This Climate Goes to Eleven
- Gerard H. Roe and Marcia B. Baker, "Why Is Climate Sensitivity So
Unpredictable?", Science 318
(2007): 629--632 [no free
- Abstract: Uncertainties in projections of future climate change
have not lessened substantially in past decades. Both models and observations
yield broad probability distributions for long-term increases in global mean
temperature expected from the doubling of atmospheric carbon dioxide, with
small but finite probabilities of very large increases. We show that the shape
of these probability distributions is an inevitable and general consequence of
the nature of the climate system, and we derive a simple analytic form for the
shape that fits recent published distributions very well. We show that the
breadth of the distribution and, in particular, the probability of large
temperature increases are relatively insensitive to decreases in uncertainties
associated with the underlying climate processes.
Roe and Baker's argument is simple but ingenious and compelling. The
climate system contains a lot of feedback loops. This means that the ultimate
response to any perturbation or forcing (say, pumping 20 million years of
accumulated fossil fuels into the air) depends not just on the initial
reaction, but also how much of that gets fed back into the system, which leads
to more change, and so on. Suppose, just for the sake of things being
tractable, that the feedback is linear, and the fraction fed back
is f. Then the total impact of a perturbation J is
J + Jf + Jf2 + Jf3 + ...
The infinite series of tail-biting feedback terms is in fact
, and so can be summed up if f
is less than 1:
is greater than 1, then the feedback goes off to infinity; I'll
come back to that.)
If we knew the value of the feedback f, we could predict the
response to perturbations just by multiplying them by 1/(1-f) —
call this G for "gain". What happens, Roe and Baker ask, if we do not
know the feedback exactly? Suppose, for example, that our measurements are
corrupted by noise --- or even, with something like the climate,
that f is itself stochastically fluctuating. The distribution of
values for f might be symmetric and reasonably well-peaked around a
typical value, but what about the distribution for G? Well, it's
nothing of the kind. Increasing f just a little increases
G by a lot, so starting with a symmetric, not-too-spread distribution
of f gives us a skewed distribution for G with a heavy right
To illustrate, here is a histogram made from drawing 10,000 random numbers
from a Gaussian distribution with mean 0.6 and a standard deviation of 0.1;
think of these as the values of f, the strength of the feedback.
It's pretty symmetric: the values in the bottom quarter run from about 0.25 to
0.53, a spread of 0.28, and those in the top quarter run from 0.67 to 0.96, a
spread of 0.29.
And here is the histogram of the corresponding values of G,
the over-all gain.
It's not symmetric at all: the bottom quarter of values run from 1.3 to 2.1, a
range of 0.8, but the top quarter runs from 3.0 to almost 26. The probability that the gain is substantially above
the median or even mean value is quite substantial, because of this skew to the right.
If we can reduce the uncertainty in f, then of course the
distribution of G will also narrow, but much more slowly than you
might think. Here is what things look like when the standard deviation
of f is cut by a half, to 0.05:
and the corresponding plot for G
The top quartile now starts at 2.7, rather than 3.0, and runs only up to 5,
rather than 26, but that just means that things might just be twice as bad as
the base-line projection, rather than ten times as bad. We would have to
incredibly accurately for the uncertainty in G
to be a
matter of "plus or minus", rather than "to within a factor of".
A few points seem worth making here.
- This is all assuming straightforward linear feedbacks, which are on the
whole positive. This means it applies to any feedback system. Very trivially,
it'll make a nice homework problem.
- Linear feedback is at best an approximation. It would be interesting to
see what happens if one includes nonlinear feedback, including the possibility
for temporarily runaway positive feedback.
(Snowball earth and
- Roe and Baker claim that the uncertainty in f is dominated by the
uncertainty in the least-certain feedback process, even if it doesn't make a
dominant contribution to the total feedback. This sounds like it assumes
linearity; I'd like to know whether it's still true in a strongly nonlinear
- Even if we're in an approximately linear feedback regime, there is a lot of
plausibility to the idea that the feedback factor is actually stochastically
fluctuating --- or least, that it fluctuates in response to forces which change
too fast to be systematically predicted from other parts of a climate model,
which comes to much the same thing.
- I can't see how adding fancy nonlinear
dynamics could make things any more predictable.
- Finally, the whole analysis can be repeated using confidence intervals
rather than random variables. It'd be very interesting to think about how to
frequentist prediction intervals here.
- I am not sure whether this qualifies as an instance of
the pattern "positive
feedback leading to highly skewed distributions". Maybe there's a common
mathematical core to this and e.g. Simon's work
distributions, but if so it's not leaping out at me.
In short: the fact that we will probably never be able to precisely predict
the response of the climate system to large forcings is so far from being a
reason for complacency it's not even funny.
Update, 27 November: I should have linked to the discussion
of this paper over
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