The setting for the development of ice cover in lakes is the annual evolution of the temperature structure of lake water. In most lakes during the summer, a layer of warm water of lower density lies above colder water below. In late summer, as air temperatures fall, this top layer begins to cool. After it has cooled and has reached the same density as the water below, the water column becomes isothermal (i.e., there is a uniform temperature at all depths). With further cooling, the top water becomes even denser and plunges, mixing with the water below, so that the lake continues to be isothermal but at ever colder temperatures. This process continues until the temperature drops to that of the maximum density of water (about 4°C, or 39°F). Further cooling then results in expansion of the space between water molecules, so that the water becomes less dense.
This change in density tends to create a new stratified thermal structure, this time with colder, lighter water on top of the warmer, denser water. If there is no mixing of the water by wind or currents, this top layer will cool to the freezing point (0°C, or 32T). Once it is at the freezing point, further cooling will result in ice formation at the surface. This layer of ice will effectively block the exchange of energy between the cold air above and the warm water below; therefore, cooling will continue at the surface, but, instead of dropping the temperature of the water below, the heat losses will be manifested in the production of ice.
The simple logic outlined above suggests that water at some depth in lakes during the winter will always be at 4°C (39°F), the temperature of maximum density, and indeed this is often the case in smaller lakes that are protected from the wind. The more usual scenario, however, is that wind mixing continues as the water column cools below 4°C, thereby overcoming the tendency toward density stratification. Between 4 and 0°C (39 and 32T), for example, the density difference might be only 0.13 kilogram per cubic metre (3.5 ounces per cubic yard). Eventually some particular combination of cold air temperature, radiation loss, and low wind allows a first ice cover to form and thicken sufficiently to withstand wind forces that may break it up. As a result, even in fairly deep lakes the water temperature beneath the ice is usually somewhere below
4°C and quite often closer to o°C. The temperature at initial ice formation may vary from year to year depending on how much cooling has occurred before conditions are right for the first initial cover to form and stabilize. In some large lakes, such as Lake Erie in North America, wind effects are so great that a stable ice cover rarely forms over the entire lake, and the water is very near o°C throughout the winter.
Before ice can form, water must supercool and ice crystals nucleate. Homogeneous nucleation (without the influence of foreign particles) occurs well below the freezing point, at temperatures that are not observed in water bodies. The temperature of heterogeneous nucleation (nucleation beginning at the surface of foreign particles) depends on the nature of the particles, but it is generally several degrees below the freezing point. Again, supercooling of this magnitude is not observed in most naturally occurring waters, although some researchers argue that a thin surface layer of water may achieve such supercooling under high rates of heat loss. Nucleation beginning on an ice particle, however, can take place upon only slight supercooling, and it is generally believed that ice particles originating from above the water surface are responsible for the initial onset of ice on the surface of a lake.
Once ice is present, further formation is governed by the rate at which the crystal can grow. This can be very fast. On a cold, still night, when lake water has been cooled to its freezing point and then slightly supercooled on the surface, it is possible to see ice crystals propagating rapidly across the surface. Typically, this form of initial ice formation is such that the crystal c-axes are vertically oriented—in contrast to the usual horizontal orientation of the c-axis associated with later thickening. Under ideal conditions these first crystals may have dimensions of one metre (about three feet) or more. An ice cover composed of such crystals will appear black and very transparent.
If the lake surface is exposed to wind, the initial ice crystals at the surface will be mixed by the agitating effects of wind on the water near the surface, and a layer of small crystals will be created. This layer will act to reduce the mixing, and a first ice cover will be formed consisting of many small crystals. Whether it is composed of large or small crystals, the ice cover, until it grows thick enough to withstand the effects of later winds, may form and dissipate and re-form repeatedly. On larger lakes where the wind prevents a stable ice cover from initially forming, large floes may be formed, and the ice cover may ultimately stabilize as these floes freeze together, sometimes forming large ridges and piles of ice. Ice ridges generally have an underwater draft several times their height above water. If they are moved about by the wind, they may scour the bottom in shallower regions. In some cases — particularly before a stable ice cover forms—wind mixing may be sufficient to entrain ice particles and supercooled water to considerable depths. Water intakes tens of metres deep have been blocked by ice during such events.
Once an initial layer of ice has formed at the lake surface, further growth proceeds in proportion to the rate at which energy is transferred from the bottom surface of the ice layer to the air above. Because at standard atmospheric pressure the boundary between water and ice is at 0°C (32 °F), the bottom surface is always at the freezing point. If there is no significant flow of heat to the ice from the water below, as is usually the case, all the heat loss through the ice cover will result in ice growth at the bottom. Heat loss through the ice takes place by conduction; designated ^ in the figure, it is proportional to the thermal conductivity of the ice (ki) and the temperature difference between the bottom and the top surface of the ice (Tm - Ts), and is inversely proportional to the thickness of the ice (h). Heat loss to the air above (also designated occurs by a variety of processes, including radiation and convection, but it may be characterized approximately by a bulk transfer coefficient (H ) times the difference between the surface x ia'
temperature of the ice and the air temperature (Ts - Ta).
(In practice, the top surface of an ice layer is not at the air temperature but somewhere between the air temperature and the freezing point. The exact figures are rarely available, but fortunately the top surface temperature, Ts, is not needed for analysis.)
Assuming that the heat flow through the ice equals the heat flow from the surface of the ice to the air above, the following formula for the thickening of ice may be fashioned:
In this formula h is the thickness of the ice, T is the
' a air temperature, Tm is the freezing point, k is the thermal conductivity of ice (2.24 watts per metre kelvin), pi is the density of ice (916 kg per cubic metre [57 pounds per cubic foot]), L is the latent heat of fusion (3.34 x 105 joules per kilogram), and t is the time since initial ice formation. The exact value of the bulk transfer coefficient (H. )
ia depends on the various components of the energy budget, but it usually falls between 10 and 30 watts per square metre kelvin. Higher values are associated with windy conditions and lower values with still air conditions, but, with other information unavailable, a value of 20 watts per square metre kelvin fits data on ice growth quite well. The formula is particularly useful in predicting growth when the ice cover is thin. The first growth rate of the ice cover is proportional to the time since formation; as the ice thickens, however, the top surface temperature more closely approaches the air temperature, and growth proceeds proportional to the square root of time.
If there is a snow layer on top of the ice, it will offer a resistance to the flow of heat from the bottom of the ice surface to the air above. In this case, the incremental thickening rate (that is, the incremental thickening [dh] in an incremental time period [dt]) may be predicted by the following formula:
where h is now the ice thickness with thermal conductiv-
i ity k, andhsisthe snowthicknesswiththermalconductivity ks. The thermal conductivity of snow depends on its density. It is greater at higher densities, ranging from about 0.1 to 0.5 watt per metre kelvin at densities of 200 to 500 kg per cubic metre (12 to 31 pounds per cubic foot), respectively
When the weight of a snow cover is sufficient to overcome the buoyancy of the ice supporting it, it is usual for the ice to become submerged and for water to flow through cracks in the ice and saturate the snow, which then freezes. This mode of ice growth is different from that analyzed above, but it is quite common, and the ice so formed is known as snow ice. At typical snow densities, a layer of snow about one-half the thickness of the supporting ice will result in the formation of snow ice layers.
As the ice thickens, there is a tendency for crystals with a horizontal c-axis orientation to wedge out adjacent crystals with a vertical c-axis orientation and so become larger in diameter with depth. The resulting structure is one of adjacent columns of single crystals and is termed columnar ice. When a very thin section of the ice is cut and examined with light through crossed polaroid sheets, the crystal structure is clearly seen.
As the temperature of the surrounding air and liquid water increases, water sequestered as solid ice reverts to its liquid state. At the microscopic level, such a reversion occurs by thinning and rotting. Wholesale melting over a lake's surface is influenced by the lake's relationship with its shoreline, warm water from rivers, and the amount of solar radiation it can absorb.
In the spring, when average daily air temperatures rise above the freezing point, ice begins to decay Two processes are active during this period, a dimensional thinning and a deterioration of the ice crystal grains at their boundaries. Thinning of the ice layer is caused by heat transfer and by melting at the top or bottom surface (or both). Deterioration, sometimes called rotting or candling because of the similarity of deteriorating ice crystals to an assembly of closely packed candles, is caused by the absorption of solar radiation. When energy from the Sun warms the ice, melting begins at the grain boundaries because the melting point there is depressed by the presence of impurities that have been concentrated between crystal grains during the freezing process.
Rotting may begin at the bottom or at the top, depending on the particular thermal conditions, but eventually the ice rots throughout its thickness. This greatly reduces the strength of the ice, so that rotten ice will support only a fraction of the load that solid, unrot-ted ice will support. Thinning and deterioration may occur simultaneously or independently of each other, so that sometimes ice thins without internal deterioration, and sometimes it deteriorates internally with little or no overall thinning. However, both processes usually occur before the ice cover finally breaks up.
Deteriorating ice has a gray, blotchy appearance and looks rotten. Because rotting takes place only by absorption of solar radiation, it progresses only during daylight hours. In addition, the presence of snow or snow ice, which either reflects most solar radiation or absorbs it rapidly in a thin layer, acts to prevent rotting of the ice below until the snow has been completely melted.
Melting of lake ice usually occurs first near the shorelines or near the mouths of streams. At these points of contact with inflowing warm water, the ice melts faster than it does at central lake locations, where most melting is caused by the transfer of heat from the atmosphere. Estimates of the rate at which thinning of the main ice cover occurs are usually based on a temperature index method in which a coefficient is applied to the air temperature above freezing.
Water temperature beneath the ice usually reaches its coldest at the time of freeze-up and then gradually warms throughout the winter. The warming is caused by the absorption of some solar radiation that has penetrated the ice cover, by the release of heat that has been stored in bottom sediments during the previous summer, and by warm water inflows. In deep lakes such warming is slight, while in shallow lakes it may amount to several degrees. After snow on the ice has melted in the spring, more solar radiation penetrates the ice cover, so that significant warming may occur. The mixing of warmed water with
deteriorated ice is responsible for the very rapid clearing of lake ice at the end of the melt season. On most lakes, the timing of the final clearing of ice is remarkably uniform from year to year, usually varying by less than a week from the long-term average date of clearing.
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