THIS POST IS A STUDY OF THE CARBON BUDGET MATHEMATICS OF CLIMATE SCIENCE AND A CONSOLIDATION OF RELATED POSTS ON THIS TOPIC.

PART-1: THE THEORY OF ANTHROPOGENIC GLOBAL WARMING

(1) Fossil Fuel Emissions: Anthropogenic global warming (AGW) theory says that our carbon cycle flows are balanced in nature but our use of fossil fuels injects external carbon that was removed from the carbon cycle millions of years ago, into this delicately balanced carbon cycle of nature and causes atmospheric CO2 to rise. The data do show that atmospheric CO2 has been rising since 1958 and this dataset is presented as evidence that fossil fuels cause atmospheric CO2 concentration to go up.

(2) The Tyndal Heat Trap: The theory of the equilibrium surface temperature of the earth rests on the heat balance of incident solar radiation, its absorption and re-radiation at infrared frequencies by the surface of the earth, and a recycle loop in infrared radiation implied by their absorption and re-radiation in the atmosphere attributed mostly to water and carbon dioxide. The surface temperature is therefore higher than it would have been without the recycle loop. It is therefore proposed that as fossil fuels cause the CO2 concentration of the atmosphere to go up, the corresponding surface temperature will go up accordingly. Therefore, higher and higher atmospheric CO2 concentration caused by fossil fuel emissions cause higher and higher surface temperature that is understood as global warming and the role of humans in providing the fossil fuel emissions included in the phrase anthropogenic global warming or AGW. This theory of warming implies that global mean temperature is proportional to the logarithm of atmospheric CO2 concentration and that therefore the rate of warming is proportional to the rate of change in the logarithm of atmospheric CO2 concentration. The regression coefficient that relates temperature to the logarithm of atmospheric CO2 concentration is described in climate science as Climate Sensitivity or ECS.

(3) Climate Action: Climate scientists used climate models to determine that this rate of warming cannot be allowed to continue because it will melt polar ice sheets and cause devastating sea level rise and when the mean global surface temperature reaches a critical value of 2C or perhaps even 1.5C above the reference pre-industrial temperature, positive climate feedbacks will accelerate the rate of warming and and cause the rate of warming to snowball out of control causing a collapse of human civilization, the end of life on earth, and the end of the planet. It is therefore imperative that we take climate action. Climate action means that humans must reduce and eventually eliminate fossil fuel emissions by switching to renewable energy like wind and solar. It is important that climate action taken reduces emissions to zero before we reach the critical amount of warming since pre-industrial of 1.5C because beyond that climate action will not be effective because the feedback acceleration mechanism will take over from fossil fuel emissions as the main driver of climate change. To ensure this timing is critical and the tool we must use to ensure this timing is to construct and strictly follow carbon budgets.

(4) The Carbon Budget: In climate science, the carbon budget is constructed with the TCRE. The carbon budget is the amount of cumulative global fossil fuel emissions that can be emitted for a given amount of warming. It is derived from the finding by climate science that cumulative annual warming is proportional to cumulative emissions with a near perfect statistically significant linear relationship. This proportionality is called the Transient Climate Response to Cumulative Emissions or TCRE, sometimes abbreviated to TCR. Its mean value is estimated to be 1.72C per teratonne of cumulative emissions with a 95% confidence interval of 0.88C to 2.52C. {Reto Knutti Reference cited below}. The dependence of the carbon budget on the TCRE raises some statistical issues about the carbon budget and invalidates its assumed climate action implication.

(5) MATHMATICS OF THE CARBON BUDGET: ISSUE#1: A time series of the cumulative values of another time series has neither time scale nor degrees of freedom. Therefore it does not contain useful information about the variables that it apparently represents. The correlation between cumulative annual warming and cumulative annual emissions is therefore spurious and it has no interpretation in terms of the phenomena that the raw data represent. Details of these statistical issues are provided in related posts on this site where it is shown that the apparent correlation is not responsiveness of temperature to emissions but a creation of a convenient sign pattern. The sign pattern is that emissions are always positive and during a time of warming the annual warming rates are mostly positive. LINK: https://tambonthongchai.com/2018/05/06/tcre/ . The role of the sign pattern in the creation of this spurious correlation is illustrated in another related post where we show that not just emissions but any variable with positive values creates just as strong if not a stronger correlation with temperature where temperature is best understood in this context as cumulative annual warming. LINK: https://tambonthongchai.com/2018/12/03/tcruparody/ . A demonstration of this statistical fallacy in the TCRE is provided in the Youtube video below.

(5) MATHMATICS OF THE CARBON BUDGET: ISSUE#2: Yet another issue in the TCRE and in carbon budgets constructed with the TCRE is that climate science uses two very different mathematical constructs to relate warming to emissions. In the ECS construct, we find that temperature is proportional to the logarithm of atmospheric CO2 concentration. Also we find in climate science theory that atmospheric CO2 concentration is proportional to cumulative emissions. These relationships imply that in the ECS theory of AGW, temperature is a logarithmic function of cumulative emissions. But in the TCRE we find a linear relationship between temperature and cumulative emissions. This contradiction creates a mathematical inconsistency explained in a related post: LINK: https://tambonthongchai.com/2020/08/26/a-mathematical-inconsistency/ . The essential part of this post is reproduced below.

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(6) THE REMAINING CARBON BUDGET PUZZLE IN CLIMATE SCIENCEAs described in related posts on this site: LINK: https://tambonthongchai.com/2020/10/13/carbon-budget-mystery/ The use of the TCRE to construct carbon budgets has created a vexing and mysterious anomaly that has frustrated climate scientists. The anomaly, described as the mystery of the remaining carbon budget, is that the carbon budget does not accumulate uniformly through the budget period so that at any time in the middle, the remaining carbon budget computed by subtraction does not equal the carbon budget re-computed with the TCRE. The explanation for that anomaly is described in detail in the related post and it is this. Recall that the TCRE is a creation of sign patterns where emissions are always positive and during a time of warming, annual warming values are mostly positive. The value of the TCRE is a related to the fraction of the annual warming values that are positive. This fraction is unlikely to be the same in the two parts of the climate budget period being compared. That is, the time span of the remaining carbon budget contains a different ratio of positive annual warming values than there was in the full span. However, in climate science what we find is that this anomaly is interpreted as some kind of complexity in the climate system that can be resolved with Earth System Models of greater complexity. However, as we note in the analysis of the Friedlingstein paper: LINK: https://tambonthongchai.com/2020/04/09/climate-statistics/ 

“Climate science has misinterpreted anomalies created by statistical errors as a climate science issue that needs to be resolved with climate models of greater complexity. In this context we find that their struggle with the remaining carbon budget puzzle demonstrates a failure of climate science to address statistical issues of the TCRE in terms of statistics. This failure has led them down a complex and confusing path of trying to find a climate science explanation of the remaining carbon budget anomaly that was created by statistical errors. The research paper presented below serves as an example of this kind of climate research. The real solution to the remaining carbon budget puzzle is to understand the statistical flaws in the TCRE correlation and to stop using it.”

Reto Knutti - Home | Facebook
RETO KNUTTI

AbstractAn 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. {[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}.

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