Mathematics of Sea Ice

K. M. Golden

Department of Mathematics

University of Utah

1 Introduction

Among the large-scale transformations of theEarth’s surface that are apparently due to globalwarming, the sharp decline of the summer Arc-tic sea ice pack is probably the most dramatic.For example, the area of the Arctic Ocean cov-ered by sea ice in September of 2012 was lessthan half of its average over the 1980’s and 1990’s.While global climate models generally predict de-clines in the polar sea ice packs over the 21st cen-tury, these precipitous losses have significantlyoutpaced most projections. Here we will showhow mathematics is being used to better under-stand the role of sea ice in the climate system andimprove projections of climate change. In partic-ular, we will focus on how mathematical mod-els of composite materials and statistical physicsare being used to study key sea ice structuresand processes, and represent sea ice more rig-orously in global climate models. Also, we willbriefly discuss these climate models as systems ofpartial differential equations (PDEs) solved usingcomputer programs with millions of lines of code,on some of the world’s most powerful computers,with particular focus on their sea ice components.

1.1 Sea Ice and the Climate System

Sea ice is frozen ocean water, which freezes at atemperature of about −1.8 C, or 28.8 F. As amaterial, sea ice is quite different from the glacialice in the world’s great ice sheets covering Antarc-tica and Greenland. When salt water freezes, theresult is a composite of pure ice with inclusions ofliquid brine, air pockets, and solid salts. As thetemperature of sea ice increases, the porosity orvolume fraction of brine increases. The brine in-clusions in sea ice host extensive algal and bacte-rial communities which are essential for support-ing life in the polar oceans. For example, krill feedon the algae, and in turn support fishes, penguins,seals, and Minke whales, and up the food chainto the top predators – killer whales, leopard seals,

and polar bears. The brine microstructure alsofacilitates the flow of salt water through sea ice,which mediates a broad range of processes, suchas the growth and decay of seasonal ice, the evolu-tion of ice pack reflectance, and biomass build-up.

As the boundary between the ocean and atmo-sphere in the polar regions of Earth, sea ice playsa critical role as both a leading indicator of cli-mate change, and as a key player in the global cli-mate system. Roughly speaking, most of the solarradiation which is incident on snow-covered seaice is reflected, while most of the solar radiationwhich is incident on darker sea water is absorbed.The sea ice packs serve as part of Earth’s polarrefrigerator, cooling it and protecting it from ab-sorbing too much heat from sunlight. The ratioof reflected sunlight to incident sunlight is calledalbedo. While the albedo of snow-covered ice isclose to 1 (larger than 0.8), the albedo of sea wa-ter is close to zero (less than 0.1).

Ice-albedo feedback. As warming tempera-tures melt more sea ice over time, fewer brightsurfaces are available to reflect sunlight, moreheat escapes from the ocean to warm the atmo-sphere, and the ice melts further. As more ice ismelted, the albedo of the polar oceans is lowered,leading to more solar absorption and warming,which in turn leads to more melting, in a positivefeedback loop. It is believed that this so-calledice-albedo feedback has played an important rolein the recent dramatic declines in summer Arcticsea ice extent.

Thus even a small increase in temperature canlead to greater warming over time, making thepolar regions the most sensitive areas to climatechange on Earth. Global warming is amplified inthe polar regions. Indeed, global climate modelsconsistently show amplified warming in the highlatitude Arctic, although the magnitude variesconsiderably across different models. For exam-ple, the average surface air temperature at theNorth Pole by the end of the 21st century is pre-dicted to rise by a factor of about 1.5 to 4 timesthe predicted increase in global average surfaceair temperature.

While global climate models generally predictdeclines in sea ice area and thickness, they havesignificantly underestimated the recent dramaticlosses observed in summer Arctic sea ice. Im-

1

2

submillimeter centimeters meters kilometers

Figure 1: Sea ice exhibits composite structure on length scales over many orders of magnitude. From left to right:

the sub-millimeter scale brine inclusions in sea ice; pancake ice in the Southern Ocean with microstructural scale

on the order of tens of centimeters; melt ponds on the surface of Arctic sea ice with meter scale microstructure;

ice floes in the Arctic Ocean on the kilometer scale.

proving projections of the fate of Earth’s sea icecover and its ecosystems depends on a better un-derstanding of important processes and feedbackmechanisms. For example, during the melt sea-son the Arctic sea ice cover becomes a complex,evolving mosaic of ice, melt ponds, and open wa-ter. The albedo of sea ice floes is determined bymelt pond evolution. Drainage of the ponds, witha resulting increase in albedo, is largely controlledby the fluid permeability of the porous sea ice un-derlying the ponds. As ponds develop, ice–albedofeedback enhances the melting process. More-over, this feedback loop is the driving mechanismin mathematical models developed to address thequestion of whether we have passed a so-calledtipping point or critical threshold in the declineof summer Arctic sea ice. Such studies often fo-cus on the existence of a saddle node bifurcationin dynamical system models of sea ice coverageof the Arctic Ocean. In general, sea ice albedorepresents a significant source of uncertainty inclimate projections and a fundamental problemin climate modeling.

Multiscale structure of sea ice. One ofthe fascinating, yet challenging aspects of mod-eling sea ice and its role in global climate is thesheer range of relevant length scales of structure,over ten orders of magnitude from the submillime-ter scale to hundreds of kilometers. In Figure 1we show examples of sea ice structure illustratingsuch a range of scales. Modeling sea ice on a largescale depends on some understanding of the phys-ical properties of sea ice at the scale of individualfloes, and even on the submillimeter scale sincethe brine phase in sea ice is such a key deter-

minant of its bulk physical properties. Today’s

climate models challenge the most powerful su-per computers to their fullest capacity. However,even the largest computers still limit the horizon-tal resolution to tens of kilometers and requireclever approximations and parameterizations tomodel the basic physics of sea ice. One of thecentral themes of this article is how to use infor-mation on smaller scales to predict behavior onlarger scales. We observe that this central prob-lem of climate science shares commonality withthe key challenges, for example, in theoreticalcomputations of the effective properties of com-posites.

Here we’ll explore some of the mathematicsused in studying sea ice and its role in the cli-mate system, particularly through the lens of seaice albedo and processes related to its evolution.

2 Global Climate Models and Sea Ice

Global climate models, also known as GeneralCirculation Models (GCM’s), are systems ofPDEs derived from the basic laws of physics,chemistry, and fluid motion. They describe thestate of the ocean, ice, atmosphere, and land,as well as their interactions. The equationsare solved on very powerful computers using 3-dimensional grids of the air-ice-ocean-land sys-tem, with horizontal grid sizes on the order oftens of kilometers. Consideration of general cli-mate models will take us too far afield, but herewe will briefly consider the sea ice components ofthese large scale models.

The polar sea ice packs consist primarily ofopen water, thin first-year ice, thicker multiyearice, and pressure ridges created by ice floes col-

3

Figure 2: Different factors contributing to the evolu-

tion of the ice thickness distribution g(h, x, t).

liding with each other. The dynamic and ther-modynamic characteristics of the ice pack dependlargely on how much ice is in each thickness range.One of the most basic problems in sea ice mod-eling is thus to describe the evolution of the icethickness distribution (ITD) in space and time.The ITD g(x, t, h)dh is defined (informally) asthe fractional area covered by ice in the thicknessrange (h, h + dh) at a given time t and locationx. The fundamental equation controlling the evo-lution of the ITD, which is solved numerically insea ice models, is

∂g

∂t= − ∇ · (gu) −

∂

∂h(βg) + Ψ

where u is the horizontal ice velocity, β is therate of thermodynamic ice growth, and Ψ is aridging redistribution function which accounts forchanges in ice thickness due to ridging and me-chanical processes, as illustrated in Figure 2.

The momentum equation, or Newton’s secondlaw for sea ice, can be deduced by considering theforces on a single floe, including interactions withother floes,

mDu

Dt= ∇ ·σ + τa + τw −mαn × u −mg∇H,

where each term has units of force per unit areaof the sea ice cover, m is the combined mass ofice and snow per unit area, τ a and τ w are windand ocean stresses, and D

Dt = ∂∂t + u · ∇ is the

material or convective derivative. This is a twodimensional equation, obtained by integrating the3D equation through the thickness of the ice inthe vertical direction.

The strength of the ice is represented by theinternal stress tensor σij. The other two termson the right hand side are, in order, stresses dueto Coriolis effects and the sea surface slope, wheren is a unit normal vector in the vertical direction,α is the Coriolis parameter, H describes the seasurface, and in this equation g is the accelerationdue to gravity.

The temperature field T (x, t) inside the sea ice(and snow layer), which couples to the ocean be-low and the atmosphere above through appropri-ate boundary conditions, satisfies an advectiondiffusion equation,

∂T

∂t= ∇ · (D(T ) ∇T ) − v · ∇T,

where D = K/ρC is the thermal diffusivity ofsea ice, K is its thermal conductivity, ρ its bulkdensity, C the specific heat, and v is an averagedbrine velocity field in the sea ice.

The bulk properties of low Reynolds numberflow of brine of viscosity η through sea ice canbe related to the geometrical properties of theporous brine microstructure using homogeniza-

tion theory. The volume fractions of brine andice are φ and 1 − φ. The local velocity and pres-sure fields in the brine satisfy the Stokes equa-tions for incompressible fluids, where the lengthscale of the microstructure (for example, the pe-riod in periodic media) is ε. Under appropri-ate assumptions, in the homogenization limit asε −→ 0, the averaged velocity v(x) and pressurep(x) satisfy Darcy’s law and the incompressibilitycondition,

v = −1

ηk ∇p, ∇ · v = 0. (1)

Here, k is the permeability tensor, with verticalcomponent kzz = k in units of m2. The perme-ability k is an example of an effective or homog-

enized parameter. The existence of the homoge-nized limits v,k, and p in (1) can be proven underbroad assumptions, such as for media with inho-mogeneities that are periodic or have translationinvariant statistics.

Obtaining quantitative information on k orother effective transport coefficients such as elec-trical or thermal conductivity and how they de-pend on, say, the statistical properties of the mi-crostructure, is a central problem in the theory

4

of composites. A broad range of techniques havebeen developed to obtain rigorous bounds, ap-proximate formulas and general theories of effec-tive properties of composite and inhomogeneousmedia, in terms of partial information about themicrostructure. This problem is, of course, quitesimilar in nature to the fundamental questionsof calculating bulk properties of matter from in-formation on molecular interactions – central tostatistical mechanics.

We note that it is also the case that one ofthe fundamental challenges of climate modeling ishow to rigorously account for sub-grid scale pro-cesses and structures in models whose grid sizeis far greater than the scale of relevant phenom-ena. For example, obviously it is not realisticto account for every detail of the sub-millimeterscale brine microstructure in sea ice in a GCM!However, the volume fraction and connectednessproperties of the brine phase control whether ornot fluid can flow through the ice. The on–off

switch for fluid flow in sea ice, known as therule of fives (see below), in turn controls suchcritical processes as melt pond drainage, snowice formation (where sea water percolates up-ward, floods the snow layer on the sea ice sur-face and subsequently freezes), the evolution ofsalinity profiles, and nutrient replenishment. Itis the homogenized transport coefficient – the ef-fective fluid permeability – which is incorporatedinto sea ice and climate models to account forthese and related physical and biogeochemicalprocesses. This effective coefficient is a well de-fined parameter (under appropriate assumptionsabout the microstructure) which captures the rel-evant microstructural transitions and determineshow a number of sea ice processes evolve. In thisexample, we will see that rigorous mathematicalmethods can be employed to analyze effective seaice behavior on length scales much greater thanthe submillimeter scale of the brine inclusions.

3 Mathematics of Composites

Here we give a brief overview of some of the math-ematical models and techniques that are used instudying the effective properties of sea ice.

3.1 Percolation Theory

Percolation theory was initiated in 1957 with theintroduction of a simple lattice model to studythe flow of air through permeable sandstones usedin miner’s gas masks. In subsequent decades,this theory has been used to successfully model abroad array of disordered materials and processes,including flow in porous media like rocks andsoils, doped semiconductors, and various typesof disordered conductors like piezoresistors, ther-mistors, radar absorbing composites, carbon nan-otube composites, and polar firn. The originalpercolation model and its generalizations havebeen the subject of intensive theoretical investi-gations, particularly in the physics community.One reason for the broad interest in the perco-lation model is that it is perhaps the simplest,purely probabilistic model which exhibits a typeof phase transition.

The simplest form of the lattice percolationmodel is defined as follows. Consider thed−dimensional integer lattice Z

d, and the squareor cubic network of bonds joining nearest neigh-bor lattice sites. We assign to each bond a con-ductivity σ0 > 0 (not to be confused with thestress tensor above) with probability p, meaningit is open (black), and a conductivity 0 with prob-ability 1− p, meaning it is closed. Two examplesof lattice configurations are shown in Figure 3,with p = 1/3 in (a) and p = 2/3 in (b). Groupsof connected open bonds are called open clusters.In this model there is a critical probability pc,0 < pc < 1, called the percolation threshold, atwhich the average cluster size diverges and an in-finite cluster appears. For the two dimensionalbond lattice pc = 1/2. For p < pc the density ofthe infinite cluster P∞(p) is 0, while for p > pc,P∞(p) > 0 and near the threshold,

P∞(p) ∼ (p− pc)β , p→ p+

c ,

where β is a universal critical exponent, that is, itdepends only on dimension and not on the detailsof the lattice. Let x, y ∈ Z

d and τ (x, y) be theprobability that x and y belong to the same opencluster. The correlation length ξ(p) is the meandistance between points on an open cluster, and isa measure of the linear size of finite clusters. Forp < pc, τ (x, y) ∼ e−|x−y|/ξ(p), and ξ(p) ∼ (pc −p)−ν diverges with a universal critical exponent

5

10 pp

1

10 ppc

ξ

c

8P

10 ppc

*

p = 1/3 p = 2/3

(a) (b) (c) (d) (e)

Figure 3: The two dimensional square lattice percolation model below its percolation threshold of pc = 1/2 in

(a) and above it in (b). (c) Divergence of the correlation length as p approaches pc. The infinite cluster density

of the percolation model is shown in (d), and the effective conductivity is shown in (e).

ν as p→ p−c , as shown in Figure 3 (c).The effective conductivity σ∗(p) of the lattice,

now viewed as a random resistor (or conductor)network, defined via Kirchoff’s laws, vanishes forp < pc like P∞(p) since there are no infinite path-ways, as shown in Figure 3 (e). For p > pc,σ∗(p) > 0, and near pc,

σ∗(p) ∼ σ0(p − pc)t, p→ p+

c ,

where t is the conductivity critical exponent, with1 ≤ t ≤ 2 in d = 2, 3, and numerical val-ues t ≈ 1.3 in d = 2 and t ≈ 2.0 in d = 3.Consider a random pipe network with effectivefluid permeability k∗(p) exhibiting similar behav-ior k∗(p) ∼ k0(p−pc)

e, where e is the permeabilitycritical exponent, with e = t for lattices Both tand e are believed to be universal − they dependonly on dimension and not on the type of lat-tice. Continuum models, like the so-called Swiss

cheese model, can exhibit non–universal behaviorwith exponents different from the lattice case ande 6= t.

3.2 Analytic Continuation and

Spectral Measures

Homogenization is where one seeks to find a ho-mogeneous medium which behaves macroscopi-cally the same as a given inhomogeneous medium.The methods are focused on finding the effectiveproperties of inhomogeneous media such as com-posites. We will see that the spectral measure ina Stieltjes integral representation for the effectiveparameter provides a powerful tool for upscalinggeometrical information about a composite intocalculations of effective properties.

We now briefly describe the analytic contin-

uation method (ACM) for studying the effec-

tive transport properties of composite materials.This method has been used to obtain rigorousbounds on effective transport coefficients of two-component and multicomponent composite ma-terials. The bounds follow from the special an-alytic structure of the representations for the ef-fective parameters, and partial knowledge of themicrostructure, such as the relative volume frac-tions of the phases in the case of composite media.The ACM was later adapted to treating the effec-tive diffusivity of passive tracers in incompressiblefluid velocity fields.

We consider the effective complex permittiv-ity tensor ε

∗ of a two-phase random medium, al-though the method applies to any classical trans-port coefficient. Here, ε(x, ω) is a (spatially) sta-tionary random field in x ∈ R

d and ω ∈ Ω, whereΩ is the set of all geometric realizations of themedium, which is indexed by the parameter ωrepresenting one particular realization, and theunderlying probability measure P is compatiblewith stationarity.

As in sea ice, we assume we are dealing with atwo-phase locally isotropic medium, so that thecomponents εjk of the local permittivity tensorε satisfy εjk(x, ω) = ε(x, ω)δjk, where δjk is theKronecker delta and

ε(x, ω) = ε1χ1(x, ω) + ε2χ2(x, ω). (2)

Here, εj is the complex permittivity for mediumj = 1, 2 and χj(x, ω) is its characteristic function,equaling 1 for ω ∈ Ω with medium j at x, and 0otherwise, with χ2 = 1 − χ1.

When the wavelength is much larger than thescale of the composite microstructure, the prop-agation properties of an electromagnetic wave ina given composite medium are determined by the

6

Figure 4: Ocean swells propagating through a vast

field of pancake ice in the Southern Ocean off the

coast of East Antarctica (photo by K. M. Golden).

These long waves do not “see” the individual floes,

whose diameters are on the order of tens of centime-

ters. The bulk wave propagation characteristics are

largely determined by the homogenized or effective

rheological properties of the pancake/frazil conglom-

erate on the surface.

quasi-static limit of Maxwell’s equations

∇×E = 0, ∇ · D = 0, (3)

where E(x, ω) and D(x, ω) are stationary electricand displacement fields, respectively, related bythe local constitutive equation D(x) = ε(x)E(x),and ek is a standard basis vector in the kth di-rection. The electric field is assumed to be unitstrength on average, with 〈E〉 = ek, where 〈·〉denotes ensemble averaging over Ω or spatial av-erage over all of R

d. The effective complex per-mittivity tensor ε

∗ is defined by

〈D〉 = ε∗〈E〉. (4)

which is an homogenized version of the local con-stitutive relation D = εE.

For simplicity, we focus on one diagonal co-efficient ε∗ = ε∗kk, with ε∗ = 〈εE · ek〉. Bythe homogeneity of ε(x, ω) in (2), ε∗ depends onthe contrast parameter h = ε1/ε2 and we definem(h) = ε∗/ε2, which is a Herglotz function thatmaps the upper half h–plane to the upper halfplane, and is analytic off (−∞, 0].

The key step in the method is obtaining aStieltjes integral representation for ε∗. This in-tegral representation arises from a resolvent rep-resentation of the electric field E = s(sI −

Γχ1)−1ek, where Γ = ∇(∆−1)∇· acts as a pro-

jection from L2(Ω, P ) onto the Hilbert space ofcurl-free random fields, and ∆−1 is based onconvolution with the free space Green’s functionfor the Laplacian ∆ = ∇2. Consider the func-tion F (s) = 1 − m(h), s = 1/(1 − h), which isanalytic off [0, 1] in the s-plane. Then writingF (s) = 〈χ1[(sI − Γχ1)

−1ek] · ek〉 yields

F (s) =

∫ 1

0

dµ(λ)

s− λ, (5)

where µ(dλ) = 〈χ1Q(dλ)ek ·ek〉 is a positive spec-

tral measure on [0, 1] and Q(dλ) is the (unique)projection valued measure associated with thebounded, self-adjoint operator Γχ1.

Equation (5) is based on the spectral theoremfor the resolvent of the operator Γχ1. It providesa Stieltjes integral representation for the effectivecomplex permittivity ε∗ which separates the com-ponent parameters in s from the complicated geo-metrical information contained in the measure µ.(Extensions of (5) to multicomponent media withε = ε1χ1 + ε2χ2 + ε3χ3 + · · ·+ εnχn involve sev-eral complex variables.) Information about thegeometry enters through the moments

µn =

∫ 1

0

λndµ(λ) = 〈χ1[(Γχ1)nek] · ek〉,

n = 0, 1, 2, . . .. For example, the mass µ0 is givenby µ0 = 〈χ1ek · ek〉 = 〈χ1〉 = φ, where φ is thevolume or area fraction of material of phase 1,and µ1 = φ(1−φ)/d if the material is statisticallyisotropic. In general, µn depends on the (n+ 1)–point correlation function of the medium.

Computing the spectral measure µ for a givencomposite microstructure first involves discretiz-ing a two phase image of the composite into asquare lattice filled with 1’s and 0’s correspond-ing to the two phases. Then the key operator Γχ1,which depends on the geometry via χ1, becomesa self adjoint matrix. The spectral measure maybe calculated directly from the eigenvalues andeigenvectors of this matrix.

4 Applications to Sea Ice

4.1 Percolation Theory

Given a sample of sea ice at temperature Tin degrees Celsius and bulk salinity S in parts

7

-15 C, = 0.033° -3 C, = 0.143° -6 C, = 0.075°φ φ φT = T = T =

0.05 0.10 0.15 0.20 0.25

brine volume fraction φ

2 x 10-10

3 x 10-10

4 x 10-10

1 x 10-10

ver

tica

l per

mea

bil

ity k

(m

)

2

0 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1-15

-14

-13

-12

-11

-10

-9

-8

-7

x = log( φ − φ )y

= l

og

kc

(d) (f )

ve

rtic

al c

orr

ela

tio

n le

ng

th(v

oxe

ls)

(e)

0.05 0.100

brine volume fraction φ

φc

(a) (b) (c)

φc

Figure 5: 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 × 8 × 2 mm sub-volumes (a)–(c) illustrate a pronounced change in the micro-scale

morphology and connectivity of the brine inclusions during warming. (d) Data for the vertical fluid permeability

k taken in situ on Arctic sea ice, displayed on a linear scale. (e) Divergence of the brine correlation length in the

vertical direction as the percolation threshold is approached from below. (f) Comparison of Arctic permeability

data in the critical regime (25 data points) with percolation theory in (7). 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 φc = 0.05.

per thousand (ppt), the brine volume fractionφ is given (approximately) by the equation ofFrankenstein and Garner,

φ =S

1000

(

49.185

|T |+ 0.532

)

. (6)

As temperature increases for fixed salinity, thevolume fraction φ of liquid brine in the ice alsoincreases. The inclusions become larger andmore connected, as illustrated in Figure 5 (a)–(c),which shows images of the brine phase in sea ice(in gold) obtained from X-ray tomography scansof sea ice single crystals.

As the connectedness of the brine phase in-creases with rising temperature, the ease with

which fluid can flow through sea ice – its fluidpermeability – should increase as well. In fact,sea ice exhibits a percolation threshold, or criticalbrine volume fraction φc, or critical temperatureTc, below which columnar sea ice is effectively im-permeable to vertical fluid fluid flow, and abovewhich the ice is permeable, and increasingly soas temperature rises. This critical behavior offluid transport in sea ice is illustrated in Figure 5(d). The data on the vertical fluid permeabilityk(φ) display a rapid rise just above a thresholdvalue of about φc ≈ 0.05 or 5%, similar to theconductivity (or permeability) in Figure 3 (e).This type of behavior is also displayed by dataon brine drainage, with the effects of drainage

8

Rm

R p

(a) (b) (c)

Figure 6: (a) A powder of large polymer spheres mixed with smaller metal spheres. (b) When the powder is

compressed, its microstrucure is similar to that of sea ice in (c).

shutting down for brine volume fractions belowabout 5%. Roughly speaking, we can refer tothis phenomenon as the on–off switch for fluidflow in sea ice. Through the Frankenstein–Garnerrelation in Equation 6, the critical brine volumefraction φc ≈ 0.05 corresponds to a critical tem-perature Tc ≈ −5C, for a typical salinity of 5ppt. Thus, this important threshold behavior hasbecome known as the rule of fives.

In view of this type of critical behavior, it isreasonable to try and find a percolation theo-retic explanation. However, with pc ≈ 0.25 forthe d = 3 cubic bond lattice, it was apparentthat key features of the geometry of the brine mi-crostructure in sea ice were being missed by lat-tices. The threshold φc ≈ 0.05 was identified withthe critical probability in a continuum percola-tion model for compressed powders which exhibitmicrostructural characteristics similar to sea ice.The identification explained the rule of fives, aswell as data on algal growth and snow-ice pro-duction. The compressed powders, shown in Fig-ure 6, were used in the development of so-calledstealthy or radar absorbing composites.

When we applied the compressed powdermodel to sea ice, we had no direct evidence thatthe brine microstructure undergoes a transitionin connectedness at a critical brine volume frac-tion. This lack of evidence was due partly to thedifficulty of imaging and quantitatively charac-terizing the brine inclusions in three dimensions,particularly the thermal evolution of their con-nectivity. Through X-ray computed tomographyand pore structure analysis we have now ana-lyzed critical behavior of the thermal evolutionof brine connectedness in sea ice single crystals,over a temperature range from −18 C to −3 C.

We have mapped three dimensional images of thepores and throats in the brine phase onto graphsof nodes and edges, and analyzed their connectiv-ity as functions of temperature and sample size.Realistic network models of brine inclusions canbe derived from porous media analysis of 3-Dmicro-tomography images. Using finite-size scal-ing techniques largely confirms the rule of fives,and that sea ice is a natural material that exhibitsa strong anisotropy in percolation thresholds.

Now we consider the application of percola-tion theory to understanding the fluid permeabil-ity of sea ice. In the continuum, the permeabil-ity and conductivity exponents e and t can takenon-universal values, and need not be equal, suchas for the three dimensional Swiss cheese model.Continuum models have been studied by map-ping to a lattice with a probability density ψ(g)of bond conductances g. Non-universal behaviorcan be obtained when ψ(g) is singular as g → 0+.However, for a lognormal conductance distribu-tion arising from intersections of lognormally dis-tributed inclusions, as in sea ice, the behavior isuniversal. Thus e ≈ 2 for sea ice.

The permeability scaling factor k0 for sea iceis estimated using critical path analysis. For me-dia with g in a wide range, the overall behavioris dominated by a critical, bottleneck conductancegc, the smallest conductance such that the criticalpath g : g ≥ gc spans the sample. With mostbrine channel diameters between 1.0 mm and 1.0cm, spanning fluid paths have a smallest, charac-teristic radius rc ≈ 0.5 mm, and we estimate k0

by the pipe-flow result rc2/8. Thus

k(φ) ∼ 3 (φ−φc)2 × 10−8 m2, φ→ φ+

c . (7)

In Figure 5 (f), field data with φ in [0.055, 0.15],

9

just above φc ≈ 0.05, are compared with (7),showing close agreement. The striking result thatfor sea ice, e ≈ 2, the universal lattice value inthree dimensions, is due to the general lognormalstructure of the brine inclusion distribution func-tion. The general nature of our results suggeststhat similar types of porous media, such as salineice on extraterrestrial bodies, may also exhibituniversal critical behavior.

4.2 Analytic Continuation

4.2.1 Bounds on the Effective Complex

Permittivity

Bounds on ε∗, or F (s), are obtained by fixings in (5), varying over admissible measures µ (oradmissible geometries), such as those that satisfyonly

µ0 = φ. (8)

and finding the corresponding range of values ofF (s) in the complex plane. Two types of boundson ε∗ are obtained. The first bound R1 assumesonly that the relative volume fractions p1 = φ andp2 = 1−p1 of the brine and ice are known, so that(8) is satisfied. In this case, the admissible set ofmeasures forms a compact, convex set. Since (5)is a linear functional of µ, the extreme values ofF are attained by extreme points of the set ofadmissible measures, which are the Dirac pointmeasures of the form p1δz. The values of F mustlie inside the circle p1/(s− z),−∞ ≤ z ≤ ∞, andthe region R1 is bounded by circular arcs, one ofwhich is parametrized in the F –plane by

C1(z) =p1

s− z, 0 ≤ z ≤ p2. (9)

To display the other arc, it is convenient to usethe auxiliary function

E(s) = 1 −ε1ε∗

=1 − sF (s)

s(1 − F (s)), (10)

which is a Herglotz function like F (s), analytic off[0, 1]. Then in the E–plane, we can parametrizethe other circular boundary of R1 by

C1(z) =p2

s− z, 0 ≤ z ≤ p1. (11)

In the ε∗–plane, R1 has vertices V1 = ε1/(1 −C1(0)) = (p1/ε1 + p2/ε2)

−1 and W1 = ε2(1 −

C1(0)) = p1ε1+p2ε2, and collapses to the interval

(p1/ε1 + p2/ε2)−1 ≤ ε∗ ≤ p1ε1 + p2ε2 (12)

when ε1 and ε2 are real, which are the classicalarithmetic (upper) and harmonic (lower) meanbounds, also called the elementary bounds. Thecomplex elementary bounds (9) and (11) are op-timal and can be attained by a composite of uni-formly aligned spheroids of material 1 in all sizescoated with confocal shells of material 2, and viceversa. These arcs are traced out as the aspect ra-tio varies.

If the material is further assumed to be sta-tistically isotropic, i.e., ε∗ik = ε∗δik, then µ1 =φ(1 − φ)/d must be satisfied as well. A conve-nient way of including this information is to usethe transformation,

F1(s) =1

p1−

1

sF (s). (13)

The function F1(s) is, again, a Herglotz functionwhich has the representation

F1(s) =

∫ 1

0

dµ1(z)

s− z. (14)

The constraint µ1 = φ(1 − φ)/d on F (s) is thentransformed to a restriction of only the mass, orzeroth moment µ1

0 of µ1, with

µ10 = p2/p1d. (15)

Applying the same procedure as for R1 yields aregion R2, whose boundaries are again circulararcs. When ε1 and ε2 are real with ε1 ≥ ε2,the region collapses to a real interval, whoseendpoints are known as the Hashin–Shtrikmanbounds. We remark that higher–order correlationinformation can be conveniently incorporated byiterating (13).

4.2.2 Inverse Homogenization

It has been shown that the spectral measure µ,which contains all geometrical information abouta composite, can be uniquely reconstructed ifmeasurements of the effective permittivity ε∗ areavailable on an arc in the complex s−plane. Ifthe component parameters depend on frequencyω (not to be confused with realizations of the ran-dom medium above), variation of ω in an interval

10

(b) percolation model p = 0.5(a) percolation model p = 0.25

λλ0 0.5 1 0 0.5 1

(d) spectral measure p = 0.5(c) spectral measure p = 0.25

µ(λ)µ(λ)

0

1

2

3

0.2

0.4

0.6

0.8

0

Figure 7: Realizations of the two dimensional lattice percolation model are shown in (a) and (b), and the

corresponding spectral functions (averaged over 5000 random realizations) are shown in (c) and (d). In (c),

there is a spectral gap around λ = 1, indicating the lack of long-range order or connectedness. The gap collapses

in (d) when the percolation threshold of p = pc = 0.5 has been reached, and the system exhibits long-range

connectedness. Note the difference in vertical scale in the graphs in (c) and (d).

(ω1, ω2) gives the required data. Reconstructingµ can be reduced to an inverse potential problem.Indeed, F (s) admits a representation through alogarithmic potential Φ of the measure µ,

F (s) =∂Φ

∂s, Φ(s) =

∫ 1

0

ln |s− z| dµ(z), (16)

where ∂/∂s = (∂/∂x − i ∂/∂y) The potential Φsolves the Poisson equation ∆Φ = −ρ, where ρ(z)is a density on [0, 1]. A solution to the forwardproblem is given by the Newtonian potential withµ(dz) = ρ(z)dz. The inverse problem is to findρ(z) (or µ) given values of Φ, ∂Φ /∂n, or ∇Φ.

The inverse problem is extremely ill-posed andrequires regularization to develop a stable numer-ical algorithm.

When frequency ω varies across the interval(ω1, ω2), the complex parameter s traces an arc Cin the s-plane. Let A be the integral operator in

(16), Aµ = ∂∂s

∫ 1

0ln |s−λ| dµ(λ), mapping the set

of measures M[0, 1] on the unit interval onto theset of derivatives of complex potentials definedon a curve C. To construct the solution we con-sider the problem of minimizing ‖Aµ− F ‖2 overµ ∈ M, where ‖·‖ is the L2(C)−norm, F (s) is thethe measured data, and s ∈ C. The solution does

11

( ) ( )( )

Figure 8: (a) Shading shows the solution to Laplace’s equation within the Antarctic MIZ (ψ) on 26 August

2010, and black curves show MIZ width measurements following the gradient of ψ (only a subset shown for

clarity). (b) Same as (a), but for the Arctic MIZ on 29 August 2010. (c) Width of the July-September MIZ

for 1979-2011 (red curve). Percentiles of daily MIZ widths are shaded dark gray (25th to 75th) and light gray

(10th to 90th). Results are based on analysis of satellite-derived sea ice concentrations from the National Snow

and Ice Data Center.

not depend continuously on the data, and regu-larization based on constrained minimization isneeded. Instead of minimizing ‖Aµ − F ‖2 overall functions in M, it is performed over a con-vex subset satisfying J(µ) ≤ β, for a stabilizingfunctional J(µ) and some β > 0. The advantage

of using quadratic J(µ) = ‖Lµ‖2, is the linearityof the corresponding Euler equation resulting inefficiency of the numerical schemes. However, thereconstructed solution necessarily possesses a cer-tain smoothness. Nonquadratic stabilization im-poses constraints on the variation of the solution.The total variation penalization, as well as a non-negativity constraint, does not imply smoothness,permitting more general recovery, including theimportant Dirac measures.

We have also solved exactly a reduced inversespectral problem by bounding the volume frac-tion of the constituents, an inclusion separationparameter q, and the spectral gap of Γχ1. We de-veloped an algorithm based on the Mobius trans-formation structure of the forward bounds whoseoutput is a set of algebraic curves in parameterspace bounding regions of admissible parametervalues. These results advance the development oftechniques for characterizing the microstructureof composite materials, and have been applied to

sea ice to demonstrate electromagnetically thatthe brine inclusion separations vanish as the per-colation threshold is approached.

5 Geometry of the Marginal Ice Zone

Dense pack ice transitions to open ocean over aregion of broken ice termed the marginal ice zone(MIZ)—a highly dynamic region where the icecover lies close to an open ocean boundary andintense atmosphere–ice–ocean interactions takeplace. The width of the MIZ is a fundamen-tal length scale for polar dynamics in part be-cause it represents the distance over which oceanwaves and swell penetrate into the sea ice cover.Wave penetration can break a smooth ice layerinto floes, meaning the MIZ acts as a buffer zonethat protects the stable morphology of the innerice. Waves also promote the formation of pancakeice, as shown in Figure 4. Moreover, the width ofthe MIZ is an important spatial dimension of themarine polar habitat and impacts human accessi-bility to high latitudes. Using a conformal map-ping method to quantify MIZ width (see below),a dramatic 39% widening of the summer ArcticMIZ, based on three decades of satellite–deriveddata (1979-2012), was reported.

12

Fra

cta

l D

ime

nsi

on

100 1 2 3 4

1010 10 105

10

Area (m )2

2

1

(c)

(d)

(e)

(a) (b)

100 1 2 3 4

1010 10

10

1

2

3

4

10

10

10

105

1010

0

Area (m )

Pe

rim

ete

r (

m)

2

Figure 9: (a) Area–perimeter data for 5,269 Arctic melt ponds, plotted on logarithmic scales. The slope

transitions from about 1 to 2 at a critical length scale of around 100 square meters. (b) Melt pond fractal

dimension D as a function of area A, computed from the data in (a), showing the transition to complex ponds

with increasing length scale. Ponds corresponding to the three red stars in (a), from left to right, are shown in

(c), (d), and (e), respectively. The transitional pond in (d) has horizontal scale of about 30 m.

Challenges associated with objective measure-ment of the MIZ width include the MIZ shape,which is in general not geodesically convex, asillustrated by the example shaded white in Fig-ure 8 (a). Sea ice concentration (c) is usedhere to define the MIZ as a body of marginalice (0.15 ≤ c ≤ 0.80) adjoining both pack ice(c > 0.80) and sparse ice (c < 0.15). To define anobjective MIZ width applicable to such shapes, anidealized sea ice concentration field ψ(x, y) satis-fying Laplace’s equation within the MIZ,

∇2ψ = 0, (17)

was introduced. We use (x, y) to denote a point intwo dimensional space, and it is understood thatwe are working on the spherical Earth. Bound-ary conditions for (17) are ψ = 0.15 where MIZborders a sparse ice region and ψ = 0.80 wherethe MIZ borders a pack ice region. The solutionsto (17) for the examples in Figure 8 (a) and (b)are illustrated by color shading. Any curve γ or-thogonal to the level curves of ψ and connectingtwo points on the MIZ perimeter (a black fieldline through the gradient field ∇ψ as in Figure8 (b)) is contained in the MIZ, and its lengthprovides an objective measure of MIZ width (`).Defined in this way, ` is a function of distancealong the MIZ perimeter (s) from an arbitrarystarting point, and this dependence is denoted` = `(s). Analogous applications of Laplace’sequation have been introduced in medical imag-ing to measure the width or thickness of humanorgans.

Derivatives in (17) were numerically approx-imated using second-order finite differences, andsolutions were obtained in the data’s native stere-ographic projection since solutions of Laplace’sequation are invariant under conformal mapping.For a given day and MIZ, a summary measure ofMIZ width (w) can be defined by averaging ` withrespect to distance along the MIZ perimeter:

w =1

LM

∫

M

`(s) ds, (18)

where M is the closed curve defining the MIZperimeter and LM is the length of M . Averagingw over July-September of each available year re-veals the dramatic widening of the summer MIZ,as illustrated in Figure 8 (c).

6 Geometry of Arctic Melt Ponds

From the first appearance of visible pools of wa-ter, often in early June, the area fraction φ of seaice covered by melt ponds can increase rapidly toover 70% in just a few days. Moreover, the accu-mulation of water at the surface dramatically low-ers the albedo where the ponds form. There is acorresponding critical drop-off in average albedo.The resulting increase in solar absorption in theice and upper ocean accelerates melting, possiblytriggering ice-albedo feedback. Similarly, an in-crease in open water fraction lowers albedo, thusincreasing solar absorption and subsequent melt-ing. The spatial coverage and distribution of melt

13

(a) (b) (c)

(d) (e) (f )

Figure 10: Evolution of melt pond connectivity and color coded connected components. (a) disconnected ponds,

(b) transitional ponds, (c) fully connected melt ponds. The bottom row shows the color coded connected

components for the corresponding image above: (d) no single color spans the image, (e) the red phase just

spans the image, (f) the connected red phase dominates the image. The scale bars represent 200 meters for (a)

and (b), and 35 meters for (c).

ponds on the surface of ice floes and the open wa-ter between the floes thus exerts primary controlof ice pack albedo and the partitioning of solarenergy in the ice-ocean system. Given the criti-cal role of ice-albedo feedback in the recent lossesof Arctic sea ice, ice pack albedo and the forma-tion and evolution of melt ponds are of significantinterest in climate modeling.

While melt ponds form a key component of theArctic marine environment, comprehensive obser-vations or theories of their formation, coverage,and evolution remain relatively sparse. Availableobservations of melt ponds show that their arealcoverage is highly variable, particularly for firstyear ice early in the melt season, with rates ofchange as high as 35% per day. Such variabil-ity, as well as the influence of many competingfactors controlling melt pond and ice floe evolu-tion, makes the incorporation of realistic treat-ments of albedo into climate models quite chal-lenging. Small and medium scale models of meltponds which include some of these mechanismshave been developed, and melt pond parameteri-zations are being incorporated into global climatemodels.

The surface of an ice floe is viewed here as a two

phase composite of dark melt ponds and whitesnow or ice. The onset of ponding and the rapidincrease in coverage beyond the initial thresh-old is similar to critical phenomena in statisticalphysics and composite materials. It is natural,therefore, to ask if the evolution of melt pond ge-ometry exhibits universal characteristics that donot necessarily depend on the details of the driv-ing mechanisms in numerical melt pond models.Fundamentally, the melting of Arctic sea ice is aphase transition phenomenon, where a solid turnsto liquid, albeit on large regional scales and overa period of time which depends on environmen-tal forcing and other factors. We thus look forfeatures of melt pond evolution that are mathe-matically analogous to related phenomena in thetheories of phase transitions and composite mate-rials. As a first step in this direction, we considerthe evolution of geometric complexity of Arcticmelt ponds.

By analyzing area−perimeter data from hun-dreds of thousands of melt ponds, we have dis-covered an unexpected separation of scales, wherethe pond fractal dimension D exhibits a transi-tion from 1 to 2 around a critical length scaleof 100 square meters in area, as shown in Fig-

14

ure 9. Small ponds with simple boundaries co-alesce or percolate to form larger connected re-gions. Pond complexity increases rapidly throughthe transition region and reaches a maximum forponds larger than 1000 m2 whose boundaries re-semble space filling curves with D ≈ 2. Theseconfigurations affect the complex radiation fieldsunder melting sea ice, the heat balance of seaice and the upper ocean, under-ice phytoplanktonblooms, biological productivity, and biogeochem-ical processes.

Melt pond evolution also appears to exhibita percolation threshold, where one phase in acomposite becomes connected on macroscopicscales as some parameter exceeds a critical value.An important example of this phenomenon inthe microphysics of sea ice (discussed above),which is fundamental to the process of melt ponddrainage, is the percolation transition exhibitedby the brine phase in sea ice, or the rule of fives

discussed in section 4.1. When the brine volumefraction of columnar sea ice exceeds about 5%, thebrine phase becomes macroscopically connectedso that fluid pathways allow flow through theporous microstructure of the ice. Similarly, evencasual inspection of the aerial photos in Figure10, shows that the melt pond phase of sea iceundergoes a percolation transition where discon-nected ponds evolve into much larger scale con-nected structures with complex boundaries. Con-nectivity of melt ponds promotes further meltingand break-up of floes, as well as horizontal trans-port of meltwater and drainage through cracks,leads, and seal holes.

Further Reading

1. Avellaneda, M. and A. Majda. 1989, Stieltjesintegral representation and effective diffusivitybounds for turbulent transport. Phys. Rev.Lett. 62:753–755.

2. Bergman, D. J. and D. Stroud. 1992, Physi-cal properties of macroscopically inhomogeneousmedia. Solid State Phys. 46:147–269.

3. Cherkaev, E. 2001, Inverse homogenization forevaluation of effective properties of a mixture.Inverse Problems 17:1203–1218.

4. Eisenman, I. and J. S. Wettlaufer. 2009, Nonlin-ear threshold behavior during the loss of Arcticsea ice. Proc. Natl. Acad. Sci. 106(1):28–32.

5. Feltham, D. L. 2008, Sea ice rheology. Ann. Rev.Fluid Mech. 40:91–112.

6. Flocco, D., D. L. Feltham, and A. K. Turner.2010, Incorporation of a physically based meltpond scheme into the sea ice component of aclimate model. J. Geophys. Res. 115:C08,012(14 pp.), doi:10.1029/2009JC005,568.

7. Golden, K. M. 2009, Climate change and themathematics of transport in sea ice. Notices ofthe American Mathematical Society 56(5):562–584 and issue cover.

8. Hohenegger, C., B. Alali, K. R. Steffen, D. K.Perovich, and K. M. Golden. 2012, Transition inthe fractal geometry of Arctic melt ponds. TheCryosphere 6:1157–1162.

9. Hunke, E. C. and W. H. Lipscomb. 2010, CICE:the Los Alamos Sea Ice Model Documentationand Software Users Manual Version 4.1 LA-CC-06-012, t-3 Fluid Dynamics Group, Los AlamosNational Laboratory.

10. Milton, G. W. 2002, Theory of Composites.Cambridge: Cambridge University Press.

11. Orum, C., E. Cherkaev, and K. M. Golden.2011, Recovery of inclusion separations instrongly heterogeneous composites from effectiveproperty measurements. Proc. Roy. Soc. Lon-don A pp. doi:10.1098/rspa.2011.0527, pp. 1–27, online November 16.

12. Perovich, D. K., J. A. Richter-Menge, K. F.Jones, and B. Light. 2008, Sunlight, water, andice: Extreme Arctic sea ice melt during the sum-mer of 2007. Geophys. Res. Lett. 35:L11,501,doi:10.1029/2008GL034,007.

13. Stauffer, D. and A. Aharony. 1992, Introductionto Percolation Theory, Second Edition. London:Taylor and Francis Ltd.

14. Stroeve, J., M. M. Holland, W. Meier, T. Scam-bos, and M. Serreze. 2007, 2007: Arctic sea icedecline: Faster than forecast. Geophys. Res.Lett. 34:L09,591, doi: 10.1029/2007GL029,703.

15. Strong, C. and I. G. Rigor. 2013, Arcticmarginal ice zone trending wider in summerand narrower in winter. Geophys. Res. Lett.40(18):4864–4868.

16. Thomas, D. N. and G. S. Dieckmann, editors.2009, Sea Ice, 2nd Edition. Oxford: Wiley-Blackwell.

17. Thompson, C. J. 1988, Classical EquilibriumStatistical Mechanics. Oxford: Oxford Univer-sity Press.

18. Torquato, S. 2002, Random Heterogeneous Ma-terials: Microstructure and Macroscopic Proper-ties. New York: Springer-Verlag.

19. Untersteiner, N. 1986, The Geophysics of SeaIce. New York: Plenum.

20. Washington, W. and C. L. Parkinson. 2005,An Introduction to Three-Dimensional ClimateModeling, 2nd Edition. Herndon, VA: Univer-sity Science Books.