The Tropopause Height

Let's have another look at what Roy Spencer writes about the GGH:

6) The tropospheric temperature lapse rate would not exist without the greenhouse effect. While it is true that convective overturning of the atmosphere leads to the observed lapse rate, that convection itself would not exist without the greenhouse effect constantly destabilizing the lapse rate through warming the lower atmosphere and cooling the upper atmosphere. Without the destabilization provided by the greenhouse effect, convective overturning would slow and quite possible cease altogether. The atmosphere would eventually become isothermal, as the full depth of the atmosphere would achieve the same temperature as the surface through thermal conduction; without IR emission, the middle and upper troposphere would have no way to cool itself in the face of this heating. This scenario is entirely theoretical, though, and depends upon the atmosphere absorbing/emitting absolutely no IR energy, which does not happen in the real world.

The first thing worth to say about this is that it is a correct account of the GGH. For some reason skeptics have had a tremendously hard time to understand the significance of this. The lukewarmers have been very anxious to keep this secret for understandable reasons. Why? Elementary, because it provides a test which could potentially lead to a falsification of the theory.

Expressed in other terms, what he says is that if the atmosphere was depleted of greenhouse gases the tropopause would be at the surface. Remember, the tropopause marks the beginning of the stratosphere which is the isothermal layer above the troposphere. 

Now please have a look at the following fact sheet:

Tropopause Values

	T K	P mBar	gm/cm3	H Km	Vsound m/s	rad years	phase °
Venus	250.0	100.000	2.1173e-04	5.307	246.4	0.026	14.83
Earth	217.0	100.000	1.6051e-04	6.358	295.4	0.043	14.96
Mars	140.0	0.010	3.7809e-08	7.070	185.0	0.000	0.01
Jupiter	110.0	140.000	3.3983e-05	16.984	768.8	2.273	50.28
Saturn	85.0	100.000	3.0224e-05	33.083	676.8	9.677	64.10
Uranus	53.0	110.000	5.7414e-05	21.770	515.1	46.340	73.81
Neptune	54.0	200.000	1.0246e-04	17.579	519.9	63.132	67.26
Titan	70.0	100.000	4.9261e-04	14.949	167.7	9.506	89.96

Almost consistently, we see that the tropopause is situated at a pressure of about or slightly above 100 mB. (Mars beeing a notable exception, on the other hand it has a very thin atmosphere so the data might not be that reliable).

What does this mean. Well, the pressure is a direct measure of the amount of atmospheric mass above that level. Hence, it corroborates the assumption that it is the total mass, rather than the amount of GHGs that determine the amount of heat trapping in the atmosphere. 

Dispelling the Smoke Screen

I am delighted to find the following paper by Joseph Reynen. Although I don't quite agree with the main conclusion of the paper, I think its Ansatz is very sound and instructive and in certain respects similar to the model I proposed in the post "A Discrete Model Atmosphere". Given this ground work I therefore believe we could be just a skip and a hop away from altogether ridding the climate discussion from one of its most stupidifying elements. I'm talking about the mysterious notion of "backradiation".

What is backradiation? I don't know. The first guess would be that it is the occurrence of electromagnetic radiation directed downwards in the atmosphere. In that case I guess we could all agree that it exists. However, some people claim that "backradiation" violates the second law of thermodynamics. What they probably mean is that the heating of a warmer object by a colder object through backradiation violates the second law. But what about obstruction to cooling through backradiation, does that violate the second law? Certainly a colder object can slow down the cooling of a warmer object. Let me give an example:

If I go outside when the temperature is 20 degrees C I loose heat at a slower rate than if I go outside when the temperature is -10 degrees C. Why is it so? In both cases my body temperature is higher than the surrounding temperature. 

Models versus reality

The exact microphysical processes that determine the rate of cooling of objects are presently unknown to humans, any attempt to model this from first principles would be a tremendously difficult task, (yet climate scientists think they can do it). Instead we usually resort to empirical studies and heuristic arguments. Empirical evidence tells us that the rate of cooling is proportional to the temperature difference. How can we explain this? One way to think about it is that each object have a certain thermal "impact" on its surrounding proportional to its temperature, but, in the competition the warmer object wins with a margin proportional to the temperature difference. This "impact" is of course composed of several elements, but there is no ground to argue that radiation cannot be part of this.

From a mathematical point of view one can of course ask the question. Why do you represent the heat flow as a two-stream model instead of a one-stream model? A fair question, but there may be technical reasons for this as I will try to explain later. But the important thing to remember is that neither the two-stream model nor the one-stream model has any direct relation to the actual microphysical processes. 

Smoke screens

What has caused this obsession with backradiation? Part of the reason I think are two smoke screens

1. Cartoons

2. Simplified mathematical models

In the cartoons the supposedly non-trivial contribution to the thermodynamics by greenhouse gases are respresented by a downward arrow often called backradiation. Hence, If you don't believe that greenhouse gases add anything non-trivial as regards to thermodynamics it is therefore natural to question the downward arrow. This is, however, a trap. First of all you should never even consider arguing about a cartoon. Whenever a greenhouse defender shows you a cartoon you say:

Please, show me the equations for the temperature profile of the entire atmosphere!

As regards the simplified mathematical models, which you can find even in supposedly rigorous books like that of Goody and Yung, the trap mainly lies in wrongly or vaguely specified boundary conditions. But this is a bit too technical to be covered here. Just recently, after having dug in the literature for quite some time, and after some creative work of my own I think I finally understood what the Greenhouse Gas Hypothesis (GGH) really is.

Why does the GGH violate the second law of thermodynamics? 

In my opinion, not all the specific cases included in the GGH violate the second law. At least not in a loose sense. One example of this is the so called "grey" atmosphere. Why do I think so? Because in the grey gas model there is no cooling of the upper atmosphere. The most flagrant violation of the second law, I think, is the simultaneous warming of the lower atmosphere together with the cooling of the upper atmosphere that supposedly occurs in a "real" atmosphere. After having read Pierrehumbert's text on the subject I can perhaps guess how this occurs in the model. It is because an uneven distribution of carbon dioxide and water vapor. Water, (due to the formation of clouds?), resides mainly the lower part of the atmosphere. The upper part, however, populated by CO2 backradiates in the CO2 spectrum to the lower part, which in turn, cunningly, radiates the energy back mainly in the H2O spectrum which CO2 cannot absorb. Wicked!

So in summary: 

The grey gas model does not violate the second law (in a loose sense)

Yet...

The grey gas model is built on a two-stream model

Consequence:

People think that GGH violates the second law (because of the cooling of the upper atmosphere). The grey gas model is kept hidden. People look at cartoons and see a downward arrow. People draw the conclusion that the downward arrow violates the second law.

Gotcha!!!

Unfortunately, however, people have now invested so much prestige in this "back-radiation" issue that I'm not sure if we can get out of this mess in the near future, though I am hopeful. 

What traps heat?

Let's look upon it this way. What is heat? Answer: It is microscopic motion. What traps motion? Answer: Mass. If you through a tennis ball into the wall the ball bounces back. If there is no wall the ball continues. Why? Because a massive object can absorb a lot of momentum without at the same time absorb a lot of energy. This is because momentum scales linearly with the velocity but energy scales with the velocity squared. Radiation obeys the law of conservation of momentum. QED.

Solution?

I think that the grey gas model can actually be useful, after one major adjustment: Replace the absorption coefficient related to the greenhouse concentration with the total mass. In doing so, the grey gas model violates the second law even less, because now it looks as a straight forward heat transport problem.

Why is the two-stream model useful?

The usefulness primarily lies in that it facilitates the modelling of the escape of heat to outer space. Please have a look at "A Discrete Model Atmosphere". If anyone comes up with a "one-stream" representation for this problem: Be my guest, I won't complain. But personally I don't think that it matters that much.

  

A stepwise axiomatic approach

This, I guess, is in part a response to Postma's latest article which I find well written and addresses several key questions concerning the GE. In particular I like the approach which starts off with simple reasoning and simple equations, adding details when necessary, rather than a heavy first principles approach from the very start. I will not go into direct polemic, instead I will try to summarize my own thoughts in what I call a "stepwise axiomatic approach". The present version of it is probably not the most succinct, but I think that this or similar approaches could be helpful in disentangling disagreements, to see where the viewpoints really depart.

Consider the following set of assumptions (axioms), each of which may be subject to amendments or cancellation. (Some are inconsistent with each other and are therefore numbered.)

A.  For the atmosphere the ideal gas law T = PV/nR holds true with an accuracy sufficient for our considerations. 

B. Thermometers measure temperature with an accuracy sufficient for our considerations.

C. There is always a (net) heat flow from higher to lower temperature

D1. The incoming sunlight is a heat flow

D2. The incoming sunlight is not a heat flow but should instead be considered as an energy source (thereby functionally equivalent to a radiator, burning of fuel. radioactive decay and so on)

Axiom C is one formulation of the second law of thermodynamics. Obviously we have to choose between D1 and D2 so lets start with D1.

Consider the set of assumptions (A, B, C, D1). Here we encounter a problem. Since the upper atmosphere is colder than the surface (B) assumption C says that there must be a net heat flow between the surface and the upper atmosphere. However, D says that there is no such net heat flow. (We have neglected the fact that the sunlight is asymmetrically distributed, but adding the asymmetry doesn't help, think about it.) So how do we resolve this? A significant number of people resolve this by an amendment to the second law of thermodynamics, stating that in a gravity field there need not be a net heat flow from higher to lower temperature. In the end they usually arrive at the following formula for the lapse rate:

The details of the reasoning behind this, and its problems, can not be covered here, instead I refer to other articles (for example on this blog). An alternative is to abandon D1 in favour of D2.

Consider the set of assumptions (A,B,C,D2).  In this case the lapse rate simply pops out in the form of the heat equation:

where F is the solar forcing. If this simple observation has evaded you until now don't be ashamed. It took me months. The heat equation I think provides a powerful tool to analyze certain topics infected with misunderstandings. I believe for example that back-radiation (in the most general sense) is neatly incorporated in the left hand side of the above equation, see for example my post "The heat equation revisited". It is not necessary to incorporate back-radiation for example in the following way:

where B is a fictitious extra forcing term supposedly arising from back-radiation. Moreover, if we test one of the predictions of the GE theory, namely that an atmosphere without greenhouse gases becomes isothermal. Given the set of assumptions (A,B,C,D2) that would correspond to an atmosphere with infinite thermal conductivity. Now ask yourself the question: Does dry air have infinite thermal conductivity?

It should also be noted that the formula for the adiabatic lapse rate may become useful also under the set of assumptions (A,B,C,D2) but then in the context of the limit it puts on convection, see previous posts. 

In summary, I think that an approach like this can resolve many misunderstandings. Try it! 

Pierrehumbert explains the cooling of the Stratosphere

Upon further reading of Pierrehumbert's "Principles of Planetary Climate" I found the following passage explaining the cooling of the stratosphere:

Note especially that he writes 

"Moreover, as the optical thickness of the atmosphere is increased, the stratosphere actually

becomes colder than even the semigrey skin temperature; this contrasts with the grey case, where

the temperature of the uppermost part of the atmosphere always approaches the skin temperature, regardless of how optically thick the rest of the atmosphere is."

The boldfaced statement seems to be what I concluded myself in the previous post "A discrete model atmosphere", thus it is not news to Pierrehumbert. However, there are a lot of questions urging for an answer now. For example:

1. In my very first post "What is the greenhouse effect" I calculated the temperature profile using the formulas provided by Goody and Yung where the stratospheric cooling fell out quite simply apparently without any assumption about a "grey" or "non-grey" atmosphere. Hence, are these formulas valid at all? Are there several greenhouse effects, in that case, which is the IPCC one? 

2. If there is no cooling of the stratosphere for a "grey" atmosphere, then why is it generally portayed as a triviality, taking as an example Spencers "blanket analogy". (Maybe we have to assume a non-grey blanket?) 

3. Where is the observational evidence for all these claims? As far as I can see, the stratospheric temperature of most planets can be calculated from the incident solar energy without any special consideration of the spectral properties of the constituent gases.

A Discrete Model Atmosphere

(A few additional comments since this post seems to have attracted certain attention. Although it might look odd, in reality it describes an ordinary diffusion process. The corresponding differential equation is described on the post "A new attempt". Here, the thermal conductivity k is normalized to 1 but if you want to vary it all you have to do is to divide the forcing vector F by k. The derivation of this I leave to the reader.)


This post is a bit technical but if you bear with me I hope it might shed some light on certain important topics. The reason why I have constructed a discrete model is that by doing so I have avoided the tricky problem of specifying the boundary conditions, which you are invariably faced with if you formulate the model in terms of differential equations. The model which I am going to present admittedly bear some SUPERFICIAL ressemblance with the greenhouse effect, but one of the questions I am asking is if it ACTUALLY reproduces the greenhouse effect. Take a look at the diagram below:

The atmosphere has been discretized into a finite number of layers each having its own "weight" w1, w2 and so on. The arrows may represent either a "real" or a "formal/mathematical" energy flow. What do I mean by this? There is actually a point in not specifying this at the moment. It could be interpreted as if the "weight" of the atmospheric layer is simply a function of the amount of greenhouse gases present and that the arrows represent upward or downward radiation. On the other hand, if we allow a certain "flexibility" in our thinking, the "weight" could be interpreted as a function of the actual weight (mass) of the layer and the arrows merely illustrate a "formal" energy conserving representation of the obstruction to cooling caused by these layers. Although that may sound like mumbo-jumbo in your ears, somehow we have to account for the eventual energy loss to outer space, and I haven't found any better way to do it than this. Now for the technicalities:

The amount of energy absorbed per unit time by each layer is the amount of incident energy times its wight. (Hence, the wight must be less than 1). Each layer loses energy per unit time proportional to its temperature AND weight. This loss is equally distributed upwards and downwards. Keep in mind that the energy that is not absorbed by the layer is free to pass on to the next and so on. (This is accounted for by inserting a factor (1-w) for each intermediate layer). Below is a Mathematica worksheet where the energy balance equations for the STATIONARY state of a model with four layers are written down and solved. Staionary means that all of the layers have reached energy balance, hence, the amount of energy lost per unit time equals the amount of energy absorbed per unit time. The left hand sides represents the energy lost and the right hand sides contains the absorbed energy.

This model is linear in temperature, unlike mainstream climatology where one is instead obsessed with the SB-law. A linear dependence on temperature may very well be physically motivated, especially if we are dealing with ideal gases, though admittedly, here it is primarily employed for computational efficiency, things would become very much more cumbersome with a T4-dependence. If the wight of the layers are assumed to decline exponentially with altitude, from the analytical and numerical solutions we see not only the formation of a stratosphere but we can also infer with almost certainty (although I have no strict proof for this) that the temperature approaches the finite value F/2 at the top of the atmosphere, where F is the solar forcing at the surface. This can be verified also with a numerical algorithm solving the equations for a model atmosphere discretized in 60 layers as is shown below (The Python-script for this is appended at the end)

What is then the main conclusion to be drawn from this. The most obvious one is that this model does not predict a cooling of the stratosphere upon an increases in the weight-function. On the contrary it either produces a warming of the entire atmosphere upon an increase in the weight or solar forcing, or conversely a cooling of the entire atmosphere upon a decrease in solar forcing or weight. In other words, the temperature gradient does not pivot, and this holds true even if the weight is interpreted as the amount of greenhouse gases present. (Admittedly it also somewhat contradicts some of my previous statements/guesses, which the reader can verify on his/her own.) 

How the cooling of the stratosphere comes about in the mainstream models is very obscure. From the most simple equations I can see that this follows from foolishly applying a Dirichlet boundary condition at the surface rather than at the top of the atmosphere, but how it emerges in their time-stepping algorithms remains a mystery to me. One of the lessons to be learned, however, is that there is much mathematical subtlety involved and that any "wordy" explanations from people like Roy Spencer must now be replaced by reproducible algorithms/equations.

Moving on, one could now ask if the model presented above could actually be useful as a model atmosphere. My answer to that is: Yes maybe, if the weight is interpreted as a function of the TOTAL MASS. The reason why I believe this is that:

1. It predicts an elevation in surface temperature as a function of atmospheric mass

2. It predicts the formation of a stratosphere whose temperature is a function of solar forcing alone

3. It can easily be extended so as to reproduce a thermosphere higher up if you add solar forcing terms to the uppermost layers.


Below is the Python code for your enjoyment:

***

import numpy as np
import matplotlib.pyplot as plt

interval = 6
meshsize = 0.1
N = int (interval/meshsize)

weight = np.zeros(N)
weight[0] = 1


for idx in range(1,N):

    weight[idx] =  np.exp(-idx*meshsize)*meshsize
    


A = np.zeros([N,N])
    
for idx1 in range(N):
    for idx2 in range(N):
        if idx1 == idx2: 
            if idx1 == 0:
                A[idx1,idx2] = -1
            else:
                A[idx1,idx2] = -2
            
        else:
            A[idx1,idx2] = weight[idx2]

            if idx2>idx1:

                for idx3 in range(idx1+1, idx2):
                    A[idx1,idx2] = A[idx1,idx2]*(1-weight[idx3])

            else:
                for idx3 in range(idx2+1, idx1):
                    A[idx1,idx2] = A[idx1,idx2]*(1-weight[idx3])    
            
F = np.zeros(N)
F[0] = 1



temp = np.linalg.solve(A,-F)
print temp

x = np.arange(0,interval,meshsize)

plt.plot(x, temp)
plt.ylabel('Temperature')
plt.xlabel('Position')
plt.show()

***

Ray Pierrehumbert explains the Radiative Transfer Equations

In order to elucidate the reasoning behind the differential equation yielding the "grey atmosphere" radiative lapse rate I now take the liberty to present some extracts from Ray Pierrehumbert's first online distribution of the book "Principles of Planetary Climate" dating November 11, 2009. In my view, this is the only understandable derivation that I have found. Take special notice of the formulas (4.6), (4.7), (4.8) and (4.29). Try to reproduce these with your own hand. I believe I have verified that, assuming that the net radiative flux q_T = (I+) - (I-) is independent of altitude, formula (4.29) yields the differential equation appearing first in the Ramanthan/Coakley review paper, except for some numerical factor which probably comes from some angular integral.

A 1978 review paper

Some day ago I came across the following review paper from 1978. Compared to other papers on the subject it appears to be relatively accessible and it contains a lot of references. In particular notice the very first formula describing radiative equilibrium of a "grey" atmosphere. Apparently it originates from a 1958 paper by a guy called Ambartsumyan. That you didn't know, did you?

I am personally convinced that the devil lies somewhere in the equations, and climatologists have indeed made a very good job making these equations as inaccessible as possible, so that virtually nobody can quote them, even less interpret them. But I'll see what information I can extract from it.