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In this article, scientists have proposed a way to regrow the Arctic icecap by pumping and freezing arctic seawater. As far as I can gather, they suggest that if seawater is pumped up onto depressions in the remaining ice, the seawater will freeze and thicken the floating ice. My question by analogy is: if I place a large ice cube in a glass and then add a smaller amount of nearly freezing cold seawater to the glass, will the seawater freeze, or will the ice cube melt?

Does it depend on the relative quantities of ice and seawater? Does the fact that seawater freezes at a lower temperature than pure water have any relevance?

I guess it does not matter if the seawater freezes or not. If you pump water out of the ocean onto land, that will reduce the ocean's level (slightly), which I guess is the end result they are looking for. However, if you pump the water onto floating ice, the ice will sink a little bit and displace more water, and that would make no change to the sea level.

Would the ideas suggested in the article make any significant and cost-effective reduction in the rise of sea levels worldwide?

The following points are a summary of the comments and answers so far (and some related questions) which I think are pertinent:

  1. Ice floating in pure water does not change the level when it melts.

  2. Ice floating in seawater does slightly increase the sea level, but the effect is slight.

  3. The main cause of sea level rise appears to be thermal expansion of seawater with increasing temperatures.

  4. Ice that is on land, such as in parts of Greenland and Antarctica and on mountain tops, obviously does raise sea levels when it melts.

  5. Removing water from the sea and supporting the water on top of ice that is floating on the sea does not change the sea level at all.

  6. Increasing the area of floating ice does increase the amount of sunlight that is reflected and indirectly may reduce temperatures and the rate of sea level rise (slightly).

I think the observation by Dale M in his answer that pumping water up onto the surface of ice exposes the water to very cold air and would freeze the pumped seawater is very valid. I also think the surface ice is most likely below $0 \ ^{\circ} \text{C}$ and if the mass of ice is greater than that of the pumped water, the water will lose more thermal energy than the ice gains. It does seem possible that it is technically possible to freeze the water by pumping it onto the surface of the ice.

However, pumping the seawater onto the top of the mostly floating ice in the Arctic seems to be the least effective place to carry out the operation. See point 5.

P.S. One thing no one has mentioned so far is the heat created when creating the energy to pump the water. The article does mention using a hydrogen powered generator, presumably to reduce pollution from burning fossil fuels, but no mention is made of the heat put into the atmosphere from burning fuel to power the generator or the heat put into the pumped water due to pumping it. I guess the power source would have to be solar (probably not very practical when there is very little sunlight for about 6 months) or wind power.

KDP
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The air is at sub-zero temperature

The ice is at zero [actually, slightly below due to the salt content], but the air can be less than zero. We don’t get any more ice out of the air because there’s almost no water vapour in it. The water under the ice is above freezing and the ice acts as an insulator between it and the air. If you take the water from underneath, where it is insulated from the sub-zero temperature of the air, and put it on top, the heat will move from the water to the air and cause the water to freeze.

Changing the amount of sea ice has a negligible effect on sea water levels because the floating ice displaces roughly the same amount of water as it did when it was liquid. However, Arctic sea ice is a big store of ‘cold’ and it keeps Arctic air and sea temperatures low. If it goes, then the summer maximums of both could be above freezing. This will increase the melting of non-sea ice, that is, glaciers that are resting on land like the Greenland ice shelf. Dumping that ice in the oceans will raise sea levels because it isn’t floating. It’s also fresh water from snow not frozen sea water so it will change the salinity.

In addition, the absence of sea ice will affect the ocean currents. This may allow warm water flows like the Gulf stream to bring more heat in from the Tropics.

Dale M
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The purpose of the endeavor is not to change sea levels; as you correctly realize, floating ice does not change that (much). A change in sea level would also require moving much larger amounts of water than in this proposal, even if it were on land: The Greenland and Antarctic ice sheet losses are in the order of magnitude of 475 billion metric tons per year — a gigaton is a cubic kilometer — but raise the sea level by only about 1.3 mm per year.1

The purpose of the plan is to prevent the Arctic sea ice from disappearing, which would have a number of undesired effects. Chiefly among them is the change in albedo, the "brightness" of the surface: Water is much darker than snow and ice, meaning it absorbs most of the sunlight hitting it which is then transformed into heat. Already, Arctic summers have seen record levels of ice reduction. This is a runaway effect, i.e., a feedback loop, because warmer water leads to less ice which leads to even warmer water.

The idea of the scientists, as detailed in a Wired Middle East article and the home page of a startup company, is to make the ice thicker in winter so that it lasts longer and covers more area even during the summer's minimum, thus breaking the vicious circle. It is essentially the same principle snow cannons use on ski slopes, just on a seasonal instead of a diurnal time scale: Use cold night/winter temperatures to freeze excess water to create a sustained snow/ice cover during the warm day/summer period. The pumped water is essentially artificial (albeit salt water) precipitation; the Arctic ice cover would be much thicker and more resilient if the Arctic had more winter precipitation, but it is a cold desert.

Shallow water on the ice surface freezes readily in winter while the open sea at the edge of the ice shelf is moved around and mixed by currents and wind with "warmer" water from below which slows its freezing.

My two cents are though that the energies involved both in the irradiation by sunlight and the sea currents are prohibitively large; any human intervention is nothing but the famous drop on the hot stove. But then, some smart interventions (like deploying aerosols to reduce surface irradiation and warming) may be efficient enough to make a difference.


1 This can easily be computed by distributing the 475 cubic kilometers uniformly over the sea surface, 361 million square kilometers: A bit more than 1 cubic kilometer of water per million square kilometers, i.e. a bit more than 1 millionth kilometer of water level, which is the mentioned millimeter.