Lost in semantics

Following the long argument about "back-radiation" and its newly invented derivative "back-conduction" I have started to speculate if most of the disagreement cannot be traced back to more or less semantic confusions and misunderstandings. First of all, below I have listed the various modes of heat transfer most commonly thought of:

Radiation
Conduction
Convection

Now consider instead the following list

Diffusive heat transfer
Convective heat transfer

What is the difference? I would say that the first list ties on to the "actual" physical mechanisms, whereas the second list is a classification into different mathematical forms. Convection probably belongs to convective heat transfer, conduction is usually thought of as diffusive, but what about radiation? My guess is that radiation should be considered diffusive too. Now let's add some more confusion: 

Thermal conductivity

The very name seems to imply that it refers only to conduction. But let's suppose you want to experimentally measure the thermal conductivity of a gas. It is possible to tell the molecules "Hey, guys! Could you stop radiating for a while, I only want to measure the conductive heat transfer." Of course it is impossible, yet we stick to the misnomer "conductivity". Now let's move on

Back-radiation

This is perhaps one of the most infuriating concepts of modern time. Who invented it? I don't know. If we look again at the first list, radiation occurs as one particular heat transfer. Hence, accepting back-radiation we also ought to be able to speak of

Back-conduction

But then the protagonists of back-radiation say "Hey, wait a minute, when I speak of back-radiation I am simply speaking of down welling electromagnetic radiation which we can measure". Ok, so in order to avoid confusion let's call it back-photons instead:

Back-photons

(People who don't like photons can instead think of "Back-electromagnetic rays".) But here comes the final nail in the coffin:

Back-phonons

What do you say now? "Well, well. Ok. But photons doesn't stick out their fingers to measure the surrounding temperature".

I rest my case.  

Derivation of the "isothermal" column

Here I outline what I believe to be a conclusive argument showing that the "kinetic" temperature is constant with height for the canonical ensemble of an ideal gas in a gravitational field. It is not taken from any "authoritative" source, hence, I make the reservation for errors. Recall the Boltzmann factor

which is the relative probability to find a single particle at height h with speed v when the system has reached equilibrium, that is, maximum Gibbs entropy. (Notice that this kind of factorization can not be done for the micro-canonical ensemble.) Now I define the "kinetic temperature" in the following way:

The reason for this notation is of course that there already exists a temperature T pertaining to the system as a whole. Let N be the total number of particles, using the Boltzmann factor the kinetic temperature can be calculated as follows:

From this point it is very easy to show that Tk is independent of h, which I leave as an exercise. It can also be shown that with this definition we have that

Reply to Cotton

By reason of a comment made by Doug Cotton on one of my earliest and most read posts "On the temperature profile of an ideal gas under the force of gravity" I will here elaborate further on this important and thought provoking topic. There are many things to say about this, first of all though, I am going to address the seemingly never ending discussion about which temperature profile that maximizes the Gibbs entropy of an ideal gas in a gravitational field.

Note first that what we are discussing is an idealized theoretical construct which relies on a postulate called the "ergodic hypothesis". What that is will be left for later. Most importantly, however, just because you have a theoretical model in your hands it is not self-evident that this model can be applied to any specific problem in the real world. Physicist seems to be especially vulnerable to this kind of confusion between model and reality, not least when it comes to thermodynamics; It is not only in climate science that researchers complain about the arrogant thermometers who never seem to measure the correct temperature.

Ludwig Wittgenstein said that the purpose of philosophy was to clear out linguistic misunderstandings. Karl Popper, on the other hand, was somewhat more skeptic about the usefulness of this principle in science where he instead advocated that a scientific theory is nothing else than the set of its predictions. He saw a danger in the possible infinite regression of constantly dwelling over definitions. In this case, however, I think that we can indeed resolve matters with the Wittgensteinian approach.

Consider an arbitrary energetically isolated (adiabatic) thermodynamic system divided into two parts. Moreover, we impose the two measures entropy (S) and temperature (T) on this system: (S1, T1) for the first half and (S2, T2) for the second half. The entropy is supposed to be a so called "extensive" thermodynamic variable which means that the total entropy S is the sum of the entropies of the two subsystems respectively:

S = S1 + S2

The same does not hold for the temperature which is a so called "intensive" thermodynamic variable. Indeed, unless we are in equilibrium it is not even defined for the system as a whole. Now we define the temperature for each subsystem by means of the entropy as follows:

1/T = dS/dE

Or put into words, the inverse of the temperature is the (infinitesimal) rate of change of entropy per unit change of energy. Now suppose the following:

1. The system as a whole has reached a maximum possible entropy given the available energy.

2. T1 != T2  (Arbitrarily we may assume that T1 > T2)

Now imagine that subsystem 1 looses a small amount of energy \Delta E to subsystem 2. The total change in entropy can then be calculated as follows

In other words: Without adding any external energy to the system as a whole we have increased the total amount of entropy thus contradicting the first assumption.

Notice the very limited number of prerequisites. We didn't say anything about an ideal gas nor anything about a gravitational field. We didn't even define the entropy other than assuming that it was extensive. This is in fact so banal that we immediately realize the following: The modern definition of temperature is constructed in such a way that the two statements

1. The (energetically isolated) system has reached a maximum possible entropy given the fixed amount of energy

2. The temperature is the same everywhere in the system

are logically equivalent. 

The paper of Coombes and Laue and related articles

According to the above analysis a "paradox" can only arise in the minds of physicists who tacitly introduce another definition of temperature (in this case for an ideal gas) and then assume that by some logical necessity this new definition must be the same as the old one. This is what I believe has happened here, that is why I gave this other definition of temperature the name "kinetic temperature" which is the ensemble average of the kinetic energy per constituent particle (omitting constant factors). If one performs the calculations one can indeed show that for the canonical ensemble of an ideal gas in the absence of a gravitational field the temperature (as defined by the Gibbs entropy) is indeed the kinetic temperature. The question you may now ask is the following

Do the distributions maximizing the Gibbs entropy for the canonical ensemble of an ideal gas in a gravitational field imply a uniform (constant) kinetic temperature?

This question has been analyzed rigorously and the answer appears to be yes. Nowhere does Cotton present any calculation or reference to show the opposite. The key to intuitively grasp this is to realize that the gravity make both the density and the pressure decline with height according to the barometric formula from which it follows that the temperature must be uniform. One should remember though that the isothermal column isn't the only hydrostatically stable column, but it is the one that maximizes the entropy. 

There is a small mistake in the Coombes and Laue paper. They talk about an adiabatically closed system when, in fact, the derivation they are referring to pertains to the canonical ensemble. The canonical ensemble is not energetically closed but may exchange energy with an external reservoir kept at a constant temperature. The statistical ensemble corresponding to an energetically isolated system is instead called the "microcanonical ensemble". In fact, the microcanonical ensemble of an ideal gas in a gravitational field was dealt with relatively recently by S. Velasco et al. They concluded that in this case the kinetic temperature was not uniform for finite systems. The practical implications of this result is almost zero though. First of all there is no reason to assume that the atmosphere is an energetically isolated system, moreover  the distributions in the microcanonical ensemble approaches those of the canonical ensemble very quickly in the thermodynamic limit. There is, however, an important didactic value in the sense that it shows that the "absolute" temperature need not be the same as the kinetic temperature for all systems.

The experiments of Graeff

These experiments pose a serious challenge to the standard wisdom if one assumes that the thermometer does indeed measure the kinetic temperature of the gas rather than simply "its own temperature". Since this needs to be verified separately I introduced another "definition" of temperature called the "empirical temperature" which is simply the reading of some particular thermometer. One of the most difficult problems with these kind of experiments is the question of how you verify that your system has reached equilibrium (that there is no tiny heat transport from the warmer to the colder parts). This must be taken on faith. In any case, the only thing these experiments can possibly show is that the ideal gas model with maximum Gibbs entropy is not valid for this experimental setup. Perhaps this is the case for the atmosphere as well.

Other "derivations" of the lapse rate

The "ergodic hypothesis" upon which the Gibbs entropy is based can be formulated like this

All microstates with equal energy are equally probable. 

Somewhere on the path this statement seems to have got confused with something like

The total energy density is the same everywhere

leading to derivations based on the following assumption

potential energy + kinetic energy = constant

which (almost) leads to the adiabatic lapse rate. A simple thought experiment tells us that this cannot possibly hold for a finite atmosphere in an infinite space since that would imply an infinite amount of energy in the system. Also here we touch the problem of how to treat the atmospheric boundary. 

  

The paper of Hans Jelbring and related articles

I and many other owe a great deal to Hans Jelbring for having brought to our attention the issue of the temperature gradient and its relevance to the climate debate. His various statements do however come in a rather incoherent form and do not, as yet, in my mind constitute a physical model of the atmosphere. The theory seems to come in two separate parts, on the one hand an assumption about a "static" gravitationally induced lapse rate and on the other hand a conjecture that the atmospheric mass is the single most important parameter in determining the elevation in surface temperature of the planets, as expressed in the title of his paper "Greenhouse Effect as a function of Atmospheric Mass". In hindsight one can see that there is something, perhaps unwittingly, catchy about this title. Notice in particular that it doesn't say "Pressure induced Greenhouse Effect" or anything like that, more about this soon. People who cannot tolerate any dissent to the greenhouse gas dogma immediately assume though that what is implied is some kind of pressure induced effect against which they can use a plethora of arguments including "the second law of thermodynamics" and "static air pressure cannot create heat" etc. There might be some justification for these arguments but it is somewhat ironic to see the same people embrace a theory that states that instead of gravity some magic gases create a temperature gradient in a system which would be isothermal in their absence  What appears to be missing though in the Jelbring theory is an incorporation of the solar forcing (F) and a theory of the atmospheric boundary layer to produce a formula with which one can calculate the temperature field.   

"Pressure/Gravity effect" versus "Blanket effect"

As I have argued on several posts there is another approach to the atmospheric mass conjecture which I here call the "blanket effect". Put in simple terms the atmosphere acts as a blanket whose effective "thickness" (L) is determined by its mass. If we treat the incoming sunlight as an energy source and assume, at a certain pressure, there is some effective boundary layer (whose temperature we put to zero for simplicity) then we can derive the following simple but illuminating formula:

T = F*L/k

where T is the temperature, F the solar forcing, L the effective length or thickness of the atmospheric blanket above that altitude and k is the thermal conductivity. The concept of effective atmospheric thickness can be made more rigorous which is done for example in the post "A Discrete Model Atmosphere" where the boundary layer is also taken care of. Since both the pressure and the "effective blanket thickness" are both proportional to the mass it is very easy to mix these up and confuse correlation with causation.

I am not completely alien to the concept of a pressure induced effect though. If we take the sun as an extreme example, I guess no physicist would consider treating the core plasma as an ideal gas.

Some answers to Doug Cotton's comments

"(a) There is no issue about the top of atmosphere or the stratosphere or thermosphere. These are regions where new absorption of incident Solar radiation dominates the much slower process of diffusion of kinetic energy. Also, the thermal gradient (aka "lapse rate') is -g/Cp where Cp is specific heat. But specific heat is only constant at a constant pressure and temperature. So the gradient approaches a limit of zero and never goes negative."

The latter part of this comment is the most confusing since it seems to assume that Cp approaches infinity. I cannot imagine the physical conditions under which it would do that. 

"(b) The paper by Coombes et al simply is not based on the Second Law of Thermodynamics. They quite incorrect assume thermal equilibrium is implied by that law. It is not, and nothing in the law implies that it must be. The law states the thermodynamic equilibrium will evolve in a state of maximum accessible entropy."

In order to understand this comment I assume that Cotton advocates a second law of thermodynamics stating that

The spontaneous tendency of any thermodynamic system is to evolve towards an equilibrium characterized by a maximum accessible entropy

As I argued in the very beginning there exists a definition of temperature which makes the conditions "maximum entropy" and "constant temperature" logically equivalent. The quantity which I guess that Cotton is more interested in is the "kinetic" temperature. I state that

Cotton is correct in asserting that the second law he assumes valid does not imply a constant kinetic temperature for all systems.

However

Analysis has shown that for the canonical ensemble of an ideal gas in a gravity field it does.

Summary

What appears to be missing in Cottons arguments is some kind of physical model, especially one that incorporates the solar forcing and takes care of the atmospheric boundary level. Moreover I conclude that his claim that the adiabatic lapse rate maximizes the Gibbs entropy of a column of air in a gravity field are unsubstantiated.      

Issues parallel to AGW

For the first time in this blogg's history I will go off topic. Or maybe not. Some time ago I became aware of another scientific issue, the HIV-AIDS hypothesis, which ressembles the AGW issue in many respects. The big HIV hype occurred at around 1990 which was before the internet age. That might explain why the dissidents in this field of science is less known to the general public. However, since there is now a pretty well-developed AGW skeptic network on the internet, I thought we might give our fellow deniers in that other field some extra airing. Medicine is not my area of expertise, hence I will not be able to contribute anything original and instead leave it to you to judge for yourself and find additional sources of information. 

This is something of an introduction/teaser

A more technical presentation questioning the very existence of a retro-virus HIV can be found below

Triviality made non-trivial

Recently, John O'Sullivan and Claes Johnson have discussed the apparent division of the climate debate participants into three categories: The alarmists, the lukewarmers and the deniers. I myself find the distinction between lukewarmers and deniers very interesting for several reasons though, as pointed out, for an outsider it might be difficult to really grasp what the fuss is all about. The reason for this is probably because the discussion has largely been focused on climate politics rather than the politics of science, a distinction which I will try to clarify in the following. Suppose you are interested in climate politics, then the relevant question to ask would be something like "What will be the temperature change from a doubling of CO2". If asked to a lukewarmer the answer would probably be something between 0.5 and 1 degrees warming whereas a denier might end up in a small interval around 0 degrees. Ok, so why all this rancour? The answer, I think, is spelt "the greenhouse effect". The discussion becomes difficult since the very definition of this effect is evasive, but let's try to list some candidates:

A1. The Tyndall Effect. An experiment reproducible in the lab where, by using electrically driven equipment, you selectively excite the degrees of freedom of GHGs causing heating.

A2. The thermodynamic effect of GHGs in the atmosphere, whatever it might be.

A3. The thermodynamic effect of backradiation on the surface temperature, whatever it might be.

A4. The difference between the measured surface temperature and a hypothetical "black-body temperature", whatever the cause.

A5. The alteration of the atmospheric temperature profile as outlined in climatology textbooks and quantified by the use of software like MODTRAN.

Lukewarmers tend to get both irate and personally insulted any time some denier has the audacity to question the well-posedness of this effect. This attitude of course has a very detrimental effect on climate science since any one who proposes some new theory about the atmosphere must relate that theory to some mysterious concept called "the greenhouse effect" that is very ill-defined but nevertheless cannot be questioned. Even the demarkation of denial is not very clear. Anthony Watts seems to be open to discussing the "magnitude" of this effect whereas Fred Singer takes a more orthodox stance: Those who question the ability of these gases to elevate the surface temperature by 33 degrees are also to be considered deniers. (As a side remark, the lukewarers seem to be very certain of what is going to happen if we lower the concentration of CO2 though the effect of an increase is, for some reasons, very uncertain). This attitude is so puzzling that it calls for an explanation. So let's try that too, why do people become lukewarmers? We may speculate in the following reasons:

B1. Credibility reasons

The issue of climate politics is so important that you, for credibility, cannot afford to associate yourself with people whom you percieve as cranks or whom someone else might percieve as cranks.

B2. Social reasons

In order to be able to hang around with the "big" guys, like Lindzen, Singer and Spencer, you have to play the game and not offend their basic beliefs.

B3. Scientific prestige

You don't want to admit that what you once believed was fundamentally misguided.

B4. Political reasons

This could be viewed as a kind of "damage controle". You realise that the scientific establishment has made a huge blunder but want to save them from complete humiliation. Hence, it is better to blame everything on computer models and some convenient patsy like Michael Mann.

B5. Religious reasons

The greenhouse effect has become like an icon for you and if it disappeared it would feel like a deep personal loss and shatter your entire belief system. The greenhouse effect has made the atmosphere "non-trivial" for you. 

The reason the lukewarmers would give themselves is probably something similar to the first one, since it would be impossible to admit any of the other. But is this credible? I don't think so. First of all, if you really believe in your arguments the mature way to deal with the situation is simply to, in a calm and pedagogical way, explain any misconceptions the deniers might have. On the contrary, it seems as if they desperately want to avoid the entire discussion, ignore it to death, which makes some of the other reasons more plausible. Moreover, they themselves constantly make attempts to make the deniers look like cranks. Here is where the proliferation of definitions of the GHE becomes handy, they may claim that:

C1. The greenhouse effect can be experimentally verified (i.e. the Tyndall effect can be experimentally verified).

C2. Deniers claim that GHGs have no thermodynamic effect at all.

C3. Deniers claim that backradiation does not exist or does not have any thermodynamic effect.

This is, of course, an active and deliberate way of discrediting the deniers. Constantly they evade the difficult position, namely that of defending the actual greenhouse gas hypothesis (A5). I will try to argue that all of the three smear tactics above (C1-C3) can be dealt with in a more or less straighforward way.

D1. The Tyndall gas effect is real. But what is the significance of this?. If you use precision instruments driven by electricity you can do a lot of things, for example, run computers and refrigerators. The real earth atmosphere system is not electrically driven and there are many other processes to consider.

D2. We do not claim that GHGs have no thermodynamic effect. In fact, many deniers believe that the atmospheric mass causes an incresed surface temperature, and since GHGs are massive particles they contribute to this effect.

D3. This is perhaps the most delicate point. Many deniers have been tempted to cook up very complicated arguments to explain why radiation from a colder atmosphere on a warmer surface has no effect at all. If you support the idea that the atmosphere does indeed have some effect on the temperature this position becomes problematic. Some people have speculated in some kind of pressure induced effect and the obvious question then becomes: Cannot the radiation pressure of CO2 contribute to this effect? If, like me, you instead envision some combined diffusion-convection process where the atmospheric mass acts as a blanket, then backradiation also have a perfectly natural role to play. Thus, I can't see any alternative than to say that backradiation does exist and that it has a thermodynamic effect.

This might come as a surprise to some people. Have I just acknowledged the greenhouse effect?Certainly not. The greenhouse effect is a ghost that can neither be verfied nor denied. As concerns backradiation, on the other hand, after many months of brooding I have come to the following tentative conclusion:

Backradiation is trivial

As simple as that. What the lukewarmers want you to believe is that it is not. Triviality made non-trivial.          

Claes Johnson's problem formulation

I just discovered that Claes Johnson has already formulated a version of the atmospheric problem in terms of the compressible Euler equations with heat source/sink:

I few notes are in place for those who (like me) are newcomers to this field. I guess it is essential to consider the compressible Euler/N-S equations, that is, you allow a variable density and solve for the primitive variables (density, velocity, temperature) . The first equation expresses conservation of mass, the second conservation of momentum and the third conservation of energy. Note that thermal conduction and kinetic viscosity are absent here, which may be justified since these are small for air. More important is perhaps the heat source/sink q, which CJ assumes adds energy to the lower layers and removes energy from the top layers. 

In other words, how you specify the function q(z,t) more or less defines the model. The questions is therefore: How does CJ define this function?

Combining Heat Transfer and Navier-Stokes equations

In this document is described how one can combine the heat equations including convection with the Navier-Stokes equations:

However, as far as I can see, for our purposes we also need to include some mechanism by which the energy is lost to space. Otherwise we need to invent some fictitious "Top of the Atmosphere" which is precisely what I want to avoid.

I think this could be accomplished by including an energy sink S at all altitudes which ought to be a function of the density, the temperature and not to forget the total mass above that altitude. In other words, the more mass aloft the less heat is lost directly into space.

So there you have it. Now we only need someone who can solve all these equations..