The best part of my department’s weekly seminar is meeting people. Last week, Dr. Les Butler from the chemistry department at Louisiana State University discussed using X-ray interferometry to get even more detail of an object using X-rays. Mostly, I loved it because there wasn’t too much chemistry involved (remember, I teach in the Chemistry and Physics Department). Outside of the talk, he shared this awesome question that he uses to help people solve problems:

Suppose the ice cap on Antarctica melted and raised the sea level by 1 meter. How much energy would it take to put all this water back on top of the ice cap?

Like I said, it’s a great question.

Antarctica Ice Sheet

What do we know about the ice in Antarctica (and try saying ice in Antarctica three times fast)? Usually I would estimate some numbers, but I don’t have a good feel for the amount of ice involved. Let’s use Wikipedia:

• The ice sheet covers 14 million square kilometers.
• The volume is 26.5 million cubic kilometers.
• If this ice melted, sea level would rise 58 meters.

I also should note the difference between antarctic and arctic ice. The Arctic floats, so when it melts (and it does melt), it doesn’t raise the sea level since it displaces water as it floats. Actually, there is a cool experiment you can try. Place a large ice cube in a glass of water. Mark the water level and check the level after the ice melts. You should find that the water level is nearly constant (it might decrease due to evaporation). But Antarctic ice rests on land. When this stuff melts, it will raise the sea level. That’s bad.

OK, but if I know the volume of Antarctica’s ice and the surface area, I can estimate the height of this ice shelf.

If I want to return the water from sea level to the top of the ice sheet, it would have to go about 2 kilometers plus the height of the land. An average land elevation of 2.5 km means you’d need to move the water around 4.5 km.

Oh, one more estimate. What if only some of the ice melts, raising sea level 1 meter (instead of 58)? How much would this decrease the height of the ice sheet? Here we can use a little proportional reasoning. If 2 km of ice leads to 58 meter water level rise, then a 1 meter rise would be 1/58 of the total ice cap.

If you melted enough ice to raise the sea level 1 meter, it would reduce the height the ice—but not by much. So, I will assume the ice sheet height remains constant at 4.5 km above sea level.

Energy Needed to Raise the Water

There is all this water down here (on the ocean) and I want it up there (on top of the ice). How do you do that? Well, there are several ways. I could just get a bucket and and carry it, or fly it up there with an airplane or pump it with a pump. But no matter how it gets up there, it’s going to take energy.

There is more than one way to deal with energy, but the simplest is to consider gravitational potential energy. As an object moves vertically upward near the surface of the Earth, gravitational potential energy increases. Assuming a constant gravitational field, this change in potential energy is:

With a gravitational field value of g = 9.8 N/kg, it would take around 10 Joules of energy to lift a 1 kilogram object 1 meter. So, lifting 1 kilo of water 4.5 km would require 44,100 Joules. But what is the mass of 1 meter of sea water? If I assume oceans cover 70.9 percent of the Earth and Earth has a radius of 6.37 x 106 meters, this would give a 1 meter depth a volume of 3.62 x 1014 m3. With a water density of 1000 kg/m3, this is a total water mass of 3.62 x 1017 kg.

Using that mass of water and raising it to the top of the ice sheet would require 1.6 x 1022 Joules. OK, now for some fun homework questions.

• What if you used solar power the size of Antarctica to raise this water? How long would it take? If you like, you can estimate 1000 Watts/m2 for the solar panels, but it would surely be lower than that because of the low angle of sunlight at the poles.
• What kind of power source would you need to raise this water in just one year? In his original question, Dr. Butler converted this power to number of nuclear submarines.
• Suppose the water is distributed evenly to every human on the planet. How much water would each human have to bring to the top of the ice sheet?
• Obviously, one way to get the water up there is to make it snow on top of the ice pack. Estimate how long it would have to snow (the maximum snow rate you can imagine) to do the job.

Sometimes it’s more fun to come up with questions than to answer them.

A Free Energy Method

OK, I have an idea that could make this work. It’s based on the energy needed to melt ice. Just hear me out. If I take 1 kg of ice and thoroughly melt it, it would require 334,000 Joules (we call this the latent heat of fusion for water). But what if I wanted to freeze it? In theory, I could get this energy from the water turning to ice.

You need 44,100 Joules per kilogram of water to get it to the top of the ice sheet, but you get 334,000 Joules per kilogram by freezing water. Boom. This will save the planet (or at least the coastal cities in danger of flooding). But how would this work? I don’t know. Maybe you could make something like a steam engine, but instead of steam use some thing that turns to a gas at a temperature of lower than 0°C. That way you could use liquid water to boil your liquid and turn some turbine. I will leave the details to an engineer.

Oh, one other idea. If you think you can get a lot of energy when water turns to ice, that’s true. But what about water vapor condensing to liquid? That gives 2.3 million Joules per kilogram. That’s way more energy. So, what if you had some system that condensed water out of the air? You probably wouldn’t even need to lift this water mass since you can take it from the air at the top of the ice sheet. But I guess this idea is silly. It’s essentially the same as what we call snowing.