The basic mathematical model of heat conduction/diffusion takes the following form with Q(x) rate of heat flow at point x (in one dimension for simplicity) and U(x) temperature

- Q(x) = - kappa (U(x+h)-U(x))/h (Fourier's Law)
- (Q(x) - Q(x-h))/h = f(x) (1st Law of Thermodynamics = energy balance)

where kappa is a positive coefficient, f(x) a heat source and h is a small (computational or physical) mesh size.

Setting kappa = 1 and f = 0 for simplicity (imposing some inhomogeneous boundary conditions), we obtain the following equation for U:

- (-U(x+h) + 2U(x) - U(x-h))/hh = 0. (1)

Suppose now that we forget the derivation of this formula and write it as:

- 2U(x)/hh = (U(x+h) + U(x-h))/hh (2)

with the following interpretation

- (U(x+h) + U(x-h))/hh = "incoming temperature" from x+h and x-h to x
- 2U(x)/hh = "outgoing temperature" from x to x+h and x-h.
- "incoming temperature" = "outgoing temperature".

Then "incoming temperature" would be like "backradiation". So far, so good.

But what is then the physical meaning of "incoming/outgoing temperature"? Is there really a physical "flow of temperature"? Is "flow of temperature" the same as "flow of heat"?

No! To understand this we rewrite the equation in computational form suitable for

explicit iteration from U(n,x) to U(n+1,x) with U(0,x) given:

- U(n+1,x) = U(n,x) - alpha (-U(n,x+h) + 2U(n,x) - U(n,x-h))/hh (3)

where alpha is a positive parameter to choose. This corresponds to time-stepping towards an equilibrium solution.

With alpha = 1 and U(n+1,x) = U(n,x) we then get back (1) along with the interpretation (2). But alpha =1 is too big for convergence of the iteration: Convergence requiers that that alpha ~ hh; more specifically the iteration convergence if alpha is smaller than hh/4, say alpha = hh/8. In this case (3) takes the form

- U(n+1,x) = U(n,x) - 1/8 (-U(n,x+h) + 2U(n,x) - U(n,x-h)) (4)

or

- U(n+1,x) = U(n,x) - h/8 (Q(n,x) - Q(n,x-h)). (5)

We can now interprete the equations (4) or (5) as expressing a form of physics effectively

establishing an equilibrium solution by iteration or time-stepping. In particular we can

interprete (5) as a (mild) relaxation decreasing U(n+1,x) as compared to U(n,x) if Q(n,x) is bigger than Q(n,x-1), that is if more flow is going out than in at x.

We next compare with (2):

- 2U(x)/hh =(U(x+h)+ U(x-h))/hh

with "out/inflow of temperature" 2U(x)/hh and (U(x+h)+ U(x-h))/hh. We now come to the punchline: There is no physics behind this massive "flow of temperature" because as h gets smaller and smaller, the "flow of temperature" increases without bound.

We can thus speak about flow of heat, but not "flow of temperature" with enormous amounts

of temperature flowing in and out (as h gets small), because this would corresponds to an unstable unobservable unphysical process.

In short: heat flows from hot to cold but there is no physics corresponding to a flow from cold to hot analagous to "backradiation". Heat can radiate from a hot to a cold body, but not from a cold to a hot. In other words:

- Backradiation corresponds to an unstable unobservable unphysical process.

You can make an analogy in economics with massive taxes (going to infinity) from the people and massive benefits (going to infinity) back to the people. Great idea, but unfortunately unstable!

Temperature is not an intensive quantity and cannot "flow"; as far as I know there is no informed argument based on the "flow" of temperature so the present article appears to be aimed at a straw man.

SvaraRaderaIn the net, radiative energy can only go from a hot body to a cold body. No argument there. But if there are two situations, one with a warm body and a cold body, and another with the same two bodies plus a body of intermediate temperature, the net radiation from the warm body will decrease in the second scenario relative to the first.

There is nothing mysterious or arcane about it. All we are saying is that a non-IR-transparent atmosphere constitutes a body of intermediate temperature between that of the surface and that of space. Decreasing the IR transparency of the atmosphere increases the influence of this body. I fail to understand all the effort to make this difficult in principle. The calculations are difficult but the principle is trivial.

Claes,

SvaraRadera"Realclimate" just put up a post trying to explain the IPCC un-physics you may want to have a look at:

http://www.realclimate.org/index.php/archives/2010/07/a-simple-recipe-for-ghe/

Yes backradiation is a straw man. Of course introducing a third body

SvaraRaderaof intermediate temperature in bewteen two bodies, can decrease

the net radiation from the hot body, like another blanket. Still without

any backradiation.

Seems to me you have indirectly derived the 2nd law of thermodynamics, which states the total entropy must increase and heat doesn't flow from cold to hot (without work input).

SvaraRaderaThe AGW "hot spot" would require a decrease in total entropy and thus violate the 2nd law.

"Of course introducing a third body of intermediate temperature in bewteen two bodies, can decrease the net radiation from the hot body, like another blanket. Still without any backradiation."

SvaraRaderaWell, I am glad you see this, since it's the only point you need to understand how the greenhouse effect works.

Still, your model is peculiar.

Imagine two bodies of very similar temperature at a finite distance apart, subject to small stochastic forcings. How do they negotiate which is the donor and which the recipient at any given instant? Suppose there is relative acceleration so that the instant is not well-defined?

The only way I can imagine to make sense of this is that each body radiates independent of the other. The net transfer is from the warmer to the cooler, but each is radiating constantly. Anyway, this interpretation is consistent with the actual phenomena. Yours requires a very complex flow of information among inert objects as they negotiate which is the donor, which the recipient, and what the rate of exchange should be.

I agree that the nature of the negotiation is unclear, but so is the backradiation, in particular through a continuum like the atmosphere.

SvaraRaderaThe negotiation in conduction/diffusion is local and thus simpler to visualize. Negotiation at distance is not so easy to understand, but

all "action at distance" is difficult. Believing that the action is transmitted

by "particles" is a cheap way out of this; it remains to explain the nature of the particles and how they travel.

Total pressure = Mass aloft X gravitation

SvaraRaderaPressure from virtual photons + Pressure from real photons = Total pressure

I would say it is as simple as that, trivial in other words.

What is virtual photon? What is a real photon? How do they travel?

SvaraRaderaTrivial?

Well, you can imagine it as a computer game where some of the agents can through boomerangs while some other can only punch you with their fists. However, all processes are governed by the law of conservation of momentum which tells us that the boomerangs does not introduce anything qualitatively different, only speed up the process towards thermal equilibrium.

SvaraRaderaGravitation is an example of apparent action at distance. To believe that

SvaraRaderathe action is performed by graviton particles is naive, as far as I can understand. Another approach is suggested in the knol http://knol.google.com/k/the-hen-and-the-egg-of-gravitation#

I didn't say anything about gravitons. I just assume that gravity causes pressure which must be compensated by other forces if the atmosphere is not to collapse.

SvaraRaderaThe particles are photons and they travel in the usual photonic way, of course.

SvaraRaderaIt seems to me that you are all convinced that the atmospheric problem is essentially a problem about heat conduction from earths surface to some mysterious non-existing radiating shell placed at some arbitrary height in the atmosphere, and which in some mysterious way maintains a temperature substantially lower that the earths surface temperature. Could it on the other hand be that the atmosphere is essentially an isothermal ensemble but that thermometers give different readings dependent on height? Could this be the root of all disagreement? I admit that I don't have any detailed analysis to support this but the idea does not seem so far fetched to me. Even in a hypothetically isothermal atmosphere the thermometer reading would likely decrease since the density of the gas must eventually go to zero. As far as I can see, the atmosphere does not have any well-defined boundary.

SvaraRaderaWhat does travel in the usual photonic way mean? Of course a photon

SvaraRaderabehaves photonically, by that is circular.

To say that photons behave photonically is circular, not a scientific argument.

SvaraRaderaAs Tobis already pointed out there is no such thing as flow of temperature, Claes just takes a term in an equation and gives it a fancy name. This imaginary flow has nothing to do with backradiation that is a real, physical process.

SvaraRaderaIf Claes wants to propose a theory different from the established theory of blackbody radiation he should come up with some better arguments, and explain where his theory gives different results from the traditional one.

Photons behaving photonically; isn't that circular?

SvaraRaderaI sense you have some suspicion about photons that I don't really understand. They are not incomprehensible, as you mentioned, Newton postulated light as particles. But that is not the important thing. If you don't trust photons solve Maxwells Magnetohydrodynamics and you will probably come to the same conclusion. It is the preservation laws that are important. Radiation has pressure, proved in experiments.

SvaraRaderaAnders,

SvaraRaderaI have read on other blogs that there is a controversy about measured temperatures having an artifact related to air density, but don't have any links to scientific papers on this...

MS,

SvaraRaderaThanks, that's interesting and good news. I have wanted to discuss that issue with experts on experimental thermodynamics for a time now, hope we can make the ball rolling.

Radiation may have pressure, but I do not see why I have to believe in

SvaraRaderaphoton particles of light or phonon particles of sound, or gravitons

transmitting gravitation. The whole idea of particles as transmitters of

forces seems utterly strange. Wave mechanics is enough I believe.

"I do not see why I have to believe in photon particles of light"

SvaraRaderaEver heard of the photoelectric effect?

Of course: Read http://www.csc.kth.se/~cgjoh/ambsblack.pdf

SvaraRaderaWell, I am not at all convinced by that. As far as I can see you have presented no observations which contradict the established scientific theory that can be explained by your own theory, nor any testable predictions of your own theory. You just have some philosophical dislike of the idea that electromagnetic radiation can behave like a particle. That you feel that way doesn't change anything about the real world.

SvaraRaderaIt seems that you also doubt particles when you say that electromagnetic waves/radiation "behaves like particles". So it "is" a wave "behaving" like a particle? Maybe we agree? If someone "behaves like a fool", does it mean the person "is a fool"?

SvaraRaderaThis fatuous argument is still rattling on, here and elsewhere. The Kiehl & Trenberth diagram doesn't show any net flow from colder atmosphere to warmer surface, in fact just the opposite. The argument put forward by Claes Johnson and Alan Siddons is indeed a "straw man". I am no supporter of the "enhanced greenhouse" theory, but I want to see the argument against it proved by logic and science, not unscientific bluster.

SvaraRaderaNet radiative energy flow between two bodies is defined as the difference between the radiative fluxes of those two bodies. The presence of the cooler body reduces the energy flow from the hotter. That is fundamental. If the energy flow is reduced, the cooling rate of the hotter body is reduced, and if the that body continues to receive energy from an external source (the sun) its temperature will rise until radiative balance is restored.

Nothing in this process violates any law of thermodynamics, and it's defined by the Stefan-Boltzmann law, the equation for which has two terms, one for each body, defining the radiation fluxes. Proponents of this "straw man" conveniently ignore this and display a fundamental lack of understanding of the science.

You say yourself that there is a net flow from hot to cold, and thus we agree. To say that a cold body is warming a hot body is misleading

SvaraRaderasince no net heat is transported from the cold to the hot. I don't understand why you get so upset over these fundamenta.

Claes, your physics is 100 years out of date. Read up on wave-particle duality, please.

SvaraRaderaAnd I don't understand why you continue to dispute really basic stuff that schoolchildren can understand. You're wearing clothes, I presume? They are keeping you warmer than you would otherwise be. And yet they are colder than you, and so according to your views, they cannot warm you. Do you see the absurdity?

Nonsense! Schrödinger did not like particles and neither do I. And I wear clothes to keep me warm.

SvaraRaderaClaes Johnson said

SvaraRaderaYou say yourself that there is a net flow from hot to cold, and thus we agree. To say that a cold body is warming a hot body is misleading

since no net heat is transported from the cold to the hot. I don't understand why you get so upset over these fundamenta.

"To say that a cold body is warming a hot body is misleading"

But WHO has said that? Kiehl & Trenberth don't.

OK, fine then.

SvaraRadera