onsdag 30 mars 2011

A note on Thermal Diffusivity

I would like to bring to your attention a thermodynamic property which I havn't given very much attention until recently. It is the thermal diffusivity, that is the rate at which a temperature change will spread in the medium. Quoting from the G&T paper:

In particular you should pay notice to the density term occuring in the expression for the thermal diffusivity. In many cases the density appears to be a trecherous quantity. The reduction in density with height explains why the equilibrium state may very well be isothermal. Ironically however, if we adopt the non-equilibrium approach this non-uniformity might very well explain why the lapse rate is steeper closer to the surface than higher up. Maybe it could even explain why the stratosphere is stratisfied: Since the density is so low the thermal diffusivity is so high that any temperature gradient will quickly disappear. At least it is worth to think about.

lördag 19 mars 2011

Equilibrium contra Non-Equilibrium

In order to properly understand some of the aspects of the greenhouse debate, it is important to make clear the concept of equilibrium contra non-equilibrium. No real world system is ever in equilibrium, there are always disturbances. In the case of the earth these disturbances are, among many others, the rotation of the earth, the non-uniformly distributed sunlight, the moon, etc etc. So why do we use the concept of equilibrium at all. The way I see it is that the equilibrium is the imaginary state that the system strives towards but never reaches. And in order no know how the non-equilibrium state evolves we need to know its ultimate goal. If there is a temperature gradient, the system will strive to erradicate it as fast as it can. Or will it?...

Let's make things simple for a while. Let's take away all non-uniformities that create disturbances in the earth's thermodynamic system. No rotation, uniform sunlight, uniform oceans, no poles, no equator, no moon. What would it be like? The greenhouse hypothesis says that this system would be in equilibrium, but the equilibrium would be profoundly affected by the amount of greenhouse gases present in the system. The fact that things will be slighty different with a change of composition of the atmosphere is of no surprise, but the strange thing is that GHEH implies that this equilibrium would be characterized by a temperature gradient (or a "radiative-convective equilibrium") and that this temperature gradient can be considered caused by the greenhouse gases. Strange isn't it, an equilibrium with a temperature gradient, like a refrigerator working without electricity. But hey, so what, the real world atmospheric temperature gradient is indisputable. Or maybe it isn't?..

GHE proponents have of course developed a cunning way to get out of this dilemma. They argue that an equilibrium with a temperature gradient is not a violation of the 2nd law, since the earth is not in equilibrium anyway. Wicked isn't it :)

But let's not argue about that now. What do the skeptics say? Some skeptics say that this equilibrium which I sketched on before will indeed be characterized by a temperature gradient but it will be caused by gravity. Since the total energy of each molecule follows a Boltzmann distribution the molecules at higher altitudes will be slower than those at lower altitudes giving rise to a temperature gradient. Convincing isn't it? The only problem is that it is wrong, as was shown long ago. Given the assumptions, gravity doesn't cause a non-uniform temperature, instead it creates a non-uniform chemical potential. To make things even more complicated (and interesting) the chemical potential is temperature dependent. (Check out the previous post "On the temprature distribution of an ideal gas under the force of gravity"). This cannot be dismissed easily since even under the Navier-Stokes approach to fluid mechanics, the isothermal air parcel is a very stable and physical concept that can be extended to an infinite space domain.

There is however another aproach. One could claim that there is an atmospheric lapse rate because the earth is not in equilibrium and thats that. At the moment, Claes Johnsson and I discuss the consequences of this assumption. In this case the incoming sunlight is treated as an energy source heating the surface and the important question becomes how the thermal transport properties are affected by an increase in the optical activity of the atmosphere, characterized by a absorption/emission parameter A. Does an increase in A lead to larger radiative heat transport and flatter lapse rate or does it lead to increased isolation and a steeper lapse rate. The first corresponds to cooling, the latter to warming. A simple observation could give a hint to the answer to that question, consider two bodies with temperatures T1 and T2 that radiate against each other with an intensity AT^4. The net heat transfer is thus

 dQ = A(T1^4 - T2^4)

The heat transfer increases as we increase A, or what is your opinion? The thriller continues...

måndag 14 mars 2011

Folklores in Physics

When I did dimploma work in mathematics, on one occasion my supervisor said to me: "In mathematics and science there are things called folklores". What he referred to in his field was the existence of theorems that everyone went around believing that someone had proved but which in reality nobody had ever proved. We are  not talking about grand famous conjectures like the Riemann hypothesis but typically minor theorems in some new emerging field. In particular he remembered a conference where somebody, lets call him Y, went before the audience and announced that he was going to prove Theorem X and almost everyone in the audience started laughing. Theorem X was a folklore, but only Y knew about it.

During recent years I think I have discovered a folklore in Physics. I'm not talking about the Greenhouse Effect, physicists don't know anything about that, and besides, it is not really the kind of folklore we are discussing here. I'm talking about the folklore about the atmospheric temperature lapse rate. Almost every physicist, chemist, (and zoologist) think that it is simple and was solved long ago. Or put in other words: Nowadays every Tom, Dick and Harry thinks that somebody else knows how to derive it, but they are wrong. 

If you ask a physicist, if you get any answer at all he or she will probably say that it follows from fundamental gas laws. If pushed on the details about which of the infinite number of lapse rates you can derive from the fundamental gas laws he or she will probably say, look in the litterature, Gibbs must have solved it. But Gibbs never solved it, and neither did anyone else. The problem was discussed intensively long ago, but was forgotten and left unsolved by physicists. This void was then filled with the Greenhouse Effect, but they never told us about it...