The True Energy Balance of the Earth+Atmosphere

Update 10 March, 2012: I have realized (on my own) that my blackbody understanding was wrong, and why my physical interpretation of the facts of my Venus/Earth comparison is nevertheless correct. I have posted on this at "My Own Blackbody Error". This post no longer reflects my scientific position, and should not be taken as such.

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UPDATE 03 March, 2012: I have obtained a better value for the mass mean temperature in the stratosphere [T(strat)] used in this article. From computerized numerical integration of the relevant stratosphere formulas (for T(h) and ρ(h)) in the 1976 US Standard Atmosphere, I found T(strat)=223.9K, not the 240K used in the article below. This does not change the analysis here or its conclusions substantially, as I will allow others to verify; it basically only means a slightly larger amount of solar absorption by the surface (from 50-52% of the incident solar, in the original article here, to perhaps as high as 54%). I also obtained a more precise mass mean temperature in the troposphere, through finer numerical integration on the computer, which merely confirms the 259.3K used here.
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When I posted the landmark article, "Venus: No Greenhouse Effect", which fundamentally corrects climate science, in November, 2010, I hoped that other competent scientists would quickly see the validity of my contribution and use it to re-establish climate science on a sound basis, stop the insanity of a politically promulgated, false consensus, and also establish my scientific worth to the world, thus allowing my far greater discoveries to gain serious attention and the worldwide recognition they deserve.

But that has not happened. I have seen NO positive advance in climate understanding since I proved, with the definitive facts, that there is no greenhouse effect, of increasing atmospheric temperature with increasing atmospheric carbon dioxide.

So here is yet another advance of my own, elementary but decisive.

In my recent post, "On the Fundamental Warming of the Atmosphere", I summarized the energy taken up, from the incident solar radiation, by the Earth-plus-atmosphere system, using widely used numbers (see, for example, the Kiehl-Trenberth energy budget). Based upon my simple Venus/Earth analysis, the relevant fractions are:

Incident solar energy = 100%

Solar energy reflected = 30%

Solar energy directly absorbed by surface = 50%

Solar energy directly absorbed by atmosphere = 20%

Of the 70% that goes to heating either the surface or the atmosphere, 2/7 directly heats the atmosphere, and 5/7 directly heats the surface. In the new understanding provided by my Venus/Earth comparison, the atmosphere and surface are separately and independently heated by incoming solar energy, and the governing hydrostatic temperature lapse rate structure of the troposphere moderates any inequality in their heating, with upward heat transport, "down" the temperature gradient, contributing to no further warming of the atmosphere, in my view.

The question should quickly have been taken up by any real climate experts, or competent scientists: Can the above energy balance be quantitatively validated (or invalidated)? The answer is yes, it can be validated.

The consensus claims that the radiating temperature of the system is 255K, based upon using "incident minus reflected" solar intensity in the Stefan-Boltzmann formula:

342 W/m^2 (mean incident) - 30% reflected = 239 W/m^2 (mean absorbed)

which gives a mean radiating temperature of 255K. (I am using whole numbers as much as possible throughout, reflecting the uncertainty I have seen in the numbers used by others.)

I know this 255K is wrong, and the radiating, or equivalent blackbody, temperature, is to be found from the incident radiation alone:

342 W/m^2, giving an equivalent blackbody temperature of 279K,

BUT THE CONSENSUS CALLS THIS RIDICULOUS, and everyone, from alarmists to the mildest skeptics, seems to accept the consensus radiation numbers, and believes 255K is the physical fact.

Here is what I find:

Incoming solar energy is absorbed in both the stratosphere and the troposphere, as well as by the surface. Because my Venus/Earth comparison looked at the Earth troposphere region, and got such definitive results, I first left out the stratosphere as being of minor importance, considered the troposphere the sole relevant atmosphere, and considered the following:

If we know the mean temperature of each of the two absorbing systems, atmosphere and surface, we can determine the mean temperature of the total surface-plus-atmosphere system, in the following way:

fraction of absorbed power given to atmosphere = f1 (= 2/7)

fraction of absorbed power given to surface = f2 (= 5/7)

mean temperature of atmosphere (troposphere) = T(trop)

mean temperature of surface =T(surf)

mean temperature of surface-plus-troposphere system = T(all)

where the consensus says T(all) is 255K, and I (and other critics of the consensus) say it should be 279K.

Then

f1 x T(trop) + f2 x T(surf) = T(all) (Equation 1)

Physically, this means that the fraction of the total power retained in the total system by a given subsystem, times that subsystem's mean temperature, gives that subsystem's contribution to the mean temperature of the whole.

For T(trop), I calculated the mass mean temperature of the troposphere, according to the formula:

T(trop) = Integral [T(h)dm] / Integral [dm] (Equation 2)

where h = altitude above the surface, and

dm = element of atmospheric mass,

with dm = ρ(h) dV, ρ = density and dV = element of volume,

and dV = volume of spherical shell of thickness dh, radius = Re + h
with Re, the radius of the Earth.

The following equations hold in the Standard Atmosphere troposphere:

T(h) = To x (1 - 6.5 h/To)

ln[ρ/ρ(o)] = 4.2559 x ln(1 - 6.5 h/To)

I plugged these into Equation 2 and solved it numerically, by dividing the troposphere between 0 and 11 km into spherical shells of thickness 0.2km, and found:

T(trop) = 259.3K (NOT 255K, note)

T(surf) = 288.15K, so Equation 1 becomes

(2/7)259.3K + (5/7)288.15K = 279.9K = 280K approximately.

Compare this to 279K obtained by using the Stefan-Boltzmann formula with INCIDENT mean solar intensity. I consider this a quantitative verification of my usage of the Stefan-Boltzmann formula, and a disproof of the consensus usage.

But WARNING:

If one uses 255K for T(trop)--as many do in their narratives, both consensus and skeptics--in the above equation, one gets

(2/7)255K + (5/7)288.15K = 278.7K,

an apparently more exact result for the true radiating temperature of the total system--by a mere 1.2K, however, which I suspect is well within the uncertainty of the calculation. And using 255K for the troposphere alone is inconsistent with making that the radiating temperature of the whole Earth-atmosphere system, and still makes the consensus wrong in its usage of the Stefan-Boltzmann formula, since 279K is still clearly the true effective radiating temperature of the total system, not 255K.

Even more importantly, from the use of Equation 1 it should be clear that the warming of the atmosphere must be independent of the warming of the surface, since the coefficients f1 and f2 on the left-hand side of the equation are independent of my Venus/Earth findings (they represent the actual solar power absorbed by the troposphere and the surface, as shown in the Kiehl-Trenberth Earth Energy Budget), and the above verification is thus an independent check upon my Venus/Earth conclusion (that is, it uses my conclusion, as expressed by Equation 1, with real-world numbers). So the atmosphere IS fundamentally warmed only by direct absorption of a fraction of the incident solar radiation (and the other upward heat transfers from the surface, in the K-T Energy budget, DO NOT CONTRIBUTE TO THE FUNDAMENTAL WARMING OF THE ATMOSPHERE--the incident 20% directly absorbed by the atmosphere does the job).

Now let's do the same budget calculation, but with the stratosphere properly included, so that we have an equation with three absorbing terms:

f1 T(strat) + f2 T(trop) + f3 T(surf) = T(all),

where f1 etc. are the fractions of the total absorbed radiation in each subsystem.

I do not have what I consider a definitive number for T(strat), the mean temperature in the stratosphere, so I estimated it from a graph of the TEMPERATURE PROFILE of the atmosphere, as a function of altitude (I leave it to the reader as an independent exercise, to check my work). I estimated T(strat) = 240K (+/- 5K, judging from what I could glean from a short internet search on the topic). The stratosphere absorbs ultraviolet (UV)--as much as 95% of the incident solar UV, internet sources say--and UV comprises between 8% and 8.5% of the incident solar energy, so I first took 8% to be the fraction of the incident solar absorbed by the stratosphere, leaving 12% to warm the troposphere and about 50% to warm the surface. The absorption equation becomes:

(8/70)x240K + (12/70)x259.3K + (50/70)x288.15K = 277.7K = 278K

I have provided this sample calculation to show that, given the uncertainties in the absorption numbers, one can still see that the true radiating temperature of the Earth-plus-atmosphere system MUST BE 279K, not 255K.

One can also play with the above equation (with 279K, on the right-hand side, as a physical constraint). Using numbers in the Kiehl-Trenberth energy budget, the reflected solar is 107 W/m^2, which is 31.3% of the incident 342 W/m^2. This makes the total absorption, spread over the stratosphere, troposphere, and surface, 68.7% of the total mean solar, rather than the 70% I used earlier. This gives us some idea of the uncertainties involved in the numbers used by different sources.

Using a reasonable variation in the absorbing fractions attributed to each of the three subsystems of stratosphere, troposphere, and surface, and using a total absorbed percentage between, say, 68.5% and 70% of the incident solar, I found 7-8% stratosphere, 10-12% troposphere, and 50-52% surface to be the most likely range for the solar fractions absorbed by each subsystem. For example, for a total of 68.7% of incident solar absorbed, as in the K-T Earth energy budget, I found 7.5% in the stratosphere, 10% in the troposphere, and 51.2% at the surface to give particularly good results:

(7.5/68.7) 240K + (10/68.7) 259.3K + (51.2/68.7) 288.15K = 278.7K

but I consider that just a good educated guess, and about as good as one can do with the numbers provided by climate science so far.

Summarizing: My evaluation of the fundamental warming of the atmosphere, separately from the warming of the surface, and both by direct absorption of entirely different fractions of the incident solar radiation, is quantitatively verified by a correct usage of the blackbody formula (to obtain 279K for the radiating temperature of the Earth-atmosphere system).