Human Math and Arctic Sea Ice

Over the last year and a half, instead of following baseball and football, I have been following the measurements of Arctic sea ice. Will it entirely melt away at minimum? If so, when? Will it continue to melt past that point? Will the Arctic reach the point of being effectively ice-free year-round? If so, when? Will Arctic sea ice set a new record low in extent this year? In area? In volume? By the way, the answers to the last three questions in 2011 are already, by some measures, yes, yes, and yes.

In the process, I have become entirely addicted to the Arctic Sea Ice web site (neven1.typepad.com), the Arctic sea ice equivalent of a fantasy football league. There, I and other scientist and math wannabees can monitor and pit themselves against near-experts, and plunder much of the same data available to scientists to try our own hand at understanding the mechanics of the ebb and flow of this ice, or to do our own predictions. Periodically, scientific research pops up that deepens our understanding in a particular area, or challenges some of our assumptions. Strange weather patterns occur as we near minimum, making the final outcome unpredictable. In fact, this year we still don’t know whether the game is over for the year or not – whether we are going into overtime.

But what disturbs me about the glimpse I am getting into science, this late in life, is that I keep seeing half-veiled glimpses of what I might call “data snow blindness,” not just from us newbies, but from some of the scientific research noted in the site. By that I mean that those who analyze Arctic sea ice data tend to get so wrapped up in elaborating the mathematical details of, say, a decrease in extent at minimum and how short-term and long-term changes in sea-ice albedo, the North Atlantic Dipole Anomaly, cloud cover, and increased Arctic surface temperatures can predict this minimum, that they seem to forget what extent data really conveys and does not convey, and how other factors not yet clearly affecting extent data should play an increasing role in future extent predictions. They keep talking about “tipping points” and “equilibria” and “fundamental changes in the system” that do not appear to be there. And so, what surfaces in the media, with few if any exceptions, gives an entirely misleading picture of what is to come.

So let me lay out my understanding of what the data is really saying, and how people seem to be going wrong. Bearing in mind, of course, that I’m not the greatest mathematician in the world, either.

The Model
I find that the easiest way to visualize what’s going on with Arctic sea ice is to think of a giant glass full of water, with an ice cube floating on top that typically almost but not quite fills the top of the glass. In the summer, the sun shines on the ice and top of the water, adding heat on top; in winter, that heat turns off. In summer and winter, the water below the ice is being heated, although a little less so in winter.

This does not quite complete the picture. You see, water is constantly flowing in from the south on one side and out to the south on another. As it hits the ice, it freezes, and moves throughout the cube until it exits on the other side – an average of about five years, apparently. These currents, too, act to break the ice apart as it is melting in summer, so (especially on the edge) there are lots of little “cubelets”. Where the ice is solidly packed together in the glass, then “area” – the amount of ice we see from above – is the same as “extent” – the region in which there is enough ice to be easily detectable. But where there are cubelets, then extent is much greater than area – as much as 70% more, according to recent figures.

One more important point: the depth of Arctic sea ice is not at all the same everywhere. This was true when the first nuclear sub approached the Pole underwater decades ago, and it is true as the Polarstern measures depth from above this year. The age of the ice, the point in the year in which it first froze, and where it is in relation to the Pole all factor in; but even in small regions, “thickness” varies. Generally, less than half the ice is almost exactly the same thickness as the average, with lots above and lots below. I find it useful to think of a normal curve of sea-ice thickness more or less peaking at the average.

In a state of “equilibrium” such as apparently has existed for perhaps 5-10 million years, the Arctic cube fills at least 70% of the Arctic in summer and just about all of the Arctic in winter. In summer, the sun melts the top and sides of the cube. However, since we only see the top, which only peeks out a little bit from the top of the water, we don’t see the way that the bottom of the cube rises in the water – the “balance point” between “salt” water below and “no-salt” ice above goes up. In the same way, in winter, we don’t see the way that the balance point goes down deeper.

Now suppose the globe starts warming. As it happens, it warms faster in the Arctic than in the equator, and it warms in both the near-surface air and in the water that is flowing through the Arctic underneath the ice, year-round. Estimates are that the air temperatures in summer in or near the Arctic have reached more than 10 degrees higher, and the rate of rise is accelerating; likewise, the ocean temperature is up more than a degree, and the rate of rise is accelerating. What we would expect to happen is that there is less and less ice during the summer – and also less and less ice during the winter, because the water underneath that cube is warmer, and the “balance point” is higher. And the rate we would be losing ice would be accelerating.

The Measures
Now how do we measure what’s going on with the ice? We point our satellites at the Arctic, and we measure in three ways: extent, area, and volume. Extent and area we have explained above; volume is area times average thickness.

To get extent, we divide the Arctic into regions, and then into subregions. If, say, more than 15% of the subregions in a region show ice, then that’s part of the ice extent. The area is the extent times the percent of subregions that show all ice (this is a very simplified description). There are difficulties with this, such as the fact that satellites tend to find it difficult to distinguish between melt ponds at the top of the ice during melting and melting of the ice “all the way down”; but it appears that those cause only minor misestimations of “actual” extent and area. Storms that wash over “cubelet” ice can cause temporary fairly large downward jumps in what the satellite perceives as ice, and hence in area; but again, this goes away as the storm subsides. All in all, the measuring system gives a pretty good picture (by now) of what’s going on with area and extent.

Volume is much more difficult to get at – because it’s very hard to get an accurate picture of thickness from a satellite. Instead, we create a model of how thick each part of the ice should be at any time of the year, and then check it out “on foot”: walking around on the ice (or icebreaking) and testing.

Now here’s what most people either don’t know or don’t think about: that model has been tested against samples almost constantly since it was first developed decades ago, and it is pretty darn accurate. When it has said, this year, that ice is as little as a meter thick near the North Pole, vessels went out and, lo and behold, found a large patch of ice 0.9 meters thick near the North Pole. That’s clearly not something that many scientists were anticipating six years ago as likely – but the model was predicting it.

It’s the Volume, Stupid
All right, let’s get back to the model. Suppose we have equilibrium, and we start an accelerating rise in air and ocean temperatures. What changes would we expect to see in real extent, area, and volume year to year?

Well, think about what’s happening. The water heating is nibbling away at the “balance point” from beneath, and at the edges. The air heating is nibbling away at the top of the ice during the summer, and expanding the length of the summer a bit, and heating the surface water at the edges a bit. So it all depends on the importance of water heating vs. air heating. If water heating has little effect compared to air heating, what happens to volume is not too far from what happens to area and extent: They start plummeting earlier and go farther down, but during the winter, when just about the whole basin gets frozen again, they go back to about where they were – a little less, because the air is more often above freezing at the very edge of the ice even in the depth of winter.

Now suppose water heating has a major role to play. This role shows up mostly as “bottom melt”, it shows up year-round in about the same amounts, and it shows up almost completely, until nearly all ice is melted, in volume. So what you’ll see, when you look at extent, area, and volume, is that volume goes down for quite a while before it’s really clear that area and extent are changing, and then area and extent start going down at minimum but not much if at all at maximum, and then the melt moves the balance point up and up and some of that normal curve starts reaching “negative thickness” as some of that melt reaches the surface of the ice and that means area starts moving down fast, and extent with it, and then in a year or two half of the ice reaches “negative thickness” and it seems like the area is cut in half and the extent by one-third, and the same thing happens the next year, while the rate of volume decrease actually starts slowing, and then in two more years there is effectively less than 1% of the ice area at maximum showing up at minimum, and less than 15% of the extent.

This is where lots of folks seem to fail to understand the difference between a measure and what it’s measuring. If you look at the volume curve, until the last few years before it reaches zero it seems to be accelerating downward – then it starts flattening out. But what’s actually happening is that some parts of the ice have reached the point where they’re melting to zero – others aren’t, because there’s a great variation in ice thickness. If we represented those parts of the ice that have melted as “negative thickness”, then we would continue to see a “volume” curve accelerating downward. Instead, we represent them as “zero thickness”, and the volume delays going to zero for four or five years.

So what are our measures telling us about the relative importance of water and air heating? First of all, the volume curve is going down, and going down at an accelerating rate. From 1980 (start of the model) to 2005, average thickness at minimum area (a good proxy for thickness at ice minimum), went from more than 3 meters to a little more than 2 meters. From 2006 to 2010, it has gone from there to a little more than 1 meter. If it were to follow the same path until 2013 or 2014, then “volume” would go to zero then at minimum, with the model’s measures of volume going to zero perhaps 3 years later, as explained in the last paragraph.

But there’s another key fact about volume: volume at maximum went down by about the same amount (15 million cubic meters) from 1980 to 2010 as volume at minimum (14 million cubic meters). In other words, instead of volume springing back during the winter almost to what it was 30 years ago, it’s almost exactly tracking the loss of volume the rest of the year. The only way this happens is if water melting from the bottom is a major part of the loss of volume from year to year.

Let’s sum up. The samples show that volume is going down at an accelerating rate throughout the year. The accelerating drop in volume shows that bottom melt is driving accelerating loss of Arctic sea ice year-round. Thus, we can predict that area and extent will soon take dramatic drops not obvious from the area and extent data, and that these will approach zero in the next 7 years or so. In other words, the key to understanding what’s going on, and what should happen next, is as far more the measure of volume – rightly understood – than area or extent.

Where’s The Awareness?
And yet, consistently, scientists and wannabes alike seem to show a surprising degree, not just of disagreement with these projections, but also of a seeming lack of awareness that there should be any issue at all. Take some of the models that participants in Arctic Sea Ice are using to predict yearly minimum extent and area. To quote one of them, “volume appears to have no effect.” The resulting model kept being revised down, and down, and down, as previous years that had shorter summers, slower rates of melt for a given weather pattern, and less thin ice, proved bad predictors.

Or, take one of the recent scientific articles that showed – a very interesting result – that if, for example, the Arctic suddenly became ice free right now, it would snap back to “equilibrium” – the scientist’s phrase – within a few years. But why was the scientist talking about “equilibrium” in the first place? Why not “long-term trend”? It is as if the scientist was staring at the volume figures and saying, well, they’re in a model, but they’re not really nailed down, are they, so I’ll just focus on area and extent – where the last IPCC report talked about zero ice in 2100, maybe. So suppose we substituted the volume’s “long-term trend” for “equilibrium”, what would we expect? Why, that a sudden “outlier” dip in volume, area, or extent would return, not to the same level as before, but to where the long-term accelerating downward curve of projected volume would say it should return. And, sure enough, the “outlier” volume dip in 2006-2007 and extent dip in 2007 returned only partway to the 2005 level – in 2009 -- before resuming their decrease at an accelerating rate.

And that brings us to another bizarre tendency: the assumption that any declines in area and extent – and, in the PIOMAS graph, in volume – should be linear. Air and ocean temperatures are rising at an accelerating rate; so an accelerating downward curve of all three measures is more likely than a linear decline. And, sure enough, since 2005, volume has been consistently farther and farther below PIOMAS’ linear-decline curve.

Then there’s “tipping point”: the idea that somehow if Arctic sea ice falls below a certain level, it will inevitably seek some other equilibrium than the one it used to have. Any look at the volume data tells you that there is no such thing as a tipping point there. Moreover, a little more thought tells you that as long as air and ocean temperatures continue their acceleration upwards, there is no such thing as a new equilibrium either, short of no ice year-round – which is a kind of equilibrium, if you don’t consider the additional heat being added to the Arctic afterwards. Sooner or later, the air and ocean temperature at all times in winter reaches above minus 4 degrees C, so that water can’t expel its salt and freeze. And that’s not hypothetical; some projections have that happening by around 2060, if not before.

And, of course, there’s “fundamental change in the system”, which implies that somehow, before or after now, the Arctic climate will act in a new way, which will cause a new equilibrium. No; the long-term trend will cause basic changes in climate, but will not be affected by them in a major way.

Follow-On Failures
The result of this data snow blindness is an inability to see the possibility – and the likelihood -- of far faster additional effects that spread well beyond the Arctic. For instance, the Canadian Prime Minister recently welcomed the idea that the Northwest Passage is opening in late summer, because Canada can skim major revenues off the new summer shipping from now on. By the data I have cited above, it is likely that all the Arctic will be ice-free in part of July, August, and September by 2020, and most of the rest of the year by 2045, and very possibly by 2035. Then shippers simply go in one end of the Arctic via the US or Russia, and the other end via the Danes (Greenland) or Russia, and bypass Canada completely.

And then there’s the Greenland land ice. Recent scientific assessments confirm that net land ice loss has doubled each decade for the last three decades. What the removal of Arctic sea ice means is the loss of a plug that was preventing many of the glaciers from sliding faster into the sea. Add a good couple of degrees of increased global warming from the fact the Arctic sea ice isn’t reflecting light back into space any more, and you have a scenario of doubling net Greenland land ice loss for the next three decades, as well. This, in turn, leads to a global sea ice rise much faster than the 16 feet (or, around the US, maybe 20 feet) rise that today’s most advanced models project by assuming that from now on, Greenland land ice loss will be rising linearly (again, why linearly? Because they aren’t thinking about the reality underlying the data). And, of course, the increased global warming will advance the day when West Antarctica starts contributing too.

The Net-Net: The Burden of Proof

As a result of these instances of data snow blindness, I continually hear an attitude that runs something like this: I’m being scientifically conservative. Show me why I shouldn’t be assuming a possibility of no long-run change, or a new equilibrium short of year-round lack of Arctic ice, or a slow, linear descent such as shows up in the area and extent figures.

What I am saying -- what the data, properly understood, are saying to me – is that, on the contrary, the rapid effects I am projecting are the most likely future outcomes – much more rapid declines in area and extent at minimum in the very near future, and an ice-free Arctic and accelerating Greenland land ice loss over the 20-30 years after. Prima facie, the “conservative” projections are less likely. The burden of proof is not on me; it’s on you, to disprove the null hypothesis.
So what about it, folks? Am I going to hear the same, tired arguments of “the data doesn’t show this yet” and “you have to prove it”, or will you finally start panicking, like those of us that understand what the data seem to be saying? Will you really understand which mathematics applies to the situation, or will you entrance yourselves with cookbook statistics and a fantasy world of “if this goes on”? It may be lonely out here; but it’s real.