GSP,Thomas:2003:SI}.
The solid ice phase 
occupies the ice grain space 
$\Omega_i \subset \Omega$, 
having a relative volume fraction $1-\phi$. 
%Let $(\Theta,P)$ be a probability space characterizing the porous medium,
%where $\Theta$ is the set of realizations of the random medium and $P$
%is a probability measure on $\Theta$. 
%For any realization $\omega \in \Theta$,
Let $\chi(x)$ be the indicator function 
of the brine pore space, with
%\begin{equation}
$\chi(x) = 1$ if $x \in \Omega_b$ and
$\chi(x) = 0$ if $x \in \Omega_i$.
The porous microstructure is assumed to have
translation invariant statistics for given $T$ and $S$, 
so that an {\it ergodic hypothesis} is satisfied, 
with equivalent ensemble and volume averaging (in the infinite volume limit), 
denoted by $\langle \cdot \rangle$ \cite{Torquato:RHM-02}.


%For an appropriate probability space with translation
%invariant statistics characterizing
%the random medium, an {\it ergodic hypothesis} is satisfied,
%so that ensemble averaging over
%realizations is equivalent to an infinite
%volume limit $|\Omega| \rightarrow \infty$ of 
%an integral average over $\Omega \subset {\mathbb{R}}^3$,
%denoted by $\langle \cdot \rangle$
%\cite{Torquato:RHM-02}. 


%We are interested in sea ice as a porous medium
%The sea ice temperature $T$ and bulk salinity $S$ determine
%its brine volume fraction $\phi$ 
%\cite{Weeks:1986:GSP,Thomas:2003:SI}.
%Within a given depth range in
%an ice sheet, the microstructural statistics can be quite
%uniform over many meters horizontally, so that the 
%conditions above are met. 
As with many other porous media \cite{Torquato:RHM-02}, 
there is a microscopic length scale $\ell$ associated
with the medium.
For example, the scale
over which the two point correlation function obtained
from $\chi(x)$ varies, is a good measure of this length, which
is small compared to
a typical macroscopic length scale $L$, such as a sample size
or thickness of a statistically homogeneous layer.
Then the parameter $\epsilon=\ell/L$ is small, and 
one is interested in obtaining the effective fluid transport behavior
in the limit as $\epsilon \rightarrow 0$. 
To obtain such information,
the method of {\it two-scale homogenization} 
or {\it two-scale convergence}
\cite{Torquato:RHM-02,Hornung:1997}
has been developed in various forms,
based on the identification of two scales:
a slow scale $x$ and a fast scale $y=x/\epsilon$.
The velocity and pressure fields in the brine,
$v^{\epsilon}(x)$ and $p^{\epsilon}(x)$ 
with $x \in \Omega_b$, are assumed to depend on these
two scales $x$ and $y$. The idea is to {\it average},
or {\it homogenize} over the fast scale 
using expansion in powers of $\epsilon$, 
which leads to a simpler equation describing the overall 
behavior of the flow.


The velocity and pressure in the brine phase
satisfy the Stokes equations,
\begin{equation}      \label{stokes1}
\nabla p^{\epsilon} = \mu \Delta v^{\epsilon}, \;\;\;
\nabla \cdot v^{\epsilon} = 0, 
                \;\;\;\;\;\; x \in \Omega_b(\omega),
\end{equation} 
with the no-slip 
boundary condition on the pore surface
$v^{\epsilon}(x)= 0$,
$x \in \partial \Omega_b.$
%A force acting on the medium such as gravity 
%can be incorporated into $p^{\epsilon}$.
The first equation in (\ref{stokes1}) 
is the steady state fluid momentum 
equation in the zero Reynolds number limit.
The macroscopic equations can be derived through a two-scale
expansion, and by
averaging the leading order term $v_0$ of the velocity
over $y$, one obtains 
\begin{equation}       \label{darcy1}
 v(x)=-\frac{1}{\mu} \; \kk \; \nabla p(x), \;\;\;\;
 \nabla \cdot v(x)=0,
\end{equation} 
where $p(x)=p_0(x)$, the leading order term in the 
expansion of $p^{\epsilon}(x)$, and 
$\kk$
is the effective permeability tensor.
The first equation in (\ref{darcy1}) is known as Darcy's law and 
the second is the macroscopic
incompressibility condition \cite{Hornung:1997}. 
%These macroscopic equations 
%were obtained by Allaire
%\cite{Hornung:1997} for periodic media 
%through an appropriate limit as $\epsilon \rightarrow 0$.
We shall be interested
in the permeability in the vertical direction $k_{zz}=k$,
in units of m$^2$.









\begin{figure}                \centering
\includegraphics{vertical_pipes.eps}
   \caption{(a) The conductivity of a two phase composite
     in a vertical electrical field is maximized by laminate 
     structures parallel to the field. (b) The vertical fluid 
     permeability of a porous medium is maximized by vertical pipes
     with appropriate radii. }         
    \label{vertical_pipes}
      
\end{figure}






To motivate our results,
consider the effective conductivity tensor $\bsig^*$ of a 
two phase composite with electrical 
conductivities $\sigma_1$ and $\sigma_2$ in the volume 
fractions $\phi$ and $1-\phi$. The defining equations 
here are $J=\sigma E$, where $J$ is the current density,
$E$ is the electric field, and $\sigma(x)$ is the local
conductivity taking the values $\sigma_1$ and $\sigma_2$.
The local fields satisfy
\begin{equation}
\nabla \times E=0, \;\;\;\; \nabla \cdot J = 0, 
\end{equation}
and the averaged equation 
$\langle J \rangle = \bsig^* \langle E \rangle$
defines $\bsig^*$, analogous to Darcy's law.
Optimal upper and lower bounds on a diagonal component $\sigma^*$ of
$\bsig^*$, such as in the vertical direction, 
known as the classical arithmetic and harmonic mean bounds
(or elementary bounds)
were obtained almost one hundred years ago by Wiener,
\begin{equation}           \label{elementary_bounds}
\frac{1}{\frac{\phi}{\sigma_1}+\frac{1-\phi}{\sigma_2}} \;\;
\leq \;\; \sigma^* \;\; \leq \;\;
\phi \sigma_1 + (1-\phi) \sigma_2.
\end{equation}
The bounds are optimal since laminates parallel to the field
realize the upper bound, as shown in Figure \ref{vertical_pipes} (a),
while laminates perpendicular to the field realize the lower bound.


For the fluid problem, we ask what configuration of the pores
maximizes $k$. 
%but a length scale not present for
%electrical conduction now enters the problem.
%The electrical conductance $g_e$ of a cylinder of unit length
%and radius $r$ is $g_e=\pi r^2 \sigma$, where $\sigma$ 
%is the conductivity of the material. On the other hand,
%the corresponding fluid conductance $g_f$ of a pipe of unit length
%and inner radius $r$ is $g_f=\pi r^4/8 \mu$. 
%The corresponding {\it fluid} conductivity
%$\sigma_f=r^2/8 \mu$ thus depends upon the cross sectional 
%area of the pipe.
Intuitively, the best arrangement to maximize $k$
%which is analogous to $\mu \sigma^*_f$ with $\sigma^*_f$ the
%effective fluid conductivity,
is in vertical pipes of appropriate radii, as shown in 
Figure \ref{vertical_pipes} (b), where in the fluid problem
a length scale not present for electrical conductivity arises.
This difference can be seen from the observation that
the electrical conductance $g_e$ of a cylinder of unit length
and radius $r$ is $g_e=\pi r^2 \sigma$, where $\sigma$ 
is the conductivity of the material, while
the fluid conductance $g_f$ of a pipe of unit length
and inner radius $r$ is $g_f=\pi r^4/8 \mu$. 
%Viewing sea ice as a two phase composite with the
%fluid conductivities $\sigma_2=0$ for the ice phase 
%and $\sigma_1=r^2/8 \mu$ for uniformly sized brine 
%inclusions, yields from the
%upper bound in (\ref{elementary_bounds})
In \cite{Golden:MM-801} we obtained a 
{\it pipe bound}, $k \leq  \phi  [a(\phi)]^2/8$, 
with mean brine inclusion horizontal 
cross-sectional radius 
$a(\phi) = 7 \times 10^{-5} + (1.6 \times 10^{-4})\phi$ 
meters. 
%This pipe bound corresponds to the arithmetic bound in 
%(\ref{elementary_bounds}) by taking $\sigma_2=0$
%for impermeable ice and $\sigma_1=$ 
The simple pipe bound, based on uniformly sized pipes
whose radii increase with $\phi$ or $T$, 
captures lab data
\cite{Golden:MM-801}. However, when considering Arctic field
data where a broader range of inclusion length scales
is relevant, 
%a few large connected brine structures can
%dominate a measurement and 
some of the data lie outside
the bound. We must consider generalizations which allow
for a distribution of radii. The derivation of such bounds
\cite{Torquato:PRL-2004,Golden:MM-801,Golden:2007:GRL}
employs diffusion processes in the pore space. 




Consider the problem of a tracer
diffusing in the fluid phase 
$\Omega_b \subset \Omega \subset {\mathbb{R}}^3$
of a porous medium. The tracer 
reacts with partially or completely absorbing {\it traps}
on the boundary $\partial \Omega_b$ of the pore space. 
%The traps themselves may represent particulate 
%sediment entrained
%in the sea ice having nutritive organic content
%for enzymatic foraging by bacteria \cite{Golden:MM-801}.
Let $c(x,t)$ with $x \in \Omega_b$
be the time dependent concentration of the 
reactant governed by the diffusion equation and boundary condition
\begin{equation}      \label{diffusion1}
  \frac{\partial c}{\partial t} = D \Delta c + G,
                \;\; x \in \Omega_b,
\;\;\;\;\;\;
D \frac{\partial c}{\partial n} +  \eta c =0,
                \;\; x \in \partial \Omega_b,
\end{equation} 
where $D$ is the diffusion coefficient of the reactant
in the fluid, $\eta$ is a positive surface reaction
rate constant, $G$ is a generation rate of reactant per unit
trap-free volume, and $n$ is the unit outward
normal from the pore space.
%As a particle diffuses in the trap-free region, or the pore space
%$\Omega_b(\omega)$, it is absorbed on the boundary of the solid
%matrix with a probability related to the surface rate constant $\kappa$.
%A dimensionless surface rate constant 
%$\eta^* = \eta \ell / D$,
%where $\ell$ is a characteristic pore length, can be used 
%to distinguish
%different regimes of influence: diffusion-controlled for
%$\eta^* \gg 1$, and reaction-controlled for
%$\eta^* \ll 1$. 
%In the diffusion-controlled regime, a reactant will typically diffuse 
%in the pore space for much longer than the characteristic time 
%associated with the surface reaction. In the limit of infinite
%surface reaction $\eta \rightarrow \infty$, the boundary
%condition becomes a perfectly absorbing 
%{\it Dirichlet condition}, with $c=0$ on $\partial \Omega_b$ 
%wherever the traps are located. 
%In the reaction-controlled regime, 
%and a reactant will typically diffuse 
%in the pore space for a relatively short time before
%if hits a trap and has a chance to react. 
%For vanishing surface reaction rate $\kappa = 0$,
%the boundary condition
%becomes a perfectly reflecting {\it Neumann condition},
%with $\frac{\partial c}{\partial n}=0$.
In the diffusion-controlled limit \cite{Torquato:PRL-2004}, where
a reactant will typically diffuse 
in the pore space for much longer than the characteristic time 
associated with the surface reaction,
we consider the steady state problem 
where the rate of removal of the 
reactant by absorbing
boundaries is exactly 
compensated by the production
rate per unit volume $G$ of the reactant.

The parameter of interest is the
{\it trapping constant},
\begin{equation}      \label{diffusion6}
{\gamma}^{-1}= \langle u \chi \rangle,
\end{equation}
where $u$ is a scaled concentration field
satisfying $\Delta u = -1$ in $\Omega_b$
and $u=0$ on the pore boundaries, obtained
from a two-scale expansion of $c$.
The trapping constant is inversely related to 
the {\it mean survival time} $\tau$ for diffusion
in the pore space, 
$
\tau = 1/\gamma \phi D.
$
Using a variational definition for the
trapping constant with an appropriately constructed
trial field yields
a general two-point {\it void lower bound} 
\cite{Torquato:PRL-2004}
\begin{equation}      \label{diffusion7}
\gamma \geq \frac{(1-\phi)^2}{{\ell}_P^2}, 
\end{equation}
where ${\ell}_P$ is a length scale defined
in terms of the two-point correlation function of the pores.
%and $\phi_S$ is the trap volume fraction.
%It is somewhat surprising that effective properties
%characterizing diffusion in a porous medium, such as $\gamma$,
%are closely related to properties characterizing 
%bulk transport from flow. In fact,
%for any ergodic porous medium, possibly anisotropic,
%$ \kk \; \leq \; \gamma^{-1} \II$,
%or equivalently, $(\gamma^{-1}\II - \kk)$ is a positive definite, symmetric,
%second order tensor \cite{Torquato:RHM-02}.









In the theory of composite materials, the Hashin-Shtrikman
bounds \cite{Torquato:RHM-02,Golden:MM-801,Gully:2007:Physica} 
are important in providing
optimal estimates on effective transport properties using 
the volume fraction information as well as 
isotropy of the material. These bounds are optimal, 
in that they can be attained
by actual geometries, namely, hierarchical arrays of 
coated spheres in $d=3$ or coated cylinders in
$d=2$, filling all space.
The two-point void lower bound in (\ref{diffusion7}) can be evaluated 
\cite{Torquato:PRL-2004} for Hashin-Shtrikman
coated cylinder geometries,
i.e., brine-filled vertical ice pipes for the sea ice problem,
with appropriate distributions of inclusion radii $R_I$,
as shown in Figure \ref{logarithmic_pipe_bounds} (a).
Using an inequality relating the permeability tensor to the
trapping constant \cite{Torquato:RHM-02}
yields an optimal, rigorous upper bound,
\begin{equation} \label{permbound2}
k \leq \frac{\phi \langle R_I^4 \rangle}{8  \langle R_I^2  \rangle}\;.
\end{equation}
Measured brine inclusion cross-sectional areas $A$ 
have been found to obey a lognormal distribution
\cite{Golden:2007:GRL}.
Then $z=\ln{A}$ has a normal probability density
with mean $\alpha$ and variance $\rho^2$,
\begin{equation}                  \label{lognormal_distribution}
P(z) = %(2 \pi \rho^2)^{-1/2}
     \frac{1}{\sqrt{2 \pi \rho^2}} \;
         e^{-(z-\alpha)^2/2 \rho^2}.
\end{equation}
We evaluated \cite{Golden:2007:GRL}
the moments in (\ref{permbound2})
using the lognormal distribution, yielding 
\begin{equation}                  \label{pipe_bound_sigma}
k(\phi) \leq   \frac{\phi}{8 \pi} \; 
\langle A(\phi) \rangle \; e^{\rho^2}.
%a(\phi) = 7 \times 10^{-5} + (1.6 \times 10^{-4})\phi \;\; \mbox{m}.
\end{equation}
With variance $\rho^2 \approx 1$
and $\langle A(\phi) \rangle=\pi a^2(\phi)$ as above,
the {\it lognormal pipe bound} in (\ref{pipe_bound_sigma})
captures our Arctic field data, as well as the lab data, 
as shown in Figure \ref{logarithmic_pipe_bounds} (b).
The field data were taken near Barrow, 
Alaska in a hydrological bail test, where 
a cylindrical hole is drilled, 
a tube is inserted to seal off horizontal flow, and time
series measurements of the water level
are used to estimate the permeability of the ice
under the hole.




\begin{figure}                \centering
\includegraphics{logarithmic_pipe_bounds.eps}
   \caption{(a) Hashin-Shtrikman coated cylinders provide an
      optimal geometry which attain the two-point void upper bound 
      for the vertical fluid permebaility. (b) The lognormal
       pipe bounds capture all our lab and field data on the vertical
       permeability of sea ice.}         
    \label{logarithmic_pipe_bounds}
        
\end{figure}




\vspace{8ex}



\eject


\noindent
{\bf \Large{Predictions for the Fluid Permeability of Sea Ice}}

\vspace{2ex}


%\section{Predictions for the Fluid Permeability of Sea Ice}


\noindent
{\bf \large{Percolation theory for sea ice}}


%\subsection{Percolation Theory for Sea Ice}


Percolation theory 
\cite{Stauffer-92,Torquato:RHM-02,Hornung:1997}
has been used to model disordered
materials where the connectedness of one phase
dominates effective behavior. Examples range
from porous rocks and bubbly glacial ice to 
semiconductors and carbon nanutube composites.
Consider the square ($d=2$) or cubic ($d=3$) network of bonds 
joining nearest neighbor sites on the integer lattice 
%${\mathbb{Z}}^d$.
${\mathbb{Z}}^d$.
%In the random pipe network,
We'll initially consider the problem of electrical transport 
through the network, and then fluid transport. 
The bonds are assigned electrical conductivities
$\sigma_0 > 0$ (open) or 0 (closed) with probabilities $p$ 
and $1-p$. 
Groups of connected open bonds 
are called open clusters, and the average cluster size
grows as $p$ increases.
The striking feature of this model is that 
there is a critical probability $p_c$, $0<p_c<1$,
called the {\it percolation threshold}, 
where an infinite cluster of open bonds first appears.
In $d=2$, $p_c = \frac{1}{2}$, and in $d=3$,
$p_c \approx 0.25$. 






\begin{figure}                \centering
\includegraphics{percolation_theory.eps}
   \caption{The two dimensional square bond lattice below
        its percolation threshold $p_c=1/2$ in (a), 
        and above its threshold in (b). }         
    \label{percolation_theory}
        
\end{figure}






The infinite cluster density 
$P_{\infty}(p)$ is defined as the probability that the origin 
(or any point, by translation invariance)
is contained in the infinte cluster, whose graph is 
shown in Figure \ref{percolation_graphs} (b).
The two-point function $\tau(x,y)$ measures the probability
that two points $x$ and $y$ belong to the same open
cluster. As $|x-y| \rightarrow \infty$, 
$\tau(x,y) \sim exp(-|x-y|/\xi)$, where $\xi$ is called the 
correlation length, which is a measure of the scale of connectedness. 
The correlation length diverges as $p$ increases toward $p_c$
and the infinite cluster forms, as shown in Figure
\ref{percolation_graphs} (a).
In the neighborhood of $p_c$, $P_{\infty}(p)$ and $\xi(p)$
exhibit power laws, with 
\begin{equation}
P_{\infty}(p) \sim (p-p_c)^{\beta}, \;\; p \rightarrow p_c^+,
\;\;\;\;\;\;\;\;
\xi(p) \sim (p_c - p)^{-\nu}, \;\; p \rightarrow p_c^-,
\end{equation}
where $\beta$ and $\nu$ are {\it universal} critical exponents which
are believed to depend only on dimension, and not on the details
of the lattice. In $d=3$, $\beta \approx 0.41$ and $\nu \approx 0.88$
\cite{Stauffer-92}.






\begin{figure}                \centering
\includegraphics{inf_cluster_dens_corr_length_graphs.eps}
   \caption{(a) The correlation length diverges as
       $p \rightarrow p_c^-$. (b) The infinite cluster
      density $P_{\infty}(p)$ vanishes below $p_c$, and takes off
      with power law behavior above $p_c$. (c) Below $p_c$,
       there is no bulk transport, and above $p_c$
       the tranpsort coefficient takes off with a different 
        power law.}         
    \label{percolation_graphs}
        
\end{figure}






Let $\sigma^*(p)$ be the effective conductivity 
of this random resistor network
in the vertical direction, as defined by Kirchoff's laws
in the infinite volume limit
\cite{Golden:PRL-2923}.  
For $p < p_c$, $\sigma^*(p)=0$, as shown in 
Figure \ref{percolation_graphs} (c). For $p>p_c$ near the threshold,  
$\sigma^*(p)$ exhibits power law behavior,
\begin{equation}                    \label{critical_conductivity}
\sigma^*(p) \sim \sigma_0(p-p_c)^t\;,\;\;\;\; p \rightarrow p_c^+,
\end{equation}
where $t$ is the conductivity critical exponent.
%There was debate concerning the value of $t$ in three dimensions  
%over roughly three decades, 
%with a wide range of published 
For lattices, %$p_c$ depends on the type of lattice, while 
$t$ is believed to be universal, 
depending only on $d$.
The corresponding fluid permeability $\kappa^*(p)$ of the lattice, 
where the bonds are open or closed pipes of fluid conductivity 
$\kappa_0/\mu=r_0^2/8\mu$, behaves like 
\begin{equation}                    \label{critical_permeability}
\kappa^*(p) \sim \kappa_0(p-p_c)^e\;,\;\;\;\; p \rightarrow p_c^+,
\end{equation}
where $e$ is the fluid permeabilty exponent.
For lattices, it is believed that $e=t$
\cite{Torquato:RHM-02}.
In $d=2$, $t \approx 1.3$,
and in $d=3$ \cite{Stauffer-92,Torquato:RHM-02},
$t \approx 2.0$.  
There is also a rigorous 
bound \cite{Golden:PRL-2923} that $t \leq 2$.
%The graph of $\kappa(p)$ in $d=2$ is illustrated 
%in the inset in Figure \ref{perc_field_data}.










In view of the apparent critical behavior 
of brine drainage in columnar sea ice depending on whether
$\phi$ is above or below about 0.05
\cite{Weeks:1986:GSP}, it was reasonable to try and
find a percolation theoretic explanation. However, 
with $p_c \approx 0.25$ for the $d=3$ cubic bond lattice,
it was apparent that key features of the geometry of
the brine microstructure in sea ice were being missed by
lattices. In \cite{Golden:S-2238} $\phi_c \approx 0.05$ 
was identified with the critical probability in a continuum
percolation model for compressed powders \cite{Kusy:N-58}
which exhibit microstructural characteristics 
similar to sea ice. The identification explained 
the rule of fives, as well as data on algal growth 
and snow-ice production. 
The compressed powders, shown in 
Figure \ref{science_figure_powder}, were used
in the development of so-called {\it stealthy} or 
radar absorbing composites. 








\begin{figure}                \centering
\includegraphics{science_figure_powder.eps}
   \caption{(a) A powder of large polymer spheres mixed with 
               smaller metal spheres \cite{Kusy:N-58}. 
              (b) When the powder is
              compressed, its microstrucure is similar
             to that of sea ice in (c). }         
    \label{science_figure_powder}
        
\end{figure}







When we applied the compressed powder model to sea ice
about a decade ago, we had no {\it direct} evidence
that the brine microstructure undergoes a transition
in connectedness at a critical brine volume fraction.
This lack of evidence was due partly to the difficulty
of imaging and quantitatively characterizing the brine 
inclusions in three dimensions, particularly the 
thermal evolution of their connectivity. 
Through X-ray computed tomography and pore structure 
analysis
\cite{Golden:2007:GRL}
we have now analyzed critical behavior of the 
thermal evolution of brine connectedness 
in sea ice single crystals,
over a temperature range from $-18^{\circ}$ C to 
$-3^{\circ}$ C.
We have mapped three dimensional images
of the brine phase onto graphs and analyzed
their connectivity as functions of temperature and sample size.
Realistic network models of brine inclusions can be derived from porous 
media analysis of 3-D micro-tomography images. Approaches include the 
well-established 3DMA package (3-Dimensional Medial Axis,
see \cite{Golden:2007:GRL} for example),
%e.g \cite{Lindquist:1999:PCE,Lindquist:2000:JGR,Prodanovic:2007:AWR} 
and the developmental tight 
dual model of Glantz and Hilpert. 
%We have worked closely with developers of both packages 
%\cite{Pringle:2008,Pringle:2006:EOS,Golden:2007:GRL}.
In both cases, a 3-D graph of nodes and edges is derived from the pores and throats. In general terms, the steps are: (1) represent the pore space by a polyhedral complex; (2) decompose this digital pore space into pores and throats, and compute individual characteristic radii; (3) map these to nodes and edges, tagged with their respective radii. 
%3DMA applies a traditional 
%voxel-based analysis. In TDM, an interpolation scheme 
%and a 3-D Delaunay tessellation are applied to create a more general polyhedral 
%decomposition of the pore space. 
%Polygonal throats are resolved as sets 
%of adjoining 2-D planes (rather than collections of voxels as in 3DMA), 
%and a mapping is created from 3-D pores to 0-D nodes and quasi 2-D 
%throats to 1-D branches. 3DMA achieves essentially the same from the 
%voxel representation.
The resulting graphs enable imbibition and 
drainage characterization, 
connectivity and critical path analyses (see below). 








\begin{figure}                \centering
\includegraphics{brine_connectivity.eps}
   \caption{X-ray CT volume rendering of brine layers 
        within a lab-grown sea-ice single crystal 
       with S = 9.3 ppt.  The (non-collocated) 
      8 $\times$ 8 $\times$ 2 mm 
      sub-volumes illustrate a pronounced change in 
     the micro-scale morphology and connectivity of 
     the brine inclusions %as the porosity $\phi$ increases 
    during warming.}
%www.ams.sunysb.edu/$~$lindquis/3dma/3dma$_$rock/3dma$_$rock.html].
      
    \label{brine_connectivity}
        
\end{figure}









An example of how brine morphology and
connectivity evolves with temperature
is shown in Figure \ref{brine_connectivity}.
X-ray CT has provided new insight into 
small-scale, intra-crystalline brine layer 
connectivity. Our work
provides the first direct observation of 
an onset in connectivity over this scale 
($\approx$ 1 cm). Notably, 
the thermal evolution of pore morphology 
that we observe is more complex than the classic 
model \cite{Thomas:2003:SI,Weeks:1986:GSP},
a transition from brine sheets to 
cylindrical to spheroidal inclusions,
which underlies a range of porosity models 
\cite{Thomas:2003:SI,Weeks:1986:GSP}.













While percolation theory has been applied 
to disordered composites for fifty years, 
key functions such as the percolation probability (which 
equals 0 for $p < p_c$ and equals 1 for $p > p_c$), 
the infinite cluster density, 
the correlation length, 
and their critical exponents, have been computed 
only for idealized models, like the square or cubic lattice, 
and {\it not} for real materials.
In general, it is difficult not only to image
the microstructure of complex disordered media, but 
to conveniently vary the volume fraction of the connected phase
in which one is interested, such as the pore space in rocks or 
the doping in semiconductors. However, for sea ice, by changing
the temperature of a single sample we can significantly 
vary the brine volume fraction.  
We have computed these functions
from our three dimensional imaging of the pore-space
evolution (in a submitted paper by Pringle, Miner, Eicken and Golden). 
For example, from analysis of the two-point
connectivity function, we have generated a graph illustrating the 
divergence of the 
correlation length quite similar to 
Figure \ref{percolation_graphs} (a). 

We have also used our data to find anisotropic 
percolation thresholds using {\it finite-size scaling}.
Rather than attempting to compute very sensitive
parameters such as the percolation threshold or critical
exponents from real data and its variation in volume fraction, 
finite-size scaling is a technique
from physics where one exploits the dependence of 
functions of interest on sample size to obtain such results.
In our case, a key relation is 
\begin{equation}
P_{\infty}(p_c,L) \sim L^{-\beta / \nu}, \;\;\;\;\; 
L \rightarrow \infty.
\end{equation}
In single crystals of sea ice the brine microstrucure is highly
anisotropic, with inclusions usually arranged in layers.
Using three different approaches, 
we have employed finite-size scaling to obtain  
anisotropic critical porosities of
$4.6 \pm 0.7 \%$ in the vertical direction, and
laterally, $9 \pm 2 \%$ parallel to
the layers and $14 \pm 4 \%$ perpendicular to them.
The value of about 4.6\% in the vertical direction
for single crystals
should be smaller than the threshold of 
about 5\% for bulk connectivity,
which it is.




Now we consider the application of percolation theory 
to understanding the fluid permebaility of sea ice.
In the continuum \cite{Stauffer-92,Torquato:RHM-02}, 
the permeability and conductivity
exponents $e$ and $t$ can take 
non-universal values, and need not be equal,
such as for the three dimensional Swiss cheese model.
%The critical properties of transport in the 
Continuum models have been
studied by mapping to a lattice 
with a probability density $\psi(g)$ of bond conductances $g$.
Non-universal behavior    %\cite{Torquato:RHM-02,Halperin:PRL-2391}
can be obtained when $\psi(g)$ is singular as $g \rightarrow 0^+$. 
%such as $P(g)~\sim~{g}^{-\alpha}$, $\alpha >0$
However, for a lognormal conductance
distribution arising from intersections of 
lognormally distributed inclusions, as in sea ice,
the behavior is {\it universal} \cite{Golden:2007:GRL}. 
Thus $e \approx 2$ for sea ice.




The permeability scaling factor $k_0$ for sea ice,
analogous to $\kappa_0$ in (\ref{critical_permeability}),
is estimated
using critical path analysis (see the article by the author in 
\cite{Hornung:1997}). 
%\cite{Ambegaokar:PRB-2612,Friedman:WRR-1703,Golden:HOM-99}.   
For media with $g$ in a wide range,     %of bond conductances $g$,
the overall behavior is dominated by
a critical, {\it bottleneck} conductance $g_c$,
%If we identify in a sample the bond with the largest conductance,
%then the next largest, and so on, 
%we eventually reach a 
the smallest conductance such that the critical path 
$\{ g : g \geq g_c \}$ spans the sample. 
%This theory quantifies the notion that {\it bottlenecks}
%control macroscopic flow.
%and originated in the study of hopping conduction 
%in semiconductors \cite{Stauffer-92,Golden:HOM-99}.
%In warm sea ice with $\phi$ well above $5\%$, the
%fluid network %of fluid carrying pathways
%can be thought of as a ``full'' lattice at $p=1$
%with $\kappa(p=1)=\kappa_0$.  
%The spanning fluid paths in warm sea ice have 
%a smallest, characteristic 
%radius $r_c$, and we estimate $\kappa_0$
%by ${r_c}^2/8$.
%where the fluid conductivity of 
%a pipe of radius $r$ is $r^2/8 \eta$. 
With most brine channel 
diameters \cite{Weeks:1986:GSP}
between 1.0 mm and 1.0 cm,
spanning fluid paths 
have a smallest, characteristic 
radius $r_c \approx 0.5$ mm, 
and we estimate $k_0$ by the pipe-flow 
result ${r_c}^2/8$.
%A similar result is obtained from analyzing how
%to facilitate fluid connectivity
%through the cellular substructure of columnar
%sea ice \cite{Weeks:1986:GSP}.
%with plate-like grains of ice
%separating layers of brine inclusions
%and average ice-platelet thicknesses 
%ranging \cite{Weeks:1986:GSP} 
%from about 0.5 mm to 1.0 mm.
Thus 
%$\kappa_0  \approx  {r_c}^2/8  
%            \approx  3.1  \times  10^{-8}\; \mbox{m}^2$,
%for $\phi$ in the critical region above $\phi_c$,
\begin{equation}                 \label{perc_theory}
k(\phi) \; \sim \; 3 \; (\phi - \phi_c)^2 \; 
\times  10^{-8}\;\; \mbox{m}^2,
\;\;\;\; \phi \rightarrow \phi_c^+.
\end{equation}
In Figure \ref{perc_field_data} (c), field data with $\phi$
in $[0.055, 0.15]$, just above 
$\phi_c \approx 0.05$ \cite{Golden:S-2238},
are compared with (\ref{perc_theory}), showing close
agreement. 
A best fit for $e$ of 1.94 
was obtained from other
data. 
The striking result 
that for sea ice, $e \approx 2$, 
the universal {\it lattice} value in three dimensions 
\cite{Torquato:RHM-02,Golden:PRL-2923}, is
due to the general lognormal 
structure of the brine inclusion 
distribution function.
The general nature of our results suggests that 
similar types of porous media, such as saline ice on 
extraterrestrial bodies,
may also exhibit universal critical behavior.











\begin{figure}                \centering
\includegraphics{Figure2_big.eps}
   \caption{(a) Data for $k$ taken {\it in situ} 
            on Arctic sea ice, displayed
            on a linear scale. (b) The effective permeability 
            $\kappa^*(p)$
            in a lattice percolation model, 
             displayed on a linear scale.
            (c) Comparison of Arctic permeability data in the 
            critical regime 
            (25 data points) with
            percolation theory in (\ref{perc_theory}).
            In logarithmic variables the predicted line has the equation 
            $y=2 \; x -7.5$, 
           while a best fit of the data yields  $y=2.07 \;x -7.45$,
            assuming \cite{Golden:S-2238} 
           $\phi_c = 0.05$.}         
    \label{perc_field_data}
        
\end{figure}




\vspace{2ex}



\noindent
{\bf \large{Hierarchical models}}

\vspace{1ex}


%\subsection{Hierarchical models}

To model $k(\phi)$ over all porosities, not just near the
percolation threshold, we consider features 
of the brine phase present over the full range $-$
some degree of connectivity, particularly
on small scales, and self-similarity. 
The basic unit of sea ice microstructure 
is an ice grain surrounded by brine and ice, 
having substructure itself on smaller scales. 
%with connected units forming brine structures over larger scales. 
Hierarchical, self-similar models
of spheres or other grains surrounded by smaller spheres,  
and so on, with brine of conductivity $\sigma_w$ 
in the pore spaces,
have been used to describe the 
transport properties of sedimentary rocks  
\cite{Wong:PRB-6606}. %Sheng:PRB-4507,Wong:PRB-6606}.
% and the related shrinking pipe model
%\cite{Wong:PRB-6606} maintain connectivity of the fluid phase 
%to zero porosity, and 
%and capture the basic features
%of sea ice microstructure.
Such microstructures are similar to those shown in 
Figure \ref{logarithmic_pipe_bounds} (a), except
in this case the basic elements are brine coated spheres of ice.
Effective medium theory 
gives Archie's law $\sigma = \sigma_w \phi^m$ 
as the conductivity of the model with
exact results for the exponent $m$,
such as $m=3/2$ for spherical grains.
The corresponding permeability exponent 
\cite{Wong:PRB-6606} is $2m$,
so that with $k_0$ as above and $m=3/2$,
the simplest hierarchical model yields
\begin{equation}           \label{hier_perm}
k(\phi) = 
           3 \; \phi^3 \; \times  10^{-8}\;\; \mbox{m}^2.
\end{equation}
\begin{figure}                \centering
\includegraphics{Figure4.eps}
   \caption{(a) Comparison of Arctic permeability data 
            (37 data points) with
            our theoretical prediction in (\ref{hier_perm}).
            In logarithmic variables the predicted line has the equation 
            $y=3 x -7.5$, 
           while a best fit of the data yields $y=3.05 x -7.50$.
           Lab data 
           on artificially grown sea ice \cite{Freitag:1999:HPA}
           give a best fit 
           of $y=3.1 x - 7.7$ \cite{Golden:2007:GRL}.       
           (b) Comparison of lab data with
           numerical simulation.}         
    \label{hier_numer}
        
\end{figure}
In Figure \ref{hier_numer} (a) the full {\it in situ} data set 
is compared to this theoretical prediction, with close agreement. 
%The figure also illustrates how 
%percolation theory provides a more detailed or ``higher order''
%approximation in the critical regime.
%It is interesting to note that in examining 
From photomicrographs 
of marine sand microstructures, 
compacted platey sand appears the most similar to the 
cellular substructure of sea ice.
In this case,
$2m \approx 3.04$,
which is closer to the best fit of 
3.05 from our field data.
For calcite aggregates and Fontainebleau sandstone,
a cubic power law like (\ref{hier_perm}) agrees closely 
with permeability data for porosities above about 10\%,
with an exponent $2m \approx 3.05$
for Fontainebleau sandstone.



Our results show that the 
parameters characterizing
fluid transport
in sea ice are almost the same as in some
crustal, sedimentary rocks.
%Not only do sea ice and such crustal rocks
Both types of media
exhibit critical behavior near a
percolation threshold, with $\phi_c \approx 5\%$
%about 5\% porosity 
for sea ice, 
and $\phi_c \approx 4\%$
%about 4\% porosity 
for calcite aggregates and
Fontainebleau sandstone.
Moreover, even though the porous microstructures in these
rocks are quite different from sea ice,
the permeability critical exponent in both 
cases falls into the lattice universality class.
%with a value of about 2. 
The permeability exponent of 3 for the hierarchical model 
%characterizing the permeability over a larger
%range of porosities, not just near $\phi_c$,
is also about the same in both cases.
It is interesting to note that 
a variation of four orders of magnitude in the 
permeability of calcite aggregates 
is achieved with 
temperatures ranging from 
633 K to 833 K,
and with confining pressures %ranging 
from 200 MPa to 300 MPa. For sea ice, 
a similar variation in permeability
%of four orders of magnitude
occurs over a temperature range of about 15 K.






\begin{figure}                \centering
\includegraphics{pipe_network.eps}
   \caption{ (a) Random pipe or resistor network. 
             (b) Close-up of a node and adjoining ducts. }         
    \label{pipes}
        
\end{figure}





\vspace{2ex}



\noindent
{\bf \large{Numerical model for fluid flow through sea ice}}

\vspace{1ex}




%\subsection{Numerical model for fluid flow through sea ice}


In \cite{Zhu:AG-129} we developed a 
two dimensional random pipe network to simulate
fluid flow through a porous medium, as
illustrated in Figure \ref{pipes}.
%and for simulating  
%the permeability of sea ice
%as a function of $T$ or $\phi$.
For sea ice the cross-sectional 
areas $A$ of the pipes are chosen 
from the lognormal distribution 
in (\ref{lognormal_distribution}), 
with $\langle A(\phi) \rangle$ as before
and $\rho^2 = 1$. 
The system is equivalent to a resistor network, which 
was solved using a fast multigrid method. 
While the model compared well with lab data for 
$\phi > 0.15$, 
it overestimated $k(\phi)$
for $\phi < 0.15$ where 
percolation effects, which were ignored, 
become significant.
In accordance with our X-ray CT data,
we introduced
random {\it disconnections} 
so that the network
falls apart as $\phi$ decreases. The results then
agree well with the lab data, as shown in 
Figure \ref{hier_numer} (b).




As discussed above, heat is exchanged between the 
ocean and atmosphere in the polar regions through sea ice. 
The thermal conductivity of sea ice is thus an important
transport coefficient entering into climate models.
We will not discuss our work in this area, but simply 
mention that the methods for studying transport 
we have outlined here can be applied to bounding and 
numerically calculating this important sea ice parameter.
One particularly interesting aspect of the thermal transport
problem is that when brine flows through sea ice, such as
for temperatures above the percolation threshold, there
is an advective term in the heat equation. A velocity field
helping to move heat around can enhance measured thermal
conductivity, which has been observed in Antarctic sea ice. 
In conjunction with our fluid permeability work, we have
been theoretically and numerically modeling such processes.









\vspace{3ex}

\eject

\noindent
{\bf \Large{Sea Ice Remote Sensing: \\
Monitoring the Impact of Climate Change}}

\vspace{2ex}






%\section{Sea Ice Remote Sensing: Assessing the Impact of Climate Change}

\noindent
Because of the global nature of monitoring 
earth's sea ice packs, information is usually obtained via 
remote sensing from platforms on satellites, aircraft
and ships 
\cite{Lubin:2006,Carsey:1992,Golden:TGRS-1655,Golden:TGRS-1675}. 
The physics underlying the problem of remotely sensing
sea ice concerns the interaction of electromagnetic
waves with a polycrystalline composite 
of four components: pure ice, brine inclusions, air 
pockets, and solid salt deposits \cite{Weeks:1986:GSP,Thomas:2003:SI}.
Here we'll focus on the two main constituents in sea ice, 
namely, pure ice with brine inclusions.
One of the grand challenges of sea ice remote sensing is to
accurately recover the thickness distribution of the pack. 
Assessing the impact of global warming on the polar regions 
involves monitoring not only the ice extent, but the ice volume,
which requires knowledge of ice thickness.
While microwaves in the GHz frequency range have been 
commonly used in many applications, due to the absorbing
nature of the brine phase and the short wavelengths involved,
they typically can only penetrate into the upper portions
of a sea ice floe. Lasers, such as on ICESat illustrated in 
Figure \ref{sea_ice_remote_sensing} (a), have been used to
bounce off the ice or snow surface, and when compared with 
returns from the sea surface in open leads, can provide 
thickness estimates. Changes in ice sheet elevation 
in Antarctica and Greenland can also be similarly obtained. 




\begin{figure}                \centering
\includegraphics{sea_ice_remote_sensing.eps}
   \caption{(a) NASA's Ice, Cloud and Land Elevation Satellite (ICESat)
              over Antarctica. It bounces laser pulses off the 
             surfaces of ice sheets and sea ice to measure
            changes in thickness and monitor climate change.
           (b) An electromagnetic induction device for measuring 
              sea ice thickness, called the ``Worbot'' after 
            Australian sea ice scientist Tony Worby. It is 
         mounted here on the icebreaker {\it Aurora Australis},
       in the Antarctic sea ice pack, September 2007.    }         
    \label{sea_ice_remote_sensing}
        
\end{figure}







Recently there has been increasing interest in
using low frequency electromagnetic induction devices
to estimate sea ice thickness 
\cite{Worby:1999:CRST}, as
shown in Figure \ref{sea_ice_remote_sensing} (b).
Electromagnetic fields in the MHz range, 
with wavelengths on the scale of meters 
are used to probe the air-ice-ocean interface and estimate
the thickness of the ice layer. 
The technique relies on a time varying primary 
magnetic field (generated by a transmitter coil) 
inducing eddy currents in the seawater beneath the 
comparatively resistive ice. The secondary magnetic 
field produced is sensed by a receiver coil, 
determining an apparent conductivity which results 
essentially from an integration over the vertical 
distance between the instrument and induced currents. 
Accurate interpretation of the data relies on understanding 
how the conductivity of the sea ice, or more generally
it complex permittivity, depends on the
microstructural characteristics. For example, current
methods of obtaining thickness estimates assume a 
single value for the conductivity. However, this
key parameter can vary by over two orders of magnitude
with depth throughout a given floe. Since typical sizes of the 
brine inclusions are on the submillimeter scale, the 
electromagnetic behavior of the sea ice can be treated using
the quasistatic approximation, where the wavelength
is much longer than the microstructural scale. 
Then the electromagnetic response of sea ice can be 
characterized by its effective 
complex permittivity tensor $\beps^*$.
For a given diagonal component in a particular direction, 
the real part corresponds to the polarizability, or dielectric
constant in that direction, and the imaginary part corresponds to
the conductivity in that direction.  
For example, the reflection coefficient of a wave incident on a
flat sea ice surface is determined by the mismatch in 
complex permittivity between the sea ice and the air above.


There has been considerable work in the past on estimating
and bounding $\beps^*$ for sea ice, particularly in the 
microwave region
%\cite{Addison:JAP-3114,Hoekstra:JGR-492,
\cite{Carsey:1992,Golden:TGRS-1655}.
%\cite{Winebrenner:MRSSI-137}
We'll discuss here some of our work on bounding
$\beps^*$, particularly in this low frequency range,
although the mathematics is quite general. We'll also 
discuss how we've been inverting these bounds and
other information to 
recover microstructural characteristics, which is 
important in monitoring fluid and thermal transport 
in sea ice. In this connection, there are new methods,
such as cross-borehole tomography developed by M. Ingham
and colleagues, which show promise
in reconstructing the permittivity profile of sea ice,
and in turn, the state of the ice itself.









Here we review our work on theoretical bounds 
on the principal components of $\beps^*$ 
with extensive outdoor tank 
and {\it in situ}
data on $\beps^*$ at 50 MHz, 
consisting of thousands of measurements,
taken with a capacitance probe
\cite{Gully:2007:Physica}.
In particular, we compare the low frequency measurements
with the Bergman-Milton bounds 
\cite{Golden:TGRS-1655,Gully:2007:Physica}
for the complex permittivity
of a two phase composite with known constituent 
permittivities
$\epsilon_1$ and $\epsilon_2$ and volume 
fractions $p_1$ and $p_2$. 
There are two types of these bounds which we employ.
The first type assumes only the permittivity and volume 
fraction information, and are known as the 
complex elementary or arithmetic and harmonic
mean bounds.
The second type further assumes statistical isotropy 
in the composite microstructure, and are known as 
the complex Hashin-Shtrikman bounds.


All the data considered are situated well
inside the complex elementary bounds.
However, there are naturally occurring anisotropies in the brine
microstructure, such as its preferred vertical orientation 
\cite{Weeks:1986:GSP,Thomas:2003:SI}, which become
more pronounced above the critical brine volume fraction 
of about 5\% where the brine phase percolates
\cite{Golden:S-2238}. Also, preferred azimuthal alignment
of the c-axes of individual crystals within the 
horizontal plane has long been observed in the Arctic,
and was present in the
Arctic sea ice whose permittivity was measured
in one of the data sets considered here.
Preferred c-axis alignment is attended by 
anistropy in the brine microstructure, and
anisotropic behavior of the complex permittivity
within the horizontal plane. 
Comparing the different data sets with 
the isotropic bounds yields interesting and useful 
insights about sea ice microstructure, its evolution
with temperature, and its electromagnetic properties.
We also invert the complex bounds to obtain
rigorous estimates of the brine volume fraction
from the permittivity data
\cite{Golden:TGRS-1675}.


\vspace{2ex}



\noindent
{\bf \large{Bounds on the complex permittivity of sea ice}}

\vspace{1ex}


%\subsection{Bounds on the complex permittivity of sea ice}


Let us briefly describe the analytic continuation method
for studying the effective properties of composite materials
\cite{Golden:TGRS-1655,Gully:2007:Physica}.
Let the local complex permittivity
$\epsilon(x)$ be a spatially stationary random field
in $x\in \mathbb{R}^d$, for an appropriate probability space 
representing the set of realizations of the random medium.
We assume  $\epsilon(x)$ for 50 MHz 
takes the values $\epsilon_1=63.3+i1930$ in brine
and $\epsilon_2=3.06$ in ice, and write
%\begin{equation}                                 \label{eq2.1}
$\epsilon = \epsilon_1 \chi + \epsilon_2 (1-\chi),$
%\end{equation}
where $\chi(x)$ is again the characteristic function of the brine phase. 
Let $E (x)$ and $D(x)$ be the stationary
random electric and displacement fields satisfying
the constitutive law
%\begin{equation}                                 \label{eq2.2}
$    D (x)= \epsilon (x) E (x) $
%\end{equation}
and the equations
\begin{equation}                                 \label{eq2.3}
\nabla \times E (x) = 0, \;\;\;\;\
\nabla \cdot D (x) = 0, 
\end{equation}
with
%\begin{equation}                                   \label{eq2.5}
$\langle E (x) \rangle = e_k,$
%\end{equation}
where $e_k$ is a unit vector in the $k^{th}$ direction for
some $k = 1, \ldots, d$, and $\langle \cdot \rangle$ means
an ensemble average or spatial average over all of
$\mathbb{R}^d$.

The effective complex permittivity tensor ${\beps}^*$
is defined by
\begin{equation}                                    \label{eq2.6}
    \langle D \rangle= {\beps}^* \langle E \rangle.
\end{equation}
For simplicity, we focus on one diagonal coefficient
$\e* = \e*_{kk}$. Due to the homogeneity of effective parameters,
$\e*(\lambda \epsilon_1, \lambda \epsilon_2)=\lambda
\e*(\epsilon_1,\epsilon_2)$, $\e*$ depends only on the ratio
$h = \epsilon_1 / \epsilon_2$, and we define
$m(h) =\e* / \epsilon_2$.
The two main properties of $m(h)$ are that it is analytic
off $(- \infty, 0]$ in the $h$--plane, and that it
maps the
upper half plane to the upper half plane,
so that it is an example of a Herglotz, or Stieltjes function.
The key step in the
analytic continuation method is obtaining
an integral representation for $\e*$.

It is more convenient to work with 
the function 
%\begin{equation}                                    \label{eq2.7}
$F(s) = 1 - m(h)$, where
$s = 1/(1-h)$,
%\end{equation}
which is analytic off $[0,1]$ in the $s$--plane.  
It was proven that $F(s)$ has the 
powerful representation
\begin{equation}                                     	\label{eq2.8}
    F(s)=\int_0^1 {\frac{d\mu(z)}{s-z}}\;, \qquad  s\notin[0,1]\;,
\end{equation}
where $\mu$ is a positive measure on $[0,1]$
(not to be confused with the viscosity from before).
Formula (\ref{eq2.8}) separates the
parameter information in $s$
from information about the mixture geometry 
contained in $\mu$, a spectral measure
of the operator $\Gamma\chi$, where
$\Gamma=\nabla(-\Delta)^{-1}\nabla \cdot$.
Statistical assumptions about the 
geometry are incorporated into $\mu$ via its moments
%\begin{equation}                                   \label{eq2.9}
$\mu_n = \int^1_0 z^n d \mu (z)$,
which can be calculated from the
correlation functions of the random medium, with 
$\mu_n = (-1)^n \langle \chi 
[( \Gamma \chi )^n e_k ] \cdot e_k \rangle$.
%For the complex elementary bounds it is assumed that 
%we know only $\mu_0 = p_1$, and if the medium
%is statistially isotropic we know
%$\mu_1 = p_1 p_2 / d$ as well.



Bounds on $\e*$, or $F(s)$, are obtained by fixing
$s$ in (\ref{eq2.8}), varying over admissible measures $\mu$ (or
admissible geometries), such as those that satisfy only $\mu_0=p_1$,
and finding the corresponding range of values of
$F(s)$ in the complex plane. Two types of bounds on $\e*$ are obtained.
The first bound $R_1$ assumes only that the relative
volume fractions $p_1$ and $p_2 = 1 - p_1$ of the ice and brine are
known, with $\mu_0=p_1$ satisfied.
In this case, the admissible set of measures forms a compact, convex
set $\mathcal{M}_0$. 
Since (\ref{eq2.8}) is a linear functional of $\mu$, the extreme 
values of $F$ are attained by extreme points of $\mathcal{M}_0$,
which are the Dirac point measures $p_1\delta_z$.
The values of $F$ lie inside 
%the circle $p_1/(s-z), -\infty \leq z \leq \infty$, and
the region $R_1$ bounded by circular arcs,
called the complex elementary bounds.
The formulas can be found in \cite{Golden:TGRS-1655,Gully:2007:Physica}.
This region collapses to the interval
%\begin{equation}                                        \label{eq2.16}
$(p_1/\epsilon_1 +
p_2/\epsilon_2)^{-1} \leq \e* \leq p_1
\epsilon_1 +p_2 \epsilon_2$
%\end{equation}
when $\epsilon_1$ and $\epsilon_2$ are real,
which are the arithmetic (upper) and harmonic (lower) mean bounds.  
The complex elementary bounds 
are optimal and can be attained by 
coated ellipsoidal geometries as the aspect ratio
varies.
If the material is further assumed to be statistically
isotropic, i.e., $\epsilon^*_{ik} = \e* \delta_{ik}$,
then $\mu_1 = p_1 p_2 / d$ must be satisfied as well.
A smaller bounding region $R_2$ can then be obatined,
which are complex versions of the Hashin-Shtrikman bounds.
Extensions of the analytic continuation method and bounds 
to multicomponent media involve techniques of several complex variables.











Now we compare three extensive sets of data
on the complex permittivity of sea ice with the
bounds.
Measurements
were taken in land-fast ice in the Chukchi Sea 
near Barrow, Alaska (14,004 data points)
and in McMurdo Sound, 
Antarctica (2,382 data points)
and in artificial, 
young sea ice in an outdoor tank in Fairbanks, Alaska
(6,403 data points).
%Originally developed for 
%non-destructive, {\it in situ} 
%measurements of soil moisture, 
The permittivity
was measured by deploying an array of Stevens Water Monitoring
Systems Hydraprobes. Each Hydraprobe
is a coaxial probe with a central tine
surrounded by three equally spaced outer tines,
aligned horizontally,
as shown in the photo inset 
in Figure \ref{elem_barrow_color} (a).
The three outer tines are held at
ground potential, and a voltage is applied to the central
tine at 50 MHz frequency, resulting in a circularly
polarized wave with electric field predominantly 
in the plane perpendicular to the tines.
The complex permittivity measured by the probe
represents an average of the components of
$\beps^*$ in this plane. 




\begin{figure}                \centering
\includegraphics{elem_barrow_color.eps}
   \caption{ (a) Comparison of data 
     on $\epsilon^*$ at 50 MHz 
     for sea ice grown in an ice tank
     with the elementary bounds, showing excellent 
     correspondence.
     Data in a brine porosity interval
     signified by a given color lie inside
     the region of the same color. 
     The data colored gray have porosities
     in the interval $[0.049,0.051]$, close to the percolation threshold. The 
     capacitance probe is shown on the right. 
    (b) Matrix-particle bounds on $\epsilon^*$ (solid curves),
     along with the complex elementary and Hashin-Shtrikman bounds 
     (dotted curves), compared with 4.75 GHz data in red.  }         
    \label{elem_barrow_color}
        
\end{figure}




In Figure \ref{elem_barrow_color} (a) the full data set 
for the Fairbanks ice tank experiment is displayed
along with a series of regions $R_1(\phi)$, for the
indicated values of the brine porosity $\phi$. 
The data has been grouped into intervals
of porosity, $[0.02,0.03]$, $[0.03,0.04]$, ...,
$[0.08,0.09]$, with a different color 
for each interval. The corresponding elementary
bound has the same color. All the data
in each interval lie inside the corresponding bound.
The data marked with the gray color lies 
in the porosity interval $[0.049,0.051]$, which
corresponds to the percolation threshold
of about 5\% conjectured in \cite{Golden:S-2238}.



It is useful to note that
tighter bounds \cite{Golden:TGRS-1655} on $\epsilon^*$
can be obtained if the material is known to have a 
matrix-particle structure with
separated inclusions, like sea ice at temperatures colder than its critical
temperature. In this case, the support of $\mu$ 
lies in an interval $[s_m,s_M]$, $0<s_m<s_M<1$.
The more separated the inclusions, the smaller the support
interval, and the tighter the bounds. Comparison of higher frequency
4.75 GHz data (red dots) 
on the complex permittivity $\epsilon^*$ of sea ice 
for $\epsilon_1=51+i45$ and $\epsilon_2=3.1$,
with a series of bounds \cite{Golden:TGRS-1655}
is shown in Figure \ref{elem_barrow_color} (b).
The outer dotted curves are complex arithmetic and 
harmonic mean bounds, and the inner curves are Hashin-Shtrikman bounds. 
The solid bounds are matrix-particle versions.
The parameter $q$ measures inclusion separation, 
and can be bounded using permittivity data.








\begin{figure}                \centering
\includegraphics{violation_isotropic_perc.eps}
   \caption{ Comparison of ice tank data 
     on the complex permittivity of sea ice
     at 50 MHz over ranges of $\phi$ with the elementary (outer)
     and isotropic (inner) bounds.
     The data violate, or lie outside, the isotropic bounds as
     brine volume increases through the percolation threshold 
     and an anisotropic, connected brine structure is formed.  }         
    \label{violation_isotropic_perc}
        
\end{figure}





In Figure \ref{violation_isotropic_perc}
we display a series of comparisons of the 50 MHz
data in the porosity intervals shown, with
corresponding elementary bounds 
$R_1(\phi_{max})$ and 
the isotropic bounds $R_2(\phi_{max})$, where 
$\phi_{max}$ is the largest porosity 
in each interval. The development of anisotropy
in the dielectric measurements, or violation 
of the isotropic bounds, as brine volume increases
through 5\%, is consistent with 
an increasingly connected and vertically 
elongated brine microstructure.
This behavior in the data would likely
be accentuated if the electric field were
oriented predominantly in the vertical direction. 






\vspace{2ex}



\noindent
{\bf \large{Inverse bounds for structural parameters}}

\vspace{1ex}




%\subsection{Inverse bounds for structural parameters}








\begin{figure}                \centering
\includegraphics{inverse_bounds.eps}
   \caption{ Comparison of the actual brine porosity $\phi$ 
            with time to the 
        inverse lower bounds, 
   % $\phi_1^{\ell}$ and $\phi_2^{\ell}$, for
          for the ice tank data in (a) 
        and for the Barrow data in (b).   }         
    \label{inverse_bounds}
        
\end{figure}







\noindent 
The objective of inverse bounds is to use data about 
the electromagnetic response of sea ice to recover 
information about its structural parameters, in particular 
its brine volume fraction $\phi$, or porosity.  
As mentioned above, the porosity $\phi$ can be written 
as a function of the bulk 
ice salinity $S$ and temperature $T$, as derived by semi-empirical 
approaches from the sea ice phase relations.
Given salinity information, for example, 
inverting for porosity yields temperature information.
The inverse method 
\cite{Golden:TGRS-1675} we use here
yields intervals of uncertainty for the brine
volume fraction $\phi$.  
Given an observed value of the complex 
permittivity $\e*$, $\phi$ 
is increased until the 
value of $\e*$ touches one boundary of 
the region $R_1$ described in the previous section,
and is then decreased until the value touches the 
other boundary.
This procedure gives a range of 
values $\phi^{\ell}_1 \leq \phi \leq \phi^u_1$, with
\begin{equation}                                 \label{eq13}
   \phi^{\ell}_1 = |f|^2 \frac{Im(\overline{s})}{Im(f)}\;,
 \;\;\;\;\;\;\;                                
   \phi^u_1 = 1 -  \frac{ |g|^2 Im(\overline{t}) }{Im(g)}\;,
\end{equation}
where $f$ is the known value of $F(s)$ 
and $g$ is the known value of $G(t)=1-\e*/\epsilon_1$
with $t=1-s$.
If the material is 
further assumed to be statistically isotropic, 
second order inverse bounds 
$\phi^{\ell}_2 \leq \phi \leq \phi^u_2$
can be obtained. 
We mention recent progress by E. Cherkaev and colleagues
on developing methods for 
reconstructing $\mu$ from bulk property
measurements. 
Such methods will likely be useful in obtaining more
detailed information about sea ice microstructure,
as they have been for bone microstructure, and in monitoring
osteoporosis \cite{Bonifasi-Lista:2005:IBM}.







\begin{figure}                \centering
\includegraphics{bone.eps}
   \caption{Bone porosity can be estimated from 
     torsional modulus data
   using reconstruction of the spectral measure, similar to
estimating brine porosity in sea ice from permittivity data. 
Such work will help
  in monitoring osteoporosis \cite{Bonifasi-Lista:2005:IBM}. The
  math doesn't care if it's sea ice or bone! }         
    \label{bone}
        
\end{figure}










In Figure \ref{inverse_bounds} (a), the actual 
brine porosity $\phi(t)$ as a function of time $t$
for the ice tank experiment is compared
with the lower inverse bound $\phi_1^{\ell}$.
Due to the high contrast in the materials,
the corresponding upper bound is quite large
and currently provides little information.
We see that the actual data lies above the 
lower bound, and that 
the variations in the reconstructed
lower bound follow the variations in the 
actual porosity. In Figure \ref{inverse_bounds} (b),
the actual brine porosity for the Barrow data
is compared with the lower inverse bounds
$\phi_1^{\ell}$ and $\phi_2^{\ell}$.
The inverse bounds are obeyed, and the
variations in $\phi(t)$ are reflected in the 
reconstructions.









%\section{Measurements of fluid and electrical transport in Antarctic sea ice}



\vspace{2ex}

\eject

\noindent
{\bf \Large{Measurements of fluid and electrical transport in 
           Antarctic sea ice}}

\vspace{1ex}





%\begin{figure}                \centering
%\includegraphics{SIPEX.eps}
%   \caption{ (a) Satellite image of the region off the coast
%   of East Antarctica where the SIPEX expedition took place.
%   The red arrows roughly indicate the ship track of the Australian
%    icebreaker {\it Aurora Australis}, 
%   shown in (b).   }         
%    \label{SIPEX}
        
%\end{figure}






\noindent
During September and October of 2007, 
Adam Gully (University of Utah
math student)
and I participated in a
six week voyage into the Antarctic sea ice pack.
The Sea Ice Physics and Ecosystem eXperiment (SIPEX),
conducted off the coast of East Antarcica, 
%as shown in Figure \ref{SIPEX}, 
was a major component of Australia's project contributions
to the worldwide International Polar Year activities of 
2007-2008. 
An extensive science program conducted by 45 scientists 
from 10 nations was led jointly by the Antarctic Climate \& 
Ecosystems Cooperative Research Centre (ACE CRC) 
and the Australian Antarctic Division. 
Valuable information was obtained that will help 
develop more accurate global climate models and 
will also help us understand how polar ecosystems 
might respond to future climate change.





Researchers employed a suite of cutting-edge technologies, 
including airborne laser altimetry and radar, and an 
under-ice remotely operated vehicle (ROV).
For the first time ever in Antarctica, helicopter flights 
covered thousands of kilometers tracking over the sea ice, 
gathering large-scale information about 
the height of the snow and ice surfaces above sea level.
The helicopter data were complemented by information 
gathered from ice coring and other work directly on the ice. 
The combined information will help validate satellite 
altimetry data, which will eventually be used to monitor 
changes in sea ice thickness around Antarctica.








\begin{figure}                \centering
\includegraphics{permeability.eps}
   \caption{ The author taking a sea ice core in (a), and
making a permeability measurement in (b), by measuring the 
 rate at which sea water fills in the partial hole through the ice. 
 %(c) Sea water is just about to flood the surface through the
 %  permeability hole. (d) The Wenner array for surface impedance
 %  tomography used to reconstruct the conductivity profile of sea ice.
 (c) Adam Gully (right) sets up a Wenner array, while Kazu Tateyama takes
   thickness measurements with an EM induction instrument. (d)
   The Wenner array was adapted to take direct measurements of the 
 vertical conductivity of extracted sea ice cores.  }         
    \label{permeability}
        
\end{figure}







Biologists combined classical ice coring techniques 
with the ROV to study the 
underside of the sea ice, to map the distribution of 
sea ice algae. The ROV observations showed high 
concentrations of Antarctic krill living on the underside of 
the sea ice as well as in cracks between ice floes. 
Other scientists worked on ice algae physiology and 
sea ice biogeochemistry to better understand processes 
in the Antarctic sea ice zone during the transition 
from winter to summer. 
Oceanographers studied the water mass properties 
and currents beneath the sea ice and found that, 
contrary to what was expected, the patterns of 
sea ice drift appear to be affected more by 
ocean currents than by wind. 









Gully and I conducted field studies of the 
fluid and electrical transport properties of first year sea ice, 
in conjunction with our mathematical modeling work. We made the 
first fluid permeability measurements of Antarctic pack ice, 
and conducted fluid tracer experiments on extracted blocks of 
sea ice. In close connection with these fluid measurements, 
we used surface impedance tomography via a Wenner electrode array 
to reconstruct the conductivity profile of the ice. 
We also 
modified our Wenner configuration to directly measure the 
vertical conductivity component of extracted cores - a first 
for sea ice. These measurements will allow the fluid permeability 
data to be directly related to sea ice electrical properties. 
Our electrical work, joint with voyage leader Tony Worby, 
was done in the same exact locations and configurations as
EM induction thickness measurements conducted by Japanese
sea ice scientist Kazu Tateyama for later correlation.
We are still in the process of analyzing the data and preparing it
for publication, so our results will be presented elsewhere.



\vspace{4ex}



\noindent
{\bf \Large{Conclusions and future challenges in sea ice modeling}}

\vspace{2ex}



%\section{Conclusions and future challenges in sea ice modeling}

%\vspace{1ex}









\begin{figure}                \centering
\includegraphics{arctic_ice_predictions.eps}
   \caption{ Coupled climate model simulations from the Community Climate System Model, version 3, showing September sea ice concentration averaged for three different time periods (left to right) 1990-1999, 2010-2019, and 2040-2049. The white line indicates the simulated ice extent, defined as the 15\% ice concentration contour. This and other similar climate models suggest significant sea ice loss will occur over the next 50 years, and near ice-free September conditions may be reached in the Arctic by the mid-21$^{\mbox{st}}$ 
century. Some scientists predict such a decline over an even shorter period of time.
(Courtesy of Marika Holland.) }         
    \label{arctic_ice_predictions}
        
\end{figure}





\noindent
Numerous challenges remain in the modeling of sea ice 
for climate applications. Many important processes are only 
crudely represented, such as snow-ice formation, 
albedo evolution, 
aspects of the snow pack overlying the ice cover, 
and the parameterization of ice strength and 
dynamic-driven ice ridging and rafting. 
Figure \ref{arctic_ice_predictions} shows dire
model predictions for the decline of the summer Arctic sea ice
pack, and much work remains to refine these 
predictions based on our increasing knowledge
of important processes in sea ice, and how the pack
interacts with the ocean and atmosphere.


Our work on fluid transport is central
to a broad range of problems in
the geophysics and biology of sea ice,
yet fundamental studies have been lacking.
We have developed a unified theory
of the fluid permeability of sea ice and
its dependence on microstructural evolution
with temperature. The theory ties together 
different approaches to estimating effective
composite behavior, and closely captures
laboratory and Arctic field data.
We have also made progress in CT imaging 
of sea-ice microstructure, quantifying 
the thermal evolution of pore connectivity 
and morphology at the (sub)granular scale, 
which in turn has informed the theoretical 
advances. Our work on sea ice electrical properties
helps provides a theoretical basis for new
techniques in remote sensing, in particular
those which help to monitor the impact of climate change.







Our work demonstrates
how to link information about the micro-scale structure
to macroscopic transport properties, which in turn
constrain global-scale processes of climatological
and biological importance. The results yield simple 
parameterizations of fluid transport through
sea ice in terms 
of porosity $\phi$, and therefore 
the state variables $T$ and $S$, and
can help to improve the representation of 
sea ice in climate, biogeochemical and geophysical models.
Future studies will have to examine how these findings impact the 
simulation of heat and fluid flow through sea ice in large-scale 
models, which currently oversimplify such processses.










\begin{figure}                \centering
\includegraphics{closing_shots.eps}
   \caption{ Antarctica is not only of great scientific
            interest, but is a place of unique beauty
            and exciting adventures. }         
    \label{closing_shots}
        
\end{figure}









\vspace{3ex}






\noindent {\bf Acknowledgements:} We thank
Hajo Eicken, Daniel Pringle, 
Jingyi Zhu, Amy Heaton, Adam Gully, Megan Morris, 
Ali Jabini, 
%Christian Sampson, Lars Backstrom, Jeremy Miner, 
and Tony Worby
for their collaboration on the work
reviewed here. 
We would also like to thank
Donald Perovich, Marika Holland, Steve Ackley, 
Elizabeth Hunke, and Cecilia Bitz, for 
helpful discussions and input, and for providing 
some of the figures shown here.
We are grateful for the support provided by the Division of
Atmospheric Sciences, the Division of Mathematical
Sciences, and the VIGRE and REU Programs 
at the US National Science Foundation (NSF), as
well as the Australian Antarctic Division.
%A number of the students involved in the research
%presented here were supported by the NSF Research 
%Experiences for Undergraduates (REU) Program,
%as well as an NSF VIGRE grant to the University of Utah 
%Math Department, and we thank NSF for 
%the unique opportunities it has made possible
%for these students. In addition to supporting 
%Gully's travel to Antarctica, six undergraduates
%have particpated in trips to the Arctic for their
%work on sea ice with the author, for which they
%are grateful.
%Part of the work presented
%was also supported by a grant of 
%High Perfomance Computing (HPC) resources 
%from the Arctic Region Supercomputing Center
%at the University of Alaska Fairbanks, and we thank them as well.


\vspace{4ex}


\noindent
{\bf To find out more:} If you are interested in finding
out more about math and sea ice, you can visit the author's 
web page www.math.utah.edu/$\sim$golden. There you will also
be able to link to a video on the SIPEX expedition to
Antarctica produced by the author, located under
www.math.utah.edu/$\sim$golden/media.html. 
A list of additional references for this article is located
under www.math.utah.edu/$\sim$golden/publications.html.
If you are interested
in learning about how math is being used to address the 
big questions in climate science, including more about sea ice, 
you are encouraged to visit the Math Awareness Month April 2009
website at www.mathaware.org on ``Mathematics and Climate.''




\vspace{3ex}


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