To serve as a basis for CO2 alarmism (together with back radiation), IPCC put together the following formula with inspiration from the Beer-Lambert law (exponential transport decrease in an absorbing medium):

- $\Delta RF = 5.35\ln (C/C_0)$

where $RF$ is Radiative Forcing from CO2 as a greenhouse gas, $\Delta RF$ is additional forcing caused by a change of concentration of atmospheric CO2 from $C_0$ (preindustrial) to a present $C$. Doubled concentration from preindustrial level would then cause an additional (warming) forcing of about $4$ Watts/m2, which is translated to 1C warming by Stefan-Boltzmann, which by feed back can become anything you like 1-5C with 5C utterly alarming! Note that without the 1C from the formula, feedback has nothing start from and so it is an absolutely crucial element of CO2 alarmism!

In CALCULATION OF ATMOSPHERIC RADIATIVE FORCING (WARMING EFFECT) OF CARBON DIOXIDE AT ANY CONCENTRATION by H. Douglas Lightfoot and Orval A. Mamer an effort is made to improve the formula from the starting point:

*Even though there is***no theoretical basis**for the Beer-Lambert formula, ∆RF = αln(C/Co), it has been accepted by the scientific community as a reasonable approximation.

*In this paper we propose an improved mathematical approximation that, like the Beer-Lambert law, has***no theoretical basis**.

Does a formula with no theoretical basis have a serious defect? Not always, since a formula may capture observations as some form of condensed experience. The validity of the formula can then be checked against observation/experience, and so be falsified.

However, if there is no way to check the validity of the formula by observation, and this is the case with the above formula, then lack of theoretical basis is a serious defect because theory is what remains if you take away observation.

So IPCC relies in its prediction of the warming effect of doubled CO2 on a formula which has neither theoretical basis nor observational support. You can say that this is shaky. How can you proclaim Net Zero with all its devastating consequences from nothing? It seems you must have some hidden agenda. You can learn about the agenda in new best seller The Truth about Energy, Global Warming, and Climate Change: Exposing Climate Lies in an Age of Disinformation by Jerome R. Corsi.

0.74112805469855 K, to be exact... if one assumes emissivity of 1.

SvaraRadera0.791648256271 K, to be exact... if one assumes emissivity of 0.93643 (ref: NASA ISCCP program).

In reality, that thermalization is increasing CAPE (Convective Available Potential Energy), which increases convective (and thus evaporative, given that greater convection will convect more water vapor) removal of energy from the surface. The only thing that happens with an increased atmospheric CO2 concentration is a reduction in extinction depth for the wavelengths that CO2 absorbs... given that the extinction depth is already ~10.4 m, that radiation which is already being thermalized (sometimes... see below) at 420 ppm will be likewise thermalized at all higher atmospheric concentrations, just in a shorter distance. Thus, the atmosphere is opaque at 14.98352 µm... and that means absorption is saturated. There's more CO2 than radiation, the CO2 can only thermalize (sometimes... see below) what's available, a higher concentration of CO2 can't thermalize any more if there isn't any more.

Now, the climastrologists claim (in the Kiehl-Trenberth graphic and other similar graphics) that surface radiant exitance is 390 W m-2.

https://i.imgur.com/RJMT26o.png

Energy removal by convection and evaporation is ~76.2% of all surface energy removal.

https://andymaypetrophysicist.files.wordpress.com/2020/09/figure-2.png

If 390 W m-2 radiant exitance is only 23.8% of surface energy removal, then total surface energy removal (by convection, latent heat and radiation) must be 1638.65546 W m-2.

And that's ludicrous. They assume that only radiative emission removes surface energy, which is how they claim this 'forcing'. They want you to believe that energy removal from the surface of the planet (1638.65546 W m-2) is higher than the solar irradiance at the atmosphere / space interface (1361 W m-2).

When the climastrologists speak of thermalization, they speak as though that's what occurs under all circumstances... but that's only half the story. They've hidden a huge lie (CAGW) behind a half-truth (thermalization via v-t (vibrational to translational) collisional processes) while conveniently ignoring the other half of the story ('dethermalization' via t-v collisional processes) and thus denying 2LoT and the Equipartition Theorem.

{ continued... }

Note here that the term 'transition temperature' is not used in relation to phase change, but to a change in the role of the given molecular species from net cooling to net warming or vice versa.

SvaraRaderaClimate alarmists claim that CAGW (Catastrophic Anthropogenic Global Warming) can occur because the CO2 molecule absorbs 14.98352 µm radiation, becomes vibrationally excited in one of its bending modes, collides with a nitrogen or oxygen molecule, and imparts that vibrational energy to the translational energy of the other molecule via a process known as collisional de-excitation, thereby increasing atmospheric temperature.

The climate alarmists claim that this process occurs under all circumstances. This represents a violation of 2LoT and the Equipartition Theorem.

CO2 is a dual-role molecule, just as all molecules capable of emitting radiation are.

The 'transition temperature' of any given molecular species is dependent upon the differential between:

1) the combined translational mode energy of two colliding molecules,

-and-

2) the lowest excited vibrational mode quantum state energy of the radiative molecule.

When 2) > 1), energy flows from vibrational mode to translational mode, which is a warming process.

When 1) > 2), energy flows from translational mode to vibrational mode, which is a cooling process.

Below ~288 K, the vibrational mode quantum state energy of CO2's lowest excited vibrational mode quantum state, CO2{v21(1)}, is higher than the average combined translational mode energy of two colliding atmospheric molecules, therefore the 2nd Law of Thermodynamics and the Equipartition Theorem dictate that energy will flow from vibrational mode to translational mode.

The increase in kinetic energy of atmospheric molecules represents an increase in temperature.

Above ~288 K, the Maxwell-Boltzmann Speed Distribution Function dictates that enough of the atmospheric molecules carry sufficient combined translational mode energy upon molecular collision to begin significantly vibrationally exciting CO2's lowest excited vibrational mode quantum state.

A graphic, showing the percentage of molecules which carry sufficient kinetic energy at 288 K to excite CO2{v21(1)}

https://i.imgur.com/CxVTcro.png

The conversion of translational mode to vibrational mode energy is, by definition, a cooling process.

This increases the time duration during which CO2 is vibrationally excited and therefore the probability that it will radiatively emit. The resultant radiation which is emitted to space is, by definition, a cooling process.

This 'transition temperature' at which CO2 changes from being a net-warming to a net-cooling molecule is ~288 K, with CO2 acting more and more in its net-cooling mode as temperature increases.

Thus CO2 is physically incapable of causing catastrophic warming, and indeed is a net atmospheric coolant above its transition temperature, in accord with 2LoT and the Equipartition Theorem.

The same concept applies for all molecules capable of emitting radiation. The only thing that changes is the transition temperature at which any given molecular species changes roles from net-cooling to net-warming or vice versa, because each molecular species has different vibrational mode quantum state energy.

The full story: In an atmosphere sufficiently dense such that collisional energy transfer can significantly occur, all radiative molecules play the part of atmospheric coolants at and above the temperature at which the combined translational mode energy of two colliding molecules exceeds the lowest excited vibrational mode quantum state energy of the radiative molecule. Below this temperature, they act to warm the atmosphere via the mechanism the climate alarmists claim happens all the time, but if that warming mechanism occurs below the tropopause, the net result is an increase of Convective Available Potential Energy (CAPE), which increases convection, which is a net cooling process.

SvaraRaderaIn other words: below ~288 K, CO2 does indeed cause warming via the mechanism described above. But above ~288 K, the population of atoms/molecules with translational mode energy sufficient to begin significantly vibrationally exciting CO2 increases, increasing the time duration during which CO2 is vibrationally excited and therefore the probability that the CO2 will radiatively emit. The conversion of translational mode to vibrational mode energy is, by definition, a cooling process. The emission of the resultant radiation to space is, by definition, a cooling process.

As CO2 concentration increases, the population of CO2 molecules able to become vibrationally excited increases, thus increasing the number of CO2 molecules able to radiatively emit, thus increasing photon flux, thus increasing energy emission to space.

As temperature increases, the population of vibrationally excited CO2 molecules increases, thus increasing the number of CO2 molecules able to radiatively emit, thus increasing photon flux, thus increasing energy emission to space.

-----

For CO2, with a molecular weight of 44.0095 amu, at 288 K the molecule will have:

Most Probable Speed {(2kT/m)^1/2} = 329.8802984961799 m/s

Mean Speed {(8kT/pm)^1/2} = 372.23005645833854 m/s

Effective (rms) Speed {(3kT/m)^1/2} = 404.0195258297897 m/s

For N2, with a molecular weight of 28.0134 amu, at 288 K the molecule will have:

Most Probable Speed {(2kT/m)^1/2} = 413.472552224243 m/s

Mean Speed {(8kT/pm)^1/2} = 466.55381409564717 m/s

Effective (rms) speed {(3kT/m)^1/2} = 506.3983877978326 m/s

-----

CO2_KE = ((1/2) m (v · v))

CO2_KE = ((1/2) * 7.307948764374951e-26 kg * (404.0195258297897 m/s * 404.0195258297897 m/s))

CO2_KE = 5.9644473243674682571545362758031e-21 J

N2_KE = ((1/2) m (v · v))

N2_KE = ((1/2) * 4.651734100954141e-26 kg * (506.3983877978326 m/s * 506.3983877978326 m/s))

N2_KE = 5.9644378149782481931358080148627e-21 J

The nearly imperceptible differential (9.5093892200640187282609404e-27 J) in kinetic energy is due to rounding errors.

E = h c / λ

λ = h c / E

λ = 299792458 m s-1 * 6.62607015e−34 J s / 5.9644473243674682571545362758031e-21 J

λ = 3.3304776605761909112261530127719e-5 m = 33.304776605761909112261530127719 µm

Thus kinetic energy at exactly 288 K is equivalent to the energy of a 33.305 µm photon.

If two molecules collide, their translational energy is cumulative, dependent upon angle of collision.

Simplistically, for a head-on collision between only two molecules, this is described by the equation:

KE = (1/2 mv^2) [molecule 1] + (1/2 mv^2) [molecule 2]

KE * 2 = 1.1928894648734936514309072551606e-20 J

λ = 299792458 m s-1 * 6.62607015e−34 J s / 1.1928894648734936514309072551606e-20 J

= 1.665238830288095455613076506386e-5 m = 16.65238830288095455613076506386 µm

That's equivalent to a 16.652 µm photon.

{ continued }

You may surmise, “But at 288 K, the combined kinetic energy of two molecules in a head-on collision isn't sufficient to excite CO2's lowest vibrational mode quantum state! It requires the energy equivalent to a 14.98352 µm photon to vibrationally excite CO2, and the combined translational mode energy of two molecules colliding head-on at 288 K is only equivalent to the energy of a 16.652 µm photon!”

SvaraRaderaTrue, but you've not taken into account that 288 K is an average temperature (the mean of the kinetic temperatures of the individual atoms and molecules)... the Maxwell-Boltzmann Speed Distribution Function gives a wide translational mode equilibrium distribution.

A graphic, showing the percentage of molecules which carry sufficient kinetic energy at 288 K to excite CO2{v21(1)}

https://i.imgur.com/CxVTcro.png

For CO2{v21(1)}

---------------

We can equate translational mode energy (J) to vibrational mode energy (J) via the following:

KE = (1/2) m (v · v)

E = h c / λ

v = √((2 h c) / (m λ))

= √((2 * 299792458 m s-1 * 6.62607015e−34 J s) / (7.307948764374951e-26 kg * 1.498352e-5 m)

= √3.9728917142978574e-25 J m / 1.0949879646998736580752e-30 kg m

= √3.9728917142978574e-25 kg m^2 s-2 / 1.0949879646998736580752e-30 kg

= 602.34968995236079225502989764642 m/s

= 640.156048 K

Also remember that CO2 has three CO2{v2} vibrational mode quantum states that are nearly degenerate: CO2{v21(1)}, CO2{v22(2)} and CO2{v23(3)}. IOW, CO2 is capable of absorbing (or emitting) three photons of nearly the same energy if those vibrational mode quantum states are excited (via either absorption of radiation or via t-v collisional processes).

CO2{v21(1)} 14.98352 µm = 14983.52 nm = 1.498352e-5 m = 667.4 cm-1

CO2{v22(2)} 14.97454 µm = 14974.54 nm = 1.497454e-5 m = 667.8 cm-1

CO2{v23(3)} 14.96782 µm = 14967.82 nm = 1.496782e-5 m = 668.1 cm-1

This implies that as temperature increases, the population of CO2 molecules excited via translational-vibrational mode (t-v) collisions increases. This increases the time duration of CO2 vibrational mode quantum state excitation and therefore the probability that CO2 will radiatively emit. This increases the population of CO2 molecules which can radiatively cool the atmosphere.

The conversion of translational mode to vibrational mode energy is, by definition, a cooling process. The emission of the resultant radiation to space is, by definition, a cooling process.

As CO2 concentration increases, the population of CO2 molecules able to become vibrationally excited increases, thus increasing the number of CO2 molecules able to radiatively emit, thus increasing photon flux, thus increasing energy emission to space.

As temperature increases, the population of vibrationally excited CO2 molecules increases, thus increasing the number of CO2 molecules able to radiatively emit, thus increasing photon flux, thus increasing energy emission to space.

IOW, as atmospheric concentration of a radiative polyatomic molecular species increases, it moderates temperature closer to the 'transition temperature' of that molecular species.

The Beer-Lambert Law implies that attenuation is the negative decadic logarithm of the transmittance... in other words, for instance, in the atmosphere, over the extinction depth for 14.98352 µm waveband, which is ~10.4 m, ~50% of extinction will occur within the first 1.04 m, then 50% of extinction of the remainder will occur over the second 1.04 m, then 50% of extinction of the remainder will occur over the third 1.04, so on and so forth.

SvaraRaderaFrom my prior writings...

If 'backradiation' from CO2 atmospheric emission causes catastrophic anthropogenic global warming, where is this 'backradiation' coming from?

The near-surface extinction depth is ~10.4 m at current atmospheric CO2 concentration. The troposphere is essentially opaque to 13.98352 µm to 15.98352 µm radiation.

CO2's absorption of IR in the troposphere below CO2's 'transition temperature' (that temperature range at which the molecule begins switching from a net cooling (radiative) to a net warming (thermalization) role and vice versa, as explicated in another post) thermalizes that radiation, increasing CAPE (Convective Available Potential Energy), which increases convection to the upper atmosphere (carrying with it the latent and specific heat of polyatomic molecules... more polyatomic molecules will carry more energy

andwill more readily emit that energy in the upper atmosphere), which is acooling process.Mean free path length for radiation increases exponentially with altitude and vice versa due to air density changing inversely exponentially with altitude, thus the net vector for radiation in the 13.98352 - 15.98352 µm band is upward, so the majority of 'backradiation' which could possibly reach the surface would be from that very thin layer of atmosphere which is within ~10.4 m of the surface, and the great majority of that energy is being thermalized and convected below CO2's 'transition temperature'. So where's this 'backradiation' energy coming from that's going to cause catastrophic anthropogenic global warming?

At 287.64 K (the latest stated average temperature of Earth) and an emissivity of 0.93643 (calculated from NASA's ISCCP program, data collected 1983-2004), integrated radiance from 13.98352 µm - 15.98352 µm is 10.8773 W/sr-m^2.

Thus the

maximumthat CO2 could absorb in the troposphere would be 10.8773 W/sr-m^2,if all CO2 were in the CO2{v20(0)} vibrational mode quantum state.Remember that a molecule which has vibrational mode quantum states already excited which are resonant with an inciding photon will not absorb that radiation (unless there are degenerate vibrational mode quantum states which are not excited)... for a solid or opaque liquid, it will be reflected, for a gas it will at most be scattered (if the photon enters the EM field of the bound electron, the phase of the bound electron and the photon can be shifted... no energy is transferred in such a case. This phase shift changes photon vector.) and at least be non-interactive.

{ continued... }

While the Boltzmann Factor calculates that 10.816% of CO2 will be excited in one of its {v2} vibrational mode quantum states at 288 K, the Maxwell-Boltzmann Speed Distribution Function shows that ~24.9% will be excited. This is higher than the Boltzmann Factor because faster molecules collide more often, weighting the reaction cross-section more toward the higher end.

SvaraRaderaThus that drops to 8.1688523 W/sr-m^2 able to be absorbed. That's for

allCO2, natural and anthropogenic... anthropogenic CO2 accounts for ~3.63% (per IPCC AR4) of total CO2 flux, thus anthropogenic CO2 can only absorb 0.29652933849 W/sr-m^2.CO2 absorbs ~50% within 1 meter, thus anthropogenic CO2 will absorb 0.148264669245 W/m^2 in the first meter, and the remainder 0.148264669245 W/m^2 within the next ~9 meters.

CO2 will absorb this radiation regardless of any increase in atmospheric concentration... the extinction depth is ~10.4 m at 14.98352 µm wavelength, reducing to ~9.7 m for a doubling of CO2 atmospheric concentration. Any tropospheric thermalization which would occur at a higher CO2 atmospheric concentration is already taking place at the current concentration. Thus the net effect of CO2 thermalization is an increase in CAPE (Convective Available Potential Energy), which increases convective transport to the upper atmosphere, which is a cooling process.

The tropospheric thermalization is saturated. Even a doubling of CO2 doesn't appreciably reduce extinction depth at the band centered around 14.98352 µm. But the upper-atmospheric radiative shedding of energy to space is

notsaturated... and more CO2 molecules will cause more upper-atmosphericcooling, increasing buoyancy of lower-atmosphere air and thus increasing convection.An increased CO2 atmospheric concentration will emit more radiation in the upper atmosphere (simply because there are more relatively-higher-molar-heat-capacity molecules absorbing energy in the lower atmosphere, more relatively-higher-molar-heat-capacity molecules convectively transporting energy to the upper atmosphere, and more molecules capable of emitting radiation in the upper atmosphere), thus more radiation will be emitted to space, and that represents a loss of energy to the system known as 'Earth', which is a cooling process.

Here's something neat... so I make a point to learn at least one new thing each day. Today I learned the Japanese Multiplication Method, which comes in handy for calculating arbitrarily large numbers when you don't have a calculator, but you do have a whiteboard, chalkboard or pen and paper.

SvaraRaderahttps://i0.wp.com/www.thedigiteachers.com/wp-content/uploads/2020/09/Japanese-Multiplication.jpg

For instance, just to test whether I could do it, I multiplied 12345 * 54321 to arrive at the correct result of 670592745, without using a calculator, and faster than I could have done it via long multiplication.

It's so easy that one can do two and three digit multiplication in one's head with this method if one can hold the image of the cross-hatch in mind while one is doing the math of carrying digits. I suspect those 'human calculators' we used to see on TV do something similar, just that somehow it's built-in to their brains, whereas we mere mortals have to practice it quite a bit to even get to two or three digit multiplication.

LOL@Klimate Katastrophe Kooks wrote:

SvaraRadera"A graphic, showing the percentage of molecules which carry sufficient kinetic energy at 288 K to excite CO2{v21(1)}

https://i.imgur.com/CxVTcro.png"

Do keep in mind that as the proportion of CO2 in one of its excited vibrational mode quantum states increases, this has the same effect as regards thermalization as a reduction in CO2 atmospheric concentration would... those vibrationally excited CO2 molecules cannot absorb resonant radiation equal to the quantum state energy at which they are already excited (unless a degenerate or nearly degenerate vibrational mode quantum state which is not yet excited exists). That radiation passes the vibrationally-excited CO2 molecule by, with at most a slight change in photon phase (slight scattering) and bound electron phase, and with at least no interaction whatsoever In any case, no energy is transferred.

CO2 has three such nearly-degenerate vibrational mode quantum states for its {v2} vibrational mode quantum state:

CO2{v20(0)} ↔ CO2{v21(1)}: 14.98352 µm

CO2{v21(1)} ↔ CO2{v22(2)}: 14.97454 µm

CO2{v22(2)} ↔ CO2{v23(3)}: 14.96782 µm

There is also the 9.4 µm or 10.4 µm de-excitation pathways (dependent upon isotopic composition of the CO2 molecule and which vibrational mode quantum state it de-excites to) for CO2{v3(1)}.

This is the same pathway used by CO2 lasers, because N2{v1(1)} is very nearly perfectly resonant with CO2{v3(1)} when accounting for N2 anharmonicity, centrifugal distortion and vibro-rotational interaction... the method of excitation is different (for atmospheric CO2{v3(1)}, it is excited via collision of N2 with solar insolation excited O3, then the vibrationally-excited N2 colliding with CO2; in a CO2 laser, CO2{v3(1)} is excited via electron impact), but the energy flow from N2{v1(1)} to CO2{v3(1)} is the same in both cases.

https://web.archive.org/web/20190702035205if_/https://i.imgur.com/Lj8WbrW.png

Adapted from image at: https://web.archive.org/web/20190528024004/http://www.barrettbellamyclimate.com/page15.htm