The Science of Doom

There are many misconceptions about how atmospheric processes work, and one that often seems to present a mental barrier is the idea of How much work can one molecule do?

This idea – presented in many ways – has been a regular occurence in comments here and it also appears in many blogs with eloquent essays on the “real role” of CO2 in the atmosphere, usually unencumbered by any actual knowledge of the scientific discipline known as physics.

Well, we all need mental images of how invisible or microscopic stuff really works.

When we consider CO2 (or any trace gas) absorbing longwave radiation the mental picture is first of trying to find a needle in a haystack.

And second, we found it, but it’s so tiny and insignificant it can’t possibly do all this work itself?

How much can one man or woman really do?

This article is really about the second mental picture, but a quick concept for the first mental picture for new readers of this blog..

Finding a Needle in a Haystack

Think of a beam of energy around 15.5μm. Here is the graph of CO2 absorption around this wavelength. It’s a linear plot so as not to confuse people less familiar with log plots. Water vapor is also plotted on this graph but you can’t see it because the absorption ability of water vapor in this band is so much lower than CO2.

CO2 absorption, 15.4-15.6um, linear, from spectralcalc.com

The vertical axis down the side has some meaning but just think of it for now as a relative measure of how effective CO2 is at each specific wavelength.

Here’s the log plot of both water vapor and CO2. You can see some black vertical lines – water vapor – further down in the graph. Remember as you move down each black horizontal grid line on the graph the absorption ability is dropping by a factor of 100. Move down two black grid lines and the absorption ability has dropped by a factor of 10,000.

CO2 absorption - log graph - 15.4-15.6um, from spectralcalc.com

Now, I’ll add in the absorption ability of O2 and N2 – the gases that make up most of the atmosphere – check out the difference:

O2 and N2 added..

Spectralcalc wouldn’t churn anything out – nothing in the database.

15.5μm photons go right through O2 and N2 as if they didn’t exist. They are transparent at this wavelength.

So, on our needle in the haystack idea, picture a field – a very very long field. The haystacks are just one after the other going on for miles. Each haystack has one needle. You crouch down and look along the line of sight of all these haystacks – of course you can only see the hay right in front of you in the first one.

Some magic happens and suddenly you can see through hay.

Picture it..  Hay is now invisible.

Will you be able to see any needles?

That’s the world of a 15.5μm photon travelling up through the atmosphere. Even though CO2 is only 380ppm, or around 0.04% of the atmosphere, CO2 is all that exists for this photon and the chances of this 15.5μm photon being absorbed by a CO2 molecule, before leaving this world for a better place, is quite high.

In fact, there is a mathematical equation which tells us exactly the proportion of radiation of any wavelength being absorbed, but we’ll stay away from maths in this post. You can see the equation in CO2 – An Insignificant Trace Gas? Part Three. And if you see any “analysis” of the effectiveness of CO2 or any trace gas which concludes it’s insignificant, but doesn’t mention this equation, you will know that it is more of a poem than science. Nothing wrong with a bit of poetry, if it’s well written..

Anyway, it’s just a mental picture I wanted to create. It’s not a perfect mental picture and it’s just an analogy – a poem, if you will. If you want real science, check out the CO2 – An Insignificant Trace Gas Series.

CO2 – The Stakhanovite of the Atmospheric World?

Back in the heady days of Stalinist Russia a mythological figure was created (like most myths, probably from some grain of truth) when Aleksei Stakhanov allegedly mined 14 times his quote of coal in one shift. And so the rest of the workforce was called upon to make his or her real contribution to the movement. To become Stakhanovites.

This appears to be the picture of the atmospheric gases.

Most molecules are just hanging around doing little, perhaps like working for the _____ (mentally insert name of least favorite and laziest organization but don’t share – we try not to offend people here, except for poor science)

So there’s a large organization with little being done, and now we bring in the Stakhanovites – these champions of the work ethic. Well, even if they do 14x or 100x the work of their colleagues, how can it really make much difference?

After all, they only make up 0.04% of the workforce.

But this is not what the real atmosphere is like..

Let’s try and explain how the atmosphere really works, and to aid that process..

A Thought Experiment

For everyone thinking, “there’s only so much one molecule can do”, let’s consider a small “parcel” of the atmosphere at 0°C.

We shine 15.5μm radiation through this parcel of the atmosphere and gradually wind up the intensity. Because it’s a thought experiment all of the molecules involved just stay around and don’t drift off downwind.

The CO2 molecules are absorbing energy – more and more. The O2 and N2 molecules are just ignoring it, they don’t know why the CO2 molecules are getting so worked up.

What is your mental picture? What’s happening with these CO2 molecules?

a) they are just getting hotter and hotter? So the O2 and N2 molecules are still at 0°C and CO2 is at first 10°C, then 100°C, then 1000°C?

b) they get to a certain temperature and just put up a “time out” signal so the photons “back off”?

c) other suggestions?

The Real Atmosphere – From Each According to His Ability, To Each According to His Need

What is the everyday life of a molecule like?

It very much depends on temperature. The absolute temperature of a molecule (in K) is proportional to the kinetic energy of the molecule. Kinetic energy is all about speed and mass. Molecules zing around very fast if they are at any typical atmospheric temperature.

Here’s a nice illustration of the idea (from http://www.chem.ufl.edu/~itl/2045/lectures/lec_d.html).

At sea level, a typical molecule will experience around 10¹⁰ (10 billion) collisions with other molecules every second. The numbers vary with temperature and molecule.

Think of another way – at sea level 8×10²³ molecules hit every cm² of surface per second.

Every time molecules collide they effectively “share” energy.

Therefore, if a CO2 molecule starts getting a huge amount of energy from photons that “hit the spot” (are the right wavelength) then it will heat up, move even faster, and before it’s had time to say “¤” it will have collided with other molecules and shared out its energy.

This section of the atmosphere heats up together. CO2 can keep absorbing energy all day long even as a tiny proportion of the molecular population. It takes in the energy and it shares the energy.

If we can calculate how much energy CO2 absorbs in a given volume of the atmosphere we know that will be the energy absorbed by that whole volume of atmosphere. And therefore we can apply other well-known principles:

• heating rates will be determined by the specific heat capacity of that whole volume of atmosphere
• re-radiation of energy will be determined by the new temperature and ability of each molecule to radiate energy at wavelengths corresponding to those temperatures

Conclusion

The ability of a CO2 molecule to be “effective” in the atmosphere isn’t dependent on its specific heat capacity.

Molecules have embraced “communism” – they share totally, and extremely quickly.

Update – New post on the related topic of understanding the various heat transfer components at the earth’s surface – Sensible Heat, Latent Heat and Radiation

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One commenter asked about CO2 absorption in the solar spectrum.

If CO2 absorbs incoming solar radiation then surely an increase in CO2 will reduce incoming radiation and balance any increase in longwave radiation.

The important factor is the usual question of quantifying the different effects.

Let’s take a look.

CO2 absorption in the 0.17-5um band, with solar spectrum overlaid

The CO2 absorption spectrum is from the line list browser of the recommended spectralcalc.com. The line list only goes down to 0.17μm (170nm), hence the reason for the graph not starting at 0.0μm.

The solar radiation is overlaid. Well, more accurately, the Planck function for 5780K is overlaid (simply drawn using Excel). Note that the CO2 absorption spectrum is on a log graph, while the radiation is on a linear graph. For those not so familiar with logarithmic graphs, the peak absorption around 4.3μm is 10^-18, while the two peak absorptions just below 1μm are at 10^-26 – which is 100,000,000 less.

The value of seeing the solar radiation spectrum overlaid is it enables you to see the relative importance of each absorption area of CO2. For example, the solar radiation between 2 – 4μm is only 5% of the solar radiation, so any absorption by CO2 will be quite limited.

Here’s the comparison with the important 15μm band of CO2. A 6μm width is shown, overlaid (blue line) with the 12-18um longwave radiation of a 288K (15°C) blackbody:

CO2 absorption in the 12-18um band, with terrestrial spectrum overlaid

Just a little explanation of this graph and how to compare it to the solar version.

The average surface temperature of the earth is 15ºC, and it emits radiation very close to blackbody radiation (watch out for a dull post on Emissivity soon).

The proportion of radiation of a 288K blackbody between 12-18μm is 28%. What we want to do is enable a comparison between the CO2 absorption of solar radiation and terrestrial radiation.

Averaged across the globe and the year the incoming solar radiation at the top of atmosphere (TOA) is 239 W/m2 and the radiation from the earth’s surface is 396 W/m2. This works out to 65% higher, but so as not to upset people who don’t quite believe the earth’s surface radiation is higher than incoming solar radiation I simply assumed they were equal and scaled this section of the earth’s terrestrial radiation to about 28% of the solar radiation on the earlier graph. We are only eyeballing the two graphs anyway.

So with this information digested, the way to compare the two graphs is to think about the absorption spectra of CO2 simply being scaled by the amount of radiation shown overlaid in both cases.

As you can see the amount of absorption by CO2 of solar radiation is a lot less than the absorption of longwave radiation. Remember that we are looking at the log plot of absorption.

Is That the Complete Story?

Really, it’s more complicated, as always with atmospheric physics. There’s nothing wrong with taking a look at the approximate difference between the two absorption spectra, but luckily someone’s already done some heavy lifting with the complete solution to the radiative transfer equations using line by line calculations. For more on these equations, see the CO2 – An Insignificant Trace Gas series, especially Part Three, Four and Five.

The paper with the heavy lifting is Radiative forcing by well-mixed greenhouse gases: Estimates from climate models in the IPCC AR4 by W.D. Collins (2006). There’s a lot in this paper and aspects of it will show up in the long awaited Part Eight of the CO2 series and also in Models, On – and Off, the Catwalk.

Solving these equations is important because we can look at the absorption spectrum of CO2 in the 15μm band, but then we have to think about the absorption already taking place and what change in absorption we can expect from more CO2. Likewise for the solar spectrum.

Here are the two graphs, which include other important trace gases, as well as the impact of a change in water vapor. Note the difference in vertical axis values – the forcing effect of these gases on solar radiation has to be multiplied by a factor of 1000 to show up on the graph. The blue lines are CO2.

Net absorption of solar radiation by various "greenhouse" gases

Longwave radiative forcing from increases in various "greenhouse" gases

You can also see that the CO2 absorption in shortwave is across quite narrow bands (as well as being scaled a lot lower than terrestrial radiation) – therefore the total energy is less again. The vertical scale is energy per μm..

From these calculations we can see that with a doubling of CO2 there will be a very small impact on the radiation received at the surface, but a comparatively huge increase in longwave radiation retained – “radiative forcing” at the tropopause (the top of the troposphere at 200mbar).

So Is That the Complete Story?

Not quite. If trace gases in the atmosphere absorb solar radiation, is that so different from the surface absorbing solar radiation?

Or to put it another way, if the radiation doesn’t strike the ground, where does it go? It’s still absorbed into the climate system, but in a different location (somewhere in the atmosphere).

But as one commenter said:

The other point [this one] you make is simply not true and/or also not proven. There is only so much energy that can be taken up by a molecule.

This is a theme that has arrived in various comments from various posts. So the concept of How much work can one molecule do? is worth exploring in a separate post.

Hopefully, it’s clear from what is presented here that increases in CO2 absorption of the solar radiation are very small compared with absorption of longwave radiation.

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In Part One, we introduced some climate model basics, including uses of climate models (not all of which are about “projecting” the future).

And we took at a look at them in their best light – on the catwalk, as it were.

Well, really, we took a look at the ensemble of climate models. We didn’t actually see a climate model at all..

Ensembles

The overall evaluation in Part One was the presentation of a “multi-model mean” or an ensemble. An ensemble can be the average of many models, or the average of one model run many times, or both combined.

We will return to more discussion about the curious nature of ensembles in a later post. Just as a starter, two observations from the IPCC.

IPCC AR4 in Chapter 8, Climate Models and their Evaluation, comments:

There is some evidence that the multi-model mean field is often in better agreement with observations than any of the fields simulated by the individual models (see Section 8.3.1.1.2), which supports continued reliance on a diversity of modelling approaches in projecting future climate change and provides some further interest in evaluating the multi-model mean results.

and a little later:

Why the multi-model mean field turns out to be closer to the observed than the fields in any of the individual models is the subject of ongoing research; a superficial explanation is that at each location and for each month, the model estimates tend to scatter around the correct value (more or less symmetrically), with no single model consistently closest to the observations. This, however, does not explain why the results should scatter in this way.

One interpretation of this would be:

We like ensembles because they give more accurate results, but we don’t really understand why..

A subject to come back to, now it’s time for a real model..

Step Forward Climate Model “Cici” – CCSM3

CCSM3, “Cici”, is the model from NCAR (National Center for Atmospheric Research) in the USA. Out of all the GCMs discussed in the IPCC AR4, Cici has the “best curves” – the highest resolution grid. Well, she comes from the prestigious NCAR..

The model’s vital statistics – first the atmosphere:

• top of atmosphere = 2.2 hPa (=2.2mbar), this is pretty much the top of the stratosphere, around 50km
• grid size = 1.4° x 1.4° (T85)
• number of layers vertically = 26 (L26)

second, the oceans:

• grid size = 0.3°–1° x 1°
• number of vertical layers = 40 (L40)

The vital statistics give a quick indication of the level of resolution in the model. And there are also model components for sea ice and land. The model doesn’t need the infamous “flux adjustment” which is the balancing term for energy, momentum and water between the atmosphere and oceans required in most models to keep the two parts of the model working correctly.

The CCSM3 model is described in the paper: The Community Climate System Model Version 3 (CCSM3) by W.D. Collins et al, Journal of Climate (2006). The source code and information about the model is accessible at http://www.ccsm.ucar.edu/models/.

And for those who love equations, especially lots of vector calculus, take a look at the 220 page technical document on CAM3, the atmospheric component.

It will be surprising for many to learn that just about everything on this model is out in the open.

CCSM3 Off the Catwalk – Hindcast Results

As with the multi-model means results in Part One we will take a look through a similar set of results for CCSM3.

Annual temperature

CCSM3 Annual Land & Sea Temperature Actual (top) vs Model (bottom)

Cici looks pretty good.

Details – The HadISST (Rayner et al., 2003) climatology of SST for 1980-1999 and the CRU (Jones et al., 1999) climatology of surface air tempeature over land for 1961–1990 are shown here. The model results are for the same period of the CMIP3 20th Century simulations. In the presence of sea ice, the SST is assumed to be at the approximate freezing point of sea water (–1.8 °C).

However, it’s hard to tell looking at two sets of absolute values, so of course we turn to the difference between model and reality.

Annual Temperature – Model Error

Model simulations of annual average temperature less observed values for Cici and for the “ensemble” or multi-model mean:

Annual Temperatures - Simulated minus observed for CCSM3 and the ensemble

In terms of absolute error around the globe, Cici and the ensemble are very close (using the Anglotzen statistical method).

We could note that even though the values are “close”, there are areas where Cici – and the ensemble – don’t do so well. In Cici’s case southern Greenland and the Labrador Sea, which might be very important for predicting the future of the thermohaline circulation. And both are particularly bad for Antarctica, a general problem for models.

To give an idea of the variation of models, here are all of the models reviewed by the IPCC in AR4 (2007):

Annual Temperature - Simulated minus observed - All models

The top right is Cici (red circle). It’s clear that Cici is a supermodel..

Standard Deviation of Temperature

The standard deviation of temperature – “over the climatological monthly mean annual cycle ” – simulated less observed for Cici and the ensemble. We could describe it as how good is the model at working out how much temperature actually varies over the year in each location?

First, however, to make sense of the “error” of model less actual, we need to know what actual values look like:

Standard Deviation of Temperature over the climatological monthly mean annual cycle

As we would expect, the oceans show a lot less temperature variation than the land and around the tropics and sub-tropics the variation is close to zero.

Now let’s take a look at the model less actual, or “model error”:

Std Deviation of Temperature - Simulated minus observed for CCSM3 and the ensemble

We can see that Cici has some problems in modeling temperature variation especially under-estimating the actual variation around northern Russia and Canada and over-estimating the variation in the Middle East and Brazil. The ensemble appears to be in slightly better shape here.

Of course, these areas are where the largest temperature variation takes place.

Diurnal Range of Land Temperature

As before, first the actual values:

Annual average of diurnal temperature range over land

And now the model less actual, or “model error”:

Diurnal temperature range over land - Actual less Model for CCSM3 and ensemble

We can see a lot of areas where the model error is quite large, usually corresponding to larger measured values. In the case of Greenland, for example, the annual average diurnal temperature range is over 20°C, while the model under-estimates this by more than 10°C. Given the legend the error might be as big as the actual value..

We can also see that on average Cici under-estimates the diurnal temperature range, and the ensemble is closer to neutral but still appears to under-estimate.

Here’s another comparison which demonstrates the problem of all the models vs observation:

Diurnal temperature range vs latitude - Observed compared with all models

The black line is the observed value. We can see that all of the models except for one are definitely under-estimating, and none of the models are particularly close to the observed values.

Now we can get to see more fundamental values.

Reflected Solar Radiation

This value is essential for calculating the basic radiation budget for the earth.

First the actual values as measured by ERBE (1985-1989):

Average Reflected Solar Radiation, ERBE

And now the model error – model less actual:

Reflected Solar Radiation - Actual less Model for CCSM3 and ensemble

The ensemble is definitely better than Cici. Cici has some large errors, for example, North Africa, Pacific Ocean and the Western Indian Ocean where the model error seems to be up to half of the actual value.

If we look at the values averaged by latitude the results appear a little better:

Reflected Solar Radiation vs latitude - Observed compared with all models

But the deviations give us a better view:

Reflected Solar Radiation vs latitude - Model error for all models

Note Cici in the solid blue line. The ensemble is proving to be the pick of the bunch..

So the model’s ability to simulate reflected solar radiation is much better by latitude than by location. But most or all of the models have significant discrepancies even when averaged over each latitude.

Outgoing Longwave Radiation

The other side of the radiation budget, first the actual ERBE measurement (1985-1989):

Measured Outgoing Longwave Radiation, ERBE

And now the model error – model less actual:

OLR - Actual less Model for CCSM3 and ensemble

As with reflected SW radiation, the ensemble performs better than Cici. So while measured values are in the range of 200-300 W/m2, Cici has some areas where the (absolute) error is in excess of 30W/m2.

Looking at the OLR values averaged by latitude, the results appear a little better:

OLR vs latitude - Observed compared with all models

And the deviations, or model error:

OLR vs latitude - Model error for all models

Rainfall

Measured from CMAP, 1980-1999:

Rainfall 1980-1999

Units are in cm of rainfall per year. And now the model error – model less actual:

Rainfall - Actual less Model for CCSM3 and ensemble

Once again the ensemble outshines Cici. There are some substantial errors in the areas where rainfall is high.

As with some of the previous model results, if we look at the model vs observed by latitude the picture is somewhat better:

Rainfall vs latitude - Observed vs all models

Humidity

Lastly, we will take a look at specific humidity. First the “measured”, as recalculated by ERA-40:

Specific Humidity in g/kg vs Latitude and Altitude, from ERA40

Observed annual mean specific humidity in g/kg, averaged zonally, 1980-1999. Note that the vertical axis is pressure on the left in mbar and km in height on the right.

And now the model error – but this time in % = (model – actual)/actual x 100:

Specific Humidity - % Error for CCSM3 and ensemble

Once again the ensemble appears to outperform Cici. And both, but especially Cici, have problems in the top half of the troposphere (around 500-200mbar) with 20-50% error in some regions in Cici’s case.

Conclusion

This has been a quick survey of model results for different parameters across the globe, but averaged annually, compared with observations.

In Part One, we saw the ensemble in its best light. But when we take a look at a real model, the supermodel Cici, we can see that she has a lot of areas for improvement.

There’s lots more to investigate about models, all to come in future parts of this series.

As always, comments and questions are welcome, but remember the etiquette.

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In the previous article in this series, The Earth’s Energy Budget – Part Two we looked at outgoing longwave radiation (OLR) and energy imbalance. At the end of the article I promised that we would look at problems of measuring things and albedo but much time has passed, promises have been forgotten and the fascinating subject of how the earth really radiates energy needs to be looked at.

If you are new to the idea of incoming (absorbed) solar radiation being balanced by OLR, or wonder how the solar “constant” of 1367W/m² can be balanced by the earth’s OLR of 239W/m² then take a look at Part One and Part Two.

Introduction

If you’ve read more in depth discussions about energy balance or CO2 “saturation” you might have read statements like:

More absorption by CO2 causes emission of radiation to move to higher, colder layers of the atmosphere

If these kind of comments confuse you, sound plain wrong, or cause you to furrow your brow because “it sounds like it’s probably right but what does it actually mean?” – well, hopefully some enlightenment can be found.

Effective Radiation

The sun’s core temperature is millions of degrees but we see a radiation from the sun that matches 5780K – its surface temperature:

Solar Radiation, top of atmosphere and at earth's surface, Taylor (2005)

In this figure there are two spectra: the top one is how the sun’s radiation looks before it reaches the top of the earth’s atmosphere – contrasted with the dotted line of a “blackbody” – or perfect radiator – at 5780K (5507°C for people new to Kelvin or absolute temperature).

The bottom one – of less interest for this article – is how the sun’s radiation looks at the earth’s surface after the atmosphere has absorbed at various wavelengths.

Why don’t we see a radiation spectrum from the sun that matches millions of degrees?

If we measure the upward longwave radiation from the earth’s surface at 15°C we see an effective “blackbody” radiator of 288K (15°C). But why don’t we see a radiation spectrum of 5000K – the temperature somewhere near the core?

The answer to both questions is that radiation from the hotter inner areas of these bodies gets completely absorbed by outer layers, which in turn heat up and radiate at lower temperatures. In the case of the sun, the radiation spectrum includes hotter areas below the surface that are not absorbed at some wavelengths, as well as the surface itself.

In the case of the earth it’s really the top skin layer that emits longwave radiation.

So when we measure the radiation from the earth with a surface temperature of 15°C (288K) we know we will see a longwave radiation that matches this 288K. This will be a total energy radiated of 390W/m² with the peak wavelength of 10.1μm. The temperature below the surface is irrelevant.

(Well, it’s not really irrelevant. The hotter layers below warm up the layers above – through conduction and radiation).

This is what the radiation looks like:

Blackbody Radiation at 15'C or 288K

This assumes an emissivity of 1. The emissivity of the surface of the earth varies slightly but is close to 1, typically around 0.98. Watch out for a dull post on emissivity at some stage..

At the top of atmosphere, as many know, the OLR is around 239W/m². For those confused by how it can be 390W/m² at the surface and 239W/m² at the top, the answer is due to absorption and re-radiation of longwave radiation by trace gases – the “greenhouse” effect. See the CO2 – An Insignificant Trace Gas? series, and especially Part Six – Visualization and CO2 Can’t Have that Effect Because.. if you don’t understand or agree with these well-proven ideas.

If the earth’s atmosphere was completely transparent to longwave radiation this spectrum would look exactly the same at the earth’s surface and at the top of atmosphere (TOA).

Here’s what it does look like with some typical blackbody radiation curves overlaid:

Outgoing longwave radiation at TOA, Taylor (2005)

(Note that the spectrum is shown in wavenumber in cm^-1. For convenience I added wavelength in μm under the wavenumber axis. Wavelength in μm = 10,000/wavenumber).

For energy balance – if the earth is not warming up or cooling down – we would expect the earth to radiate out the same amount of energy that it absorbs from the sun. That amount is 239W/m², which equates to an average temperature of 255K (-18°C).

As the text for this graphic shows, when the energy under the curve is integrated this is what it comes to! But as you can see the actual spectrum is not a “blackbody curve” for 255K. So let’s take a closer look.

Everything Gets Through or Nothing Gets Through – a Few Thought Experiments

Imagine a world where the upwards longwave radiation from the earth’s surface didn’t get absorbed by any gases in the atmosphere.

Most people are familiar with that thought experiment – it’s a staple of the most basic radiation model in climate science. The radiation at the top of the atmosphere would look like this (the top graph):

255K radiation, 100% transmittance

This is the blackbody radiation at 255K (-18°C) with 100% “transmittance” through the atmosphere. The area under the curve, if we extend it out to infinity, is 239W/m².

And of course, because the radiation hadn’t been absorbed or attenuated in any way, the temperature at the earth’s surface would also be 255K. Chilly.

Now let’s think about what would happen if the atmosphere allowed radiation only through the “atmospheric window” and everywhere else the transmittance was zero:

323K radiation through a perfect "atmospheric window", 8-14um

The bottom graph shows how the transmittance of the atmosphere varies with wavelength in this thought experiment.

The top graph in this case is the blackbody radiation from 323K (50°C) only allowed through between 8-14μm. The energy under the curve is 239W/m². (Note the higher values on the vertical scale compared with the earlier graphs).

So if the atmosphere absorbed all of the surface radiation below 8μm and above 14μm the earth’s surface would heat up until it reached 50°C (323K). Why? Because if the temperature was only 15°C the amount of energy radiated out would only be 141W/m². More energy coming in than going out = earth heats up. The surface temperature would keep heating up until eventually 239W/m² made it out through the atmospheric window – which is 50°C.

Closer to The Real World – Illustration of Radiation from Multiple Layers in the Atmosphere

Even in the atmospheric window some radiation is absorbed, i.e. the transmittance is not 1. But let’s assume for sake of argument it is 1. So energy in the 8-14μm band just passes straight through the atmosphere. It’s still a thought experiment.

Lots of gases absorb at lots of wavelengths – which makes thinking about it as a whole very difficult. So we’ll just assume that the rest of the atmosphere outside the atmospheric window all shares the same absorption characteristics – that is, every wavelength is identical in terms of absorption of radiation.

Now let’s try and consider what really happens in the atmosphere. Each “layer” of the atmosphere radiates out energy according to the temperature in that layer. For reference, here is the temperature (and pressure) at different heights:

Atmospheric Temperature & Pressure Profile, Bigg (2005)

The highlighted area at the bottom – the troposphere – is the area of interest. This is where most of the atmosphere (by molecules and mass) actually resides.

In our thought experiment radiation from the surface (outside the atmospheric window) gets completely absorbed by the atmosphere, or at least the amount that gets through is very small. Taken to the extreme we would get the result shown a few graphs earlier where the surface temperature rises up to 50’C.

But just because surface radiation doesn’t get out doesn’t mean that radiation from the atmosphere can’t get out.

Each layer of the atmosphere radiates according to its temperature. Even if the atmosphere’s transmittance is zero when considering the entire thickness of the atmosphere, there will be some layer where radiation starts to get through.

This is partly because there is less atmosphere to absorb the closer we get to the “top”. And also because as we get higher in the atmosphere it gets thinner. Less molecules to absorb radiation. Even if some gas is a fantastically good absorber of energy, there must be a point where radiation is hardly absorbed. For example, at the top of the stratosphere, about 50km, the pressure is around 1mbar – 1000x less than at the surface. At the top of the troposphere (the tropopause) the pressure is around 200mbar – 5x less than at the surface.

The challenge in thinking about the atmosphere radiating is that unlike the surface of the earth where all radiation is emitted from the very surface, instead radiation is emitted from lots of different layers:

• Higher up – less absorption, more radiation makes it through
• Lower down – more absorption, less radiation makes it through

But let’s still keep it simple and think about the surface temperature being our standard 15°C (288K) and the atmospheric window letting through everything between 8-14μm. This means 141W/m² makes it out through this window.

If we have energy balance, the OLR = 239W/m2 in total = 141 (through the atmospheric window) + 98 (radiated from the atmosphere at some height, wavelengths outside 8-14μm).

What temperature equates to this layer in the atmosphere? Well, assuming no absorption above this radiating layer (not really the case), and only radiation outside 8-14μm, the temperature of the atmosphere would have to be 219K, or -54°C. Take a look back at the temperature profile above – this is pretty much the top of the troposphere, around 11km.

Remember that this isn’t exactly how radiation gets radiated out to space – it doesn’t come from one “skin layer”. We might consider that if the transmittance of the atmosphere is 1 at this height, then maybe at 10km the transmittance is 0.8 and at 9km the transmittance is 0.5, and at 8km the transmittance is 0.1..

So each layer is radiating energy, with higher layers being colder but more of their radiation getting through, and lower layers being warmer – so radiating a higher amount – but less of their radiation getting through.

For many people reading, this is a straightforward concept, why so long.. for others it might still seem tough to grasp..

So here is a sample radiation diagram with illustrative values only (and values a little different from above):

Radiation from different heights in the atmosphere, illustrative values only

What the diagram shows is the radiation outside the 8-14μm band. That’s because in our thought experiment the 8-14μm band doesn’t absorb any radiation (and therefore can’t radiate in this band either).

Take the top layer at 11km. If we calculate the blackbody radiation of 219K (-54°C) and exclude radiation in the 8-14μm band the radiation is 98W/m². Then the grey block above with “0.6” is the atmosphere above with transmittance of 0.6, so the radiation actually getting through from this layer to the top of atmosphere is 59W/m². Similarly for the two other layers (with different values).

In total, the energy leaving the top of atmosphere (outside of the atmospheric window) is 98W/m². (It’s just a coincidence that this is the value of the top layer before any absorption). And inside the atmospheric window was the number we already calculated of 141W/m², so the total OLR is 239W/m².

Of course, we all know the real atmosphere is much more complex with lots of different absorption at different wavelengths. But hopefully this “intermediate” example help to explain how the atmosphere radiates out energy.

So finally, onto the real point..

What Happens with More Absorbing Gases?

Remember how this long post started..

If you’ve read more in depth discussions about energy balance or CO2 “saturation” you might have read statements like:

More absorption by CO2 causes emission of radiation to move to higher, colder layers of the atmosphere

Now, maybe this kind of statement will make more sense.

In our model – our thought experiment – above, we had a uniform absorber of radiation outside the atmospheric window. Suppose we increase the amount of this absorber – the skies open and someone pours some more in and stirs it around. Let’s say the amount increases by 10%.

Well, take a look back at the last diagram. See the transmittance values for each layer in the atmosphere – 0.6 at 11km high, 0.25 at 10km high and 0.1 at 9km high.

Regardless of how realistic these actual numbers are, increasing the amount of absorbing gas by 10% will automatically mean that each of the transmittance numbers is reduced by 10%. And so less radiation makes it out to the TOA (top of atmosphere).

Effectively because lower layers are contributing less energy out through TOA the effective radiating height has moved up. It’s not because some directive has been passed down from a higher authority. And it’s not because one layer has stopped and another layer has taken over.

It’s just that lower layers contribute less, so the “average radiating height” is now higher and colder.

(Note: it might look at first sight that the average height is still the same even though the amount of radiation has reduced. This is not really the case, see the note at end).

In our particular example what would happen is that the OLR would reduce from 239W/m² down to 141+98*0.9=229 W/m². So the surface would warm up and this would warm up each layer of the atmosphere until eventually a new hotter steady state was reached.

Conclusion

This has been a long post to try and create more of an understanding of how the earth actually radiates energy, and why more of any trace gas increases the “greenhouse” effect.

It does it because more “absorbing gases” reduce the amount of radiation that can make it out from lower layers in the atmosphere. These lower layers are hotter and radiate much more energy. Proportionately more energy will then be radiated from higher layers which are colder, and therefore these radiate less energy.

It’s a not a mystical force that raises the “effective radiating height” in the atmosphere. But the effective radiating height does increase.

Note

In the example above, the three layers together contributed 98W/m² at TOA. That is an “effective temperature” of 219K – remembering that we are excluding radiation from the 8-14μm window. If we reduce the radiation from these three layers by 10%, we now have 89W/m² which is about 212K – effectively radiating from a colder level in the atmosphere.

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General Circulation Models or Global Climate Models – aka GCMs – often have a bad reputation outside of the climate science community. Some of it isn’t deserved. We could say that models are misunderstood.

Before we look at models on the catwalk, let’s just consider a few basics

Introduction

In an earlier series, CO2 – An Insignificant Trace Gas we delved into simpler numerical models. These were 1d models. They were needed to solve the radiative transfer equations through a vertical column in the atmosphere. There was no other way to solve the equations – and that’s the case with most practical engineering and physics problems.

Here’s a model from another world:

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”.

Finite element analysis is used successfully in lots of areas of practical problem solving, many orders simpler of course, than GCMs.

Uses of Models

One use of models is to predict, no project, future climate scenarios. That’s the one that most people are familiar with. And to supply the explanation for recent temperature increases.

But models have more practical uses. They are the only way to provide quantitative analysis of certain situations we want to consider. And they are the only way to test our understanding of the causes of past climate change.

Analysis

On this blog one commenter asked about how much equivalent radiative forcing would be present if all the Arctic sea ice was gone. That is, with no sea ice, there is less reflection of solar radiation. So more absorption of energy – how do we calculate the amount?

You can start with a very basic idea and just look at the total area of Arctic sea ice as a proportion of the globe, and look at the local change in albedo from around 0.5-0.8 down to 0.03-0.09, multiply by the current percentage area in sea ice to find a number in terms of the change in total albedo of the earth. You can turn that into the change in radiation.

But then you think a little bit deeper and want to take into account the fact that solar radiation is at a much lower angle in the Arctic so the first number you got probably overstated the effect. So now, even without any kind of GCM, you can simply use the equation for the reduction in solar insolation due to the effective angle between the sun and the earth:

I = S cos θ – but because this angle, θ, changes with time of day and time of year for any given latitude you have to plug a straightforward equation into a maths program and do a numerical integration. Or write something up in Visual Basic or whatever your programming language of choice is. Even Excel might be able to handle it.

This approach also gives the opportunity to introduce the dependence of the ocean’s albedo on the angle of sunlight (the albedo of ocean with the sun directly overhead is 0.03 and with the sun almost on the horizon is 0.09).

This will give you a better result. But now you start thinking about the fact that the sun’s rays are travelling in a longer path through the atmosphere because of the low angle in the sky.. how to incorporate that? Is it insignificant or highly significant? Perhaps including or not including this effect would change the “radiative forcing” by a factor of two? (I have no idea).

So if you wanted to quantify the positive feedback effect of melting ice your “model” starts requiring a lot more specifics. Atmospheric absorption by O2 and O3 depending on the angle of the sun. And the model should include the spatial profile of O3 in the stratosphere (i.e., is there less at the poles, or more).

It’s only by doing these calculations that the effect of sea ice albedo can be reliably quantified. So your GCM is suddenly very useful – essential in fact.

Without it, you would simply be doing the same calculations very laboriously, slowly and less accurately on pieces of paper. A bit like how an accounts department used to work before modern PCs and spreadsheets. Now one person in finance can do the job of 10 or 20 people from a few decades ago. Without an accountant someone can just change an exchange rate, or an input cost on a well-created spreadsheet and find out the change in cash-flow, P&L and so on. Armies of people would have been needed before to work out the answers.

And of course, the beauty of the GCM is that you can play around with other factors and find out what effect they have. The albedo of the ocean also changes with waves. So you can try some limits between albedo with no waves and all waves and see the change. If it’s significant then you need a parameter that tells you how calm or stormy the ocean is throughout the year. And if you don’t have that data, you have some idea of the “error”.

Everyone wants their own GCM now..

Of course, in that thought experiment about sea ice albedo we haven’t calculated a “final” answer. Other effects will come into play (clouds).. But as you can see with this little example, different phenomena can be progressively investigated and reasonably quantified.

Past Climate

Do we understand the causes of past climate change or not? Do the Milankovitch cycles actually explain the end of the last ice age, or the start of it?

This is another area where models are invaluable. Without a GCM, you are just guessing. Perhaps with a GCM you are guessing as well, but just don’t know it.. A topic for another day.

Common Misconception

The idea floats around that models have “positive feedback” plugged into them. Positive feedback for those few who don’t understand it.. increases in temperature from CO2 will induce more changes (like melting Arctic sea ice) that increase temperature further.

Unless it’s done very secretly, this isn’t the case. The positive feedbacks are the result of the model’s output.

The models have a mixed bag of:

• fundamental equations – like conservation of energy, conservation of momentum
• parameterizations – for equations that are only empirically known, or can’t be easily solved in the “grid” that makes up the 3d “mesh” of the GCM

More on these important points in the next post.

“Necessary but Not Sufficient”

A last comment before we see them on the catwalk – the catwalk “retrospective” – is that models matching the past is a necessary but not sufficient condition for them to match the future. However, it is – or it would be – depending on what we find.. a great starting point.

Models On the Catwalk

20th century temperature hindcast vs actual - ensemble

Most people have seen this graph. It comes from the IPCC AR4 (2007).

The IPCC comment:

Models can also simulate many observed aspects of climate change over the instrumental record. One example is that the global temperature trend over the past century (shown in Figure 1) can be modeled with high skill when both human and natural factors that influence climate are included.

In summary, confidence in models comes from their physical basis, and their skill in representing observed climate and past climate changes. Models have proven to be extremely important tools for simulating and understanding climate, and there is considerable confidence that they are able to provide credible quantitative estimates of future climate change, particularly at larger scales. Models continue to have significant limitations, such as in their representation of clouds, which lead to uncertainties in the magnitude and timing, as well as regional details, of predicted climate change. Nevertheless, over several decades of model development, they have consistently provided a robust and unambiguous picture of significant climate warming in response to increasing greenhouse gases.

Now of course, this is a hindcast. Looking backwards. One way to think about a hindcast is that it’s easy to tweak the results to match the past. That’s partly true and, of course, that’s how the model gets improved- until it can match the past.

The other way to think about the hindcast is that it’s a good way to test the model and find out how accurate it is.

The model gets to “past predict” many different scenarios. So if someone could tweak a model so that it accurately ran temperature patterns, rainfall patterns, ocean currents, etc – if it can be tweaked so that everything in the past is accurate – how can that be a bad thing? Also the model “tweaker” can change a parameter but it doesn’t give the flexibility that many would think. Let’s suppose you want to run the model to calculate average temperatures from 1980-1999 (see below) so you put your start conditions into the model, which are values for 1980 for temperature and all other “process variables” and crank up the model.

It’s not like being able to fix up a painting with a spot of paint in the right place – it’s more like tuning an engine and hoping you win the Dhaka rally. After you blew the engine halfway through you get to do a rebuild and guess what to change next. Well, analogies – just illustrations..

Obviously, these results would need to be achieved by equations and parameterizations that matched the real world. If “tweaking” requires non-physical laws then that would create questions. Well, more on this also in later posts.

More model shots.. The top graphic is the one of interest. This is actual temperature (average 1980-1999) in contours with the shading denoting the model error (actual minus model values). Light blue and light orange (or is it white?) are good..

Actual 1980-1999 temperature with shading denoting model error (top graphic)

The model error is not so bad. Not perfect though. (Note that for some reason, not explained, the land temperature average is over a different time period than sea surface temperatures).

Temperature range:

1980-1999 Temperature range in each location and Model error in temperature range

The standard deviation in temperature gives a measure of the range of temperatures experienced. The colors on the globe indicate the difference between the observed and simulated standard deviation of temperatures.

Simplifying, the light blue and light orange areas are where the models are best at working out the monthly temperature range. The darker colors are where the models are worse. Looks pretty good.

Rainfall:

Actual Rainfall vs Model Rainfall, 1980-99

This one is awesome. Remember that rainfall is calculated by physical processes. Temperature, available water sources, clouds, temperature changes, winds, convection..

Ocean temperature:

Ocean potential temperature and model error 1957-1990

Ocean potential temperature, what’s that? Think of it as the real temperature with unstable up and down movements factored out, or read about potential temperature.. Note that the contours are the measurements (averaged over 34 years) and the shaded colors are the deviations of actual – model. So once again the light blue and light orange are very close to reality, the darker colors are further away from reality.

This one you would expect to be easier to get right than rainfall, but still, looking good.

Conclusion

It’s just the start of the journey into models. There will be more, next we will look at Models Off the Catwalk. So if you have comments it’s perhaps not necessary to write your complete thoughts on past climate, chaos.. Interesting, constructive and thoughtful comments are welcome and encouraged, of course. As are questions.

Hopefully, we can avoid the usual bunfight over whether the last ten years actual match the model’s predictions. Other places are so much better for those “discussions”..

Update – Part Two now published.

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We cover some basics in this post. The subject was inspired by one commenter on the blog.

• When we look at a “radiative forcing” what does it mean?
• What immediate and long-term impact does it have on temperature?
• What is the new equilibrium temperature?

Radiative Forcing

The IPCC, drawing on the work of many physicists over the years, states that the radiative forcing from the increase in CO2 to about 380ppm is 1.7 W/m². You can see how this is all worked out in the series CO2 – An Insignificant Trace Gas.

What is “radiative forcing”? At the top of atmosphere (TOA) there is an effective downward increase in radiation. So more energy reaches the surface than before..

Thermal Lag

If you put very cold water in a pot and heat it on a stove, what happens? Let’s think about the situation if the water doesn’t boil because we don’t apply so much heat..

Simple Thermal Lag

I used simple concepts here.

T= water temperature and the starting temperature of the water, T (t=0) = 5°C

Air temperature, T1 = 5°C

Energy in per second = constant (=1000W in this example)

Energy out per second = h x (T – T1), where h is just a constant (h=20 in this example)

And the equation for temperature increase is:

Energy per second, Q = mc.ΔT

m = mass, and c= specific heat capacity (how much heat is required to raise 1kg of that material by 1’C) – for water this is 4,200 J kg^-1 K^-1. I used 1kg.

ΔT is change in temperature (and because we have energy per second the result is change in temperature per second)

The simple and obvious points that we all know are:

• the liquid doesn’t immediately jump to its final temperature
• as the liquid gets closer to its final temperature the rate of temperature rise slows down
• as the temperature of the liquid increases it radiates or conducts or convects more energy out, so there will be a new equilibrium temperature reached

In this case, the heat calculation is by some kind of simple conduction process. And is linearly proportional to the temperature difference between the water and the air.

It’s not a real world case but is fairly close – as always, simplifying helps us focus on the key points.

What might be less obvious until attention is drawn to it (then it is obvious) – the final temperature doesn’t depend on the heat capacity of the liquid. That only affects how long it takes to reach its equilibrium – whatever that equilibrium happens to be.

Heating the World

Suppose we take the radiative forcing of 1.7W/m² and heat the oceans. The oceans are the major store of the climate system’s heat, around 1000x more energy stored than in the atmosphere. We’ll ignore the melting of ice which is a significant absorber of energy.

Ocean mean depth = 4km (4000m)  – the average around the world

Only 70% of the earth’s surface is covered by ocean and we are going to assume that all of the energy goes into the oceans so we need to “scale up” – energy into the oceans =  1.7/0.7 = 2.4 W/m² going into the oceans.

The density of ocean water is approximately 1000 kg/m³ (it’s actually a little more because of salinity and pressure..)

Each square meter of ocean has a volume of 4000 m³ (thinking about a big vertical column of water), and therefore a mass of 4×10⁶ kg.

Q = mc x dT

Q is energy, m is mass, c is specific heat capacity = 4.2 kJ kg^-1 K^-1,
dT = change in temperature

We have energy per second (W/m²), so change in temperature per second, dT = Q/mc

dT per second = 2.4 / (4×10⁶ x 4.2×10³)

= 1.4 x10^-10 °C/second

dT per year = 0.004 °C/yr

That’s really small! It would take 250 years to heat the oceans by 1°C..
Let’s suppose – more realistically – that only the top “well-mixed” 100m of ocean receives this heat, so we would get (just scaling by 4000m/100m):

dT per year = 0.18 ‘C per year.

An interesting result, which of course, ignores the increase in heat lost due to increased radiation, and ignores the heat lost to the lower part of the ocean through conduction.

If we took this result and plotted it on a graph the temperatures would just keep going up!

Calculating the new Equilibrium Temperature

The climate is slightly complicated. How do we work out the new equilibrium temperature?

Do we think about the heat lost from the surface of the oceans into the atmosphere through conduction, convection and radiation? Then what happens to it in the atmosphere? Sounds tricky..

Fortunately, we can take a very simple view of planet earth and say energy in = energy out. This is the “billiard ball” model of the climate, and you can see it explained in CO2 – An Insignificant Trace Gas – Part One and subsequent posts.

What this great and simple model lets us do is compare energy in and out at the top of atmosphere (TOA). Which is why “radiative forcing” from CO2 is “published” at TOA. It helps us get the big picture.

Energy radiated from a body per unit area per second is proportional to T⁴, where T is temperature in Kelvin (absolute temperature). Energy radiated from the earth has to be balanced by energy we absorb from the sun.

This lets us do a quick comparison, using some approximate numbers.

Energy absorbed from the sun, averaged over the surface of the earth, we’ll call it P_old = 239 W/m².

Surface temperature, we’ll call it T_old = 15°C = 288K

If we add 1.7W/m² at TOA what does this do to temperature? Well, we can simply divide the old and new values, making the equation slightly easier..

(T_new/T_old)⁴ =P_new/P_old

So T_new=288 x (239+1.7/239)^1/4

Therefore, T_new = 288.5K or 15.5°C   – a rise of 0.5°C

I don’t want to claim this represents some kind of complete answer, but just for some element of completeness, if we redo the calculation with the radiative forcing for all of the “greenhouse” gases, excluding water vapor, we have a radiative forcing of 2.4W/m².

Tnew = 288.7 or 15.7°C   – a rise of 0.7°C.

(Note for the purists, I believe the only way to actually calculate the old and new surface temperature is using the complete radiative transfer equations, but the results aren’t so different)

Conclusion

The aim of this post is to clarify a few basics, and in the process we looked at how quickly the oceans might warm as a result of increased radiative forcing from CO2.

It does demonstrate that depending on how well-mixed the oceans are, the warming can be extremely slow (250 years for 1°C rise) or very quick (5 years for 1°C rise).

So from the information presented so far, temperatures we currently experience at the surface might be the new equilibrium from increased CO2, or a long way from it – this post doesn’t address that huge question! Or any feedbacks.

What we ignored in the calculation of temperature rise was the increased energy lost as the temperature rose – which would slow the rise down (like the heated water in the graph). But at least it’s possible to get a starting point.

We can also see a rudimentary calculation of the final increase in temperature – the new equilibrium – as a result of this forcing (we are ignoring any negative or positive feedbacks).

And the new equilibrium doesn’t depend on the thermal lag of the oceans.

Of course, calculations of feedback effects in the real climate might find thermal lag parameters to be extremely important.

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