# What happens to water level when ice XII melts?

There exists this famous idea that if all floating icebergs melt, water level will stay the same (because the water replaced by ice is the volume of the melted ice). Now,

1. Is this always so, if you allow more exotic forms of ice (take e.g. ice XII with density 1.29) ?
2. Is this always so on/in earth ?

EDIT:

I was looking a possibility that you would have water as a liquid that contains pieces of water as a solid. No land is assumed to exist here. "Ice" is always floating or sunken(if possible).

To reformulate, can Ice be sunken in water in any circumstance (pressure, temperature)?

EDIT 2:

Wikipedia: "Ice VII has a density of about 1.65 g cm-3 (at 2.5 GPa and 25°C)".

And water as an incompressible liquid has a density of 1 g cm-3. This means that ice VII is at the bottom of the "lake" and will expand when it melts. Is this true? And are there environments in earth where this (or similar) can happen?

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It's certainly possible for ice to sink in water under the right conditions. The diagram this section of Wikipedia's ice page will show you the conditions under which the various types of ice can form. Most of the "exotic" ones such as XII will form only at pressures greater than around 200MPa. These high-pressure forms are all denser than water, so they would sink to the bottom. This means that they would displace less liquid than their weight, so melting them would result in an increase of the surface level.

Earth's oceans aren't deep enough for these types of ice to form. The pressure at the bottom of the Mariana trench is about 100MPa. Since pressure increases linearly with depth, the oceans would need to be around twice as deep in order for this to happen.

However, there is a solid form of water that does sink in Earth's oceans. This is methane clathrate, which is a crystalline solid consisting of methane molecules surrounded by water ones. It can form at pressures found in the ocean, and there's rather a lot of it in sediments below the sea floor. Although methane clathrates are denser than water, I don't know what the effect on the sea level would be if they melted. This is because when it melts the methane is released as gas and bubbles to the surface, and I don't know whether the volume increase due to the melting is bigger or smaller than the volume decrease due to the methane escaping into the atmosphere. (If large quantities of clathrate did melt then the direct effect on sea level would be the least of our worries, because methane is a very powerful greenhouse gas and there's an awful lot more of it locked up in clathrates than we currently release industrially. This could actually happen.)

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(Assuming that you don't heat enough to change the density of the pre-existing liquid water)

Any floating body displaces as much liquid as its weight. So, let's say we have 5g of ice. Part of it will be above water. The part that is below the water level will have a net volume equal to the volume of 5g of liquid water. So, if the ice melts, it will become 5g of liquid water, and fill up the space snugly. Thus there is no net change in the water level.

This will happen regardless of the density of the ice you're referring to. It will work even if the ice has pockets of water in it. As long as the floating body is capable of completely transforming into liquid water by a pure physical reaction, there will never be a change in water level. This is because Archimedes' principle works for nonuniform bodies. Replace "ice" above with "ice-water conglomerate" in my above explanation, it will still work.

I don't understand what you mean by "in/on earth"--this is true anywhere with gravity. Note that, as Rory mentioned, land-based ice isn't "floating", so this won't work.

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1. If the ice floats (and not falls to the bottom), then it is completely irrelevant what density it has. It displaces exactly the volume of the water that corresponds to its mass, and since ice melts into the water it will not change the level. Let me just supplement you that the reported density of ice XII 1.29 g/cm$^3$ is for the temperature of 127 K, I would expect much smaller density at 273 K.

2. I guess the problem of the sea-level rise is connected to ice that does not float (e.g. glaciers). There are impressive quantities of non-floating ice on Antarctica.

Two main factors contributed to observed sea level rise.[6] The first is thermal expansion: as ocean water warms, it expands.[7] The second is from the contribution of land-based ice due to increased melting. The major store of water on land is found in glaciers and ice sheets.

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By definition, as long as you are only talking about floating ice, the water level must stay the same when the ice melts - it is already displacing it's mass of water, whether it is solid or liquid.

(For land based ice, please have a look at my answer to this question over on Skeptics)

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