The global temperature is inherently unstable
Let me make an immediate apology about the title of this post: it would be more correct to have called it “the global temperature is inherently bistable” . But it seems like bad form to use terms in the title that may not be immediately understood by your audience.
Up until quite recently, I must admit, I have completely shied away from any discussions about global warming. It has been such an emotionally- and morally-charged debate during the past few years that only very rarely is a fact presented without an accompanying censure or call to arms. This creates a particularly bad environment for the advancement of scientific debate, where dissenters are immediately shouted down because of a strong political/social pressure to reach consensus (not that I can’t see why; there is much more urgency to global warming than to most scientific riddles). The problem is that the fundamental mechanism through which scientific ideas advance is disagreement. If scientists aren’t allowed to argue with each other, then they arrive at the truth much more slowly.
Of course, the biggest barrier for me personally was that I didn’t know anything reliable about global warming. So a few weeks ago, while I was at the American Physical Society meeting in Portland, I went to a couple of hours’ worth of talks about global warming and climate research. The one that interested me most was a talk by Jonathan Katz from Washington University. The main purpose of his talk was to discuss the so-called “geoengineering” solutions to global warming. But along the way he made an interesting point that I’d like to repeat: the global temperature is inherently unstable. That is, even without human intervention, the global temperature has more than one fixed point, and over time it will flip back and forth from one to the other.
I certainly don’t mean to present myself as any kind of climate expert (I am obviously far from it), but for the first time I find myself capable of presenting a coherent, quantitative argument about climate change and I think it’s worth sharing. Most of what I’m going to say in this post is outlined in this fairly simple paper on the arxiv.
The short list of things we definitely know about global warming
The first thing that Katz did in his talk was review the climate change facts about which we are certain. It was a pretty short list, and it basically goes like this:
- The global temperature has been rising over the last hundred years. (On a year-to-year or decade-to-decade scale, though, fluctuations in temperature are too strong to reveal any trends)
- Humans have increased the amount of in the atmosphere. (See the fairly dramatic graph on the right, which shows levels beginning to take off right at the onset of the industrial revolution)
- Throughout the last million years, levels and global temperature have had a strong positive correlation.
- There has been no increase in the rate of natural disasters in the last half-century.
Point #4 isn’t really about climate change, but I include it just to refute one of the more irresponsible claims you hear sometimes: namely, that man-made global warming has caused more hurricanes and earthquakes and tsunamis and such. It isn’t true, because the rate of hurricanes and earthquakes and tsunamis has not been increasing.
The interesting points, however, are 2 and 3. If humans are definitely increasing levels, and if all throughout history levels and global temperature haven risen and fallen together, then doesn’t that mean that anthropogenic emissions are driving the global temperature increase?
Not necessarily. I don’t mean to preach, but no one graduates from high school without hearing that “correlation does not imply causation”. In other words, we don’t know whether levels cause the temperature to increase, or whether an increase in temperature causes levels to go up. It may be that there is some entirely different mechanism which causes the global temperature to change, and that increases in global temperature cause more to be released into the atmosphere. We just don’t know. It’s true that is a greenhouse gas, but it’s a relatively weak one, and it remains far from clear whether the increase in we are seeing could be responsible for the rise in temperature.
What we do know, however, is that the global temperature has fluctuated quite a bit without any human intervention. Below is a graph of the global average temperature during the last half-million years or so. You’ll see that it tends to fluctuate back and forth between two values: something about six degrees colder than the present (an “ice age”) and something that is perhaps a few degrees warmer (a “hot interglacial”). In this sense the global temperature is “bistable”: it tends to flop back and forth from one level to another.
What Katz suggested in his talk, and what I’m going to explain here, is a straightforward (if extremely simplified) mechanism by which this kind of behavior can be explained. What it implies is that the climate can make dramatic changes all on its own, and we need to be prepared for these changes even if we are not the ones causing them.
How the Earth gets and retains heat: a simplified model
Writ large, the heating and cooling of the Earth is actually pretty simple. The Earth gets all its heat from sunlight and it loses all its heat by radiating into space. So the power input to a unit area of the Earth’s surface is some constant (which should be averaged over a daily rotation and a yearly orbit) while the radiated power follows the Stefan-Boltzmann law: ; is called the Stefan-Boltzmann constant.
If you were describing the temperature of, say, the moon, then the problem of planetary temperature could be solved by this simple description alone. You would just need to equate and and then you could solve for the temperature of the moon’s surface, which is perfectly stable. That is, the temperature of the moon is the one at which the power it radiates into space is equal to the power it absorbs from the sun, so that the surface temperature of the moon remains constant.
In describing the Earth, however, there is the complication of the “greenhouse effect”. That is, the Earth’s atmosphere retains some of that radiated heat rather than sending it all back into space. In other words, some fraction of the energy radiated by the Earth gets saved, so that the total power output of the earth is not but . The fraction of the radiation that gets “blocked” by the atmosphere depends in general on the thermal properties of all its constituent gases, so in general is some complicated function of temperature .
Nonetheless, a simple description of the blocking of heat by the atmosphere can be concocted by considering only the strongest of the greenhouse gases: water vapor. We don’t generally talk about water vapor as a greenhouse gas because its presence has nothing to do with human activity. Even if humans decided to pump huge amounts of water vapor into the atmosphere, the amount of water vapor in the atmosphere would quickly equilibrate by precipitation. Nonetheless, water vapor is by far the gas which is most responsible for absorbing radiated heat from the Earth. So from here on I’m going to ignore all other gases and focus on absorption of heat by water vapor.
It is a strange fact about water that it is almost completely transparent to sunlight, while it is pretty opaque to most other wavelengths of radiation (see the graph above). This is sort of a sobering thought: it means that if our eyes were adapted to see at pretty much any other range of wavelengths, then a glass of water (and the atmosphere itself) would look completely opaque. What it implies for climate, though, is that the amount of water vapor in the air has almost no effect on the amount of sunlight that reaches the Earth’s surface. The peak of the solar radiation spectrum is at a wavelength of about 500 nm, squarely in the visible range, so sunlight passes through the atmosphere largely unhindered.
The radiation from the Earth’s surface, however, is at a very different wavelength (about 10 micrometers) so it is strongly affected by the amount of water vapor in the atmosphere. And the amount of water vapor in the atmosphere is dependent on the temperature. So while it’s fair to say that the energy we get from the sun is unaffected by temperature, the fraction that we retain is strongly temperature-dependent.
The fraction can be estimated by a couple of arguments about evaporation and vapor pressure, but I will just present the following as empirical statements:
where is the vapor pressure of water and is the enthalpy of evaporation of a water molecule. The value of at the Earth’s current average temperature (about 17 degrees Celsius) is about 0.3.
These statements can be added to our description of radiated power to make an estimate of the total power (input – output) received by the Earth as a function of temperature. The total power determines the rate of temperature change , so finally we get that the temperature of the Earth is governed by
Here, is the heat capacity (per unit area) of the Earth. With a few rough estimates, we can plot the rate of temperature change as a function of average temperature :
Bistability of the global climate
The graph above shouldn’t be taken as a serious numerical prediction, since there are a lot of rough estimates that go into it, but it does demonstrate some important features. First of all, it shows two points where the temperature would be unchanging (). The first is at about 6 degrees, and the second at about 17. While both of these are technically “fixed points”, only one of them is actually stable: the lower one. For example, imagine that the Earth’s average temperature were less than 6 degrees. Then would be positive, which means the temperature would increase until it hit 6 degrees. Similarly, if were between 6 and 17 degrees, then would be negative, which means that the temperature would fall until it hit 6 degrees. This is what I call an “ice age”: the temperature would be stable at some relatively low value. If the temperature is lower than 6, the radiation from the sun is larger than the radiation of the earth, and the temperature goes up. If the temperature is higher than 6 and lower than 17, then the radiation from the earth is larger than the radiation from the sun and the temperature goes down again.
But if the temperature rises above 17, then the existence of significant amounts of water vapor in the atmosphere allows the Earth to retain much of the energy it would have otherwise radiated into space. At such large temperatures, increasing the temperature also causes a substantial increase in the amount of water vapor, which ends up driving the temperature even higher. This means that if the temperature is above 17, then it will keep rising until the Earth settles into some relatively hot state, where most of the Earth’s radiated energy is blocked from escape by water vapor (corresponding to or so, where the model above breaks down). Such a hot temperature is also relatively stable, and is what I have called a “hot interglacial”.
This is the take-away message: that the Earth’s climate has two stable points. One, an “ice age”, is when the temperature is low and there is little retention of heat by the atmosphere. The second, a “hot interglacial”, is when the temperature is high but most of the heat that would be radiated away is saved by the atmosphere. The planet may assume either of these points, and the geological record seems to indicate that it tends to switch from one to the other every fifty thousand years or so. The exact cause of the “switching” is probably too sophisticated to be understood at the level I’m exploring here, but I can imagine that some large, unpredictable event like a volcanic eruption or a meteorite strike might provide enough of a disturbance to push the planet from a cold era to a hot era, or vice-versa.
Being prepared for both extremes
In his talk at the APS meeting, Katz explained the conclusion above and used it to justify the need for “geoengineering” solutions to global warming — that is, artificial manipulation of the atmosphere in order to reduce retention of thermal radiation or increase reflectivity to sunlight. I don’t know enough to advocate for or against any particular approach, or even to tell how urgently such solutions are needed (the graph above suggests that the process of “flopping” from one fixed point to another would take about 100 years). But I can say this much: if humans stick around long enough, they will need to be prepared for dramatic changes in the climate, whether or not they are the cause of it.