Err, wow…. I think. Wayne Jackson has been mostly absent over the last six months but obviously not idle. Now he has dropped in on an old thread of Stephen Wilde’s and left this remarkable comment:
Sorry it has taken me six months to reply to this thread Stephen but it has taken a LOT of digging to get myself a handle on what really matters in planetary atmosphere temperature profiles. I have picked up a lot of clues from many including gallopingcamel, Nikolov & Zeller, Miskolczi, Ramanathan and the CERES team, even Huffman, but none of them seemed to give the complete, 100%, answer but each had hints in their texts.
“The derivation of the adiabatic lapse rate is taught in high schools and provides an explanation of the observed temperature gradient in Earth’s atmosphere.”
Absolutely. I have come to realize that a lapse rate is a terrible metric to use if you are striving to describe an atmosphere, it is the pressure ratio between two layers and the mean heat capacity ratio (λ=Cp/Cv) adjusted for the natural greenhouse function (1 minus 1/3) that exactly describes each atmosphere. The GH adjustment is therefore G=2/3 for optically thick atmospheres.
This line of thought derives from some topics: heat capacity ratio (λ), degrees of freedom of molecules (DOF), heat capacities (Cp & Cv), speed of sound in gases, potential temperature ( Φ=T(P0/P)^(R/Cp) ), and equipartition of energy per mode bi-directionally, always.
Take earth by the ’76 Standard Atmosphere or the ISA, at 11020 meters the temperature there and upward is 216.65K. The pressure here is 22629 Pa. Our atmosphere is predominately composed of linear di and tri atomic molecules setting the Cp/Cv to be right at 1.4 decreased very slightly for the IR active components and most importantly at their concentration.
Mean DOF (degrees of freedom) = 5.008
λ = 1+2/DOF
(note: DOF = 2/( λ-1)
P0 = 22629
216.65‹K› * (101325 / P0)^(G * (λ-1)/λ)
Mean DOF = 5.77
λ = 1+2/DOF
P0 = 1‹atm›
338.3‹K› * (92‹atm› / P0)^(G * (λ-1)/λ)
( By the way, the 5.77 DOF was from some actual lab experiments to measure CO2’s effective Cp. This derived value came from a Cp/Cv of approximately within error of 1.3466. Remember, CO2 is linear and has no permanent moment dipole)
You can plug in any pressure level for either atmosphere and you will find the temperature at that pressure level will always very closely match (+/- 0.1°C) either Earth’s standard atmosphere or Venus’s International Reference Atmosphere (VIRA) very closely, too close to ignore. I digitized a plot of the VIRA atmosphere from a peer reviewed paper and the values are close to 735.1K at surface at 92‹atm› and the other end at 60 km and 23093 Pa where the temperature is 262.5‹K›.
Caveat One: It seems logical to me that this relation with a G of 2/3 should only hold for atmospheres where the per-line tau or optical thickness is greater than one, maybe even much greater than one. Earth’s mean tau is 1.87 and it seems to hold by this relationship so that seems to give us a somewhat scale of magnitude.
Caveat Two: Why the G=2/3? In one respect that seems logical also. The surface has an energy pressing upward and at most lines are all absorbed at some level, matters not where, and that absorption likewise presses both downward and upward to space. You get two pressures within the atmosphere and one pressure to space giving you the 2/3. That’s a bit simplistic but that’s one explanation. The other might be that horizontal radiation within the atmosphere itself is always bound and guaranteed absorption since horizontally all strong radiation lines are homogenous. That would give 4/6th or 2/3rd, six being possible directions in three dimensional space and all other radiation not included in this 4/6th is destined to escape upward, the 1/3rd portion. All of this would require proper differential analysis of course but each in its own right seems logical macroscopically viewing the atmosphere as a single element.
Caveat Three: Does this hold for each and every thick planetary atmosphere?
Don’t know about Saturn and Jupiter yet, and here is why you don’t want to get dragged into thinking that lapse rates are anything but descriptive (not causal) in nature. Both of those gas giants have a much lower environmental lapse rate but due to their huge size and gravitational pull the pressure ratios across levels are much further apart in kilometers than on the Earth or Venus and so even though pressure level to pressure level ratios are close to the Earth and Venus, looking at temperatures, the lapse rates are far apart in value. Watch out.
Another thing I have learned is that just by knowing the speed of sound and the mean molar mass of an atmosphere you can, along with Van der Waals adjustments, get the empirical λ which is much better than relying on idealistic theoretical values. Maybe one day NASA can lower an explosive charge to explode on the surface and actually measure Venus’s mean speed of sound vertically across its atmosphere so we can get as actual mean λ and DOF which would add credence to this.
And that brings me to why CO2 levels have but a minuscule influence on Earth’s mean surface temperature, the 5.008 might increase to say something like 5.018 due to the doubling of CO2 from 0.039% but that only budges the surface temperature, and get this, downward by less than 0.1°C. This assumes the isothermal portion of the lower stratosphere maintains the same temperature.
You might wonder how the exponent R/Cp in the potential temperature equation ended up as (λ-1)/ λ.
Cp = λnR/(λ-1)
We are dealing molar so n=1, drop it, and therefore flip Cp:
1/Cp = (λ-1)/(λR)
R/Cp = (λ-1)/λ
Add the G and it at least matches the entire profiles for both Earth and Venus with correct heat capacity ratios applied and that is where I became VERY interested in this relationship, very curious as to why. I thought you and gallopingcamel might be curious too.
Something seems amiss; something besides CO2 increased the temperature in the last few decades. Could it have been the sun after all in some manner?
Maybe this might give someone the push to read and help me understand this process a bit deeper for I’m rather new to meteorology, thermodynamics and climatology.
If someone knows why this relationship exists please let me know.
But, I am still digging along this line of evidence.
One more note on G=2/3. Ramanathan(1987?) listed the g at 0.332, Miskolczi (2007) lists the g of 0.333. I just took the more than evident of g=exactly 1/3 to give the G as 1-1/3 or 2/3. Assuming here a simple geometric explanation which may not be really true.