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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
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, #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$, we have constructed a theory that is able to ex-
plain the radiating fingerlike patterns on lake ice that we call
lake stars. The model yields a prediction for the wavelength
of the most unstable mode as a function of various physical
parameters that agrees with field observations. Proof of con-
cept experiments revealed the robustness of the fingering
pattern, and to leading order the results also agree with the
model. There is substantial scatter in the data, and the overall
comparison between field observations, model, and experi-
ment demonstrates the need for a comprehensive measure-
ment program and a fully nonlinear theory which will yield
better quantitative comparisons. However, the general pre-
dictions of our theory capture the leading-order features of
the system.
ACKNOWLEDGMENTS
We thank K. Bradley and J. A. Whitehead for laboratory
and facilities support and D. H. Rothman for helpful com-
ments. This research, which began at the Geophysical Fluid
Dynamics summer program at the Woods Hole Oceano-
graphic Institution, was partially funded by National Science
Foundation !NSF" Grant No. OCE0325296, NSF Grant No.
OPP0440841 !J.S.W.", and Department of Energy Grant No.
DE-FG02-05ER15741 !J.S.W.". V. C. T. acknowledges finan-
cial support from NSF.
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#11$ 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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A two-dimensional model for the evolution of the fire line – the interface between burned and unburned regions of a wildfire – is formulated. The fire line normal velocity has three contributions: (i) a constant rate of spread representing convection and radiation effects; (ii) a curvature term that smooths the fire line; and (iii) a Stefan-like term in the direction of the oxygen gradient. While the first two effects are geometrical, (iii) is dynamical and requires the solution of the steady advection–diffusion equation for oxygen, with advection owing to a self-induced ‘fire wind’, modelled by the gradient of a harmonic potential field. The conformal invariance of this coupled pair of partial differential equations, which has the Péclet number $\textit {Pe}$ as its only parameter, is exploited to compute numerically the evolution of both radial and infinitely long periodic fire lines. A linear stability analysis shows that fire line instability is possible, dependent on the ratio of curvature to oxygen effects. Unstable fire lines develop finger-like protrusions into the unburned region; the geometry of these fingers is varied and depends on the relative magnitudes of (i)–(iii). It is argued that for radial fires, the fire wind strength scales with the fire's effective radius, meaning that $\textit {Pe}$ increases in time, so all fire lines eventually become unstable. For periodic fire lines, $\textit {Pe}$ remains constant, so fire line stability is possible. The results of this study provide a possible explanation for the formation of fire fingers observed in wildfires.
Preprint
OCCURRENCE OF ICE PHENOMENA IN THE WATER BODIES OF THE SILESIAN UPLAND IN THE ANTHROPOPRESSURE CONDITIONS (PhD Thesis) An aim of the study was to determine characteristics and conditions of occurrence of ice phenomena in the water bodies on the Silesian Upland and also to determine a role of anthropopressure in their forming (a course). The studies included 39 water bodies, diversified in terms of their origin, location, morphometric and hydrological characteristics, quality of retained waters and an impact of anthropopressure. Research studies were made within 2010,2011 and 2012 hydrological years. The studies included lots of aspects. Those studies included: bathymetric plans 33 out of 39 test objects, measurements of the selected parameters of quality of waters retained in the basins ( a temperature, oxygenation, oxygen concentration, electric conductivity, a surface water reaction, and α chlorophyll concentrations), the examinations of the water temperature, the oxygenation and the oxygen concentrations in the profiles located over so called the deepest point, continuous monitoring of the thermal waters of the basin retaining thermally contaminated water was conducted in a continuous way and measurements of spatial differentiation of the surface water temperature in this water body were made, the studies also included measurements and observations of ice cover thickness, an ice type and ice cover forms, measurements of spatial differentiation of ice thickness, taking samples of the ice cover what enabled to determine an ice structure in a precise way, recording of ice cover thermal conditions at different depths, determining a ice cover formation degree of the water bodies by tele-detection data and an analysis of a course of the ice cover formation degree of the Kozłowa Góra body in the multiannual period 1964–2012. Figures of the daily changes in the ice cover thickness obtained by the measurements and interpolations juxtaposed with the daily figures of the air temperature from the meteorological station of Earth Sciences Faculty of University of Silesia in Sosnowiec served as the basis for determining a relationship between those two variables. By a correlation coefficient there was examined an impact of morfometric characteristics of the water bodies in question such as: a basin area (A), its maximum height (Hmax) and mean depth (H av), a volume of currently retained water (V) and an exposure indicator (Ie) at: the beginning of the ice phenomena (IPB), the end of the ice phenomena (IPE), a number of days with the ice phenomena (IPN), the commencement of occurrence of the full ice cover (FCB), the end of occurrence of the full ice cover (FCE) and a number of the days with the ice cover in a particular investigation season. An cluster analysis was used to check if the objects selected for the studies constitute a uniform group in terms of the thermal regime and occurrence of the ice phenomena. Eight parameters characterizing the ice regime and the thermal conditions of the water bodies in question during three ice cover formation periods were used to perform the analysis. Those were: the maximum daily ice increment (MDII), the minimum water temperature (Tmin), the maximum water temperature (Tmax), the mean water temperature (Tmean.), the maximum ice thickness (MIT), the mean ice thickness (MIT), a number of the days with the ice phenomena (IPN) and a number of the days with the ice cover (ICN). Two clusters: S1 and S2 were accepted a priori on basis of the variables mentioned above. 39 water bodies on the Silesian Upland including lots of aspects of specificity of occurrence of the ice phenomena were investigated, those studies enabled to formulate a series of conclusions corresponding to the aims defined in the in the introduction part and verifying the main hypothesis. On basis of the calculated correlation coefficients of the daily air temperature changes with the daily ice thickness changes, it can be stated that in the case of 36 out of 39 water bodies they were significant (from moderate to strong), what can indicate their natural or quasi-natural ice regime. A course of the ice phenomena in that group of the water bodies was varied, mainly by the environmental factors, which can include e.g. a hydrological type of a water region, its location and the morfometric parameters. Such relationships were not present (Pod Borem water body) or were very weak (e.g. Somerek, Sośnica-Makoszowy) and statistically insignificant only in the case of mining water bodies burdened with thermal pollutants. The values of the correlation coefficients of the mean daily air temperature with the ice thickness changes were diversified in particular seasons, what resulted mainly from differences of snow thickness accumulated on the ice. The weakest reactions of the ice thickness to the air temperature changes were found in the first research season when the snow cover was the thickest and remained for the longest time. It confirms a significant role of snow in insulating of the ice cover from atmospheric conditions. An analysis of relationships between the morfometric characteristics of the water bodies and a course of the ice phenomena showed that there were vital and statistically significant relationships between the water reservoir basins and commencement of the ice cover formation. The mean depth and reservoir capacity should be included into the most important morfometric characteristic resulting in commencement of the ice phenomena. The bigger they were the later the lake ice and the ice cover appeared in the water regions. However, the morfometric parameters of the water bodies did not have an influence on ice melting in the water bodies because a rate of its melting depended only on external factors, which include intensity of solar radiation reaching ice and an air temperature course. Performance of the multivariate cluster analysis by the k-means on basis of eight parameters concerning the ice phenomena and the thermal conditions of the water bodies enabled to separate two sub-groups in the set in question. The first cluster included the mining water bodies: Pod Borem, Somerek and Sośnica-Makoszowy, and other water bodies belonged to the other cluster. Both groups differ immeasurably in terms of the characteristic integrated in the statistical analysis. The first group was represented by the water bodies characterized by water anthropometry and a very weak development of the ice phenomena, the other water regions were concentrated in the second group, of which ice cover formation and water thermal conditions depend mainly on the natural environment conditions, at a slight influence of human activity. In majority of the water bodies in question all phases of the ice cover formation were observed: shore ice, a partial, incomplete and complete ice cover, and also ice floats and icings present in the littoral zone. A number of days with the ice phenomena in particular winter periods were diversified, being on average: 91, 100 and 85, in the first, second and third winter season, respectively. The ice thickness was characterized by the similar differentiation. The mean maximum ice thickness was: 23,0 cm, 19,2 cm and 31,0 cm, at the mean values at a level of 13,4 cm, 13,1 cm and 12,3 cm in the first, second and third winter period, respectively. The differentiation of duration of the ice phenomena and ice thickness resulted mainly from the varied atmospheric conditions in the particular study seasons. From the examinations of the vertical structure of the ice cover of the water bodies, it results that characteristic layers, formed at different ice formation stages (ice stratigraphy) can be differentiated in their structure. At first so called primary ice is formed, and this ice depending on the conditions in which it is formed, is characterized by horizontal (crystalline ice formed at windless weather) or random crystal packing (murky ice, formed in the conditions of intensive wind overturns or snowfalls while freezing). Column crystalline ice, characterized by the vertical big crystal packing freezes to the primary ice from the bottom. Imposed ice, also called snow ice, is formed as the last one. From the investigations of the spatial diversification of the ice cover thickness made for 30 water bodies it results that the ice thickness differences within the particular water regions increase during winter. The main factor which causes an increase in the ice cover thickness differentiation is non-uniform freezing of snow ice (imposed) in the particular sectors of the water bodies. Secondary and local significance has a water circulation and heat release from sediments. From the ice cubature calculations it results that the ice covers comprise from 3,7% to 70,0% of current capacity of the water regions, what results mainly from their morfometric characteristics. Ice temperature recording during two research seasons showed that the snow accumulating on the ice cover played a key role in shaping its thermal conditions, resulting in dynamics of daily increases in its thickness. The snow lying on the ice cover insulates it effectively from an influence of atmospheric conditions, causing slowing down ice thickness increases from the bottom, even during freezing temperatures. Prolonged stay periods of the ice cover on the water bodies have a significant influence on the conditions present under the ice. The ice cover formation insulates the water body from the atmosphere, what prevents from significant cooling down of water. In the seasons when the ice covered is formed very quickly the water bodies had a much higher temperature than during the winters in which the ice cover formation was preceded by repeated mixing of limnic waters at the temperature oscillating 0°C. Ice cover formation of the water regions has an influence on ecological conditions present in the water bodies during winter, contributing to significant decreases in the oxygen concentrations. The ice cover, adequately snow lying on it, leads to inhibiting solar radiation to lake waters and limiting a photosynthesis process in the water bodies, what together with oxygen consumption in the chemical and bio-chemical changes results in a decrease in the amount of the oxygen dissolved in the water, including hypoxia. During long, frosty and snowy winters when the ice cover was for at least several dozen days, in the small shallow water bodies a decrease in the oxygen concentration was big enough to cause mass fish death (so called winter fish kill). The ice cover lying on the biggest water bodies on the Silesian Upland (e.g. Dzierżno Duże, Kuźnica Warężyńska, Kozłowa Góra, Pogoria III), plays a significant morfogenetic role, transforming the shore zones as a result of a thermal and mechanical pressure. The thermal changes in the ice volume leading to changes in its volume and pushing the ice cover to the shore and hummocky ice of small sizes formed during windy periods caused destabilization of the shore zone, including infrastructure elements. Recreational use of the water bodies also in the winter period is characteristic for the Silesian Upland. The ice covers of the water regions in questions were mainly used for recreational purposes (ice fishing, cross-country skiing, skating, kite surfing, ice surfing, ice diving) to a lesser degree for economic purposes ( fishing of plankton organisms). Aerial photographs and satellite images can be a useful source of data, owing to which spatial determination of a rate of freezing and ablation of big water bodies, for which observations from a shore do not comprise the entire area of a water region, can be determined. Owing to the tele-detection products a detailed analysis of a course of cracks and clefts, formed in the ice cover and location of places in which morass is formed yearly (e.g. on the water bodies of Żabie Doły, Łąka), can be made. An analysis of the data concerning the time variation of the ice cover formation of the Kozłowa Góra water body showed that in period 1964–2012 – a reaction to the air temperature increase of 1,3°C in the winter half-year – there was a slight decrease in of the maximum (about 8,2 cm) and mean ice thickness (about 5,6 cm) an a number of the days with the ice cover (about 36 days). Those trends can be one of the signs of climate changes taking place nowadays.
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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.
Chapter
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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