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\begin{document}


\begin{center}

{\bf Electrical Signature of Brine Percolation in Sea Ice}


\vspace{2ex}




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)



H. Eicken, University of Alaska Fairbanks, Fairbanks, AK, 99775-7320, USA



A. Gully, University of Utah, Department of Mathematics,
           155 S 1400 E, RM 233, Salt Lake City, UT 84112-0090, USA.



M. Ingham, School of Chemical and Physical Sciences, 
Victoria University of Wellington,P.O. Box 600, 
Wellington, NZ


K. A. Jones, School of Chemical and Physical Sciences, 
Victoria University of Wellington,P.O. Box 600, 
Wellington, NZ


J. Lin, University of Utah, Department of Mathematics,
           155 S 1400 E, RM 233, Salt Lake City, UT 84112-0090, USA.


J. E. Reid, Groundprobe Geophysics, 1/288 Victoria Rd., 
Malaga 6090, AU

C. S. Sampson, University of Utah, Department of Mathematics,
           155 S 1400 E, RM 233, Salt Lake City, UT 84112-0090, USA.



A. P. Worby, Australian Antarctic Division and 
ACE CRC, University of Tasmania, Private Bag 80, Hobart, 7001, AU



**********************************


Kenneth M. Golden, \hspace{-1.3ex}%
Department of Mathematics, University of Utah, USA,
Hajo Eicken, \hspace{-1.3ex}%
University of Alaska Fairbanks, USA\hspace{-0.4ex},
Adam Gully, \hspace{-1.3ex}%
University of Utah, Department of Mathematics, USA\hspace{-0.4ex},
Malcolm Ingham, \hspace{-1.2ex}%
School of Chemical and Physical Sciences,
Victoria University of Wellington, NZ\hspace{-0.4ex},
Keleigh A. Jones, \hspace{-1.2ex}%
School of Chemical and Physical Sciences, 
Victoria University of Wellington, NZ\hspace{-0.4ex},
Joyce Lin, \hspace{-1.2ex}%
University of Utah, Department of Mathematics, USA\hspace{-0.4ex},
James E. Reid, \hspace{-1.05ex}%
Groundprobe Geophysics, Malaga, AU\hspace{-0.4ex},
Christian S. Sampson, \hspace{-1.3ex}%
University of Utah, Department of Mathematics, USA\hspace{-0.4ex},
and 
Anthony P. Worby \hspace{-1.3ex}%
Australian Antarctic Division and ACE CRC,
University of Tasmania, AU


\end{center}




%Malcolm Ingham\footnote{School of Chemical and Physical Sciences, 
%Victoria University of Wellington, P.O. Box 600, 
%Wellington, NZ}, Keleigh A. Jones\footnotemark[3], \\
%Joyce Lin\footnotemark[1], 
%James E. Reid\footnote{Groundprobe Geophysics, 1/288 Victoria Rd., 
%Malaga 6090, AU}, 
%Christian S. Sampson\footnotemark[1], 
%Anthony P. Worby\footnote{Australian Antarctic Division and 
%ACE CRC, University of Tasmania, Private Bag 80, Hobart, 7001, AU}












%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{abstract}
Fluid flow through sea ice governs a broad range
of geophysical and biological processes in the polar marine
environment.
For example, the evolution of melt ponds and sea ice 
albedo, which is important in climate modeling, 
is constrained by drainage through the
porous brine microstructure.
However, for brine volume
fractions below about 5\%, columnar sea ice is effectively 
impermeable to fluid flow.
In two different
experiments conducted in the Arctic and Antarctic,
we have found that this critical fluid transition
is signaled electrically by
sea ice resistivity rising sharply over three orders
of magnitude near the brine connectivity threshold.
The data are accurately explained 
by percolation theory, with the same 
universal critical exponent which 
captures fluid behavior. 
These results enables us to connect specific electrical signatures to 
important transport processes such as melt pond drainage, 
CO$_2$ pumping, salt budget evolution,
and the flux of nutrients which sustain biomass build-up.
 

\end{abstract}










%\keywords{sea ice | percolation | fluid transport | electrical transport}





%\section*{Introduction}




%\baselineskip24pt

%\setlength{\baselineskip}{13pt}



%title page, abstract, introduction, results, discussion, 
%materials and methods, acknowledgments, references,




%\setcounter{page}{1} 
%\pagestyle{plain} 



\section{Introduction}





Polar sea ice is a key component
of Earth's climate system, and
a leading indicator of climate change
\cite{Thomas:2008:SI,Serreze:S-1533,Boe:2009:NG}.
%\cite{Untersteiner:1986-GSI,Jeffries:1998:ASI,Untersteiner:1990:SDA,%
%Kattenberg:1986:GSP,Serreze:2000:OER,Sturm:SA-2003,Perovich:1999:YIG}.
As a material sea ice is a composite of pure ice
with brine and air inclusions. The brine microstructure
hosts extensive algal and bacterial communities 
which sustain life in the polar oceans
\cite{Thomas:2008:SI,Fritsen:S-782,%
Lizotte:1998:ASI,Eicken:PB-3,Krembs:PNAS:2011}. 
While global climate models predict declines 
in sea ice area and thickness, 
they have significantly underestimated the recent loss  
in Arctic summer ice extent \cite{Boe:2009:NG}. 
We focus here on sea ice transport processes
which are critical to the role that sea ice 
plays in climate, and can lead to more realistic 
representations of sea ice in climate models.
In particular, we investigate the electrical behavior
of sea ice associated with fluid transport phenomena
and microstructural transitions.



Improving projections of the fate of Earth's sea ice
cover and its ecosystems depends on 
a better understanding of important processes and feedback mechanisms.
During the melt season the Arctic sea ice cover 
becomes a complex, evolving mosaic of ice,
melt ponds, and open water. 
While snow and ice reflect most incident sunlight, 
melt ponds and ocean absorb most of it. The reflectance
or albedo of sea ice floes is determined by melt pond evolution 
\cite{Perovich:2002:JGR,Scott:2010:JGR,Polashenski:JGR:2012}.
Drainage of the ponds, with a resulting increase in albedo,
is largely controlled by the fluid permeability
of the porous sea ice underlying the ponds 
\cite{Eicken:JGR-C08007,Golden:2007:GRL}.
As ice recedes with melting, more water surface
is exposed, which increases solar absorption,
%As melting increases, so does solar absorption, 
leading in turn to more melting, and so on. This 
{\it ice--albedo feedback} has played a significant 
role in the decline of the summer Arctic 
ice pack \cite{Perovich:GRL-2007,Perovich:GRL-2008}.
Moreover, it the driving mechanism in 
mathematical models of whether we have passed 
a {\it tipping point} or {\it critical threshold}
%or {\it saddle-node bifurcation}
in the decline of summer Arctic sea ice
\cite{Eisenman:PNAS-2009,Notz:PNAS-2009}.
Sea ice albedo also represents a significant source 
of uncertainty in climate projections
and a fundamental problem in climate modeling
\cite{Flocco:2010:JGR,Scott:2010:JGR,%
Pedersen:2009:JGR,Polashenski:JGR:2012}.



Fluid flow through sea ice also governs the 
evolution of the salt budget and salinity 
profiles \cite{Thomas:2008:SI},
%\cite{Weeks:1986:GSP,Eicken:GMP-02}, 
convection-enhanced thermal transport
\cite{Lytle:JGR-8853},
ocean-ice-atmosphere CO$_2$ exchanges \cite{Rysgaard:JGR:2009},
and the build-up of algal biomass fueled by fluxes of nutrients
\cite{Thomas:2008:SI,Fritsen:S-782}.
%Eicken:PB-3}.
It also drives
snow-ice formation, which accounts for 
a significant portion of the ice produced in the Southern Ocean
\cite{Maksym:JGR:2008}. 
Sea water percolates upward
through the porous brine microstructure, flooding the 
snow layer, which subsequently freezes.
%This process may become
%important in the Arctic with increased precipitation
%and thinning ice, so that floes are more susceptible to flooding.






















While fluid flow is substantially restricted for 
brine volume fractions $\phi$ below about 5\%, columnar sea ice
is increasingly permeable
for $\phi$ above 5\% \cite{Golden:S-2238}.
For a typical bulk salinity
of 5 ppt, the critical porosity
$\phi_c \approx 5\%$ corresponds to a temperature 
$T_c \approx -5^{\circ}$ C, which is known as the {\it rule of fives}.
This critical behavior of the fluid permeability
%which constrains the above processes,
results from a connectivity or percolation threshold
in the brine microstructure
\cite{Golden:S-2238,Golden:2007:GRL,Pringle:JGR:2009}.
If the fluid transport properties of sea ice
can be linked to its electrical properties, 
which is the aim of this paper,
then new approaches can be brought to bear
in monitoring the state of sea ice.
For example, it could open the door to the development 
of sensors to enhance existing buoy networks, provide 
information on key ice processes, and improve 
integration with satellite data.




\begin{figure}      \centering
 \includegraphics[width=85mm]{Figure_1.eps}
\caption{{\bf(a)} A Wenner electrode array 
is configured to measure the vertical conductivity 
of Antarctic sea ice, by inserting the four probes into an
extracted ice core. 
{\bf(b)} A current $I$ is injected into the core 
through the outer electrodes C1 and C2. 
The potential difference $\Delta V$ resulting from the current flow 
is measured by the inner electrodes P1 and P2. 
The ratio $\Delta V/I$ is the resistance $R$ in ohms. 
Here the electrode spacing is $L=10$ cm and $a=10$ cm.
{\bf(c)} A cross-borehole array is frozen into
Arctic sea ice. The DC resistivity profile was tomographically 
reconstructed in the volume enclosed by the electrode 
strings. One of the strings, with 
10 cm separation of the plates, is shown in {\bf(d)}.}


\label{Figure_1}

\end{figure}



The electrical conductivity of sea ice has been studied 
over the past five decades 
\cite{Fujino:1963:LTS,Addison:JAP-3114,Thyssen:1974:P,%
Buckley:CRST:1986,Reid:2006:AG,Ingham:2008:CRST}.
However, there have been no observations of critical behavior in 
electrical properties corresponding to the 
microstructural transition %articulated by 
encapsulated in the rule of fives.
Here we report on two types of experiments where 
electrical resistivity data clearly display
critical behavior at the brine percolation threshold.
The mathematical description we develop provides 
a rigorous link between fluid and electrical
transport in sea ice, with both displaying
the same type of universal critical behavior,
thus laying the foundation for
the techniques referred to above.
In fact, we further develop this foundation by
partitioning the range of resistivity values
of our data
into intervals which correspond to distinct regimes
of fluid permeability characteristics and 
related process behavior, such as melt pond development, 
and fluxes of nutrients and CO$_2$.



One of the goals of this work is to obtain data on the 
linkages between electrical and hydraulic properties, yielding 
information about key microstructural variables in the 
process \cite{Wong:PT:1988}. 
The value of such an approach lies in the potential to 
then extract information about other key variables describing 
the state of sea ice, e.g., pertaining to its rheology or 
potential to harbor microbial communities, from measurements 
of electric properties which are more easily obtained, e.g., 
from {\it in situ} drifting sensors that can monitor the evolution 
of sea ice non-destructively (Figure 1 d). 



The findings presented here also have implications for
measuring ice thickness, an important
gauge of the impact of global warming.
Not only is thickness data
important in comparing climate model predictions to 
observed behavior,
but in specifying the initial conditions necessary 
for long-term numerical simulations.
Promising techniques for advanced airborne or surface-based 
measurements of ice thickness depend on the interaction of 
electromagnetic (EM) fields with sea ice.
%Almost all methods for obtaining such data
%depend on the interaction of electromagnetic (EM) fields 
%679 words
%with sea ice.
For example, there has been significant interest in the 
development of 
EM induction devices
\cite{Haas:2004:GRL,Reid:2006:AG}
mounted on ships, planes and helicopters.
These techniques, and the interpretation of the data
to obtain thickness information,
rely on knowledge of the electrical properties of sea ice,
and how they vary with depth, temperature, salinity, 
and ice type. The results presented here shed significant
light on such issues.




\section{Results}





Sea ice is an anisotropic composite with vertically
elongated brine inclusions and
corresponding anisotropy in the effective fluid 
permeability and electrical conductivity tensors.
%For processes of interest to climate and ecosystem studies
%such as the drainage of melt ponds, fluid flow in the vertical 
%direction is most relevant; thus our interest here in the 
%vertical electrical conductivity.
Most methods for measuring sea ice conductivity 
involve indirect or inverse techniques, such as 
surface-based geoelectric profiling using a Wenner array 
of electrodes \cite{Fujino:1963:LTS,Thyssen:1974:P,%
Buckley:CRST:1986,Reid:2006:AG,Ingham:2008:CRST,Sampson:DSR:2011}. 
Generally with these methods
the vertical conductivity
$\sigma^*_v$ 
is inherently mixed with the horizontal components.
Here we are most interested in $\sigma^*_v$ due to 
its connection with vertical fluid flow.
%and the vertical component of the fluid permeability tensor.







During the
Sea Ice Physics and Ecosystem 
Experiment (SIPEX) in September and October of 2007,
we made {\it direct}
measurements of $\sigma^*_v$ 
in Antarctic pack ice by adapting
a four probe Wenner array 
for use in cylindrical ice cores,
as shown in Figure 1 a and b. 
The study area was located off the coast of 
East Antarctica, between
115$^{\circ}$ E and 130$^{\circ}$ E, and 
64$^{\circ}$ S
and 66$^{\circ}$ S. 
At 8 of the 15 ice stations along the cruise track
of the Australian icebreaker {\it Aurora Australis},
we extracted vertical cores from thin
first-year sea ice, with lengths ranging 
from 34 cm to 86 cm. We obtained 26 averaged data
points from 67 raw measurements of the resistance 
between the inner probes. We also measured
temperature and salinity profiles for the cores,
so that the data on $\sigma^*_v$ could be viewed as a function 
of brine volume fraction $\phi$
(see Methods).















In the Arctic, we used the technique of 
cross-borehole DC resistivity 
tomography
\cite{Ingham:2008:CRST,Jones:JGR:2010},
as shown in Figure 1 c and d. 
The ice is probed in its natural state,
utilizing two or four vertical strings of 
electrodes frozen into the ice. It has been
shown that this method can be used to 
derive the horizontal component of the 
anisotropic resistivity profile. 
Moreover, it has recently been demonstrated that the
vertical component of $\bsig^*$ can be obtained
as well \cite{Jones:JGR:2010,Ingham:2008:CRST}.
If a minimum of four electrode strings are used, 
the geometric mean of the vertical and horizontal 
components of $\bsig^*$ can be derived, along with
the horizontal component \cite{Ingham:2008:CRST},
yielding the vertical component.


Measurements of the temporal variation 
in the resistivity structure of first-year Arctic
sea ice through spring warming have been made 
approximately 1 km off the coast of 
Barrow, Alaska at 71$^{\circ}$ 21$^\prime$ 
56.45$^{\prime \prime}$ N, 
156$^{\circ}$ 32$^\prime$ 39.01$^{\prime \prime}$ W. 
Electrode strings were installed in landfast 
first year ice in late January 2008.  
Cross-borehole measurements were made on 
6 separate occasions between early April 
and mid June 2008, allowing both the horizontal 
and vertical components of the ice resistivity 
to be derived. 
A sea ice mass balance site and
an ice core sampling program
at the same 
location \cite{Druckenmiller:CRST:2009}
provided ice temperature and 
salinity data, allowing the variation in 
resistivity structure to be correlated 
with brine volume fraction $\phi$.






Lattice and continuum percolation theories 
\cite{Stauffer-92}
have been used to model a broad range of disordered
materials where the connectedness of one phase
dominates effective transport behavior. 
%Examples include rocks, 
%semiconductors, thin films, glacial ice, 
%polycrystalline metals, 
%radar absorbing coatings, and 
%carbon nanotube 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 consider the problem of electrical transport 
%through the network. 
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.
In this model
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 $d=3$,
$p_c \approx 0.25$. 
Typical configurations for the $d=2$ square lattice
above and below the threshold
are shown in Figure 2 a and b.




\begin{figure}          \centering
 \includegraphics[width=8.5cm]
{Figure_2.eps}
\caption{The two dimensional square bond lattice below
        its percolation or connectivity threshold $p_c=1/2$ 
        in {\bf(a)}, 
        and above in {\bf(b)}.
       Below $p_c$,
       there is no bulk transport, and above $p_c$
       the effective conductivity takes off with 
       power law behavior, as shown in {\bf(c)}. 
       In {\bf(d)} the effective resistivity diverges as $p$
       approaches $p_c$ from the right, with a vertical
       asymptote at $p=p_c$. 
       We display the vertical resistivity
       formation factor data 
       from the Antarctic in {\bf(e)}  
       and the Arctic in {\bf(f)}, along with
       the same prediction from percolation theory in each. 
       Both data and theory
       exhibit divergent behavior as $\phi$ approaches
       $\phi_c \approx 0.05$ from the right, 
       with a vertical asymptote at $\phi = \phi_c$,
       electrically signaling the transition to relatively impermeable ice. }

\label{Figure_2}
\end{figure}






Let $\sigma^*(p)$ be the effective conductivity 
of the network 
in the vertical direction \cite{Stauffer-92}.
For $p < p_c$, $\sigma^*(p)=0$, as shown in 
Figure 2 c. For $p>p_c$ and near $p_c$,  
$\sigma^*(p)$ 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 $d$.
In $d=2$, $t \approx 1.3$,
and in $d=3$,
$t \approx 2.0$ \cite{Stauffer-92}.  
There is also a rigorous 
bound \cite{Golden:PRL-2923} that $1 \leq t \leq 2$
in $d=2$ and $d=3$.
Since $\sigma^*(p) \rightarrow 0$ as $p \rightarrow p_c^+$,
the effective resistivity $\rho^*(p)=1/\sigma^*(p)$ diverges
as $p \rightarrow p_c^+$, with
%$\rho^*(p) \sim \rho_0(p-p_c)^{-t}$, 
%$\rho_0 = 1/\sigma_0$, and
a vertical asympote at $p=p_c$, as shown in 
Figure 2 d.
It should be remarked that for samples of 
two phase composites with finite component resistivities, 
like sea ice,
%with component resistivities
%$\rho_0$ and $\rho_1 \gg 1$ with $\rho_0 \ll \rho_1$,
the behavior only approximates the asymptote, and for
$p < p_c$, $\rho^*$ remains finite.
%with  $\rho^* < \rho_1$
%and $\rho^* \rightarrow \rho_1$ as $p \rightarrow 0^+$.




The fluid permeability $\kappa^*(p)$ 
corresponding to (\ref{critical_conductivity}), 
where the open bonds are pipes of fluid conductivity 
$\kappa_0/\eta=r_0^2/8\eta$ and radius $r_0$, behaves like 
%\begin{equation}                    \label{critical_permeability}
$\kappa^*(p) \sim \kappa_0(p-p_c)^e$ as $p \rightarrow p_c^+$,
%\end{equation}
with $e$ the fluid permeability exponent and $\eta$ the fluid viscosity.
For lattices, it is believed \cite{Stauffer-92} 
that $e=t$.
In the continuum, 
the 
exponents $e$ and $t$ can take 
non-universal values, and need not be equal,
such as for the three dimensional Swiss cheese model
\cite{Halperin:PRL-2391,Stauffer-92}.
However, for 
lognormally distributed inclusions, as in sea ice,
the behavior is {\it universal} 
\cite{Golden:2007:GRL,Berkowitz:TPM-275}. 
Thus for sea ice, $t=e \approx 2$.


In order to use percolation theory to quantitatively describe
the vertical conductivity $\sigma^*_v(\phi)$,
and to provide
a link between fluid and electrical transport in sea ice,
we recall our result \cite{Golden:2007:GRL} 
for the vertical fluid permeability 
\begin{equation}                 \label{perc_theory}
k_v^*(\phi) \; \sim \; 3 \; (\phi - \phi_c)^2 \; 
\times  10^{-8}\;\; \mbox{m}^2,
\;\;\;\; \phi \rightarrow \phi_c^+.
\end{equation}
The scaling factor $k_0=3 \times 10^{-8}$ 
is estimated using critical path analysis  
\cite{Stauffer-92,Friedman:WRR-1703}. 
The effective behavior of media with a broad
range of local conductances is dominated by a 
critical {\it bottleneck} conductance related to
the minimal radius in a connected pathway 
of appropriate scale.
To relate $\sigma^*_v$ to $k^*_v$, we use 
the following relation
from critical path analysis \cite{Friedman:WRR-1703}.
With $r_{c}$ denoting the critical radius for our 
centimeter scale electrical
experiments, then 
\begin{equation}            \label{cond_perm_relation}
k_v^* = \frac{r_{c}^2}{8} \; \frac{\sigma^*_v}{\sigma_b}\;,
\end{equation}
where $\sigma_b$ is the conductivity of brine,
which depends \cite{Stogryn:IEEE-523} on temperature $T$. 
%via $\sigma_b=-T \exp(0.5193+0.08755T)\;
%\Omega^{-1} \; \mbox{m}^{-1}$.
By measuring the radii of vertical pathways 
in X-ray tomography 
images \cite{Golden:2007:GRL,Pringle:JGR:2009}, we
estimate a range in mm of
%$r_{c}$ (mm) to be in the range
$0.1   \leq  r_{c}  \leq  0.2$.


It is useful to consider
the vertical conductivity formation factor 
%\begin{equation}
$F=\sigma^*_v/\sigma_b$,
%\end{equation}
which removes the dependence of the effective parameter
on the changing conductivity of the brine, 
and depends only on the pore
volume fraction and geometry.
In view of (\ref{critical_conductivity})
and (\ref{cond_perm_relation}),
$F(\phi) \; \sim \; F_0 \; (\phi - \phi_c)^2$ as
$\phi \rightarrow \phi_c^+$, where
$F_0 = 8k_0/r_c^2$.
The estimates of 0.1 mm to 0.2 mm 
for $r_{c}$ yield a range for $F_0$ of $6 \leq F_0 \leq 24$.




In order to compare our conductivity 
measurements with percolation theory, we must
exclude data below $\phi_c \approx 0.05$ \cite{Golden:2007:GRL}, 
since the theory
is only valid for $\phi > \phi_c$.
It is more illustrative to display the data in terms
of the reciprocal $G = 1/F=\rho_v^*/\rho_b$, which is the 
vertical resistivity formation factor.
In Figure 2 e and f we show the two data
sets from the Antarctic and Arctic. By fixing
the exponent $t=2$ and the threshold value $\phi_c = 0.05$
in the above expression for $F(\phi)$,
a statistical best fit of the data yields a value
of $F_0\approx 9$, which lies inside our predicted range, so that
\begin{equation}           \label{perc_formation2}
F(\phi) \; \sim \; 9 \; (\phi - 0.05)^2, \;\;\;
      \phi \rightarrow \phi_c^+.
\end{equation}

We see that the data agree well with the theory, and that they both
exhibit divergent behavior with a vertical asymptote at the
percolation threshold. Moreover, 
%in Figure \ref{Figure_3} (F) the Antarctic data is displayed
in the logarithmic variables 
$x=\log{(\phi-0.05)}$ and $y=\log{F}$, the
line predicted by percolation theory in (\ref{perc_formation2}) is
$y=2x+\log{F_0}$, with $\log{F_0}=0.95, F_0=9$. 
Critical path analysis yields the bounds 
$0.8 \leq \log{F_0} \leq 1.4$, and the statistical 
best fit for the Antarctic data in f is $y=1.99x+0.93$,
where 0.93 lies inside these bounds.  
In logarithmic variables, the standard error of the regression
is 0.38 for the Arctic data and 0.22 for the Antarctic
data (that is, approximately 68\% of the Antarctic data is within 0.22
of the regression line). The increased scatter in the Arctic data
is not surprising given the substantial inverse computation required to 
obtain the formation factor data.

%It is important to remark that sea ice is 
%a real composite where the brine microstructure
%can be highly variable,
%particularly over small scales relevant to electrical measurements.
%We would thus not necessarily expect data for $\phi < 5\%$ to be 
%closely captured by extrapolation of percolation formulas
%to such situations. Nevertheless, we would expect a dominant feature
%of data on sea ice resistivity {\it vs.} $\phi$
%to be a rapid ascent as $\phi \rightarrow \phi_c^+$. 




To model $\sigma_v^*(\phi)$ over all porosities, 
we consider features 
of the brine phase present over the full range $-$
some degree of small-scale connectivity, 
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 models
of spheres or other grains surrounded by smaller spheres,  
and so on, with brine in the pore spaces \cite{Golden:2007:GRL},
were used to model $k_v^*(\phi)$.
The simplest model yields a 
result of $k_v^*(\phi)=k_0 \, \phi^3$. 
Via (\ref{cond_perm_relation}) we obtain an Archie's
law result of $F(\phi)=F_0 \, \phi^3$. A statistical
best fit of our Antarctic data yields a value
of $F_0 \approx 16$, which is in the 
estimated range. In Figure 3 a,
our Antarctic data is
shown along with fits derived from both models, 
and in b, Arctic permeability data \cite{Golden:2007:GRL}
is shown relative to predictions from both models.






\begin{figure}        \centering
\includegraphics[width=8cm]{Figure_3.eps} 
\caption{{\bf(a)} Antarctic field data on the vertical formation 
factor $F=\sigma_v^*/\sigma_b$ is compared with
the hierarchical model  
$F(\phi)=F_0\phi^3$.
The prediction of percolation theory is also shown.
{\bf(b)} Comparison of Arctic fluid permeability data with
    the hierarchical model, along
     with percolation theory \cite{Golden:2007:GRL}. 
      In both figures
     percolation theory captures the trend of the data
     in the percolation regime more closely than Archie's law. }
\label{Figure_3}
\end{figure}





\begin{figure}        \centering
\includegraphics[width=8cm]{Figure_4.eps} 
\caption{Cross-borehole tomographic reconstructions
of the vertical resistivity formation factor
for Arctic sea ice before {\bf(a)} and after {\bf(b)}
melt pond formation. The evolution of 
resistivity structure seen here is consistent with 
warming of the ice, thus increasing the fluid permeability
and facilitating the infiltration of meltwater into the 
upper layer of sea ice from the surface.
To help connect the electrical properties of sea ice
to its important processes, 
the range of the resistivity formation factor $G$
can be divided into five regimes: 
$G > 625$ (ice impermeable enough to allow ponds 
to grow for surface ablation rates 10 cm/d or 
larger for a critical pore radius of 0.1 mm);
$125 < G \leq 625$, blue (ice impermeable enough to allow 
ponds to grow for surface ablation rates 
between 10 and 50 cm/d for a critical pore radius of 0.1 mm);
$31.3 < G \leq 125$, green (at formation factors of 31.3 or 
larger ice is impermeable from the perspective 
of CO$_2$ exchange and build-up of nutrients and biomass 
in the ice \cite{Rysgaard:JGR:2007}, and sufficiently
impermeable to fluid drainage to support surface ponding);
$1 < G \leq 31.3$, orange (highly permeable ice that allows for 
CO$_2$ pumping and build-up of nutrients and biomass);
$G \leq 1$, red (assumed to be free water column).
Only the most resistive ice $G > 625$ is not shown 
in the figure. }
\label{Figure_4}
\end{figure}













Figure 4 illustrates how we can derive information about 
the permeability structure and relevant transport processes 
from resistivity soundings of Arctic sea ice with {\it in situ} 
electrode strings \cite{Jones:JGR:2010}. 
Thus, the different formation 
factor regimes shown correspond to 
different permeability classes, with the lowermost ice 
layers permeable enough to allow for gas and nutrient 
exchange conducive to biomass build-up and CO$_2$ pumping
\cite{Rysgaard:JGR:2007}, 
based on a critical permeability 
of $4 \times 10^{-11}$ m$^2$, 
corresponding to a resistivity formation factor of 31.3 
for $r_c = 0.1$ mm. This permeable base layer increases in 
vertical extent as the ice warms and thins due to bottom 
and surface melt. The ice interior is permeable enough to 
allow for meltwater flushing and reduction of ice salinity 
at surface ablation rates of 10 cm/d or less even prior to 
the onset of melt \cite{Freitag:JG-349}, corresponding 
to a resistivity formation factor of 625. High resistivity 
formation factors near the top in Figure 4 b are in part 
explained by such percolation of freshwater below accumulations 
of surface melt water.












It has been demonstrated in field experiments conducted
in both the Arctic and Antarctic that sea ice exhibits
critical behavior in its electrical transport properties
at a percolation threshold. Such behavior provides the
electrical signature of a key transition
in fluid transport properties, known as the 
{\it rule of fives}, which determines whether or
not fluid can flow through sea ice.
This transition constrains
a broad range of processes which are important in
the geophysics and biology of the polar regions.
The phenomenon is explained theoretically using
percolation theory, which provides a universal
power law describing the data from both poles,
as well as a rigorous link between the 
fluid and electrical transport properties of sea ice.
Our findings open the door to a new generation of
techniques for {\it in situ} analysis and remote monitoring
of transport processes which are critical to 
improving projections of the future trajectory
of the polar ice packs.

%\vspace{5ex}

%\noindent Acknowledgements are in \cite{Golden:acknowledge:2010}.




\begin{figure}     \centering
 \includegraphics[width=8.5cm]{Figure_5.eps}
\caption{Comparison of field lines for a parallel plate configuration 
in (a) with those for a four probe Wenner array in (b). }


 \label{Figure_5}

\end{figure}




\vspace{3ex}



\section{Methods}

\vspace{1ex}

Traditional surface-based geophysical 
techniques for {\it in situ} measurements, such as the
Wenner array used for surface impedance tomography, 
have been used to study the resistivity profile of sea ice,
particularly its anisotropic structure
\cite{Fujino:1963:LTS,Thyssen:1974:P,%
Buckley:CRST:1986,Reid:2006:AG,Ingham:2008:CRST,Sampson:DSR:2011}.
During SIPEX we adapted the 
Wenner array to directly measure
the vertical component of the anisotropic conductivity tensor
$\bsig^*$ of sea ice.
At 8 of the 15 ice stations along the cruise track
of the Australian icebreaker {\it Aurora Australis},
we extracted vertical cores from thin
first-year sea ice, with lengths ranging 
from 34 cm to 86 cm.
Thermistor probes were inserted into small holes drilled every 5 cm. 
We used a Wenner electrode array along sections 
of the cores, connected to a YEW 
Earth Resistance Tester operating at 38 Hz, as indicated in
%Holes which were slightly smaller than the 
%thin cylindrical probes 
%were drilled into the ice to ensure a tight fit
%of the probes, and a good electrical connection.  
Figure 1 a and b.
This set-up yields the resistance
along the axis of the cylindrical
ice core between probes P1 and P2, corresponding to
the vertical direction {\it in situ}, with 
$a=L=10$ cm 
(or $a=L=5$ cm in some cases).
After the temperature and resistance measurements were taken,
which took about 10 to 20 minutes,
we cut each core into 10 cm sections which were later
melted, so that we could obtain bulk salinity 
measurements for each section. The 
temperature and salinity measurements allowed us to calculate
a brine volume fraction profile for each core
\cite{Eicken:GMP-02}.










Plate electrodes in contact with the ends of a cylinder
generate parallel field lines which make measuring
the conductivity 
of the cylinder material relatively straightforward,
as illustrated in Figure 5 a.
To assess the accuracy of our four probe method, the commercial package
Comsol 3.5a was used to create a finite element 
model of cylindrical sea ice cores 0.09~m in diameter 
and 0.5~m in length. Four metal probes of 0.004~m 
in diameter and 0.09~m in length were inserted 
approximately 0.07~m into the core, similar to 
Figure 1 b. When the 
current is injected through the outer probes instead of parallel plates, 
as in Figure 5 b,
the nearby field lines show significant curvature. 
However, in the boxed measurement region in Figure 5 b 
where the inner probes are located, 
the field lines are relatively straight, 
thus minimizing the error between the actual conductivity
of the material and what is measured by the array.
Numerical simulations show that 
if the outer probes are 5 cm or more from 
the inner measurement region, this error is less than
8.5\%, and is less than 1.5\% if the distance is 10 cm or more,
as for much of our data.


When extracting a sea ice core to measure its properties, 
loss of brine is a principal concern. However, for our
experiments we did not see any evidence of significant 
brine loss during the relatively short measurement
periods with air temperatures ranging from about $-6^{\circ}$ C
to $-18^{\circ}$ C (with most below $-9^{\circ}$ C). 
Moreover, the probes are inserted deep into the core, minimizing
contact with potential brine surface films. Our numerical simulations
and these observations establish the Wenner array 
as a viable field method for 
{\it direct} resistivity measurements.















\vspace{5ex}


\noindent
{\bf Acknowledgments:}
We are grateful for the support provided by the Division of Mathematical
Sciences (DMS), the Arctic Natural Sciences (ARC) Program, and the Office of 
Polar Programs (OPP) at the US National Science Foundation (NSF)
through grants 
DMS-0537015, DMS-0940249,
and ARC-0934721.
Joyce Lin was supported by an NSF Postdoctoral
Fellowship through a VIGRE grant (DMS-0602219)
to the Department of Mathematics at the University of Utah.
Adam Gully and Christian Sampson were partially supported by
the NSF Research 
Experiences for Undergraduates (REU) Program through the 
VIGRE grant, and graduate support through VIGRE.
This work was also supported in part by the 
Australian Government through the Antarctic Climate and 
Ecosystems Cooperative Research Centre.
Finally, we thank the crew of the
{\it Aurora Australis} for their help and support 
during the SIPEX Antarctic expedition.











\bibliographystyle{plain}
\bibliography{golden,cmg,opp08}









\end{document}







\begin{abstract}
Fluid flow through sea ice mediates a broad range
of geophysical and biological processes in the polar marine
environment.
For example, the evolution of melt ponds and sea ice 
albedo, which is important in climate modeling, 
is constrained by drainage through the
porous brine microstructure.
Fluid flow also facilitates snow-ice formation,
%\cite{Maksym:JGR:2008}, 
the evolution of the salt budget,
%\cite{Thomas:2008:SI}, 
%convection-enhanced thermal transport,
%\cite{Lytle:JGR-8853},
%ocean-ice-atmosphere 
%CO$_2$ exchange, 
%\cite{Rysgaard:JGR:2007},
and biomass build-up sustained by nutrient fluxes.
%\cite{Thomas:2008:SI,Fritsen:S-782}.
However, for brine volume
fractions below about 5\%, columnar sea ice is effectively 
impermeable to fluid flow,
which controls these processes.
%\cite{Golden:S-2238,Golden:2007:GRL,Pringle:JGR:2009}.
In two different
experiments conducted in the Arctic and Antarctic,
we have found that this critical transition in 
fluid flow 
%at the brine connectivity threshold
exhibits a strong electrical signature, with 
sea ice resistivity sharply rising over three orders
of magnitude near the brine connectivity threshold.
The data are accurately explained 
by percolation theory, with the same 
universal critical exponent of 2 which 
captures the behavior of the 
fluid permeability. 
%\cite{Golden:2007:GRL}.
%The data also indicate marked changes in the 
%conductivity profile with the onset of melt ponds. %surface ponding.
Our results demonstrate that classical lattice models
of phase transitions in statistical physics can help unravel the
complexity of transport in this
multiscale random medium.
The theory enables electrical classification of sea ice layers in 
terms of their fluid flow properties, 
thus connecting specific electrical signatures to 
important transport processes such as melt pond drainage, 
CO$_2$ pumping, and nutrient fluxes.
Our findings lay the foundation 
for electromagnetic monitoring of 
transport phenomena in sea ice,
which can help track key transitions in the 
state of polar sea ice and improve projections 
of its fate and impact on ecosystems. 
\end{abstract}













Finally, in Figure 4 we show
cross-borehole tomographic reconstructions
of the vertical resistivity formation factor
for Arctic sea ice. In a, the 
profile was obtained before the onset 
of melt pond formation. The ice is cold and
electrically resistive. In b, the profile
was obtained well after a melt pond had formed.
The ice is warmer and significantly more conductive.
By 16-17 June the ice had not 
only thinned but had ablated from the surface. 
The top 2 electrodes in each string were in air, 
and the third was in melt water, so that
the top 0.3 m of the profile are blank. 
Further, the very high resistivity in the next 
0.1-0.2 m likely results from 
fresh melt water percolating 
downwards, replacing the brine. This leads to
values of the resistivity in this top layer 
roughly two to three times higher than 
before the formation of the melt pond. In the bottom 
0.1 m or so, we also see highly conductive ice,
likely due to high porosity and sea water infiltration.










%although accurate measurements of permeability in sea ice 
%are difficult and time consuming. % \cite{Freitag:JG-349}.
If the fluid transport properties of sea ice
can be linked to its electrical properties, 
which is the aim of this paper,
then new approaches can be brought to bear
in monitoring the state of sea ice.
%which are critical to studies of climate change 
%and ecosystem response. 
For example, it could open the door to the development 
of sensors to enhance existing buoy networks, provide 
information on key ice processes, and improve 
integration with satellite data. 





Although the two data sets were obtained 
in different types of sea ice
and by quite different methods, 
they agree closely and are both accurately captured 
with the same critical exponent
from lattice percolation theory
used to predict fluid permeability 
\cite{Golden:2007:GRL}.
Moreover, the Arctic data was taken
in late spring
and shows the transformation in electrical properties 
associated with onset of surface melt pond development.






The findings presented here also have implications for
measuring ice thickness, 
a key variable in tracking the response of the ice cover 
to changes in external forcing, such as global warming.
Not only is thickness data
important in comparing climate model predictions to 
observed behavior,
but in specifying the initial conditions necessary 
for long-term numerical simulations. 
Key methods for obtaining such data
depend on the interaction of electromagnetic (EM) fields 
%679 words
with sea ice.
For example, there has been significant interest in the 
development of 
EM induction devices
\cite{Haas:2004:GRL,Reid:2006:AG}
mounted on ships, planes and helicopters.
These techniques, and the interpretation of the data
to obtain ice thickness,
rely on knowledge of the electrical properties of sea ice,
and how they vary with depth, temperature, salinity, 
and ice type. The results presented here shed significant
light on such issues.






Traditional surface-based geophysical 
techniques for {\it in situ} measurements, such as the
Wenner array used for surface impedance tomography, 
have been used to study the resistivity profile of sea ice,
particularly its anisotropic structure
\cite{Fujino:1963:LTS,Thyssen:1974:P,%
Buckley:CRST:1986,Ingham:2008:CRST,Sampson:DSR:2011}.






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Lattice and continuum percolation theories 
\cite{Stauffer-92}
have been used to model disordered
materials where the connectedness of one phase
dominates effective behavior. 
Examples include rocks, 
semiconductors, thin films, glacial ice, 
polycrystalline metals, 
radar absorbing coatings, and 
carbon nanotube 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 will initially consider the problem of electrical transport 
through the network, and then the relation to 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 $d=3$,
$p_c \approx 0.25$. 
Typical configurations for the $d=2$ square lattice
above and below the threshold
are shown in Figure \ref{combined_data}. 












%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%








To date, columnar microstructures have received 
disproportionate attention, mostly due to their prevalence 
in Arctic sea ice and their importance in undisturbed ice growth 
\cite{Weeks:1986:GSP,Light:2003:ETM}.
Granular ice, however, is predominant in the Antarctic 
\cite{Jeffries:1994:AJ,Worby:1995:RR,Eicken:1998:AARS} 
and exhibits substantially different pore microstructure, 
with a near-complete lack of intragranular inclusions 
and a film of brine enveloping individual grains. 








Our findings about this
on$-$off switch for fluid transport in sea ice shed new light on our 
understanding of some key processes in global climate, biogeochemical 
and remote sensing models. 


Fluid flow through porous sea ice mediates a broad range of geophysical 
and biological processes, such as surface flooding and snow-ice formation, 
the evolution of ice albedo and salinity profiles, and nutrient 
replenishment for algal and bacterial communities.
Fluid flow through porous sea ice mediates a broad range of geophysical 
and biological processes, such as surface flooding and snow-ice formation, 
the evolution of ice albedo and salinity profiles, and nutrient 
replenishment for algal and bacterial communities. We have made the first measurements 
of fluid permeability in Antarctic pack ice, and find that the predominance 
of granular ice there leads to strikingly different fluid transport behavior 
than in the Arctic. In particular, granular ice exhibits a percolation threshold,
effectively the on$-$off switch for fluid transport, for brine volume fractions 
around 10\%, rather than 5\% observed in the columnar ice which dominates in the Arctic.
Processes controlled by this switch are critical to investigations of climate change,
and the response to polar ecosystems.






Polar sea ice is a critical component of 
earth's climate system, 
both as an indicator and agent of climatic change 
\cite{Serreze:S-1533,Thomas:2003:SI}.  %Jeffries:1998:ASI
It is also a primary habitat for microbial
communities which sustain the rich foodwebs
of the polar oceans 
\cite{Thomas:2003:SI,Fritsen:S-782}.
%As a material, sea ice is a porous composite of
%ice with brine, air and salt inclusions.
The geophysical and biological importance 
of sea ice is largely derived from its porous composite 
structure
%a porous composite %nature %of ice with brine inclusions 
\cite{Thomas:2003:SI,Weeks:1986:GSP}.
%\cite{Thomas:2003:SI,Fritsen:S-782}.      %Lizotte:1998:ASI,
%As a material, sea ice is a porous composite of ice with brine, air, 
%and salt inclusions \cite{Weeks:1986:GSP,Eicken:GMP-02}.
%whose microstructure depends strongly on temperature. 
Pivotal to understanding and predicting the 
behavior of sea ice in the environment is its 
permeability $k$, which controls bulk flow and 
depends on the fluid volume fraction $\phi$ and 
microstructural characteristics. %Ice permeability and brine flow
Brine transport facilitates %a variety of processes, 
sea ice production from freezing of flooded ice
surfaces, %\cite{Maksym:2001:PCE},
%Ackley:AJUS-133, 
convection-enhanced thermal transport,
%\cite{Lytle:JGR-8853,Trodahl:JGR-11358}, 
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}. 
For example, summer sea-ice albedo, a key parameter
in the global climate system, 
depends on surface melt-pond hydrology whose evolution
is controlled largely by ice permeability
\cite{Eicken:JGR-C08007}.
The transport of nutrients
and enzymes through the pore matrix of sea ice is
essential for sustaining life in this extreme environment
\cite{Thomas:2003:SI,Fritsen:S-782}. 
%fundamental, life sustaining processes. 
Furthermore, through its control of the presence and 
nature of liquid water at the ice surface, 
$k(\phi)$ is a key variable in sea-ice remote 
sensing, a principal tool for 
%observing high-latitude climate change.
monitoring global warming.

Yet, despite its importance, the  
problem of understanding how $k$ depends on 
the brine microstructure 
%and how the pore structure evolves with temperature,
%and to measurements of $k(\phi)$,
has received relatively little attention
\cite{Thomas:2003:SI,Eicken:JGR-C08007,Golden:S-2238,%
Petrich:CRST-131,Wettlaufer:JFM-291,%
Oertling:JGR-2004,Ono:AG-298,Freitag:1999:HPA}.
Here we address this fundamental issue 
theoretically,
using rigorous bounds, percolation and hierarchical
models, and numerical simulation, and experimentally with
laboratory and Arctic field data on $k(\phi)$, obtaining
close agreement. We also develop
micro-scale imaging methods based on 
X-ray computed tomography (CT) and 
pore structure analysis \cite{Prodanovic:JCIS-282}
to provide detailed pictures of the brine
microstructure and its evolution with temperature,
along with quantitative measurements of 3-D
pore connectivity. Our microstructural
imaging results support the 
theory and help to interpret the permeability data.
Moreover, they 
illuminate important deviations from the classic 
sea-ice microstructural model 
\cite{Thomas:2003:SI,Weeks:1986:GSP},
and provide the first direct, quantitative evidence 
for a transition in connectedness which
was conjectured here \cite{Golden:S-2238}
nearly a decade ago.








%Fluid transport phenomena in sea ice can 
%now be incorporated into global climate, 
%algal growth, remote sensing, and other types of models.







%A key observation \cite{Weeks:1986:GSP} 
%about fluid transport in sea ice
%which must be addressed in a theoretical treatment 
%is that 
It has long been observed \cite{Weeks:1986:GSP} that 
columnar sea ice is effectively impermeable to brine
transport for $\phi$ less than about $5\%$, yet is 
permeable for $\phi$ above $5\%$.
%While columnar sea ice is effectively impermeable 
%to brine drainage  %and desalination
%for $\phi$ less than about $5\%$, 
%desalination from brine transport \cite{Weeks:1986:GSP} 
%becomes significant
%as measured by higher rates of change of salinity,
%for $\phi$ above $5\%$.
For a bulk salinity $S$ of 5 parts per thousand (ppt),
the critical porosity $\phi_c \approx 5\%$   
corresponds to a critical temperature $T_c \approx -5^{\circ}$C,
via equations 
relating $\phi$ to $T$ and $S$
\cite{Thomas:2003:SI,Weeks:1986:GSP} .   
This {\it rule of fives} was discussed \cite{Golden:S-2238} 
in terms where $\phi_c$ was identified with
the critical probability in a {\it continuum} percolation model 
for a compressed powder        % \cite{Kusy:N-58}
exhibiting microstructural 
characteristics qualitatively similar 
to sea ice.
In percolation theory,
the permeability just above a threshold 
can be characterized by a critical 
exponent $e$ and a scaling factor.
In the continuum, $e$ can vary depending on local properties,
yet for lattice systems it is believed to be {\it universal}, 
depending only
on dimension \cite{Torquato:RHM-02,Halperin:PRL-2391}.
Here we find 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},
due to the general lognormal 
structure of the brine inclusion 
distribution function \cite{Perovich:JGR-18327}.
The general nature of our results suggests that 
similar types of porous media, such as saline ice on 
extraterrestrial bodies \cite{Thomas:S-641},
may also exhibit universal critical behavior.
%Nevertheless, our work here
%provides a framework for addressing such problems.

In fact, we find that the 
parameters characterizing
fluid transport
in sea ice are almost the same as in some
crustal, sedimentary rocks
\cite{Knackstedt:PRE-R5181}.
%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. 
Another permeability exponent discussed below 
%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 
\cite{Knackstedt:PRE-R5181}
is achieved with 
temperatures ranging from 
633 K to 833 K,
and with confining pressures %ranging 
from 200 MPa to 300 MPa. For the {\it in situ} sea ice 
data considered here, a similar variation
in permeability
%of four orders of magnitude
occurs over a temperature range of about 15 K.



%\section*{Fluid flow through porous sea ice}


\vspace{1ex}


%\noindent
%{\bf  Fluid flow through porous sea ice.}
We consider low Reynolds number flow of
%of a fluid with viscosity $\eta$
%through a porous medium.
%in a region $\Omega$, a subset of three dimensional space,
%$\Omega \subset {\mathbb{R}}^3$. 
brine of viscosity $\eta$ through sea ice.
The volume fractions of brine and ice are $\phi$ and $1-\phi$. 
%$\Omega_b \subset \Omega$, 
%has volume fraction $\phi$, and the ice $1-\phi$.
%$\Omega_i \subset \Omega$, 
%with a relative volume fraction $1-\phi$. 
The velocity and pressure fields in the brine
%$\vv_b(\xx)$ and pressure $p_b(\xx)$
%in the brine for $\xx \in \Omega_b$ 
satisfy the Stokes equations for incompressible fluids.
%satisfy the Stokes and incompressibility equations.
%$\nabla p_{b} - \eta {\nabla}^2 \vv_{b} = \ff$ and 
%$\nabla \cdot \vv_{b} = 0$, 
%where $\ff(\xx)$ is a force acting on the medium such as gravity.
%We are interested in sea ice as a porous medium
%for a given temperature and salinity. At a given depth within
%the ice, its microstructural characteristics can be relatively 
%uniform over many meters. 
Under appropriate assumptions %on the microstructure 
\cite{Torquato:RHM-02},
%such as translation invariant statistics,
the homogenized velocity $\vv(\xx)$ and pressure $p(\xx)$
%obtained through averaging or limiting procedures,  
%can be obtained in scaled limiting or averaging procedures, 
%such as taking the microstructural scale
%to zero or the volume to infinity \cite{Hornung:1997,Sahimi-95}. 
%They 
satisfy Darcy's law 
%Hornung:1997,
%\begin{equation}
$ {\vv} = - \kk \; \nabla p /\eta$
%\end{equation}
and 
$
\nabla \cdot \vv = 0, 
$
where $\kk$ is the permeability tensor, 
with vertical permeability k$_{zz}=k$ in units of m$^2$.
Transport depends on pore-size statistics, and in sea ice
%An important way of characterizing the porous medium, sea ice, is as follows.
the inclusion cross-sectional areas $A$ 
%in mm$^2$
%of the inclusions 
(above a cut-off) are lognormally distributed
\cite{Perovich:JGR-18327}. Then $Z=\ln{A}$ has a normal probability density
with mean $\mu$ and variance $\sigma^2$,
\begin{equation}                  \label{lognormal_distribution}
P(Z) = %(2 \pi \rho^2)^{-1/2}
     \frac{1}{\sqrt{2 \pi \sigma^2}} \;
         e^{-(Z-\mu)^2/2 \sigma^2}.
\end{equation}


The bulk of sea-ice brine resides in 
intra-crystalline layers whose connectivity 
is expected to underpin the development of 
larger-scale permeability. 
As a first step to obtain quantitative 
information about pore connectivity across 
all relevant scales, we tracked the thermal 
evolution of the connectivity 
of the intragranular 
inclusions from X-ray CT of 
laboratory-grown sea-ice single crystals
\cite{Pringle:EOS-2006} 
(doped with CsCl to improve ice/brine 
contrast), as shown in 
Figure \ref{pipebound_sigma} (a)
[J. Miner, {\it et al.}, in preparation].





\begin{figure}                \centering
\includegraphics{Figure1.eps}
   \caption{(a) X-ray CT images ($\approx 10$ mm across)
              of the brine microstructure of sea ice
            for two values of the porosity $\phi$.
            As temperature and $\phi$ increase, 
           so do inclusion sizes and connectivity.
            (b) Comparison of {\it in situ} data on 
            $k$ (m$^2$) for 
            Arctic sea ice (37 diamonds) 
            and lab data \cite{Freitag:1999:HPA}
           for artificially grown sea ice 
            (20 circles) with
            the upper bound in (\ref{pipe_bound_sigma}).}         
    \label{pipebound_sigma}
        
\end{figure}






%\vspace{1ex}



%\noindent
%{\bf Bounds.}
%Knowing the range of bulk properties of composite materials
%under microstructural constraints provides benchmarks 
%which are useful in assessing
%data, and in understanding how effective 
%behavior depends on mixture geometry.  
Rigorous bounds on bulk properties 
of composites provide benchmarks
which are useful in assessing data, and in connecting effective behavior
to mixture geometry \cite{Torquato:RHM-02}. 
In previous work \cite{Golden:MM-801} we found a {\it pipe bound}, 
$k \leq  \phi a^2(\phi) /8$ for 
porous media with uniformly sized inclusions, 
where the cross-sectional 
radius $a(\phi) = 7 \times 10^{-5} + (1.6 \times 10^{-4})\phi$ m
increases according to measurements of pore sizes with temperature
\cite{Perovich:JGR-18327,Bock:2005}.
%Under the assumption of uniformly sized inclusions, which increase in size with
%$\phi$ or $T$ according to mean sizes measured 
%in sea ice \cite{Perovich:JGR-18327,Bock:2005}, 
%$k \leq  \phi [a(\phi)]^2/8$ with
%$a(\phi) = 7 \times 10^{-5} + (1.6 \times 10^{-4})\phi$ m.
%This bound assumes a  medium with 
%uniformly sized inclusions,
%can be attained by vertical pipes,
%assuming uniformly sized inclusions,
The pipe bound is a special case of optimal void bounds
\cite{Torquato:PRL-2004,Torquato:RHM-02},
%and is a fluid analog of the arithmetic mean upper bound
%on the effective electrical or thermal conductivity
%of a two phase composite \cite{Torquato:RHM-02}.
%with known constituent conductivites and volume fractions 
and captures data on $k$
for centrifuged cores of
artificially grown sea ice \cite{Freitag:1999:HPA,Golden:MM-801}.
However, when compared with {\it in situ} data 
on Arctic sea ice with more microstructural 
variations and larger pores, often found in the 
form of vertical arrays or channels, %though full 
%connectivity is typically limited to lowermost ice layers), 
some data exceed the bound.







Here we evaluate the 
%moments $\langle R^2 \rangle$ and $\langle R^4 \rangle$ in the
more general void bounds \cite{Torquato:PRL-2004} 
$k \leq \frac{\phi}{8} \langle R^4 \rangle /   \langle R^2  \rangle $
for the lognormal
distribution of brine inclusion cross sectional areas 
$A=\pi R^2$ in (\ref{lognormal_distribution}).
%A short calculation of 
%The moments $\langle R^2 \rangle$ and $\langle R^4 \rangle$ 
%of the inclusion radii $R$ can be calculated 
%by exploiting the fact that 
Since $\ln{A^2}=2\ln{A}$ is normally
distributed with mean $2\mu$ and variance $4 \sigma^2$, a short calculation
%of the moments for the inclusion radii $R$
yields             %an extra factor of $\exp(\rho^2)$ in the pipe bound,  
\begin{equation}                  \label{pipe_bound_sigma}
k(\phi) \leq   \frac{\phi}{8 \pi} \; \langle A(\phi) \rangle \; e^{\sigma^2}.
%a(\phi) = 7 \times 10^{-5} + (1.6 \times 10^{-4})\phi \;\; \mbox{m}.
\end{equation}
With variance \cite{Perovich:JGR-18327} $\sigma^2 \approx 1$
and $\langle A(\phi) \rangle=\pi a^2(\phi)$ as above,
the {\it lognormal pipe bound} in (\ref{pipe_bound_sigma})
captures the field data, as well as the lab data,
as shown in Figure \ref{pipebound_sigma} (b). 
The field data were taken near Barrow, 
Alaska in a hydrological bail test
\cite{Freitag:JG-349,Eicken:JGR-C08007}. 
A cylindrical hole is drilled, 
a tube is inserted to seal off horizontal flow, and time
series measurements of the water level %(resolution $\pm$5 mm)
are used to estimate the permeability of the ice
under the hole.  % using Darcy's Law.
The total error for the derived permeability 
is estimated at $< 50 \%$ for both the 
lab and field measurements,
with significantly less relative error.
The $\phi$ values for the lab data  
are {\it effective porosities}
\cite{Freitag:1999:HPA,Petrich:CRST-131}.
For the range considered
(except one point),
the effective porosity differs only negligibly from
the total porosity \cite{Petrich:CRST-131}.
The result in (\ref{pipe_bound_sigma}) 
provides bounds on the trapping constant and
mean survival time for diffusion 
which can react with the pore boundaries 
\cite{Torquato:RHM-02,Golden:MM-801,Torquato:PRL-2004}.
Such processes arise in bacterial foraging and
NMR imaging \cite{Torquato:RHM-02,Golden:MM-801,Bock:2005}.






\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,Petrich:CRST-131} 
           $\phi_c = 0.05$.}         
    \label{perc_field_data}
        
\end{figure}





%\vspace{1ex}




%\noindent
%{\bf Percolation.} 
Percolation theory 
\cite{Torquato:RHM-02,Halperin:PRL-2391,Golden:PRL-2923} %,Golden:HPM-27
%Broadbent:PCPS-629,
has been used to model disordered
materials where the connectedness of one phase
dominates effective behavior.
Consider the square ($d=2$) or cubic ($d=3$) network of bonds 
joining nearest neighbor sites on the integer lattice ${\mathbb{Z}}^d$.
%In the random pipe network,
The bonds are assigned fluid conductivities
$\kappa_0 > 0$ (open) or 0 (closed) with probabilities $p$ 
and $1-p$. 
%Groups of connected open bonds 
%are called open clusters. 
%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, connected set of open bonds first appears.
In $d=2$, $p_c = \frac{1}{2}$, and in $d=3$,
$p_c \approx 0.25$. 
Let $\kappa(p)$ be the permeability 
of this random network
in the vertical direction.  
For $p < p_c$, $\kappa(p)=0$. For $p>p_c$, near the threshold  
$\kappa(p)$ exhibits power law behavior,
\begin{equation}                    \label{critical_permeability}
\kappa(p) \sim \kappa_0(p-p_c)^e\;,\;\;\;\; p \rightarrow p_c^+,
\end{equation}
where $e$ is the permeability 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 
$e$ is believed to be {\it universal}, 
depending only on $d$, and is equal to $t$,
the lattice electrical conductivity exponent
\cite{Torquato:RHM-02,Halperin:PRL-2391,Golden:PRL-2923,Berkowitz:TPM-275}.
%In $d=2$ \cite{Stauffer-92,Sahimi-95}, $t \approx 1.3$.
In $d=3$, it is believed \cite{Torquato:RHM-02}
that $t \approx 2.0$, 
and there is 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 the continuum, $e$ %and $t$ can take 
can take non-universal values, %and need not be equal,
such as for the three dimensional Swiss cheese model
\cite{Torquato:RHM-02,Halperin:PRL-2391,Berkowitz:TPM-275}.
%The critical properties of transport in the 
Continuum models have been
studied 
\cite{Torquato:RHM-02,Halperin:PRL-2391,Berkowitz:TPM-275} 
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 universal, which has been verified numerically 
\cite{Berkowitz:TPM-275}. Thus $e \approx 2$ for sea ice.














We estimate the scaling factor $k_0$ for sea ice
using critical path analysis
\cite{Friedman:WRR-1703,Golden:HOM-99}. 
%\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,Petrich:CRST-131},
are compared with (\ref{perc_theory}), showing close
agreement. 
A best fit for $e$ of 1.94 
was obtained from other
data \cite{Ono:AG-298}. 




X-ray CT has provided new insight into 
small-scale, intra-crystalline brine layer connectivity. 
Figure \ref{brine_micro_evol} (a) 
illustrates the thermal evolution of inclusion 
morphology and connectivity, 
the latter being quantified in (b) and (c)
\cite{Prodanovic:JCIS-282}.
Figure \ref{brine_micro_evol} (b)
provides the first direct observation of 
an onset in connectivity over this scale 
($\approx$ 1 cm). Consistent with percolation 
theory, this appears to occur at a lower 
porosity than the predicted threshold for 
bulk connectivity ($\phi_c \approx 0.05$). Notably, 
the thermal evolution of pore morphology 
in (a) 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}.
These images provide insight into and 
constraints for more detailed modeling 
of micro-scale inclusion connectivity. 
A similar analysis of large-scale pore 
networks remains challenging, but is 
inherently reflected in the {\it in situ} 
permeability data.




\begin{figure}                \centering
\includegraphics{Figure3_big.eps}
   \caption{ (a) 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 the step-wise warming cycle.
    (b) and (c) Vertical connectivity of inclusions 
   %a lab-grown sea-ice single crystal (S = 9.3 ppt, analyzed volume 
   over a cylinder of height 8 mm and diameter 21 mm.
   The fractional connectivity is the proportion of inclusions
intersecting the upper surface which are also connected to the
lower surface. The values
$\phi_c \approx 0.05$ and $T_c \approx -10^{\circ}$ C are 
the predicted \cite{Golden:S-2238,Petrich:CRST-131} bulk connectivity
thresholds for sea ice with $S=9.3$ ppt. }
%www.ams.sunysb.edu/$~$lindquis/3dma/3dma$_$rock/3dma$_$rock.html].
      
    \label{brine_micro_evol}
        
\end{figure}






%\vspace{1ex}

%\noindent
%{\bf Hierarchical models.}
To model $k(\phi)$ over all porosities, 
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{Sen:G-781,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.
Effective medium theory 
\cite{Sen:G-781,Wong:PRB-6606}
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}
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 \cite{Jackson:G-1250}, 
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\%
\cite{Knackstedt:PRE-R5181}, with an exponent $2m \approx 3.05$
for Fontainebleau sandstone.






\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 $y=3.05 x -7.50$.
           The lab data 
           on artificially grown sea ice \cite{Freitag:1999:HPA}  
           give a best fit 
           of $y=3.1 x - 7.7$.       
           (b) Comparison of lab data with
           numerical simulation.}         
    \label{hier_numer}
        
\end{figure}








%\vspace{1ex}

%\noindent
%{\bf Numerical simulation.}
In previous work \cite{Zhu:2006} a 
two dimensional pipe network was developed to simulate
fluid flow through a porous medium.
%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 $\sigma^2 = 1$. 
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 Figure \ref{brine_micro_evol} (b),
here we introduce
random {\it disconnections} 
so that the fluid network
falls apart as $\phi$ decreases, 
and mostly so 
by $\phi_c \approx 0.05$. The results then
agree well with the lab data, as shown in 
Figure \ref{hier_numer} (b).


The hierarchical model, characterized by the line in 
Figure \ref{hier_numer} (a),
provides perhaps the coarsest
estimate of sea ice permeability which still captures 
the overall trend of the data for the full range 
of relevant porosities.
%With some degree of
%connectivity to zero porosity, this model slightly overestimates
%the permeability for very low porosities, 
The percolation model, as shown in Figure \ref{hier_numer} (a), 
can be viewed as providing
a higher order approximation for the permeability
in the critical regime, 
which more accurately reflects our observations of 
brine connectivity. The numerical model is sufficiently
robust to incorporate the results of further
investigations of the brine microstructure and 
its evolution with temperature \cite{Pringle:EOS-2006}.







%\vspace{1ex}

%\noindent
%{\bf Conclusions.}
Brine transport is central
to a broad range of problems in
the geophysics and biology of sea ice,
yet fundamental studies have been lacking.
Here 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. We have found that the fluid transport
properties of sea ice bear a striking resemblance
to those of some crustal rocks. Our methods demonstrate
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. Finally,
our results yield simple 
parameterizations of fluid transport through
sea ice in terms of 
of porosity $\phi$, and therefore 
the state variables $T$ and $S$.
These findings can help to improve the representation of 
sea ice in climate, biogeochemical and geophysical models.


\vfill

\eject


%\vspace{1ex}

\noindent {\bf Acknowledgements:} We thank
Stephen Ackley, Johannes Freitag,
Victoria Lytle, Donald Perovich, Ma\v{s}a
Prodanovi\'{c},
Michael Shelley, and Po-zen Wong for 
helpful conversations. 
We are grateful for the support provided by the Division of
Atmospheric Sciences and the Division of Mathematical
Sciences at the US National Science Foundation (NSF),
through the Collaboration in Mathematical Geosciences
(CMG) Program.
Amy Heaton gratefully acknowledges
the support of the NSF Research Experiences 
for Undergraduates (REU) Program.
CT data were analyzed with 3DMA software obtained at 
[Lindquist, W. B., {\it et al.}, 2006,
%$ 
%\mbox{www.ams.sunysb.edu/}^~
%\mbox{lindquis/3dma/3dma-rock/3dma-rock.html} 
%$].
www.ams.sunysb.edu/  \\ $^{\sim}$%
lindquis/3dma/3dma$_{-}$rock/3dma$_{-}$rock.html].


%\eject


%\setcounter{page}{1} 
%\pagestyle{plain} 


%\vspace{2ex}


\bibliographystyle{naturemag}

\bibliography{golden,cmg,opp08}


\eject


\pagestyle{empty}



\hspace{30.2em}\today
~~

\vspace{1ex}

\noindent
To the Editor:

\vspace{2ex}

\noindent
Attached please find a copy of our paper,
``Thermal evolution of permeability 
and microstructure in sea ice,''
by K. M. Golden, H. Eicken, A. L. Heaton, J. Miner, 
D. Pringle, and J. Zhu.
We would like to submit this paper for publication 
as a Report in {\it Science}. Below are the two 
summaries requested in the submission guidelines.


\vspace{1ex}

\noindent
{\bf Summary 1.}
Sea ice is an indicator of climate change, and
hosts extensive microbial communities. %which support the polar ecosystems. 
Bulk fluid flow through sea ice, controlled by its permeability, 
facilitates numerous processes in the polar environment.
%is necessary for a broad range of geophysical and biological processes,
%facilitates the formation, maintenance, and melting 
%of the sea ice packs, and helps to sustain life in this extreme environment. 
%and how it depends on the microstructure.
%However, little is known about the permeability and its
%temperature dependence.
We have developed a rigorous theory for the fluid permeability
of sea ice which agrees closely with field and lab data,
and is based on unprecedented characterizations of the
brine microstructure and its temperature evolution.
Fluid transport in sea ice can now be incorporated more readily
into climate, oceanographic, and biological
models. Applying   
mathematics of composites and condensed matter physics 
to a central, interdisciplinary problem impacting polar
geophysics and biology, will likely be of interest
to a wide range of scientists, engineers, and 
mathematicians.




\vspace{1ex}

\noindent
{\bf Summary 2.}
Earth's sea ice packs can be viewed as ``canaries in the coal mine''
for global warming. They also serve as habitats for
microbial communities supporting the delicate polar ecosystems. 
Fluid transport through sea ice
facilitates a broad range of processes involved in the
formation, maintenance, and melting of the packs,
and in assessing climate change.  
By carrying nutrients and enzymes through the ice, 
brine flow is essential to sustaining life in this 
extreme environment. We present fundamental studies 
of how the permeability of sea ice, which controls
fluid transport, depends on the brine microstructure
and its evolution with temperature. 
The results can be broadly applied, from incorporation into
global climate, oceanographic, and biophysical models, to 
astrobiological inquiries.


%providing a framework for assessing sea ice on Europa as a potential habitat for life.



\vspace{2ex}

\noindent
Our submission includes this cover letter, the PDF file
containing the paper, and one other PDF file with a supporting
manuscript.
We estimate about 1600 words of text in the paper,
including about 150 words on methods, and there
are 4 figures comprising 13 separate panels
(individual images or graphs) included.
The supporting manuscript is entitled,
``A network model for fluid transport through sea ice,''
by J. Zhu, A. Jabini, K. M. Golden, H. Eicken and M. Morris,
{\it Annals of Glaciology}, 2007, in press (in
galley proof form, which contains only a few minor, stylistic errors).
It provides the basis for one of the numerous 
results presented here. However, the result
in this paper specifically goes beyond what was reported 
in the supporting manuscript, and addresses a key problem
that was raised there.

\vspace{2ex}

\noindent
We would like to thank you for your consideration
of our submission to {\it Nature}.





\vspace{2ex}


\noindent
Sincerely yours,

\vspace{2ex}

\noindent
Kenneth M. Golden \\
Professor of Mathematics

\vspace{2ex}

\noindent
University of Utah\\
Department of Mathematics\\
155 S 1400 E RM 233\\
Salt Lake City, UT 84112-0090 USA

\vspace{1ex}

\noindent
golden@math.utah.edu   \\  %\hspace{4ex}
801-581-6176 (office)  \hspace{4ex} 801-581-4148 (fax)

 

%\eject

%\vspace{10ex}

%\begin{figure}

%  \hspace{8ex} 
% \epsfxsize=5.8in
% \epsfxsize=4.5in
% \epsfbox{fig2.eps} 

%\caption{Comparison of the microstructures of (a) compressed
%         powder of large polymer particles of radius $R_p$
%         and small metal particles of radius $R_m$ 
%         \protect\cite{Malliaris:JAP-614}, and (b) sea ice
%         \protect\cite{Arcone:JGR-14281}. }
 
%\end{figure}



%\begin{figure}

%  \hspace{16ex} 
% \epsfxsize=4.5in

% \epsfxsize=3.5in
%  \hspace{16ex}
% \epsfbox{fig3.eps} 


%  \epsfxsize=5in
%  \hspace{5ex}
%  \epsfbox{fig4.eps} 
 



%  \epsfxsize=5in
%   \hspace{5ex} \epsfbox{fig5.eps} 

%  \vspace{2ex}
 
 
 


%\input epsf
%\begin{figure}

%  \epsfxsize=5in
%  \hspace{5ex}
%  \epsfbox{fig4.eps} 
 

%\caption{Brine volume contours in sea ice during the
%         HIHO HIHO experiment in the East Antarctic sea ice
%         zone.  An abrupt increase in air temperature 
%         late on day 231 resulted in the entire 
%         sea ice layer having brine volumes above
%         the percolation threshold of $p_c \approx 5\%$,
%         rendering it permeable to brine transport.
%         Brine then infiltrated the surface, and the
%         resulting slush layer later froze, forming
%         snow ice.}
  
%\end{figure}


%\hspace{120ex}
%\cite{Acknowledgements}





\end{document}










\begin{figure}
\begin{center} \includegraphics[height=50mm]{resistmeas2}
\caption{Calculated resistivities of simulated sea ice 
cores as a function of the distance between the inner and 
outer probes. Each simulation measurement is marked 
with a circle, a best fit line indicating the overall 
trend. As the distance between the current probes gets large, 
the measurements approach the true value indicated by the 
horizontal line. 
\label{resistance}}
\end{center}
\end{figure}

















goldenms.pdf
goldensupp.pdf


One sentence summary:

We present fundamental studies of fluid transport
in sea ice which impact our understanding of key
processes in polar climatology and biology, and 
discover that sea ice is remarkably similar to 
crustal rocks.




Names, affiliations, and e-mail addresses 
of five potential referees. 


Donald Perovich, Willie Weeks, Bill Lipscomb,
Jean-Louis Tison


Any suggested cover illustrations. 




which must be addressed in a comprehensive accounting is

whose microstructure depends strongly on temperature. 





The polar sea ice packs play a key role in earth's ocean-climate 
system, and are sensitive indicators of climatic change. As a 
material, sea ice is a porous composite of ice with brine, air, 
and salt inclusions, whose microstructural properties depend 
strongly on temperature. The transport of brine, which carries 
salt, heat, and nutrients through sea ice, controls a broad range 
of geophysical, oceanographic, and biological processes. However, 
measurements of the fluid permeability of sea ice are sparse and 
little is known theoretically. We give mathematical formulations 
of the two key problems of fluid transport in sea ice, bulk flow 
of brine, and diffusion of dissolved substances such as pollutants 
or bacterial enzymes. We present a comprehensive theory for the 
fluid permeability of sea ice, based on rigorous bounds, continuum 
percolation theory, hierarchical models, network simulation, and 
microstructural analysis. Our theoretical results closely capture 
laboratory and Arctic field data.





Our results yield a simple parameterization of the permeability
in terms of the sea ice state variables, salinity $S$ and 
temperature $T$. Now fluid transport phenomena in sea ice can 
be readily incorporated into, say, global climate and 
biological growth models.






The transport of brine 
%carrying salt, heat, and nutrients
through sea ice facilitates 
%plays a critical role in
a wide range of geophysical, oceanographic, and biological
processes 
\cite{Weeks:1986:GSP,Thomas:2003:SI,Fritsen:S-782}.
%Lizotte:1998:ASI,
%Lytle:JGR-8853,Trodahl:JGR-11358,Carmack:GSI-641,Maksym:2001:PCE}.
%Hosseinmostafa:JEA-421}. 
However, measurements of the fluid permeability $\kappa$ of sea ice are 
sparse \cite{Ono:AG-298,Freitag:1999:HPA}, 
%Saito:LTS-55,
and little is known theoretically. 
The brine microstructure exhibits a marked transition 
in macroscopic connectedness at a critical value $\phi_c \approx 5\%$ of the
brine volume fraction $\phi$,
called the {\it percolation threshold}$\;$ \cite{Golden:S-2238}.
Above $\phi_c$, 
%$\kappa(p) \sim \kappa_0(p-p_c)^e$,
the rapid rise of $\kappa(\phi)$ can be characterized
by a critical exponent $e$ and a scaling factor $\kappa_0$.
In general, 
$e$ can take a range of values in the continuum, depending
on the microstructure.  
Here we use
%with lognormally distributed brine inclusions 
%\cite{Perovich:JGR-18327}, 
continuum
percolation theory 
\cite{Halperin:PRL-2391,Berkowitz:TPM-275} to find
%and critical path analysis \cite{Friedman:WRR-1703,Golden:HOM-99}
that for sea ice $e \approx 2$, 
the {\it universal} exponent for lattices
\cite{Stauffer-92,Golden:PRL-2923},
and critical path analysis \cite{Friedman:WRR-1703,Golden:HOM-99}
to estimate 
$ \kappa_0 \approx 3  \times  10^{-8}\; \mbox{m}^2$.
While measurements of $\kappa$ taken on
Arctic sea ice in-situ naturally have considerable scatter,
%due primarily to natural microstructural inhomogeneities,
our theoretical calculations of $\kappa_0$ and
$e$ capture the data's scale and trend.
A statistical best fit of our data in logarithmic variables
matches the theory to within a relative error of $4\%$.
%in the variables
%$y=\log{\kappa}$ and $x=\log{(\phi - \phi_c)}$ yields
%the line $y=2.07x-7.45$, compared to our prediction
%of $y=2x-7.5$. 
%and a $\log - \log$ equation of $y=2.0x-7.5$.
%Arctic field data yield
%a best fit of 2.07. 
%The critical properties of sea ice near $\phi_c$ are described by
%Percolation theory applies near $\phi_c$,
%While percolation theory
%applies only near $\phi_c$, 
The most general features of sea ice as a porous medium 
%over relevant temperature and length scales
%present over all brine volumes 
over all $\phi$, not just near $\phi_c$,
are reflected in
a class of self-similar models 
\cite{Sen:G-781,Sheng:PRB-4507,Wong:PRB-6606}
%for which effective medium theory
%is exact, 
which maintain connectivity of the fluid phase
to zero porosity. 
%using our calculation of $\kappa_0$ and 
The simplest of these models predicts power law behavior 
which matches
%corresponding 
a logarithmic best fit of our data to within a relative error of $2\%$,
and another data set \cite{Freitag:1999:HPA} to within $3\%$.
%line of $y=3z-7.5$ with $z=\log{(\phi)}$, compared to a statistical
%best fit of $y=3.05z-7.50$.
We also obtain a rigorous, general upper bound on $\kappa$ which 
successfully captures
97\% of our field and laboratory data points.

%yields
%the line $y=2.07x-7.45$, compared to our prediction
%of $y=2x-7.5$. 

%displays power law dependence of $\kappa(\phi)$
%with an exponent of 3  
%Arctic field data yield a best fit of 3.05.
%Due to the generality of some of our results,
%it is reasonable to expect them to apply as well to sea ice  
%potentially found on other planets or moons.

%$\kappa(p) \sim \kappa_0 p^{2m}$
%with $m=3/2$ and a $\log - \log$ 
%line $y=3x-7.5$.
%Arctic field data yield
%a best fit of $y=3.05x-7.50$. 
%over the full range of volume fractions from $p=3 \%$ to $p=24 \%$
%yields an exponent of $3.05$.  
%Due to the generality 
%of our results, it is anticipated that they will apply as well to 
%sea ice potentially found on other 
%planets or moons, such as Europa.

\vspace{1.5ex}


%Earth's sea ice packs form the boundary layer
%between the ocean and atmosphere in the polar regions,
%mediating the transference of heat, radiation, momentum, and gases.


As the boundary layer between the ocean and atmosphere 
in the polar regions, the sea ice packs 
form a key component of earth's climate system,
%play a key role in global
%climate and as an indicator of climatic change
%\cite{Untersteiner:1990:SPS,Curry:1995:EIA},
and serve as an important habitat for the rich microbial
communities of the polar oceans
\cite{Thomas:2003:SI,Fritsen:S-782}.
%Lizotte:1998:ASI
Sea ice is distinguished from other natural
porous media 
%such as rocks, soils, and bones 
in that its properties
can change so dramatically
with temperature. 
For example, over a temperature range of approximately
$15^{\circ}$C, fluid 
permeabilities measured in sea ice under
natural conditions range over more than six orders of
magnitude, which is almost the
entire range found in other geologic media.
%\cite{Freeze:1995:G}. 
%Porosities vary almost over the same
%range of magnitudes, as do pore sizes 
%\cite{Perovich:JGR-18327,Perovich:1991:SDM,Eicken:2000:NMR}.
%The motion of brine through sea ice 
%depends upon the connectivity of the inclusions and
%plays a critical role 
Brine transport is essential to a variety of processes, such as 
sea ice production through freezing of flooded ice
surfaces \cite{Maksym:2001:PCE},
%Ackley:AJUS-133, 
convection-enhanced thermal transport 
\cite{Lytle:JGR-8853,Trodahl:JGR-11358}, 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}.
Microwave signatures of sea ice 
%during remote sensing
can be affected not only by changes in the 
microstructure with temperature but by 
flooding and freezing of sea ice surfaces 
\cite{Hosseinmostafa:JEA-421,Golden:TGRS-1675}.
%,Golden:TGRS-1675}. Lytle:AJUS-125
Moreover, the transport of nutrients, sediments, and biomass
through connected pore spaces are
fundamental, life sustaining processes. 
%dependent upon microstructural connectedness on a number of length scales. 
For example, the growth rate of a convection-fueled 
algal bloom in the Weddell Sea fell off rapidly after 
the inhabited ice layer
was rendered effectively impermeable to nutrient replenishment
during the fall freeze-up \cite{Fritsen:S-782,Golden:S-2238}.
Recently there has been speculation 
as to whether possible life forms on ice-covered 
extraterrestrial bodies such as Europa have carved out
strategies for survival reminiscent of
the bacterial and algae species found in sea ice
\cite{Thomas:S-641}.
%might serve as proxies for possible life forms 
%it may be possible forlife to exist in the ice
%covering an alien sea on Europa. 
Our findings here provide a framework for 
the analysis of fluid transport problems in sea ice, whether
they arise in geophysical, climatological, oceanographic,
biological, or astrobiological settings.


 




++++++++++++++++++++++


Sea ice is a porous composite whose microstructure depends
sensitively on temperature. It mediates interactions between the 
ocean and atmosphere in the polar regions, and is an
indicator of global climate change. Sea ice also
hosts extensive microbial communities which support the 
polar ecosystems. Fluid transport through sea ice
facilitates a broad range of processes involved in the
formation, maintenance, and melting of the sea ice packs.
By carrying nutrients and enzymes, brine flow is essential
to sustaining life in this extreme environment.
Fluid transport through sea ice is determined by its
properties as a porous medium, namely, its vertical 
fluid permeability $k$. 










It has long been observed that sea ice freshens with age as brine drains 
out through the porous microstructure. The input of brine into the upper
ocean during sea ice formation, and of fresh water during sea ice melting
depend upon the fluid transport properties of the ice.  


Observations by early polar  that , fluid transport phenomena have 
been central to 



it has long been understood that fluid brine drainage, etc.
are important inhow sea ice interacts with the upper ocean
more recently it has become apparent that fluid transport
processes are central to 


our results are derivede ffrom first principles
given that there are no adjustable parameters, it is 



______________________________________________________________



For example, the effective conductivity $\sigma$ in the vertical direction of
a composite of two materials with
conductivities $\sigma_1$ and $\sigma_2$ and volume fractions $p_1$ and $p_2$
%The effective conductivity $\sigma$ of the composite in the vertical direction
obeys \cite{Torquato:RHM-02}
%cannot exceed the arithmetic mean of the component conductivities \cite{Torquato:RHM-02}, 
$\sigma \leq p_1 \sigma_1 +p_2 \sigma_2$. 
This optimal, {\it arithmetic mean} upper bound can be 
attained by vertically arranged layers or cylinders,
regardless of their relative sizes. 
%and has been known since the early 1900's.




In porous media, the permeability involves a length scale 
not present in analogous electrical or thermal problems. 
For example, the electrical conductance %$g_e$ 
of a cylinder of a material with conductivity $\sigma_1$
of radius $r$ and unit length
is $\pi r^2 \sigma_1$. 
%On the other hand,
The corresponding fluid conductance 
for Poiseuille flow in a pipe %$g_f$ 
of inner radius $r$ and unit length is $\pi r^4/8 \eta$.
Then the fluid conductivity, the analog of $\sigma_1$,
is $r^2/8 \eta$, which is proportional to the pipe's 
cross sectional area.






%the permeability of 
%many pipes of small radius is less than that
%of a few pipes of larger radius occupying the 
%same volume fraction. 

   


old abstract


Monitoring polar sea ice is central to gauging
the impact of climate change. Most techniques
for obtaining information about sea ice
depend on its electrical properties. In two types of
experiments conducted in the Arctic and Antarctic,
we find that sea ice effectively has an 
{\it on$-$off} switch for electrical conductivity at a 
critical brine volume fraction of about 5\%. 
Above this threshold value,
the conductivity is accurately described by percolation theory 
with a universal critical exponent, and by exploiting
theoretical relations between 
electrical and fluid transport. 
These results open the door to electrically monitoring 
key processes in sea ice, such as melt pond evolution, 
snow-ice formation, CO$_2$ exchanges, and nutrient replenishment
for microbial communities.
They also affect the interpretation of ice thickness data.




\begin{center}

{\Large {\bf Electrical Signature of Brine Percolation 
in Sea Ice}}

\vspace{5ex}

Kenneth M. Golden\footnote{Department of Mathematics, 
University of Utah, 155 S 1400 E, RM 233, Salt Lake City, UT 84112-0090 USA}, 
Hajo Eicken\footnote{
University of Alaska Fairbanks, Fairbanks, AK, 99775-7320, USA}, 
Adam Gully\footnotemark[1], 
Malcolm Ingham\footnote{School of Chemical and Physical Sciences, 
Victoria University of Wellington, P.O. Box 600, 
Wellington, NZ}, Keleigh A. Jones\footnotemark[3], \\
Joyce Lin\footnotemark[1], 
James E. Reid\footnote{Groundprobe Geophysics, 1/288 Victoria Rd., 
Malaga 6090, AU}, 
Christian S. Sampson\footnotemark[1], 
Anthony P. Worby\footnote{Australian Antarctic Division and 
ACE CRC, University of Tasmania, Private Bag 80, Hobart, 7001, AU}


\end{center}















%\cite{Thomas:2003:SI,Fritsen:S-782,Lizotte:1998:ASI,Eicken:PB-3}. 
While global climate models predict declines in sea ice area and thickness, 
they have significantly underestimated recent loss  
in Arctic summer ice extent \cite{Boe:2009:NG}. 

 





It has been observed that
for brine volume fractions $\phi$ below about 5\%, columnar sea ice





















 




For example, the evolution of melt ponds and 
summer ice albedo is constrained by drainage
through porous sea ice,
%\cite{Eicken:JGR-C08007}, 
where 
ice-albedo feedback is believed to play a 
key role in the decline of summer Arctic sea ice.
%\cite{Perovich:GRL-2007}.
Fluid flow also facilitates 
snow-ice formation,
%\cite{Maksym:JGR:2008}, 
the evolution of the salt budget,
%\cite{Thomas:2008:SI}, 
convection-enhanced thermal transport,
%\cite{Lytle:JGR-8853},
%ocean-ice-atmosphere 
CO$_2$ exchange, 
%\cite{Rysgaard:JGR:2007},
and biomass build-up sustained by nutrient fluxes.
%\cite{Thomas:2008:SI,Fritsen:S-782}.








Improving projections of the fate of Earth's ice
packs and their complex ecosystems depends on 
a better understanding of key processes and feedback mechanisms.
For example, the evolution of melt ponds and 
summer ice albedo is constrained by drainage
through porous sea ice \cite{Eicken:JGR-C08007}. 
It is believed that ice-albedo feedback has played a 
key role in the decline of summer Arctic sea ice
\cite{Perovich:GRL-2008}.
Fluid flow through sea ice mediates the evolution of
salinity profiles \cite{Thomas:2008:SI},
%\cite{Weeks:1986:GSP,Eicken:GMP-02}, 
convection-enhanced thermal transport
\cite{Lytle:JGR-8853},
ocean-ice-atmosphere CO$_2$ exchanges \cite{Rysgaard:JGR:2009},
and biomass build-up fueled by nutrient fluxes
\cite{Thomas:2008:SI,Fritsen:S-782}.
%Eicken:PB-3}.
It also enables
snow-ice formation, where sea water floods the 
ice surface and then freezes, 
accounting for more than a quarter of the ice 
produced in the Southern Ocean
\cite{Maksym:JGR:2008}. 
%This process may become
%important in the Arctic with increased precipitation
%and thinning ice, so that floes are more susceptible to flooding.













