During the recent (2015) Paris climate negotiations there was a lot of talk in the media about "limiting the global temperature increase to 2 degrees Celsius". I keep wondering about what this value of 2 degrees really means, and what can we read from this value. It is said to refer to some sort of global average temperature. How is this average calculated, how meaningful can it be?

Temperature is what physicists call an intensive quantity, as opposed to extensive quantities which are additive. Thus there is no physically natural way to define an "average temperature". Any averaging process will necessarily be arbitrary and ad hoc. It can be done through space and through time, with various sampling frequencies in both, and with various averaging methods (arithmetic, geometric, various weighted versions, etc.). It is indeed possible to construct pathological functions $f(x)$ the estimated average of which will be wildly different based on the averaging method, or even based on something as innocent looking as the sampling frequency.

Of course these difficulties don't mean that any average of an intensive quantity is meaningless. It is completely reasonable to try to mathematically formalize the intuitive concept of warming. But they do mean that it has to be done carefully. It is also quite striking for those of us who are not climate scientist that the amount of warming we are talking about is on a scale (0-2 C) that is much smaller than the temperature variation we might experience during a time scale as short as a day or distance scales as short as a few hundred of kilometers.

So, to finally come to concrete questions:

  • How is this average calculated exactly? Why is the specific method used considered to be reasonable?
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    $\begingroup$ You can find details about how global mean temperature is computed here ncdc.noaa.gov/monitoring-references/faq/anomalies.php $\endgroup$ – Isopycnal Oscillation Dec 15 '15 at 19:32
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    $\begingroup$ Energynumbers has a point. Voting to close, but I'll gladly support re-opening once it only asks one question. There are multiple good questions here, so please feel free to ask them seperately! Cheers. $\endgroup$ – Semidiurnal Simon Dec 16 '15 at 16:22
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    $\begingroup$ @SimonW & EnergyNumbers Thanks for the comment. I think the questions are strongly related. I don't think it would have been appropriate to ask them separately. In fact I don't even think of these as two separate question. I wrote them as such to articulate them better. My focus is mainly on the second one, but that one makes little sense without involving the first one as well. $\endgroup$ – Shoebox Dec 16 '15 at 17:29
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    $\begingroup$ I would accept an answer to the second one with only as much explanation about the first as necessary. If you still think they should be separated, then I can separate out the first one into another post. $\endgroup$ – Shoebox Dec 16 '15 at 17:30
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    $\begingroup$ @SimonW I think the way to separate the Qs yet keep them related is to edit this down to #1, and ask another stack Q covering #2. There can be a link in each stack Q referring to the other. $\endgroup$ – Aabaakawad Dec 18 '15 at 19:07

It is also quite striking for those of us who are not climate scientist that the amount of warming we are talking about is on a scale (0-2 C) that is much smaller than the temperature variation we might experience during a time scale as short as a day or distance scales as short as a few hundred of kilometers.

TLDR, but here goes:

There's a few factors to consider regarding the relatively tiny 2 degrees C and I'm not sure how to explain this without being long, but obviously you're correct, we hardly notice a 2 degree change ourselves. But if you look at the total energy of Earth you can get a slightly better picture.

The numbers below are only approximations but should be close enough to give an idea of how the trapped heat plays in the Earth's atmosphere and oceans.

The sun shines on the Earth about 1.8*10^17 Joules per second, or about 5.6*10^24 joules per year. Source. Remember that number, I'll reference it a few times.

The rough division is about 48% of that hits the surface, 23% is absorbed in the atmosphere, 29% is reflected back into space. Source. Now, because the Earth is in largely in a stable equilibrium where energy in roughly equals energy out, the Earth radiates about the equivalent of 71% of that 5.6*10^24 joules per year that the sun shines on it, the other 29% being reflected.

This will be obvious to some but I'll mention it anyway. Global warming has nothing to do with the sun giving the Earth more energy. It has to do with the Earth radiating less energy into space, at least until the Earth warms enough for a new equilibrium to re-stabilize the energy in = energy out. Raising the concentration of greenhouse gas changes the equilibrium of heat transfer. It's the same principal as putting on a sweater.

Using the same link as above, 59 of that 71% radiates from Earth's upper atmosphere and the remaining 12% from the surface. For greenhouse gas driven climate change to work, the only way that can work is that the 59% needs to drop (OK, not the only way, you can also play with albedo). But heat is trapped in the lower atmosphere, the upper atmosphere cools and the 59% drops a little lower. Cloud cover and changes in Albedo such as snow cover, ocean cover and green vs desert are factors too, (the 29% mentioned above), but that gets enormously complicated, and beyond the scope of anything I can try to explain, so I'm going to ignore that for now and just look at the 59% of the 5.6*10^24 joules per year.

Thermal radiation increases at roughly the 4th power of the temperature, so if we use the average temperature on Earth at about 288 Kelvin and add 2 degrees to that, the radiation difference between 288 Kelvin and 290 Kelvin using the 4th power rule is a 2.8% increase, or an increase from 12% to 12.34% of the 5.6*10^24. To keep the energy balance consistent, when a new equilibrium is reached, the 59% drops to 58.66% with 0.34% being "new heat" due to the additional greenhouse gas. This change presumably has already happened. In fact, today, the heat balance is out of equilibrium which is why the Earth is warming. The Earth is taking in more heat than it's giving back into space.

0.34% of 5.6*10^24 = 1.9*10^22 joules. By comparison, the combined energy use of all man-kind is about 1/40th of that, or about 5*10^20th Joules. Source. So, from a human perspective it's an enormous amount of energy but to the something the size of the Earth it's quite small. Needless to say, the 0.34% estimate is pretty rough, so take the numbers below with some flexibility, but they're also, at least somewhat in the range.

Second, we have to consider where the heat is going. By most estimates, more than 90% of this goes into the oceans and about 90% or a bit more of the heat is going into the oceans with some melting ice, some going into the ground and melting tundra and only a small percentage going into the atmosphere where surface temperatures are calculated.

Source, Source and Source (The first mentions a very specific 93.4% into the oceans and 2.3 into the Atmosphere. The last 2 mention "more than 90%" and "about 90%", so there might be some uncertainty on the precise ratio and I'm going to round up to 3%. Feel free to re-run the numbers with 2.3%, 4%, whatever you like.

3% of 1.9*10^22 = 5.8*10^20 (not a typo, it's actually 1.93)

So the atmosphere, by rough estimates, adds about 5.8*10^20 joules of energy as a result of the increase in CO2. So, to estimate the expected warming from that amount of energy:

The mass of the Atmosphere is about 5.1*10^21 grams. Source. About 4.2 joules can heat 1 gram of water 1 degree C. Source, and the specific heat of the atmosphere is about 1/4.1 that of water, Source. Of ocean water, slightly lower, I'm going to estimate 1/4.0

so, doing the maths: (4.0/4.2)*(5.1*10^21/5.8*10^20), my roughest of calculations says that we'd need about 9 years of trapped heat to warm the atmosphere 1 degree C. If we use skeptical sciences 2.3% number, that ups it to about 11.5 years, and both estimates are obviously too aggressive. The observed surface temperature warming is closer to 1 degree every 40-50 years, give or take, though the warming on land is faster than the warming over water. But my estimate can't help but be quite rough. It excludes albedo and cloud formation entirely, which is important, but trying to calculate is hugely complex and far above my pay-grade. It's difficult to calculate the effects of clouds on global warming even with super-computers.

Now, obviously ocean currents, dictate varying amounts of heat exchanged between the ocean and the atmosphere, so one year to the next can vary by significantly more than 1/9th of 1 degree, but that's movement of heat, not new heat added to the system.

Now if I calculate how long it would take to heat the oceans, and as I understand it, the deep oceans aren't warming, but the surface and sometimes called "mid-oceans" or mid-depth are warming, (articles here and here

Using the numbers above, for 1 mile deep using with the estimate of about 33 times the heat going into the oceans than into the atmosphere, reapplying the specific heat of water to atmosphere ratio of 4 to 1 and estimating the mass of 1 mile of ocean water equivalent to about 160 atmospheres, adjusting for 71% of surface cover, it comes to a rough estimate of 124 years for my very rough heat trapping estimate to warm 1 mile of oceans about 1 degree C. I couldn't find any precise warming in the oceans, but 1 degree of warming, 1 mile deep in 124 years sounds roughly in the neighborhood of what the experts expect. (I think).

So, long story short, while local temperature can change significantly over different parts of the day, global average temperature isn't nearly so easily changed. On a single day (using numbers above), the Earth gets 71% of 5.6*10^24/365 or about 1.1*10^22 joules, and an amount quite close to that radiates away from the earth every day too. Based on my very rough estimate, the difference between heat the Earth takes in and heat the Earth radiates is about 0.3%. On individual days you might get a fair bit bigger than a 0.3% difference, but there's still not enough heat added by 1 day of sunlight to significantly change the Earth's global average temperature.

Now, obviously, the Earth's surface warms measurably under daylight but that's because 48% of the suns energy reaches the surface directly and the heat is localized, and while 1/2 of the Earth's surface warms the other half cools, so it tends to average out. There's insufficient energy in sunlight to make much change to global average temperature over a short period of time.

With that in mind, the 2 degrees that the IPCC discusses, which includes ocean heat trapping, is quite significant. It more likely works out to about 3 degrees on land, 1.5 degrees over water, with rising oceans due to both thermal expansion and melting ice. Warmer oceans means bigger hurricanes. It's easy to say "2 degrees won't matter", but the models say it actually might. It's a pretty significant variation. It also takes quite a bit of time for the 40% increase in CO2 to create that much warming, which is why the projections of 2 degrees warming go out to about 2100.

Another way to point out lack of global temperature variation is to look at the charts for global average temperature. while, I couldn't find any charts for daily variation of average global temperatures, I found some monthly variation charts here. Source. Assuming the monthly averages have gotten more accurate in recent decades, The average global temperature change month to month is about 0.13 degrees C over land and about 0.095 degrees C over oceans since 1980, with the largest month to month variation about 0.47 degrees C over land 0.36 over oceans. There's probably some rounding errors in there, but there's clearly a degree of consistency to global average surface temperature and even more consistancy to surface temperature over the oceans, and that's exactly what we should expect, having looked at the heat in and heat out.

Local affects are driven by circulation, but circulation is simply the moving of heat from one place to another. Circulation isn't the creation of heat but the behavior of heat and differences in air pressures and it doesn't measurably add any heat to the system.

Corrections to my too long answer welcome.

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