The grounding line with nowhere to stop

The floating ice that still raises the sea worked out what an ice sheet adds to the ocean when it goes, and found that what counts is the ice above flotation: the part of a marine ice sheet that the rock is holding up rather than the sea. It ended by noting that the arithmetic of flotation says nothing about when, and that the rate is set by how fast the grounding lines — the lines where ice resting on rock lifts off and begins to float — move inland. That is where this essay begins. The same principle of Archimedes that fixed the sea-level arithmetic also fixes how much ice crosses a grounding line, and with it whether a grounding line can stand still at all.

The answer depends on the shape of the rock. Over much of West Antarctica the bed lies hundreds of metres below sea level at the coast and deepens towards the interior, reaching more than two kilometres below the sea in the deepest basins. Johannes Weertman argued in 1974 that an ice sheet whose grounding line rests on such a bed cannot be in a stable steady state; John Mercer drew the consequence for West Antarctica in 1978; and Christian Schoof put the argument on a firm footing in 2007, with a formula for the ice flux across a grounding line that turns it into a calculation a single figure can carry. The figures here use the bed, densities and constants of the Marine Ice Sheet Model Intercomparison Project, MISMIP, a shared test case that ice-sheet models are checked against: an idealised continent whose bed falls below the sea, rises again in a broad hump, and then drops away to the deep ocean.

Buoyancy sets the thickness at the line #

A slab of ice resting on rock below sea level is a body sitting on the bottom of a tank, and the block the water does not lift found what that means: the water presses on its sides and its top, but where rock and ice are in contact, the water cannot get underneath to push up. The rock carries the weight. Thin the ice, and at some thickness the water’s push from below, wherever it can reach, would exceed the ice’s weight; the ice lifts off and floats. The grounding line is the place where that happens, and at that place the ice is exactly as thick as it must be to just touch the bottom:

$$\rho_i h_f = \rho_w d \quad\Rightarrow\quad h_f = \frac{\rho_w}{\rho_i}\,d,$$

with $d$ the depth of the water and $\rho_i$, $\rho_w$ the densities of ice and seawater. For ice of 900 kg/m³ in water of 1,000, the values the benchmark uses, the thickness at the line is 1.11 times the depth; for real glacier ice and seawater, 917 and 1,028, it is 1.12. A grounding line in water 723 metres deep has ice 803 metres thick crossing it, with a ninth of that, 80 metres, standing above the sea.

It is the principle the weight of the water that is not there derived from the pressures on a body’s faces, applied at the one place where the faces change from rock to water. And it is not a statement about how much snow falls on the ice sheet, or how fast the ice flows, or how warm it is. It is Archimedes’ principle applied at one place, and it holds whatever the rest of the ice sheet is doing. Everything else about the grounding line follows from combining it with one other fact.

The numbers it gives for West Antarctica are large. Where the bed lies 2 kilometres below the sea, the ice at a grounding line there would be 2.24 kilometres thick, and every metre of it below the waterline is ice whose weight the rock, not the sea, is carrying — the ice above flotation that the sea-level arithmetic counted. Where the bed lies 600 metres down, the ice at the line is 670 metres thick. At the grounding line the depth of the bed is the thickness of the ice, multiplied by a ratio of densities that nothing in the climate changes.

Twice the depth, twenty-seven times the ice #

The other fact is how ice flows. Ice creeps under its own weight, with a strain rate that rises as the cube of the stress, Glen’s law; and where it slides over a wet bed the friction rises only weakly with speed. Near a grounding line the ice changes from resting on the bed, where it is held back by friction, to floating, where nothing holds it but the stretching of the shelf beyond. Schoof showed that over a short boundary layer at the line, the ice’s flow adjusts so that the flux across the line depends only on the thickness there, as a power:

$$q \propto h_f^{(m + n + 3)/(m + 1)},$$

with $n = 3$ for Glen’s law and $m = 1/3$ for a commonly used sliding law. The exponent is $19/4$. Ice flux rises with grounding-line thickness faster than any other quantity in the problem.

Put flotation into it and the flux is a steep function of the water depth at the grounding line. Doubling the depth doubles the thickness, and doubling the thickness multiplies the flux by $2^{19/4}$, almost twenty-seven. A grounding line in 500 metres of water exports a few hundredths of a cubic kilometre of ice a year for each kilometre of coast; one in two kilometres of water exports some seven hundred times as much. Depth is the dial, and buoyancy is what connects the dial to the flow.

What falls and what leaves #

A steady ice sheet must send across its grounding line all the snow that falls on the grounded part upstream. With snow accumulating at a rate $a$ over the distance $x_g$ from the ice divide to the line, the balance is

$$a\,x_g = q(x_g),$$

where the right side is the flux at whatever depth the bed has at $x_g$. The left side rises steadily with distance. The right side follows the bed: it rises where the water deepens seaward and falls where the water shallows. The next figure draws both against position for the benchmark sheet.

The curves cross three times, and the crossings are not alike. At 1,376 kilometres the bed deepens seaward. Push the grounding line a little further out and it sits in deeper water, its ice is thicker, its export rises more steeply than the snowfall, and the sheet loses ice until the line comes back. Pull it a little inland and the opposite happens: the export falls below the snowfall, the sheet thickens, and the line advances. That is a stable equilibrium. At 800 kilometres the same holds.

At 1,124 kilometres the bed deepens inland, and everything reverses. Pull the grounding line inland and it sits in deeper water — that is what a bed deepening inland means — so the ice there must be thicker to touch the bottom, and the export at the line goes up, while the snowfall upstream has gone down. The sheet loses more than it gains and the line retreats further, into deeper water still. Push it seaward and it exports less than it gains and advances. Any small displacement grows. The equilibrium exists on paper and nowhere else.

This is the same structure as the diver that the depth past which it must sink trimmed to hang at one depth in a tank: a balance that exists, with a restoring force of the wrong sign. There, a diver pushed deeper had its gas squeezed and lost buoyancy, and so sank further. Here, a grounding line pushed inland finds deeper water, exports more, and so retreats further. In both, the one quantity that sets the force rises in the direction that makes the displacement worse.

The test is the one why a ship comes back upright applied to a hull: not whether a balance exists, but how the imbalance changes when the system is displaced from it. A ship is stable when heeling it moves the centre of buoyancy further out than the centre of gravity, so that the righting moment grows with the heel. A grounding line is stable when moving it seaward raises the export faster than it raises the snowfall upstream, so that the imbalance grows against the displacement. The quantity that decides is a slope — of the righting arm for the ship, of the export curve for the ice sheet — and its sign at the balance point is everything. The log that floats on its corner found a beam with more than one balanced attitude and only some of them stable; the ice sheet has more than one balanced position, and which are stable is decided by the rock.

The criterion is general, and it is sharper than the benchmark makes it look. Wherever the bed deepens inland, the flux at the line rises as the line retreats, while the snowfall it must balance falls; unless something else intervenes, no steady grounding line can sit on any such slope.

Advance and retreat take different paths #

The rate factor $A$ in Glen’s law measures how readily the ice flows. Warmer ice is softer, and so, for the purposes of the flux formula, is ice sliding on a slipperier bed or no longer held back by a confining shelf. Changing $A$ moves the export curve up or down against the fixed snowfall line, and the balanced positions move with it.

The result is an S. For stiff ice, at the left, there is one equilibrium, far out on the seaward side. For soft ice, at the right, there is one far inland. In between, from $A = 5.0 \times 10^{-26}$ to $2.1 \times 10^{-25}$, there are three: two stable, on either side of the inland-deepening stretch, and the unstable one on it. Start from stiff ice and soften it slowly, and the grounding line creeps inland along the upper branch until the branch ends, at the right edge of the window, where its equilibrium merges with the unstable one and both vanish. There is then nowhere nearby to stop, and the line must cross the whole inland-deepening stretch to the lower branch. Stiffen the ice again and the line does not come back along the same path. It creeps seaward along the lower branch, all the way to the left edge of the window, before that branch ends and it jumps back out.

This is the shape the boiling curve that turns back found in heat transfer from a hot surface: a curve that folds over itself, with a middle branch that is never seen and two outer branches between which the system jumps, at different places going up and going down. For an ice sheet the consequence is that its present position is not a function of its present climate. A sheet that has crossed an inland-deepening basin under one set of conditions will not return when those conditions are restored. It needs them pushed well past where they were.

There is evidence that West Antarctica has been round such a loop. During the last interglacial, about 125,000 years ago, the world was slightly warmer than before industrialisation and the sea stood six to nine metres higher than today, more than Greenland and the warming of the ocean together seem able to account for. Sediment cores drilled through the ice shelves of the Ross Sea hold the remains of open-water organisms from warm periods further back, in the Pliocene and the early Pleistocene, when the ice must have been gone from places it now covers to a depth of hundreds of metres. And in 2023 a study of the genomes of a small Antarctic octopus found that populations now separated by the whole width of the West Antarctic ice sheet had exchanged genes directly about 125,000 years ago, as though a seaway had opened through where the ice now stands. The ice sheet has evidently collapsed and regrown, and the regrowth needed the much colder conditions of a glacial period to push it back out across the deep basins — the lower branch of the S, followed all the way to its end.

Slow, then fast, then slow #

The balance tells where a grounding line can rest. It does not say how fast one moves when it cannot, because that needs a model of how losing ice at the line turns into thinning, and thinning into retreat, over the whole sheet. The last figure uses the crudest model that has the right ingredients — the flux at the line, the snowfall upstream, and a profile for the grounded ice so that a change in volume becomes a change in position — to watch the jump happen.

The retreat has three parts. For the first thousands of years the grounding line moves slowly, at tens of metres a year, because it is still on bed that shallows inland: each step back takes it into shallower water, the export falls, and the imbalance shrinks. Once it reaches the inland-deepening stretch the imbalance grows instead, and the retreat accelerates — to about two hundred metres a year in this model — and carries on past the end of the deepening stretch, because there is still no balanced position until the line reaches the inner stable state at 722 kilometres. There the bed shallows inland again, the export falls to the snowfall, and the line stops.

The model’s timings are its own. It has no ocean, no ice shelf, and an ice profile drawn rather than computed, and real grounding lines on the Amundsen Sea coast have been measured retreating at hundreds of metres to kilometres a year, driven by warm ocean water melting the ice from below. What the model does carry is the shape of the record, and it is a shape that matters for anyone watching a glacier: a slow start says nothing about how fast the middle will be, because the speed is set by the bed the line has not yet reached.

What the figures leave out #

The figures use a flowline: one dimension, a glacier of infinite width. In a real basin the ice is confined between walls and, often, held back by a floating shelf jammed against islands and the sides of a bay, which pushes back on the grounding line. That buttressing changes the flux formula, and Hilmar Gudmundsson and colleagues showed in 2012 that strong enough buttressing can hold a grounding line stable even on a bed that deepens inland. The instability is a property of an unbuttressed line; whether a given glacier is in that state depends on its shelf, which is why the loss of ice shelves is watched so closely.

The bed is fixed. Under a retreating ice sheet the rock rebounds as its load is removed, and in parts of West Antarctica the mantle beneath is soft enough for the rebound to be fast — centimetres a year — which raises the bed under the grounding line and can slow or halt a retreat. The sea level is fixed, though the retreat itself raises it, and the ice’s own gravity, which draws the sea towards it, falls as the ice is lost, lowering the sea nearby. And the flux law assumes a sliding law with a single exponent; other laws, in which friction is limited where the water pressure beneath the ice is high, change the exponent but not its sign. The instability survives all of these as a tendency, and each of them can hold it off.

Still open: whether Thwaites has already started #

The Amundsen Sea sector of West Antarctica, draining into the sea through the Thwaites and Pine Island glaciers, rests on a bed that deepens inland for hundreds of kilometres. Its grounding lines have retreated measurably since satellite radar began recording them in the 1990s, and in 2014 Ian Joughin and colleagues, and separately Eric Rignot and colleagues, argued that the retreat of Thwaites may already be self-sustaining — that the grounding line has reached the inland-deepening part of its bed and that the rest of the retreat will follow without further warming, over centuries. Others point to ridges in the bed that could pin the line for a time, to the remaining buttressing from the eastern ice shelf, and to the rebound of the bed, any of which could slow it greatly. A second proposed instability, in which ice cliffs left by collapsing shelves are too tall to support their own weight and fail, would make the retreat faster still, and whether it operates in nature is contested. Between them, these decide whether West Antarctica contributes a few tens of centimetres to sea level this century or considerably more — set beside what the rise a joule of heat buys from warming the ocean alone — and the answer is not yet known.

The statement buoyancy makes is short. At a grounding line the ice is exactly thick enough to touch the bed, $\rho_w/\rho_i$ times the water’s depth; the ice crossing the line goes as the 19/4 power of that thickness, so a line in twice the depth exports 26.9 times the ice; and on a bed that deepens inland a retreat therefore raises the export while it lowers the snowfall upstream, so the benchmark sheet has no stable grounding line between 974 and 1,265 km and, once softened past $2.1 \times 10^{-25}$, retreats across the whole stretch from 1,376 to 722 km. Archimedes fixes the thickness at the edge, and the shape of the rock decides whether the edge can stay.