Frequently Made Errors in Climate Science - The Greenhouse Effect - Comments

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haruspex
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haruspex submitted a new PF Insights post

Frequently Made Errors in Climate Science - The Greenhouse Effect

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Continue reading the Original PF Insights Post.
 
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  • #2
Buzz Bloom
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Hi haruspex:

I very much like your Insight article. I have a question though.

The article says:
If we treat the Earth as emitting and absorbing radiation as a “black body”, ignoring the atmosphere, and treating incoming light as spread evenly over the whole Earth’s surface all the time, we can calculate the equilibrium temperature as -18C. At that temperature, black body radiation would balance insolation.

Adding a non-greenhouse atmosphere, e.g. pure nitrogen, doesn’t change this.​
I understood that N2 scatters blue light, and this is what makes the sky look blue. Wouldn't the scattering of blue photons by N2 then reduce the number of blue photons that hit the earth, since about half the scattered photons return to space? Wouldn't this reduce the equilibrium average temperature of the surface compared with no atmosphere?

Regards,
Buzz
 
  • #3
haruspex
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Hi haruspex:

I very much like your Insight article. I have a question though.

The article says:
If we treat the Earth as emitting and absorbing radiation as a “black body”, ignoring the atmosphere, and treating incoming light as spread evenly over the whole Earth’s surface all the time, we can calculate the equilibrium temperature as -18C. At that temperature, black body radiation would balance insolation.

Adding a non-greenhouse atmosphere, e.g. pure nitrogen, doesn’t change this.​
I understood that N2 scatters blue light, and this is what makes the sky look blue. Wouldn't the scattering of blue photons by N2 then reduce the number of blue photons that hit the earth, since about half the scattered photons return to space? Wouldn't this reduce the equilibrium average temperature of the surface compared with no atmosphere?

Regards,
Buzz
Yes, that's true too. From what I read, the scattering of green light in air at 1atm. is 10-5m-1, and blue light about double that. From that I calculate that about 4% of green light and 8% of blue light would never make it through. Taking the mean power loss as 6%, that would knock 1.5% off the surface temperature, or 4o K.

Edit: I forgot to allow for IR, barely scattered at all. So I would think the power lost would be no more than 4% in total.
 
  • #4
alw34
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What conclusions can be drawn from such 'frequent errors". Who do you think is making such errors and what are the effects? Thanks.
 
  • #5
haruspex
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What conclusions can be drawn from such 'frequent errors". Who do you think is making such errors and what are the effects? Thanks.
They are errors I have come across in posts (not just in these forums), and, just occasionally, errors I have made myself.
This particular FME post is a bit different from my others (Friction, Moments, etc.) in that there is reason to suspect that some people making climate science errors do so disingenuously.
 
  • #6
alw34
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some people making climate science errors do so disingenuously.

Oh, the horror!! Yet, one must secure government grants or perish.

In most scientific models, a single false prediction invalidates the model.

Climate models seem immune from such scrutiny.
 
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  • #7
haruspex
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Oh, the horror!! Yet, one must secure government grants or perish.

In most scientific models, a single false prediction invalidates the model.

Climate models seem immune from such scrutiny.
You are wandering off the topic of what errors are commonly made and what the correct versions are. If you wish to engage in a discussion on the politicisation of science; on the relationship between scepticism, consensus, scientific progress and public policy; or on the use of models of complex systems, let's do that as a private dialogue.
 
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  • #8
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The Insight article is excellent. But I can't help making a small correction to this statement:
"Note: If something is a negative feedback it acts to dampen change; it cannot make the change go in reverse."
In systems with time delays, accumulated effects, and limiting functions, negative feedback can cause cyclic behavior that includes periodic reverse effects.
The dynamics of the system we are talking about is very complicated and includes all those features.

That being said, if the negative feedback causes a reverse effect at one time, there will probably be a contributing effect later. So it might fool us now and cause a catastrophe later.
 
  • #9
D H
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Minor comment: Please replace that image that depicts a 5250 °C blackbody curve with something else. The Sun's effective blackbody temperature is not 5250 °C, or 5523 kelvins. It is 5777 or 5778 kelvins, depending on who you read. Siting that image has become a "frequently made error in climate science." This is a good example of why wikipedia is not a good source.

Otherwise, this is a very nice insight article.
 
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  • #10
haruspex
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Minor comment: Please replace that image that depicts a 5250 °C blackbody curve with something else. The Sun's effective blackbody temperature is not 5250 °C, or 5523 kelvins. It is 5777 or 5778 kelvins, depending on who you read. Siting that image has become a "frequently made error in climate science." This is a good example of why wikipedia is not a good source.

Otherwise, this is a very nice insight article.
Thanks D H, I was not aware that was wrong. But I like the image, much clearer than most, so I've just added a note. (Is the curve wrong, or just the number? The peak seems to be in about the same place as in other images I found.)
 
  • #11
mheslep
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Regardless of the current uncertainty on the subject, why no inclusion of the possibility of clouds in Negative Feedbacks? The CFMIP modeled feedback is "approximately zero", but this is not the same as saying there is none.

IPCC AR5 7.2.5.4: "The key physics is in any case not adequately represented in climate models. Thus this particular feedback mechanism is highly uncertain"
 
  • #12
haruspex
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Regardless of the current uncertainty on the subject, why no inclusion of the possibility of clouds in Negative Feedbacks? The CFMIP modeled feedback is "approximately zero", but this is not the same as saying there is none.

IPCC AR5 7.2.5.4: "The key physics is in any case not adequately represented in climate models. Thus this particular feedback mechanism is highly uncertain"
I included some examples. I made no claim it was exhaustive. There are several more I could have listed (positive and negative). My purpose was to illustrate what is meant by a feedback.
The Insights post concerns aspects of climate science that are well settled and uncontentious yet often misunderstood or misrepresented. I have no plan to get into the more debatable areas. That would be beyond my expertise.
Last I checked, the general view was that whether clouds act mainly as negative or positive feedbacks depended on altitude, but I don't recall which way it worked or whether there was consensus on which dominates.
 
  • #13
mheslep
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I included some examples. I made no claim it was exhaustive. ...
Sure, then perhaps say so? "divided into negative and positive feedbacks, such as ...". Or, "...for example"
 
  • #14
haruspex
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Sure, then perhaps say so? "divided into negative and positive feedbacks, such as ...". Or, "...for example"
Done.
 
  • #15
FactChecker
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I included some examples. I made no claim it was exhaustive. There are several more I could have listed (positive and negative). My purpose was to illustrate what is meant by a feedback.
The Insights post concerns aspects of climate science that are well settled and uncontentious yet often misunderstood or misrepresented. I have no plan to get into the more debatable areas. That would be beyond my expertise.
Last I checked, the general view was that whether clouds act mainly as negative or positive feedbacks depended on altitude, but I don't recall which way it worked or whether there was consensus on which dominates.
Negative feedback is more complicated than you imply. It can cause dynamic instability. It depends on the dynamics of the system, which is very complicated in this case.
 
  • #16
haruspex
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Negative feedback is more complicated than you imply. It can cause dynamic instability. It depends on the dynamics of the system, which is very complicated in this case.
To address your earlier comment I already added a clause on time delayed feedbacks. If you feel it is necessary to mention instability specifically I'm happy to add that.
Thanks.
 
  • #17
FactChecker
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To address your earlier comment I already added a clause on time delayed feedbacks. If you feel it is necessary to mention instability specifically I'm happy to add that.
Thanks.
No. I like your insight post. Thanks for the addition. I think what you have is as good as it can reasonable be. Negative feedback such a can of worms that it's not realistically possible to do much better.
 
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  • #18
BillyT
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… I understood that N2 scatters blue light, and this is what makes the sky look blue. ... Buzz
I think that is miss leading. Even if the atmosphere were argon, the sky would still be blue. On the scale of longer wave lengths, say red light wave length, there is nearly a constant number of atoms (or molecules) in a cube with edge of length equal to red wave lengths. But in a cube with half that edge length (1/8 as many molecules inside) the statistical fluctuation as a percentage in number of molecules inside the cube are more significant. The scattering is due to the greater varying index of refraction that small volumes have. Nothing to due with air being mostly N2.
 
  • #19
klimatos
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Derek,

Excellent post--although I have a few quibbles. You probably would have addressed them if you were given more space, but here goes:

In your point 6, you state:

1) The Sun warms the Earth’s surface.
2) Conduction and evaporation carry this heat energy to the air at the Earth’s surface.
3) Some conduction, but mostly convection, carries the heat energy higher.

All of these statements are misleading in some respect:

1) Some two-thirds of the radiant energy reaching the surface of the Earth comes from atmospheric molecules—mostly water molecules, the remaining third comes from the Sun. These atmospheric water molecules, of course, got their energy from the Sun in the first place. If you had simply said that, “The Sun warms the Earth”, your statement would be correct.

2) More than 90% of the heat transmitted from the surface to the atmosphere comes in the form of terrestrial infrared radiation. Again, mostly radiation from water molecules. Conduction and enthalpic cycling (condensation, not evaporation) play only very minor roles.

3) Once again, radiation plays the dominant role. Air is simply an extremely poor conductor of heat. An air mass has less thermal energy and a much lower temperature after it has been forced up than it had at its lower elevation. How does this “carry the heat energy higher”?
 
  • #20
haruspex
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2) More than 90% of the heat transmitted from the surface to the atmosphere comes in the form of terrestrial infrared radiation. Again, mostly radiation from water molecules. Conduction and enthalpic cycling (condensation, not evaporation) play only very minor roles.
You're right, I missed a chunk. But I believe the fraction reradiated is more like 80% (Kiehl and Trenberth, 1997). Most of that is radiated back down again, so convection is (just) the major escape route through the troposphere.
 
  • #21
klimatos
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Haruspex,

Like you, I much admire the 1997 paper by Kiehl and Trenberth. I think it should be in every atmospheric scholar's sourcebook. I believe it to be the standard to which all subsequent global heat budget studies should be compared. Which brings me to another quibble:

In your point 7, you object to the statement, “CO2 is insignificant compared with H2O as a greenhouse gas”. I would object right along with you, but I don’t know any serious scholars of the atmosphere who would make such an unsupported statement. On the other hand, a good many atmospheric scholars would strongly agree that it is a less significant greenhouse gas. Kiehl and Trenberth [1997] are among them. They call the role of water vapor “dominant”.

1. Water vapor accounts for more than 40 Wm-2 of absorption of short-wave (solar) radiation. Carbon-dioxide accounts for less than 1 Wm-2 (none, under cloudy skies).

2. The vast majority of photons emitted by terrestrial IR sources are emitted by water molecules. Atmospheric water molecules can absorb all of these photons, atmospheric carbon-dioxide can absorb only a small fraction (the “overlap” wavelengths). Carbon-dioxide plays no significant role in the emission of terrestrial IR.

3. At 4,000 ppm compared to 400 ppm, water vapor is ten times more abundant in our atmosphere than is carbon-dioxide. In your argument you state, “Increasing the level of a relatively rare greenhouse gas has more effect than increasing the level of a more common species.” This statement is true only if the more common gas is approaching its optical saturation density. This is not true for water vapor. This fact is attested to by the common observation that ground fogs and mists are rapidly dissipated by direct sunlight.
 
  • #22
Buzz Bloom
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I think that is miss leading. Even if the atmosphere were argon, the sky would still be blue.
Hi BillyT:

Thanks for your post. I am always interested in learning something new. Can you recommend a reference that explains more about this phenomenon?

Regards,
Buzz
 
  • #23
BillyT
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Hi BillyT: Thanks for your post. I am always interested in learning something new. Can you recommend a reference that explains more about this phenomenon? Regards, Buzz
Read the last paragraph of right column, page 269 here:
http://homepages.wmich.edu/~korista/colors_of_the_sky-Bohren_Fraser.pdf That gives the Einstein / Smoluchowski theory which does not even require discrete molecules - only density fluxuation, that cause refractive index spatial variation.
Prior to that Einstein / Smoluchowski theory, the scatering of molecules was used (and still usually is) to explain why the sky is blue. Basically if a plane EM wave is passing thur a dense medium the E-field will oscillate the bound electrons, and an accelerated electron produces radiation which combines with the incident radiation to change the net phase of radiation field. (Why the refractive index exist.*) This electronic radiation goes in all direction in a plain normal to the oscillator motion, but that of many electrons all driven in phase (dense medium & plain wave) only constructively interfer in the forward (or backward, but 180 scatter is weak) direction. I.e. in dense medium of uniform density there is no scattering, but add dynamic density variation on the scale of the wave length you do get scattering.

These small scale density fluxuation are percentage wise much larger in a rarified gas. So most of the scattering of blue light takes place high up in the atmosphere, but not so high that there is little there. Lower in the armosphere, most of the scattering is by small particles (dust etc.).

* As the earlier theory which does consider the individual atomic / molecular scattering gives almost the correct results, few of the papers found by Google have much to say about Einstein / Smoluchowski theory, but it clearly does not depend upon the existance of atoms or molecules - Certainly not what they are. Why I could say that an argon amosphere would also make the sky blue.
 
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  • #24
Buzz Bloom
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That gives the Einstein / Smoluchowski theory which does not even require discrete molecules - only density fluxuation, that cause refractive index spatial variation.
Hi @BillyT:

Thanks very much for the reference. I am a bit confused by your quote relative to what I read in the reference.

The quote above seems to be saying that the Einstein / Smoluchowski theory about fluxuations explains the blue sky. The following is a quote from the reference.
Einstein stated that "it is remarkable that our theory does not make direct use of the assumption of the discrete distribution of matter" . . .
There is no dispute, then, about what really causes the sky to be blue? Is it "really" scattering by molecules? Or, if you want to be more precise, scattering by electrons. Even more precise, scattering by bound electrons: free electrons would not give us a blue sky. The preposition "by" indicates an agent, and molecular scattering is the agent responsible for the sky's blueness.​
This seems to contradict your quote's suggestion that molecules are not responsible for the blue sky. Your quote seems to support the Einstein / Smoluchowski theory, and the references seems to say it's wrong. Or have I misinterpreted what you said? Or what the article says?

ADDED
Now that I have read the whole article, I get that any molecule of sufficiently small size, e.g. argon, would also make the sky blue, provided such molecules have no specific energy levels in its spectrum that would absorb blue light. So the reason that NO2 makes the sky blue is that NO2 makes up most (~80%) of the atmosphere. Since O2 makes up most of the remainder, I am curious. Does O2 also scatter light with smaller wavelengths like blue and violet?

Regards,
Buzz
 
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  • #25
BillyT
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Hi @BillyT:

Thanks very much for the reference. I am a bit confused by your quote relative to what I read in the reference.

The quote above seems to be saying that the Einstein / Smoluchowski theory about fluxuations explains the blue sky. The following is a quote from the reference.
Einstein stated that "it is remarkable that our theory does not make direct use of the assumption of the discrete distribution of matter" . . .
There is no dispute, then, about what really causes the sky to be blue? Is it "really" scattering by molecules? Or, if you want to be more precise, scattering by electrons. Even more precise, scattering by bound electrons: free electrons would not give us a blue sky. The preposition "by" indicates an agent, and molecular scattering is the agent responsible for the sky's blueness.​
This seems to contradict your quote's suggestion that molecules are not responsible for the blue sky. Your quote seems to support the Einstein / Smoluchowski theory, and the references seems to say it's wrong. Or have I misinterpreted what you said? Or what the article says?

ADDED
Now that I have read the whole article, I get that any molecule of sufficiently small size, e.g. argon, would also make the sky blue, provided such molecules have no specific energy levels in its spectrum that would absorb blue light. So the reason that NO2 makes the sky blue is that NO2 makes up most (~80%) of the atmosphere. Since O2 makes up most of the remainder, I am curious. Does O2 also scatter light with smaller wavelengths like blue and violet? Regards, Buzz
Note I said:
"EM wave is passing thur a dense medium the E-field will oscillate the bound electrons, and an accelerated electron produces radiation which combines with the incident radiation to change the net phase of radiation field. (Why the refractive index exist.*) "

So yes there must be some matter (especially bound electrons) to make a refractive index. O2, N2,NO2, & argon all supply "bound electrons" for the incident sunlight's E-field to shake.

If the bound electronic density were exactly the same every where, so would be the index of refration and there would be only forward (and bckward) scattering. What Einstein / Smoluchowski theory does is to note that all you need is variation of the index of refraction on wave lenght scale. - you don't really need to get into the facts relating to how the index of refraction is created, but that was the approach used to explain blue sky originally. Einstein / Smoluchowski skipped the real details about atoms having bound electrons, but they must be present to make the index of refracion, and if their density is varying on the wave length scale, then so will th index of refraction. - One is a high-level POV and the standard, older one is a more detailed POV, but they are essentially the same.

Sort of like the difference between: The IC motor makes torque that turns the wheels vs The vaporized gasoline volume and presure greatly increases as it oxidizes and they forces the pistons down ..."
 
  • #26
Danielvr
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Hi, a noob here, so please bear with me as I show my ignorance :)

Anyone who has ever spent some time in the depths of a coalmine knows that it can get pretty hot down in Earth's crust. (much) Deeper still, the core of our planet has even been calculated to be 10,800 degrees hot (as hot as the Sun's surface!). Some 90% of the energy required to maintain this temperature comes from the decay of radioactive elements, and if I understand correctly, it is eventually dissipated through (infrared?) radiation at Earth's surface. I read that the amount of heat caused by this radiation is almost the same as the total heat measured emanating from the planet (http://phys.org/news/2006-03-probing-earth-core.html). [Broken]

I wonder how all this figures in climate models. Climate scientists always seem to talk about light (energy) coming into our athmosphere from the Sun and then being converted into infrared light, which may or may not be trapped as heat in the athmosphere. But how about the (apparently) much more significant amount of infrared radiation coming from within - do today's models account for that at all?

I read the third section above, "Black Body Earth", which speaks of an -18 degree equilibrium, but I don't think I understand. Does that or does that not account for this heat from within the planet? I don't think it does, or else there wouldn't be places on Earth where it gets even colder in Winter. Or, maybe those colder places can be explained by regional differences in heat conductivity of the planet's mantle and crust?

Thanks in advance for any enlightening comments!
 
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  • #27
Bandersnatch
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I wonder how all this figures in climate models.
This is a negligible effect - while it gets very hot deep inside the planet, this heat is not well conducted towards the surface, and contributes only a minuscule part of the heat the surface receives from the sun.
See here:
https://en.wikipedia.org/wiki/Earth's_internal_heat_budget

I read the third section above, "Black Body Earth", which speaks of an -18 degree equilibrium, but I don't think I understand. Does that or does that not account for this heat from within the planet? I don't think it does, or else there wouldn't be places on Earth where it gets even colder in Winter. Or, maybe those colder places can be explained by regional differences in heat conductivity of the planet's mantle and crust?
The equilibrium temperature is the uniform temperature the whole surface would need to have in order to reradiate the specified amount of energy. You can have places during winter, at high latitudes, or simply at night with lower temperatures, as long as there are places with higher temperatures elsewhere.
I.e., it has to be compared with the averaged global temperature, not with specific individual instances.
 
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  • #28
Buzz Bloom
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O2, N2,NO2, & argon all supply "bound electrons" for the incident sunlight's E-field to shake.
Hi @BillyT:

Thanks for your post answering my question. I apologize for misinterpreting your
"EM wave is passing thur a dense medium the E-field will oscillate the bound electrons, and an accelerated electron produces radiation which combines with the incident radiation to change the net phase of radiation field."​
The technical aspects of E-field and change to the net phase of the radiation field are a bit over my head. In the context of the other quote, I was confused and thought you were saying the Einstein / Smoluchowski theory was a sufficient explanation.

Regards,
Buzz
 
  • #29
Jeff Rosenbury
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Light does not strike the earth evenly. Light at the equator (on average) hits at a 90º angle and has full effect. But toward the poles this drops off. Meanwhile heat energy radiates more or less uniformly.

This causes the poles to be cooler than the equator.

It also means melting icecaps decrease the albedo actually increases the blackbody radiation all other things being equal (which they are not of course).

This is complex because of the earths tilt on its axis affects the radiation patterns which interacts with the seasonal changes in albedo. Add clouds, water vapor, and seasonal temperature variations to get a poorly understood mix of feedback loops.

I know it's poorly understood since few people seem able to remember the sun doesn't shine in the arctic in the winter. I've been there. It's dark for months at a time.
 
  • #30
klimatos
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Jeff Rosenbury (post 29) said:
"Meanwhile heat energy radiates more or less uniformly." Not true, not even close. In order for this to be true, the two major radiators, the Earth's surface and the greenhouse gases would have to be almost isothermal. They are far from that. In actuality, both the wavelengths and the intensities of global terrestrial radiation vary tremendously from place to place and from time to time.

"This causes the poles to be cooler than the equator." Not true. The poles are cooler because the average angle of incidence of insolation is lower. This, in turn, is due to the fact that the Earth is a sphere.

" . . . actually increases the blackbody radiation . . ." The Earth emits no blackbody radiation. The Earth emits as a greybody. Many scholars put the coefficient of emissivity at about 0.95.

" . . .to get a poorly understood mix of feedback loops." It is actually fairly well understood, and whole library shelves of books and tens of thousands of scholarly articles have been written on the subject. I refer you to almost any Climatology 101 course.

I, too, have been to the Arctic (Thule, Greenland)--although during the summer months. We enjoyed continuous daylight.
 
  • #31
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Water has a 4 times higher heat capacity than CO2. Also the heat capacity of air is higher than of CO2. Similar is the course of the conductivity of heat with the lowest amount by CO2 (Datas of the Handbook of Chemistry and Physics 14 -24, 81. Ed., 2000/2001). There is a wide range of overlapping of the absorption of the infrared between water and CO2. Mostly in the atmosphere is a higher concentration of water than of CO2. Therefore water should have a higher radiation force than CO2. By most processes, where humans produce CO2 and many others much water gets in the atmosphere. The contribution of water of the greenhouse effect is about 60%. Therefor is to avoid the production of steam and heat in first line and not 0,04 % of CO2.
 
  • #32
mheslep
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Water has a 4 times higher heat capacity than CO2. Also the heat capacity of air is higher than of CO2. .
Heat capacity of the molecule is no more relevant to radiative forcing than is the heat capacty of glass in a greenhouse. The ability of CO2 to scatter certain infrared wavelengths is the issue, that is, to stop some infrared from escaping to space.
 
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  • #33
BillyT
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Water has a 4 times higher heat capacity than CO2. ...
I doubt that as H2O is not a linear molecule as 0-C-O is. Thus with an extra non-zero monent of inertia for storing energy in rotation H2O vapor should have slightly greater means of storing energy when heated say, 0.1 C degree.

Water has the two Hs on the same side of the O, with 105 degree angle between them and having partially lost their electron are with positive charged and the O is negatively charged. - Why water molecule has a permanet dipole and is such a powerful radiator and absorber IR compared to CO2 or more graphically, 0-C-O.
 
  • #34
Jeff Rosenbury
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Jeff Rosenbury (post 29) said:
"Meanwhile heat energy radiates more or less uniformly." Not true, not even close. In order for this to be true, the two major radiators, the Earth's surface and the greenhouse gases would have to be almost isothermal. They are far from that. In actuality, both the wavelengths and the intensities of global terrestrial radiation vary tremendously from place to place and from time to time.

"This causes the poles to be cooler than the equator." Not true. The poles are cooler because the average angle of incidence of insolation is lower. This, in turn, is due to the fact that the Earth is a sphere.

" . . . actually increases the blackbody radiation . . ." The Earth emits no blackbody radiation. The Earth emits as a greybody. Many scholars put the coefficient of emissivity at about 0.95.

" . . .to get a poorly understood mix of feedback loops." It is actually fairly well understood, and whole library shelves of books and tens of thousands of scholarly articles have been written on the subject. I refer you to almost any Climatology 101 course.

I, too, have been to the Arctic (Thule, Greenland)--although during the summer months. We enjoyed continuous daylight.
If the changing polar albedo effect is well understood, why do educated climatologists continually refer to a lowered polar albedo as absorbing more heat as a positive feedback?

The polar regions still have a low "average angle of incidence of insolation" (which is what I thought I said). That means little light coming in.

On the greybody thing, I did equate mistakenly albedo with thermal emissivity which is only a weak approximation. Albedo refers to the incoming light (visible and short wave infarred) and emissivity the outgoing light (long wave infarred). The two aren't the same, particularly for liquid water which is very complex (varying with surface conditions like wind/wave conditions). But the reciprocity principal means darker surfaces which absorb more also emit more greybody radiation (though that changes with wavelength).

Still, a look at outgoing longwave maps makes it clear that ice is a poorer emitter than water.

None of your comments change the point that polar regions emit more heat than they absorb, and polar albedo is not a clear positive feedback.
 
  • #35
BillyT
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If the changing polar albedo effect is well understood, why do educated climatologists continually refer to a lowered polar albedo as absorbing more heat as a positive feedback? ...
Beause it is. The greater heat absorbed by lower albedo melts more ice and snow. lowering the albedo even more as bare (darker) spots develope.

Also often there develope pools of water in low spots and the albedo there is lowered too, compared to the ice under the bottom of the pool.

It is even more comlex as for hundreds of years (or more) soot from forest fires has been falling on the snow but is soon covered when the snow is not melting, as it is today. Now hundreds or years of accumuated soot are being exposed - that drops the albedo from about 0.9 to 0.1 so a nine fold increase in the solar heating occurs - your will be hard pressed to find any more strong positive feed back than that.
This video discusses the "lakes" forming on the ice increasing solar absorption: http://cdn.rollingstone.com/feature/greenland/video/lakes.webm
If you watch this UCLA video / discusion below, not only will you better understand the positive feed back of melt water effects but also as their sensor moves over the ice, you will see it is quite dark in some areas with decades of accumulated soot are now exposed on the surface.
Even without opening this video, you can see the ice/ snow surface is far from the white you expect for snow. Again, I challenge you to find any Postive Feedback as strong as the one happening NOW in Greenland's ice sheet.
 
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