## 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"

IanCat 16:42 PM on 29 February, 2012_{s}=f(S_{0}, P_{s })). This analysis is fundamentally flawed. In the previous section they've demonstrated precisely that the ideal gas law is a good model for the atmosphere, which implies that there is a strong relationship between surface temperature and pressure. If they don't remove this effect, they are essentially asking how temperature depends on temperature itself, rendering their analysis moot. I have to check the above carefully, though my intuition tells me that this is not right) Even ignoring the above, they have eight data points, where two (mercury and the moon) is fitted automatically with the choice of function. They are left with 6 data points, and they have FOUR free parameters in their best fit. If this is not curve fitting, what is?Jose_Xat 16:59 PM on 29 February, 2012Chris Coloseat 17:20 PM on 29 February, 2012Tom Curtisat 17:54 PM on 29 February, 2012_{TE}P_{s}= T_{s}/T_{gb}Substituting into equation (8), we obtain: T_{s}= 25.3966(S_{0}+ 0.0001325)^{0.25}* T_{s}/T_{gb}Dividing both sides by T_{s}/T_{gb}, we have (a) T_{gb}= T_{s}= 25.3966(S_{0}+ 0.0001325)^{0.25}Substituting from equation (2), we then have (b) 2/5 * {(S_{0}+ 0.0001325) * (1 - A_{gb})/(es)}^{0.25}= 25.3966(S_{0}+ 0.0001325)^{0.25}where A stands for albedo (alpha), e stands for emissivity (epsilon), and s stands for the Stefan-Boltzmann constant (sigma). Hence we have: (c) [{(S_{0}+ 0.0001325) * (1 - A_{gb})/(es)}^{0.25}]/(S_{0}+ 0.0001325)^{0.25}= 63.4915 Cancelling out and distributing, we have (d) (1 - A_{gb})^{0.25}/(e^{0.25}*s^{0.25}) = 63.4915 and hence (e) (1 - A_{gb})^{0.25}/e^{0.25}=~= 0.8239 or, by raising both sides to the fourth power: (f) (1-A_{gb})/e =~= 0.4607 We are told that (g) A_{gb}=~= 0.125, and hence (h) 1/e =~= 0.5265 or (i) emissivity approximately equals 1.9 which is impossible. OK, maths is not my strong suite, so it is entirely possible I have made an algebraic error above. If so, could somebody please point it out to me. But if not, why are we paying attention to this ridiculous theory which can be true only of the laws of thermodynamics are false.Jose_Xat 18:43 PM on 29 February, 2012Jose_Xat 18:53 PM on 29 February, 2012Tom Curtisat 19:58 PM on 29 February, 2012_{gb}. What is more, their figure 5 plots the ratio of T_{s}to T_{gb}. Hence T_{gb}is definitely part of the new theory. The only definition they provide for it, however, is in equation (2) and related discussion. Therefore, that is not a flaw in my algebra. It is at best a flaw in their presentation, such that they do not have a theory until they provide us with an alternative definition of T_{gb}, assuming, of course, no other error in my algebra.Riccardoat 23:46 PM on 29 February, 2012Tom Curtisat 02:00 AM on 1 March, 2012_{gb}assumes no thermal inertia, and no thermal distritution through conduction (or any other means). That is as unrealistic an ideal case as the standard "effective temperature", which in effect assumes absorptivity of 1, and perfect heat distribution so that no point on the surface has a different temperature than any other point. The practical importance of the effective temperature is that it is amaximummean surface temperature that can be achieved without a green house effect. Conversely, T_{gb}calculates a theoretical minimum temperature for a body without an atmosphere. As such it isverysurprising that Nikolov and Zeller report an observed mean surface temperature for the moon equal to their calculated mean surface temperature, ie, 154.3 K (table 1). Pressed to justify this figure on WUWT, Nikolov justifies the value by appeal to Vasavada et al, 1999 who reportmodelledmean lunar surface temperatures of 220 K at the equator (figure 3), and 130 K at 85 degrees north (figure 4) (also reported by wikipedia). Calculating the mean as (140 plus 140 plus 220)/3, yields a mean surface temperature of 167 K, which is probably an underestimate. Nikolov also appeals to Diviner, which reports a mean equatorial temperature of 206 K, and a mean polar temperature of 98 K (mean of equator plus two poles: 134 K, but the averaging method leaves much to be desired) Nikolov himself calculates the mean form the diviner data as (100+206)/2 or 153 K. These very low values contradict the subsurface measurements at the Apollo 15 (26 degrees North) and Apollo 17 (20 degrees North) sites. They show a subsurface temperatures approaching 253 K (Apollo 15) and 257 K (Apollo 17) conservatively estimated, showing these temperatures to be the mean surface temperature for those sites, ie, in a sub-equatorial region the mean temperature is at least 30 degrees K higher than estimated by the model, and nearly 50 K higher than estimated by Diviner. Consistent with that, Daniel Harris cites Peter Eckart ("The Lunar Base Handbook", 2nd Ed 2006) to the effect that the polar region (excluding shaded craters) has a mean of 220 K, while equatorial regions have a mean of 255K. (Rough estimate of the combined mean: 230 K). I have also seen figures of 243 K and 250 K cited with dubious provenance. I have been unable to find relevant figures from the Chang E-1 satelliteThe important thingabout this is that while Nikolov and Zeller report anobservedmean surface temperature of the moon as being 154.3 degrees C, clearly the observation made in determining that value was that that was what their theory predicted. In some circles reporting theoretical predictions as being observed results is frowned upon. That Nikolov and Zeller are prepared to do so, however, calls into question the remarkable "predictive accuracy" of their theory as shown in their figure 5. It may well be that for many values, and not just for the Moon, what is "accurately predicted" is just the value calculated in the prediction.IanCat 02:06 AM on 1 March, 2012Tom Curtisat 02:33 AM on 1 March, 2012IanCat 03:46 AM on 1 March, 2012Jose_Xat 04:48 AM on 1 March, 2012Jose_Xat 05:56 AM on 1 March, 2012Riccardoat 06:07 AM on 1 March, 2012Jose_Xat 06:33 AM on 1 March, 2012Response:[DB] Inflammatory snipped.

Riccardoat 07:34 AM on 1 March, 2012gallopingcamelat 09:12 AM on 1 March, 2012Response:[DB] Imputations of impropriety and fraud snipped. Please pay more than a passing nod to the Comments Policy of this site.

Jose_Xat 09:40 AM on 1 March, 2012Moderator Response:[JH] Please refrain from using this comment thread as your personal scratch pad.Jose_Xat 09:56 AM on 1 March, 2012scaddenpat 10:20 AM on 1 March, 2012Jose_Xat 12:26 PM on 1 March, 2012Moderator Response:[JH] Suggest that you do your calculations off-line and keep your comments focused. People are just not inclined to read lengthy rambling posts.gallopingcamelat 17:03 PM on 1 March, 2012KRat 17:07 PM on 1 March, 2012Jose_X- I will(putting my 2 cents in, if that)note that it is a bit difficult to follow a stream of consciousness post. It would be easier to follow if you were to note yourassertion(s), followed by the backing. Otherwise it's a bit difficult to identify any issues you might raise. That said - Huffman appears to think that plate tectonics is nonsense(not the best recommendation), N&Z have apparently used basic curve fitting to re-derive the S-B relationship without acknowledging it, PV=nRT requires aseparatetemperature driven by convection and radiative physics to set the pressures, etc.Even WUWThas pointed out N&Z issues with significant justification. Which aspect(s) of those postings do you consider significant issues with basic radiative physics?KRat 17:19 PM on 1 March, 2012(over-attributed to larger frequencies within the timeframe thatof the period under analysis, with no predictive power whatsoever - correlation without causation, unable to make any predictions as it is not based upon anyhappento somewhat match selected astronomical periods)physicsrelated to the system. That's fine as long as the inputs/forcings do not change, but cannot predict future(or past, as shown above)behavior if they do, for example as CO2 levels change. I'll note that the quadratic term in Scafetta's most recent work(oddly without emphasis in his papers)does roughly correspond to CO2 forcing. With no conclusions drawn, or analysis applied...Where is your skepticism?KRat 17:27 PM on 1 March, 2012(for reasons not clearly justified, or understood here, as it leads to a worse fit)he changes to cyclic+linear again. Either way, his model fails in backprojection, includes no physics, but is simply correlation without causation, and hencehas no predictive power.Physics: Good predictions Good stats: Reasonable predictions ifSince CO2 forcings are constantly changing, ENSO, solar, and volcanic forcings are not correlated,nothing changesBad stats: Better off repeatedly flipping a coinScafetta's work is simply bad statistics.gallopingcamelat 17:38 PM on 1 March, 2012gallopingcamelat 17:54 PM on 1 March, 2012Response:[DB]

"The IPCC models suffer from confirmation bias because they assume that CO2 is responsible for most of the recent warming while ignoring the evidence that 50-80% is attributable to natural causes."Unsupported assertion. It is incumbent upon you to now provide links to peer-reviewed articles published in reputable journals that document that the evidence you mentions both exists and that the IPCC has ignored it.

You will be held accountable for the above statement.

Riccardoat 18:29 PM on 1 March, 2012Jose_Xat 00:59 AM on 2 March, 2012Jose_Xat 01:08 AM on 2 March, 2012KRat 02:07 AM on 2 March, 2012"The main failure has to do with no reasonable guesstimate for a trend curve going far back. This failure is independent of frequency analysis."I would greatly disagree. There are now about a dozen decent proxy reconstructions of temperature for the last thousand years or so - with no major disagreements. As shown in the comparison here, Scafetta's cycles divergedrasticallyoutside their training interval. They fail in hindcasting, which provides zero evidence that they will succeed in forecasting. Frequency analysiscanbe helpful in attribution and identification of causal relationships - but it cannot stand alone. You need to follow up by examining the physics. Scafetta performed a very basic frequency analysis on a certain period of one temperature record(not crosschecking against more than one temperature record, incidentally), made some very odd data processing choices(there's a frequency peak at ~4 years, which he, and then fits those frequencies to various astronomic periodsdoes not discuss- but he runs the temperature data through a 4-year smoothing, which eliminates it!)without a causal link. That's about as straightforward a case of Correlation without Causation as it gets. Going on the physics, on the other hand (as in Lean and Rind 2008, and in Foster and Rahmstorf 2011), including radiative physics: start with a causal link, examine the time evolution of the forcings for attribution, and from that determine the influence and weighting of various inputs - that has both hindcast and forecast capabilities. And, given that said attributionsdoaccount for the evolution of the temperature record given our knowledge of the forcings(within quite small variations), Occam's Razor indicates thatinvoking mysterious cyclic influences via unsupported linkages is both unnecessary and foolish.KRat 02:11 AM on 2 March, 2012Jose_Xat 03:01 AM on 2 March, 2012