September 9, 2010

This Week's Finds (Week 302)

John Baez

In "week301" I sketched a huge picture in a very broad brush. Now I'd like to start filling in a few details: not just about the problems we face, but also about what we can do to tackle them. For the reasons I explained last time, I'll focus on what scientists can do.

As I'm sure you've noticed, different people have radically different ideas about the mess we're in, or if there even is a mess.

Maybe carbon emissions are causing really dangerous global warming. Maybe they're not — or at least, maybe it's not as bad as some say. Maybe we need to switch away from fossil fuels to solar power, or wind. Maybe nuclear power is the answer, because solar and wind are intermittent. Maybe nuclear power is horrible! Maybe using less energy is the key. But maybe boosting efficiency isn't the way to accomplish that.

Maybe the problem is way too big for any of these conventional solutions to work. Maybe we need carbon sequestration: like, pumping carbon dioxide underground. Maybe we need to get serious about geoengineering — you know, something like giant mirrors in space, to cool down the Earth. Maybe geoengineering is absurdly impractical — or maybe it's hubris on our part to think we could do it right! Maybe some radical new technology is the answer, like nanotech or biotech. Maybe we should build an intelligence greater than our own and let it solve our problems.

Maybe all this talk is absurd. Maybe all we need are some old technologies, like traditional farming practices, or biochar: any third-world peasant can make charcoal and bury it, harnessing the power of nature to do carbon sequestration without fancy machines. In fact, maybe we need go back to nature and get rid of the modern technological civilization that's causing our problems. Maybe this would cause massive famines. But maybe they're bound to come anyway: maybe overpopulation lies at the root of our problems and only a population crash will solve them. Maybe that idea just proves what we've known all along: the environmental movement is fundamentally anti-human.

Maybe all this talk is just focusing on symptoms: what we need is a fundamental change in consciousness. Maybe that's not possible. Maybe we're just doomed. Or maybe we'll muddle through the way we always do. Maybe, in fact, things are just fine!

To help sift through this mass of conflicting opinions, I think I'll start by interviewing some people.

I'll start with Nathan Urban, for a couple of reasons. First, he can help me understand climate science and the whole business of how we can assess risks due to climate change. Second, like me, he started out working on quantum gravity! Can I be happy switching from pure math and theoretical physics to more practical stuff? Maybe talking to him will help me find out.

So, here is the first of several conversations with Nathan Urban. This time we'll talk about what it's like to shift careers, how he got interested in climate change, and issue of "climate sensitivity": how much the temperature changes if you double the amount of carbon dioxide in the Earth's atmosphere.

JB: It's a real pleasure to interview you, since you've successfully made a transition that I'm trying to make now — from "science for its own sake" to work that may help save the planet.

I can't resist telling our readers that when we first met, you had applied to U.C. Riverside because you were interested in working on quantum gravity. You wound up going elsewhere... and now you're at Princeton, at the Woodrow Wilson School of Public and International Affairs, working on "global climate change from an Earth system perspective, with an emphasis on Bayesian data-model calibration, probabilistic prediction, risk assessment, and decision analysis". That's quite a shift!

I'm curious about how you got from point A to point B. What was the hardest thing about it?

NU: I went to Penn State because it had a big physics department and one of the leading centers in quantum gravity. A couple years into my degree my nominal advisor, Lee Smolin, moved to the Perimeter Institute in Canada. PI was brand new and didn't yet have a formal affiliation with a university to support graduate students, so it was difficult to follow him there. I ended up staying at Penn State, but leaving gravity. That was the hardest part of my transition, as I'd been passionately interested in gravity since high school.

I ultimately landed in computational statistical mechanics, partly due to the Monte Carlo computing background I'd acquired studying the dynamical triangulations approach to quantum gravity. My thesis work was interesting, but by the time I graduated, I'd decided it probably wasn't my long term career.

During graduate school I had become interested in statistics. This was partly from my Monte Carlo simulation background, partly from a Usenet thread on Bayesian statistics (archived on your web page), and partly from my interest in statistical machine learning. I applied to a postdoc position in climate change advertised at Penn State which involved statistics and decision theory. At the time I had no particular plan to remain at Penn State, knew nothing about climate change, had no prior interest in it, and was a little skeptical that the whole subject had been exaggerated in the media ... but I was looking for a job and it sounded interesting and challenging, so I accepted.

I had a great time with that job, because it involved a lot of statistics and mathematical modeling, was very interdisciplinary — incorporating physics, geology, biogeochemistry, economics, public policy, etc. — and tackled big, difficult questions. Eventually it was time to move on, and I accepted a second postdoc at Princeton doing similar things.

JB: It's interesting that you applied for that Penn State position even though you knew nothing about climate change. I think there are lots of scientists who'd like to work on environmental issues but feel they lack the necessary expertise. Indeed I sometimes feel that way myself! So what did you do to bone up on climate change? Was it important to start by working with a collaborator who knew more about that side of things?

NU: I think a physics background gives people the confidence (or arrogance!) to jump into a new field, trusting their quantitative skills to see them through.

It was very much like starting over as a grad student again — an experience I'd had before, switching from gravity to condensed matter — except faster. I read. A lot. But at the same time, I worked on a narrowly defined project, in collaboration with an excellent mentor, to get my feet wet and gain depth. The best way to learn is probably to just try to answer some specific research question. You can pick up what you need to know as you go along, with help. (One difficulty is in identifying a good and accessible problem!)

I started by reading the papers cited by the paper upon whose work my research was building. The IPCC Fourth Assessment Report came out shortly after that, which cites many more key references. I started following new articles in major journals, whatever seemed interesting or relevant to me. I also sampled some of the blog debates on climate change. Those were useful to understand what the public's view of the important controversies may be, which is often very different from the actual controversies within the field. Some posters were willing to write useful tutorials on some aspects of the science as well. And of course I learned through research, through attending group meetings with collaborators, and talking to people.

It's very important to start out working with a knowledgeable collaborator, and I'm lucky to have many. The history of science is littered with very smart people making serious errors when they get out of their depth. The physicist Leo Szilard once told a biologist colleague to "assume infinite intelligence and zero prior knowledge" when explaining to him. The error some make is in believing that intelligence alone will suffice. You also have to acquire knowledge, and become intimately familiar with the relevant scientific literature. And you will make mistakes in a new field, no matter how smart you are. That's where a collaborator is crucial: someone who can help you identify flaws in arguments that you may not notice yourself at first. (And it's not just to start with, either: I still need collaborators to teach me things about specific models, or data sets, that I don't know.) Collaborators also can help you become familiar with the literature faster.

It's helpful to have a skill that others need. I've built up expertise in statistical data-model comparison. I read as many statistics papers as I do climate papers, have statistician collaborators, and can speak their own language. I can act as an intermediary between scientists and statisticians. This expertise allows me to collaborate with some climate research groups who happen to lack such expertise themselves. As a result I have a lot of people who are willing to teach me what they know, so we can solve problems that neither of us alone could.

JB: You said you began with a bit of skepticism that perhaps the whole climate change thing had been exaggerated in the media. I think a lot of people feel that way. I'm curious how your attitude evolved as you began studying the subject more deeply. That might be a big question, so maybe we can break it down a little: do you remember the first thing you read that made you think "Wow! I didn't know that!"?

NU: I'm not sure what was the first. It could have been that most of the warming from CO2 is currently thought to come from feedback effects, rather than its direct greenhouse effect. Or that ice ages (technically, glacial periods) were only 5-6 °C cooler than our preindustrial climate, globally speaking. Many people would guess something much colder, like 10 °C. It puts future warming in perspective to think that it could be as large, or even half as large, as the warming between an ice age and today. "A few degrees" doesn't sound like much (especially in Celsius, to an American), but historically, it can be a big deal — particularly if you care about the parts of the planet that warm faster than the average rate. Also, I was surprised by the atmospheric longevity of CO2 concentrations. If CO2 is a problem, it will be a problem that's around for a long time.

JB: These points are so important that I don't want them to whiz past too quickly. So let me back up and ask a few more questions here.

By "feedback effects", I guess you mean things like this: when it gets warmer, ice near the poles tends to melt. But ice is white, so it reflects sunlight. When ice melts, the landscape gets darker, and absorbs more sunlight, so it gets warmer. So the warming effect amplifies itself — like feedback when a rock band has its amplifiers turned up too high.

On the other hand, any sort of cooling effect also amplifies itself. For example, when it gets colder, more ice forms, and that makes the landscape whiter, so more sunlight gets reflected, making it even colder.

Could you maybe explain some of the main feedback effects and give us numbers that say how big they are?

NU: Yes, feedbacks are when a change in temperature causes changes within the climate system that, themselves, cause further changes in temperature. Ice reflectivity, or "albedo", feedback is a good example. Another is water vapor feedback. When it gets warmer — due to, say, the CO2 greenhouse effect — the evaporation-condensation balance shifts in favor of relatively more evaporation, and the water vapor content of the atmosphere increases. But water vapor, like CO2, is a greenhouse gas, which causes additional warming. (The opposite happens in response to cooling.) These feedbacks which amplify the original cause (or "forcing") are known to climatologists as "positive feedbacks".

A somewhat less intuitive example is the "lapse rate feedback". The greenhouse effect causes atmospheric warming. But this warming itself causes the vertical temperature profile of the atmosphere to change. The rate at which air temperature decreases with height, or lapse rate, can itself increase or decrease. This change in lapse rate depends on interactions between radiative transfer, clouds and convection, and water vapor. In the tropics, the lapse rate is expected to decrease in response to the enhanced greenhouse effect, amplifying the warming in the upper troposphere and suppressing it at the surface. This suppression is a "negative feedback" on surface temperature. Toward the poles, the reverse happens (a positive feedback), but the tropics tend to dominate, producing an overall negative feedback.

Clouds create more complex feedbacks. Clouds have both an albedo effect (they are white and reflect sunlight) and a greenhouse effect. Low clouds tend to be thick and warm, with a high albedo and weak greenhouse effect, and so are net cooling agents. High clouds are often thin and cold, with low albedo and strong greenhouse effect, and are net warming agents. Temperature changes in the atmosphere can affect cloud amount, thickness, and location. Depending on the type of cloud and how temperature changes alter its behavior, this can result in either positive or negative feedbacks.

There are other feedbacks, but these are usually thought of as the big four: surface albedo (including ice albedo), water vapor, lapse rate, and clouds.

For the strengths of the feedbacks, I'll refer to climate model predictions, mostly because they're neatly summarized in one place:

Section 8.6 of the Intergovernmental Panel on Climate Change Fourth Assessment Report, Working Group 1 (AR4 WG1).

There are also estimates made from observational data. (Well, data plus simple models, because you need some kind of model of how temperatures depend on CO2, even if it's just a simple linear feedback model.) But observational estimates are more scattered in the literature and harder to summarize, and some feedbacks are very difficult to estimate directly from data. This is a problem when testing the models. For now, I'll stick to the models — not because they're necessarily more credible than observational estimates, but just to make my job here easier.

Conventions vary, but the feedbacks I will give are measured in units of watts per square meter per kelvin. That is, they tell you how much of a radiative imbalance, or power flux, the feedback creates in the climate system in response to a given temperature change. The reciprocal of a feedback tells you how much temperature change you'd get in response to a given forcing.

Water vapor is the largest feedback. Referring to this paper cited in the AR4 WG1 report:

• Brian J. Solden and Isaac M. Held, An assessment of climate feedbacks in coupled ocean-atmosphere models, Journal of Climate 19 (2006), 3354-3360.

you can see that climate models predict a range of water vapor feedbacks of 1.48 to 2.14 W/m2/K.

The second largest in magnitude is lapse rate feedback, -0.41 to -1.27 W/m2/K. However, water vapor and lapse rate feedbacks are often combined into a single feedback, because stronger water vapor feedbacks also tend to produce stronger lapse rate feedbacks. The combined water vapor+lapse rate feedback ranges between 0.81 to 1.20 W/m2/K.

Clouds are the next largest feedback, 0.18 to 1.18 W/m2/K. But as you can see, different models can predict very different cloud feedbacks. It is the largest feedback uncertainty.

After that comes the surface albedo feedback. Its range is 0.07 to 0.34 W/m2/K.

People don't necessarily find feedback values intuitive. Since everyone wants to know what that means in terms of the climate, I'll explain how to convert feedbacks into temperatures.

First, you have to assume a given amount of radiative forcing: a stronger greenhouse effect causes more warming. For reference, let's consider a doubling of atmospheric CO2, which is estimated to create a greenhouse effect forcing of 4±0.3 W/m2. (The error bars represent the range of estimates I've seen, and aren't any kind of statistical bound.) How much greenhouse warming? In the absence of feedbacks, about 1.2±0.1 °C of warming.

How much warming, including feedbacks? To convert a feedback to a temperature, add it to the so-called "Planck feedback" to get a combined feedback which accounts for the fact that hotter bodies radiate more infrared. Then divide it into the forcing and flip the sign to get the warming. Mathematically, this is....

JB: Whoa! Slow down! I'm glad you finally mentioned the "Planck feedback", because this is the mother of all feedbacks, and we should have talked about it first.

While the name "Planck feedback" sounds technical, it's pathetically simple: hotter things radiate more heat, so they tend to cool down. Cooler things radiate less heat, so they tend to warm up. So this is a negative feedback. And this is what keeps our climate from spiralling out of control.

This is an utterly basic point that amateurs sometimes overlook — I did it myself at one stage, I'm embarrassed to admit. They say things like:

"Well, you listed a lot of feedback effects, and overall they give a positive feedback — so any bit of warming will cause more warming, while any bit of cooling will cause more cooling. But wouldn't that mean the climate is unstable? Are you saying that the climate just happens to be perched at an unstable equilibrium, so that the slightest nudge would throw us into either an ice age or a spiral of ever-hotter weather? That's absurdly unlikely! Climate science is a load of baloney!"

(Well, I didn't actually say the last sentence: I realized I must be confused.)

The answer is that a hot Earth will naturally radiate away more heat, while a cold Earth will radiate away less. And this is enough to make the total feedback negative.

NU: Yes, the negative Planck feedback is crucial. Without this stabilizing feedback, which is always present for any thermodynamic body, any positive feedback would cause the climate to run away unstably. It's so important that other feedbacks are often defined relative to it: people call the Planck feedback λ0, and they call the sum of the rest λ. Climatologists tend to take it for granted, and talk about just the non-Planck feedbacks, λ.

As a side note, the definition of feedbacks in climate science is somewhat confused; different papers have used different conventions, some in opposition to conventions used in other fields like engineering. For a discussion of some of the ways feedbacks have been treated in the literature, see:

• J. R. Bates, Some considerations of the concept of climate feedback, Quarterly Journal of the Royal Meteorological Society 133 (2007), 545-560.

JB: Okay. Sorry to slow you down like that, but we're talking to a mixed crowd here.

So: you were saying how much it warms up when we apply a radiative forcing F, some number of watts per square meter. We could do this by turning up the dial on the Sun, or, more realistically, by pouring lots of carbon dioxide into the atmosphere to keep infrared radiation from getting out.

And you said: take the Planck feedback λ0, which is negative, and add to it the sum of all other feedbacks, which we call λ. Divide F by the result, and flip the sign to get the warming.

NU: Right. Mathematically, that's

T = -F/(λ0+λ)


λ0 = -3.2 W/m2/K

is the Planck feedback and λ is the sum of other feedbacks. Let's look at the forcing from doubled CO2:

F = 4.3 W/m2.

Here I'm using values taken from Soden and Held.

If the other feedbacks vanish (λ=0), this gives a "no-feedback" warming of T = 1.3 °C, which is about equal to the 1.2 °C that I mentioned above.

But we can then plug in other feedback values. For example, the water vapor feedbacks 1.48-2.14 W/m2/K will produce warmings of 2.5 to 4.1 °C, compared to only 1.3 °C without water vapor feedback. This is a huge temperature amplification. If you consider the combined water vapor+lapse rate feedback, that's still a warming of 1.8 to 2.2 °C, almost a doubling of the "bare" CO2 greenhouse warming.

JB: Thanks for the intro to feedbacks — very clear. So, it seems the "take-home message", as annoying journalists like to put it, is this. When we double the amount of carbon dioxide in the atmosphere, as we're well on the road to doing, we should expect significantly more than the 1.2 degree Celsius rise in temperature than we'd get without feedbacks.

What are the best estimates for exactly how much?

NU: The IPCC currently estimates a range of 2 to 4.5 °C for the overall climate sensitivity (the warming due to a doubling of CO2), compared to the 1.2 °C warming with no feedbacks. See Section 8.6 of the AR4 WG1 report for model estimates and Section 9.6 for observational estimates. An excellent review article on climate sensitivity is:

• Reto Knutti and Gabriele C. Hegerl, The equilibrium sensitivity of the Earth's temperature to radiation changes, Nature Geoscience 1 (2008), 735-748.

I also recommend this review article on linear feedback analysis:

• Gerard Roe, Feedbacks, timescales, and seeing red, Annual Reviews of Earth and Planetary Science 37 (2009), 93-115.

But note that there are different feedback conventions; Roe's λ is the negative of the reciprocal of the Soden & Held λ that I use, i.e. it's a direct proportionality between forcing and temperature.

JB: Okay, I'll read those.

Here's another obvious question. You've listed estimates of feedbacks based on theoretical calculations. But what's the evidence that these theoretical feedbacks are actually right?

NU: As I mentioned, there are also observational estimates of feedbacks. There are two approaches: to estimate the total feedback acting in the climate system, or to estimate all the individual feedbacks (that we know about). The former doesn't require us to know what all the individual feedbacks are, but the second allows us to verify our physical understanding of physical feedback processes. I'm more familiar with the total feedback method, and have published my own simple estimate as a byproduct of an uncertainty analysis about the future ocean circulation:

• Nathan M. Urban and Klaus Keller, Probabilistic hindcasts and projections of the coupled climate, carbon cycle and Atlantic meridional overturning circulation system: a Bayesian fusion of century-scale observations with a simple model, Tellus A, July 16, 2010.

I will stick to discussing this method. To make a long story short, the observational and model estimates generally agree to within their estimated uncertainty bounds. But let me explain a bit more about where the observational estimates come from.

To estimate the total feedback, you first estimate the radiative forcing of the system, based on historic data on greenhouse gases, volcanic and industrial aerosols, black carbon (soot), solar activity, and other factors which can change the Earth's radiative balance. Then you predict how much warming you should get from that forcing using a climate model, and tune the model's feedback until it matches the observed warming. The tuned feedback factor is your observational estimate.

As I said earlier, there is no totally model-independent way of estimating feedbacks — you have to use some formula to turn forcings into temperatures. There is a balance between using simple formulas with few assumptions, or more realistic models with assumptions that are harder to verify. So far people have mostly used simple models, not only for transparency but also because they're fast enough, and have few enough free parameters, to undertake a comprehensive uncertainty analysis.

What I've described is the "forward model" approach, where you run a climate model forward in time and match its output to data. For a trivial linear model of the climate, you can do something even simpler, which is the closest to a "model independent" calculation you can get: statistically regress forcing against temperature. This is the approach taken by, for example:

• Piers M. de F. Forster and Jonathan M. Gregory, The climate sensitivity and its components diagnosed from Earth radiation budget data, Journal of Climate 19 (2006), 39-52.

In the "total feedback" forward model approach, there are two major confounding factors which prevent us from making precise feedback estimates. One is that we're not sure what the forcing is. Although we have good measurements of trace greenhouse gases, there is an important cooling effect produced by air pollution. Industrial emissions create a haze of aerosols in the atmosphere which reflects sunlight and cools the planet. While this can be measured, this direct effect is also supplemented by a far less understood indirect effect: the aerosols can influence cloud formation, which has its own climate effect. Since we're not sure how strong that is, we're not sure whether there is a strong or a weak net cooling effect from aerosols. You can explain the observed global warming with a strong feedback whose effects are partially cancelled by a strong aerosol cooling, or with a weak feedback along with weak aerosol cooling. Without precisely knowing one, you can't precisely determine the other.

The other confounding factor is the rate at which the ocean takes up heat from the atmosphere. The oceans are, by far, the climate system's major heat sink. The rate at which heat mixes into the ocean determines how quickly the surface temperature responds to a forcing. There is a time lag between applying a forcing and seeing the full response realized. Any comparison of forcing to response needs to take that lag into account. One way to explain the surface warming is with a strong feedback but a lot of heat mixing down into the deeper ocean, so you don't see all the surface warming at once. Or you can do it with a weak feedback, and most of the heat staying near the surface, so you see the surface warming quickly. For a discussion, see:

• Nathan M. Urban and Klaus Keller, Complementary observational constraints on climate sensitivity, Geophysical Research Letters 36 (2009), L04708.

We don't know precisely what this rate is, since it's been hard to monitor the whole ocean over long time periods (and there isn't exactly a single "rate", either).

This is getting long enough, so I'm going to skip over a discussion of individual feedback estimates. These have been applied to various specific processes, such as water vapor feedback, and involve comparing, say, how the water vapor content of the atmosphere has changed to how the temperature of the atmosphere has changed. I'm also skipping a discussion of paleoclimate estimates of past feedbacks. It follows the usual formula of "compare the estimated forcing to the reconstructed temperature response", but there are complications because the boundary conditions were different (different surface albedo patterns, variations in the Earth's orbit, or even continental configurations if you go back far enough) and the temperatures can only be indirectly inferred.

JB: Thanks for the summary of these complex issues. Clearly I've got my reading cut out for me.

What do you say to people like Lindzen, who say negative feedbacks due to clouds could save the day?

NU: Climate models tend to predict a positive cloud feedback, but it's certainly possible that the net cloud feedback could be negative. However, Lindzen seems to think it's so negative that it makes the total climate feedback negative, outweighing all positive feedbacks. That is, he claims a climate sensitivity even lower than the "bare" no-feedback value of 1.2 °C. I think Lindzen's work has its own problems (there are published responses to his papers with more details). But generally speaking, independent of Lindzen's specific arguments, I don't think such a low climate sensitivity is supportable by data. It would be difficult to reproduce the modern instrumental atmospheric and ocean temperature data with such a low sensitivity. And it would be quite difficult to explain the large changes in the Earth's climate over its geologic history if there were a stabilizing feedback that strong. The feedbacks I've mentioned generally act in response to any warming or cooling, not just from the CO2 greenhouse effect, so a strongly negative feedback would tend to prevent the climate from changing much at all.

JB: Yes, ever since the Antarctic froze over about 12 million years ago, it seems the climate has become increasingly "jittery":

As soon as I saw the incredibly jagged curve at the right end of this graph, I couldn't help but think that some positive feedback is making it easy for the Earth to flip-flop between warmer and colder states. But then I wondered what "tamed" this positive feedback and kept the temperature between certain limits. I guess that the negative Planck feedback must be involved.

NU: You have to be careful: in the figure you cite, the resolution of the data decreases as you go back in time, so you can't see all of the variability that could have been present. A lot of the high frequency variability (< 100 ky) is averaged out, so the more recent glacial-interglacial oscillations in temperature would not have been easily visible in the earlier data if they had occurred back then.

That being said, there has been a real change in variability over the time span of that graph. As the climate cooled from a "greenhouse" to an "icehouse" over the Cenozoic era, the glacial-interglacial cycles were able to start. These big swings in climate are a result of ice albedo feedback, when large continental ice sheets form and disintegrate, and weren't present in earlier greenhouse climates. Also, as you can see from the last 5 million years:

the glacial-interglacial cycles themselves have gotten bigger over time (and the dominant period changed from 41 to 100 ky).

As a side note, the observation that glacial cycles didn't occur in hot climates highlights the fact that climate sensitivity can be state-dependent. The ice albedo feedback, for example, vanishes when there is no ice. This is a subtle point when using paleoclimate data to constrain the climate sensitivity, because the sensitivity at earlier times might not be the same as the sensitivity now. Of course, they are related to each other, and you can make inferences about one from the other with additional physical reasoning. I do stand by my previous remarks: I don't think you can explain past climate if the (modern) sensitivity is below 1 °C.

JB: I have one more question about feedbacks. It seems that during the last few glacial cycles, there's sometimes a rise in temperature before a rise in CO2 levels. I've heard people offer this explanation: warming oceans release CO2. Could that be another important feedback?

NU: Temperature affects both land and ocean carbon sinks, so it is another climate feedback (warming changes the amount of CO2 remaining in the atmosphere, which then changes temperature). The ocean is a very large repository of carbon, and both absorbs CO2 from, and emits CO2 to, the atmosphere. Temperature influences the balance between absorption and emission. One obvious influence is through the "solubility pump": CO2 dissolves less readily in warmer water, so as temperatures rise, the ocean can absorb carbon from the atmosphere less effectively. This is related to Henry's law in chemistry.

JB: Henry's law? Hmm, let me look up the Wikipedia article on Henry's law. Okay, it basically just says that at any fixed temperature, the amount of carbon dioxide that'll dissolve in water is proportional to the amount of carbon dioxide in the air. But what really matters for us is that when it gets warmer, this constant of proportionality goes down, so the water holds less CO2. Like you said.

NU: But this is not the only process going on. Surface warming leads to more stratification of the upper ocean layers and can reduce the vertical mixing of surface waters into the depths. This is important to the carbon cycle because some of the dissolved CO2 which is in the surface layers can return to the atmosphere, as part of an equilibrium exchange cycle. However, some of that carbon is also transported to deep water, where it can no longer exchange with the atmosphere, and can be sequestered there for a long time (about a millennium). If you reduce the rate at which carbon is mixed downward, so that relatively more carbon accumulates in the surface layers, you reduce the immediate ability of the ocean to store atmospheric CO2 in its depths. This is another potential feedback.

Another important process, which is more of a pure carbon cycle feedback than a climate feedback, is carbonate buffering chemistry. The straight Henry's law calculation doesn't tell the whole story of how carbon ends up in the ocean, because there are chemical reactions going on. CO2 reacts with carbonate ions and seawater to produce bicarbonate ions. Most of the dissolved carbon in the surface waters (about 90%) exists as bicarbonate; only about 0.5% is dissolved CO2, and the rest is carbonate. This "bicarbonate buffer" greatly enhances the ability of the ocean to absorb CO2 from the atmosphere beyond what simple thermodynamic arguments alone would suggest. A keyword here is the "Revelle factor", which is the relative ratio of CO2 to total carbon in the ocean. (A Revelle factor of 10, which is about the ocean average, means that a 10% increase in CO2 leads to a 1% increase in dissolved inorganic carbon.)

As more CO2 is added to the ocean, chemical reactions consume carbonate and produce hydrogen ions, leading to ocean acidification. You have already discussed this on your blog. In addition to acidification, the chemical buffering effect is lessened (the Revelle factor increased) when there are fewer carbonate ions available to participate in reactions. This weakens the ocean carbon sink. This is a feedback, but it is a purely carbon cycle feedback rather than a climate feedback, since only carbonate chemistry is involved. There can also be an indirect climate feedback, if climate change alters the spatial distribution of the Revelle factor in the ocean by changing the ocean's circulation.

For more on this, try Section 7.3.4 of the IPCC AR4 WG1 report and Sections 8.3 and 10.2 of:

• J. L. Sarmiento and N. Gruber, Ocean Biogeochemical Dynamics, Princeton U. Press, Princeton, 2006.

JB: I'm also curious about other feedbacks. For example, I've heard that methane is an even more potent greenhouse gas than CO2, though it doesn't hang around as long. And I've heard that another big positive feedback mechanism might be the release of methane from melting permafrost. Or maybe even from "methane clathrates" down at the bottom of the ocean! There's a vast amount of methane down there, locked in cage-shaped ice crystals. As the ocean warms, some of this could be released. Some people even worry that this effect could cause a "tipping point" in the Earth's climate. But I won't force you to tell me your thoughts on this — you've done enough for one week.

Instead, I just want to make a silly remark about hypothetical situations where there's so much positive feedback that it completely cancels the Planck feedback. You see, as a mathematician, I couldn't help wondering about this formula:

T = -F/(λ0+λ)

The Planck feedback λ0 is negative. The sum of all the other feedbacks, namely λ, is positive. So what if they add up to zero? Then we're be dividing by zero! When I last checked, that was a no-no.

Here's my guess. If λ0+λ becomes zero, the climate loses its stability: it can drift freely. A slight tap can push it arbitrarily far, like a ball rolling on a flat table.

And if λ were actually big enough to make λ0+λ positive, the climate would be downright unstable, like a ball perched on top of a hill!

But all this is only in some linear approximation. In reality, other effects will eventually kick in. For example, even in the case of a runaway greenhouse effect (as on Venus), the temperature won't rise indefinitely. An object radiates power proportional to the fourth power of its temperature, so the Planck feedback will eventually step in and keep the Earth from heating up beyond a certain point.

NU: Yes, we do have to be careful to remember that the formula above is obtained from a linear feedback analysis. For a discussion of climate sensitivity in a nonlinear analysis to second order, see:

• I. Zaliapin and M. Ghil, Another look at climate sensitivity, Nonlinear Processes in Geophysics 16 (2010), 113-122.

JB: Hmm, there's some nice catastrophe theory in there — I see a fold catastrophe in Figure 5, which gives a "tipping point".

Okay. Thanks for everything, and we'll continue next week!

For more discussion go to my blog, Azimuth.

The significant problems we have cannot be solved at the same level of thinking with which we created them. - Albert Einstein

© 2010 John Baez