Content uploaded by J. S. Wettlaufer
Author content
All content in this area was uploaded by J. S. Wettlaufer
Content may be subject to copyright.
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