Skip to content

The global temperature is inherently unstable

April 7, 2010

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:

  1. 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)
  2. Humans have increased the amount of \textrm{CO}_2 in the atmosphere.  (See the fairly dramatic graph on the right, which shows \textrm{CO}_2 levels beginning to take off right at the onset of the industrial revolution)
  3. Throughout the last million years, \textrm{CO}_2 levels and global temperature have had a strong positive correlation.
  4. 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 \textrm{CO}_2 levels, and if all throughout history \textrm{CO}_2 levels and global temperature haven risen and fallen together, then doesn’t that mean that anthropogenic \textrm{CO}_2 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 \textrm{CO}_2 levels cause the temperature to increase, or whether an increase in temperature causes \textrm{CO}_2 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 \textrm{CO}_2 to be released into the atmosphere.  We just don’t know.  It’s true that \textrm{CO}_2 is a greenhouse gas, but it’s a relatively weak one, and it remains far from clear whether the increase in \textrm{CO}_2 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 P_{sun} (which should be averaged over a daily rotation and a yearly orbit) while the radiated power follows the Stefan-Boltzmann law: P_{rad} = \sigma T^4\sigma 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 P_{sun} and P_{rad} and then you could solve for the temperature of the moon’s surface, which is perfectly stable.  That is, the temperature T 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 f_b of the energy radiated by the Earth gets saved, so that the total power output of the earth is not P_{rad} but P_{rad} (1 - f_b).  The fraction of the radiation f_b that gets “blocked” by the atmosphere depends in general on the thermal properties of all its constituent gases, so in general f_b is some complicated function of temperature f_b(T).

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.

The absorptivity of water

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 P_{sun} we get from the sun is unaffected by temperature, the fraction f_b that we retain is strongly temperature-dependent.

The fraction f_b can be estimated by a couple of arguments about evaporation and vapor pressure, but I will just present the following as empirical statements:

f_b \propto p_v(T)^{1/2},

p_v(T) \propto T^{3/2} \textrm{exp}(-\Delta H/k_B T),

where p_v is the vapor pressure of water and \Delta H is the enthalpy of evaporation of a water molecule.  The value of f_b 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 dT/dt, so finally we get that the temperature of the Earth is governed by

C_p \cdot dT/dt = P_{sun} - \sigma T^4 (1 - f_b(T)).

Here, C_p is the heat capacity (per unit area) of the Earth.  With a few rough estimates, we can plot the rate of temperature change dT/dt as a function of average temperature T:


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 T would be unchanging (dT/dt = 0).  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 dT/dt would be positive, which means the temperature would increase until it hit 6 degrees.  Similarly, if T were between 6 and 17 degrees, then dT/dt 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 f_b > 0.7 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.

14 Comments leave one →
  1. Jason A. permalink
    April 7, 2010 11:29 pm

    “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.”

    There’s a difference between honest dissent and denial/obstruction, though. I don’t think you’d argue that we’ll reach the truth faster if we stop ‘shouting down’ the people who think the universe is less than 10,000 years old, to use an extreme example.

    “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.”

    I don’t think this is a fair statement. What do you mean by weak? That there’s not very much of it compared to, say, water vapor? Water vapor accounts for at least 60% of the total greenhouse effect, but it’s a feedback rather than a forcing, like you said (precipitation). CO2’s contribution to the total greenhouse effect is something around 25-30% (while being a couple orders of magnitude less abundant than H2O). But it’s contribution to climate forcing (which drives the system to change) is much higher due to the slow carbon cycle, it hangs around in the atmosphere for a long time once you put it up there. It’s pretty clear that the current warming is due to a greenhouse gas forcing, since the lower atmosphere is warming while the upper is cooling, and CO2 is the most significant gas when it comes to forcing.

    It’s true that climate has changed many times without human intervention, it’s the speed of the change we’re seeing now that’s anomalous.

    It’s more than just trying to draw causation from correlation. It’s possible to estimate the climate sensitivity due to CO2, which is the amount of temperature increase associated with a given CO2 increase. Then you can take the amount of human-contributed CO2 in the atmosphere (by isotopic signature) and find the expected amount of warming. It agrees with observations.

    • gravityandlevity permalink*
      April 8, 2010 5:33 pm

      Thanks Jason, I appreciate your perspective. I didn’t mean to say that I don’t believe in global warming, or that I think it’s a political hoax. I just wanted to point out that our understanding of climate is primitive, and that complicated cycling is possible without human intervention.

      “It’s more than just trying to draw causation from correlation. It’s possible to estimate the climate sensitivity due to CO2, which is the amount of temperature increase associated with a given CO2 increase.”

      It was my understanding that this was still very much an open question, and that the “climate sensitivity to CO2” is still mostly beyond our ability to predict. If you have any good references that suggest otherwise, feel free to post them here.

    • Richard permalink
      June 5, 2010 7:52 am


      I don’t think that is right. As far as I can find about 95% of the greenhouse effect is from water vapour. This, and the small contribution of CO2, fits well with the absorption spectra of the two molecules. H2O absorbs strongly over a broad spectrum; CO2 absorbs strongly over a narrower band, mostly at the same frequencies as H2O. Since water vapour accounts for about 1% of the atmosphere on average, and CO2 only 0.038% this supports the idea of water as a really dominant greenhouse gas.

  2. yogi permalink
    April 8, 2010 5:27 am

    You make a rational argument which is definitely not new.

    But that doesn’t mean you won’t be crucified.

    Politics and science have always mixed to some degree but the climate warming debate has to be one of the worst cases in recent memory.

  3. April 8, 2010 4:42 pm

    Thank you for your very (undeservedly) complimentary discussion of my oversimplified little paper. The complete web reference is Some comments on climate, very much in the spirit of this blog entry, are in

  4. gravityandlevity permalink*
    April 8, 2010 5:36 pm

    A word of warning about the first graph, which shows CO2 levels: the bottom of the y-axis is not at zero. So what looks like a 4x increase in CO2 is really only about 30%.

  5. Michael B. permalink
    April 14, 2010 12:55 am

    Nice, clean, parsimonious model! And a lucid explanation of climate change.

    A colleague recently pointed me to this carbon bathtub model in National Geographic.

    It would be interesting to see these two models combined into a very simple but engaging explanation of how CO2 emissions affect climate change.

    • gravityandlevity permalink*
      April 14, 2010 11:30 am

      Thanks for the link; it’s a good, simple explanation of the basic perceived problem. That is, indeed, the reason why people are so nervous right now. We have inflow of CO2 into the atmosphere that is much larger than outflow, and this will be hard to reverse.

      But the problem, as I perceive it, is that no one knows exactly what this will imply. Will it make the temperature rise until we bake ourselves to death? Will it just push us to a different fixed point in the temperature (where, say, water vapor levels adjust to keep the temperature mostly constant)? Or are we already at a mostly-stable upper limit of the temperature?

      At the moment, I am willing to say that there is an urgent need for more climate research, and a need to begin preparing for different scenarios. But I am not willing to say that we are on the brink of disaster.

  6. brenatevi permalink
    April 17, 2010 4:09 pm

    Thank you for posting this. I think I’m going to shout this from the rooftops. I’ve always been a Global Warming doubter, although in certain circles I might possibly be considered a denier, simply because I don’t think the evidence is strong enough to say that we “beyond a shadow of doubt” causing global warming, as certain portion… no, let me change that, as a large portion of the scientific and political community wants us to believe.

    It’s nice seeing a scientific voice explaining that I’m not insane.

  7. May 27, 2010 3:13 am

    If only more people would hear this.

  8. Richard permalink
    June 5, 2010 8:06 am


    As for the certainty of there actually being any significant, consistent warming over the last century that is far from confirmed, it is even starting to look dubious. There are dreadful problems with the data used to make this claim, especially in the US, Canada, Russia and Australia. By chance these are some of the areas showing most warming.

    For example in the US 90% of the temperature stations are sited so as to give error bars larger than a degree. The effect being claimed is less than this. In Canada a single station is used to represent a vast area, most of the land mass in fact. This sensor has recently thrown up some errors, which are still included in the data set. They were only noticed because the errors were so large (and positive, I might add). In Russia only sites which showed warming were included in the data sets; the Russian government have themselves pointed out that if all data are included no warming is shown. In Australia all the data have all been adjust to show more warming. Those who did so refuse to explain why except in one case. In that case they simply adjust a temperature sensor that had been moved in elevation by the ISA lapse rate over that elevation change – a completely arbitrary adjustment as ISA is a standard for calibrating instruments and standardising experiments, not a representation of reality even in the free atmosphere, let alone near the ground, let alone in a sunny country near the coast!

    These four areas alone represent a huge land area, and most of the land area that is said to be warming. There are further problems with much of the data, lack of data, lack of precision, movement of stations, urban heat island effect and so on. So even your basic assumption, that the Earth warmed over the last century is not supported by sound evidence.

  9. coolstar permalink
    July 13, 2011 12:16 pm

    You really need to read and have more exposure to real climate modellers. There’s in fact almost ZERO doubt in the community that the increase of CO2 in the atmosphere has contributed to the rise in temperature. No reasonable interpretation of the science can come to any other conclusion (in fact, it’s impossible to explain how such an increase would NOT lead to a global rise in temperature). Sometimes correlation DOES mean there’s a causal relationship. The cliche’s is valid only when there’s no plausible physical mechanism for the correlation, such as sunspot number and Dow Jones index. Your estimate of the “flopping” timescale is off by about three orders of magnitude. As clearly shown in the ice core data in your first plot, no flip-flop on a 100 year time scale has EVER been observed. Speaking as another professional physicist (of the astro variety) I’m very disappointed by this post.

    • gravityandlevity permalink*
      July 13, 2011 12:52 pm

      Hi coolstar. I didn’t mean to dispute that there is some causal relationship between CO2 in the atmosphere and global temperature. It’s just that, from what I’ve heard, it is far from clear that this relationship is strong enough on its own to account for the change we are actually seeing. If I am wrong about this (which is very possible), I would appreciate any explanation or references you can give.

      As far as the “flopping” time scale, I think this is a mistake of language. I didn’t mean to say that the period of the flopping is about 100 years. I meant to say that once a “flop” has begun the process of changing from low to high or vice-versa takes about 100 years. Once the upper/lower plateau is reached, it is indeed stable for about 100,000 years, as you say.


  1. See?! I’m not completely insane! « Bren’s Mental Dump

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )


Connecting to %s

%d bloggers like this: