The floating ice that still raises the sea

There is a calculation every physics student does once and remembers: a glass is filled to the brim with water and an ice cube floats in it, a little of the ice standing above the rim. When the ice melts, does the glass overflow? It does not. The level stays exactly where it was, and the argument is two lines long. A floating body displaces its own weight of the liquid it floats in — that is the weight of the water that is not there, which is not an approximation but a consequence of how pressure varies with depth. So the ice cube is displacing a volume of water whose weight equals its own. When it melts it becomes water of exactly that weight, which occupies exactly that volume, and fills exactly the hole it used to make.

The same argument is often extended to the polar seas. Sea ice and the floating ice shelves around Antarctica are already afloat, so — the argument runs — their melting cannot raise the sea, and only the ice resting on land matters. The conclusion is very nearly right, and how nearly is worth computing, because the two-line argument has a hidden assumption, and the ice in the glass and the ice in the ocean differ in exactly that assumption.

The assumption in the two lines #

The argument used one fluid twice. The ice displaces its weight of the liquid in the glass, and it melts into water; the volumes are equal only if those are the same substance. When they differ, the level after melting changes by

$$\Delta V = \frac{m}{\rho_{\text{w}}} - \frac{m}{\rho},$$

the meltwater’s volume minus the volume the ice displaced, where $m$ is the ice’s mass, $\rho_{\text{w}}$ the density of fresh water and $\rho$ the density of the liquid it floats in. If the liquid is denser than water, the ice was displacing a smaller volume of a heavier liquid, and its meltwater overfills the hole. If the liquid is lighter than water — but still denser than ice, or the ice would sink — the meltwater underfills it, and the level falls.

The curve passes through zero at fresh water and nowhere else. Seawater, about 2.8 per cent denser than fresh water, gives a rise of a fifth of a millimetre in a glass — too small to see against a meniscus, but not zero. Saturated brine, 20 per cent denser, gives more than a millimetre. Castor oil, at 961 kg/m³, is one of the few common liquids in which ice floats and which is lighter than water, and in it the level falls: the meltwater sinks to the bottom as a separate layer, smaller in volume than the oil it was displacing.

Nothing in this is subtle once the formula is written down, and nothing in the original argument was wrong. It answered a question about ice in fresh water. The ocean asks a different question — and so does a salt lake, where a fresh layer can sit on brine for years, as the pond that is hottest at the bottom found, and ice melting into the top of it melts into its own kind.

The stone in the ice cube #

There is a second way to break the two-line argument, and it is a famous puzzle in disguise. Freeze a small stone into the ice cube. The cube still floats, its waterline set by the same balance that rights a ship and decides how deep it rides, and while it floats it displaces water weighing as much as the ice and the stone together. When the ice melts the stone sinks to the bottom, where it displaces only its own volume. The level falls by the difference between the stone’s weight in water and its volume.

An air bubble frozen into the ice changes nothing, and neither does a chip of wood: both float once released and go on displacing their own weight, exactly as before. Anything denser than water lowers the level, and the denser it is, the more — but never by more than the volume of water equal to its weight, the dashed line, approached by an object of infinite density and no volume at all.

The puzzle in disguise is the boat in the swimming pool: someone in a rowing boat throws a heavy iron anchor overboard, and the question is whether the pool’s level rises or falls. In the boat, the iron displaced its weight in water — several litres for a steel anchor. On the bottom it displaces its volume — under a litre. The level falls. Many people who should know better answer the other way, picturing the splash, and the argument that settles it is the same one that settles the ice cube.

Both puzzles say the same thing in different words. Something held up by a floating body displaces its weight of water; something resting on the bottom displaces its volume. Melting, sinking and throwing overboard all move a weight from the first kind of support to the second. That distinction — whether a load is floating or resting on something — is the whole of what the rest of this essay is about, scaled up from a glass to a continent.

Weighed in salt water, measured out as fresh #

The ice in the polar seas floats in seawater, at about 35 grams of salt per kilogram, and its density at 0 °C is 1028 kg/m³. Ice from a glacier or an ice shelf is fresh: it formed from compacted snow and contains almost no salt. It displaces its weight of seawater and melts into its weight of fresh water, which is 2.83 per cent larger in volume.

The meltwater does not stay fresh. It mixes into the sea, and mixing salt water and fresh water is not perfectly additive in volume: the mixture occupies very slightly less than its parts did separately, because the density of seawater is not a linear function of its salinity. The mixture heavier than either water found the same curvature, in temperature rather than salt, making two water masses denser together than either alone. Here it takes back a small part of the excess. The volume a kilogram of fresh water adds once mixed into the sea is its partial specific volume, $v - S\,dv/dS$, computed from the international equation of state for seawater, and the excess falls from 2.83 to 2.75 per cent. Mixing matters, but not much.

So floating fresh ice is not neutral. It is a source of sea-level rise about a thirty-sixth as large, per unit of ice, as ice that was never afloat — not because it was out of the water, but because the water it was displacing was salt. Jenkins and Holland pointed this out in a short paper in 2007, noting that the argument everyone had been using applied exactly only to a freshwater ocean.

Only the ice above flotation fills the sea #

The more consequential version of the same reasoning concerns ice that is not floating at all. Much of the Antarctic ice sheet rests on rock that is below sea level, in places by more than a kilometre, because the ice’s own weight has pushed the crust down and because the rock was low to begin with. Ice resting on a bed below sea level is partly standing in for seawater. If it thinned enough, it would float, and from then on it would displace its own weight of sea.

A column of ice on a bed at depth $b$ below sea level floats once it is thinner than $\rho_{\text{sea}} b / \rho_{\text{ice}}$ — 561 metres of ice on a bed 500 metres down, 1121 on a bed a kilometre down. Ice up to that thickness is, in sea-level terms, already in the ocean: it occupies a space that seawater would occupy if the ice were not there, and if it melted the sea would flow in to fill that space. Melting it adds only the freshwater excess, the small slope at the left of the figure. Only the ice above that thickness — above flotation, in glaciologists’ term — raises the sea when it melts, by 0.916 cubic metres of water per cubic metre of ice, the ratio of ice’s density to the volume a kilogram of meltwater takes up in the sea.

This is the stone in the ice cube again. The ice below flotation is like the part of the floating cube’s weight that its own volume of water accounts for; the ice above flotation is like the stone, a weight resting on the bottom. When the ice melts, the sea fills the hole left by the first part and gains the water of the second.

The figure that summarises an ice sheet’s threat to the coasts is therefore its volume above flotation, not its volume. For the West Antarctic Ice Sheet, much of whose bed lies hundreds of metres below sea level, that figure is about 3.3 metres of global sea level, by the estimate of Bamber and colleagues in 2009 — substantially less than the same ice would represent if it sat on land above the sea. It is also the calculation that tells a glaciologist where an ice sheet’s edge must be. The grounding line, where the ice lifts off the bed and becomes a floating shelf, sits exactly where the ice thins to its flotation thickness, as the log that floats on its corner would put it: where the weight of the ice equals the weight of the water it could displace. The pressure at the base of the ice, which the pressure that only knows depth fixes from the column above it, is exactly the pressure the sea would exert there at that point, and the ice lifts off when the second exceeds the first.

A thousand cubic kilometres, three ways #

Put numbers on the three kinds of ice and spread the water over the world’s ocean, about 361 million square kilometres.

A thousand cubic kilometres of ice on land, above the sea, raises it by two and a half millimetres. The same ice floating as a shelf raises it by 0.068 millimetres. Sea ice raises it by less again, because sea ice is not fresh: when seawater freezes, most of the salt is pushed out, but some is trapped in pockets of brine, and first-year sea ice holds several grams of salt per kilogram. Its meltwater is a little salty, and its excess over the displaced volume is correspondingly smaller.

The Arctic’s winter sea ice holds of the order of twenty thousand cubic kilometres. If all of it melted and never refroze, the sea would rise by about a millimetre — nothing beside the other terms in the budget, which is the real content of the familiar statement that sea ice does not raise the sea. The floating ice shelves of Antarctica hold several hundred thousand cubic kilometres, and their complete loss would add a few centimetres by this mechanism alone, about four by Jenkins and Holland’s estimate. The grounded ice behind them holds the equivalent of nearly sixty metres.

Rising when it floats, not when it melts #

There is a corollary that is easy to get backwards. A glacier flowing off a continent into the sea crosses its grounding line and becomes part of a floating shelf, and eventually breaks off as an iceberg. At which moment does it raise the sea?

At the grounding line, not at melting. While the ice rests on the bed, the water it would displace if afloat is not displaced — it is the rock that is holding the ice up. The moment the ice floats, it displaces its weight of seawater, and the sea level rises by that much at once. When the iceberg later drifts north and melts, it adds only the 2.75 per cent excess. So the rate at which the Antarctic ice sheet raises the sea is set by the flux of ice across its grounding lines, and the eventual melting of the shelves and icebergs is almost irrelevant to the volume.

This is also why the floating shelves matter so much more than their own sea-level contribution suggests. A shelf confined in a bay, scraping against its sides and resting on shoals, pushes back on the glacier feeding it — it buttresses the grounded ice and slows its flow across the grounding line. When the Larsen B shelf on the Antarctic Peninsula disintegrated in a few weeks in 2002, the glaciers that had fed it accelerated, some to several times their earlier speed, and began discharging grounded ice into the sea. The shelf’s own melting added a negligible amount; its absence let the grounded ice behind it run.

What the glass in a kitchen actually shows #

Anyone who tries the seawater version of the experiment at home will find the change hard to see, and the reason is the one effect left out of every figure here: temperature. The ice cube’s melting takes heat from the water around it — 334 joules per gram, as the heat that changes no temperature found — so 30 grams of ice in a 200-gram glass of water at 20 °C leaves the whole glass at about 7 °C. Water contracts as it cools through that range by about one part in ten thousand per degree, so the glass’s contents shrink by about a third of a cubic centimetre, lowering the level in a 7 cm glass by nearly a tenth of a millimetre.

That is less than half the 0.21 mm rise from the salt, but of the same order and of the opposite sign, and a meniscus a millimetre high hides both. Leave the glass for an hour to come back to room temperature and the thermal contraction is undone, and the salt effect remains. The ocean, which is nearly at the melting point already and enormous compared with any iceberg, sees only the salt effect.

Why the ice is there to float at all #

None of this would arise if ice behaved like most solids and sank in its own melt. It floats because water expands by about nine per cent on freezing, the same negative volume change that the melting curve that leans the wrong way found behind the backward slope of water’s melting line. An iceberg in seawater floats with $917/1028$, or 89.2 per cent, of its volume below the surface and 10.8 per cent above; the same iceberg in a freshwater lake would show only 8.3 per cent. The familiar “nine-tenths below” is a statement about salt water.

The fraction is exact only for a body that does not compress. Real ice shelves carry a layer of compacted snow, firn, tens of metres thick and much less dense than solid ice, and it is the column’s mean density that sets how high it rides — the same reckoning the balloon that floats at a density made for a gas bag, with the column’s weight compared against the water it pushes aside. A body whose density depends on how deep it is pushed has a different stability again, as the depth past which it must sink found for a compressible diver; ice is too stiff for that to matter, but firn is not, and satellite estimates of shelf thickness from the height of the surface depend on getting the firn right.

The sea itself is not quite incompressible either. The sea that stands lower because water gives found that compression under its own weight lowers the ocean surface by about thirty metres, and that any water added to the ocean is compressed a little as it is buried. The figures here count volumes at the surface, which is where a change in sea level is read.

A level that settles at once, in a sea that does not #

Every figure here is an equilibrium: the ice has melted, the water has mixed, and the level has settled. None shows the ocean’s dynamic response. Meltwater from Antarctica does not spread instantly over the globe; it is carried by currents, and the gravitational pull of the ice sheet on the surrounding sea — which piles water up against a large ice sheet — is released as the ice is lost, so that sea level near the ice actually falls and the rise is concentrated far away. The figures treat the ocean as a single basin with a single level.

The density of seawater has been taken at 0 °C and at the surface; the polar ocean beneath an ice shelf is a little colder than that and under pressure, which changes the excess by a few hundredths of a per cent. The ice’s density has been fixed at 917 kg/m³, the value for solid ice; real ice shelves have a layer of compacted snow, or firn, on top that is much less dense, which changes their flotation thickness and must be allowed for when the thickness is inferred from the height of the surface above the sea. The sea ice salinity of 6 g/kg is typical of first-year ice and too high for old ice, which drains its brine over successive summers and becomes nearly fresh.

And the column figure takes the bed as fixed. The rock under an ice sheet rises when the ice is removed — slowly, over thousands of years, as the mantle flows back — and where the bed rises, the space the sea must refill shrinks. In some parts of West Antarctica the rebound is fast enough to matter within a century, and whether it can slow the retreat of the ice is one of the questions the next figure would need to answer.

Still open: how fast the grounding lines retreat #

The arithmetic of flotation fixes what the ice will add when it goes. It says nothing about when. The rate is set by how fast the grounding lines move inland, and where the bed beneath the ice deepens inland — as it does under much of West Antarctica — a retreating grounding line moves onto deeper rock, where the ice must be thicker to stay grounded, so it carries more ice across the line and may retreat further. Whether that feedback is already running away in the Amundsen Sea sector, as some models and observations of the Thwaites and Pine Island glaciers suggest, or whether buttressing, bed rebound and the roughness of the bed will slow it, is the central uncertainty in projections of sea level for the coming centuries.

The ice cube in the glass carries the principle without any of that. A floating body displaces its weight; one resting on the bottom displaces its volume; and whatever moves a load from one kind of support to the other changes the level. The two-line argument is correct for the glass it was written about. The sea is salt, and the ice in it is partly standing on rock — and in those two differences are the 2.75 per cent and the 3.3 metres.