By Robert G. Brown, Duke University (elevated from a WUWT comment)
I spent what little of last night that I semi-slept in a learning-dream state chewing over Caballero’s book and radiative transfer, and came to two insights. First, the baseline black-body model (that leads to T_b = 255K) is physically terrible, as a baseline. It treats the planet in question as a nonrotating superconductor of heat with no heat capacity. The reason it is terrible is that it is absolutely incorrect to ascribe 33K as even an estimate for the “greenhouse warming” relative to this baseline, as it is a completely nonphysical baseline; the 33K relative to it is both meaningless and mixes both heating and cooling effects that have absolutely nothing to do with the greenhouse effect. More on that later.
I also understand the greenhouse effect itself much better. I may write this up in my own words, since I don’t like some of Caballero’s notation and think that the presentation can be simplified and made more illustrative. I’m also thinking of using it to make a “build-a-model” kit, sort of like the “build-a-bear” stores in the malls.
Start with a nonrotating superconducting sphere, zero albedo, unit emissivity, perfect blackbody radiation from each point on the sphere. What’s the mean temperature?
Now make the non-rotating sphere perfectly non-conducting, so that every part of the surface has to be in radiative balance. What’s the average temperature now? This is a better model for the moon than the former, surely, although still not good enough. Let’s improve it.
Now make the surface have some thermalized heat capacity — make it heat superconducting, but only in the vertical direction and presume a mass shell of some thickness that has some reasonable specific heat. This changes nothing from the previous result, until we make the sphere rotate. Oooo, yet another average (surface) temperature, this time the spherical average of a distribution that depends on latitude, with the highest temperatures dayside near the equator sometime after “noon” (lagged because now it takes time to raise the temperature of each block as the insolation exceeds blackbody loss, and time for it to cool as the blackbody loss exceeds radiation, and the surface is never at a constant temperature anywhere but at the poles (no axial tilt, of course). This is probably a very decent model for the moon, once one adds back in an albedo (effectively scaling down the fraction of the incoming power that has to be thermally balanced).
One can for each of these changes actually compute the exact parametric temperature distribution as a function of spherical angle and radius, and (by integrating) compute the change in e.g. the average temperature from the superconducting perfect black body assumption. Going from superconducting planet to local detailed balance but otherwise perfectly insulating planet (nonrotating) simply drops the nightside temperature for exactly 1/2 the sphere to your choice of 3K or (easier to idealize) 0K after a very long time. This is bounded from below, independent of solar irradiance or albedo (or for that matter, emissivity). The dayside temperature, on the other hand, has a polar distribution with a pole facing the sun, and varies nonlinearly with irradiance, albedo, and (if you choose to vary it) emissivity.
That pesky T^4 makes everything complicated! I hesitate to even try to assign the sign of the change in average temperature going from the first model to the second! Every time I think that I have a good heuristic argument for saying that it should be lower, a little voice tells me — T^4 — better do the damn integral because the temperature at the separator has to go smoothly to zero from the dayside and there’s a lot of low-irradiance (and hence low temperature) area out there where the sun is at five o’clock, even for zero albedo and unit emissivity! The only easy part is to obtain the spherical average we can just take the dayside average and divide by two…
I’m not even happy with the sign for the rotating sphere, as this depends on the interplay between the time required to heat the thermal ballast given the difference between insolation and outgoing radiation and the rate of rotation. Rotate at infinite speed and you are back at the superconducting sphere. Rotate at zero speed and you’re at the static nonconducting sphere. Rotate in between and — damn — now by varying only the magnitude of the thermal ballast (which determines the thermalization time) you can arrange for even a rapidly rotating sphere to behave like the static nonconducting sphere and a slowly rotating sphere to behave like a superconducting sphere (zero heat capacity and very large heat capacity, respectively). Worse, you’ve changed the geometry of the axial poles (presumed to lie untilted w.r.t. the ecliptic still). Where before the entire day-night terminator was smoothly approaching T = 0 from the day side, now this is true only at the poles! The integral of the polar area (for a given polar angle d\theta) is much smaller than the integral of the equatorial angle, and on top of that one now has a smeared out set of steady state temperatures that are all functions of azimuthal angle \phi and polar angle \theta, one that changes nonlinearly as you crank any of: Insolation, albedo, emissivity, \omega (angular velocity of rotation) and heat capacity of the surface.
And we haven’t even got an atmosphere yet. Or water. But at least up to this point, one can solve for the temperature distribution T(\theta,\phi,\alpha,S,\epsilon,c) exactly, I think.
Furthermore, one can actually model something like water pretty well in this way. In fact, if we imagine covering the planet not with air but with a layer of water with a blackbody on the bottom and a thin layer of perfectly transparent saran wrap on top to prevent pesky old evaporation, the water becomes a contribution to the thermal ballast. It takes a lot longer to raise or lower the temperature of a layer of water a meter deep (given an imbalance between incoming radiation) than it does to raise or lower the temperature of maybe the top centimeter or two of rock or dirt or sand. A lot longer.
Once one has a good feel for this, one could decorate the model with oceans and land bodies (but still prohibit lateral energy transfer and assume immediate vertical equilibration). One could let the water have the right albedo and freeze when it hits the right temperature. Then things get tough.
You have to add an atmosphere. Damn. You also have to let the ocean itself convect, and have density, and variable depth. And all of this on a rotating sphere where things (air masses) moving up deflect antispinward (relative to the surface), things moving down deflect spinward, things moving north deflect spinward (they’re going to fast) in the northern hemisphere, things moving south deflect antispinward, as a function of angle and speed and rotational velocity. Friggin’ coriolis force, deflects naval artillery and so on. And now we’re going to differentially heat the damn thing so that turbulence occurs everywhere on all available length scales, where we don’t even have some simple symmetry to the differential heating any more because we might as well have let a five year old throw paint at the sphere to mark out where the land masses are versus the oceans, and or better yet given him some Tonka trucks and let him play in the spherical sandbox until he had a nice irregular surface and then filled the surface with water until it was 70% submerged or something.
Ow, my aching head. And note well — we still haven’t turned on a Greenhouse Effect! And I now have nothing like a heuristic for radiant emission cooling even in the ideal case, because it is quite literally distilled, fractionated by temperature and height even without CO_2 per se present at all. Clouds. Air with a nontrivial short wavelength scattering cross-section. Energy transfer galore.
And then, before we mess with CO_2, we have to take quantum mechanics and the incident spectrum into account, and start to look at the hitherto ignored details of the ground, air, and water. The air needs a lapse rate, which will vary with humidity and albedo and ground temperature and… The molecules in the air recoil when the scatter incoming photons, and if a collision with another air molecule occurs in the right time interval they will mutually absorb some or all of the energy instead of elastically scattering it, heating the air. It can also absorb one wavelength and emit a cascade of photons at a different wavelength (depending on its spectrum).
Finally, one has to add in the GHGs, notably CO_2 (water is already there). They have the effect increasing the outgoing radiance from the (higher temperature) surface in some bands, and transferring some of it to CO_2 where it is trapped until it diffuses to the top of the CO_2 column, where it is emitted at a cooler temperature. The total power going out is thus split up, with that pesky blackbody spectrum modulated so that different frequencies have different effective temperatures, in a way that is locally modulated by — nearly everything. The lapse rate. Moisture content. Clouds. Bulk transport of heat up or down via convection. Bulk transport of heat up or down via caged radiation in parts of the spectrum. And don’t forget sideways! Everything is now circulating, wind and surface evaporation are coupled, the equilibration time for the ocean has stretched from “commensurate with the rotational period” for shallow seas to a thousand years or more so that the ocean is never at equilibrium, it is always tugging surface temperatures one way or the other with substantial thermal ballast, heat deposited not today but over the last week, month, year, decade, century, millennium.
Yessir, a damn hard problem. Anybody who calls this settled science is out of their ever-loving mind. Note well that I still haven’t included solar magnetism or any serious modulation of solar irradiance, or even the axial tilt of the earth, which once again completely changes everything, because now the timescales at the poles become annual, and the north pole and south pole are not at all alike! Consider the enormous difference in their thermal ballast and oceanic heat transport and atmospheric heat transport!
A hard problem. But perhaps I’ll try to tackle it, if I have time, at least through the first few steps outlined above. At the very least I’d like to have a better idea of the direction of some of the first few build-a-bear steps on the average temperature (while the term “average temperature” has some meaning, that is before making the system chaotic).
rgb
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Bart says:
January 14, 2012 at 11:19 am
wayne says:
January 13, 2012 at 8:00 pm
” Latitudinal, the spacing is every three degrees. “
To get equal area coverage of a sphere, you should not space uniformly by latitude, but by vertical height. That is why you had to do the workaround you describe at January 13, 2012 at 8:35 pm. Remember, the area element is cos(theta)*d_phi*d_theta, where theta is the latitude. Transform to a new set of coordinates with z = sin(theta) and the area element becomes dz*d_phi.
— — —
Hi Bart. Thanks. I did realize that slicing even heights for the latitudes, as computing areas for spherical caps or latitude bands, is much more elegant and really more proper mathematically. I was just not sure at the moment that there might not be some math later to give me problems, like finding the proper cell centers and areas. I just stayed with what I could visualize and knew I could get there correctly though it was the long way around. Heh, one of my past supervisors said I always did look at all problems upside down and backwards, always wondered if that is not why I ended up with all of the very hard programming tasks that none of the other forty didn’t even want to think of touching. But I will look into switching to that form now that I have a set of numbers to compare against for correctness. Thanks again.
Bart says:
January 14, 2012 at 11:19 am
wayne says:
January 13, 2012 at 8:00 pm
” Latitudinal, the spacing is every three degrees. “
To get equal area coverage of a sphere, you should not space uniformly by latitude, but by vertical height. That is why you had to do the workaround you describe at January 13, 2012 at 8:35 pm. Remember, the area element is cos(theta)*d_phi*d_theta, where theta is the latitude. Transform to a new set of coordinates with z = sin(theta) and the area element becomes dz*d_phi.
— — —
Hi Bart. Thanks. I did realize that slicing even heights for the latitudes, as computing areas for spherical caps or latitude bands, is much more elegant and really more proper mathematically. I was just not sure at the moment that there might not be some math later to give me problems, like finding the proper cell centers and areas. I just stayed with what I could visualize and knew I could get there correctly though it was the long way around. Heh, one of my past supervisors said I always did look at all problems upside down and backwards, always wondered if that is not why I ended up with all of the very hard programming tasks that none of the other forty didn’t even want to think of touching. But I will look into switching to that form now that I have a set of numbers to compare against for correctness. Thanks again.
Really very sorry Mods, it looks like there’s an italic problem in my post,
Myrrh says:
January 14, 2012 at 2:04 pm
It looks like I’ve put in an open italic where there shouldn’t be one, in front of the following :
“I asked you for proof of this. You haven’t actually come back with any proof. You don’t appear to have noticed that you haven’t come back with any proof or any detail of the experiments you say there’s an abundance of. ”
which comes after the blockquote beginning:
apologies, please would you fix.
[Reply: this is a common WordPress glitch. It’s not the fault of any commenters. Fixed now. ~dbs, mod.]
Brilliant, you worked it out before I’d even seen it. Many thanks.
davidmhoffer says:
January 14, 2012 at 11:57 am
What we need (I think) is to go back to as granular a copy of the data as possible, daily at a minimum if not hourly, and convert each and every value for T in each and every grid cell to P via SB Law.
Once that is done, THEN average P over the course of a day and year and the globe and graph THAT trend. We could then look at the change in ACTUAL DIRECTLY MEASURED w/m2 and see what change has occurred over the course of the temperature record. I will bet that if there is a significant increase in w/m2 at all, the trend will be well under what we see by trending T calculated from a global average of an annual average of a daily average which tells us NOTHING about the change in energy balance of the planet over the course of time because the only way to calculate that is from the trend in average P, the trend in T is USELESS and misleading.
———————-
I did something like this for one of the global SURFRAD stations monitoring solar radiation, downwelling IR, etc. over one 24 hour period.
http://www.srrb.noaa.gov/surfrad/index.html
The radiation flows do not match what the actual surface temperature is doing. The 2 metre air temperature changes by absolutely miniscule amounts on a per second basis while the radiation flows indicate temperatures should change by a wide margin.
http://img140.imageshack.us/img140/4109/tablemountainall.png
http://img12.imageshack.us/img12/3225/tablemountainnets.png
So, I did some further calculations and realized this is the general condition all over the world and there is even a greater out-of-balance for water surfaces, the tropopause etc. The changes in energy content are so small per second, that the radiation model does not work. We have to move down to another level to make sense of it.
Robert: “Integrating over a sphere is a pain in the ass, in other words (been there, done that).”
My first was just about two month’s ago, monte carlo, and the next a week ago, cellular, and I have to agree, many choices and tradeoffs and hard to change tracks once the train is moving. Definitely not excel territory. 🙂
Did you see the results from a first run per your laid out cases of hypothetical planets?
http://wattsupwiththat.com/2012/01/12/earths-baseline-black-body-model-a-damn-hard-problem/#comment-863484
(C&D has wrong descripts, skip on down a few comments for details)
LazyTeenager says:
January 12, 2012 at 6:32 pm
******************
Hey Lazy, try your demonstration at 3 AM and tell me how it works out. For both cases. Inquiring minds want to know the out come of the real world test.
P.S Are you really a checkout clerk?
[REPLY: No he’s not. Don’t you think, maybe, that is a snark too far? -REP]
George E. Smith; says:
January 14, 2012 at 3:02 pm
So the Planck formula, and the S-B result are useful starting points to investigate emission from real objects. No real object obeys either the Planck radiation law, or the S-B law. In particular, the earth’s moon, is not even approximately close to being able to absorb ALL EM radiation that falls on it so the moon doesn’t obey either Planck or S-B, but despite Myrhr’s declaration; it doesn’t “junk” S-B, nor does NASA
🙂 Well, whatever else it did, my choice of wording spurred you into an excellent, but what do I know?, explanation…
From which I can now see that NASA indeed did not junk SB and Planck, but used these imaginary constructs as a base for their calculations of the moon, much as those working with gases will begin with the ideal gas law which likewise is purely imaginary and which no real gas obeys and by stages add back all that is missing.
I think now this is of like ilk as I have previously pointed out elsewhere, that NASA has changed its story to fit in with AGW. NASA used to teach that the heat we feel from the Sun was the invisible thermal infrared, now it teaches that this doesn’t even reach us and has given the properties of heat to visible light, in the ‘shortwave in, thermal out’ meme of the energy budget that heats their imaginary land and oceans, which it can’t do in the real world.
So, what this is, is another example of the AGWScience Fiction meme producing department’s fiddling with physics to promote their truly fictitious ‘greenhouse gas warming’ and generally dumb down the education system, as it follows the same pattern I found in discovering that ideal gas law was used as proof that carbon dioxide diffused spontaneously into the atmosphere and could accumulate for hundreds and thousands of years regardless that its real gas weight, attraction etc. precluded such a thing happening.
So, it seems, that AGWSF uses SB&Planck in raw form without any tweaking for reality around us just as carbon dioxide, oxygen and nitrogen are designated the fictitious ideal gas by them and used without tweaking to explain such things as ‘well-mixed’. It’s not that NASA junked it then, but that now NASA teaches something different, hence all the confusion.
From the confusion by using the basic ideal gas, we now have taught that the atmosphere around us is empty space and these molecules zip around at vast speeds thoroughly mixing, that is, ideal gas without volume, gravity, attraction etc. And a whole generation of children have been educated into thinking that it is the atmosphere around us, because it is taught so, as the PhD in physics teaching the subject explained to me. Which is why I call it an imaginary world, because based on the imaginary ideal gas law without any tweaking.
But it has to be that to get rid of the Water Cycle..
[This teacher was first adamant that carbon dioxide could not separate out from the atmosphere in which it had spontaneously thoroughly mixed as per ideal gas law. I couldn’t quite believe that he thought this, let alone taught it, (he said he’d fail me for my views if taking the exams he set and marked), so I proposed a thought experiment to make sure I really understood what he was saying.
At first he adamantly denied that carbon dioxide could separate out of the atmosphere to pool at the ground after becoming thoroughly mixed as per AGWSF, but after I had shown him real world examples of carbon dioxide displacing air to pool on the ground, breweries etc., he conceded that it did ‘come down’ but created a novel explanation for it, that somehow it brought down the whole package of air it was in with it…
Anyway, the thought experiment was simply to imagine a room in which an amount of carbon dioxide had pooled on the floor and what happened next without any change to the conditions in which the carbon dioxide had pooled, no work done such as opening windows or putting on a fan.
He said that the carbon dioxide would rise spontaneously as per ideal gas and quickly diffuse into the atmosphere so thoroughly mixing that it would take a great amount of work to separate it out again.
I said it would remain pooled on the floor because it was heavier than air and so would not spontaneously rise into the atmosphere of the room without any work done.]
So, OK, how does AGWSF use the imaginary basic SB&Planck and what kind of world does that create if based without tweaking on this imaginary blackbody…?
Bill Illis;
The radiation flows do not match what the actual surface temperature is doing. >>>
Thanks for those graphs! Actually, that’s pretty much what I would expect to see. In fact, if you sift through N&Z, that’s what they say to expect as well.
The surface temperature can’t possibly correlate well to insolation and net radition in the short term. The surface temperature is modified by net radiation, conductance, convection, and most importantly, heat capacity.
Your graphs show exactly why applying “average P” to arrive at “average T” for the earth makes no sense what so ever. The theoretical black body temperature of the earth is governed by P which varies by hundreds of w/m2 every day. If earth was a perfect black body that would mean that T would also vary by in excess of 300 degrees K every day. Instead, it varies by only a few degrees because heat capacity establishes a central range that the fluctuation can never stray very far from, and that value is further mitigated by convection and conduction which must speed up due to higher insolation and fall of during periods of low insolation (hence night time cooling slows down to a crawl even though insolation is zero and the theoretical black body of earth would be approaching absolute zero at night).
Earth surface as a whole must redistribute energy via ALL possible mechanisms until thermal equilibrium is reached. If the GHG’s were not present, the redistribution must still end up in the same place, just more gets done by other means. It doesn’t matter to the end state if the GHG’s are there or not, the redistribution stabilizes at equilibrium, it just perhaps doesn’t get there as quickly.
But changing the mass of the atmosphere DOES change things. It changes the heat capacity of the atmosphere and it changes conductance and convection. These factors in turn govern the distribution curve that results in equilibrium, and hence the observed average temperature canbe calculated from simply surface pressure and insolation.
Dr. Brown,
The solid Earth does some of the integrals:
ftp://ftp.iers.org/products/eop/long-term/c04_08/iau2000/eopc04_08_IAU2000.62-now
ftp://ftp.iers.org/products/geofluids/atmosphere/aam/GGFC2010/AER/
“Apart from all other reasons, the parameters of the geoid depend on the distribution of water over the planetary surface.” — N.S. Sidorenkov
Sidorenkov, N.S. (2005). Physics of the Earth’s rotation instabilities. Astronomical and Astrophysical Transactions 24(5), 425-439.
http://images.astronet.ru/pubd/2008/09/28/0001230882/425-439.pdf
Gross, R.S. (2007). Earth rotation variations – long period. In: Herring, T.A. (ed.), Treatise on Geophysics vol. 11 (Physical Geodesy), Elsevier, Amsterdam, in press, 2007.
http://geodesy.eng.ohio-state.edu/course/refpapers/Gross_Geodesy_LpER07.pdf
http://geodesy.geology.ohio-state.edu/course/refpapers/Gross_Geodesy_LpER07.pdf
–
You mention the coupling of evaporation & wind.
Visualizing Shared Patterns:
1. Column-integrated Water Vapor Flux with their Convergence:
http://i51.tinypic.com/126fc77.png
Compare 1 with 2 & 3:
2. Near-Surface (850hPa) Wind:
http://i52.tinypic.com/nlo3dw.png
3. Near-Surface (850hPa) Wind — Polar View:
http://i54.tinypic.com/29vlc0x.png
Compare with wind-driven ocean gyres:
4. Wind-Driven Ocean Currents:
http://upload.wikimedia.org/wikipedia/commons/6/67/Ocean_currents_1943_%28borderless%293.png
Note the place of 2 & 3 (near-surface wind) in 5:
5. Zonal Wind Vertical Profile:
http://i51.tinypic.com/34xouhx.png
For another perspective on 5’s westerlies (mid-latitude ~200hPa spots), see 6 & 7:
6. 200hPa Wind:
http://i52.tinypic.com/zoamog.png
7. 200hPa Wind — Polar View:
http://i52.tinypic.com/cuqyt.png
Credit: Climatology animations have been assembled using JRA-25 Atlas [ http://ds.data.jma.go.jp/gmd/jra/atlas/eng/atlas-tope.htm ] images. JRA-25 long-term reanalysis is a collaboration of Japan Meteorological Agency (JMA) & Central Research Institute of Electric Power Industry (CRIEPI).
Regards.
davidmhoffer: Don’t be silly, I said “at equilibrium” half a dozen times in first comment.
Ah. All of your subsequent comments apply to equilibria. I missed that.
Since the earth has never been in equilibrium and isn’t now, and isn’t projected to be (that is, no one has supplied any kind of proof that the dynamic system of the earth climate is capable of equilibrium), then you are not arguing about anything that actually exists. Dr. Brown’s post summarized the reasons why the climate system is not known accurately enough to predict what effects CO2 accumulation will have, and your comments have all been beside the point.
davidmhoffer: 4. Averaging T to understand a change in energy balance is hopelessly useless.
I wrote that as well. However, computing the spatio-temporally averaged mean T is useful for estimating the net warming or cooling over a given time span, such as since 1850. Estimating the effect of CO2 on past or future warming isn’t possible right now. Claiming that you know that CO2 has no effect in the recent past, and can’t have an effect in the near future, if that’s what you have been trying to establish, is not substantiable.
Setpic Matthew;
I wrote that as well. However, computing the spatio-temporally averaged mean T is useful for estimating the net warming or cooling over a given time span, such as since 1850. >>>
No! it does not! The energy balance can be completely unchanges down to the last joule but a change in the distribution (cooler tropics warmer arctic for example) will show up as a positive temperature trend if you average T, but the total watts in and the total watts out haven’t changed.
If the earth is “warming up” and by “warming up” we mean retaining more energy than it is losing, averaging T may STILL give you the wrong answer. It is possible to have an average of T that is increasing while the energy retained is decreasing and vice versa, which I’ve shown via example calculations on several occassions.
You CANNOT average T and calculate P, nor may you average P and calculate T! Doing so arrives at meaningless numbers from which only incorrect conclusions may be drawn.
Septic Matthew;
Dr. Brown’s post summarized the reasons why the climate system is not known accurately enough to predict what effects CO2 accumulation will have, and your comments have all been beside the point.>>>
If you can say that, then I’m sorry to say that you’ve completely missed the point.
Point him to the definition of an intensive variable.
Mark
This has been a very instructive interchange but it appears to be petering out -> finally! I can rarely spend much time reading blogs and comments except late at night and I could not spend the time on this one to get more involved. However, the experience has sharpened my understanding of the problems with the debate itself, not just the science. There are too many terms, labels, pre-accepted conclusions, incomplete physics, improperly applied mathematical relationships, shortcuts, and limited models upon which people on both sides, both amateur and professional, base their arguments. Even more noticeable is that some of these things apparently mean different things to different people participating in the same discussion. Nevertheless, if the foundation of a model is weak, no amount of arguing will make the result true, even if the argument is won. In seems that winning the argument is becoming more important than the truth of understanding. This “debate club” mentality has completely overwhelmed political discussion in the US and is now entering the realm of science as well. Let’s get it right, not win arguments. I was pleasantly surprised at the mostly calm interchange of comments to this post but that may have been due to the moderator! That said, I am no saint myself. I can sometimes get it embarrassingly wrong and I have in the past publicly crashed and burned!
I am not being critical of the commenters to this blog, which no doubt range from interested non-technical people to highly educated scientists. Obviously Dr. Brown in his posting and everyone who commented are passionate and want to get it right. I am critical more of the climate community itself which supports many “facts” which would not pass muster in a 2nd year undergraduate engineering class (Yes, I am an engineer!) or directly contradict 100 years of research by geologists, a similarly well developed scientific discipline. For instance, pick up a geology guide at the Yellowstone National Park Visitors’ Center and read what the Grand Teton glaciers did during the Little Ice Age that did not occur.
Below are my responses to those who thought enough of my posts to comment on them for better or worse:
Tom Folkert:
In answer to my question about where there is surface cooling to counter the Arctic heating in order to make the average rate of change, Tom said:
>>The short answer is that that place is the “top of the atmosphere”.
>>Adding more CO2 raises the average altitude for emission of IR. Higher
>>altitudes are colder, so the CO2 will emit less IR.
Thanks for pointing out that I was not thinking 3-dimensionally about the temperature problem. I am a pilot and I should have known better. Nevertheless, I do not think you can claim that surface heating can be countered by top-of-the-atmosphere cooling to make the average. First, the average is calculated only from surface temperatures and is not integrated over the entire height of the atmosphere. It is correct to compare top-of-the-atmosphere photonic emission with surface temperature in a balance equation? They are two different things. Secondly, since the entire argument about radiative balance means that energy reaching the surface must also leave the earth’s emissions sphere at the same location it entered, you cannot make the argument that the top of the atmosphere cools as the surface heats up. That implies, from an engineering point of view, that the thermal conductance of the atmosphere decreases as its temperature goes up. That would form an unstable system resulting in the surface temperature running away while the top of the atmosphere freezes, clearly something that is not physically realizable. Third, if the equal-intensity emission surface rises in altitude, it by definition still emits the same amount of energy so the energy transfer has not changed. You actually can make the argument that the atmosphere got hotter. If you mark a coordinate on the emission surface above my favorite spot over the equator, after that surface rises to a higher altitude because of heat absorption by carbon dioxide in the lower atmosphere, the temperature at my fixed coordinate, at the original altitude but now below the emission surface, most likely increased. The upper atmosphere got hotter, not cooler. Finally, the photonic energy emitted by the carbon dioxide comes from the change in a vibrational state of the molecule, not from a change in its temperature, which as you know is the linear velocity of the molecule. That the carbon dioxide is cooler (lower linear velocity) when the iso-emission surface rises should not change the emission statistics for that vibration state. Am I wrong about that?
The iso-emission surface is a mathematical concept that I suspect was created to be a summary of a variety of mechanisms, a summary that could then be used to make communication more efficient. It is my observation that it has created confusion. I prefer to reduce things to the absolute. For instance, the iso-emission surface is often described as a sphere representing the altitude at which 240W/m2 is emitted from the earth. It is a sphere only if you use the average emission value. In reality it should touch the earth’s surface at the poles (assuming that the earth’s axis is not tilted and energy balance is achieved only by local outbound radiation) and reaches very high in the atmosphere over the equator where the incoming sunlight reaches 1366W/m2. If you do construct a spherical emitting surface at some fixed altitude above the earth, it will have a different temperature at each point and will emit at a different intensity at each point. That sphere will form a potential surface. The derivative of a potential surface, that is its local slope, defines the “force” between two points on that surface that transfers energy from one point to the other. In non-technical terms, the same thing happens that I described in my first post: at a specified altitude, the potential surface will have slope from the equator to the poles, forcing energy to flow to the poles to be radiated .
Tom, your observation has made clear to me a very important point I had not considered before and have never seen mentioned: Energy comes into the system photonically and can only leave the system photonically because hot gas molecules cannot leave Earth’s gravitational field like steam leaves a boiling pot of water. No matter how the sloar energy bounces around inside our system, to leave our system it must excite a state in a molecule, a liquid, or a solid that can convert it back to a photon.
David M Hoffer:
>>If you scroll back upthread to Joe’s first comment on the issue,
>>you’ll see that he pretty much nailed exactly where the “break even”
>>latitude is just from theoretical physics.
Thanks!
Robert Brown of Duke:
My model did not get more complex than a non-rotating black body plus the differential heat load on the globe due to the local slope of the globe. I did not begin to add the levels of complexity you mentioned in your post but already the boundary limits of my world differ from the climate models. You were very right!
Tom Folkert
>>So if you had a container with a mixture of hot N2 & CO2,
>>the N2 could lose some energy via radiation, but it would lose
>>more by transferring energy to CO2 via collisions, and then
>>having the CO2 radiate the energy to space.
I think you are right about nitrogen’s rate of radiation. I ran across information about carbon dioxide lasers when researching another topic and found that they are like a fluorescent light bulb but with nitrogen and carbon dioxide inside the tube. The electron current hits the nitrogen molecules and excites them. The nitrogen molecules do not readily emit but do lose their energy to the carbon dioxide molecules by collision. The carbon dioxide emits the energy as photons, albeit at the famous 10 micron wavelength!
This description raises the following question: if the atmosphere is populated with excited carbon molecules and IR energy of the same wavelength travels only upwards from the surface, is there any stimulated emission from the excited carbon dioxide? I think the yes or no of this question is determined solely by how long the atmospheric carbon dioxide stays excited before colliding with another molecule and de-energizes. If the answer is yes, it would mean that a cubic meter of atmosphere does not emit its IR radiation in all directions equally but instead preferentially in the direction of space. It may be only a very small percentage or even a fraction of a percent but it would change the outcome of the energy balance equations for the green house effect.
Good night everyone! I enjoyed the discussions!
davidmhoffer: No! it does not! The energy balance can be completely unchanges down to the last joule but a change in the distribution (cooler tropics warmer arctic for example) will show up as a positive temperature trend if you average T, but the total watts in and the total watts out haven’t changed.
If the earth is “warming up” and by “warming up” we mean retaining more energy than it is losing, averaging T may STILL give you the wrong answer. It is possible to have an average of T that is increasing while the energy retained is decreasing and vice versa, which I’ve shown via example calculations on several occassions.
Ah yes, I have written myself that the distribution can change. If that was the point that I missed, I do agree that I missed it.
dmh, it looks like I shall have to reread everything that you wrote.
dmh: From Joe’s comment upthread:
“The primary mechanism responsible for maintaining the incredible 4-billion-year stability of our system must be the physical transfer of heat from the equator to the poles where it is radiated away.”
Precisely. The laws of thermodynamics require this statement to be true. The amount of net loss of energy in high latitudes must balance, to the last photon, the net absorption in low latitudes. Not only does this satisfy the laws of thermodynamics, it also explains why the concentration of GHG’s in the atmosphere is, in fact, immaterial.
The “stability” is not “equilibrium”, and includes sufficient variation to include climates that are hostile to human civilization. The laws of thermodynamics do not require the balance that you claim, as they only require that all the heat be accounted for; you can have transient net influxes and net efluxes.
Septic Matthew;
The “stability” is not “equilibrium”, and includes sufficient variation to include climates that are hostile to human civilization. The laws of thermodynamics do not require the balance that you claim, as they only require that all the heat be accounted for; you can have transient net influxes and net efluxes.>>>>
For the second time, please note the number of time I said “at equilibrium” in my comments.
Further, please note the title of the thread which begs the question as to how to calculate the blackbody temperature of the earth. Calculating blackbody is, by definition, a calculation of the equilibrium state. Further, the laws of thermodynamics must absolutely be satisfied under ALL circumstances, they don’t get an exemption for transient states. These include transient states in which some of the energy is being asorbed (for example) by heat capacity during the transient state. The numbers must balance to satisfy the laws of thermodynamics even when some of the numbers come from processes that do not exist in equilibrium states.
I guess it depends what you are setting out to prove as to whether all the effort is worthwhile. perhaps Hans Jelbring made a smart move with his model atmosphere and isometrically heated planet surface in his 2003 paper:
http://tallbloke.wordpress.com/2012/01/01/hans-jelbring-the-greenhouse-effect-as-a-function-of-atmospheric-mass/
At least if he is right you won’t have to worry about including radiative effects of GHG’s
OK, here’s my first referee comment to this paper (which I would not have accepted). He creates an artificial adiabatic planet. He asserts that the atmosphere is in thermal equilibrium. He explicitly states indeed that the average kinetic energy of the molecules of the atmosphere is a constant throughout. Then he proceed to derive a lapse rate for this atmosphere.
No. Equipartition states that T is proportional to the internal energy per molecule. If the atmosphere is in equilibrium, it is at a uniform temperature T. Period. In fact, this is the zeroth law of thermodynamics, the definition of thermal equilibrium.
http://en.wikipedia.org/wiki/Zeroth_law_of_thermodynamics
See especially the discussion on thermal equilibrium wherein it is proven that if two systems i and j (or two parts of one system) are in thermal equilibrium, T_i = T_j. End of story. So there can be no adiabatic lapse rate for a gas in thermal equilibrium, that is, a gas in a container, no matter how you apply external forces or accelerate the system to create a pressure gradient. It is entirely possible to have a gas in a vertical column where the pressure drops off with height all at the same temperature, and if the gas is in equilibrium it is at the same temperature.
Thus I don’t understand Jelbring’s argument from the beginning. He asserts equilibrium, but then imposes an adiabatic lapse rate in temperature that contradicts equilibrium, at least thermal equilibrium. To put it another way, the molecules of gas at the cooler temperature have to have a completely different maxwell-boltzmann distribution of temperatures, do they not, with completely different average speeds? They are not in thermal equilibrium.
If you have a thermal gradient in any system with a mechanism for the transfer of energy, you have a heat transfer from hotter to colder in any steady state solution to the problem. This too is a law of thermodynamics (or nearly so) — the second law. Heat is irreverisbly transferred only from hot (higher T) reservoirs to colder (lower T) reservoirs, not the other way around. If it ever goes the other way around, you can create heat engines that violate the second law. In the case of your adiabatic lapse rate atmosphere, one could just put an electrical thermocouple between the upper and lower atmosphere and generate electricity forever, right? Gravity somehow causes the temperature to re-equilibrate with a gradient, but the gradient will drive a current that makes a resistive wire hot in one place.
So Jelbring’s paper egregiously violates the laws of thermodynamics on page 2 or 3. Not just a little, mind you — a lot. Enough so that I can build a perpetual motion machine quite easily (and, mentally, just did).
The only way you can have an adiabatic lapse rate is if you have a differential delivery of heat to the upper and lower surfaces that maintains them at different temperatures. You have to have a constant flow of heat, and the system is never in global thermal equilibrium, quite the contrary.
This is where any static model for a thermal gradient to the atmosphere fails before it gets off of the ground. The thermal gradient is not due to statics. It is caused by heat flow, and if the lapse rate is a vertical decrease in temperature, that means that heat is always flowing from the hot lower to the colder upper.
“Thermodynamics” is tricky for open systems. You can talk about local thermal equilibrium — a sufficiently large volume of well mixed air or water might have a “temperature” relative to a thermometer placed in contact only with that limited volume — but if there is heat flow and thermal gradients, it isn’t in thermodynamic equilibrium globally. Theoretical computations of e.g. adiabatic lapse rate do not assume global thermal equilibrium, they assume local thermal equilibrium and adiabatic expansion consistent with the pressure. I’m not certain that they are completely consistent, but perhaps they are sufficiently so, given the relative timescales in question.
I have much the same issues with N&Z, BTW. I’d have to see how the temperature differential is maintained that doesn’t involve differential radiative cooling rates (greenhouse) somewhere in the system and that doesn’t allow me to build a perpetual motion machine across the thermal gradient. Gravitation is reversible and can never replenish energy in a steady state system, provide additional free energy to drive an engine. You need heat input from the Sun, and you need direct atmospheric cooling to keep the top of the atmosphere cooler than the surface.
I think that both Jelbring and N&Z have things upside down, especially with regard to venus. Venus’s atmosphere is optically thick in pretty much all relevant wavelengths. The atmosphere itself reaches thermal equilibrium with a thermal gradient.
rgb
Robert Brown says:
January 16, 2012 at 9:37 am
Thanks, Robert, With great trepidation, I must disagree with you.
Consider a gas in a kilometre-tall sealed container. You say it will have no lapse rate, so suppose (per your assumption) that it starts out at an even temperature top to bottom.
Now, consider a collision between two of the gas molecules that knocks one molecule straight upwards, and the other straight downwards. The molecule going downwards will accelerate due to gravity, while the one going upwards will slow due to gravity. So the upper one will have less kinetic energy, and the lower one will have more kinetic energy.
After a million such collisions, are you really claiming that the average kinetic energy of the molecules at the top and the bottom of the tall container are going to be the same?
I say no. I say after a million collisions the molecules will sort themselves so that the TOTAL energy at the top and bottom of the container will be the same. In other words, it is the action of gravity on the molecules themselves that creates the lapse rate.
What am I missing here?
Many thanks for all your contributions to the discussion,
w.
Robert, one further thought. You might think that this would be a violation of the conservation of energy, because you can use that temperature difference to do work.
But if you do so, you remove energy from the tall container, so it is not a perpetual motion machine. You still have to add energy to the system to get continuous work out of it.
w.
davidmhoffer: Further, the laws of thermodynamics must absolutely be satisfied under ALL circumstances, they don’t get an exemption for transient states.
I didn’t say that the laws of thermodynamics were ever not satisfied. I said that Earth may transiently receive more energy from the sun than it radiates to the rest of space.
No matter how many times you used the word “equilibrium” it is clear from your writing that you address non-equilibrium cases — unless you are asserting that the earth has actually been in equilibrium.
Further, please note the title of the thread which begs the question as to how to calculate the blackbody temperature of the earth. Calculating blackbody is, by definition, a calculation of the equilibrium state.
This is one of the ways in which the calculations can not be assumed to be accurate for the earth; that is, inaccuracies of at least a few percent are introduced by using equilibrium approximations to model processes on earth. The temperature of the earth, its mean and its particular temperature in each place, fluctuates. This indicates that the earth system is not in thermal equilibrium. Sometimes it is unclear whether you really mean “equilibrium” in place of “steady state” or “within the basin of attraction”. In equilibrium there is no heat flow; in steady state the inflows equal the outflows, for the whole earth as well as for every 3D region of it; within the basin of attraction you can have heat flows (and temperature changes) within a range (like claimed for earth above) without exact periodicity, and without steady-state for any duration of time. Strictly speaking, we can not know for sure that we are within a basin of attraction, but the earth climate system is surely never at equilibrium or in steady state.
Robert Brown repeated a common misunderstanding at 1008, 13Jan2012 with: Anything with a temperature radiates.
The better way to write it is: The Stefan-Boltzmann equation assigns a radiation temperature vector equivalent and proportional to the radiation intensity vector of any black body that radiates.
Horses come before carts. S-F simply assigns a measurable radiating temperature vector to a mathematically defined black body radiating vector.
George E Smith got it right at 1502, 14Jan12. He does not confuse radiation temperature vectors with thermal temperature scalars.
The language in the GHG theory debate is appallingly confusing to laymen. As Prof Richard Lindzen, MIT Meteorologist, is so fond of saying, the science of meteorology is indeed in its infancy.
This professional chemical process control system engineer says the quality of the multivariable dynamic matrix model relating independent inputs like fossil fuel combustion rate and solar incidence to atmospheric CO2 content and both thermal and radiant temperatures of Earth’s surface and atmosphere are utterly inadequate for designing Earth’s thermostat. Control engineers developed the commercial art of rigorous and empirical modeling to suit an engineering purpose in the 1970’s; GHG theorists are apparently oblivious to its existence. I proved this chemical process system is not measurable, unobservable and uncontrollable in 1997.
I have not found a rebuttal to my WUWT essay of 0718, 13Jan12 explaining why the 33C GHG effect is a whatchamacallit.
@Willis
@Robert
If you gentlemen will pardon me for intruding between you, I don’t think there’s a conflict between the two notions, and I’m referring here to the real world, not the one that Willis has going on the other thread. ☺
Anyway, whether by conduction or convection or intelligent design (!), if the column of air has arrived at a stable state with hotter stuff at the bottom and cooler stuff at the top, it will only stay in that condition if continually heated from the bottom, thus satisfying Robert’s need for a perpetual energy flow in order to maintain the gradient. If the column didn’t have that energy input, it would gradually lose energy and settle to the ground until limited by increasing density. I don’t quarrel about the system not being in TD equilibrium, it won’t be, but it will be ‘stable’ as long as the energy keeps coming.
By the way, I have greatly enjoyed Robert’s many contributions of late. I am about to start back to classes a week from today, and as luck would have it (well, careful planning actually), I’m teaching a section of Thermodynamics. I may “poach” some of the turns of phrase that I’ve seen you use. ☺
/dr.bill