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Chris Reynolds

SATire (Jens),

The maths describing things like whether there is a tipping point in Arctic sea ice is common to that found in engineering examining similar problems.

e.g. Eisenman, 2012, "Factors controlling the bifurcation structure of sea ice retreat"

The problem is that, as in engineering, the problem is so complex and the unknowns so numerous, that one has to see what happens. In engineering that's a case of - "Well, lets switch on the power and see if it works." In the case of the Arctic this translates to - "Well, let's just be patient and see what happens."

There are seriously smart people working on the Arctic problem. Many of them are coming round to anticipating a rapid collapse, though a fewer number are coming out and saying so. The professional conservatism of scientists means they tend to sit back and watch when they suspect something astonishing will happen. If they can't crack the problem, I'm sure it can't be cracked.

We really don't know, which means we just have to see what happens.


Rob, my apologies for any offence caused. That was not intended and I think it is really great to see you try such calculations.

>"it would be more helpful if you point out where I go wrong in reasoning, assumptions and math in that post"

Sadly I fear my abilities are unable to keep up with you.

If we go back to my post next but one after your Dec 15 posts. I am wondering if it is necessary to present a few (maybe just one per season) calculations for different times of year with a higher temperature in summer that grows in Autumn and declines in winter/spring.

Unfortunately, I am lacking in ability to see how to convert your linear/sqrt ice thickness into a heat flow measure. Consequently, I am unsure how well this is incorporated into your calculations or whether it isn't necessary/appropriate.

Maybe such a seasonal review of what is happening isn't needed because I am not following very well.


Dear Chris,

thank you for the link to the very interesting paper - it is engineering-like as I asked for. Unfortunately, it is not very helpful for my intention. Maybe I was not clear enough. The basic calculations above in the thread (e.g. by Rob and Wipneus) probably have more potential to convince poeple, if an ice-free arctic in 2015 could be predicted right now in an article. That surely would draw attention to the authors, if it finaly happens.

Since we have only one earth, just sitting back and lock is actually not a feasible option. In engineering we can deal with such things when we build the really big machines. We urgently need some kind of risk management and we need to know the save margins to decide in a proper way.

What we actually need to know are the propabilities for the earth running in "expensive" states. I am not talking about a sea-level rise of 1 m or so - that would be in the margins of todays european costal protection. But to give a probability for 5 m sea-level rise could allow poeple to decide between well known costs for additional costal protection and alternative CO2-reduction. To exclude 20m sea-level rise could also help, because that would be clearly to much to deal with - of-shore cities are quite expensive.

An ice-free arctis is not directly expensive, but a proper forecast would give confidence to the authors of said forecast, because attention will be there. If those authors could manage to give propabilities for serios sea-level rise inculding some of the other feedbacks (permafrost, methan under the ocean...) one could estimate the costs of the thread and could compare that to the cost of avoiding it. E.g. 10% probability for 20 m would convince everybody to stopp burning carbon. 10% probability for 5 m would perhaps be tight. Since we have only one trial the things should be worked out very carefully. Or do we really have to wait until the new algea in the soon to be ice-free arctic ocean have eaten all the CO2 down to 300ppm again after a wrong decision?

Ian Mc

First post, so apologies up front if necessary. I wanted to add my 2c to the Predict Next Year's figure pool.

As a lot of predictions based on the more usual mathematcial functions are around. I went with an ARIMA model (http://en.wikipedia.org/wiki/Autoregressive_integrated_moving_average).

The model is ARIMA(4,2,0) and, while I'm not completely happy about it, it follows the data fairly well. Predictions (95%CI) in 1000km3 are,

2013 3.06 (0.18,5.94)
2014 1.52 (0.00,5.03)
2015 0.42 (0.00,4.69)
2016 0.00 (0.00,5.34)

So a 5% chance of an ice free Arctic next year rising to 50% by 2016.

As an aside the autocorrelogram of the differenced Sept Mean values showed that, as expected, this year's value is the best predictor of the next year's. However a smaller knot around 11/12/13/14 years might fit nicely with the solar cycle (proviso that there are 3 full cycles in the data).

Happy Holidays
Ian Mc


The economical aspects of climate change have been reviewed in several well-known studies, FI the Stern Review, now 6 years ago.
Since I haven’t read it myself, I can imagine it would be worth while to do that. It could provide some standards on how to work out the status quo now that we’re further down the line.

As Chris has explained, the situation is quite intricate. Mankind is very good at solving practical problems on a one-by-one basis. This one OTOH is complex and seems to be far out of reach of any ‘engineering’.

I can’t see how an ice-free Arctic could be called ‘not directly expensive’, other than one would consider climate change as just alike one of everyday’s societal challenges.

It is not.

Instead of dealing with engineerable risks, we’re confronted by a situation that has phylosophical, even existential format.

To illustrate it crudely in basic IT-symbolics: 1 = cut GHG = chance / 0 = out, over.

Rob Dekker

Apologies accepted, crandles.
The purpose of my post was to assess how much an energy "perturbation" ripples through the current Arctic over a full year, using current observations to assess the impact on winter freezing and summer thawing, so as to assess 'stability' of the Arctic at this point in time.

However, others here also complained that they could not follow my reasoning, so I'll have to find a better way to explain myself.

Meanwhile Chris Reynolds gave this great reference to Eisenman 2012, and I think that paper is much more interesting, and much greater depth, and much better written, than my two-paragraph deliberations on energy perturbations.
Very interesting discussion in that paper regarding the stability of ice covered state, an ice free minimum state, the stability of an ice free summer, and an year-around ice free Arctic.

So may I suggest we use that paper as foundation for further discussion on the stability of the Arctic ?

Finally, with 2012 drawing to a close, an historic year in which snow cover as well as ice cover anomalies broke all previous records, and thus once again the Arctic appears to be more sensitive than even the most aggressive models seem to suggest, again we wonder about the future.

In that respect, let me close with what I found the most convincing "climate change" graphic of 2012 : Kinnard et al 2011, as updated with 2012 data :

We all know what is going to happen, and that the biggest changes are still ahead of us. We just don't know how fast Nature will adjust to our grand experiment.


Werther, I think you are totaly right with your statements:

(Werther): Instead of dealing with engineerable risks, we’re confronted by a situation that has phylosophical, even existential format.

To illustrate it crudely in basic IT-symbolics: 1 = cut GHG = chance / 0 = out, over.

Unfortunately, while it was possible to get the people in northern Europe with that argument it failed in Northern America so far. The Stern-Review was obviously not convincing there, too. I think, those poeple need the "money-argument" badly, but maybe you know that folks better than me.

In Germany we face now the danger, that we loose the poeple in the next few years. After paying significant investments (houses insulation and renewable energy paid by taxes and electricity bill) poeple are getting frustrated paying the bill alone and in vain.

So we need North America in the boat soon to save the momentum. Once it is lost, poeple in northern Europe will just increase the levee a bit and start watching the world getting warmer.


Larger image:

Max and min PIOMAS volume can be extrapolated using gompertz fit but that doesn't have a physical basis. So can we extimate the volume reduction during the melt season (for simplicity called 'melt' and volume increase (for simplicity called freeze)?

The fit of melt volume shown above is largely a linear function of a 92 day average CT area for May, June and July. The weigtings are a sine wave that peaks on 21 June and troughs at 21 Dec. The less ice area there is, the more open ocean there is to absorb heat. Using a linear function of such heat absorbed for volume of ice melted seems reasonably physically based. I am also assuming deviations from trend in max volume are noise that reverses itself.

So my formula for melt volume is
-.01319 * (92DWAA) + 26.855 + (Max vol- Max Vol fit)

Larger image:

This appears to show that the 92 day weighted area average (92DWAA) can be estimated as a linear function of the maximum volume. This seems reasonable - the more volume of ice there is at maximum, the harder it will be to clear areas of ice so there is a larger area of ice.

The projected 92DWAA shown is calculated as
16.135 * Max volume + 309.2

Judging from this, there does not (at least yet) seem to be any reason to use any more complicated function of the potential variables that could be used.

What I am less clear about is what shape function to use to project the increase in volume as the minimum volume declines. A linear function of the reciprocal of minimum ice volume seems too optimistic.

Any suggestion?


The lower the minimum ice volume, the more heat energy there is likely to be in the water so a later start to the freeze but lower ice volume mean faster heat flow. Also more GHG reduces the maximum ice volume that can be reached.

An alternative to trying to model the delay and the heat flow is to suggest the past is a good guide to how the maximum reduces through two influences of higher GHG and delayed start to freeze. Using a tangent extrapolation of the gompertz fit of maximum ice volume seems likely to overestimate the freeze.

So the question is whether this is still sufficiently based on physical causes to say such an extrapolation has physical underpinning?


Hi Crandles,

I would go with Wipneus' exponential for the min. volume because melt rate in summer would increase with free ocean area. If you ignore icebergs origin from land you could skip the Gompertz-Tail.

Therefore, freeze would be maximal when min. volume becomes zero. After that point freeze and max volume are same line. If max. volume also would show an exponential decay (that is exactly the question how things will turn out), the difference would be the delta between freeze and melt lines. If an ice-free arctic is now natures all-year-goal, everything would be zero after that point. So linear trend is unlikely in nature - it is just a good short term approximation.


Hi SATire,

There is very little difference between exponential and gompertz fits. I think it is much more preferable to use gompertz fit so that I am not a piori assuming a fast end to the remaining ice by the shape of curve I am using.

However, what I am getting at here is that scientists don't like either exponential or gompertz extrapolations because of the lack of physical basis for such assumed shapes. So I am trying to move to modeling the changes in ice volume with a physical basis of what happens to the energy budget. The aim is to blunt the argument that extrapolations do not have a physical basis.

The modeled melt volume seems to work well - not only is the general pattern achieved but the peaks and troughs seem fairly well matched. I would expect better evidence of the relation of melt volume with area can be obtained by looking at data on a regional level. With more evidence it may turn out that a linear fit of volume^0.66667 or some other power might do as well or better. However, for the moment, with a linear fit of volume apparently working so well it seems a bad idea to add extra free parameters.

It is getting a freeze volume projection that is physically based that I am mainly looking for help with.

Yes, I agree with you that the maximum function is unlikely to be linear or even smooth when minimum ice reaches zero. So if I go with a linear tangent extrapolation of gompertz fit of maximum, the analysis using that system should be restricted to when zero minimum is reached and what happens before then. Long extrapolations outside the range of data available are always very dodgy so what happens after zero minimum probably shouldn't be looked at this way anyway.



from a physicist point of view I do not like linear functions - only as a first approximation of anything ;-) The exponential decay is based on a physical argument (rate of something proportional to a quantity). The Gompertz-function is nearly an expontial - if you ignore the tail and I think you should in order to ignore ice from land swimming in the sea as its physical basis. That part is not froozen in winter...

The big questions is, if maximum ice volume decays exponentially - rate equations become difficult. I would guess a lot of exponentials of both signs are in the game and it is far from clear, which one are dominant at which stage. But the linear function is out for sure, because freeze will max. and finaly will become zero ;-)


Yes, I agree that the linear model is wrong. But then all models are wrong but some models are useful. So the question is, is there a more useful fit shape than my suggestion of linear as a first approximation. If your guess is lots of exponentials of both signs and far from clear which are dominant then that seems like a guess which isn't very useful.


Or maybe you are usefully telling me to give up because it cannot be done?

Certainly a new snow cover feedback could become important. Where ice is so thin, snow cannot collect on the ice so there is less insulation and more ice can grow faster. Extrapolation tends to assume if we haven't seen it yet, it won't happen at all.

That does seem a valid criticism of attempts at extrapolation.


hmm? The "lot of exponentials" was only ment for the max. ice volume after the time the min. volume already hit zero. I would guess a Gompertz there in average - no ice in warm winters and some ice in cold winters averaging to that tail. But that is a cruel guess. That Eisenman-paper describes nicely the many things to consider - unfortunately without a crude solution, of course.

But until summer ice is completely gone - the exponential should work fine - wasn't that your intention of your picture? So my suggestion for that range is just to skip the tail from Gompertz ;-)


"to skip the tail from Gompertz" to describe the min. ice volume. Sorry for being unclear again.


Sorry for misunderstanding your 'lots of exponentials'.

>"The exponential decay is based on a physical argument (rate of something proportional to a quantity)."

We would need negative rate of change of ice volume to be proportional to reciprocal of volume? But to suggest that is physically based we would have to describe why. I don't really expect that to drop out of the analysis.

Suggesting that heat flow is inversely proportional to volume or thickness seems over-optimistic to me. I imagine something more like: There is an equilibrium thickness and heat flow is proportional to the difference between actual thickness and the equilibrium thickness.

In the centre of the pack where there is plenty of time to approach the equilibrium thickness, then the equilibrium level is the only thing that matters and that will depend on GHG levels. Towards edges of maximum pack size, there probably isn't time to approach to equilibrium thickness so a delayed start to the freeze up matters.

So we end up with part of the maximum volume depending almost entirely on GHG levels and for remaining part of the maximum volume, both GHGs and minimum volume &/or water temperatures matter.

That doesn't seem like a situation where the decline would be as fast as exponential to me. Quadratic or cubic formula seem more possible than exponential. However, other explanations of the major dependencies may well be better.


Maybe I can not really get your point. Instead of guessing, I am trying to give you mine ;-)

Simple approximation of albedo-feedback kicking the summer ice minimum (linear rate equation): The ice is eaten by heat absorbed in open water --> the rate of decline of ice-volume is proportional to negative ice-volume --> exponential decline.

For winter ice maximum the radiative cooling, the humidity and the winds are important and ocean currents play a major role --> forget about such simple formulars, the exponential may describe a trend but that would be purely by accident.

When you talked about equilibrium you lost me totally - we are not at equilibrium right now. We now have nearly pre-industrial clima with mid-pliocene CO2-forcing heading towards miocene CO2-forcing. Clima is now starting to swing to the new equilibrium - first round will be ice-free arctic, second round thawing of greenland and siberian permafrost - gaining some extra momentum from methane for a short time until we all like to swim in the arctic basin. Third round maybe thawing of antarctica and some extra water for our off-shore cities. After that we will have equilibrium again - a different one. It is easy to describe the equilibrium states using history - but the dynamics of the transition is rather complex, because we kick it so hard and fast and we can not find the timescales in history, because neither man nor dinosaurs acted like us before.


Sorry I wasn't clear about what I meant by equilibrium. I was talking about an equilibrium ice thickness if winter continued indefinitely. If the ice is thin then the heat loss is at a greater rate than it is being supplied from below and extra ice form. If the ice is thicker then heat builds up and melts ice.

Above equilibrium is only considering thermal growth of ice. There is also mechanical thickening from crushing and slabbing which quite possible depends on the mass of ice around but the dominant process is the thermal thickening so I was concentrating on just that.

>"For winter ice maximum the radiative cooling, the humidity and the winds are important and ocean currents play a major role --> forget about such simple formulars, the exponential may describe a trend but that would be purely by accident."

Certainly the rate of radiative cooling matters and depends a lot on ice volume. Even if in some cases winter may be long enough for the thermal equilibrium thickness for winter to be approached, the equilibrium thickness depends on GHG levels, rate of upward head flow from below which depends on ocean currents and also air circulation from lower latitudes. So in reality almost everything depends on everything.

Does this mean I should give up or should I hope to pick out the really important dominant processes and see if approximating those gives a good fit?

If GHG level are increasing in a reasonably linear manner and this determines the level of humidity that can be supported, then perhaps the changes in these factors like these and air and ocean current patterns are similar from year to year.

Anyway FWIW, using that linear extrapolation of maximum curve, ie same change as from 2011 to 2012 in the gompertz fit occurs in each subsequent year my projection runs as follows:

Year Max Vol Melt Min vol Freeze
2012 21.951 18.105 3.846 17.533
2013 21.379 18.227 3.151 17.379
2014 20.531 18.408 2.123 17.501
2015 19.624 18.601 1.024 17.682
2016 18.705 18.796 -0.091 17.875

Zero summer ice reached by 2016 despite the fairly conservative linear extrapolation of the max volume curve.


Doh, got that projection wrong. Try again:

Year Max Vol Melt Min vol Freeze
2012 21.951 18.105 3.846 17.379
2013 21.225 18.260 2.965 17.534
2014 20.499 18.414 2.085 17.688
2015 19.773 18.569 1.204 17.843
2016 19.047 18.723 0.324 17.997
2017 18.321 18.878 -0.557 18.152

2017 before zero summer ice reached.


New study in GRL finds that first year ice absorbs 50% more solar radiation then muli year ice.

“a continuation of the observed sea-ice changes will increase the amount of light penetrating into the Arctic Ocean, enhancing sea-ice melt and affecting sea-ice and upper-ocean ecosystems.”

According to the NYT
"The authors of the new study say their data allows them to predict that the trend will only accelerate as the feedback loop continues."



I previously said: But to suggest [exponential decline] is physically based we would have to describe why. I don't really expect that to drop out of the analysis.

I now think it does seem to drop out of the analysis, albeit with other additional terms.

I ended up with
Melt = -0.01319 (16.135 * Max Vol + 309) +26.855
Simplifies to
Melt = -0.2128 * Max Vol + 22.777

I think through my approximations the freeze volume also ends up with steady fall and a part that depends on the minimum volume.


yes crandles, as I mentioned January 02, 2013 at 14:13, freeze would be maximal just when min. volume becomes zero. After that point freeze and max volume are same line - whatever that line is.

Because the exponential is the solution of above mentioned most simple albedo-feedback rate equation (actually exponential growth of the volume loss) I would ask what is actually the physical meaning of your analyis. Slowing of albedo-feedback as min. volume approaches zero? What could make the min. volume line to take a sudden turn left? I am not convinced.


>What could make the min. volume line to take a sudden turn left?

An feedback effect which we haven't yet seen in action. I gave an example earlier. If ice is thick enough in autumn it will accumulate snow which is a good insulator and keeps the freeze volume down. If the minimum volume is low enough snow won't collect on sea surface and this allow more ice to form as more heat can pass through ice than through ice and snow.

The purpose of gompertz rather than exponential was so that I didn't rule out a long tail of slowly reducing ice. So now I can claim that I started a piori with a function that allowed for possibility of a long tail of slowly reducing ice and it is the data that it seems to be the data that has ruled this out not the a priori assumption. That isn't quite perfect as that snow feedback might start to have noticable effect only at lower ice volumes than we have reached. But I have put in some conservative assumptions in to compensate for such unknowns.

Bifurications are possible so ice free before 2017 is still possible. What I am trying to do is use conservative assumptions and show ice is unlikely to last beyond 2017 plus perhaps a couple of years for noise.

Not only that but also minimize any criticism that maybe my model is wrong or maybe the shape of the extrapolation is wrong because I haven't got a physical basis for the extrapolation.

Maybe for a best guess, exponential+other components functions would be appropriate because of the way I see such equations dropping out of the analysis.

Anyway the way to show there won't be a sudden turn to the left is to start by assuming there is. That job isn't fully done, particularly re people who think that snow feedback effect might be important. However, proofs aren't part of science only math, we should follow the analysis wherever it takes us without wishing it to show one way or another. If it shows that the best case isn't very good, sorry but so be it. Better to have the advance warning than not.(i.e. You seem to have been misinterpreting my aim.)


So you disagree on the exponential solution of the albedo-feedback rate equation for min. ice volume but instead you model a "feedback effect which we haven't yet seen in action" to make the analysis looking more smooth, somehow nice and convincing? Sorry, I am out of this. Maybe I really have misinterpreted your aim.


But I actually agree that an exponential with other linear components looks more likely to be the form of the solution.

I don't think I have attempted to model a feedback that I haven't yet seen (how would I do that?), just made some allowances in case there are some other feedbacks that have been excluded by my simplifying assumptions.

If you want to know my aim in finer detail beyond to further the science/ability to predict sea ice: I am wondering if it is possible to rule out the 2020s', the 2030s' and later dates for first year without summer sea ice that that models show and scientists tend to quote but only so far as the science will allow me to do so. If the analysis showed those dates were entirely possible then there wouldn't be a problem like this to solve.

Only by making allowance for noise and other feedbacks that might exist to make the date later, can I credibly make a claim to show dates later than 2020 are very unlikely.


Kaufman et al 2009 produced a proxy temperature record of last 2000 years and found last decade was warmest. Unfortunately that record is only decadally resolved. Is it likely there is some record with annual resolution since 1979?

Would Kaufman be the person to ask?

Would upward heat flux over arctic area be likely to be related to average of last two or three years or more like 30? years or some other period?

Perhaps areas near the entry of Atlantic water would be related to average of past two or three years while areas further away would be related to longer averages. So some sort of average weighted towards more recent years.

If aiming for that sort of average, perhaps there is sufficient resolution in the proxy data in recent years to test if this has any correlation with maximum ice volumes?



To reaffirm the "hole" in the polar sea ice concentration and thickness imagery from NCOF is not real but a model anomaly, this is the response I received from the UK Met:

"The hole is not real - it is due to a combination of a gap in the satellite cover and a modelling problem. The model should be able to cope with the satellite gap, but is not doing this as well as it should. We are making a change later this month to rectify the problem, and the hole will no longer be present."

The upcoming change in the NCOF/Met modeling will be from the Louvaine le Neuve Ice Model (LIM), developed in Belgium. This is the model which has the problem. The change is a switch to the CICE model as a Met modeling basis.

According to the "My Ocean" Service Desk, the change may be coming on February 5th.

I am updating the A4R websites today, the CH4 and CO2 imagery are begin updated first, then the sea ice data.

The METOP 2 IASI data set is not complete, there has been a lapse in updating since the 10th. The NOAA OSPO have been notified.

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