FIGURE 1: PROPORTIONALITY OF CUMULATIVE WARMING TO CUMULATIVE EMISSIONS

FIGURE 2: CUMULATIVE VALUES OF SIGN CONSTRAINED RANDOM NUMBERS

FIGURE 3: UNCONSTRAINED CUMULATIVE VALUES OF RANDOM NUMBERS

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RELATED POSTS 

A MATHEMATICAL INCONSISTENCY

WILL EMISSION REDUCTION CHANGE THE RATE OF WARMING?

[THE CARBON BUDGETS OF CLIMATE SCIENCE]  

[THE REMAINING CARBON BUDGET ANOMALY EXPLAINED]

TCRU: A PARODY OF THE TCRE

THE TCRE CARBON BUDGET

STATISTICAL FLAWS CREATE CLIMATE SCIENCE CONFUSION

THE REMAINING CARBON BUDGET IS A CREATION OF STATISTICAL ERRORS

BACKGROUND INFORMATION

1. The essential theory of climate change that serves as the rationale for the overhaul of the energy infrastructure to renewable and non-fossil fuel sources is that the artificial production of carbon dioxide by humans from previously sequestered carbon deep under the ground causes a perturbation of nature’s delicately balanced carbon cycle and climate system with possibly catastrophic consequences in terms of artificial global warming caused by rising atmospheric CO2 concentration and its GHG forcing of surface temperature (Callendar 1938, Hansen 1981, Lacis 2010).
2. Increasing GHG forcing of surface temperature by atmospheric CO2 is described in terms of its absorption of long wave radiation from the earth’s surface. This mechanism implies a linear relationship between surface temperature and the logarithm of atmospheric CO2 concentration. The regression coefficient of this line is described as a sensitivity (ECS) of temperature to atmospheric CO2 concentration. When it is multiplied by the logarithm of two, it yields the increase in temperature in Celsius units for each doubling of atmospheric carbon dioxide. This relationship is found in climate models that contain the the theoretical mechanisms described and has been estimated as mean ECS = 3C with 90% CI = [1.5-4.5]C (Charney 1979, IPCC 2007).
3. The ECS sensitivity serves as the link between artificial carbon dioxide emissions from fossil fuels and warming with the argument that these emissions cause atmospheric CO2 to rise and that rise in turn causes warming due to the GHG forcing of atmospheric CO2 concentration described by the ECS. However, the search for empirical evidence of the ECS in the observational data generated such a large range of values that the so called “uncertainty problem” placed this line of reasoning in doubt. The uncertainty problem and possible reasons for it are presented in a related post. [ECS: Equilibrium Climate Sensitivity ].
4. The ECS uncertainty issue in climate science severely weakened the rationale for the principal policy implication of climate change research, that of reducing fossil fuel emissions to avoid catastrophic runaway warming by way of GHG forcing. This crisis in climate science created an urgent research agenda to resolve this core issue in the policy implications of climate change.

TCRE TO THE RESCUE 

1. A breakthrough came in 2009 when three papers, all of them published in 2009, proposed an alternate way to connect emissions to warming. All three of them (Matthews, Gregory, Allen) proposed the same new causal connection between emissions and warming.  Here we will describe the Matthews paper. The description applies equally and exactly to all three papers. These papers, and the Matthews paper in particular, created a sensation in climate science as the community of scientists heaved a sigh of relief having re-established empirical evidence of human cause, crucial to the primary agenda of climate change research – that of reducing fossil fuel emissions and thereby moving the energy infrastructure away from fossil fuels.
2. In 2009, Damon Matthews (and co-authors) submitted a paper to the journal Nature with an amazing discovery that could rescue climate science from the climate sensitivity uncertainty problem (Matthews et al 2009). He found a near perfect linear relationship and correlation between cumulative fossil fuel emissions and surface temperature and proposed that the OLS linear regression coefficient of this line, expressed in Celsius units of warming per TTC (teratonnes of carbon equivalent) of cumulative emissions serves as an alternate measure of the effect of emissions on warming and a new rationale for attenuating warming by cutting fossil fuel emissions. The coefficient was named Transient Climate Response to Cumulative Emissions and was christened with the acronym TCRE and it quickly came to be seen as the replacement for the failed ECS parameter.
3. The TCRE became a sensational success. In accepting the (Matthews, 2009) paper for publication the editor of Nature wrote a congratulatory editorial comment (here he uses the acronym CCR (Carbon Climate Response) for TCRE). “To date, efforts to describe and predict the climate response to human CO2 emissions have focused on climate sensitivity: the equilibrium temperature change associated with a doubling of CO2. But recent research has suggested that this ‘Charney’ sensitivity, so named after the meteorologist Jule Charney who first adopted this approach in 1979, may be an incomplete representation of the full Earth system response, as it ignores changes in the carbon cycle, aerosols, land use and land cover. Matthews et al. propose a new measure, the carbon-climate response, or CCR. Using a combination of a simplified climate model, a range of simulations from a recent model inter-comparison, and historical constraints, they find that independent of the timing of emissions or the atmospheric concentration of CO2 emitting a trillion tonnes of carbon will cause 1.0 – 2.1 C of global warming, a CCR value that is consistent with model predictions for the twenty-first century.”
4. The success of the TCRE as a foundational concept in relating climate change to emissions and thereby in developing carbon emission budgets for 2ºC and 1.5ºC, and their appropriate climate action plan and emission guidelines, led to its adoption by the IPCC in its AR5 report and also in its emergency call to the 1.5ºC carbon budget in the SR15 report of 2018.
5. The initial value of 90%CI TCRE = [1.0 – 2.1]ºC per teratonne of cumulative emissions reported by Matthews was found to be very stable with later works reporting values within a narrow range of TCRE = [0.75-2.5] quite unlike the large range of ECS values listed in a related post [The Greenhouse Effect of Atmospheric CO2]. The achievement was heralded by climate scientists around the world with a 2017 paper by converted ECS advocate Reto Knutti calling for discarding the ECS altogether and replacing it with the strong and stable TCRE (Knutti 2017). In the paper titled “Beyond Equilibrium Climate Sensitivity”, he wrote “Estimates of the transient climate response are better constrained by observed warming and are more relevant for predicting warming over the next decades. Newer metrics relating global warming directly to the total emitted CO₂ show that in order to keep warming to within 2 °C, future CO₂ emissions have to remain strongly limited, irrespective of climate sensitivity being at the high or low end”.

STATISTICS OF THE TCRE

1. A statistical detail overlooked in the rush to TCRE as a replacement for the failed ECS parameter is that correlations between cumulative values of time series data are spurious.
2. Since temperature is cumulative warming, the TCRE derives from a spurious correlation between cumulative values. A regression coefficient that depends on a spurious correlation does not have an interpretation because a time series of cumulative values of another time series contains neither time scale not degrees of freedom as explained in this related post [LINK] .
3. Figure 1 above shows the strong positive correlation between cumulative emissions and cumulative warming used by climate science and by the IPCC to track the effect of emissions on temperature and to derive the “carbon budget” for various acceptable targets of warming such as 2C and 1.5C.
4. These so called carbon budgets then serve as policy tools for international climate action agreements and climate action imperatives of the United Nations. And yet, all such budgets are numbers with no interpretation in the real world because they are derived from spurious correlations.
5. Figure 2 is a video of the correlation between random emissions and random warming (in the left frame) and the correlation between their cumulative values (in the right frame). In this demonstration, the random numbers fit the sign pattern seen in emissions that are always positive, and warming which are mostly positive.
6. When this pattern is enforced a strong correlation is seen between the two series of the cumulative values of random numbers – as in the TCRE papers listed below. Under these conditions, a statistically significant and stable TCRE regression coefficient can be computed from the data but this coefficient has no interpretation because its only information content is that the source data follow similar sign patterns such that emissions are always positive and annual warming are mostly positive.
7. Figure 3 shows that when this sign convention is not enforced, the correlation falls apart and provides strong support for the interpretation of strong correlations between cumulative  values in terms of sign patterns and not in terms of a real relationship between the the variables being studied. Some of the frames in the Figure 3 video will show a positive correlation when a sign pattern forms by chance in unconstrained random number generation.
8. The role of the sign pattern is demonstrated in a related post where it is shown that any variable with that sign pattern, even UFO sightings, work just as well as fossil fuel emissions. The only thing fossil fuel emissions and UFO sightings have in common is that there values are always positive. [LINK}
9. Yet another issue with the TCRE is a mathematical inconsistency with the theory of anthropogenic global warming in terms of the Greenhouse Effect of atmospheric CO2 described in a related post [LINK]. Briefly, the Greenhouse Effect implies a logarithmic relationship between cumulative emissions and warming. The linear relationship of the TCRE is inconsistent with the theory for which it claims to serve as empirical evidence.

CONCLUSION

We conclude from this analysis that the only information content of strong correlations between cumulative values of time series data is that they happen to follow certain sign patterns. The further interpretation of these correlations and regression coefficients in terms of human cause of warming and in terms of carbon budgets for 1.5C and 2C is not possible. Climate science has fallen afoul of fundamental statistical considerations in the use of the specious TCRE metric not only to validate cause and effect in natural phenomena but also as a policy tool for setting carbon budgets.  A more fundamental issue with regard to the TCRE is that it is unrelated to the climate science theory that relates warming to emissions. The theory of anthropogenic global warming is a causation sequence from fossil fuel emissions to rising atmospheric CO2 concentration and from there by way of the greenhouse effect of atmospheric CO2 to higher temperatures. There is no role for a TCRE parameter in this theory as highlighted in its mathematical inconsistency in a related post: [LINK] . The statistical issue with respect to only positive values for emissions is demonstrated in a related post where it is shown that not just emissions but any variable with only positive values works just as well, [LINK]

A YOUTUBE VIDEO DEMONSTRATION OF THE SPURIOUSNESS OF CORRELATIONS BETWEEN CUMULATIVE  VALUES OF TIME SERIES DATA. The red lines show correlations between the source data. The blue lines show correlations between their cumulative values. In the left panel the data are unrelated. In the right panel various degrees of correlation were inserted into the data.

TCRE BIBLIOGRAPHY

BACKGROUND PAPERS

1. 1938: Callendar, Guy Stewart. “The artificial production of carbon dioxide and its influence on temperature.” Quarterly Journal of the Royal Meteorological Society 64.275 (1938): 223-240. By fuel combustion man has added about 150,000 million tons of carbon dioxide to the air during the past half century. The author estimates from the best available data that approximately three quarters of this has remained in the atmosphere. The radiation absorption coefficients of carbon dioxide and water vapor are used to show the effect of carbon dioxide on “sky radiation.” From this the increase in mean temperature, due to the artificial production of carbon dioxide, is estimated to be at the rate of 0.003°C. per year at the present time. The temperature observations a t zoo meteorological stations are used to show that world temperatures have actually increased at an average rate of 0.005°C. per year during the past half century.
2. 1979: Charney, Jule G., et al. Carbon dioxide and climate: a scientific assessment. National Academy of Sciences, Washington, DC, 1979. Incontrovertible evidence’ human activity is changing the atmosphere. A wait-and-see policy means waiting until it is too late. When will CO2 double? 2030—2050. All models mutually supporting: 5 of 5 models predict warming. Upper bound from H1: +3.5° (Over-estimation of water vapor feedback) Lower bound from M series: +2° (Under-estimation of water vapor feedback) Best estimate of surface temperature change? +3° degrees (probable error of ±1.5°) A historical document: Jules Charney and Jimmy Carter. Conclusions are comforting for scientists and disturbing for policy makers. Conclusions have generally held: +2.6°—4.1° clustering around +3°C.
3. 1981: Hansen, James, et al. “Climate impact of increasing atmospheric carbon dioxide.” Science 213.4511 (1981): 957-966. The global temperature rose by 0.2°C between the middle 1960’s and 1980, yielding a warming of 0.4°C in the past century. This temperature increase is consistent with the calculated greenhouse effect due to measured increases of atmospheric carbon dioxide. Variations of volcanic aerosols and possibly solar luminosity appear to be primary causes of observed fluctuations about the mean trend of increasing temperature. It is shown that the anthropogenic carbon dioxide warming should emerge from the noise level of natural climate variability by the end of the century, and there is a high probability of warming in the 1980’s. Potential effects on climate in the 21st century include the creation of drought-prone regions in North America and central Asia as part of a shifting of climatic zones, erosion of the West Antarctic ice sheet with a consequent worldwide rise in sea level, and opening of the fabled Northwest Passage.
4. 2007: Change, Intergovernmental Panel On Climate. “Climate change 2007: The physical science basis.” Agenda 6.07 (2007): 333. Changes in the atmospheric abundance of greenhouse gases and aerosols, in solar radiation and in land surface properties alter the energy balance of the climate system. These changes are expressed in terms of radiative forcing which is used to compare how a range of human and natural factors drive warming or cooling influences on global climate. Since the Third Assessment Report (TAR), new observations and related modelling of greenhouse gases, solar activity, land surface properties and some aspects of aerosols have led to improvements in the quantitative estimates of radiative forcing.

TCRE PAPERS

1. 2009: Matthews, H. Damon, et al. “The proportionality of global warming to cumulative carbon emissions.” Nature 459.7248 (2009): 829.  ABSTRACT: The global temperature response to increasing atmospheric CO2 is often quantified by metrics such as equilibrium climate sensitivity and transient climate response1. These approaches, however, do not account for carbon cycle feedbacks and therefore do not fully represent the net response of the Earth system to anthropogenic CO2 emissions. Climate–carbon modelling experiments have shown that: (1) the warming per unit CO2 emitted does not depend on the background CO2 concentration2; (2) the total allowable emissions for climate stabilization do not depend on the timing of those emissions3,4,5; and (3) the temperature response to a pulse of CO2 is approximately constant on timescales of decades to centuries3,6,7,8. Here we generalize these results and show that the carbon–climate response (CCR), defined as the ratio of temperature change to cumulative carbon emissions, is approximately independent of both the atmospheric CO2 concentration and its rate of change on these timescales. From observational constraints, we estimate CCR to be in the range 1.0–2.1 °C per trillion tonnes of carbon (Tt C) emitted (5th to 95th percentiles), consistent with twenty-first-century CCR values simulated by climate–carbon models. Uncertainty in land-use CO2 emissions and aerosol forcing, however, means that higher observationally constrained values cannot be excluded. The CCR, when evaluated from climate–carbon models under idealized conditions, represents a simple yet robust metric for comparing models, which aggregates both climate feedbacks and carbon cycle feedbacks. CCR is also likely to be a useful concept for climate change mitigation and policy; by combining the uncertainties associated with climate sensitivity, carbon sinks and climate–carbon feedbacks into a single quantity, the CCR allows CO2-induced global mean temperature change to be inferred directly from cumulative carbon emissions.
2. 2010: Lacis, Andrew A., et al. “Atmospheric CO2: Principal control knob governing Earth’s temperature.” Science 330.6002 (2010): 356-359. Ample physical evidence shows that carbon dioxide (CO₂) is the single most important climate-relevant greenhouse gas in Earth’s atmosphere. This is because CO₂, like ozone, N₂O, CH₄, and chlorofluorocarbons, does not condense and precipitate from the atmosphere at current climate temperatures, whereas water vapor can and does. Noncondensing greenhouse gases, which account for 25% of the total terrestrial greenhouse effect, thus serve to provide the stable temperature structure that sustains the current levels of atmospheric water vapor and clouds via feedback processes that account for the remaining 75% of the greenhouse effect. Without the radiative forcing supplied by CO₂ and the other noncondensing greenhouse gases, the terrestrial greenhouse would collapse, plunging the global climate into an icebound Earth state.
3. 2011: Raupach, Michael R., et al. “The relationship between peak warming and cumulative CO2 emissions, and its use to quantify vulnerabilities in the carbon-climate-human system.” Tellus B: Chemical and Physical Meteorology 63.2 (2011): 145-164. Interactions between the carbon cycle, climate and human societies are subject to several major vulnerabilities, broadly defined as factors contributing to the risk of harm from human-induced climate change. We assess five vulnerabilities: (1) effects of increasing CO₂ on the partition of anthropogenic carbon between atmospheric, land and ocean reservoirs; (2) effects of climate change (quantified by temperature) on CO₂ fluxes; (3) uncertainty in climate sensitivity; (4) non-CO₂radiative forcing and (5) anthropogenic CO₂ emissions. Our analysis uses a physically based expression for T_p(Q_p), the peak warming T_p associated with a cumulative anthropogenic CO₂ emission Q_p to the time of peak warming. The approximations in this expression are evaluated using a non-linear box model of the carbon-climate system, forced with capped emissions trajectories described by an analytic form satisfying integral and smoothness constraints. The first four vulnerabilities appear as parameters that influence T_p(Q_p), whereas the last appears through the independent variable. In terms of likely implications for T_p(Q_p), the decreasing order of the first four vulnerabilities is: uncertainties in climate sensitivity, effects of non-CO₂ radiative forcing, effects of climate change on CO₂fluxes and effects of increasing CO₂ on the partition of anthropogenic carbon.
4. 2013: Gillett, Nathan P., et al. “Constraining the ratio of global warming to cumulative CO2 emissions using CMIP5 simulations.” Journal of Climate 26.18 (2013): 6844-6858. The ratio of warming to cumulative emissions of carbon dioxide has been shown to be approximately independent of time and emissions scenarios and directly relates emissions to temperature. It is therefore a potentially important tool for climate mitigation policy. The transient climate response to cumulative carbon emissions (TCRE), defined as the ratio of global-mean warming to cumulative emissions at CO₂doubling in a 1% yr⁻¹ CO₂ increase experiment, ranges from 0.8 to 2.4 K EgC⁻¹ in 15 models from phase 5 of the Coupled Model Intercomparison Project (CMIP5)—a somewhat broader range than that found in a previous generation of carbon–climate models. Using newly available simulations and a new observational temperature dataset to 2010, TCRE is estimated from observations by dividing an observationally constrained estimate of CO₂-attributable warming by an estimate of cumulative carbon emissions to date, yielding an observationally constrained 5%–95% range of 0.7–2.0 K EgC⁻¹.
5. 2014: Raupach, Michael R., et al. “Sharing a quota on cumulative carbon emissions.” Nature Climate Change 4.10 (2014): 873.  Any limit on future global warming is associated with a quota on cumulative global CO₂ emissions. We translate this global carbon quota to regional and national scales, on a spectrum of sharing principles that extends from continuation of the present distribution of emissions to an equal per-capita distribution of cumulative emissions. A blend of these endpoints emerges as the most viable option. For a carbon quota consistent with a 2 °C warming limit (relative to pre-industrial levels), the necessary long-term mitigation rates are very challenging (typically over 5% per year), both because of strong limits on future emissions from the global carbon quota and also the likely short-term persistence in emissions growth in many regions.
6. 2014: Friedlingstein, Pierre, et al. “Persistent growth of CO 2 emissions and implications for reaching climate targets.” Nature geoscience 7.10 (2014): 709. Efforts to limit climate change below a given temperature level require that global emissions of CO₂ cumulated over time remain below a limited quota. This quota varies depending on the temperature level, the desired probability of staying below this level and the contributions of other gases. In spite of this restriction, global emissions of CO₂ from fossil fuel combustion and cement production have continued to grow by 2.5% per year on average over the past decade. Two thirds of the CO₂ emission quota consistent with a 2 °C temperature limit has already been used, and the total quota will likely be exhausted in a further 30 years at the 2014 emissions rates. We show that CO₂emissions track the high end of the latest generation of emissions scenarios, due to lower than anticipated carbon intensity improvements of emerging economies and higher global gross domestic product growth. In the absence of more stringent mitigation, these trends are set to continue and further reduce the remaining quota until the onset of a potential new climate agreement in 2020. Breaking current emission trends in the short term is key to retaining credible climate targets within a rapidly diminishing emission quota.

1. 2014: Allen, Myles R., and Thomas F. Stocker. “Impact of delay in reducing carbon dioxide emissions.” Nature Climate Change4.1 (2014): 23. Recent downward revisions in the climate response to rising CO₂ levels, and opportunities for reducing non-CO₂ climate warming, have both been cited as evidence that the case for reducing CO₂ emissions is less urgent than previously thought. Evaluating the impact of delay is complicated by the fact that CO₂ emissions accumulate over time, so what happens after they peak is as relevant for long-term warming as the size and timing of the peak itself. Previous discussions have focused on how the rate of reduction required to meet any given temperature target rises asymptotically the later the emissions peak. Here we focus on a complementary question: how fast is peak CO₂-induced warming increasing while mitigation is delayed, assuming no increase in rates of reduction after the emissions peak? We show that this peak-committed warming is increasing at the same rate as cumulative CO₂ emissions, about 2% per year, much faster than observed warming, independent of the climate response.
2. 2014: Herrington, T., and K. Zickfeld. “Path independence of climate and carbon cycle response over a broad range of cumulative carbon emissions.” Earth System Dynamics 5.2 (2014): 409-422. Recent studies have identified an approximately proportional relationship between global warming and cumulative carbon emissions, yet the robustness of this relationship has not been tested over a broad range of cumulative emissions and emission rates. This study explores the path dependence of the climate and carbon cycle response using an Earth system model of intermediate complexity forced with 24 idealized emissions scenarios across five cumulative emission groups (1275–5275 Gt C) with varying rates of emission. We find the century-scale climate and carbon cycle response after cessation of emissions to be approximately independent of emission pathway for all cumulative emission levels considered. The ratio of global mean temperature change to cumulative emissions – referred to as the transient climate response to cumulative carbon emissions (TCRE) – is found to be constant for cumulative emissions lower than ∼1500 Gt C but to decline with higher cumulative emissions. The TCRE is also found to decrease with increasing emission rate. The response of Arctic sea ice is found to be approximately proportional to cumulative emissions, while the response of the Atlantic Meridional Overturning Circulation does not scale linearly with cumulative emissions, as its peak response is strongly dependent on emission rate. Ocean carbon uptake weakens with increasing cumulative emissions, while land carbon uptake displays non-monotonic behavior, increasing up to a cumulative emission threshold of ∼2000 Gt C and then declining.
3. 2014: Krasting, J. P., et al. “Trajectory sensitivity of the transient climate response to cumulative carbon emissions.” Geophysical Research Letters 41.7 (2014): 2520-2527. The robustness of Transient Climate Response to cumulative Emissions (TCRE) is tested using an Earth System Model (Geophysical Fluid Dynamics Laboratory‐ESM2G) forced with seven different constant rates of carbon emissions (2 GtC/yr to 25 GtC/yr), including low emission rates that have been largely unexplored in previous studies. We find the range of TCRE resulting from varying emission pathways to be 0.76 to 1.04°C/TtC. This range, however, is small compared to the uncertainty resulting from varying model physics across the Fifth Coupled Model Intercomparison Project ensemble. TCRE has a complex relationship with emission rates; TCRE is largest for both low (2 GtC/yr) and high (25 GtC/yr) emissions and smallest for present‐day emissions (5–10 GtC/yr). Unforced climate variability hinders precise estimates of TCRE for periods shorter than 50 years for emission rates near or smaller than present day values. Even if carbon emissions would stop, the prior emissions pathways will affect the future climate responses.
4. 2015: MacDougall, Andrew H., and Pierre Friedlingstein. “The origin and limits of the near proportionality between climate warming and cumulative CO2 emissions.” Journal of Climate 28.10 (2015): 4217-4230. The transient climate response to cumulative CO₂ emissions (TCRE) is a useful metric of climate warming that directly relates the cause of climate change (cumulative carbon emissions) to the most used index of climate change (global mean near-surface temperature change). In this paper, analytical reasoning is used to investigate why TCRE is near constant over a range of cumulative emissions up to 2000 Pg of carbon. In addition, a climate model of intermediate complexity, forced with a constant flux of CO₂ emissions, is used to explore the effect of terrestrial carbon cycle feedback strength on TCRE. The analysis reveals that TCRE emerges from the diminishing radiative forcing from CO₂ per unit mass being compensated for by the diminishing ability of the ocean to take up heat and carbon. The relationship is maintained as long as the ocean uptake of carbon, which is simulated to be a function of the CO₂ emissions rate, dominates changes in the airborne fraction of carbon. Strong terrestrial carbon cycle feedbacks have a dependence on the rate of carbon emission and, when present, lead to TRCE becoming rate dependent. Despite these feedbacks, TCRE remains roughly constant over the range of the representative concentration pathways and therefore maintains its primary utility as a metric of climate change.
5. 2015″ Goodwin, Philip, Richard G. Williams, and Andy Ridgwell. “Sensitivity of climate to cumulative carbon emissions due to compensation of ocean heat and carbon uptake.” Nature Geoscience 8.1 (2015): 29. Climate model experiments reveal that transient global warming is nearly proportional to cumulative carbon emissions on multi-decadal to centennial timescales^1,2,3,4,5. However, it is not quantitatively understood how this near-linear dependence between warming and cumulative carbon emissions arises in transient climate simulations^6,7. Here, we present a theoretically derived equation of the dependence of global warming on cumulative carbon emissions over time. For an atmosphere–ocean system, our analysis identifies a surface warming response to cumulative carbon emissions of 1.5 ± 0.7 K for every 1,000 Pg of carbon emitted. This surface warming response is reduced by typically 10–20% by the end of the century and beyond. The climate response remains nearly constant on multi-decadal to centennial timescales as a result of partially opposing effects of oceanic uptake of heat and carbon⁸. The resulting warming then becomes proportional to cumulative carbon emissions after many centuries, as noted earlier⁹. When we incorporate estimates of terrestrial carbon uptake¹⁰, the surface warming response is reduced to 1.1 ± 0.5 K for every 1,000 Pg of carbon emitted, but this modification is unlikely to significantly affect how the climate response changes over time. We suggest that our theoretical framework may be used to diagnose the global warming response in climate models and mechanistically understand the differences between their projections.
6. 2016: MacDougall, Andrew H. “The transient response to cumulative CO 2 emissions: a review.” Current Climate Change Reports2.1 (2016): 39-47. The transient climate response to cumulative CO₂ emissions (TCRE) is a metric of climate change that directly relates the primary cause of climate change (cumulative CO₂ emissions) to global mean temperature change. The metric was developed once researchers noticed that the cumulative CO₂ versus temperature change curve was nearly linear for almost all Earth system model output. Here, recent literature on the origin, limits, and value of TCRE is reviewed. TCRE appears to emerge from the diminishing radiative forcing per unit mass of atmospheric CO₂ being compensated by diminishing efficiency of ocean heat uptake and the modulation of airborne fraction of carbon by ocean processes. The best estimate of the value of TCRE is between 0.8 to 2.5 K EgC⁻¹. Overall, TCRE has been shown to be a conceptually simple and robust metric of climate warming with many applications in formulating climate policy.
7. 2016: Leduc, Martin, H. Damon Matthews, and Ramón de Elía. “Regional estimates of the transient climate response to cumulative CO 2 emissions.” Nature Climate Change 6.5 (2016): 474. The Transient Climate Response to cumulative carbon Emissions (TCRE) measures the response of global temperatures to cumulative CO₂emissions^1,2,3,4. Although the TCRE is a global quantity, climate impacts manifest predominantly in response to local climate changes. Here we quantify the link between CO₂ emissions and regional temperature change, showing that regional temperatures also respond approximately linearly to cumulative CO₂ emissions. Using an ensemble of twelve Earth system models, we present a novel application of pattern scaling^5,6 to define the regional pattern of temperature change per emission of CO₂. Ensemble mean regional TCRE values range from less than 1 °C per TtC for some ocean regions, to more than 5 °C per TtC in the Arctic, with a pattern of higher values over land and at high northern latitudes. We find also that high-latitude ocean regions deviate more strongly from linearity as compared to land and lower-latitude oceans. This suggests that ice-albedo and ocean circulation feedbacks are important contributors to the overall negative deviation from linearity of the global temperature response to high levels of cumulative emissions. The strong linearity of the regional climate response over most land regions provides a robust way to quantitatively link anthropogenic CO₂ emissions to local-scale climate impacts.
8. 2017: Ehlert, Dana, et al.”The sensitivity of the proportionality between temperature change and cumulative CO2 emissions to ocean mixing.” Journal of Climate 30.8 (2017): 2921-2935. The ratio of global mean surface air temperature change to cumulative CO₂ emissions, referred to as transient climate response to cumulative CO₂ emissions (TCRE), has been shown to be approximately constant on centennial time scales. The mechanisms behind this constancy are not well understood, but previous studies suggest that compensating effects of ocean heat and carbon fluxes, which are governed by the same ocean mixing processes, could be one cause for this approximate constancy. This hypothesis is investigated by forcing different versions of the University of Victoria Earth System Climate Model, which differ in the ocean mixing parameterization, with an idealized scenario of 1% annually increasing atmospheric CO₂ until quadrupling of the preindustrial CO₂ concentration and constant concentration thereafter. The relationship between surface air warming and cumulative emissions remains close to linear, but the TCRE varies between model versions, spanning the range of 1.2°–2.1°C EgC⁻¹ at the time of CO₂ doubling. For all model versions, the TCRE is not constant over time while atmospheric CO₂ concentrations increase. It is constant after atmospheric CO₂ stabilizes at 1120 ppm, because of compensating changes in temperature sensitivity (temperature change per unit radiative forcing) and cumulative airborne fraction. The TCRE remains approximately constant over time even if temperature sensitivity, determined by ocean heat flux, and cumulative airborne fraction, determined by ocean carbon flux, are taken from different model versions with different ocean mixing settings. This can partially be explained with temperature sensitivity and cumulative airborne fraction following similar trajectories, which suggests ocean heat and carbon fluxes scale approximately linearly with changes in vertical mixing.
9. 2017: Millar, Richard J., et al. “Emission budgets and pathways consistent with limiting warming to 1.5 C.” Nature Geoscience10.10 (2017): 741. The Paris Agreement has opened debate on whether limiting warming to 1.5 °C is compatible with current emission pledges and warming of about 0.9 °C from the mid-nineteenth century to the present decade. We show that limiting cumulative post-2015 CO₂ emissions to about 200 GtC would limit post-2015 warming to less than 0.6 °C in 66% of Earth system model members of the CMIP5 ensemble with no mitigation of other climate drivers. We combine a simple climate–carbon-cycle model with estimated ranges for key climate system properties from the IPCC Fifth Assessment Report. Assuming emissions peak and decline to below current levels by 2030, and continue thereafter on a much steeper decline, which would be historically unprecedented but consistent with a standard ambitious mitigation scenario (RCP2.6), results in a likely range of peak warming of 1.2–2.0 °C above the mid-nineteenth century. If CO₂ emissions are continuously adjusted over time to limit 2100 warming to 1.5 °C, with ambitious non-CO₂ mitigation, net future cumulative CO₂ emissions are unlikely to prove less than 250 GtC and unlikely greater than 540 GtC. Hence, limiting warming to 1.5 °C is not yet a geophysical impossibility, but is likely to require delivery on strengthened pledges for 2030 followed by challengingly deep and rapid mitigation. Strengthening near-term emissions reductions would hedge against a high climate response or subsequent reduction rates proving economically, technically or politically unfeasible.
10. 2017: MacDougall, Andrew H., Neil C. Swart, and Reto Knutti. “The uncertainty in the transient climate response to cumulative CO2 emissions arising from the uncertainty in physical climate parameters.” Journal of Climate 30.2 (2017): 813-827. An emergent property of most Earth system models is a near-linear relationship between cumulative emission of CO₂ and change in global near-surface temperature. This relationship, which has been named the transient climate response to cumulative CO₂ emissions (TCRE), implies a finite budget of fossil fuel carbon that can be burnt over all time consistent with a chosen temperature change target. Carbon budgets are inversely proportional to the value of TCRE and are therefore sensitive to the uncertainty in TCRE. Here the authors have used a perturbed physics approach with an Earth system model of intermediate complexity to assess the uncertainty in the TCRE that arises from uncertainty in the rate of transient temperature change and the effect of this uncertainty on carbon cycle feedbacks. The experiments are conducted using an idealized 1% yr⁻¹ increase in CO₂ concentration. Additionally, the authors have emulated the temperature output of 23 models from phase 5 of the Climate Model Intercomparison Project (CMIP5). The experiment yields a mean value for TCRE of 1.72 K EgC⁻¹ with a 5th to 95th percentile range of 0.88 to 2.52 K EgC⁻¹. This range of uncertainty is consistent with the likely range from the Fifth Assessment Report of the Intergovernmental Panel on Climate Change (0.8 to 2.5 K EgC⁻¹) but by construction underestimates the total uncertainty range of TCRE, as the authors’ experiments cannot account for the uncertainty from their models’ imperfect representation of the global carbon cycle. Transient temperature change uncertainty induces a 5th to 95th percentile range in the airborne fraction at the time of doubled atmospheric CO₂ of 0.50 to 0.58. Overall the uncertainty in the value of TCRE remains considerable.
11. 2017: Turner, Katherine, Ric Williams, and Andreas Oschlies. “Scenario dependency of the transient climate response to cumulative emissions.” EGU General Assembly Conference Abstracts. Vol. 19. 2017. The transient climate response to emissions (TCRE), in relating surface temperature changes to cumulative carbon emissions, provides a means of estimating carbon budgets from global warming benchmarks. Current Earth System Model results indicate that the TCRE is linear and scenario-independent. We explore the sensitivity of the TCRE to scenario and model parameter uncertainties using 8 configurations of the UVic Earth System Model of Intermediate Complexity, forced by 2 twenty-first-century emissions scenarios (RCP 4.5 and 8.5). We find that the TCRE is higher under RCP 4.5 than 8.5 by 0.3-0.8 K/1000 Pg C and shows opposing nonlinear tendencies in these scenarios: an increase of 0.15-0.5 K/1000 Pg C over RCP 4.5 and a decrease of 0-0.7 K/1000 Pg C over RCP 8.5. These differences are robust across model configurations with perturbed land and ocean parametrizations and are the result of the decreased efficiency of heat transport into the deep ocean under decelerating emissions.
12. 2017: Munshi, Jamal. “Limitations of the TCRE: Transient Climate Response to Cumulative Emissions.” (2017). Observed correlations between cumulative emissions and cumulative changes in climate variables form the basis of the Transient Climate Response to Cumulative Emissions (TCRE) function. The TCRE is used to make forecasts of future climate scenarios based on different emission pathways and thereby to derive their policy implications for climate action. Inaccuracies in these forecasts likely derive from a statistical weakness in the methodology used. The limitations of the TCRE are related to its reliance on correlations between cumulative values of time series data. Time series of cumulative values contain neither time scale nor degrees of freedom. Their correlations are spurious. No conclusions may be drawn from them.  [LINK]  
13. 2017: Knutti, Reto, Maria AA Rugenstein, and Gabriele C. Hegerl. “Beyond equilibrium climate sensitivity.” Nature Geoscience10.10 (2017): 727. Equilibrium climate sensitivity characterizes the Earth’s long-term global temperature response to increased atmospheric CO₂concentration. It has reached almost iconic status as the single number that describes how severe climate change will be. The consensus on the ‘likely’ range for climate sensitivity of 1.5 °C to 4.5 °C today is the same as given by Jule Charney in 1979, but now it is based on quantitative evidence from across the climate system and throughout climate history. The quest to constrain climate sensitivity has revealed important insights into the timescales of the climate system response, natural variability and limitations in observations and climate models, but also concerns about the simple concepts underlying climate sensitivity and radiative forcing, which opens avenues to better understand and constrain the climate response to forcing. Estimates of the transient climate response are better constrained by observed warming and are more relevant for predicting warming over the next decades. Newer metrics relating global warming directly to the total emitted CO₂ show that in order to keep warming to within 2 °C, future CO₂ emissions have to remain strongly limited, irrespective of climate sensitivity being at the high or low end.
14. 2018: Millar, Richard J., and Pierre Friedlingstein. “The utility of the historical record for assessing the transient climate response to cumulative emissions.” Phil. Trans. R. Soc. A 376.2119 (2018): 20160449. The historical observational record offers a way to constrain the relationship between cumulative carbon dioxide emissions and global mean warming. We use a standard detection and attribution technique, along with observational uncertainties to estimate the all-forcing or ‘effective’ transient climate response to cumulative emissions (TCRE) from the observational record. Accounting for observational uncertainty and uncertainty in historical non-CO₂ radiative forcing gives a best-estimate from the historical record of 1.84°C/TtC (1.43–2.37°C/TtC 5–95% uncertainty) for the effective TCRE and 1.31°C/TtC (0.88–2.60°C/TtC 5–95% uncertainty) for the CO₂-only TCRE. While the best-estimate TCRE lies in the lower half of the IPCC likely range, the high upper bound is associated with the not-ruled-out possibility of a strongly negative aerosol forcing. Earth System Models have a higher effective TCRE range when compared like-for-like with the observations over the historical period, associated in part with a slight underestimate of diagnosed cumulative emissions relative to the observational best-estimate, a larger ensemble mean-simulated CO₂-induced warming, and rapid post-2000 non-CO₂ warming in some ensemble members.
15. 2018: Sokolov, Andrei, et al. “Evaluation of transient response of climate system based on the distribution of climate system parameters constrained by observed climate change.” EGU General Assembly Conference Abstracts. Vol. 20. 2018. Transient climate response (TCR, i.e., temperature change in the time of CO2 doubling) and transient climate response to cumulative carbon emission (TCRE, defined as the ratio of surface warming to cumulative implied carbon emissions at the time of CO2 doubling) are often used to quantify climate system response to a non-stationary forcing. TCR and TCRE are not directly observable characteristics of the climate system and their available estimates are obtained using results of simulations with climate models and, in case of TCRE, estimates of historical carbon emissions. In this study, we present estimates for TCR and TCRE obtained in the simulations with the MIT Earth System Model of intermediate complexity (MESM). First, the MESM was used to create a joint probability distribution for climate system parameters that define climate system response to the external forcing (e.g. climate sensitivity and rate of ocean heat uptake). This distribution was calculated by comparing results from a large ensemble of historical MESM simulations with available observations for changes in surface air temperature and oceanic heat content. To evaluate the estimated distribution, we carried out an ensemble of historical (1861-2010) simulations using 400 samples of climate parameters and examined where observations appeared within the distribution. Distributions for TCR and TCRE were calculated from an ensemble of 400 runs in which MESM was forced by increasing CO2 concentration. In our simulations, the median value of TCR (1.7K) is close to that of the CMIP5 models (1.8K). Simultaneously, the 90% probability range of TCR (1.4 – 2.0K) is significantly narrower than estimates based on CMIP5 models (1.2 – 2.4K). The relatively narrow range of TCR in our simulations is explained, in part, by the correlation between climate sensitivity and the rate of oceanic heat uptake imposed by observations. In the MESM simulations, the values of TCRE vary (90% range) from 1.3 to 2.0 K/ EgC, a similar range from 1% per year CO2 increase experiment with CMIP5 models is 0.8-2.4K/EgC. At the same time an observationally constrained 5%-95% range, obtained by Gillett et al. (2013), using CMIP5 simulations and observed temperature is 0.7-2.0K/ EgC. We also present results on dependency of TCR and TCRE on the rate of CO2 increase.
16. 2018: Katavouta, Anna, et al. “Reconciling Atmospheric and Oceanic Views of the Transient Climate Response to Emissions.” Geophysical Research Letters (2018). The Transient Climate Response to Emissions (TCRE), the ratio of surface warming and cumulative carbon emissions, is controlled by a product of thermal and carbon contributions. The carbon contribution involves the airborne fraction and the ratio of ocean saturated and atmospheric carbon inventories, with this ratio controlled by ocean carbonate chemistry. The evolution of the carbon contribution to the TCRE is illustrated in a hierarchy of models: a box model of the atmosphere‐ocean and an Earth system model, both integrated for 1,000 years, and a suite of Earth system models integrated for 140 years. For all models, there is the same generic carbonate chemistry response: An acidifying ocean during emissions leads to a decrease in the ratio of the ocean saturated and atmospheric carbon inventories and the carbon contribution to the TCRE. Hence, ocean carbonate chemistry is important in controlling the magnitude of the TCRE and its evolution in time. Plain Language Summary: The increase in surface temperature with the amount of carbon emitted to the atmosphere depends on the uptake and storage of heat and carbon. Ocean heat uptake acts to strengthen surface warming, as the ocean becomes more stratified in time. Carbon uptake by the ocean and terrestrial system acts to weaken surface warming by removing carbon from the atmosphere. The proportionality of surface warming to carbon emissions may be written in terms of a thermal contribution multiplied by a carbon contribution. The carbon contribution depends on the increase in the atmospheric carbon inventory plus the maximum amount of carbon that the ocean may hold. To understand the role of ocean chemistry, we diagnose the response of climate models of differing complexity over centennial and millennial timescales. In all the models, there is a similar carbon response: During emissions, the ocean surface acidifies and the maximum amount of carbon that the ocean can hold decreases, which weakens the carbon contribution to the proportionality of surface warming to carbon emissions. Hence, ocean carbonate chemistry is important in controlling the proportionality of surface warming to carbon emissions and its evolution in time.
17. 2018: Matthews, H. Damon, et al. “Focus on cumulative emissions, global carbon budgets and the implications for climate mitigation targets.” Environmental Research Letters 13.1 (2018): 010201. The Environmental Research Letters focus issue on ‘Cumulative Emissions, Global Carbon Budgets and the Implications for Climate Mitigation Targets‘ was launched in 2015 to highlight the emerging science of the climate response to cumulative emissions, and how this can inform efforts to decrease emissions fast enough to avoid dangerous climate impacts. The 22 research articles published represent a fantastic snapshot of the state-or-the-art in this field, covering both the science and policy aspects of cumulative emissions and carbon budget research. In this Review and Synthesis, we summarize the findings published in this focus issue, outline some suggestions for ongoing research needs, and present our assessment of the implications of this research for ongoing efforts to meet the goals of the Paris climate agreement.

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