Guest essay by Joe Born
Lead author Christopher Monckton turned down my request for further information about how the Table 2 “transience fraction” values in Monckton et al., “Why Models Run Hot: Results from an Irreducibly Simple Climate Model,” were obtained from a Gerard Roe paper’s Fig. 6. Those values are implausible, so in this post we will follow Lord Monckton’s suggestion that we obtain our own values. We will then use the results to address some misconceptions about feedback and sensitivity to assumed stimuli.
The innovation in Monckton et al.’s model of the relationship between stimulus and response was its separating a linear model’s step response into magnitude and shape factors. Since by definition , this makes the relationship between feedback and equilibrium response value apparent because , where is the no-feedback value, styled the “Planck climate-sensitivity parameter” in their paper.
“Transience fraction” is the name they gave that shape component : it is a normalized step response. And Roe’s Fig. 6 depicts responses to a forcing step. So one might speculate that Monckton et al. obtained their Table 2 transience-fraction values by normalizing those Fig 6 values.
However, that graph contains only one or arguably three curves, whereas Monckton et al.’s Table 2 contains values for five. Rather than explain in response to my consequent inquiry how the authors might have accomplished this loaves-and-fishes feat, Lord Monckton simply stated: “The table was derived from a graph in Gerard Roe’s magisterial paper of 2009 on feedbacks and the climate. Far from obscuring anything, we had made everything explicit.”
Lord Monckton also issued an invitation: “If Mr Born disagrees with Dr Roe’s curve, he is of course entirely free to substitute his own.” I reluctantly accept that invitation, not because I know of any reason to reject (or accept) Dr. Roe’s curves but because the values that Monckton et al. inferred from them are implausible.
Fig. 1 depicts the step responses of which Monckton et al.’s Table 2 are normalized versions. That is, it shows the results of multiplying each row’s entries by the corresponding , where and is the value their table lists as . (Although Monckton et al. also use the notation “,” the values seem to be the only ones they use, so we will simply use as our feedback-coefficient symbol.)
Consider that plot’s various values for years. Suppose that 25 years after a step in forcing we have observed a 0.24 K temperature increase. By consulting that plot, we conclude that the feedback is a modest , implying an equilibrium climate sensitivity of 1.4 K. If the increase were 0.29 K instead, we would infer a rather larger feedback coefficient, which implies a larger equilibrium climate sensitivity, 2.0 K. But if the temperature increase were still greater, 0.31 K, we would infer—no feedback at all. And that implies equilibrium climate sensitivity of only 1.2 K.
How’s that again? A higher current temperature increase implies a lower equilibrium climate sensitivity?
Theoretically, that result is not completely impossible; if you make different system assumptions for different feedback levels, you can get results like that. But it is far from self-evident that the Roe paper from which Monckton et al. purportedly derived their Table 2 dictates so implausible a result. So, to use Monckton et al.’s model but employ more-plausible behavior, we resort to rolling our own version of Table 2.
To that end we will model the system in the manner that Fig. 2 depicts. We use the notation in the forward block as shorthand for its input-output relationship. Dr. Roe said his results came from a model that included diffusion, but for the sake of simplicity we’ll assume a lumped-parameter model. We arbitrarily adopt a third-order relationship between that block’s output and its input :
For reasons that will become apparent in due course we adopt the following coefficient values: , , , , and .
Fig. 3 depicts the response that such a system exhibits when its input is a unit step: it asymptotically approaches a linear increase without bound. This is what we would expect of a system that steadily receives heat but sheds no heat in response.
To obtain the step responses for the various levels of feedback in Monckton et al.’s Table 2, we add feedback, as our Fig. 2’s feedback block indicates. This is a simple matter of replacing with and replacing with for respective values of total-feedback coefficient :
Fig. 4 depicts the solutions to that equation for a unit step in forcing and shows that they approximate all Monckton et al. Table 2 values except the implausible entries. This is not entirely coincidental, since the open-loop equation’s coefficients were so chosen as to make the normalized solution to the corresponding closed-loop equation approximate the entries in one of Table 2’s rows.
The values that correspond to the Fig. 4 solutions take the form
For the feedback values in Monckton et al.’s Table 2, the coefficients are as follows, with the ’s given in years:
Recall that the total-feedback values we used to obtain those values were sums of the climate-science-style feedback values and the negative reciprocal of what Monckton et al. call the “Planck climate-sensitivity parameter.” Lord Monckton has <a href=http://wattsupwiththat.com/2015/03/16/where-the-complex-climate-models-go-wrong/#comment-1887123>objected</a> to my having done so: “The misinterpretation – commonplace, but wrong – is in his assumption that the Planck parameter is a feedback just like all the others. No. In fact, it plays a special role in the determination of climate sensitivity, is not summed with the true feedbacks but is, uniquely, instead multiplied by their sum, is separately multiplied by the initial forcing to determine zero-feedback warming, again unlike any of the true feedbacks. . . .”
That objection is largely conceptual rather than substantive. Response curves exactly the same as Fig. 4’s would have resulted if the calculations had been based on the block diagram of Fig. 5 rather than that of Fig. 2. Note that, as Lord Monckton prefers, the feedback-block legend in Fig. 5 is simply rather than . Note also that the legend on Fig. 5’s forward block is instead of : Figs. 2 and 5 have different relationships between forward-block output and input. Instead of the open-loop equation given above for , the equation for is the one that results from the above closed-loop equation when is set to zero: its step response is the solid black curve in Fig. 4.
Now, that response’s value is , so the loop gain is indeed : in a sense, as Lord Monckton says, the “Planck parameter . . . is not summed with the true feedbacks but is . . . instead multiplied by their sum.” But that characterization results merely from the way in which we defined the system and from the level of abstraction we used. The Fig. 5 diagram is merely a higher-level version of Fig. 6, which shows that Fig. 5’s forward block is itself a feedback system. And by the distributive property Fig. 6 is equivalent to Fig. 2, from which we derived our transience-fraction values. So, as far as Monckton et al.’s linearized model goes, little depends on whether the “Planck parameter” is thought of as a feedback quantity: Lord Monckton has objected to form, not substance.
It bears emphasis at this point that Lord Monckton was correct in saying that “in all runs of our model that concerned equilibrium sensitivity (and most of them did) [the transience fraction ] is simply unity”: the relationship between feedback and equilibrium sensitivity does not depend on Monckton et al.’s having gotten the transience fraction right. Moreover, Monckton et al. were modest in their claims for the model, referring to the “rough and ready fashion” in which it represented forcings and feedbacks and pointing out that the model’s use is “narrowly focused on determining the transient and equilibrium responses of global temperature to specified radiative forcings and feedbacks in a simplified fashion.”
The model is advisedly simplified, intended to be simple enough that the authors could commend its use to “those who have no access to or familiarity with the general-circulation models,” just “a working knowledge of elementary mathematics and physics.” And Lord Monckton quite understandably turned down requests for computer code on the basis that users could readily implement the model on a pocket calculator and thereby “be determining climate sensitivity more reliably than the [IPCC] in minutes.”
But the latter feature is the result of the fact that their model simply multiplies the time- value of the stimulus by the time- value of the step response rather than, as would be more conventional, convolving entire time sequences. And, at least as implemented, that feature appears to have resulted from failure to heed Mark Twain’s admonition that we “be careful to get out of an experience only the wisdom that is in it and stop there.”
In response to my observation that, as he put it, “we relied on a model generated by a step-function representing the effects of a sudden pulse in CO2 concentration rather than one in which concentration increases by little and little,” Lord Monckton admitted that they had indeed relied on a step stimulus. But he justified that approach with wisdom he had gotten out of a related experience:
“I had first come across the problem of stimuli occurring not instantaneously but over a term of years when studying the epidemiology of HIV transmission. My then model, adopted by some hospitals in the national health service, overcame the problem by the use of matrix addition, but sensitivity tests showed that assuming a single stimulus all at once produced very little difference compared with the time-smeared stimulus, merely displacing the response by a few years. Similar considerations apply to the climate.”
Even for “determining the transient and equilibrium responses of global temperature,” that experience may have had rather less wisdom than Lord Monckton supposed.
Fig. 7 illustrates transient climate response, i.e., the temperature increase that will have resulted at the time when CO2 concentration has doubled after increasing by 1% per year. The legend in the upper left gives the equilibrium climate sensitivities that Monckton et al.’s Table 2 feedback values imply: it gives the changes in temperature that would ultimately result if that doubled concentration remained in perpetuity. For the corresponding systems, whose step responses Fig. 4 depicts, Fig. 7’s solid curves represent the responses to an instantaneous CO2-concentration doubling, and its dashed curves represent corresponding responses to doubling over about 70 years ().
If after the 70 years in which CO2 concentration has doubled we observe the 1.7 K temperature change marked by an “x” on that the plot, then the equilibrium climate sensitivity inferred by the Monckton et al. approach, i.e., from multiplying the current stimulus value by the current step-response value, would be the 3.4 K associated with the solid curve closest to the marked value at 70 years. But the equilibrium climate sensitivity inferred from the more-conventional approach of convolving the stimulus’s derivative with respective step responses would be more like the 12.4 K associated with the closest dashed curve.
There are those who would consider such a difference significant.
Again, this does not mean that the Monckton et al. model is completely unworkable. In particular, I know of no specific problem with the results it gives for the , i.e., unity-transience-fraction, case. Moreover, although we have seen that the Monckton et al. approach has significant problems in principle, it would be hard to tell without a more-detailed account of underlying data and assumptions how significantly those problems may have affected the numerical results at which Monckton et al. arrived in practice.
Also, although we have seen that the difference between Monckton et al.’s simple multiplication and the more-accurate, conventional convolution approach can be significant, a simple-multiplication approach may still be justified if the expected forcing trajectories are obliging enough. If the trajectories that the forcings follow are expected to be roughly linear, for example, one could adopt values based on a ramp response rather than the step response from which Monckton et al. obtained theirs.
But it is not clear that those conditions apply. In short, Monckton et al.’s model has limitations of which “the user manual for the simple model, bringing it within the reach of all who have a working knowledge of elementary mathematics and physics” omits warnings the reader could have profited from. At the very least, users should exercise caution when they use the model’s values.