## 13 April 2009

### Ice and sea level

Melting sea ice or ice shelves can indeed change sea level. It turns out that I was probably the first person to compute by how much the sea ice can do so, and there's a story for tomorrow about why I wasn't the person to publish this in the scientific literature even though I had the answer more than a decade before the next person to look at the problem.

The first peer-reviewed investigation was published by Peter D. Noerdlinger and Kay R. Brower, in The Geophysical Journal International, 170, pp. 145-150, 2007 The melting of floating ice raises the ocean level. The DOI is 10.1111/j.1365-246X.2007.03472.x You can get the authors' copy at the linked title. They included a simple experimental demonstration as well, which I hadn't done. I also didn't think about the ice shelf contribution, which turns out to be 10 times larger than sea ice's. Oh well.

The wrong answer on this question is to say that a floating body displaces its own mass, so when it melts, the water level is unchanged. Now, as this is partly quoting Archimedes and he was an awfully bright guy, there's at least good company.

The reason it fails (and, by the way, it isn't clear that Archimedes didn't know about this) is that what is melting when we melt sea ice or ice shelves is not the same stuff as what it is floating in -- sea water. Sea water is salty, about 3.5% salt. Sea ice is fairly fresh, about 0.5% salt. And ice shelves are completely fresh. If we were melting ice shelf in to fresh water, the level would indeed not change. You can test this with a glass of water and some ice cubes. To find just what happens when you melt frozen stuff that's floating in a liquid, you have to do the math. I've got it in my Sea Level Change FAQ. The result is, sea level rises slightly. I found a few millimeters (about 4) for sea ice. Noerdlinger and Brower found a few centimeters (also about 4) for ice shelves. It isn't much, but it isn't exactly zero.

A way to think about what happens, minus most of the math, is to envision a block of ice floating in the ocean. The density of ice is lower than ocean water -- about 917 kg per cubic meter for ice, versus 1028 kg per cubic meter for ocean water. The difference is why '90'% of an ice berg is below the surface. Conversely, 10% is above the surface. It's actually (1028-917)/1028 above the surface, 10.8%. Now melt the fresh ice, but don't let it spread out or mix with the ocean. That gives us a material with density near 1000 kg per cubic meter. That blob, for the same reason that the ice was floating in the first place, sits (1028 - 1000)/1028 of itself above the water level -- 2.7%. That bit sticking above water level means that melting such ice does contribute to sea level change. If the density of the stuff you melt is different than the stuff it is floating in, you can indeed have a rise (or fall, if the melt is denser than what it floats in) in fluid level.

This example is also why I, in particular, and scientists more generally, want you to 'show me the math'. The thing is, when I first wrote the sea level FAQ, I made the same error as everybody else was making. It was an early commentator (Rick Chappell) who told me I was wrong (he didn't have the math, but did have the above principle) that prompted me to work out the math in detail. I was doing show just to show him in detail that he was wrong. In fact, at the end of my calculation, I'd shown myself that I was wrong. So updated the FAQ and my thanks to Rick. Merely asserting principles would not have changed anybody's mind. The problem being that, although a principle might be true (if I jump, the earth moves the other way), the effect might be too tiny to worry about. It's only when we do the math, get quantitative, that we can decide whether something is too tiny or not.

A slightly different example, prompted by this point going back to ancient Greece. There is an Aristotelean (or at least his time) principle that "Nature abhors a vacuum." Now, for many purposes in daily life, this is not a bad principle. If you try making a vacuum, you'll need to do things to shield it against nature trying to fill it back up. On the other hand, it was also an argument that there were no such things as atoms. The argument being, if there were atoms (discrete bits of matter) then there would be gaps between the atoms. But those gaps would be vaccum -- and 'nature abhors vacuum'. Now that we know that there are indeed atoms, the principle applies, but it is directed to how nature responds, and in which cases.

chriscolose said...

Just out of curiosity, how would someone work out (back-of-the-envelope) the sea level rise from an ice sheet for a given mass (volume?) loss? For instance, how could I get "7 meter sea level rise" for the Greenlabnd ice sheet. Should be a relatively simple algebraic relationship.

Robert Grumbine said...

Fairly easy to do back of the envelope (or Fermi estimate).

For starting purposes, I'll take the density of ice and sea water to be the same. They're not, ice is about 10% less dense. But 10% is close.

The area of the ocean is about 350 million km^2. To raise sea level by 1 m, we need to add 350 million km^2 * 1 m of water. If we have a slab of ice sitting on land, and it were 350 meters thick, then we'd only need one that was 1 million km^2 in area. Greenland is actually about 2.2 million km^2. So each meter of global sea level means about 170 meters of elevation of the Greenland ice sheet.

So 7 meters of sea level means an average elevation/thickness loss of about 1100 meters over the entire area of Greenland.

Greenland's peak thickness is well over 3000 meters (over 2 miles). But not all the area is ice covered, though, granted not much. And not all the volume of the ice cap is above sea level -- the mass of the ice sheet is so great that the crust has sunk below sea level in some places.

Hank Roberts said...

http://www.nature.com/nature/journal/v458/n7240/full/nature07933.html

excerpts and links to what's available for those like me without subscription access:

Most curious to hear something from someone who knows the science here.

Anonymous said...

My understanding is that sea level rise from melting ice (from all sources) would not be evenly spread over the world's oceans.
The earth's spin will tend to force water to the equator.
Has this density difference been taken into consideration?
These small differences are being applied to a massive amount of ice.

TomG

Jay Alt said...

Even the 'simplest' calculations of sea level rise from earlier decades contain at least 10 components that influence the result. Tides, temps, winds, . . .

Here is a very nice resource from James Titus, who examined these issues in depth for EPA.

http://users.rcn.com/jtitus/

http://www.climatesciencewatch.org/index.php/csw/details/epa-titus/

EliRabett said...

Asserting the principle is enough for most physicists. It's a culture difference. Engineers OTOH don't believe in principles.

Robert Grumbine said...

TomG: You're right that circulation effects will be important in deciding where the water that melts from ice will wind up. It also applies to how fast it will get there. The Noerdlinger+Brouwer paper mentions some of this. (Had I been writing then -- which Noerdlinger did invite me to -- there'd have been more.)

But remember, the mass of ice is large compared to you and me, but not so impressive compared to the ocean.

Eli: Depends on your engineer and your scientist. I've seen it go the other way. There is a strain some engineers pick up of being overly dogmatic. Some scientists, particularly physicists, do as well.

Alastair said...

Engineers believe that they are right. Scientists believe that everyone else is wrong.

Cheers, Alastair.

Alastair said...

Here is an example of where a scientist believed everyone else (well me) was wrong.

It was reported that the Arctic sea level had been falling whereas every where else the sea level is rising. This was repeated on RealClimate here.

I pointed out in comment #19 that melting sea ice could be the reason for that. I seem not to have explained myself well there so I will try again here.

The sea level before the ice melts is at the top of a column which consists of salt water with ice above. Above the surface level there is more ice.

When the ice melts then the column will consists of only salt water. The top of the column below the surface was low density ice and after melting it will be replaced by denser salt water which will require less height to maintain the same pressure at the sea bed.

If it has a higher pressure at the sea bed it will undercut adjacent water columns and sink until the pressure is equal.

So not only does melting sea ice raise sea levels globally it will reduce sea levels in the Arctic.

Using the principles outlined in your FAQ, I did the calculations in the RealClimate post, which I won't repeat here since you as a scientist will not believe me anyway :-)

Cheers, Alastair.

PS Did I mention I have a degree in Electrical Engineering?

Cam said...

Great blog! this statement "If we were melting ice shelf in to fresh water, the level would indeed not change." However is wrong! the reason is that the portion of the floating object in the upper fluid (air in this case) ALSO experiences an upthrust equal to the weight of fluid displaced. Melting icecubes in a glass of water therefore causes the level to go up! The "math" is here:

Physics Education: 2001 vol. 36 (1) pp. 75-76

Note that had I known this in highschool I would have gotten one more question wrong in a physics exam since most textbooks are wrong about this too!

Robert Grumbine said...

Thanks Cam. I've sent the author a request for further explanation here, or a copy of his paper (since I can't get it from my libraries). The effect must be far smaller than the salinity effect. But I need more detail to compute it.

Alastair said...

I've just worked out what Cam is getting at.

The weight of water displaced by an ice cube in a glass of water is not the weight of the ice cube. It is actually the weight of the ice cube less the weight of air it is displacing. Since roughly only 1/10th of the ice cube is in air, and the density of air is much less than water or ice, then the increase in water level will be very small.

HTH,

Cheers, Alastair.

Anonymous said...

Vernon says:

I agree that there is a problem with Archimedes principle, namely that ice is fresh water and the sea is salt water. A study by Peter D. Noerdlinger and Kay R. Brower, in The Geophysical Journal International, 170, pp. 145-150, 2007 “The melting of floating ice raises the ocean leve”l says it does but I think there is an error. I could be wrong but here is what I think actually happens. I am not addressing grounded ice. I used 2000 numbers Arctic sea ice and got a sea ice volume of 42,500km3. I then took the total volume of sea water globally, 1,320,000,000km3. Now to be fair I used the numbers from the study. I then did the following calculations:

42500km3 Total volume of arctic sea ice
1320000000km3 Total volume of sea water

1.026 Specific Gravity Sea water
0.919 Specific Gravity Sea Ice

38067.74km3 Displacement of the sea ice
46245.92km3 Displacement of melted sea ice
8178.18km3 Additional volume above sea ice when melted

1320008178km3 Increase in Sea water volume
0.00062% Volume increase expressed as a percentage
361000000km2 Surface area of the Sea
3.656509695km Average depth
0.02265m Increase in average depth or 2.27cm

However, when we figure in the mixing of fresh water to saltwater into a homogeneous mixture, the specific gravity goes from 1.026 to 1.025999089 which is not much.

1320001172km3 expansion due to lower specific gravity
38067.72km3 volume of the ice at the new specific gravity

As you can see, the new volume of the melted ice once mixed is only 0.0338km3 greater than the displacement.

The increase in sea level is 0.00325m or 3.25mm.

What it means is that there is an initial rise until the mixing is complete and then the difference in sea level height is 3.25mm. Now I could have looked up the Antarctic numbers for sea ice but since the preponderance is in the Arctic, I did not bother. Take a look, I dont see where I am wrong but maybe you'll spot something.

Robert Grumbine said...

Vernon, I can't see where you think Noerdlinger or I made our error(s). We arrived at 3-4 mm sea level rise for melting sea ice, and you arrive at 3.25. One thing we can assure you of is that you don't have that many significant digits.

In carrying out your calculation, you do have a pair of offsetting errors. On one hand, you compute the volume of melted sea ice as being greater than the volume of sea ice in the first place. It'll be less, of course, as the melt water will be near 1005 kg/m^2 vs. the 910 or so of sea ice.

On the other hand, you also add a step beyond what Noerdlinger and I did -- of mixing the sea ice melt water throughout the entire ocean, instantaneously. That reduces the sea level effect. But it's physically seriously incorrect. The melt water first sits on the surface of the ocean. It takes work (wind mixing -- a new process beyond the melting) to mix it in to the water column. And it takes time -- hundreds of years for current circulation rates -- for a surface event to be mixed throughout the whole volume of the ocean.