I wanted to mention a Thomas Stocker paper called The Closing Door of Climate Targets. I came across it thanks to Paul Price, on Twitter. It’s an attempt to illustrate what we would need to do in terms of emission reductions to achieve certain targets, and what targets become unachievable if we wait too long. It’s quite a nice little calculation since it is quite simple, but one should always be a little careful given the uncertainties in climate sensitivity and carbon cycle feedbacks.
The basic idea is to assume that our emssions grow at some rate 
until a time 
and then are reduced at a rate 
. One can therefore express the emissions as
where 
is the emissions at time 
, 
is the emissions today (about 9.3 GtC/yr), is taken in Stocker (2013) to be 1.8%/year, and is the rate at which emissions are reduced. The next step is to determine the cumulative emissions, because the amount of warming depends almost linearly on cumulative emissions. The cumulative emissions are
which can be solved to give
where 
is the cumulative emissions today (530 GtC).
The equilibrium warming due to cumulative emissions 
is somewhere between 
and 
oC per 1000 GtC (Equilibrium Climate Senstivities of between 1.7oC and 4.8o – if I’ve done the conversion correctly. I also think this includes some carbon cycle feedback uncertainty). To get the equilibrium warming you then use
.
You can also estimate an effective minimum temperature limit by assuming that the fastest emission reduction rate would fix the cumulative emissions at the level when 
. I think there is a slight subtlety that the paper doesn’t quite get; if emissions were halted completely, then atmospheric concentrations would drop and warming would largely cease; warming to equilibrium requires reducing emissions to the level where atmospheric concentrations are fixed. This, however, requires such a low level of emissions, that it is probably a reasonable representation of the minimum level of warming. The result is
![\Delta T_{min} = \beta \left[ C_o + \dfrac{1}{r} E_o \left(\exp^{r ( t_1 - t_o)} - 1 \right) \right] = \beta C_1.](https://s0.wp.com/latex.php?latex=%5CDelta+T_%7Bmin%7D+%3D+%5Cbeta+%5Cleft%5B+C_o+%2B+%5Cdfrac%7B1%7D%7Br%7D+E_o+%5Cleft%28%5Cexp%5E%7Br+%28+t_1+-+t_o%29%7D+-+1+%5Cright%29+%5Cright%5D+%3D+%5Cbeta+C_1.&bg=ffffff&fg=333333&s=0&c=20201002)
So, now you have a set of fairly basic equations that you can use to determine relationships between rates of emission reduction, start year of emission reduction, and climate sensitivity. For example, the figure on the right shows the contours of peak warming plotted against start year and rate of emission reduction, assuming 
oC per 1000 GtC, and 
%. Even now, to keep warming to 2oC would require reductions of > 3% per year. Beyond 2030, it becomes virtually impossible.
You can also consider the rate of emission reduction that would be required to meet a particular target. This is shown in the left hand panel of the figure below. For example, waiting till 2040 before starting to reduce emissions, would require annual reductions of 10% to keep warming below 2oC. It also shows what isn’t possible. For example, beyond 2050, 2oC is no longer possible. The right panel shows the minimum achievable target, for different 
values, and for different rates of emission increase .
There are of course a lot of parameters that one can play with. You can try different climate sensitivities, , and different rates of emission increase, . If you want to consider transient, rather than equilibrium, response then use a range for of between 1oC and 2.1oC per 1000 GtC. Most of the numbers used here are probably reasonable middle values, so one should probably think of the results here as giving us about a 50% chance of keeping warming below some level. The main point, though, is that the longer we wait, the more extreme the emission reductions will need to be if we realise that we should be aiming to keep warming below some preferred level, and the more likely it becomes that some targets will effectively be no longer possible.
So, now you have a set of fairly basic equations that you can use to determine relationships between rates of emission reduction, start year of emission reduction, and climate sensitivity. For example, the figure on the right shows the contours of peak warming plotted against start year and rate of emission reduction, assuming 
...and Then There's Physics 
So what do we think is feasible I really wonder. Political scientists should really try to answer that because all of these projections on offer in the reports dont really answer the question.
Kevin Anderson paints a grim picture in his video essentially saying that neo classical economists have their finger on the,pulse and reductions of 1% per annum are about it.
Pete,
It wouldn’t surprise me if 1% per year will be about it. If the analysis here has merit, then we’re looking at just about keeping it below 3oC if we start now and if Equilibrium Sensitivity is around 2.5oC. I’m neither confident that we will start now nor that Equilibrium Sensitivity is below 2.5oC.
“think of the results here as giving us about a 50% chance of keeping warming below some level.”
Sorta like playing Russian Roulette with 3 rounds instead of one in the cylinder.
If your traveling at 28 mph towards a bridge abutment, and you apply the brakes 1000 feet away, everything’s fine – unless you’re driving a supertanker. They take about 4 miles to stop, not counting the thousand feet from the bridge to the bow. At 35Gt/per year emissions driving the global economy, stopping or turning it is like steering a 41000 foot long ultratanker; We need to have already applied full reverse and left rudder ten years ago.
well First you have to stop current emission from rising and then reduce emissions so lets be generous and say emissions are rising at 2% per annum currently and then need to fall by 2% per annum – so that 4% in real terms which is a massive ask. Its all a bit grim
Along these lines from KA.
Pete,
Thanks, I’ll have a look at that. In a sense, it’s even more massive, because we’re going to want emissions to actually drop, while increasing how much energy we generate every year.
In real terms its a long shot to get below 3C I would suggest
The International Energy Agency has reported that in 2014 CO2 emissions stalled for the first time in 40 years that emissions stayed level w/o an economic downturn. http://www.iea.org/newsroomandevents/news/2015/march/global-energy-related-emissions-of-carbon-dioxide-stalled-in-2014.html
I hope this is peak CO2. I also hope that we go from no growth in CO2 emissions to rapid decrease in CO2 emissions.
As I was looking for a free .pdf, I see he gave a lecture that is on the YouTubes.
Best,
D
Jim… in that mix is low fuel prices driving up demand. I still hope for the best.
Dumb question: Shouldn’t there be some kind of sink term to model emissions leaving the atmosphere? Or is the effect small enough to ignore for current purposes?
Probably demonstrating extreme cluelessness, but one reason I ask is because this data http://www.esrl.noaa.gov/gmd/ccgg/trends/global.html shows great volatility in the magnitude of annual increases in atmospheric CO2 concentration, at least over the last 10 yrs. On a percentage basis, the annual changes seem to be in the 0.5% – 0.7% range.
Sorry, posted too quickly; add:
… versus the 9.3 / 530 = 1.8% current annual emissions/cum emissions numbers given here.
Ummm. I probably need to read more.
Szilard,
That’s essentially included in the climate sensitivity term (1.3 to 3.9oC per 1000 GtC). I think this also includes some uncertainty related to what fraction of the emissions remain in the atmosphere.
Also, the percentage in the post refers to the rate at which the emissions are increasing (for example (E(2015)-E(2014))/E(2014) * 100) not the percentage of the total emissions to date.
@Szilard
If I understand you right, you’re referring to the seasonal variation, which is regular as clockwork each year. This is due to the fact that most of the land—and therefore most of the vegetation—is in the northern hemisphere. The annual variation is caused by the increase in plant growth every Spring, followed by the die-off every autumn. This may help: https://www.youtube.com/watch?v=x1SgmFa0r04
ATTP / Szilard …
“I think there is a slight subtlety that the paper doesn’t quite get; if emissions were halted completely, then atmospheric concentrations would drop and warming would largely cease; warming to equilibrium requires reducing emissions to the level where atmospheric concentrations are fixed. ”
The temperature increase estimate based on ECS is providing a measure of where the (atmospheric surface) warming will get to over time, so “largely cease” is I assume shorthand for “not increase eventual warming”, but of course even if we stopped burning fossil fuels today, we are not now at equilibrium, the energy imbalance remains, so warming continues for a long while yet. We still have choices (however problematic) to reduce carbon in the atmosphere (CC*) so today’s concentration is a major, but not the only factor.
Also, I the short term (5-10 years) my understanding is that roughly 50% of emitted CO2 will be absorbed in upper ocean, but the 50% increment left in the atmosphere will take 100s/1000s of years to be captured by natural processes, due in part to the very slow processes between upper ocean and the bulk below; and slow geological processes; etc.
So taking account of both these ‘fast’ and ‘very slow’ sink processes into a single ECS – if that is what is being done – is possibly a source of confusion, especially for the uninitiated, or the easily confused 🙂
szilard – the annual variation that John correctly mentions (worth looking up Keeling Curve to see this effect), is not what I am referring to. I was referring to the medium to longer term trend.
Oilman,
Yes, quite. US and UK oil consumption is surging right now. I suspect the same is likely to be true for much of the OECD. It’ll be intersting to see just how much oil consumption surges now that demand destruction is no longer an issue. All the predictions of “peak demand” are looking a little silly at the moment. This time it looks like we’ll chew through the excess more quickly though, compared with the 80’s and 90’s. However, cheap oil seems to be being enjoyed by the advanced economies.
As for the ‘peak CO2’ calls, they’re based on a single annual datapoint and dodgy Chinese coal data (see here: http://www.reuters.com/article/2015/09/16/us-climatechange-carbon-china-exclusive-idUSKCN0RF1QT20150916 ). There’s at least 0.5% error in the global CO2 measurement (BP got a 0.5% increase 2014-15) and I suspect it should be a few percentrage points higher, especially given the magnitude of recent historical revisions to Chinese data. The same is probably true with global GDP figures (again see China, plus OECD economies frequently revise up or down by a few fractions of a %), so all the calls that last year we somehow grew the global economy while emissions stalled should be taken with an entire cellar of salt. As far as I can tell the logic behind calling a peak/decoupling in global CO2 is no different from the logic of calling a “pause” in global warming. There’s simply no statisticaly valid argument to back it up.
Richard,
The subtlety is that if we halted emissions completely, then the carbon sinks would take up CO2 at a rate that meant that we would essentially cease warming. We wouldn’t warm to the equilibrium temperature of the concentration at which we stopped emitting, but the transient temperature.
Actually there is – I think – an additional subtlety to the above. If we were to stop emitting, at the time at which we stopped emitting, the NH would be closer to equilibrium than the SH. That means that then NH would actually cool, while the SH warmed. However, we would settle at close to the transient response for the atmospheric CO2 concentration at the time at which we stopped emitting, not the equilibrium response for this CO2 concentration.
at a rate commensurate with the short term 50% part that is not evident in the calculation above?? Very surprising that there is no committed warming …
It’s explained in the Steve Easterbrook post. However, it requires essentially no emissions after some time. Even if we reduced emissions by 90%, there would still be committed warming.
ATTP,
I dont think this is working how you figure it out, because if we cut emissions, we cut also aersol-emission and because they are net negative forcing, the planet would be first start to warm from a little bit to strong warming (depends on how negativ Aerosolforcing really is).
Another point would be some possible countereffects between the hemispheres related to their Imbalance via meriodonal heat transport.
My favorite is by Full-Cut-Emissions: A while (10y or more) global warming, then a longer phase of stagnation (on both hemispheres) to the point where the slow reduction in Forcing comes near to oceans eqibrilium, then slow cooling.
Christian,
Yes, you’re right that if there are also aerosols being emitted and we cut all emissions, then you’d see the carbon sinks reducing the GHG forcing, but also a reduction in the negative aerosol forcing, so there would be some committed warming in that case.
I did mentioned the hemisphere issue a few comments ago, though.
Sam taylor: Rig count is still down, and shale oil was where US oil growth was, so I suspect prices might rebound sooner, and start reducing consumption. Live in hope.
ATTP … Thanks. In terms of the inertia in the system that manifests itself elsewhere …
… in case anyone says “phew” when we finally stop adding to atmosphere, worth reading IPCC AR5 SPM 2.4 …
“Stabilisation of global average surface temperature does not imply stabilization for all aspects of the climate system. Shifting biomes, soil carbon, ice sheets, ocean temperatures and associated sea level rise all have their own intrinsic long timescales which will result in changes lasting hundreds to thousands of years after global surface temperature is stabilised”
And as Prof Alley colourfully said … “1C is a big insult to a glacier!”
ATTP, you write: “the amount of warming depends almost linearly on cumulative emissions”.
For the record, whilst this may be true for simulations by most current Earth system models, it is an entirely model dependent result. So please don’t present it as if a fact. If one builds a model with a low ECS, and moderate climate-cycle feedbacks, warming peaks immediately if emissions cease and declines quite rapidly thereafter. Which would happen in the real climate system is not as yet known, of course.
==> “Which would happen in the real climate system is not as yet known, of course.
This is true. For all we know, unicorns might fall from the skies if emissions cease.
Nic,
Yes, I realise it is not a fact. So, for clarity, our current understanding is that it depends almost linearly on temperature.
However, given the sites, and organisation, that you associate with, the idea that you can come here and tone troll me is utterly amazing. What the hell are you playing at? Do you have no self-awareness whatsoever? Do you really not get the irony of you writing appallingly dishonest posts at Climate Audit, commenting at Bishop Hill, and writing reports for a pseudo-denial organisation like the GWPF, and then coming here and suggesting that maybe I should have qualified myself a bit more carefully than I did. You really do need to look at how you present and defend your own work before coming here and tone trolling me. Seriously; WTF!!!!!
In fact, a thoughtful and decent response might actually be in order, because I really cannot believe that someone like yourself, who seems completely unwilling to acknowledge possible issues with your own work, can have just done what you’ve done.
In fact,
If we knew the real climate system, we wouldn’t be having this debate.
I’ll also ask Nic a variant of the question I’ve asked him before. What if ECS is not low and carbon sinks do not rapidly draw down atmospheric CO2 (as is expected)? What then? Also, what is the relevance of the possibility that ECS might be low?
Nic,
” If one builds a model with a low ECS, and moderate climate-cycle feedbacks, warming peaks immediately if emissions cease and declines quite rapidly thereafter. ”
Thats not true, its depends on how far away is the climate at these moment from your eqibrilium state, its the same if the ECS is low, its only the response which is indeed reduced. The only think that work it out as you said, is when ECS and TCR would be nearly the same, but if there is a gape between them, your statements is untrue.
I thought sensitivity was an emergent property and that physically plausible GCMs do not exhibit very low sensitivity because it is physically implausible.
Hence the abiding problem with lukewarmerism and palaeoclimate behaviour.
ATTP,
Be not so rude to him, because that is what “skeptics” want, later then, they tell others, mi mi mi, ATTP was so nasty to me, bit i only…
Dont let you played in this way
Re the no-aerosol upsurge if all anthropogenic GHG emissions somehow stopped tomorrow (well, in 2000), here’s a pic from Meinshausen 2005:
(According to Mark Lynas that dissertation was the basis of NEF’s almost defunct One Hundred Months campaign.)
Indeed, and – given the heat capacity of the oceans – the likelihood that the ECS and TCR are almost the same is probably vanishingly small.
Christian,
Yes, you’re probably right. If Nic wants to come back and actually engage in a serious discussion, I’m happy to engage more thoughtfully. I just have little interest in pandering to people who seem to expect others to behave in ways that they aren’t themselves. He could try correcting some of the crap on BH and what’s presented by the GWPF before coming here and being pedantic.
In 2005, not 2000.
Vinny,
Well, that seems roughly consistent with what we both said. If we halt all emissions, including aerosols, then we’d see warming initially, and then cooling back towards the transient response at the time when emissions ceased.
Just trying to be helpful, ATTP.
Vinny,
I would expect nothing else 😉
ATTP,
Thermal Inerita is huge (well to see on the slow energy accumulation in the oceans), the ratio TCR/ECS never comes close to 1.
On the Topic “Nic”
As you said. And all this wishful thinking that climate sensivity is low, i cant hear it anymore, its like in a Church and everyday, they pray that sensivity have to be low and ignore, that sensivtiy also can be high and such People i cant trust that say with certain, the sensivity is low/high.
But the hard truth is, we cannot exclude lower or higher sensivity, but this is not a reason for say “sensivity is low” but this is exactly what he (and others) do… but this is another story and not the topic here.
Christian,
Indeed, another question that I would quite like to hear Nic discuss is the relationship between his estimates of TCR and ECS. IIRC, his best estimate for the TCR is around 1.4oC. His best estimate for the ECS is around 1.6oC. However, these are independent best estimates (i.e., from the distributions for TCR and ECS independently). Given the larger thermal inertia, this ratio (1.4/1.6) seems implausible. Of course, both the TCR and ECS have ranges, but it would seem that one could somehow combine them (and take the thermal inertia into account) to try and get a better sense of the best estimates. It’s getting late, so I may not have expressed that as clearly as I would have liked.
ATTP,
Agree at this points, but you have also to realize that all thos Value is estimate by observational datasets and estimates of Forcing Data, i have corrected Heatuptake, Coverage-Bias and possible (to less solar and more aersol-forcing by vulcans) and i come up to arround ECS: arround 2.5K with methode of N&C(2014)
Another point should be the to simple Heatuptake is such methodes like N&C (2014) because its only 1 dimension and makes no difference between upper and depper ocean and then you get a bias, let me short explain:
The Methode only accounts for Fullsystem-Heatuptake(0m to the deepest Layer) but not in the Layers atself, but this does matter, because the upper Layer is coupled to the atmosphere, so there is in real climate state a differenz if there is heatuptake in upper or deeper Layer, so if you heatuptake is more in upper layer, the atmosphere warms faster in relation to you heatuptake on the deeper layern. This means, you got more surface warming by the same heat-uptake if the heatuptake is more in the upper and less in the depper layer.
So this mean for methodes like N&C(2014) that they have to serparte the heatuptake in their calculations. And this is why their ratio of TCR/ECS is biased, we see this now, heatuptake this and last year beginns to gain in the upper Layern, but upper+deeper Layer Uptake is nearly the same like the years before and surface warming accelerated now (to now there is not so much El-Nino in the Surface Values that this can only explain the warming)
Ok, have to go to bed, if i have time tomorrow i would explain it better like this, its to late for me in 5h i have to get up
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So using this formula can you work out the following:
A rich economy has high emissions but low rate of increase. A poor economy is the reverse. At what point should both parties agree it is better to build new infra-structure in the poor economy rather than the rich one. ie the rich country builds the poor country’s power plants because this will have more effect on reducing emissions than investing in their own country.
mdenison,
I suspect that that may illustrate exactly the type of problem we face.
[Mod : I was going to post your comment and respond, but it’s sufficiently ranty that I won’t bother. You could try again, if you really wanted to, but it would need to be somewhat more measured than your first attempt. I’m sure Nic Lewis is perfectly capable of defending himself if he wanted to do so.]
Just so I am clear then … The assumption about linear climate sensitivity is not a fact then ? And that a much lower sensitivity than assumed in these models is perfectly possible ? Wouldn’t that help to explain the actual fact that global average temperatures have not risen as predicted ? And also mean we may not have a problem to solve ?
Maybe I am missing something, but wouldn’t that be something to get clear up front ??
ATTP: ” – given the heat capacity of the oceans –”
Good to see the heat capacity of the oceans being considered. Next step: Consider the integration of additional solar input to the oceans during the second half of the C20th while the solar output was higher than the long term average over several decades and low cloud fraction diminished 1980-2000 (According to ISCCP data and Palle et al earthshine data).
Then it might finally be realised that successively declining peak solar cycle amplitudes don’t necessarily indicate a lack of solar forcing involvement in rising SSTs.
[Mod: Sorry, not really any better. Probably save ourselves both time if you simply stopped now.]
ImranCan,
We do know that the feedback response is almost certainly not strictly linear. However, for small changes in temperature, this appears to be a reasonable approximation. However, there is also evidence that the temperature change isn’t linear (see here for example). This non-linearity, however, does not necessarily mean that the feedbacks are themseves non-linear, but could be simply reflecting regional differences in warming rates (see here). I also wrote about the possibility of non-linear feedbacks here.
It would be wonderful if we could clear this up and know in advance what the feedback response will be. It’s essentially one of the things that we can’t know with certainty.
tallbloke,
If you think that people have been ignoring it, you really haven’t been concentrating.
Why? The change of energy in the oceans is largely set by the planetary energy imbalance, not simply by integrating solar input.
To be honest, I have no great interest in a lengthy discussion with someone who seems to dispute the basic Greenhouse effect. You’re welcome to do so, but you don’t have to do it here.
Christian,
It seems that your suggestion may have been rather prescient, given the recent Bishop Hill post complaining about my lack of civility.
I think there is likely to be more than a single dominant component in planetary energy balance.
I agree that a simple integration of solar data is unlikely to give exactly the right result without taking other factors into consideration. However, the fact that it has been hypothesised by serious mainstream climatologists like Kevin Trenberth that the ‘missing heat’ has been transferred from the surface to the deep ocean is a clear indication that the shallow depth used in GCM’s to compute the ‘residence time’ of fluctuating input anomalies is incorrect.
Internal luni-solar tides mix ocean water to far greater depths, and this means the data needs to be low pass filtered (integrated) in order to correctly model oceanic responses to energy input variation on multidecadal timescales. So pointing to diminishing solar cycle max amplitudes from 1958 and saying this shows the Sun wasn’t involved in rising ocean heat content won’t cut it.
Solar activity levels were above the longterm series average all the way from 1934 to 2003.
tallbloke,
Integrating the solar data is unlikely to give anything close to the right answer.
This has little to do with what you’re suggesting. This relates to variability in the rate at which energy is transported to the deep ocean (and hence, variability in the rate of surface warming).
ATTP,
As i said, in a simple manner, if you have no arguments, you have to win the folks by showing how mad the enemy is. And thats the way how “skeptics” try to win the normal folks.
Its very simple but works
TB
Except that they may well not have been, according to the latest revision of the SSN proxy in Clette et al. (2014):
Earlier, you mentioned the ISCCP cloud data which Evan et al. (2007) points out should not be used as evidence of changing trends in cloud cover:
You also threw in the Earthshine studies (Palle 2004; Palle 2008) but neglected to mention that these are outliers – nobody else can find the increased albedo from 1999 – 2003 and that includes the satellited observations such as MODIS, CERES etc. It’s also worth remembering that Palle (2008) corrected for instrument bias in the earlier study and showed a much reduced trend in albedo compared to the 2004 paper.
> It seems that your suggestion may have been rather prescient […]
As if AT has not been warned before.
Is Einstein wrong?
http://www.npr.org/sections/13.7/2011/09/28/140839445/is-einstein-wrong
Yes, my ClimateBallTM is a little rusty 🙂
Got your very own thread! Impressive.
It’s interesting to me is how much they care what you do and don’t do. One would think that given that they consider you uncivil, irrelevant, wrong, unintelligent, propagandistic, and unfair (as the result of deleting comments), they wouldn’t even be paying attention. There is a certain beauty to that kind of illogic.
Of course, as usual the reverse logic would apply.
Which, IMO, all kinda shows what this is mostly all about, if you get my drift.
Joshua,
I tried to avoid pointing that particular irony out 🙂
> they wouldn’t even be paying attention
Some do, and otters warned against it, e.g.:
http://www.bishop-hill.net/blog/2015/10/5/puffed-rice.html?postSubmitted=true¤tPage=2#comments
(However, RichardT seems to disagree.)
By chance our Beloved Bishop’s choir remains silent about AT’s main point.
Milllyun years of evo.
Joshua,
You do know the original name of this blog?
AT,
it’s a shame you’ve not managed a dialogue with Nic. He’s far and away the most impressive contrarian IMHO but nevertheless very limited. James Annan showed the way to play.
http://www.climatedialogue.org/climate-sensitivity-and-transient-climate-response/
vtg,
It is a pity, and it is partly my own fault for having less patience that I probably should have had. However, Nic also has a tendency to be pedantic and to focus on minor issues that he regards as wrong, rather than addressing the broader point. I also think the discussion we had with Nic about his use of a Jeffrey’s prior didn’t help.
James at least has actual credentials in this field; I’m just a blogger 🙂
AT,
Yes, contrarians can be a pain in the arse. I was amazed you managed to reach an accommodation with RPS, for instance.
I mainly don’t bother these days. Plus, I’m not even a blogger 🙂
VTG –
==> “You do know the original name of this blog?”
Not sure I get your point…but if I do…then that’s what I meant about the reverse logic also being true: It’s interesting to me how invested people are in discussions with or talking about people that they think are obviously wrong, that they think will never be convinced of their obvious wrongness, that they think are immoral and/or unethical, etc. And what adds to the beauty is that no matter how much energy they invest in that process, nothing changes in the slightest.
What’s that about?