Trying it the other way, by setting K so that AA=ED brought the temperatures at the ground into near equality. So I think I’m onto something here.

In any event, posting like this and having to correct my self at every turn is turning into a novel. I will try to prepare a draft of a paper of the semi-grey model I am developing and post it into a site, that way I can make revisions to it as it gets tested and validated. It just takes time and a lot of work. Hopefully something useful will come of this.

Yes, but how can I argue with results? Figures 21 and 22 of “The greenhouse effect and the spectral decomposition of the clear-sky terrestrial radiation” by Miskolczi and Mlynczak compares the various ways to calculate the ground temp and skin temp, and at first sight, adding the extra exp(-tau) really seems to improve things. That would be no different than adding a term or constant heuristically, and seeing if it improves a correlation, regardless of how it was come about. I just have not been able to reproduce it.

In any event, I thought that if I made the ground temperature equal to the bottom atmospheric layer temperature to remove the discrepancy, then I could get closer to that equation. So I added a K term taking energy from the ground and having it completely absorbed in the bottom layer. And I was still able to get an exact result. I still find it hard to believe, but I guess that how the math works out.

To my surprise, the ground temperature did not change. It was completely invariant to convection. The bottom air layer temp did change and became equal to the ground temp, and the temp profile changed, but the top layer temp also remained constant. Amazingly this means that you could have an adiabatic rate and the ground temperature would still be correctly estimated from a model in radiative equilibrium with no convection. I used to think that maybe one could find the height of the tropopause first and then work downwards to where the ground temperature would be from the lapse rate, but it looks like its the other way around: The ground temp is defined by the total optical thickness only, and then the temp decreases from that fixed point upwards towards the tropopause.

SOD: I’ve meaning to ask you, in your rte program you adjusted the height to equipartition the pressure.. why did you make that choice? did it help in some calculation?. I just realized that dP = -rho*g*dz is dP=-dm/Area*g or basically you equipartion the mass just as I did.. just curious.

]]>The part I still need to figure out is were the term exp(-tau) in the equation f = 1/2 + tau/2 + exp(-tau) comes from…

It comes from Miskolczi’s incorrect derivation in Appendix B of M2007. If you (properly) treat Bo as a constant and Bg as a variable, you get a different value. It’s all there in the comment thread somewhere.

]]>It turns out that the Radiative terms all can be ratioed by the incoming long wave radiation So*(1-A)/4 which in equilibrium is equal to OLR. I did all my analysis with a ‘total’ emissivity et, but in terms of total optical thikness tau it is:

tau = -ln(1-et) or et = 1-e^-tau

I went ahead and tried to cast my results in the variables provided by Milkolczi, here are some:

SG/OLR = 1 + tau/2

ST/OLR = e^(-tau)*(1+tau/2)

EU/OLR = 1 – e^(-tau)*(1+tau/2)

AA/OLR = (1-e^-tau))*(1+tau/2)

ED/OLR = tau/2

Green house factor

GH/OLR = SG/OLR -1 = tau/2

Normalized green house factor

Gn/OLR = 1 – 1/(SG/OLR) = (SG-1)/SG = (tau/2)/(1+tau/2)

Temperature at the top of the atmosphere

Ttop = ( 1/2 * OLR / boltzman )^1/4

Temperature at the bottom of the atmosphere

Tbot = ( (1/2 + tau/2) * OLR / boltzman )^1/4

Temperature at the ground (eg ~ 1 )

Tg = ( (1 + tau/2)/eg * OLR / boltzman )^1/4

We see that there is a discontinuity from the ground temperature to the bottom air temperature (no convection), that there is a finite top atmosphere temperature, that the mentioned temperatures and the greehouse factor are completely defined by the optical thickness (so tau is equivalent to GH). Also we see that AA is not equal to ED.

Now, by having an auxiliary equation for height, it turns out that if the boundary condition for the the last layer is zero pressure, then that layer becomes of infinite height (I would put an arbitrarily small pressure as boundary to avoid that) so I guess this would be an infinite atmosphere, granted with miniscule zero density and pressure at the top. The part I still need to figure out is were the term exp(-tau) in the equation f = 1/2 + tau/2 + exp(-tau) comes from and how does it relate to my model, as it seems to be key in Miskolczi’s derivation, bringing AA ~ ED as well as bringing the bottom atmospheric temperature equal to ground temp.

Any information on the difference between his ‘semi-infinite atmosphere’ and his ‘bounded or semi-tranparent’ atmosphere, and physical meening of the extra term will be appreciated it.

Ok I know there is a complicated derivation for it, but I did not get my results from any of those formulas, just by the application of transmissibility = exp(-ko* density * delta length) in exact closed form. I would like to see if I can reproduce his equation using my method.

]]>Bo must be a constant because the math requires it. When you integrate a function, say x, the result is the integral of the function plus a constant.

∫xdx = x²/2 + C

C must always be constant because otherwise the derivative of the integral wouldn’t be equal to the original function.

d(x²/2 + C)dx = x

dC/dx ≡ 0

Yet Miskolczi in Appendix B of M2007 treats the integration constant Bo as if it were a variable and treats the variable Bg as if it were a constant. You can’t do that and claim your results are correct. It’s the equivalent of saying 2 + 2 = whatever I want it to be at any given time.

If you actually do the calculations based on Miskolczi’s equations, the results violate the original boundary conditions.

I+(0) ≠ H/π

The derivation is incorrect.

]]>One problem is that B0 is a constant which is determined – and required – by the First Law of Thermodynamics.

If a solution conflicts with this important law I think we can all agree that the solution is in error?

So the fundamental step (*assuming that there was a point to developing the source function for the semi-gray atmosphere within the general RTE solution..*) would be to demonstrate why B0 varying didn’t violate the first law of thermodynamics. That will be difficult, I think.

The global average relative humidity at 400 mbars is about 35%. What thermodynamic principle limits it to only 35%? This is much lower than the saturation limit, so that is not the reason. If it is not limited by a property of water (the saturation limit), then it must be limited by an energy restraint.

The conclusion doesn’t follow at all. But the topic is not a simple one.

The explanations in Water Vapor Feedback & Global Warming by Held & Soden (2000) are worth considering. (The whole paper should be read by everyone interested in this subject).

Convection takes parcels of air upwards – and if this was the only process then the relative humidity (above the boundary layer) would be at 100%.

But air rising is balanced by air that is descending. In fact, in the tropics deep convection accounts for ≅ 10% of the area, and subsiding air accounts for ≅ 90%. (The ascending air is moving a lot faster than the descending air so mass balance does occur).

The descending air was saturated at its point of coldest ascent. So as it descends its relative humidity drops – and in the sub-tropics this can result in relative humidity less than 10%.

Air mixes. So ascending air and descending air produce an average of relative humidity decreasing with height. The observed features can be approximately produced by climate models. You can see an example in the Held & Soden paper.

]]>But I think you misunderstood the purpose of taking the derivative of equation B7. Referring to “The most efficient cooling of the clear atmosphere requires a total optical depth that maximizes Bo”, you said “If someone wants to explain what thermodynamic principle create the first statement – I would be delighted.”

The global average relative humidity at 400 mbars is about 35%. What thermodynamic principle limits it to only 35%? This is much lower than the saturation limit, so that is not the reason. If it is not limited by a property of water (the saturation limit), then it must be limited by an energy restraint.

Miskolczi postulates that the amount of water vapour is determined by the requirement of the “maximum entropy principle” such that the optical depth adjusts to maximize the OLR, and Bo for any given surface temperature. He also postulates that the cloud cover is determined by the maximum entropy principle (page 19, M2007).

The Maximum entropy principle is an extension of the second law. It can be stated as “A system will select the path or assemblage of paths out of available paths that minimizes the potential or maximizes the entropy at the fastest rate given the constraints”. See:

http://www.lawofmaximumentropyproduction.com/

This implies that the atmosphere will maximize the Bo by adjusting water vapour amount for any given Bg.

Instead of saying the amount of GHG is what is currently there plus what we add plus feedbacks, we say the amount of GHG is that which maximizes the Bo for each possible surface temperature. This requires calculating dBo/dtau, keeping Bg constant. Therefore, in equation B10, your “correction” in red is not required.

]]>Eli,

A consequence of LTE is that temperature measured by any means is the same. In plasma emission spectroscopy, LTE doesn’t exist, radiative energy loss is quite large, for example, and different methods of measuring temperature give different results.

]]>Ken Gregory, Thanks for the corrections, I have fixed the article.

]]>But the point is still that d(Bo)/dτ is always zero, i.e. Bo is a constant. Without a theory that requires a constant optical depth or whatever ratio of fluxes to remain constant, all the messing about with NCEP/NCAR reanalysis numbers is completely meaningless. Those ratios only apply to conditions near present conditions, not to conditions where, say, CO2 is doubled.

]]>