## A bit more about committed warming

On a number of occasions I’ve pointed out that our warming committment is not actually the equilibrium temperature to our current atmospheric concentration because, if we halted all emissions, atmospheric CO2 would drop as the natural sinks took up more and more of our emissions. I should stress, though, that the long-term atmospheric concentration is limited by the Revelle factor, which I discuss here and here – we would expect about 20% to 30% of our emissions to remain in the atmosphere for thousands of years.

As a rough approximation (which I try to illustrate in this post) the committed warming is comparable to the transient response at the time when we stop all emissions (i.e., the transient response to the peak atmospheric CO2 concentration, or – equivalently – the peak change in forcing). This is also discussed in this Realclimate post, this Steve Easterbrook post and in this paper.

However, it is more complicated than suggested by the above, mainly because we’re not simply emitting CO2; there are also aerosols and short-lived greenhouse gases. In this comment Thorsten Mauritsen mentions that he and Robert Pincus have recently published a paper that looks at this. The paper is called Committed warming inferred from observations.

They essentially use an energy balance (or, observationally-based) approach to quantify the equilibrium climate sensitivity (ECS) and the transient climate response (TCR) and then use these to estimate the committed warming under a number of different scenarios. The two main equations they use are

$T_{commit} = T + [Q + \delta F - F_{aero} - F_{SLCF}] \dfrac{ECS}{F_{2x}},$

and

$T_{commit} = T + [\delta F - F_{aero} - F_{SLCF}] \dfrac{TCR}{F_{2x}}.$

In the above $Q$ is the current planetary energy imbalance, $F_{aero}$ is the current aerosol forcing (which is negative), $F_{SLCF}$ is the current forcing due to short-lived greenhouse gases, and $\delta F$ is the small change in forcing between the period considered in the study (centred on 2010) and about now (2016).

The 5 scenarios they consider are, equilibrium warming with everything remaining constant at today’s values (i.e., $F_{aero}$ and $F_{SLCF}$ both 0 in the top equation), equilibrium warming but taking into account that aerosols will precipitate quite quickly (i.e., include non-zero $F_{aero}$ in top equation), equilibrium warming taking into account both the removal of the aerosol forcing and that due to short-lived GHGs (i.e., include both non-zero $F_{aero}$ and $F_{SLCF}$ in top equation), equilibrium warming this century (replace $ECS$ with $TCR$ in the top equation), and committed warming taking into account continued ocean uptake of CO2 (bottom equation with all terms non-zero).

Credit: Mauritsen & Pincus, Nature, 2017

The results are shown in the figure on the right. On long enough timescales, we would expect something close to the result represented by $e$. This takes into account continued CO2 uptake by the oceans, precipitation of aerosols, and the decay of short-lived GHGs, but allows for a small amount of committed warming mainly due to the forcing adjustment from emissions between the middle of their time period (2010) and about now (2016). However, we might expect aerosols to precipitate faster than the decay of the slow-lived GHGs, which would then produce some short-term warming that then decays towards the value represented by $e$ (i.e., tending towards something between $b$ and $c$ and then tending towards $e$). The key point, though, is that our committed warming is smaller than would be inferred from the equilibrium response to current concentrations, even if you take aerosols and short-lived GHGs into account.

A few additional comments. The range of warming is mainly due to the quite large uncertainty in the aerosol forcing (which impacts the observationally-based estimates for the ECS and TCR). However, it is becoming increasingly clear that large aerosol forcings (very negative) are probably unlikely. Hence, the range may be smaller than indicated in the figure. On the other hand, there is increasing evidence that observationally-based climate sensitivity estimates are probably lower than the Earth’s true sensitivity. This is partly because the temperature datasets tends to suffer from coverage bias and miss some of the warming at high latitudes. There’s also the issue that combining sea surface temperatures and air temperatures may slightly under-estimate the warming, that forcings may have different efficacies, that the pattern of sea surface warming can impact the forced response, and that feedbacks may increase slightly as we approach equilibrium (although this may not be too big an issue if the committed warming is quite low).

This may indicate that these results slightly under-estimate the committed warming, but that the higher ends of the range may be less likely that these results suggests. However, I still think that a key point is that at any time our committed warming is closer to the transient response to the forcing at that time, than to the equilibrium response, with some caveats related to forcing due to aerosols and short-lived greenhouse gases. This essentially means that any amount of emission reductions can have an impact, even on relatively short timescales.

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### 84 Responses to A bit more about committed warming

1. Chubbs says:

Unfortunately we also have committed emissions from existing fossil fuel-powered powerplants, homes, cars, etc. that will take decades to replace and also possibly from permafrost and other natural systems that are not in equilibrium.

2. Indeed, as the end of the RealClimate post says

However, the practical implication of this reframing is small. We are clearly not going to get to zero emissions any time soon, and even the 60-70% cuts required to stabilise concentrations initially seem a long way off. Thus as a practical matter, it doesn’t really matter whether the inertia is climatic or societal or technological or economic because the globe will continue to warm under all realistic scenarios (what we do have a possible control over is the magnitude of that warming). Thus further adaptation measures will still be needed.

However, I do still think it’s important to recognise what our actual committment is and also to recognise that (as this paper indicates) emission reductions can have an impact even on short timescales.

3. As Chubbs alludes to, this carbon cycle system is not just about physics, it is also about global biology on a warmed planet. I think there is insufficient reason to be confident that the natural carbon sinks can and will function as the historic record indicates they have and as you suggest they will in the opening paragraph of this post. You might be right, but I am not confident about the natural carbon cycle function in response to the amazing pulse of CO2 that we have injected into the atmosphere.

When you say committed warming, is this just another way of discussing the time lag between emissions and the related warming? I have normally seen that discussed as in the range of 10 to 50 years. If I understand your post, you think the range should be discussed as 0 to 50 years because you support a model that suggests there is no lag. Do I have this right as to your position on this?

4. small,

As Chubbs alludes to, this carbon cycle system is not just about physics, it is also about global biology on a warmed planet. I think there is insufficient reason to be confident that the natural carbon sinks can and will function as the historic record indicates

I think we are pretty confident about ocean uptake. Less so about the land biosphere. However, if the committed warming is similar to the transient response to the peak forcing, then this may not be very different to what we’ve already experienced (or will have experienced before we get emissions to zero).

When you say committed warming, is this just another way of discussing the time lag between emissions and the related warming? I have normally seen that discussed as in the range of 10 to 50 years. If I understand your post, you think the range should be discussed as 0 to 50 years because you support a model that suggests there is no lag. Do I have this right as to your position on this?

No, I think my general view is similar to what is presented in paper which suggests that the peak warming occurs about 10 years after the emission (i.e., if you try to tie warming to specific emission, then the warming due to each packet of emission occurs after about 10 years). Hence, there is a bit of a lag, but not nearly as large as the lag associated with equilibrium warming if concentrations were stabilised (rather than emissions halted).

5. Paul Williams says:

If you look at the current natural absorption rate of Carbon (last several months since the El Nino influence has faded), it is up to about 6.0 Gts Carbon annualized right now.

That is the highest number ever and is a value that is way higher than any assumed in your CO2 models. It wouldn’t take long to get rid of the 250 GTs excess CO2 in the atmosphere from 280 ppm level with that type of natural absorption (note 290 or 300 ppm would be more realistic to get to than 280 ppm).

6. Paul,
The problem though is that natural sinks would not continue to take it up at that rate if we were to halt all emissions. It would initially decay exponentially with a timescale in excess of 100 years. However, there is also the Revelle factor which limits how much CO2 can be taken up by the oceans. A Revelle factor of 10 means that the fractional change in atmospheric CO2 will be 10 times the fractional change in dissolved carbon in the oceans. This means that some fraction of our emissions will remain in the atmosphere for thousands of years (to be drawn down very slowly by weathering). The fraction that remains in the atmosphere does depend on how much we emit, but for the amounts we are likely to emit it is probably between 20% and 30%. You can try paper if you want to know more.

7. I use diffusion of dopants into a semiconductor as an almost perfect analogy to incorporation of CO2 into the ocean. It’s almost perfect, as the only part missing in a classical semiconductor diffusion problem is the variability of diffusivity due to differing pathways. However, if you account for this variation, you can get arbitrarily close to the heuristic Berne model, which is most often referred to when describing the long tails of CO2 residence time.

8. Paul is also confused about the idea that you can “just” remove the excess atmospheric CO2 and thereby get atmospheric concentrations back to (close to) pre-industrial.

But that’s not the way it works at all.

Even if you assume that the terrestrial biosphere would continue to drawdown, say, 6 GtC in the absence of our fossil fuel and other forcing – an implausible outcome, but assume it anyway… Even in that situation, as you started to drawdown atmospheric carbon, the ocean-atmosphere CO2 dynamic must change and compensating pulses of CO2 would be released from the ocean to the atmosphere. Pretty basic and well-known stuff. You would need to remove far more than just the current atmospheric CO2 excess stock to get back to a given concentration.

Furthermore, there is the whole implausibility of the the terrestrial biosphere obediently storing an excess, say, >800 GtC stock relative to pre-industrial when atmospheric concentrations would have (hypothetically) returned to pre-industrial. Frankly, doesn’t pass the laugh test, let alone have any scientific robustness…

9. Jai Mitchell says:

Since CH4 is the largest contributor to SLCPs and our current process of mitigation is to shift to CH4 as a primary power source, it seems that the assumption that SLCPs will NOT go down as fast, or faster than, aerosols is a good one. In fact, it is likely that in the near term SLCPs may increase, especially given https://www.theguardian.com/environment/2017/sep/29/methane-emissions-cattle-11-percent-higher-than-estimated

However, in none of these analyses shown are they including warming soil carbon cycle feedbacks, nor are they including the fact that forest feedbacks are already turning positive, both from human activity and from stressors in rainfall patterns (tropics) and heat & insect stress (boreal) https://www.carbonbrief.org/tropical-forests-no-longer-carbon-sinks-because-human-activity

The study is the best that I have seen besides not including these feedback responses and not including the SO2 reduction impact on IPO (and subsequent additional carbon cycle feedbacks from tropical forest and peat systems). These are very complex systems and it is very difficult to capture all of the responses.

10. thank you, I got it. peak warming occurring about 10 years is what I have usually read with outliers from 0 to 50 years. The zero number makes no sense to me, but I am not a serious number cruncher, I just survey a lot of data and blog posts and try to figure out what sound like it is most likely to be correct. The range that I have adopted for discussion purposes is 10 to 25 years. I am pretty sure that is the correct range because it happens to be identical to the commonly accepted range for angels dancing on the head of a pin. In either instance, once we stop emitting CO2 or can persuade angels to stop twirling on pinhead, we will be able to determine a precise number.

11. Jai Mitchell says:

P.S.

Arctic albedo under a slightly warmer world.

12. SteveF says:

“I think we are pretty confident about ocean uptake”

You might be interested in this current project led by Andy Watson:

http://gtr.rcuk.ac.uk/projects?ref=NE%2FP015042%2F1

Not sure if anything has been published yet.

13. Jai,

Since CH4 is the largest contributor to SLCPs and our current process of mitigation is to shift to CH4 as a primary power source, it seems that the assumption that SLCPs will NOT go down as fast, or faster than, aerosols is a good one.

What I was suggesting was that if we halted the emission of aerosols and short-lived GHGs then the aerosols would precipitate faster than the short-lived GHGs would decay. I agree that if we actively reduce the emission of short-lived GHGs, without reducing the emission of aerosols, then the short-lived GHG forcing could reduce faster than the aerosol forcing.

14. small,

thank you, I got it. peak warming occurring about 10 years is what I have usually read with outliers from 0 to 50 years. The zero number makes no sense to me, but I am not a serious number cruncher

What it’s really saying is that if you consider a single pulse of emission, then it will be taken up by the natural sinks at a rate that means that the warming due to that specific pulse of emission peaks about 10 years after the emission occurs. The reason we keep warming is not really because we’re still warming due to emissions that occurred many decades ago, but because we keep emitting CO2 into the atmosphere.

15. SteveF,
Thanks, that looks interesting. I’ll try to have a closer look when I get a chance.

16. Everett F Sargent says:

Natural Variability and Anthropogenic Trends in the Ocean Carbon Sink (circa 2017 (print) or 2016 (online))

Click to access ARMS_proofs.pdf

See Figure 1 and section 3 “BASICS OF THE OCEAN CARBON SINK”

From the Summary Notes section (1st point) …

The underlying chemistry and basic mechanisms of the ocean carbon sink are well understood. Biological mechanisms are critical to the ocean’s natural ability to sequester carbon in the deep ocean, but the ocean carbon cycle’s response to the anthropogenic perturbation of rapid growth in pCO2atm is dominated by the solubility effect as (delta)pCO2 becomes more negative.
(where negative means pCO2ocean – pCO2atm is < zero).

I found this paper just now via ATTP's Archer (2009) link via Google Scholar. 🙂

17. angech says:

rustneversleeps says: October 1, 2017 at 4:22 pm

“Even if you assume that the terrestrial biosphere would continue to drawdown, say, 6 GtC in the absence of our fossil fuel and other forcing. Even in that situation, as you started to drawdown atmospheric carbon, the ocean-atmosphere CO2 dynamic must change and compensating pulses of CO2 would be released from the ocean to the atmosphere. Pretty basic and well-known stuff. You would need to remove far more than just the current atmospheric CO2 excess stock to get back to a given concentration.”

Whilst admiring your determination to keep the status quo I fear there is a flaw in your logic. If, you remove the current excess CO2 stock and CO2 was returned from the ocean that CO2 excess would in turn be drawn down. At some point in time we would reach the tortoise and hare figure of getting back to a given concentration.

” there is the whole implausibility of the the terrestrial biosphere obediently storing an excess, say, >800 GtC stock relative to pre-industrial when atmospheric concentrations would have (hypothetically) returned to pre-industrial.”

Um, we are using fossil fuels which the terrestrial biosphere conveniently put away Millenia ago for use in those later times when the when the atmosphere had returned to post aggressive carbon deposition.

18. angech,

At some point in time we would reach the tortoise and hare figure of getting back to a given concentration.

Yes, but this will take longer than 100000 years. There’s no claim that we will never return to pre-industrial levels. The issue is that it will remain elevated for a very, very long time (something like 20-30% of what we’ve emitted will remain in the atmosphere for thousands of years).

Um, we are using fossil fuels which the terrestrial biosphere conveniently put away Millenia ago for use in those later times when the when the atmosphere had returned to post aggressive carbon deposition.

We’re burning these fossil fuels at a rate much, much faster than the rate at which it was deposited (I forget the numbers, but it’s something like what we burn in a year took 10000 years to deposit as fossils).

19. KHome1990 says:

I have a couple questions in relation to this, and also another post you had referenced on Twitter:

1. Your committed estimates suggest total committed warming is not 2C yet. Excluding step e (long term natural carbon uptake), what do you think is the PPM level of CO2 in which we would hit 2C of warming at equilibrium? Basically, the CO2 level where averaging steps b/c would yield a 2C estimate.

2. Also, how is committed centennial warming only 1.3C (at step d)? We already have warmed 1.1C (20/15/2016 minus small El Nino influence, also close to where we are in 2017 so far). You show net increase from aerosol and SLCF changes alone is 0.24C. Aerosols/SLCFs are relatively fast feedbacks, they’d both be complete well before the end of this century if we were to stop emitting today. Thus already commits us to 1.34C this century before CO2 is even considered at all – above what you get in step d.

20. 1. It depends on what the ECS actually is, but if we assume 3oC, then we want to solve the following.

$\Delta T = 2 = \dfrac{\Delta F}{F_{2x}} ECS,$

for $\Delta F$. If $F_{2x}$ is 3.7 W/m^2 and $ECS$ is 3K, then $\Delta F$ is 2.5W/m^2.

We can then use

$\Delta F = 5.35 \ln \left( \dfrac{C}{C_o}\right),$

to find the conentration $C$, given that $C_o = 280ppm$. It would be about 450ppm. Would be slightly higher if the ECS is lower, and slightly lower if the ECS is higher.

2. That’s because we’re assuming that we halt everything now. In that case, the aerosols precipitate and the short-lived GHGs decay and we tend towards an equilibrium warming of about 1.5K, but this will take longer than a century and will slow as we approach equilibrium and so we will have warmed slightly less than this by the time we get to 2100.

Something to bear in mind is that this is a slightly unrealistic scenario of halting all emissions now. However, it does illustrate the relationships between committed warming, emissions, and the different types of forcings (CO2, aerosols, and short-lived GHGs).

21. angech says:

“the long-term atmospheric concentration is limited by the Revelle factor, which I discuss here and here – we would expect about 20% to 30% of our emissions to remain in the atmosphere for thousands of years.
We’re burning these fossil fuels at a rate much, much faster than the rate at which it was deposited (I forget the numbers, but it’s something like what we burn in a year took 10000 years to deposit as fossils).”
Big numbers, big concept.
The extra amount we produce is still reasonably small compared with the overall yearly carbon cycle.
Whether the sinks are capable of coping is one of the questions. The fact that after several [many] severe outgassing of CO2 events in the past we are still here indicates to me that the biosphere and the earth sea chemical mixes have shown the needed resilience.
We are in uncharted waters.

22. angech,
No, the carbonate chemistry of seawater is well understood. There is a limit to how much it can take up. It’s almost certain that some fraction of our emissions will remain in the atmosphere for thousands of years. There is also paleo evidence that supports this (i.e., past events where enhancements in atmospheric CO2 have remained for thousands of years).

23. verytallguy says:

Whether the sinks are capable of coping is one of the questions. The fact that after several [many] severe outgassing of CO2 events in the past we are still here indicates to me that the biosphere and the earth sea chemical mixes have shown the needed resilience.

Short version: this is utter bollocks.

Longer version: the “resilience” of the biosphere in previous events is irrelevant to the issues at hand. This can be easily demonstrated by a reductio ad absurdum: would we be happy if all land mammals were extinct by 2100? Such an outcome would not compromise the continuance of the biosphere in any way, yet I would suggest might nevertheless be seen as, well, sub-optimal.

24. @angech: “I fear there is a flaw in your logic.”

Actually, I am not “logicing” my way to the real life consequences. As ATTP mentions, the ocean carbonate chemistry is well understood. And, for a given pulse of atmospheric CO2 – whether that pulse is positive OR negative (i.e. an increase or a decrease), about 80% of it “decays” (e.g. to *OR* from the ocean) after about 500 years.

It looks like this (Figure A). Note that the green line is a *negative* pulse:

As far as the terrestrial biosphere ability to draw down the roughly 650 (and growing) GtC we have net emitted during since agriculture or the industrial revolution, take your pick… For about 10,000 years an atmosphere with about 280 ppm CO2 was in rough equilibrium with a biosphere that held X GtC. There is *NO* understood reason to indicate that the carbon cycle would somehow take a brief couple-of-centuries excursion from 280ppm to 450ppm and back to 280ppm as the terrestrial biosphere miraculously found a way to now store X + ~800 GtC long-term. How would that happen? Where? Why?

Another “why” question: Why do you so often just make things up in your head and present them as credible “just so” (yet counterfactual) representations of well-understood real-world phenonmenon.

* from http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0081648, but any basic carbon cycle text or literature confirms the same thing.

25. KHome1990 says:

For #2, how much additional warning would be caused by CO2 alone this century if all emissions were halted? The issue I get is we have 1.1C warming already, and combination of aerosols and SLCFs will net an additional 0.24C. So that puts at 1.34C this century before even considering CO2. That means additional warming this century from CO2 would have to be zero to arrive at step d, which doesn’t make sense.

3. Also, for all these estimates, how much additional warning must be added to account for methane and other non-SL GHGs?

26. KHome1990,

For #2, how much additional warning would be caused by CO2 alone this century if all emissions were halted? The issue I get is we have 1.1C warming already, and combination of aerosols and SLCFs will net an additional 0.24C. So that puts at 1.34C this century before even considering CO2. That means additional warming this century from CO2 would have to be zero to arrive at step d, which doesn’t make sense.

Actually it does make sense. If we halted all emissions, then CO2 will be taken up by the natural sinks (oceans mainlym but also biosphere) in way that essentially cancels further warming. There are some complications due to the different hemispheres (the NH is closer to equilibrium than the SH due to having more land mass) but – on average – if we halt all emissions then there is little committed warming due to CO2. Try looking at the posts and article that I link to at the end of the second paragraph of the post.

27. KHome1990,
Actually maybe I misunderstood your question. They’re basing this on HadCRUT4, which does not show as much as 1.1oC (more like 0.95oC), so in their model there is some warming due to CO2.

28. KHome1990,
Maybe a simple answer to your question about (d) is that on long timescales the dominant forcing is CO2 (aerosols precipitate and short-lived GHGs decay). Today we’re at 400ppm, so if that is fixed, then the change in forcing is

$\Delta F = 5.35 \ln\left( \dfrac{400}{280} \right) = 1.91 Wm^{-2}.$

If the ECS is 3K, then this would lead to an equilibrium warming of $3 \times 1.91/3.7 = 1.55K$. This, however, will take longer than a century, so by 2100 we would expect it to be slightly lower then this.

29. Kyle Armour says:

Nice post. Here’s my take on committed warming inferred from observations (AR4 numbers) from several years ago: http://faculty.washington.edu/karmour/papers/ArmourRoe_GRL2011.pdf . It’s a slightly more straightforward calculation than than Mauritsen and Pincus used, but it produces very similar answers when updated to AR5 numbers.

Regarding the key point that at any time our committed warming is closer to the transient response to the forcing at that time, than to the equilibrium response: This is likely true, but the caveats related to aerosols are significant and can’t be ignored; if aerosol forcing is at the high end of the AR5 estimate (which is correlated with a high climate sensitivity) then we may be committed to much more warming, while if it’s at the low end then we could committed to even less warming than the transient response.

30. Kyle,
Thanks, I hadn’t seen that paper of yours; I’ll have a good look at it. I was well aware that there is a large aerosol forcing uncertainty, but am only just starting to appreciate the impact that this has on our warming commitments.

31. Thorsten Mauritsen says:

Kyle, yes we do handle the ‘double’ effect of aerosol forcing, see the paper discussion around equation 2. A high aerosol forcing demands a high TCR, and in addition an extra forcing when the aerosol is removed. Thus, the upper end of case b is much higher than that of case a. One can also see in Figure 3 from the shading how high TCR, and therefore high commitments, are all associated with strong aerosol cooling.

KHome99 and ATTP, also the HadCRUT dataset gives nearly ~1.1 K in year 2016. For this kind of work one needs an expectation value, and one cannot base this on the average of a single year which is affected by internal variability (a strong El Nino). Even the eleven years we use, 2005-2015, is on the edge of too short and probably a few hundreds too cold, but nevertheless has the advantage that it is centred on the end of the AR5 forcing time series and the same as the latest planetary imbalance estimate. It is important that these are congruent.

32. KHome1990 says:

In all of these, I’m not considering step e yet (i.e. natural heat uptake), only interested on warming/equilibrium estimates prior to that effect.

1. I’ve seen the HadCRUT4, relative to what baseline done it have 0.95C? It had said 2015 and 2016 were >1C from pre-industrial in its data. Relative to 1961-1990 (the baseline used in the data itself), 2016 is just under +0.8C (so assume +0.75C when removing El Nino). So what baseline is the +0.95C relative to, and are the warming amounts given in the figure in this article relative to that same baseline?

2. Also, are there any GHGs of significance besides CO2 that aren’t considered to be a SLCF (i.e. methane, etc.)? If there are, how much needs to be added to committed warming estimates to account for the concentrations of those GHGs already in the atmosphere? Or is every other GHG besides CO2 handled as a SLCF?

3. Lastly, some GHGs like methane can decompose into CO2. What would CO2 concentration be if we stopped emitting, but all SLCFs decomposed and released resultant CO2?

33. KHome1990 says:

Thorsten, as for how much we’ve already warmed, I always will use the value of the current warmest year on record. It’s rare to have a year that’s much cooler than that without a La Nina. Even after 1998, most subsequent years that didn’t have a La Nina were only slightly cooler than 1998, even if they weren’t new records. (i.e. 2001-2004 and 2006, for instance)

34. Thorsten,

KHome99 and ATTP, also the HadCRUT dataset gives nearly ~1.1 K in year 2016.

Thanks. I was thinking of the likely decadal average which (IIRC) the recent Millar et al. paper suggests would be around 0.95C, but I didn’t make that very clear.

KHome,

I’ve seen the HadCRUT4, relative to what baseline done it have 0.95C? It had said 2015 and 2016 were >1C from pre-industrial in its data.

As Thorsten points out, HadCRUT4 might show 1.1 average over 2016, but you should really consider a longer time period so as to try and average out some of the variability (ENSO events, for example). The number I was quoting was something like the value used in the recent Millar et al. paper in which they estimate that the average for the decade to 2020 will probably be around 0.95C.

Also, are there any GHGs of significance besides CO2 that aren’t considered to be a SLCF (i.e. methane, etc.)?

I don’t know. Maybe someone else knows the answer to this.

Lastly, some GHGs like methane can decompose into CO2. What would CO2 concentration be if we stopped emitting, but all SLCFs decomposed and released resultant CO2?

My understanding is that some will produce extra CO2, but I think the net effect of this is small. Again, someone else might have a more definitive answer to this.

35. KHome,

Thorsten, as for how much we’ve already warmed, I always will use the value of the current warmest year on record.

The problem (I think) with doing this is that you can get quite a large amount of internally-driven variability on short timescales (years). Hence, if you simply use values for a single year (temperature, change in forcing, system heat uptake rate) then you can over-estimate (or underestimate) things like the TCR and ECS. It’s better to use values averaged over a longer timescale (about a decade) so as to try and average over this internally-driven variability.

36. dikranmarsupial says:

angech wrote “The extra amount we produce is still reasonably small compared with the overall yearly carbon cycle.”

Perhaps the oldest and most basic carbon cycle canard stretches its wings again. Whether anthropogenic emissions are large or small compared with the magnitude of environmental fluxes is entirely irrelevant, what matters is whether they are large or small compared with the difference between total natural emissions and total natural uptake as that is what governs the rise or fall of atmospheric CO2 concentrations.

angech followed this by “Whether the sinks are capable of coping is one of the questions.”

ATTP wrote “No, the carbonate chemistry of seawater is well understood.”

Indeed, there was an excellent article or two on this not so long ago on this very blog. Pity angech seems to have learned so little from it, given he featured repeatedly in the comments (not uniformly to his credit).

angech continued “The fact that after several [many] severe outgassing of CO2 events in the past we are still here indicates to me that the biosphere and the earth sea chemical mixes have shown the needed resilience.”

The flaw in that argument is fairly obvious, yes, the carbon cycle dealt with the outgassing, but on what timescale (thousands of years) and at what increase in global temperatures (IIRC paleoclimate estimates of ECS based on such events tends to be at the higher end of the IPCC range, but I am happy to stand corrected).

followed by a soundbite: “We are in uncharted waters.”

Sadly this sort of thing doesn’t encourage me to go back to commenting on climate blogs on a regular basis.

37. Thorsten Mauritsen says:

KHome1990, we assign CH4, CO and NOx as short-lived climate forcers. For CH4 we separate the emissions from fossil fuels from other sources, such as agriculture etc. We consider the fact that these are reactive gases which cause other greenhouse gases, such as ozone.

As for the baseline temperature, we are well aware this is an issue. We chose to use 1850-99, and there is an argument in the Method section:

“The choice of 1850–1899 as the pre-industrial reference in equations (1)–(5) is a compromise of availability of temperature observations and having boundary conditions representative of pre-industrial conditions, for example, a volcanic forcing close to the long-term mean. Yet, industrialization had already commenced during this period, and relative to 1750 there was a positive total forcing of 0.15 W m 2 . On the contrary, the period 2005–2015 had relatively little volcanic activity, and counting on future volcanic forcing at the level of the past forcing would yield less commitment. Together with a low e cacy of volcanic forcing32 these two e ects closely cancel.”

I hope this helps.

38. dikranmarsupial says:

I wrote “what matters is whether they are large or small compared with the difference between total natural emissions and total natural uptake”

I forgot to mention, they are large by comparison, approximately twice as large (hence an airborne fraction of approximately 1/2), which is why anthropogenic emissions are currently the primary factor governing changes in atmospheric CO2 concentrations.

39. KHome1990 says:

ATTP, first of all do you agree reasonably well with the central estimates of committed warming provided in the figure? (I had previously thought it was actually your own figure, which I why I hadn’t asked.)

As for time period to average, maybe it’s okay to knock 0.05C off the warmest year on record if it was an El Nino year (so you’d knock that off 2016), but the issue I get with using a decade for the current TCR level is that it will almost always underestimate the TCR at the present year. It’s extremely unlikely the last few years of a 10-year period will not be warmer than the first few years of the same 10-year period, unless natural forcings were strongly warming (El Nino) early in the decade and strongly cooling (La Nina plus volcanoes) late in the decade. In this particular case, 2015 and 2016 are so substantially warmer than all the previous warmest years (with 2017 headed to only be perhaps slightly cooler) that an average including any years prior to 2015 (or at least 2014) now seems irrelevant for current TCR.

Also, considering all GHGs plus aerosols, what level of CO2 would be most likely to result in an equilibrium of 2C? (I know you had estimated 450 ppm before, but that was considering only CO2 and not effects from SLCFs plus aerosols.)

Lastly, what’s the likely warming in 2050, relative to pre-industrial, if CO2 doubling occurs in 25-30 years (+5-6 PPM/year, leading to doubling in the 2040s)? I’m not looking for just the TCR; I’m looking for the level of warming after all other factors have also been considered (like aerosols, SLCFs, and climate feedbacks).

Thorsten, do all committed warming figures take into account non-SLCF GHGs that will be released into the atmosphere when existing SLCFs decay? Do they take climate feedbacks into account? And lastly, how much extra CO2 (in PPM) will SLCF decay add into the atmosphere even if we emit nothing?

40. Thorsten Mauritsen says:

KHome1990, I am reluctant to answering your questions as I have a feeling we are not using the same concepts, eg. concerning what are climate feedbacks, or TCR. I believe, however, that you will find answers to your questions by reading the paper and the methods section. There are also references directly to the relevant sections of AR5, wherein the forcing from reactive gases is estimated, and on my Web page you will find an archive with all data and scripts needed to reproduce the study. Our thinking is that this will make it easy for people to test alternative ideas, include more years, etc.

I hope this helps.

41. KHome1990,
I’m also not sure how to answer your questions either. I think this is all pretty clear. What happens in future is going to mostly depend on what we do in future (what do we emit and how much). So, there isn’t really a simple answer to some of what you’re asking.

42. Steven Mosher says:

” and on my Web page you will find an archive with all data and scripts needed to reproduce the study. Our thinking is that this will make it easy for people to test alternative ideas, include more years, etc.”

Bravo

Fascinating conceptual overview of the five schools of thought in 'open science' https://t.co/pAYP41xiVn #openscholarship pic.twitter.com/Cx24UIjzxR— Jon Tennant (@Protohedgehog) October 1, 2017

43. Willard says:

44. Everett F Sargent says:
45. verytallguy says:

46. KHome1990 says:

ATTP, I’m basically looking at two simple scenarios with two simple resultant questions. The first is if no more carbon is emitted, how much warming is likely by 2050? The second is the same, except in a scenario where we hit 560 PPM around 2045 with annual increase around 5 PPM over the next three decades. In both cases I want a best estimate of the amount of warming that would likely be realized by 2050 over pre industrial with appropriate considerations for SLCFs, aerosols, and positive feedbacks (such as ice melt) taken into account.

47. KHome1990,
There aren’t really exact answers to your questions, mainly because we don’t precisely know climate sensitivity. If we stopped emitting everything, then we would tend to towards e in the figure. However, by 2050, all the aerosol forcing would probably be gone, but not all of the shirt-lived GHG forcing would be gone, so it would probably be somewhat higher and coming down. Of course, there is a range for this warming and it could, or lower, with the caveat that the high end of the range is probably less likely than indicated because the very negative aerosols forcings are probably less likely than we had thought.

If, however, we actually hit 560ppm, then we would expect to be close to the transient response to a doubling of atmospheric CO2 (so, probably somewhere close to 2oC).

Maybe if you explain why you would like answers to these questions, I can better see what you’re getting at. I think you can probably work out the basic answers by just looking at the various projections that are available via the IPCC AR5 (and also using what’s in the paper discussed in this post).

48. There’s an interesting Climate Lab Book Guest Post that discusses the chance of staying below 1.5oC, but also mentions Thorsten’s recent paper and the impact of aerosols and short-lived GHGs.

49. I suspect this paragraph in the Climate Lab Book post is a key one with respect to KHome1990’s questions.

To summarise, the extra warming after all CO2 emissions cease could be close to zero. The two well-known positive contributions (ocean heating and aerosol reductions) are compensated by CO2 uptake and rapid methane reductions to first approximations, as already reported in the AR5 Chapter 12. However, the uncertainties on these cancellations are very large and taken together enhance the chances that the 1.5 °C target is out of reach already, depending mainly on the strength of the present-day aerosol cooling.

50. KHome1990 says:

As for ceasing emissions, that pretty much covers it. As for transient warming if we double CO2, I’ve heard that depends when it is doubled (i.e. if CO2 was doubled in 2030 that’s different from if it was doubled in 2070). So I don’t get how just saying it would be 2C without specifying a year makes sense.

My case is simply assume CO2 is doubled in 2045 – what would the transient response in 2045 be (would you still think close to 2C), assuming ECS is 3C-3.5C range? What would the additional warming be relative to 2016 (as opposed to total warming over pre-industrial)? But make sure not just CO2 warming through 2045 is taken into account, but also best estimates of aerosol effects that would be associated with that rate of CO2 increase, plus SLCFs and climate feedback such as ice melt. All I’m asking for is best estimates, I know there’s uncertainty with any of these numbers.

51. angech says:

Re Kevin and ATTP’s points of view. In terms of committed warming given the operant conditions and the lack of commitment I doubt that any figure short of a 3.2 C total by 2100 is possible under current scenarios.
Let us hope I am partly right about carbon sinks and that the ECS is lower than assumed in those conditions.
KHome1990 given a 0.3 C per decade 33 years might basically bring a 1.0 C rise by 2050 under both your scenarios. Given a couple of hundred years you would see quite a divergence.

52. Jai John Mitchell says:

The Mauritsen & Pincus graphic in this blog post is haunting and has been following me for a few days. I always knew that higher SO2 negative forcing implied higher ECS (and vice-versa) but have not seen it so clearly laid out what the implications are at our current 490 ppmv CO2e forcing.

Now that ECS is indicating that it will be on the higher end of the IPCC spectrum (slightly more than 4.5) then the SO2 forcing parameter is closer to -1.0 W/m^2 and our current locked in warming is about +2.7 C above pre-industrial (if you include arctic albedo induced warming and carbon cycle emissions from warming soils and forests after >2.0C).

This doesn’t mean that there is no hope, only that we have to do everything humanly possible to prevent further emissions and greater warming.

53. Jai John Mitchell says:

P.S. it would have been nice if Mauritsen & Pincus had put 66th percentile bars on their graphic.

54. Willard says:

That’s all he won, alas. And he still does not seem to realize it’s his own job to give people a real sense of addressing the scale and the timeframe of a 2C target.

55. KHome1990,

As for transient warming if we double CO2, I’ve heard that depends when it is doubled (i.e. if CO2 was doubled in 2030 that’s different from if it was doubled in 2070). So I don’t get how just saying it would be 2C without specifying a year makes sense.

My understanding is that if we’re considering the a difference of a few decades, then it doesn’t make much difference. What mostly matters is how much we emit, not how fast we do it.

But make sure not just CO2 warming through 2045 is taken into account, but also best estimates of aerosol effects that would be associated with that rate of CO2 increase, plus SLCFs and climate feedback such as ice melt.

I don’t know our future aerosol and SLCF forcing projections, and I don’t think anyone knows them. It depends very much on whay pathway we follow. They are, however, likely to become less important as the CO2 forcing becomes more and more dominant (aerosols precipitate and SLCFs decay, while CO2 accumulates). Essentially, I don’t think there is an easy answer to what you’re asking, unless you specify what you think the aerosol and SLCF forcing will likely be.

56. Jai,

Now that ECS is indicating that it will be on the higher end of the IPCC spectrum (slightly more than 4.5)

Where are you getting this from? I certainly haven’t seen much to indicate that there is increasing evidence that the ECS will be on the higher end of the IPCC spectrum.

it would have been nice if Mauritsen & Pincus had put 66th percentile bars on their graphic.

I think the graph shows the median plus the $2 \sigma$ range. 66% would be $1 \sigma$ so you could roughly get that by halving the range.

57. KHome1990 says:

Assume aerosols and SLCFs either increase/decrease through 2045 at same growth/decay rates as average over past 5-10 years; CO2 reaches 560 PPM in 2045. What is best estimate in that case of warming in 2045 relative to both 2016 (+1.1C) and pre industrial? As for feedbacks, mainly it’s things like sea ice melt or cloud feedbacks I want to be sure are considered, if possible at least.

58. KHome1990 says:

angech, where do you get a warning rate of 0.3C/decade from? Also that means we are still about 75 years (2090 or so) away from the 3.2C you mention this century. That’s if we basically go business as usual. If we start to change at any point at least in the next 15 years, I’d think we could reduce to 0.2C by 2050 which would leave us just under 3C end of century assuming there isn’t further slowing if rate late century.

59. KHome,

Assume aerosols and SLCFs either increase/decrease through 2045 at same growth/decay rates as average over past 5-10 years

If that were the case, then I think one could assume that they roughly cancel each other and that most of the warming would be due to the increase in CO2. However, I think this is probably not a reasonable assumption, since aerosols precipitate and, hence, the aerosol forcing shouldn’t continue growing at the same rate (especially if we use less coal). Also, short-lived GHGs decay and also maybe won’t grow at the same rate (not sure about this though). So, maybe one should assume slightly more warming than CO2 only.

Again, I don’t think there is an easy way to answer your question. I don’t think I can do any better than I have. The IPCC RCP scenarios probably also provide approximate answers.

60. Jai John Mitchell says:

ATTP,

Fully 2/3 of California’s black carbon emissions comes annually from forest fires. with 1.3 times as much black carbon from fires as from fossil fuel combustion in North America (see end of section 2.2 https://www.fs.fed.us/nrs/pubs/jrnl/2014/nrs_2014_liu_001.pdf ) Since these are going to increase in a warming world, the black carbon emissions going forward in a best case scenario will only slightly go down. For Methane, the shift toward this fuel source and away from coal will both increase fugitive emissions, and reduce SO2 emissions, additionally, increased emissions from frozen soils is being observed now, though desiccation of some tropic regions may reduce natural emissions quite a bit. So, in the most optimistic case, SLCPs may reduce their forcing impact by approx 50% in the next 30 years, with SO2 emissions being reduced much more rapidly over that time.

re: ECS ~4.5 (note: All papers must be considered together with compounding effects and implications)
http://onlinelibrary.wiley.com/doi/10.1002/2015GL065911/full (explicit)
http://www.nature.com/nature/journal/v533/n7603/full/nature17423.html (Eocene with slow and carbon cycle feedbacks removed)
http://science.sciencemag.org/content/352/6282/224 (explicit)
http://www.annualreviews.org/doi/abs/10.1146/annurev-earth-100815-024150 (inferred)
http://onlinelibrary.wiley.com/doi/10.1002/2015GL064119/abstract (explicit)
http://www.nature.com/nature/journal/v505/n7481/full/nature12829.html (explicit – increased lower bound to 3.0)
http://journals.ametsoc.org/doi/10.1175/JCLI-D-15-0234.1 (tropical super greenhouse effect analysis – inferred when taken in context with cloud dynamic analyses above)
https://link.springer.com/article/10.1007/s40641-015-0021-7 (inferred – shows model error has led to underestimate of ECS)
and (of course)
http://www.nature.com/nclimate/journal/v6/n10/full/nclimate3079.html (Armour 2016 – explicit)
http://www.realclimate.org/index.php/archives/2016/01/marvel-et-al-2015-part-1-reconciling-estimates-of-climate-sensitivity/ (marvel 2016 – shows regional observation constraints raise ECS with greater effect on studies with higher ECS)

Taken together, the shift in observational constraints (regional forcing impacts, cloud effects, ITCZ shifting, Hadley Cell shifting and upper tropospheric tropical water vapor impacts from increased atm mixing) Supports the work of modelers who are using multiple lines of evidence to show that individual PPE constraints were overstated on the low-end side and these both work to verify the paleoclimate work (specifically the Eocene and Pliocene analyses that show ESS at > 6.0K)

h/t to ASLR at Arctic Sea Ice Forum for tirelessly documenting the body of work on the subject over these last 5 years. https://forum.arctic-sea-ice.net/index.php?action=profile;area=showposts;u=40

61. Jai John Mitchell says:

P.S.
my first red flag on the regional implications of SO2 forcing impacts being underrepresented in the models came after reading this important work from 2001. http://www.kenrahn.com/DustClub/Articles/Xu%202001%20Abrupt%20change%20in%20climate.pdf
I have followed your work for about 5 years now and respect it very much. If I could make a humble request, I would ask that you give this a thorough review and do a blog post on the implications of these observations. I would be very interested in what you come up with.

62. KHome1990,
Below is a figure that might help. It shows the cumulative emissions for the 4 different Representative Pathways (RCPs) and shows how the warming depends more on how much we emit, rather than on how fast. It also shows atmospheric concentrations (bubbles) and each dot on each line is a decade. The range is represented by the width band.

63. Jai,
But some of those papers are simply reconciling observationally-based estimates (which suggest a best estimate of 2oC, or slightly lower) and paleo/model estimates (which suggest a range from about 2 to 4.5oC, with a best estimate close to 3oC). That doesn’t really seem to justify claiming that it will be on the higher end of the IPCC spectrum (slightly more than 4.5) (maybe I’m misunderstanding what you’re suggesting?).

64. Everett F Sargent says:

Figure 2.3 | Global mean surface temperature increase as a function of cumulative total global carbon dioxide (CO2) emissions from various lines of evidence. Multi-model results from a hierarchy of climate carbon-cycle models for each Representative Concentration Pathway (RCP) until 2100 are shown (coloured lines). Model results over the historical period (1860 to 2010) are indicated in black. The coloured plume illustrates the multi-model spread over the four RCP scenarios and fades with the decreasing number of available modelsin RCP8.5. Dots indicate decadal averages, with selected decades labelled. Ellipses show total anthropogenic warming in 2100 versus cumulative CO2 emissions from 1870 to 2100 from a simple climate model (median climate response) under the scenario categories used in WGIII. Temperature values are always given relative to the 1861–1880 period, and emissions are cumulative since 1870. Black filled ellipse shows observed emissions to 2005 and observed temperatures in the decade 2000–2009 with associated uncertainties.(WGI SPM E.8, TS TFE.8, Figure 1, TS.SM.10, 12.5.4, Figure 12.45, WGIII Table SPM.1, Table 6.3)

http://ipcc.ch/report/ar5/syr/

Click to access SYR_AR5_FINAL_full_wcover.pdf

(page 63)

The black line and shaded area is supposed to be 1% CO2 increase per annum (atmospheric) AFAIK.

Yes.

“CMIP5 simulations with 1% yr–1 CO2 increase only are illustrated by the dark grey area (range definition similar to RCPs above) and the black thin line (multimodel average).”

Click to access WG1AR5_TS_FINAL.pdf

(page 104 TFE.8, Figure 1)

65. I’ve never quite understood the grey region because the 1% per year simulations refer to how the atmospheric concentration changes, not emissions (I think). So, putting it on a plot of emissions versus temperature would seem to require some kind of assumed relationship between emissions and concentrations.

66. Okay, I think I know what they’ve done to produce the grey region – they’ve based it on the equivalent concentrations in the bubbles.

67. Everett F Sargent says:

Jai John Mitchell,

It’s a review paper with dozens of ECS papers/estimates (it even cites the Harde 2017 paper I mentioned as a ‘missive missile that misses massively’ in the Harde thread).

68. Jai John Mitchell says:

ATTP,

The range of estimates of ECS on the low-end are dominated by observational estimates. The reconciling of these with centennial-based feedbacks in models shows that the lower end range is not applicable. similarly, http://www.nature.com/nature/journal/v505/n7481/full/nature12829.html shows that the new lower bound is at or higher than 3.0.

The majority of other papers I posted show that the modeled outputs hold assumptions that are proven to be understating the recently (last 3 years) observed changes. Together they show that the true likely range of ECS, with centennial timescale feedbacks, is between 3.5 and 6.0 with a central estimate of 4.7 (or so – after 2.0C warming above pre-industrial is attained)

69. Everett F Sargent says:

Please do a complete prior art search, because just picking all the high estimate papers is called something (which is NOT what Knutti et. al. (2017) did AFAIK)

From the SOM of Beyond equilibrium climate sensitivity (Knutti et. al. (2017)) …

Click to access ngeo3017-s1.pdf

Paleoclimate proxies and modelling: Four of 26 with a stated mean/median above 4.5C
Reviews, theory, combined lines of evidence: Zero for 28 with a stated mean/median above 4.5C
Inferred from GCMs and Constrained with climatology: Two for 41 with a stated mean/median above 4.5C
Constraints from the observed warming in response to forcing: One for 56 with a stated mean/median above 4.5C

So, in total, seven for 151 with a stated mean/median above 4.5C
Six for 151 with a stated mean/median below 1.5C

http://www.nature.com/ngeo/journal/v10/n10/extref/ngeo3017-s2.xlsx

70. Jai,
I don’t think that there is much chance of ECS being as high as you suggest. I think there are indications that the observationally-based estimates are biased low, but that simply brings them into line with the other estimates which suggest that it is probably above 2C and probably below 4.5C.

71. As far as the ECS goes, the TCR-to-ECS ratio is largely set by the rate at which energy is taken up by the ocean. Given how much is being taken up at the moment, it seems unlikely that this ratio can be smaller than 0.6 (and, similarly, unlikely that it is bigger than 0.8). A reasonable estimate might be around 0.7. Therefore, if ECS is as high as 4.5C, then that implies a TCR of about 3C. This is above the IPCC likely range and suggest that some kind of natural influence is suppressing about 50% of the forced warming. Given that we think it extremely unlikely that natural influences can have produced more than 50% of the observed warming, it seems equally unlikely that it could have suppressed 50% of the forced warming.

72. Everett F Sargent says:

Well RealClimate has another post on the Millar paper …
http://www.realclimate.org/index.php/archives/2017/10/1-5oc-geophysically-impossible-or-not/

I’m sort of not believing how high HadCRUT and GISSTEMP are in the upper of the two plots shown above.

73. Willard says:

KevinA’s three points are pure gold:

74. Jai John Mitchell says:

thanks for that Everett,

However, any count that includes Lewis or Curry (or monkton for that matter) is invalid.

http://science.sciencemag.org/content/358/6359/101
WOODS HOLE, Mass. — After 26 years, the world’s longest-running experiment to discover how warming temperatures affect forest soils has revealed a surprising, cyclical response: Soil warming stimulates periods of abundant carbon release from the soil to the atmosphere alternating with periods of no detectable loss in soil carbon stores. **Overall, the results indicate that in a warming world, a self-reinforcing and perhaps uncontrollable carbon feedback will occur between forest soils and the climate system**, adding to the build-up of atmospheric carbon dioxide caused by burning fossil fuels and accelerating global warming. The study, led by Jerry Melillo, Distinguished Scientist at the Marine Biological Laboratory (MBL), appears in the October 6 issue of Science.

75. angech says:

Disappointing that some of my comments have not passed moderation. Perhaps fair in not wanting to offend new posters but it seems that snark is usually directed one way only. The better way to handle this would be to moderate or withdraw such comments on both sides. Or ask me to repost in a better way. Sensitivity indicates nerves being hit.
One problem is that I expect comments, generally, to get through as they usually do. When they don’t and I have spent time and effort trying to get a view point as plainly and clearly expressed as possible it is hard to reproduce a second time. I do now, as a consequence, try to click and save my comments so as to conserve my thoughts. But I forget some times.
My comments on how to assess cumulative warming, sorry Everett I know that is a term used by others and only repeated by you, and expansive ECS, sorry Jai, will be lost to posterity but do not lose their message.
If people wish to make policy statements that is fine. Correcting misunderstandings that are used to push policy should be encouraged in a blog that supports scientific integrity, on both sides.

76. angech,
Not really meant to play the ref, but some of your comments just don’t really make any sense.

77. angech,

Or ask me to repost in a better way. Sensitivity indicates nerves being hit.

Okay, yes, do this. Shorter, more specific, and try to avoid making it seem that you’re simply mocking the person you’re responding to.

78. angech says:

ATTP
Points taken.
At any time when my post is too aggressive or snarky I would prefer it removed anyway than leave it up as a reminder of my lapses.

79. KHome1990 says:

The IPCC graph suggests we hit 2C of warming at about 530 PPM of CO2. But it suggests a relatively low end ECS, only about 2.3C. If ECS is 4-4.5C instead, how much warming do we see concurrent with 530 PPM?

Jai – when you say a central estimate of ECS is 4.7C after 2.0C warning is attained – are you suggesting total ECS is 6.7C? And what fraction of the ECS is likely to be realized on a transient timescale (i.e. when doubling is attained)?

80. KHome1990,

But it suggests a relatively low end ECS, only about 2.3C. If ECS is 4-4.5C instead, how much warming do we see concurrent with 530 PPM?

I think you’re confusing transient and equilibrium responses. What’s (mostly) relevant in these scenarios (as I try to explain in the post) is the transient response, not the equilibrium response.

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