The Science of Doom

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:

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:

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.

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:

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.

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)