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# Star patterns on lake ice

Authors:
• Yale University and Nordic Institute for Theoretical Physics

## Abstract and Figures

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.
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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., #57$", but work on the formation process itself has been solely heuristic. Knight #5$ 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$* a 0 k ! ln!a 0 " 1 + . !17" Equation !17" can be approximated in 0- x. 1 as , ) a 0 ln 2 !a 0 "S x!1 x", !18" where x(k ! ln!a 0 "/ a 0 . The stability curve !17" and the approximation !18" are plotted in Fig. 3. The essential features of !17" are a maxi- mum in the range 0' k ! 'a 0 /ln!a 0 ", zero growth rate at k ! =a 0 /ln!a 0 ", and a linear increase in stability with k ! for large k ! . The long-wavelength cutoff is typical of systems with a Péclet number, here with the added effect of latent heat em- bodied in the Stefan number. This demonstrates the compe- tition between the advection and diffusion of heat and mo- mentum !in a harmonic pressure ﬁeld", the former driving the instability and the latter limiting its extent. The maximum growth rate occurs at approximately k max ! ) a 0 2 ln!a 0 " , !19" with !nondimensional" growth rate , max ) a 0 4S ln 2 !a 0 " . !20" Translating !19" and !20" back into dimensional quanti- ties, we ﬁnd that the most unstable mode has angular size given by ( deg = 720 ° ! u 0 r 0 * r 0 a 0 + ln * r 0 a 0 + , !21" and has growth rate given by , dim = u 0 4Sr 0 ln 2 !r 0 /a 0 " * a 0 r 0 + . !22" III. EXTRACTING INFORMATION FROM FIELD OBSERVATIONS Field observations of lake stars cannot be controlled. A reasonable estmate for r 0 is the radius of the wetted !snow" region around the lake stars, and observations #57$ 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
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, #57$and hence 0.07.r ! /r 0 .0.15. We note that Eqs. !21" and !22" are more sensitive to a 0 /r 0 than a 0 or r 0 independently #12$. 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 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$" may more closely mirror ﬁeld observations, the present state of the latter does not warrant that level of detail. Instead, we examine the model physics through a simple proof of concept experiment described presently. IV. DEMONSTRATING LAKE STARS IN THE LABORATORY A 30-cm-diameter circular plate is maintained below freezing !) 0.5 ° C", and on top of this we place a 0.51 cm deep layer of slush through which we ﬂow 1 °C water. Given the technical difﬁculties associated with its pro- duction, the grain size, and hence the permeability of the slush layer is not a controlled variable. This fact inﬂuences our results quantitatively. In 14 runs we varied the initial size of the water-ﬁlled central hole !a 0 ", that of the circular slush layer !r 0 ", and the ﬂow rate !Q", which determines u 0 . The ﬂow rate is adjusted manually so that the water level !h 0 " in the central hole remains constant #13$. 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
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.
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#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".
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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 #57$and hence 0.3 . r LS /r 0 .0.6. #13$ 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
... The field driving the phase change is then different (temperature and not the concentration) but the positive feedback mechanism is the same: a small bump on an advancing interface between melted and unmelted regions will focus the flow and thus lead to the increased melting in its vicinity. This instability leads to the formation of star-like patterns on frozen lakes and rivers (Fig. 5)-whenever a hole forms in the ice cover, relatively warm (at 4 • C) water flows to the surface and melts the snow layer [33,34]. In the photographs in Fig. 5 the flow lines of the water are also beautifully marked on the ice, which clearly shows the focusing of the flow at the tips of the fingers. ...
... In the photographs in Fig. 5 the flow lines of the water are also beautifully marked on the ice, which clearly shows the focusing of the flow at the tips of the fingers. Tsai and Wettlaufer [34,35] proposed a mathematical model of the advancing melting front which is essentially equivalent to Eqs. (27)-(30) with the only difference that an infinite permeability of the upstream part has been assumed. The dispersion relation for this model can then be obtained directly by applying Eq. (66) with = 0, i.e., ...
... Note that this is different from the dispersion relation reported in Refs. [34,35] due to a rather subtle point, which seems to be overlooked in these works. Namely, the driving force for the instability is the temperature profile in the upstream phase, which is a solution of the convection-diffusion Eq. (53). ...
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Glaciological observations of under-flooding suggest that fluid-induced hy-draulic fracture of an ice sheet from its bed sometimes occurs quickly, possibly driven by turbulently flowing water in a broad sheet flow. Taking the approximation of a fully turbulent flow into an elastic ice medium with small fracture toughness, we derive an approximate expression for the crack-tip speed, opening displacement and pressure pro-file. We accomplish this by first showing that a Manning-Strickler channel model for re-sistance to turbulent flow leads to a mathematical structure somewhat similar to that for resistance to laminar flow of a power-law viscous fluid. We then adapt the plane-strain asymptotic crack solution of Desroches et al. [1994] and the power-law self-similar so-lution of Adachi and Detournay [2002] for that case to calculate the desired quantities. The speed of crack growth is shown to scale as the overpressure (in excess of ice over-burden) to the power 7/6, inversely as ice elastic modulus to the power 2/3, and as the ratio of crack length to wall roughness scale to the power 1/6. We tentatively apply our model by choosing parameter values thought appropriate for a basal crack driven by the rapid drainage of a surface meltwater lake near the margin of the Greenland Ice Sheet [Das et al., 2008]. Making various approximations perhaps relevant to this setting, we estimate fluid inflow rate to the basal fracture and vertical and horizontal surface dis-placements, and find order-of-magnitude agreement with observations by Das et al. [2008] associated with lake drainage. Finally, we discuss how these preliminary estimates could be improved.
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Evidence is presented that convection as well as pressure may be involved in the formation of the cell‐like melt centers that are often seen in ice sheets formed over natural waters. Pressure due to the weight of snow supplies the primary force necessary to cause the flow of water producing the melting. Convective overturning of the water under the ice and the consequent differential melting of the underside of the ice are thought to influence the position at which the melt centers become established. Temperature data are given showing the presence of convective overturning of shallow water under ice due to solar radiation absorption there. Pressure measurements are also given, indicating the change in the equilibrium water level with reference to a 25‐cm thick ice cover immediately after a heavy snowfall. © 1965, by the Association for the Sciences of Limnology and Oceanography, Inc.
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It has long been recognized that lake ice is composed of two layers: a lower, relatively clear layer of “black” ice that forms by downward freezing, and an upper, bubbly layer of “white” ice that forms when snow falls onto an ice sheet and depresses it to the point where water floods the surface, making slush that later freezes solid [1, 2, 3]. The ice on a given lake or pond, at a given time, may be all black or mostly white or any proportion between. The whiteness is caused by light scattering from small air bubbles. The black ice may contain air bubbles too, but they are larger and are aligned vertically, with the freezing direction.
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