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Formation of slush on floating ice
Cold Regions Science and Technology, 15 (1988) 33-38 Elsevier Science Publishers B.V., Amsterdam - - Printed in the Netherlands
33
FORMATION OF SLUSH ON FLOATING ICE Charles A, Knight National Center for Atmospheric Research*, Boulder, Colorado 80307 (U.S.A.) (Received June 3, 1987; accepted in revised form August 10, 1987)
ABSTRACT Slush layers often form after snowfall on floating ice sheets, producing after they freeze, the upper layer of white, bubbly ice that is often found. The stability conditions for slush layers and snow on top of floating ice are examined in some detail the patterns of flooding and the reasons for them are discussed, and it is pointed out that it may be fairly common for snow layers on floating ice to be unstable with respect to slush formation. I f there were some reason to desire thicker ice layers, slush formation couM be stimulated rather easily in some conditions by making holes in the ice. In some other conditions, the instability is probably conditional in that some flooding must take place before further flooding can be spontaneous.
INTRODUCTION
ICE-SLUSH-SNOW STABILITY
This paper is an extension of a recent report (Knight, 1987) that focused upon the radiating, branching channels that form in slush on freshwater lakes when snow on the lake ice is flooded with the lake water from beneath. It is well known that the water beneath the ice tends to be above its melting point, because convective mixing in cooling from above can only operate to produce an isothermal profile at + 4 ° C, the temperature o f m a x i m u m density. Downward cooling to temperatures below *The National Center for Atmospheric Research is sponsored by the National Science Foundation.
0165-232X/88/$03.50
+ 4 °C is by relatively slow heat conduction.** Thus, flow of water from below into the snow and slush is often accompanied by melting, and the melting causes the striking radiating patterns. The process appears to be essentially like the chemical dissolution in flow through porous aggregates that produces fractal patterns (Daccord, 1987; Daccord and Lenormand, 1987). The same or similar features in slush or slush-free ice have been explained other ways, by convection (Katsaros, 1981 ) or by cracking ( G o w and Govoni, 1983 ), but it seems clear that most of them are due to the flooding process with melting (Woodcock, 1965; Tokairin, 1977; Knight, 1987). That controversy - - to the extent that it is one - - is not pursued here. Rather we concentrate first on the stability relations: given snowfall on a floating ice sheet, what degree of slush formation is stable? This leads to recognition that instability is probably quite common.
© 1988 Elsevier Science Publishers B.V.
The gravitational stability of a floating ice-slush-snow sheet is perhaps best treated in terms o f observable starting conditions: and ice thickness Tice and a snowfall depth SF, the snow depth on dry ground. In order to solve the problem, one must **Note: Woodcock (1965) and Tokairin (1977), among many others, present temperature profiles in the water beneath ice cover on ponds. If first freeze-up is in windy conditions, the stirring may cool most of the water to nearly 0°C before an ice sheet forms. Also, since normal sea water does not have a temperature of maximum density above its freezing point, hole formation and flooding of snowcover are not to be expected to be nearly as prominent on sea ice as on freshwater ice.
34 specify the capillary rise of water in snow CR and the volume reduction that occurs when snow is soaked with water to form slush. A number of observations show this shrinkage to be substantial and fairly rapid, much more rapid than the natural densification of the dry snow (Knight, 1987; some of the evidence will be discussed below). Thus when ice is covered by both slush and snow, we set
I SF=-9.18T, ce +220.6cm
50
--
. . . .
(
p. . . .
.
Pslush = P~ce Pice -t- \ 1 -- Pice /| Pwat~r,
(2)
but with the shrinkage N
Pslush=Np. . . . + ( 1
/
40
')I
I
/
N~w
) Pwater,
(3)
•
t .a
.,I
." "L, /
:
O 0
," /
t
/
1
/h 20
/ //
-20cm ''e
/_z']
v/~i
/L/
ITslu,h =0 i
~i
,I ~I i
30
/'/
L ~ i
4// 'Z9 I I0
/
/ " j/"
',
•/", IO
" / / /" ' ' /
I 40
Is 50
Tice: ICE THICKNESS (era) Fig. 1. The stable thicknesses of slush and snow layers on a floating ice sheet are shown as a function of the ice thickness and the total, dry snowfall. Assumptions are Ps,o,= 0.1 Mg m -3, the capillary rise of water in snow is 2 cm, and when snow becomes soaked, the ice framework shrinks to 1/3 its former volume (see text).
and ice and snow only. The general equations for the dividing lines are S F - CRpwat~r- Ti~ (Pwater--Pice) 0.0908 p . . . .
which transforms to Pslush = 1Mg m -3 _ 0.0908 Np . . . .
/ ~ ~1
t
)Ocm
/ / / /// .
",//
~I
(1)
where T is the thickness o f a layer (Tsnow is the thickness of snow above the slush here) and Nis the shrinkage factor. The variables in the stability problem are the bulk densities Psnow and Ps].sh, CR and N, as well as the initial variables Ti~ and SF. Assuming no volume change of the snow framework when it is transformed to slush ( N = 1 ), no trapping of air bubbles, and no melting,
p.
',// 'L
]'slush =
15cm
30
SF= Tsnow+ NTslush
ll//l'/ //LII
IIIIL
(5)
(4) and
taking pic~=0.9168 and Pwate~=lMg m -3, and rounding. It seems clear that Ps,ow, CR and N are all significantly variable in nature. While there seem to be no values of CR and N in the literature, observations in the field and some simple laboratory experiments put CR in the vicinity of 2 cm. While Nis not a constant, of course (it is even less constant than CR), field observations show the shrinkage to be very considerable over times of a few hours, and we take N = 3 as a realistic value for the purpose of illustration. Observational justification for these values is given in a later section. On the basis o f these assignments, as well as setting p . . . . = 0.1 Mg m - 3, Fig. 1 presents the equilibrium snow and slush thicknesses as a function of T ~ and SF. The field is divided into three areas: from left to right, ice and slush only; ice, snow and slush;
SF (=Tsnow) -CRpwater + Tice
(Pwater--Pice)
Psnow
(6)
The equations that describe the ice, snow and slush thicknesses inside the "three-phase" field are Tslush = T s n o w
P.... Pwater --Pslush
T~ice -Pwater ---Pice Pwater --Pslush
+CR
Pw~te~
(7)
Pwater -- Pslush
combined with eqns. (1) and (4). The results in Fig. 1 - - especially the substantial thicknesses of slush that are stable in very common conditions of Tice and S F - - are surprising at first,
35 and two physical influences need to be appreciated to make these results intuitive. First, the density of slush is so close to that o f water that a very large thickness of slush below the hydrostatic water level is required to produce any substantial buoyancy. Second, CR = 2 cm is very important. It means that, assuming stability (minimizing the sum o f the surface and the gravitational potential energies), any snow within 2 cm of the hydrostatic water level must be transformed to slush. How stability is, or is not, achieved when snow falls on floating ice is discussed in the next section, but one simple possibility is worth noting here, in connection with Fig. 1. If the ice sheet is envisioned as having holes through it, but of such a size that the capillary rise in them is negligible, then no slush actually forms until the snowfall reaches the depth corresponding to the dotted line through the origin with slope 0.83, so that the water level is at the ice surface. Then the amount of slush formed corresponds to that along that line: about 6.5 cm for any ice thickness greater than about 22 cm.
ACHIEVING EQUILIBRIUM The flooding or "slushing" in nature, when it does occur, is from around the edge o f the ice sheet the shore of the pond or lake - - and through holes in the ice. The lateral extent of slushing from any one source of water is usually a metre or less, in the writer's experience, with examples as big as several tens of metres seen only rarely. Most water transport in the larger areas o f slush from single holes is along the radiating, branching melt channels. Without melting, slushing would occur to a lesser extent; probably much less, since melting provides both the holes in the ice and the channels for the water to spread more quickly through the snow. In the vicinity o f Boulder, Colorado, on the plain east of the Rocky Mountains, many of the snowfalls occur at cold front passages after warm weather. Preexisting ice sheets are often warm and quite permeable, so that there are probably small holes, sometimes called veins, along the crystal boundary intersection lines. The vertical, tubular air bubbles that sometimes form as the ice freezes also can be parts of channels. As the weight o f snow forces the
Snow
Ice Water \
~
~ ~ ~
; s'''`" /1"
_,~\
\
Fig. 2. Schematic of slushing by the flooding of snow on an ice sheet, from the shore. warmer water up from under the ice through these tiny holes, there is a fundamental instability. Even if there are a lot of small holes at the start, the largest of them will carry the most water flow, hence suffer the most melting, and hence get larger at a faster rate than the smaller ones. The holes that develop this way end up typically one to several cm in diameter, spaced from about 10 cm to a few metres apart; closer together the thinner the ice. The feedback that causes this instability-- the bigger the hole the more flow through it, and the more flow the more melting and enlargement of the hole - - is quite similar in principle to that which localizes the outward flow of water in the branching channels in the slush. It is interesting that the discrete holes, with associated cellular and radiating patterns in the slush, can also form from snowfall on a layer of floating slush, with no ice sheet being present at all. As snow falls onto a pre-existing slush layer, the water is forced up through the slush, and even in this case the flow becomes localized by melting as discrete holes about a centimetre in diameter. When the ice is thick or the top of the ice sheet is substantially below the freezing point at the time the snow starts falling, there may be no way for slushing to occur at first except at the edge of the pond. This instability can last for days or entire seasons, but often it is slowly relieved in the following way. The first flooding comes from the edges. A stage in the process, with the thickness variations exaggerated for illustration, is shown in Fig. 2. After a band of slush one to several metres wide forms along the shore, the next stage can be formation of holes in the ice in the region indicated by A in Fig. 2. Another strip of snow is transformed to slush by water flowing through these new holes, and the process repeats itself all the way to the center of the pond (Fig. 3). When Tice and SF are both of the order o f 10
36
::::
:
?i? i~ .......
Z m
Fig. 3. Sawhill Ponds, Boulder, Colo., 18 Jan. 1987, 0900. The snowfall was about 15 cm on the night of 15 Jan. At the time of the photo, undisturbed snow on the ice was about 9 cm deep, and the ice was about 8 cm thick. The large, circular area of slush was from a hole drilled 24 hours previously. Note the three stages of slushing from the shore, contrasting because of frost formation on the frozen slush surface. The temperature between snowfall and this time was continually below freezing. cm, the rate o f progress o f the slushing is typically a few tens o f metres a day. The rate depends u p o n how fast the holes can form, which is probably quite sensitive to the temperature gradient in the water beneath the ice, and might be sensitive also to the crystal texture o f the ice, a m o n g other things. N e w hole f o r m a t i o n at the edge o f a flooded region is presumably encouraged for two reasons. First, the slush w a r m s the layer o f ice to the melting point throughout. Secondly, the extra weight on the flooded portion o f the ice depresses the ice sheet and m a y greatly increase the pressure difference across it at the edge o f the flooded area, leading to faster enlargement o f any tiny hole. It is perhaps worth remarking that the finding that white ice can be thicker a r o u n d the edges o f a lake ( A d a m s and Roulet, 1980, 1984) m a y be a result o f this edge-to-center progression o f slushing, as well as the extra snow thickness near lake shores, caused by drifting. W h e n Tice and S F are such that slush f o r m a t i o n is favored energetically but no holes f o r m to allow it to happen, one can encourage the process by drilling holes (Woodcock, 1965). We have done this also, and c o m e back the next day to find the hole
Fig. 4. Sawhlll Ponds, Boulder, Colo., 16 Jan. 1987, 0900. Slushing feature from a hole drilled about 17 hours previously in ice 6.5 cm thick with about 10 cm snow cover. The hole enlarged from about 2 to 4 cm in diameter, and the melt channels extended about two meters out from the hole. The frozen, top layer of the slush was removed around the hole to make the hole and the melt channels more visible. The upper layer of the ice sheet was bubbly, so melting reveals darker ice beneath. The slush adhered to the upper ice layer, and channels in it are visible too, on the piece of the ice crust turned upside down near the blade of the ice axe, 1 m long.
greatly enlarged and slush developed a r o u n d it, with the radiating, branching melt channels (Figs. 3, 4). I f one desired to encourage ice formation, both by f o r m i n g slush on top o f the ice where it can then freeze and by getting rid o f some or all o f the snow, which is an excellent insulator, a convenient m e t h o d is to use salt pellets to "drill" the holes. In one case a hole with associated flooding was started overnight in ice 16 c m thick, using a 5 g pellet o f NaC1. A laboratory study would be required to determine exactly when conditions are appropriate and what size o f salt pellets would be needed in different regimes o f Tice a n d temperature. Observations to date have not shown clearly whether the hydrostatic water level has to be at the u p p e r ice surface before the flooding can start naturally, or whether a capillary rise within the ice its e l f - say along grain boundaries or the veins where they intersect - - can deliver water into the snow. I f the water level must be at the ice surface, as seems probable, then the region in Fig. 1 between the solid line b o u n d a r y o f the Tsl,sh----0 region and the dotted
37
Snow Slush
Thin Water
Ice
Water
C
Slush
Layer/L--
-----
I
T Icm I0 crn
~cm
j
~
Water Boundary
water
(a)
Water
j
ly Ice Layer
Fig. 5. Measured profile through snow and slush at the edge of a pond, with slow flooding from right to left. Note the different horizontal and vertical scales.
Frozen Slush Slush
-'~___~- ~ line through the origin is all conditionally unstable when there is ice and snow only, with no slush. Some flooding is needed to weigh down the ice before further flooding can be spontaneous.
O B S E R V A T I O N A L B A S I S FOR C R ~ 2 AND N~ 3
CM
Figure 5 shows the results of a typical set of measurements across the slush-snow boundary, as has been shown schematically in Fig. 2. Note that the horizontal scale is a factor of ten different from the vertical. Thus in reality the step in the snow surface is much less pronounced than shown, but it is still distinctly a step. The measurements were made by carefully cutting a trench in the snow and slush so as to disturb the wall as little as possible, photographing the wall at a low angle, and measuring from the photographs. N is at least 3 in this case, by comparing Tsnowat the far left ( = SF) with Tstush at the far right. N can be smaller in between, but that represents a time-dependent, transient condition as the water percolates from right to left in the figure. The measurements are within about + 0.5 cm because of the way they were made and, in the case of the slush-snow interface, because the interface itself has a small-scale relief (irregularity) o f about 0.5 cm. In cutting trenches of this kind in slush, it was found that when Tslus h w a s less than about 2 cm, water did not flow out to cover the ice in the trench. With Tslushgreater than about 2 cm, water immedi-
\Bubbly Ice Layer Ice I cm
H (b)
Water
Fig. 6. Cross sections through the artificial flooding feature produced by drilling a hole through the ice, shown in Fig. 4. The details are in part schematic (see text). ately and rapidly flowed from the slush onto the ice surface. Thus CR ~ 2 cm. The same result was obtained in the laboratory. When the water at 0 °C is poured into a transparent container of undisturbed snow, the level to which the snow becomes obviously soaking wet is about 2 cm above the hydrostatic water level. Figure 6 gives two schematics of the artificiallyinduced slushing feature of Fig. 4. 10 cm of snow on 6.5 cm of ice transformed to about 4 cm of slush on the ice overnight. Again, N is about 3. The temperature fell to about - 15 ° C and about 1 cm at the top of the slush had frozen the next morning (Fig. 4). The cross section through the central hole (Fig. 6 ( a ) ) shows the extent to which the melting enlarged the hole that was drilled originally. Figure 6(b) is a cross section of one of the radiating channels, showing the melting into the preexisting ice. (This rounded groove in the pre-existing, fiat ice surface has always been found under the radiating channels, and seems by itself to be unassailable proof of melting.) A thin water layer is shown between
38 the slush and the upper surface o f the original ice sheet, because when pieces of the upper, frozen slush layer were broken free and removed, the slush adhered to them, separating cleanly from the ice beneath. The thickness of this water layer was not measured. However, if the ice framework in a horizontal layer of slush does shrink, its buoyancy will keep it in contact with the upper ice layer. Neumann (1958) observed a similar layering that also might be explained this way.
DISCUSSION The roles of snow and slush in lake ice formation have been discussed before and the importance of slush formation is well appreciated (Shaw, 1965; Jones, 1969; Adams and Roulet, 1980, 1984). The motive of the present study has been to contribute to understanding the physics o f the flooding process. Neither the importance o f the capillary rise in snow to the equilibrium thickness of slush on lakes nor the probably c o m m o n instability toward slush formation on frozen lakes seem to be well known; and the posibility o f inducing slushing with salt pellets has not been suggested before, to the writer's knowledge.
ACKNOWLEDGMENTS Assistance in the field, discussions with, and comments from A. Weinheimer are greatly appreciated, as is the manuscript preparation by F. Huth.
REFERENCES
Adams, W.P. and N.T. Roulet (1980). Illustration of the roles of snow in evolution of the winter cover of a lake. Arctic, 33(1): 100-116. Adams, W.P. and N.T. Roulet (1984). Sampling of snow and ice on lakes. Arctic, 37(3): 270-275. Colbeck, S.C. (1974). The capillary effects on water percolation in homogeneous snow. J. Glaciology, 13(67): 85-97. Colbeck, S.C. (1986). Statistics of coarsening in water-saturated snow. Acta Metall., 34(3): 347-352. Daccord, G. (1987). Chemical dissolution of a porous medium by a reactive fluid. Phys. Rev. Letters, 58(5): 479-482. Daccord, G. and Lenormand, R. ( 1987). Fractal patterns from chemical dissolution. Nature, 325:41-43. Gow, A.J. and J.W. Govoni (1983). Ice growth on Post Pond, 1973-1982. CRREL Report 83-4 (U.S. Army Cold Regions Research and Engineering Laboratory, Hanover, N.H.), 25 pp. Jones, J.A.A. (1969). The growth and significance of white ice at Knob Lake, Quebec. Canadian Geographer, XIII(4): 354-372. Katsaros, K.B. (1981). Convection patterns in a pond. Bull. Amer. Meteor. Soc., 62 (10): 1446-1453. Knight, C.A. (1987). Slush on lakes. In: D.E. Loper (Ed.), Structure and Dynamics of Partially Solidified Systems, Martinus Nijhoff, Dordrecht, pp. 455-465. Neumann, H-G. ( 1958). Zellmuster auf der oberfliiche eines teiches. Beitr~igeZur Phys. der Atmos.: 30 (4): 246-253. Shaw, J.B. (1965). Growth and decay of lake ice in the vicinity of Schefferville (Knob Lake), Quebec. Arctic, 18(2): 123-131. Tokairin, A. (1977). Lake ice. Kodansha, Tokyo, Japan, 103 pp. (in Japanese). Woodcock, A.H. (1965). Melt patterns in ice over shallow waters. Limnol. Oceanogr., 10: R290-R297.
33
FORMATION OF SLUSH ON FLOATING ICE Charles A, Knight National Center for Atmospheric Research*, Boulder, Colorado 80307 (U.S.A.) (Received June 3, 1987; accepted in revised form August 10, 1987)
ABSTRACT Slush layers often form after snowfall on floating ice sheets, producing after they freeze, the upper layer of white, bubbly ice that is often found. The stability conditions for slush layers and snow on top of floating ice are examined in some detail the patterns of flooding and the reasons for them are discussed, and it is pointed out that it may be fairly common for snow layers on floating ice to be unstable with respect to slush formation. I f there were some reason to desire thicker ice layers, slush formation couM be stimulated rather easily in some conditions by making holes in the ice. In some other conditions, the instability is probably conditional in that some flooding must take place before further flooding can be spontaneous.
INTRODUCTION
ICE-SLUSH-SNOW STABILITY
This paper is an extension of a recent report (Knight, 1987) that focused upon the radiating, branching channels that form in slush on freshwater lakes when snow on the lake ice is flooded with the lake water from beneath. It is well known that the water beneath the ice tends to be above its melting point, because convective mixing in cooling from above can only operate to produce an isothermal profile at + 4 ° C, the temperature o f m a x i m u m density. Downward cooling to temperatures below *The National Center for Atmospheric Research is sponsored by the National Science Foundation.
0165-232X/88/$03.50
+ 4 °C is by relatively slow heat conduction.** Thus, flow of water from below into the snow and slush is often accompanied by melting, and the melting causes the striking radiating patterns. The process appears to be essentially like the chemical dissolution in flow through porous aggregates that produces fractal patterns (Daccord, 1987; Daccord and Lenormand, 1987). The same or similar features in slush or slush-free ice have been explained other ways, by convection (Katsaros, 1981 ) or by cracking ( G o w and Govoni, 1983 ), but it seems clear that most of them are due to the flooding process with melting (Woodcock, 1965; Tokairin, 1977; Knight, 1987). That controversy - - to the extent that it is one - - is not pursued here. Rather we concentrate first on the stability relations: given snowfall on a floating ice sheet, what degree of slush formation is stable? This leads to recognition that instability is probably quite common.
© 1988 Elsevier Science Publishers B.V.
The gravitational stability of a floating ice-slush-snow sheet is perhaps best treated in terms o f observable starting conditions: and ice thickness Tice and a snowfall depth SF, the snow depth on dry ground. In order to solve the problem, one must **Note: Woodcock (1965) and Tokairin (1977), among many others, present temperature profiles in the water beneath ice cover on ponds. If first freeze-up is in windy conditions, the stirring may cool most of the water to nearly 0°C before an ice sheet forms. Also, since normal sea water does not have a temperature of maximum density above its freezing point, hole formation and flooding of snowcover are not to be expected to be nearly as prominent on sea ice as on freshwater ice.
34 specify the capillary rise of water in snow CR and the volume reduction that occurs when snow is soaked with water to form slush. A number of observations show this shrinkage to be substantial and fairly rapid, much more rapid than the natural densification of the dry snow (Knight, 1987; some of the evidence will be discussed below). Thus when ice is covered by both slush and snow, we set
I SF=-9.18T, ce +220.6cm
50
--
. . . .
(
p. . . .
.
Pslush = P~ce Pice -t- \ 1 -- Pice /| Pwat~r,
(2)
but with the shrinkage N
Pslush=Np. . . . + ( 1
/
40
')I
I
/
N~w
) Pwater,
(3)
•
t .a
.,I
." "L, /
:
O 0
," /
t
/
1
/h 20
/ //
-20cm ''e
/_z']
v/~i
/L/
ITslu,h =0 i
~i
,I ~I i
30
/'/
L ~ i
4// 'Z9 I I0
/
/ " j/"
',
•/", IO
" / / /" ' ' /
I 40
Is 50
Tice: ICE THICKNESS (era) Fig. 1. The stable thicknesses of slush and snow layers on a floating ice sheet are shown as a function of the ice thickness and the total, dry snowfall. Assumptions are Ps,o,= 0.1 Mg m -3, the capillary rise of water in snow is 2 cm, and when snow becomes soaked, the ice framework shrinks to 1/3 its former volume (see text).
and ice and snow only. The general equations for the dividing lines are S F - CRpwat~r- Ti~ (Pwater--Pice) 0.0908 p . . . .
which transforms to Pslush = 1Mg m -3 _ 0.0908 Np . . . .
/ ~ ~1
t
)Ocm
/ / / /// .
",//
~I
(1)
where T is the thickness o f a layer (Tsnow is the thickness of snow above the slush here) and Nis the shrinkage factor. The variables in the stability problem are the bulk densities Psnow and Ps].sh, CR and N, as well as the initial variables Ti~ and SF. Assuming no volume change of the snow framework when it is transformed to slush ( N = 1 ), no trapping of air bubbles, and no melting,
p.
',// 'L
]'slush =
15cm
30
SF= Tsnow+ NTslush
ll//l'/ //LII
IIIIL
(5)
(4) and
taking pic~=0.9168 and Pwate~=lMg m -3, and rounding. It seems clear that Ps,ow, CR and N are all significantly variable in nature. While there seem to be no values of CR and N in the literature, observations in the field and some simple laboratory experiments put CR in the vicinity of 2 cm. While Nis not a constant, of course (it is even less constant than CR), field observations show the shrinkage to be very considerable over times of a few hours, and we take N = 3 as a realistic value for the purpose of illustration. Observational justification for these values is given in a later section. On the basis o f these assignments, as well as setting p . . . . = 0.1 Mg m - 3, Fig. 1 presents the equilibrium snow and slush thicknesses as a function of T ~ and SF. The field is divided into three areas: from left to right, ice and slush only; ice, snow and slush;
SF (=Tsnow) -CRpwater + Tice
(Pwater--Pice)
Psnow
(6)
The equations that describe the ice, snow and slush thicknesses inside the "three-phase" field are Tslush = T s n o w
P.... Pwater --Pslush
T~ice -Pwater ---Pice Pwater --Pslush
+CR
Pw~te~
(7)
Pwater -- Pslush
combined with eqns. (1) and (4). The results in Fig. 1 - - especially the substantial thicknesses of slush that are stable in very common conditions of Tice and S F - - are surprising at first,
35 and two physical influences need to be appreciated to make these results intuitive. First, the density of slush is so close to that o f water that a very large thickness of slush below the hydrostatic water level is required to produce any substantial buoyancy. Second, CR = 2 cm is very important. It means that, assuming stability (minimizing the sum o f the surface and the gravitational potential energies), any snow within 2 cm of the hydrostatic water level must be transformed to slush. How stability is, or is not, achieved when snow falls on floating ice is discussed in the next section, but one simple possibility is worth noting here, in connection with Fig. 1. If the ice sheet is envisioned as having holes through it, but of such a size that the capillary rise in them is negligible, then no slush actually forms until the snowfall reaches the depth corresponding to the dotted line through the origin with slope 0.83, so that the water level is at the ice surface. Then the amount of slush formed corresponds to that along that line: about 6.5 cm for any ice thickness greater than about 22 cm.
ACHIEVING EQUILIBRIUM The flooding or "slushing" in nature, when it does occur, is from around the edge o f the ice sheet the shore of the pond or lake - - and through holes in the ice. The lateral extent of slushing from any one source of water is usually a metre or less, in the writer's experience, with examples as big as several tens of metres seen only rarely. Most water transport in the larger areas o f slush from single holes is along the radiating, branching melt channels. Without melting, slushing would occur to a lesser extent; probably much less, since melting provides both the holes in the ice and the channels for the water to spread more quickly through the snow. In the vicinity o f Boulder, Colorado, on the plain east of the Rocky Mountains, many of the snowfalls occur at cold front passages after warm weather. Preexisting ice sheets are often warm and quite permeable, so that there are probably small holes, sometimes called veins, along the crystal boundary intersection lines. The vertical, tubular air bubbles that sometimes form as the ice freezes also can be parts of channels. As the weight o f snow forces the
Snow
Ice Water \
~
~ ~ ~
; s'''`" /1"
_,~\
\
Fig. 2. Schematic of slushing by the flooding of snow on an ice sheet, from the shore. warmer water up from under the ice through these tiny holes, there is a fundamental instability. Even if there are a lot of small holes at the start, the largest of them will carry the most water flow, hence suffer the most melting, and hence get larger at a faster rate than the smaller ones. The holes that develop this way end up typically one to several cm in diameter, spaced from about 10 cm to a few metres apart; closer together the thinner the ice. The feedback that causes this instability-- the bigger the hole the more flow through it, and the more flow the more melting and enlargement of the hole - - is quite similar in principle to that which localizes the outward flow of water in the branching channels in the slush. It is interesting that the discrete holes, with associated cellular and radiating patterns in the slush, can also form from snowfall on a layer of floating slush, with no ice sheet being present at all. As snow falls onto a pre-existing slush layer, the water is forced up through the slush, and even in this case the flow becomes localized by melting as discrete holes about a centimetre in diameter. When the ice is thick or the top of the ice sheet is substantially below the freezing point at the time the snow starts falling, there may be no way for slushing to occur at first except at the edge of the pond. This instability can last for days or entire seasons, but often it is slowly relieved in the following way. The first flooding comes from the edges. A stage in the process, with the thickness variations exaggerated for illustration, is shown in Fig. 2. After a band of slush one to several metres wide forms along the shore, the next stage can be formation of holes in the ice in the region indicated by A in Fig. 2. Another strip of snow is transformed to slush by water flowing through these new holes, and the process repeats itself all the way to the center of the pond (Fig. 3). When Tice and SF are both of the order o f 10
36
::::
:
?i? i~ .......
Z m
Fig. 3. Sawhill Ponds, Boulder, Colo., 18 Jan. 1987, 0900. The snowfall was about 15 cm on the night of 15 Jan. At the time of the photo, undisturbed snow on the ice was about 9 cm deep, and the ice was about 8 cm thick. The large, circular area of slush was from a hole drilled 24 hours previously. Note the three stages of slushing from the shore, contrasting because of frost formation on the frozen slush surface. The temperature between snowfall and this time was continually below freezing. cm, the rate o f progress o f the slushing is typically a few tens o f metres a day. The rate depends u p o n how fast the holes can form, which is probably quite sensitive to the temperature gradient in the water beneath the ice, and might be sensitive also to the crystal texture o f the ice, a m o n g other things. N e w hole f o r m a t i o n at the edge o f a flooded region is presumably encouraged for two reasons. First, the slush w a r m s the layer o f ice to the melting point throughout. Secondly, the extra weight on the flooded portion o f the ice depresses the ice sheet and m a y greatly increase the pressure difference across it at the edge o f the flooded area, leading to faster enlargement o f any tiny hole. It is perhaps worth remarking that the finding that white ice can be thicker a r o u n d the edges o f a lake ( A d a m s and Roulet, 1980, 1984) m a y be a result o f this edge-to-center progression o f slushing, as well as the extra snow thickness near lake shores, caused by drifting. W h e n Tice and S F are such that slush f o r m a t i o n is favored energetically but no holes f o r m to allow it to happen, one can encourage the process by drilling holes (Woodcock, 1965). We have done this also, and c o m e back the next day to find the hole
Fig. 4. Sawhlll Ponds, Boulder, Colo., 16 Jan. 1987, 0900. Slushing feature from a hole drilled about 17 hours previously in ice 6.5 cm thick with about 10 cm snow cover. The hole enlarged from about 2 to 4 cm in diameter, and the melt channels extended about two meters out from the hole. The frozen, top layer of the slush was removed around the hole to make the hole and the melt channels more visible. The upper layer of the ice sheet was bubbly, so melting reveals darker ice beneath. The slush adhered to the upper ice layer, and channels in it are visible too, on the piece of the ice crust turned upside down near the blade of the ice axe, 1 m long.
greatly enlarged and slush developed a r o u n d it, with the radiating, branching melt channels (Figs. 3, 4). I f one desired to encourage ice formation, both by f o r m i n g slush on top o f the ice where it can then freeze and by getting rid o f some or all o f the snow, which is an excellent insulator, a convenient m e t h o d is to use salt pellets to "drill" the holes. In one case a hole with associated flooding was started overnight in ice 16 c m thick, using a 5 g pellet o f NaC1. A laboratory study would be required to determine exactly when conditions are appropriate and what size o f salt pellets would be needed in different regimes o f Tice a n d temperature. Observations to date have not shown clearly whether the hydrostatic water level has to be at the u p p e r ice surface before the flooding can start naturally, or whether a capillary rise within the ice its e l f - say along grain boundaries or the veins where they intersect - - can deliver water into the snow. I f the water level must be at the ice surface, as seems probable, then the region in Fig. 1 between the solid line b o u n d a r y o f the Tsl,sh----0 region and the dotted
37
Snow Slush
Thin Water
Ice
Water
C
Slush
Layer/L--
-----
I
T Icm I0 crn
~cm
j
~
Water Boundary
water
(a)
Water
j
ly Ice Layer
Fig. 5. Measured profile through snow and slush at the edge of a pond, with slow flooding from right to left. Note the different horizontal and vertical scales.
Frozen Slush Slush
-'~___~- ~ line through the origin is all conditionally unstable when there is ice and snow only, with no slush. Some flooding is needed to weigh down the ice before further flooding can be spontaneous.
O B S E R V A T I O N A L B A S I S FOR C R ~ 2 AND N~ 3
CM
Figure 5 shows the results of a typical set of measurements across the slush-snow boundary, as has been shown schematically in Fig. 2. Note that the horizontal scale is a factor of ten different from the vertical. Thus in reality the step in the snow surface is much less pronounced than shown, but it is still distinctly a step. The measurements were made by carefully cutting a trench in the snow and slush so as to disturb the wall as little as possible, photographing the wall at a low angle, and measuring from the photographs. N is at least 3 in this case, by comparing Tsnowat the far left ( = SF) with Tstush at the far right. N can be smaller in between, but that represents a time-dependent, transient condition as the water percolates from right to left in the figure. The measurements are within about + 0.5 cm because of the way they were made and, in the case of the slush-snow interface, because the interface itself has a small-scale relief (irregularity) o f about 0.5 cm. In cutting trenches of this kind in slush, it was found that when Tslus h w a s less than about 2 cm, water did not flow out to cover the ice in the trench. With Tslushgreater than about 2 cm, water immedi-
\Bubbly Ice Layer Ice I cm
H (b)
Water
Fig. 6. Cross sections through the artificial flooding feature produced by drilling a hole through the ice, shown in Fig. 4. The details are in part schematic (see text). ately and rapidly flowed from the slush onto the ice surface. Thus CR ~ 2 cm. The same result was obtained in the laboratory. When the water at 0 °C is poured into a transparent container of undisturbed snow, the level to which the snow becomes obviously soaking wet is about 2 cm above the hydrostatic water level. Figure 6 gives two schematics of the artificiallyinduced slushing feature of Fig. 4. 10 cm of snow on 6.5 cm of ice transformed to about 4 cm of slush on the ice overnight. Again, N is about 3. The temperature fell to about - 15 ° C and about 1 cm at the top of the slush had frozen the next morning (Fig. 4). The cross section through the central hole (Fig. 6 ( a ) ) shows the extent to which the melting enlarged the hole that was drilled originally. Figure 6(b) is a cross section of one of the radiating channels, showing the melting into the preexisting ice. (This rounded groove in the pre-existing, fiat ice surface has always been found under the radiating channels, and seems by itself to be unassailable proof of melting.) A thin water layer is shown between
38 the slush and the upper surface o f the original ice sheet, because when pieces of the upper, frozen slush layer were broken free and removed, the slush adhered to them, separating cleanly from the ice beneath. The thickness of this water layer was not measured. However, if the ice framework in a horizontal layer of slush does shrink, its buoyancy will keep it in contact with the upper ice layer. Neumann (1958) observed a similar layering that also might be explained this way.
DISCUSSION The roles of snow and slush in lake ice formation have been discussed before and the importance of slush formation is well appreciated (Shaw, 1965; Jones, 1969; Adams and Roulet, 1980, 1984). The motive of the present study has been to contribute to understanding the physics o f the flooding process. Neither the importance o f the capillary rise in snow to the equilibrium thickness of slush on lakes nor the probably c o m m o n instability toward slush formation on frozen lakes seem to be well known; and the posibility o f inducing slushing with salt pellets has not been suggested before, to the writer's knowledge.
ACKNOWLEDGMENTS Assistance in the field, discussions with, and comments from A. Weinheimer are greatly appreciated, as is the manuscript preparation by F. Huth.
REFERENCES
Adams, W.P. and N.T. Roulet (1980). Illustration of the roles of snow in evolution of the winter cover of a lake. Arctic, 33(1): 100-116. Adams, W.P. and N.T. Roulet (1984). Sampling of snow and ice on lakes. Arctic, 37(3): 270-275. Colbeck, S.C. (1974). The capillary effects on water percolation in homogeneous snow. J. Glaciology, 13(67): 85-97. Colbeck, S.C. (1986). Statistics of coarsening in water-saturated snow. Acta Metall., 34(3): 347-352. Daccord, G. (1987). Chemical dissolution of a porous medium by a reactive fluid. Phys. Rev. Letters, 58(5): 479-482. Daccord, G. and Lenormand, R. ( 1987). Fractal patterns from chemical dissolution. Nature, 325:41-43. Gow, A.J. and J.W. Govoni (1983). Ice growth on Post Pond, 1973-1982. CRREL Report 83-4 (U.S. Army Cold Regions Research and Engineering Laboratory, Hanover, N.H.), 25 pp. Jones, J.A.A. (1969). The growth and significance of white ice at Knob Lake, Quebec. Canadian Geographer, XIII(4): 354-372. Katsaros, K.B. (1981). Convection patterns in a pond. Bull. Amer. Meteor. Soc., 62 (10): 1446-1453. Knight, C.A. (1987). Slush on lakes. In: D.E. Loper (Ed.), Structure and Dynamics of Partially Solidified Systems, Martinus Nijhoff, Dordrecht, pp. 455-465. Neumann, H-G. ( 1958). Zellmuster auf der oberfliiche eines teiches. Beitr~igeZur Phys. der Atmos.: 30 (4): 246-253. Shaw, J.B. (1965). Growth and decay of lake ice in the vicinity of Schefferville (Knob Lake), Quebec. Arctic, 18(2): 123-131. Tokairin, A. (1977). Lake ice. Kodansha, Tokyo, Japan, 103 pp. (in Japanese). Woodcock, A.H. (1965). Melt patterns in ice over shallow waters. Limnol. Oceanogr., 10: R290-R297.