Global Average Temperatures

[Admiral_Bobski]

Okay, here's a starting point for anyone who is interested.

I did some back-of-envelope calaculations (don't worry: it was a used envelope, so I was recycling!) and came up with this little sum which is laden with assumptions and almost certainly wrong. But let's have a look at it anyway.

Svante Arrhenius came up with his Greenhouse Law which, when simplified, looks like this:

ΔF = α ln(C/C₀)

where ΔF is the change in temperature, α is (I think) the climate sensitivity (which would be a constant) and ln(C/C₀) is the natural log of current CO2 divided by "Start Point" CO2.

So let's have a little fun rejigging the equation. We want to find the climate sensitivity, so:

α = ΔF /ln(C/C₀)

Now let's plug some numbers in. The IPCC's lower estimate for temperature change caused by a doubling of CO2 is 2C (their high is 4.5C). We shall assume, for the sake of easy numbers, that pre-industrial CO2 levels were at 250ppm (an estimate about 10-20ppm too low, but bear with me).

So, according to these figures, if CO2 were to reach 500ppm then we would have seen a total rise of 2C. Plugging these values into the equation give us a value for α:

α = 2/ln(500/250)

α = 2/ln(2)

α ≈ 2.885

Now we know what alpha is, let's work backwards. Now, obviously we can't divide 0 by another number, so let's see what temperatures would be like with a CO2 concentration of just one part per million:

ΔF = α ln(C/C₀)

ΔF = 2.885 x ln(1/250)

Gives us a change in temperature of -15.93C

Which means that with virtually all the CO2 in the atmosphere gone we would have an average global temperature of about -2C.

(If the climate sensitivity was 4.5C, out of interest, the temperature change would be -35.85C, giving an average global temperature of around -22C, which is lower than the Earth's temperature should be without an atmosphere.)

It also means that even if we were to reach the dizzy heights of 2000ppm of CO2 we would see a maximum temperature increase of almost exactly 6C from pre-industrial levels (only another 5 or so to go then!).

(Climate sensitivity of 4.5C changes this figure to about 13.5C)

So according to this calculation, CO2 would be responsible for fully half of the difference in temperature between an Earth with an atmosphere and an Earth without in the lowest climate sensitivity estimate. (In the higher estimate of 4.5C, CO2 is responsible for over 100% of the difference, which tells me one of four things - either my calculations are wrong, the IPCC's higher estimate is wrong, the estimate of Earth's temperature sans atmosphere is wrong, or Arrhenius was wrong. Anyone care to check my maths for errors? There's bound to be a mistake in there somewhere!)

Anyway, I've rambled on for long enough already, so it's over to you! If you're interested...

CB

EDIT - A quick note about the Arrhenius equation above. I copied and pasted it from the Wiki page and it didn't come out quite right. (C/C) is supposed to read as (C/C₀).

Edited November 13, 2009 by Captain_Bobski

[AtlanticFlamethrower]

Hope I'm not crowding out smarter people by replying.

α = 2/ln(500/250)

α = 2/ln(2)

α ≈ 2.885

I can't get to the result.

What is the value of n?

[VillagePlank]

Here's the paper.

The part we are looking at is at the top of page 267 where he describes the formula in English, not in mathematics: Thus if the quantity of carbonic acid increases in geometric progression, the augmentation of the temperature will increase nearly in arithmetic progression He never published the derivation.

Is is probably better written for modern audiences as ...

ΔT = α log(C/C₀)

where one might read it as the change in temperature is proportional to some constant multiplied by the logarithm of the observed quantity of CO₂ divided by the starting quantity of CO₂

Anyway, first things first - the constant, α - cannot really be a constant. What happens when someone turns out the lights ... erm ... I mean the sun - does the relationship still hold? Of course not, so, there is tacit assumption that, all other things being equal (which can only be produced in a lab) that there is relationship between the change of CO₂ and the change of temperature, T.

So ... onto the gas laws: what happens when you add more of a gas to a finite amount of space with a constant pressure ...

EDIT log means log10 in this context.

Edited November 14, 2009 by VillagePlank

[Gray-Wolf]

The reason for my 'big brush' , and probably his lab, is to stop all the distractions of 'the real world' to enable you to first see if the relationship holds.When the 'all other things being equal ' bit is removed you can then look at how our natural complexities impact what should be occuring?

C-Bobs 6c calcs do seem to mirror some of the higher CO2 figures we have for the ETM though.

[VillagePlank]

The reason for my 'big brush' , and probably his lab, is to stop all the distractions of 'the real world' to enable you to first see if the relationship holds.When the 'all other things being equal ' bit is removed you can then look at how our natural complexities impact what should be occuring?

C-Bobs 6c calcs do seem to mirror some of the higher CO2 figures we have for the ETM though.

Well, as a thought experiment turn off the sun, and reason through to see if temperature increases when the CO₂ content is increased.

[Admiral_Bobski]

Here's the paper.

The part we are looking at is at the top of page 267 where he describes the formula in English, not in mathematics: Thus if the quantity of carbonic acid increases in geometric progression, the augmentation of the temperature will increase nearly in arithmetic progression He never published the derivation.

Is is probably better written for modern audiences as ...

ΔT = α log(C/C₀)

...

EDIT log means log10 in this context.

Very quick post because I've just woken up (shocking, I know!), so please forgive my brevity

The Arrhenius equation doesn't use log 10 but the natural log (which is written as ln, which is a single term, hence there is no "n" in the equation).

Using log10 will produce a different result.

On the other hand, VP is correct in saying that the term α can't be a constant, which rather complicates the issue. What, therefore, do they mean when they say that the climate sensitivity to a doubling of CO2 is 2-4.5C?

CB

[Gray-Wolf]

Well, as a thought experiment turn off the sun, and reason through to see if temperature increases when the CO₂ content is increased.

Don't we run into Boyles law if we go too far with the increasing of the gas and force temps up anyway?

I know what you are saying and of course a primary energy source to drive the heating needs to be there ,unless you are a gas giant and have fantastic pressures within your lower atmosphere (but not quite a star).

[Methuselah]

Well, as a thought experiment turn off the sun, and reason through to see if temperature increases when the CO₂ content is increased.

That's an interesting idea, VP...My own (limited) view on it is this:

Assumption #1: Solar output remains constant;

Assumption #2: heat from Earth's internal radioisotopes remains constant;

Assumption #3: the effective temperature of space remains constant;

Assumption #4: atmospheric GHG concentration is variable...

Question: will the Earth's surface temperature vary as a result of the above (GHG) variability?

Answer: yes, of course it will.

Reason: the spectroscopic characteristics of GHGs will alter the effective temperature of the planet - Stephan-Boltzmann???

[Admiral_Bobski]

Don't we run into Boyles law if we go too far with the increasing of the gas and force temps up anyway?

I know what you are saying and of course a primary energy source to drive the heating needs to be there ,unless you are a gas giant and have fantastic pressures within your lower atmosphere (but not quite a star).

Boyle's Law may have some bearing on the situation, but it is probably negligible. To the uninitiated, Boyle's law says that temperature and pressure are proportional to one another in a closed system with a fixed volume.

If the Earth's temperature increases then the gas in the atmosphere expands, increasing the atmospheric pressure. Of course this only works if the atmosphere can't expand anywhere (and the amount of warming we're talking about would increase either the volume or pressure of the atmosphere by the tiniest of amounts).

Furthermore, the atmosphere isn't a completely closed system - atmospheric gases escape into space all the time. The reason we still have an atmosphere is because these lost particles are replaced by new ones. (Did the particles fall or were they pushed?!)

It would be interesting to see if there has been any detectable change in atmospheric pressure over the last 150 years. Anyone...?

CB