<blockquote>Despite being asked several times to apply it to the neutral atmosphere they have totally failed.</blockquote>
<b>(1)</b> Here are links to comments above which have already provided explanations of our methods generalised to apply for a neutral atmosphere:
I showed how it applies to a neutral atmosphere in <a href="http://judithcurry.com/2011/08/13/slaying-the-greenhouse-dragon-part-iv/#comment-144127" rel="nofollow">this comment.</a> above (Nov 29). I explained again with emphasis on the neutral dry atmosphere in <a href="http://judithcurry.com/2011/08/13/slaying-the-greenhouse-dragon-part-iv/#comment-144398" rel="nofollow">this comment</a>.
I explained the connection between Willis' simple black shell model, and a grey atmosphere radiative model in <a href="http://judithcurry.com/2011/08/13/slaying-the-greenhouse-dragon-part-iv/#comment-144788" rel="nofollow">this comment</a>. Willis' ideas don't involve any possibility of convection, so there needs to be a significant change to his methods in order to apply it to a neutral atmosphere which has been relaxed by convection to the DALR. Willis in fact is giving the first steps towards calculation of the "radiative equilibrium", which is what applies in a planet's stratosphere. I explained in the subsequent comment <a href="http://judithcurry.com/2011/08/13/slaying-the-greenhouse-dragon-part-iv/#comment-144790" rel="nofollow"> how it is generalized to account for the effects of convection in the troposphere, and in particular how to handle the DALR neutral atmosphere.
The simpler radiative case, however, was enough to show the errors in G&T's invocations of the second law; which is why we used it in our paper.
I gave an example of a well established program which uses these methods, including the refinements I have mentioned in the comments linked above, to get to a meaningful model of the troposphere on Earth: MODTRAN.
MODTRAN gives results for the emission spectrum and its intensity which are indeed quite close to empirical observations. There are other programs which give further refinements; although they all still use success layers and calculate radiation between them. (This is simply stepwise numeric integration.)
<b>(2)</b> Our models and comments are simply explaining conventional physics that is used without quibble even in your own references:
We've shown that your own reference describes the atmospheric greenhouse effect in the same terms as we do; including the 33K temperature difference between what is expressed at the surface and what is expressed out to space. It does not go into the calculation methods applied in models; but the authors are actually well established experts on model calculations, all of which do use stepwise numeric integrations up the vertical column to get quantified results; and you could confirm this if you asked them or even simply took the time to read more of their publications.
<b>(3)</b> Here is the key point of difference which is the main hurdle in your way for comprehending conventional models for calculating atmospheric temperature profiles -- INCLUDING surface temperature.
The key stumbling point seems to be your claim that Cp captures all the important radiative properties of the atmosphere, and that the lapse rate alone is sufficient to explain surface temperatures.
Interestingly, if you look at your own reference again (<a href="http://www-as.harvard.edu/education/brasseur_jacob/ch2_brasseurjacob_Jan11.pdf" rel="nofollow">Brasseur and Jacob</a>), most of the discussion is on the stratosphere, where lapse rates are much steeper than adiabatic, and indeed reverse as you get up to the ozone layer. In these cases, the temperature profiles are given by radiative equilibrium; and Cp is not used at all. The absorption/emission spectrum is key.
Every working model able to explain Earth's surface temperature uses the absorption and emission of IR within the atmosphere. Cp alone doesn't cut it.
<b>(4)</b> It is you, not us, who is failing to answer straightforward questions.
We HAVE answered questions; you don't agree with the answers, which is your prerogative.
Could you please answer this straightforward question (given now for the third time). It's deliberately open ended to avoid presuming your answer, but I think anyone reading on who looks into the physics required to get an answer will be able to see the importance of greenhouse effects.
<i>Convection, and energy moved by latent heat of evaporation and condensation, all carry energy up into the atmosphere. Where does that energy end up? How does the atmosphere get rid of the energy it is gaining from convection?</i>
If you dispute the premise of the first sentence, just say so; otherwise do please indicate how the atmosphere sheds this energy, in your view.
<b>Postscript</b>. This isn't about CAGW. The issue here is the basis for Earth's surface temperature right now; not about identification and comparison of all the many processes that might or might not cause Earth's surface temperature to rise or fall a few degrees. It's all about why the Earth's surface is at about 15C right now; a temperature at which thermal emissions carry much more power than the solar input.
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