The lake that has to cool all the way down first

The layer a parcel cannot leave took stratification as given, a stack with the light on top that resists being stirred. The lake that holds its gas until it does not found one way that stack can undo itself. This essay is about a lake in autumn, and about one small numerical oddity of water that shapes it. A pond and a very deep lake on the same line of latitude, in the same weather, will behave differently through a winter, and for a reason that is hidden from anyone who only looks at the surface. A person standing on the shore can see only one thing that differs, and it is the single fact they would not guess matters: how much water there is underneath.

The oddity is that cold water is not the densest water. Almost every other liquid contracts all the way to freezing. Water contracts down to a little below four degrees, then expands.

The amount is tiny: about 0.13 kilograms per cubic metre, on a thousand. In a world without gravity no one would notice it. But the whole of convection works on tiny density differences, as the water that warms on the way down and many others found, and what it matters to here is not how big the difference is but which way it goes. Above 3.98 °C, cooling makes water heavier. Below it, cooling makes water lighter. So a parcel of surface water that cools will either sink or stay, according to which side of the line it is on.

The reason lies with the structure of ice. Each molecule in ice is held in an open tetrahedral cage by hydrogen bonds, with a great deal of empty space between. In the liquid, those bonds are forever breaking and re-forming; there are clusters of ice-like order, more of them at lower temperatures. As water is cooled, two effects compete. Ordinary thermal contraction brings molecules closer. The growth of open clusters pushes them apart. At 3.98 °C the two are exactly balanced, and below that the clusters win. That balance is why the number is what it is, and calculations of it from first principles have never been easy.

What a lake does when it gets cold #

Consider a lake at the end of summer. A warm layer sits on top, perhaps 18 °C, about ten metres deep, and below it cold water, perhaps 6 °C, has been sitting since last spring. Autumn comes, the air is cold, and heat leaves at the surface.

Cooling makes the top layer denser, so it sinks, stirring until it is well mixed, as the plume that stops below the surface showed in a different setting. As it cools, the mixed layer eats down into what lies beneath. When the warm layer has cooled to match the cold layer, the whole lake is one temperature, and the barrier between them is gone. In a lake with a wind on it this is often a dramatic event. It is called overturn, and it brings up nutrient-rich water from the bottom and takes oxygen down, and a lake can change colour in days. Cooling and mixing go on, and now they take the whole depth with them. The entire lake cools as one.

Up to this point, nothing about the behaviour is peculiar to water. But at 3.98 °C, the whole lake reaches its maximum density. Now a little more cooling at the surface makes that thin top layer lighter, not heavier. It stays where it is. The next loss of heat falls only on that layer, which cools further and is lighter still. All the machinery of convection has stopped. Nothing beneath it takes any further part.

The surface then falls from four degrees to zero very fast. In 85 days it dropped from 18 to 4, because in that time it was taking everything underneath with it. In 5 more it drops the last four degrees, and ice appears. The whole depth was reduced to four degrees first, and only the top metre or two, mixed by wind, had any part in the last act. A thin skin of cold water floats on a lake that is all at the same temperature, and the first ice, when it forms, rests on water a few degrees above zero. Underneath, the bottom stays near four degrees for the whole winter. This is a large part of why fish survive it, though the common story that they would otherwise be frozen solid is an exaggeration. A lake freezing from the bottom up would be a problem, but the top-down freezing has more to do with ice floating than with the density maximum.

The same weather and very different winters #

This is what it means for the surface to be misleading. Look at three lakes under the same steady cooling.

The shared behaviour of the first two months is that they all cool only their warm surface layer. Then the curves part where each lake’s whole column comes in. Sixty metres has a large store of heat at six degrees to get rid of, and so it hangs near four for a long time. At the plateau, an observer on the shore sees nothing change for weeks: same surface temperature, same steel-grey water. Then in a few days the surface crashes through to freezing. The shape of the curve is really a measure of depth. Add the fact that real lakes have a mean depth that varies across their surface, and it follows that a lake will often freeze first over its shallow bays and last over its deep basin, even though the surface temperature everywhere seemed the same through the autumn.

The depth of the lake thus sets how much of the winter is needed. Take a representative steady cooling and ask how long it takes.

The mean depth of Lake Erie is 19 m and it commonly freezes almost completely in a cold winter. Lake Ontario, on the same chain with similar air, has a mean depth of 86 m and rarely does, and Superior at 147 m has not been observed to do so completely except in the most extreme years, with heavy ice coverage only. This figure’s number of about 40 m is not a sharp limit, as real lakes have very unsteady heat loss and very strong winds that keep a cold surface layer mixed down. Large cold lakes are also well known for keeping their shipping lanes open through winters that freeze their small neighbours. But the order of magnitude is right and the explanation does not need ocean-scale effects. A deep lake is storing the summer down to four degrees and cannot be rid of it in time.

That is why Baikal is a puzzle: it has a mean depth of over 700 m and it freezes every winter. The reason is that the density maximum depends on pressure. It falls by about 0.02 °C per bar, so water deep in Baikal has its maximum at a lower temperature than water at the surface, and a layer near the top reaches its maximum long before the great depths do. This separates the lake’s upper water from its lower and limits how far down the autumn overturn has to go. The depth that has to be cooled is then much less than the depth of the lake. So the figure’s rule is right about what limits ice and wrong about how much of the lake counts, in the special case of a lake so deep that pressure matters.

Why the sea is different #

Salt changes the arithmetic, and the change is one of the more elegant things in oceanography. Dissolving salt in water lowers the temperature of maximum density, and it also lowers the freezing point. But it lowers them at very different rates. The maximum falls by about 0.2 °C per gram of salt per kilogram of water and the freezing point by only 0.055.

They cross at a salinity of 24.6 grams per kilogram. That line divides the world’s waters, not at a point on any map but in the arithmetic of two curves. Seawater, at about 35, is well beyond it. For such water the density maximum lies below the freezing point, which is to say it is not reached. As long as the water is liquid, cooling makes it heavier. A cold surface always sinks. So there is no temperature at which a surface layer in the open sea can quietly float on warmer water, and the entire column would need to be taken down to freezing, about −1.9 °C, before ice could form. The North Atlantic is several kilometres deep; a winter would not touch that.

And yet sea ice forms every year over millions of square kilometres. The explanation is salt again, working in a different way. Salinity changes density far more strongly than temperature does in cold water. In the Arctic, a layer of relatively fresh water, from rivers and melting ice, sits on the saltier water beneath, in a boundary called the halocline, with a density step that cooling cannot overcome. Overturning reaches only the bottom of that layer. So the depth that has to be cooled to freezing is set by the fresh layer’s thickness, a matter of tens of metres at most. Where the fresh cap is missing or thin, as in the Greenland Sea, the water overturns to great depths instead and no ice forms at all over that part of the sea. That deep overturning is a major driver of the ocean’s circulation, and it is helped along by the salt that sinks through a stable sea, in places where warm salty water lies over cold fresh.

The Baltic is on the other side of the line. At about 7 grams per kilogram in its surface water, it has a maximum density near 2 °C, marked on the figure above. It freezes the way a lake does, with an overturn that stops at its own low maximum, and with a stronger halocline below. It is a rare example of a large sea that plays by a lake’s rules, and that fact has mattered to shipping there for a long time.

Two waters that are denser when mixed #

A curved density function does something else, and it has no parallel in a straight one. Take equal parts of water at 2 °C and at 6 °C. They have nearly the same density, as they sit about equally far from the maximum on either side. Mix them, and the result is water at 4 °C, which is denser than either. So the two can float side by side without any tendency to overturn, and if they meet and mix, the mixture sinks. That is called cabbeling. The mixture heavier than either water treated the case for sea water, in which temperature and salt together produce a similar effect, and in which it matters for how Antarctic water sinks. In a lake, it has its most visible form in the thermal bar described below.

The curve has another effect on what happens once a lake has frozen. With the top sealed by ice, the water beneath stops feeling the wind. Heat continues to reach it from the sediments, a store from summer that leaks up at about a watt per square metre, and sunlight comes through clear ice. The water below is at nearly four degrees at the bottom and zero just under the ice, so it is a stable stack on which the layer a parcel cannot leave operates. Thickening of the ice from then on follows the slow square-root law of the ice that grows more slowly the thicker it gets, since the ice itself now insulates. The lake is well sealed. A metre of ice on a lake passes less heat than a very cold wind could take directly, and that is a good part of why a lake that is frozen on top so rarely freezes to its bottom. Only a pond of a metre or two, in the coldest climates, freezes all the way down.

A lake’s stable winter stack and a summer one do not look the same, and neither needs the same kind of stirring to break. Both can show a remarkable contrast with what the pond that is hottest at the bottom relies on, where a strong gradient of salt holds hot water beneath cold, so a stack can be held stable by composition in the face of temperature. This lake has only the curvature of a single density curve to work with. The whole of the annual pattern of mixing, and with it most of what lakes do for living things, depends on two crossings of the same line at 3.98 °C, in spring and in autumn.

The result is that lakes have been classified by how often they turn over. A dimictic lake crosses that line twice a year. A warm monomictic lake does not freeze and turns over once in winter. A cold monomictic one stays under four degrees all year and turns over once, in summer. Meromictic lakes have a permanent stack of salt, and their deep water may stay isolated for centuries.

Ice from water that is not ready #

The sum assumes a certain neatness that is not always there. The figures use a still lake with wind only to mix a top two-metre layer, heat lost only through the surface and a single column with no sideways flow. Real lakes have rivers, which are often warmer or colder, and wind that can drive currents down the whole depth of a shallow one. Sun shining through ice warms the water directly below it in late winter, so that a thin layer warmer than four degrees can even form beneath the ice, in a pattern opposite to what the simple picture would suggest.

A large lake also shows an effect that a column cannot. Shallow coastal water, being of smaller depth, reaches 3.98 °C first. As it goes on cooling it is less dense than the slightly warmer water further out, and where the two meet there is a line of water at the maximum density that sinks. It is called a thermal bar, and it can run for hundreds of kilometres parallel to the shore, moving slowly offshore. For weeks it blocks the mixing of nearshore water with the water of the centre of the lake. Rivers’ pollutants can get trapped against the coast during that time, and it is a real consideration in managing a lake such as Ontario.

The surface-layer treatment also says nothing about how ice grows once it has formed, a question the ice that grows more slowly the thicker it gets took up, and it treats heat loss as constant, whereas in a real lake it falls when the surface has cooled close to the temperature of the air and changes completely under a layer of snow. Those are matters of magnitude for a forecast, and a limnologist would need all of them. The structure that the figures show is independent of them. A surface below four degrees, with a whole lake sitting beneath it, is possible only once the whole lake has been through four.

Still open: what a warmer winter does to a lake’s turnover #

Many temperate lakes turn over twice a year, once in autumn as described and again in spring, when the surface warms up through four degrees and the column is again uniform. Those two overturns are the lake’s main means of bringing oxygen down to its deep water. In a warming climate, some lakes that have always frozen are ceasing to, and at that point a lake that mixed twice a year mixes once, or in a deep lake, not fully at all. Lake Geneva’s deep water has gone without a complete overturn for over a decade at a time. Records show deep-water oxygen falling in many such lakes. How far this trend goes for any given basin depends on wind and on how the depth profile distributes heat, and it is not a matter that a single-column model can decide. Whether the very largest deep lakes would stop their deep overturns altogether, and what that would do to their ecosystems, is under study with long-term monitoring that is, for the largest lakes, barely half a century old.

Water is densest at 3.98 °C, so a lake must cool as a whole down to that value before its surface can fall further: 85 days for a 30 m column at 100 W/m², then 5 days to ice; above 24.6 grams of salt per kilogram, the density maximum falls below the freezing point, and the sea has no such stopping place and freezes only where a fresher layer floats on top. A difference of one part in eight thousand settles who freezes, as a lake with a cap of gas was settled by a slightly different density. The habit worth carrying from this is to ask, of a surface that seems to be doing nothing, what lies underneath that the surface is waiting for.