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\begin{center}

{\bf PROJECT SUMMARY}

\vspace{3ex}

{\large Multiscale Models of Melting Arctic Sea Ice}




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During the Arctic melt season, the sea ice surface undergoes a 
remarkable transformation from vast expanses of snow covered ice 
to beautiful mosaics of ice and melt ponds. 
Small, disconnected ponds on the ice surface grow and coalesce 
to form much larger connected structures with complex 
boundaries.  Melt pond area fraction $\phi$ can undergo 
a critical transition with a rapid rise from 
0\% to more than 70\% in just a few days. 


Sea ice albedo, 
a key parameter in climate modeling, is 
determined by the 
evolution of melt pond and ice floe configurations. 
Ice-albedo feedback has played a major role 
in the recent declines of the summer Arctic sea ice pack. 
However, understanding the evolution of melt ponds and sea ice 
albedo remains a significant challenge to improving 
climate models. In fact, not including
the significant effects of melt ponds in the last generation of these 
models is a key factor in explaining why the observed losses
outpaced the projections.



Viewed from high above,
the sea ice surface can be thought of as a two phase composite
of ice and melt water. The boundaries between the two phases
evolve with increasing complexity and a rapid onset of 
large scale connectivity, or percolation of the melt phase.  
Pond fractal dimension
transitions from about 1 to 2 around a critical length scale of 
100 square meters in area. This type of behavior 
is similar to what is observed in 
phase transformations in statistical physics
and geometrical transitions in composite materials. 


We propose here to study the evolution of melt pond geometry
on Arctic sea ice, and to develop multiscale models of 
pond structure which can provide fundamental
new insights into the melting process and the evolution
of sea ice albedo.
We will explore how mathematical models arising in 
statistical mechanics, composite materials, 
and pattern formation can
be used to efficiently characterize and quantify
melt pond structure.
Such systems often exhibit universal behavior
%where order parameters such as the effective conductivity
%of a composite or the magnetization of an Ising ferromagnet
%are described near their thresholds
described by critical exponents depending only on 
dimension and not on the details of the system. 
Developing such theories for melt ponds 
will provide novel ways of looking at the 
evolution of Arctic melting and ice albedo, as
well as a rigorous framework for finding 
large scale properties from local information. 
Key components of our proposed research include: 




\vspace{-1ex}

\begin{enumerate}
\item [$\bullet$] We will investigate
the transition in fractal geometry 
of melt ponds, and develop mathematical models
which capture this striking behavior. Related
studies will focus on the melt pond percolation threshold
and the onset of large scale connectedness.

\vspace{-1.2ex}


\item [$\bullet$] Models of
statistical physics and composite materials will be used 
to explore universal properties of melt pond evolution. 
Theories of {\it homogenization} and {\it upscaling} 
provide a framework for the 
calculation of effective properties
relevant to climate models.


\vspace{-1.2ex}


\item [$\bullet$] 
Mathematical findings and tools 
developed in this project will be exploited to improve 
the representation of melt ponds in high-resolution 
sea ice models.



\vspace{-1.2ex}


\item [$\bullet$] 
We will gather as much melt pond imagery as possible
from a wide range of sources, and develop image analysis
techniques to support the mathematical investigations, 
such as mapping complex melt ponds onto networks of
nodes and edges.


\end{enumerate}










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The rich behavior we observe for melt pond evolution 
is similar to phase transitions in
statistical mechanics
and composite materials. 



which can improve the treatment of sea ice albedo in climate models.





Such systems often display universal 
critical behavior 












The rich behavior we observe for melt pond evolution 
is similar to phase transitions in
statistical mechanics
and composite materials. 
Such systems often exhibit universal critical behavior,
where order parameters such as the effective conductivity
of a composite or the magnetization of an Ising ferromagnet
are described near their thresholds
by critical exponents depending only on 
dimension and not on the details of the system. 
Developing such theories for melt ponds 
will provide fundamental new insights into the 
evolution of ice pack albedo and Arctic melting, as
well as a rigorous framework for finding 
large scale properties from local information. 
Key components of our proposed research include: 










We propose to investigate key processes and feedback
mechanisms to improve 
our understanding of the role of sea ice in the climate
system, as well as 
projections of the fate of Arctic sea ice.
In particular, our focus will be on the formation and evolution 
of sea ice melt ponds, as well as ice$-$albedo feedback.





As the Arctic warms in the spring, small melt ponds on the 
surface of sea ice grow and coalesce to form much larger 
connected structures with complex boundaries.   
Melt pond area fraction can exhibit critical behavior
with a rapid rise from 0\% to over 70\%
in just a few days
with a corresponding critical drop in albedo.
Analysis of surface images suggests that 
melt pond formation resembles a percolation process, with 
the onset of large scale connectivity and complex fractal geometry
beyond a critical threshold. 
%In fact, area {\it vs.} perimeter data for melt ponds
%indicate a transition from simple shapes
%to highly ramified regions whose boundaries are similar
%to plane-filling curves.
%At a critical length scale, the fractal dimension 
%of the boundaries appears to transition 
%from 1 (Euclidean) to about 2. 


The precipitous decline of the summer Arctic sea ice 
pack is a strong signal that 
Earth's climate is changing.
Most current climate models, however, have significantly 
underestimated this retreat.









The rich behavior we observe for melt pond evolution 
is similar to phase transitions in
statistical mechanics
and composite materials. 
Such systems often exhibit universal critical behavior,
where order parameters such as the effective conductivity
of a composite or the magnetization of an Ising ferromagnet
are described near their thresholds
by critical exponents depending only on 
dimension and not on the details of the system. 
Developing such theories for melt ponds 
will provide fundamental new insights into the 
evolution of ice pack albedo and Arctic melting, as
well as a rigorous framework for finding 
large scale properties from local information. 
Key components of our proposed research include: 

\vspace{-1ex}

\begin{enumerate}
\item [$\bullet$] We will investigate
the complex fractal boundaries
of melt ponds, and develop mathematical models
of their geometry. 

\vspace{-1.2ex}


\item [$\bullet$] Theories of phase transitions in 
statistical physics and homogenization for
composite materials will be used 
to explore melt pond evolution. These theories will 
be used to calculate effective properties 
relevant to climate models
such as ice pack albedo.


\vspace{-1.2ex}


\item [$\bullet$] 
A dynamical systems model of ice$-$albedo feedback 
for a composite of ice floes and open water 
will be developed. 



\vspace{-1.2ex}


\item [$\bullet$] 
We will conduct related field research on melt pond evolution
with aerial photos, and investigations of critical behavior
in the porous
micro- and macro-structure of the ice pack, 
and its transport properties relevant to 
fluid flow and melt pond drainage. 


\end{enumerate}









