Doing the math

 

Introduction

Some gases in the atmosphere are transparent for visible light but not for infrared light. The visible sunlight that hits the ground is warming it up, but the IR-light from the ground is prevented from directly radiating back to space because of these gases. This causes a warming of the atmosphere close to the ground. It would be much colder on Earth without this effect, called the greenhouse effect.

In the last million years, when human beings have been on Earth, the concentration of the greenhouse gas Carbon dioxide has varied between 160 ppm and 300 ppm (parts per million), low concentration during Ice ages and high during the warming periods as now.

Carbon dioxide is the gas of life, it's captured from the atmosphere by green plants, building them up, and return to the atmosphere via the exhauled air when animals eats the plants. However, there is Carbon stored underground, from plants and animals living in the past, that is not a part of the process no more and taking those up in air increase the concentration of Carbon dioxide and cause global warming to some extent.

The Keeling curve

In 1958 the concentration of Carbon dioxide began to be measured at the observatorium of Mauna Loa. It appeared that there was a seasonal variation and a growing trend.

 

The raw data can be found at NOAA.

(For the moment at row data  in text form) 

Raw data shows that there are an increasing growth of the concentration and that the increment correspond well to the use of fossil fuels.

 

Since the Keeling curve has a growing annual growth it isn't appropriate to approximate it with an exponential function. The fact that the emission of Carbon dioxide isn't growing from zero ppm leads to the model:

Non-linear LSM regression leads to

and an annual mean trend graph of the Keeling curve which obviously is far more appropriate:

Red marks is annual average from raw data. The conclusion must be that the raw data ppm minus 256 ppm are well approximated with an exponential function with a fixed growth of 0.0162 which give a doubling time 43 years. 

Global warming

In the 1800s scientists discovered that an atmosphere with more Carbon dioxide should be warmer than an atmosphere with less Carbon dioxide, near the surface. Svante Arrhenius discovered an approximate formula

If the concentration increase from C1 to C2 the temperature will approximately increase from T1 to T2. K is a constant depending on the climate sensitivity, the temperature raise from a doubling of the concentration. Due to IPCC the climate  sensitivity is 3.0 ± 1.5 °C. Therefore K ≈ 3.0/ln(2) or K=4.3 ± 2.3.

Data from five international meteorological institutes gives temperature anomalies:

The trend for the Keeling curve above and Arrhenius formula prompts for doing a long-term forecast (here K is set to 3.6, corresponding to climat sensitivity 2.5 C):

So 'business as usual' would give around 0.25 + 3.25 = 3.5 °C higher average temperature until year 2100 than under the pre-industrial era.

Politics

Since the trend of the human emission part of the Keeling curve is stable there have been no effect from the political promises and commitments so far.

From the Keeling curve approximation and the Arrhenius approximative formula it follows that 'business as usual' would give a convex temperature curve (red). If all countries decided to freeze their emission to the 2021 level, the result would be a concave temperature curve (blue).