The heat equation revisited

Let's recall the heat equation in one spatial dimension:

dU/dt = D*d^2U/dx^2

Here U is the heat-content or temperature at a given position and time, and D is the diffussivity constant. In order to completely solve the heat equation one needs to specify boundary conditions which might depend on various energy sources et.c., but we will not be so much concerned with that here. Often one is interested in finding a stationary solution that does not change in time, and for such a solution we have that dU/dt = 0 everywhere. The stationary one-dimensional heat equation is thus D*d^2U/dx^2 = 0.

Now let's try to construct a simple radiation model for an almost ideal gas. Many of the thermodynamic properties of an ideal gas, such as the energy content and pressure, are proportional to temperature so let's assume that we can interchangeably speak about the heat content and temperature. Suppose that the radiation is proportional to the temperature U and also proportional to a parameter A which is a measure of the emmisivity. Let's discretize the position variable with an index n. To start with, also suppose that every layer absorbs all incoming radiation (an unphysical assumption that we will later relax). If we now try to find a stationary solution, that is a solution that does not change in time, then there is no build-up or loss of energy and hence we must have balance between incoming and outgoing radiation. Now take the perspective of the layer at position n

In words:

Heat absorbed from the adjacent layers - Heat lost by radiation = 0

In numbers:

AU(n-1) + AU(n+1) - 2AU(n) = 0

Notice in particular the "back-radiation" term AU(n+1). However, what we have just written is simply the discrete form of the stationary heat equation:

A*d^2U/dx^2 = 0,

now with the diffusivity constant simply given by the emissivity parameter A. Relaxing the assumption that all incoming radiation is absorbed leads in its simplest form to the model I described in the post "A simple radiation model". One can of course take into account all sorts of other circumstances, for example a position dependent thickness of the gas and so on, but very quicly such more elaborate models become analytically intractable and one would probably need to use computers. 

However, the main message is that a model which incorporates "back-radiation" does not necessarily lead to a greenhouse effect.

A comment on Claes' answer to Roy

In a recent post Claes Johnson attempts to answer the following question from Roy Spencer:

How does the surface 'know' how opaque the atmosphere is before it 'decides' at what rate it should emit IR?

There is one point which I would like to make here which seem to have escaped many GHE-skeptics. In a previous post I constructed a simple radiation model, which does not necessarily come close to the real situation, but which nevertheless highlights something important. In the model in question no part of the system 'knows' what goes on anywhere else, the only things each part knows is its own temperature and absorptivity. The model also contains 'backradiation'. Moreover, the 'backradiation' taken alone does in fact slow down the rate of cooling in the system. The question is now: Does the model reproduce anything like the greenhouse effect? The answer is: It doesn't. In the model the temperature lapse rate flattens as the absorptivity/emmisivity increases with the consequence that the system cools, which is a clear deviation from the so called GHE.

The reason for this is probably the following: The amount of backradition can never exceed the radiation that at the same time is lost to outer space. Thus the backradiation cannot trap energy in the system since it is always associated with an 'out-radiation' that is equally big. 

The real difference with the GHE and reality is thus probably much more subtle than both Roy and Claes wants it to appear. To be honest, I have not quite understood the supposed mechanism of GHE although I have tried, but maybe I will succeed in the future to completely disentagle the mathematical structure of it. I very much encourage the mathematically inclined audience to also make such an attempt, since the present 'wordy' discussion on 'backradition' has not managed to clear the confusion. 

My suspicion though is that the greenhouse effect is based on a mechanism of reflection rather than absorption-thermalization-thermal reemission. Hence it is formally more akin to radiation pressure, but that remains to be clarified. Good luck.

Cloud or No Cloud

Much confusion surrounding the debate concerning the fundamentals of the greehouse theory, in my opinion can often be traced back to a careless hopping between concepts from equilibrium- and non equilibrium thermodynamics respectively. Here I will present a simple thought experiment that might highlight this issue. Consider a body which by regulated inner chemical reactions maintains a temperature of 37 degrees Celsius. Now consider two situations:

1. The body is placed in vacuum (outer space)

2. The body is surrounded by a nitrogen cloud which has a temperature of 20 degrees Celsius.

In case 1 it is obvious that the body will radiate energy to space at a rate which depends on its temperature. The exact law governing the magnitude of this radiation is not really important here, but let's suppose that it follows a T^4 law. But what about case 2? It seems obvious that the body will transport heat to the gas due to the temperature difference but how much will it radiate? Greenhouse theory given an unequivocal answer to that: It will still radiate as much as it did in vacuum, and since there are no greenhouse gases around there will be no back-radiation. In case 2 we are thus left with the conclusion that the body loses more heat to the surrounding than it did in case 1. But does this really make sense? I can think of several ways to argue that the body will in fact not radiate nearly as much as it did in vacuum, and I will for the moment leave it to the reader to think about this. If, however, you believe that the surrounding gas will reduce the amount of radiation it does have implications for the greenhouse effect, it means that the application of an equilibrium radiation law, like the Stefan-Boltzmann law, to a surface which is not equilibrated to the surrounding gas (with or without greenhouse gases) is invalid.

If you are still hesitant I might just through out the question quite bluntly:

Which would you like to try, cloud or no cloud?

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...

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...

The photon gas versus the ideal gas, round 2

As a warmup I thought we could have a look at the following paradoxical behaviour of two reservoirs of ideal gas put in a gravity field and allowed to exchange both energy and matter:

The yellow numbers indicate the kinetic energy of the molecules and the "temperature" standing to the right is simply the ensemble average of the kinetic energy. Now imagine that one of the molecules in the upper reservoir jumps down:

Taking into account that the molecule gains some kinetic energy during the fall we arrive at:

Look what happened! Energy went from lower temperature to higher temperature, thereby equating the temperature of both reservoirs. Pretty queer, huh. But if something is screwing things up here it is certainly not greenhouse gases. In a way this illustrates an important difference between the photon gas and the ideal gas concerning the relationship between internal energy and temperature:

As we can see, for the photon gas the internal energy density is simply a constant times the temperature to the power of four, period. For the ideal gas the situation is different:

Notice the "n", which is the particle density. Hence, in contrast to the photon gas, the ideal gas can have an arbitrarily nonuniform distribution of energy but at the same time be isothermal. Of course this is of no concern for climate science since in their world the atmosphere is a photon gas and an ideal gas at the same time. That is also pretty queer.

Misuses of the complexity argument

In many discussions you encounter the argument "the climate is too complex to be properly understood". But what does it really mean and why do people use it. I think that in many cases it is used to avoid some uncomfortable or difficult question. The more interesting question would be: "What aspects of climate are too complex to be properly understood". Let me give a few examples:

1. Why is it brighter during daytime than during night? Is that complex or simple?

2. Why is it warmer during summer than during winter? Is that complex or simple?

3. Why is the average temperature lower at 5 km altitude than at the surface? Is that complex or simple?

The last question has apparently shown to be too complex for physicists, but maybe it is in fact simple, that we are just too stupid to realize it? Who knows. Maybe the difficulty lies in the fact that the third question posed can under certain circumstances turn almost meaningless. Below I will give two examples of common usage of the complexity argument, one good and one bad.

1. Climatologists claim that the climate is largely determined by the greenhouse effect. They try to prove this hypothesis by running computer simulations and compare them with temperature records of the past, but in reality small temperature variations are too complex to model so the greenhouse hypothesis is still unproven.

2. Nobody has denied the greenhouse effect, but we don't know its future magnitude because climate is so complex. It can be plus 1 or plus 4 or maybe plus 100 degrees or perhaps even -1, who knows, climate is so complex so we shouldn't do anything anyway.

Which one of the above do you think is best?