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The Science article The average U.S. family destroys a football field's worth of Arctic sea ice every 30 years states:

Every additional metric ton of carbon dioxide (CO2) puffed into the atmosphere appears to cost the Arctic another 3 square meters of summer sea ice.

which carries a link to the abstract of a peer-reviewed Science paper Observed Arctic sea-ice loss directly follows anthropogenic CO2 emission. However the abstract says (in part):

Since climate-model simulations of the sea-ice loss differ substantially, we here use a robust linear relationship between monthly-mean September sea-ice area and cumulative CO2 emissions to infer the future evolution of Arctic summer sea ice directly from the observational record. The observed linear relationship implies a sustained loss of 3 ± 0.3 m2 of September sea-ice area per metric ton of CO2 emission.

My question is about the simplification of linking tons of CO2 to square meters of ice, and without reference to time. The abstract seems to simply say that there appears to be a linear relationship, and since models disagree, then this is a "good enough" way to think about it.

Is it so simple - if in 2017 no CO2 were burned the sea ice would return to normal? No. The summer sea ice would likely continue to decrease for decades. An analogy might be births and deaths of a population. While there might be a rough (or "robust") correlation between birth rate and death rate, a cessation of births in a given year would have almost no impact on death rate for perhaps a half-century.

Is there any mathematics that justifies this simple "robust linear relationship" to be causal? What does "directly follows" mean?

I'm not in any way questioning the causality between CO2 and a wide variety of changes in climate, including reduction in summer sea-ice. I'm only questioning the original quote and the quantitative relationship: 1 metric ton of CO2 costs 3 square meters of summer sea ice.

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above x2: Graphics from Science Magazine.

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    $\begingroup$ It looks as if they just divided two quantities (output/input) that are related through a chain of actual processes, but ignoring what happens in those processes. That would be very hard to do because input and output differ widely in time and space, so the intermediate steps are diverse. This is almost a case of correlation without causation. $\endgroup$ – Jan Doggen Nov 4 '16 at 11:12
  • $\begingroup$ The articles say every metric ton of CO2 reduces about 3 ± 0.3 m^2 of sea ice, you say km in your question. 3 square meters per metric ton of CO2 sounds in the range of plausible. $\endgroup$ – userLTK Nov 5 '16 at 7:30
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    $\begingroup$ I worry about giving advice on a site that discourages conversation, but my feeling is leave the check and vote up in the answer, but correct the title, maybe make a footnote that a correction was made. $\endgroup$ – userLTK Nov 5 '16 at 8:06
  • $\begingroup$ @userLTK thank you for pointing out the error. I've fixed the title to agree with the units in the quotes - square meters, not square kilometers. $\endgroup$ – uhoh Nov 5 '16 at 10:18
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First lets take the question at face value and do the simple comparison. I will take the years 1979 and 2016 as arbitrary example, over a 37 year period - the longest for which accurate data are available. During that time the minimum extent of Arctic sea ice, defined as the area of Ocean with at least 15% of sea ice (as determined by satellite) decreased from 6.93 to 4.14 million square kilometers. That is, a decrease of 2.79 M Km2. The actual ice loss was much greater because of ice thinning, but that is incidental to the question. During the same period the CO2 emissions by the USA increased by about 160 M tonnes. So on the face of it, assuming a direct linear relationship, 1 tonne of emitted carbon from America results in an average of 0.029 km2 reduction in ice, or about 3 hectares (Not Km2! let's get the units correct!).

Of course, as every scientist knows, a correlation - even a good correlation, doesn't necessarily imply causation. In this case it would be correct to say that every tonne of burned coal eventually contributes towards the loss of Arctic ice, but the whole process is much more complex, with an abundance of other contributing factors, feedback processes, and non-linear relationships coming into play. Among these complicating factors are Arctic ice thinning, changing Arctic albedo, changing ocean-scale oscillations, Other country emissions, inter-annual changes in thermal inertia (land, ice, sea and atmosphere), and high-level atmospheric chemistry (volcanic gasses, aerosols, etc.). So although there is an element of truth in your article, it would be naive to imply a direct relationship.

To keep an eye on how the Arctic ice is responding to climate change I highly recommend http://nsidc.org/arcticseaicenews/

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    $\begingroup$ Sea ice loss should be run against global CO2 production over the period of time, not just US, so I don't like the 3 hectares estimate, but otherwise, I agree with your answer. Also, the article says square meters, not square km. That's 6 orders of magnitude difference. $\endgroup$ – userLTK Nov 5 '16 at 7:39
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    $\begingroup$ userLTK - yes of course it shouldn't just be about USA emissions - I'm merely answering in the same spirit as the question. $\endgroup$ – Gordon Stanger Nov 5 '16 at 9:36
  • $\begingroup$ @GordonStanger thank you for your answer! The link to NSIDC is quite helpful. Please note I've just fixed the title to agree with the quotes - square meters, not square kilometers. $\endgroup$ – uhoh Nov 5 '16 at 10:17
  • $\begingroup$ @GordonStanger I think the journal article is relating the absolute global CO2 emitted from 1979-2016 (about 1 trillion tons) to the arctic ice loss (2.79 trillion sq. meters) to get 3 square meters per year. You are getting a factor of 10,000 greater loss per ton (3 hectares per ton = 30,000 sq. m. per ton) by looking at US change in CO2/year over the interval. $\endgroup$ – DavePhD May 18 '17 at 15:17

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