If we were to assume two hypothetical scenarios of CO2 increase, which would cause more warming?

I think most of you are forgetting that radiated power goes as T^4, so if the planet is already very warm (as in the 10,000 ppm C0 scenario), the relative change in temperature by doubling CO2 is much less than for a cold planet. This is because as T increases, it takes far less change in T to get a particular power radiated (so the increase in T to get an increase of 3.7 W/m^2 radiated is smaller as T increases). Given that, the increase in surface temperature would be larger for 100->200 than for 10,000 to 20,000 although the surface temperature will be warmer for the latter. But you asked which would get warmer, which I interpreted in a relative sense, meaning the smaller concentration increase makes it warmer even though the radiative forcing is the same in both cases.

edit: You can't make sense of this problem unless you assume there are no other forcings beside CO2. If you start including water vapor, aerosols, etc. then you need to specify the initial conditions more precisely (this is why it is useless, as most skeptics do, to talk about paleo levels of CO2 being relevant to today's climate: it doesn't matter that atmospheric CO2 was 10,000 ppm tens of millions of years ago because it has no relevance to what will happen if we double CO2 today). Anyway, given the solar forcing for Earth, these concentrations of CO2 all lead to very cold planets so any water that exists will be ice, and you can reasonably neglect water vapor in the calculation since the vapor pressure of ice is a lot less than water.

Anyway, Stefan-Boltzmann works very well. Increase the downwelling longwave radiative flux by X W/m^2 and the surface temperature at equilibrium will rise so that the new temperature will obey

sigma*(T_new)^4 = sigma*(T_old+dT)^4 = F_old + X

where F_old is the radiative forcing before the increase in the longwave downwelling flux. That formula will hold no matter if there are feedbacks that determine X or not since it merely describes how surface temperature has to change (on average) for a system in radiative equilibrium. Your problem really can't be done because you don't know F_old for all the conditions you've described (and, more importantly, dT, the temperature change, is a function of F_old and T_old), and arbitrarily making F_old all the same for the different conditions isn't really right from a physical standpoint since planets don't really work that way (you can't really get the same surface temperature for different longwave forcings unless other things have changed which makes it somewhat academic to even discuss the problem (this goes back to why 10,000 ppm 100 million years ago is irrelevant)). It's more useful, in my opinion, to assume a constant downwelling shortwave flux (what you get for Earth), and then compute the increase in X for each CO2 concentration. In that case, as I discuss above, doubling CO2 for a low net forcing increases temperature more than doubling at high concentrations.

Last thought: I assumed a planet with no water, no dust, and only CO2 and the balance of the atmosphere being non-radiative, say helium to avoid anyone from bringing collision-induced dipoles into play. This is really the only way I think it makes sense to talk about the problem, since in that case the initial conditions are relatively well defined and the temperature will go up or down only in response to the shortwave forcing and the radiative forcing from CO2. Otherwise, you have to know how much water vapor is in the atmosphere to begin with, how albedo is distributed etc, and how these feedbacks change with an increase in the CO2 forcing. It's almost impossible to do even if you specify "conditions like the present-day atmosphere" since the feedbacks aren't known precisely enough (see the figure on the uncertainties in the IPCC AR4 executive summary, for instance). Finally, it doesn't matter whether you are talking about warming the atmosphere or surface, the fact is that the temperature has to increase (in my simple scenario above) by whatever amount you need to give the equivalent extra power out of the top. If the atmosphere is already radiating 300 W/m^2 and you increase the CO2 forcing by 4 W/m^2, then it has to go to 304 W/m^2 to reattain radiative equilibrium. That temperature increase is different than the temperature increase you would need for the same increase in forcing if the previous radiative equilibrium was at 250 W/m^2 and the new one was at 254 W/m^2. All I'm saying is you have to know the initial conditions to solve this problem, and just specifying the starting and ending CO2 concentrations is not enough information to give a quantitative answer unless you simplify the problem along the lines I state above.

Interesting question. My guess would be that 3 would cause the most warming, followed by 1 then 2. You'd probably have to write your own code to answer quantitatively - I don't think standard radiative transfer models like MODTRAN include high pressure effects you'd see with such high concentrations in scenario one and perhaps three. I know for certain the log approximation wouldn't hold.

The inequalities tell you nothing, however.

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That's what I had assumed, but i found the physical implications of your scenarios more interesting than the intended question. The former I'll be mulling over for a while.

With the revised scenarios it becomes 3, 2=1, and the inequalities would still tell you nothing.

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Edit2:

Gcnp makes an interesting point. Isn't the issue, though, that this radiative forcing is imposed at the TOA? The simplified method for determining no-feedback response is to differentiate SB with respect to temperature, taking the inverse and setting T to the effective temperature of the planet. Since Teff is determined by incoming solar, and assuming that remains constant, shouldn't the no-feedback response remain the same (assuming equal radiative forcing for each doubling)? If we assume a roughly constant, linear lapse rate, do starting surface temps matter so much?

The surface energy budget is profoundly more complicated, and it's not immediately obvious to me that an imposed forcing of 4 Wm-2 at the TOA would always equate to a 4 Wm-2 forcing at the surface, or ever.

Maybe I'm misunderstanding your point?

Edit3

RC has an interesting post which explains the issues with using the surface budget only:

http://www.realclimate.org/index.php/archives/2010...

I agree that F varies with T^4, but I'm still not understanding you, so if you can explain just a little more, I'd appreciate it.

I have two main objections (valid or not):

1) The Planck response seems insensitive to surface temperatures -- dT/dF = 1/(4*sigma*Teff^3), where Teff=255 K. 

2) The 4 Wm-2/2xCO2 is valid for the TOA only. Your equation mistakenly assumes that any increase in downwelling radiation will be compensated for by an increase in surface temperatures, but as the RC article explains, not all energy transport at the surface is radiative, and neither will the increased opacity in IR always translate to increased downwelling radiation at the surface (see tropics). The surface may only warm because the atmosphere above it warms.

If the Earth is emitting at 240Wm-2 (255K), and an imposed forcing reduces that by 4Wm-2 (now emitting at 235Wm-2 or 254K), then the atmosphere, which warms roughly as a unit, must warm to remove that imbalance -- the atmosphere "dragging" surface temps along with it.

Is my confusion due to my atmosphere being too grey?

Scenario 2 will cause just as much warming as scenario 1....

Right after you magically change CO2 to have 100 times more greenhouse reactivity.

C'mon, who do you think you're fooling? All your doing is considering the ratio of increase, and ditching everything else that matters in the argument, namely the values themselves. We live in a world that does have physical limitations, constant chemical properties, and set of physics not subject to change that rules existance.

Go ahead and step in an enclosed area with a CO2 concentration of scenario 1. Since you think ignoring plausibility is perfectly fine in the real world then you should be okay. Don't worry about considering red herring arguments. Dizziness and vomiting doesn't put that on a high priority.

Using the simplified first-order approximation expression for carbon dioxide from Wikipedia:

5.35 * LN(then/now) Watts per square meter.

I find:

20,000 ppm results in an extra 3.7 Watts per square meter heating over what it would be at 10,000 ppm 200 ppm results in an extra 3.7 Watts per square meter heating over what it would be at 100 ppm

(Yes, heating can be measured in Watts.)

Now for Scenario 3:

10,100 ppm results in an extra 24.7 watts per square meter heating over what it would be at 100 ppm

So what temperature change does that translate to? Ah! THAT is the question. That is where the arguments start.

(EDIT)

DOH! I didn't copy the 1st answer. It just took me that long to look it up and do the calculations. Honest!

To answer your question, we need to estimate carbon dioxide's radiative forcing under both scenarios. We can use the following formula to do so:

∆F = 5.35 ln(C/Co)

In this formula, 'C' is the current concentration of atmospheric carbon dioxide, and 'Co' is the reference concentration.

Using this formula, we realize that carbon dioxide would have a radiative forcing of 3.8 W/m^2 in both scenarios one and two. Therefore, these scenarios would result in exactly the same amount of warming.

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In your third scenario, carbon dioxide would have a radiative forcing of 24.7 W/m^2. Thus, this scenario would result in the largest temperature change by far.

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I don't quite understand what the question is driving at. Obviously the more CO2 increases the worse off we are, regardless of how straightforward or convoluted the semantics stating the increase.

At any rate, Jimz lies flagrantly, recklessly and stupidly on this page (as usual): "CO2 has not been a main driver of temperature change in the past but alarmists ignore the past and pretend this time is different, because this time, they are alive and things are different. There is the evil capitalism and oil companies now that didn't exist before and they must be responsible."

The truth, here, is that a hundred years of real science has established that CO2 increases from human fossil fuel burning are changing global climate: http://www.aip.org/history/climate/index.htm

Most of this history occurred before phony scientist JimZ even heard of the subject of global warming, let alone started his Sarah Palin idiocy and fake-conservative kook copy-cat routine on YA.

I realize you would love to turn the atmosphere into a simple litle 4th grade math formula. The problem is that it isn't. Math belongs in math. If the real world doesn't behave like your simple formula, what is the point? CO2 has not been a main driver of temperature change in the past but alarmists ignore the past and pretend this time is different, because this time, they are alive and things are different. There is the evil capitalism and oil companies now that didn't exist before and they must be responsible. . Alarmists see everything in terms of CO2. To most of the rest of us, that is asinine. IMO, it is because aramists must demonize it to push thier petty political agendas.

Nowhere have you established that any change in CO2 concentration will cause any change in temperature.

"... that behave more closely to the logarithmic relationship between concentration and temperature..."

What logarithmic relationship????

CO2 concentrations have been increasing and the temps have been decreasing (some would say level, but they are ignoring the data fudging that is the norm these days) for 15 years.

Which "logarithmic relationship" are you referring to, the one where temps increase with CO2 increases, or the one where temps decrease with increases in CO2?

Trees are good.

Trees need CO2

Therefore, CO2 is good.