Pekka Pirilä on Energy

Over on Judith Curry’s blog, a somewhat random discussion started. It started with Fred Moolten posting an off-hand comment, and he later expounded upon his point. The basic idea is by just looking at a red tulip, one can gain information on the color of crows. If one wanted to know whether or not all crows are black, they could find out without ever looking at a crow.

This is counter-intuitive, and a number of disagreements sprang up. I decided to try to write a clear explanation as to why Fred’s argument was correct, but since the thread was already cluttered and this was all off-topic anyway, I decided to post my explanation elsewhere. The host here invited me to post my thoughts, and so here it is an explanation of why you can look at anything and gain information about anything else.

The argument being tested is All objects in group X have property y.

It is accepted there are a finite number of objects, and they are non-changing. The group which contains all of these objects will be referred to as group Z. Since there are a finite number of objects, there are a finite number of objects possessing property y, and there are a finite number of objects not possessing property y. They will be referred to as groups A and B respectively. The two groups are disjoint.

Accepting the above, we know the argument is true if group X is a subset of group A. If any member of group X is a member of group B, the argument is false.

Assuming no other knowledge, the chance of any member of X being a member of B is B/Z. This chance increases as the size of X increases giving the total chance of the argument being true as (B/Z)^X. The chance of the argument being false is 1 – (B/Z)^X.

Given these simple equations, certain observations can be made. Z has a positive correlation with the chance of the argument being true. B has a negative correlation (as does X). This means if B decreases in size, the chance of the argument being true increases. This means if members of B can be determined not to be members of group X (thus being removed from the analysis), the chance of the argument being true increases.

This can be directly related to crows and their colors. Group X is crows, and y is the color black. Group B is all non-black objects, including red tulips. Observing a red tulip means it can be removed from group B for this analysis, and thus, the size of group B is decreased. This increases the chance of the argument (all crows are black) being true.

Of course, there is little practical value in examining flowers when you want information about birds. The amount of information gained is infinitesimally small, so it’s basically unnoticeable. It certainly doesn’t compare to the fact we can find pictures of white crows with a quick Google search. On the other hand, it is a lot more fun.

Hopefully this can help clear up things for people. If there is anything you are confused about, or if you think I’ve missed something, feel free to let me know!

I realized an example might help explain why this works the way it works:

Imagine you have ten colored balls. You know three of the balls are red. You also know four of the balls have magnets in them. Suppose you are asked what the probability is that all the red balls have magnets in them. It’s a simple enough problem to solve.

Now suppose somebody hands you one of the blue balls, and you find it has a magnet in it. They then ask you, “What is the probability all the red balls have magnets in them?” Obviously you won’t give the same answer as before. You now know there are only three magnets amongst the nine remaining balls, so you’ll change your answer accordingly. The same is true if the blue ball you were handed didn’t have a magnet inside it (you’d get different results, of course).

So even without ever examining a red ball, you can gain information about the red balls. In the same way, you can gain information about crows by only looking at flowers. The only difference is how much information you can gain.

Upon reading that response again, I realize my e-mailed response was completely wrong. I misunderstood the point of the response, so what I said probably made no sense. Now that I’ve reread it, I’ll try responding again.

Nothing I’ve said relies on a change in the expected size of A. The question is merely, “Is X a subset of A?” Ruling out values in B doesn’t change the size of A, it just increases the chances of values lying within A. You are keeping one group constant while decreasing the size of the other. Taken to the extreme, group B could be reduced to 0 (an empty set). At this point, all of X must inherently fall within A, and thus the argument is proven to be true.

The reason this works is even though A hasn’t been affected, Z has been. The chance of X residing in A is related to the proportion of A/Z. Since Z shrinks with each tulip observed (remember, Z = A + B), the chance of X residing in A increases. If you put numbers into the problem, this becomes easy to see:

There are fifty black objects (A = 50). There are 50 non-black objects (B = 50). There are 20 crows (X = 20). The chance of a single crow being non-black is A/(A + B) or 50/100, or 50%. The chance of no crow being black is 50%^20.

Now then, if you see a red tulip, there are only 49 non-black objects left. If you observe 24 more red tulips, there are only 25 non-black objects left. This means the chance of a single crow being non-black is 50/(25 + 50), or 2/3. The chance of no crow being black is 67%^20.

The same principle applies no matter how large the sets involved are.

Hi Pekka, hi Brandon.

Brandon, please consider the following example.

Let’s say I know that there are 4 objects (for simplicity), each a tulip or a crow, but I don’t know how many red objects there are. To make it simpler, let’s also say I know that there are 2 tulips and 2 crows.

This means there are 16 possible worlds I might be in.

Using a 4-letter shorthand to denote the different worlds (the first letter denoting the colour of tulip 1, second letter the colour of tulip 2, third letter the colour of crow 1, and fourth letter the colour of crow 2, where letter R is red, and B is black), the 16 different possible worlds are :

01 RRRR
02 RRRB
03 RRBR
04 RRBB
05 RBRR
06 RBRB
07 RBBR
08 RBBB
09 BRRR
10 BRRB
11 BRBR
12 BRBB
13 BBRR
14 BBRB
15 BBBR
16 BBBB

In four of the worlds (04, 08, 12 and 16), all crows are black.

Before I make any observations, I don’t know which world I’m in. Assuming an equal likelihood of being in any of the worlds, I can say that the probability that I’m in a world where all crows are black is 4/16 = 1/4

Now I make my first observation. Let’s say the object is a red tulip. I now know that I’m not in worlds 13, 14, 15, or 16, so I’m in one of 12 possible worlds, in 3 of which (04, 08 and 12) all crows of black. So after my first observation, I can say that the probability that I’m in a world where all crows are black is 3/12 = 1/4 … no change.

I make my second observation. Let’s say the object is another red tulip. I now know that I’m not in worlds 05, 06, 07, 08, 09, 10, 11 and 12 either, so I’m in one of 4 possible worlds, in 1 of which (04) all crows are black. So after my second observation, I can say that the probability that I’m in a world where all crows are black is 1/4 … still no change.

Therefore finding two red tulips has not changed the probability of being in a world where all crows are black.

Let’s now go through the other possible observations.

Say my first observation is a black tulip. I now know that I’m not in worlds 01, 02, 03 or 04, so I’m in one of 12 possible worlds, in 3 of which (08, 12 and 16) all crows of black. Probability that I’m in a world where all crows are black is 3/12 = 1/4 … no change.

Let’s say my second observation is also a black tulip. I now know that I’m not in worlds 05, 06, 07, 08, 09, 10, 11 and 12 either, so I’m in one of 4 possible worlds, in 1 of which (16) all crows are black. Probability that I’m in a world where all crows are black is 1/4 … no change.

Let’s say instead that, following that 1st black tulip, my second observation is a red tulip. I now know that I’m not in worlds 13, 14, 15, or 16 either, so I’m in one of 8 possible worlds, in 2 of which (08 and 12) all crows are black. Probability that I’m in a world where all crows are black is 2/8 = 1/4 … no change.

Or let’s say that my first observation was a red tulip (13, 14, 15, and 16 eliminated as before, and prob. still 1/4), and my second observation is a black tulip. I now know that I’m not in worlds 01, 02, 03 and 04 either, so I’m in one of 8 possible worlds, in 2 of which (08 and 12) all crows are black. Probability that I’m in a world where all crows are black is 2/8 = 1/4 … no change.

So for all possible observations when examining the two tulips, the probability that I’m in a world where all crows are black remains 1/4, unchanged from the time before I’d made any observations.

Therefore examining tulips doesn’t change the probability that I’m in an world where all crows are black – it provides me with no extra information about the colours of crows.

I agree with Pekka Pirilä’s explanation of the problem, and it is why I tried to be explicit about all assumptions I was making. Changing the assumptions, unsurprisingly, gives different results. In oneuniverse’s example, he discarded an assumption which has been present in the examples given by Fred Moolten and myself. That is, we assumed there were no empty sets. In other words, we know there must be at least one of each class.

Discarding that assumption renders the rest of the problem moot as it allows for the possibility of vacuous truths. The uncertainty introduced by those is what causes oneuniverse to conclude no information is gained, and the removal of them is why Pekka Pirilä’s follow-up does show information gained.

As for the condemnation of inductive reasoning, I don’t agree using it is “impossible” in any situation. The less rigorous an experiment, the more difficult inductive reasoning will be, but I don’t see it ever being impossible. That leaves aside the fact even if you can’t draw any firm conclusions based upon inductive reasoning, you can still apply it to a situation. There is nothing wrong with reaching the conclusion, “I don’t know.”

As for the climate discussion, I’m not sure how you would apply this problem to it, hence why I said it wasn’t significant. It obviously has some relevance, as would any issue about reasoning, but I can’t think of any real parallels in climate science. Do you have any particular ones in mind?

Oh well, I wasn’t going to preempt more comment space on Judy Curry’s blog for this, because it was already too long, but since the discussion continues here, and is entertaining, I’ll add my two cents.

First, this issue is an exercise in logic, not ornithology. Neither Hempel nor anyone else is going to suggest observing all non-black objects to determine the blackness of crows, but as the wikipedia article indicates, the counter-intuitive nature of the conclusion that seeing a non-black non-crow or raven is confirmatory comes from the intuitive belief that it adds zero evidence, in contrast to the logical inference that it adds an extremely tiny amount of evidence that is not zero. To sum it up, it has zero practical value in evaluating the color of crows, but that is not the same as claiming it has no zero evidentiary weight. It does. Discussions of its practical significance for crow color would therefore not be germane.

The Hempel conclusion continues to appear irrefutable to me under the conditions he specified. As far as I know, this conclusion has not subsequently been refuted in a general sense by anyone over the course of more than 60 years, despite many analyses, but instead, discussants have described the underlying assumptions and in some cases suggested that different conclusions could emerge with different starting assumptions.

Two assumptions seem particularly worth noting. The first is that crows exist, but in the world we inhabit, that doesn’t seem to be a problem. The second involves the categories within which we investigate the black crow hypothesis. Four categories might be investigated: (A) All crows. (B) All non-crows. (C) All black objects. (D) All non-black objects. The blackness of all crows can be proved or falsified in theory by category A or D. However, that is not true for either B or C. B is of particular interest – looking specifically at all non-crows cannot tell us anything about the blackness of crows.

Here is where some ambiguity arises. When we observe a red tulip, are we operating in the informative category D (non-black objects) or the non-informative category B (non-crows)?

Since practicality is irrelevant, we must specify what assumptions we use in this hypothetical scenario. Hempel is correct when we specify that we have noted the color of an object either before or simultaneous with an observation of its nature – crow or non-crow. It’s red, and yes, it’s also a tulip, and so we have eliminated one possible example of red crows from category D. On the other hand, if we observe what is clearly a tulip at a distance without discerning its color, subsequently discovering its color will not put it into category D, but into the non-informative category B. This distinction does emphasize the arbitrary nature of the problem and the fact that it has no practical real-world application for understanding crows. It does not, in my view, invalidate the basic logical principle of equivalence that underlies the correctness of Hempel’s reasoning, as long as we specify that we are asking whether a red object we observe is a crow or something else. To pursue these distinctions further would, in my opinion, put too much weight on the practical importance of our observations, which we already agree would be negligible.

A final word about the different scenarios from oneuniverse and from Pekka for distribution of colors, crows, and tulips. Utilizing Pekka’s hypothetical scenarios, 3/14 involve all black crows, they are equally probable, and the probability of all black crows is 3/14, or 6/15 if we observe a red tulip. Here is where the Hempel logic clearly shows its value. Imagine that we start with those 14 possibilities, and then draw 1000 samples from this population of objects, and – Not A Single Red Crow is Observed. Is the probability of all black crows still only 3/14? This is a point that I tried to clarify in the Climate Etc discussion but probably didn’t explain too well. If we don’t know whether all crows are black, the evidence gained from our observations affects any prior probabilities we might have assumed. Even 1000 observations wouldn’t prove the non-existence of a red crow, but as has been discussed many times, confirmation isn’t the same as proof. Observing 1000 objects without seeing a non-black crow, while seeing many non-black objects of other types, would confirm (very very slightly) the blackness of all crows even though it obviously wouldn’t prove it. In Pekka’s example, even observing one red tulip is confirmatory, by increasing the prior probability from 3/14 to a posterior probability of 6/15.

@Pekka Pirilä

“This faulty alternative is of course that of trying to make conclusions on colors of crows from observations concerning tulips. No reasonable real world setup allows for this interpretation.”

We can’t draw conclusions about the color of crows by observing tulips, but we can by observing red objects. I agree it’s not a “real world” problem, because no-one would try to ascertain whether crows are black by that approach, but I see no flaw in the logic itself. I’ll see if I can think of a plausible hypothetical real world example where the same logic would have practical implications, based on the equivalence principle, but most real world problems are solved more directly, rather than by equivalence, so it probably won’t be easy.

@Pekka Pirilä

Pekka – I’m continuing this discussion simply because it’s enjoyable, and not with the expectation that proofs can emerge on a subject as amibiguous as this one.

You continue to state in one way or another that the “efficiency” of the equivalence principle in the Hempel paradox is “exactly zero”, but the Hempel argument, of course, is that it “seems” to be exactly zero because it is too small to have any practical utility or to be measurable, but is in fact very slightly greater than zero. Of course, the efficiency of observing crows directly would clearly be greater than that of observing non-black objects, and so each crow observation would carry more statistical weight than each non-black observation. Some inadvertent degree of non-randomness might reduce statistical power, but should not invalidate the principle unless observations of non-black objects deliberately avoided crows – the paradox as described implies that no attempt is made to avoid crow observations, and in fairness to Hempel and in the interest of seeing the paradox as an interesting exercise in inductive reasoning, we shouldn’t impose an additional arbitrary condition of selectively excluding crow observations. As far as I can see, the Hempel logic leads to the conclusion he draws, unless a specific flaw can be found, and none has yet been identified that I’m aware of. The hypothetical example you gave above with 14 different combinations appears to support Hempel, since observing a red tulip increased the probability that all crows are black.

Here is a more “realistic” hypothetical I’ve conjured up, suggested by the recent killing of Osama Bin Laden. I don’t know how realistic it actually is, but it may not totally misconstrue what our CIA in the U.S. does to avert terrorist operations. It goes as follows:

The CIA has been compiling a list of individuals requiring special surveillance because they pose a greater than average threat of engineering a major terrorist bomb plot. The list has now grown to 25,000, and is too large for adequate surveillance of all these individuals, so the CIA needs to reduce its size by removing names of individuals who don’t pose a major threat. There is evidence that participants in bomb plots have been trained at terrorist training camps, and the CIA hypothesizes that only trained individuals pose a major threat. Via equivalence, this translates into untrained = non-dangerous, or at least not dangerous enough for the limited special surveillance resources to be spent on them

Of the 25,000 listed individuals, 534 were known to have been trained, and 10,502 were known to be untrained. No data are available for the remainder. During the interval of interest, 31 major bomb attacks were attempted, and it is assumed that additional attempts are being plotted, although the number is unknown. Of these, 4 were attempted by individuals among the 534 who were trained. The remaining 27 were attempted by individuals known not to be on the list, and their training status is unknown. The evidence provides some correlation between training and bomb threat, but are these individuals more dangerous than the untrained ones? The CIA then reviews the 10,502 untrained individuals and finds that none of the 31 attempts involved any of them. It concludes that its use of limited surveillance resources would be optimized by removing them from the list, on the grounds that the data support the hypothesis that only trained individuals pose a bomb threat, while acknowledging that exceptions might be possible – i.e., the evidence provided confirmation but not proof. It also concludes that more effort must be applied to ascertaining the training status of individuals on the list.

Regarding the red tulip analogy, “untrained” = red, and “not involved in a bomb attempt” = tulip. Note that the analogy involves data focused on untrained status as the independent variable rather than on “not involved in bomb attempt”. Simply analyzing the latter (24,996 on the list plus unquantified millions not on the list) would not have provided evidence tending to exclude the possibility that untrained individuals were dangerous.

This is the best analogy I can imagine at the moment. It is slightly better constrained than Hempel in limiting the total number of individuals involved, which is why it works better, but I think the main principles are similar enough to illustrate the point about equivalence.

I realize that it’s possible to contrive scenarios where the equivalence principle might not work, but under all reasonable assumptions that the Hempel scenario is what it seems to be – a non-black object is observed either by chance or because one is on the alert for non-black objects, and turns out not to be a crow – it does appear to work.

@fredmoolten

Here’s another hypothetical A = B equivalence example more relevant to climate. I’ll phrase it in the “B” form:

Large changes in the concentration of a single atmospheric component that do not reduce emissions to space in the 12 to 18 um region as a fraction of total OLR are not rises in CO2 concentration.

I’m confused. How do you justify saying, “As the original question is not explicitly fully defined, it becomes fully defined only when supplemented with sufficient implied assumptions…”? Just what part of the problem lacks definition? Just what assumptions are being supplemented?

As far as I can tell, the formulation I provided is perfectly well-defined, and I can’t see any absent assumptions in the most recent example from Fred Moolten.

@Pekka Pirilä

“unjustified assumption that…. observing a red tulip would decrease the expected number of other red objects. ”

I believe that as long as red objects are finite in number, observing one reduces the number left to observe. If that is a false assumption, then I’ll admit the remainder of the Hempel argument is wrong, but I’ll proceed below based on the truth of that proposition.

Rather than repeat previous points, I’ll leave the main argument for others to review from earlier comments. However, I do want to revise a previous statement I made regarding timing. Previously, I suggested that observing an object to be red and observing it also to be a tulip simultaneously or later supported the black crow hypothesis, but that observing it first to be a tulip (i.e., a non-crow) before determining its color eliminated any information to be gained by determining its color at a later time. I now suggest that the timing doesn’t matter, and that as long as a tulip turns out to be red, it supports black crows. This gets back to my first sentence in this comment about finite red objects. If a tulip is later found to be red, it reduces (I believe) the number of red objects that could later turn out to be crows. If it turns out to be some other color, the number of remaining red objects remains unchanged. Interestingly, if it turns out to be black, it supports the hypothesis that all crows are non-black. We happen to know that some crows are black, and so that hypothesis would eventually be falsified by observing a black crow, but if we knew nothing about crow color, the non-existence of black crows would be supported every time a black object turned out not to be a crow – or at least the equivalence principle would lead to that conclusion.

One other point relates to your statement, “I have stated that the non-observation of red crows in some place over some period decreases their likelihood, but for that it does not matter, whether a tulip that we might see during that period is red or black.”

I agree with the first part, but only if the non-observation of red crows occurs when things are being observed. If you are not doing any observing at all (e.g., you’re asleep), the non-observation of red crows is uninformative. If you are observing things, it is informative, but that means that you must be observing things that are not red crows. I stated above why I thought the color of an observed tulip (non-black vs black) affected the probability of black or non-black crows. Obviously, observing a black crow would be even more informative.

I can see that this issue is never going to be resolved, but it’s a nice form of mental exercise to analyze all the possibilities, including the effects of unstated assumptions. It’s also a relief to have an intelligent discussion that isn’t plagued by accusations about Climategate and the alleged sins of the “alarmists”.

Pekka – I need to review the note you linked to in more detail, but I have two questions – one related to the note and the second to my question on Climate etc.

1. You state, “At a differentially earlier time t – dt the same particles were at the vertical coordinate z – w dt and their vertical velocity was w + g dt where g is the gravitational acceleration. The influence of the change of velocity to the vertical coordinate is second order in time differential and can be neglected.”

What happens if you don’t neglect the change in velocity in calculating back to their initial velocities in the path they are taking? If the downward and upward molecules crossing a particular level, z, have the same mean velocity, and you consider that it has changed since they began their journey, doesn’t this require that the velocities of the downward molecules were less earlier and that of the upward molecule greater earlier? This would be consistent with a lower temperature above and a higher one below for the velocities to become equal at z.

2. What is your view on the significance of the increase in potential energy involved in moving heat away from the planet surface, against gravity, so as to establish an isothermal profile? How does this affect the entropy calculation for the entire system- planet plus atmosphere?

@Pekka Pirilä
It is, perhaps time to put the Moolten ‘model’ to rest. Fred has described what mathematically represents a constant density system, at variance with either an isothermal or adiabatic profile. His picture is not unlike that used to calculate the thermal conductivity of ideal gases as described in the chapter on kinetic theory of gases in any respectable physical chemistry text, e.g. A.J. Rutgers. Here, however, the widths of the regions above and below the hypothetical surface equal the mean free paths and differ when there is a density gradient. The net result of such a calculation is that the thermal conductivity is independent of density (Maxwell’s ‘paradox’). As density is that parameter which interacts with the gravitational field, …, etc.

The Verkley-Gerkema reference you’ve provided cites isothermal solution by Gibbs, Maxwell, and Boltzmann. For current textbook references, Landau & Lifshitz, Statistical Mechanics, paragraphs 25, 38. Pierrehumbert actually says “This is the essence of the explanation for why temperature decreases with height: turbulent stirring relaxes the troposphere towards constant theta, yielding the dry adiabat.” RTP omits pointing out, however, that stirring involves doing work on the system, work which is dissipated and eventually radiated to space by a steady-state system. It should come as no surprise that the work of stirring by convection and radiation combined adds up to 240W/m2.

@quondam

Quondam – I don’t believe you’ve carefully read what I wrote, which has nothing to do with constant density – i.e., it does not require molecular density below and above a layer z, described by Pekka to be equal. It does require net mean mass flux and kinetic energy flux across the layer to be zero. I’ve read the reference you cite, and others, which differ among themselves in their conclusions, but I don’t think any specifically addresses or controverts the points I raise regarding coupling to the planetary surface and the increase in the potential energy of atmospheric heat required to proceed from adiabatic to isothermal. At this point, it appears reasonable to conclude that an adiabatic profile best describes the equilibrium state of an atmosphere coupled to the surface, even if that might not be true of an atmosphere in isolation, but I remain tentative about this and interested in seeing my points addressed. The dissipative element is irrelevant, I believe, if we are considering a non-emissive atmosphere.

I believe that RTP’s description that you quoted involves the reduction in lapse rate toward an adiabat rather than an increase in lapse rate from isothermal to adiabatic. Elsewhere, he discusses how an adiabatic profile will be expected regardless of whether heat flow through the system is large or negligible.

I know this is repetitive, but because we may be talking at cross purposes, I’ll rephrase my question in relationship to some of the foregoing.
Consider three adjacent altitudes: z, z+, and z-, where z+ is lower (higher density) and z- higher (lower density) than z. Within any time interval, dt, a fraction of molecules from z- (Fz-) will cross z downward, and a smaller fraction from z+ (Fz+) will cross z upward. The size of the fractions will differ (so that the absolute number of molecules is the same) but their mean kinetic energies must be equal. The mean kinetic energy of Fz- will equal its energy at altitude z-, plus added energy from gravity. If we now look at the same number of z- molecules that have traveled in some other direction from z- during dt, will they not have a lower mean kinetic energy (for example, molecules traveling upward will gain potential energy at the expense of kinetic energy)? The converse applies to altitude z+, where Fz+ kinetic energy will be less than the same number of z+ molecules traveling in a different direction. When averaged over all directions, will the mean kinetic energy of Fz+ not underestimate the energy of all molecules at z+, and the mean kinetic energy of Fz- not overestimate the energy of all molecules at z-? If so, z+ must be warmer than z- for their energies at z to be equal. Why does this not require an adiabatic profile in the absence of some constantly applied upwardly directed external force to counteract gravity and accelerate molecules upward rather than isotropically? If such a force exists, what is its source?

I’ll try to address the Second Law question in more detail later. I see no violation, but for the moment, remember that transferring heat will change the heights and pressures within the two columns in a non-equilibrium direction.

Fred,
I may have misinterpreted your description in the Climate Etc. thread. From your comment there, I read: “However, while the Boltzmann distribution describes the mean, we can’t assume that in a gravitational field, the energy distribution of the molecules that were traveling downward was equal to those had traveled horizontally or upward from the same starting level. If that is true, it must be shown rather than assumed.”

If I’m interpreting this statement as you intend, you are asking why the expectation value for the vertical velocity of particle in a gravitational field could not be negative. If that were the case, the center of mass of the entire system of particles would share this velocity.

@Pekka Pirilä
Pekka – I’m not sure I follow your logic. Consider an even lower level, z++, as far below z+ as z is above it. In any interval, the number of molecules crossing z++ from z+ will of course exceed the number from z+ crossing z, because the slower molecules will not reach z. However, the mean energy of those crossing z++ from z+ will be its kinetic energy plus potential energy, and the same applies to those crossing z from z+. The z++ molecules will have lost potential energy while the z molecules will have gained it. If they started from z+ with the same total energy, how can they cross those two boundaries with the same kinetic energy, or with their kinetic energy at z+?

@quondam
Quondam – My surmise is the following. In a gravitational field, the mass of a gas column will tend to move downward unless something stops it. I presume that “something” is the planetary surface, which must be exerting a net upward force. Averaged over the sphere of a planet, this implies a compressive force from the atmosphere matched by an expansive counterforce from the planet, reflected in an upward force from surface molecules in equilibrium with the atmosphere. At the surface/gas interface, the upward and downward directed energies must be equal, and in a static situation, surface temperatures should reflect that equality. If we start at zero net energy flux there, my reasoning described above simply applies the same logic at every higher level by requiring that temperature differences exert an expansionary upward tendency that offsets gravity. If we fail to consider the surface/gas relationship, it seems to me, at least at first glance, that an isothermal gas column (mean molecular kinetic energy independent of altitude) requires molecules that move down to the bottom to return upward in the face of gravity with the same energy they had downward, but without a boost from the surface. (I do need to think a bit more about the last sentence, but that is my first intuition).

@Pekka Pirilä
Pekka – I’m trying to discern specifically where our disagreement lies, since we agree that compared with molecules moving downward from a point, those moving upward will be slowed in traveling a specified distance dz, and a disproportionate number will fail to reach it before losing their upward momentum. I assume though that there is disagreement about those molecules that do travel the distance. Are you saying that at level z, although the number of molecules sufficiently energetic to traverse z from z+ will be smaller than the number from z+ needed to traverse z++ (as we agree), the mean kinetic energies of the molecules that do traverse those levels will nevertheless be equal? That the energies would be equal seems to me to be wrong, even for arbitrarily small dz, but it may not be what you are saying. If it is, then using your term w for vertical velocity, wouldn’t we have to look at dw/dz and apply it to the Boltzmann distribution to determine the deviation, if any, from an isotropic distribution of kinetic energies in molecules distant from z in different directions? Intuitively, it seems to me that if we reduce the number of molecules successfully crossing z from below by slowing all upward molecules, those that do cross will also have a reduced mean kinetic energy, but perhaps that can be shown not to be true.

Pekka,
I was interested in your comments at Climate etc about bias and cherrypicking in IPCC WG2.
You say this is “a major problem of the IPCC WG2″,
“this problem with WG2 is so severe that I cannot judge at all, what I should take seriously, and what is spurious consequence of the bias in research.”
“The report of the environmental group distorts the message of the original paper, and the IPCC report distorts the text of the report even further from the original paper.”
“I have really lost my trust in the WG2 report.”

I am collecting examples of IPCC distortions, exaggerations and bias at the site
http://sites.google.com/site/globalwarmingquestions/ipcc
So far I have concentrated on WG1, with only a few examples from WG2 that have been highlighted in the media. Your comments seem consistent with what Richard Tol found in WG3 and I and others have found in WG1.
Please could you provide some specific details of some of the examples you have found in WG2, on your blog?