## Radiative Balance, Feedback, and Runaway Warming

#### Posted on 26 February 2012 by Chris Colose

Skeptical Science has previously discussed the topic of feedbacks and why the existence of *positive feedbacks* (i.e., those feedbacks that amplify a forcing) do not necessarily lead to runaway warming, or even to an inherently "unstable" climate system. I also wrote on it at RealClimate (and Pt. 2). This was brought up again in Lord Christopher Monckton's response to SkepticalScience, where he asserted:

"First, precisely because the climate has proven temperature-stable, we may legitimately infer that major amplifications or attenuations caused by feedbacks have simply not been occurring...A climate subject to the very strongly net-positive feedbacks imagined by the IPCC simply would not have remained as stable as it has."

I wanted to revisit the subject in order to take a different approach on the subject of positive feedbacks. This involves the relationship between Earth's surface temperature and outgoing infrared radiation (the energy Earth emits to space). Determining how the outgoing longwave radiation (OLR) depends on surface temperature and greenhouse content is a fundamental determinant to any planetary climate.

I'll begin with very trivial, ideal cases, and then slightly build up in complexity in order to relate the problem to climate sensitivity. By the end, it should be clear why positive feedbacks can exist that inflate climate sensitivity but do not necessarily call for a runaway warming case. We'll also see a scenario, commonly discussed by planetary scientists, in which it *does* lead to a runaway.

First, we begin with the simplest case in which the Earth has no atmosphere and essentially acts as a perfect radiator. In this case, the outgoing radiation is given by the Stefan-Boltzmann equation OLR=σT^{4}. T is temperature. σ is a constant, so the equation means that the outgoing radiation grows rapidly with temperature, (to the power of four) as shown below.

*Figure 1: Plot of OLR vs. Surface temperature for a perfect blackbody*

In the next case, suppose that we add some CO_{2} to the atmosphere (400 ppm). The atmosphere here is completely dry (and therefore no water vapor feedback). In this example, the addition of CO_{2} will reduce the OLR for any given temperature, since the atmosphere absorbs some of the exiting energy. This is displayed with the red curve in Figure 2 (the black curve is from above for reference).

*Figure 2: Relationship between OLR and surface temperature for a blackbody (black curve) and with 400 ppm CO2 (red curve). The horizontal line is the absorbed solar radiation. *

Also plotted in Figure 2 is a horizontal line at 240 W/m^{2}, which corresponds to the amount of solar energy that Earth absorbs. In equilibrium, the Earth receives as much solar energy as it does emit infrared radiation. Therefore, in the above plot, the points at which the horizontal line intersect the black/red lines will correspond to the equilibrium climates in this model. Note that the red line makes this intersection at a higher surface temperature, which is the greenhouse effect.

Now let's step up the complexity a bit. We'll throw in some water vapor into the model, but not just a fixed amount of water vapor. This time, we'll also let the water vapor concentration increase as temperature increases. Water vapor is a good greenhouse gas, so now the infrared absorption grows with temperature. This is the water vapor feedback. The blue line in the next figure is the OLR for a planet with the same 400 ppm CO_{2}, in addition to this operating feedback.

*Figure 3: Relationship between OLR and surface temperature, as above, but with a constant relative humidity atmosphere (blue line, implying increasing water vapor with temperature)*

In this figure, we see that the OLR does not depend very much on the water vapor at low temperatures. This makes sense, because at temperatures this cold (such as during a snowball Earth), there is so little water vapor in the air. However, at temperatures similar to the modern global mean and warmer, the OLR drops tremendously and the the T^{4} dependence instead becomes much flatter. We'll get a more clear picture of that means for climate sensitivity in the next diagram.

In the next diagram, I've removed the red curve for convenience. But I've added two horizontal lines this time. You can think of this as two possible values for the incoming solar radiation.

*Figure 4: OLR vs. surface temperature for a blackbody (black curve) and an atmosphere with CO2 and a water vapor feedback (blue curve). The horizontal lines give two values for the absorbed incoming solar radiation, and the colored shapes give possible equilibrium points. On the trajectory where water vapor exists, sensitivity is enhanced because the temperature difference between the two red circles (as sunlight goes up) is greater than the difference between the two blue squares.*

To interpret Figure 4, suppose that we increase the amount of sunlight that the Earth gets, which means we jump from the red to the green line in the above figure. If the Earth were a blackbody (black curve) then the temperature change that results from this would just be the difference between the values at the two blue squares. However, in a world with a water vapor feedback, the temperature difference is given by the distance between the two red circles. We can infer from this that water vapor has increased climate sensitivity, yet it did not cause a runaway warming effect.

Now let's consider one more case. Notice in the previous diagram that at very high temperatures, the OLR starts to flatten out, and indeed eventually can become almost flat. This is due to the rapid increase in water vapor (and infrared absorption) as temperature goes up. But suppose we pump up the amount of sunlight that the Earth gets to much higher values than in the last figure. This new value is shown by the horizontal green line in the figure below.

*Figure 5: As above, but the green line corresponds to higher incoming solar radiation.*

Once again, if we follow the black curve (with no atmosphere), then we get an expected increase in temperature as the amount of sunlight goes up. But if we follow the blue curve (the system with an operating water vapor feedback), then something strange happens.

At some point the OLR becomes so flat, that it can never increase enough to match the incoming sunlight. In this case, it actually becomes impossible to establish a radiative equilibrium scenario, and the result is a *runaway greenhouse*. This is the same phenomenon planetary scientists talk about in connection with the possible evolution of Venus or exoplanets outside our solar system. The system will only be able to come back to radiative equilibrium once the rapid increase of water vapor mass with temperature ceases, which in extreme cases may not be until the whole ocean is evaporated.

From these figures, we can readily see the fallacy in "positive feedbacks imply instability" type arguments. There is in fact a negative feedback that always tends to win out in the modern climate. This is the increase in planetary radiation emitted to space as temperature goes up. Positive longwave radiation feedbacks only weaken the efficiency at which that restoring effect operates. Instead of the OLR depending on T^{4}, it might depend on T^{3.9}, or maybe even T^{3} at higher temperatures; eventually the OLR becomes independent of the surface temperature altogether. I haven't discussed shortwave feedbacks, such as the decrease in albedo as sea ice retreats. That only raises the position of the horizontal lines slightly, allowing for a warmer equilibrium point, but in no way compromises the argument.

In fact, the same sort of argument can be applied if we let the albedo vary with temperature (and so the absorbed solar radiation is no longer given by a horizontal line). The opposite extreme, a snowball Earth, can then be thought of as a competition between the decreased longwave radiation to space as the planet cools, and the increased reflection as the planet brightens (when the ice line is advancing toward the equator). As with a runaway greenhouse, it's not inevitable that this occurs, as is evident from times in Earth's history when ice advanced but did not reach the equator.

As a final note, it's worth mentioning that it is virtually impossible to trigger a true runaway greenhouse in the modern day by any practical means, at least in the sense that planetary scientists use the word to describe the loss of any liquid water on a planet. The most realistic fate for Earth entering a runaway is to wait a couple billion years for the sun to increase its brightness enough, such that Earth receives more sunlight than the aforementioned outgoing radiation limit that occurs in moist atmospheres. None of this means that climate sensitivity cannot be relatively high however.

*Note: Except for the first graph, all computations here were done using the NCAR CCM radiation module embedded within the Python Interface for Ray Pierrehumbert's supplementary online material to the textbook "Principles of Planetary Climate." The lapse rate feedback is included as an adjustment to the moist adiabat. I've assumed near-saturated conditions are maintained (constant 100% relative humidity) with temperature, although the argument is qualitatively similar with lesser RH values.*

This post has been adapted into the Advanced rebuttal to "Positive feedback means runaway warming"

Steve Lat 18:27 PM on 25 February, 2012Doug Hutchesonat 19:54 PM on 25 February, 2012Paul Dat 20:48 PM on 25 February, 2012chriskozat 22:00 PM on 25 February, 2012owl905at 23:15 PM on 25 February, 2012John Brookesat 23:20 PM on 25 February, 2012apiratelooksat50at 23:21 PM on 25 February, 2012Bob Lacatenaat 00:50 AM on 26 February, 2012Alexandreat 01:43 AM on 26 February, 2012David Lewisat 03:39 AM on 26 February, 2012David Lewisat 03:57 AM on 26 February, 2012Alexandreat 04:19 AM on 26 February, 2012Chris Coloseat 04:29 AM on 26 February, 2012Sky_Hunterat 05:05 AM on 26 February, 2012Tor Bat 05:51 AM on 26 February, 2012sauerjat 05:55 AM on 26 February, 2012Chris Colose, Well done!... I am a chemical engineer. It is beaten into our heads to relate things with graphs as doing so shows relationships so well. As of yet, I've never seenanyoneexplain 'sensitivity' so well as you have done. Sensitivity is simply the 'gain' or 'multiplier' that relates a change in an independent variable (i.e. forcing, such as CO2) to a change in the dependent variable (temp). In formula form:dT =. Essentially,Sensitivity x dCO2Sis the SLOPE on a graph. (For a graph formatted with Temp on the x- axis as in the article,Sensitivity would actually be '1/slope'.) And, WOW!, your article shows this visually so well! It clearly shows how a more horizontal sloped curve (for Flux vs Temp formatted graphs) versus a more vertically sloped curve result in HIGHER sensitivities. This is the kind of technical (but yet simple) explanation that willturn the headsof the more technically astute minds out there (assuming they are the least bit open-minded)! This is the kind of stuff that might get them to say, "Oh! Now I get it!" Like,mentioned, I would propose a@4 chriskoz and @7 PiratePART-2 follow-up articlewhich I think would knock this whole thing over the fence. Then, I would feel fully ready to explain the science of global warming amply loaded with the needed ammunition that could NOT be refuted. I can only wonder if such an addendum would also be universally helpful.Here is the suggestion:After Fig.3, I would take the article on a slightly different course. For the next graph, I would showtwo RED curves; one for 250ppm (pre-industrial) and another for 500ppm (give a hypothetical year, 2065).These CO2 values work well with the typical scientific talk ofWhy 250 & 500 curves?doublingCO2 and how such a doubling impacts temperature. Plus it shows realistic resulting temperatures for a starting point (pre-ind) CO2 and not-so-far-away (2065) CO2. ... Leave the horizontal line at 240 W/m2 (at this point in the article assume albedo differences and solar cycle variances are still neutral). With these twoCO2-onlyRED curves, I would expect the difference in equilibrium temp to be~1.0 - 1.2K, which I have read on this site as the expected dTemp for doubling of CO2 with NO other feedbacks mixed in. Then, on this same graph, addtwo BLUE curvesrepresenting the average Relative Humidity earth conditions, one at 250ppm CO2 and the other at 500ppm CO2. I assume the '500ppm w/RH' curve will simply be shifted down below the '250ppm w/RH' curve (you would know the exact particulars on this better than I). If you look at the present Fig.3, I see how, at 200K, the RED curve is shifted vertically down below the BLACK curve, and the BLUE curve (at this 200K) also "starts" at this same point (at this trace humidity state). For the 250ppm & 500ppm RED & BLUE curves, they will simply start (at the 200K temp) attwo different vertical shifts downfrom the BLACK curve, with the 500ppm curve simply being 'LOWER' than the 250ppm curve (probably by the same porportional vertical distance). Then, when these curves move to the right from this starting point (200K), they will cross the 240W/m2 horizontal line at their various equilibrium temperatures.[Slight diversion: WOW, your article is so COOL how it visually explains these changes in theWith your clear graphs, any technically oriented person wouldequilibriumpoint.instantly understand & get your point and realize its significance in a heartbeat!]Since the two BLUE curves are so much flatter (i.e. or lower angle, or read higher sensitivity), the horizontal distance between the two BLUE curves will be much greater than the horizontal distance for the RED curves, i.e. a much higher sensitivity. Therefore, the 1.0-1.2 dT for doubling of CO2 alone turns into~3.0dT when adding in the humidity feedback. ... This would really help explain the positive feedback of humidity added on top of doubling CO2 alone. After this, I would take the article the direction as proposed by. This graph@4 chriskozshow the impact on Temp caused by ancouldextremechange in solar radiation, such as during the Maunder Minimum, but I would like to see the article stay more applicable to today's typical solar cycles. Therefore, I would setup this next graph to show the resulting dT between1)a LOW POINT on our current measured SOLAR CYCLES and2)a HIGH POINT on our current SOLAR CYCLES. I don't know what the difference in average, earth-surface, net input W/m2 for these two points amounts to ... ~0.25W/m2??? But whatever it is, show these two horizontal lines super-imposed on a "detailed" view of the graph ('detailed' so to better see the dT impact). I expect this equilibrium dTemp impact to be roughly 10% (or less) than that for the CO2 doubling w/ RH impact (~3.0dT). I recall reading on this site that modern variances in solar radiation amounts to (at most) 1/10th that of human induced CO2 changes.owl905: Your point about 400ppm (or 1,000,000 ppm) is NOT a small point! And, the article probably needs to be tweaked a bit in its wording to assure absolute accuracy.The greenhouse effect of CO2 can NOT beONLYa function of its %concentration (PPM) amongst the rest of gases. Instead, it HAS to be a function of actual #moles per unit volume of CO2 in the atmosphere (or, more clearly for most people, MASS per unit volume). These two are not the same; a given PPM does NOT mean a set MASS/volume concentration. For example, imagine an atmosphere that only had 1/10 the mass (in the entire atmosphere) of N2 and O2 (these being non-greenhouse gases). Then, let's only add CO2 to this hypothetical atmosphere; thus it is the ONLY greenhouse gas in the atmosphere. But, let's add the CO2 to a %concentration of 400ppm, then obviously the mass of CO2 (per unit volume) would be 1/10 of our present atmosphere. There is NO WAY this 1/10 mass/volume CO2 atmosphere could have the same greenhouse effect as our present atmosphere. The point here is that the key driving variable HAS to be MASS/volume NOT % of molecules which, technically, is all that PPM really states. Therefore, the key variable defining greenhouse effect is not PPM, but actually mass/volume or molecules/volume (however you want to express it). But, this detail is getting TOO technical for most people to understand & it deviates from the usually 'talk' of defining the greenhouse effect as a function of PPM. So, I would simply say (for Fig.2) that you would first"flood the atmosphere with present day quantities of N2 and O2, then add CO2 to the concentration of 400PPM".Then, explain that the N2 & O2 cause NO greenhouse effect, so you have created an artifical atmosphere that has aCO2-only greenhouse effectthat would mimic today's atmospheric "concentration" of 400ppm of CO2 (this being your original intention in Fig.2) ... without explaining the detail that this also means that this hypothetical atmosphere would also have the samemass/volume concentration of CO2as today's atmosphere. This slight wording tweak ("400ppm in the same N2 & O2 atmosphere as per today's atmosphere") would then be technically fully accurate and defendable.Sorry for LONG comment. I tend to get too windy.WheelsOCat 06:15 AM on 26 February, 2012Chris Coloseat 06:33 AM on 26 February, 2012keithpickeringat 07:10 AM on 26 February, 2012From these figures, we can readily see the fallacy is "positive feedbacks imply instability" type arguments.... should read "the fallacyin"Chris Coloseat 07:12 AM on 26 February, 2012Charlie Aat 08:25 AM on 26 February, 2012owl905at 08:53 AM on 26 February, 2012Tom Curtisat 11:14 AM on 26 February, 2012Chris Coloseat 13:46 PM on 26 February, 2012Bruce A.at 03:39 AM on 27 February, 2012Chris Coloseat 04:06 AM on 27 February, 2012David Lewisat 05:20 AM on 27 February, 2012Chris Coloseat 06:02 AM on 27 February, 2012williamat 06:35 AM on 27 February, 2012Jose_Xat 16:03 PM on 27 February, 2012gallopingcamelat 16:09 PM on 27 February, 2012Jose_Xat 16:14 PM on 27 February, 2012KRat 16:19 PM on 27 February, 2012Chris Coloseat 16:33 PM on 27 February, 2012Chris Coloseat 16:35 PM on 27 February, 2012Jose_Xat 17:08 PM on 27 February, 2012Jose_Xat 18:07 PM on 27 February, 2012Jose_Xat 18:14 PM on 27 February, 2012scaddenpat 18:40 PM on 27 February, 2012gallopingcamelat 15:30 PM on 28 February, 2012Jose_Xat 15:42 PM on 28 February, 2012Jose_Xat 16:13 PM on 28 February, 2012Chris Coloseat 05:08 AM on 29 February, 2012Camburnat 05:31 AM on 29 February, 2012scaddenpat 06:59 AM on 29 February, 2012scaddenpat 07:55 AM on 29 February, 2012Camburnat 10:43 AM on 29 February, 2012Jose_Xat 11:24 AM on 29 February, 2012gallopingcamelat 15:19 PM on 29 February, 2012Jose_Xat 16:41 PM on 29 February, 2012