# Volume of melted polar ice caps

If (or rather when) the polar ice caps melt, will they occupy more, the same or less volume?

Somehow all options seem possible to me. I don't assume they are just big ice cubes floating on the poles. So, if a huge mountain of submarine non-floating ice melts, wouldn't it occupy less space? But if the mountain is above the sea level, that's a new mass of water entering the sea.

Obviously, in a real case scenario other factors would be relevant too, like warmer water occupying more space than cold water. But I'm interested in this factor isolated from the rest.

You can't treat the two ice-caps the same way.

Arctic ice (excluding Greenland) is sea-ice: it is floating in the sea. The volume of this ice is slightly more than the volume of an equivalent mass of sea-water, which is why it floats. When it melts its volume will be the same (to very good approximation: there are small effects due to differences in salinity, and possible air trapped in the ice) as the equivalent mass of sea-water. Because of Archimedes' principle there will therefore be no (again, to first-order) change in sea-level because of this.

Antarctic ice and the Greenland ice-sheet is land-ice: it is not floating on water but supported by the land. If it melts the volume change will be about the same as for arctic ice (again, to a decent first-order approximation). But when it melts it will no longer be an ice-sheet supported by land: it will be in the sea, and there will be a very significant sea-level rise as a result.

Ice-sheets have enormous thermal capacity and will take a long time to melt (if you directed all the Sun's energy which reaches the top of the atmosphere on the Earth into melting the Antarctic ice-sheet it would take about a year to do so I think). Sea-level rise will however be significant long before all the ice melts, and worse, it doesn't need to melt: it just needs to become sea-ice. We don't (to my knowledge) have models which can predict either whether that process (land ice sliding off the land in significant quantities) will happen nor its timescales.

• You're quite right, but there's another possible mechanism at work, shutdown of the Atlantic Meridional Overturning Circulation which could lead to drastic climate changes. – Mike Dunlavey Jul 28 '17 at 17:10
• @MikeDunlavey I'm not an ocean person, although I sit near some: I believe that this is one of the apocalyptic scenarios that people think is actually rather unlikely. However this is rumour and memory on my part. – tfb Jul 28 '17 at 18:44

Ice floats in fresh water. This shows that ice is less dense that water. When ice melts, it occupies less volume.

When ice floats, most of it is submerged and a little sticks up above the water. The change in volume leave just enough water to fill the submerged part of the ice.

Sea water is a little denser than fresh water. This means a little more ice sticks up above sea water. When floating sea ice melts, there is a little more than enough to fill the submerged part.

A lot of the ice in the arctic is floating on the Arctic Ocean. The rise in sea level from that would be small.

Antarctica and Greenland are another story. The ice is on land. When it melts, it flows into the ocean and raises sea level.

And as you say, raising the temperature of the top few thousand feet of the ocean will raise sea level from thermal expansion.

• Actually melting floating ice does change the water level because ice doesn't contain salt. Fresh water is less dense than salty water. Level change is minuscule. Agree on all other stuff. – Communisty Jul 21 '17 at 13:06
• @Communisty - Good point. I edited the post. – mmesser314 Jul 21 '17 at 14:18
Consider a block of ice of mass m floating on water.

From Archimedes' Principle,


$$m=\rho_{water} \times V_{submerged} \ \ \ - (1)$$

Now the mass of ice remains same after melting, therefore


$$V_{melted\ ice}\times\rho_{water} = \rho_{ice}\times V_{ice}\ \ \ -(2)$$

$$V_{melted \ ice}= \frac{\rho_{ice}\times V_{ice}}{\rho_{water}}\ \ \ -(3)$$ $$V_{melted \ ice}= \frac{m}{\rho_{water}}\ \ \ \ -(4)$$

    Substituting value of mass m from eqn. (1) in equation (4) we get :


$$V_{melted \ ice }=V_{submerged}$$

This proves that after melting the ice block(now water) will occupy the same volume inside the water as it occupied before melting. Hence there will be no rise in water level solely because of melting of the ice.

Try thinking what will happen to the level of water when a mass of density greater than water is frozen inside an ice block and the ice block melts.

## protected by Qmechanic♦Jul 21 '17 at 13:52

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