Sunday, November 30, 2014

Quick and dirty explanation of the Greenhouse Equation and theory

Here's a quick and dirty explanation of the greenhouse equation, which I hope will be helpful to understand the theory behind it. 

In the first post of this series we derived the following equation from the 1st law of thermodynamics and ideal gas law to calculate the temperature at any height in the troposphere:

T = Te + (lapse rate)*(h - he)   (1)


T = calculated T at height (h), or (s) used in equation below
Te = equilibrium temperature with the Sun (a constant)
lapse rate = -g/Cp = -gravity/heat capacity at constant pressure
he = height at the average "effective radiating level" or ERL, where T= equilibrium Te with the Sun 

Since we are calculating the gravitational greenhouse effect on the mass of the atmosphere, in order to conserve energy, one-half of the gravitational potential energy of the atmosphere has to be above the center of mass and one-half below. This point is at 1/2 of the surface pressure after a logarithmic adjustment for pressure and density with altitude. Since the surface pressure in atmosphere units is by definition = 1 atmosphere, in the greenhouse equation log(P/2) = log(1/2) below.

Again to conserve energy, the equation has to balance the local vertical equilibrium (since gravity is a vertical forcing vector) at every given local height the gravitational potential energy with the opposing thermodynamic energy of convection. Thus, the center of mass where log(P/2) must be the balance point where the two opposing forces balance, and this same point must also be at the equilibrium temperature with the Sun and near the mid-point of the adiabatic lapse rate. 

We previously calculated he to be located at the h where P=log(Ps/2) and the same point where Te = 255K = equilibrium temperature with the Sun.

The gravitational acceleration constant (g) appears twice in the equation, since to calculate the gravitational forcing we used Newton's second law of motion F=ma, which applied to the atmosphere is F=mg. The second use of the gravitational constant is from the dry adiabatic lapse rate.

Here's the greenhouse equation and my quick and dirty notes on how the components I just discussed enter into equation (1) above to calculate the temperature T at any height including the surface, as well as the entire 33C greenhouse effect, and without ever once using any radiative forcing whatsoever from greenhouse gases:


  1. A wonderful series of posts that must have involved an enormous amount of thought and hard slog over years.

    Although I knew the basic principles it is very gratifying to see someone work through the detail in such a methodical and logical fashion.

    I'm sure there are points of detail that can be picked out that may need refinement or adjustment but the basics are correct and in accordance with the science of the mid 20th century which subsequently seems to have been ignored when the faulty radiative theories came to the fore.

    Meteorologists were always aware of such matters but it all went wrong when astrophysicists entered the climate arena.

    Unfortunately, astrophysics relies heavily on radiative exchanges to analyse planets around distant suns so astrophysicists have little or no knowledge of how non radiative thermodynamics plays out within an actual atmosphere.

    The most critical error was the absence of any appreciation of the thermal effects of work done with or against gravity in convective uplift and descent.

    That initial error, compounded by people with no meteorological experience over decades has led to the current climate farrago.

    Congratulations to MS and to anyone who has been assisting him.

    Maybe we can now start to come out of the climate science Dark Age.

  2. "If scientists of the past had known that the temperature of every planet with an atmosphere rises in direct proportion to atmospheric pressure, do you suppose they would have come up with a theory that attributed heating to the presence of certain trace gases that occupy less than one percent of our atmosphere? No, of course they wouldn’’t have.

    The theory of the greenhouse effect was concocted for the purpose of explaining why the earth is warmer than predicted. Yet every planet is warmer than predicted."

    (by Alan Siddons - 'Slaying the Sky Dragon - Death of The Greenhouse Gas Theory')

    "Can anybody explain to me how such an insignificant quantity of a gaseous substance can possibly heat up the entire column of air . . . by re-radiation . . . when air does not react to radiation in the first place——check your microwave oven for proof of that. The air inside is heated off the food, not the microwaves. Just like the real world!"

    (by Hans Schreuder - 'Slaying the Sky Dragon - Death of The Greenhouse Gas Theory')

    Never were truer words spoken than the above words of Siddons and Schreuder!

  3. The strategic mistake made by the so called 'Slayers' was in insisting that there was no greenhouse effect when they only meant that there was no RADIATIVE greenhouse effect.

    Their opponents used that error to sideline them.

    The Slayers just needed to accept that there is a greenhouse effect but it is mass induced.

    That is actually implicit in most of their work but they failed to get it across in the public domain.

    1. Hi Steven,

      You and I have been on the same page, with slightly different interpretations on the "gravity thermal effect", (I have been saying that the gravity effect allows the atmosphere to retain energy, essentially the same thing, I think)

      Its good to see that MS has had the time to actually quantify the theory and put it into a simple mathematical form.

      Now, how to get this accepted in places like WUWT, Steven Goddard etc, ! ???

      Regards to all.

      Andy G55

    2. Andy G55,

      I would love to see this work get onto WUWT and Real Science, however, you do realize that this will never happen, don't you? Anthony Watts (WUWT) and Tony Heller (Steven Goddard) are sophists and much more concerned with traffic on their sites than they are with the truth.

      You want proof? Go back to Heller's site and read the number of times he stated that he was no longer going to entertain those that did not believe in the GHE (the radiative GHE as posited by the IPCC), and yet he continued posting, post after post after post, each time claiming he "had enough" and yet he continued to fuel the discussions for more than THREE WEEKS! ... Are you freaking kidding me? ... Tony Heller is a sophist piece of crap. You are never going to get him to admit to this work for what it is, never mind admitting to its credibility.

      Anthony Watts is no different and probably even worse. Hell, after countless, unarguable refutations of Willis' "steel greenhouse" garbage by Joe Postma (and others), Watts still refuses to recognize that he, Willis and Spencer go their asses handed to them SOUNDLY and irrefutably. Willis' "steel greenhouse" is about as stupid and sophistic as it gets. It was a way for Willis' et al to attempt their own version of 3-card monte. A con game! And they know it!

      Again, I would LOVE to see this at WUWT and Real Science, but I think there is a better chance that the US will be deficit free for the 2015 fiscal year.

      I would suggest however, that you hand this off to Mark Morano at Climate Depot. I will probably pass this along to Joe Postma, PSI, and John O'Sullivan myself. But if you can get this to Mark, I am pretty sure he would get this posted at least on Climate Depot and you may even get lucky enough to get Drudge to pick it up from there.

      This is REALLY good stuff here guys (and gals). Keep it up!!! .. We desperately need this sort of thing to finally crush these AGW (and GHE) con artists.

    3. squidly..

      You may have notice in one of those posts on Real Science, I stated that I had removed the Real Science bookmark from my computer.

      Tony has done some very useful work illuminating the junk that is the US temperature data. He should stick to that instead of taking the SkS censorship route ! A pity. :-(


    4. AndyG55,

      Yes, I recall reading that very comment, and to that, I agree and have done the same. Heller has actually banned me from posting. Evidently I struck a nerve with him. I made a very short little comment to him criticizing him for repeatedly stoking the flames for which he was complaining, he quickly deleted that comment and banned me from posting. To me, that said EVERYTHING that I need to know about Tony Heller. Heller is a sophist of the first order. I agree that this is a shame, as he has done a tremendous job of pointing out historical data and information, both interesting AND detrimental to the AGW cause. I wish he would have had the self discipline to confine himself to just that.

  4. Squidly, you have hit the proverbial nail squarely on the head. Those other "science" websites are just interested in how may visitors they can get. Not sure how profitable that is for them really, but I guess they have to try to get fame somehow. People rarely remember who "lost"or who was "wrong", but the "arc of history" sure remembers who was right.

    Steve Goddard was bragging about how many "hits" he got after he started his "Is the Greenhouse effect (radiative GHE) real ?" line of conversation. Yes he got a bunch of folks that agree with him to denigrate and abuse those that disagree, not much of an accomplishment really. I've been through decades of design reviews for Earth imaging satellites (and other stuff), the goal is always to get to the correct answer, trying to make others look "dumb as a box of rocks" is never the goal.

    So, in summary, lets leave those poor misguided fools alone, in a few more years they will likely look as "dumb as a box of rocks", and I apologize in advance for insulting rocks, some of my best friends are rocks and they can be quite informative if you know what questions to ask.....

    Cheers, Kevin

    1. Kevin,

      I believe in the case of Tony Heller, that it is an ego thing. If you browse his blog backwards in time, one can see a distinct and progressive pattern. Heller has really taken an authoritarian tone with everyone, especially during this past year. Perhaps he has just had too many cheerleaders in the past and cannot accept criticism or contradictory information. I noticed quite an uptick in this behavior after the past couple of Drudge links to his site. Correlation? .... could be.

      Anyway, no matter. I finally realized that I wasted too much time on his blog. His site has become such an echo chamber and a continuous repeat of the same stuff over and over again. It was a good thing for him to ban me, and a good thing for me to waste MUCH less time at his blog.

  5. I think this equation could perhaps be simplified further.

    Playing with it in Excel, I notice that the Surface Pressure can be changed to any value from zero to 1000000 and there is no change to the end result.

    Similarly, so long as gravity is not zero, any value from .001 to 1000000 also does not change the end result.

    Thank you for your work; it has been quite intriguing reading the posts related to this equation.

    1. I don't know where you are changing things but that's not what I find at all. Changing surface pressure a tiny amount e.g. 0.1 has a huge effect on the temperature calculation.

      Surface pressure in atmospheres is the 1 in the log(1/2) part of the equation

    2. If little 's' = 0, changing the gravity does not have any affect because the change on the 'g' outside the bracket cancels with the change in 'g' on the bottom of the fraction inside the bracket.

      At any other 's' value, changing gravity does have an effect.


    3. The problem I see is...

      If the gravity constant is changed, then shouldn't the surface atmospheric pressure also change?

      Or should it.?

      Even if you use the alternate equation for 'P', at 's' = 0, log(0) = 1

      Is there a problem here ?

      (been a long day down here in Aust., thinking time needed, after food, and some wine ! :-)


    4. The equation is not designed to handle things like that, so no. The equation assumes g is constant and P at surface = 1 atmosphere by definition, so that is a constant as well.

  6. MS, I think you should send your work to Dr. Robert Brown from Duke University Physics Department, who is a regular at WUWT (rgbatduke) and who's comments and posts are among the very best at that site. As Willis mentioned previously, Brown has already done related analysis:

    so it should be an easy task for him to verify or falsify your equation, or to point out problems that need to be solved.

  7. Do you have a more general form that can be used on other planets?

    That would be interesting. :-)


    1. Titan is the only Earth analog for this equation, and the theory on center of mass predicting the ERL works perfectly there as well.

  8. "Yet every planet is warmer than predicted" That's a funny way of saying a theory is not quite right and lacks a few "fudge factors" (Newton/Einstein) or a few more variables in an already complex formula. Or in paraphrasing a certain adage: "Atmosphere DOES matter" (since it has "weight" etc.)

  9. I just have a quick question:

    When I tried entering your equation in Excel, I noticed that increasing the Surface Pressure always results in the temperature decreasing. Shouldn't increased pressure result in the temperature increasing, if the temperature is based upon the mass of the atmosphere?

    As an example, just entering "2" for Surface Pressure changes the temperature from "288.478" to "254.852". For the other inputs, I used the following:

    1367 . . . . . . . . . solar constant
    1 . . . . . . . . . . . . emissivity
    5.67037E-08 . . . Stefan-Boltzmann constant
    9.81 . . . . . . . . . . gravitational constant
    0.02897106 . . . . average molar mass of the atmosphere
    0.3 . . . . . . . . . . . albedo
    1.5077 . . . . . . . . ratio of specific heat capacity of the atmosphere
    8.314462175 . . . universal gas constant
    2.718281828 . . . base of natural log

    1. No no can do. This is just one single equation designed for Earth surface P=1 atmosphere, derived on the basis that the global average surface pressure is always a constant at 1 atmosphere. If you mess around with that assumption, it changes other parts of the calculation that also assume surface P=1 atmosphere and use the ideal gas law and barometric formulas. Thus the eqn has to be modified for use on Titan or any other planet where surface P is not equal to 1 atmosphere.