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Star patterns on lake ice

Victor C. Tsai

1,

*

and J. S. Wettlaufer

2,†

1

Department of Earth & Planetary Sciences, Harvard University, Cambridge, Massachusetts 02138, USA

2

Department of Geology & Geophysics and Department of Physics, Yale University, New Haven, Connecticut 06520-8109, USA

!Received 14 February 2007; published 18 June 2007

"

Star patterns, reminiscent of a wide range of diffusively controlled growth forms from snowﬂakes to

Saffman-Taylor ﬁngers, are ubiquitous features of ice-covered lakes. Despite the commonality and beauty of

these “lake stars,” the underlying physical processes that produce them have not been explained in a coherent

theoretical framework. Here we describe a simple mathematical model that captures the principal features of

lake-star formation; radial ﬁngers of !relatively warm" water-rich regions grow from a central source and

evolve through a competition between thermal and porous media ﬂow effects in a saturated snow layer

covering the lake. The number of star arms emerges from a stability analysis of this competition and the

qualitative features of this meter-scale natural phenomenon are captured in laboratory experiments.

DOI: 10.1103/PhysRevE.75.066105 PACS number!s": 82.40.Ck, 45.70.Qj, 92.40.Vq

I. INTRODUCTION

The scientiﬁc study of the problems of growth and form

occupies an anomalously broad set of disciplines. Whether

the emergent patterns are physical or biological in origin,

their quantitative description presents many challenging and

compelling issues in, for example, applied mathematics #1$,

biophysics #2$, condensed matter #3$, and geophysics #4$,

wherein the motion of free boundaries is of central interest.

In all such settings a principal goal is to predict the evolution

of a boundary that is often under the inﬂuence of an insta-

bility. Here we study a variant of such a situation that occurs

naturally on the frozen surfaces of lakes.

Lakes commonly freeze during a snowfall. When a hole

forms in the ice cover, relatively warm lake water will ﬂow

through it and hence through the snow layer. In the process

of ﬂowing through and melting the snow, this warm water

creates dark regions. The pattern so produced looks starlike

!see Fig. 1", and we refer to it as a “lake star.” These com-

pelling features have been described qualitatively a number

of times !e.g., #5–7$", but work on the formation process

itself has been solely heuristic. Knight #5$ outlines a number

of the physical ideas relevant to the process, but does not

translate them into a predictive framework to model ﬁeld

observations. Knight’s main idea is that locations with faster

ﬂow rates melt preferentially, leading to even faster ﬂow

rates and therefore to an instability that results in ﬁngers.

This idea has features that resemble those of many other

instabilities such as, for example, those observed during the

growth of binary alloys #8$, in the ﬂow of water through a

rigid hot porous medium #9$, or in more complex geomor-

phological settings #10$, and we structure our model accord-

ingly.

Katsaros #6$ and Woodcock #7$ attribute the holes from

which the stars emanate and the patterns themselves to ther-

mal convection patterns within the lake, but do not measure

or calculate their nature. However, often the holes do not

exhibit a characteristic distance between them but rather

form from protrusions !e.g., sticks that poke through the ice

surface" #5$ and stars follow, thereby ruling out a convective

mechanism as being necessary to explain the phenomena.

The paucity of literature on this topic provides little more

than speculation regarding the puncturing mechanism but

lake stars are observed in all of these circumstances. There-

fore, while hole formation is necessary for lake-star forma-

tion, its origin does not control the mechanism of pattern

formation, which is the focus of the present work.

II. THEORY

The water level in the hole is higher than that in the wet

snow-slush layer #5$ and hence we treat this warm water #11$

region as having a constant height above the ice, or equiva-

lently a constant pressure head, which drives ﬂow of water

through the slush layer, which we treat as a Darcy ﬂow of

water at 0 °C. We model the temperature ﬁeld within the

liquid region with an advection-diffusion equation and im-

pose an appropriate !Stefan" condition for energy conserva-

tion at the water-slush interface. The water is everywhere

incompressible. Finally, the model is closed with an outer

boundary condition at which the pressure head is assumed

known.

Although we lack in situ pressure measurements, circular

water-saturated regions !a few meters in radius" are observed

around the lake stars. Hence, we assume that the differential

pressure head falls to zero somewhere in the vicinity of this

circular boundary. The actual boundary at which the differ-

ential pressure head is zero is not likely to be completely

uniform !as in Fig. 4 of Knight #5$", but treating it as uniform

is a good approximation in the linear regime of our analysis.

Finally, we treat the ﬂow as two dimensional. Thus, although

the water in direct contact with ice must be at 0 ° C, we

consider the depth-averaged temperature, which is above

freezing. Additionally, the decreasing pressure head in the

radial direction must be accompanied by a corresponding

drop in water level. Therefore, although the driving force is

more accurately described as deriving from an axisymmetric

gravity current, the front whose stability we assess is con-

*

Electronic address: vtsai@fas.harvard.edu

†

Electronic address: john.wettlaufer@yale.edu

PHYSICAL REVIEW E 75, 066105 !2007"

1539-3755/2007/75!6"/066105!5" ©2007 The American Physical Society066105-1

trolled by the same essential physical processes that we

model herein. Our analysis could be extended to account for

these three-dimensional effects.

The system is characterized by the temperature T, a Darcy

ﬂuid velocity u, pressure p, and an evolving liquid-slush

interface a. The liquid properties are

!

!thermal diffusivity",

C

P

!speciﬁc heat at constant pressure", and

"

!dynamic vis-

cosity" and the slush properties are # !permeability",

$

!solid

fraction", and L !latent heat". We nondimensionalized the

equations of motion by scaling the length, temperature, pres-

sure, and velocity with r

0

, T

0

, p

0

, and #p

0

/

"

r

0

, respectively.

Thus, our model consists of the following system of dimen-

sionless equations:

!

%

!

t

+ u · !

%

=

&

"

2

%

, r

i

' r ' a!

(

,t", !1"

%

= 0, a!

(

,t" ' r ' 1, !2"

p = 1, r

i

' r ' a!

(

,t", !3"

"

2

p = 0, a!

(

,t" ' r ' 1, !4"

! · u = 0, r

i

' r ' a!

(

,t", !5"

%u%

a

−

= %u%

a

+

, r = a!

(

,t", !6"

u = − !p, a!

(

,t" ' r ' 1, !7"

with boundary conditions

a

˙

= −

&

S

!

%

, r = a!

(

,t", !8"

%

=

&

1,

r = r

i

0, r = a!

(

,t"

0,

r = 1,

'

!9"

and

p =

&

1,

r = r

i

1, r = a!

(

,t"

0,

r = 1,

'

!10"

where !1" describes the temperature evolution in the liquid,

!4" and !5" describe mass conservation with a Darcy ﬂow !7"

in the slush, !8" is the Stefan condition, and !9" and !10" are

the temperature and pressure boundary conditions, respec-

tively !see Fig. 2". Note that !3" and !5" can both be satisﬁed

since the liquid region has an effectively inﬁnite permeabil-

ity.

The dimensionless parameters

&

and S of the system are

given by

&

(

!

u

0

r

0

and S (

$

L

C

P

T

0

, !11"

which describe an inverse Péclet number and a Stefan num-

ber, respectively. Because the liquid must be less than or

equal to 4 ° C, we make the conservative estimates that T

0

'4 ° C,

$

)0.3, and use the fact that L /C

P

)80 °C from

which we see that S) 6* 1. Using

!

)10

−7

m

2

s

−1

and the

ﬁeld observations of Knight #5$ to constrain u

0

!1 'u

0

'10 cm/h" and r

0

!0.3' r

0

'3 m", we ﬁnd that

&

'0.1+ 1.

We therefore employ the quasistationary !S*1" and large

Péclet number !

&

+1" approximations, and hence Eqs.

!1"–!10" are easily solved for a purely radial ﬂow with cy-

lindrical symmetry !no

(

dependence" and circular liquid-

slush interface a

0

. This !boundary layer" solution is

u = ur

ˆ

= −

1

ln!a

0

"

1

r

r

ˆ

, r

i

' r ' 1, !12"

FIG. 1. !Color online" Typical

lake-star patterns. The branched

arms are approximately 1 m in

length. Quonnipaug Lake, Guil-

ford, Connecticut, 8 March 2006.

(1), (3) and (5)

(2), (4) and (7)

r = r

i

r = r

0

r = a

BC at r=r

i

(9a) and (10a)

BC at r=r

0

(9c) and (10c

)

BC at r=a

(6), (8), (9b)

and (10b)

Field

Equations

Boundary Conditions

FIG. 2. Schematic of the geometry of the model. The perspec-

tive is looking down on a nascent star. The equations !refer to text

for numbering" are shown in the domains of the system where they

are applicable.

VICTOR C. TSAI AND J. S. WETTLAUFER PHYSICAL REVIEW E 75, 066105 !2007"

066105-2

p

b

=

ln!r"

ln!a

0

"

, r ) a

0

, !13"

%

0

= 1 −

*

r

a

0

+

!1/

&

"#−1/ln!a

0

"+2

&

$

, r ' a

0

, !14"

Sa

0

a

˙

0

− 1/ln! a

0

" + 2

&

= 1, !15"

where Eq. !15" has an approximate implicit solution for a

0

given by

a

0

2

4

−

1

2

a

0

2

ln!a

0

" =

t

S

. !16"

We perform a linear stability analysis around this quasi-

steady cylindrically symmetrical ﬂow. Proceeding in the

usual way, we allow for scaled perturbations in

%

and a with

scaled wave number k

!

=

&

k, nondimensional growth rate

,

,

and amplitudes f!r" and g, respectively. Keeping only terms

linear in

&

, 1/ S, and g, we solve !4" subject to !10", substitute

into !6", and satisfy !5" and !1". This gives the nondimen-

sional growth rate !

,

" as a function of scaled wave number

!k

!

":

,

=

1

2a

0

ln

2

!a

0

"S

#

,

1 + 4k

!

2

ln

2

!a

0

" − 1$

*

a

0

− k

!

ln!a

0

"

− 1

+

.

!17"

Equation !17" can be approximated in 0- x. 1 as

,

)

a

0

ln

2

!a

0

"S

x!1 − x", !18"

where x(−k

!

ln!a

0

"/ a

0

.

The stability curve !17" and the approximation !18" are

plotted in Fig. 3. The essential features of !17" are a maxi-

mum in the range 0' k

!

'a

0

/ln!a

0

", zero growth rate at k

!

=a

0

/ln!a

0

", and a linear increase in stability with k

!

for large

k

!

. The long-wavelength cutoff is typical of systems with a

Péclet number, here with the added effect of latent heat em-

bodied in the Stefan number. This demonstrates the compe-

tition between the advection and diffusion of heat and mo-

mentum !in a harmonic pressure ﬁeld", the former driving

the instability and the latter limiting its extent. The maximum

growth rate occurs at approximately

k

max

!

)

a

0

− 2 ln!a

0

"

, !19"

with !nondimensional" growth rate

,

max

)

a

0

4S ln

2

!a

0

"

. !20"

Translating !19" and !20" back into dimensional quanti-

ties, we ﬁnd that the most unstable mode has angular size

given by

(

deg

=

720 °

!

u

0

r

0

*

r

0

a

0

+

ln

*

r

0

a

0

+

, !21"

and has growth rate given by

,

dim

=

u

0

4Sr

0

ln

2

!r

0

/a

0

"

*

a

0

r

0

+

. !22"

III. EXTRACTING INFORMATION

FROM FIELD OBSERVATIONS

Field observations of lake stars cannot be controlled. A

reasonable estmate for r

0

is the radius of the wetted !snow"

region around the lake stars, and observations #5–7$ bound

the value as 1.5.r

0

.4 m. This is simply because if there

were signiﬁcant excess pressure at this point then the wetting

front would have advanced further. However, it is also pos-

sible that the effective value of r

0

, say r

0

ef f

, is less than this,

either because the wetted radius is smaller earlier in the star

formation process or because the ambient pressure level is

reached at smaller radii. Here, we take a

0

to be the radius of

the roughly circular liquid-ﬁlled region at the center of the

lake star !r

!

" as the best approximation during the initial

stages of star formation !see Fig. 4". Field observations show

that 0.1. r

!

.0.5 m, #5–7$ and hence 0.07.r

!

/r

0

.0.15.

We note that Eqs. !21" and !22" are more sensitive to a

0

/r

0

than a

0

or r

0

independently #12$. With this interpretation of

r

0

we ﬁnd a reasonable estimate of u

0

as 1.4/ 10

−5

.u

0

.2.8/ 10

−5

m/ s. Using these parameter values, the most

unstable mode should have wavelength between 8° and 130°.

Letting the number of branches be N = 360° /

(

deg

, then 3

'N '45, and we clearly encompass the observed values for

lake stars !4 ' N' 15", but note that values !N )15" are

never seen in the ﬁeld.

Despite the dearth of ﬁeld observations, many qualitative

features embolden our interpretation. For example, the stars

with larger values of a

0

/r

0

have a larger number of branches.

Moreover, for any value of a

0

/r

0

, our analysis predicts an

increase in N with r

0

and u

0

. Indeed, u

0

increases with p

0

!higher water height within the slush layer" and # !less well-

packed snow". Therefore, we ascribe some of the variability

0 0.5 1 1.5 2 2.5

3

−6

−5

−4

−3

−2

−1

0

1

k’

σ

−a

0

/ln(a

0

)

a

0

/ln

2

(a

0

)

a

0

= 0.06

a

0

= 0.5

approx

FIG. 3. !Color online" Stability curve: Nondimensional growth

rate

,

versus nondimensional wave number k

!

. Scales for the axes

are given at the upper left !

,

axis" and the lower right corners !k

!

axis".

,

is plotted for the range of plausible a

0

#dot-dashed !blue"

and dashed !red" curves$ and for the approximation !18" #solid

!green" curve$.

STAR PATTERNS ON LAKE ICE PHYSICAL REVIEW E 75, 066105 !2007"

066105-3

among ﬁeld observations to variations in these quantities

!which have not been measured in the ﬁeld" and the remain-

der to nonlinear effects. Because the dendritic arms are ob-

served long after onset and are far from small perturbations

to a radially symmetric pattern, as one might see in the initial

stages of the Saffman-Taylor instability, the process involves

nonlinear cooperative phenomena. Hence, our model should

only approximately agree with observations. Although a rig-

orous nonlinear analysis of the long-term star evolution pro-

cess !e.g., #3$" may more closely mirror ﬁeld observations,

the present state of the latter does not warrant that level of

detail. Instead, we examine the model physics through a

simple proof of concept experiment described presently.

IV. DEMONSTRATING LAKE STARS

IN THE LABORATORY

A 30-cm-diameter circular plate is maintained below

freezing !) −0.5 ° C", and on top of this we place a

0.5–1 cm deep layer of slush through which we ﬂow 1 °C

water. Given the technical difﬁculties associated with its pro-

duction, the grain size, and hence the permeability of the

slush layer is not a controlled variable. This fact inﬂuences

our results quantitatively. In 14 runs we varied the initial size

of the water-ﬁlled central hole !a

0

", that of the circular slush

layer !r

0

", and the ﬂow rate !Q", which determines u

0

. The

ﬂow rate is adjusted manually so that the water level !h

0

" in

the central hole remains constant #13$. Fingering is observed

in every experimental run, and hence we conclude that ﬁn-

gers are a robust feature of the system. Two distinct types of

ﬁngering are observed: small-scale ﬁngering !see Fig. 5" that

forms early in an experimental run, and larger channel-like

ﬁngers !see Fig. 6" that are ubiquitous at later times and

often extend from the central hole to the outer edge of the

slush. Since the channel-like ﬁngers provide a direct path for

water to ﬂow, effectively shorting the Darcy ﬂow within the

slush, their subsequent dynamics are not directly analogous

to those in natural lake stars. However, in all runs, the initial

small-scale ﬁngers have the characteristics of lake stars and

hence we focus upon them. We note that, because the larger

channel-like ﬁngers emerge out of small-scale ﬁngers, they

likely represent the nonlinear growth of the linear modes of

instability, a topic left for future study. Finally, we measure

the distance between ﬁngers !d

f

", so that for each experiment

we can calculate u

0

=Q /!2

0

r

0

h

0

",

(

calc

(

(

deg

, from Eq. !21",

and

(

obs

=180° d

f

/!

0

a

0

", and we can thereby compare ex-

periment, theory, and ﬁeld observations.

In Fig. 7 we plot

(

obs

versus

(

calc

for the various ﬁeld

0 100 200 300 400 50

0

−50

0

50

100

150

φ

ca

l

c

(deg)

φ

obs

(

deg

)

FIG. 7. !Color online" Comparison of theory, experiment, and

ﬁeld observations. Circles are ﬁeld observations !cyan, best con-

strained ﬁeld observation; black, range of plausible ﬁeld observa-

tions", triangles are experimental results !blue upward-pointing tri-

angles were unambiguous; red left-pointing triangles have channels

but show no clear small-scale ﬁngers, so channel spacing is taken

for d

f

; green right-pointing triangles were compromised by the

quality of the images". Errors are approximately 0.3 cm, 0.5 cm,

2 mm, 5 ml/min, and 0.2 cm !respectively" for the ﬁve measured

quantities. All experimental results thus have error bars of at least a

factor of 2 in the x coordinate and 30% in the y coordinate. Typical

error bars are shown on one measurement. The solid red line is the

theoretical prediction; the dotted green line is the best-ﬁt line to the

blue triangles.

r

0

r

LS

r

0

eff

r

l

FIG. 4. Schematic showing r

0

, r

0

ef f

, r

LS

, and r

!

.

FIG. 5. !Color online" Typical experimental run where small-

scale ﬁngers are present. For scale, the nozzle head has diameter of

5 mm.

VICTOR C. TSAI AND J. S. WETTLAUFER PHYSICAL REVIEW E 75, 066105 !2007"

066105-4

observations for which we have estimates of parameters, the

laboratory experiments described above, and the model #Eq.

!21"$. There is a large amount of scatter in both the experi-

mental and observational data, and the data do not lie on the

one-to-one curve predicted by the model. However, the ex-

periments are meant to demonstrate the features of the model

predictions, and the results have the correct qualitative trend

!having a best-ﬁt slope of 0.34". We also attempt to ﬁnd

trends in the experimental data not represented by the model

by comparing y (

(

obs

/

(

calc

versus various combinations of

control parameters ! (x " including r

0

, a

0

, r

0

/a

0

, r

0

u

0

,

r

0

/a

0

ln!r

0

/a

0

", and ln!r

0

/a

0

"/ !a

0

u

0

". For all plots of y vs x,

our model predicts a zero slope !and y intercept of 1". A

nonrandom dependence of y on x would point to failure of

some part of our model. Thus, to test the validity of our

model, we perform signiﬁcance tests on all nonﬂagged data

with the null hypothesis being a nonzero slope. In all cases,

the null hypothesis is accepted !not rejected" at the 95% con-

ﬁdence level. Thus, although the agreement is far from per-

fect, the simple model captures all of the signiﬁcant trends in

the experimental data.

V. CONCLUSIONS

By generalizing and quantifying the heuristic ideas of

Knight #5$, we have constructed a theory that is able to ex-

plain the radiating ﬁngerlike patterns on lake ice that we call

lake stars. The model yields a prediction for the wavelength

of the most unstable mode as a function of various physical

parameters that agrees with ﬁeld observations. Proof of con-

cept experiments revealed the robustness of the ﬁngering

pattern, and to leading order the results also agree with the

model. There is substantial scatter in the data, and the overall

comparison between ﬁeld observations, model, and experi-

ment demonstrates the need for a comprehensive measure-

ment program and a fully nonlinear theory which will yield

better quantitative comparisons. However, the general pre-

dictions of our theory capture the leading-order features of

the system.

ACKNOWLEDGMENTS

We thank K. Bradley and J. A. Whitehead for laboratory

and facilities support and D. H. Rothman for helpful com-

ments. This research, which began at the Geophysical Fluid

Dynamics summer program at the Woods Hole Oceano-

graphic Institution, was partially funded by National Science

Foundation !NSF" Grant No. OCE0325296, NSF Grant No.

OPP0440841 !J.S.W.", and Department of Energy Grant No.

DE-FG02-05ER15741 !J.S.W.". V. C. T. acknowledges ﬁnan-

cial support from NSF.

#1$ T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, J. Comput.

Phys. 169, 302 !2001".

#2$ M. P. Brenner, L. S. Levitov, and E. O. Budrene, Biophys. J.

74, 1677 !1998"; H. Levine and E. Ben-Jacob, Phys. Biol. 1,

14 !2004".

#3$ M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851

!1993"; I. S. Aranson and L. S. Tsimring, ibid. 78, 641 !2006".

#4$ N. Goldenfeld, P. Y. Chan, and J. Veysey, Phys. Rev. Lett. 96,

254501 !2006"; M. B. Short, J. C. Baygents, and R. E. Gold-

stein, Phys. Fluids 18, 083101 !2006".

#5$ C. A. Knight, in Structure and Dynamics of Partially Solidiﬁed

Systems, edited by D. E. Loper !Martinus Nijhoff, Dordrecht,

1987", pp. 453–465.

#6$ K. B. Katsaros, Bull. Am. Meteorol. Soc. 64, 277 !1983".

#7$ A. H. Woodcock, Limnol. Oceanogr. 10, R290 !1965".

#8$ M. G. Worster, Annu. Rev. Fluid Mech. 29, 91

!1997".

#9$ S. Fitzgerald and A. W. Woods, Nature !London" 367, 450

!1994".

#10$ N. Schorghofer, B. Jensen, A. Kudrolli, and D. H. Rothman, J.

Fluid Mech. 503, 357 !2004".

#11$ A ﬁnite body of fresh water cooled from above will have a

maximum below ice temperature of 4 °C.

#12$ For the later stages of growth, clearly in the nonlinear regime

not treated presently, a

0

may also be interpreted as the radius

of the lake star !r

LS

". Field observations show that 1.r

LS

.2 m #5–7$ and hence 0.3 . r

LS

/r

0

.0.6.

#13$ In many of the runs, we begin the experiment without the

central hole. In practice, however, the ﬁrst few drops of warm

water create a circular hole with radius one to three times the

radius of the water nozzle !0.5'a

0

'1.0 cm". It is signiﬁ-

cantly more difﬁcult to prepare a uniform permeability sample

with a circular hole initially present; these runs are therefore

more difﬁcult to interpret.

FIG. 6. !Color online" Typical run where channels form. This

picture is taken from the underside. Note: part of the slush broke off

when it was ﬂipped to image it. The ruler scale is in centimeters.

STAR PATTERNS ON LAKE ICE PHYSICAL REVIEW E 75, 066105 !2007"

066105-5