The Science of Doom

Recap

Part One of the series introduced the shortwave radiation from the sun, the balancing longwave radiation from the earth and the absorption of some of that longwave radiation by various “greenhouse” gases. The earth would be a cold place without the “greenhouse” gases.

Part Two discussed the factors that determine the relative importance of the various gases in the atmosphere.

Part Three and Four got a little more technical – an unfortunate necessity. Part Three introduced Radiative Transfer Equations including the Beer-Lambert Law of absorption. It also introduced the important missing element in many people’s understanding of the role of CO₂ – re-emission of radiation as the atmosphere heats up.

Part Four brought in band models. These are equations which quite closely match the real absorption of CO₂ (and the other greenhouse gases) as a function of wavelength. They aren’t strictly necessary to get to the final result, but they have an important benefit – they allow us to easily see how the absorption changes as the amount of gas increases. And they are widely used in climate models because they reduce the massive computation time that are otherwise involved in solving the Radiative Transfer Equations. The important outcome as far as CO₂ is concerned – “saturation” can be technically described.

Solving the Equations

The equations of absorption and radiation in the atmosphere – the Radiative Transfer Equations – have been known for more than 60 years. Solving the equations is a little more tricky.

Like many real world problems, the radiative processes in the atmosphere can be mathematically described from 1st principles but not “analytically” solved. This simply means that numerical methods have to be used to find the solution.

There’s nothing unproven or “suspicious” about this approach. Every problem from stresses in bridges and buildings to heat dissipation in an electronic product uses this method.

The problem of the effect of greenhouse gases in the atmosphere is formulated with a 1-dimensional model. This is the simplest approach (after the “billiard ball” model we saw in part one). But like any model there are certain assumptions that have to be made – the boundary conditions. And over the last 40 years different scientists have approached the problem from slightly different directions, making comparisons not always easy.

Because the role of CO₂ in the atmosphere is causing such concern the results of these models is consequently much more important. And so a lot of effort recently has gone into standardizing the approach. We’ll look at a few results, but first, for those who would like to visualize what modern methods of “numerical analysis” are about – a little digression.. (and for those who don’t, jump ahead to the Ramanathan.. subheading).

Digression on Numerical Methods

Stress analysis in an impeller

Here’s a visualization of “finite element analysis” of stresses in an impeller. See the “wire frame” look, as if the impeller has been created from lots of tiny pieces?

In this totally different application, the problem of calculating the mechanical stresses in the unit is that the “boundary conditions” – the strange shape – make solving the equations by the usual methods of re-arranging and substitution impossible. Instead what happens is the strange shape is turned into lots of little cubes. Now the equations for the stresses in each little cube are easy to calculate. So you end up with 1000’s of “simultaneous” equations. Each cube is next to another cube and so the stress on each common boundary is the same. The computer program uses some clever maths and lots of iterations to eventually find the solution to the 1000’s of equations that satisfy the “boundary conditions”.

In the case of the radiative transfer equations (RTE) we want to know the temperature profile up through the atmosphere. The atmosphere is divided into lots of thin slices. Each “slice” has some properties attached to it:

• gases like water vapor, CO2, CH4 at various concentrations with known absorption characteristics for each wavelength
• a temperature -unknown – this is what we want to find out
• radiation flowing up and down through the “slice” at each wavelength – unknown  – we also want to find this out

And we have important boundary conditions – like the OLR (outgoing longwave radiation) at the top of the atmosphere. We know this is about 239 W/m² (see The Earth’s Energy Budget – Part One). Using the boundary conditions, we solve the radiative transfer equations for each slice, and the computer program does this by creating lot of simultaneous equations (energy in each wavelength flowing between each slice is conserved).

Ramathan and Coakley, 1978

Why bring up an old paper? Partly to demonstrate some of the major issues and one interesting approach to solving them, but also to give a sense of history. A lot of people think that the concern over greenhouse gases is something new and perhaps all to do with the IPCC or Al Gore.

Back in 1978, V. Ramanathan and J.A. Coakley’s paper Climate Modeling through Radiative-Convective Models was published in Reviews of Geophysics and Space Physics.

It wasn’t the first to tackle the subject and points to the work done by Manabe and Strickler in 1964. By the way, V. Ramanathan is a bit of a trooper, having published 169 peer-reviewed papers in the field of atmospheric physics from 1972-2009..

I’m going to call the paper R&C – so R&C cover the detailed maths of course, but then discuss how to deal with the “problem” of convection.

In the lower part of the atmosphere heat primarily moves through convection. Hot air rises – and consequently moves heat. Radiation also transfers heat but less effectively. The last section of Part Three introduced this concept with the “gray model”. Here was the image presented:

The Gray Model of Radiative Equilibrium, from "Handbook of Atmospheric Science" Hewitt and Jackson (2003)

Remember that each section of the atmosphere radiates energy according to its temperature. So when we are solving the equations that link each “slice” of the atmosphere we have to have a term for temperature.

But how do we include convection? If we don’t include it our analysis will be wrong but solving for convection is a very different kind of problem, related to fluid dynamics..

What R&C did was to approach the numerical solution by saying that if the energy transfer from radiation at any point in their vertical profile resulted in a temperature gradient less than that from convection then use the known temperature profile at that point. And if it was greater than the temperature gradient from convection then we don’t have to think about convection in this “slice” of the atmosphere.

By the way, the terminology around how temperature falls with height through the atmosphere is called “the lapse rate” and it is about 6.5K/km.

These assumptions in the two cases didn’t mean that absorption and re-radiation were ignored in the lower part of the atmosphere – not at all. But the equations can’t be solved without including temperature. The question is, do we solve the equations by calculating temperature – or do we use an “externally imposed” temperature profile?

There is lots to digest in the paper as it is a comprehensive review. The few of interest for this post:

Doubling CO2 from 300ppm to 600ppm

• Longwave radiative forcing at the top of the troposphere – 3.9W/m2
• Surface temperature increase 1.2°C
• Result of change in radiative forcing when relative humidity stays constant (rather than absolute humidity staying constant) – surface temperature increase is doubled

(Note: this is not quite the “standardized” version of doubling considered today of 287ppm – 576ppm)

Relative Effect of CO2 and water vapor

This is under 1978 conditions of 330ppmv for CO2 and in a cloudy sky. Here they run the calculation with and without different gases and look at how much more outgoing longwave radiation there is, i.e. how much longwave radiation is absorbed by each gas. The problem is complicated by the fact that there is an overlap in various bands so there are combined effects.

• Removing CO2 (and keeping water vapor) – 9% increase in outgoing flux
• Removing water vapor (and keeping CO2) – 25% increase in outgoing flux

Everyone (= lots of people in lots of websites who probably know a lot more than me) says that this paper calculates the role of CO2 between 9% and 25% but that’s not how I read it. Perhaps I missed something.

Extract from Ramanathan & Coakley (1978) - Relative contribution of H2O, CO2 and O3

What it says to me is that overlap must be significant because if we take out water vapor it is only a 25% effect. And if we take out CO2 it is a 9% effect. (I have emailed the great V. Ramanathan to ask this question, but have not had a response so far.)

Therefore, guessing at the overlap effect, or more accurately, assigning the overlap equally between the two, water vapor has about 2.5 times the effect of CO2. As you will see in the next paper, this is about what our later results show.

So, more than 30 years ago, atmospheric physicists calculated some useful results which have been confirmed and refined by later scientists in the field.

Kiehl and Trenberth 1997

Earth’s Annual Global Mean Energy Budget by J.T.Kiehl and Kevin Trenberth was published in Bulletin of the American Meteorological Society in 1997. (The paper is currently available from this link)

The paper is very much worth a read in its own right as it reviews and updates the data at the time on the absorption and reflection of solar radiation and the emission and re-absorption of longwave radiation. (There is an updated paper – that free link currently works – in 2008 but it assumes the knowledge of the 1997 paper so the 1997 paper is the one to read).

This paper doesn’t assess the increase in radiative forcing or the consequent temperature change that might imply from the current levels of CO₂, CH₄ etc. Instead this paper is focused on separating out the different contributions to shortwave and longwave absorbed and reflected and so on.

What is interesting about this paper for our purposes in that they quantify the relative role of CO₂ and water vapor in clear sky and cloudy sky conditions.

To do the calculation of absorption and re-emission of longwave radiation they used the US Standard Atmosphere 1976 for vertical profiles of temperature, water vapor and ozone. They assumed 353ppmv of CO₂, 1.72ppmv of CH₄ and 0.31 of N₂O, all well mixed. Note that, like R&C, they assumed a temperature profile to carry out the calculations because convection dominates heat movement in the lower part of the atmosphere.

Two situations are considered in their calculations – clear sky and cloudy sky.

Let’s look at the clear sky results:

Upward Longwave Radiation, Numbers from Kiehl & Trenberth (1997)

The radiation value from the earth’s surface matches the temperature of 288K (15°C) – you can see how temperature and radiation emitted are linked in the maths section at the end of CO2 – An Insignificant Trace Gas? Part One.

The value calculated initially at the top of atmosphere was 262 W/m², the value was brought into line with the ERBE measured value of 265 W/m² by a slight change to the water vapor profile, see Note 1 at the end.

Of course, the difference between the surface and top of atmosphere values is accounted for by absorption of long wave radiation by water vapor, CO₂, etc. No surprise to those who have followed the series to this point.

By comparison the cloudy sky numbers were:

• Surface – 390W/m² (no surprise, the same 288K surface)
• TOA – 235W/m². More radiation is absorbed when clouds are present. See Note 2 at end.

Now onto the important question: of the 125W/m² “clear sky greenhouse effect”, what is the relative contribution of each atmospheric absorber?

The only way to calculate this is to remove each gas in turn from the model and recalculate.

Clear Sky

• Water vapor contributes 75W/m² or 60% of the total
• CO2 contributes 32W/m² or 26% of the total

Cloudy Sky

• Water vapor contributes 51W/m² or 59% of the total
• CO₂ contributes 24W/m² or 28% of the total

Note that significant longwave radiation is also absorbed by liquid water in clouds.

Conclusion

Using these three elements:

• the well known equations of radiative transfer (basic physics)
• the measured absorption profiles of each gas
• the actual vertical profiles of temperature and concentrations of the various gases in the atmosphere

The equations can be solved in a 1-d vertical column through the atmosphere and the relative effects of different gases can be separated out and understood.

Additionally, the effect in “radiative forcing” of the current level of CO₂ and of CO₂ doubling (compared with pre-industrial levels) can be calculated.

This radiative forcing can be applied to work out the change in surface temperature – with “all other things being equal”.

“All other things being equal” is the way science progresses – you have find a way to separate out different phenomena and isolate their effects.

The temperature increase in the R&C paper of 1.2°C only tells us the kind of impact from this level of radiative forcing. Not what actually happens in practice, because in practice we have so many other factors affecting our climate. That doesn’t mean it isn’t a very valuable result.

Now the value of radiative forcing will be slightly changed if  “all other things are not equal” but if the concentration of water vapor, CO₂, CH₄, etc are similar to our model the changes will not be particularly significant. It is only really the actual temperature profile through the atmosphere that can change the results. This is affected by the real climate of 3d effects – colder or warmer air blowing in, for example. Overall, from comparing the results of 3-d models – ie the average results of lots of 1-d models, the values are not significantly changed – more on this in a later post.

We see that CO₂ is around 25% of the “greenhouse” effect, with water vapor at around 60%.

Note that the calculation uses the “US Standard Atmosphere” – different water vapor concentrations will have a significant impact, but this is an “averaged” profile.

The only way to really determine the numbers is to run the RTE (radiative transfer equations) through a numerical analysis and then redo the calculations without each gas.

The two questions to ask if you see very different numbers is “under what conditions?” and more importantly “how did you calculate these numbers?” Hopefully, for everyone following the series it will be clear that you can’t just eyeball the spectral absorption and the average relative concentrations of the gases and tap it out on a calculator.

I thought it would be all over by Part Three, but CO₂ is a gift that keeps on giving..

Updates:

CO2 – An Insignificant Trace Gas? Part Six – Visualization

CO2 – An Insignificant Trace Gas? Part Seven – The Boring Numbers

CO2 – An Insignificant Trace Gas? – Part Eight – Saturation

See also – Theory and Experiment – Atmospheric Radiation – demonstrating the accuracy of the radiative-convective model from experimental results

Notes and References

Note 1 – As Kiehl and Trenberth explain, there are some gaps in our knowledge in a few places of exactly how much energy is absorbed or reflected from different components under different conditions. One of the first points that they make is that the measurement of incoming shortwave and outgoing longwave (OLR) are still subject to some questions as to absolute values. For example, the difference between incoming solar and the ERBE measurement of OLR is 3W/m². There are some questions over the OLR under clear sky conditions. But for the purposes of “balancing the budget” a few numbers are brought into line as the differences are still within instrument uncertainty.

Note 2 – I didn’t want to over-complicate this post. Cloudy sky conditions are more complex. Compared with clear skies clouds reflect lots of solar (shortwave) radiation, absorb slightly more solar radiation and also absorb more longwave radiation. Overall clouds cool our climate.

References

Climate Modeling through Radiative-Convective Models , V. Ramanathan and J.A. Coakley, Reviews of Geophysics and Space Physics (1978)

Earth’s Annual Global Mean Energy Budget , J.T.Kiehl and Kevin Trenberth, Bulletin of the American Meteorological Society (1997)

In the first post about CO2 I included a separate maths section which showed the energy budget for the earth and also derived how much energy we receive from the sun. A comment today reminded me that I should do a separate article about this topic. I’ve seen lots of comments on other blogs where people trip up over the basic numbers. It’s easy to get confused.

Don’t worry, there won’t be a lot of maths. This is to get you comfortable with some basics.

Energy from the Sun

It’s quite easy to derive how much energy we expect from the sun, but the good news is that since 1978 there have been satellites measuring it.

The solar “constant” is often written as S, so we’ll keep that convention. I put “constant” in quotes because it’s not really a constant, but that’s how it’s referred to. (And anyway, the changes year to year and decade to decade are very small – a subject for another post, another day).

The first important number, S = 1367 W/m²

Note the units – the amount of energy per second (the Watts) per unit area (the meters squared). By the way, sorry America, the science world moved on. We won’t convert it to ft²..

Just for illustration here’s the satellite measurements over 20 years:

Solar Radiation Received - measured by Satellites

For anyone a little confused, note that different satellites get different absolute measurements, it is the relative measurements that are more accurate.

Comparing Apples and Oranges? Surface Area vs Area of a Disc

The sun is really long way away from the earth – about 150M km (93M miles). We measure the incoming solar radiation at the top of the atmosphere in W/m².

So how much total energy can be absorbed into the earth’s climate system from this solar radiation?

Solar radiation received against a "2d disc". From Elementary Climate Physics, Taylor (2005)

Hopefully the answer will become more obvious by looking at the image above. The solar radiation from a long way away strikes the effective 2d area that the earth cuts out.

A 2d area – or a flat disc – has area, A = πr²

Therefore, the total energy received by the earth = Sπr²

[Radius of the earth = 6.37 x 10⁶ m (6,370 km) so Energy per second from the sun = 174,226,942,644,300,000 W  also written as 1.74 x 10¹⁷ W]

It’s a really big number, so to make everything easier to visualize, climate scientists generally stay with W/m2, rather than numbers like 1.74 x 10¹⁷ W.

Now the real surface area of the earth is actually, A_e= 4πr² (not πr²)

(Area of earth, A_e= 510M km², or 5.1×10¹⁴m²)

Why isn’t the energy received by the sun = S x 4πr²?

Look back at the graphic – is the sun shining equally on every part of the earth every second, for all 24 hours of the day? It’s not. It’s shining onto one side of the earth. It’s night time for half the world at any given moment.

So think of it like this – the absolute maximum area receiving the sun’s energy on average can only be half of the surface area of the earth – 2πr² (=4πr²/2)

But that’s not the end of the story. Picture someone where the sun is right down near the horizon. It’s still daytime but obviously that part of the earth is not receiving 1367W/m² – they are receiving a lot less. In fact, the only spot on earth where someone receives 1367W/m² is where the sun is directly overhead. So the effective area receiving the solar constant of 1367 W/m² can’t even be as high as 2πr².

So if the idea that solar radiation only strikes an effective area of πr² is still causing you problems, this is the concept that might help you.

Linking Incoming Solar Radiation to the Earth’s Outgoing Radiation

The earth radiates out energy in a way that is linked to the surface temperature. In fact it is proportional to the fourth power of absolute temperature.

As we think about the earth radiating out energy, it might be clearer why we labored the point earlier about the area that the sun’s energy was received over.

Take a look at that graphic again. The energy from the sun hits an effective 2d disc with area = πr².

The earth radiates out energy from its whole surface area = 4πr².

So to be able to compare “apples and oranges”, when climate scientists talk about energy balance and the climate system they usually convert radiation from the sun into the effective radiation averaged across the complete surface of the earth.

This is simply 1367/4 = 342.

The second important number, incoming solar radiation at the top of atmosphere = 342 W/m² (averaged across the whole surface of the earth).

Some energy is reflected but before we consider that note that this doesn’t mean that each square meter of the earth receives 342 W/m² – it’s just the average. The equator receives more, the poles receive less.

Albedo

Not all of this 342 W/m² is absorbed. The clouds, aerosols, snow and ice reflect a lot of radiation. Even water reflects a few percent. On average, about 30% of the solar radiation is reflected back out. A lot of slightly different numbers are used because it’s difficult to measure average albedo.

The third important number, solar radiation absorbed into the climate system = 239 W/m²

This is simply 342 * (100% – 30%). You see slightly different numbers like 236, 240 – all related to the challenges of accurate measurement of albedo.

Some of the radiation is absorbed in the atmosphere, and the rest into the land and oceans.

The Equation

Energy radiated out from the climate system must balance the energy received from the sun. This is energy balance. If it’s not true then the earth will be heating up or cooling down. Even with current concerns over global warming the imbalance is quite small. And so, as a starting point, we say that energy radiated out = energy absorbed from the sun.

Energy radiated from the earth, E_e = S (1- A) / 4  in W/m²

where A = albedo (as a number between 0 and 1, currently 0.3)

Conclusion

The solar constant, S = 1367 W/m²

The solar radiation at the top of atmosphere averaged over the whole surface of the earth = 342 W/m²

The solar radiation absorbed by the earth’s climate system = 239 W/m² (about 28% into the atmosphere and 72% into the earth’s surface of land, oceans, ice, etc)

Therefore, the approximate radiation from the earth’s climate system at the top of atmosphere also equals 239 W/m².

These numbers are useful to remember.

Update – new post The Earth’s Energy Budget – Part Two

Update – new post The Earth’s Energy Budget – Part Three

Recap

Part One opened up the topic and introduced the simple “billiard ball” or zero-dimensional analysis of the earth’s climate system. The sun radiates “shortwave” energy which is absorbed in the atmosphere and the earth’s surface. This heats up the earth’s climate system and it radiates out “longwave” energy.

The longwave energy gets significant absorption from water vapor, CO2 and methane (among other less important gases). This absorption heats up the atmosphere which re-radiates long wave energy both up and back down to the earth’s surface.

It is this re-radiation which keeps the earth’s surface at around +15°C instead of -18°C.

Part Two looked at why different gases absorb and radiate different proportions of energy – what the factors are that determine the relative importance of a “greenhouse” gas. Also why some gases like O2 and N2 absorb almost nothing in the longwave spectrum.

The Part Three introduced Radiative Transfer Equations and finished up with a look at what is called the gray model of the atmosphere. The gray model is useful for getting a conceptual understanding of how radiative transfer creates a temperature profile in the atmosphere.

However, part three didn’t finish up with enlightenment on the complete picture of CO2. The post was already long enough.

In this post we will look at “band models” and explain a little about saturation.

Band Models

Many decades ago when physicists had figured out the radiative transfer equations and filled up books with the precise and full derivations there was an obvious problem.

There was clearly no way to provide an analytical solution to how longwave radiation was absorbed and re-emitted through the atmosphere. Why? Because the actual absorption is a very complex and detailed function.

For example, as shown in an earlier post in the series, here is one part of the CO2 absorption spectrum:

From "Handbook of Atmospheric Sciences", Hewitt & Jackson 2003

The precise structure of the absorption is also affected by pressure broadening as well as a couple of other factors.

So long before powerful computers were available to perform a full 1-d model through the earth’s atmosphere, various scientists started working out “parameterizations” of the bands.

What does this mean? Well, the idea is that instead of actually having to look up the absorption at each 0.01μm of the longwave spectrum, instead you could have an equation which roughly described the effect across one part of the band.

Goody in 1952 and Malkmus in 1967 proposed “narrow band” methods. Subsequently others proposed “wide band” methods. Later researchers analyzed and improved these band paramaterizations.

Without using these parameterizations, even today, with very powerful computers, it is weeks of computational time to calculate the 1-d radiative transfer function for the atmosphere for one profile.

It’s important to note that the parameterizations can be tested and checked. Kiehl and Ramanathan did a big study in 1983 and showed that many of the models were well within 10% error compared with the detailed line by line calculations.

Here is one band model:

Random Band Model

Looks ugly doesn’t it? But it makes the calculations a million times easier than the detailed spectral lines all the way from 4μm up to 30μm.

The first term, T_Δν, is transmittance – it’s just how much radiation gets through the gas.

Transmittance = I/Io

If you don’t mind a little maths – otherwise skip to the next section

Let’s explain the equation and what it means for saturation.

First of all what are the variables?

T_Δν – is the transmittance in the spectral interval Δν. Transmittance is the fraction of radiation that passes through: 0 – no radiation gets through;  1 – all the radiation gets through.

S, α and δ are all part of the band model: S – average line strength; α – line width; δ – line spacing

u – the absorber amount in the path (this is the important one to keep an eye on)

By the definition of Transmittance,T_Δν = e^-χ, where χ is optical thickness. It’s the Beer Lambert law that we already saw in part three.

An alternative way of writing this is χ = – log (T_Δν) , that is, the optical thickness is the log of the transmittance

Well, even the tricky band model equation can be simplified..

If   Su/πα << 1 (this means if the expression on the left side is a lot less than 1 – which happens when there isn’t “very much” of the absorbing gas)

Then the above question can be simplified to:

T_Δν = exp (-Su/δ)

This means the optical thickness of the path is directly proportional to the amount of gas, u

So in part three when we looked at the Beer-Lambert law we saw this shape of the curve:

Transmittance of Radiation as "optical thickness" increases

But we couldn’t properly evaluate the expression because the absorption variable was a complex function of wavelength.

What the band model allows us to do is to say that under one condition, the weak condition, the optical thickness is a linear function of absorber amount, and therefore that the amount of radiation getting through the atmosphere – the Transmittance – follows this form: e^-χ

And in another condition, if

If   Su/πα >> 1 (much greater than 1, which means there is “lots” of the absorbing gas)

Then the band model can be “simplified” to:

T_Δν = exp (-(Su)^1/2/( δ √(πα)) )

Ok, not too easy to immediately see what is going on? But S, δ and α are constants for a given absorbing gas..

So it is easy to see what is going on:

T_Δν is proportional to exp (-u^1/2), i.e., proportional to exp (-√u)

Or as optical thickness, χ =- log (T_Δν),

Optical thickness, χ is proportional to √u

The optical thickness, in the strong condition, is proportional to the square root of the amount of the absorber.

“Saturation” and how Transmittance and Optical Thickness Depend on the Concentration of CO2

If you skipped the maths above, no one can blame you.

Recapping what we learnt there –

In the weak condition, if we double the concentration of CO2, the optical thickness doubles and in the strong condition if we increase the concentration of CO2 by a factor of 4, the optical thickness doubles

And what were the weak and strong conditions? They were mathematically defined, but keeping it non-technical: weak is “not much” CO2 and strong is “a lot” of CO2.

But we can say that in the case of CO2 (in the 15μm band) through the troposphere (lower part of the atmosphere) it is the strong condition. And so if CO2 doubled, the optical thickness would increase by √2 (=1.4).

Simple? Not exactly simple, but we made progress. Before, we couldn’t get any conceptual understanding of the problem because the absorption spectrum was lots of lines that prevented any analytical formula.

What we have achieved here is that we have used a well-proven band model and come up with two important conditions that allow us to define the technical meaning of saturation – and even better, to see how the increasing concentration of CO2 impacts the absorption side of the radiative transfer equations.

But it’s not over yet for “saturation”, widely misunderstood as it is.. Remember that absorption is just one half of the radiative transfer equations.

Before we finish up, optical thickness isn’t exactly an intuitive or common idea, and neither is e^-√χ. So here is a idea of numerically how transmittance changes under the weak and strong conditions as the concentration increases. Remember that transmittance is nice and simple – it is just the proportion of radiation that gets through the absorbing gas.

Transmittance = I/Io

Suppose our optical thickness, χ = 1.

T = 0.36        =exp(-1)

Under the weak condition, if we double our optical thickness, χ = 2;     T = 0.13    =exp(-2)

and double it again, χ = 4;     T = 0.017    =exp(-4)

Under the strong condition, double our optical thickness, χ = 2;     T = 0.24    =exp(-√2)=exp(-1.41)

and double it again, χ = 4;     T = 0.13      =exp(-√4) = exp(-2)

Note: these numbers are not meant to represent any specific real world condition. It just demonstrates the kind of change you get in the amount of radiation being transmitted as the gas concentration increases under the two different conditions. It helps you get an idea of e^-χ vs e^-√χ. Assuming that a few people would want to know..

Conclusion

To carry out the full 1-d radiative transfer equations vertically through the atmosphere climate scientists usually make use of band models. They aren’t perfect but they have been well tested against the “line by line” (LBL) absorption spectra.

Because they provide a mathematical parameterization they also allow us to see conceptually what happens when the concentration of an important gas like CO2 is increased. We can calculate the transmittance or absorptance that takes place.

It helps us understand “saturation” – which we have done by looking at the “strong” and “weak” conditions for optical thickness.

This term “saturation” is widely misused and conveys the idea that CO2 has done all its work and adding more CO2 doesn’t make any difference. As we will see in a future part of this series, due to the fact that gases that heat up also radiate, adding more CO2 does increase the radiative forcing at the surface – even if CO2 could have no more effect through the lower part of the atmosphere.

Well, that’s to come. What we have looked at here is some more detail of exactly how transmittance and optical thickness increase as CO2 increases.

The next post will look at the 1-dimensional model results..

Update – Part Five now published

Reference

CO2 Radiative Paramaterization Used in Climate Models: Comparison with Narrow Band Models and With Laboratory Data, J.T. Kiehl and V. Ramanathan (1983)

Recap

Part One of the series started with this statement:

If there’s one area that often seems to catch the imagination of many who call themselves “climate skeptics”, it’s the idea that CO2 at its low levels of concentration in the atmosphere can’t possibly cause the changes in temperature that have already occurred – and that are projected to occur in the future. Instead, the sun, that big bright hot thing in the sky (unless you live in England), is identified as the most likely cause of temperature changes.

And covered the “zero-dimensional” model of the sun and the earth. Also known as the “billiard ball” model. It was just a starting point to understand the very basics.

In Part Two we looked a little closer at why certain gases absorbed energy in certain bands and what the factors were that made them more, or less, effective “greenhouse” gases.

In this part, we are going to start looking at the “1-dimensional” model. I try and keep any maths as basic as possible and have separated out some maths for the keen students.

When you arrive at a new subject, the first time you see an analysis, or model, it can be confusing. After you’ve seen it and thought about it a few times it becomes more obvious and your acceptance of it grows – assuming it’s a good analysis.

So for people new to this, if at first it seems a bit daunting but you do want to understand it, don’t give up. Come back and take another look in a few days..

Models

If your background doesn’t include much science it’s worth understanding what a “model” is all about. Especially because many people have their doubts about GCM’s or “Global Climate Models”.

One of the ways that a model of a physics (or any science) problem is created is by starting from first principles, generating some equations and then finding out what the results of those equations are. Sometimes you can solve this set of equations “analytically” – which means the result is a formula that you can plot on a graph and analyze whichever way you like. Usually in the real world there isn’t an “analytical” solution to the problem and instead you have to resort to numerical analysis which means using some kind of computer package to calculate the answer.

The starting point of any real world problem is a basic model to get an understanding of the key “parameters” – or key “players” in the process. Then – whether you have an analytical solution or have to do a numerical analysis doesn’t really matter – you play around with the parameters and find out how the results change.

Additionally, you look at how closely the initial equations matched the actual situation you were modeling and that gives you an idea of whether the model will be a close fit or a very rudimentary one.

And you take some real-world measurements and see what kind of match you have.

Radiative Transfer

In the “zero dimensional” analysis we used a very important principle:

Energy into a system = Energy out of a system, unless the system is warming or cooling

The earth’s climate was considered like that for the simple model. And for the simple model we didn’t have to think about whether the earth was heating up, the actual temperature rise is so small year by year that it wouldn’t affect any of those results.

In looking at “radiative transfer” – or energy radiated through each layer of the atmosphere – this same important principle will be at the heart of it.

What we will do is break up the atmosphere into lots of very thin sections and analyze each section. The mathematical tools are there (calculus) to do that. The same kind of principles are applied, for example, when structural engineers work out forces in concrete beams – and in almost all physics and engineering problems.

And when we step back and try and re-analyze, again it will be on the basis of Energy in = Energy out

If you are new to ideas of radiation and absorption, go back and take a look at Part One – if you haven’t done so already.

In this first look I’ll keep the maths as light as possible and try and explain what it means. If following a little maths is what you want, there is some extra maths separated out.

First Step – Absorption

As we saw in part one, radiation absorbed by a gas is not constant across wavelengths. For example, here is CO2 and water vapor:

CO2 and water vapor absorption, by spectracalc.com from the HITRANS database

What we want to know is if we take radiation of a given wavelength which travels up through the atmosphere, how much of the radiation is absorbed?

We’ll define some parameters or “variables”.

I(λ) – The intensity, I, of radiation which is a function of wavelength, λ

I₀(λ) –  is the initial or starting condition (the intensity at the earth’s surface).

z – the vertical height through the atmosphere

n – how much of an absorbing gas is present

σ(λ) – absorption cross-section at wavelength λ (this parameter is dependent on the gas we are considering, and identifies how effective it is at capturing a photon of radiation at that wavelength)

The result of a simple mathematical analysis produces an equation that says that as you:

• increase the depth through the atmosphere that the radiation travels
• or the concentration of the gas
• or its “absorption cross-section”

Then more radiation is absorbed. Not too surprising!

When the concentration of the gas is independent of depth (or height) the mathematical result becomes:

Iz = I₀(λ).exp(-nσ(λ)z) also written as Iz = I₀(λ).e^-nσ(λ)z

This is the Beer-Lambert Law. The assumption that the number of gas molecules is independent of depth isn’t actually correct in the real world, but this first simple approximation gets us started. We could write n(z) in the equation above to show that n was a function of depth through the atmosphere.

[In the above equation, e is the natural log value of 2.781 that comes up everywhere in natural processes. To make complex equations easier to read, it is a convention to write “e to the power of x” as “exp(x)”]

Here’s what the function looks like as “nσ(λ)z” increases – I called this term “x” here in this graph.

Transmittance of Radiation as "optical thickness" (x) increases.

It’s not too hard to imagine now. I_z is the amount of radiation making it through the gas. I_z=1 means all of it got through, and I_z=0 means none of it got through.

As you increase vertical height through the gas, or the amount of the gas, or the absorption of the gas, then the amount of radiation that gets through decreases. And it doesn’t decrease linearly. You see this kind of shape of curve everywhere in nature including the radiation decay of uranium..

This result is not too surprising to most people. But it’s knowing only this part which has many confused, because the question comes – about CO2 – doesn’t it saturate?

Isn’t it true that as CO2 increases eventually it has no effect? And haven’t we already reached that point?

Excellent questions. Skip the maths derivation of this section if you aren’t interested to find out about our Second Step – Radiation

First Step – Absorption – Skip this, it’s the Maths

You can skip this if you don’t like maths.

The intensity of light of wavelength λ is I(λ). This light passes through a depth dz (“thin slice”) of an absorber with number concentration n, and absorption cross-section σ(λ), and so is reduced by an amount dI(λ) given by:

dI(λ) = -I(λ)nσ(λ)dz = I(λ)dχ                   [equation 1]

where χ is defined as optical depth. It’s just a convenient new variable that encapsulates the complete effect of that depth of atmosphere at that wavelength for that gas.

We integrate equation 1 to obtain the intensity of light transmitted a distance z through the absorber Iz(λ):

I_z(λ) = I₀(λ) exp{-∫nσ(λ)dz}                 [equation 2]
[note the integral is from 0 to z – not able to get the webpage to display what I want here]

In the case where the concentration of the absorbing gas is independent of the depth through the atmosphere the above equation is simplified to the Beer Lambert Law

Iz = I₀(λ).exp(-nσ(λ)z) also written as Iz = I₀(λ).e^-nσ(λ)z

Note that this assumption is not strictly true of the atmosphere in general – the closer to the surface the higher the pressure and, therefore, there is more of absorbing gases like CO2.

Second Step – Radiation

Once the atmosphere is absorbing radiation something has to happen.

The conceptual mistake that most people make who haven’t really understood radiative transfer is they think of it something like torchlight trying to shine through sand – once you have enough sand nothing gets through and that’s it.

But energy absorbed has to go somewhere and and in this case the energy goes into increased heat of that section of the atmosphere, as we saw in Part Two of this series.

In general, and especially true in the troposphere (the lower part of the atmosphere up to around 10km), the increased energy of a molecule of CO2 (or water vapor, CH4, etc) heats up the molecules around it – and that section of the atmosphere then radiates out energy, both up and down.

Let’s introduce a new variable, B = intensity of emitted radiation

The relationship between I (radiation absorbed) – and B (radiation emitted) – integrated across all wavelengths, all directions and all surfaces is linked through conservation of energy.

But these two parameters are not otherwise related. Making it more difficult to conceptually understand the problem.

I depends on the radiation from the ground, which in turn is dependent on the energy received from the sun and longwave radiation re-emitted back to the ground.

Whereas B is a function of the temperature of that “slice” of the atmosphere.

The equation that includes absorption and emission for this thin “slice” through the atmosphere becomes:

dI = -Inσdz + Bnσdz = (I – B)dχ  (where χ is defined as optical depth)

dI is “calculus” meaning the change in I, dz is the change in z and dχ is the change in χ, or optical thickness.

What does this mean? Well, if I could have just written down the “result” like I did in the section on absorption, I would have done, but because it has become more difficult, instead I have written down the equation linking B and I in the atmosphere..

What it does mean is that the more radiation that is absorbed in a given “slice”  of the atmosphere, the more it heats up and consequently the more that “slice” then re-emits radiation across a spectrum of wavelengths.

Solving the Equation to Find out what’s Going on

There are two concepts introduced above:

• absorption, relatively easy to understand
• emission, a little harder but linked to absorption by the concept “energy in = energy out”

From here there are two main approaches..

1. One approach is called the grey model of radiative transfer, and it uses a big simplification to show how radiation moves energy through the atmosphere.
2. The other approach is to really solve the equations using numerical analysis via computers.

The problem is that we have some equations but they aren’t simple. We saw the Beer-Lambert law of absorption links to the emission in a given section of the atmosphere, but we know that the absorption is not constant across wavelengths.

So we have to integrate these equations across wavelengths and through the atmosphere (to link radiation flowing through each “slice” of the atmosphere)

To really find the solutions – how much longwave radiation gets re-radiated back down to the earth’s surface as a result of CO2, water vapor and methane – we need a powerful computer with all of the detailed absorption bands of every gas, along with the profile of how much of each gas at each level in the atmosphere.

The good news is that they exist. But the bad news is that you can’t grab the equation and put it in excel and draw a graph – and find out the answer to that burning question that you had.. what about the role of CO2? and how does that compare with the role of water vapor?

And I still haven’t spelt out the saturation issue..

Finding out that the subject is more complex that it originally appeared is the first step to understanding this subject!

The important concept to grasp before we move on is that it is not just about absorption, it’s also about re-emission.

The Gray Model

The “gray” model is very useful because it allows us to produce a simple mathematical model of the temperature profile through the atmosphere. We can do this because instead of thinking about the absorption bands, instead we assume that the absorption across wavelengths is constant.

What? But that’s not true!

Well, we do it to get a conceptual idea of how energy moves through the atmosphere when absorption and re-emission dominate the process. We obviously don’t expect to find out the exact effect of any given gas. The gray model uses the equations we have already derived and adds the fact that the absorber varies in concentration as a function of pressure.

The Gray Model of Radiative Equilibrium, from "Handbook of Atmospheric Science" Hewitt and Jackson (2003)

The graph shown here is the result of developing the equations we have already seen, both for absorption and the link between absorption and re-emission.

The equations totally ignore convection! On the graph you can see the real “lapse rate”, which is the change in temperature with altitude. This is dominated by convection, not by radiation.

So how does the gray model help us?

It shows us how the temperature profile would look in an atmosphere with no convection and where there is significant and uniform absorption of longwave radiation.

Convection exists and is more significant than radiation in the troposphere – for moving energy around! Not for absorbing and re-emitting energy. The significance of the real “environmental lapse rate” of 6.5K/km is that it will change the re-emission profile. So it complicates the numerical analysis we need to do. It means that when the numerical analysis is done of the equations we have already derived, the real lapse rate is one more factor that has to be added to that 1-d analysis.

To get a conceptual feel for how that might change things – remember how the radiation spectrum changes with temperature – not a huge amount. So at each layer in the atmosphere the radiation spectrum using the real atmospheric temperature profile will be slightly different than using the “gray model”. But it can be taken into account.

Conclusion

This post has covered a lot of ground and not given you a nice tidy result. Sorry about that.

It’s an involved subject, and there’s no point jumping to the conclusion without explaining what the processes are. It is understanding the processes involved in radiative physics and the way in which the subject is approached that will help you understand the subject better.

And especially important, it will help you see the problems with a flawed approach. There are lots of these on the internet. There isn’t a nice tidy analytical expression which links radiative forcing to CO2 concentration, and which separates out CO2 from water vapor. But 1-d numerical models can generate reliable and believable results.

In Part Four, we will finally look at saturation, how it’s misunderstood, how much radiative forcing more CO2 will add (all other things being equal!) and how CO2 compares with water vapor.

So watch out for Part Four, and feel free to comment on this post or ask questions.

Update – Part Four is now online

Recap

Part One of the series started with this statement:

If there’s one area that often seems to catch the imagination of many who call themselves “climate skeptics”, it’s the idea that CO₂ at its low levels of concentration in the atmosphere can’t possibly cause the changes in temperature that have already occurred – and that are projected to occur in the future. Instead, the sun, that big bright hot thing in the sky (unless you live in England), is identified as the most likely cause of temperature changes.

Part One looked mainly at the radiation balance – what the sun provides (lots of energy at shortwave) and what the earth radiates out (longwave). Then it showed how “greenhouses gases” – water vapor, CO2 and methane (plus some others)  – absorb longwave radiation and re-emit radiation both up out of the atmosphere and back down to the earth’s surface. And without this absorption of longwave radiation the earth would be 35°C cooler at its surface. The post concluded with:

CO₂ and water vapor are very significant in the earth’s climate, otherwise it would be a very cold place.

What else can we conclude? Nothing really, this is just the starting point. It’s not a sophisticated model of the earth’s climate, it’s a “zero dimensional model”.. the model takes a very basic viewpoint and tries to establish the effect of the sun and the atmosphere on surface temperature. It doesn’t look at feedback and it’s very simplistic.

Two images to remember..

First, the sun’s radiated energy is mostly under 4μm in wavelength (shortwave), while the earth’s radiated energy is over 4μm (longwave), meaning that we can differentiate the two very easily:

Radiation vs Wavelength - Sun and Earth

Second, the aborption that we can easily measure in the earth’s longwave radiation from different molecules:

Radiation spectra from the earth showing absorption from atmospheric gases

Recap over.. This post was going to introduce the basic 1-d model of radiative transfer, but enough people asked questions about the absorption properties of gases that I thought it was was worth covering in more detail.. The 1-d model will have to wait until Part Three.

Why don’t the Atmospheric Gases Absorb Energy according to their Relative Volume?

Just because CO2 only consists of 0.04% of gases doesn’t mean it only contributes 0.04% of atmospheric absorption and re-emission of long wave radiation. Why is that?

Oxygen, O2, constitutes 21% of the atmophere and nitrogen, N2, constitutes 78%. Why aren’t they important “greenhouse” gases? Why are water vapor, CO2 and methane (CH4) the most important when they are present in such small amounts?

For reference, the three most important gases by volume are:

• Water vapor – 0.4% averaged throughout the atmosphere, but actual value in any one place and time varies (See note 1 at end of article)
• CO2               – 0.04% (380ppmv), well mixed (note: ppmv is parts per million by volume)
• CH4               – 0.00018% (1.8ppmv), well mixed

Now there are three factors in determining the effect of longwave absorption:

1. The amount of the gas by volume
2. How much longwave energy is radiated from the earth at wavelengths that the gas absorbs
3. The ability of the gas to absorb energy at a given wavelength

The first one is the simplest to understand. In fact, it’s knowing only this factor that causes so much confusion.

The second point is not immediately obvious, but should become clearer by reviewing the earth’s radiation spectrum:

Radiation vs Wavelength - Sun and Earth

Different amounts of energy are radiated at different wavelengths. For example, the amount of energy emitted between 10-11μm is eight times the amount of energy between 4-5μm (for radiation from a surface temperature of 15 °C or 288K).

CO2 has a wide absorption band centered around 15μm, which is where the long-wave radiation from the earth is at almost its highest level. By contrast, one of water vapor’s absorption lines is at 6.27μm – where the radiation is a slightly lower level (about 25% less) and more importantly, the other water vapor absorption lines are where the radiation is 5-10x lower intensity.

However, there is around 10x as much water vapor than CO2 in the atmosphere, which is why it is the most important greenhouse gas.

And Third, Why are Some Gases More Effective at Absorbing Longwave Energy?

Why aren’t O2 and N2 absorbers of longwave radiation?

Molecules with two identical atoms don’t change their symmetry when any rotation or vibration takes place. As a result they can’t move into different energy states.

But triatomic molecules like CO2, H2O and CH4 can bend as they vibrate. They can move into different energy states by changing their shape. Consequently they can absorb the energy from an incoming photon if its energy matches the new state.

And some molecules have many more energy states they can move into. This changes their absorption profile because their spectral breadth is effectively wider.

Here’s a graphic of one part of the actual CO2 absorption lines. Apologies for the poor quality scan..

From "Handbook of Atmospheric Sciences", Hewitt & Jackson 2003

(Note that the x-axis is “Wavenumber, cm-1”. This is a convention for spectral people. Wavenumber is the number of wavelengths present in 1cm. I added the actual wavelength underneath.)

This shows the complexity of the subject once we look at the real detail. In practice, these individual discrete absorption lines “broaden” due to pressure broadening (collisions with other molecules) and Doppler broadening (as a result of the absorbing molecule moving in the same or opposite direction to the photon of light).

However, the important point to remember is that different molecules absorb at different frequencies and across different ranges of frequencies.

This third factor is the most important in determining the absorption properties of longwave radiation.

As an interesting comparison, molecule by molecule methane absorbs about 20x as much energy as CO2. But of course it is present in much smaller quantities.

Here are water vapor and CO2 across 5-25μm from the HITRANS database:

CO2 and water vapor absorption, by spectracalc.com from the HITRANS database

See Note 2 at the end of the article.

What about Oxygen?

A digression on oxygen.. It is important in the earth’s atmosphere because it absorbs UV, but when these high energy photons from the sun interact with O2 it breaks into O+O. Then a cycle takes place where O2 and O combine to form O3 (ozone), and later O3 breaks up again. By the time the sun’s energy has reached the lower part of the atmosphere (troposphere) all of the lower wavelength energy (most of the UV) has been filtered out.

O3 itself does absorb some longwave energy, at 9.6um, but because there is so little O3 in the troposphere it is not very significant.

What Happens when a Greenhouse Gas Absorbs Energy?

Once a gas molecule has absorbed radiation from the earth it has a lot more energy. But in the lower 100km of the atmosphere, the absorbed energy is transferred to kinetic energy by collisions between the absorbing molecules and others in the layer. Effectively, it heats up this layer of the atmosphere.

The layer itself will act as a blackbody and re-radiate infrared radiation. But it re-radiates in all directions, including back down to the earth’s surface. (If it only radiated up away from the earth there would be no “greenhouse” effect from this absorption).

Conclusion

We are still on the “zero dimensional model” – some call it the billiard ball model – of the radiative balance in the earth’s climate system.

A few different factors affect the absorption of the earth’s longwave radiation by various gases.

O2 barely absorbs any (see note 2 below), and neither does N2 (nitrogen). Among the other gases – the main greenhouse gases being water vapor, CO2 and methane – we see that each one has different properties – none of which can be determined by our intuition!

Different molecules can absorb energy in certain frequencies simply because of their ability to change shape and move to different energy states. The primary property that creates a strong “greenhouse” effect is to have a strong and wide absorption around a wavelength that the earth radiates. This is centered about 10μm (and isn’t symmetrical) so the further away from the peak energy the absorption occurs, the less relevant that absorption line becomes in the earth’s energy balance.

In the next part in the series, we will look at the 1-dimensional model and also what happens when absorption in a wavelength is saturated.

Note 1 – Water Vapor ppmv: After consulting numerous reference works, I couldn’t find one which gave the averaged water vapor throughout the atmosphere, or the troposphere. The actual source for the 0.4% was Wikipedia.

Because all the reference works danced around without actually giving a number I suspect it is “up in the air”. Here is one example:

Water vapor concentration is highly variable, ranging from over 20,000 ppmv (2%) in the lower tropospherical atmosphere to only a few ppmv in the stratosphere..

Atmospheric Science for Environmental Scientists (2009) Hewitt & Jackson

There is a great application, Spectral Calc for looking at atmospheric concentrations and absorption lines. Specifically http://spectralcalc.com/atmosphere_browser gives plots of atmospheric concentration and the data agrees with the Wikipedia number given in the body of this article:

CO2 and water vapor by volume, from "Spectral Calculator" database

Averaging over the whole atmosphere, the concentration of water vapor does seem to be around 10x the CO2 value.

Note 2 – Optical Thickness: The spectral plots from the HITRANS database shown in the body of the article give the capture cross-section per mole (i.e. per “unit” of that gas, not per unit volume of the general atmosphere).

One commenter asked why another plot from a different website drawing on the same HITRANS database produced this:

Optical Thickness of O2 and water vapor from http://www.atm.ox.ac.uk

Note that I’ve adjusted the plots so that similar values on the y-axes are aligned for both graphs. And note that the vertical axis is logarithmic.

His comment was that oxygen, O2, is only maybe 1000 times lower in absorption than water vapor (10⁰ =1 vs 10³ =1000) at 6.7μm and given that O2 is 20% of the atmosphere instead of 0.4%, O2 should be comparable to water vapor as a greenhouse gas.

But in fact, this graphical plot isn’t plotting the absorption by units of molecule – instead it is plotting Optical Thickness.

This is a handy variable which we will see more of in Part Three. Optical Thickness essentially takes the value of Intensity, which is per unit of molecules,  and “integrates” that value up through the entire height of the atmosphere.

As a result it gives the picture of the complete influence of that gas at different frequencies without having to work out the relative proportions of the gas at different heights in the atmosphere.

So the example above compares the complete absorption (in a simplistic model) through the whole atmosphere, giving O2 about 3000x less effect than water vapor at 6.7μm.

Update – Part Three is now online