The brine pore space in sea ice can form complex connected structures whose geometry is critical in the governance of important physical transport processes between the ocean, sea ice, and surface. Recent advances in three-dimensional imaging using X-ray micro-computed tomography have enabled the visualization and quantification of the brine network morphology and variability. Using imaging of first-year sea ice samples at in situ temperatures, we create a new mathematical network model to characterize the topology and connectivity of the brine channels. This model provides a statistical framework where we can characterize the pore networks via two parameters, depth and temperature, for use in dynamical sea ice models. Our approach advances the quantification of brine connectivity in sea ice, which can help investigations of bulk physical properties, such as fluid permeability, that are key in both global and regional sea ice models.

The detailed microstructure of sea ice is critical in both
governing the movement of fluid between the ocean and the sea ice surface and
controlling processes such as ice growth and decay

Previous research has recognized the importance of thermally activated
percolation thresholds

Network models have successfully been employed in a variety of fields to
describe complex phenomena and predict future behavior, in particular fluid
flow in porous media

In this paper, we develop a methodology for describing the morphology and variability of brine networks in a vertical column of first-year sea ice. We construct a network model of the pore structure of sea ice and use topological techniques to characterize this brine network. This yields a set of network statistics that characterizes channels from different depths and temperature, which we can later use to inform more sophisticated models of sea ice. Future applications include refining under what conditions the “Rule of Fives” applies, predicting bulk physical properties such as heat transfer and fluid permeability, and improving the ability to describe processes such as brine drainage and desalination. This approach provides advances in quantifying the brine connectivity in sea ice, which we can then incorporate into both global and regional sea ice models.

This work will focus on two of the ice cores extracted from different
locations in the Ross Sea, Antarctica during a October–November 2012 field
campaign. The 1.78 m Butter Point ice core was collected at
77

We build on the methodology of Lieb-Lappen et al. (2017) to convert the
binarized images of the brine phase to a simplified representation as a
network. Network models are now ubiquitous across many fields as they provide
mathematical descriptions of complex phenomena that are amenable to detailed
analysis. Abstractly, a network is a collection of

Our model is, in a sense, a refinement of other network models used to study
transport within sea ice. Percolation models, such as those developed to
justify the “Rule of Fives”

To create our network, we must identify nodes and edges between them that
reflect the structure of the brine channels in the sample. To begin, we think
of a sample as a cube in three dimensional space,

Our collection of node–throat size pairs gives an approximation of the brine
structure in a sea ice sample but lacks an important feature – it does not
specify how the brine regions are connected to one another from one
horizontal slice to another. Including edges between nodes allows us to
encode this feature. We define an edge from node

In the language of networks

Sketch illustrating how the brine channel network is defined. Four horizontal two-dimensional slices are shown with lines connecting adjacent nodes (not all lines are drawn). Different colors represent different brine channels in this sample.

With the definition of the network in place, we next introduce terminology
which helps describe the evolution of a brine channel as it progresses
downwards through an ice sample. For a fixed node

With this mathematical description of the brine structure, we can calculate
statistics about its evolution. For a fixed throat size,

In summary, the set of probabilities defines a discrete-time probabilistic
process. Given a node at a particular height

We first used standard morphological metrics as defined in previous work to
describe the brine network shape and size

Definition and shape of brine channels scanned with a

Since the cooling stage did not significantly warm samples beyond

We compared the

Comparing

Average throat size

Average throat size

From the binarized images of the brine phase for the Butter Point and the
Iceberg Site ice cores, we created a mathematical network. We will use the
term

In the analysis below, we use several metrics to describe the topology of the brine network that have important fluid flow implications. We first look at the throat sizes of the brine channels to gain an insight into the quantity of fluid that can move through different regions of a given brine channel. We then look at specific branches, both in terms of the number of branches and the size of each branch to learn more about the specific pathways available for fluid movement. As part of this analysis, we incorporate our various transition probabilities to understand how the likelihood of a given branch to split into multiple branches depends upon the throat size of the parent branch. Finally, we will examine the size distribution of particular paths, looking for “pinch points” that may restrict flow and large regions that can provide maximum flow through the network. Together, these metrics will provide a detailed description of the micro-scale complexity of the brine network.

For each brine channel we calculated the average throat size

The number of branches for a particular brine channel has potentially
significant implications for fluid flow and permeability, such as influencing
the rate at which chemical species may pass through the sea ice

Throat size

Throat size

The figure shows

To gain insight into the behavior of a channel, we visualized the number of
branches and distribution of throat sizes by plotting the throat size

Next we examined the branching of particular nodes to understand the behavior
of particular fluid flow paths. Following a branch of a channel downwards, at
an individual node the branch may end, continue onwards, or split into
multiple branches. Conversely, by looking upwards, a node can be considered
to be the first in a new branch, the continuation of a branch, or the joining
point of multiple branches. Thus, for each node in a brine channel, we can
count the number of edges above (incoming) and below (outgoing) to determine
the degree of splitting or joining of branches in the channel

With knowledge of the total number of split points and join points, we then
investigated the likelihood that branching was dependent upon the throat
size. Figure

In frazil ice, the story for pockets that vanish is the same, as throat sizes become larger, they do not vanish in the next level. For remaining, splitting, and joining, however, there are new wrinkles in frazil ice relative to columnar ice. For pockets that remain, for larger throat sizes we see three types of behavior, two of which are similar to the behaviors in columnar ice (indicated by the top and bottom arrows of the second plot in the first row); however, a third behavior, where fifty percent of pockets remain, is new for frazil ice (middle arrow). This new behavior is echoed in the probabilities of splitting and joining (indicated by the middle arrows in those plots) which shows that in this regime, brine channels have a complex behavior, remaining, splitting, and joining with high frequency. This third category of behavior for large throat sizes is a signature of frazil ice.

In addition, we summed the total number of edges leaving (splits) and
entering (joins) each node over all nodes for the five largest brine channels
of each sample. Figure

Topological complexity of the five largest brine channels in each
sample for both the Butter Point (red) and the Iceberg Site (blue) cores. The
panel

Cumulative distribution functions for number of brine channels as
functions of the total number of pixels in the channel. The
panels

We next examined the fluid flow capacity of each channel by both summing the
number of pixels associated with all nodes for each channel and summing the
total throat sizes of all nodes in each channel. We note that this represents
a region larger than the pathways used for current fluid flow since many
branches do not connect the top of a sample to the bottom. However, when the
ice begins to warm and the branches become more interconnected, the process
will likely start from the existing regions containing brine. Thus, this
metric offers a starting place for comparing the capacity for fluid flow
across different samples. Figure

Cumulative distribution functions for number of brine channels as
functions of the summed throat size of all nodes in the channel. The
panels

Largest brine channel at 70 cm in the Butter Point ice core. Although this brine channel connects from top to bottom, there is not a directed path that does so. Any connecting path involves movements both upwards and downwards. One such path is highlighted in red.

Probability distributions of paths connecting the top to the bottom
for all brine channels in the Butter Point ice core. Only paths greater than
50 steps, or 750

To further assess fluid flow capabilities, we analyzed individual branches of
brine channels to isolate particular paths through the network. By
construction, moving from

We completed a similar analysis on the brine channel network, however this
time allowing for both upward and downward flow. Allowing for upward flow can
present a challenge in tracking various pathways if there is a repeating
loop. Thus, we only considered paths that reached every node but had no
loops. In the language of networks, we avoided complexities arising from
cycles by only considering different spanning trees. We used a depth-first
search algorithm to find all paths reaching the maximum vertical extent of
each channel

We used the 1753 paths of length greater than 50 steps to develop probability
distributions for basic network statistics important for fluid flow such as

The primary objective of this work has been to improve our
characterization of brine channel topology, morphology, and connectivity, in
order to provide sea ice modelers with a greater level of detail on the
factors that affect microstructural transport properties. While most
percolation models use coarse microstructural properties to form a
statistical basis for predicting connectivity, ours derives finer-grained
statistics empirically, allowing for better representation of the range of
physical properties found in sea ice of different types and conditions. We
can statistically model the evolution of brine channels as we move downwards
through the sea ice cover. Beginning with an initial brine pocket, our
estimates of the evolution probability distributions from

Overall, we observed similar morphological profiles for both first-year sea ice cores. Topological complexity had the expected C-shape profile that is consistent with complex frazil ice in the top of the core, relatively cold columnar ice below it, and increasingly warmer columnar ice at lower depths. However, we did not have good success in imaging and thresholding ice with the warmest in situ temperatures at the bottom of the core.

Our estimates of the evolution probability distributions provides a
stochastic model of brine channels within sea ice at different temperatures,
extending the percolation models described above. Their structural features
reveal the onset of transitions between different types of ice: in our
analysis, we see different statistical features that delineate frazil and
columnar ice. Further, the level of detail inherent to this technique allows
us to quantify some of the finer details of brine channel structure and
development. In addition to estimating the expected brine volume and
permeability for ice at a fixed temperature, we can see when and why
permeability arises by analyzing the probabilistic structures. For example,
Fig.

When examining the branching in brine channels, we observe that the largest channels have the greatest number of branches, but overall brine channel size does not appear to have a direct correlation with the number of branches. Brine channel size is most dependent upon the depth and consequently ice type. When a split in a brine channel does occur, it is most likely to split into two child branches, and after the split, the brine generally has access to a larger region of the sea ice than before. Starting and ending brine pocket sizes are strongly correlated with the flow capacity with larger initial/final sizes strongly indicative of increased flow. We detected pinch points in the brine channels that are critical points when determining whether the sea ice cover has crossed the percolation threshold. However, further work is needed in examining warmer ice with greater brine volume fractions.

Our framework enables us to statistically replicate the pore structure of sea
ice at different depths and temperatures. The next step for this work is to
create a brine channel network from the probability distributions presented
here. For a sample at a given depth/temperature, first an initial region at
the top of the ice would be selected with size consistent with the statistics
shown here. The brine channel could grow or shrink, split into multiple
branches, join with other branches, remain constant, or stop, all with
probabilities dependent upon the thickness, depth/temperature, and proximity
to other brine channels. The model described herein can help address
questions such as how microstructural changes may be path dependent (e.g.,
whether to consider both upwards and downwards flow), how fluid flow may vary
with depth, and what are the percolation implications of temperature
fluctuations in an ice core. In summary, we successfully developed a method
using

All data for this paper are archived with the U.S. Antarctic
Program Data Center (USAP-DC) under the DOI

The authors declare that they have no conflict of interest

This research was supported by US National Science Foundation (NSF) grant PLR-1304134. The views and conclusions contained herein are those of the authors and should not be interpreted as representing official policies, either expressed or implied, of the NSF or the United States Government. Edited by: Jennifer Hutchings Reviewed by: Christian Sampson and Malcolm Ingham