## Friday, November 28, 2014

### The Greenhouse Equation

A recent series of Hockey Schtick posts

have derived the entire ~33°C greenhouse effect as a consequence of gravitational forcing rather than radiative forcing from greenhouse gases, and entirely independent of radiative forcing from greenhouse gases. We have also determined the effective radiating height (average) or ERL in the troposphere (where T = the equilibrium temperature of Earth with the Sun), and found the ERL to be located as expected at the center of mass of the atmosphere if the ERL height and temperature are a function of mass/gravity/pressure rather than radiative forcing from greenhouse gases.

We now join the gravitational greenhouse effect to the only source of energy that the Earth receives, the Sun, and show that solar shortwave radiative forcing plus gravitational forcing calculates the Earth's surface temperature, ERL height and temperature, and the entire tropospheric temperature profile perfectly, without any contribution from greenhouse gas forcing, nor dependence on greenhouse gas concentrations, nor dependence upon emission/absorption spectra from greenhouse gases.

We show that the entire 33°C greenhouse effect that raises Earth's equilibrium temperature with the Sun of -18C or 255K up to +15C or 288K at the surface, and the temperature at any height in the atmosphere from the surface to top of the troposphere (above which the atmosphere is too thin to sustain convection), can be fully explained by the following equation, which I'm calling "the greenhouse equation": The "Greenhouse Equation" calculates temperature (T) at any location from the surface to the top of the troposphere as a function of atmospheric mass/gravity/pressure and radiative forcing from the Sun only, and without any radiative forcing from greenhouse gases. Note the pressure (P) divided by 2 in the greenhouse equation is the pressure at the center of mass of the atmosphere (after density correction), where the temperature and height are equal to the equilibrium temperature with the Sun and ERL respectively.

which solves T as a function of mass/pressure/gravity for which none of the variables are dependent upon radiative forcing from greenhouse gases, and for which the only radiative forcing we require to reproduce the entire tropospheric temperature profile is that from the Sun. Note none of the constants and variables on the right side of the greenhouse equation are related to GHG radiative forcing, and temperature does not appear on the right side of the equation and thus it can't be a tautology of temperature.

T = temperature at height (s) meters above the surface, thus at the surface s = 0

s = height in meters above the surface to calculate the temperature T, thus at the surface s=0
S = the solar constant = 1367 W/m2, derivation here
ε = emissivity = 1 assuming Sun and Earth are blackbodies
σ = the Stefan-Boltzmann constant = 5.6704 x 10-8 W m-2 K-4
g = gravitational acceleration = 9.8 m/s^2

m = average molar mass of the atmosphere = 29g/mole = 0.029kg/mole
α = albedo = 0.3 for earth

C = Cp = the heat capacity of the atmosphere at constant pressure, ~ 1.5077 average for Earth
P = surface pressure in the unit atmospheres, defined as = 1 atmosphere for latitude of Paris
R = universal gas constant = 8.3145 J/mol K
e = the base of the natural logarithm, approximately equal to 2.71828

As shown by the prior posts listed above, all of the components of this entire gravitational "greenhouse equation" were first derived from the ideal gas law, the First Law of Thermodynamics, the Stefan-Boltzmann equation, Newton's second law of motion (F = ma), and well-known barometric formulae, without ever once introducing any variables dependent upon radiative forcing from greenhouse gases.

In a prior post we determined surface pressure by the relation:

P = e^-((Mgh/(RT))  (15) where T = T at height (s) (or (h) in the prior post)

for which we substitute for P in the greenhouse equation above (note s = h from the prior post) to yield:
After plugging in the numerical values, Wolfram Alpha solves the greenhouse equation to find Earth's surface (where the height (s)=0) the temperature T is equal to 288.433K or 15.28C, which is the same as determined from satellite measurements: Note the 1000 in the numeric solution is the conversion factor between meters and kilometers
We can now use the greenhouse equation for many other purposes including determining the effect of a change in solar activity on the expected Earth surface temperature. If we increase the solar constant in the above numerical solution by 1 W/m2 from 1367 to 1368 W/m2, we find an increase in surface temperature from T=288.433K to 288.486K, an increase of 0.056C (this includes the division by 4 to convert solar insolation from a flat disk to a sphere): Note Wolfram Alpha is solving for T, providing a unique solution of T as a function of the mass/gravity/pressure forcing at each height (s) in meters above the surface (s = 0 at the surface). Each solution for T as a function of height (s) perfectly reproduces the observed tropospheric temperature profile at each height (s), from the surface to the ERL located at the atmospheric center of mass and then on up to the top of the troposphere.
Wolfram Alpha solves for the unique temperature (T) that satisfies the greenhouse equation for a given height (s)  with P substituted by e^-((Mgh/(RT)) (15) above, showing there is a unique solution of T for every value of height (s), with no T on the right-hand side of the solution for T, proving the greenhouse equation is not a tautology:

We have also demonstrated why the atmospheric mass/gravity/pressure theory of the greenhouse effect also perfectly explains the observed greenhouse effect on Titan, the closest Earth analog in our solar system, and the only planet other than Earth with an atmosphere comprised of mostly non-greenhouse gases (Titan: 98.4% Nitrogen, 0.1% hydrogen, and only 1.5% greenhouse gas methane compared to Earth's 78.09% nitrogen, 20.95% oxygen, 0.93% argon, 0.04% carbon dioxide).

Note this equation would not be expected to hold for planets with thick, opaque cloud tops, such as Venus, which in addition to heating from the bottom up due to gravity/pressure, is also heated from top down by absorption of sunlight at the TOA and probable conduction of heat downward from the opaque cloud tops. It also cannot be applied to planets with thin atmospheres such as Mars, which has a surface pressure of only ∼0.006 bar. In addition, the ideal gas law and barometric equations are only true if the heat capacity C in the greenhouse equation stays constant while the temperature changes, which is true for N2 and O2 (more than 99% of our atmosphere and 98.4% of Titan's atmosphere), but CO2 does not. Since CO2 comprises 96.5% of the Venus atmosphere, the heat capacity C (Cp) in the greenhouse equation would have to be adjusted as a function of temperature.

I welcome all suggestions and refutations of the "greenhouse equation." There can be only one valid theory of the 33C greenhouse effect, since if both the gravitational and greenhouse gas radiative forcing theories had merit, the Earth would be at least 33C warmer than the present.

1. Hi MS,
Well done Sir!

2. I like your input,even if it iis a bit heavy on fomulas but i appreciate the point
I just wonder where the green house gases fit in.
They need to have some effect positive or negative.

1. Greenhouse gases (primarily water vapor) DO have a large effect on Earth's climate:

http://hockeyschtick.blogspot.com/2014/10/greenhouse-gases-do-have-profound.html

which is cooling due to an increase of atmospheric heat capacity since the lapse rate equation

dT/dh = -g/Cp

shows that temperature T and heat capacity Cp are inversely related:

http://hockeyschtick.blogspot.com/2014/11/derivation-of-entire-33c-greenhouse.html

The fact that the dry adiabatic lapse rate is double the wet adiabatic lapse rate proves that water vapor is a negative-feedback cooling agent, not warming.

2. Actually, water vapour is thermally neutral until it condenses out whereupon some of its energy radiates out to space from the condensate.

That difference between the dry and moist lapse rates is, however, significant for another reason.

Since water vapour is lighter than air it rises further and faster than air would have done and so a larger proportion of its total energy (sensible plus latent) is converted to PE during uplift due to the extra work being done against gravity.

On the subsequent descent that additional PE derived from the latent heat of evaporation held within the vapour has to be recovered and so the dry air must warm more on the descent than it cooled on the ascent.

The big mistake that many make is that in reality a large proportion of the energy contained in water vapour as latent heat is actually returned to the surface as kinetic energy on the descent. That is what the difference between the moist and dry rates is telling us.

That is also why the Kiehl Trenberth diagram is wrong as I explain here:

http://www.newclimatemodel.com/correcting-the-kiehl-trenberth-energy-budget/

3. Water vapour does radiate without condensing out but not enough for it to lose its latent energy before it creates more PE than dry air would have done.

3. I was wondering if those differential equations could be used to show the system response to a perturbation. The perturbation being say a transfer of heat from an upper layer to a lower layer (like a radiative transfer) and see what the gas equations do to it. Would they work to move the displaced heat back to where it came from. Ie: a negative feedback.

1. I showed the example of a change in solar radiative forcing above.

As to GHG radiative forcing, the equation assumes it is zero as does the mass/gravity/pressure GHE theory.

As to water vapor (and CO2 to a tiny extent), an increase causes an increase of heat capacity (C) in the greenhouse equation above, which as you can see is inversely related to temperature (T), thus this would cause cooling instead of warming.

4. Very interesting series of posts! I have studied the seasonal variations of the two hemisperes and I have come to the conclusion that 1W/m2 increased solar radiation will give about 0.062 C higher temperatures globally, which is very close to your result. See here:

1. Thanks, I'll add your paper to my collection of low-sensitivity papers. Did you calculate a global sensitivity figure or just NH and SH?

2. I calculated NH and SH separately and then took the average of the two. But see my comment below, 0.062C is for 1W/m2 on a spherical globe.

5. Oh, please. Claiming that gravity can cause the surface temperature of a planet to rise in the absence of greenhouse gases in the atmosphere violates the Second Law of Thermodynamics. See Dr. Robert Brown's elegant proof at

http://wattsupwiththat.com/2012/01/24/refutation-of-stable-thermal-equilibrium-lapse-rates/

and my somewhat less elegant proof at

http://wattsupwiththat.com/2012/01/13/a-matter-of-some-gravity/

w.

1. Of course the atmosphere is not assumed to be in thermal equilibrium, at least certainly not by my derivations, which is driven by thermodynamic disequilibrium of convection/lapse rate/etc

Apparently you believe you've disproven the great physicist Maxwell, who was apparently the first to state the atmosphere temperature gradient is driven by pressure and would not be isothermal, and that

"In the convective equilibrium of temperature, the absolute temperature is proportional to the pressure raised to the power (γ-1)/γ, or 0,29. " referring to γ defined as the "heat capacity ratio" = Cp/Cv [ratio of specific heat capacity at constant pressure to specific heat capacity at constant volume] also referred to as the Poisson relation based on the ideal gas law & 1st law, which shows remarkable agreement with the gravito-thermal "greenhouse effect" found on all the planets with thick atmospheres [fig 5 & 6 from Nikolov and Zeller]"

Short answer? Both of these "proofs" make false assumptions about the gravitational greenhouse effect, they thus fail to refute Maxwell's book and the "greenhouse equation" derivations.

2. "violates the Second Law of Thermodynamics"

Hahahaha

The radiative theory of the GHE is the one that violates the 2nd LoT by making the absurd assumption that a cold body makes a hot body 33C hotter, disproven by Pictet's experiment 214 years ago, the 1st and 2nd laws of thermodynamics, the principle of maximum entropy production, Planck's law, the Pauli exclusion principle, and quantum mechanics. There is one and only one greenhouse effect theory compatible with all of these basic physical laws and millions of observations. Can you guess which one it is?

http://hockeyschtick.blogspot.com/2014/11/derivation-of-entire-33c-greenhouse.html

3. I've had this conversation with Willis on several occasions over at WUWT.

The flaw in his position is that he does not (will not) accept that energy diverted by conduction and convection into the adiabatic cycle between surface and atmosphere cannot simultaneously be radiated to space otherwise the atmosphere would fall to the surface.

He thinks the S-B equation requires that the surface temperature MUST lead to radiation to space equivalent to the existing 288K surface temperature.

In reality radiation to space is at 255K and the other 33K goes to conduction and convection holding the atmosphere off the surface againt gravity.

Willis said that since the exchange of energy between surface and atmosphere nets out to zero it can have no effect on surface temperature. That is incorrect. There are two processes (radiation to space and work against gravity) involved and the same parcel of energy cannot be used for both simultaneously.

Sadly, Anthony Watts and many other reputable folks just don't get it either.

The reason seems to be that none of them ever understood the true nature of adiabatic processes.

Some time ago I went into that aspect in tiresome detail here:

4. With regard to my article about adiabatic processes I refer to the point of balance within the atmosphere where a molecule contains equal PE and KE.

That equates to MS's centre of mass for the atmosphere.

In effect, MS is supplying the mathematical underpinning for the conceptual narratives in my articles.

5. Yes and also explained in the derivation of the adiabatic lapse rate in my post

First, the basic assumption can be adopted that the atmosphere, in hydrostatic terms, is a self-gravitating system in constant hydrostatic equilibrium due to the balance of the two opposing forces of gravity and the atmospheric pressure gradient, according to the equation:

dP/dz = - ρ × g (1)

where ρ is the density (mass per volume) and g the acceleration due to gravity.

http://hockeyschtick.blogspot.com/2014/11/derivation-of-entire-33c-greenhouse.html

The radiative GHE proponents have unfortunately confused cause with effect. As shown above, temperature is a function of mass/gravity/pressure, and radiance and absorption/emission spectra from GHGs a function of that temperature. Radiative GHE proponents make the absurd assumption that the whole process is driven the opposite way around:

Cold body (like CO2 at -80C for 15u emission by Wien's law) makes a hotter body at -18C much hotter by 33C all the way to 15C
And then assume that 15C temperature drives the lapse rate/convection/pressure/etc.

It's absurd and the exact opposite on all counts!

What planet to these people live on, and then come here to tell me I'm the one violating the 2nd LoT! Mindboggling.

6. "As shown above, temperature is a function of mass/gravity/pressure, and radiance and absorption/emission spectra from GHGs a function of that temperature"

Absolutely.

The thing is that incoming sunlight is constantly pumping the air off the surface (adiabatic ascent) and gravity is constantly compressing it again (adiabatic descent) and since both processes are continuous and in balance the additional thermal energy at the surface (above S-B) is constantly being replenished and is therefore lopcked into the adiabativc 'loop' which I proposed in my article.

The net effect is that incoming solar energy gets a 'free pass' straight through the system which puts energy in and energy out in balance whilst elevating surface temperature by 33C.

Once one goes back to first principles it really is very simple.

7. Willis, I have responded to your "proof" and alleged refutations with a new post:

Why the atmosphere is in horizontal thermodynamic equilibrium but not vertical equilibrium

http://hockeyschtick.blogspot.com/2014/11/why-atmosphere-is-in-horizontal.html

Hope you now understand why your "proof" is a strawman that confuses horizontal and vertical equilibrium.

-Best

6. Sorry I was too quick. 1W/m2 for the solar constant corresponds to 0.175W/m2 at the spherical earth with albedo of 0.3 (1/4*0.7 = 0.175W/m2). This means that my empirically based calculation actually gives 0.01 C when increasing the solar constant with 1W/m2.

7. Wow, a bold new theory presented to be tested through falsification unlike the CAGW stories. Wonderful to see such an effort, I hope you at some point will try to have this published so that there would be a wide debate between the two competing theories. The climate science desperately needs this kind of an approach, where an open debate is necessary!

1. Thank you, I am vetting it first in the blogosphere, and have sent it to skeptical physicists to look it over and point out any errors or tautologies, so we'll see what comes of that first.

2. Erm, why are you vetting it in the blogsphere and sending to skeptical physicists?

Shouldn't you have you should have it checked out by someone impartial?

Surely there are people out there who know their stuff and could give an honest answer one way or the other?

Personally, I have some a bit of experience with heat transfer in closed internal systems, but I wouldn't feel completely confident in saying whether you are right about this or have made an error somewhere.

Someone can though.

8. "f both the gravitational and greenhouse gas radiative forcing theories had merit, the Earth would be at least 33C warmer than the present."

Yes.

As I said elsewhere:

"t is proposed that radiatively active gases in the atmosphere cause that uplift in temperature by radiating outgoing energy back to the surface for a reduction of the rate of cooling which then causes a surface temperature enhancement.

On the other hand we have the mechanical process of lifting the atmospheric gas molecules off the ground and thereafter maintaining atmospheric height. That is governed by the Gas Laws and the energy needed in order to maintain atmospheric height also requires a surface temperature enhancement.

Obviously both processes cannot be duplicating the same effect."

from here:

http://www.newclimatemodel.com/the-gas-constant-as-the-global-thermostat/

9. MSNovember 28, 2014 at 11:46 PM says:

Of course the atmosphere is not assumed to be in thermal equilibrium, at least certainly not by my derivations, which is driven by thermodynamic disequilibrium of convection/lapse rate/etc

Apparently you believe you've disproven the great physicist Maxwell, who was apparently the first to state the atmosphere temperature gradient is driven by pressure and would not be isothermal, and that

"In the convective equilibrium of temperature, the absolute temperature is proportional to the pressure raised to the power (γ-1)/γ, or 0,29. " referring to γ defined as the "heat capacity ratio" = Cp/Cv [ratio of specific heat capacity at constant pressure to specific heat capacity at constant volume] also referred to as the Poisson relation based on the ideal gas law & 1st law, which shows remarkable agreement with the gravito-thermal "greenhouse effect" found on all the planets with thick atmospheres [fig 5 & 6 from Nikolov and Zeller]"

Short answer? Both of these "proofs" make false assumptions about the gravitational greenhouse effect, they thus fail to refute Maxwell's book and the "greenhouse equation" derivations."

Thanks, MS. I note that you have not identified any specific fault in either Dr. Brown's or my proofs. Instead, you've simply asserted that you are right. Oh, you say that there are "false assumptions about the greenhouse effect" in both our proofs ... but since neither proof has anything to do with the greenhouse effect, I fear that your uncited and unsupported assertion is meaningless.

I also note that your entire section in quotes is uncited ... although it appears at first reading to be ascribed to Maxwell, in fact we have no clue who you are quoting. And you have not given a citation to any statement by Maxwell which supports your view.

If you think there are errors in either proof, I invite you to QUOTE EXACTLY the statement(s) that you think are wrong in either proof, and let us know why you think they are in error. Your unreferenced handwaving is totally unconvincing.

Best regards,

w.

1. It isn't for MS to prove Willis wrong, it is up to Willis to prove himself right.

In order to do so Willis must show how the same parcel of energy can radiate away whilst simultaneously being engaged in recycling energy between surface and atmosphere via the adiabatic process of convection which was initially provoked by conduction from surface to atmosphere.

To do that he must breach the Laws of Thermodynamics.

2. Here's my first post on Maxwell

http://hockeyschtick.blogspot.com/2014/05/maxwell-established-that-gravity.html

and I also discuss Maxwell at the first link at the top of this post. After you tell me why your proofs disprove what Maxwell said, then maybe I'll reconsider, but the fact is this equation is entirely derived from standard barometric equations, all I did is join them together under the assumption center of mass sets the height of the ERL rather than radiative forcing, and thus where to start the lapse rate. That's all, nothing but that new single new assumption.

3. Come on Willis, why not show some respect for someone who tries to come up with a falsifiable theory. Now you two are talking past each other.

When you always demand that "If you think there are errors in either proof, I invite you to QUOTE EXACTLY the statement(s) that you think are wrong in either proof, and let us know why you think they are in error. Your unreferenced handwaving is totally unconvincing", how about showing the same respect towards MS? It should be easy task to show where his theory is at fault due to the detailed presentation of his arguements

4. Willis,

See if you can follow this one.

A planetary surface with no atmosphere on top has equilibrated with its solar input and radiates away to space just as much energy per unit of time as it gets in from its heat source. Its heat LOSS equals its heat GAIN. Its entire heat loss comes via RADIATION.

Let's say: 2 parts IN, 2 parts OUT.

We then place a massive atmosphere on top of the surface and keep the solar input to the surface constant.

What happens? The surface all of a sudden loses some of the incoming radiative heat from the Sun to this atmosphere by way of conduction > convection. This energy is then no longer available for the surface to RADIATE to space.

2 parts IN (from the Sun), 1 part OUT to the atmosphere, 1 part OUT (radiated to space).

The surface with an atmosphere on top is no longer able to maintain its radiative equilibrium with its heat source, because some of the energy coming in is now lost through other heat loss mechanisms.

It still maintains heat balance (heat OUT = heat IN), only no longer a purely RADIATIVE balance. Why? Because the surface is no longer in a purely RADIATIVE situation.

The Stefan-Boltzmann equation computes the RADIATIVE heat loss of an object.

This equals its TOTAL heat loss ONLY in a purely radiative situation, like with a BB/GB in space or a very hot object.

It applies to a pure/ideal emitter. The surface of the Earth is NOT such a pure/ideal emitter. It loses the majority of its heat via conduction/evaporation to the atmosphere.

5. If you add an atmosphere to a planet in radiation equilibrium, this simply will set the system off balance momentarily. During this time, the planet system (including the atmosphere) will radiate less to space since it is warming up the atmosphere. But once, the atmosphere is in equilibrium with the planetary surface, the planet system will radiate to space the original amount again (2 parts in your example). Thus, the planet surface is still at 255°K.

6. " But once, the atmosphere is in equilibrium with the planetary surface, the planet system will radiate to space the original amount again (2 parts in your example). Thus, the planet surface is still at 255°K. "

Not correct.

The gravitational potential energy accumulated within the atmosphere during the progress of the first convective cycle (before the adiabatic loop closes) cannot be dissipated to space because following the closing of the adiabatic loop constant pumping up of the atmosphere by incoming solar ( convective ascent) is matched by constant compression within the gravitational field (convective descent) and at that point the adiabatic loop neither adds NOR LOSES any energy.

Which, for Earth, leaves the surface 33K warmer than S-B.

Only when one reduces or stops insolation will the amount of energy in the adiabatic loop decline.

The adiabatic loop contains a permanent store of gravitational potential energy which is constsantly recycled up and down by convection and cannot be lost to space because gravitational potential energy is not heat and cannot radiate.

7. "The adiabatic loop contains a permanent store of gravitational potential energy"

Exactly, the atmosphere is a store of energy. But still, it is a store of energy just added to the earth, its still the earth's surface which is the origin of the atmosphere's temperature. You argue that the energy contained in the atmosphere (which we agree is constant) is warming the earth's surface by compression, which is simply not true. You cannot make a system (the earth) warmer by adding mass (the atmosphere) and leaving the source constant (sun) constant. You need the greenhouse mechanism to explain this.

8. "You cannot make a system (the earth) warmer by adding mass (the atmosphere) and leaving the source constant (sun) constant. You need the greenhouse mechanism to explain this."

Ohh??? Please explain how Flagstaff AZ. is always so much cooler than other close by locations (Phoenix, Tuscon, Las Vegas, Death Valley, etc.). It is because Flagstaff is higher (6,910 ft.) in the atmosphere that surrounding, low lying locations (Phoenix [1,086], Tucson [2,389], Las Vegas [2,100]).

Pikes Peak in Colorado is 14,115 ft. & 47 degF & Boulder, right close by, is 5430 ft but @64 degF.

With weather patterns being basically equal, lower elevation location will be warmer than higher elevation locations due to atmospheric density warming less quickly than lower elevation, higher density locations.

9. "You cannot make a system (the earth) warmer by adding mass (the atmosphere) and leaving the source constant (sun) constant. "

You can because more mass increases surface density which allows a larger proportion of the solar energy passing through to be retained and for longer within the conductive/convective process.

That is why large gas planets get hotter than the Earth despite receiving less sunlight.

10. MS, this is a brilliant culmination to a brilliant series of posts. This is real science being done in front of our eyes. Thanks so very much for the opportunity to see this progress in real time.

I am especially glad to see the mathematical underpinning of this culminating post --- the whole series in fact. I know Maxwell would be proud to see this work. I also know that the luke-warmers will hate it. :-)

In the end, the equation will hold up to observation of the real world planet earth ... or it will not. Time will tell, but I am looking forward to the process of validation.

Congratulations. This is an impressive intellectual work regardless of the future outcome.

1. "I also know that the luke-warmers will hate it. :-)"

Not only the luke-warmers but also the entire trillion dollar CAGW government-science-industrial complex, so I expect everyone to fight this tooth and nail LOL

Thanks again Mark for your kind words and best wishes

11. I hope I can get this comment passed the robot detector. (my eyesight hinders that a bit)

I would like to say that I offer my congratulations for doing real science and offering a real equation to back it up. In fact, the whole series has been superb. Your work is the essence of what science is supposed to be. (regardless if it stands or falls in the long run)

I also congratulate you on letting luke-warmers post here. We often are "snipped" or banned at WUWT due to the host's censorship of anything that goes against "back-radiation" as the main driver of our climate. He calls it "Slayer Nonsense". Nice to know that some places allow "Luke-Warmer Nonsense" even if the favor is not returned.

I hope to see this post much discussed. I note that the TalkShop is already discussing it. :-)

Thanks for your hard work on this topic. ~ Mark

12. First, note that according to this standard atmosphere calculator
http://www.digitaldutch.com/atmoscalc/
the altitude where P=P_surface/2 is rather at 5500m which gives a temperature of 252.4°K and thus 2.6°C below 255°C (the temperature produced by the term in the 4th root). This would change your surface temperature estimation.

In any case, in your formula you basically can use any pair of temperature and pressure (altitude) occuring in the real troposphere and you'll always more or less calculate the actual surface temperature. Note that when I speak of temperature I mean the term in the 4th root which represents 255°C. But as I said, just plug in another pair of values of the real troposphere (for example P=0.4 atm. and T=241°K) and you'll get the same result.
This means that your formula is just a complicated way to extrapolate surface temperature from an arbitrary pair of values of T and P (or height H)..

1. "blah blah...standard atmosphere calculator"

The greenhouse equation corrects for both density and pressure changes with altitude, and in fact the equation calculates the standard atmosphere within 0.28C at every single height from the surface to ~11,000 meters:

http://hockeyschtick.blogspot.com/2014/11/the-greenhouse-equation-predicts.html

"In any case, in your formula you basically can use any pair of temperature and pressure (altitude) occuring in the real troposphere and you'll always more or less calculate the actual surface temperature."

False again. Wolfram Alpha does a simultaneous solution of T on both sides of the equation to solve for the one and only one unique value of the T on both sides, and can be used to produce an even more complex derivation as shown above that has the T only on the left side. That is how Wolfram Alpha simultaneously calculates both Ts and there is one and only one T for each value of the height (s). It is NOT a tautology, and Wolfram Alpha always notifies you if you have any tautologies and irrational solutions, but there are NONE for this equation.

2. It is not a mathematical tautology, it is a physical one! The temperature function in the troposphere is a straight line. By using the actual lapse rate of 6.5°K/km (your g/C, note that you put C=1.5 in order to obtain the observed 6.5°K/km) you just need one pair of temperature and height (or temperature and pressure) to entirely define the function! You do this by using T=255°K and P=0.5 atm, but you could actually use any other pair of values occuring in the troposphere!
I call it tautology because you use real data (lapse rate of 6.5 and T at a certain P) to derive a forumula to predict this same real data. In this way you can neither prove nor disprove conventional greenhouse theory or any other explanation.

3. The symbolic solution of the equation requires NO such assumptions and is based upon dT=(-g/Cp)dh, where dT is a function of (-g/Cp)dh.

Of course, Cp varies constantly at every location on the globe between the dry rate 9.8C/km and the wet rate 5C/km, and the observed global average temporally and spatially is 6.5C/km which when used in the numerical solution perfectly reproduces the global standard average temperature profile within 0.28C!

Plug what ever value for Cp into the numerical solution you like, the symbolic equation and theory still holds and absolutely disproves the radiative GHE theory. You cannot have it both ways. Period.

13. Congratulations. Great job consistent with intuition. Changes in the circulation (decrease in solar activity) are sufficient to show the huge temperature differences between summer and winter, depending on the sunlight.

14. I am sorry, but Willis Eschenbach's understanding of basic physics is minimal. The 2nd law does is not violated by these equations.....back-radiation violates the 2nd law. The 2nd law shows how a "time" dependent equilibrium process moves from a high energy gradient to a lower equilibrium state--hot to cold. For "back-radiation" to process heat, there must be an "organized" heat gradient to push heat from hot to cold....obviously, this is nonsense, as there are no hot spots in the troposphere (organized heat) as the IPCC models predicted.

1. Back radiation actually does not violate 2nd law of thermodynamics. This law applies to the net heat transfer between 2 bodies (i.e. you need work to make NET heat flow from the cold to the warm body).
The (warmer) earth radiates to the (cooler) atmosphere which sends back part of this radiation. The net heat transfer is still from earth to atmosphere and thus no violation of the 2nd law is taking place. Hope this clarifies...

2. "Back radiation actually does not violate 2nd law of thermodynamics"

Correct, but what AGW proponents claim as the consequence of back radiation does.

They say GHGs makes the surface warmer than it otherwise would have been by impeding the flow of radiation to space.

There is no such impedance because convective overturning alters to negate the thermal effect of GHGs.

In any event, the greenhouse effect being mass induced, the thermal effect of GHGs would be insignificant even if they were right.

Once one attributes the primary greenhouse effect to mass, gravity and insolation then any further effect from GHGs becomes too small to measure.

3. "Once one attributes the primary greenhouse effect to mass, gravity and insolation then any further effect from GHGs becomes too small to measure."

Exactly. :-)

4. "There is no such impedance because convective overturning alters to negate the thermal effect of GHGs."

What do you mean by this? "Convective overturning" does not "negate" the effect, it simply adapts the atmosphere temperature to the boundary condition which is the earth. So, as the earth's surface warms due to radiation sent back, so does the atmosphere (by convective overturning, as you say). No violation of the 2nd law...

5. GHGs allow energy out to space that would otherwise have had to be returned to the surface in adiabatic descent before being radiated out to space.

That is a cooling process.

Less energy goes back to the surface in adiabatic descent but the atmosphere has then risen less high so density is higher at the surface.

Higher surface density allows a greater part of insolation to be conducted to the air which is a warming effect.

Both processes cancel out.

6. GHG's radiate uniformely in every direction.. so they also radiate back to earth, thus leading to more energy radiated to earth which increases surface temperature. Of course, the energy radiated to space is always the same (thus GHG's do not "allow energy out to space that would otherwise have had to be returned to the surface") and depends only on the distance between earth and sun (and of course the sun's power).

7. 8. Ok that's the last comment I'm allowing on this thread that tries to sell the repeatedly debunked radiative GHE, once again repeating the 2nd-law-violating CAGW nonsense that a much colder body makes a much hotter body 33C hotter. It's complete nonsense and I've just provided mathematical proof the CAGW greenhouse effect theory is complete nonsense, and the actual cause of the entire 33C GHE is mass/gravity/pressure not affected by GHGs.

The topic of this thread is the equation above. Either state specifically where the mathematical flaw is located or agree with it, but I'm not going to further discuss the CAGW GHE nonsense on this thread.

15. Well done. Adiabatic heating is a force to be reckoned with, as anyone who lives on the lee side of Rockies knows full well. We call them Chinooks and they can raise temperatures from -30 C to 5 C in a matter of hours.

16. Well done !

Your formula works and I like it. Correct me if I am wrong but I think what the formula is really saying is the following :

T(s) = Teff + (H-s)*LR

H = height where atmosphere is half that of surface pressure calculated from hydrostatic pressure distribution. This is then assumed to be the effective height of IR emission to space corresponding to Teff

I think this is a very good approximation to reality. However it does not explain the origin of the phenomena described. It is more of a phenomenological formula that explains observations. It does not mean for example that gravity generates any surface warming or for that matter that greenhouse gases are responsible either.

The effective emission height being about at a pressure of P(surface)/2 is interesting. This is the height at which there are an equal number of IR active molecules(CO2) below the level as there are above the level. This makes sense as being approximately where outgoing radiation dominates over downward radiation from above. The atmosphere thins out exponentially above.

To get a deeper understanding of the physical processes you cannot avoid doing a full radiative transfer calculation. People who tell you the greenhouse effect is obvious are lying. The details are not obvious as each molecule has a different quantum line spectrum.

17. " it does not explain the origin of the phenomena described. "

It impliedly tells us that origin.

It is the density variation with height interacting with energy from the sun.

The denser the air at the surface the more of the available solar input is diverted to conduction and convection.

The rate at which density reduces with height as a result of the sttrength of the gravitational field determines the relative proportions of KE (heat) and PE (not heat) carried by each molecule.

18. "It is the density variation with height interacting with energy from the sun."

Apart from clouds and stratospheric ozone the atmosphere absorbs very little incident solar radiation. The sun warms the surface which the radiates heat upwards. It does also conduct heat to the lower atmosphere which induces some convection. However the surface cools fastest through radiation.

I have just written a post on all this

The role of gravity in the greenhouse effect

A basic Question: Is there a lapse rate on a planet with an atmosphere consisting of 100% nitrogen or argon ?

19. It is implicit in my comment that the energy from the sun with which the density variation interacts is primarily from the sun warmed surface but also from any radiative materials in the atmosphere.

Conduction and convection alone cause the greenhouse effect and radiative materials tend to weaken convective overturning by allowing leakage of radiation to space from within the atmosphere.

20. help! am I right?: there is no explicit power to lift warm air , but it is just pushed upwards by cold/more dense air, which is pulled downwards by gravity. then there exists Arbeit= work, that poduces warmth ,continuosly through gravity(that is what Heller denies!?) so there is warming by gravity-right? and: denser air has more molekules to be energised/warmed by sun. is that an explanation without equation?- or didnt I get it right?

21. This is nonsense. Here's how you can tell:

1. there's no reference to humidity or water vapor in that equation.

2. the dry adiabatic lapse rate is approximately double the wet adiabatic lapse rate.

So, obviously, that equation cannot describe the lapse rate.

1. False. Your comment is nonsense, and here's how you can tell:

1. Of course there is no mention of water vapor or humidity because that is unnecessary to determine the atmospheric temperature profile and surface temperature.

2. Duh and guess what, the wet adiabatic lapse rate is 1/2 the dry, which means water vapor acts as a strong COOLING agent, NOT warming!

So obviously this equation is correct because it perfectly reproduces the US Standard Atmosphere lapse rate.

22. Unless you can include Venus, Jupiter, Saturn, Uranus and Neptune for calculations of atmospheric temperature using this equation, it remains incomplete. For Venus, especially where we have good physical measurements of atmospheric temperatures, this is a MUST if you wish to fully provide a robust theory.
May I suggest that rather than working from the surface up, to make your calculations, you work from the depth at which equilibrium with the solar constant is found (or perhaps solar constant minus the reflected component of the energy) and then work both up AND down, with your equation. UP to find the mean radiating height to space of the energy absorbed at maximum flux lower down. DOWN because the ideal gas laws and conversion of gravitational potential energy to kinetic energy must still apply even with zero solar input.
Explaining Venus and quantifying conditions there will correct physics, is VITAL to fully falsifying the greenhouse effect/global warming dogma.