The ice that grows more slowly the thicker it gets

The equation that only runs forwards found the diffusion equation underneath the spreading of ink and heat, and the square root of time with which anything that diffuses spreads. The summer that reaches the cellar in December drove the same equation at its boundary with the seasons and found the heat of summer arriving underground months late. It ended with a warning: where the ground freezes, the water in it absorbs or gives up latent heat at a fixed temperature, and the frozen front becomes a moving boundary that the diffusion equation alone does not describe. That, it said, is a Stefan problem.

This essay solves the simplest one, which is also the one most people have stood on. A lake in a cold spell grows ice downward from its surface, and the thickness of that ice decides whether it is safe to walk on, whether a road can be driven across it, and how much light reaches the water below. The thickness grows as the square root of time, as though it diffused, and yet the rate is set by something diffusion knows nothing about: the heat it takes to freeze water, which has to escape through the ice itself.

Why the ice is on top at all #

Before any of this can happen the lake has to begin freezing at its surface, and that it does so is a peculiarity of water. Most liquids cooled from above sink as they cool, so the whole body cools together and freezes from the bottom up. Fresh water is densest at four degrees, as the mixture heavier than either water found: as autumn cools a lake, its surface water sinks and is replaced until the whole lake reaches four degrees, and from then on further cooling makes the surface water lighter, so it stays on top and cools to freezing alone. A thin sheet of ice then forms on a calm night once the surface has cooled slightly below zero, which it must, because ice needs a seed to start and the barrier a new phase has to climb makes perfectly clean water reluctant to freeze; in a real lake dust, frost crystals falling from the air and the shore supply the seeds within a fraction of a degree. From the moment a continuous sheet forms, the surface is sealed, the water below is cut off from the cold air, and the only way for the lake to lose heat is through the ice. Stefan’s law takes over.

Where the ice forms and where its heat goes #

Water in a lake below a layer of ice sits at 0 °C at the ice’s underside — the ice and the water are in contact at their melting point, and the water below has been mixed to that temperature or is warmer still at depth, since water is densest at four degrees. The ice’s top surface is in contact with air colder than freezing. New ice can form only at the underside, where the water is, and freezing a kilogram of water there releases 334 kilojoules of latent heat. That heat has to go somewhere, and the only way out is up, through the ice, to the cold air. The heat that changes no temperature found latent heat being exchanged at a fixed temperature; here the fixed temperature is the boundary, and the heat must be conducted away from it.

Because the ice changes slowly compared with the time heat takes to cross it, the temperature inside it is very nearly a straight line from the air’s temperature at the top to zero at the bottom, and the heat flowing up through it is the conductivity times the temperature difference divided by the thickness: $k\Delta T/h$. That heat flow is exactly what freezing the next layer releases: $\rho L\,dh/dt$. Equating them,

$$\rho L\,\frac{dh}{dt} = \frac{k\,\Delta T}{h} \quad\Longrightarrow\quad h = \sqrt{\frac{2k\,\Delta T\,t}{\rho L}}.$$

That is Stefan’s law. Josef Stefan worked it out in 1889 and 1891, motivated by measurements of the growth of sea ice made by polar expeditions, and compared it with their records. The figure draws the profile at three times: the line through the ice gets shallower as the ice thickens, the same temperature difference across a longer path, so less heat escapes per hour and less new ice forms.

The first night and the rest of the winter #

The numbers make the square root vivid. Under air ten degrees below freezing, eleven centimetres of ice form in the first day, twenty-nine by the end of a week, sixty-one by the end of a month. The first day gives more than a third of the month’s ice. At twenty degrees below freezing everything is $\sqrt{2}$ larger: the extra cold buys only forty per cent more ice, because heat flow is proportional to the temperature difference and the thickness grows only as the square root of the heat removed.

The simple law assumes the ice’s top is at the air’s temperature, which it is not: heat must also pass from the ice’s surface into the air, by convection and radiation, with a resistance of its own. Adding it as a fixed heat-transfer coefficient — twenty watts per square metre per kelvin, typical of a light breeze — slows the early growth a great deal, because when the ice is thin the air’s resistance dominates, and the late growth hardly at all, because thick ice is the larger resistance. The dashed curves start linearly and then join the square root: five centimetres after the first day instead of eleven, fifty-one after a month instead of sixty-one. Real ice on a calm, clear night grows faster than the air temperature suggests, because a clear sky radiates heat away from the surface and cools it below the air.

Only the product matters #

The law contains the temperature difference and the time only as their product, $\Delta T\,t$. That means a week at twenty below grows the same ice as a fortnight at ten below, and that a whole winter of varying cold can be summarised in one number: the sum, over days, of how far the air’s average was below freezing. It is called the freezing degree-days, and it is the number lake-ice forecasters, road builders on frozen lakes and engineers of ice bridges work with.

By Stefan’s law the ice thickness is $\alpha\sqrt{\text{degree-days}}$ with $\alpha = \sqrt{2k/\rho L}$, 3.5 centimetres per square root of a kelvin-day for clear ice. Five hundred degree-days, a month at about seventeen below or two months at eight below, grow seventy-nine centimetres. Real lakes grow less, typically half to four-fifths of that coefficient, because of the snow that falls on them, the wind that stirs and cools their surfaces unevenly, and the warmer water that may lie below. The empirical coefficients lake-ice forecasters use for snow-covered lakes, windswept lakes and rivers are all versions of Stefan’s coefficient reduced by what the simple law leaves out.

What the ice will carry #

The thickness matters because of what it bears, and the bearing grows faster than the thickness. A floating ice sheet loaded at a point bends like a plate on an elastic foundation, the water beneath pushing back where the ice is pressed down, and the load it can carry before cracking grows as the square of its thickness: twice the ice, four times the load. Rules of thumb used for travel on lake ice put this as a load in kilograms of a few times the square of the thickness in centimetres for good clear ice — a person on ten centimetres, a car on twenty or more, a heavy truck on fifty or more — with large safety margins and much smaller allowances for white ice, cracks and current.

Put that beside Stefan’s law and the result is tidy: the thickness grows as the square root of the degree-days, so the square of the thickness, and with it the load the ice can carry, grows in proportion to the degree-days themselves. Every cold day adds the same increment of bearing capacity, whether it comes early in the winter on thin ice or late on thick. The ice roads of northern Canada and Russia, built each winter across frozen lakes and rivers, are opened to heavier and heavier trucks as the degree-days accumulate, and their operators often thicken them artificially by pumping lake water onto the surface to freeze from above, where the cold air is, instead of waiting for it to grow from below through the insulation of the ice already there.

Frost in the ground #

The same arithmetic, with soil in place of ice, sets how deep the ground freezes, the question the summer that reaches the cellar in December left open. Frozen soil conducts heat to the surface; the freezing front below releases the latent heat of the water in the soil’s pores; and the depth of frost grows as the square root of the freezing index, the ground’s version of degree-days. Soil holds much less water than a lake, so it has less latent heat to release per metre and freezes deeper for the same cold, which is why the Stefan number for soil is larger and engineers use a corrected version of the law, due to Berggren, that accounts for the sensible heat. Building codes in cold countries specify foundations below the frost line it predicts, because water freezing beneath a footing expands and lifts it.

A centimetre of snow, eleven of ice #

Snow on the ice is a second layer of insulation in series with the ice. Fresh snow is mostly air, and its thermal conductivity is about 0.2 watts per metre per kelvin, a tenth of the ice’s, so each centimetre of snow resists heat as much as eleven centimetres of ice. The growth law becomes $h^2/2k + sh/k_s = \Delta T\,t/\rho L$, and the effect is dramatic: sixty days at ten below grow eighty-six centimetres of bare ice, forty-seven under five centimetres of snow, thirty under ten and only eleven under thirty. A heavy snowfall early in the winter keeps a lake’s ice thin until spring, and a skating rink cleared of snow grows ice faster than the snow-covered lake around it.

Snow does a second thing that the figure does not show. Its weight presses the ice down, and if it is heavy enough water floods up through cracks onto the ice’s surface, soaks the snow and freezes as a layer of white, bubbly snow-ice on top. Lakes in snowy climates grow much of their ice from above this way, and the snow-ice is weaker than the clear ice below it. Safe-ice guidelines treat the two separately for that reason.

What the simple law leaves out, and why it is so good for water #

The quasi-steady argument assumed the ice conducts heat instantly compared with how fast it grows, so that its temperature is always a straight line. It also ignored the heat that must be removed from the ice itself as it cools: newly formed ice at the bottom is at 0 °C, and as more forms beneath it, it must cool towards the air’s temperature, giving up its own sensible heat. Franz Neumann found the exact solution, which keeps both: the thickness still grows as the square root of time, $h = 2\lambda\sqrt{\kappa t}$, but with a coefficient $\lambda$ fixed by a transcendental equation involving the Stefan number, $\mathrm{St} = c\,\Delta T/L$, the ratio of the sensible heat the ice gives up in cooling to the latent heat it gave up in forming.

For ice the Stefan number is small, because water’s latent heat is enormous compared with ice’s heat capacity: 0.063 at ten degrees below freezing. The simple law is then within one per cent of the exact one, and within four per cent even at forty degrees below. That is a property of water, not of the method. A molten metal solidifying in a mould, or the ground freezing where it holds little water, has a much larger Stefan number — less latent heat to release, more sensible heat to remove — and the simple law overestimates the growth by tens of per cent. The moving-boundary problem Stefan posed for polar ice now governs the casting of metals, the freezing of food and the growth of crystals, and in most of those the exact treatment matters.

The square root of time, for a different reason #

The square root in Stefan’s law looks like the square root of diffusion, the jiggle that proved atoms or the spread of ink, and it is worth seeing that it comes from somewhere else. A diffusing quantity spreads as $\sqrt{Dt}$ because a random walk’s displacement grows as the square root of the number of steps, the walker of the walk that comes home wandering back and forth and getting, on average, only that far. The ice grows as $\sqrt{t}$ because its growth rate is inversely proportional to its own thickness: it slows itself down. Any process in which the thing being built is the obstacle to building more grows the same way. The oxide layer on a piece of silicon heated in oxygen thickens as the square root of time once it is thick, because oxygen must diffuse through the oxide already formed to reach the silicon; a layer of corrosion product on a metal, the dried crust on a drying lake bed, a carbonated layer advancing into concrete, all show the same law for the same reason. The ice on a lake is the most visible member of the family. In every one of them the evidence is the same: double the time and the layer grows by only forty per cent, because the second half of the time is spent pushing through the layer the first half built.

What the pictures cannot show #

The figures treat a lake as a still body of water at 0 °C under ice with uniform properties and a steady air temperature. Real lakes have currents and springs that bring warmer water to the ice’s underside, which slows or reverses growth locally and makes ice thickness patchy; the air temperature varies daily, which the degree-day summary handles well and the steady curves do not; and the surface heat loss depends on wind, cloud and the sky’s radiative temperature, not a fixed coefficient. Snow’s conductivity varies from under 0.1 for fresh powder to over 0.5 for wind-packed snow, and its depth changes through the winter. The degree-day band is a typical range from practice, not a computed result. Nothing here is a guide to whether ice is safe: that depends on the ice’s quality and condition, which no growth law predicts.

Still open: how fast Arctic sea ice grows under thinner snow and warmer seas #

Sea ice is the same ordinary ice — the low-pressure form that the melting line that turns round found is less dense than its liquid, which is why it floats and why the whole problem has its cold side on top — complicated by salt, which is expelled from the growing ice as brine and makes the ice’s conductivity and the freezing point depend on its history, and by an ocean below that delivers heat to the ice’s underside. Stefan’s law, with those corrections, remains the core of the models of how much new ice forms each Arctic winter. With the Arctic warming, the ice is thinner and younger, the snow on it arrives later, and the ocean heat beneath it has risen; thinner ice grows faster, by the same square root, which partly offsets the summer losses. How large that negative feedback is, and how it interacts with changing snowfall on the ice, is among the questions about the future of Arctic sea ice that observations and models are still settling.

The habit worth carrying away is to ask what a growing layer has to pass through to keep growing. Ice forms at a lake’s underside and its latent heat must be conducted up through the ice already formed, so it slows itself: $h = \sqrt{2k\,\Delta T\,t/\rho L}$, 11 cm in the first day at 10 K below freezing and 61 cm in a month, depending only on the degree-days — and a centimetre of snow on top insulates like eleven of ice. The square root here is not a random walk but a layer that is its own obstacle.