måndag 31 januari 2011

Is the atmosphere an ideal gas, a blackbody or maybe something else?

Mixing or confusing physical concepts can be devastating. In previous posts we have seen examples of  such confusion which appear to be almost semantical. In my view, many misunderstandings could be avoided if

2. It was made clear whether the incoming sunlight should be treated as a heat flow or as an energy source.

Moreover, there seems to be a mixing of physical models. On one occasion (radiation) the atmospheric layers are treated as black-bodies, on another occasion (convective overturning, adiabat etc.) they are suddenly transformed into an almost ideal gas. If you follow the links you will discover that their thermodynamic properties, like pressure and heat capacity, differ in a qualitative way. 

However, is there any other way forward? In the end we must have a model that takes into account both radiation and kinetic energy of air molecules. Could such a model be that of a "boson gas"? The idea would be to treat all thermal excitations in the atmosphere, including photons, phonons, molecular momentum and so on as one kind of particle: boson. What would it be like?  

Since the number of bosons is not limited we would have to treat it as a grand canonical ensemble. As a matter of fact, papers have been written on this subject: 

Some excerpts:

Can anyone make sense of this?

söndag 30 januari 2011

The hypothesis of Jelbring and the rebuttal by Erren and Dietze

In 2003, Swedish climatologist Hans Jelbring proposed a theory, based mostly on heuristic lines of argument, for explaining the atmospheric lapse rate and its heating impact on the surface. In a way it could be viewed as a modern rendition of the hypothesis of Herapath. The paper, published in E&E, had the title "Greenhouse Effect as a function of Atmospheric Mass". Below is a quote of the central hypothesis:

"In an ideal gas atmosphere, the adiabatic temperature lapse rate has to be –g/cp where cp is the heat capacity of the gas (ref 2 p. 49). Theoretical calculations are well confirmed by observational evidence in the atmosphere of Earth. The adiabatic temperature lapse rate on Earth is thus –9.81/1004 = –0.0098 K/m. As James R. Holton concluded after deriving this result: “Hence, the dry adiabatic lapse rate is approximately constant throughout the lower atmosphere.” The temperature lapse rate in our model atmosphere also has to be –g/cp, since its atmosphere is organized adiabatically."

The paper was followed by a fierce rebuttal  by Hans Erren and Peter Dietze entitled "The Greenhouse Effect should not be redifined" published in the same journal. I will quote parts of the text, hopefully capturing the essence of the message:

"Hans Jelbring titles his paper "The 'Greenhouse effect' as a function of atmospheric mass" though another term would be required as by definition the Greenhouse effect (GE) is a radiative effect, i.e. warming from back-radiation to ground, which is independent of atmospheric mass and adiabatic lapse rate and thus cannot be governed in its magnitude. The GE is from infrared (IR) absorption and thermal re-emission (see chapter "Radiative Forcing" of <http://www.john-daly.com/forcing/moderr.htm>) which occurs independently from the thermodynamic processes that Hans J considers to be the only relevant ones in the troposphere. His suspicion that IPCCs Global Warming (GW) models wrongly consider all the atmospheric energy transport being radiative only, is incorrect. The basic GE modelling copes with the radiative and convective part of the energy fluxes."

It is true that the basic GE modelling takes into account a convective overturning, but the radiative and convective processes are highly interacting, see the post "What is the greenhouse effect". Furthermore:

"In the atmosphere we find a combination of convective lapse (in regions where convection is strong), a gravity lapse plus the radiative lapse. The author seems to deny atmospheric radiation and to consider greenhouse scientists who base their theory on radiative physics as to foolishly believe in conventional radiative models and HITRAN spectra which he considers to be without scientific foundation. He simply redefines the GE as the difference in temperature between the surface
and a non-existent minus 18 degC black body reference shell at an arbitrary level of altitude, resulting from the magnitude of the adiabatic lapse rate. He concludes that this lapse rate is independent of the mass and the temperature of the model atmosphere. He further concludes that the GE expressed by the lapse rate is constant and independent of radiative properties (!) of the constituents. In particular we object his statement that "the atmospheric mass exposed to a gravity field is the cause of the substantial part of GW". Strong doubts arise whether the author has understood the radiative GE at all."

What they mean with "combination of lapses" remains obscure. Notably, they also write:

"If we would assume that no GHGs exist (as they are asserted to be irrelevant) and the Earth had a resting atmosphere which is fully transparent to radiation, the ground would turn to minus 18 degC to get into radiative equilibrium with the incoming solar 240 W/m² - no matter what mass the atmosphere has and to what extent a gravity lapse rate may cause a cooler (than minus 18 °C) upper atmosphere. Upper layers would be cooler because the vertical component of the thermal molecular speed is reduced. This "gravity lapse rate" may be similar to the lapse rate g/Cp that Hans J uses. But it cannot cause a warming (relative to minus 18 degC) of the atmosphere near ground and thus a warming of the ground itself. So the adiabatic lapse rate cannot explain the plus 15 degC ground temperature and thus replace any radiative basic and anthropogenic GE. Any ground warmer than minus 18 DegC would definitely mean a perpetuum mobile permanently producing energy out of gravity. Static air pressure cannot produce permanent heat in an open non-insulated system."

It should be emphasized that the greenhouse hypothesis is not ambiguous on the (average) temperature at any altitude, it is clearly defined for any absorption/emmision parameter. Compare with the wiew of Roy Spencer

lördag 29 januari 2011

Why is something possible in Climate Science that is impossible in Nature?

One may question the greenhouse effect on the grounds that it doesn't show up in the laboratory and hasn't been able to account for any historic climate change. And this is a most valid objection, after all, ultimately science is about explaining observed phenomena. But it is often said that the greenhouse effect is a necessary consequence of "the laws of nature". Also, many people skeptical to the conclusions of the IPCC believe that CO2 will cause some warming, just not that much. It is in fact plausible that by changing the composition of the atmosphere, living in that altered atmosphere will be somewhat different from living in the old atmosphere, but that is a truism and has nothing to do with science. The question is, will this change be adequately described by the so called "greenhouse effect"?

We will now go a step further and make an attempt to figure out what went wrong. Why isn't the greenhouse effect a necessary consequence of the laws of nature?

Let's first review some basic physics. We need to distinguish bewteen organized energy, for example translational kinetic energy, and disorganized energy in the form of heat. A single particle cannot possess disorganized energy, but a collection of particles can do that. Organized energy can be transformed into heat by dissipation. We can also extract work (organised energy) from heat but the second law of thermodynamics puts a limit to that process, not all heat can be converted into work

Above you have an illustration of these concepts applied to an ideal gas enclosed in a rectangular volume. Lets now have a look at the greenhouse effect:

The arrows indicate electromagnetic radiation that is absorbed and emitted by the atmospheric layers. The atmosphere is in equilibrium when each layer emits the same amount of energy as it absorbs, and we recall that emmision is dependent on emmisivity/absorption and temperature, while absoption is not dependent on temperature. Now have a look at the first image again. See any similarities? 

Marcus Aurelius once said "Of each particular thing ask: What is it in itself, what is its nature?". So let's ask that question about the arrows in the greenhouse diagram above. To me they indicate radiation pressure. You may respond, but the pressure is so insignificant compared to the atmospheric pressure. I respond, ok let's neglect it then.

If you throw a ball up in the air it doesn't suddenly stop and convert its kinetic energy into heat energy by itself. It can only do so by dissipation, which means interaction with the environment. An atmospheric layer can of course collide with the adjacent upper layer, but that layer is busy colliding with another layer and so on ad infinitum. 

Other partly similar explanations has already been proposed by G&T (distinction between energy and heat) and Miskolci (saturated greenhouse effect). Also, check out Claes Johnson's article on blackbody radiation.

torsdag 27 januari 2011

Roy Spencer defends the Greenhouse Effect

In a blogpost from Roy Spencer published some time ago, he defends the existence of a natural greenhouse effect. The entire post can be read here. I will pick out a few passages that concerns the second law of thermodynamics. Spencer writes the following:

"A second objection has to do with the Second Law of Thermodynamics. It is claimed that since the greenhouse effect depends partly upon cooler upper layers of the atmosphere emitting infrared radiation toward the warmer, lower layers of the atmosphere, that this violates the 2nd Law, which (roughly speaking) says that energy must flow from warmer objects to cooler objects, not the other way around."

There is indeed a formulation of the 2nd law that states the following:

Heat flows spontaneously from higher to lower temperature

He goes on:

"There are different ways to illustrate why this is not a valid objection. First of all, the 2nd Law applies to the behavior of whole systems, not to every part within a system, and to all forms of energy involved in the system…not just its temperature. And in the atmosphere, temperature is only one component to the energy content of an air parcel."

What Spencer wants to say with this is somewhat obscure. The formulation I stated above can be found in standard textbooks on thermodynamics and is pretty straightforward. Furthermore he states that the 2nd law applies to all forms of energy, not just the temperature. First of all, temperature is not a form of energy but relates to energy and entropy by the formula

1/T = dS/dE.

Secondly, the energy he refers to that is not included in the "temperature", does he mean the potential energy? Probably. Well, the potential energy could be included in the heat capacity of the gas, indeed, in statistical mechanics one observes that the heat capacity of an ideal gas in a gravitational field increases to 5/2kT per constituent particle to be compared with the value 3/2kT holding without the field. This accounts for the potential energy of the gas.

Furthermore he writes:

"Secondly, the idea that a cooler atmospheric layer can emit infrared energy toward a warmer atmospheric layer below it seems unphysical to many people. I suppose this is because we would not expect a cold piece of metal to transfer heat into a warm piece of metal. But the processes involved in conductive heat transfer are not the same as in radiative heat transfer. A hot star out in space will still receive, and absorb, radiant energy from a cooler nearby star…even though the NET flow of energy will be in the opposite direction.
In other words, a photon being emitted by the cooler star doesn’t stick its finger out to see how warm the surroundings are before it decides to leave."

This is even more obscure. What precisely is the difference between conductive heat transfer and radiative heat transfer? Is it that when a hot metal plate looses heat to a colder plate it does so because it has first measured the temperature of the colder plate and concluded that it was lower that its own?

We will return to discuss this issue at length later. Stay tuned..

söndag 23 januari 2011

On the temperature profile of an ideal gas under the force of gravity

The discussion concerning the temperature profile of an ideal gas under the force of gravity has a long history. There are many excellent texts on this subject so I will mostly refer the reader to these and just add a few comments of my own.

In an old book on the history of statistical mechanics entitled "The kind of motion we call heat" you can find the following passage:

"...According to Herapath, the force of gravity by itself produces a temperature variation in a vertical column of air, namely a decrease of about 1 F for every 100 yards increase in height above the earth's surface (assuming perfectly dry air); the 'total altitude of the air' would thus be approximately 31 miles, if it terminates when the temperature has dropped to absolute zero.

Herapath's work was refused publication by the Royal Society of London but seems to have had ample publicity through its appearance in several issues of the Annals of Philosophy....

Clearly, if Herapath is right then the greenhouse hypothesis is wrong, as is explained in the post "What is the greenhouse effect". But is Herapath right? Well, the Royal Society seemed skeptical at the time. There is one problem with his hypothesis that can be expressed quite simply:

The temperature must not become negative

This is so obvious so why do I mention it? Well, if we assume that the ground temperature of the earth is on average 288 K and that the temperature decreases by 9.8 degrees C per kilometer, then the atmosphere must "end" at an altitude of approximately 30 km, otherwise the temperature will become negative.  Today we know that the atmosphere extends to much higher altitudes and that the temperature appears to be stratisfied beginning already at an altitude of 11 km. The stratification must occur at some point independent of any ozone absoption of UV-light and so on.

However, there seems to be something intuitively appealing with the assumption that temperature decreases with height independent of any motions or stirring of the air. In 1985 Coombes and Laue published a paper called "A paradox concerning the temperature distribution of a gas in a gravitational field":

Clearly, we must distinguish between the single particle kinetic energy expectation value and the ensemble average. The fact that the particles with low energy must reside at lower altitudes means that they contribute to lowering the ensemble average of the kinetic energy at these altitudes. Or put in other words, the density decreases according to the barometric formula just as the pressure does, so if the ideal gas law is to be valid the temperature must be uniform in the column.

The problem can also be treated within the context of fluid mechanics under Navier-Stokes equations, yielding some similar but also some slightly different answers.

So where do we go from here, what is the significance of the Coombes-Laue conclusion? We could make an attempt to analyze it from a logical point of view in the following way. Consider the following three definitions of temperature:

1. Absolute Temperature, denoted T

The absolute temperature could be defined using the notion of equilibrium in the following way: Whenever two systems in thermal contact with each other but isolated from the surroundings cease to exchange net heat (energy) they share some common property called temperature. The absolute temperature can be given a number according to the formula 1/T = dS/dE where S is the entropy and E is the internal energy. In order to define the entropy we need to identify the degrees of freedom of the system.

2. Kinetic Temperature, denoted T'

The kinetic temperature we define as the ensemble average of the translational kinetic energy of the molecules. This can only be applied to gases but is not excluded to ideal gases. Allowing the inclusion of water vapour and other gases with extra degrees of freedom does not pose any difficulties.

3. Empirical Temperature, denoted T''

The empirical temperature we define as the reading of some preferred thermometer. This definition differs from the others since it concerns the physical world and not "the world of ideas".

So, what Coombes and Laue showed, as well as many people before them,  was that for a particular statistical ensemble, namely the canonical ensemble of an ideal gas in a gravitational field, we have that T = T'. However, as a matter of fact this is no longer true if you consider the microcanonical ensemble, though the difference in this case may be considered as rather insignificant. In any case, the more relevant question would be the following:

Is, under all circumstances, T = T' = T'' ?

It seems as if the greenhouse hypothesis implies the following answer to the above question:

Under all circumstances we have that T' = T'', but T = T' = T'' holds only if there are no greenhouse gases around. This of course requires that we treat the incoming sunlight as a heat flow regardless of the composition of the atmosphere. It appears as if the controversy concerning the greenhouse effect in relation to the second law of thermodynamics can be formulated as a logical/semantic disagreement of the kind just described.

There are many interesting perceptions of the related problem concerning the atmosphere. The following passage was quoted in the falsification paper of Gerlich and Tscheuschner:

"Some have problems with the energy that is radiated by the greenhouse gases
towards the surface of the Earth (150W=m2 - as shown above) because this energy
flows from a colder body (approx. -40 deg C) to a warmer one (Earth's ground approx.
+15 deg C) apparently violating the second law of thermodynamics. This is
a wrong interpretation, since it ignores the radiation of the Sun (even 6000 K).
With respect to the total balance the second law is obeyed indeed."

There is something very strange about this statement since if we follow the advice given and treat the incoming sunlight as a heat flow then there is no net heat flow between the surface and the atmosphere averaged over a day-night cycle.

torsdag 20 januari 2011

What is the greenhouse effect?

If you want to debate the greenhouse effect, a good start is to know what it is. People argue whether there exists such a thing as the greenhouse effect,  however, here we will provide a simplified mathematical formulation that nevertheless captures the essential features. The background information is gathered from Goody and Yung "Atmospheric radiation", Oxford University Press.

The underlying principles are rather simple, the earths surface and the atmospheric layers radiate isotropically according to Stefan-Bolzmanns law but with variable absorption/emissivity. The radiation absorbed is instantly thermalized. Furthermore, the absorption a(z) varies with altitude z according to:

The ground temperature is given by the formula:

Here Fs is a measure of the incoming solar radiation. Finally the lapse rate is given by:

Where H is a height scale. We will now solve these equations numerically for three different values of a0. (H = 1, sigma = 1, Fs = 1 held constant).

The blue line is for a0 = 0, the green line is for a0 = 0.5 and the red line is for a0 = 1. As we can see, by increasing the absorption/emission parameter we heat the surface and cool the upper layers of the atmosphere. Note that for a0 = 0 we have an isothermal atmosphere. In more advanced models the pure radiative equilibrium is modified so that when the lapse rate exceeds a certain critical value convection sets in. These models are called radiative-convective models. A lapse rate neutrally stable to convection is characterized by a constant potential temperature and are often called adiabats. These differ depending on the amount of water vapour present. Of course, the equations presented in the beginning need some further explanation, but we will leave that for now and refer to the litterature. 

See also

Although we were able to give a mathematical illustration of the greenhouse effect in just a few lines, unfortunately the scientific debate hardly ever reaches this point. Many proponents of the theory don't seem to be willing to communicate it in the way I just did, or they might simply not know what they are talking about.

Usually, successful scientific debates are carried out in the spirit of Leibniz by the parties carefully explaining the axioms, assumptions, approximations and equations underpinning their theory/model. In the greenhouse debate though, neither part seems to be interested in such a discussion. However, from the skeptical point of view the danger of not knowing your target is obvious. You become more succeptible to the sophistry of your opponent, you are more likely to make mistakes and to focus on extraneous details rather than essentials.

Equipped with the equations and their solutions we may now disect some common pseudo descriptions of the greenhouse effect:

1) The greenhouse gases absorb the outgoing terrestrial IR-radiation thereby heating the atmosphere.

Comment: The greenhouse gases do not heat the atmosphere, they change the temperature profile of the atmosphere.

2) By adding greenhouse gases the optical thickness of the atmosphere increases, hence the effective radiating level is pushed to a higher altitude. Since the temperature decreases with increasing altitude, in order to get into equilibrium with the incoming solar radiation the temperature at this new altitude must be that of the blackbody temperature of the earth. The ground temperature is then recovered by following the adiabat.

Comment: This is more to the point but still deceptive. It is true that the red curve crosses the blue curve at a higher altitude than the green curve does, but a lot of information is left out, for example, it is because of the greenhouse effect that the temperature decreases with height in the first place. In essence, the description doesn't explain anything.