CAGW is nothing more than a complex mathematical scam... the proof

[Morpheus]

Ok, this is going to be long and in-depth... I'll do my best to lay it out clearly and concisely. It won't be easy, as there's no easy method of inputting formulae, so I'll just make them plain text, rather than LaTeX.

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The takeaways:

1) The climatologists have conflated their purported "greenhouse effect" with the Kelvin-Helmholtz Gravitational Auto-Compression Effect (aka the lapse rate).

2) The climatologists purport the causative agent for their purported "greenhouse effect" to be "backradiation".

3) The Kelvin-Helmholtz Gravitational Auto-Compression Effect's causative agent is, of course, gravity.

4) "Backradiation" is physically impossible because energy cannot spontaneously flow up an energy density gradient.

5) The climatologists misuse the Stefan-Boltzmann (S-B) equation, using the idealized blackbody form of the equation upon graybody objects, which manufactures out of thin air their purported "backradiation". It is only a mathematical artifact due to that aforementioned misuse of the S-B equation. It does not and cannot actually exist. Its existence would imply rampant violations of the fundamental physical laws.

6) Polyatomic molecules are net atmospheric radiative coolants, not "global warming" gases.  Far from the 'global warming gas' claimed by the climatologists, water acts as a literal refrigerant (in the strict 'refrigeration cycle' sense) below the tropopause. CO2 is the most prevalent atmospheric radiative coolant above the tropopause and the second-most prevalent (behind water vapor) below the tropopause. Peer reviewed studies corroborating this are referenced in the paper at the end of this post.

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CAGW (Catastrophic Anthropogenic Global Warming, due to CO2) is nothing more than a complex mathematical scam. This isn't stated as opinion, but as mathematically-precise and provable fact. It's not that we should be discussing the severity of the temperature increase due to CO2, it's that we should be laughing anyone who claims that any polyatomic molecule causes 'heat trapping' off the stage.

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Ok, the basics first...



As you can see, there are two forms of the Stefan-Boltzmann (S-B) equation... one for idealized blackbody objects, one for graybody objects.

The idealized blackbody form of the S-B equation assumes emission to 0 K and ε = 1 by the very definition of idealized blackbody objects.

Idealized Blackbody Object (assumes emission to 0 K and ε = 1 by definition):
q_bb = ε σ (T_h^4 - T_c^4) A_h
         = 1 σ (T_h^4 - 0 K) 1 m^2
         =    σ  T^4

The graybody form of the S-B equation assumes emission to > 0 K and ε < 1.

Graybody Object (assumes emission to > 0 K and ε < 1):
q_gb = ε σ (T_h^4 - T_c^4) A_h

The 'A_h' term is merely a multiplier, used if one is calculating for an area larger than unity [for instance: >1 m^2], which converts the result from radiant exitance (W m-2, radiant flux per unit area) to radiant flux (W).



Temperature is equal to the fourth root of radiation energy density divided by Stefan's Constant (ie: the radiation constant), per Stefan's Law.
e = T^4 a
a = 4σ/c
e = T^4 4σ/c
T^4 = e/(4σ/c)
T = 4^√(e/(4σ/c))
T = 4^√(e/a)

q = ε_h σ (T_h^4 – T_c^4)

[1] ∴ q = ε_h σ ((e_h / (4σ / c)) – (e_c / (4σ / c)))

Canceling units, we get J sec-1 m-2, which is W m-2 (1 J sec-1 = 1 W).
W m-2 = W m-2 K-4 * (Δ(J m-3 / (W m-2 K-4 / m sec-1)))

[2] ∴ q = (ε_h c (e_h - e_c)) / 4

Canceling units, we get J sec-1 m-2, which is W m-2 (1 J sec-1 = 1 W).
W m-2 = (m sec-1 (ΔJ m-3)) / 4

One can see from the immediately-above equation that the Stefan-Boltzmann (S-B) equation is all about subtracting the radiation energy density of the cooler object from the radiation energy density of the warmer object.

[3] ∴ q = (ε_h * (σ / a) * Δe)

Canceling units, we get W m-2.
W m-2 = ((W m-2 K-4 / J m-3 K-4) * ΔJ m-3)

One can see from the immediately-above equation that the Stefan-Boltzmann (S-B) equation is all about subtracting the radiation energy density of the cooler object from the radiation energy density of the warmer object.

You will note that σ = (a * c) / 4… the S-B Constant equals Stefan’s Constant multiplied by the speed of light in vacua divided by 4.

[4] ∴ q = (ε_h * ((a * c) / a) * Δe) / 4 = (ε_h  * c * Δe) / 4

Canceling units, we get J sec-1 m-2, which is W m-2  (1 J sec-1 = 1 W).
W m-2 =  (m sec-1 * ΔJ m-3) / 4

Note that [2] and [4] are identical, arrived at via two different avenues.

So radiant exitance at its most simplified (and thus the S-B equation at its most simplified) is just the emissivity of the warmer object (because emissivity only applies to objects which are emitting, and only the warmer object will be emitting… the colder object will be unable to emit in the direction of the warmer object because energy cannot spontaneously flow up an energy density gradient) multiplied by the speed of light in vacua, multiplied by the energy density differential, all divided by 4.

For graybody objects, it is the radiation energy density differential between warmer object and cooler object which determines warmer object radiant exitance. Warmer objects don't absorb radiation from cooler objects (a violation of 2LoT in the Clausius Statement sense and Stefan's Law); the lower radiation energy density gradient between warmer and cooler objects (as compared to between warmer object and 0 K) lowers radiant exitance of the warmer object (as compared to its radiant exitance if it were emitting to 0 K). The radiation energy density differential between objects manifests a radiation energy density gradient, each surface's radiation energy density manifesting a proportional radiation pressure.

The climatologists use:
q = σ T^4
... on graybody objects, and sometimes slap ε<1 onto that, when they should be using:
q = ε σ (T_h^4 - T_c^4)

This has the effect of artificially inflating radiant exitance of all calculated-upon objects.

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Assume:
288 K planet
255 K atmosphere

ε = 0.93643 (per NASA ISCCP program)

σ = (2 π^5 k_B^4) / (15 h^3 c^2) = 5.67037441918442945397099673188923087584012297029130e-8 W m-2 K-4
where:
σ = Stefan-Boltzmann Constant
k_B = Boltzmann Constant (1.380649e−23 J K−1)
h = Planck Constant (6.62607015e−34 J Hz−1)
c = light speed (299792458 m sec-1)

e = T^4 a
where:
e = energy density (J m-3)
T = temperature (K)
a = 4σ/c = 7.5657332500339284719430800357226e-16 J m-3 K-4

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Using the Idealized blackbody form of the S-B equation:

Surface To Atmosphere Radiant Exitance:
q = 0.93643 * 5.67037441918442945397099673188923087584012297029130e-8 W m-2 K-4 * (288^4) = 365.30616894307 W m-2

Atmosphere To Surface Radiant Exitance:
q = 5.67037441918442945397099673188923087584012297029130e-8 W m-2 K-4 * (255^4) = 239.757641819 W m-2

Net: 365.30616894307 W m-2 - 239.757641819 W m-2 = 125.54852712407 W m-2

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Now, that seems to make sense, right? Except that the 'atmosphere to surface' radiant exitance (aka 'backradiation') violates the fundamental physical laws... energy does not and cannot spontaneously flow up an energy density gradient. And applying emissivity to the idealized blackbody form of the S-B equation is a misuse of the S-B equation (remember that it assumes ε = 1).

Essentially, the climatologists are treating real-world graybody objects as though they are idealized blackbody objects... with emission to 0 K and emissivity of 1 (sometimes... other times they slap emissivity onto the idealized blackbody form of the S-B equation while still assuming emission to 0 K... which is still a misuse of the S-B equation, for graybody objects).

This essentially isolates each object into its own system so it cannot interact with other objects via the ambient EM field, which grossly inflates radiant exitance of all objects, necessitating that the climatologists carry these incorrect values through their calculation and cancel them on the back end (to get their equation to balance) by subtracting a wholly-fictive 'cooler to warmer' energy flow from the real (but far too high because it was calculated for emission to 0 K) 'warmer to cooler' energy flow.

That wholly-fictive 'cooler to warmer' energy flow is otherwise known as 'backradiation'... it is nothing more than a mathematical artifact due to that aforementioned misuse of the S-B equation.

As I show here and in the below-linked paper, the correct usage of the S-B equation for graybody objects is via subtracting cooler object energy density from warmer object energy density to arrive at the energy density gradient, which determines radiant exitance of the warmer object.

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Using the Graybody form of the S-B equation:

Surface Radiant Exitance:
q = 0.93643 * 5.67037441918442945397099673188923087584012297029130e-8 W m-2 K-4 * ( (288^4) - (255^4)) = 140.78992041451 W m-2

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Now, how can we ascertain which is correct?

Well, we can calculate the energy density of each, then perform the calculations.

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e = T^4 a
a = 4σ/c = 7.5657332500339284719430800357226e-16 J m-3 K-4

Surface Energy Density:
e = (288^4) * 7.5657332500339284719430800357226e-16 J m-3 K-4 = 5.20500290293e-6 J m-3

Atmosphere Energy Density:
e = (255^4) * 7.5657332500339284719430800357226e-16 J m-3 K-4 = 3.1989816343e-6 J m-3

Energy Density Gradient:
5.20500290293e-6 J m-3 - 3.1989816343e-6 J m-3 = 2.0060212686e-6 J m-3

Surface To Atmosphere Radiant Exitance:
σ / a * Δe * ε_h = W m-2

First, take a look at this:
σ / a = 5.67037441918442945397099673188923087584012297029130e-8 W m-2 K-4 / 7.5657332500339284719430800357226e-16 J m-3 K-4 = 74948114.5024 W m-2 / J m-3.

Well, what do you know... that's the conversion factor for radiant exitance (W m-2) and energy density (J m-3)!

It's almost as if the radiant exitance of graybody objects is determined by the energy density gradient, right?

5.67037441918442945397099673188923087584012297029130e-8 W m-2 K-4 / 7.5657332500339284719430800357226e-16 J m-3 K-4 * 2.0060212686e-6 J m-3 * 0.93643 = 140.78992041246 W m-2

Atmosphere To Surface Radiant Exitance:
5.67037441918442945397099673188923087584012297029130e-8 W m-2 K-4 / 7.5657332500339284719430800357226e-16 J m-3 K-4 * -2.0060212686e-6 J m-3 * 0.93643 = -140.78992041246 W m-2

In other words, 'Atmosphere To Surface' radiant exitance is zero (there can't be negative radiant exitance). The negative denotes that energy is flowing from surface to atmosphere.



That's nearly exact (except for rounding error... in the case of the Hyperphysics website, the rounding of σ and the rounding of the end result... in the linked paper, I account for the rounding of σ on that site and arrive at a precision of 3.8 parts per 100 trillion) to the full graybody S-B equation (because it is the full graybody S-B equation, stripped to its most basic).

The Hyperphysics website:
http://hyperphysics.phy-astr.gsu.edu/hba...an.html#c3

As you can see, the proper method of calculating radiant exitance for graybody objects is by subtracting the energy density of the cooler object from the energy density of the warmer object to arrive at the energy density gradient, which determines radiant exitance of the warmer object.

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The cooler object cannot emit in the direction of the warmer object because energy cannot spontaneously flow up an energy density gradient. To claim that it can is directly analogous to claiming that water can spontaneously flow uphill (ie: up a pressure gradient).

Most people cannot think in terms of energy, energy density and energy density gradient. We need to analogize to something they’re familiar with. Thus, just as, for instance, water only spontaneously flows down a pressure gradient, energy only spontaneously flows down an energy density gradient. That’s 2LoT in the Clausius Statement sense, in a nutshell. So one tack to take is to ask people if water can ever spontaneously flow uphill. Of course they’ll say, “No, water cannot flow uphill on its own.” Then show them dimensional analysis.

mass (M), length (L), time (T), absolute temperature (K), amount of substance (N), electric charge (Q), luminous intensity ©

We denote the dimensions like this: [Mx, Lx, Tx, Kx, Nx, Qx, Cx] where x = the number of that dimension

We typically remove dimensions which are not used.

Force: [M1 L1 T-2] /
Area: [M0 L2 T0] =
Pressure: [M1 L-1 T-2] /
Length: [M0 L1 T0] =
Pressure Gradient: [M1 L-2 T-2]

Explain to them that Pressure is Force / Area, and that Pressure Gradient is Pressure / Length. Remind them that water only spontaneously flows down a pressure gradient (ie: downhill). Then introduce energy. Tell them that energy is much like water. It requires an impetus to flow, just as water requires an impetus (pressure gradient) to flow. In the case of radiative energy, that impetus is a radiation energy density gradient, which is analogous to (and in fact, literally is) a radiation pressure gradient.

Energy: [M1 L2 T−2] /
Volume: [M0 L3 T0] =
Energy Density: [M1 L-1 T-2] /
Length: [M0 L1 T0] =
Energy Density Gradient: [M1 L-2 T-2]

Explain to them that Energy Density is Energy / Volume, and Energy Density Gradient is Energy Density / Length. Highlight the fact that Pressure and Energy Density have the same units (bolded above). Also highlight the fact that Pressure Gradient and Energy Density Gradient have the same units (bolded above).

So we’re talking about the same concept as water only spontaneously flowing down a pressure gradient (ie: downhill) when we talk of energy (of any form) only spontaneously flowing down an energy density gradient. Energy density is pressure, an energy density gradient is a pressure gradient… for energy.

In fact, the highest pressure ever attained was via lasers increasing energy density in nuclear fusion experiments.

It’s a bit more complicated for gases because they can convert that energy density to a change in volume (1 J m-3 = 1 Pa), for constant-pressure processes, which means the unconstrained volume of a gas will change such that its energy density (in J m-3) will tend toward being equal to pressure (in Pa). This is the underlying mechanism for convection. It should also have clued the climatologists in to the fact that it is solar insolation and atmospheric pressure which ‘sets’ temperature, not any ‘global warming’ gases.

Since a warmer object will have higher radiation energy density at all wavelengths than a cooler object (because remember, temperature is a measure of radiation energy density, equal to the fourth root of radiation energy density divided by Stefan’s Constant):



… ‘backradiation’ can do nothing to warm the surface because energy cannot spontaneously radiatively flow from lower to higher radiation energy density, and thus CAGW is nothing more than a complex mathematical scam perpetrated to obtain multiple billions of dollars in funding for trough-grubbing line-toeing ‘scientists’ and by perfidious politicians to push a Marxist One World Government “Build Back Better” agenda.

“But they’ve measured backradiation!”, some may claim. Yeah, no.

https://claesjohnson.blogspot.com/2011/0...meter.html

As Professor Claes Johnson shows in that article on his website, pyrgeometers (the instrument typically used to ‘measure’ backradiation) utilize the same sort of misuse of the S-B equation as the climatologists use. The bastardized form of the S-B equation used by pyrgeometers [ usually some form of q = (σ T_h^4 – σ T_c^4) or equivalently L_d = U_emf/S + σT_b, as outlined in the documentation for the instrument, with U_emf/S being negative in sign ] apriori assumes a subtraction of a wholly-fictive ‘cooler to warmer’ energy flow from the real (but far too high because it was calculated for emission to 0 K) ‘warmer to cooler’ energy flow, which as has been shown, is fallacious.

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Do remember that photons, each a quantum of energy, are considered the force-carrying gauge bosons of the EM interaction.

Going back to dimensional analysis:
We start with Energy: [M1 L2 T−2] -
Force: [M1 L1 T-2] *
Length: [M0 L1 T0] = [M0 L0 T0]

We are left with nothing on the 'transmitting' end... [M0 L0 T0]. In other words, that Energy is used to apply a Force along a Length. It’s obvious then, that if an equal and opposing Force were applied along that Length, no energy can flow… this is just as true radiatively as it is mechanically.

Aside:

As Δe → 0, ΔT → 0, q → 0. As q → 0, the ratio of graybody object total emissive power to idealized blackbody object total emissive power → 0. In other words, emissivity → 0. At thermodynamic equilibrium for a graybody object, there is no radiation energy density gradient and thus no impetus for photon generation.

As Δe → 0, ΔT → 0, photon chemical potential → 0, photon Free Energy → 0. At zero chemical potential, zero Free Energy, the photon can do no work, so there is no impetus for the photon to be absorbed. The ratio of the absorbed to the incident radiant power → 0. In other words, absorptivity → 0.

α = absorptivity = absorbed / incident radiant power
ρ = reflectivity = reflected / incident radiant power
τ = transmissivity = transmitted / incident radiant power

α + ρ + τ = 100%

For opaque surfaces τ = 0% ∴ α + ρ = 100%

If α = 0%, 0% + ρ = 100% ∴ ρ = 100% … all incident photons are reflected at thermodynamic equilibrium for graybody objects.

This coincides with standard cavity theory… applying cavity theory outside a cavity, for two graybody objects at thermodynamic equilibrium, no absorption nor emission takes place. The system reaches a state of quiescence (which is the definition of thermodynamic equilibrium). The photons remaining in the intervening space set up a standing wave, with the wavemode nodes at the object surfaces by dint of the boundary constraints. Nodes being a zero-crossing point (and anti-nodes being the positive and negative peaks), no energy can be transferred into or out of the objects. Photon chemical potential is zero, they can do no work, photon Free Energy is zero, they can do no work... there is no impetus for the photons to be absorbed. Should one object change temperature, the standing wave becomes a traveling wave with the group velocity proportional to the radiation energy density gradient and in the direction of the cooler object.

Now, obviously, if energy cannot spontaneously flow when there is zero energy density gradient (ie: at thermodynamic equilibrium), it certainly cannot spontaneously flow up an energy density gradient.

That Force applied along a Length gives us (on the 'receiving' end):
Force: [M1 L1 T-2] *
Length: [M0 L1 T0] =
Work: [M1 L2 T-2]

You'll note that Energy and Work have the same units:
Work: [M1 L2 T-2] = Energy: [M1 L2 T−2]

For those who want to put it in terms of Momentum:
Momentum: [M1 L1 T−1] *
Velocity: [M0 L1 T-1] =
Work: [M1 L2 T−2]

That means Energy Expended = Force * Length = Momentum * Velocity = Work

There's a reason for that. Free Energy is defined as that energy capable of performing work. This is reflected in the equation for Free Energy (represented here as a single object and its environment):

F = U – TS + PV
Where: F = Free Energy; U = internal energy; T = absolute temp; S = final entropy; TS = energy the object can receive from the environment; PV = work done to give the system final volume V at pressure P

If U > TS + PV, F > 0… energy must flow from object to environment.
If U = TS + PV, F = 0… no energy can flow to or from the object.
If U < TS + PV, F < 0… energy must flow from environment to object.

Of course, if we were talking about a system with only two objects with the same physical parameters and nothing else in the system, we could represent the Free Energy as: F = U_1 – U_2

Which is better represented as internal energy over volume to get energy density (since internal energy is an extensive property), converting the calculation to that of an intensive property and thus allowing us to compare dissimilar-sized objects: F = U_1 / V_1 – U_2 / V_2 = e_1 - e_2

And that’s exactly what the S-B equation does. Remember that temperature is a measure of radiation energy density, equal to the fourth root of radiation energy density divided by the radiation constant (Stefan’s Constant). Remember that I wrote above:

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q = (ε_h * c * (e_h - e_c)) / 4

Canceling units, we get J sec-1 m-2, which is W m-2 (1 J sec-1 = 1 W).
W m-2 = (m sec-1 (ΔJ m-3)) / 4

Or equivalently:
q = ε_h * σ / a * (e_h - e_c)

Canceling units, we get W m-2.
W m-2 = ((W m-2 K-4 / J m-3 K-4) * ΔJ m-3)

One can see that the S-B equation is all about subtracting the radiation energy density of the cooler object from the radiation energy density of the warmer object (to arrive at the radiation energy density gradient) because Free Energy is all about subtracting the energy density of one object from the energy density of the other object (no matter the form of that energy).

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"But how does that make CAGW a scam?", some may ask... well, because we're being lied to, based upon an unscientific premise.

The climatologists have misused the Stefan-Boltzmann (S-B) equation (and the fundamental physical laws), and in the process, have practically flipped reality on its head… polyatomics (CO2, H2O, etc.) are not "global warming gases", they are net atmospheric radiative coolants (radiative emission to space being the only way that Earth can shed energy); monoatomics (Ar) are not inert gases that have no effect upon climate, they are the actual "greenhouse gases" (because they cannot emit IR, and thus cannot shed energy to space... they dilute the radiative coolant gases); homonuclear diatomics (N2, O2) are somewhere in between… they can radiatively emit IR (and thus shed energy from the system known as 'Earth'), but only under certain conditions (collisional perturbation of their net-zero electric dipole, which is why homonuclear diatomic vibrational mode quantum states are meta-stable and relatively long-lived. Collisions happen exponentially less frequently as altitude increases), and thus are "greenhouse gases" like the monoatomics, just not to the same extent.

We live, at the planet's surface, in what can be analogized to the evaporator section of a world-sized AC unit, with polyatomics being net atmospheric radiative coolants (a higher concentration of them increases thermodynamic coupling between heat source (surface) and sink (space)), and with monoatomics and homonuclear diatomics playing the same role as non-condensable gases would play in an AC unit... diluting the polyatomic radiative gases which transit the majority of the energy, thus reducing the efficiency at which energy is transited from surface to upper atmosphere, then radiatively emitted to space.

Think about it this way... we all know the air warms up during the daytime as the planet's surface absorbs energy from the sun. Conduction of that energy when air contacts the planet’s surface is the major reason air warms up.

How does that ~99% of the atmosphere (N2, O2, Ar) cool down? It cannot effectively radiatively emit.

Convection moves energy around in the atmosphere, but it cannot shed energy to space. Conduction depends upon thermal contact with other matter and since space is essentially a vacuum, conduction cannot shed energy to space… this leaves only radiative emission. The only way our planet can shed energy is via radiative emission to space. Fully ~76.2% of all surface energy is removed via convection, advection and evaporation. The surface only radiatively emits ~23.8% of all surface energy to space. That ~76.2% must be emitted to space by the atmosphere.



Thus, common sense dictates that the thermal energy of the constituents of the atmosphere which cannot effectively radiatively emit (N2, O2, Ar) must be transferred to the so-called ‘greenhouse gases’ (CO2 being a lesser contributor below the tropopause and the largest contributor above the tropopause, water vapor being the main contributor below the tropopause) which can radiatively emit and thus shed that energy to space. Peer-reviewed studies corroborating this are referenced in the linked file below.

So, far from being ‘greenhouse gases’ which ‘trap heat’ in the atmosphere, those polyatomic radiative gases actually shed energy from the atmosphere to space. They are net atmospheric radiative coolants.

In short, in an atmosphere sufficiently dense such that collisional energy transfer can significantly occur, all polyatomic radiative molecules play the part of atmospheric radiative coolants at and above the temperature at which the combined translational mode energy of two colliding particles (atoms or molecules) exceeds the lowest excited vibrational mode quantum state energy of the radiative molecule. Below this temperature, they act to warm the atmosphere via thermalization (the mechanism the climate alarmists claim happens all the time), but if that occurs below the tropopause, the net result is an increase of Convective Available Potential Energy, which increases convection, which is a net cooling process. It is a gradation... as temperature increases, so too does the population of vibrationally excited polyatomics, and thus increases radiative emission. For CO2, that 'transition temperature' (the temperature at which the molecule transitions from being 'net warmant' to 'net coolant' and vice versa) is ~288 K.

I state "approximately 288 K" because the angle of collision of two atmospheric atoms or molecules can be from any angle, and thus it's nearly impossible to calculate an exact temperature. We could, of course, use statistical probability to lump the calculations into ranges of angles, then aggregate the results of each range of angles, but the basic idea still holds.



You will note that while the graphic above only addresses CO2 and N2, the same applies to all atmospheric gases due to the Equipartition Theorem.


The Boltzmann Factor (for air at 288 K and for the combined translation mode (kinetic) energy of two colliding particles (atoms or molecules) necessary to excite CO2's lowest vibrational mode quantum state) is ~0.8461... in other words, for every 100 air particles at the Most Probable Speed (MPS), ~84 other particles will have sufficient combined translational mode (kinetic) energy to excite CO2's lowest excited vibrational mode quantum state CO2{v21(1)} during collision.

Polyatomics are 'dual-role' gases... the role they play is dependent upon:
1) The combined translational mode (kinetic) energy of two colliding atoms or molecules.
- and -
2) The lowest excited vibrational mode quantum state energy of the polyatomic molecule.

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

When 1) > 2), energy flows from translational mode to vibrational mode, which is a cooling process. Due to the energy density gradient, that energy cannot flow back to translational mode, and is instead radiatively emitted. Emitted radiation that goes out to space represents a loss of energy for the system known as 'Earth' and is thus a cooling process.

The climatologists only told people half the story (thermalization by CO2 via vibrational mode to translation mode (v-t) collisional energy transfer processes). They didn't tell anyone about the inverse (translational mode to vibrational mode (t-v) collisional energy transfer processes, (then that energy being radiatively emitted to space)), which is a cooling process. That didn't fit their doomsaying narrative, so they left it out.

In other words, the climatologists only told people about the warming part (thermalization), not the cooling part. In order to hew to the fundamental physical laws, one must consider energy flow both to and from the CO2 molecule.

This doesn’t just apply to CO2, however. It applies to all atmospheric polyatomic molecules. In fact, far from the 'global warming gas' claimed by the climatologists, water acts as a literal refrigerant (in the strict ‘refrigeration cycle’ sense) below the tropopause:

The refrigeration cycle (Earth) [AC system]:

A liquid evaporates at the heat source (the surface) [in the evaporator], it is transported (convected) [via an AC compressor], it gives up its energy to the heat sink and undergoes phase change (emits radiation in the upper atmosphere, the majority of which is upwelling owing to the mean free path length / altitude / air density relation) [in the condenser], it is transported (falls as rain or snow) [via that AC compressor], and the cycle repeats.

That’s kind of why, after all, the humid adiabatic lapse rate (~3.5 to ~6.5 K km-1) is lower than the dry adiabatic lapse rate (~9.81 K km-1).

You will note that the dry adiabatic lapse rate is due to the monoatomics and homonuclear diatomics... we've removed in this case the predominant polyatomic which reduces lapse rate.

Remember that an actual greenhouse works by hindering convection of energy out of the greenhouse.

In an atmosphere consisting of solely monoatomics and homonuclear diatomics (ie: no polyatomic radiative molecules), the atoms / molecules could pick up energy via conduction by contacting the surface, just as the polyatomics do; they could convect just as the polyatomics do… but once in the upper atmosphere, they could not as effectively radiatively emit that energy, the upper atmosphere would warm, lending less buoyancy to convecting air, thus hindering convection… and that’s how an actual greenhouse works, by hindering convection.

For homonuclear diatomics, there would be some collisional perturbation of their net-zero electric dipole and thus some emission in the atmosphere, but by and large the atmosphere could not effectively emit (especially at higher altitudes, because the probability of collision decreases exponentially with altitude).

Thus the surface would have to radiatively emit that energy (which is currently ~76.2% of all energy removed from the surface via radiation, convection and evaporation) instead… and a higher surface radiant exitance implies a higher surface temperature.

These concepts used to be common knowledge. Somewhere along the way, the concepts got skewed to fit a particular narrative. Eventually, the concepts described herein will be common knowledge again, whereupon CAGW and its offshoots will be dumped on the midden heap of bad scientific ideas.

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Now, having gotten through all of that, there is a paper which goes into much more depth.

To date, there's not been a warmist physicist nor a climatologist who has been able to refute the scientific reality contained within that file, though several have tried. The last was physicist Bob Wentworth, who gave up after I backed him into numerous logical traps. Some have even changed their stance because of the information within that file.

That file:

  Climate Sanity.pdf (Size: 2.29 MB / Downloads: 182)

Anyone is free to use any of what I write however they wish... rewrite it to make it more easily understandable then publish it under your own name, publish it in whole or in part as-is under your own name, round-file it, whatever. I don't even want any attribution. Let's just work on destroying CAGW before it destroys our way of life.

[Billy_Bob Patriot Master]

Ok, this is going to be long and in-depth... I'll do my best to lay it out clearly and concisely. It won't be easy, as there's no easy method of inputting formulae, so I'll just make them plain text, rather than LaTeX.

CAGW is nothing more than a complex mathematical scam. This isn't stated as opinion, but as mathematically-precise and provable fact. It's not that we should be discussing the severity of the temperature increase due to CO2, it's that we should be laughing anyone who claims that any polyatomic molecule causes 'heat trapping' off the stage.

---

Ok, the basics first...



As you can see, there are two forms of the Stefan-Boltzmann (S-B) equation... one for idealized blackbody objects, one for graybody objects.

The idealized blackbody form of the S-B equation assumes emission to 0 K and ε = 1 by the very definition of idealized blackbody objects.

The graybody form of the S-B equation assumes emission to > 0 K and ε < 1.

Idealized Blackbody Object (assumes emission to 0 K and ε = 1):
q_bb = ε σ (T_h^4 - T_c^4) A_h
         = 1 σ (T_h^4 - 0 K) 1 m^2
         =    σ  T^4

Graybody Object (assumes emission to > 0 K and ε < 1):
q_gb = ε σ (T_h^4 - T_c^4) A_h

The 'Ah' term is merely a multiplier, used if one is calculating for an area larger than unity [for instance: >1 m^2], which converts the result from radiant exitance (W m-2, radiant flux per unit area) to radiant flux (W).

Temperature is equal to the fourth root of radiation energy density divided by Stefan's Constant (ie: the radiation constant).
e = T^4 a
a = 4σ/c
e = T^4 4σ/c
T^4 = e/(4σ/c)
T = 4^√(e/(4σ/c))
T = 4^√(e/a)

q = ε_h σ (T_h^4 – T_c^4)

[1] ∴ q = ε_h σ ((e_h / (4σ / c)) – (e_c / (4σ / c)))

Canceling units, we get J sec-1 m-2, which is W m-2 (1 J sec-1 = 1 W).
W m-2 = W m-2 K-4 * (Δ(J m-3 / (W m-2 K-4 / m sec-1)))

[2] ∴ q = (ε_h c (e_h - e_c)) / 4

Canceling units, we get J sec-1 m-2, which is W m-2 (1 J sec-1 = 1 W).
W m-2 = (m sec-1 (ΔJ m-3)) / 4

One can see from the immediately-above equation that the Stefan-Boltzmann (S-B) equation is all about subtracting the radiation energy density of the cooler object from the radiation energy density of the warmer object.

[3] ∴ q = (ε_h * (σ / a) * Δe) / 4

Canceling units, we get W m-2.
W m-2 = ((W m-2 K-4 / J m-3 K-4) * ΔJ m-3) / 4

You will note that σ = (a * c) / 4… the S-B Constant equals Stefan’s Constant multiplied by the speed of light in vacua divided by 4.

[4] ∴ q = (ε_h * ((a * c) / a) * Δe) / 4 = (εh  * c * Δe) / 4

Canceling units, we get J sec-1 m-2, which is W m-2  (1 J sec-1 = 1 W).
W m-2 =  (m sec-1 * ΔJ m-3) / 4

Note that [2] and [4] are identical, arrived at via two different avenues.

So radiant exitance at its most simplified (and thus the S-B equation at its most simplified) is just the emissivity of the warmer object (because emissivity only applies to objects which are emitting, and only the warmer object will be emitting… the colder object will be unable to emit in the direction of the warmer object because energy cannot spontaneously flow up an energy density gradient) multiplied by the speed of light in vacua, multiplied by the energy density differential, all divided by 4.

For graybody objects, it is the radiation energy density differential between warmer object and cooler object which determines warmer object radiant exitance. Warmer objects don't absorb radiation from cooler objects (a violation of 2LoT in the Clausius Statement sense and Stefan's Law); the lower radiation energy density gradient between warmer and cooler objects (as compared to between warmer object and 0 K) lowers radiant exitance of the warmer object (as compared to its radiant exitance if it were emitting to 0 K). The radiation energy density differential between objects manifests a radiation energy density gradient, each surface's radiation energy density manifesting a proportional radiation pressure.

The climatologists use:
q = σ T^4
... on graybody objects, and sometimes slap ε<1 onto that, when they should be using:
q = ε σ (T_h^4 - T_c^4)

This has the effect of artificially inflating radiant exitance of all calculated-upon objects.

---

Assume:
288 K planet
255 K atmosphere

ε = 0.93643 (per NASA ISCCP program)

σ = (2 π^5 k_B^4) / (15 h^3 c^2) = 5.67037441918442945397099673188923087584012297029130e-8 W m-2 K-4
where:
σ = Stefan-Boltzmann Constant
k_B = Boltzmann Constant (1.380649e−23 J K−1)
h = Planck Constant (6.62607015e−34 J Hz−1)
c = light speed (299792458 m sec-1)

e = T^4 a
where:
e = energy density
a = 4σ/c = 7.5657332500339284719430800357226e-16 J m-3 K-4

---

Using the Idealized blackbody form of S-B equation:

Surface To Atmosphere Radiant Exitance:
q = 0.93643 * 5.67037441918442945397099673188923087584012297029130e-8 W m-2 K-4 * (288^4) = 365.30616894307 W m-2

Atmosphere To Surface Radiant Exitance:
q = 5.67037441918442945397099673188923087584012297029130e-8 W m-2 K-4 * (255^4) = 239.757641819 W m-2

Net: 365.30616894307 W m-2 - 239.757641819 W m-2 = 125.54852712407 W m-2

---

Now, that seems to make sense, right? Except that the 'atmosphere to surface' radiant exitance (aka 'backradiation') violates the fundamental physical laws... energy does not and cannot spontaneously flow up an energy density gradient. And applying emissivity to the idealized blackbody form of the S-B equation is a misuse of the S-B equation (remember that it assumes ε = 1).

---

Using the Graybody form of the S-B equation:

Surface Radiant Exitance:
q = 0.93643 * 5.67037441918442945397099673188923087584012297029130e-8 W m-2 K-4 * ( (288^4) - (255^4)) = 140.78992041451 W m-2

---

Now, how can we ascertain which is correct?

Well, we can calculate the energy density of each, then perform the calculations.

---

e = T^4 a
a = 4σ/c = 7.5657332500339284719430800357226e-16 J m-3 K-4

Surface Energy Density:
e = (288^4) * 7.5657332500339284719430800357226e-16 J m-3 K-4 = 5.20500290293e-6 J m-3

Atmosphere Energy Density:
e = (255^4) * 7.5657332500339284719430800357226e-16 J m-3 K-4 = 3.1989816343e-6 J m-3

Energy Density Gradient:
5.20500290293e-6 J m-3 - 3.1989816343e-6 J m-3 = 2.0060212686e-6 J m-3

Surface To Atmosphere Radiant Exitance:
σ / a * Δe * εh = W m-2

5.67037441918442945397099673188923087584012297029130e-8 W m-2 K-4 / 7.5657332500339284719430800357226e-16 J m-3 K-4 * 2.0060212686e-6 J m-3 * 0.93643 = 140.78992041246 W m-2

Atmosphere To Surface Radiant Exitance:
5.67037441918442945397099673188923087584012297029130e-8 W m-2 K-4 / 7.5657332500339284719430800357226e-16 J m-3 K-4 * -2.0060212686e-6 J m-3 * 0.93643 = -140.78992041246 W m-2

---

In other words, 'Atmosphere To Surface' radiant exitance is zero (there can't be negative radiant exitance). The negative denotes that energy is flowing from surface to atmosphere.

As you can see, the proper method of calculating radiant exitance is by subtracting the energy density of the cooler object from the energy density of the warmer object to arrive at the energy density gradient, which determines radiant exitance of the warmer object.

The cooler object cannot emit in the direction of the warmer object because energy cannot spontaneously flow up an energy density gradient. To claim that it can is directly analogous to claiming that water can spontaneously flow uphill (ie: up a pressure gradient).

Most people cannot think in terms of energy, energy density and energy density gradient. We need to analogize to something they’re familiar with. Thus, just as, for instance, water only spontaneously flows down a pressure gradient, energy only spontaneously flows down an energy density gradient. That’s 2LoT in the Clausius Statement sense, in a nutshell. So one tack to take is to ask people if water can ever spontaneously flow uphill. Of course they’ll say, “No, water cannot flow uphill on its own.” Then show them dimensional analysis.

mass (M), length (L), time (T), absolute temperature (K), amount of substance (N), electric charge (Q), luminous intensity ©

We denote the dimensions like this: [Mx, Lx, Tx, Kx, Nx, Qx, Cx] where x = the number of that dimension

We typically remove dimensions which are not used.

Force: [M1 L1 T-2] /
Area: [M0 L2 T0] =
Pressure: [M1 L-1 T-2] /
Length: [M0 L1 T0] =
Pressure Gradient: [M1 L-2 T-2]

Explain to them that Pressure is Force / Area, and that Pressure Gradient is Pressure / Length. Remind them that water only spontaneously flows down a pressure gradient (ie: downhill). Then introduce energy. Tell them that energy is much like water. It requires an impetus to flow, just as water requires an impetus (pressure gradient) to flow. In the case of radiative energy, that impetus is a radiation energy density gradient, which is analogous to (and in fact, literally is) a radiation pressure gradient.

Energy: [M1 L2 T−2] /
Volume: [M0 L3 T0] =
Energy Density: [M1 L-1 T-2] /
Length: [M0 L1 T0] =
Energy Density Gradient: [M1 L-2 T-2]

Explain to them that Energy Density is Energy / Volume, and Energy Density Gradient is Energy Density / Length. Highlight the fact that Pressure and Energy Density have the same units (bolded above). Also highlight the fact that Pressure Gradient and Energy Density Gradient have the same units (bolded above).

So we’re talking about the same concept as water only spontaneously flowing down a pressure gradient (ie: downhill) when we talk of energy (of any form) only spontaneously flowing down an energy density gradient. Energy density is pressure, an energy density gradient is a pressure gradient… for energy.
It’s a bit more complicated for gases because they can convert that energy density to a change in volume (1 J m-3 = 1 Pa), for constant-pressure processes, which means the unconstrained volume of a gas will change such that its energy density (in J m-3) will tend toward being equal to pressure (in Pa) (see below). This is the underlying mechanism for convection. It should also have clued the climastrologists in to the fact that it is solar insolation and atmospheric pressure which ‘sets’ temperature, not any ‘global warming’ gases.

Since a warmer object will have higher radiation energy density at all wavelengths than a cooler object (because remember, temperature is a measure of radiation energy density, equal to the fourth root of radiation energy density divided by Stefan’s Constant):



… ‘backradiation’ can do nothing to warm the surface because energy cannot spontaneously radiatively flow from lower to higher radiation energy density, and thus CAGW is nothing more than a complex mathematical scam perpetrated to obtain multiple billions of dollars in funding for trough-grubbing line-toeing ‘scientists’ and to push a Marxist One World Government “Build Back Better” agenda.

“But they’ve measured backradiation!”, some may claim. Yeah, no.

https://claesjohnson.blogspot.com/2011/0...meter.html

As Professor Claes Johnson shows in that article on his website, pyrgeometers (the instrument typically used to ‘measure’ backradiation) utilize the same sort of misuse of the S-B equation as the climastrologists use. The bastardized form of the S-B equation used by pyrgeometers [ usually some form of q = (σ Th4 – σ Tc4) or equivalently Ld = Uemf/S + σTb, as outlined in the documentation for the instrument, with Uemf/S being negative in sign ] apriori assumes a subtraction of ‘cooler to warmer’ energy flow from ‘warmer to cooler’ energy flow, which as has been shown in this paper, is fallacious.

Do remember that photons, each a quantum of energy, are considered the force-carrying gauge bosons of the EM interaction.

Going back to dimensional analysis:
We start with Energy: [M1 L2 T−2] -
Force: [M1 L1 T-2] *
Length: [M0 L1 T0] = [M0 L0 T0]

We are left with nothing on the 'transmitting' end... [M0 L0 T0]. In other words, that Energy is used to apply a Force along a Length. It’s obvious then, that if an equal and opposing Force were applied along that Length, no energy can flow… this is just as true radiatively as it is mechanically.

That Force applied along a Length gives us (on the 'receiving' end):
Force: [M1 L1 T-2] *
Length: [M0 L1 T0] =
Work: [M1 L2 T-2]

You'll note that Energy and Work have the same units:
Work: [M1 L2 T-2] = Energy: [M1 L2 T−2]

For those who want to put it in terms of Momentum:
Momentum: [M1 L1 T−1] *
Velocity: [M0 L1 T-1] =
Work: [M1 L2 T−2]

That means Energy Expended = Force * Length = Momentum * Velocity = Work

There's a reason for that. Free Energy is defined as that energy capable of performing work. This is reflected in the equation for Free Energy (represented here as a single object and its environment):

F = U – TS + PV
Where: F = Free Energy; U = internal energy; T = absolute temp; S = final entropy; TS = energy the object can receive from the environment; PV = work done to give the system final volume V at pressure P

If U > TS + PV, F > 0… energy must flow from object to environment.
If U = TS + PV, F = 0… no energy can flow to or from the object.
If U < TS + PV, F < 0… energy must flow from environment to object.

Of course, if we were talking about a system with only two objects with the same physical parameters and nothing else in the system, we could represent the Free Energy as: F = U1 – U2

Which is better represented as internal energy over volume to get energy density (since internal energy is an extensive property), converting the calculation to that of an intensive property and thus allowing us to compare dissimilar-sized objects: F = U1/V1 – U2/V2 = e1 - e2

And that’s exactly what the S-B equation does. Remember that temperature is a measure of radiation energy density, equal to the fourth root of radiation energy density divided by the radiation constant (Stefan’s Constant). Remember that I wrote above:

---

q = (ε c (e_h - e_c)) / 4

Canceling units, we get J sec-1 m-2, which is W m-2 (1 J sec-1 = 1 W).
W m-2 = (m sec-1 (ΔJ m-3)) / 4

One can see that the S-B equation is all about subtracting the radiation energy density of the cooler object from the radiation energy density of the warmer object (to arrive at the radiation energy density gradient) because Free Energy is all about subtracting the energy density of one object from the energy density of the other object (no matter the form of that energy).

---

"But how does that make CAGW a scam?", some may ask... well, because we're being lied to, based upon an unscientific premise.

The climatologists have misused the Stefan-Boltzmann (S-B) equation (and the fundamental physical laws), and in the process, have practically flipped reality on its head… polyatomics (CO2, H2O, etc.) are not "global warming gases", they are net atmospheric radiative coolants (radiative emission to space being the only way that Earth can shed energy); monoatomics (Ar) are not inert gases that have no effect upon climate, they are the actual "greenhouse gases" (because they cannot emit IR, and thus cannot shed energy to space... they dilute the radiative coolant gases); homonuclear diatomics (N2, O2) are somewhere in between… they can radiatively emit IR (and thus shed energy from the system known as 'Earth'), but only under certain conditions (collisional perturbation of their net-zero electric dipole, which is why homonuclear diatomic vibrational mode quantum states are meta-stable and relatively long-lived. Collisions happen exponentially less frequently as altitude increases), and thus are "greenhouse gases" like the monoatomics, just not to the same extent.

We live, at the planet's surface, in what can be analogized to the evaporator section of a world-sized AC unit, with polyatomics being net atmospheric radiative coolants (a higher concentration of them increases thermodynamic coupling between heat source (surface) and sink (space)), and with monoatomics and homonuclear diatomics playing the same role as non-condensable gases would play in an AC unit... diluting the polyatomic radiative gases which transit the majority of the energy, thus reducing the efficiency at which energy is transited from surface to upper atmosphere, then radiatively emitted to space.

Think about it this way... we all know the air warms up during the daytime as the planet's surface absorbs energy from the sun. Conduction of that energy when air contacts the planet’s surface is the major reason air warms up.

How does that ~99% of the atmosphere (N2, O2, Ar) cool down? It cannot effectively radiatively emit.

Convection moves energy around in the atmosphere, but it cannot shed energy to space. Conduction depends upon thermal contact with other matter and since space is essentially a vacuum, conduction cannot shed energy to space… this leaves only radiative emission. The only way our planet can shed energy is via radiative emission to space. Fully ~76.2% of all surface energy is removed via convection, advection and evaporation. The surface only radiatively emits ~23.8% of all surface energy to space. That ~76.2% must be emitted to space by the atmosphere.



Thus, common sense dictates that the thermal energy of the constituents of the atmosphere which cannot effectively radiatively emit (N2, O2, Ar) must be transferred to the so-called ‘greenhouse gases’ (CO2 being a lesser contributor below the tropopause and the largest contributor above the tropopause, water vapor being the main contributor below the tropopause) which can radiatively emit and thus shed that energy to space. Peer-reviewed studies corroborating this are referenced in the linked file in the first post.

So, far from being ‘greenhouse gases’ which ‘trap heat’ in the atmosphere, those polyatomic radiative gases actually shed energy from the atmosphere to space. They are net atmospheric radiative coolants.

In short, in an atmosphere sufficiently dense such that collisional energy transfer can significantly occur, all polyatomic radiative molecules play the part of atmospheric radiative coolants at and above the temperature at which the combined translational mode energy of two colliding particles (atoms or molecules) exceeds the lowest excited vibrational mode quantum state energy of the radiative molecule. Below this temperature, they act to warm the atmosphere via thermalization (the mechanism the climate alarmists claim happens all the time), but if that occurs below the tropopause, the net result is an increase of Convective Available Potential Energy, which increases convection, which is a net cooling process. It is a gradation... as temperature increases, so too does the population of vibrationally excited polyatomics, and thus increases radiative emission. For CO2, that 'transition temperature' (the temperature at which the molecule transitions from being 'net warmant' to 'net coolant' and vice versa) is ~288 K.




You will note that while the graphic above only addresses CO2 and N2, the same applies to all atmospheric gases due to the Equipartition Theorem.

The Boltzmann Factor (for air at 288 K and for the combined translation mode (kinetic) energy of two colliding particles (atoms or molecules) necessary to excite CO2's lowest vibrational mode quantum state) is ~0.8461... in other words, for every 100 air particles at the Most Probable Speed (MPS), ~84 other particles will have sufficient combined translational mode (kinetic) energy to excite CO2's lowest excited vibrational mode quantum state CO2{v21(1)} during collision.

The climatologists only told people half the story (thermalization by CO2 via vibrational mode to translation mode (v-t) collisional energy transfer processes). They didn't tell anyone about the inverse (translational mode to vibrational mode (t-v) collisional energy transfer processes, (then that energy being radiatively emitted to space)), which is a cooling process. That didn't fit their doomsaying narrative, so they left it out.

In other words, the climatologists only told people about the warming part (thermalization), not the cooling part. In order to hew to the fundamental physical laws, one must consider energy flow both to and from the CO2 molecule.

This doesn’t just apply to CO2, however. It applies to all atmospheric polyatomic molecules. In fact, far from the 'global warming gas' claimed by the climatologists, water acts as a literal refrigerant (in the strict ‘refrigeration cycle’ sense) below the tropopause:

The refrigeration cycle (Earth) [AC system]:

A liquid evaporates at the heat source (the surface) [in the evaporator], it is transported (convected) [via an AC compressor], it gives up its energy to the heat sink and undergoes phase change (emits radiation in the upper atmosphere, the majority of which is upwelling owing to the mean free path length / altitude / air density relation) [in the condenser], it is transported (falls as rain or snow) [via that AC compressor], and the cycle repeats.

That’s kind of why, after all, the humid adiabatic lapse rate (~3.5 to ~6.5 K km-1) is lower than the dry adiabatic lapse rate (~9.81 K km-1).

You will note that the dry adiabatic lapse rate is due to the monoatomics and homonuclear diatomics... we've removed in this case the predominant polyatomic which reduces lapse rate.

Remember that an actual greenhouse works by hindering convection of energy out of the greenhouse.

In an atmosphere consisting of solely monoatomics and homonuclear diatomics (ie: no polyatomic radiative molecules), the atoms / molecules could pick up energy via conduction by contacting the surface, just as the polyatomics do; they could convect just as the polyatomics do… but once in the upper atmosphere, they could not as effectively radiatively emit that energy, the upper atmosphere would warm, lending less buoyancy to convecting air, thus hindering convection… and that’s how an actual greenhouse works, by hindering convection.

For homonuclear diatomics, there would be some collisional perturbation and thus some emission in the atmosphere, but by and large the atmosphere could not effectively emit (especially at higher altitudes, because the chance of collision decreases exponentially with altitude).

Thus the surface would have to radiatively emit that energy (which is currently ~76.2% of all energy removed from the surface via radiation, convection and evaporation) instead… and a higher radiant exitance implies a higher surface temperature.

These concepts used to be common knowledge. Somewhere along the way, the concepts got skewed to fit a particular narrative. Eventually, the concepts described herein will be common knowledge again, whereupon CAGW and its offshoots will be dumped on the midden heap of bad scientific ideas.

---

Now, having gotten through all of that, there is a paper which goes into much more depth.

To date, there's not been a warmist physicist nor a climastrologist who has been able to refute the scientific reality contained within that file, though several have tried. The last was physicist Bob Wentworth, who gave up after I backed him into numerous logical traps. Some have even changed their stance because of the information within that file.

That file:
https://tinyurl.com/ClimateSanity

Excellent Post!

I understood all of the science, and it was well put together. You laid out why the Climate Sensitivity number is wrong (thus all of the models using them are wrong).

[Billy_Bob Patriot Master]

Note:

The math will be too complex for most layman readers. I am impressed that you took the time to explain where the numbers come from.  The post is very informative as to the "scam" AND WHY IT IS A SCAM. Laying out the misuse of physical properties and conveniently ignoring half of the energy transfer process is very enlightening. Using inappropriate a Black Body, when a Grey Body is the correct one to be using, further exposes the misuse of the physics involved and is scientific fraud at a minimum.

Again, Welcome to the Forum.

Bill

[Morpheus]

Note:

The math will be too complex for most layman readers. I am impressed that you took the time to explain where the numbers come from.  The post is very informative as to the "scam" AND WHY IT IS A SCAM. Laying out the misuse of physical properties and conveniently ignoring half of the energy transfer process is very enlightening. Using inappropriate a Black Body, when a Grey Body is the correct one to be using, further exposes the misuse of the physics involved and is scientific fraud at a minimum.

Again, Welcome to the Forum.

Bill

I've been trying to simplify the paper to make it more readable and understandable by the layperson, but my brain just keeps adding complexity, and my engineer-brain has zero artistic ability (I still draw stick-figures for people, have no perspective to draw in multiple dimensions, and cannot draw a straight line freehand to save my life), so it's difficult to put the concepts to graphics (especially animated graphics, as would be required here).

If anyone wants to take a crack at it, I'd appreciate it. As I said above, anyone can feel free to use what I write however they wish, no attribution necessary.

[ChemEngineer]

Yeah but everybody knows that........

[Morpheus]

Yeah but everybody knows that........

There's some things in the paper (derivations of the S-B equation, for instance) that I don't think anyone's ever seen before. Some very interesting relations between electrical theory and thermodynamics, too. I even created a model circuit in a circuit simulator to solve a thermodynamics problem.

[Morpheus]

Yeah but everybody knows that........

There's some things in the paper (derivations of the S-B equation, for instance) that I don't think anyone's ever seen before. Some very interesting relations between electrical theory and thermodynamics, too. I even created a model circuit in a circuit simulator to solve a thermodynamics problem.

The below seems to be gaining quite a bit of traction against those who would deny the mathematics... so if you're in an online argument, hit them with what's below.

---

---

Climatologists claim that CO2 is a "global warming gas" because of their purported "greenhouse effect", which is caused by "backradiation". Disprove "backradiation" as unphysical, and the "greenhouse effect" falls, and thus falls AGW / CAGW.

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

There are two forms of the S-B equation:



[1] Idealized Blackbody Object form (assumes emission to 0 K and ε = 1 by definition):
q_bb = ε σ (T_h^4 - T_c^4)
         = 1 σ (T_h^4 - 0 K)
         =    σ  T^4

[2] Graybody Object form (assumes emission to > 0 K and ε < 1):
q_gb = ε σ (T_h^4 - T_c^4)

Climatologists misuse the S-B equation, using the idealized blackbody form of the equation upon real-world graybody objects. This essentially isolates each object into its own system so objects cannot interact via the ambient EM field, it assumes emission to 0 K, and it thus artificially inflates radiant exitance of all calculated-upon objects. Thus the climatologists must carry these incorrect values through their calculations and cancel them on the back end to get their equation to balance, subtracting a wholly-fictive 'cooler to warmer' energy flow from the real (but too high because it was calculated for emission to 0 K) 'warmer to cooler' energy flow.

The S-B equation for graybody objects isn't meant to be used to subtract a wholly-fictive 'cooler to warmer' energy flow from the real (but too high because it was calculated for emission to 0 K) 'warmer to cooler' energy flow, it's meant to be used by subtracting cooler object energy density from warmer object energy density to arrive at the energy density gradient, which determines radiant exitance of the warmer object. This is true even for the traditional form of the S-B equation, because temperature is a measure of radiation energy density, per Stefan's Law.

That wholly fictive 'cooler to warmer' energy flow is otherwise known as 'backradiation'. It is nothing more than a mathematical artifact due to the misuse of the S-B equation. It does not and cannot exist. Its existence would imply rampant violations of the fundamental physical laws.

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

Temperature is equal to the fourth root of radiation energy density divided by Stefan's Constant (ie: the radiation constant), per Stefan's Law.

e = T^4 a
a = 4σ/c
e = T^4 4σ/c
T^4 = e/(4σ/c)
T = 4^√(e/(4σ/c))
T = 4^√(e/a)

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

The traditional Stefan-Boltzmann equation for graybody objects:
q = ε_h σ (T_h^4 – T_c^4)

[1] ∴ q = ε_h σ ((e_h / (4σ / c)) – (e_c / (4σ / c)))

Canceling units, we get J sec-1 m-2, which is W m-2 (1 J sec-1 = 1 W).
W m-2 = W m-2 K-4 * (Δ(J m-3 / (W m-2 K-4 / m sec-1)))

[2] ∴ q = (ε_h c (e_h - e_c)) / 4

Canceling units, we get J sec-1 m-2, which is W m-2 (1 J sec-1 = 1 W).
W m-2 = (m sec-1 (ΔJ m-3)) / 4

One can see from the immediately-above equation that the Stefan-Boltzmann (S-B) equation for graybody objects is all about subtracting the radiation energy density of the cooler object from the radiation energy density of the warmer object.

[3] ∴ q = (ε_h * (σ / a) * Δe)

Canceling units, we get W m-2.
W m-2 = ((W m-2 K-4 / J m-3 K-4) * ΔJ m-3)

You will note that σ = (a * c) / 4… the S-B Constant equals Stefan’s Constant multiplied by the speed of light in vacua divided by 4.

[4] ∴ q = (ε_h * ((a * c) / a) * Δe) / 4 = (ε_h * c * Δe) / 4

Canceling units, we get J sec-1 m-2, which is W m-2 (1 J sec-1 = 1 W).
W m-2 = (m sec-1 * ΔJ m-3) / 4

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

The Stefan-Boltzmann equation in energy density form ([3] above):
σ / a * Δe * ε_h = W m-2

σ / a = 5.67037441918442945397099673188923087584012297029130e-8 W m-2 K-4 / 7.5657332500339284719430800357226e-16 J m-3 K-4 = 74948114.5024 W m-2 / J m-3.

Well, what do you know... that's the conversion factor for radiant exitance (W m-2) and energy density (J m-3)!

It's almost as if the radiant exitance of graybody objects is determined by the energy density gradient, exactly as I've repeatedly stated, right?

Energy can't even spontaneously flow when there is zero energy density gradient:

σ [W m-2 K-4] / a [J m-3 K-4] * Δe [J m-3] * ε_h = [W m-2]

σ [W m-2 K-4] / a [J m-3 K-4] * 0 [J m-3] * ε_h = 0 [W m-2]

... it is certainly not going to spontaneously flow up an energy density gradient.

---

Note 2LoT in the Clausius Statement sense:
"Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time."

'Heat' [M1 L2 T-2] is definitionally an energy [M1 L2 T-2] flux (note the identical dimensionality), thus equivalently:
"Energy can never flow from a colder to a warmer body without some other change, connected therewith, occurring at the same time."

That "some other change" typically being external energy doing work upon the system energy to pump it up the energy density gradient, which is what occurs in, for example, AC units and refrigerators.

Remember that temperature is a measure of energy density, equal to the fourth root of radiation energy density divided by Stefan's Constant, per Stefan's Law, thus equivalently:
"Energy can never flow from a lower to a higher energy density without some other change, connected therewith, occurring at the same time."

Or, as I put it:
"Energy cannot spontaneously flow up an energy density gradient."

My statement is merely a restatement of 2LoT in the Clausius Statement sense.

---

Do remember that a warmer object will have higher energy density at all wavelengths than a cooler object:



... so there is no physical way possible by which energy can spontaneously flow from cooler to warmer. 'Backradiation' is nothing more than a mathematical artifact due to the climatologists misusing the S-B equation.

The above completely destroys AGW and CAGW, because they are predicated upon the existence of "backradiation" (radiation spontaneously flowing up an energy density gradient) as the causative agent for the climatologists' claimed "greenhouse effect".

---

The IPCC has attempted to shift from a "backradiation" causative agent for their "greenhouse effect" to a "blanket" analogy (while still invoking "backradiation" as the causative agent of that atmospheric "blanket"), to wit:

https://www.ipcc.ch/report/ar6/wg1/downloads/outreach/IPCC_AR6_WGI_SummaryForAll.pdf

Some gases in our atmosphere – such as carbon dioxide, methane and nitrous oxide – act like an insulation blanket for the Earth. They warm the Earth by making it harder for heat to be released into outer space. Much like adding a blanket around your body heats you up and keeps you warm, or the walls of a greenhouse help keep the air inside warmer than its surroundings.

https://report.ipcc.ch/ar6/wg1/IPCC_AR6_WGI_FullReport.pdf#page=2249

Greenhouse effect: The infrared radiative effect of all infrared absorbing constituents in the atmosphere. Greenhouse gases (GHGs), clouds, and some aerosols absorb terrestrial radiation emitted by the Earth’s surface and elsewhere in the atmosphere. These substances emit infrared radiation in all directions, but, everything else being equal,the net amount emitted to space is normally less than would have been emitted in the absence of these absorbers because of the decline of temperature with altitude in the troposphere and the consequent weakening of emission. An increase in the concentration of GHGs increases the magnitude of this effect; the difference is sometimes called the enhanced greenhouse effect. The change in a GHG concentration because of anthropogenic emissions contributes to an instantaneous radiative forcing. Earth’s surface temperature and troposphere warm in response to this forcing, gradually restoring the radiative balance at the top of the atmosphere.

Except that as temperature increases, (t-v) (translational mode -to- vibrational mode) collisional energy transfer processes increase and thus spectral emission increases only because CO2 is a net atmospheric radiative coolant (transferring translational mode energy to vibrational mode energy, then radiatively emitting it). So they are attempting to claim that CO2 is a "global warming gas" and simultaneously a net atmospheric radiative coolant, a contradiction... which is why their claims make no sense upon close examination.

Their claim, "These substances emit infrared radiation in all directions" is their attempt at invoking their old "backradiation" causative agent, just hidden in more obfuscatory verbiage... as shown above, polyatomics do not "emit infrared radiation in all directions", they can only emit when that emitted energy can spontaneously flow down an energy density gradient, which means they cannot emit downwelling (except under temperature inversion conditions).

Further, their "blanket" analogy doesn't even make sense... a blanket hinders convection so the atmosphere doesn't cool you off as much. Yet they attempt to analogize the atmosphere to a blanket. Blankets don't convect (in fact, the very mechanism by which blankets work is hindrance of convection), the atmosphere does convect.

Further, adding more radiative polyatomics to the atmosphere means more emitters per parcel of air, thus that parcel can emit more radiation to space... diametrically opposite to the claims of the IPCC.

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

The CAGW (Catastrophic Anthropogenic Global Warming, due to CO2) hypothesis has been disproved... it does not reflect reality.

CAGW Is Nothing More Than A Complex Mathematical Scam... The Proof
https://www.patriotaction.us/showthread.php?tid=2711

The takeaways:
1) The climatologists have conflated their purported "greenhouse effect" with the Kelvin-Helmholtz Gravitational Auto-Compression Effect (aka the lapse rate).

2) The climatologists claim the causative agent for their purported "greenhouse effect" to be "backradiation".

3) The Kelvin-Helmholtz Gravitational Auto-Compression Effect's causative agent is, of course, gravity.

4) "Backradiation" is physically impossible because energy cannot spontaneously flow up an energy density gradient.

5) The climatologists misuse the Stefan-Boltzmann (S-B) equation, using the idealized blackbody form of the equation upon graybody objects, which manufactures out of thin air their purported "backradiation". It is only a mathematical artifact due to that aforementioned misuse of the S-B equation. It does not and cannot actually exist. Its existence would imply rampant violations of the fundamental physical laws.

6) Polyatomic molecules are net atmospheric radiative coolants, not "global warming" gases. Far from the 'global warming gas' claimed by the climatologists, water acts as a literal refrigerant (in the strict 'refrigeration cycle' sense) below the tropopause. CO2 is the most prevalent atmospheric radiative coolant above the tropopause and the second-most prevalent (behind water vapor) below the tropopause. Peer reviewed studies corroborating this are referenced in the paper at the link above.

---

---

[Morpheus]

Yeah but everybody knows that........

There's some things in the paper (derivations of the S-B equation, for instance) that I don't think anyone's ever seen before. Some very interesting relations between electrical theory and thermodynamics, too. I even created a model circuit in a circuit simulator to solve a thermodynamics problem.

The below seems to be gaining quite a bit of traction against those who would deny the mathematics... so if you're in an online argument, hit them with what's below.

---

---

Temperature is equal to the fourth root of radiation energy density divided by Stefan's Constant (ie: the radiation constant), per Stefan's Law.

e = T^4 a
a = 4σ/c
e = T^4 4σ/c
T^4 = e/(4σ/c)
T = 4^√(e/(4σ/c))
T = 4^√(e/a)

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

The traditional Stefan-Boltzmann equation for graybody objects:
q = ε_h σ (T_h^4 – T_c^4)

[1] ∴ q = ε_h σ ((e_h / (4σ / c)) – (e_c / (4σ / c)))

Canceling units, we get J sec-1 m-2, which is W m-2 (1 J sec-1 = 1 W).
W m-2 = W m-2 K-4 * (Δ(J m-3 / (W m-2 K-4 / m sec-1)))

[2] ∴ q = (ε_h c (e_h - e_c)) / 4

Canceling units, we get J sec-1 m-2, which is W m-2 (1 J sec-1 = 1 W).
W m-2 = (m sec-1 (ΔJ m-3)) / 4

One can see from the immediately-above equation that the Stefan-Boltzmann (S-B) equation for graybody objects is all about subtracting the radiation energy density of the cooler object from the radiation energy density of the warmer object.

[3] ∴ q = (ε_h * (σ / a) * Δe)

Canceling units, we get W m-2.
W m-2 = ((W m-2 K-4 / J m-3 K-4) * ΔJ m-3)

You will note that σ = (a * c) / 4… the S-B Constant equals Stefan’s Constant multiplied by the speed of light in vacua divided by 4.

[4] ∴ q = (ε_h * ((a * c) / a) * Δe) / 4 = (ε_h * c * Δe) / 4

Canceling units, we get J sec-1 m-2, which is W m-2 (1 J sec-1 = 1 W).
W m-2 = (m sec-1 * ΔJ m-3) / 4

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

The Stefan-Boltzmann equation in energy density form ([3] above):
σ / a * Δe * ε_h = W m-2

σ / a = 5.67037441918442945397099673188923087584012297029130e-8 W m-2 K-4 / 7.5657332500339284719430800357226e-16 J m-3 K-4 = 74948114.5024 W m-2 / J m-3.

Well, what do you know... that's the conversion factor for radiant exitance (W m-2) and energy density (J m-3)!

It's almost as if the radiant exitance of graybody objects is determined by the energy density gradient, exactly as I've repeatedly stated, right?

Energy can't even spontaneously flow when there is zero energy density gradient:

σ [W m-2 K-4] / a [J m-3 K-4] * Δe [J m-3] * ε_h = [W m-2]

σ [W m-2 K-4] / a [J m-3 K-4] * 0 [J m-3] * ε_h = 0 [W m-2]

... it's most certainly not going to spontaneously flow up an energy density gradient.

Denying this is denying a fundamental precept of mathematics... that anything multiplied by zero equates to zero.

And do remember that a warmer object will have higher energy density at all wavelengths than a cooler object:



... so there is no physical way possible by which energy can spontaneously flow from cooler to warmer. 'Backradiation' is nothing more than a mathematical artifact due to the climatologists misusing the S-B equation.

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

The above completely destroys AGW and CAGW, because they are predicated upon the existence of "backradiation" (radiation spontaneously flowing up an energy density gradient) as the causative agent for the climatologists' claimed "greenhouse effect".

The CAGW (Catastrophic Anthropogenic Global Warming, due to CO2) hypothesis has been disproved... it does not reflect reality.

CAGW Is Nothing More Than A Complex Mathematical Scam... The Proof
https://www.patriotaction.us/showthread.php?tid=2711

The takeaways:
1) The climatologists have conflated their purported "greenhouse effect" with the Kelvin-Helmholtz Gravitational Auto-Compression Effect (aka the lapse rate).

2) The climatologists claim the causative agent for their purported "greenhouse effect" to be "backradiation".

3) The Kelvin-Helmholtz Gravitational Auto-Compression Effect's causative agent is, of course, gravity.

4) "Backradiation" is physically impossible because energy cannot spontaneously flow up an energy density gradient.

5) The climatologists misuse the Stefan-Boltzmann (S-B) equation, using the idealized blackbody form of the equation upon graybody objects, which manufactures out of thin air their purported "backradiation". It is only a mathematical artifact due to that aforementioned misuse of the S-B equation. It does not and cannot actually exist. Its existence would imply rampant violations of the fundamental physical laws.

6) Polyatomic molecules are net atmospheric radiative coolants, not "global warming" gases. Far from the 'global warming gas' claimed by the climatologists, water acts as a literal refrigerant (in the strict 'refrigeration cycle' sense) below the tropopause. CO2 is the most prevalent atmospheric radiative coolant above the tropopause and the second-most prevalent (behind water vapor) below the tropopause.

---

---

One of the sticking points of some folks arguing against the mathematics which disprove AGW and CAGW is the Kelvin-Helmholtz Gravitational Auto-Compression Effect, the underlying mechanism of the lapse rate... they challenge by claiming that that effect only happens with stellar bodies, and does not happen on Earth.

They need to push themselves to study a bit deeper.

"Although the term ‘auto-compression’ may be unfamiliar to some, this can be seen as simply an engineering term for what meteorologists call the ‘lapse rate’ and astronomers call the ‘Kelvin-Helmholtz’ contraction."

https://article.sciencepublishinggroup.c...0170606.18

[Morpheus]

Yeah but everybody knows that........

There's some things in the paper (derivations of the S-B equation, for instance) that I don't think anyone's ever seen before. Some very interesting relations between electrical theory and thermodynamics, too. I even created a model circuit in a circuit simulator to solve a thermodynamics problem.

The below seems to be gaining quite a bit of traction against those who would deny the mathematics... so if you're in an online argument, hit them with what's below.

---

---

Temperature is equal to the fourth root of radiation energy density divided by Stefan's Constant (ie: the radiation constant), per Stefan's Law.

e = T^4 a
a = 4σ/c
e = T^4 4σ/c
T^4 = e/(4σ/c)
T = 4^√(e/(4σ/c))
T = 4^√(e/a)

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

The traditional Stefan-Boltzmann equation for graybody objects:
q = ε_h σ (T_h^4 – T_c^4)

[1] ∴ q = ε_h σ ((e_h / (4σ / c)) – (e_c / (4σ / c)))

Canceling units, we get J sec-1 m-2, which is W m-2 (1 J sec-1 = 1 W).
W m-2 = W m-2 K-4 * (Δ(J m-3 / (W m-2 K-4 / m sec-1)))

[2] ∴ q = (ε_h c (e_h - e_c)) / 4

Canceling units, we get J sec-1 m-2, which is W m-2 (1 J sec-1 = 1 W).
W m-2 = (m sec-1 (ΔJ m-3)) / 4

One can see from the immediately-above equation that the Stefan-Boltzmann (S-B) equation for graybody objects is all about subtracting the radiation energy density of the cooler object from the radiation energy density of the warmer object.

[3] ∴ q = (ε_h * (σ / a) * Δe)

Canceling units, we get W m-2.
W m-2 = ((W m-2 K-4 / J m-3 K-4) * ΔJ m-3)

You will note that σ = (a * c) / 4… the S-B Constant equals Stefan’s Constant multiplied by the speed of light in vacua divided by 4.

[4] ∴ q = (ε_h * ((a * c) / a) * Δe) / 4 = (ε_h * c * Δe) / 4

Canceling units, we get J sec-1 m-2, which is W m-2 (1 J sec-1 = 1 W).
W m-2 = (m sec-1 * ΔJ m-3) / 4

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

The Stefan-Boltzmann equation in energy density form ([3] above):
σ / a * Δe * ε_h = W m-2

σ / a = 5.67037441918442945397099673188923087584012297029130e-8 W m-2 K-4 / 7.5657332500339284719430800357226e-16 J m-3 K-4 = 74948114.5024 W m-2 / J m-3.

Well, what do you know... that's the conversion factor for radiant exitance (W m-2) and energy density (J m-3)!

It's almost as if the radiant exitance of graybody objects is determined by the energy density gradient, exactly as I've repeatedly stated, right?

Energy can't even spontaneously flow when there is zero energy density gradient:

σ [W m-2 K-4] / a [J m-3 K-4] * Δe [J m-3] * ε_h = [W m-2]

σ [W m-2 K-4] / a [J m-3 K-4] * 0 [J m-3] * ε_h = 0 [W m-2]

... it's most certainly not going to spontaneously flow up an energy density gradient.

Denying this is denying a fundamental precept of mathematics... that anything multiplied by zero equates to zero.

And do remember that a warmer object will have higher energy density at all wavelengths than a cooler object:



... so there is no physical way possible by which energy can spontaneously flow from cooler to warmer. 'Backradiation' is nothing more than a mathematical artifact due to the climatologists misusing the S-B equation.

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

The above completely destroys AGW and CAGW, because they are predicated upon the existence of "backradiation" (radiation spontaneously flowing up an energy density gradient) as the causative agent for the climatologists' claimed "greenhouse effect".

The CAGW (Catastrophic Anthropogenic Global Warming, due to CO2) hypothesis has been disproved... it does not reflect reality.

CAGW Is Nothing More Than A Complex Mathematical Scam... The Proof
https://www.patriotaction.us/showthread.php?tid=2711

The takeaways:
1) The climatologists have conflated their purported "greenhouse effect" with the Kelvin-Helmholtz Gravitational Auto-Compression Effect (aka the lapse rate).

2) The climatologists claim the causative agent for their purported "greenhouse effect" to be "backradiation".

3) The Kelvin-Helmholtz Gravitational Auto-Compression Effect's causative agent is, of course, gravity.

4) "Backradiation" is physically impossible because energy cannot spontaneously flow up an energy density gradient.

5) The climatologists misuse the Stefan-Boltzmann (S-B) equation, using the idealized blackbody form of the equation upon graybody objects, which manufactures out of thin air their purported "backradiation". It is only a mathematical artifact due to that aforementioned misuse of the S-B equation. It does not and cannot actually exist. Its existence would imply rampant violations of the fundamental physical laws.

6) Polyatomic molecules are net atmospheric radiative coolants, not "global warming" gases. Far from the 'global warming gas' claimed by the climatologists, water acts as a literal refrigerant (in the strict 'refrigeration cycle' sense) below the tropopause. CO2 is the most prevalent atmospheric radiative coolant above the tropopause and the second-most prevalent (behind water vapor) below the tropopause.

---

---

One of the sticking points of some folks arguing against the mathematics which disprove AGW and CAGW is the Kelvin-Helmholtz Gravitational Auto-Compression Effect, the underlying mechanism of the lapse rate... they challenge by claiming that that effect only happens with stellar bodies, and does not happen on Earth.

They need to push themselves to study a bit deeper.

"Although the term ‘auto-compression’ may be unfamiliar to some, this can be seen as simply an engineering term for what meteorologists call the ‘lapse rate’ and astronomers call the ‘Kelvin-Helmholtz’ contraction."

https://article.sciencepublishinggroup.c...0170606.18

Another tack the warmists try to take is to claim that energy can flow willy-nilly without regard to the energy density gradient at the quantum scale. In reality, 2LoT is rigorously observed at the quantum scale, with a whole subset family of conditions which must be met for any change of state to take place, that subset family of conditions making up the macroscopic 2LoT.

Code:

https://sci-hub.st/https://doi.org/10.1073/pnas.1411728112

[Morpheus]

So we find that "backradiation" (a mathematical artifact conjured from thin air via a misuse of the Stefan-Boltzmann (S-B) equation in the climatologist's Energy Balance Climate Models) cannot exist because energy cannot spontaneously flow up an energy density gradient without violating the fundamental physical laws. Thus the "greenhouse effect (due to backradiation)" cannot exist. Thus "greenhouse gases (due to the greenhouse effect (due to backradiation))" cannot exist.

The only effect a changing concentration of any given atmospheric gas has is its effect upon the adiabatic lapse rate. And we can calculate the exact change in the lapse rate (and thus the change in surface temperature) for any given change in concentration for any given gas. The one thing we cannot calculate yet is the change in magnitude of radiative cooling in the upper atmosphere with an increasing concentration of polyatomics, which sets the 'starting point' of the lapse rate lower, which translates down through the lapse rate to a cooler surface. We know it's a significant effect (stratospheric cooling of as much as 2 K from 1986 - 2022).

Let's calculate the Specific Lapse Rate of each constituent atomic or molecular species of the atmosphere. The Specific Lapse Rate is what the Adiabatic Lapse Rate would be if the atmosphere consisted of only that particular gas.

Idealized dry gas molar heat capacity lapse rate:
If we take ϒ = 1.404, g = 9.80665 m s-2, R = 8.31446261815324 J mol-1 K-1 and M = 28.9647 g mol-1, then:

dT / dh = -0.4/1.404 * (((28.9647 g mol-1) * 9.80665 m s-2) / 8.31446261815324 J mol-1 K-1) = -9.7330377706482238008458858152373 K km-1

The stated molar isobaric heat capacity for dry air is Cp = 7/2 R
7 / 2 * 8.31446261815324 J mol-1 K-1 = 29.10061916353634 J mol-1 K-1

∴ Molar Heat Capacity / 7 * 2 = Specific Gas Constant

dT / dh = -0.4/1.404 * (((Molar Mass) * 9.80665 m s-2) / Specific Gas Constant) = Specific Lapse Rate

The below data is taken from the model atmosphere I constructed in my paper at:

https://www.patriotaction.us/showthread.php?tid=2711

... to calculate the Specific Lapse Rate below:

Code:

Symbol: | Molar Mass:       | Molar Heat Capacity: | Specific Lapse Rate (SLR):
-------------------------------------------------------------------------------
H2      | 2.01588 g mol-1   | 28.82 J mol-1 K-1    | 0.6859482857817 K km-1
He      | 4.002602 g mol-1  | 20.7862 J mol-1 K-1  | 1.8883738683977 K km-1
H2O     | 18.01528 g mol-1  | 75.327 J mol-1 K-1   | 2.3453681364178 K km-1
CH4     | 16.04246 g mol-1  | 35.69 J mol-1 K-1    | 4.4080355942551 K km-1
N2      | 28.0134 g mol-1   | 29.12 J mol-1 K-1    | 9.4339738283240 K km-1
CO      | 28.0101 g mol-1   | 29.1 J mol-1 K-1     | 9.4393555726775 K km-1
Ne      | 20.1797 g mol-1   | 20.7862 J mol-1 K-1  | 9.5205114453312 K km-1
O2      | 31.9988 g mol-1   | 29.38 J mol-1 K-1    | 10.680770320623 K km-1
N2O     | 44.0128 g mol-1   | 38.6 J mol-1 K-1     | 11.181816712950 K km-1
CO2     | 44.0095 g mol-1   | 36.94 J mol-1 K-1    | 11.683426182319 K km-1
O3      | 47.9982 g mol-1   | 39.22  J mol-1 K-1   | 12.001569302138 K km-1
NO2     | 46.0055 g mol-1   | 37.2 J mol-1 K-1     | 12.127952596066 K km-1
SO2     | 64.0638 g mol-1   | 39.87  J mol-1 K-1   | 15.757493460485 K km-1
Ar      | 39.948 g mol-1    | 20.7862 J mol-1 K-1  | 18.846929895790 K km-1
SF6     | 146.06 g mol-1    | 93 J mol-1 K-1       | 30.187357269247 K km-1
Kr      | 83.798 g mol-1    | 20.95 J mol-1 K-1    | 39.225663804284 K km-1
I2      | 253.80894 g mol-1 | 54.43 J mol-1 K-1    | 45.728742264382 K km-1
Xe      | 131.293 g mol-1   | 21.01 J mol-1 K-1    | 61.282460659191 K km-1

Note the calculated Dry Adiabatic Lapse Rate above: -9.7330377706482238008458858152373 K km-1

The negative sign means temperature decreases with altitude. Usually we leave the negative sign off, with the knowledge that temperature decreases with altitude.

(N2) 9.433973828324 K km-1 * 0.780761158 +
(O2) 10.680770320623 K km-1 * 0.20944121395198 +
(Ar) 18.84692989579 K km-1 * 0.00934 +
(CO2) 11.683426182319 K km-1 * 0.00043 +
(CH4) 4.4080355942551 K km-1 * 0.0000018 +
(Xe) 61.282460659191 K km-1 * 0.0000000869565217391 +
(SF6) 30.187357269247 K km-1 * 0.0000000000115 +
(CO) 9.4393555726775 K km-1 * 0.00000008 +
(I2) 45.728742264382 K km-1 * 0.00000000009 +
(Kr) 39.225663804284 K km-1 * 0.000001 +
(SO2) 15.757493460485 K km-1 * 0.000000015 +
(NO2) 12.127952596066 K km-1 * 0.00000033698 +
(O3) 12.001569302138 K km-1 * 0.0000000003 +
(N2O) 11.18181671295 K km-1 * 0.00000033671 +
(Ne) 9.5205114453312 K km-1 * 0.0000182 +
(He) 1.8883738683977 K km-1 * 0.000005222 +
(H2) 0.6859482857817 K km-1 * 0.00000055 =

(N2) 7.36568033074394 +
(O2) 2.23699350189356 +
(Ar) 0.176030325226679 +
(CO2) 0.00502387325839717 +
(CH4) 0.00000793446406965918 +
(Xe) 0.00000532890962253648 +
(SF6) 0.000000000347154608596341 +
(CO) 0.0000007551484458142 +
(I2) 0.00000000411558680379438 +
(Kr) 0.000039225663804284 +
(SO2) 0.000000236362401907275 +
(NO2) 0.00000408687746582232 +
(O3) 0.0000000036004707906414 +
(N2O) 0.00000376502950541739 +
(Ne) 0.000173273308305028 +
(He) 0.00000986108834077279 +
(H2) 0.000000377271557179935 = 9.78397288330931 K km-1

Above, I've adjusted relative concentrations to arrive at 1,000,000 ppm. You'll note the model atmospheres you find online exceed 1,000,000 ppm, which is impossible and which skews results.

The differential of only 0.050935112661 K km-1 between the 9.7330377706482238008458858152373 K km-1 calculated result and the 9.78397288330931 K km-1 derived result shows we're pretty close. The first result's calculation makes assumptions that could skew the result, so it's a good bet the second result is more precise... but then, the second result still isn't exact because there are more than 17 gases in the atmosphere.

If the climate alarmists were serious about reducing temperature, they'd advocate for removing all Ar... it serves no biological purpose, it's used in industry so we need stocks of it, it has a higher concentration than CO2 and thus would be easier to remove, its removal wouldn't destroy all life on the planet (as CO2's total removal would) and its removal would lower the lapse rate (and thus cool the surface) by:

(Ar) 18.846929895790 K km-1 * 5.105 km * 0.000001 = 0.000096213577118008 K ppm-1
(Ar) 18.846929895790 K km-1 * 5.105 km * 0.009340 = 0.8986348102821 K

But wait! We also have to account for the atoms and molecules which that Ar displaces. We'll do the calculations for the three most-prevalent atomic or molecular species.

N2 | 28.0134 g mol-1 | 29.12 J mol-1 K-1 | 9.4339738283240 K km-1
(N2) 9340 ppm * 0.780761158 ppm = 7292.30921572 ppm
(N2) 780761.158 ppm + 7292.30921572 ppm = 788053.46721572 ppm
(N2) 9.433973828324 K km-1 * 5.105 km * 0.780761158 = 37.6017980884478 K
(N2) 9.433973828324 K km-1 * 5.105 km * 0.78805346721572 = 37.9529988825939 K
(N2) 37.9529988825939 K - 37.601798088447 K = 0.351200794146905 K warming

O2 | 31.9988 g mol-1 | 29.38 J mol-1 K-1 | 10.680770320623 K km-1
(O2) 9340 ppm * 0.20944121395198 ppm = 1956.18093831149 ppm
(O2) 209441.21395198 ppm + 1956.18093831149 ppm = 211397.394890292 ppm
(O2) 10.680770320623 K km-1 * 5.105 km * 0.20944121395198 = 11.4198518271666 K
(O2) 10.680770320623 K km-1 * 5.105 km * 0.211397394890292 = 11.5265132432324 K
(O2) 11.5265132432324 K - 11.4198518271666 K = 0.106661416065799 K warming

CO2 | 44.0095 g mol-1 | 36.94 J mol-1 K-1 | 11.683426182319 K km-1
(CO2) 9340 ppm * 0.00043 = 4.0162 ppm
(CO2) 430 ppm + 4.0162 ppm = 434.0162 ppm
(CO2) 11.683426182319 K km-1 * 5.105 km * 0.00043 = 0.0256468729841176 K
(CO2) 11.683426182319 K km-1 * 5.105 km * 0.0004340162 = 0.0258864147777892 K
(CO2) 0.0258864147777892 K - 0.0256468729841176 K = 0.0002395417936716 K warming

0.8986348102821 K - 0.351200794146905 K - 0.106661416065799 K - 0.0002395417936716 K = 0.440533058275724 K decrease in lapse rate

Removing all Ar would decrease the lapse rate (and thus surface temperature) by 0.440533058275724 K.

Conversely, removing all CO2 would only reduce the lapse rate (and thus surface temperature) by:

(CO2) 11.683426182319 K km-1 * 5.105 km * 0.000001 = 0.0000596438906607385 K ppm-1
(CO2) 11.683426182319 K km-1 * 5.105 km * 0.000430 = 0.0256468729841176 K

But wait! We also have to account for the atoms and molecules which that CO2 displaces. We'll do the calculations for the three most-prevalent atomic or molecular species.

N2 | 28.0134 g mol-1 | 29.12 J mol-1 K-1 | 9.4339738283240 K km-1
(N2) 430 ppm * 0.780761158 = 335.72729794 ppm
(N2) 780761.158 ppm + 335.72729794 ppm = 781096.88529794 ppm
(N2) 9.433973828324 K km-1 * 5.105 km * 0.780761158 = 37.6017980884478 K
(N2) 9.433973828324 K km-1 * 5.105 km * 0.78109688529794 = 37.6179668616258 K
(N2) 37.6179668616258 K - 37.6017980884478 K = 0.016168773178002 K warming

O2 | 31.9988 g mol-1 | 29.38 J mol-1 K-1 | 10.680770320623 K km-1
(O2) 430 ppm * 0.20944121395198 = 90.0597219993514 ppm
(O2) 209441.21395198 ppm + 90.0597219993514 ppm = 209531.273673979 ppm
(O2) 10.680770320623 K km-1 * 5.105 km * 0.20944121395198 = 11.4198518271666 K
(O2) 10.680770320623 K km-1 * 5.105 km * 0.209531273673979 = 11.4247623634523 K
(O2) 11.4247623634523 K - 11.4198518271666 K = 0.00491053628570093 K warming

Ar | 39.948 g mol-1 | 20.7862 J mol-1 K-1 | 18.846929895790 K km-1
(Ar) 430 ppm * 0.00934 = 4.0162 ppm
(Ar) 9340 ppm + 4.0162 ppm = 9344.0162 ppm
(Ar) 18.84692989579 K km-1 * 5.105 km * 0.00934 = 0.898634810282194 K
(Ar) 18.84692989579 K km-1 * 5.105 km * 0.0093440162 = 0.899021223250616 K
(Ar) 0.899021223250616 K - 0.898634810282194 K = 0.000386412968421901 K warming

0.0256468729841176 K - 0.016168773178002 K - 0.004910536285700930 K - 0.000386412968421901 K  = 0.00418115055199277 K.

Removing all CO2 would decrease the lapse rate (and thus surface temperature) by 0.00418115055199277 K.

"No one is advocating for removing all CO2 from the atmosphere! That's just ridiculous! That would kill all life on the planet!", someone will assert.

Ok, let's assume they draw CO2 down from 430 ppm to 280 ppm (150 ppm decrease). That would reduce the lapse rate (and thus surface temperature) by:

(CO2) 11.683426182319 K km-1 * 5.105 km * 0.000001 = 0.0000596438906607385 K ppm-1
(CO2) 11.683426182319 K km-1 * 5.105 km * 0.000430 = 0.0256468729841176 K
(CO2) 11.683426182319 K km-1 * 5.105 km * 0.000280 = 0.0167002893850068 K
(CO2) 11.683426182319 K km-1 * 5.105 km * 0.000150 = 0.00894658359911077 K

But wait! We also have to account for the atoms and molecules which that CO2 displaces. We'll do the calculations for the three most-prevalent atomic or molecular species.

N2 | 28.0134 g mol-1 | 29.12 J mol-1 K-1 | 9.4339738283240 K km-1
(N2) 150 ppm * 0.780761158 = 117.1141737 ppm
(N2) 780761.158 ppm + 117.1141737 ppm = 780878.2721737 ppm
(N2) 9.433973828324 K km-1 * 5.105 km * 0.780761158 = 37.6017980884478 K
(N2) 9.433973828324 K km-1 * 5.105 km * 0.7808782721737 = 37.6074383581611 K
(N2) 37.6074383581611 K - 37.6017980884478 K = 0.00564026971329668 K warming

O2 | 31.9988 g mol-1 | 29.38 J mol-1 K-1 | 10.680770320623 K km-1
(O2) 150 ppm * 0.20944121395198 = 31.416182092797 ppm
(O2) 209441.21395198 ppm + 31.416182092797 ppm = 209472.630134073 ppm
(O2) 10.680770320623 K km-1 * 5.105 km * 0.20944121395198 = 11.4198518271666 K
(O2) 10.680770320623 K km-1 * 5.105 km * 0.209472630134073 = 11.4215648049407 K
(O2) 11.4215648049407 K - 11.4198518271666 = 0.00171297777410118 K warming

Ar | 39.948 g mol-1 | 20.7862 J mol-1 K-1 | 18.846929895790 K km-1
(Ar) 150 ppm * 0.00934 = 1.401 ppm
(Ar) 9340 ppm + 1.401 ppm = 9341.401 ppm
(Ar) 18.84692989579 K km-1 * 5.105 km * 0.00934 = 0.898634810282194 K
(Ar) 18.84692989579 K km-1 * 5.105 km * 0.009341401 = 0.898769605503737 K
(Ar) 0.898769605503737 K - 0.898634810282194 K = 0.000134795221542916 warming

0.00894658359911077 K - 0.00564026971329668 K - 0.00171297777410118 K - 0.000134795221542916 K = 0.00145854089016999 K.

Reducing CO2 from 430 ppm to 280 ppm would decrease the lapse rate (and thus surface temperature) by 0.00145854089016999 K.

About 1/1000th of a degree. For trillions of dollars wasted.

So while trillions of dollars are being squandered on the non-solution of removing CO2, the above represents a lucrative business opportunity which would absolutely show profound effects upon the climate, and would provide industry with plentiful Ar gas as an additional income stream, without killing industry (and potentially all life on the planet) as would the restriction of CO2.

And do remember, removing CO2 removes the predominant upper atmospheric radiative coolant, which means the upper atmosphere warms, which translates down through the lapse rate to a warmer surface.

----------

Do keep in mind that as CO2 concentration increases it's displacing primarily homonuclear diatomics (which have a net-zero electric dipole and thus cannot emit unless that dipole is perturbed via collision... collisions occurring exponentially less frequently with altitude) and monoatomics (which have no vibrational mode quantum states and thus cannot absorb nor emit IR in any case) with a radiative polyatomic. The homonuclear diatomics and monoatomics dilute the radiative polyatomics (which radiatively cool the upper atmosphere by absorbing resonant solar IR, then re-emitting it spaceward (ie: down the energy density gradient)). Do remember that a molecule can only emit when that energy can spontaneously flow down the energy density gradient, so it's not going to be emitting earthward (ie: up the energy density gradient)... the highest probability of emission is when the molecule's dipole moment is perpendicular to the maximum energy density gradient... this increases emission probability in two ways: maximum energy density gradient; and when an atom or molecule emits a photon, there is a probability distribution of emission direction, with maximum emission probability perpendicular to the dipole moment and zero emission probability parallel to the dipole moment.

Further, given that ~49% of solar insolation is IR between 700 nm - 1 mm wavelength, increased CO2 concentration acts as a 'shield', reducing energy flux to the surface in areas where the energy density gradient between upper atmosphere and surface is such that resonant solar insolation in the IR band could reach the surface.

"But why doesn't CO2 act as a 'shield' to warm the surface by blocking outgoing terrestrial IR radiation, just as it acts as a 'shield' to cool the planet by blocking incoming solar IR insolation?", someone may ask. Well, it would, if the atmosphere below the tropopause didn't convect... but it does convect, which carries that energy away from the surface. In fact, given that surface-level atmospheric attenuation length at 14.98352 µm is ~10.4 m, that means the atmosphere is essentially opaque at that wavelength... there are more CO2 molecules able to absorb terrestrial radiation at that wavelength than there is terrestrial radiation at that wavelength... and that terrestrial radiation is absorbed, then thermalized via v-t (vibrational mode -to- translational mode) collisional energy transfer, which increases CAPE (Convective Available Potential Energy), which increases convection, which drags with it water vapor (which has a comparatively much higher thermal capacity), thus removing more energy from the surface. This isn't anything new, the increase of CAPE occurs at essentially the same magnitude for any CO2 concentration higher than ~50 ppm. There simply isn't any more resonant IR energy available to thermalize and thus increase CAPE.

But while absorption and thermalization at the surface is saturated, in the upper atmosphere, a higher concentration of emitters means the atmosphere is more capable of emitting, and that energy must flow out to space due to the energy density gradient and the MFPL / altitude / air density relation.

So an increasing CO2 concentration is very, very slightly increasing the lapse rate, while displacing lower-DOF non-radiative monoatomics and homonuclear diatomics with a radiative polyatomic which increases thermodynamic coupling between heat source (ie: the surface) and sink (ie: space). So while the lapse rate will be very, very slightly higher, the starting temperature for the lapse rate (in the upper atmosphere) will be lower, which translates down through the lapse rate to a cooler surface.

Because we don't live in a "greenhouse", we live in what can be analogized to a world-sized AC unit... the surface would be akin to the AC unit's evaporator section (ie: the heat source); the atmosphere would be akin to the AC unit's working fluid; convection would be akin to the AC compressor (ie: the motive force to move the working fluid); and space would be akin to the AC unit's condenser section (ie: the heat sink). For the same reason that we use complex high-DOF polyatomic CFC, HFC or HCFC molecules as a refrigerant in (and evacuate low-DOF non-condensable gases from) an AC unit, the higher the DOF of any given constituent molecule in the atmosphere, the more tightly it thermodynamically couples heat source (surface) and heat sink (space).

Now remember that radiative emission from the atmosphere doesn't occur at any one specific altitude, it's a whole-atmosphere phenomenon... which is why CO2 is the most predominant atmospheric radiative coolant above the tropopause; and the second-most predominant atmospheric radiative coolant (behind water vapor) below the tropopause.


The image above is from a presentation given by Dr. Maria Z. Hakuba, an atmospheric research scientist at NASA JPL.


The image above is adapted from the Clough and Iacono study, Journal Of Geophysical Research, Vol. 100, No. D8, Pages 16,519-16,535, August 20, 1995.

Note that the Clough & Iacono study is for the atmospheric radiative cooling effect, so positive numbers at right are cooling, negative numbers are warming.

And, given that the lapse rate is 'anchored' at TOA (that altitude at which the atmosphere effectively becomes transparent to any given wavelength of radiation... it varies for different wavelengths; the mean TOA being the mean of all wavelength-specific TOAs), a cooler upper atmosphere will translate down to the surface via the Kelvin-Helmholtz Gravitational Auto-Compression Effect (aka the lapse rate) to also cause a cooler surface.

And finally, keep in mind that the ~33 K Kelvin-Helmholtz Gravitational Auto-Compression Effect (aka the average humid Adiabatic Lapse Rate) has nothing to do with any "greenhouse effect" nor any "greenhouse gases" as claimed by the climatologists, it is strictly a manifestation of translational mode (kinetic) energy in the z-axis DOF being converted to gravitational potential energy with altitude (and vice versa), that change in z-axis DOF kinetic energy equipartitioning to the other 2 DOF upon subsequent collision, which changes air temperature (because temperature in this respect is a measure of the kinetic energy of the air atoms and molecules), which is why temperature decreases as altitude increases (and vice versa).

The climatologists misuse the Stefan-Boltzmann (S-B) equation in their Energy Balance Climate Models (EBCMs... which I prove above using the Kiehl-Trenberth 'Earth Energy Balance' graphic, which is a representation of the mathematics in their EBCM).

That misuse of the S-B equation uses the idealized blackbody form of the equation upon real-world graybody objects. It assumes emission to 0 K and thus artificially inflates radiant exitance of all calculated-upon objects. Thus the climatologists must carry these incorrect values through their calculations and cancel them on the back end, subtracting a wholly-fictive 'cooler to warmer' energy flow from the real (but too high because it was calculated for emission to 0 K) 'warmer to cooler' energy flow.

That wholly-fictive 'cooler to warmer' energy flow is otherwise known as "backradiation"... it is merely a manifestation of the misuse of the S-B equation, it does not and cannot actually exist. Its existence would imply rampant violations of the fundamental physical laws.

And the S-B equation for graybody objects is meant to be used by subtracting cooler object energy density from warmer object energy density to arrive at the energy density gradient, which determines radiant exitance of the warmer object.

Now, the climatologists claim that wholly-fictive "backradiation" causes their claimed "greenhouse effect", which they use to designate polyatomics (CO2, CH4, H2O, etc.) as "greenhouse gases". You will note the climatologists had no choice but to use the polyatomics... monoatomics have no vibrational mode quantum states and thus cannot emit (nor absorb) IR in any case; and homonuclear diatomics have a net-zero electric dipole which must be perturbed via collision for the molecule to emit (or absorb)... and collisions occur exponentially less frequently as altitude increases.

But water, a polyatomic which the climatologists claim is a "greenhouse gas" is such an effective atmospheric radiative coolant that it drastically reduces the dry Adiabatic Lapse Rate and acts as a literal refrigerant (in the strict 'refrigeration cycle' sense) below the tropopause:

The refrigeration cycle (Earth) [AC system]:

A liquid evaporates at the heat source (the surface) [in the evaporator], it is transported (convected) [via an AC compressor], it gives up its energy to the heat sink and undergoes phase change (emits radiation in the upper atmosphere, the majority of which is upwelling owing to the energy density gradient and the mean free path length / altitude / air density relation) [in the condenser], it is transported (falls as rain or snow) [via that AC compressor], and the cycle repeats.

That’s kind of why, after all, the humid Adiabatic Lapse Rate (~3.5 to ~6.5 K km-1) is lower than the dry Adiabatic Lapse Rate (~9.81 K km-1). The humid Adiabatic Lapse Rate is the Dry Adiabatic Lapse Rate minus the radiative cooling by water.

So one can see right away that the claim of the climatologists doesn't even make sense, and is in fact diametrically opposite to reality... because the easiest lie to tell is an inversion of reality... the climatologists needn't invent new physics to explain their blather, and most people cannot tell the difference between reality and flipped-causality anyway because they're scientifically-illiterate.

You will also note that the climatologists had no choice but to conflate their "greenhouse effect" with the Kelvin-Helmholtz Gravitational Auto-Compression Effect (aka the lapse rate)... they knew their "greenhouse effect" was fictional (because it's based upon the wholly-fictive "backradiation", which is nothing more than a mathematical artifact conjured out of thin air by the misuse of the S-B equation in the climatologist's Energy Balance Climate Models) and thus couldn't show any effect whatsoever, so they conflated it to a known physical process which shows the effect they want.

Now, they use this claimed "greenhouse effect" to claim certain of those "greenhouse gases" will cause CAGW (Catastrophic Anthropogenic Global Warming, due to CO2).

And they use that to claim we must reduce CO2 emission, which they use as the basis for all of the offshoots of AGW / CAGW (net zero, carbon credit trading, carbon capture and sequestration, replacing reliable baseload electrical generation with intermittent 'renewables', etc.)

The entirety of it is predicated upon that mathematical fraudery in the misuse of the S-B equation. It is all nothing more than a complex mathematical scam.

[Morpheus]

Thus, the entirety of CAGW describes a physical process which is physically impossible. The only effect upon surface temperature of any given change in concentration of any given constituent atomic or molecular species of the atmosphere is a change in the Adiabatic Lapse Rate.

We can calculate the Specific Lapse Rate (what the Adiabatic Lapse Rate would be if the atmosphere consisted of only that gas) for any gas:

Idealized dry gas molar heat capacity lapse rate:
If we take ϒ = 1.404, g = 9.80665 m s-2, R = 8.31446261815324 J mol-1 K-1 and M = 28.9647 g mol-1, then:

dT / dh = -0.4/1.404 * (((28.9647 g mol-1) * 9.80665 m s-2) / 8.31446261815324 J mol-1 K-1) = -9.7330377706482238008458858152373 K km-1

The stated molar isobaric heat capacity for dry air is Cp = 7/2 R
7 / 2 * 8.31446261815324 J mol-1 K-1 = 29.10061916353634 J mol-1 K-1

∴ Molar Heat Capacity / 7 * 2 = Specific Gas Constant

dT / dh = -0.4/1.404 * (((Molar Mass) * 9.80665 m s-2) / Specific Gas Constant) = Specific Lapse Rate

Symbol: Molar Mass: Molar Heat Capacity: Specific Lapse Rate (SLR):
H2 | 2.01588 g mol-1 | 28.82 J mol-1 K-1 | 0.6859482857817 K km-1
He | 4.002602 g mol-1 | 20.7862 J mol-1 K-1 | 1.8883738683977 K km-1
H2O | 18.01528 g mol-1 | 75.327 J mol-1 K-1 | 2.3453681364178 K km-1
CH4 | 16.04246 g mol-1 | 35.69 J mol-1 K-1 | 4.4080355942551 K km-1
N2 | 28.0134 g mol-1 | 29.12 J mol-1 K-1 | 9.4339738283240 K km-1
CO | 28.0101 g mol-1 | 29.1 J mol-1 K-1 | 9.4393555726775 K km-1
Ne | 20.1797 g mol-1 | 20.7862 J mol-1 K-1 | 9.5205114453312 K km-1
O2 | 31.9988 g mol-1 | 29.38 J mol-1 K-1 | 10.680770320623 K km-1
N2O | 44.0128 g mol-1 | 38.6 J mol-1 K-1 | 11.181816712950 K km-1
CO2 | 44.0095 g mol-1 | 36.94 J mol-1 K-1 | 11.683426182319 K km-1
O3 | 47.9982 g mol-1 | 39.22  J mol-1 K-1 | 12.001569302138 K km-1
NO2 | 46.0055 g mol-1 | 37.2 J mol-1 K-1 | 12.127952596066 K km-1
SO2 | 64.0638 g mol-1 | 39.87  J mol-1 K-1 | 15.757493460485 K km-1
Ar | 39.948 g mol-1 | 20.7862 J mol-1 K-1 | 18.846929895790 K km-1
SF6 | 146.06 g mol-1 | 93 J mol-1 K-1 | 30.187357269247 K km-1
Kr | 83.798 g mol-1 | 20.95 J mol-1 K-1 | 39.225663804284 K km-1
I2 | 253.80894 g mol-1 | 54.43 J mol-1 K-1 | 45.728742264382 K km-1
Xe | 131.293 g mol-1 | 21.01 J mol-1 K-1 | 61.282460659191 K km-1

---

Note the calculated Dry Adiabatic Lapse Rate above: -9.7330377706482238008458858152373 K km-1

The negative sign means temperature decreases with altitude. Usually we leave it off.

(N2) 9.433973828324 K km-1 * 0.780761158 +
(O2) 10.680770320623 K km-1 * 0.20944121395198 +
(Ar) 18.84692989579 K km-1 * 0.00934 +
(CO2) 11.683426182319 K km-1 * 0.00043 +
(Ne) 9.5205114453312 K km-1 * 0.0000182 +
(He) 1.8883738683977 K km-1 * 0.000005222 +
(CH4) 4.4080355942551 K km-1 * 0.0000018 +
(Kr) 39.225663804284 K km-1 * 0.000001 +
(H2) 0.6859482857817 K km-1 * 0.00000055 +
(NO2) 12.127952596066 K km-1 * 0.00000033698 +
(N2O) 11.18181671295 K km-1 * 0.00000033671 +
(Xe) 61.282460659191 K km-1 * 0.0000000869565217391 +
(CO) 9.4393555726775 K km-1 * 0.00000008 +
(SO2) 15.757493460485 K km-1 * 0.000000015 +
(O3) 12.001569302138 K km-1 * 0.0000000003 +
(I2) 45.728742264382 K km-1 * 0.00000000009 +
(SF6) 30.187357269247 K km-1 * 0.0000000000115 =

(N2) 7.36568033074394 +
(O2) 2.23699350189356 +
(Ar) 0.176030325226679 +
(CO2) 0.00502387325839717 +
(Ne) 0.000173273308305028 +
(He) 0.00000986108834077279 +
(CH4) 0.00000793446406965918 +
(Kr) 0.000039225663804284 +
(H2) 0.000000377271557179935 +
(NO2) 0.00000408687746582232 +
(N2O) 0.00000376502950541739 +
(Xe) 0.00000532890962253648 +
(CO) 0.0000007551484458142 +
(SO2) 0.000000236362401907275 +
(O3) 0.0000000036004707906414 +
(I2) 0.00000000411558680379438 +
(SF6) 0.000000000347154608596341 = 9.78397288330931 K km-1

I've adjusted relative concentrations to arrive at 1,000,000 ppm. The model atmospheres you find online exceed 1,000,000 ppm, which is impossible and skews results.

The differential of only 0.050935112661 K km-1 between the 9.7330377706482238008458858152373 K km-1 calculated result and the 9.78397288330931 K km-1 derived result shows we're pretty close. The first result's calculation makes assumptions that could skew the result, so it's a good bet the second result is more precise... but then, the second result still isn't exact because there are more than 17 gases in the atmosphere.

The dry Adiabatic Lapse Rate is typically stated as "~9.81 K km-1", so we're very close to that.

---

Let's increase CO2 concentration by 10x, from 430 ppm to 4300 ppm (3870 ppm increase).

(CO2): 11.683426182319 K km-1 * 5.105 km * 0.000001 = 0.0000596438906607385 K ppm-1
(CO2): 11.683426182319 K km-1 * 5.105 km * 0.004300 = 0.256468729841176 K
(CO2): 11.683426182319 K km-1 * 5.105 km * 0.000430 = 0.0256468729841176 K
(CO2): 11.683426182319 K km-1 * 5.105 km * 0.003870 = 0.230821856857058 K

But wait! We also have to account for the atoms and molecules which that CO2 displaces. We'll do the calculations for the three most-prevalent atomic or molecular species.

N2 | 28.0134 g mol-1 | 29.12 J mol-1 K-1 | 9.4339738283240 K km-1
(N2) 3870 ppm * 0.780761158 = 3021.54568146 ppm
(N2) 780761.158 ppm - 3021.54568146 ppm = 777739.61231854 ppm
(N2) 9.433973828324 K km-1 * 5.105 km * 0.780761158 = 37.601798088447 K
(N2) 9.433973828324 K km-1 * 5.105 km * 0.77773961231854 = 37.456279129845 K
(N2) 37.601798088447 K - 37.456279129845 K = 0.145518958601 K cooling

O2 | 31.9988 g mol-1 | 29.38 J mol-1 K-1 | 10.680770320623 K km-1
(O2) 3870 ppm * 0.20944121395198 = 810.53749899416 ppm
(O2) 209441.21395198 ppm - 810.53749899416 ppm = 208630.67645298 ppm
(O2) 10.680770320623 K km-1 * 5.105 km * 0.20944121395198 = 11.419851827166 K
(O2) 10.680770320623 K km-1 * 5.105 km * 0.20863067645298 = 11.37565700054 K
(O2) 11.419851827166 K - 11.37565700054 K = 0.044194826625 K cooling

Ar | 39.948 g mol-1 | 20.7862 J mol-1 K-1 | 18.846929895790 K km-1
(Ar) 3870 ppm * 0.00934 = 36.1458 ppm
(Ar) 9340 ppm - 36.1458 ppm = 9303.8542 ppm
(Ar) 18.84692989579 K km-1 * 5.105 km * 0.00934 = 0.8986348102821 K
(Ar) 18.84692989579 K km-1 * 5.105 km * 0.0093038542 = 0.895157093566402 K
(Ar) 0.8986348102821 K - 0.895157093566402 K = 0.00347771671569797 K cooling

0.230821856857058 K - 0.145518958601 K - 0.044194826625 K - 0.00347771671569797 K = 0.03763035491536 K increase in lapse rate.

Increasing CO2 concentration by 10x causes only 0.03763035491536 K increase in the lapse rate. Any climate model prognosticating a higher number is flat out wrong.

And that doesn't even take into account the radiative cooling at the top of the atmosphere (where the lapse rate is 'anchored'), due to more radiative polyatomics per parcel of air (remember, that radiative polyatomic (CO2) is primarily displacing a nonradiative monoatomic (Ar) and effectively-nonradiative homonuclear diatomics (N2, O2)), which translates down through the lapse rate to a cooler surface. The lapse rate starts at a lower temperature (in the upper atmosphere), so it ends at a lower temperature (at the surface).

---

Let's remove all Ar:

(Ar) 18.846929895790 K km-1 * 5.105 km * 0.000001 = 0.000096213577118008 K ppm-1
(Ar) 18.846929895790 K km-1 * 5.105 km * 0.009340 = 0.8986348102821 K

But wait! We also have to account for the atoms and molecules which that Ar displaces. We'll do the calculations for the three most-prevalent atomic or molecular species.

N2 | 28.0134 g mol-1 | 29.12 J mol-1 K-1 | 9.4339738283240 K km-1
(N2) 9340 ppm * 0.780761158 = 7292.30921572 ppm
(N2) 780761.158 ppm + 7292.30921572 ppm = 788053.46721572 ppm
(N2) 9.433973828324 K km-1 * 5.105 km * 0.780761158 = 37.6017980884478 K
(N2) 9.433973828324 K km-1 * 5.105 km * 0.78805346721572 = 37.9529988825939 K
(N2) 37.9529988825939 K - 37.601798088447 K = 0.351200794146905 K warming

O2 | 31.9988 g mol-1 | 29.38 J mol-1 K-1 | 10.680770320623 K km-1
(O2) 9340 ppm * 0.20944121395198 = 1956.18093831149 ppm
(O2) 209441.21395198 ppm + 1956.18093831149 ppm = 211397.394890292 ppm
(O2) 10.680770320623 K km-1 * 5.105 km * 0.20944121395198 = 11.4198518271666 K
(O2) 10.680770320623 K km-1 * 5.105 km * 0.211397394890292 = 11.5265132432324 K
(O2) 11.5265132432324 K - 11.4198518271666 K = 0.106661416065799 K warming

CO2 | 44.0095 g mol-1 | 36.94 J mol-1 K-1 | 11.683426182319 K km-1
(CO2) 9340 ppm * 0.00043 = 4.0162 ppm
(CO2) 430 ppm + 4.0162 ppm = 434.0162 ppm
(CO2) 11.683426182319 K km-1 * 5.105 km * 0.00043 = 0.0256468729841176 K
(CO2) 11.683426182319 K km-1 * 5.105 km * 0.0004340162 = 0.0258864147777892 K
(CO2) 0.0258864147777892 K - 0.0256468729841176 K = 0.0002395417936716 K warming

0.8986348102821 K - 0.351200794146905 K - 0.106661416065799 K - 0.0002395417936716 K = 0.440533058275724 K decrease in lapse rate

Removing all Ar would decrease the lapse rate (and thus surface temperature) by 0.440533058275724 K.

---

Let's reduce CO2 concentration from 430 ppm to 280 ppm:

(CO2) 11.683426182319 K km-1 * 5.105 km * 0.000001 = 0.0000596438906607385 K ppm-1
(CO2) 11.683426182319 K km-1 * 5.105 km * 0.000430 = 0.0256468729841176 K
(CO2) 11.683426182319 K km-1 * 5.105 km * 0.000280 = 0.0167002893850068 K
(CO2) 11.683426182319 K km-1 * 5.105 km * 0.000150 = 0.00894658359911077 K

But wait! We also have to account for the atoms and molecules which that CO2 displaces. We'll do the calculations for the three most-prevalent atomic or molecular species.

N2 | 28.0134 g mol-1 | 29.12 J mol-1 K-1 | 9.4339738283240 K km-1
(N2) 150 ppm * 0.780761158 = 117.1141737 ppm
(N2) 780761.158 ppm + 117.1141737 ppm = 780878.2721737 ppm
(N2) 9.433973828324 K km-1 * 5.105 km * 0.780761158 = 37.6017980884478 K
(N2) 9.433973828324 K km-1 * 5.105 km * 0.7808782721737 = 37.6074383581611 K
(N2) 37.6074383581611 K - 37.6017980884478 K = 0.00564026971329668 K warming

O2 | 31.9988 g mol-1 | 29.38 J mol-1 K-1 | 10.680770320623 K km-1
(O2) 150 ppm * 0.20944121395198 = 31.416182092797 ppm
(O2) 209441.21395198 ppm + 31.416182092797 ppm = 209472.630134073 ppm
(O2) 10.680770320623 K km-1 * 5.105 km * 0.20944121395198 = 11.4198518271666 K
(O2) 10.680770320623 K km-1 * 5.105 km * 0.209472630134073 = 11.4215648049407 K
(O2) 11.4215648049407 K - 11.4198518271666 = 0.00171297777410118 K warming

Ar | 39.948 g mol-1 | 20.7862 J mol-1 K-1 | 18.846929895790 K km-1
(Ar) 150 ppm * 0.00934 = 1.401 ppm
(Ar) 9340 ppm + 1.401 ppm = 9341.401 ppm
(Ar) 18.84692989579 K km-1 * 5.105 km * 0.00934 = 0.898634810282194 K
(Ar) 18.84692989579 K km-1 * 5.105 km * 0.009341401 = 0.898769605503737 K
(Ar) 0.898769605503737 K - 0.898634810282194 K = 0.000134795221542916 warming

0.00894658359911077 K - 0.00564026971329668 K - 0.00171297777410118 K - 0.000134795221542916 K = 0.00145854089016999 K.

Reducing CO2 from 430 ppm to 280 ppm would decrease the lapse rate (and thus surface temperature) by 0.00145854089016999 K.

[ChemEngineer]

Terribly long, excessively boring.  If Sunset Tommy read all of it, he is surely the only one.
  
In a nutshell, a couple of illustrations should suffice much better than Analysis Paralysis:

           

[Morpheus]

Terribly long, excessively boring.  If Sunset Tommy read all of it, he is surely the only one.

Science is not here to entertain you, it is here to inform you, and often requires verbosity and lengthy maths to do so.

[Billy_Bob Patriot Master]

Let's make this easy...

One:  CO2 once excited and holding energy become opaque transparent to further energy. This is called energy saturation.

Two:  Once saturated CO2 is incapable of further energy retention.

Three:  CO2 itself is incapable of warming due to its short energy retention span. Thus, it must pass on warming energy by collision or emission. 

Four:  Microcirculations (Thunderstorms) move the heat that is retained above cloud barrier where this energy is released to space..

In short: The solar energy increase is paralleled in blackbody output. This means there is very little to no warming. Ther EREB Satellite showed this problem many years ago and disproved the CAGW hypothesis and also laid waste to the modeling as an utter failure.

   

[ChemEngineer]

Terribly long, excessively boring.  If Sunset Tommy read all of it, he is surely the only one.

Science is not here to entertain you, it is here to inform you, and often requires verbosity and lengthy maths to do so.

You miss the point that ONLY those who are as technically proficient at all this material as the author was ca POSSIBLY be "informed" by it.  
I had no intention of wading through mountains of calculations, graphs and dialogue when it can be neatly summarized with far less pain and effort, which will convince far more people who need to be convinced.

http://Climate-Change-Cult.blogspot.com

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