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!