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Discussion > That CO2 thing again..

michael hart,

"I rarely, if ever, get much traction when posing questions that start with "...but much of the atmosphere is not in equilibrium, so would CO2 not actually increase radiative losses to space in certain regions such as, say, sub-tropical regions where there is 'adiabatic' warming of low-humidity air..."

Is that included in what you think is worth a separate discussion, or is it covered in your blog-posts mentioned above"

I think I understand what you are getting at. It is very relevant to this discussion, rather than being a "feedback" or "sensitivity" question.

Of course, the climate is never in equilibrium. One place, one day, one month, one season, one year - all are in constant change. The "new equilibrium" is more of a conceptual device to help people grasp the inevitable response of climate to radiative forcing (prior to any feedback).

Let's take a region of descending sub-tropical air, which is dry and warm. The same applies to that region of air as to any other region - when you increase GHGs in the atmosphere, especially in the mid- to upper-troposphere, the OLR reduces. Of course, if you take a location where the temperature profile is surface =300K, lapse rate =7K/km and CO2 =390ppm, then this location warms so surface =305K, lapse rate =6.5K/km and now CO2 = 400ppm the OLR will have increased.

Was this the idea you were getting at? I might have misunderstood. So if this is the way you were thinking about the problem then I'm sure I can complete the picture and explain. If I have misunderstood, can you put some numbers into an example?

Sep 2, 2014 at 9:48 AM | Unregistered CommenterScience of Doom

Mike -

Are you absolutely sure that if there were no "greenhouse gases" in the atmosphere then there would be no atmosphere?

I think you perhaps meant to write "...there would be no greenhouse effect?"

If the only gases in the atmosphere were gases that do not interact with radiation then, yes, I'm pretty sure that the greenhouse effect would not exist - almost by definition. All the incoming energy from the Sun would have to be lost to space via direct radiation from the ground (or what would pass for ground in the absence of water and organic matter - rock and sand presumably).

But if there were no CO₂, H₂O, CH₄, O₃ in the atmosphere, the world would be different in all sorts of ways - no oceans for a start.

A question I have never seen discussed is what would happen if all the CO₂ could be magic-ed out of the atmosphere. Would things carry on much as before climate-wise? (Of course, plants would have a hard time and all things that depend on them, so that things would soon start to change drastically in all sorts of ways.)

Sep 2, 2014 at 9:59 AM | Registered CommenterMartin A

Martin A
I'm probably being a bit pedantic but that quote read as if you were saying that without greenhouse gases there would be no atmosphere.
But it's still a point of view. Huffman's argument in essence is that the temperature on Venus and on Earth is a function of atmospheric pressure. Allowing for that and relative distance from the sun the temperature of both planets is what you would expect (or at least in the proper relation to each other) without any need to invoke "greenhouse gases".
I'm not for one minute saying he's right but is he wrong about the temperature relationship between the two and if he isn't and given what we know about the composition of the atmosphere of the two planets why is he wrong about the other?
Mostly the answers I get when I pose that question are that "everyone knows" he's wrong about the greenhouse gases but no-one is prepared to give a straight answer about the other half!

Sep 2, 2014 at 11:31 AM | Registered CommenterMike Jackson

Mike -

I said "if the Earth's atmosphere contained zero greenhouse gases, then it would not exist so far as Earth is concerned."

I can see why you thought I meant the atmosphere would not exist. I meant it to be read as:

"if the Earth's atmosphere contained zero greenhouse gases, then the greenhouse effect would not exist so far as Earth is concerned."

As I said before somewhere, H D Huffman's argument seems to be:

(1) Allowing for the differing distances from the Sun, Venus and Earth have approximately the same profile of temperature vs. pressure (pressure being a function of altitude).

(2) Therefore the greenhouse effect does not exist.

That (2) follows from (1) may be obvious to D L Huffman. But I don't see any reason whatever why (2) should follow from (1). IN the absence of a step-by-step explanation, I assume it's bollocks to say that (2) necessarily follows from (1)

[I assume he's correct about (1). It's the sort of thing I'd have checked up on when I first read it, but I don't remember if I did. If he were wrong about (1), no doubt the experts at SkS would have rapidly pointed that out.

I'd be interested to see an explanation for what Huffman has noticed. Even if it's merely a coincidence.]

Sep 2, 2014 at 12:10 PM | Registered CommenterMartin A

The profile of temperature vs height (the "lapse rate") depends on a few factors that are nothing to do with radiation or the greenhouse effect.
This is atmospheric physics 101 as found in pretty much any textbook. It wasn't discovered by anyone recently. It has been known about for a very long time.

Strictly speaking this is the adiabatic lapse rate. Expanding into lower pressure means work is done. This energy has to come from somewhere so the parcel of air cools. Likewise for the reverse.

So you can determine what happens to the temperature of a parcel of air when it is raised 1km. That determines the change. What determines the actual temperature?

If you pump up a bike tire quickly it heats up. If you leave it alone, even though it is at high pressure it cools down. Pressure does not cause temperature. How does the surface of the earth radiate an average of almost 400 W/m2 when the climate system only absorbs 240 W/m2 from the sun? The fact that rising air cools and descending air warms doesn't explain this observed effect.

Sep 2, 2014 at 12:38 PM | Unregistered CommenterScience of Doom

SoD, I think it should be “environment lapse rate”, as adiabatic lapse rate really applies to a rising parcel of air, the “adiabatic” being heat loss by pressure reduction only, all other potential heat losses being ignored. The ELR is what you would experience as you rise through the air (not with it). I can’t remember the actual numbers, but the ALR tends to be lower than the ELR, which is why a thermal continues to rise, as, despite cooling, it remains warmer than the air around it. When the ELR is zero, or positive, you get thermal inversion, and the rising plume of smoke, vapour, whatever, suddenly flattens out, thus trapping the moisture, heat and pollution, giving you hot, muggy conditions.

However, there being an environment lapse rate does indicate that there could be some validity to the theory, even if not fully confirming it.

Sep 2, 2014 at 2:07 PM | Registered CommenterRadical Rodent

How does the surface of the earth radiate an average of almost 400 W/m2 when the climate system only absorbs 240 W/m2 from the sun?
Excellent question, SoD. What is the answer? Since the only source of energy which the earth has is the sun where does the extra 160w/m2 come from?

Martin A
I think I provided an answer to the apparent gap in Huffman's reasoning earlier. My understanding of what he is saying is that if the relative temperatures of the earth and Venus can be accounted for by distance from the sum and atmospheric pressure then CO2 is irrelevant given the massive difference between the concentrations on the two planets.
I don't seriously know of anyone who disputes the greenhouse effect but only to the extent that it is a useful shorthand for laymen (like me) to understand why we don't all freeze to death. I'm inclined to the view that one of the problems we face is that the over-simplification of the climate system (things like the greenhouse effect being "like a blanket" and treating the earth as a black body because it makes all the calculations easier) has created a lot of the confusion and disagreement about what the climate actually is and does.
Even as we discuss on this thread we find phrases and ideas being bandied about that I'm pretty certain the bandiers (!) themselves — me included — don't fully understand and can't agree on the definition of!

Sep 2, 2014 at 2:08 PM | Registered CommenterMike Jackson

SoD

So you can determine what happens to the temperature of a parcel of air when it is raised 1km. That determines the change. What determines the actual temperature?

I imagine that lapse rate becomes a bit more complicated when the parcel of air at ground level contains water vapour which condenses out on the way up as the parcel rises. Though presumably this is well studied and understood.

Sep 2, 2014 at 2:12 PM | Registered CommenterMartin A

Since the only source of energy which the earth has is the sun where does the extra 160w/m2 come from?

To answer on sod's behalf - if I may - it's energy (from the sun) that the earth did not manage to get rid of at the first attempt - it came bouncing back - so it is being got rid of again.


the bandiers (!) themselves — me included — don't fully understand and can't agree on the definition of!

There are some people who know what they know and they know what they don't know. There are some others who are unsure of both those things.

I think I provided an answer to the apparent gap in Huffman's reasoning earlier. My understanding of what he is saying is that if the relative temperatures of the earth and Venus can be accounted for by distance from the sum and atmospheric pressure then CO2 is irrelevant given the massive difference between the concentrations on the two planets.

But could it not equally be argued that the similar temperature profiles mean that both planets benefit from the greenhouse effect so the existence of the effect is confirmed?

If there were a third planet whose atmosphere were a mixture of argon and nitrogen - non greenhouse gases - that also followed the same profile, I'd begin to agree that it raised some questions. But I have tried to see it but I just don't see why what he has observed should mean that the greenhouse effect does not exist.

Sep 2, 2014 at 2:33 PM | Registered CommenterMartin A

One reason I am suspicious of the greenhouse effect is that, if you were to have a greenhouse that comprised of a web of glass totalling just 1% of the area covered, it would not be a very good greenhouse, with the heat disappearing quickly through the 99% not covered. However, I can accept it as a simplification in an attempt towards understanding. In the same way, I can also accept the idea of the atmosphere as a “blanket” covering the Earth, shielding us from the worst of the heat, and keeping what heat that did come in, erm, in – indeed, the blanket analogy could probably be closer to the truth. On a calm, clear day, we will all have seen the mirage-pools created as the air in contact with the sun-warmed ground heats, changing its density, thus its refractive index. When disturbed by a passing breeze, the mirage disappears as the air is dispersed in the surrounding air, and the heat balances out. Over different surfaces, there is different heating, thus thermals form over darker ground, as the heated air rises, and cooler air moves over to replace it. Could it be that the heating is not so much the “trapping” of the heat by the greenhouse gasses, but more likely the conduction of heat from hot surfaces to the not-very-radiant nitrogen and oxygen that makes up the vast bulk of the atmosphere? This would explain the difference in temperatures between latitudes; the incidence of the sunlight through the atmosphere is no different in any latitude, it is the incidence on the ground that is. Indeed, one could argue that sunlight has more atmosphere to travel through in the higher latitudes thus giving a greater chance for the GHGs to “trap” their heat, so the poles should be warmer that the tropics! We know this is not so, as it is that the sunlight has less atmosphere to travel through in the tropics before it heats the ground that the tropics are so warm (as well as striking it in more concentrated form). The blanket analogy becomes more apposite on a cloudy day, with the cloud reflecting the heat back to space, the ground underneath not being heated, thus cloudy days tend to be cooler days; conversely, what heat there is cannot be radiated out during the night, and cloudy nights tend to be warmer nights. In deserts, where there is considerably less water, the air can heat quite dramatically during the day, and cool rapidly during the night (I have heard that the rocks can often fracture from the thermal shock, but have no evidence of it; similarly, I have also heard that the Arabs used to have wall-enclosed courtyards with dark tiling that they flooded in the evening, and harvested ice in the morning, but have not been able to confirm that, either). Whichever it may be – blanket or greenhouse – it would seem that it is the presence of water in the atmosphere that has the most observable impact on heating and heat-retention, as water is the single component that is in different concentrations in the two scenarios mentioned, CO2 concentrations being notionally constant in both.

Sep 2, 2014 at 3:02 PM | Registered CommenterRadical Rodent

Martin A: take a look at the difference in concentrations of CO2 (the principle “greenhouse gas” in these discussions); on Earth, it is 0.04% (approaching dangerous levels, according to some); on Venus, it is 97%. Earth’s atmosphere is 99% non-greenhouse gasses; why do you need a third planet?

A parcel of air containing water is only relevant to the lapse rate when it has cooled to its dew-point; it then has a “saturated” adiabatic lapse rate (SALR – also known as “moist” ALR), which is about half its “dry” lapse rate, as the condensing water releases its heat.

Sep 2, 2014 at 3:13 PM | Registered CommenterRadical Rodent

Martin A
I think RR has given the answer. As I've said above I don't dispute the greenhouse effect, ie, that the atmosphere provides the earth with a temperature range amd atmosphere that makes it habitable. I don't think Huffman is disputing that though I suspect that, like others, he's not keen on that analogy.
What he is saying is that given the very, very wide difference in CO2 concentration between the two planets then CO2 cannot (if No. 1 in his hypothesis is correct) have a material effect on the temperature of earth's atmosphere.

Your other answer worries me. 240 w/m^ comes in. There is only 240w/m^ in the system. It can bounce about to its heart's content. It is still only part of that 240 w/m^ input. I don't thinkl I'm alone any more in being extremely sceptical about "back radiation" which performs much the same function as CO2 in the Climate Bible. Without them there is no panic, no agw, and nothing but natural variation.

Sep 2, 2014 at 4:47 PM | Registered CommenterMike Jackson

RR - are we talking about whether the greenhouse effect exists at all, or whether a small addition of carbon dioxide will raise the global temperature a few degrees? Mike Jackson suggests it is the latter.

Either way, I still don't understand how DHH's observation makes it apparent that either is nonsense. Maybe I'm slow but I still think there should be a succesion of logical steps that even I could follow.

why do you need a third planet? Because both Earth and Venus have atmospheres with greenhouse gases. If there were a planet whose atmosphere had no greenhouse gas content yet it behaved the same as Earth and Venus, then that would make one start to question whether greenhouse gases have any effect on anything. Does that answer make sense?

Mike Jackson - don't let my answer worry you. You read a lot of bollocks about "back radiation warming the planet" from people who should know better. All the warming was done by the incoming 240W/m² as the planet warmed up to reach its steady-state condition. Once it has reached equilibrium, a total of 240W/m² arrives from the Sun and 240W/m² departs from the upper atmosphere to outer space.

The 160W/m² is book-keeping. The walls of your living room are giving out oodles of radiation at each other but it all sums to zero and does not cause you any anxiety, I am sure.

It needs more than a couple of lines to explain clearly but it can be explained using basic principles of physics with no hocus-pocus whatever needed.

Some other time I'll have a go at writing it up.

Sep 2, 2014 at 6:00 PM | Registered CommenterMartin A

Martin A
I'll have an other go because what I'm saying is clear to me so I'm evidently not explaining it right. (Where have I heard that one before).
Please feel free to correct me if what I'm saying is not what you understand Huffman to be proposing.

1. If Venus and Earth were equidistant from the Sun and their atmospheric pressure were the same their GAT would be the same. This is my understanding of Huffman's argument. The difference in temperature is due to a combination of the different distance from the Sun and the different atmospheric pressure.
2. The composition of those two atmospheres is therefore irrelevant. The greenhouse effect is (theoretically) the same for both planets and since one is 90+% O2 and N and the other is 95% CO2 CO2's influence on temperature has got to be minimal.

As I said, I don't necessarily agree with this but it seems to me that the hypothesis is straightforward enough.

As for the other I'm even more stunned especially since my understanding of "book-keeping exercises" is that they are a means of transferring non-existent income/expenditure to a different budget. The books still have to balance and 240 on one side and 400 on the other — result, as Micawber said, misery. I look forward to the explanation!

Sep 2, 2014 at 6:19 PM | Registered CommenterMike Jackson

Martin why do you have trouble disagreeing with "unquantified" statements even when they are clearly wrong? That is illogical. And pointing to some local decrease in temperature as if it refutes the global effect of CO2 is like, when it is pointed out that you are badly sunburnt, 'refuting' the assertion by pointing to the white strip underneath your watch strap. You'd be more credible if you just admitted that it is indeed possible to question an unquantified statement.

Mike: "There is only 240w/m^ in the system." No, the 240w/m^2 is the flow entering (and leaving) the top of atmosphere not the energy "in the system". The 400w/m^2 is the flow at the surface. The difference is because there is energy contained (dare I say trapped?) within the Earth 'system'.

Sep 2, 2014 at 6:21 PM | Unregistered CommenterRaff

As usual, Raff, what you say makes no sense whatsoever.

Martin: I think it is whether “greenhouse gasses” are actually a reality, outside the laboratory. I suggest that the evidence is that they are not; while certain gasses do show signs of “absorbing” radiative heat in the laboratory, that does not mean that the effect will be significant “in the wild”, so to speak. It is possible that the problem is likely to be considerably more complex than we are exploring, at present, though the end result could also be so obvious when it is found, we will say, “How could we miss that!” (A mantra that has been common throughout history.) As I said, earlier, Venus has an atmosphere that is at least 97% “GHG”; Earth has an atmosphere that is about 99% non-GHG; why do you balk at comparing the two? Especially as we are constantly being told that we should fear a rise of 0.01% CO2 as the results would be catastrophic! Venus has an atmosphere that contains 96.96% more CO2 by concentration than the Earth’s, yet its temperature at Earth-normal pressure is what could be expected of Earth’s, should it be the distance that Venus is from the Sun; therefore, the “greenhouse effect” is actually minimal.

Sep 2, 2014 at 8:56 PM | Registered CommenterRadical Rodent

I still find it fascinating that it is considered possible to create energy!

240w/m^2 in,at the top and **INCREASED** to 400w/m^2 at the surface!

If we could just tap this energy source we would have free clean energy for a long long time.

:-)

Sep 2, 2014 at 9:15 PM | Unregistered CommenterSteve Richards

Just dropping by for 60 seconds...

RR "Earth has an atmosphere that is about 99% non-GHG"

No. Earth's atmosphere is only about 97% non-GHG. water vapour is a GHG .

If (I said IF) the greenhouse effect were saturated in both planets you'd expect the GHE to be similar between the two even if one is 97% co2 and the other is 3% h2o + 0.01%co2. It needs a calculation to show that they should be different if the GHE is real. I havent seen one


I don't balk at comparing the two - I just don't see how without either a detailed logical argument or a detailed mathematical calulation hdh's observation tells us anything one way or the other.

Sep 2, 2014 at 10:57 PM | Registered CommenterMartin A

Mike Jackson:

"SoD. What is the answer? Since the only source of energy which the earth has is the sun where does the extra 160w/m2 come from?"

This is a common reaction to one of the most basic radiative heat transfer questions.

The reaction is "you have created energy.. I'll just point that out and so I've already proved my case, nothing more to do!"

Well, if I can just grab you for a few minutes, I'll explain.
1. No energy is created
2. It's just like lagging a pipe but with radiation rather than conduction
3. How this "created energy" can't be used

1. No energy is created.
In A Challenge for Bryan – The Solution (which I had to provide because Bryan wasn't prepared or able to provide the solution) I give the simplest example possible of a "radiation shield" where adding this barrier increases the surface temperature.

The solution is calculated from the first law of thermodynamics (no energy is created or destroyed) and the Stefan Boltzmann equation of radiation.

So how, using conservation of energy, do we prove that a body is radiating more than its "internal energy source". Simple. The radiation shield stops energy leaving which means energy accumulates, which means temperature increases.

In the new equilibrium condition the body radiates exactly the amount of energy it receives - that is, a) its internal energy source + b) the radiation back from the shield.

If I am wrong it should be the work of a few minutes to provide the alternative correct solution. Interestingly, Bryan "the greenhouse effect violates the second law of thermodynamics" agreed with the result after it was produced. I have produced a few examples similar to this before - no one ever provides alternative equations.

Why not? If it violates the first law of thermodynamics, can't anyone demonstrate that energy has been created? Answer, because no energy has been created.

I also provided a graph of energy leaving the system vs time. You can easily see where the "new invented energy" comes from. (Energy retained).

2. It's just like lagging a pipe but with radiation rather than conduction.
If you lag a pipe you reduce heat loss and increase the internal temperature. No one has a problem with that. Reduce the conductivity and less heat leaves, so temperature must increase.
What is different about radiation? There is a 2 way transfer and it is outside most people's common experience, otherwise the broad brush idea is exactly the same. The radiation shield has prevented energy leaving, therefore the temperature has increased.

3. Steve Richards says "If we could just tap this energy source we would have free clean energy for a long long time."
No. If you "tap into it" you will decrease the surface temperature of the earth. Think of it like this. For a while temperature increased until energy leaving the system matched the energy absorbed by the system. That equilibrium happens to have the surface of the earth at a higher temperature than the system. Why does anyone imagine that the surface temperature of the earth has to be the same as the temperature of the climate system? Does the internal temperature of a pipe have to be at the same temperature as the outside of the lagging?

If you connected one end of a conductive rod to the surface of the earth and the other end to say an orbiting cold asteroid then you would cause the surface temperature to decrease. You only get to use energy once.

It's very simple.

My challenge to you - find the created energy in the example for Bryan and add your comments to that article, or here.

Sep 3, 2014 at 12:00 AM | Unregistered CommenterScience of Doom

Mike Jackson:

"I don't think I'm alone any more in being extremely sceptical about "back radiation""

You aren't. But that doesn't mean anything.

What are you skeptical about? Its existence, or the reason for its existence?

In a 3 part series I show long term measurements from around the globe of flux, spectra of this back radiation and lots more - Back Radiation

Let me know after you have read this if you are skeptical of back radiation.

Sep 3, 2014 at 12:06 AM | Unregistered CommenterScience of Doom

Radical Rodent,

On the lapse rate, I went into a lot of detail in a few articles on this subject.

Density, Stability and Motion in Fluids – some basics about instability

Potential Temperature – explaining “potential temperature” and why the “potential temperature” increases with altitude

Temperature Profile in the Atmosphere – The Lapse Rate – Lots more about the temperature profile in the atmosphere

The environmental lapse rate is what exists in practice. The adiabatic lapse rate tells you what happens if you move a parcel of air quickly. In the tropics you get something very close to the adiabatic lapse rate. This is because convection is caused by strong solar heating of the surface. Outside of the tropics you get a much more complicated environment because there is not so much strong convection. In the polar regions you get everything from temperature inversions, i.e. surface temperature colder than the atmosphere above, through to the adiabatic lapse rate.

Hopefully the above articles explain everything that needs explaining about this subject - except for the important bit that no one understands, which is what the actual lapse rate will turn out to be when there isn't much convection.

Sep 3, 2014 at 12:18 AM | Unregistered CommenterScience of Doom

Mike Jackson - SoD has answered comprehensively if you take the time to work through what he has said and pointed to.

Here are a few lines from an Amazon review of "Slaying the Sky Dragon" which may possibly help explain how the surface of the Earth radiates more W/m² than eventually escape to space.

It's talking about the highly simplified model where the Earth is represented as a grey body which is surrounded by a thin layer of dense greenhouse gas. (Grey for visible light, rather than black, because 30% of the light from the Sun is reflected.)

(...) This chapter starts with a diagram from a Washington University course. Many readers of this review will be familiar with similar diagrams showing:
- 342 W/m² arriving as solar radiation
- 102.6 W/m² being reflected to space immediately
- 239.4 W/m² continuing downwards and then warming the earth (taken to be 240 W/m² in the text)

The earth then re-radiates 240 W/m², as it is in thermal equilibrium.

It is assumed that the 240 W/m² radiation leaving the earth is absorbed by an atmospheric layer and re-radiated 50% upward and 50% downward, so that 120 W/m² goes to space and 120 W/m² goes back to earth, where it is absorbed and the 120 W/m² is then re-radiated.

I see no problem with this. Of the 120 W/m² re-radiated by the earth, 60 W/m² returns again, then 30 W/m², and so on. So we have going spaceward 120 + 60 + 30 +... = 240 W/m². This is the same as originally arrived at the surface, so things are in equilibrium, with as much power being radiated spaceward as originally arrived at the surface directly from the sun.

Nothing wrong with that that I can see. The earth's temperature is being maintained but it is not receiving any additional heat from anywhere - including the cooler greenhouse gases above its surface so there is no need to argue that cold objects do not heat hotter objects. (...)

*All* the heating originally done to warm up the Earth until it reached its equilibrium temperature was done by incoming sunlight, even though some radiation was bouncing back and forward between the Earth and the greenhouse gas shell as this went on.

Does the explanation here make things any clearer? If not, then best to disregard it, rather than get bogged down in several explanations at once.

Sep 3, 2014 at 9:11 AM | Registered CommenterMartin A

Raff

The difference is because there is energy contained (dare I say trapped?) within the Earth 'system'.
Where from? And if it's trapped, how come it's getting out?

Sep 3, 2014 at 9:18 AM | Registered CommenterMike Jackson

I see no problem with this. Of the 120 W/m² re-radiated by the earth, 60 W/m² returns again, then 30 W/m², and so on. So we have going spaceward 120 + 60 + 30 +... = 240 W/m².

WRONG! You never reach the initial figure of you continually halve the remainder. Infinitesimal the difference may eventually be but the explanation is just not correct. Neither does it appear to take any account of convection or conduction, the two other "channels" which move heat around a system. How does the fact that Earth is a "water planet" affect absorption? How does the existence of air currents affect the movement of heat around the atmosphere?

My problem is that everything I read is highly theoretical and appears to take little or no account of the real world with the result that we end up with Trenberth's "missing heat" because he and you and SoD are convinced it has to be there because the theory says it's there.

SoD
I will go and read your thesis on Back Radiation but I am still going to need a deal of convincing that 1 watt of energy, having done its job, is capable of doing the same job again and again until it finally manages to escape into space. (I know that is a rather simplistic way of putting it but I hope ot gets the idea across.)
And I am going to take even more convincing that increasing the concentration of CO2 from .003% to .004% is going to make a damn of a difference anyway.

Sep 3, 2014 at 9:45 AM | Registered CommenterMike Jackson

1 = (¹/₂) / (1 - ¹/₂) = ¹/₂ + ¹/₄ + ¹/₈ + ¹/₁₆ + ¹/₃₂ + ... = 1

Use of the word "trapped" is a matter of choice. I had a brief discussion with Raff on that.

My view was that if a succession of mice are running through a pipe to get into my barn, it does not make sense to say that the mice are "trapped" in the pipe even though there are always (say) three mice in the pipe. Or that the cars travelling along the M4 are "trapped" on the motorway.

Likewise for heat which is constantly exiting while constantly being replaced.

But if someone wants to talk that way, they are free to do so.

Sep 3, 2014 at 10:13 AM | Registered CommenterMartin A