Reflections on Monckton et al.'s 'Transience Fraction'

Guest essay by Joe Born

In Monckton et al., “Why Models Run Hot: Results from an Irreducibly Simple Climate Model,” the manner in which the authors used their so-called transience fraction raised questions in the minds of many readers. The discussion below tells why. It will show that the Monckton et al. paper obscures the various factors that should go into selecting that parameter, and it will suggest that the authors seem to have used it improperly. It will also explain why so many object to their discussion of electronic circuits.

The discussion below will not deal with how well the model performs or whether the Monckton et al. paper interpreted the IPCC reports correctly. It will be limited to basic feedback principles that are obvious to most engineers and to not a few scientists. But there are circumstances in which stating the obvious is helpful, and I believe that Monckton et al. have presented us one.

Equation 1 of the Monckton et al. paper provides us laymen with a handy back-of-the-envelope model by which we can perform sanity checks on things we hear about the climate system. If we concentrate on equilibrium values and assume that carbon dioxide is the only driver, we can drop the ‘s from that equation’s penultimate line and take the and parameters to be unity to obtain:

The expression on the right can be recognized as the solution to the equation illustrated by Fig. 1, namely, .

Here is a temperature-change response to the initial radiation imbalance , or “forcing,” that would result from a carbon-dioxide-concentration-caused optical-density increase. The optical-density increase raises the effective altitude—and, lapse rate being what it is, reduces the effective temperature—from which the earth radiates into space, so less heat escapes, and the earth warms. The ‘s represent departures from a hypothetical initial equilibrium state of zero net top-of-the-atmosphere radiation, and the forcing is considered to keep the same value so long as the increased carbon-dioxide concentration does, even if the consequent temperature increase has eliminated the initial radiation imbalance and thus returned the system to equilibrium.

Without any knock-on effects, or “feedback,” the response would simply be , where is a coefficient widely accepted to be approximately 0.32 . The forcing produced by a carbon-dioxide-concentration increase from to is stated by the last line of Monckton et al.’s Equation 1 to be , where it is widely accepted that , i.e., that a doubling of the CO2 concentration would cause a forcing of about .

So the model’s user can readily see the significance of the main controversial parameter, namely, the feedback coefficient , which represents knock-on effects such as those caused by the consequent increases in water vapor, the resultant reduction in lapse rate, etc. In particular, the user can see that if were positive enough to make close to unity—i.e., to make the right-hand-side expression’s denominator close to zero—the global temperature would be highly sensitive to small variations in various parameters.

Fig. 2 depicts this effect: approaches infinity as approaches , i.e., as approaches unity. (That plot omits values that exceed unity; for reasons we discuss below, Monckton et al.’s discussion of that regime in connection with electronic circuits is questionable.)

The quantities discussed so far are those that occur at equilibrium, i.e., in the condition that prevails after a given forcing has been constant for a long enough time that transient effects in the response have died out. To arrive at a value for times when the forcing has not been remained unchanged long enough to reach equilibrium, the model includes a “transience fraction” to represent the ratio that the response at time bears to the equilibrium value. Other subscript ‘s are added to indicate that for different times the various quantities’ effective values may differ. Finally, to arrive at the response to all forcings, a coefficient representing the ratio that carbon-dioxide forcing bears to all forcings is included:

As we mentioned above, the transience fraction is of particular interest. As the Monckton et al. paper’s Table 2 shows, the ratio that the response at time bears to its equilibrium value depends not only on time but also on the feedback coefficient . Of course, it would be too complicated for us to investigate how such a dependence arises in the climate models on which the IPCC depends. But we can get an inkling by so modifying the block diagram of Fig. 1 as to incorporate a simple “one-box” (single-pole) time dependence.

Fig. 3 depicts the resultant system. The bottom box bears the legend “,” which in some circles means that the rate at which that box’s output changes is the product of its input and a heat capacity . (The is the complex frequency of which Laplace transforms are functions, but we needn’t deal with that here; suffice it to say that division by in the complex-frequency domain corresponds to integration in the time domain.)

What the diagram says is that a sudden drop in the amount of radiation escaping into space causes the temperature response to rise as the integral of the stimulus divided by . That temperature rise both increases the radiation escape by and partially offsets that radiation escape by .

Now, Fig. 3 can justly be criticized for wildly conflating time scales; it does not reflect the fact that the speed with which the surface temperature would respond to optical density alone is much greater than, say, the speed of feedback due to icecap-caused albedo changes. But that diagram is adequate to illustrate certain basic feedback principles.

The output of the Fig. 3 system is a solution to the following equation:

For example, if equals zero before and it equals thereafter, that solution for is:

where . That is, the equilibrium value of is the same as it was before we added the time dependence, but the added time dependence shows that the equilibrium value is approached asymptotically.

Fig. 4 depicts the solution for several values of feedback coefficient . What it shows is that a greater feedback coefficient yields a higher temperature output .

Another way of looking at the response is to separate its shape from its amplitude, and that brings us to transience fraction , which is our principal focus. Fig. 5 depicts this quantity, which is the ratio at time of the response to its equilibrium value. That plot shows that, although greater feedback results in a greater equilibrium temperature, it also results in the equilibrium value’s being reached more slowly.

Of course, those plots give the relationship between current and equilibrium output only for our toy, one-box model. Monckton et al. instead employed the relationship set forth in their Table 2 and depicted by the dashed lines in Fig. 6 below. In a manner that their paper does not make entirely clear, they inferred the Table 2 relationship from a paper by Gerard Roe, who explored feedback and depicted in his Fig. 6 (similar to Monckton et al.’s Fig. 4) how a “simple advective-diffusive ocean model” responds to a step in forcing for various values of feedback coefficient.

As Fig. 6 above shows, the Monckton et al. values initially rise more quickly, but then approach unity more slowly, than the ones that result from our Fig. 3 one-box model. As to the specifics of his model, Roe merely referred to a pay-walled paper, but in light of his describing that model as having a “diffusive” aspect we might compare the Table 2 values with the behavior of, say, a semi-infinite slab’s surface, as Fig. 7 does. Except for the value, the curves are similar over the illustrated time interval, but the slab thermal diffusivity used to generate Fig. 7’s solid curves was about 2000 times that of water, so the nature of the Roe model remains a mystery. Monckton et al. may have had a reason for following Roe’s model choice instead of any other, but they did not share that reason with their readers. For all we can tell, that choice was arbitrary.

More troubling, though, was the fact that they chose only a single transience-fraction curve for each value of total feedback, whereas we would expect that the curve would additionally depend on other factors. Let’s return to a simple lumped-parameter model like Fig. 3 to discuss what some of those factors may be.

Recall that in Fig. 3 the two feedback boxes were the same except for their values and ; the feedbacks they represent did not operate over different time scales. But the IPCC models can be expected to employ feedbacks that operate with different delays. Feedback effects such as water vapor may act quickly, whereas the albedo effects of melting icecaps may become manifest only over long time intervals.

To illustrate such a difference, we divide the feedback represented by Fig. 3’s upper feedback box into two portions, as Fig. 8 illustrates: and , . The legend in the uppermost box means that its output asymptotically approaches times the input with a time constant of . In other words, if that box’s input were a step from zero to at time zero, its output would be at time .

Now we’ll compare the responses of Fig. 8-type systems that differ not only in feedback but also in the portion of the feedback that operates with a greater delay. Fig. 9 compares such different systems’ responses, and we see that, as we expect, the magnitude of the higher-feedback system’s response is greater. In contrast to what we saw before, though, Fig. 10’s comparison of the systems’ curves shows that it is the higher-feedback system that responds more quickly. This tells us that the curve depends not only on the value of total feedback but also on the nature of that feedback’s particular constituents. And it raises the question of what feedback-speed mix Monckton et al. assumed.

Or maybe it raises the question of just how simple their model is to use. Let’s return to that model and note the dependencies on :

Of the five subscript ’s, three simply represent the time dependence of the stimulus or response, leaving the subscripts on the feedback coefficient and transience fraction to represent the time dependence of the model itself. That equation might initially suggest a rather complicated relationship: transience fraction depends not only on time but also on the feedback coefficient—which itself depends on time.

But Monckton et al.’s Table 2 suggests that the relationship not quite as convoluted as all that: the transience fraction actually depends not on time-variant feedback but only on , the value that the feedback coefficient reaches after all feedbacks have completely kicked in. One may therefore speculate that, although the transience-fraction function depends on the feedback’s ultimate value, that function was not intended to account for feedback time variation; one might speculate that the feedback time function serves that purpose.

But that would make the §4.8 discussion of the transience fraction puzzling, since it begins with the observation that “feedbacks act over varying timescales from decades to millennia” and goes on to explain that “the delay in the action of feedbacks and hence in surface temperature response to a given forcing is accounted for by the transience fraction .” So Monckton et al. did not make it clear just where the feedback’s time variation should go. Also, separating the feedback’s final value from its time variation in the manner we just considered doesn’t work out mathematically, particularly in the early years of the stimulus step.

And that brings us to another problem. Note that the forcing used as the stimulus by the Roe paper from which Monckton et al. obtained their Table 2 values was a step function: the forcing took a single step to a new value at and then maintained that value. That’s the type of stimulus we have tacitly assumed in the discussion so far. But the CO₂ forcing in real life has been more of a ramp than a step, so we would expect the function to differ from what we have considered previously.

In Fig. 11 the dotted curves represent step and ramp stimuli, while the solid curves represent a common system’s corresponding curves. Obviously, the values are lower for the ramp response than for the step response.

For all that is apparent, though, Monckton et al. failed to make this distinction. In their §7 and Table 4 they appear to use the step-response values of their Table 2 to model the response to a forcing that rose between 1850 and the present, and that forcing was not a step; it was more like a ramp. The Table 2 values could have been used properly, of course, by convolving them with the forcing’s time derivative, but nothing in the Monckton et al. paper suggested employing such an approach—which, in any event, does not lend itself well to pocket-calculator implementation.

Moreover, it’s not clear how Monckton et al.’s §7 statement that “the 0.6 K committed but unrealized warming mentioned in AR4, AR5 is non-existent” was arrived at. That section refers to their Table 4, which shows that the values computed for the model result from multiplication by a transience fraction , supposedly taken from Table 2. The values 0.7, 0.6, and 0.5 respectively given in Table 4 for f values 1, 1.5, 2.2 suggest that in fact the central estimate leaves (1 – 1/0.6) * 0.8 = 0.53 K of warming yet to be realized.

So Monckton et al. have chosen a family of curves based on a model cited by Roe that for all they’ve explained is no better than the toy models of Figs. 3 and 8. Those curves apparently result from applying to that model a step stimulus rather than the more ramp-like stimulus that carbon-dioxide enrichment has caused. Although their discussion did refer to the fact that some feedbacks operate more slowly than others, they did not clearly tell where to incorporate the mix of feedback speeds to be assumed. And, as we just observed, it’s not clear that they properly used the transience-fraction curves they did choose in concluding that “the 0.6 K committed but unrealized warming mentioned in AR4, AR5 is non-existent.” In short, their selection and application of values are confusing.

That doesn’t mean that their model lacks utility. If one keeps in mind that various factors Monckton et al. do not discuss affect the curve, their model can afford insight into various effects that we laymen hear about. In particular, it can help us assess the plausibility of various claimed feedback levels. A particularly effective use of the model is set forth in their §8.1. If the authors’ representation of IPCC feedback estimates is correct, their model helps us laymen appreciate why the IPCC’s failure to reduce its equilibrium climate-sensitivity estimate requires explanation in the face of reduced feedback estimates. And note that §8.1 doesn’t depend on at all.

Despite the confusion caused by Monckton et al.’s discussion of , therefore, Monckton et al. have provided a handy way for us laymen to perform sanity checks. And their model helps us understand their reservations regarding the plausibility of significantly positive feedback. It shows that, generally speaking, one would expect high positive feedback to cause relatively wild swings, whereas the earth’s temperature has remained within a narrow range for hundreds of thousands of years.

Unfortunately, they compromised their argument’s force with an unnecessary discussion of electronics that did more to raise questions than to persuade. Specifically, their paper says:

“In Fig. 5, a regime of temperature stability is represented by , the maximum value allowed by process engineers designing electronic circuits intended not to oscillate under any operating conditions.”

Although that lore may make sense in some contexts, it’s quite arbitrary; parasitic reactances and other effects can result in unintended oscillation even in amplifiers designed to employ negative values of . Even worse is the following:

“Also, in electronic circuits, the singularity at , where the voltage transits from the positive to the negative rail, has a physical meaning: in the climate, it has none.”

And Lord Monckton expanded upon that theme as follows

“Thus, in a circuit, the output (the voltage) becomes negative at loop gains >1.”

Although one can no doubt conjure up a situation in which such a result would eventuate, it’s hardly the inevitable consequence of greater-than-unity loop gains. To see this, consider the circuit of Fig. 12.

The amplifier in that drawing generates an output that within the amplifier’s normal operating range equals the product of its open-loop gain and the difference between signals received at its inverting (-) and non-inverting (+) input ports. In the illustrated circuit the non-inverting input port receives a positive fraction of the output from a resistive voltage-divider network so that, in the absence of the capacitor, the non-inverting port’s input would be . A negative voltage at the inverting port would result in a positive output voltage, which, because it is positively fed back, would tend to make the output even more positive than the open-loop value

Now, that is not a particularly typical use of feedback. More typically feedback is designed to be negative because the open-loop gain undesirably depends on the input—i.e., the amplifier is nonlinear—yet for (and typically is much, much larger than the value 5 we use below for purposes of explanation), the closed-loop gain : the relationship is nearly linear despite the amplifier’s nonlinearity.

In principle, though, if is independent of input, there are no stray reactances to worry about, we are sure that the feedback coefficient will not change, the lark’s on the wing, etc., etc., then there is no reason why positive feedback cannot be used. If and , for example, the loop gain would be +0.5, which would make the output : that feedback would double the gain.

But in that example the loop gain is less than unity. What about , which makes negative? I.e., what about the situation in which Monckton et al. tell us that “in a circuit, the output (the voltage) becomes negative”? Well, despite what they say, it doesn’t necessarily become negative.

To see that, let’s change the feedback coefficient to 0.4 and keep amplifier open-loop gain equal to 5 so that the loop gain , i.e., so that the loop gain exceeds unity. And let’s make the inverting port’s input a time step from 0 volt to -0.1 volt. That value is inverted and amplified, the result appears as the output , and an attenuated version of that output appears at the non-inverting input port.

But propagation from output port to input port is not instantaneous, and, to enable us to observe what may happen during that propagation (and avoid tedious transmission-line math), we have exaggerated the inevitable time delay by placing a capacitor in the feedback circuit. (In a block diagram like those above, the legend on the feedback-circuit block would accordingly be ).

As Fig. 13’s top plot shows, the output is initially (-0.1)(-5) = 0.5 volt and then rises exponentially as the feedback operates. If the amplifier had no limits, the output would grow without bound; despite what Monckton et al. say about in electronic circuits, the output would not go negative.

But we have assumed for Fig. 13 that the amplifier does have limits: its output is limited to less than 15 volts. Accordingly, the output goes no higher than 15 volts even though the signal at the non-inverting input port still increases for a time after the output reaches that limit. Still, the output does not go negative.

We could characterize that effect as the total loop gain’s decreasing to just under unity, as the middle plot illustrates, or as the small-signal loop gain’s falling abruptly to zero, which the bottom plot shows. (That is, no input change that doesn’t raise above +3 volts would result in any output change at all.) No matter how we characterize it, though, the formula doesn’t apply in this case.

Why? Because it’s the solution to an equation that says the output is equal to the product of (1) the amplifier gain and (2) the sum of the input and a fraction of the output. And in the case that equation is never true: delay prevents from ever catching up to until the amplifier gain has so decreased that the loop gain no longer exceeds unity.

Now, none of this contradicts Monckton et al.’s main point. Increasingly positive loop gains make a system more sensitive to variations in parameters such as open-loop gain and feedback coefficient , so in light of the earth’s relatively narrow temperature range it’s unlikely that climate feedbacks are very positive—if they are positive at all. But the authors would have made their point more compellingly if they had avoided the circuit-theory discussion. And they would have made their model more accessible if their discussion of the transience fraction hadn’t raised so many questions.