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% Author names in capital letters:
\authorrunninghead{GOLDEN ET AL.}

% Shorter version of title entered in capital letters:
\titlerunninghead{Electrical Signature of Percolation in Sea Ice}

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

\begin{document}

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%  TITLE
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\title{Electrical Signature of the Percolation Threshold  %Transition 
in Sea Ice}
%
% e.g., \title{Terrestrial ring current:
% Origin, formation, and decay $\alpha\beta\Gamma\Delta$}
%

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%  AUTHORS AND AFFILIATIONS
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%Use \author{\altaffilmark{}} and \altaffiltext{}

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% matching \altaffiltext will appear at bottom of page.


\vspace{3ex}



\authors{K. M. Golden\altaffilmark{1},
H. Eicken\altaffilmark{2}, A. Gully\altaffilmark{1},
M. Ingham\altaffilmark{3}, K. A. Jones\altaffilmark{3}, J. Lin\altaffilmark{1}, J. E. Reid\altaffilmark{4}, C. S. Sampson\altaffilmark{1}, and A. P. Worby\altaffilmark{5}}




\vspace{65ex}





\altaffiltext{1}{University of Utah, Department of Mathematics, 
155 S 1400 E, RM 233, Salt Lake City, UT 84112-0090, USA. (golden@math.utah.edu)}

\altaffiltext{2}{University of Alaska Fairbanks, Fairbanks, AK, 99775-7320, USA.}

\altaffiltext{3}{School of Chemical and Physical Sciences, 
Victoria University of Wellington,P.O. Box 600, 
Wellington, NZ.}

\altaffiltext{4}{Groundprobe Geophysics, 1/288 Victoria Rd., 
Malaga 6090, AU.}

\altaffiltext{5}{Australian Antarctic Division and 
ACE CRC, University of Tasmania, Private Bag 80, Hobart, 7001, AU.}

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\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
exhibits a strong electrical signature, with
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 permeability. 
These results enable us to connect specific electrical profiles to 
important transport processes such as melt pond drainage, 
CO$_2$ pumping,
and the flux of nutrients which sustain biomass build-up.
\end{abstract}

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%  TEXT
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\noindent
Key Points: 

\noindent

\begin{enumerate}
\item  Data on the electrical resistivity of sea ice were taken in the Arctic and Antarctic 
\noindent
\item  A strong electrical response near the brine percolation threshold 
      is observed and explained theoretically 
\noindent
\item  The results enable remote monitoring of key sea ice processes and transitions %governed by fluid transport

\end{enumerate}


\vspace{-1ex}



\section{Introduction}





Polar sea ice is a key component
of Earth's climate system, and
a leading indicator of climate change
\citep{Thomas:2008:SI,Serreze:S-1533}.
%,Boe:2009:NG}.
%\citep{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 phase
hosts extensive microbial communities 
which sustain life in the polar oceans
\citep{Thomas:2008:SI,Fritsen:S-782}.
%Lizotte:1998:ASI,Eicken:PB-3,
%Krembs:PNAS:2011}.
Fluid flow through
the porous microstructure mediates key processes
impacting the 
climatology and biology of sea ice.
Improving projections of the fate of Earth's sea ice
cover and its ecosystems depends on 
a better understanding of these important processes and feedback mechanisms.

%For example, 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. 

For example, the evolution of sea ice albedo represents 
a fundamental problem in climate modeling
and a significant source of uncertainty in climate projections
\citep{Flocco:2010:JGR,%Scott:2010:JGR,%
%Pedersen:2009:JGR,
Polashenski:JGR:2012}.
The albedo of sea ice floes is 
determined by melt pond evolution 
\citep{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 
\citep{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 \citep{Perovich:GRL-2007}.%,Perovich:GRL-2008}.

















Fluid flow through sea ice governs the 
evolution of the salt budget and 
salinity 
profiles \citep{Thomas:2008:SI},
%\citep{Weeks:1986:GSP,Eicken:GMP-02}, 
convection-enhanced thermal transport
\citep{Lytle:JGR-8853},
ocean-ice-atmosphere CO$_2$ exchanges \citep{Rysgaard:JGR:2009},
and the build-up of algal biomass fueled by nutrient fluxes
\citep{Thomas:2008:SI,Fritsen:S-782}.
%Eicken:PB-3}.
It also drives
snow-ice formation, accounting for 
a significant portion of the ice produced in the Southern Ocean
\citep{Maksym:JGR:2008}. 
Sea water percolates upward
through the porous microstructure, flooding the 
snow layer, and subsequently freezing.






While fluid flow is substantially restricted for 
brine volume fractions $\phi$ below about 5\%, columnar sea ice
is increasingly permeable
for $\phi$ above 5\% \citep{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. 
This critical behavior of the fluid permeability,
which is known as the {\it rule of fives},
results from a connectivity or percolation threshold
in the brine microstructure
\citep{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 
\citep{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 
\citep{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. 
Our results indicate that such information
could potentially be obtained from measurements 
of electric properties via
{\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
\citep{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.


\vspace{-2ex}

\section{Measuring the electrical properties of sea ice}

%\vspace{-1ex}



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 \citep{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.






%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
%\citep{Fujino:1963:LTS,Thyssen:1974:P,%
%Buckley:CRST:1986,Reid:2006:AG,Ingham:2008:CRST,Sampson:DSR:2011}.
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.
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.
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).
We obtained 26 averaged data
points from 67 raw measurements of the resistance 
between the inner probes.
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
\citep{Eicken:GMP-02}.
In the auxiliary material we demonstrate that the Wenner
array deployed along the core axis is a viable field method for 
measuring conductivity, yielding values
of the vertical component very close 
to the results of classical, parallel plate experiments. 



In the Arctic, we used the technique of 
cross-borehole DC resistivity 
tomography
\citep{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 been demonstrated that the
vertical component of {\boldmath${\sigma}^*$} can be obtained
as well \citep{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 {\boldmath${\sigma}^*$} can be derived, along with
the horizontal component \citep{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 \citep{Druckenmiller:CRST:2009}
provided ice temperature and 
salinity data, allowing the variation in 
resistivity structure to be correlated 
with brine volume fraction $\phi$.

















\vspace{-2ex}


\section{Modeling the electrical conductivity of sea ice}



Lattice and continuum percolation models 
\citep{Stauffer-92}
have been used to study a broad range of disordered
materials where the connectedness of one phase
dominates effective transport behavior. 
In sea ice, the fluid and electrical transport properties
are largely determined by the connectedness of the brine phase
$-$ an electrically conducting fluid.
Here we briefly describe a lattice percolation 
model which provides the theoretical framework
for predicting the electrical conductivity 
of sea ice, as well as its fluid permeability. This model,
and how it is adapted to the microstructure of sea ice, is 
covered in more detail in the auxiliary material.

Consider the two dimensional square network of bonds
(edges) joining nearest neighbor sites (vertices)
in the integer lattice ${\mathbb{Z}}^2$, 
as shown in Figure 2 a and b.
The bonds are assigned electrical conductivities
$\sigma_0 > 0$ (open) or 0 (closed) with probabilities $p$ 
and $1-p$, so that a relative proportion $p$ of 
the bonds are open (on average). 
The {\it percolation threshold} $p_c$ is the 
smallest value of $p$ for which an infinite, connected
cluster of open bonds forms.
In two dimensions ($d=2$), $p_c = \frac{1}{2}$, and in three ($d=3$),
$p_c \approx 0.25$. 
For $p$ just above the percolation threshold, $p > p_c$, 
where conducting pathways span the infinite lattice, 
the effective or bulk
conductivity $\sigma^*(p)$ takes off with power law behavior,
\begin{equation}                    \label{critical_conductivity}
\sigma^*(p) \sim \sigma_0(p-p_c)^t\;,
\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 dimension and not, for example, 
on whether the lattice is square or triangular.
In $d=2$, $t \approx 1.3$,
and in $d=3$,
$t \approx 2.0$ \citep{Stauffer-92}.  



In applying percolation theory to sea ice, 
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 $\sigma_b$ of the brine.
%and depends only on the pore
%volume fraction $\phi$ and pore geometry.
In view of (\ref{critical_conductivity}),
%and (\ref{cond_perm_relation}),
\begin{equation}                    \label{formation_factor}
F(\phi) \; \sim \; F_0 \; (\phi - \phi_c)^2,
\end{equation}
where $\phi_c \approx 0.05$ \citep{Golden:S-2238,Golden:2007:GRL,Pringle:JGR:2009}, 
and $t \approx 2.0$, the $d=3$ universal lattice value. 
Even though sea ice is a continuum, its
lognormally distributed pores lead to 
universal critical behavior \citep{Golden:2007:GRL,Berkowitz:TPM-275}.

The scaling factor $F_0$ is obtained
by relating the electrical conductivity to the fluid permeability
of sea ice obtained in \citep{Golden:2007:GRL},
through a critical (or {\it bottleneck}) radius $r_c$ \citep{Friedman:WRR-1703}. 
By measuring the radii of vertical pathways 
in X-ray tomography 
images \citep{Golden:2007:GRL,Pringle:JGR:2009}, we
estimate a range in mm of
%$r_{c}$ (mm) to be in the range
$0.1   \leq  r_{c}  \leq  0.2$, yielding 
a range for $F_0$ of $6 \leq F_0 \leq 24$.
These relations between 
fluid and electrical transport in sea ice are
developed in the auxiliary material.

\section{Comparison of theory and data}


In order to compare our conductivity 
measurements with percolation theory, we must
exclude data below $\phi_c \approx 0.05$ \citep{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. As the conductivity $F$
becomes very small near $\phi_c$, its reciprocal $G$ becomes very
large, with its behavior approximating a vertical asymptote 
near $\phi = \phi_c$.
In Figure 2 c and d 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}
$F(\phi) \; \sim \; 9 \; (\phi - 0.05)^2$.
%\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 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
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 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 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 \citep{Golden:2007:GRL},
were used to model $k_v^*(\phi)$.
The simplest model yields a 
result of $k_v^*(\phi)=k_0 \, \phi^3$ and
%Via (\ref{cond_perm_relation}) we obtain 
an Archie's
law $F(\phi)=F_0 \, \phi^3$. 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 \citep{Golden:2007:GRL}
is shown relative to predictions from both models.







\vspace{-1ex}

\section{Discussion}






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 \citep{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
\citep{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 \citep{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.




\vspace{-1ex}

\section{Conclusions}





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 can
improve projections of the fate
of Earth's ice packs and the response of polar ecosystems.

%\vspace{5ex}

%\noindent Acknowledgements are in \citep{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}




%%% End of body of article:

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%  ACKNOWLEDGMENTS


\begin{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.
\end{acknowledgments}

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\vspace{-3ex}



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%  \providecommand{\doi}{doi:\discretionary{}{}{}\begingroup
%  \urlstyle{rm}\Url}\fi
%
%\bibitem[{\textit{Atkinson and Sloan}(1991)}]{AtkinsonSloan}
%Atkinson, K., and I.~Sloan (1991), The numerical solution of first-kind
%  logarithmic-kernel integral equations on smooth open arcs, \textit{Math.
%  Comp.}, \textit{56}(193), 119--139.
%
%\bibitem[{\textit{Colton and Kress}(1983)}]{ColtonKress1}
%Colton, D., and R.~Kress (1983), \textit{Integral Equation Methods in
%  Scattering Theory}, John Wiley, New York.
%
%\bibitem[{\textit{Hsiao et~al.}(1991)\textit{Hsiao, Stephan, and
%  Wendland}}]{StephanHsiao}
%Hsiao, G.~C., E.~P. Stephan, and W.~L. Wendland (1991), On the {D}irichlet
%  problem in elasticity for a domain exterior to an arc, \textit{J. Comput.
%  Appl. Math.}, \textit{34}(1), 1--19.
%
%\bibitem[{\textit{Lu and Ando}(2012)}]{LuAndo}
%Lu, P., and M.~Ando (2012), Difference of scattering geometrical optics
%  components and line integrals of currents in modified edge representation,
%  \textit{Radio Sci.}, \textit{47},  RS3007, \doi{10.1029/2011RS004899}.

%\end{thebibliography}

%Reference citation examples:

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%...has been shown [\textit{Kilby et al.}, 2008].
%...has been shown [{\textit  {Lewin}}, 1976; {\textit  {Carson}}, 1986; {\textit  {Bartholdy and Billi}}, 2002; {\textit  {Rinaldi}}, 2003].
%...has been shown [e.g., {\textit  {Lewin}}, 1976; {\textit  {Carson}}, 1986; {\textit  {Bartholdy and Billi}}, 2002; {\textit  {Rinaldi}}, 2003].

%...as shown by \citet{jskilby}.
%...as shown by \citet{lewin76}, \citet{carson86}, \citet{bartoldy02}, and \citet{rinaldi03}.
%...has been shown \citep{jskilbye}.
%...has been shown \citep{lewin76,carson86,bartoldy02,rinaldi03}.
%...has been shown \citep [e.g.,][]{lewin76,carson86,bartoldy02,rinaldi03}.
%
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% DO NOT use other cite commands (e.g., \cite, \citeyear, \nocite, \citealp, etc.).

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\end{article}

%% Enter Figures and Tables here:

% When submitting articles through the GEMS system:
% COMMENT OUT ANY COMMANDS THAT INCLUDE GRAPHICS.
%
% DO NOT USE \psfrag or \subfigure commands.
%
% Figure captions go below the figure.
% Table titles go above tables; all other caption information
%  should be placed in footnotes below the table.




%\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}



% DRAFT figure/table, including eps graphics
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% \begin{figure}
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% \end{table*}
%
% ---------------
% EXAMPLE TABLE
%
%\begin{table}
%\caption{Time of the Transition Between Phase 1 and Phase 2\tablenotemark{a}}
%\centering
%\begin{tabular}{l c}
%\hline
% Run  & Time (min)  \\
%\hline
%  $l1$  & 260   \\
%  $l2$  & 300   \\
%  $l3$  & 340   \\
%  $h1$  & 270   \\
%  $h2$  & 250   \\
%  $h3$  & 380   \\
%  $r1$  & 370   \\
%  $r2$  & 390   \\
%\hline
%\end{tabular}
%\tablenotetext{a}{Footnote text here.}
%\end{table}

% See below for how to make landscape/sideways figures or tables.









\begin{figure}      \centering
 \includegraphics[width=14cm]{Figure_1.eps}
% \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}


\begin{figure}          \centering
 \includegraphics[width=12.5cm]{Figure_2.eps}
% \includegraphics[width=8.5cm]{Figure_2.eps}
\caption{The two dimensional square 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(c)}  
       and the Arctic in {\bf(d)}, 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}

\begin{figure}        \centering
\includegraphics[width=14cm]{Figure_3.eps} 
%\includegraphics[width=12cm]{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 \citep{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=10cm]{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 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 connect the electrical properties of sea ice
to its important processes, 
the range of the resistivity formation factor $G$
is 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 \citep{Rysgaard:JGR:2007}, and sufficiently
impermeable to 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.}
\label{Figure_4}
\end{figure}



\eject


\eject


\section{Auxiliary Material: Percolation Theory for Sea Ice}


Lattice and continuum percolation theories 
\citep{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 \citep{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_2}
\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 $d=3$,
$t \approx 2.0$ \citep{Stauffer-92}.  
There is also a rigorous 
bound \citep{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.
For 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 \citep{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
\citep{Halperin:PRL-2391,Stauffer-92}.
However, for 
lognormally distributed inclusions, as in sea ice,
the behavior is {\it universal} 
\citep{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 \citep{Golden:2007:GRL} 
for the vertical fluid permeability 
\begin{equation}                 \label{perc_theory_2}
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  
\citep{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 \citep{Friedman:WRR-1703}.
With $r_{c}$ denoting the critical radius for our 
centimeter scale electrical
experiments, then 
\begin{equation}            \label{cond_perm_relation_2}
k_v^* = \frac{r_{c}^2}{8} \; \frac{\sigma^*_v}{\sigma_b}\;,
\end{equation}
where $\sigma_b$ is the conductivity of brine,
which depends \citep{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 \citep{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 $\phi_c \approx 0.05$ \citep{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 c and d 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_2}
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 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
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 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 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 \citep{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 \citep{Golden:2007:GRL}
is shown relative to predictions from both models.





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.







\begin{figure}          \centering
 \includegraphics[width=12cm]
{Figure_5.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 $p=p_c$. 
        }

\label{Figure_2}
\end{figure}










\begin{figure}     \centering
 \includegraphics[width=14cm]{Figure_6.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}







\end{document}

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%\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}











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 \citep{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
\citep{Halperin:PRL-2391,Stauffer-92}.
However, for 
lognormally distributed inclusions, as in sea ice,
the behavior is {\it universal} 
\citep{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 \citep{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  
\citep{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 \citep{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 \citep{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 \citep{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 $\phi_c \approx 0.05$ \citep{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 c and d 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 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
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 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 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 \citep{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 \citep{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 \citep{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 \citep{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}













