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The temperature of the atmosphere has increased since anthropogenic activities became significant in the 1960’s. This is clear from a world-wide system of measurements sometimes, confusingly, referred to as “surface” temperatures, but which were taken at a height of 1.2 metres above the Earth’s surface. Therefore, the temperatures apply to the atmosphere, not the land surface. The data for the anomalies is provided by NOAA, Ref (1), and is plotted in the graph by BLACK “positive signs”.

The temperature of the atmosphere increases only because it receives energy. The long-established ** Kinetic Theory of Gases** shows that the increase in temperature is proportional to the increase in energy, and provides an equation relating these two variables. This anthropogenic energy can come from absorption by atmospheric carbon dioxide, and from emission of primary energy caused by the burning of fossil fuels, or by nuclear power. In fact, any source other than renewable energy, such as solar, wind and wave.

__Carbon Dioxide__Carbon dioxide is known to absorb electromagnetic energy at certain infrared wavelengths, and the amount absorbed is proportional to the concentration in the atmosphere. This has increased almost linearly during the anthropogenic period, as shown in Ref (2). The absorbed energy is rapidly added to the existing energy in the atmosphere, by “thermalization” of the excited carbon dioxide molecules, and a small proportion, only a few percent, is retained in the atmosphere. Therefore, the aggregate of added energy increases non-linearly.

As shown in the figure, for the linear increase in concentration of carbon dioxide, the

**in energy absorbed and retained in the atmosphere each year is constant. Let this annual increase be referred to as the “**

__increase__**”.**

__absunit__So, in 1981, Year 1, n=1, the increase in absorbed energy is 1 absunit, and so the total absorption is 1 absunits.

In Year 2, n=2, the increase is 1, and so the total absorption is 2 absunits. But, this is added to the total from Year 1, making the aggregate = 3 absunits.

In Year 3, n=3, the increase is 1, and so the total absorption is 3 absunits. But , this is added to the total for Year 2, making the aggregate = 6 absunits.

The table below provides the aggregates of Atmosphere Energy for 47 years.

A curve can be generated from this data for the relation between the temperature anomaly and carbon dioxide absorption in the atmosphere. This is shown in GREEN in the graphs. The position against the temperature anomaly axis is explained later. **Primary Energy**The data for the annual global primary energy consumption are provided by “Our World in Data”, Ref (3); this has been converted into the yearly

**aggregate energy anomaly**stating from zero in 1980, up to the latest figure available as at 2019. Then the data was converted into Joules/m

^{2 }to enable use of the “standard column” of the atmosphere based upon a surface area of 1 m

^{2}.

The Kinetic Theory of Gases states that E = 3kT/2 where k is Boltzmann’s Constant

k = 1.381 * 10

^{-23 }

E is the energy in Joules

T is the Absolute Temperature

The temperature anomalies were then calculated accordingly, and the points plotted in RED squares in the graph.

__Carbon Dioxide and Primary Energy__The sum of the two separate effects, the Total, is shown as the BLUE curve.

The carbon dioxide curve was extended from 2015 onwards, by means of recent work by Van Wijngaarden and William Happer, Ref (4). They showed that at a CO2 concentration of 400ppm which was reached in 2015, the absorption of infrared energy at the relevant CO2 wavelengths was virtually complete. Therefore, no extra energy can be absorbed by CO2 even if further such gas emissions may occur. However, absorption will continue to occur at the level at 400ppm. So, the temperature will continue to rise at a constant rate, and a straight line will result as shown.

The work of Van Wijngaarden and William Happer is supported by Schack, Ref (5) and Schildknecht, Ref (6).

**The energy retained in the atmosphere is only a small proportion of the total supply, and it is necessary to establish the retention factor. The method used is given in Appendix 1, and a value of 0.092 was found**

__The atmosphere energy retention factor__

__Results__

The combined effect of the two retained energies is shown as the Total curve, in BLUE, and this curve was fitted to the measured temperature anomaly points, BLACK + signs. This was determined by the value of the atmosphere retention factor.

The anomalies due to the Primary Energy are fixed, but those for carbon dioxide depend upon the value of the absunit. Please refer to Appendix 1.

** The Future**The carbon dioxide curve has been extended as a straight line as already explained. This is valid providing that the atmospheric concentration is not reduced below 400ppm.

If the primary energy continues the recent trend, the Total curve may be extended linearly.

This predicts a temperature anomaly of approximately 1.2 degC above the 1980 level by 2040. More temperature points would be helpful, but this takes time.

The estimated rate of temperature rise is 0.027 degC per year, from 2015 onwards, when the anomaly was 0.56 degC above 1980.

** Conclusion** There are two causes of the increase in atmosphere temperature. Carbon dioxide It is not sufficient simply to stop further emissions into the atmosphere, because absorption will still continue at the 2015 rate.

The concentration must be

**reduced,**ideally to about 300ppm.

Primary energy This has more than twice the effect of carbon dioxide.

To whatever use this energy may be put, the retained portion will eventually end up as heat energy in the atmosphere.

Therefore, in order to prevent further warming from this source, it is necessary to change entirely to renewables; solar, wind and wave. But, NOT nuclear!

**Appendix 1**

**Consider the energy aggregates from 1980 onwards.**

__To determine the atmosphere energy retention factor, a__

Let P be the increase in aggregate Primary Energy emitted into the atmosphere, from 1980.

Let C be the increase in aggregate energy absorbed by carbon dioxide and then re-emitted into the atmosphere, from 1980.

Let R be the increase in total aggregate energy

**, from 1980.**

__retained in the atmosphere__In each case, let the subscripts 0 and 5 denote the years 2000 and 2015 respectively.

( Ex. P

_{5}is the aggregate primary energy for 2015 ).

From the given references, and subsequent conversions, we have the following data. The temperature anomalies were taken from the fitted Total energy curve, rather than the basic measurements, because of their inevitable erratic nature.

The aggregate energies R

_{5 }and R

_{0 }were calculated by means of the Kinetic Theory of Gases.

ΔT degC R J/m

^{2 }(*10

^{6}) Pr.En J/m

^{2 }(*10

^{7 })

2015 0.56 R

_{5 }= 3.8629 P

_{5 }= 3.02

2000 0.25 R

_{0}= 1.7245 P

_{0}= 1.48 ___________________________________________________

The Kinetic Theory states that for

**a gas such as carbon dioxide which has 3 atoms per molecule, the change of energy ΔE is related to the change of temperature ΔT by the equation:**

__one molecule__ΔE = 3k.ΔT/2 where k = Boltzmann’s Constant, 1.381*10

^{-23 }

But there are 3.33*10

^{29 }nitrogen and oxygen molecules in the standard atmospheric column.

Therefore, the aggregate energy for the whole column is

R

_{5}= 1.5*1.381*10

^{-23 }* 0.56 * 3.33*10

^{29 }Joules/m

^{2}

= 3.8629*10

^{6 }Joules/m

^{2}

Similarly, R

_{0}= 1.7245*10

^{6 }Joules/m

^{2}

The

**total energy**comprises the Primary Energy from fossil fuels and nuclear power, plus the carbon dioxide absorbed energy from the radiation from the Earth’s surface.

This total energy is emitted into the atmosphere, but only a small proportion is retained.

Let a = the atmosphere energy retention factor. This must be determined.

Therefore, in 2015 a(P

_{5}+ C

_{5}) = R

_{5 }……………………..(1)

and in 2000 a(P

_{0}+ C

_{0}) = R

_{0 }……………………..(2)

Divide (1) by (2) P

_{5}+ C

_{5}= R

_{5}……………………….(3)

P

_{0}+ C

_{0}R

_{0 }

The aggregate values of the Primary Energies P

_{5}and P

_{0}are given above.

It is required to find the values for the

**carbon dioxide**energies C

_{5 }and C

_{0 }.

The aggregate atmosphere energies C

_{5 }and C

_{0}are shown in the table in terms

of the “absunit”.

Let u = the value of 1 absunit

From the table, at 2015 where n = 35 there are 630 absunits, so C

_{5 }= 630u

and at 2000 where n = 20 there are 210 absunits, so C

_{0 }= 210u

Substitute these values in (3)

P

_{5}+ 630u = R

_{5}……………………………………….(4)

P

_{0}+ 219u R

_{0 }

With a little algebra, we find the value of the absunit for this calculation

u = P

_{0 }.R

_{5 }– P

_{5 }.R

_{0}

630.R

_{0 }– 210.R

_{5}

**……………………(5)**

__u = 1.8496 * 10__^{4 }Joules/m^{2}From (1) a(P

_{5}+ C

_{5}) = R

_{5 }and C

_{5 }= 630u

Hence, we find

__a = 0.092__

**The atmosphere energy retention factor is 0.092, or 9.2 %**

__The 91% of the energy which leaves the atmosphere is thought to go into the oceans, and this is supported by NOAA, at Climate.gov, Ref (7). In order for the usual temperature difference between atmosphere and ocean to be maintained, it is necessary for the proportion of the anthropogenic energy entering the ocean to be the greater because of the oceans’ larger thermal capacity.__

__Appendix 2To determine the energy absorbed from the atmosphere__

Let P be the increase in aggregate Primary Energy emitted into the atmosphere, from 1980.

Let C be the increase in aggregate energy absorbed by carbon dioxide and then re-emitted into the atmosphere, from 1980.

Let R be the increase in total aggregate energy

**, from 1980.**

__retained in the atmosphere__Let a be the atmospheric energy retention factor.

In each case, let the subscripts 0 and 5 denote the years 2000 and 2015 respectively.

( Ex. P_{5} is the aggregate primary energy for 2015 ).

By definition, the energy retained in the atmosphere is equal to the basic energy input multiplied by the **atmospheric energy retention factor. **

So, a( P + C ) = R

Therefore, by the end of 2000, C

_{0 }= (R

_{0 }/a) – P

_{0 }

_{ }

and by the end of 2015, C_{5 } = (R_{5 }/a) – P_{5}

So, with reference to Appendix 1, by the end of 2015,

C_{5 } = (3.8629*10^{6 }) – 3.02*10^{7 }

9.2*10^{-2}

__C _{5} = 1.1788*10^{7 } J/m^{2}__

This is the basic aggregate “carbon energy” accrued from 1980 to 2015.

Similarly, the basic aggregate “carbon energy” accrued from 1980 to 2000 is

__C___{0}= 0.3945*10^{7 }J/m^{2}Let ΔC

_{(5-0)}be the increase in the carbon aggregate from 2000 to 2015.

So, ΔC

_{(5-0)}= C

_{5 }– C

_{0}

=

__7.843*10__

^{6 }J/m^{2}Similarly, ΔP

_{(5-0)}= (3.02 – 1.48)*10

^{7}

**=**Therefore, the total increase in basic aggregate energy input to the atmosphere from 2000 to 2015 is ΔP

__1.54*10__^{7 }__J/m__

^{2}_{(5-0)}

_{ }+ ΔC

_{(5-0)}

ΔP

_{(5-0) }+ ΔC

_{(5-0)}= 1.54*10

^{7}

**+ 0.7843*10**

^{7}

**J/m**

^{2}

**=**

__2.3243*10__

^{7}**But, the aggregate energy retained, from the table in Appendix 1, is R**

_{5}– R

_{0}

= 2.1384*10

^{6 }J/m

^{2}

The difference is the energy returned to the Earth’s surface,

which is (2.3243 – 0.2138)*10

^{7}

= 2.1105*10

^{7 }J/m

^{2}

**This was in 15 years (3.1536 *10**

^{7}seconds per year), so the average rate of return to the surface was

__44.6 mW/m__

References^{2}References

(1)https://www.ncdc.noaa.gov/cag/global/time-series/globe/land_ocean/ann/

Select: Timescale Annual; Start Year 1970; End Year 2022; Surface Land and Ocean; Plot

(2) https://www.climate.gov/news-features/understanding-climate/climate-change-atmospheric-carbon-dioxide

(3) https://ourworldindata.org/energy-production-consumption,

(4) Van Wijngaarden and Happer, https://arxiv.org/abs/2006.03098

(5) Schack, Phys.Blatter,

**28**26 (1972)

(6) Schildknecht, arXiv:2004.0 0708v2 [physics.ao-ph] 5 Aug 2020

(7) https://www.climate.gov/news-features/understanding-climate/climate-change-ocean-heat-content