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All IPCC definitions taken from Climate Change 2007: The Physical Science Basis. Working Group I Contribution to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, Annex I, Glossary, pp. 941-954. Cambridge University Press.

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An exponential increase in CO2 will result in a linear increase in temperature

What the science says...

Despite the logarithmic relationship between CO2 and surface temperatures, atmospheric CO2 levels are rising so fast that unless we dramatically decrease our emissions, global warming will accelerate over the 21st Century.

Climate Myth...

An exponential increase in CO2 will result in a linear increase in temperature

There is a logarithmic relationship between radiative forcing (which is directly proportional to the change in surface temperature at equilibrium) and the atmospheric CO2 increase.  Note that we are not currently at equilibrium as there is a planetary energy imbalance, and thus further warming 'in the pipeline' from the carbon we've already emitted.  Therefore, estimates of the rate of warming due to CO2 thus far will will be underestimates, unless accounting for this 'warming in the pipeline'.

This logarithmic relationship means that each doubling of atmospheric CO2 will cause the same amount of warming at the Earth's surface.  Thus if it takes as long to increase atmospheric CO2 from 560 to 1120 parts per million by volume (ppmv) as it did to rise from 280 to 560 ppmv, for example, then the associated warming at the Earth's surface will be roughly linear.  So the question then becomes, how fast do we expect atmospheric CO2 to rise over the next century?

How Fast will Atmospheric CO2 Rise?

The IPCC addressed this question by examining a number of different anthropogenic emissions scenarios.  Scenario A1F1 assumes high global economic growth and continued heavy reliance on fossil fuels for the remainder of the century.  Scenario B1 assumes a major move away from fossil fuels toward alternative and renewable energy as the century progresses.   Scenario A2 is a middling scenario, with less even economic growth and some adoption of alternative and renewable energy sources as the century unfolds.  The projected atmospheric CO2 levels for these scenarios is shown in Figure 1.

Figure 1: Atmospheric CO2 concentrations as observed at Mauna Loa from 1958 to 2008 (black dashed line) and projected under the 6 IPCC emission scenarios (solid coloured lines). (IPCC Data Distribution Centre)

In short, following the 'business as usual' approach without major steps to move away from fossil fuels and limit greenhouse gas emissions, we will likely reach 850 to 950 ppmv of atmospheric CO2 by the year 2100.  It will have taken approximately 200 years (from 1850 to 2050) for the first doubling of atmospheric CO2 from 280 to 560 ppmv, but it will only take another 70 years or so to double the levels again to 1120 ppmv.  This will result in an accelerating rate of global warming, not a linear rate.  Under Scenarios A2 and A1F1, the IPCC report projects that the global temperature in 2095 will be 2.0–6.4°C above 1990 levels (2.6-7.0°C above pre-industrial), with a best estimate of 3.4 and 4.0°C warmer (4.0 and 4.6°C above pre-industrial average surface temperatures), respectively. 

Figure 2: Global surface temperature projections for IPCC Scenarios. Shading denotes the ±1 standard deviation range of individual model annual averages. The orange line is  constant CO2 concentrations at year 2000 values. The grey bars at right indicate the best estimate (solid line within each bar) and the likely range.  (Source: IPCC).

Life in the Fast Lane

Some skeptics have claimed that these projected amounts of warming have not been borne out in the surface temperature changes over the past decade.  But there are many factors which impact short-term global temperatures, which may conceal the long-term warming caused by increasing atmospheric CO2.  So if we want to know if the IPCC projections are realistic, rather than examining noisy short-term temperature data, we should examine how much atmospheric CO2 is increasing.

When we look at this data, we find that observed CO2 emissions in recent years have actually been tracking close to or above the worst case (A1F1) scenario.

Figure 3: Observed global CO2 emissions from fossil fuel burning and cement production compared with IPCC emissions scenarios. The coloured area covers all scenarios used to project climate change by the IPCC (Copenhagen Diagnosis).

What Lies Ahead

So if we continue in a business-as-usual scenario, we should expect to see atmospheric CO2 levels accelerate rapidly enough to more than offset the logarithmic relationship with temperature, and cause the surface temperature warming to accelerate as well.  Cllaims of a linear increase in temperature ignore that in the 'business as usual' scenario, we are currently on pace to double the current atmospheric CO2 concentration (390 to 780 ppmv) within the next 60 to 80 years, and we have not yet even come close to doubling the pre-industrial concentration (280 ppmv) in the past 150 years.  Thus the exponential increase in CO2 will outpace its logarithmic relationship with surface temperature, causing global warming to accelerate unless we take serious steps to reduce greenhouse gas emissions.  In fact, to continue the current rate of warming over the 21st Century, we would need to achieve IPCC scenario B1 - a major move away from fossil fuels toward alternative and renewable energy.

Last updated on 28 January 2011 by dana1981.

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Comments

Comments 1 to 8:

  1. dagold,

    [Answer to dagold's question from here.]

    There are a few answers to your question, each of them playing a part, but I think the biggest answer is simply that warming is not and cannot be expected to be linear. The current rate of warming will increase in some decades, decrease in others, and the final equilibrium temperature of the planet will not necessarily be reached at the same point in time that a doubling is reached... warming will continue beyond that (for how long, it's hard to say)... although, too, since any response to CO2 is logarithmic, in that sense warming will slow, not speed, as we reach a true doubling of atmospheric CO2.

    Take, for example, the Arctic feedback. As the summer ice melts, the open water absorbs rather than reflects incoming sunlight back into space. At the moment, this feedback is minimal because the system has not gained enough energy to melt Arctic ice by enough, soon enough in the spring/summer months, to have that much effect. When that point is reached, however, and accelerates, we can probably expect even faster warming.

    In a nutshell, there are a lot of feedbacks that are not simple and linear. Some will occur in steps (like a sudden increased release of methane gas from bogs and oceans) or wait to be triggered at certain tipping points. Some of these may not occur for decades, others perhaps not even for a century (such as the CO2 feedback from a transition of large swaths of the Amazon rainforest, God forbid, to savanna).
  2. This is an easy misconception to make. However, we must look at CO2 as a function of time, which over the long run is strongly concave up.



    Temperature as a function of time can also be shown to be concave up - rates of change that increase.



    So 0.18 deg C/decade over the past 35 years can be said to be just the tip of the warming iceberg.
  3. The second plot in Muoncounter's post is the first time that I've actually seen the different sensitivities plotted against time, in the context of realised carbon emissions.

    It's interesting to consider that if the effect of 20th century aerosol release is as significant as some suggest, and if it is accounted for in the trajectory, then the actual climate sensitivity would seem to follow closely the path of a 3 degrees sensitivity that is usually given as the most likely response to a doubling of atmospheric carbon dioxide.
  4. BernardJ#3: "20th century aerosol release"

    Nope, the curves are CO2 radiative forcing only, calculated from a best fit to CO2 as a function of time and applying the usual formula.
  5. Hi Muoncounter.

    Yes, I understand the nature of the three curves. The point I was trying to make was that if the real-data plot was distilled to a trajectory representing only the CO2-forced component, would it (or would it not) more closely follow the 3 degree sensitivity, than it does with overlying feedbacks/forcings included?
  6. There's another factor at work here: As temperature rises, CO2 sinks become less and less effective, so the relationship between total CO2 emissions and CO2 concentration is exponential over a scale of decades to centuries.

    This exponential relationship between emissions and concentration essentially cancels the logarithmic relationship between concentration and warming, so that the relationship between total emissions and warming is more or less linear.

    See Matthews et al 2009:

    http://www.nature.com/nature/journal/v459/n7248/abs/nature08047.html

    and more discussion at ClimateSight:

    http://climatesight.org/2012/05/16/cumulative-emissions-and-climate-models/
  7. I was curious to see if global warming trend acceleration was noticeable in the Foster and Rahmstorf Trend Calculator. I used the GISS dataset option and Excel to plot 17 year trend values over 16 years (ie from 1979-1996 to 1994-2011). Here it is:

    Image and video hosting by TinyPic

    Excel's TREND function calculated the increase of these trends to be 0.0579C per decade per decade.

    Another way to think of that acceleration is every 21 months the per decade rate increases by 0.01C

    Extrapolating that rise to a 2012 midpoint indicates that the current rate of warming is around 0.26C per decade, which compared to the often quoted average of 0.17C per decade (over the period 1979-2011) is rather alarming.

    Now I'm just a humble engineer with only a basic grasp of statistics and climate science, so I'll concede that maybe 17 year trends over 16 consecutive years is a bit too short or not enough data to pass a significance test, but the Foster and Rahmstorf datasets are far less noisy than the raw datasets that require 30 year trends.
  8. Turns out GISS gave the highest warming acceleration. Here are my results for all the datasets using the same range and method as before:

    GISS 0.0579C per decade per decade, or +0.01C per decade every 21 months.
    UAH 0.0531C per decade per decade, or +0.01C per decade every 23 months.
    CRU 0.0301C per decade per decade, or +0.01C per decade every 40 months.
    NCDC 0.0273C per decade per decade, or +0.01C per decade every 44 months.
    RSS 0.0062C per decade per decade, or +0.01C per decade every 194 months.

    So while acceleration is apparent, it does look like there is not enough datapoints yet to get a value that all the datasets can agree on.

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