The Science of Doom

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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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:

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

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:

Radiative Forcing vs CO2 concentration, Myhre et al (1998)

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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