# A MATHEMATICAL INCONSISTENCY

Posted August 26, 2020

on:**THIS POST IS A COMPARISON OF THE ECS AND TCR, THE TWO METHODS USED INTERCHANGEABLY IN CLIMATE SCIENCE TO RELATE WARMING TO FOSSIL FUEL EMISSIONS. THE COMPARISON IS MADE IN TERMS OF MATHEMATICAL CONSISTENCY.**

**THE EQUILIBRIUM CLIMATE SENSITIVITY ECS**

The **ECS **measure of the impact of fossil fuel emissions on warming holds that atmospheric CO2 concentration at any given time is a linear function of cumulative emissions and that surface temperature is a logarithmic function of atmospheric CO2 concentration. These two relationships imply that surface temperature is a logarithmic function of cumulative emissions. That in turn implies that the amount of warming caused by a given level of cumulative emissions is the difference between the logarithms of the two cumulative emissions before and after.

**TRANSIENT CLIMATE RESPONSE TO CUMULATIVE EMISSIONS TCR**

The **TCR **measure of the impact of fossil fuel emissions on warming holds that the amount of warming is a linear function of cumulative emissions. This linearity is mathematically inconsistent with the **ECS **measure which implies that the amount of warming is proportional to the difference between the logarithms of the cumulative emissions before and after the period of warming under study.

**IMPLICATIONS OF THIS MATHEMATICAL INCONSISTENCY **

The mathematical inconsistency described above shows that the significant research effort in climate science to resolve the **ECS **and **TCR **measures of anthropogenic warming in terms of fossil fuel emissions with Earth System Models {**ESM**} is not possible because the two methods of computing the impact of emissions on temperature are not mathematically consistent.

**CONFIRMATION BIAS, CIRCULAR REASONING, AND MATHEMATICAL IMPOSSIBILITY OF EARTH SYSTEM MODELS {ESM}**

In a related post on **ESM **{https://tambonthongchai.com/2020/08/25/earth-system-models-and-carbon-budgets/} we argue that the **ESM **construction procedure of beginning with the **TCR **warming derived from cumulative emissions and then explaining the result from an arbitrarily expanded list of **ECS **climate drivers is a form of circular reasoning. In this post we find that a further and more serious flaw in this procedure is the mathematical impossibility of the **ESM **exercise of reconciling **ECS **and **TCR** .

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