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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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

TCRE TO THE RESCUE

STATISTICS OF THE TCRE

Figure 3 shows that when this sign convention is not enforced, the correlation falls apart

When this pattern is enforced a strong correlation is seen between the two series of the cumulative values of random numbers –

These so called carbon budgets then serve as policy tools for international climate action agreements and climate action imperatives of the United Nations

a time series of cumulative values of another time series contains neither time scale not degrees of freedom

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 .

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.

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

TCRE PAPERS

In the absence of more stringent mitigation, these trends are set to continue and further reduce the remaining quota

Efforts to limit climate change below a given temperature level require that global emissions of CO 2 cumulated over time remain below a limited quota

15 models from phase 5 of the Coupled Model Intercomparison Project (CMIP5)

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

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 CO 2 levels, and opportunities for reducing non-CO 2 climate warming, have both been cited as evidence that the case for reducing CO 2 emissions is less urgent than previously thought. Evaluating the impact of delay is complicated by the fact that CO 2 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 CO 2 -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 CO 2 emissions, about 2% per year , much faster than observed warming, independent of the climate response.

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.

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.

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 CO 2 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 CO 2 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 CO 2 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 CO 2 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.

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 simulations6,7. 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 carbon 8. 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.

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 CO 2 emissions (TCRE) is a metric of climate change that directly relates the primary cause of climate change (cumulative CO 2 emissions) to global mean temperature change. The metric was developed once researchers noticed that the cumulative CO 2 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 CO 2 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.

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 CO 2 emissions 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 CO 2 emissions and regional temperature change, showing that regional temperatures also respond approximately linearly to cumulative CO 2 emissions . Using an ensemble of twelve Earth system models , we present a novel application of pattern scaling5,6 to define the regional pattern of temperature change per emission of CO 2 . 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 p rovides a robust way to quantitatively link anthropogenic CO 2 emissions to local-scale climate impacts.

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 CO 2 emissions, referred to as transient climate response to cumulative CO 2 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 CO 2 until quadrupling of the preindustrial CO 2 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 CO 2 doubling . For all model versions, the TCRE is not constant over time while atmospheric CO 2 concentrations increase. It is constant after atmospheric CO 2 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.

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 CO 2 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 CO 2 emissions are continuously adjusted over time to limit 2100 warming to 1.5 °C, with ambitious non-CO 2 mitigation , net future cumulative CO 2 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.

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 CO 2 and change in global near-surface temperature. This relationship, which has been named the transient climate response to cumulative CO 2 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 CO 2 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 CO 2 of 0.50 to 0.58. Overall the uncertainty in the value of TCRE remains considerable.

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.

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]

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 CO 2 concentration. 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 CO 2 show that in order to keep warming to within 2 °C, future CO 2 emissions have to remain strongly limited, irrespective of climate sensitivity being at the high or low end.

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-CO 2 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 CO 2 -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 CO 2 -induced warming, and rapid post-2000 non-CO 2 warming in some ensemble members.

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.

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.