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ChrisReynolds

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Re: The Slow Transition
« Reply #100 on: August 18, 2014, 06:57:15 PM »
So many people to answer, I'll go through a narative with names in the text.

Wipneus presents the plot I was thinking of (thanks!), but I seem to have mis-remembered the degree to which there is a region of later melt outs along the transpolar drift. Doesn't affect my argument on that issue above, just means I can't rely on that plot as evidence.

Viddaloo shows a graph of percent melt out. I've previously done a lot of work on that using PIOMAS gridded data. The key graphic, based on a similar approach in GCM data (Keene et al IIRC - not sure on that), is here.

That's from this post:
http://dosbat.blogspot.co.uk/2013/12/go-on-say-something-outrageous.html
With which I no longer agree, but there are regional plots of the above approach on that page. Viddaloo's plot  shows how as the thickest MYI ice has declined it has allowed a greater percentage of total volume to be lost during the melt season.

Taking grid boxes over 2.2m in April as likely to be MYI, and that under 2.2m likely to be FYI, I've previously plotted the volumes of grid boxes above and below 2.2m.
http://1.bp.blogspot.com/-dtylZuUZTQA/U3JuStMr6eI/AAAAAAAAAXo/4TNvDYH3EpU/s1600/Nosedive.png
Note that FYI increases as MYI falls.

Michael Hauber makes a point I broadly agree with. I've said up thread * that I think there may be oscillation of MYI, and this implies memory. When I talk about the decline of MYI meaning memory is lost, it is the long term memory of ice many years old that is constantly being added to. That's what has allowed the decline of MYI as seen in the graphic linked to in the previous paragraph. But as FYI takes over, it's response to perturbations is rapid, so it diminishes the long term memory of the ice (Bitz & Roe). *Actually I said it at the end of this blog post:
http://dosbat.blogspot.co.uk/2014/08/july-status-2014.html

Cesium's first two graphs are interesting, but I've already made clear my interpretation of that data that there is actually a more modest increase from 1978 to 2006, with a step jump in 2007 and another step in 2010.
http://forum.arctic-sea-ice.net/index.php/topic,933.msg32494.html#msg32494
(I know those are in terms of percent volume loss, and my graphs are in terms of volume but I think the same argument covers the different metric).

The last two plots of volume gain shown by Cesium are covered by the same argument as above (step jumps in 2007 and 2010). But it is worth noting (IMO) the difference between whole Arctic and Central Arctic. In the whole Arctic, the thinning after the 2007 volume loss event affects largely the peripheral seas of the Arctic Ocean. In the Central Arctic it isn't until the 2010 volume loss that there is a 'lasting' effect - It may not last beyond this year!

I've been thinking about a series of posts on my idea (which is really just re-stating what the modellers have been saying + an obervation). As part of that, it has been bugging me that my explanations regards memory are a bit crap, and the Bitz & Roe is a bit mathematically heavy. So I'm pondering doing a 'toy' model to demonstrate it.

Let me know if I've overlooked anyone's points.

cesium62

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Re: The Slow Transition
« Reply #101 on: August 18, 2014, 08:02:10 PM »
Cesium's first two graphs are interesting, but I've already made clear my interpretation of that data that there is actually a more modest increase from 1978 to 2006, with a step jump in 2007 and another step in 2010.
http://forum.arctic-sea-ice.net/index.php/topic,933.msg32494.html#msg32494
(I know those are in terms of percent volume loss, and my graphs are in terms of volume but I think the same argument covers the different metric).

The last two plots of volume gain shown by Cesium are covered by the same argument as above (step jumps in 2007 and 2010). But it is worth noting (IMO) the difference between whole Arctic and Central Arctic. In the whole Arctic, the thinning after the 2007 volume loss event affects largely the peripheral seas of the Arctic Ocean. In the Central Arctic it isn't until the 2010 volume loss that there is a 'lasting' effect - It may not last beyond this year!

Let me know if I've overlooked anyone's points.


Just to clarify: my "point" to the first two graphs were to (1) see if I could reproduce vidaloo's graph (I'm not sure if I did), and (2) suggesting that vidaloo might find it interesting to compare and contrast the whole arctic averages with the central arctic averages, which is more or less what this thread is about.

I don't think I had a point to the second two graphs, other than that you pointed out that the increase in FYI volume was interesting so I thought I'd toss together the graphs for that.  If I keep practicing, maybe someday I'll produce a useful graph too.  ;-)

By the way...  I'm sticking to my guns for a fast pre-2020 melt out of the arctic despite the fact that every single graph I look at supports Chris' argument for a Slow Transition.  I never let facts get in the way of a dramatic story line.   ;D

viddaloo

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Re: The Slow Transition
« Reply #102 on: August 18, 2014, 09:56:15 PM »
Just to clarify: my "point" to the first two graphs were to (1) see if I could reproduce vidaloo's graph (I'm not sure if I did), and (2) suggesting that vidaloo might find it interesting to compare and contrast the whole arctic averages with the central arctic averages, which is more or less what this thread is about.

You very much did, and much better than my preliminary/unfinished graph, so I thank you for it. At the moment I don't have the knowledge or inspiration to go into the different Arctic areas, so I'll just focus on the ice as a whole for a while, keeping things simple. Like you I'm afraid our transition will NOT be a slow one, as the steep increase in Loss % also suggests. I think we'll see a couple more 2007s and 2010s — ie step increases — and then the 100% mark will be met, going down to 95% perhaps in one of the following years as a natural variation before staying at 100% permanently from a decade after the first Zero Ice Year and forward. Focus will then shift to Zero October Ice, Zero August Ice etc, and to the by then rapid fall in April maximum. By the time we have Zero April Ice, we'll have other problems to solve....
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ChrisReynolds

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Re: The Slow Transition
« Reply #103 on: August 18, 2014, 09:57:08 PM »
I'd love you to be right. A fast crash would be exciting beyond words.  8)

Last year in April (that long ago???) I was playing around with a toy model.
http://dosbat.blogspot.co.uk/2013/04/long-tail-or-fast-crash.html

I've just been playing around with a new one using extent instaed of CT Area and using a normal distributon of random variation for the melt season - which seems to be largely governed by weather. The equations are a bit different, but the results are similar.

I need to examine the distribution of residuals from the melt season loss equation, because when I use the full std deviation I occasionally get unfeasible rising exponentials. But when I half the SD it behaves better.

What's interesting is the even with the full SD most runs drop to around 5k km^3 and stay there long term - the average september volume for 2007 to 2013 is about 5k Km^3.

I'll post tomorrow and can upload the Excel spreadsheet if anyone's interested.

four consecutive runs are attached...

Comradez

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Re: The Slow Transition
« Reply #104 on: August 20, 2014, 09:33:13 PM »
It seems to me like the area numbers for the beginning of June are going to have to come down before we can reasonably expect ice-free Septembers.  The peak sun needs some low albedo surface area to work on, and that will require lower June 1st sea ice area numbers. 

I have found that the average arctic sea ice CT area ice decline from June 1st to September 1st was about 6.177 million km^2.  Interestingly, if 2014 ends up at about 3.8 million km^2 by September 1st as I expect at this point, then it will fall right around this average with a decline of 6.184 million km^2.  This has actually been an average, not that bad melt season when comparing to the 1979-2014 baseline, although it of course has seemed way below average compared to the 2007-2014 baseline to which we have psychologically adjusted to expect nowadays.  (The average decline between June 1st and September 1st for 2007-2014 has been 6.824 million km^2). 

The highest decline from June 1st to September 1st in the record was, of course, 2014 at 7.434 million km^2. 

The conclusion I draw from this is that, considering that the melt weather in 2012 was pretty much ideal, CT sea ice area has to be less than 8.5 million km^2 by June 1st to give any year a realistic shot at getting below 1 million km^2 by the September minimum.  (This 1 million km^2 mark is, of course, what many people consider to be a fair definition of "practically ice-free"). 

I have made a chart with some linear extrapolations of June 1st and Sept 1st CT areas: 





What I have concluded from this is that:

A.  To the naked eye, the linear extrapolations seem to fit with the existing data very well, especially after 2013 and 2014, and especially the June 1st data.  It is looking more and more like 2012 was a fluke bad year, and the underlying trend really is linear rather than exponential.

B. According to this linear extrapolation, June 1st CT area will not be approaching that critical 8.5 million km^2 mark on a regular basis until about 2050, thus precluding September minima under 1 million km^2 even under subsequently ideal summer melt conditions.  That said, there could be some years that dip down below this June 1st trend line, but the critical thing is that for a <1 million km^2 melt-out to occur significantly before 2050, BOTH the June 1st area reading would have to be below extrapolation (implying above-average melt weather before June 1st), AND the melt weather would have to continue to be optimal, or else, even if you get down to 8.5 million km^2 by June 1st, if you only have average melt weather after that and lose 6.1 million km^2 or even 6.8 million km^2 after that, you still only make it down to 2.4 million km^2 or 1.7 million km^2.  To get a <1 million km^2 melt-out before 2050, BOTH the pre-June 1st and post-June 1st melt season would have to have above-average melt weather. 

Therefore, I would place my bets on the earliest outlier <1 million km^2 September melt-out happening no earlier than 2035, with regular <1 million km^2 September melt-outs not occurring until at least 2045. 

viddaloo

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Re: The Slow Transition
« Reply #105 on: August 20, 2014, 10:07:37 PM »
For the record: The June 1 to September 1st period has an all time high of 79.9% meltdown — 15659 km3 — in the extreme year of 2012, which is up from 76.4% the year before. Both percentages are relative to the amount of ice present on June 1st. June–August (inclusive, same as your period) thus also has the current record for 3 months melt, at almost 80% of the volume.
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Comradez

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Re: The Slow Transition
« Reply #106 on: August 21, 2014, 12:58:25 AM »
Interesting chart, viddaloo!  It's good to keep the volume measurements in mind as well. 

Now, I got to thinking, if 8.5 million km^2 is the critical area measurement to get down to by June 1st, where can we reasonably expect that 8.5 million km^2 to be?

We can expect the arctic basin to still have about 4 million km^2 of that 8.5 right off the bat, even in a good year (for melt). 

The Greenland sea does not change a lot (export-dominated), so we can bank at least 0.3 million km^2 for that. 

Hudson Bay tends to show very little variation year-to-year (usually around 1 million km^2 around June 1st), but it is conceivable that an early snowmelt could start an early aggressive melt there, so let's bank 0.7 million km^2 for there. 

The Canadian Archipelago usually does not melt out much before June 1st (usually about 0.55 million km^2 by June 1st), but let's be charitable and assume a bit of early melting and bank 0.5 million km^2 for the CAA.

Baffin Bay is usually about 0.6 million km^2 by June 1st, but let's be charitable and bank only 0.4 million km^2 for there. 

The Barents Sea usually has about 0.2 million km^2 by June 1st, but some of that is probably export.  In any case, let's bank 0.1 million km^2 for the Barents. 

So far we have, in millions of km^2:
CAB:  4.0
Greenland:  0.3
Hudson:  0.7
CAA:  0.5
Baffin:  0.4
Barents:  0.1
Total:  6.0

That leaves us with only 2.5 million km^2 to play with for all of the following: 
*Okhotsk
*St. Lawrence
*Bering
*Chukchi
*Beaufort
*ESS
*Laptev
*Kara

I think we have to assume pretty much right off the bat that Okhotsk, St. Lawrence, and even Bering need to be totally ice-free by June 1st to have any shot at squeezing under the quota.  So that leaves us with:
*Chukchi - 0.6 max
*Beaufort - 0.5 max
*ESS - 0.9 max
*Laptev - 0.7 max
*Kara - 0.8 max
Which, together, have about 3.5 million km^2 in total area. 

We would need about 1 million km^2 of this 3.5 million km^2, or about 30% of these seas, melted by June 1st, along with reasonably aggressive melts in the other more peripheral seas, to meet the target of 8.5 million km^2. 

I'd say it could be doable in a really freak year.  BUT, there would need to be early polynyas in ALL of these 5 main peripheral arctic basins by June 1st - sort of like 2014 had in the Laptev and Chukchi, or like 2012 had in the Kara and Beaufort - but ALL AT THE SAME TIME. 

To produce that by June 1st, I guess you would need a combination of aggressive northern hemisphere snowmelt and aggressive Fram export to thin out the CAB enough to give some room for offshore winds to create some polynyas.

In other words, it would take the perfect storm of ingredients, much more extreme than 2012, to get a melt season down to 8.5 million km^2 by June 1st, at least where we currently stand on the trend.  I'd say, by 2035 there could be occasional years, though, where we start geting these ingredients to come together. 

Anyways, that's what 8.5 million km^2 on June 1st would look like, in case you were curious.

Steven

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Re: The Slow Transition
« Reply #107 on: August 21, 2014, 09:05:19 PM »
Last year in April (that long ago???) I was playing around with a toy model...
I've just been playing around with a new one...

...most runs drop to around 5k km^3 and stay there long term


It looks like the ice volume in those runs does not converge to zero, eventually?

So it is necessary to modify the model, in order to make it more realistic?

In the toy model in your 2013 blog post, the volume loss during the melt season depends only on the preceding April volume (plus a random term for the unpredictable weather).  And the volume gain during the freezing season depends only on the preceding September volume (plus a random term).

But there seems to be no explicit time-dependence in the model.  So the model runs do not "know" whether the year is 2020 or 2040 or 2060 etc.  Perhaps with an extra time-dependent term the model would be more realistic?

ChrisReynolds

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Re: The Slow Transition
« Reply #108 on: August 21, 2014, 09:11:38 PM »
Last year in April (that long ago???) I was playing around with a toy model...
I've just been playing around with a new one...

...most runs drop to around 5k km^3 and stay there long term


It looks like the ice volume in those runs does not converge to zero, eventually?

So it is necessary to modify the model, in order to make it more realistic?

In the toy model in your 2013 blog post, the volume loss during the melt season depends only on the preceding April volume (plus a random term for the unpredictable weather).  And the volume gain during the freezing season depends only on the preceding September volume (plus a random term).

But there seems to be no explicit time-dependence in the model.  So the model runs do not "know" whether the year is 2020 or 2040 or 2060 etc.  Perhaps with an extra time-dependent term the model would be more realistic?


Quite right, it is very naive, being based solely on previous behaviour. The decline in winter thermodynamic equilibrium thickness that will continue with AGW is not shown, because it hasn't yet happened and the preferrential decline of thicker MYI has swamped that signal meaning it plays no role in the derivation of the equations.

However the equations are based on the full period, yet the outcome is a levelling. I find that intriguing, but still place most weight on my arguments in the 'Slow transition' blog post, rather than on this 'model'.

crandles

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Re: The Slow Transition
« Reply #109 on: August 22, 2014, 12:27:12 AM »

Quite right, it is very naive, being based solely on previous behaviour. The decline in winter thermodynamic equilibrium thickness that will continue with AGW is not shown, because it hasn't yet happened and the preferrential decline of thicker MYI has swamped that signal meaning it plays no role in the derivation of the equations.

However the equations are based on the full period, yet the outcome is a levelling. I find that intriguing, but still place most weight on my arguments in the 'Slow transition' blog post, rather than on this 'model'.

>"because it hasn't yet happened"

Are you sure that is true rather than just we haven't got a good measure of the rate of thinning?

I thought someone had found an old paper suggesting first year, un-deformed ice grew up to 3m thick. It doesn't do that any more.

Also the piomas data you provided for thinning by thickness showed numbers as appended. While cells upto 1.1m show hardly any thinning, cells 1.4 to 1.9m thinned by 20 to 30cm over the approx 20 year time difference and greater thinning for ice just over 2m.

Just because PIOMAS provides this measure of about 1cm thinning per year for ice that was 1.4 to 1.9m thick, doesn't mean this measurement is real or that it will continue at that rate. Nevertheless I would say:

1. It is very likely that FYI thinning is happening.
2. We don't have a reliable measurement of rate of thinning. (Maybe experts know more than us.)
3. A rather wild estimate of 1cm thinning per year might be more realistic than assuming no thinning.


If first year, un-deformed ice that grew to 2m thick in the 1986 to 1995 period, now grows to 1.7m thick and over next 10 years thins to 1.6m thick, won't this have a noticeable effect not only on ensuring that more of it melts out each year but also that areas will generally melt out earlier?


Upto thick, Thinning, No of cells
0.05   0.012625   544
0.1   0.039512   379
0.15   0.052066   290
0.2   0.054725   270
0.25   0.024543   187
0.3   0.033331   160
0.35   0.033191   148
0.4   0.016789   156
0.45   0.000714   161
0.5   -0.00793   140
0.55   0.023515   145
0.6   -0.00319   137
0.65   0.030771   143
0.7   0.016611   133
0.75   0.046062   122
0.8   0.017486   133
0.85   0.036872   146
0.9   -0.00289   155
0.95   0.017624   155
1   0.024052   154
1.1   0.016633   332
1.2   0.114822   285
1.3   0.197466   246
1.4   0.214475   317
1.5   0.233391   465
1.6   0.225696   573
1.7   0.236094   452
1.8   0.265252   389
1.9   0.281049   397
2   0.348766   353
2.1   0.470283   387
2.2   0.469196   347
2.3   0.483397   357
2.4   0.521745   458
2.5   0.563506   571
2.6   0.544019   722
2.7   0.555709   551
2.8   0.621021   641
2.9   0.713077   558
3   0.718345   483
3.25   0.820449   766
3.5   0.944344   585
3.75   1.079212   469
4   1.134068   450
5.56   1.401121   1734

ghoti

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Re: The Slow Transition
« Reply #110 on: August 22, 2014, 03:48:55 AM »
I'd guess the reduced thickening should be well correlated to freezing degree days. I don't know where to find that data but I bet we'd see a decreasing trend.

viddaloo

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Re: The Slow Transition
« Reply #111 on: August 22, 2014, 04:47:10 AM »
In the toy model in your 2013 blog post, the volume loss during the melt season depends only on the preceding April volume (plus a random term for the unpredictable weather).  And the volume gain during the freezing season depends only on the preceding September volume (plus a random term).


According to this graph based on total Arctic ice volume (PIOMAS), the Winter gain % is very much dependent on Summer loss %, but not the other way around. That is, Summer loss % does NOT depend on how much was regained during the Winter, nor on the amount of ice available in April. This means Spring & Summer are behind the wheel, and Autumn & Winter are merely reacting to what's been done to the ice during the melt.
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ChrisReynolds

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Re: The Slow Transition
« Reply #112 on: August 22, 2014, 07:21:52 PM »
Sorry Crandles,

I did allude to decline in TET as having reduced "that will continue", what I mean is it hasn't been a strong factor governing the fits to the scatter plots used. And as I say the decline in MYI has swamped the lesser decline in TET.

I think the 3m thickening is over several years. If I recall correctly even papers as early as Thorndike 1975 refer to 2m as the typical thickness for first year undeformed ice in a winter. Actually their model produces an average thickness of 246cm after two years, which they conclude is similar to observed thickness.

In 1981 on the coast of Baffin Bay ice was observed to grow to around 1.5m thick.
http://www.igsoc.org:8080/journal/27/96/igs_journal_vol27_issue096_pg315-330.pdf
PIOMAS April 2014 thickness is similar.

However interannual variation in temperature causes substantial variation in thickness.
http://www.bio.gc.ca/science/research-recherche/ocean/ice-glace/documents/prinsen34.pdf
Prinsberg finds that this is due to thermodynamic thickening changes due to weather (warmer/colder)
Both histograms show two thick ice classes separated in both years by 40cm with a shift to thinner ice in 2008 relative to 2004. In 2004, the ice class peaks were centred at 1.3 and 1.7 m, while in 2008 they were centred at 1.0 and 1.4 m. A thinner ice class peak at 0.6m was also present in 2004.

This suggests that the warming in the Arctic winter must have reduced TET.

Viddaloo,

I agree that winter volume gain is occurring in response to loss of ice in the summer, but only because as summer losses have increased so September thickness and area (or extent) has been reduced. For

However I disagree that April volume plays no role in the summer melt. Firstly this is from first principles  - with thinner ice the amount of open water that can be revealed with a certain summer thinning increases non-linearly.

cesium62

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Re: The Slow Transition
« Reply #113 on: August 26, 2014, 10:09:34 AM »
Speaking of heat flow...

Most of the arctic regions are fully melting out each year.  The CAA and CAB are not yet ice free, and the Greenland Sea is kept fed with ice blowing out of the CAB.  The other seas are not only melting out, the total volume of ice in each of these seas is dropping.  This implies that when the random heat wave rolls through one of the seas over the course of a melting season, there is plenty of heat to melt all the ice in the sea, plus some excess heat that, with the right weather conditions, can help melt out the CAB.

We can ignore the southern seas: the oshkotsk, the bering, hudson bay...  These waters are isolated from the arctic, and their excess heat may not propagate as well.  For the remaining seas, if we look at the kinds of volume of ice that melted before the seas fully melted out and we project forward a couple of years of additional heat gain, and if we project forward a couple of years of additional ice loss, we can estimate an excess heat available metric in terms of a volume of ice that could be melted if it were present.

This amount of excess heat won't be available in any place every year, but there is a reasonable chance that the right set of weather conditions could bring the excess heat to each of the northern seas in a single year.

Eyeballing the volume graphs for the various seas, I come up with:

beaufort:  300 km^3 excess volume melting heat capacity
chukchi:   350
ESS:         500
Laptev:     250
Kara:        400
Barents:    700
Greenland: -100
CAA:          -250
CAB:         -4000
                ------
                1850 km^3 excess heat in areas that completely melt out.

Given the right weather conditions over the next few years, the September minimum might drop down to 1,850 km^3, which would get rid of 50% of the remaining multi-year ice.  Another hot summer could blow through within five years after that when the not-so-northern arctic seas have even less ice and the overall temperature is yet higher still.

The model I'm proposing is:
* We linearly fit winter maximum ice volume to each of the northern arctic seas listed above with a vertical error term that captures how far actual volume has been vertically above or below the trend line.  We justify the linear fit as global temperatures and global sea temperatures are believed to be growing roughly linearly at this time.  (Arguably, winter ice volumes start dropping super-linearly once they melt out, so the linear trend may be reasonably conservative.)
* We linearly fit summer melting ice volume for each sea in a similar fashion.  However, we only use points before the sea melted out completely.  We can't tell how much heat was available when the sea has melted out completely.
* We assume that excess heat from one of the seas near the CAB can easily transfer its excess heat to the CAB.  This might happen via heat staying in the nearby sea and ice blowing out of the CAB into the adjacent sea.  It might happen via warm winds blowing up from the south or via ocean circulation.
* For each year, we can use a monte carlo simulation to estimate the probability that the arctic will be ice free that year.  For each sea, we pick a random ice volume for that sea based on the previously computed trendline and error bar.  We also pick a random available melting heat in a similar fashion.  We add the volumes and count the number of trials for each year where the simulated arctic melted out.
* As a variant, we can look at a model where the weather is correlated across the seas.  We pick a random number between -1 and 1 representing the location on the error bars where we will pick volumes.  We separately pick one random number that will be used to choose ice volume from the various trendlines, and one random number that will be used for heat volumes.

[We should keep in mind that first year ice is slightly saltier than multi year ice.  Once a full melt out occurs, the replacement ice should be a bit easier to melt out, which should help us be slightly conservative underestimating the excess heat.  New ice would also form more slowly, helping to explain why winter ice volumes would drop super-linearly.]

I'll have to toss the code for that together one of these days...

jdallen

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Re: The Slow Transition
« Reply #114 on: August 26, 2014, 10:25:20 PM »
Seemed like a good place to toss this in for discussion.

A Wild Card is about to be played in our Arctic melting game.  Effect will be short term for certain, but may be worthy of examination as to its potential impact.

http://volcanocafe.wordpress.com/2014/08/24/bardarbunga-nature-of-the-beast/
This space for Rent.

cesium62

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Re: The Slow Transition
« Reply #115 on: August 27, 2014, 02:29:33 AM »
So I coded up my simple model.  The green curve represents PIOMAS Arctic Ocean September ice volume.  The blue, red, and yellow lines represent the 25th, 50th, and 75th percentile model-estimated volume.  The purple line represents the model-predicted probability all ice will melt out.  ("Arctic Ocean" is here roughly defined as the oceans north of 70 degrees including the Chukchi; I added Baffin Bay to my list above.)

The correlated weather model looks particularly unrealistic.

I personally like the probability contours.

In the above, each year is pretty much independent of every other year.  The next thing I'd want to do is model the ice growth differently.  If a warm year comes along and wipes out a bunch of really thick ice, the volume of ice the next year will likely be below trend.  I would want to split the volume of a region up into multiple buckets, where each bucket contains the volume of ice in a small thickness range.  The melt season would melt an equal volume of ice in each thickness range (except that if all the ice melts out in a range, the excess heat moves off to the next range).  During the freeze season, ice is added back separately to each bucket: buckets with thicker ice add back less thickness.

But, for this, I probably have to go start crunching gridded data for myself.  And it's probably time for me o move to the "homebrew: build your own arctic ice model" thread...


ChrisReynolds

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Re: The Slow Transition
« Reply #116 on: August 27, 2014, 08:58:35 PM »
Seemed like a good place to toss this in for discussion.

A Wild Card is about to be played in our Arctic melting game.  Effect will be short term for certain, but may be worthy of examination as to its potential impact.

http://volcanocafe.wordpress.com/2014/08/24/bardarbunga-nature-of-the-beast/


As an 'awe of nature' junkie, a massive volcanic eruption would make up for this year's dull season on the ice.

ChrisReynolds

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Re: The Slow Transition
« Reply #117 on: August 27, 2014, 09:00:03 PM »
Cesium,

I'll comment over the weekend when I'm not so tired I can't follow the post describing the model.

DavidR

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Re: The Slow Transition
« Reply #118 on: August 28, 2014, 03:53:13 AM »
Here's an interesting observation. 
There have been seven record lows in SIE since 1980, (1985, 1990, 1995, 2002, 2005, 2007, 2012).  In six out of seven of these cases the record year has been the third in a run of unusually  hot years in the GISS temperature record for the area north of 64N.
http://data.giss.nasa.gov/gistemp/tabledata_v3/ZonAnn.Ts+dSST.txt.
(anomalies in 100s of a degree)
1985 - first three consecutive years above 30+ anomaly.
1990 - first  three consecutive years above 40+ anomaly.
1995 - three years above 40+ with a combined record of 250+ and 1995 having a record 145+
2002 - first three consecutive years above 100+
2005 - first  three years with total anomaly above 400 with 2005  a record 201+
2007 - first three consecutive years above 165+
2012 - first three consecutive years  above 180+

2013 dropped to  an anomaly of 125+, the third lowest since 2000,  so perhaps its not that  surprising that  we didn't see a record this year.

The increase in the anomaly for records has been increasing at  about  35 / 100s every  five years. However between 2007 and 2012 it  increased by  only 6 /100s.
« Last Edit: August 28, 2014, 05:18:45 AM by DavidR »

cesium62

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Re: The Slow Transition
« Reply #119 on: August 28, 2014, 11:44:51 AM »
Cesium,

I'll comment over the weekend when I'm not so tired I can't follow the post describing the model.

I'm not taking myself too seriously here.  I just talk out loud to organize my thoughts.  I know I'm not adding much to the conversation, and hope you all don't mind me chatting here too much.  It's pretty quiet so I don't think I'm too bothersome yet...

ChrisReynolds

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Re: The Slow Transition
« Reply #120 on: August 28, 2014, 06:20:57 PM »
Cesium,

I'll comment over the weekend when I'm not so tired I can't follow the post describing the model.

I'm not taking myself too seriously here.  I just talk out loud to organize my thoughts.  I know I'm not adding much to the conversation, and hope you all don't mind me chatting here too much.  It's pretty quiet so I don't think I'm too bothersome yet...

Not at all. My three equation 'toy model' is just playing around with an idea and seeing where it leads. It's such 'play' that can test and improve understanding.

David R,

Thanks for that, the trouble can be trying to tie all these relationships together. In this instance the linkage between AGW and sea ice loss seems to be the box that fits into, but it might have a place in the prediction methods box.

DavidR

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Re: The Slow Transition
« Reply #121 on: August 29, 2014, 02:36:02 AM »
Chris,
I think there is quite a bit of useful information in these figures.
  • The arctic has continued to  warm rapidly despite the 'global hiatus'. 
  • You need a series of warm years to get a record, 2014 would have to  have been at  least one full degree warmer than 2011 to  get a new record.
  • 2010 and 2011  were both warmer than 2012, and 2013 was nearly  three quarters of a degree cooler than both 2010 and 2011.
  • The fact that  2012 and 2013 were cooler than 2010 and 2011 suggests an explanation for the leveling off of the PIOMAS data.
  • 2011 and 2012 were both La Nina years suggesting there is maybe a one year delay translating global temperature variations into the Arctic presumably  through water temperatures.
All interesting avenues for further thought.

plinius

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Re: The Slow Transition
« Reply #122 on: August 29, 2014, 03:40:32 PM »
Gee, that's wrong. The arctic has warmed rapidly connected to the pseudo-hiatus (where should the hiatus be? I have never seen one).
I cannot see why one needs a series of warm years for a record. 2014 had extremely low Fram strait transport. This is NOT a temperature problem. Also I cannot see how that little change in "global" temperatures by ENSO should translate directly into arctic sea ice. Happy to believe in effects on AO and NAO, but this is NOT a trivial temperature effect.


Chris,
I think there is quite a bit of useful information in these figures.
  • The arctic has continued to  warm rapidly despite the 'global hiatus'. 
  • You need a series of warm years to get a record, 2014 would have to  have been at  least one full degree warmer than 2011 to  get a new record.
  • 2010 and 2011  were both warmer than 2012, and 2013 was nearly  three quarters of a degree cooler than both 2010 and 2011.
  • The fact that  2012 and 2013 were cooler than 2010 and 2011 suggests an explanation for the leveling off of the PIOMAS data.
  • 2011 and 2012 were both La Nina years suggesting there is maybe a one year delay translating global temperature variations into the Arctic presumably  through water temperatures.
All interesting avenues for further thought.

DavidR

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Re: The Slow Transition
« Reply #123 on: August 30, 2014, 08:15:32 AM »
Plinius,
As pointed about above, all 7 of the record low years since 1979 have occurred during the third of three warm years in the region 90N-64N. There have been no groups of three warm years that have not seen record lows.  This may not be 'proof' that  you  need a series of warm years to get a record but  it is a pretty good indicator.

As for describing the temperature change as 'little',  a quarter or half of a degree on average over a year for a significant portion of the globe is a significant amount of extra heat.  Compare that change to the average rate of change under AGW of 0.15 degrees a decade.

crandles

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Re: The Slow Transition
« Reply #124 on: August 30, 2014, 01:54:04 PM »
As pointed about above, all 7 of the record low years since 1979 have occurred during the third of three warm years in the region 90N-64N. There have been no groups of three warm years that have not seen record lows.

Actually 1997 was end of 3 years over anomaly of 80/100.

There are no records without fitting one of your two rules and 7 out of 8 times either of the rules are true a record is reached, does seem pretty good.

It does seem the rules are somewhat arbitrary and this has worked fortuitously. eg since 110 106 127 produced a record, why not also one year later with 106 127 143? This seems like higher temperatures but doesn't fit the rules.

Similarly 1988 to 1991 has 74 41 63 76.

With a short record, it is dangerous to extrapolate complex rules when the items may be just associative.

Despite such qualms about how well your rules appear to work, it does feel as if there may be something in the idea that a record tends to occur after more than one warm year.

If monthly zonal mean temperatures were available, it might be possible to use for prediction.

ChrisReynolds

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Re: The Slow Transition
« Reply #125 on: August 30, 2014, 05:43:46 PM »
I'm in the process of adding Arctic Ocean thickness to my thickness data derived from PIOMAS gridded data. Just been looking at the data and here's a resulting plot that is relevant to this thread.



Plot of April PIOMAS average thickness for the Arctic Ocean (blue), together with the cumulative thickness change of thickness (red).

Arctic Ocean is; Beaufort, Chukchi, East Siberian, Laptev, Kara, Barents, Greenland, Central, Canadian Arctic Archipelago.
« Last Edit: August 30, 2014, 06:05:52 PM by ChrisReynolds »

DavidR

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Re: The Slow Transition
« Reply #126 on: August 30, 2014, 06:19:49 PM »
As pointed about above, all 7 of the record low years since 1979 have occurred during the third of three warm years in the region 90N-64N. There have been no groups of three warm years that have not seen record lows.

Actually 1997 was end of 3 years over anomaly of 80/100.

There are no records without fitting one of your two rules and 7 out of 8 times either of the rules are true a record is reached, does seem pretty good.

It does seem the rules are somewhat arbitrary and this has worked fortuitously. eg since 110 106 127 produced a record, why not also one year later with 106 127 143? This seems like higher temperatures but doesn't fit the rules.

Similarly 1988 to 1991 has 74 41 63 76.

With a short record, it is dangerous to extrapolate complex rules when the items may be just associative.

Despite such qualms about how well your rules appear to work, it does feel as if there may be something in the idea that a record tends to occur after more than one warm year.

If monthly zonal mean temperatures were available, it might be possible to use for prediction.
Agreed the correlation is not perfect.  1997 was much cooler than 1995 and I have no  doubt that  weather has an effect that hasn't been taken into  account in this analysis. Nevertheless I think the relationship is strong enough to  say that there is a correlation between a run of hot  years and a record low extent.  And that  the temperature is rising  rapidly. The fourth hot year in a sequence has never produced a record low and I  can't explain that.

1988-1991 had a record low extent  in 1990 as did 1995-1997 in 1995. so the claim that  no three year warm period occurred without   stands up, it is just that it  wasn't always the last year  of the three.     

Wipneus

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Re: The Slow Transition
« Reply #127 on: August 30, 2014, 06:46:38 PM »

 the cumulative thickness change of thickness (red).


The integral (cumulative) of the differential (change) of a function is the function itself, apart from a constant (mathematics).

So why are you confusing the graphic with the same function twice?

ChrisReynolds

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Re: The Slow Transition
« Reply #128 on: August 30, 2014, 07:07:12 PM »

 the cumulative thickness change of thickness (red).


The integral (cumulative) of the differential (change) of a function is the function itself, apart from a constant (mathematics).

So why are you confusing the graphic with the same function twice?

It just showed the unusual levelling better in a graphical sense.

OSweetMrMath

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Re: The Slow Transition
« Reply #129 on: August 30, 2014, 08:25:17 PM »
Chris,

I think you're missing Wipneus' point. Your graph shows the same data twice, just with different vertical scales. One version may show your claim better than the other, but that doesn't change the fact that putting the same data up twice in one graph is confusing.

crandles

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Re: The Slow Transition
« Reply #130 on: August 30, 2014, 09:31:44 PM »
The graphic of the data alone could suggest:

ArcticThicknessstepdown by crandles2011

I think you are on stronger ground with your calculations of how low it could go.

ChrisReynolds

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Re: The Slow Transition
« Reply #131 on: August 31, 2014, 08:26:28 AM »
Chris,

I think you're missing Wipneus' point. Your graph shows the same data twice, just with different vertical scales. One version may show your claim better than the other, but that doesn't change the fact that putting the same data up twice in one graph is confusing.

Nope. I don't miss his point, my calculus may be rusty but with  degree in electronics I have done calculus. But OK, if people find it confusing I won't use it. A major advantage of diff/int is that it centres the early period of negligible change early in the series on zero, without any arbitrary scaling.

The same technique used on September extent shows that extent falls away from zero around 1995 - when Lindsay & Zhang hypothesise that the ice/ocean albedo feedback took off.

Crandles, possibly so.

cesium62

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Re: The Slow Transition
« Reply #132 on: September 03, 2014, 08:34:00 AM »
I wanted to jot down some items regarding albedo...

Over in models and math, seaicesailor writes:

"""
Back to PIOMAS: exponential fitting might work initially, due to initial albedo amplification so dV/dt = k*(Vo - V) (suppose thickness constant). However there are such simple facts that make the crossing of exponential fitting with V=0 curve invalid to predict year of ice-free. As the ice shrinks more every summer, the heat in Arctic ocean is less efficiently transferred to remaining ice, so the eq becomes dV/dt = k* V^p * (Vo-V), where p is>0. So much for exponential fitting. This eq leads to a volume never crossing x-axis but approaching 0 asymptotically. Then there is the issue of constant thickness. It seems plausible that thickness of remaining ice reaches a constant or even increases since remaining ice probably thick MYI for quite a while. That would lead to a volume much more slowly approaching V=0, until a very hot year sends everything to hell. Because that is the most important fact: earth is warming constantly (hiate apart, CO2 unstoppable unfortunately)
"""

I like the way that's stated.  Albedo feedbacks give us a sigmoid curve to add into a linear CO2 curve.  Can we decide whether we are in the exponential decay portion of the albedo curve?  or have we by now advanced to the linear portion of the curve?  (I suspect we haven't advanced to the leveling off phase of the curve.)  Also, how strong is the albedo feedback relative to the CO2 advance?

http://arctic-news.blogspot.com/p/arctic-sea-ice.html is vaguely related.  It states that albedo feedback is around 0.45 w/m^2 vs 0.48w/m^2 for methane and 1.66w/m^2 for CO2 (using 2011, 2005, and 2005 data).  But that's over the entire northern hemisphere.

The referenced abstract (http://www.nature.com/ngeo/journal/v4/n3/full/ngeo1062.html) suggests that sea ice reduced albedo accounts for about 0.225 w/m^2 averaged over the entire northern hemisphere.  If I concentrate the sea ice albedo effect into the area north of 60 degrees north and guess at insolation effects... I get something like albedo having a current warming effect of 1.6w/m^2 in the arctic ocean.   I suppose that might be the right order of magnitude, but it would be nice to have a more authoritative figure.

ChrisReynolds

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Re: The Slow Transition
« Reply #133 on: September 03, 2014, 08:24:30 PM »
The equations befuddle me, it's been decades since I did my degree.

dV/dt = k*(Vo - V)

dV/dt = k* V^p * (Vo - V),

OK dV/dt - rate of change of volume w.r.t. time.
k - constant of proportionality?
V - volume at time 't'?
Vo - start volume??
p - power to which V is raised??

Perhaps it will become clear when the final two terms are explained. But...

I don't get why, when allowing for efficiency of heat transfer the V^p factor is included.

(Hang on, is it because... model the pack as a circle of radius r, reduce r and thickness linearly an the result is a power law decline, not a linear one.)

OSweetMrMath

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Re: The Slow Transition
« Reply #134 on: September 05, 2014, 01:01:25 AM »
Since no one else has responded to ChrisReynolds, I can at least talk about the math, if not the physical models. (I am not the source of these models and this discussion should not be taken as an endorsement of these models.)

My reading on all of the variables agrees with yours. It might be better to think of Vo as a "maximum" volume, rather than the "start" volume. The basic argument for the exponential loss in ice is that as the ice melts, the albedo decreases, which increases the heat absorption, which increases the melt rate.

dV/dt is the change in volume, so will be negative for melting ice. (Vo - V) represents the difference between the maximum ice volume and the current ice volume, and so is a measure of the heat absorption in the exponential model. If the current volume V were equal to the maximum volume Vo, this model predicts that the ice volume would be unchanging. This also implies that if V were greater than Vo, the ice volume would be increasing, leading to an ice age, I suppose.

k is the constant of proportionality, which you could fit to the data. For the signs to work out, k should be negative, but this isn't a big deal.

This model is unphysical because when V is zero, that implies a large rate of loss of volume, which does not make sense. In principle, when the volume is zero, the rate of change of the volume must be nonnegative. It's reasonable to conclude the model is nonsense, regardless of how well it has fit the data up to this point.

To fix this, you can add the V^p term.

You can argue for the V^p term from a physical standpoint, in that heat transfer is proportional to surface area. Therefore, with smaller surface area, there is less heat transfer and therefore a slower rate of ice melt. Volume has a power law relationship with surface area, but the power depends on the shape of the ice. (If the ice is a cube, the power is 2/3. If the ice is a flat sheet and melts by losing thickness, the power is 0. The effective power could be anything in between, or potentially even greater than this.) With a sufficiently complex physical model, it may be possible to derive p from the model. Otherwise, you could again try to fit it to the data.

The mathematical advantage of the V^p term is that if p is positive, then as the volume goes to zero, the rate of melt must also go to zero. Compared to the first model, this at least makes sense in that you can't melt ice you don't have, and this model will not predict that the ice continues to melt after it ceases to exist.

The solution to the first differential equation is an exponential, familiar from Wipneus' graphs. I'm not sure if the second equation has a convenient closed form, and if so, if anyone is actively using it. While I still wouldn't take its predictions seriously without further physical justifications, it wouldn't hurt to look at it.

ChrisReynolds

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Re: The Slow Transition
« Reply #135 on: September 05, 2014, 06:23:39 PM »
Thanks for the helpful explanation MrSweetMath, very much appreciated.

ChrisReynolds

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Re: The Slow Transition
« Reply #136 on: September 06, 2014, 09:48:55 AM »
Cesium,

Sorry, I forgot to get back to you.

The idea of taking the (long term) linear trend of GW and using that to establish energy gain is quite neat. Correlated vs Un-correlated weather - could you detail what you mean by that?

It is worth noting the following. After Lindsay & Zhang's finding that the current period of volume decline started around 1995: The 1995 to 2013 September volume has a linear fit with an R2 of 0.9, and a zero crossing in 2025 (2022 to 2030), this might in part account for the probability distribution for total melt out.

I've PM'd you as I'm so late replying.

SATire

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Re: The Slow Transition
« Reply #137 on: September 06, 2014, 12:42:08 PM »
This model is unphysical because when V is zero, that implies a large rate of loss of volume, which does not make sense. In principle, when the volume is zero, the rate of change of the volume must be nonnegative. It's reasonable to conclude the model is nonsense, regardless of how well it has fit the data up to this point.
OSweetMrMath - from physical point of view that model is simple but not non-sense. It is a phase transition from a situation with sea ice to a situation without. It is quite typical that at such phase transistions your model is not continuously differentiable. So - I see no problem with that point.

SATire

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Re: The Slow Transition
« Reply #138 on: September 06, 2014, 12:58:29 PM »
The equations befuddle me, it's been decades since I did my degree.

dV/dt = k*(Vo - V)
Chris - that is close but not right. The loss (-dV/dt) is proportional to open area - which one may assume proportional (thus the k) to some Vo-V. However - the meaning of that Vo is now a bit unclear - but you may fit that ;-)

So the differential equation describing "volume loss is due to albedo feedback" reads

-dV/dt=k(Vo-V)

and you will see that putting V=Vo(1 - exp(kt)) solves that differential equation.

To get a function that makes physically sense - e.g. no negative ice-volume you set the function V=0 for all t with 1<exp(kt). So - if the ice is gone it will not come back due to that albedo increase. So any effect in winter is neglected by this model...

ChrisReynolds

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Re: The Slow Transition
« Reply #139 on: September 07, 2014, 09:11:10 AM »
Thanks SATire.

cesium62

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Re: The Slow Transition
« Reply #140 on: September 07, 2014, 09:56:56 AM »
Cesium,

Sorry, I forgot to get back to you.

The idea of taking the (long term) linear trend of GW and using that to establish energy gain is quite neat. Correlated vs Un-correlated weather - could you detail what you mean by that?

It is worth noting the following. After Lindsay & Zhang's finding that the current period of volume decline started around 1995: The 1995 to 2013 September volume has a linear fit with an R2 of 0.9, and a zero crossing in 2025 (2022 to 2030), this might in part account for the probability distribution for total melt out.

I've PM'd you as I'm so late replying.

"correlated weather" looks at what would happen if the weather in all regions was correlated; if they were all likely to experience sunny weather together or all likely to experience cloudy weather together.  "Uncorrelated weather" is each region has its own weather completely unrelated to any other region.

For correlated weather, we pick a random number and use that random number in all regions.  For uncorrelated weather, we pick a separate random number for each region.

Correlated weather looks pretty bogus and can be ignored.

ChrisReynolds

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Re: The Slow Transition
« Reply #141 on: September 07, 2014, 11:51:36 AM »
Cesium,

I'm not sure the idea of weather being uncorrelated at region level is really sound. There might be an argument for two sets of random numbers, a whole Arctic (correlated) element plus a regional dither might be more realistic.

crandles

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Re: The Slow Transition
« Reply #142 on: September 07, 2014, 12:34:08 PM »
Cesium,

I'm not sure the idea of weather being uncorrelated at region level is really sound. There might be an argument for two sets of random numbers, a whole Arctic (correlated) element plus a regional dither might be more realistic.

I was thinking if winds are from the South somewhere then there are other places where winds are from the North. However I am not sure what weather conditions you are building in, if you are just doing levels of sunshine and counting on winds in different areas effectively cancelling then uncorrelated may be reasonably OK. It is then the frequencies of the sunshine levels that needs to be realistic - getting two different years to be as far apart as they usually are might be difficult?

OSweetMrMath

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Re: The Slow Transition
« Reply #143 on: September 07, 2014, 11:50:37 PM »
This model is unphysical because when V is zero, that implies a large rate of loss of volume, which does not make sense. In principle, when the volume is zero, the rate of change of the volume must be nonnegative. It's reasonable to conclude the model is nonsense, regardless of how well it has fit the data up to this point.
OSweetMrMath - from physical point of view that model is simple but not non-sense. It is a phase transition from a situation with sea ice to a situation without. It is quite typical that at such phase transistions your model is not continuously differentiable. So - I see no problem with that point.

If you want to argue that the first model is really

dV/dt = k(Vo - V) when V > 0
and
dV/dt = 0 when V = 0

so there's a discontinuity in the derivative when V=0, I'm not really going to contest that. My bias is in favor of a continuous derivative, but this doesn't have to be true.

On the other hand, if we allow for a discontinuity, who's to say the rule shouldn't be

dV/dt = k(Vo - V) when V > a
and
dV/dt = cV when V < a

(k and c are negative constants) or something? You could go as far as arguing that the first model predicts a collapse in the ice volume following 2012, and that didn't happen, so maybe we're already below the threshold a.

I don't seriously believe this, but when the physical justification for a mathematical model is fairly weak, it's easy to come up with alternative models, and hard to argue for the superiority of any particular model in predicting future values.

My biases with any of these models are, first, avoid overinterpreting them. (In particular, do not take predictions of future values very seriously.) Second, when evaluating mathematically simple models, smooth models should be considered more simple than unsmooth models. ("Smooth" implies that the function and some number of derivatives of the function are continuous. The number of required derivatives is deliberately vague.)

cesium62

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Re: The Slow Transition
« Reply #144 on: September 08, 2014, 08:58:04 AM »
Cesium,

I'm not sure the idea of weather being uncorrelated at region level is really sound. There might be an argument for two sets of random numbers, a whole Arctic (correlated) element plus a regional dither might be more realistic.

I was thinking if winds are from the South somewhere then there are other places where winds are from the North. However I am not sure what weather conditions you are building in, if you are just doing levels of sunshine and counting on winds in different areas effectively cancelling then uncorrelated may be reasonably OK. It is then the frequencies of the sunshine levels that needs to be realistic - getting two different years to be as far apart as they usually are might be difficult?

Chris:  It's interesting to think about how to build that intermediate model.

Crandles:  For this model, the assumption is that "weather", of whatever kind, is constant random noise.  (And evenly distributed within the min/max given by historical data since 1978.  e.g. There are no 100 year events other than those recently seen.)

SATire

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Re: The Slow Transition
« Reply #145 on: September 08, 2014, 01:06:38 PM »
Second, when evaluating mathematically simple models, smooth models should be considered more simple than unsmooth models. ("Smooth" implies that the function and some number of derivatives of the function are continuous. The number of required derivatives is deliberately vague.)
OSweetMrMath - even the slope of most simple linear fit is not smooth (= "not continuously differentiable" (math language)). Of course, since if you hit zero ice the differential equation must change and so the slope is different.

A simple analogue: A falling apple - the slope of the function position vs. time will change in the moment it hits the floor. And it makes no sense to introduce some forces decelerating the apple before touch down to get a "soft landing". Furthermore - if you measure heigth vs. time you will recognice, that the apple bounces and so it will stay much longer in few mm height than predicted by the model "free fall". You will see a "long slow tail" but there is no new physics behind - it just bounces because there is a floor and some additional force there. Similar things may happen with sea-ice. And there is also statistics involved: The error bars are significant but the volume can not go below zero - so statistically all parts of error bars below zero are cut and measurements will result in more values above zero than predicted by the plain model. If you respect such behavior (of a "positive definite function" (math language) like volume), then it will be more difficult to reject the exponential model or the linear model even if you observe the tail - so, if you can not reject the simple model why introducing a more complicated one with strange forces just above the floor?

Edit: Please do not misunderstand me: I think it is very likely that the exponential can be excluded in some years by fitting and also, that some "slow transistion" due to other feedbacks is possible. But those things are not modeled right now and we have to do that work properly. And any personal bias must be avoided, if you want to convince anybody.
« Last Edit: September 08, 2014, 01:19:27 PM by SATire »

crandles

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Re: The Slow Transition
« Reply #146 on: September 08, 2014, 03:13:55 PM »
If you want to argue that the first model is really

dV/dt = k(Vo - V) when V > 0
and
dV/dt = 0 when V = 0

so there's a discontinuity in the derivative when V=0, I'm not really going to contest that. My bias is in favor of a continuous derivative, but this doesn't have to be true.

On the other hand, if we allow for a discontinuity, who's to say the rule shouldn't be

dV/dt = k(Vo - V) when V > a
and
dV/dt = cV when V < a

(k and c are negative constants) or something? You could go as far as arguing that the first model predicts a collapse in the ice volume following 2012, and that didn't happen, so maybe we're already below the threshold a.

There is an obvious physical reason for a discontinuity when the ice volume hits 0. What would be the reason at Volume=a when a is positive?

If you were suggesting c was positive such that we got an equilibrium around V=a then that would seem un-physical.

I prefer smooth models without discontinuities, but we haven't enough models to show the range of possible outcomes.

So far discussed

1. Exponential increase in melt volume
and
2. As the volume goes to zero, the rate of melt also goes to zero. (Gompertz shape tending to 0 volume).

But these are not the only options. I think the more likely scenario is

3. Gompertz shape tending to a negative volume. Obviously you cannot actually have negative volume so you either view 0 volume as causing a discontinuity or as the negative volume representing a quantity of heat above that necessary to melt all the ice (such that it would melt that negative volume if it were present).

How would I create a model of this?

First I would have a model of maximum ice volume. This would mainly have a gompertz shape as the MYI rapidly disappears then slows up as we get to having little MYI left. However I think FYI thickness is also thinning. So the gompertz shape should tend to a slowly declining volume trend:

e.g. something like
1.8m * area of arctic ocean +  volume of ice currently in thicknesses less than 1.8m

The 1.8m should decline by say 1cm per year.

I would then want to model ice melt volume as increasing as the maximum volume declines.

I see no reason why the rate of volume melt should approach 0 as the volume approaches 0. I don't think that should preclude a gompertz like shape.


OSweetMrMath

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Re: The Slow Transition
« Reply #147 on: September 08, 2014, 06:11:55 PM »
Second, when evaluating mathematically simple models, smooth models should be considered more simple than unsmooth models. ("Smooth" implies that the function and some number of derivatives of the function are continuous. The number of required derivatives is deliberately vague.)
OSweetMrMath - even the slope of most simple linear fit is not smooth (= "not continuously differentiable" (math language)). Of course, since if you hit zero ice the differential equation must change and so the slope is different.

I hesitated to use the word "smooth" for this very reason. The meaning of "smooth" depends on mathematical context. It often means that all derivatives of a function exist and are continuous, which is way more smoothness than I'm requiring. Of some relevance to the current discussion, in the context of nonparametric regression, "smooth" (or "sufficiently smooth") often means something like two continuous derivatives, and with the integral of the square of the second derivative sufficiently small.

If I had meant continuously differentiable, I would have said so. I specifically meant something stronger than continuously differentiable, and specifically defined "smooth" as having several continuous derivatives. As in, if the second derivative is not continuous, I would probably not consider the function to be smooth. If the third derivative is not continuous, I probably would consider the function to be smooth.

A function which is linear above zero and constant at zero is an excellent example of a function I would consider to be not smooth, and therefore not simple. Thank you for providing it.

I think we can agree that "simple", as I am using it here, is not "math language". (In the context of Lebesgue integration, for example, a "simple function" has a math meaning, but that is clearly not the meaning we are using here.) As such, we can disagree about what it means. You may consider a function of the form y = -mx +b, x < b/m, and y = 0, x > b/m to be simple, but I fundamentally disagree.

Also, my use of the word "bias" might have been misleading. I don't mean "error in thinking which is likely to cause wrong conclusions" or something. I mean conclusions which I have come to through experience, which I believe to be correct, but I am unlikely to convince others of the correctness by argument alone.

This started with ChrisReynolds' question about the meaning and interpretation of the two differential equations. Again, I don't think either equation is reliable for predicting ice volume, but I do think smoothness is one argument in favor of the second equation rather than the first. You are welcome to not agree.

SATire

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Re: The Slow Transition
« Reply #148 on: September 08, 2014, 06:58:08 PM »
This started with ChrisReynolds' question about the meaning and interpretation of the two differential equations. Again, I don't think either equation is reliable for predicting ice volume, but I do think smoothness is one argument in favor of the second equation rather than the first. You are welcome to not agree.

Thank you. I gladly continue disagreeing, since I have no reason to change my mind: It is ok beeing not smooth while hitting the floor ;-)

But for your problem p not zero but e.g. p=3/2 in that
-dV/dt=V^p (1-V)
I would like to point you to that wolfram alpha, which can do a lot of integration for you:
http://www.wolframalpha.com/input/?i=dy%2Fdx%3Dy^%285%2F2%29-y^%283%2F2%29
(nasty function that is...)

OSweetMrMath

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Re: The Slow Transition
« Reply #149 on: September 08, 2014, 08:42:20 PM »
The graph from Wolfram Alpha is somewhat misleading. The function Wolfram Alpha plots is

dy/dx = y^5/2 - y^3/2

so in terms of the original form, Vo = 1, k = 1, and p = 3/2. We are interested in the region where y < 1. All this graph shows is that when y(0) > 1, the function runs off to infinity. When y < 1, the function is a sigmoid, so considerably less nasty.

I've attached a numerical simulation of the above function from R, with y(0)=0.99. Note that the exact shape of the curve is dependent on the parameters k and p, but the basic sigmoid shape will exist for any positive p and k.

Some brief experimentation with plots of dy/dx = k(y^(p+1) - y^p) shows that k controls how fast the function declines, with larger values producing faster declines, and p controls the size of the tail, with larger values producing larger tails. (This should not be a surprise, because with p = 0, we are back to an exponential with no tail at all.)
« Last Edit: September 08, 2014, 08:53:59 PM by OSweetMrMath »