The Science of Doom

Recap

In Part Five we finally got around to seeing our first calculations by looking at two important papers which used “numerical methods” – 1-dimensional models – to calculate the first order effect from CO2. And to separate out the respective contribution of water vapor and CO2.

Both papers were interesting in their own way.

The 1978 Ramanathan and Coakley paper because it is the often cited paper as the first serious calculation. And it’s good to see the historical perspective as many think scientists have been looking around for an explanation of rising temperatures and “hit on” CO2. Instead, the radiative effect of CO2, other trace gases and water vapor has been known for a very long time. But although the physics was “straightforward”, solving the equations was more challenging.

The 1997 Kiehl and Trenberth paper was discussed because they separate out water vapor from CO2 explicitly. They do this by running the numerical calculations with and without various gases and seeing the effects. We saw that water vapor contributed around 60% with CO2 around 26%.

I thought the comparison of CO2 and water vapor was useful to see because it’s common to find people nodding to the idea that longwave from the earth is absorbed and re-emitted back down (the “greenhouse” effect) – but then saying something like:

Of course, water vapor is 95%-98% of the whole effect, so even doubling CO2 won’t really make much difference

The question to ask is – how did they work it out? Using the complete radiative transfer equations in a 1-d numerical model with the spectral absorption of each and every gas?

Of course, everyone’s entitled to their opinion.. it’s just not necessarily science.

The “Standardized Approach”

In the calculations of the “greenhouse” effect for CO2, different scientists approached the subject slightly differently. Clear skies and cloudy skies, for example. Different atmospheric profiles. Some feedback from the stratosphere (higher up in the atmosphere), or not. Some feedback from water vapor, or not. Different band models (see Part Four). And also different comparison points of CO2 concentrations.

As the subject of the exact impact of CO2 – prior to any feedbacks – became of more and more concern, a lot of effort went into standardizing the measurement/simulation conditions.

One of the driving forces behind this was the fact that many different GCMs (Global Climate Models) produced different results and it was not known how much of this was due to variations in the “first order forcing” of CO2. (“First order forcing” means the effect before any feedbacks are taken into account). So different models had to be compared and, of course, this required some basis of comparison.

There was also the question about how good band models were in action compared with line by line (LBL) calculations. LBL calculations require a huge computational effort because the minutiae of every absorption line from every gas has to be included. Like this small subset of the CO2 absorption lines:

Band models are much simpler, and therefore widely used in GCMs. Band models are “paramaterizations”, where a more complex effect is turned into a simpler equation that is easier to solve.

Averaging

Does one calculation of CO2 radiative forcing from an “average atmosphere” gives us the real result for the whole planet?

Asking the question another way, if we calculate the CO2 radiative forcings from all the points around the globe and average the radiative forcing do we get the same result as one calculation for the “average atmosphere”.

This subject was studied in a 1998 paper: Greenhouse gas radiative forcing: Effects of average and inhomogeneities in trace gas distribution, by Freckleton et al. They ran the same calculations with 1 profile (the “standard atmosphere”), 3 profiles (one tropical plus a northern and southern extra-tropical “standard atmosphere”), and then by resolving the globe into ever finer sections.

The results were averaged (except the single calculation of course) and plotted out. It was clear from this research that using the average of 3 profiles – tropical, northern and southern extra-tropics – was sufficient and gave only 0.1% error compared with averaging the calculation at 2.5% resolution in latitude.

The Standard Result

The standard definition of radiative forcing is:

The change in net (down minus up) irradiance (solar plus longwave; in W/m²) at the tropopause after allowing for stratospheric temperatures to readjust to radiative equilibrium, but with surface and tropospheric temperatures and state held fixed at the unperturbed values.

What does it mean? The extra incoming energy flow at the top of atmosphere (TOA) without feedbacks from the surface or the troposphere (lower part of the atmosphere). The stratospheric adjustment is minor and happens almost immediately (there are no oceans to heat up or ice to melt in the stratosphere unlike at the earth’s surface). Later note added – “almost immediately” in the context of the response of the surface, but the timescale is the order of 2-3 months.

The common CO2 doubling scenario, from pre-industrial, is:

278ppm -> 556 ppm

And the comparison to the present day, of course, depends on when the measurement occurs but most commonly uses the 278ppm value as a comparison.

IPCC AR4 (2007)  pre-industrial to the present day (2005),  1.7 W/m²

IPCC AR4 (2007)  doubling CO2,  3.7 W/m²

Just for interest.. Myhre at al (1998) calculated the effects of CO2 – and 12 other trace gases – from the current increases in those gases (to 1995). They calculated separate results for clear sky and cloudy sky. Clear sky results are useful in comparisons between models as clouds add complexity and there are more assumptions to untangle.

They also ran the calculations using the very computationally expensive Line by Line (LBL) absorption, and compared with a Narrow Band Model (NBM) and Broad Band Model (BBM).

CO2 current (1995) compared to pre-industrial, clear sky – 1.76W/m², cloudy sky 1.37W/m²

(The NBM and BBM were within a few percent of the LBL calculations).

There are lots of other papers looking at the subject. All reach similar conclusions, which is no surprise for such a well-studied subject.

Where does the IPCC Logarithmic Function come from?

The 3rd assessment report (TAR) and the 4th assessment report (AR4) have an expression showing a relationship between CO2 increases and “radiative forcing” as described above:

ΔF = 5.35 ln (C/C₀)

where:

C₀ = pre-industrial level of CO2 (278ppm)
C = level of CO2 we want to know about
ΔF = radiative forcing at the top of atmosphere.

(And for non-mathematicians, ln is the “natural logarithm”).

This isn’t a derived expression which comes from simplifying down the radiative transfer equations in one fell swoop!

Instead, it comes from running lots of values of CO2 through the standard 1d model we have discussed, and plotting the numbers on a graph:

From New estimates of radiative forcing due to well mixed greenhouse gases, Myhre et al, Geophysical Research Letters (1998).

The graph reasonably closely approximates to the equation above. It’s very useful because it enables people to do a quick calculation.

E.g. CO2 = 380ppm, ΔF = 1.7W/m²

CO2 = 556ppm, ΔF = 3.7 W/m²

Easy.

Benefit of Using “Radiative Forcing” at TOA (top of atmosphere)

First of all, we can use this number to calculate a very basic temperature increase at the surface. Prior to any feedbacks – or can we? [added note, James McC kindly pointed out that my calculation of temperature is wrong and so maybe it is too simplistic to use this method when there is an absorbing and re-transmitting atmosphere in the way. I abused this approach myself rather than following any standard work. All errors are mine in this bit – we’ll let it stand for interest. See James McC’s comments in About this Blog)

In Part One of this series, in the maths section at the end (to spare the non-mathematically inclined), we looked at the Stefan-Boltzmann equation, which shows the energy radiated from any “body” at a given temperature (in K):

Total energy per unit area per unit time, j = εσT⁴

where ε= emissivity (how close to a “blackbody”: 0-1), σ=5.67×10^-8 and T = absolute temperature (in K).

The handy thing about this equation is that when the earth’s climate is in overall equilibrium, the energy radiated out will match the incoming energy. See The Earth’s Energy Budget – Part Two and also Part One might be of interest.

We can use the equations to do a very simple calculation of what ΔF = 3.7W/m² (doubling CO2) means in terms of temperature increase. It’s a rough and ready approach. It’s not quite right, but let’s see what it churns out.

Take the solar incoming absorbed energy of 239W/m² (see The Earth’s Energy Budget – Part One) and comparing the old  (only solar) – and new (solar + radiative forcing for doubling CO2 values), we get:

T_new⁴/T_old⁴ = (239 + 3.7)/239

where T_new = the temperature we want to determine, T_old = 15°C or 288K

We get T_new = 289.1K or a 1.1°C increase.

Well, the full mathematical treatment calculates a 1.2°C increase – prior to any feedbacks – so it’s reasonably close.

[End of dodgy calculation that when recalculated is not close at all. More comments when I have them].

Secondly, we can compare different effects by comparing their radiative forcing. For example, we could compare a different “greenhouse” gas. Or we could compare changes in the sun’s solar radiation (don’t forget to compare “apples with oranges” as explained in The Earth’s Energy Budget – Part One). Or albedo changes which increase the amount of reflected solar radiation.

What’s important to understand is that the annualized globalized TOA W/m² forcing for different phenomena will have subtly different impacts on the climate system, but the numbers can be used as a “broad-brush” comparison.

Conclusion

We can have a lot of confidence that the calculations of the radiative forcing of CO2 are correct. The subject is well-understood and many physicists have studied the subject over many decades. (The often cited “skeptics” such as Lindzen, Spencer, Christy all believe these numbers as well). Calculation of the “radiative forcing” of CO2 does not have to rely on general circulation models (GCMs), instead it uses well-understood “radiative transfer equations” in a “simple” 1-dimensional numerical analysis.

There’s no doubt that CO2 has a significant effect on the earth’s climate – 1.7W/m² at top of atmosphere, compared with pre-industrial levels of CO2.

What conclusion can we draw about the cause of the 20th century rise in temperature from this series? None so far! How much will temperature rise in the future if CO2 keeps increasing? We can’t yet say from this series.

The first step in a scientific investigation is to isolate different effects. We can now see the effect of CO2 in isolation and that is very valuable.

Although there will be one more post specifically about “saturation” – this is the wrap up.

Something to ponder about CO2 and its radiative forcing.

If the sun had provided an equivalent increase in radiation over the 20th century to a current value of 1.7W/m², would we think that it was the cause of the temperature rises measured over that period?

Update – CO2 – An Insignificant Trace Gas? Part Eight – Saturation is now published

References

Greenhouse gas radiative forcing: Effects of average and inhomogeneities in trace gas distribution, Freckleton at al, Q.J.R. Meteorological Society (1998)

New estimates of radiative forcing due to well mixed greenhouse gases, Myhre et al, Geophysical Research Letters (1998)

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After posting some comments on various blogs and seeing the replies I realized that a page like this was necessary.

For people who’ve just arrived at this page, you might be asking:

What effect?

-which in itself is one of the most important questions, but let’s not jump ahead..

The background is the series CO2 – An Insignificant Trace Gas? and especially the last post – which maybe should have come earlier! – CO2 – An Insignificant Trace Gas? Part 6 – Visualization

If you take a quick look at that last post you will find a few simple measurements that demonstrate that CO2 and other “greenhouse” gases have an effect at the earth’s surface.

What Effect?

In brief, simply that CO2 and other greenhouse gases add a “radiative forcing” to the earth’s surface. A “radiative forcing” means more energy and, therefore, heating at the earth’s surface. And more CO2 will increase this slightly.

At this stage, we have said nothing about feedback effects or even the end of the world.. The series on CO2 is simply to unravel its effect on global temperatures all other things being equal. Which of course, they are not! But we have to start somewhere.

Here are two graphics from Part Six showing energy up and energy down that are the basis for many many questions..

“TOA” = top of atmosphere.

The simple story these two graphics outline is that the earth radiates “longwave radiation” from its surface (because it has been heated by “shortwave radiation” from the sun).

The radiation from the earth’s surface is a lot more than the radiation leaving the atmosphere. Where does it go?

And why do we measure longwave radiation downwards at the earth’s surface. Where does that come from? And why do the wavelengths match those of CO2, methane and so on?

The answer – CO2 and other “greenhouse” gases absorb longwave radiation and re-emit radiation, both up (which continues on its journey out of the atmosphere) and down. The downward component increases the temperature at the earth’s surface.

The story sparked many questions on other blogs..

Questions like these are great, they clarify for me the common problems people have in understanding the “greenhouse” effect (always in quotes because it’s not really like a greenhouse at all!)

I’m not writing to try and change people’s minds. I’m writing for people who are asking questions and want to understand the subject. The only two things I ask:

• Be prepared to think it over
• If you have questions or comments, please ask these questions or make these comments (just remember the etiquette)

1. The Downwards Radiation is probably from the Sun

One commenter said:

You cite measurements of downward radiation. Were those measurements taken during the day or at night? Your link doesn’t say, and the answer is extremely critical to your argument.

I wasn’t clear why it really mattered so I asked. The response from the commenter was:

More than half of what we receive from the sun is already in the IR, so a daytime measurement is just measuring spectral lines by shining a light source through a gas. Anyone could do that in a lab with just air. The energy measured is just solar energy.

Anyone who has read the CO2 series on this blog, even just Part One, will have their hands in the air already..

Log plot of solar radiation vs terrestrial radiation by wavelength. The solar radiation is amount absorbed (i.e. takes into account typical albedo) and received at 45°.

Linear plot of the same data.

• 99% of the sun’s radiation has a wavelength less than 4μm
• 99.9% of the earth’s radiation has a wavelength greater than 4μm

There is almost no overlap, so if we measure what we conventionally call longwave radiation (>4μm) we know it comes from the earth. And if we measure what we conventionally call shortwave radiation (<4μm) we know if comes from the sun.

This simple fact is an amazing help in understanding the climate! But most people don’t know it!

Two related points have arisen, one of which is alluded to in the question above:

1. “Half of the radiation from the sun is infra-red, therefore..”

True but a red-herring. Infra-red means longer wavelength than visible light. Greater than 0.7μm. Not greater than 4μm.

2. “The sun’s energy is way way higher, so even though only 1% of its energy is greater than 4μm this will still overwhelm the earth’s energy above 4μm.”

This is true when we look at the energy at its source, but only a two billionth of the sun’s total energy is received by the earth. Alternatively, considering the radiation per m² the solar radiation is reduced by a factor of 46,000 (as a result of the inverse square law) by the time it reaches the earth.

The total energy from the sun’s radiation (at the earth’s surface) is very similar to the total energy radiated from the earth. (Actually no surprise otherwise the earth would rapidly heat up!)

For more on this, see The Sun and Max Planck Agree – Part Two.

2. Energy has to Balance (at the Top of Atmosphere)

If you measure all the energy going in at TOA vs all the energy going out at TOA, you will find that they net to zero over time.

This is true. Everyone agrees. The “greenhouse” gas theory doesn’t make a claim that contradicts this well-established fact. In fact, it relies on it!

Let’s clarify the numbers because I gave the “clear sky” results in the graphic, but most of the time it’s cloudy and then the numbers are lower.

The average over the globe and over a year at the top of atmosphere for incoming and outgoing radiation is about 240W/m². Strictly speaking it is incoming radiation absorbed, because about 30% of “incoming” is reflected by clouds and the earth’s surface. Check out the numbers in The Earth’s Energy Budget. This is all measured by satellites.

Note – what is very important about this is that the radiation in and out at the top of atmosphere balance. Down at the earth’s surface many other effects are going on – convection, latent heat (evaporation and condensation of water) as well as radiation. Energy in = Energy out is true everywhere that no heating or cooling is going on. But it’s not necessarily true that Radiation in = Radiation out at the surface or in the atmosphere, as other ways exist of losing or gaining energy. At the top of atmosphere there is no convection and no water vapor, so energy can only be moved by radiation.

Hopefully that makes sense. Read on..

3. The most popular – You are “Creating” Energy!

Since there is no NEW energy being put into the system, and the amount of energy being put in will, over the long term, equal exactly the amount of energy coming out, all you get at most is a short term fluctuation. If I am wrong, then you have invented perpetual motion.

This is a common theme and a recurring one. Many people think that the theory is effectively claiming that CO2 is creating energy.

Obviously that doesn’t happen.

Therefore, QED, the “greenhouse” effect doesn’t occur! The defence rests.

Well.. let’s take a look.

I’ll first give an analogy. This is an illustration not a proof.

You have a house without a roof. It has a heater on the floor and there aren’t any other sources of energy. The temperature being measured is around 10°C, it’s a bit chilly. Someone puts a roof on the house, what happens to the temperature? It goes up, maybe now it is 15°C.

No, it can’t have gone up. The roof doesn’t create energy so the temperature must still be 10°C!

Of course, no one reading this is confused. But when I gave that example I had people still trying to demonstrate that this analogy wasn’t valid.

Suppose there was no energy source. The roof – or insulation – wouldn’t create any heat.

True – and if the earth had no sun heating it, CO2 wouldn’t have any heating effect at the earth’s surface either.

What is the theory claiming for CO2?

1. It isn’t creating energy
2. It isn’t adding energy to the climate system
3. It is absorbing and re-emitting “energy”

So instead of all the radiation from the earth’s surface simply heading up and out of the top of atmosphere, instead, some proportion is being “redirected” back down to the earth’s surface.

Like a roof but different.

The point is, there is no violation of energy conservation or any other law of thermodynamics.

The longwave energy being re-emitted back down to the earth’s surface – as you can see in the 2nd graphic above – simply increases the surface temperature. It increases the surface temperature above what it would be if this effect didn’t exist. (Like a roof on a house).

4. Your Radiation Numbers are Wrong

Referring to my “Upwards longwave radiation from the surface of the earth is around 390W/m².”

Nyet. At 0 C, radiance is about 320 watts/m². At 30 C, its about 550 watts/m². You can’t just average the numbers from low to high across the globe and get the right answer either. you get a curve with a peak or high about mid day, but you also get a curve with a peak at the equator as compared to the poles. The average between the lows and the highs is NOT the average of the curve.

This is a very good point and worth covering in a little detail.

How do we come up with the number 390W/m² in the first place?

There is a relationship between temperature and radiation, which is very well established, known as the Stefan-Boltzman law. You can see it at the start of the maths section in Part One.

Energy radiated is proportional to the 4th power of (absolute) temperature

Yuck. Before you skip forward, here are some example numbers (I used the amazing and recommended  spectralcalc.com).

• -20°’C (253K) or -4°F – 232 W/m²
• -10°C (263K) or 14°F- 271 W/m²
• 0°C (273K) or 32°F – 315 W/m²
• 10°C (283K) or 50°F – 364 W/m²
• 20°C (293K) or 68°F – 418 W/m²
• 30°C (303K) or 86°F – 477W/m²

So our commentor was correct as to the method. If you want to work out how much energy is radiated from the surface of the earth, you can’t just assume you can use the earth’s average annual global temperature (15°C) to get the average radiation of 390W/m².

This is true because the relationship is non-linear – see how the radiation increases more and more for the same 10°C or 10K rise in temperature.

Luckily this work has already been done for us, it doesn’t actually change the result that much.

Average annual global radiation from the earth’s surface = 396 W/m² (See note 1 at end)

What’s very interesting about this number is that it is nowhere near 240W/m² – that number would represent a temperature of -18°C (about 0°F).

So in fact, energy radiated upwards from the earth’s surface is a lot higher than energy radiated out of the top of atmosphere. What’s going on?

5. If your numbers are correct, which I doubt, the earth will ignite

I’m not going to go check your numbers but just consider what you are saying. Your claim is that 156 w/m² is being retained as extra energy kept inside the atmosphere over the long term. If you are right the planet should ignite in a few days.

So, we saw that the energy out of the system – from the top of atmosphere – is only 240W/m²

And energy radiated up from the earth’s surface is 396W/m². So if the claim is correct that this “missing energy” is re-radiated back down to the surface, then simple arithmetic demonstrates that the energy will keep “piling up” and the earth will ignite.

Obviously that won’t happen. QED, the theory or the measurements are wrong. The defence rests.

Except.. let’s look a bit closer. The measurements are right, of course. So in fact, anyone disputing the theory needs their own theory to explain the numbers..

If we add extra radiation to the surface of the earth what happens? Simple – it heats up. As the surface heats up it radiates more energy back out. So it keeps heating up until the energy being lost is balanced by the energy coming in.

The point at which the earth’s temperature will stop changing is the value at which the outgoing radiation from the top of the atmosphere is balanced by the sun’s incoming radiation absorbed.

Well, that’s why the earth’s temperature is not -18°C. With no greenhouse effect it would be.

If there was nothing absorbing the upwards longwave radiation, and re-radiating some of it downwards, the radiation from the surface of the earth would only be 240W/m2 – a surface temperature of around -18°C (0°F).

Conclusion

To many people it seems like a wacky theory easily refuted by common sense, the basic laws of thermodynamics or the fact that the earth hasn’t apparently heated up for a decade. (I didn’t comment on that one, the climate is very complex, many factors affect climate).

The theory wasn’t invented by the IPCC or Al Gore (he only invented the internet). And it wasn’t formed from a desire to understand why the earth warmed up over much of the 20th century.

The theory was developed by physicists going back to the start of the 20th century (well, probably before but I’m haven’t studied the history of the subject). Thousands of physicists have studied the subject, dissected it, written papers on it and improved on it.

Even the many “skeptics” of what has become known as AGW in its IPCC form are not skeptics of these concepts. (e.g. Lindzen, Roy Spencer, John Christy)

I don’t want to try and pull the “argument from authority” because I don’t really accept it myself. But pause for thought if you are still not convinced, if you still think this theory magically creates energy from somewhere or violates the 2nd law of thermodynamics – and ask yourself:

If 99.99% of physicists past and present believe this effect is real and measurable, how likely is it that none of them realized there is a basic error in the theory?

Note 1

The value of 396W/m2 is calculated in Trenberth and Kiehl’s 2008 update to their 1997 paper: Earth’s Annual Global Mean Energy Budget. In the 2008 paper they comment that the upwards radiation from the surface cannot be assumed by averaging the temperature arithmetically and then calculating the radiation. So they take data on the surface temperature around the globe and re-calculate. Depending on exactly the method the values come in at 396.4, 396.1, 393.4. They stick with 396W/m².

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Introduction

There are two themes in current “consensus” climate science. Perhaps it’s not apparent that they are contradictory..

• One side – climate is predictable
• The other – “tipping” points ahead, perhaps very close

This subject isn’t easy to untangle and no one really knows what the answer is.

What this post is about is one of those “tipping” points and how complex climate really is. This post is about the thermohaline circulation, also known in shorthand as the THC.

“Thermohaline” sounds like a tough concept to understand – but although the concepts involved don’t require any special science knowledge they aren’t immediately obvious.

Thermo – relates to temperature, and Haline relates to Saline, or Salt..

Energy Balance and the “Conveyor Belt”

When you consider the difference between the incoming solar radiation and the outgoing longwave radiation by latitude you start to realize how the earth’s climate moves heat around – more specifically how heat moves from the equator to the poles:

This graphic demonstrates the calculated solar radiation in – versus latitude, against the radiation out from the earth’s surface (longwave radiation).

In short, the equator receives a lot more energy compared with the poles because the sun is – comparatively – overhead a lot more. Therefore, the atmosphere and the oceans transport heat from the equator to the poles.

Here’s another graphic of the same imbalance:

Interestingly, the oceans and the atmosphere share the heat transfer more or less 50/50:

Now we see that the ocean takes a big role in moving energy to the poles, let’s look a closer look at what drives the ocean currents..

Temperature.. and Salt

Two obvious factors that push the ocean currents around are winds and the coriolis effect. The coriolis effect has to do with the fact that earth is rotating. Every explanation I have seen of it seems like a recipe for confusion if you haven’t already spent some time on it, so I’m not going to repeat that problem.. (take a look at the link above if you want to understand it better).

But the major factor that drives ocean currents is density. What determines density? Temperature and salinity:

The first time you see this kind of chart your eyes glaze over and you quickly move on to the next section.

But let’s try and make it easier to understand:

Here’s one example – cold water, almost freezing, becomes less saline. This might happen if a lot of ice was melting. See the change in density – 1.026 to 1.003kg/m³ – it doesn’t seem like much but it is very significant.

If you take the reverse direction, as water freezes it leaves most of its salt behind, so the water becomes much more saline and the density goes in the other direction.

The Thermohaline Circulation

Cold and high salinity water has a higher density than any other ocean water and so it sinks. There are two places that most ocean water sinks – around Greenland and at the Wedell Sea in Antarctica. Here is a simplistic representation of global ocean circulation:

One consequence of the THC is that warm surface water moves from the equator up to norther Europe. If that “global conveyor” didn’t exist then northern Europe would be much colder.

The driving force is the very cold very saline water that sinks rapidly just south of Greenland.

The Tipping Point

As the world heats up, which it is currently doing (in a broad sense, see Note 1), Arctic sea ice and the Greenland ice sheet are melting more rapidly.

At some point, the amount of melt water – low salinity water – will probably change the balance of the THC and send the system into reverse.

All the evidence is that this has happened before.

When – and if – it does happen, the heat conveyor will turn off, northern Europe will cool down and the Arctic and Greenland will refreeze.

As that happens, positive feedback from ice albedo and then from water vapor will keep driving the temperatures in the colder direction.

Well, who knows exactly what will happen? Or when.

In some follow on posts we will look at this in some more detail. Currently, GCMs (general circulation models) do not have a “tipping point” for the THC, they just show a weakening.

Conclusion

Our understanding of ocean dynamics do not require any new development of physics but we do require a lot more data. Temperature and salinity throughout the oceans has started to be provided through the Argo project. How much ice is melting is – surprisingly – a difficult subject.

Solving the equations of motion for the oceans requires temperature and salinity data as well as the meltwater component.

The possibility that the THC will change direction is a huge issue in the predictability of future climate.

If it does “switch” there will be significant climate effects. We can’t assume this effect will have a happy ending.

Note 1: The world has been heating up – broadly speaking, as there have been some ups and downs – since the end of the last ice age, 18,000 years ago. Sea levels have risen around 120m. And in the last 100 years the earth’s surface temperature has increased by around 0.7°C.

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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

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:

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.

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:

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)

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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:

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?

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

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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)

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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:

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.

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 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

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