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And to clarify: Evilreductionist = David Gould. For some reason, I was not able to log in as Evilreductionist from my home computer last night.
Timothy Chase, Yes, you are correct: feedbacks may already be built in. But a linear function is, imo, the simplest function to begin with. As to making it more physical, that was not really my intention. All I wanted to do when I first started looking at this was to try to work out how on earth linear temperature increase could lead to parabolic ice decline. As it turned out, the very first function I tried - the linear one - led to a parabolic result without me having to worry about inserting any feedback terms specifically. And as for using ice melt in my function as a proxy for temperature, that is what I do. As the starting assumption was that there is a linear relationship between the two, there is no need to have a starting temperature function.
In other words, when there was no warming, there is no parabola - just a horizontal line. When the warming starts, the shape of the ice decline is a parabola curving downwards. When the cooling starts, the shape of the ice increase is a parabole curving upwards.
Wipneus, I understand that if you extrapolate a parabola into negative years then that happens. However, mathematically, whatever year you wish to start at you simply define that as 'year zero'. Then that issue goes away. Usually, the year that you start is simply one where the melt and the refreeze were equal - in other words, an equilibrium world. Then you model heating (or cooling) from there. The simplest way to model what happens in a cooling world is to set the world to a cooling world (which just means changing the sign).
Wipneus, I am not sure that I understand. If I change the sign of the slope to negative (indicating a linearly cooling world) sea ice increases, as expected. Could you clarify your criticism?
Superman, The simplest model of ice melt in a linearly warming environment produces a quadratic decline in the amount of ice at the end of each melt season. You do not need to include any feedbacks, positive or negative, to get this result. Hence, given that the amount of ice is declining in a quadratic fashion there is no evidence that any feedbacks, positive or negative, are yet operating. You can test this yourself. I can email you the excel spreadsheet that shows what happens to ice volume in an environment in which temperature is linearly increasing. Or you can start with these basic assumptions: 1.) Refreeze always equals some value, F. 2.) Melt equals some value M + yZ, where y is year and Z is the slope of the increased melt each year. Note each year slightly more ice melts - a linear increase, mapping directly to a linear increase in temperature. Put this in an excel spreadsheet. Iterate for a number of years. Plot the graph of minimum volume. You will see that it declines quadratically. No feedbacks required.
My understanding of 'the rot' is that it was caused by ever increasing temperatures in the Arctic. I do not think that the collapse was anything special - one year, there was going to be a dive in volume, simply because of variables around the trend - just as for a short period volume appeared relatively stable, even though there was a long-term and rapid decline in progress. We are not really going to learn too much more than 'the more heat you push at ice, the faster it melts'. There may be things to learn about how that heat was distributed, but the basic reason for why we are in the state we are now is that heat melts ice. Now, it could be that the reason many models did not pick up on the rapid decline is that scientists were trying to take into account all kinds of potential negative and positive feedbacks, feedbacks that in the real world simply have not had time to kick in. While you do need to pay close attention to the trees a lot of the time, sometimes the forest is worth looking at, too.
And I keep saying 'exponential' when I mean 'quadratic'. Sigh.
I have to say that I do not think that we have yet experienced the kinds of feedbacks that seem to be discussed here, apart from temperature amplification due to albedo changes. I have said this before, but a simple model of ice melt with linearly increasing temperatures produces an exponential decline in ice volume. We are observing both linearly increasing temperatures and an exponential decline in volume. No other parameters or feedbacks are needed to model this. Further, the time period over which all this has occurred is very, very short. For many feedback processes, there simply has not been enough time for them to kick in in a significant way. I understand that the simple model is likely to be wrong in important respects. But it seems to be working over the short time period that we have observed.
And Chris Reynolds nailed it with the 1.4 metre thickness calculation.
My simple projection was that this year should hit around 3.2. I think that that is going to be about right. This is just a quadratic projection, which matches my understanding of the very basic physics.
Toggle Commented Aug 30, 2012 on ASI 2012 update 10: (wh)at a loss at Arctic Sea Ice
Sorry - 25 August.
Piomas - via Tamino - has come out early. The last few days were: 2012 229 4.098 2012 230 4.032 2012 231 3.963 2012 232 3.907 2012 233 3.828 2012 234 3.772 2012 235 3.719 2012 236 3.651 2012 237 3.629 2012 238 3.599 I think this is only up to 26 August.
Transient sensitivity has a central value of 2 degrees. And according to UAH, the temperature above the Arctic ocean has been increasing by 0.53 degrees per decade for the last 33 years, which translates into an increase of around 1.7 degrees.
Toggle Commented Jun 21, 2012 on Fringe fries at Arctic Sea Ice
I know that the current rate of ice melt is exciting, but as of yet it does not seem to me that we have too much clear evidence that this year will break the record. After all, this time last year ice was at a record low and it did not break the record. Weather still has a huge role to play in all this. I think that all we can say at this stage is that the thin late season ice is melting very rapidly, which is something that we expected in any case.
Toggle Commented Jun 14, 2012 on Fringe fries at Arctic Sea Ice
D'oh: right the first time on the equation ...
Toggle Commented Mar 19, 2012 on March 2012 Open Thread at Arctic Sea Ice
Correction to the equation: C is the *maximum* ice value at t = 0.
Toggle Commented Mar 19, 2012 on March 2012 Open Thread at Arctic Sea Ice
To add some more rough maths to the picture, if we assume: 1.) an ongoing energy imbalance of 0.5 watts per square metre; 2.) roughly 10^7 seconds in the melt season; and 3.) roughly 10^12 square metres in the Arctic; then we end up with around 5 * 10^19 additional joules being pumped into the Arctic every year. It takes around 3 * 10^17 joules to melt one cubic kilometre of ice. This indicates that there is more than enough extra energy per year to melt around 25 additional cubic kilometres, plus some for ocean heating plus some for atmospheric heating. And if we plug in proper figures (around 17,500 for the volume at the end of the melt season of year 0) to the equation, 25 is close to the number that matches what we have seen in the PIOMAS volume curve.
Toggle Commented Mar 19, 2012 on March 2012 Open Thread at Arctic Sea Ice
The equation (not exponential - polynomial) that falls out is: -At^2 + At + C where A is half the annual increase in melt and C is the minimum ice value at t = 0, with t the number of years since t = 0. As an example, starting with an annual maximum of 35,000 cubic kilometres and an annual minimum of 25,000 cubic kilometres and increasing melt by 10 cubic kilometres per year sees zero summer sea ice in 71 years. Increasing the melt by 30 cubic kilometres per year instead sees zero summer sea ice in 41 years.
Toggle Commented Mar 19, 2012 on March 2012 Open Thread at Arctic Sea Ice
The exponential curve falls mathematically out of the physics of a simplified ice melt model. If temperature in the Arctic is increasing linearly then it can be assumed that the amount of ice lost each melt season will also increase linearly, as that is how the physics of melting ice works. If we assume that there is no change to the freeze - in other words, the same amount of ice freezes each year - then we end up with an exponential decline in end of summer ice volume. Try it in excel or in R. There is no feedback necessary for this to occur. All it requires is a linear increase in temperature to cause a linear increase in ice melt. Note that this is a very simplified model. But the fact that the simple model automatically pops out an exponential result is pretty interesting when that is what we are observing ...
Toggle Commented Mar 19, 2012 on March 2012 Open Thread at Arctic Sea Ice
It seems to me that Neven is right: in the short term, sea ice rarely behaves as we expect it to based on the data that we have. And this is, obviously, because there is so much that we do not know about what is going on in the Arctic. There are 23 more days in August, so a lot may happen. But I think a record is unlikely at this stage of the game.
From a physical perspective, if the amount of energy in the Arctic increases linearly, ice volume should decline in a parabolic way (ie, as a 2nd order polynomial function). I would suggest that the temperature data that we have for the Arctic (for example, via UAH here: shows that energy in the Artic is indeed increasing in linearly. So the only thing that would prevent a parabolic decline would be if there was something that prevented this increasing energy from mixing well across the whole Arctic. While I can certainly see sheltered bays and narrow channels - such at those in the Canadian archipelago - being barriers to energy mixing, there is nothing of the kind in the central Arctic basin. Unless such a physical mechanism is presented, I am betting on parabolic decline in Arctic ice volume.
Toggle Commented May 18, 2011 on Trends in Arctic Sea Ice Volume at Arctic Sea Ice
Patrick, interesting stuff. :) Is there still an ice bridge in the Nares? If so, how long do you think that it is likely to remain? [i'm currently having trouble accessing your blog, so you might reply here] David
Toggle Commented Mar 23, 2011 on Ice In Baffin Bay at Arctic Sea Ice
Looks good. I was hoping that you would do this. While I check out your ice graphs page every day, sometimes the link disappears from my browser window, making it necessary for me to google the site. A small delay, but for an addict that delay is painful indeed! ;) And now I will no longer have to suffer. Fantastic. :)
Toggle Commented Mar 22, 2011 on New design at Arctic Sea Ice
re albedo and predictions for a rapid collapse, if you run the prediction model for last year's numbers, what happens? Does it come close to matching what we saw?