[Original paper: “Srikanth Toppaladoddi & J. S. Wettlaufer (2015) “Theory of the sea ice thickness distribution” *Physical Review Letters ***115 **148501]

Most people have a pretty good idea that the polar ice caps are melting. While things in the Antarctic are a bit less straightforward than the Arctic, it’s safe to say we’re losing ice in both. In the Arctic we’re losing it especially fast; nearly half of the ice cover that was there in the 1970’s is gone today.

When we talk about things like sea level rise, though, it’s not the *extent*, i.e. areal coverage of ice, that matters: it’s the *volume*. Water trapped in ice is water that’s not flooding our cities. It’s easy to measure the extent of the ice with satellites – you just look down and see where there’s ice – but as you might imagine it’s a lot trickier to recover the 3D information of how thick ice is, especially when you’re trying to do it for the entire Arctic or Antarctic. The thickness of ice can change a lot over small distances, too, so it’s hard to justify taking a handful of samples and hoping they represent an entire ice sheet.

{Fig. 1 – Arctic ice is pretty, but maybe its thickness is hard to guess by eye}

What we need, then, is a *theory* for the *distribution* of how thick the ice is. ‘Distribution’ here means we could predict the chances a randomly chosen ice chunk is likely to be however thick, analogously to predicting when we roll two dice that they sum to 7, or any other number.

If we know the total area covered by an ice sheet, and we know what percentage of the total ice cover is a given thickness, we can just add up all the thicknesses and figure out how much ice is there. Additionally, if we can develop a theory like this, we can then test it against existing observations. If it matches well, we can say not only how much ice there is, but why!

The authors of the rather short-yet-dense paper we’re discussing today have developed such a theory.

*Building a theory*

The thickness of ice is a mishmash product of melting, freezing, rafting, ridging, and the many other processes that affect ice. The ice pack thins by melting or thickens by freezing similarly over big distances; it’d be strange to find, if I were sitting on some melting ice, that the ice 10 meters away from me wasn’t melting – there aren’t many shady trees in Fig. 1, after all, so the sunlight is pretty even, and sunlight is what melts things. What makes the thickness hard to figure out is that on top of the melting and freezing there are all these little mechanical crunches and collisions that shuffle chunks of ice around in the pack; if one chunk of ice slides over another, all of a sudden that ice is twice as thick as the ice right around it.

The key observation the authors make, though, is that the mechanical processes that deform ice, e.g. the rafting and ridging, happen a lot faster than the melting and freezing. Based on this observation, they can treat this shuffling of ice around as random, and apply the vast mathematical theory of random processes, originally developed for the random shuffling of molecules in a gas. Once they do this, the math becomes somewhat straightforward, and they end up with a formula for the distribution of ice thickness, which incorporates a parameter for the vigorousness of the mechanical shuffling and another for the relative influence of melting/freezing.

*Testing the theory*

How does the theory compare with observations? Using a large set of observations from the Arctic, this is what it looks like:

{Fig. 2 – The theory seems to work impressively well; shown is a typical fit, where the blue is the theoretical guess, and the red is observed thicknesses. The *x*-axis is the thickness of the ice and the *y-*axis is the probability that a randomly chosen chunk of ice is that thick.}

That’s pretty good, if you ask me! Even the very worst fit out of all the observation sets isn’t bad. The parameter values for different months change a bit, corresponding sensibly to more melting in warmer months and more freezing in colder months, but the distribution makes sense.

This theory then gives us both i) a novel way to take satellite images of ice cover and determine how much ice there is beneath it, ii) a rationale for why ice thickness is the way that is, namely that lots of processes are happening but the large-scale ones of melting and freezing happen slower than the small-scale ones of mechanically shuffling ice around. That’s pretty cool.

The theory isn’t without limitations, of course; the assumption that the theory is based on is a bit idealized, and the fit isn’t always perfect to observations (no theory really is). That said, the paper just came out, and besides being plausible also appears to work excellently. The possibilities for where one could go with it are expansive, and it definitely marks a step forward in our understanding of polar ice cover.

Cilmate change deniers often forget that it is the volume, rather than area covered, of ice that matters; even if there were twice as much area covered by ice in the Arctic, it wouldn’t really matter if it were all a centimeter thick. This misconception makes it all the more important to improve how we measure sea ice thickness, for which this theory is a powerful tool.

Cael was once told by a professor that applied mathematicians are ‘intellectual dilettantes,’ which has been a proud self-identification for Cael since that moment. Cael is a graduate student at MIT & Woods Hole, & studies the ocean from a mathematical perspective; right now Cael is trying to figure out how detailed our measurements of phytoplankton communities can be if we detect them from space. Otherwise, Cael plays accordion, gardens, & reads instead of sleeping like it’s still fifth grade.

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