I have been doing some work on Arctic sea ice volume, trying to determine whether a second order polynomial function had a physical basis. And I have discovered that it does. While others have obviously already worked this out, it is new to me, and thus at least a little bit exciting. :)

To look at this, what I did was sit and think about what would happen in an Arctic that was melting, and write down a few things.

The first thing that I thought of was that there are two significant parts to the Arctic year - the melt and the freeze. Using the values generated by Frank http://snipt.org/xwgn, I determined that over the period of the model (and, yes, PIOMAS is *not* data, but a model, but it does not matter for the purposes of this exercise) there was an increase in the amount of ice melting each year and a decrease in the amount of ice freezing each year. This increase and decrease were each moving in a linear fashion. It was difficult for me to see how a second order polynomial function could emerge from these linear functions. Silly me, as we will see.

So I set up a model that mirrored these linear changes in melt and freeze, and then looked at the yearly totals at maximum and minimum that resulted. Graphing these totals, I found that the declines in each perfectly followed a second order polynomial function ... What an earth was going on here?

I tried various values for the change constants in both the melt and the freeze periods, but always ended up with second order polynomial functions. So I decided to investigate this function a little more by differentiating it and seeing if the resultant function related in any way to the change constants.

And, of course, it did. What I found was that the differentiated function for the decline in ice volume at the end of the melt season was - with X years - always:

- (Melt Constant + Freeze Constant)* X + (Melt Constant + Freeze Constant)/2

The differentiated function for the decline in ice volume at the end of the freeze season was - with X years - always:

- (Melt Constant + Freeze Constant)* X + (3*Melt Constant + Freeze Constant)/2

Why these particular functions? The constants in them result from the 1/2 years offset between the two seasons. The Melt Constant + Freeze Constant is simply the total yearly change - the two constants added.

So integrating this returns us to our second order polynomial. And why do we integrate? Because the reductions in ice volume in any year are *summed* to the reductions in ice volumes of all previous years. And a sum function is an integral.

In other words, we do not start from scratch each year: each year, we are melting from a lower volume of ice and freezing from a lower volume of ice.

Basically, what it means is that if melting and freezing change in a linear fashion then we get a second order polynomial function for the ice volume totals.

And is there a physical basis for such a linear increase and decrease? Of course: the linear increase in energy, as measured through linear temperature change, in the Arctic due to rising CO2.

Which points to a dramatic crash in Arctic ice volume, and thus area and extent, over the next few years. Indeed, using PIOMAS, further modelling suggests that zero volume will be reached at the end of the melt period in 2018 at the latest, with it occurring possibly as early as 2013.

My projections are:

Year Volume (cubic kilometres)

2011 -> 3744

2012 -> 2853

2013 -> 1935

2014 -> 990

2015 -> 18

2016 -> -981

2017 -> -2007

2018 -> -3060

(all values have a two deviation error range of +/- 2445)

## Thursday, May 19, 2011

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