Why a melting iceberg doesn't raise the sea

Fill a glass to the very brim with water, then crowd it with ice cubes until they stand heaped above the rim, glistening and proud of the surface. Now the obvious happens in your head before it happens on the counter: as the ice melts, all that ice standing above the water has to go somewhere, and the only somewhere is down into the glass, and the glass is already full, so it will spill. You might even slide a saucer underneath to catch it. Come back in an hour. The ice is gone, the water is exactly at the brim, and the saucer is dry. Not almost dry — dry. The heaped-up ice that so plainly had to overflow the glass melted down into it and raised the level by nothing at all.

The expectation that it should spill is not foolish; it is arithmetic. You can see ice above the waterline. That ice is water, more or less, in a stiff white form, and it is not yet part of the pool in the glass. Add it in and surely the pool grows. The mistake is subtle and it hides in the word add. The ice above the line is not water waiting to be added. It has already been counted. A floating thing pushes down on the water it sits in, and the water pushes back up hard enough to hold the thing's whole weight — that is what floating is, the exact balance of those two pushes. To hold up the cube, the water has had to get out of the way of the cube's submerged part, and getting out of the way means rising. So the level in the glass already stands higher, by the moment you drop the ice in, than it would if the ice weren't there — and it stands higher by an amount that answers to the entire weight of the cube, the proud part included. The part above the surface is not uncounted surplus. It is the reason the part below is riding as deep as it is.

Then the melt makes the two halves of the bookkeeping meet. Weigh the cube: call it a certain mass of water, frozen. When it melts, that mass does not change — you cannot melt matter away — so you end with exactly that mass of liquid water. And here is the hinge: a floating object displaces a weight of water equal to its own weight, which for water floating in water is the same as saying it displaces its own melted volume. The hole the submerged part was holding open in the pool is precisely the size of the puddle the whole cube collapses into. The ice that stood above the line melts down and drains into the space its own underside had already cleared, fills it to the brim, and stops. The surplus you saw and the shrinkage you didn't were the same quantity all along, approaching the waterline from opposite sides, and they cancel to nothing. It does not matter how the cube is shaped, or how much of it happened to stick up, or how many cubes you crowd in. The level does not creep up a little and it does not have to be measured carefully to see that it didn't. It is held at zero by an identity.

Why an identity, and not just a close call? Because the two things that cancel are not two facts. They are one fact seen twice. Ice floats with about a tenth of itself above the water because ice is about a tenth less dense than water — the same tenth. The freeboard, the bright lip standing above the surface, is not extra water; it is the visible receipt of the density deficit, the amount by which a given mass of ice is puffed up larger than the same mass of liquid. Melting is just cashing that receipt: the puffed-up solid relaxes back into the denser liquid, losing exactly the tenth of its bulk that had been standing above the line, and settling into exactly the volume it was displacing below it. High rider, big shrinker; low rider, small shrinker; and the two are locked together because they are both just how much lighter than water this ice is. There is no slack in it for the level to move.

Two honest edges, because the clean glass is not the whole ocean. The first: the glass is exact only because the meltwater and the water it joins are the same stuff. In the sea they are not. An iceberg and floating sea ice are fresh — or nearly — and fresh water is a touch less dense than salt water, so a berg's meltwater takes up a little more room than the salt water it displaced while floating. So melting the world's floating ice does raise the sea, by a hair: an effect real enough to have been measured and modelled, and small enough to come out in centimeters were every floating scrap of ice on the planet to melt. The household version is exact; the ocean carries a fine correction the glass doesn't. I would rather name that than sell you the tidy version and have a later reader catch the seam.

The second edge is the one that matters, and it inverts the reassurance you might be tempted to take from all this. None of the above says the seas are not rising. They are. It says only that the floating ice is not what raises them — the icebergs and the sea ice and the ice shelves already afloat can vanish and the tide will barely notice, for the reason the glass gives. What raises the sea is the ice that is not floating: the sheets of Greenland and Antarctica and the mountain glaciers, resting on rock, water that was never in the ocean and that runs into it new. That is added water, uncounted water, the real thing the glass's ice only pretended to be. And beside it, quieter and nearly as large, is a rise with no ice in it at all — warm water simply takes up more room than cold, so the ocean swells as it heats, from the surface down through its whole depth. Between the land ice arriving and the sea itself expanding, the water climbs. The berg calving off the shelf and melting on the evening news is the most photographed part of the process and close to the least consequential part of the sum.

So the surprise here is not that a cause runs backward. It is that a change you brace for does not come — that two quantities you would never think to set against each other, how high the ice rides and how far it shrinks, turn out to be the same number wearing two faces, and they meet at the waterline and make nothing happen. The ocean keeps a stricter set of books than we do. The ice we can see, standing above the line and seeming to owe the pool its bulk, has already paid. It is the ice we cannot see — the ice that was never in the water to begin with — that comes due.