April | 2011 | The Science of Doom

Archive for April, 2011

Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Twelve – Curve of Growth

Posted in Atmospheric Physics on April 30, 2011| 9 Comments »

One question that has intrigued me for a while – how much of the transmittance (and change in transmittance) from CO2 in the atmosphere is caused by weak lines, and how much is caused by the “far wings” of individual lines.

Take a look back at Part Nine. Here is the calculated change in transmittance from the surface to the tropopause for a doubling of CO2 from pre-industrial levels:

Figure 1

We can see that there is significant change outside of the central range of wavenumbers. (For reference, 667 cm^-1 = 15 μm).

Here is another of the graphics – showing how an individual line absorbs across a range of wavenumbers:

Figure 2

There are many lines in the HITRAN database for CO2 (over 300,000 lines), many of them weak.

So back to my original question – is it the “wings” = line broadening, of the individual lines, or the weaker lines that have the biggest effect? And how do we quantify it?

The Curve of Growth

The Curve of Growth is about the change in transmittance with an increase in path length (specifically the increase in number of absorbing molecules in the path).

The reason the problem is not so simple is because each line has a line shape that absorbs across a range of wavelengths.

We know that transmittance, t=e^-τ – this is the Beer-Lambert law. The spreading out of the line means that the reduction in transmittance as the path increases isn’t as strong as predicted by a simple application of the Beer-Lambert law.

Here is a calculation of this for a typical tropospheric condition (note 1) – for one isolated absorption line with its line center at 1000 cm^-1:

Figure 3

Each curve represents a doubling of the number of absorbing molecules in the path. So, if this was at constant density, each curve represents a doubling of the path length.

What does this graph show?

At the line center, an increase in the path length soon causes saturation. But out in the “wings” of the absorption line the reduction in transmittance changes much more slowly.

With some maths we can show that for very small paths (note 2), the absorptance is linearly proportional to the number of absorbers in the path.

And with some more maths we can show that for very large paths (note 2), the absorptance increases as the square root of the number of absorbers in the path.

Here is another way to view the “curve of growth”. I haven’t seen it shown this way before:

Figure 4

This is the same scenario, but this time the x-axis (bottom axis) has the effective path length, while each curve represents a different distance from the line center. The legend is showing the distance from the line center compared with the “line width”. So “10” = 1 cm^-1 from the line center, while “0” is the line center.

Notice the difference at a mass path x absorption coefficient = 10⁰ (=1) where the line center has a transmittance of 0.05 while at 1 cm^-1 distance from the line center the line has a transmittance of 0.97.

What happens when we look at the effect of the whole line?

Here is a curve of the equivalent line width, W as the path increases. This equivalent line width is just the total effect across enough bandwidth to take into account the effects of the far wings of the line (note 3):

Figure 5

The 2nd graph is the important one. This shows the power relationship between W and Su, as Su increases – where Su = absorption coefficient x mass of molecules in the path. S is the absorption coefficient, and u is the mass.

For a given line, S is a constant, and so an increase in Su means an increase in the number of molecules in the path.

So if we believe that W = (Su)^x, how do we determine x?

We can do it by taking the log of both sides: log(W)=x.log(Su), so by calculating the slope of this log relationship we can see how this value, x, changes.

The 2nd graph shows:

at very small paths, the line strength is proportional to Su – because x = 1
at very large paths, the line strength is proportional to √(Su) – because x = 0.5

This is simply backing up by numerical calculation the earlier claim: “With some maths we can show..”

So if you want to understand “the curve of growth” you need to understand that at very small optical thickness the “equivalent line” grows in linear proportion to the mass of molecules in the path, and with very large optical thickness the “equivalent line” grows in proportion to the square root.

And probably the easiest way to see it conceptually is to take another look at Figure 3.

Line Wings in Practice

So back to my original question. Is it the weak lines or the wings of the strong lines that has the effect?

The first step is to calculate the effect of the line wings. I took the MATLAB program developed for Part Nine and made a few changes. The original program had simply applied the equation for line shape out to the edge of the region under consideration. In this version I added a new factor, c_a, which “chopped” the line shape. The factor c_a limits the line shape for each line to the line center, v₀ ± (c_a * line width).

The line width is a parameter in the HITRAN database and is a measure of the shape of the line, approximately the value where the strength has fallen to half the peak value.

The original program was run for two values of atmospheric CO2: 280ppm (pre-industrial); and 560ppm (doubling of CO2) – with 15 layers and a calculation every 0.01 cm^-1 across the band of 500 cm^-1 – 850 cm^-1. These are called the “Standard” results. And for these, and all following simulations, we are only considering the main isotopologue of CO2, which accounts for over 98% of atmospheric CO2.

Then the revised program was run for a number of values of c_a: 5000, 1000, 100 & 10.

Given that the typical line width is 0.05 – 0.1 cm^-1 this means that each line is considered between two extremes: across 250-500 cm^-1 down to only 0.5 – 1 cm^-1.

As expected, with c_a = 5000, the difference between that and the Standard is almost nil. And as c_a is reduced the differences increase:

Figure 6

By the time ca = 100, the differences are noticeable, and at ca = 10, the differences in some parts of the band are huge. The above result (figure 6) is dominated by the ca=10 result so here is ca=100 separately:

Figure 7

So this demonstrates that even when the line shape is “limited” to 100x the line width, there is still a noticeable effect in the transmittance calculation for the troposphere.

For completeness, here is the same comparison at figure 6, but for 560ppm:

Figure 8

Across this band, the standard case at 280ppm: t = 0.4980, and at 560ppm, t = 0.437.

Here is how the mean transmittance changes as we crop the line shape. Cropping the line shape means that we make the atmosphere more transparent.

Figure 9

We can see from figure 9 that the change in transmittance from 280 – 560 ppm is of a similar magnitude at each artificial “cropping” of the line shape.

Weak Lines

So let’s take a look at the effect of weak lines. For this case I used the same MATLAB program developed for Part Nine but with a user-defined selection of lines. For example, the top 10% of lines by strength.

Here is the transmittance change @ 280ppm through the troposphere, for all lines (standard) less the transmittance of the strongest 10% of lines:

Figure 10

And here is the graph of all lines – the strongest 1%:

Figure 11

And here is the mean transmittance for both 280ppm and 560ppm as only the strongest lines are considered:

Figure 12

It’s clear that the weakest 90% of lines have virtually no effect on the transmittance of the atmosphere. There are a lot of very weak lines in the HITRAN database.

The calculated transmittance for 100% of the lines at 280ppm = 0.4974 and for the top 10% = 0.4994 – meaning that the top 10% of lines account for 99.6% of the transmittance.

At 560ppm the top 10% of lines account for 99.3%.

Conclusion

Some of this analysis is of curiosity value only.

However, it is very useful to understand the “curve of growth” – and to realize how absorptance increases as the mass in the path increases.

And it’s at least interesting to see how the “far wings” of the individual lines have such an effect on the transmittance through the atmosphere. Even “cropping” the effect at 100x the line width has a significant effect on the atmospheric transmittance.

And for the question posed at the beginning, both the weak lines and the far wings of individual lines have an effect on the total atmospheric transmittance.

Many people have appreciated the massive absorption at the peak of the CO2 band (around 15 μm). But as we have seen in earlier parts of this series (and as shown in Figure 1), it is towards the “edges” of the band where the largest changes take place as CO2 concentration increases.

Remember as well that total transmittance is not really a complete picture of radiative transfer in the atmosphere. The atmosphere also emits radiation, and so the temperature profile of the atmosphere is just as important for seeing the whole picture.

Other articles in the series:

Part One – a bit of a re-introduction to the subject.

Part Two – introducing a simple model, with molecules pH2O and pCO2 to demonstrate some basic effects in the atmosphere. This part – absorption only.

Part Three – the simple model extended to emission and absorption, showing what a difference an emitting atmosphere makes. Also very easy to see that the “IPCC logarithmic graph” is not at odds with the Beer-Lambert law.

Part Four – the effect of changing lapse rates (atmospheric temperature profile) and of overlapping the pH2O and pCO2 bands. Why surface radiation is not a mirror image of top of atmosphere radiation.

Part Five – a bit of a wrap up so far as well as an explanation of how the stratospheric temperature profile can affect “saturation”

Part Six – The Equations – the equations of radiative transfer including the plane parallel assumption and it’s nothing to do with blackbodies

Part Seven – changing the shape of the pCO2 band to see how it affects “saturation” – the wings of the band pick up the slack, in a manner of speaking

Part Eight – interesting actual absorption values of CO2 in the atmosphere from Grant Petty’s book

Part Nine – calculations of CO2 transmittance vs wavelength in the atmosphere using the 300,000 absorption lines from the HITRAN database

Part Ten – spectral measurements of radiation from the surface looking up, and from 20km up looking down, in a variety of locations, along with explanations of the characteristics

Part Eleven – Heating Rates – the heating and cooling effect of different “greenhouse” gases at different heights in the atmosphere

Part Twelve – The Curve of Growth – how absorptance increases as path length (or mass of molecules in the path) increases, and how much effect is from the “far wings” of the individual CO2 lines compared with the weaker CO2 lines

And Also –

Theory and Experiment – Atmospheric Radiation – real values of total flux and spectra compared with the theory.

References

The HITRAN 2008 molecular spectroscopic database, by L.S. Rothman et al, Journal of Quantitative Spectroscopy & Radiative Transfer (2009)

Notes

Note 1 – As you can see in Part Nine, the line shape differs as the pressure reduces.

Note 2 – Technically speaking, for “very small paths” we are really considering the case where optical thickness, τ <<1 (very much less than 1). And for “very large paths” we are considering the case where optical thickness, τ>>1 (very much greater than 1).

Note 3 – For a line strength = W, if we want to calculate absorptance, a = 1 – t, where t= transmittance, across any given band, Δv, the calculation is very simple:

a = 1-t = W / Δv

Read Full Post »

The Mystery of Tau – Miskolczi – Part Three – Kinetic Energy

Posted in Basic Science, Commentary, Debunking Flawed "Science" on April 26, 2011| 123 Comments »

In Part One we looked at the calculation of total atmospheric optical thickness.

In Part Two we looked at the claim that the surface and atmosphere exchanged exactly equal amounts of energy by radiation. A thermodynamics revolution if it is true, as the atmosphere is slightly colder than the surface. This claim is not necessary to calculate optical thickness but is a foundation for Miskolczi’s theory about why optical thickness should be constant.

In this article we will look at another part of Miskolczi’s foundational theory from his 2007 paper, Greenhouse Effect in Semi-Transparent Planetary Atmospheres, Quarterly Journal of the Hungarian Meteorological Service.

For reference of the terms he uses, the diagram from the 2007 paper:

From Miskolczi (2007)

Figure 1

On pages 6-7, we find this claim:

Regarding the origin, E_U is more closely related to the total internal kinetic energy of the atmosphere, which – according to the virial theorem – in hydrostatic equilibrium balances the total gravitational potential energy. To identify E_U as the total internal kinetic energy of the atmosphere, the E_U = S_U / 2 equation must hold.

Many people have puzzled over the introduction of the virial theorem (note 1), which relates total kinetic energy of the atmosphere to total potential energy of the atmosphere. Generally, there is a relationship between potential energy and kinetic energy of an atmosphere so I don’t propose to question it, we will accept it as a given.

By the way, on the diagram S_U = S_G, i.e. S_U = upwards radiation from the surface. And E_U = upwards radiation from the atmosphere (cooling to space).

Kinetic Energy of a Gas

For people who don’t like seeing equations, skip to the statement in bold at the end of this section.

Here is the equation of an ideal gas:

pV = nkT (also written as pV = NRT) [1]

where p = pressure, V = volume, n = number of molecules, k = 1.38 x 10^-23 J/K = Boltzmann’s constant, T = temperature in K

This equation was worked out via experimental results a long time ago. Our atmosphere is a very close approximation to an ideal gas.

If we now take a thought experiment of some molecules “bouncing around” inside a container we can derive an equation for the pressure on a wall in terms of the velocities of the molecules:

pV = Nm<v_x²> [2]

where m = mass of a molecule, <v_x²> = average of v_x², where v_x = velocity in the x direction

Combining [1] and [2] we get:

kT = m<v_x²>, or

m<v_x²>/2 = kT/2 [3]

The same considerations apply to the y and z direction, so

m<v²>/2 = 3KT/2 [4]

This equation tells us the temperature of a gas is equal to the average kinetic energy of molecules in that gas divided by a constant.

For beginners, the kinetic energy of a body is given by mv²/2 = mass x velocity squared divided by two.

So temperature of a gas is a direct measure of the kinetic energy.

The Kinetic Error

So where on earth does this identity come from?

..To identify E_U as the total internal kinetic energy of the atmosphere..

E_U is the upwards radiation from the atmosphere to space.

To calculate this value, you need to solve the radiative transfer equations, shown in Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations. These equations have no “analytic” solution but are readily solvable using numerical methods.

However, there is no doubt at all about this:

E_U ≠ 3kT_A/2 [5]

where T_A = temperature of the atmosphere

that is, E_U ≠ kinetic energy of the atmosphere

As an example of the form we might expect, if we had a very opaque atmosphere (in longwave), then E_U = σT_A⁴ (the Stefan-Boltzmann equation for thermal radiation). As the emissivity of the atmosphere reduces then the equation won’t stay exactly proportional to the 4th power of temperature. But it can never be linearly proportional to temperature.

A Mystery Equation

Many people have puzzled over the equations in Miskolczi’s 2007 paper.

On p6:

The direct consequences of the Kirchhoff law are the next two equations:
E_U = F + K + P (M5)
S_U − (F⁰ + P⁰ ) = E_D − E_U (M6)

Note that I have added a prefix to the equation numbers to identify they as Miskolczi’s. As previously commented, the P term (geothermal energy) is so small that it is not worth including. We will set it to zero and eliminate it, to make it a little easier to see the problems. Anyone wondering if this can be done – just set F’ = F⁰ + P⁰ and replace F⁰ with F’ in the following equations.

So:

E_U = F + K (M5a)
S_U − F⁰ = E_D − E_U (M6a)

Please review figure 1 for explanation of the terms.

If we accept the premise that A_A = E_D then these equations are correct (the premise is not correct, as shown in Part Two).

M5a is simple to see. Taking the incorrect premise that surface radiation absorbed in the atmosphere is completely re-emitted to the surface: therefore, the upward radiation from the atmosphere, E_U must be supplied by the only other terms shown in the diagram – convective energy plus solar radiation absorbed by the atmosphere.

What about equation M6a? Physically, what is the downward energy emitted by the atmosphere minus the upward energy emitted by the atmosphere? What is the surface upward radiation minus the total solar radiation?

Well, doesn’t matter if we can’t figure out what these terms might mean. Instead we will just do some maths, using the fact that the surface energy must balance and the atmospheric energy must balance.

First let’s write down the atmospheric energy balance:

A_A + K + F = E_U + E_D [10] – I’m jumping the numbering to my equation 10 to avoid referencing confusion

This just says that Surface radiation absorbed in the atmosphere + convection from the surface to the atmosphere + absorbed solar radiation in the atmosphere = energy radiated by the atmosphere from the top and bottom.

Given the (incorrect) premise that A_A = E_D, we can rewrite equation 10:

K + F = E_U [10a]

We can see that this matches M5a, which is correct, as already stated.

So first, let’s write down the surface energy balance:

F⁰ – F + E_D = S_U + K [11]

This just says that Solar radiation absorbed at the surface + downward atmospheric radiation = surface upward radiation + convection from the surface to the atmosphere.

Please review Figure 1 to confirm this equation.

Now let’s rewrite equation 11:

S_U – F⁰ = E_D – F – K [11a]

and inserting eq 10a, we get:

S_U – F⁰ = E_D -E_U [11b]

Which agrees with M6a.

And as an aside only for people who have spent too long staring at these equations – re-arrange the terms in 11b:

Su – Ed = F0 – Eu; The left side is surface radiation – absorbed surface radiation in the atmosphere (accepting the flawed premise) = transmitted radiation. The right side is total absorbed solar radiation – upward emitted atmospheric radiation. As solar radiation is balanced by OLR, the right side is OLR – upward emitted atmospheric radiation = transmitted radiation.

Now, let’s see the mystery step :

In Eq. (6) S_U − (F⁰ + P0 ) and E_D − E_U represent two flux terms of equal magnitude, propagating into opposite directions, while using the same F0 and P0 as energy sources. The first term heats the atmosphere and the second term maintains the surface energy balance. The principle of conservation of energy dictates that:
S_U − (F⁰) + E_D − E_U = F⁰ = OLR (M7)

This equation M7 makes no sense. Note that again I have removed the tiny P0 term.

Let’s take [11b], already demonstrated (by accepting the premise) and add (E_D -E_U) to both sides:

S_U – F⁰ + (E_D – E_U) = E_D – E_U+ (E_D -E_U) = 2(E_D -E_U) [12]

So now the left side of eq 12 matches the left side of M7.

The M7 equation can only be correct if the right side of eq 12 matches the right side of M7:

2(E_D -E_U) = F⁰ [13] – to be confirmed or denied

In concept, this claim is that downward radiation from the atmosphere minus upward radiation from the atmosphere = half the total planetary absorbed solar radiation.

I can’t see where this has been demonstrated.

It is not apparent from energy balance considerations – we wrote down those two equations in [10] and [11].

We can say that energy into the climate system = energy out, therefore:

F⁰ = OLR = E_U + S_T [14] (atmospheric upward radiation plus transmitted radiation through the atmosphere)

Which doesn’t move us any closer to the demonstration we are looking for.

Perhaps someone from the large fan club can prove equation 7. So many people have embraced Miskolczi’s conclusion that there must be a lot of people who understand this step.

Conclusion

I’m confused about equation 7 of Miskolczi.

Running with the odds, I expect that no one will be able to prove it and instead I will be encouraged to take it on faith. However, I’m prepared to accept that someone might be able to prove that it is true (with the caveat about accepting the premise already discussed).

The more important point is equating the kinetic energy of the atmosphere with the upward atmospheric radiation.

It’s a revolutionary claim.

But as it comes with no evidence or derivation and would overturn lots of thermodynamics the obvious conclusion is that it is not true.

To demonstrate it is true takes more than a claim. Currently, it just looks like confusion on the part of the author.

Perhaps the author should write a whole paper devoted to explaining how the upwards atmospheric flux can be equated with the kinetic energy – along with dealing with the inevitable consequences for current thermodynamics.

Update 31st May: The author confirmed in the ensuing discussion that equation 7 was not developed from theoretical considerations.

References

Greenhouse Effect in Semi-Transparent Planetary Atmospheres, Miskolczi, Quarterly Journal of the Hungarian Meteorological Service (2007)

Notes

Note 1 – A good paper on the virial theorem is on arXiv: The Virial Theorem and Planetary Atmospheres, Victor Toth (2010)

Read Full Post »

The Mystery of Tau – Miskolczi – Part Two – Kirchhoff

Posted in Basic Science, Commentary, Debunking Flawed "Science" on April 25, 2011| 87 Comments »

In Part One we looked at the usefulness of “tau” = optical thickness of the atmosphere.

Miskolczi has done a calculation (under cloudless skies) of the total optical thickness of the atmosphere. The reason he is apparently the first to have done this in a paper is explained in Part One.

The 2010 paper referenced the 2007 paper, Greenhouse Effect in Semi-Transparent Planetary Atmospheres, Quarterly Journal of the Hungarian Meteorological Service.

The 2010 paper suggested an elementary flaw, but referenced the 2007 paper. The 2007 paper backed up the approach with the same apparently flawed claim.

The flaw that I will explain doesn’t affect the calculation of optical thickness, τ. But it does appear to affect the theoretical basis for why optical thickness should be a constant.

First, the graphic explaining the terms is here:

From Miskolczi (2007)

Figure 1

The 2010 paper said:

One of the first and most interesting discoveries was the relationship between the absorbed surface radiation and the downward atmospheric emittance. According to Ref. 4, for each radiosonde ascent the
E_D = A_A = S_U – S_T = S_U(1− exp(−τ_A)) = S_U(1− T_A ) = S_U.A (5)
relationships are closely satisfied. The concept of radiative exchange was the discovery of Prevost [17]. It will be convenient here to define the term radiative exchange equilibrium between two specified regions of space (or bodies) as meaning that for the two regions (or bodies) A and B, the rate of flow of radiation emitted by A and absorbed by B is equal to the rate of flow the other way, regardless of other forms of transport that may be occurring.

Ref. 4 is the 2007 paper, which said:

According to the Kirchhoff law, two systems in thermal equilibrium exchange energy by absorption and emission in equal amounts, therefore, the thermal energy of either system can not be changed. In case the atmosphere is in thermal equilibrium with the surface, we may write that..

What is “thermal equilibrium“?

It is when two bodies are in a closed system and have reached equilibrium. This means they are at the same temperature and no radiation can enter or leave the system. In this condition, energy emitted from body A and absorbed by body B = energy emitted from body B and absorbed by body A.

Kirchhoff showed this radiative exchange must be equal under the restrictive condition of thermal equilibrium. And he didn’t show it for any other condition. (Note 2).

However, the earth’s surface and the atmosphere are not in thermal equilibrium. And, therefore, energy exchanged between the surface and the atmosphere via radiation is not proven to be equal.

Dr. Roy Spencer has a good explanation of the fallacy and the real situation on his blog. One alleged Miskolczi supporter took him to task for misinterpreting something – here:

With respect, Dr Spencer, it is not reasonable, indeed it verges on the mischievous, to write an allegation that Miskolczi means that radiative exchange is independent of temperature. Miskolczi means no such thing. To make such an allegation is to ignore the fact that Miskolczi uses the proper laws of physics in his calculations. Of course radiative exchange depends on temperature, and of course Miskolczi is fully aware of that.

and here:

..Planck uses the term for a system in thermodynamic equilibrium, and the present system is far from thermodynamic equilibrium, but the definition of the term still carries over..

I couldn’t tell whether the claimed “misinterpretation” by Spencer was of the real law or the Miskolczi interpretation. And this article will demonstrate that the proper laws of physics have been ignored.

And I have no idea whether the Miskolczi supporter represented the real Miskolczi. However, a person of the same name is noted by Miskolczi for his valuable comments in producing the 2010 paper.

Generally when people claim to overturn decades of research in a field you expect them to take a bit of time to explain why everyone else got it wrong, but apparently Dr. Spencer was deliberately misinterpreting something.. and that “something” is very clear only to Miskolczi supporters.

After all, the premise in the referenced 2007 paper was:

According to the Kirchhoff law, two systems in thermal equilibrium exchange energy by absorption and emission in equal amounts, therefore, the thermal energy of either system can not be changed. In case the atmosphere is in thermal equilibrium with the surface, we may write that..

Emphasis added.

So if the atmosphere is not in thermal equilibrium with the surface, we can’t write the above statement.

And as a result the whole paper falls down. Perhaps there are other gems which stand independently of this flaw and I look forward to a future paper from the author when he explains some new insights which don’t rely on thermodynamic equilibrium being applied to a world without thermodynamic equilibrium.

Thermodynamic Equilibrium and the Second Law of Thermodynamics

If you put two bodies, A & B, at two different temperatures, T_A and T_B, into a closed system then over time they will reach the same temperature.

Let’s suppose that T_A > T_B. Therefore, A will radiate more energy towards B than the reverse. These bodies will reach equilibrium when T_A = T_B (note 1).

At this time, and not before, we can say that ” ..two systems in thermal equilibrium exchange energy by absorption and emission in equal amounts”. (Note 2).

Obviously, before equilibrium is reached more energy is flowing from A to B than the reverse.

Non-Equilibrium

Let’s consider a case like the sun and the earth. The earth absorbs around 240 W/m² from the sun. The sun absorbs a lot less from the earth.

Let’s just say it is a lot less than 1 W/m². Someone with a calculator and a few minutes spare can do the sums and write the result in the comments.

No one (including of course the author of the paper) would suggest that the sun and earth exchange equal amounts of radiation.

However, they are in the condition of “radiative exchange”.

The Earth’s Surface and the Atmosphere

The earth’s surface and the bottom of the atmosphere are at similar temperatures. Why is this?

It is temperature difference that drives heat flow. The larger the temperature difference the greater the heat flow (all other things remaining equal). So any closed system tends towards thermal equilibrium. If the earth and the atmosphere were left in a closed system, eventually both would be at the same temperature.

However, in the real world where the climate system is open to radiation, the sun is the source of energy that prevents thermal equilibrium being reached.

The bottom millimeter of the atmosphere will usually be at the same temperature as the earth’s surface directly below. If the bottom millimeter is stationary then it will be warmed by conduction until it reaches almost the surface temperature. But 10 meters up the temperature will probably reduce just a little. At 1 km above the surface the temperature will be between 4 K and 10 K cooler than the surface.

Note: Turbulent heat exchange near the surface is very complex. This doesn’t mean that there is confusion about the average temperature profile vs height through the atmosphere. On average, temperature reduces with height in a reasonably predictable manner.

Energy Exchanges between the Earth’s Surface and the Atmosphere

According to Miskolczi:

A_A = E_D [4]

Referring to the diagram, A_A is energy absorbed by the atmosphere from the surface, and E_D is energy radiated from the atmosphere to the surface.

Why should this equality hold?

The energy from the surface to the atmosphere = A_A+ K (note 3), where K is convection.

The energy absorbed in total by the atmosphere = A_A + K + F, where F is absorbed solar radiation in the atmosphere.

The energy emitted by the atmosphere = E_D + E_U , where E_U is the energy radiated from the top of the atmosphere.

Therefore, using the First Law of Thermodynamics for the atmosphere:

A_A + K + F = E_D + E_U + energy retained

i.e., energy absorbed = energy lost – energy retained

No other equality relating to the atmospheric fluxes can be deduced from the fundamental laws of thermodynamics.

In general, because the atmosphere and the earth’s surface are very close in temperature, A_A will be very close to E_D.

It is important to understand that absorptivity for longwave radiation will be equal to emissivity for longwave radiation (see Planck, Stefan-Boltzmann, Kirchhoff and LTE), therefore, if the surface and the atmosphere are at the same temperature then the exchange of radiation will be equal.

Where does the atmosphere radiate from, on average? Well, not from the bottom meter. It depends on the emissivity of the atmosphere. This varies with the amount of water vapor in the atmosphere.

The atmospheric temperature reduces with height- by an average of around 6.5 K/km – and unless the atmospheric radiation was from the bottom few meters, the radiation from the atmosphere to the surface must be lower than the radiation absorbed from the surface by the atmosphere.

If radiation was emitted from an average of 100 m above the surface then the effective temperature of atmospheric radiation would be 0.7 K below the surface temperature. If radiation was emitted from an average of 200 m above the surface then the effective temperature of atmospheric radiation would be 1.3 K below the surface temperature.

Mathematical Proof

For people still thinking about this subject, a simple mathematical proof.

Temperature of the atmosphere, from the average height of emission, T_a

Temperature of the surface, T_s

Emissivity of the atmosphere = ε_a

Absorptivity of the atmosphere for surface radiation = α_a

If T_a is similar to T_s then ε_a ≈ α_a (note 4).

(In the paper, the emissivity (and therefore absorptivity) of the earth’s surface is assumed = 1).

Surface radiation absorbed by the atmosphere, A_A = α_aσT_s⁴ .

Atmospheric radiation absorbed by the surface, E_D = ε_aσT_a⁴ .

Therefore, unless T_a = T_s, A_A ≠ E_D .

If Roy Spencer’s experience is anything to go by, I may now be accused of deliberately misunderstanding something.

Well, words can be confused – even though they seem plain enough in the extract shown. But the paper also asserts the mathematical identity:

A_A = E_D [4]

I have demonstrated that:

A_A ≠ E_D [4]

I don’t think there is much to be misunderstood.

Two bodies at different temperatures will NOT exchange exactly equal amounts of radiation. It is impossible unless the current laws of thermodynamics are wrong.

As a more technical side note.. because ε_a ≈ α_a and not necessarily an exact equality, it is possible for the proposed equation to be asserted in the following way:

A_A = E_D if, and only if, the following identity is always true, α_a(T_s)σT_s⁴ = ε_a(T_a)σT_a⁴ .

Therefore:

T_s/T_a = (ε_a(T_a)/α_a(T_s))^1/4 [Equation B]

– must always be true for equation 4 of Miskolczi (2007) to be correct. Or must be true over whatever time period and surface area his identity is claimed to be true.

Another quote from the 2007 paper:

The popular explanation of the greenhouse effect as the result of the LW atmospheric absorption of the surface radiation and the surface heating by the atmospheric downward radiation is incorrect, since the involved flux terms (A_A and E_D) are always equal.

Emphasis added.

Note in Equation B that I have made explicit the dependence of emissivity on the temperature of the atmosphere at that time, and the dependence of absorptivity on the temperature of the surface.

Emissivity vs wavelength is a material property and doesn’t change with temperature. But because the emission wavelengths change with temperature the calculation of ε_a(T_a) is the measured value of ε_a at each wavelength weighted by the Planck function at T_a.

It is left as an exercise for the interested student to prove that this identity, Equation B, cannot always be correct.

The “Almost” Identity

In Fig. 2 we present large scale simulation results of AA and ED for two measured diverse planetary atmospheric profile sets. Details of the simulation exercise above were reported in Miskolczi and Mlynczak (2004). This figure is a proof that the Kirchhoff law is in effect in real atmospheres. The direct consequences of the Kirchhoff law are the next two equations:

EU = F + K + P (5)
SU − (F0 + P0 ) = ED − EU (6)

The physical interpretations of these two equations may fundamentally change the general concept of greenhouse theories.

From Miskolczi (2007)

Figure 2

This is not a proof of Kirchhoff’s law, which is already proven and is not a law that radiative exchanges are equal when temperatures are not equal.

Instead, this is a demonstration that the atmosphere and earth’s surface are very close in temperature.

Here is a simple calculation of the ratio of A_A:E_D for different downward emitting heights (note 5), and lapse rates (temperature profile of the atmosphere):

Figure 3

Essentially this graph is calculated from the formula in the maths section and a calculation of the atmospheric temperature, T_a, from the height of average downward radiation and the lapse rate.

Oh No He’s Not Claiming This is Based on Kirchoff..

Reading the claims by the supporters of Miskolczi at Roy Spencer’s blog, you read that:

Miskolczi is not claiming that A_A = E_D by asserting (incorrectly) Kirchhoff’s law
Miskolczi is claiming that A_A = E_D by experimental fact

So the supporters claim.

Read the paper, that’s my recommendation. The 2010 paper references the 2007 paper for equation 4. The 2007 paper says (see larger citation above):

..This figure is a proof that the Kirchhoff law is in effect in real atmospheres..

In fact, this is the important point:

Anyone who didn’t believe that it was a necessary consequence of Kirchhoff would be writing the equations in the maths section above (which come from well-proven radiation theory) and realizing that it is impossible for A_A = E_D.

And they wouldn’t be claiming that it demonstrated Kirchhoff’s law. (After all, Kirchhoff’s law is well-proven and foundational thermodynamics).

However, it is certain that on average E_D < A_A but very close to A_A.

Hence the Atmospheric Window Cooling to Space Thing

From time to time, Miskolczi fans have appeared on this blog and written interesting comments. Why the continued fascination with the exact amount of radiation transmitted from the surface through the atmospheric window?

I have no idea whether this point is of interest to anyone else..

One of the comments highlighted the particular claim and intrigued me.

Yes, indeed, that’s right: Simpson discovered the atmospheric window in 1928. It was not till the work of Miskolczi in 2004 and 2007 that it was discovered that practically all the radiative cooling of the land-sea surface is by radiation direct to space.

Apart from the (unintentional?) humor inherent in the Messianic-style claim, the reason why this claim is a foundational point for Miskolczi-ism is now clear to me.

If exactly all of the radiation absorbed by the atmosphere is re-radiated to the surface and absorbed by the surface (A_A = E_D) then these points follow for certain:

radiation emitted by the atmosphere to space = convective heat from the surface into the atmosphere + solar radiation absorbed by the atmosphere
total radiative cooling to space = radiation transmitted through the atmospheric window + convective heat plus solar radiation absorbed by the atmosphere

A curiosity only.

Changing the Fundamental View of the World

Miskolczi claims:

The physical interpretations of these two equations may fundamentally change the general concept of greenhouse theories.

He is being too modest.

If it turns out that A_A = E_D then it will overturn general radiative theory as well.

Or demonstrate that the atmosphere is much more opaque than has currently been calculated (for all of the downward atmospheric radiation to take place from within a few tens of meters of the surface).

This in turn will require the overturning of some parts of general radiative theory, or at least, a few decades of spectroscopic experiments, which consequently will surely require the overturning of..

Conclusion

How is it possible to claim that A_A = E_D and not work through the basic consequences (e.g., the equations in the maths section above) to deal with the inevitable questions on thermodynamics basics?

Why claim that it has fundamentally changed the the general concept of the inappropriately-named “greenhouse” theory when it – if true – has overturned generally accepted radiation theory?

Perhaps α(λ) ≠ ε(λ) and Kirchhoff’s law is wrong? This is a possible consequence. (In words, the equation says that absorptivity at wavelength λ is not equal to emissivity at wavelength λ, see note 4).
Or perhaps the well-proven Stefan-Boltzmann law is wrong? This is another possible consequence.

Interested observers might wonder about the size of the error bars in Figure 2. (And for newcomers, the values in Figure 2 are not measured values of radiation, they are calculated absorption and emission).

As already suggested, perhaps there are useful gems somewhere in the 40 pages of the 2007 paper, but when someone is so clear about a foundational point for their paper that is so at odds with foundational thermodynamic theory and the author doesn’t think to deal with that.. well, it doesn’t generate hope.

Update 31st May – the author comments in the ensuing discussion that Aa=Ed is an “experimental” conclusion. In Part Four I show that the “approximate equality” must be an error for real (non-black) surfaces, and Ken Gregory, armed with the Miskolczi spreadsheet, later confirms this.

References

Greenhouse Effect in Semi-Transparent Planetary Atmospheres, Miskolczi , Quarterly Journal of the Hungarian Meteorological Service (2007)

The Stable Stationary Value of the Earth’s Global Average Atmospheric Planck-Weighted Greenhouse-Gas Optical Thickness, Miskolczi, Energy & Environment(2010)

The Theory of Heat Radiation, Max Planck, P. Blakiston’s Son & Co (1914) : a translation of Waermestrahlung (1913) by Max Planck.

Notes

Note 1 – Of course, in reality equilibrium is never actually reached. As the two temperatures approach each other, the difference in energy exchanged is continually reduced. However, at some point the two temperatures will be indistinguishable. Perhaps when the temperature difference is less than 0.1°C, or when it is less than 0.0000001°C..

Therefore, it is conventional to talk about “reaching equilibrium” and no one in thermodynamics is confused about the reality of the above point.

Note 2 – Max Planck introduces thermodynamic equilibrium:

Note 3 – Geothermal energy is included in the diagram (P⁰). Given that it is less than 0.1 W/m² – below the noise level of most instruments measuring other fluxes in the climate – there is little point in cluttering up the equations here with this parameter.

Note 4 – Emissivity and absorptivity are wavelength dependent parameters. For example, snow is highly reflective for solar radiation but highly absorbing (and therefore emitting) for terrestrial radiation.

At the same wavelength, emissivity = absorptivity. This is the result of Kirchhoff’s law.

If the temperature of the source radiation for which we need to know the absorptivity is different from the temperature of the emitting body then we cannot assume that emissivity = absorptivity.

However, when the temperature of source body for the radiation being absorbed is within a few Kelvin of the emitting body then to a quite accurate assumption, absorptivity = emissivity.

For example, the radiation from a source of 288K is centered on 10.06 μm, while for 287 K it is centered on 10.10 μm. Around this temperature, the central wavelength decreases by about 0.035 μm for each 1 K change in temperature.

An example of when it is a totally incorrect assumption is for solar radiation absorbed by the earth. The solar radiation is from a source of about 5800 K and centered on 0.5 μm, whereas the terrestrial radiation is from a source of around 288 K and centered on 10 μm. Therefore, to assume that the absorptivity of the earth’s surface for solar radiation is equal to the emissivity of the earth’s surface is a huge mistake.

This would be the same as saying that absorptivity at 0.5 μm = emissivity at 10 μm. And, therefore, totally wrong.

Note 5: What exactly is meant by average emitting height? Emitted radiation varies as the 4th power of temperature and as a function of emissivity, which itself is a very non-linear function of quantity of absorbers. Average emitting height is more of a conceptual approach to illustrate the problem.

Read Full Post »

The Mystery of Tau – Miskolczi

Posted in Atmospheric Physics, Commentary on April 22, 2011| 60 Comments »

Many people have requested an analysis of Miskolczi’s theories.

I start with his more recent paper: The Stable Stationary Value of the Earth’s Global Average Atmospheric Planck-Weighted Greenhouse-Gas Optical Thickness, Energy & Environment (2010).

It’s an interesting paper and clearly Miskolczi has put a lot of time and effort into it. I recommend people read the paper for themselves, and the link above provides free access.

The essence of the claim is that the optical thickness of the earth’s atmosphere is a constant – at least over the last 60 years – where water vapor cancels out any change from CO2. So if more CO2 increases the optical thickness, then the optical thickness from water vapor will reduce.

In his paper he make this statement:

Unfortunately no computational results of E_U, S_T, A, T_A and τ_A can be found in the literature, and therefore our main purpose is to give realistic estimates of their global mean values, and investigate their dependence on the atmospheric CO2 concentration.

Among the terms noted in this quote, τ_A is the optical thickness of the atmosphere.

As we delve into the paper, hopefully the reasons why this value isn’t calculated in any papers will become clear. In fact, the first question people should be asking themselves is this:

If the result is of significant importance why has no one else calculated this parameter before?

There are thousands of papers about radiative transfer, CO2 and water vapor.

Why has no one (apparently) published their calculations of the globally averaged optical thickness of the atmosphere and how it has changed over time?

There is a reason..

What is Optical Thickness?

You can find a more complete explanation of optical thickness in Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations, which I definitely recommend reading even though it has many equations. (Actually, because it has many equations..)

Because optical thickness isn’t an obvious parameter, let’s start with a simpler property called transmittance.

Transmittance is the proportion of radiation which is transmitted through a body (in this case, the atmosphere). We will use the letter “t” to refer to it.

t has a value between 0 and 1. Slightly more formally, we can write 0 ≤ t≤ 1.

For t = 1, the body is totally transparent to incident radiation.

For t = 0, the body is totally opaque and absorbs all incident radiation.

For non-scattering atmospheres (note 1), absorptance, a = 1- t, which means that whatever is not absorbed gets transmitted. This is simple enough, and everyone would expect this from the First Law of Thermodynamics.

Now for optical thickness. We will use τ for this parameter. τ is the Greek letter “tau”.

The Beer-Lambert law says that the transmittance of a beam of radiation:

t = exp(-τ)

The “exp” is a mathematical convention for “e to the power of”. So this can alternatively be written as:

t = e^-τ

Which means that when τ = 1, t = 0.36; when τ = 2, t = 0.14; and when τ = 10, t = 0.000045.

Optical thickness is tedious to calculate because the properties of each gas vary strongly with wavelength.

In brief, for each molecule at each wavelength, the total optical thickness is equal to the total number of molecules in the path x the absorption coefficient (which is a function of wavelength).

So optical thickness is a very handy parameter. Calculating it does take some work and a pre-requisite is a database of all the spectroscopic values for each molecule – as well as knowing the total amount of each gas in the path we want to calculate.

Absorption and Emission

The atmosphere absorbs and also emits.

Absorption, as we have just seen, is a function of the total amount of each gas (in a path) as well as the properties of each gas.

And, in case it is not obvious, the total radiation absorbed is also a function of the intensity of radiation travelling through the body that we want to calculate. This is because absorption = incident radiation x absorptance.

What about emission?

Emission of radiation is a function of the temperature of the atmosphere, as well as its emissivity, ε. This parameter emissivity is equal to the absorptivity or absorptance, of a body at any given wavelength – or across a range of wavelengths. This is known as Kirchhoff’s law.

Emission = ε . σT⁴ in W/m², where T is the temperature of the atmosphere at that point.

If we want to calculate the radiative transfer through the atmosphere we need both terms.

Here is a simple example of why. Readers who followed the series Understanding Atmospheric Radiation and the “Greenhouse” Effect will remember that I introduced a simple atmosphere with two molecules, pCO2 and pH2O. These had a passing resemblance to the real molecules, but had properties that were much simpler, for the purposes of demonstrating some important aspects of how radiation interacts with the atmosphere.

This following example has three scenarios. Each scenario has the same total amount of water vapor through the atmosphere, but a different profile vs height. These are shown in the graph:

Figure 1

The bottom graph shows the top of atmosphere (TOA) flux from each of the three scenarios.

If we calculated the total transmittance through the atmosphere it would be the same in each scenario (update: correction – see Ken Gregory’s point below). Because the optical thickness is the same. The optical thickness is the same because the total number of pH2O molecules in the path is the same.

Yet the TOA flux is very different.

This is because where the atmosphere emits from is very important in calculations of flux. For example, in the case of the 3rd scenario, the TOA flux is lower because more of the water vapor is at colder temperatures, and less is at hotter temperatures.

From Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations:

dI_λ/dτ = I_λ – B_λ(T) [12]

which is also known as Schwarzschild’s Equation – and is the fundamental description of changes in radiation as it passes through an absorbing (and non-scattering) atmosphere. B_λ(T) = the Planck function, which is a function of temperature. And the subscript λ in each term identifies the wavelength dependence of this equation.

For the mathematically minded, it will be clear reviewing the above equation that total optical thickness tells you less than you need. As the location of optical thickness varies, if temperature varies (which it does in the atmosphere) then you can get different results for the same optical thickness.

That is, the simulations above demonstrate what is clear, and easily provable, from the form of the fundamental equation.

This is why papers on total optical thickness of the atmosphere over time are hard to come by. It is of curiosity value only.

What About Methane, Nitrous Oxide and Halocarbons?

The total optical thickness of the atmosphere is not just determined by water vapor and CO2. If the atmosphere has an invariant optical thickness then surely all molecules should be included?

According to WM Collins and his co-authors (2006):

The increased concentrations of CO2, CH4, and N2O between 1750 and 1998 have produced forcings of +1.48, +0.48, and +0.15 W m, respectively [IPCC, 2001]. The introduction of halocarbons in the mid-20th century has contributed an additional +0.34 Wm for a total forcing by WMGHGs of +2.45Wm with a 15% margin of uncertainty.

I’m sure someone with enough determination can find some results for the changes in the radiative forcing from CH4 and N2O between 1950 and 2010. But this at least demonstrates that there is some significant absorption characteristics for other molecules. After all, halocarbons have added a quarter of the longer term CO2 increase in radiative forcing from CO2 (from 1750 to the present day) in just half a century.

So if total optical thickness from CO2 and water vapor has stayed constant over 60 years then surely total optical thickness must have increased?

This is not mentioned in the paper and seems to be a major blow to the not-particularly-useful result calculated.

Update, 31st May: Ken Gregory, a Miskolczi supporter armed with the spreadsheet of calculations, says that minor gases were kept constant. So Part Six demonstrates my basic calculations of optical thickness changes due to CO2 and some minor gases.

Cloudy Thinking

Miskolczi says:

In all calculations of A, T_A, t_A, and of the radiative flux components, the presence or absence of clouds was ignored; the calculations refer only to the greenhouse gas components of the atmosphere registered in the radiosonde data; we call this the quasi-all-sky protocol. It is assumed, however, that the atmospheric vertical thermal and water vapor structures are implicitly affected by the actual cloud cover, and that the atmosphere is at a stable steady state of cloud cover.

Assumed but not demonstrated.

Clouds have a huge impact on the radiative (and convective) heat transfers in the atmosphere. From Clouds and Water Vapor – Part One:

Clouds reflect solar radiation by 48 W/m² but reduce the outgoing longwave radiation (OLR) by 30 W/m², therefore the average net effect of clouds – over this period at least – is to cool the climate by 18 W/m².

Are they constant?

Here is a snapshot from Vardavas & Taylor (2007):

From Vardavas & Taylor (2007)

Figure 2

Another important point – given the non-linearity of the equations of radiative transfer, even if the cloud cover stayed at a constant global percentage but the geographical distribution changed, the optical thickness of the atmosphere cannot be assumed constant.

Here are some values of cloud emissivity from Hartmann (1994):

From Hartmann (1994)

Figure 3

Just for some perspective, as emissivity reaches 0.8, τ = 1.6; with emissivity = 0.9, τ = 2.3. And Miskolczi calculates the global average optical thickness of the atmosphere – without clouds – at 1.87.

At the end of his paper, Miskolczi concludes:

Apparently, the global average cloud cover must not have a dramatic effect on the global average clear-sky optical thickness..

I can’t understand, from the paper, where this confidence comes from.

Conclusion

There is more in the paper, including some very suspect assumptions about radiative exchange. However, six out of the 19 references in the paper are to Miskolczi himself and the fundamental equations brought up for energy balance (where radiative exchange is referenced) rely on his more lengthy 2007 paper, Greenhouse effect in semi-transparent planetary atmospheres.

I will try to read this paper before commenting on these energy balance equations.

However, the key points are:

optical thickness of the total atmosphere is not a very useful number
the useful headline number has to be changes in TOA flux, or radiative forcing, or some value which expresses the overall radiative balance of the climate system (update: see this comment for the correct measure)
optical thickness calculated as constant over 60 years for CO2 and water vapor appears to prove that total optical thickness is not constant due to increases in other well-mixed “greenhouse” gases
clouds are not included in the calculation, but surely overwhelm the optical thickness calculations and cannot be assumed to be constant

Other Articles in the Series:

Part Two – Kirchhoff – why Kirchhoff’s law is wrongly invoked, as the author himself later acknowledged, from his 2007 paper

Part Three – Kinetic Energy – why kinetic energy cannot be equated with flux (radiation in W/m²), and how equation 7 is invented out of thin air (with interesting author comment)

Part Four – a minor digression into another error that seems to have crept into the Aa=Ed relationship

Part Five – Equation Soufflé – explaining why the “theory” in the 2007 paper is a complete dog’s breakfast

Part Six – Minor GHG’s – a less important aspect, but demonstrating the change in optical thickness due to the neglected gases N2O, CH4, CFC11 and CFC12.

Further Reading:

References

The Stable Stationary Value of the Earth’s Global Average Atmospheric Planck-Weighted Greenhouse-Gas Optical Thickness, Miskolczi, Energy & Environment (2010)

Radiative forcing by well-mixed greenhouse gases: Estimates from climate models in the Intergovernmental Panel on Climate Change (IPCC) Fourth Assessment Report (AR4), Collins et al, JGR (2006)

Radiation and Climate, Vardavas & Taylor, Oxford University Press (2007)

Global Physical Climatology, Hartmann, Academic Press (1994) – reviewed here

Notes

Note 1 – For longwave radiation (>4 μm), scattering is negligible in the atmosphere.

Read Full Post »

Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Eleven – Heating Rates

Posted in Atmospheric Physics on April 21, 2011| 38 Comments »

The subject of atmospheric heating rates is one which is worth spending time on.

What is a heating rate?

To see the usefulness of a heating rate let’s consider the per capita income of a country.

Per capita income compares the ratio of total $ to the total population. If we compare the total income of China to the total income of Laos we don’t have a useful comparison. If we compare the per capita income of China to the per capita income of Laos.. well, who knows whether we have a meaningful comparison – but at least we have something more useful. Something more relevant.

Energy absorbed in a layer of the atmosphere causes heating at a certain rate. Energy lost from a layer of the atmosphere causes cooling at a certain rate.

Heating rates tell us something different from total energy lost or gained. Suppose a 1m layer of the atmosphere gains 1,000 J/m², what will the temperature change be?

The specific heat capacity of the atmosphere at constant pressure is 1005 J/(K.kg) – which means it takes just over 1,000 J to lift the temperature of 1 kg of the atmosphere by 1K (=1°C).

However, the atmospheric density decreases with height:

Figure 1

At the surface, where pressure = 1000 mbar, the density = 1.2 kg/m³.

So 1,000 J/m² lifts the temperature of a 1m layer of the atmosphere at the surface by 0.83 K (calculated by ΔT=1,000/[1.2 x 1005]) .

At the top of the stratosphere, near 50km where the pressure = 1 mbar, the density = 0.0016 kg/m³.

Here, 1,000 J/m² lifts the temperature of a 1 m layer of the atmosphere by 620 K (calculated by ΔT=1,000/[0.0016 x 1005]).

So it’s a bit like sharing out the total income of China among the residents of Laos.

That’s why heating rates are useful – they relate the amount of energy with the amount of atmosphere.

More CO2 equals More Absorption but More CO2 equals More Emission

One of the confusing aspects in atmospheric radiation comes as people start to consider the fact that the atmosphere emits as well as absorbs.

So more radiatively-active gases (=”greenhouse” gases) causes more absorption? Or more emission? Or doesn’t one just balance out the other and so there is no change?

There are legitimate questions to ask.

The only way to answer these questions is to solve the Schwarzschild equation – see Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations.

What we will do is first of all review the heating/cooling rates vs altitude and try and understand some features qualititively.

Here is the right way to think about the problem:

Absorption at any given wavelength depends on:

the quantity of gases that absorb at that wavelength
the effectiveness of each gas at absorbing at that wavelength
the “amount” of radiation travelling through that part of the atmosphere (note 1).

Emission at any wavelength depends on:

the quantity of gases that absorb at that wavelength (and therefore emit at the same wavelength)
the effectiveness of each gas at absorbing (and therefore emitting) at that wavelength
the temperature of the gas

In the case of shortwave (=solar radiation) the atmosphere absorbs but does not emit. This is because the atmosphere is not hot enough to radiate significantly below 4 μm, see The Sun and Max Planck Agree – Part Two.

In the case of longwave (= terrestrial / atmospheric radiation) absorption is from radiation from above and below. But usually the radiation from below is a lot higher than from above. This is because the earth’s surface emits close to blackbody radiation (the surface has a very high emissivity over all wavelengths), and the atmosphere (which doesn’t emit as a blackbody) is hotter closer to the surface.

Doesn’t a Heating or Cooling Rate Mean that the Atmosphere is Heating up or Cooling Down?

No.

The sun heats the atmosphere (a heating rate), but the atmosphere radiates to space (a cooling rate), and also convection moves heat through the troposphere.

We can still have a heating rate, a cooling rate and convective heat transfer while the atmosphere is in approximate energy balance (=not changing in temperature). If the temperature of one part of the atmosphere is not changing then these will sum to zero.

So heating rates vs height give us insight into the strength of these effects, and we can break the effects up between the responsible gases (water vapor, CO2, ozone, etc).

Heating Rates

From the always excellent Grant Petty, A First Course in Atmospheric Radiation, the solar heating of the atmosphere, for a standardized tropical atmosphere:

From Petty (2006)

Figure 2 – Solar heating

If we showed total energy absorbed at each height in the atmosphere, then the troposphere would overwhelm the stratosphere (upper atmosphere). But because we are showing energy absorbed in proportion to the density of the atmosphere, the upper atmosphere appears more important.

We see that ozone causes the highest heating rate in the stratosphere, whereas water vapor causes the highest heating rate in the troposphere, and CO2 has a very small effect.

The water vapor heating rate is – of course – concentrated in the bottom few km of the atmosphere because water vapor is concentrated here.

Most of the absorbed solar radiation is absorbed by the earth’s surface. The surface absorption is not shown in this graph. In turn, the surface heats the atmosphere primarily through convection. The convective heat transfer is also not shown.

Now let’s look at longwave heating (cooling) rates for a few different regions:

From Petty (2006)

Figure 3

We see that the heating rates are mostly negative, meaning that these are really cooling rates. Most of the atmosphere is cooling via longwave radiation. However, one small part of the atmosphere experiences a heating rate due to longwave radiation – the tropical tropopause.

Why?

The tropopause is the coldest part of the atmosphere – the top of the troposphere and bottom of the stratosphere. And the coldest part of the atmosphere radiates less than it absorbs.

Let’s see the breakdown of cooling rates by individual gases:

From Petty (2006)

Figure 4

We see that water vapor has a peak longwave cooling at around 3 km and another maximum at 10 km. The lower peak is caused by the “continuum” – also shown separately on the graph – which we will return to shortly.

Ozone shows a heating rate in the stratosphere below 30km. If we had the graph extend up to the top of the stratosphere, around 50km, we would see ozone with a cooling to space higher up.

We also see that CO2 has a very small cooling effect until we get into the stratosphere. Generally, each layer experiences a very small heating/cooling effect from CO2 because CO2 has a such a strong absorption that energy is absorbed from layers very close by – which are at very similar temperatures. The tropopause is the coldest part of the atmosphere so absorbs a little more radiation via CO2 than it emits – consequently a small heating effect.

As we get up into the stratosphere we see a progressively stronger cooling to space from CO2. In part, this is because of the reduction in pressure broadening at lower pressures = higher altitudes (see Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Nine). This effect causes the absorptivity of CO2 to reduce at higher altitudes meaning that the radiation from the layer below can “get through” to space.

The Continuum

Water vapor has an unusual absorption profile. In the so-called “atmospheric window” of 8-12 μm where there are no strong absorption bands from any atmospheric molecule (apart from ozone at 9.6 μm), water vapor still absorbs. Here the absorption coefficient is proportional to the atmospheric pressure.

In general, the more we have of a particular gas, the more absorption. This is expressed in the Beer-Lambert law.

But for other gases, the absorption coefficient is a constant for a given wavelength – not proportional to pressure. In the Beer-Lambert law we multiply the mass of absorbing molecules in the path by the absorption coefficient to find the optical thickness (note 2).

For the water vapor continuum the absorption coefficient is proportional to pressure. And absorptivity is a function of the absorption coefficient and the total mass in the path (which is proportional to pressure). Therefore, total absorption in the continuum band is a very strong function of pressure.

Water vapor concentration is concentrated at lower levels in the atmosphere so the total absorption due to the water vapor continuum falls off very quickly with height.

This is why the lower peak cooling rate occurs. The absorption by water vapor due to the continuum above 3 km is very small – so around 3km the atmosphere (in these wavelengths) can very effectively cool to space. Other bands of water vapor absorb more strongly, so effective cooling to space doesn’t really begin until the concentration of water vapor drops to very low values. Hence the second peak at 10 km.

Stratospheric Cooling

A long time we had a look at Stratospheric Cooling. This strange phenonemon is expected from more CO2 in the atmosphere. All other things being equal, the troposphere will warm and the stratosphere will cool.

Radiative-convective models predict this. Once you’ve got to grips with basic radiation in the atmosphere, it is easy to see why the troposphere will warm.

But why will the stratosphere cool?

Some will look at Figure 4 and say “ah ha“. More CO2 will move the CO2 line over the left and so that’s why the stratosphere will cool.

As a cautionary note, the heating rate at level z is equal to:

From Petty (2006)

Figure 5

Where among other terms, the italicized “T” is the band-averaged transmittance between z and z’, and the integrals are (obviously to the mathematicians) for each “level” (note 3) between the surface and z, or between the top of atmosphere (∞) and z..

If we went through this equation we would find that there are competing terms – terms which represent absorption of energy from other parts of the atmosphere (heating), and terms which represent emission of energy from this layer (cooling). Increasing CO2 increases absorption in the stratosphere. Increasing CO2 increases emission from the stratosphere.

Given that radiative-convective models predict stratospheric cooling we can say confidently that more CO2 will move the cooling curve in figure 4 to the left in the stratosphere (note 4). So emission will be higher than absorption.

However, we haven’t developed an intuitive understanding of why. At least, I haven’t.

To develop an intuitive understanding I would need the solution of these equations for a variety of conditions, and after playing around with changed parameters and reviewing results it would all start to make sense. That’s what I would hope.

But that’s just me. Others can perhaps just see it all dance out of the equations in a flash (think – the crazy one in The Hangover in the casino). Or out of the fundamental physics.

Conclusion

Heating rates help give insight into how the atmosphere absorbs and emits radiation from different “greenhouse” gases at different levels.

Generally the peak cooling rates for each band occur when that band “thins out” enough in the layers above to allow significant radiation to space, rather than just to the level immediately above.

Convection is the most important mechanism for moving heat in the troposphere (but not the stratosphere).

This article hasn’t considered convection at all – which just demonstrates the ongoing plot to hide the importance of convection. Once people realize how important convection is, radiative heating and radiative cooling to space will be.. the same.