Does Melting Sea Ice Raise Sea Levels?

Up until now I have unthinkingly assumed that melting sea ice doesn't change sea levels. The reason is a basic principle of physics: Archimedes' Principle, which says that a floating object displaces its own weight of liquid. The idea is that when the ice melts, it will exactly fill in the 'space' that the ice block made in the water, thus leaving the water level unchanged.

But there's a fly in the ointment: when sea ice freezes, it preferentially expels salt, in the process becoming purer than the sea water it is floating in. Pure water is less dense than salty water, so when the ice melts it will overflow the 'hole in the water' that the ice had occupied, and when it overflows, it raises the water level.

That's true, and that's a very interesting application of physics and a lesson to think precisely about the physics of any situation. In other words, as physics, it is fun and interesting.

But let's see how this is being used. At the alarmist site Skeptical Science, which purports to 'expose' the fallacies of skeptical views on global warming, they start out:

Sea level rise due to floating ice?

It is widely believed that melting of floating sea ice does not contribute to sea level rise. Is this really true?

And physorg.com starts out their article like this:

Melting of Floating Ice Will Raise Sea Level

When ice on land slides into the ocean, it displaces ocean water and causes sea level to rise. People believe that when this floating ice melts, water level doesn’t rise an additional amount because the freshwater ice displaces the same volume of water as it would contribute once it melts. Similarly, people also think that when ocean water freezes to form sea ice and then melts, the water is merely going through a change of state, so it won’t affect sea level. However, in a visit to NSIDC in May, Dr. Peter Noerdlinger, a professor at St. Mary’s University in Nova Scotia, Canada, suggested otherwise.

Scared yet? It would seem this is yet another hairy scary danger caused by CO₂ to add to all the rest of the usual suspects. But read on. Buried in the text you will find some remarkable numbers. The former site quotes "1.6% of current sea level rise (about 3.1 mm per year) is caused by loss of sea ice. The other tells us "melt water from sea ice and floating ice shelves could add 2.6% more water to the ocean than the water displaced by the ice, or the equivalent of approximately 4 centimeters (1.57 inches) of sea-level rise."

So what is really going on here?Tracking down the scientific papers at the root of this business is at first confusing. One of them (Jenkins and Holland)¹ gives a figure of 5.2 millimetres for the sum of Arctic and Antarctic sea ice. But Noerdlinger and Brower's paper³ gives a figure of 47 millimetres for the melting of the "total floating ice volume". What's the difference?

Well we can start by doing a quick sum of our own. The volume of the world's sea ice is about 75,000 km³. (See 2.) The density of ice is about 0.92 (of the density of fresh water), so this ice would melt into a volume of 69,000 km³. The total area of the oceans is 361,000,000 km². If this ice were land ice, it would raise sea levels by 69,000 / 361,000,000, which equals about 19cm. Really? Even if it were not floating, if the entire melt volume added entirely to sea level rise, we are only talking about a bit over seven inches! But it is floating, so let's check how much it will raise sea level.

The density of sea water at the sea surface is 1027kg/m³.

So 1000kg of sea water occupies 1000/1027m³ = 0.9737m³.

69,000 km³ of melted ice (assuming all fresh water) has mass of 69,000 * 10⁹ kg, which, as ice, displaces its own weight in sea water, which is 0.9737m^3 * 69000 * 10⁹ m³ of sea water, which equals  67,185 * 10⁹ m³.

But being fresh water, it will occupy 69,000 * 10⁹ m³, The fresh water 'doesn't quite fit' in the salt water 'hole' by this much: (69,000 - 67,185) * 10⁹ m³, or  1,815 * 10⁹ m³. Once again, dividing this by the area of the oceans gives  1,815 / 361,000,000 = 5.03 millimetres.

So yes, the fuss really is about a mere 5mm.

But where does the 47 mm come from?

"Sea ice" is frozen sea water, frozen and melting yearly during the summer / winter heating and cooling cycle. But this is not the only floating ice. The Antarctic ice shelves are actually land ice that has been forced out onto the sea by the pressure of ice building up behind it. Ice shelves are therefore part of an approximately steady state process as snow (which was originally sea water that evaporated) falls in the interior and builds up the mass of interior land-based ice. But their total volume is about ten times that of sea ice, and if all that melted, that is where the 47mm comes from.

The annoying thing about all this

I don't know about you, but 47 millimetres doesn't exactly strike terror into my heart or freeze the marrow with fear of puppy dogs drowning as the oceans break their banks. And remember, the ice shelves cannot all melt within a few years. They are colossal blocks of continental ice, which would, in the most drastic scenarios, take decades or longer to melt. But even in this worst of all scenarios, the 47mm, or under two inches, is the once-only total melt figure. Once that has melted, nothing more can be contributed from floating ice.

In other words, the "rising seas from floating ice melt" is yet another "cry wolf" beat-up.

Here's the basic problem. This article was inspired by a post on Wattsupwiththat about melting sea ice. In that post, it was stated that melting ice doesn't raise sea levels. All hell broke loose in the comments as the warmist trolls let fly with these 'devastating facts' about the truth of sea ice melt.

But this is completely unfair for this reason. If you ask a scientist for his opinion, as a scientist, on whether any thing X can, or could, or might, influence some other thing Y, he will almost certainly say yes, regardless of what X and Y are. This is because of a number of reasons. First, almost any behaviour is strictly consistent with almost any theory. Is relativity consistent with elephants flying around the room? Yes. It's so unlikely we can be sure it won't happen, but yes, it is possible.

Secondly, almost nothing at all has no effect. Will CO₂ cause warming? Yes. Will warming cause some trouble somewhere for someone? Yes. Are humans changing the amount of CO₂? Yes. Will an ant breathing change the amount of CO₂? Yes. No matter what the question about X influencing Y, a careful scientist is almost compelled to answer yes. But the important thing is: by how much?

So we see why it is so very easy for the alarmists to get big majorities in favour of their views by writing slanted questionnaires. Q1: Are humans adding to the amount of CO₂? Q2: Is CO₂ heating the atmosphere? Q3: Are there dangers from planetary warming? Any careful scientist will answer yes to all three questions, even if he is completely convinced that the alarmist theory is nonsense (as it indeed is).

So here, with sea levels, we have yet another case of this. Scientists distinguish effects by what they call 'order'. The first order effect of melting sea ice on sea levels is zero. Contrary to the shrill alarmists who attack every speck of a defect in articles on WUWT, that is a perfectly correct answer - to first order! But there is a second order effect: Yes, to second order, a small effect exists. Amusingly, the Jenkins and Holland paper concerns even smaller effects, third order effects that actually reduce further the already tiny second order effects we have been examining here.

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1 Melting of floating ice and sea level rise.  Adrian Jenkins and David Holland.
http://efdl.cims.nyu.edu/publications/refereed/grl_melt_floating_ice_07.pdf

2 On Sea Ice  By W. F. Weeks, W. D. Hibler (III.),
http://books.google.com.au/books?id=9S55O6WzuL8C&pg=PA1&lpg=PA1&dq=total+worldwide+sea+ice+volume&source=bl&ots=6SBdMIM93K&sig=w7E4fw-jAJmZ8IFMLZp9e2U-GuU&hl=en&ei=uaOTTsnsFcnRrQeY8tiVBg&sa=X&oi=book_result&ct=result&resnum=10&ved=0CFsQ6AEwCTgK#v=onepage&q&f=false

 3 The Melting of Floating Ice Raises the Ocean Level. Peter D. Noerdlinger & Kay R. Brower