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The Atmosphere Cools to Space by CO2 and Water Vapor, so More GHGs, More Cooling!

Posted in Atmospheric Physics, Basic Science, Debunking Flawed "Science" on August 12, 2014| 50 Comments »

The atmosphere cools to space by radiation. Well, without getting into all the details, the surface cools to space as well by radiation but not much radiation is emitted by the surface that escapes directly to space (note 1). Most surface radiation is absorbed by the atmosphere. And of course the surface mostly cools by convection into the troposphere (lower atmosphere).

If there were no radiatively-active gases (aka “GHG”s) in the atmosphere then the atmosphere couldn’t cool to space at all.

Technically, the emissivity of the atmosphere would be zero. Emission is determined by the local temperature of the atmosphere and its emissivity. Wavelength by wavelength emissivity is equal to absorptivity, another technical term, which says what proportion of radiation is absorbed by the atmosphere. If the atmosphere can’t emit, it can’t absorb (note 2).

So as you increase the GHGs in the atmosphere you increase its ability to cool to space. A lot of people realize this at some point during their climate science journey and finally realize how they have been duped by climate science all along! It’s irrefutable – more GHGs more cooling to space, more GHGs mean less global warming!

Ok, it’s true. Now the game’s up, I’ll pack up Science of Doom into a crate and start writing about something else. Maybe cognitive dissonance..

Bye everyone!

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Halfway through boxing everything up I realized there was a little complication to the simplicity of that paragraph. The atmosphere with more GHGs has a higher emissivity, but also a higher absorptivity.

Let’s draw a little diagram. Here are two “layers” (see note 3) of the atmosphere in two different cases. On the left 400 ppmv CO2, on the right 500ppmv CO2 (and relative humidity of water vapor was set at 50%, surface temperature at 288K):

Cooling-to-space-2a

Figure 1

It’s clear that the two layers are both emitting more radiation with more CO2.More cooling to space.

For interest, the “total emissivity” of the top layer is 0.190 in the first case and 0.197 in the second case. The layer below has 0.389 and 0.395.

Let’s take a look at all of the numbers and see what is going on. This diagram is a little busier:

Cooling-to-space-3a

Figure 2

The key point is that the OLR (outgoing longwave radiation) is lower in the case with more CO2. Yet each layer is emitting more radiation. How can this be?

Take a look at the radiation entering the top layer on the left = 265.1, and add to that the emitted radiation = 23.0 – the total is 288.1. Now subtract the radiation leaving through the top boundary = 257.0 and we get the radiation absorbed in the layer. This is 31.1 W/m².

Compare that with the same calculation with more CO2 – the absorption is 32.2 W/m².

This is the case all the way up through the atmosphere – each layer emits more because its emissivity has increased, but it also absorbs more because its absorptivity has increased by the same amount.

So more cooling to space, but unfortunately more absorption of the radiation below – two competing terms.

So why don’t they cancel out?

Emission of radiation is a result of local temperature and emissivity.

Absorption of radiation is the result of the incident radiation and absorptivity. Incident upwards radiation started lower in the atmosphere where it is hotter. So absorption changes always outweigh emission changes (note 4).

Conceptual Problems?

If it’s still not making sense then think about what happens as you reduce the GHGs in the atmosphere. The atmosphere emits less but absorbs even less of the radiation from below. So the outgoing longwave radiation increases. More surface radiation is making it to the top of atmosphere without being absorbed. So there is less cooling to space from the atmosphere, but more cooling to space from the surface and the atmosphere.

If you add lagging to a pipe, the temperature of the pipe increases (assuming of course it is “internally” heated with hot water). And yet, the pipe cools to the surrounding room via the lagging! Does that mean more lagging, more cooling? No, it’s just the transfer mechanism for getting the heat out.

That was just an analogy. Analogies don’t prove anything. If well chosen, they can be useful in illustrating problems. End of analogy disclaimer.

If you want to understand more about how radiation travels through the atmosphere and how GHG changes affect this journey, take a look at the series Visualizing Atmospheric Radiation.

 

Notes

Note 1: For more on the details see

Note 2: A very basic point – absolutely essential for understanding anything at all about climate science – is that the absorptivity of the atmosphere can be (and is) totally different from its emissivity when you are considering different wavelengths. The atmosphere is quite transparent to solar radiation, but quite opaque to terrestrial radiation – because they are at different wavelengths. 99% of solar radiation is at wavelengths less than 4 μm, and 99% of terrestrial radiation is at wavelengths greater than 4 μm. That’s because the sun’s surface is around 6000K while the earth’s surface is around 290K. So the atmosphere has low absorptivity of solar radiation (<4 μm) but high emissivity of terrestrial radiation.

Note 3: Any numerical calculation has to create some kind of grid. This is a very course grid, with 10 layers of roughly equal pressure in the atmosphere from the surface to 200mbar. The grid assumes there is just one temperature for each layer. Of course the temperature is decreasing as you go up. We could divide the atmosphere into 30 layers instead. We would get more accurate results. We would find the same effect.

Note 4: The equations for radiative transfer are found in Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations. The equations prove this effect.

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A Challenge for Bryan – The Solution

Posted in Basic Science, Debunking Flawed "Science" on August 11, 2014| 420 Comments »

In A Challenge for Bryan I put up a simple heat transfer problem and asked for the equations. Bryan elected not to provide these equations. So I provide the answer, but also attempt some enlightenment for people who don’t think the answer can be correct.

As DeWitt Payne noted, a post with a similar problem posted on Wattsupwiththat managed to gather some (unintentionally) hilarious comments.

Here’s the problem again:

Case 1

Spherical body, A, of radius ra, with an emissivity, εa =1. The sphere is in the vacuum of space.

It is internally heated by a mystery power source (let’s say nuclear, but it doesn’t matter), with power input = P.

The sphere radiates into deep space, let’s say the temperature of deep space = 0K to make the maths simpler.

1. What is the equation for the equilibrium surface temperature of the sphere, Ta?

Case 2

The condition of case A, but now body A is surrounded by a slightly larger spherical shell, B, which of course is itself now surrounded by deep space at 0K.

B has a radius rb, with an emissivity, εb =1. This shell is highly conductive and very thin.

2a. What is the equation for the new equilibrium surface temperature, Ta’?

2b. What is the equation for the equilibrium temperature, Tb, of shell B?

 

Notes:

The reason for the “slightly larger shell” is to avoid “complex” view factor issues. Of course, I’m happy to relax the requirement for “slightly larger” and let Bryan provide the more general answer.

The reason for the “highly conductive” and “thin” outer shell, B, is to avoid any temperature difference between the inside and the outside surfaces of the shell. That is, we can assume the outside surface is at the same temperature as the inside surface – both at temperature, Tb.

This kind of problem is a staple of introductory heat transfer. This is a “find the equilibrium” problem.

How do we solve these kinds of problems? It’s pretty easy once you understand the tools.

The first tool is the first law of thermodynamics. Steady state means temperatures have stabilized and so energy in = energy out. We draw a “boundary” around each body and apply the “boundary condition” of the first law.

The second tool is the set of equations that govern the movement of energy. These are the equations for conduction, convection and radiation. In this case we just have radiation to consider.

For people who see the solution, shake their heads and say, this can’t be, stay on to the end and I will try and shed some light on possible conceptual problems. Of course, if it’s wrong, you should easily be able to provide the correct equations – or even if you can’t write equations you should be able to explain the flaw in the formulation of the equation.

In the original article I put some numbers down – “For anyone who wants to visualize some numbers: ra=1m, P=1000W, rb=1.01m“. I will use these to calculate an answer from the equations. I realize many readers aren’t comfortable with equations and so the answers will help illuminate the meaning of the equations.

I go through the equations in tedious detail, again for people who would like to follow the maths but don’t find maths easy.

Case 1

Energy in, Ein = Energy out, Eout  :  in Watts (Joules per second).

Ein = P

Eout = emission of thermal radiation per unit area x area

The first part is given by the Stefan-Boltzmann equation (σTa4, where σ = 5.67×10-8), and the second part by the equation for the surface area of a sphere (4πra²)

Eout = 4πra² x σTa4 …..[eqn 1]

Therefore, P = 4πra²σTa4 ….[eqn 2]

We have to rearrange the equation to see how Ta changes with the other factors:

Ta = [P / (4πra²σ)]1/4 ….[eqn 3]

If you aren’t comfortable with maths this might seem a little daunting. Let’s put the numbers in:

Ta = 194K (-80ºC)

Now we haven’t said anything about how long it takes to reach this temperature. We don’t have enough information for that. That’s the nice thing about steady state calculations, they are easier than dynamic calculations. We will look at that at the end.

Probably everyone is happy with this equation. Energy is conserved. No surprises and nothing controversial.

Now we will apply the exact same approach to the second case.

Case 2

First we consider “body A”. Given that it is enclosed by another “body” – the shell B – we have to consider any energy being transferred by radiation from B to A. If it turns out to be zero, of course it won’t affect the temperature of body A.

Ein(a) = P + Eb-a ….[eqn 4], where Eb-a is a value we don’t yet know. It is the radiation from B absorbed by A.

Eout(a) = 4πra² x σTa4 ….[eqn 5]- this is the same as in case 1. Emission of radiation from a body only depends on its temperature (and emissivity and area but these aren’t changing between the two cases)

– we will look at shell B and come back to the last term in eqn 4.

Now the shell outer surface:

Radiates out to space

We set space at absolute zero so no radiation is received by the outer surface

Shell inner surface:

Radiates in to A (in fact almost all of the radiation emitted from the inner surface is absorbed by A and for now we will treat it as all) – this was the term Eb-a

Absorbs all of the radiation emitted by A, this is Eout(a)

And we made the shell thin and highly conductive so there is no temperature difference between the two surfaces. Let’s collect the heat transfer terms for shell B under steady state:

Ein(b) = Eout(a) + 0  …..[eqn 6] – energy in is all from the sphere A, and nothing from outside

             =  4πra² x σTa4 ….[eqn 6a] – we just took the value from eqn 5

Eout(b) = 4πrb² x σTb4 + 4πrb² x σTb4 …..[eqn 7] – energy out is the emitted radiation from the inner surface + emitted radiation from the outer surface

                = 2 x 4πrb² x σTb4 ….[eqn 7a]

 And we know that for shell B, Ein = Eout so we equate 6a and 7a:

4πra² x σTa4 = 2 x 4πrb² x σTb4 ….[eqn 8]

and now we can cancel a lot of the common terms:

ra² x Ta4 = 2 x rb² x Tb4 ….[eqn 8a]

and re-arrange to get Ta in terms of Tb:

Ta4 = 2rb²/ra² x Tb4 ….[eqn 8b]

Ta = [2rb²/ra²]1/4 x Tb ….[eqn 8b]

or we can write it the other way round:

Tb = [ra²/2rb²]1/4 x Ta ….[eqn 8c]

Using the numbers given, Ta = 1.2 Tb. So the sphere is 20% warmer than the shell (actually 2 to the power 1/4).

We need to use Ein=Eout for the sphere A to be able to get the full solution. We wrote down: Ein(a) = P + Eb-a ….[eqn 4]. Now we know “Eb-a” – this is one of the terms in eqn 7.

So:

Ein(a) = P + 4πrb² x σTb4 ….[eqn 9]

and Ein(a) = Eout(a), so:

P + 4πrb² x σTb4 = 4πra² x σTa4  ….[eqn 9]

we can substitute the equation for Tb:

P + 4πra² /2 x σTa4 = 4πra² x σTa4  ….[eqn 9a]

the 2nd term on the left and the right hand side can be combined:

P = 2πra² x σTa4  ….[eqn 9a]

And so, voila:

T’a = [P / (2πra²σ)]1/4 ….[eqn 10] – I added a dash to Ta so we can compare it with the original value before the shell arrived.

T’a = 21/4 Ta   ….[eqn 11] – that is, the temperature of the sphere A is about 20% warmer in case 2 compared with case 1.

Using the numbers, T’a = 230 K (-43ºC). And Tb = 193 K (-81ºC)

Explaining the Results

In case 2, the inner sphere, A, has its temperature increase by 36K even though the same energy production takes place inside. Obviously, this can’t be right because we have created energy??.. let’s come back to that shortly.

Notice something very important – Tb in case 2 is almost identical to Ta in case 1. The difference is actually only due to the slight difference in surface area. Why?

The system has an energy production, P, in both cases.

  • In case 1, the sphere A is the boundary transferring energy to space and so its equilibrium temperature must be determined by P
  • In case 2, the shell B is the boundary transferring energy to space and so its equilibrium temperature must be determined by P

Now let’s confirm the mystery unphysical totally fake invented energy.

Let’s compare the flux emitted from A in case 1 and case 2. I’ll call it R.

  • R(case 1) = 80 W/m²
  • R(case 2) = 159 W/m²

This is obviously rubbish. The same energy source inside the sphere and we doubled the sphere’s energy production!!! Get this idiot to take down this post, he has no idea what he is writing..

Yet if we check the energy balance we find that 80 W/m² is being “created” by our power source, and the “extra mystery” energy of 79 W/m² is coming from our outer shell. In any given second no energy is created.

The Mystery Invented Energy – Revealed

When we snapped the outer shell over the sphere we made it harder for heat to get out of the system. Energy in = energy out, in steady state. When we are not in steady state: energy in – energy out = energy retained. Energy retained is internal energy which is manifested as temperature.

We made it hard for heat to get out, which accumulated energy, which increased temperature.. until finally the inner sphere A was hot enough for all of the internally generated energy, P, to get out of the system.

Let’s add some information about the system: the heat capacity of the sphere = 1000 J/K; the heat capacity of the shell = 100 J/K. It doesn’t much matter what they are, it’s just to calculate the transients. We snap the shell – originally at 0K – around the sphere at time t=100 seconds and see what happens.

The top graph shows temperature, the bottom graph shows change in energy of the two objects and how much energy is leaving the system:

Bryan-sphere

At 100 seconds we see that instead of our steady state 1000W leaving the system, instead 0W leaves the system. This is the important part of the mystery energy puzzle.

We put a 0K shell around the sphere. This absorbs all the energy from the sphere. At time t=100s the shell is still at 0K so it emits 0W/m². It heats up pretty quickly, but remember that emission of radiation is not linear with temperature so you don’t see a linear relationship between the temperature of shell B and the energy leaving to space. For example at 100K, the outward emission is 6 W/m², at 150K it is 29 W/m² and at its final temperature of 193K, it is 79 W/m² (=1000 W in total).

As the shell heats up it emits more and more radiation inwards, heating up the sphere A.

The mystery energy has been revealed. The addition of a radiation barrier stopped energy leaving, which stored heat. The way equilibrium is finally restored is due to the temperature increase of the sphere.

Of course, for some strange reason an army of people thinks this is totally false. Well, produce your equations.. (this never happens)

All we have done here is used conservation of energy and the Stefan Boltzmann law of emission of thermal radiation.

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A Challenge for Bryan

Posted in Basic Science, Debunking Flawed "Science" on August 7, 2014| 105 Comments »

Bryan needs no introduction on this blog, but if we were to introduce him it would be as the fearless champion of Gerlich and Tscheuschner.

Bryan has been trying to teach me some basics on heat transfer from the Ladybird Book of Thermodynamics. In hilarious fashion we both already agree on that particular point.

So now here is a problem for Bryan to solve.

Of course, in Game of Thrones fashion, Bryan can nominate his own champion to solve the problem.

Case A

Spherical body, A, of radius ra, with an emissivity, εa =1. The sphere is in the vacuum of space.

It is internally heated by a mystery power source (let’s say nuclear, but it doesn’t matter), with power input = P.

The sphere radiates into deep space, let’s say the temperature of deep space = 0K to make the maths simpler.

1. What is the equation for the equilibrium surface temperature of the sphere, Ta?

Case B

The condition of case A, but now body A is surrounded by a slightly larger spherical shell, B, which of course is itself now surrounded by deep space at 0K.

B has a radius rb, with an emissivity, εb =1. This shell is highly conductive and very thin.

2a. What is the equation for the new equilibrium surface temperature, Ta’?

2b. What is the equation for the equilibrium temperature, Tb, of shell B?

 

Notes:

The reason for the “slightly larger shell” is to avoid “complex” view factor issues. Of course, I’m happy to relax the requirement for “slightly larger” and let Bryan provide the more general answer.

The reason for the “highly conductive” and “thin” outer shell, B, is to avoid any temperature difference between the inside and the outside surfaces of the shell. That is, we can assume the outside surface is at the same temperature as the inside surface – both at temperature, Tb.

For anyone who wants to visualize some numbers: ra=1m, P=1000W, rb=1.01m

This problem takes a couple of minutes to solve on a piece of paper. I suspect we will wait a decade for Bryan’s answer. But I love to be proved wrong!

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Does the surface temperature change with “back radiation”? Kramm vs Gerlich

Posted in Basic Science, Commentary, Debunking Flawed "Science" on July 20, 2014| 113 Comments »

In The “Greenhouse” Effect Explained in Simple Terms I list, and briefly explain, the main items that create the “greenhouse” effect. I also explain why more CO2 (and other GHGs) will, all other things remaining equal, increase the surface temperature. I recommend that article as the place to go for the straightforward explanation of the “greenhouse” effect. It also highlights that the radiative balance higher up in the troposphere is the most important component of the “greenhouse” effect.

However, someone recently commented on my first Kramm & Dlugi article and said I was “plainly wrong”. Kramm & Dlugi were in complete agreement with Gerlich and Tscheuschner because they both claim the “purported greenhouse effect simply doesn’t exist in the real world”.

If it’s just about flying a flag or wearing a football jersey then I couldn’t agree more. However, science does rely on tedious detail and “facts” rather than football jerseys. As I pointed out in New Theory Proves AGW Wrong! two contradictory theories don’t add up to two theories making the same case..

In the case of the first Kramm & Dlugi article I highlighted one point only. It wasn’t their main point. It wasn’t their minor point. They weren’t even making a point of it at all.

Many people believe the “greenhouse” effect violates the second law of thermodynamics, these are herein called “the illuminati”.

Kramm & Dlugi’s equation demonstrates that the illuminati are wrong. I thought this was worth pointing out.

The “illuminati” don’t understand entropy, can’t provide an equation for entropy, or even demonstrate the flaw in the simplest example of why the greenhouse effect is not in violation of the second law of thermodynamics. Therefore, it is necessary to highlight the (published) disagreement between celebrated champions of the illuminati – even if their demonstration of the disagreement was unintentional.

Let’s take a look.

Here is the one of the most popular G&T graphics in the blogosphere:

From Gerlich & Tscheuschner

From Gerlich & Tscheuschner

Figure 1

It’s difficult to know how to criticize an imaginary diagram. We could, for example, point out that it is imaginary. But that would be picky.

We could say that no one draws this diagram in atmospheric physics. That should be sufficient. But as so many of the illuminati have learnt their application of the second law of thermodynamics to the atmosphere from this fictitious diagram I feel the need to press forward a little.

Here is an extract from a widely-used undergraduate textbook on heat transfer, with a little annotation (red & blue):

From Incropera & DeWitt (2007)

From “Fundamentals of Heat and Mass Transfer” by Incropera & DeWitt (2007)

Figure 2

This is the actual textbook, before the Gerlich manoeuvre as I would like to describe it. We can see in the diagram and in the text that radiation travels both ways and there is a net transfer which is from the hotter to the colder. The term “net” is not really capable of being confused. It means one minus the other, “x-y”. Not “x”. (For extracts from six heat transfer textbooks and their equations read Amazing Things we Find in Textbooks – The Real Second Law of Thermodynamics).

Now let’s apply the Gerlich manoeuvre (compare fig. 2):

Fundamentals-of-heat-and-mass-transfer-post-G&T

Not from “Fundamentals of Heat and Mass Transfer”, or from any textbook ever

Figure 3

So hopefully that’s clear. Proof by parody. This is “now” a perpetual motion machine and so heat transfer textbooks are wrong. All of them. Somehow.

Just for comparison, we can review the globally annually averaged values of energy transfer in the atmosphere, including radiation, from Kiehl & Trenberth (I use the 1997 version because it is so familiar even though values were updated more recently):

From Kiehl & Trenberth (1997)

From Kiehl & Trenberth (1997)

Figure 4

It should be clear that the radiation from the hotter surface is higher than the radiation from the colder atmosphere. If anyone wants this explained, please ask.

I could apply the Gerlich manoeuvre to this diagram but they’ve already done that in their paper (as shown above in figure 1).

So lastly, we return to Kramm & Dlugi, and their “not even tiny point”, which nevertheless makes a useful point. They don’t provide a diagram, they provide an equation for energy balance at the surface – and I highlight each term in the equation to assist the less mathematically inclined:

Kramm-Dlugi-2011-eqn-highlight

 

Figure 5

The equation says, the sum of all fluxes – at one point on the surface = 0. This is an application of the famous first law of thermodynamics, that is, energy cannot be created or destroyed.

The red term – absorbed atmospheric radiation – is the radiation from the colder atmosphere absorbed by the hotter surface. This is also known as “DLR” or “downward longwave radiation, and as “back-radiation”.

Now, let’s assume that the atmospheric radiation increases in intensity over a small period. What happens?

The only way this equation can continue to be true is for one or more of the last 4 terms to increase.

  • The emitted surface radiation – can only increase if the surface temperature increases
  • The latent heat transfer – can only increase if there is an increase in wind speed or in the humidity differential between the surface and the atmosphere just above
  • The sensible heat transfer – can only increase if there is an increase in wind speed or in the temperature differential between the surface and the atmosphere just above
  • The heat transfer into the ground – can only increase if the surface temperature increases or the temperature below ground spontaneously cools

So, when atmospheric radiation increases the surface temperature must increase (or amazingly the humidity differential spontaneously increases to balance, but without a surface temperature change). According to G&T and the illuminati this surface temperature increase is impossible. According to Kramm & Dlugi, this is inevitable.

I would love it for Gerlich or Tscheuschner to show up and confirm (or deny?):

  • yes the atmosphere does emit thermal radiation
  • yes the surface of the earth does absorb atmospheric thermal radiation
  • yes this energy does not disappear (1st law of thermodynamics)
  • yes this energy must increase the temperature of the earth’s surface above what it would be if this radiation did not exist (1st law of thermodynamics)

Or even, which one of the above is wrong. That would be outstanding.

Of course, I know they won’t do that – even though I’m certain they believe all of the above points. (Likewise, Kramm & Dlugi won’t answer the question I have posed of them).

Well, we all know why

Hopefully, the illuminati can contact Kramm & Dlugi and explain to them where they went wrong. I have my doubts that any of the illuminati have grasped the first law of thermodynamics or the equation for temperature change and heat capacity, but who could say.

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The “Greenhouse” Effect Explained in Simple Terms

Posted in Atmospheric Physics, Basic Science, Debunking Flawed "Science" on June 26, 2014| 591 Comments »

Over the last few years I’ve written lots of articles relating to the inappropriately-named “greenhouse” effect and covered some topics in great depth. I’ve also seen lots of comments and questions which has helped me understand common confusion and misunderstandings.

This article, with huge apologies to regular long-suffering readers, covers familiar ground in simple terms. It’s a reference article. I’ve referenced other articles and series as places to go to understand a particular topic in more detail.

One of the challenges of writing a short simple explanation is it opens you up to the criticism of having omitted important technical details that you left out in order to keep it short. Remember this is the simple version..

Preamble

First of all, the “greenhouse” effect is not AGW. In maths, physics, engineering and other hard sciences, one block is built upon another block. AGW is built upon the “greenhouse” effect. If AGW is wrong, it doesn’t invalidate the greenhouse effect. If the greenhouse effect is wrong, it does invalidate AGW.

The greenhouse effect is built on very basic physics, proven for 100 years or so, that is not in any dispute in scientific circles. Fantasy climate blogs of course do dispute it.

Second, common experience of linearity in everyday life cause many people to question how a tiny proportion of “radiatively-active” molecules can have such a profound effect. Common experience is not a useful guide. Non-linearity is the norm in real science. Since the enlightenment at least, scientists have measured things rather than just assumed consequences based on everyday experience.

The Elements of the “Greenhouse” Effect

Atmospheric Absorption

1. The “radiatively-active” gases in the atmosphere:

  • water vapor
  • CO2
  • CH4
  • N2O
  • O3
  • and others

absorb radiation from the surface and transfer this energy via collision to the local atmosphere. Oxygen and nitrogen absorb such a tiny amount of terrestrial radiation that even though they constitute an overwhelming proportion of the atmosphere their radiative influence is insignificant (note 1).

How do we know all this? It’s basic spectroscopy, as detailed in exciting journals like the Journal of Quantitative Spectroscopy and Radiative Transfer over many decades. Shine radiation of a specific wavelength through a gas and measure the absorption. Simple stuff and irrefutable.

Atmospheric Emission

2. The “radiatively-active” gases in the atmosphere also emit radiation. Gases that absorb at a wavelength also emit at that wavelength. Gases that don’t absorb at that wavelength don’t emit at that wavelength. This is a consequence of Kirchhoff’s law.

The intensity of emission of radiation from a local portion of the atmosphere is set by the atmospheric emissivity and the temperature.

Convection

3. The transfer of heat within the troposphere is mostly by convection. The sun heats the surface of the earth through the (mostly) transparent atmosphere (note 2). The temperature profile, known as the “lapse rate”, is around 6K/km in the tropics. The lapse rate is principally determined by non-radiative factors – as a parcel of air ascends it expands into the lower pressure and cools during that expansion (note 3).

The important point is that the atmosphere is cooler the higher you go (within the troposphere).

Energy Balance

4. The overall energy in the climate system is determined by the absorbed solar radiation and the emitted radiation from the climate system. The absorbed solar radiation – globally annually averaged – is approximately 240 W/m² (note 4). Unsurprisingly, the emitted radiation from the climate system is also (globally annually averaged) approximately 240 W/m². Any change in this and the climate is cooling or warming.

Emission to Space

5. Most of the emission of radiation to space by the climate system is from the atmosphere, not from the surface of the earth. This is a key element of the “greenhouse” effect. The intensity of emission depends on the local atmosphere. So the temperature of the atmosphere from which the emission originates determines the amount of radiation.

If the place of emission of radiation – on average – moves upward for some reason then the intensity decreases. Why? Because it is cooler the higher up you go in the troposphere. Likewise, if the place of emission – on average – moves downward for some reason, then the intensity increases (note 5).

More GHGs

6. If we add more radiatively-active gases (like water vapor and CO2) then the atmosphere becomes more “opaque” to terrestrial radiation and the consequence is the emission to space from the atmosphere moves higher up (on average). Higher up is colder. See note 6.

So this reduces the intensity of emission of radiation, which reduces the outgoing radiation, which therefore adds energy into the climate system. And so the climate system warms (see note 7).

That’s it!

It’s as simple as that. The end.

A Few Common Questions

CO2 is Already Saturated

There are almost 315,000 individual absorption lines for CO2 recorded in the HITRAN database. Some absorption lines are stronger than others. At the strongest point of absorption – 14.98 μm (667.5 cm-1), 95% of radiation is absorbed in only 1m of the atmosphere (at standard temperature and pressure at the surface). That’s pretty impressive.

By contrast, from 570 – 600 cm-1 (16.7 – 17.5 μm) and 730 – 770 cm-1 (13.0 – 13.7 μm) the CO2 absorption through the atmosphere is nowhere near “saturated”. It’s more like 30% absorbed through a 1km path.

You can see the complexity of these results in many graphs in Atmospheric Radiation and the “Greenhouse” Effect – Part Nine – calculations of CO2 transmittance vs wavelength in the atmosphere using the 300,000 absorption lines from the HITRAN database, and see also Part Eight – interesting actual absorption values of CO2 in the atmosphere from Grant Petty’s book

The complete result combining absorption and emission is calculated in Visualizing Atmospheric Radiation – Part Seven – CO2 increases – changes to TOA in flux and spectrum as CO2 concentration is increased

CO2 Can’t Absorb Anything of Note Because it is Only .04% of the Atmosphere

See the point above. Many spectroscopy professionals have measured the absorptivity of CO2. It has a huge variability in absorption, but the most impressive is that 95% of 14.98 μm radiation is absorbed in just 1m. How can that happen? Are spectroscopy professionals charlatans? You need evidence, not incredulity. Science involves measuring things and this has definitely been done. See the HITRAN database.

Water Vapor Overwhelms CO2

This is an interesting point, although not correct when we consider energy balance for the climate. See Visualizing Atmospheric Radiation – Part Four – Water Vapor – results of surface (downward) radiation and upward radiation at TOA as water vapor is changed.

The key point behind all the detail is that the top of atmosphere radiation change (as CO2 changes) is the important one. The surface change (forcing) from increasing CO2 is not important, is definitely much weaker and is often insignificant. Surface radiation changes from CO2 will, in many cases, be overwhelmed by water vapor.

Water vapor does not overwhelm CO2 high up in the atmosphere because there is very little water vapor there – and the radiative effect of water vapor is dramatically impacted by its concentration, due to the “water vapor continuum”.

The Calculation of the “Greenhouse” Effect is based on “Average Surface Temperature” and there is No Such Thing

Simplified calculations of the “greenhouse” effect use some averages to make some points. They help to create a conceptual model.

Real calculations, using the equations of radiative transfer, don’t use an “average” surface temperature and don’t rely on a 33K “greenhouse” effect. Would the temperature decrease 33K if all of the GHGs were removed from the atmosphere? Almost certainly not. Because of feedbacks. We don’t know the effect of all of the feedbacks. But would the climate be colder? Definitely.

See The Rotational Effect – why the rotation of the earth has absolutely no effect on climate, or so a parody article explains..

The Second Law of Thermodynamics Prohibits the Greenhouse Effect, or so some Physicists Demonstrated..

See The Three Body Problem – a simple example with three bodies to demonstrate how a “with atmosphere” earth vs a “without atmosphere earth” will generate different equilibrium temperatures. Please review the entropy calculations and explain (you will be the first) where they are wrong or perhaps, or perhaps explain why entropy doesn’t matter (and revolutionize the field).

See Gerlich & Tscheuschner for the switch and bait routine by this operatic duo.

And see Kramm & Dlugi On Dodging the “Greenhouse” Bullet – Kramm & Dlugi demonstrate that the “greenhouse” effect doesn’t exist by writing a few words in a conclusion but carefully dodging the actual main point throughout their entire paper. However, they do recover Kepler’s laws and point out a few errors in a few websites. And note that one of the authors kindly showed up to comment on this article but never answered the important question asked of him. Probably just too busy.. Kramm & Dlugi also helpfully (unintentionally) explain that G&T were wrong, see Kramm & Dlugi On Illuminating the Confusion of the Unclear – Kramm & Dlugi step up as skeptics of the “greenhouse” effect, fans of Gerlich & Tscheuschner and yet clarify that colder atmospheric radiation is absorbed by the warmer earth..

And for more on that exciting subject, see Confusion over the Basics under the sub-heading The Second Law of Thermodynamics.

Feedbacks overwhelm the Greenhouse Effect

This is a totally different question. The “greenhouse” effect is the “greenhouse” effect. If the effect of more CO2 is totally countered by some feedback then that will be wonderful. But that is actually nothing to do with the “greenhouse” effect. It would be a consequence of increasing temperature.

As noted in the preamble, it is important to separate out the different building blocks in understanding climate.

Miskolczi proved that the Greenhouse Effect has no Effect

Miskolczi claimed that the greenhouse effect was true. He also claimed that more CO2 was balanced out by a corresponding decrease in water vapor. See the Miskolczi series for a tedious refutation of his paper that was based on imaginary laws of thermodynamics and questionable experimental evidence.

Once again, it is important to be able to separate out two ideas. Is the greenhouse effect false? Or is the greenhouse effect true but wiped out by a feedback?

If you don’t care, so long as you get the right result you will be in ‘good’ company (well, you will join an extremely large company of people). But this blog is about science. Not wishful thinking. Don’t mix the two up..

Convection “Short-Circuits” the Greenhouse Effect

Let’s assume that regardless of the amount of energy arriving at the earth’s surface, that the lapse rate stays constant and so the more heat arriving, the more heat leaves. That is, the temperature profile stays constant. (It’s a questionable assumption that also impacts the AGW question).

It doesn’t change the fact that with more GHGs, the radiation to space will be from a higher altitude. A higher altitude will be colder. Less radiation to space and so the climate warms – even with this “short-circuit”.

In a climate without convection, the surface temperature will start off higher, and the GHG effect from doubling CO2 will be higher. See Radiative Atmospheres with no Convection.

In summary, this isn’t an argument against the greenhouse effect, this is possibly an argument about feedbacks. The issue about feedbacks is a critical question in AGW, not a critical question for the “greenhouse” effect. Who can say whether the lapse rate will be constant in a warmer world?

Notes

Note 1 – An important exception is O2 absorbing solar radiation high up above the troposphere (lower atmosphere). But O2 does not absorb significant amounts of terrestrial radiation.

Note 2 – 99% of solar radiation has a wavelength <4μm. In these wavelengths, actually about 1/3 of solar radiation is absorbed in the atmosphere. By contrast, most of the terrestrial radiation, with a wavelength >4μm, is absorbed in the atmosphere.

Note 3 – see:

Density, Stability and Motion in Fluids – some basics about instability
Potential Temperature – explaining “potential temperature” and why the “potential temperature” increases with altitude
Temperature Profile in the Atmosphere – The Lapse Rate – lots more about the temperature profile in the atmosphere

Note 4 – see Earth’s Energy Budget – a series on the basics of the energy budget

Note 5 – the “place of emission” is a useful conceptual tool but in reality the emission of radiation takes place from everywhere between the surface and the stratosphere. See Visualizing Atmospheric Radiation – Part Three – Average Height of Emission – the complex subject of where the TOA radiation originated from, what is the “Average Height of Emission” and other questions.

Also, take a look at the complete series: Visualizing Atmospheric Radiation.

Note 6 – the balance between emission and absorption are found in the equations of radiative transfer. These are derived from fundamental physics – see Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations – the equations of radiative transfer including the plane parallel assumption and it’s nothing to do with blackbodies. The fundamental physics is not just proven in the lab, spectral measurements at top of atmosphere and the surface match the calculated values using the radiative transfer equations – see Theory and Experiment – Atmospheric Radiation – real values of total flux and spectra compared with the theory.

Also, take a look at the complete series: Atmospheric Radiation and the “Greenhouse” Effect

Note 7 – this calculation is under the assumption of “all other things being equal”. Of course, in the real climate system, all other things are not equal. However, to understand an effect “pre-feedback” we need to separate it from the responses to the system.

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Radiative Atmospheres with no Convection

Posted in Atmospheric Physics, Basic Science on June 21, 2014| 89 Comments »

If we open an introductory atmospheric physics textbook, we find that the temperature profile in the troposphere (lower atmosphere) is mostly explained by convection. (See for example, Things Climate Science has Totally Missed? – Convection)

We also find that the temperature profile in the stratosphere is mostly determined by radiation. And that the overall energy balance of the climate system is determined by radiation.

Many textbooks introduce the subject of convection in this way:

  • what would the temperature profile be like if there was no convection, only radiation for heat transfer
  • why is the temperature profile actually different
  • how does pressure reduce with height
  • what happens to air when it rises and expands in the lower pressure environment
  • derivation of the “adiabatic lapse rate”, which in layman’s terms is the temperature change when we have relatively rapid movements of air
  • how the real world temperature profile (lapse rate) compares with the calculated adiabatic lapse rate and why

We looked at the last four points in some detail in a few articles:

Density, Stability and Motion in Fluids – some basics about instability
Potential Temperature – explaining “potential temperature” and why the “potential temperature” increases with altitude
Temperature Profile in the Atmosphere – The Lapse Rate – lots more about the temperature profile in the atmosphere

In this article we will look at the first point.

All of the atmospheric physics textbooks I have seen use a very simple model for explaining the temperature profile in a fictitious “radiation only” environment. The simple model is great for giving insight into how radiation travels.

Physics textbooks, good ones anyway, try and use the simplest models to explain a phenomenon.

The simple model, in brief, is the “semi-gray approximation”. This says the atmosphere is completely transparent to solar radiation, but opaque to terrestrial radiation. Its main simplification is having a constant absorption with wavelength. This makes the problem nice and simple analytically – which means we can rewrite the starting equations and plot a nice graph of the result.

However, atmospheric absorption is the total opposite of constant. Here is an example of the absorption vs wavelength of a minor “greenhouse” gas:

From Vardavas & Taylor (2007)

From Vardavas & Taylor (2007)

Figure 1

So from time to time I’ve wondered what the “no convection” atmosphere would look like with real GHG absorption lines. I also thought it would be especially interesting to see the effect of doubling CO2 in this fictitious environment.

This article is for curiosity value only, and for helping people understand radiative transfer a little better.

We will use the Matlab program seen in the series Visualizing Atmospheric Radiation. This does a line by line calculation of radiative transfer for all of the GHGs, pulling the absorption data out of the HITRAN database.

I updated the program in a few subtle ways. Mainly the different treatment of the stratosphere – the place where convection stops – was removed. Because, in this fictitious world there is no convection in the lower atmosphere either.

Here is a simulation based on 380 ppm CO2, 1775 ppb CH4, 319 ppb N2O and 50% relative humidity all through the atmosphere. Top of atmosphere was 100 mbar and the atmosphere was divided into 40 layers of equal pressure. Absorbed solar radiation was set to 240 W/m² with no solar absorption in the atmosphere. That is (unlike in the real world), the atmosphere has been made totally transparent to solar radiation.

The starting point was a surface temperature of 288K (15ºC) and a lapse rate of 6.5K/km – with no special treatment of the stratosphere. The final surface temperature was 326K (53ºC), an increase of 38ºC:

Temp-profile-no-convection-current-GHGs-40-levels-50%RH

Figure 2

The ocean depth was only 5m. This just helps get to a new equilibrium faster. If we change the heat capacity of a system like this the end result is the same, the only difference is the time taken.

Water vapor was set at a relative humidity of 50%. For these first results I didn’t get the simulation to update the absolute humidity as the temperature changed. So the starting temperature was used to calculate absolute humidity and that mixing ratio was kept constant:

wv-conc-no-convection-current-GHGs-40-levels-50%RH

Figure 3

The lapse rate, or temperature drop per km of altitude:

LapseRate-noconvection-current-GHGs-40-levels-50%RH

Figure 4

The flux down and flux up vs altitude:

Flux-noconvection-current-GHGs-40-levels-50%RH

Figure 5

The top of atmosphere upward flux is 240 W/m² (actually at the 500 day point it was 239.5 W/m²) – the same as the absorbed solar radiation (note 1). The simulation doesn’t “force” the TOA flux to be this value. Instead, any imbalance in flux in each layer causes a temperature change, moving the surface and each part of the atmosphere into a new equilibrium.

A bit more technically for interested readers.. For a given layer we sum:

  • upward flux at the bottom of a layer minus upward flux at the top of a layer
  • downward flux at the top of a layer minus downward flux at the bottom of a layer

This sum equates to the “heating rate” of the layer. We then use the heat capacity and time to work out the temperature change. Then the next iteration of the simulation redoes the calculation.

And even more technically:

  • the upwards flux at the top of a layer = the upwards flux at the bottom of the layer x transmissivity of the layer plus the emission of that layer
  • the downwards flux at the bottom of a layer = the downwards flux at the top of the layer x transmissivity of the layer plus the emission of that layer

End of “more technically”..

Anyway, the main result is the surface is much hotter and the temperature drop per km of altitude is much greater than the real atmosphere. This is because it is “harder” for heat to travel through the atmosphere when radiation is the only mechanism. As the atmosphere thins out, which means less GHGs, radiation becomes progressively more effective at transferring heat. This is why the lapse rate is lower higher up in the atmosphere.

Now let’s have a look at what happens when we double CO2 from its current value (380ppm -> 760 ppm):

Temp-profile-no-convection-doubled-GHGs-40-levels-50%RH

Figure 6 – with CO2 doubled instantaneously from 380ppm at 500 days

The final surface temperature is 329.4, increased from 326.2K. This is an increase (no feedback of 3.2K).

The “pseudo-radiative forcing” = 18.9 W/m² (which doesn’t include any change to solar absorption). This radiative forcing is the immediate change in the TOA forcing. (It isn’t directly comparable to the IPCC standard definition which is at the tropopause and after the stratosphere has come back into equilibrium – none of these have much meaning in a world without convection).

Let’s also look at the “standard case” of an increase from pre-industrial CO2 of 280 ppm to a doubling of 560 ppm. I ran this one for longer – 1000 days before doubling CO2 and 2000 days in total- because the starting point was less in balance. At the start, the TOA flux (outgoing longwave radiation) = 248 W/m². This means the climate was cooling quite a bit with the starting point we gave it.

At 180 ppm CO2, 1775 ppb CH4, 319 ppb N2O and 50% relative humidity (set at the starting point of 288K and 6.5K/km lapse rate), the surface temperature after 1,000 days = 323.9 K. At this point the TOA flux was 240.0 W/m². So overall the climate has cooled from its initial starting point but the surface is hotter.

This might seem surprising at first sight – the climate cools but the surface heats up? It’s simply that the “radiation-only” atmosphere has made it much harder for heat to get out. So the temperature drop per km of height is now much greater than it is in a convection atmosphere. Remember that we started with a temperature profile of 6.5K/km – a typical convection atmosphere.

After CO2 doubles to 560 ppm (and all other factors stay the same, including absolute humidity), the immediate effect is the TOA flux drops to 221 W/m² (once again a radiative forcing of about 19 W/m²). This is because the atmosphere is now even more “resistant” to the escape of heat by radiation. The atmosphere is more opaque and so the average emission of radiation of space moves to a higher and colder part of the atmosphere. Colder parts of the atmosphere emit less radiation than warmer parts of the atmosphere.

After the climate moves back into balance – a TOA flux of 240 W/m² – the surface temperature = 327.0 K – an increase (pre-feedback) of 3.1 K.

Compare this with the standard IPCC “with convection” no-feedback forcing of 3.7 W/m² and a “no feedback” temperature rise of about 1.2 K.

Temp-profile-no-convection-280-560ppm-CO2-40-levels-50%RH

Figure 7 – with CO2 doubled instantaneously from 280ppm at 1000 days

Then I introduced a more realistic model with solar absorption by water vapor in the atmosphere (changed parameter ‘solaratm’ in the Matlab program from ‘false’ to ‘true’). Unfortunately this part of the radiative transfer program is not done by radiative transfer, only by a very crude parameterization, just to get roughly the right amount of heating by solar radiation in roughly the right parts of the atmosphere.

The equilibrium surface temperature at 280 ppm CO2 was now “only” 302.7 K (almost 30ºC). Doubling CO2 to 560 ppm created a radiative forcing of 11 W/m², and a final surface temperature of 305.5K – that is, an increase of 2.8K.

Why is the surface temperature lower? Because in the “no solar absorption in the atmosphere” model, all of the solar radiation is absorbed by the ground and has to “fight its way out” from the surface up. Once you absorb solar radiation higher up than the surface, it’s easier for this heat to get out.

Conclusion

One of the common themes of fantasy climate blogs is that the results of radiative physics are invalidated by convection, which “short-circuits” radiation in the troposphere. No one in climate science is confused about the fact that convection dominates heat transfer in the lower atmosphere.

We can see in this set of calculations that when we have a radiation-only atmosphere the surface temperature is a lot higher than any current climate – at least when we consider a “one-dimensional” climate.

Of course, the whole world would be different and there are many questions about the amount of water vapor and the effect of circulation (or lack of it) on moving heat around the surface of the planet via the atmosphere and the ocean.

When we double CO2 from its pre-industrial value the radiative forcing is much greater in a “radiation-only atmosphere” than in a “radiative-convective atmosphere”, with the pre-feedback temperature rise 3ºC vs 1ºC.

So it is definitely true that convection short-circuits radiation in the troposphere. But the whole climate system can only gain and lose energy by radiation and this radiation balance still has to be calculated. That’s what current climate models do.

It’s often stated as a kind of major simplification (a “teaching model”) that with increases in GHGs the “average height of emission” moves up, and therefore the emission is from a colder part of the atmosphere. This idea is explained in more detail and less simplifications in Visualizing Atmospheric Radiation – Part Three – Average Height of Emission – the complex subject of where the TOA radiation originated from, what is the “Average Height of Emission” and other questions.

A legitimate criticism of current atmospheric physics is that convection is poorly understood in contrast to subjects like radiation. This is true. And everyone knows it. But it’s not true to say that convection is ignored. And it’s not true to say that because “convection short-circuits radiation” in the troposphere that somehow more GHGs will have no effect.

On the other hand I don’t want to suggest that because more GHGs in the atmosphere mean that there is a “pre-feedback” temperature rise of about 1K, that somehow the problem is all nicely solved. On the contrary, climate is very complicated. Radiation is very simple by comparison.

All the standard radiative-convective calculation says is: “all other things being equal, an doubling of CO2 from pre-industrial levels, would lead to a 1K increase in surface temperature”

All other things are not equal. But the complication is not that somehow atmospheric physics has just missed out convection. Hilarious. Of course, I realize most people learn their criticisms of climate science from people who have never read a textbook on the subject. Surprisingly, this doesn’t lead to quality criticism..

On more complexity  – I was also interested to see what happens if we readjust absolute humidity due to the significant temperature changes, i.e. we keep relative humidity constant. This led to some surprising results, so I will post them in a followup article.

Notes

Note 1 – The boundary conditions are important if you want to understand radiative heat transfer in the atmosphere.

First of all, the downward longwave radiation at TOA (top of atmosphere) = 0. Why? Because there is no “longwave”, i.e., terrestrial radiation, from outside the climate system. So at the top of the atmosphere the downward flux = 0. As we move down through the atmosphere the flux gradually increases. This is because the atmosphere emits radiation. We can divide up the atmosphere into fictitious “layers”. This is how all numerical (finite element analysis) programs actually work. Each layer emits and each layer also absorbs. The balance depends on the temperature of the source radiation vs the temperature of the layer of the atmosphere we are considering.

At the bottom of the atmosphere, i.e., at the surface, the upwards longwave radiation is the surface emission. This emission is given by the Stefan-Boltzmann equation with an emissivity of 1.0 if we consider the surface as a blackbody which is a reasonable approximation for most surface types – for more on this, see Visualizing Atmospheric Radiation – Part Thirteen – Surface Emissivity – what happens when the earth’s surface is not a black body – useful to understand seeing as it isn’t..

At TOA, the upwards emission needs to equal the absorbed solar radiation, otherwise the climate system has an imbalance – either cooling or warming.

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On Uses of A 4 x 2: Arrhenius, The Last 15 years of Temperature History and Other Parodies

Posted in Atmospheric Physics, Basic Science, Debunking Flawed "Science" on June 15, 2014| 60 Comments »

As a friend of mine in Florida says:

You can’t kill stupid, but you can dull it with a 4×2

Some ideas are so comically stupid that I thought there was no point writing about them. And yet, one after another, people who can type are putting forward these ideas on this blog.. At first I wondered if I was the object of a practical joke. Some kind of parody. Perhaps the joke is on me. But, just in case I was wrong about the practical joke..

 

If you pick up a textbook on heat transfer that includes a treatment of radiative heat transfer you find no mention of Arrhenius.

If you pick up a textbook on atmospheric physics none of the equations come from Arrhenius.

Yet there is a steady stream of entertaining “papers” which describe “where Arrhenius went wrong”, “Arrhenius and his debates with Fourier”. Who cares?

Likewise, if you study equations of motion in a rotating frame there is no discussion of where Newton went wrong, or where he got it right, or debates he got right or wrong with contemporaries. Who knows? Who cares?

History is fascinating. But if you want to study physics you can study it pretty well without reading about obscure debates between people who were in the formulation stages of the field.

Here are the building blocks of atmospheric radiation:

  • The emission of radiation – described by Nobel prize winner Max Planck’s equation and modified by the material property called emissivity (this is wavelength dependent)
  • The absorption of radiation by a surface – described by the material property called absorptivity (this is wavelength dependent and equal at the same wavelength and direction to emissivity)
  • The Beer-Lambert law of absorption of radiation by a gas
  • The spectral absorption characteristics of gases – currently contained in the HITRAN database – and based on work carried out over many decades and written up in journals like Journal of Quantitative Spectroscopy and Radiative Transfer
  • The theory of radiative transfer – the Schwarzschild equation – which was well documented by Nobel prize winner Subrahmanyan Chandrasekhar in his 1952 book Radiative Transfer (and by many physicists since)

The steady stream of stupidity will undoubtedly continue, but if you are interested in learning about science then you can rule out blogs that promote papers which earnestly explain “where Arrhenius went wrong”.

Hit them with a 4 by 2.

Or, ask the writer where Subrahmanyan Chandrasekhar went wrong in his 1952 work Radiative Transfer. Ask the writer where Richard M. Goody went wrong. He wrote the seminal Atmospheric Radiation: Theoretical Basis in 1964.

They won’t even know these books exist and will have never read them. These books contain equations that are thoroughly proven over the last 100 years. There is no debate about them in the world of physics. In the world of fantasy blogs, maybe.

There is also a steady stream of people who believe an idea yet more amazing. Somehow basic atmospheric physics is proven wrong because of the last 15 years of temperature history.

The idea seems to be:

More CO2 is believed to have some radiative effect in the climate because of the last 100 years of temperature history, climate scientists saw some link and tried to explain it using CO2, but now there has been no significant temperature increase for the last x years this obviously demonstrates the original idea was false..

If you think this, please go and find a piece of 4×2 and ask a friend to hit you across the forehead with it. Repeat. I can’t account for this level of stupidity but I have seen that it exists.

An alternative idea, that I will put forward, one that has evidence, is that scientists discovered that they can reliably predict:

  • emission of radiation from a surface
  • emission of radiation from a gas
  • absorption of radiation by a surface
  • absorption of radiation by a gas
  • how to add up, subtract, divide and multiply, raise numbers to the power of, and other ninja mathematics

The question I have for the people with these comical ideas:

Do you think that decades of spectroscopy professionals have just failed to measure absorption? Their experiments were some kind of farce? No one noticed they made up all the results?

Do you think Max Planck was wrong?

It is possible that climate is slightly complicated and temperature history relies upon more than one variable?

Did someone teach you that the absorption and emission of radiation was only “developed” by someone analyzing temperature vs CO2 since 1970 and not a single scientist thought to do any other measurements? Why did you believe them?

Bring out the 4×2.

Note – this article is a placeholder so I don’t have to bother typing out a subset of these points for the next entertaining commenter..

Update July 10th with the story of Fred the Charlatan

Let’s take the analogy of a small boat crossing the Atlantic.

Analogies don’t prove anything, they are for illustration. For proof, please review Theory and Experiment – Atmospheric Radiation.

We’ve done a few crossings and it’s taken 45 days, 42 days and 46 days (I have no idea what the right time is, I’m not a nautical person).

We measure the engine output – the torque of the propellors. We want to get across quicker. So Fred the engine guy makes a few adjustments and we remeasure the torque at 5% higher. We also do Fred’s standardized test, which is to zip across a local sheltered bay with no currents, no waves and no wind – the time taken for Fred’s standarized test is 4% faster. Nice.

So we all set out on our journey across the Atlantic. Winds, rain, waves, ocean currents. We have our books to read, Belgian beer and red wine and the time flies. Oh no, when we get to our final destination, it’s actually taken 47 days.

Clearly Fred is some kind of charlatan! No need to check his measurements or review the time across the bay. We didn’t make it across the Atlantic in less time and clearly the ONLY variable involved in that expedition was the output of the propellor.

Well, there’s no point trying to use more powerful engines to get across the Atlantic (or any ocean) faster. Torque has no relationship to speed. Case closed.

Analogy over.

 

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Ghosts of Climates Past – Part Four – Understanding Orbits, Seasons and Stuff

Posted in Basic Science, Climate History on October 14, 2013| 12 Comments »

In Part Three we had a very brief look at the orbital factors that affect solar insolation.

Here we will look at these factors in more detail. We start with the current situation.

Seasonal Distribution of Incoming Solar Radiation

The earth is tilted on its axis (relative to the plane of orbit) so that in July the north pole “faces” the sun, while in January the south pole “faces” the sun.

Here are the TOA graphs for average incident solar radiation at different latitudes by month:

From Vardavas & Taylor (2007)

From Vardavas & Taylor (2007)

Figure 1

And now the average values first by latitude for the year, then by month for northern hemisphere, southern hemisphere and the globe:

TOA-solar-total-by-month-and-latitude-present

Figure 2

We can see that the southern hemisphere has a higher peak value – this is because the earth is closest to the sun (perihelion) on January 3rd, during the southern hemisphere summer.

This is also reflected in the global value which varies between 330 W/m² at aphelion (furthest away from the sun) to 352 W/m² at perihelion.

Eccentricity

There is a good introduction to planetary orbits in Wikipedia. I was saved from the tedium of having to work out how to implement an elliptical orbit vs time by the Matlab code kindly supplied by Jonathan Levine. He also supplied the solution to the much more difficult problem of insolation vs latitude at any day in the Quaternary period, which we will look at later.

Here is the the TOA solar insolation by day of the year, as a function of the eccentricity of the orbit:

Daily-Change-TOA-Solar-vs-Eccentricity-2

Figure 3 – Updated

The earth’s orbit currently has an eccentricity of 0.0167. This means that the maximum variation in solar radiation is 6.9%.

Perihelion is 147.1 million km, while aphelion is 152.1 million km. The amount of solar radiation we receive is “the inverse square law”, which means if you move twice as far away, the solar radiation reduces by a factor of four. So to calculate the difference between the min and max you simply calculate: (152.1/147.1)² = 1.069 or a change of 6.9%.

Over the past million or more years the earth’s orbit has changed its eccentricity, from a low close to zero, to a maximum of about 0.055. The period of each cycle is about 100,000 years.

Here is my calculation of change in total annual TOA solar radiation with eccentricity:

Annual-%Change-TOA-Solar-vs-Eccentricity

Figure 4

Looking at figure 1 of Imbrie & Imbrie (1980), just to get a rule of thumb, eccentricity changed from 0.05 to 0.02 over a 50,000 year period (about 220k years ago to 170k years ago). This means that the annual solar insolation dropped by 0.1% over 50,000 years or 3 mW/m² per century. (This value is an over-estimate because it is the peak value with sun overhead, if instead we take the summer months at high latitude the change becomes  0.8 mW/m² per century)

It’s a staggering drop, and no wonder the strong 100,000 year cycle in climate history matching the Milankovitch eccentricity cycles is such a difficult theory to put together.

Obliquity & Precession

To understand those basics of these changes take a look at the Milankovitch article. Neither of these two effects, precession and obliquity, changes the total annual TOA incident solar radiation. They just change its distribution.

Here is the last 250,000 years of solar radiation on July 1st – for a few different latitudes:

TOA-Solar-July1-Latitude-vs0-250k-499px

Figure 5 – Click for a larger image

Notice that the equatorial insolation is of course lower than the mid-summer polar insolation.

Here is the same plot but for October 1st. Now the equatorial value is higher:

TOA-Solar-Oct1-Latitude-vs0-250k-499px

Figure 6 – Click for a larger image

Let’s take a look at the values for 65ºN, often implicated in ice age studies, but this time for the beginning of each month of the year (so the legend is now 1 = January 1st, 2 = Feb 1st, etc):

TOA-Solar-65N-bymonth-vs0-250k-lb-499px

Figure 7 – Click for a larger image

And just for interest I marked one date for the last inter-glacial – the Eemian inter-glacial as it is known.

Come up with a theory:

  • peak insolation at 65ºN
  • fastest rate of change
  • minimum insolation
  • average of summer months
  • average of winter half year
  • average autumn 3 months

Then pick from the graph and let’s start cooking.. Having trouble? Pick a different latitude. Southern Hemisphere – no problem, also welcome.

As we will see, there are a lot of theories, all of which call themselves “Milankovitch” but each one is apparently incompatible with other similarly-named “Milankovitch” theories.

At least we have a tool, kindly supplied by Jonathan Levine, which allows us to compute any value. So if any readers have an output request, just ask.

One word of caution for budding theorists of ice ages (hopefully we have many already) from Kukla et al (2002):

..The marine isotope record is commonly tuned to astronomic chronology, represented by June insolation at the top of the atmosphere at 60′ or 65′ north latitude. This was deemed justified because the frequency of the Pleistocene gross global climate states matches the frequency of orbital variations..

..The mechanism of the climate response to insolation remains unclear and the role of insolation in the high latitudes as opposed to that in the low latitudes is still debated..

..In either case, the link between global climates and orbital variations appears to be complicated and not directly controlled by June insolation at latitude 65’N. We strongly discourage dating local climate proxies by unsubstantiated links to astronomic variations..

[Emphasis added].

I’m a novice with the historical records and how they have been constructed, but I understand that SPECMAP is tuned to a Milankovitch theory, i.e., the dates of peak glacials and peak inter-glacials are set by astronomical values.

Articles in the Series

Part One – An introduction

Part Two – Lorenz – one point of view from the exceptional E.N. Lorenz

Part Three – Hays, Imbrie & Shackleton – how everyone got onto the Milankovitch theory

Part Five – Obliquity & Precession Changes – and in a bit more detail

Part Six – “Hypotheses Abound” – lots of different theories that confusingly go by the same name

Part Seven – GCM I – early work with climate models to try and get “perennial snow cover” at high latitudes to start an ice age around 116,000 years ago

Part Seven and a Half – Mindmap – my mind map at that time, with many of the papers I have been reviewing and categorizing plus key extracts from those papers

Part Eight – GCM II – more recent work from the “noughties” – GCM results plus EMIC (earth models of intermediate complexity) again trying to produce perennial snow cover

Part Nine – GCM III – very recent work from 2012, a full GCM, with reduced spatial resolution and speeding up external forcings by a factors of 10, modeling the last 120 kyrs

Part Ten – GCM IV – very recent work from 2012, a high resolution GCM called CCSM4, producing glacial inception at 115 kyrs

Pop Quiz: End of An Ice Age – a chance for people to test their ideas about whether solar insolation is the factor that ended the last ice age

Eleven – End of the Last Ice age – latest data showing relationship between Southern Hemisphere temperatures, global temperatures and CO2

Twelve – GCM V – Ice Age Termination – very recent work from He et al 2013, using a high resolution GCM (CCSM3) to analyze the end of the last ice age and the complex link between Antarctic and Greenland

Thirteen – Terminator II – looking at the date of Termination II, the end of the penultimate ice age – and implications for the cause of Termination II

References

Last Interglacial Climates, Kukla et al, Quaternary Research (2002)

Modeling the Climatic Response to Orbital Variations, John Imbrie & John Z. Imbrie, Science (1980)

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CERES, AIRS, Outgoing Longwave Radiation & El Nino

Posted in Basic Science, Measurement on February 7, 2013| 21 Comments »

Measurements of outgoing longwave radiation (OLR) are essential for understanding many aspects of climate. Many people are confused about the factors that affect OLR. And its rich variability is often not appreciated.

There have been a number of satellite projects since the late 1970’s, with the highlight (prior to 2001) being the five year period of ERBE.

AIRS & CERES were launched on the NASA AQUA satellite in May 2002. These provide much better quality data, with much better accuracy and resolution.

CERES has three instruments:

  • Solar Reflected Radiation (Shortwave): 0.3 – 5.0 μm
  • Window: 8 – 12 μm
  • Total: 0.3 to > 100 μm

AIRS is an infrared spectrometer/radiometer that covers the 3.7–15.4 μm spectral range with 2378 spectral channels. It runs alongside two microwave instruments (better viewing through clouds): AMSU is a 15-channel microwave radiometer operating between 23 and 89 GHz; HSB is a four-channel microwave radiometer that makes measurements between 150 and 190 GHz.

From Aumann et al (2003):

The simultaneous use of the data from the three instruments provides both new and improved measurements of cloud properties, atmospheric temperature and humidity, and land and ocean skin temperatures, with the accuracy, resolution, and coverage required by numerical weather prediction and climate models.

Among the important datasets that AIRS will contribute to climate studies are as follows:

  • atmospheric temperature profiles;
  • sea-surface temperature;
  • land-surface temperature and emissivity;
  • relative humidity profiles and total precipitable water vapor;
  • fractional cloud cover;
  • cloud spectral IR emissivity;
  • cloud-top pressure and temperature;
  • total ozone burden of the atmosphere;
  • column abundances of minor atmospheric gases such as CO, CH, CO, and N2O;
  • outgoing longwave radiation and longwave cloud radiative forcing;
  • precipitation rate

More about AIRS = Atmospheric Infrared Sounder, at Wikipedia, plus the AIRS website.

More about CERES = Clouds and the Earth’s Radiant Energy System, at Wikipedia, plus the CERES website – where you can select and view or download your own data.

How do CERES & AIRS compare?

CERES and AIRS have different jobs. CERES directly measures OLR. AIRS measures lots of spectral channels that don’t cover the complete range needed to just “add up” OLR. Instead, OLR can be calculated from AIRS data by deriving surface temperature, water vapour concentration vs height, CO2 concentration, etc and using a radiative transfer algorithm to determine OLR.

Here is a comparison of the two measurement systems from Susskind et al (2012) over almost a decade:

Susskind-CERES-vs-AIRS-2012

From Susskind et al (2012)

Figure 1

The second thing to observe is that the measurements have a bias between the two datasets. But because we have two high accuracy measurement systems on the same satellite we do have a reasonable opportunity to identify the source of the bias (total OLR as shown in the graph is made of many components). If we only had one satellite, and then a new satellite took over with a small time overlap any biases would be much more difficult to identify. Of course, that doesn’t stop many people from trying but success would be much harder to judge.

In this paper, as we might expect, the error sources between the two datasets get considerable discussion. One important point is that version 6 AIRS data (prototyped at the time the paper was written) is much closer to CERES. The second point, probably more interesting, is that once we look at anomaly data the results are very close. We’ll see a number of comparisons as we review what the paper shows.

The authors comment:

Behavior of OLR over this short time period should not be taken in any way as being indicative of what long-term trends might be. The ability to begin to draw potential conclusions as to whether there are long-term drifts with regard to the Earth’s OLR, beyond the effects of normal interannual variability, would require consistent calibrated global observations for a time period of at least 20 years, if not longer. Nevertheless, a very close agreement of the 8-year, 10-month OLR anomaly time series derived using two different instruments in two very different manners is an encouraging result.

It demonstrates that one can have confidence in the 1° x 1° OLR anomaly time series as observed by each instrument over the same time period. The second objective of the paper is to explain why recent values of global mean, and especially tropical mean, OLR have been strongly correlated with El Niño/La Niña variability and why both have decreased over the time period under study.

Why Has OLR Varied?

The authors define the average rate of change (ARC) of an anomaly time series as “the slope of the linear least squares fit of the anomaly time series”.

Susskind-2012-table-1

We can see excellent correlation between the two datasets and we can see that OLR has, on average, decreased over this time period.

Below is a comparison with the El Nino index.

We define the term El Niño Index as the difference of the NOAA monthly mean oceanic Sea Surface Temperature (SST), averaged over the NOAA Niño-4 spatial area 5°N to 5°S latitude and 150°W westward to 160°E longitude, from an 8-year NOAA Niño-4 SST monthly mean climatology which we generated based on use of the same 8 years that we used in the generation of the OLR climatologies.

From Susskind et al (2012)

From Susskind et al (2012)

Figure 2

It gets interesting when we look at the geographical distribution of the OLR changes over this time period:

From Susskind et al (2012)

From Susskind et al (2012)

Figure 3 – Click to Enlarge

We see that the tropics have the larger changes (also seen clearly in figure 2) but that some regions of the tropics have strong positive values and other regions have strong negative values. The grey square square centered on 180 longitude is the Nino-4 region. Values as large as +4 W/m²/decade are found in this region. And values as large as -3 W/m²/decade are found over Indonesia (WPMC region).

Let’s look at the time series to see how these changes in OLR took place:

Susskind-Time-Series-2012

Figure 4 – Click to Enlarge

The main parameters which affect changes in OLR month to month and year to year are a) surface temperatures b) humidity c) clouds. As temperature increases, OLR increases. As humidity and clouds increase, OLR decreases.

Here are the changes in surface temperature, specific humidity at 500mbar and cloud fraction:

From Susskind (2012)

From Susskind (2012)

Figure 5 – Click to Enlarge

So, focusing again on the Nino-4 region, we might expect to find that OLR has decreased because of the surface temperature decrease (lower emission of surface radiation) – or we might expect to find that the OLR has increased because the specific humidity and cloud fraction have decreased (thus allowing more surface and lower atmosphere radiation to make it through to TOA). These are mechanisms pulling in opposite directions.

In fact we see that the reduced specific humidity and cloud fraction have outweighed the effect of the surface temperature decrease. So the physics should be clear (still considering the Nino-4 region) – if surface temperature has decreased and OLR has increased then the explanation is the reduction in “greenhouse” gases (in this case water vapor) and clouds, which contain water.

Correlations

We can see similar relationships through correlations.

The term ENC in the graphs stands for El Nino Correlation. This is essentially the correlation of the time-series data with time-series temperature change in the Nino-4 region (more specifically the Nino-4 temperature less the global temperature).

As the Nino-4 temperature declined over the period in question, a positive correlation means the value declined, while a negative correlation means the value increased.

The first graph below is the geographical distribution of rate of change of surface temperature. Of course we see that the Nino-4 region has been declining in temperature (as already seen in figure 2). The second graph shows this as well, but also indicates that the regions west and east of the Nino-4 region have  a stronger (negative) correlation than  other areas of larger temperature change (like the arctic region).

The third graph shows that 500 mb humidity has been decreasing in the Nino-4 region, and increasing to the west and east of this region. Likewise for the cloud fraction. And all of these are strongly correlated to the Nino-4 time-series temperature:

From Susskind (2012)

From Susskind et al (2012)

Figure 6 – Click to expand

For OLR correlations with Nino-4 temperature we find a strong negative correlation, meaning the OLR has increased in the Nino-4 region. And the opposite – a strong positive correlation – in the highlighted regions to east and west of Nino-4:

From Susskind (2012)

From Susskind (2012)

Figure 7 – Click to expand

Note the two highlighted regions

  • to the west: WPMC, Warm Pool Maritime Continent;
  • and to the east: EEMA, Equatorial Eastern Pacific and Atlantic Region

We can see the correlations between the global & tropical OLR and the OLR changes in these regions:

Susskind-2012-table-2

Figure 8 – Click to expand

Both WPMC and EEPA regions together explain the reduction over 10 years in OLR. Without these two regions the change is indistinguishable from zero.

Conclusion

This article is interesting for a number of reasons.

It shows the amazing variability of climate – we can see adjacent regions in the tropics with completely opposite changes over 10 years.

It shows that CERES gets almost identical anomaly results (changes in OLR) to AIRS. CERES directly measures OLR, while AIRS retrieves surface temperature, humidity profiles, cloud fractions and “greenhouse” gas concentrations and uses these to calculate OLR.

AIRS results demonstrate how surface temperature, humidity and cloud fraction affect OLR.

OLR has – over the globe – decreased over 10 years. This is a result of the El-Nino phase – at the start of the measurement period we were coming out of a large El-Nino event, and at the end of the measurement period we were in a La Nina event.

The reduction in OLR is explained by the change in the two regions identified, which are themselves strongly correlated to the Nino-4 region.

References

Interannual variability of outgoing longwave radiation as observed by AIRS and CERES, Susskind et al, Journal of Geophysical Research (2012) – paywall paper

AIRS/AMSU/HSB on the Aqua Mission: Design, Science Objectives, Data Products, and Processing Systems, Aumann et al, IEEE Transactions on Geoscience and Remote Sensing (2003) – free paper

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Visualizing Atmospheric Radiation – Part One

Posted in Basic Science on January 3, 2013| 45 Comments »

This could be considered as a continuation of the earlier series – Atmospheric Radiation and the “Greenhouse” Effect – but I’ve elected to start a new series.

It’s clear that many people have conceptual problems with the subject of what is, in technical terms called radiative transfer. That is, how radiation travels through the atmosphere and is affected by the atmosphere.

Radiation and Gas in a Box

First, let’s consider what happens as we shine an intense beam of infrared radiation at a narrow range of wavelengths (let’s say somewhere in the region of 15μm) through a box of CO2 gas at room temperature:

Atmospheric-radiation-2

Figure 1

The red arrow on the left is the incident radiation. The graph indicates the spectrum. The spectrum on the right is made up of two main parts:

  • the transmitted radiation – the incident radiation attenuated by the absorbing gas
  • the emitted radiation due to the temperature and emissivity of the gas at these wavelengths

The yellow spectrum shows what we would measure from one of the sides. Note that the transmitted radiation that goes from left to right has no effect on this yellow spectrum (except in so far as absorption of the incident radiation affects the temperature of the gas).

If we increase the length of the box (left to right) – and keep the density the same – the transmitted radiation from the right side would decrease in intensity. If we reduce the length of the box (again, same density) the transmitted radiation from the right would increase in intensity.

But the emitted radiation from the top is only dependent on the temperature of the gas and its emission/absorption lines.

And the temperature of the gas is of course affected by the balance between absorbed and emitted radiation as well as any heat transfer from the surroundings via convection and conduction.

Hopefully, this is clear. If anyone thinks this simple picture is wrong, now is the time to make a comment. Confusion over this part means that you can’t make any progress in understanding atmospheric radiation.

Scattering is insignificant for longwave radiation (4μm and up). Stimulated emission is insignificant for intensities seen in the atmosphere.

Radiation in the Atmosphere

How does radiation travel through the atmosphere?

Atmospheric-radiation-1

Figure 1

The idea shown here is a spectrum of radiation at different wavelengths incident on a “layer” of the atmosphere (see note 1). The atmosphere has lots of absorption lines of many different strengths. As a result the transmitted radiation making it out of the other side is some proportion of the incident radiation. The proportion varies with the wavelength.

The atmosphere also emits radiation, and the emission lines are the same as the absorption lines. More about that in Planck, Stefan-Boltzmann, Kirchhoff and LTE.

However, the emission depends on the temperature of the gas in the layer (as well as the absorption/emission lines). But the absorption depends on the intensity of incident radiation (as well as the absorption/emission lines), which in turn depends on the temperature of the source of the radiation.

So in almost every case, the sum of transmitted plus emitted radiation is not equal to the incident radiation. By the way, the spectrum at the top is just a raggedy freehand drawing to signify that the outgoing spectrum is not like the incoming spectrum. It’s not meant to be representative of actual intensity vs wavelength.

And – it’s a two way street. I only showed one half of the story in figure 1. The same physics affects downward radiation in exactly the same way.

Considering One Wavelength at a Time

To calculate the actual transmission of radiation through the layer we simply work out the transmissivity, tλ, of the layer at each wavelength, λ (tλ simply indicates that t will vary for each value of λ we consider). We do that by looking up values calculated by spectroscopic professionals. These values are per molecule, or per kg of particular molecules so we need to find out how much of each absorbing gas is present.

1. The incident radiation making it through the layer = Iλ x tλ – for example, it could be 90% making it through, or 20%.

2. The “new” radiation emitted from each side of the layer equals the “Planck blackbody function at the temperature of the layer and the wavelength of interest” x Emissivity of the gas at that wavelength.

In case people are interested this can be written as Bλ(T).ελ, where ελ = emissivity at that wavelength, and Bλ(T) is the “Planck function” at that temperature and wavelength. Well, emissivity = absorptivity (at the same wavelength) and absorptivity = 1-transmissivity, so the same equation can be written as Bλ(T).(1-tλ).

Digression

Perhaps (the Planck function showing up in an equation) this is where many blogs (Parady blogs?) get the idea, and promote and endorse the idea, that climate science depends on the assumption that the atmosphere emits as a blackbody. There are some cases where the atmospheric emission is not far from the “blackbody assumption” (e.g., in clouds), but that is due to reality not assumption. There is no “blackbody assumption for the atmosphere” in climate science. But there is a movement of people who believe it to be true.

Misleadingly, they like to be known as “skeptics”.

End of digression..

Doing the Calculation

So it’s not really that hard to understand how radiation travels through the atmosphere. It is difficult to calculate it, mostly due to having to read a million absorption lines, figure out the correct units, get a model of the atmosphere (temperature profile + concentration of different “greenhouse” gases at each height), write a finite element program and work out a solution.

But that is just tedious details, it’s not as hard as having to understand general relativity or potential vorticity.

One important point – it is not possible to do this calculation in your head. If you think you have done it in your head, even to a close approximation, please go back and read this section again, then look up all of the absorption lines.

Still convinced – post your answer in a comment here.

In the next article I’ll explain radiative-convective models and show some results from the atmospheric model I built in MATLAB which uses the HITRAN database. Now that we have a model which calculates realistic values for emission, absorption and transmission we can slice and dice the results any way we want.

Does water vapor mask out the effects of CO2? What proportion of radiation is transmitted through the atmospheric window? What is the average emission height to space?

Related Articles

Part Two – some early results from a model with absorption and emission from basic physics and the HITRAN database

Part Three – Average Height of Emission – the complex subject of where the TOA radiation originated from, what is the “Average Height of Emission” and other questions

Part Four – Water Vapor – results of surface (downward) radiation and upward radiation at TOA as water vapor is changed

Part Five – The Code – code can be downloaded, includes some notes on each release

Part Six – Technical on Line Shapes – absorption lines get thineer as we move up through the atmosphere..

Part Seven – CO2 increases – changes to TOA in flux and spectrum as CO2 concentration is increased

Part Eight – CO2 Under Pressure – how the line width reduces (as we go up through the atmosphere) and what impact that has on CO2 increases

Part Nine – Reaching Equilibrium – when we start from some arbitrary point, how the climate model brings us back to equilibrium (for that case), and how the energy moves through the system

Part Ten – “Back Radiation” – calculations and expectations for surface radiation as CO2 is increased

Part Eleven – Stratospheric Cooling – why the stratosphere is expected to cool as CO2 increases

Part Twelve – Heating Rates – heating rate (‘C/day) for various levels in the atmosphere – especially useful for comparisons with other models.

Notes

Note 1 – What is a layer of atmosphere? Isn’t the thickness of this layer somewhat arbitrary? What if we change the thickness? And doesn’t radiation go in all directions, not just up?

These are all good questions.

In typical physics terms the actual equation of “radiative transfer” is a differential equation, which expresses continual change. In practical terms, solving a differential equation in most real world cases requires a numerical solution which has finite thicknesses for each layer.

People trying to solve these kind of problems usually check what happens to the solution as they go for more of thinner layers vs less of thicker layers. There is a trade-off between accuracy and speed.

Radiation does go in all directions. The plane parallel assumption has very strong justification and – in simple terms – mathematically resolves to a vertical solution with a correction factor. You can see the plane parallel assumption and the derivation of the equations of radiative transfer in Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations.

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