By Joe Born

1. Introduction

By presenting actual calculation results from a specific feedback-system example, the plots below will put some graphical meat on the verbal bones of Nick Stokes’ recent “Demystifying Feedback” post.

I heartily agree with the main message I took from Mr. Stokes’ post: although some climate equations may be similar to certain equations encountered in, say, electronics, it’s not safe to import electronics results that the climate equations don’t intrinsically dictate. But I’m less convinced that Mr. Stokes succeeded in removing the mystery from feedback. I’m reminded of what a professor said over a half century ago in one of those compulsory science courses: “Don’t just scope it out; work it out.”

What the professor meant is that we humans tend to overestimate our abilities to intuit an equation’s implications. Actual calculations routinely reveal that the equation doesn’t mean what we had thought. That can be true even of equations as simple as the equilibrium scalar feedback equation at issue here.

Except for folks who have significant experience in working through feedback systems, for example, readers may not take as much meaning as might be hoped from abstract statements such as the following:

“One thing that is important is that you keep the sets of variables separate. The components of x₀ satisfy a state equation. The perturbation components satisfy equations, but are proportional to the perturbation. You can’t mix them. This is the basic flaw in Lord Monckton’s recent paper.”

Working through an actual example could provide more insight. And an occasion to do just that is presented by Christopher Monckton’s claim that feedback theory imposes (what we’ll call) the entire-signal rule:

“[S]uch feedbacks as may subsist in a dynamical system at any given moment must perforce respond to the entire reference signal then obtaining, and not merely to some arbitrarily-selected fraction thereof.”

Critics like Mr. Stokes and Roy Spencer have disputed that rule. And, indeed, there are good practical reasons in climate science for treating feedback as something that’s responsive only to changes rather than to entire quantities. Yet, if we instead accept Lord Monckton’s entire-signal rule for the sake of argument and work through its implications, we can gain more insight into questions like what “you can’t mix them” really means.

So in what follows we’ll accept that rule and define an example feedback system in which the feedback responds to the entire output rather than only to perturbations. And we’ll observe the rule’s implications by working through the system’s responses to a range of inputs.

In the process we’ll juxtapose the small- and large-signal versions of metrics like “feedback fraction” and “system-gain factor” to reveal the latent ambiguities with which they afflict feedback discussions. We’ll also see examples of how easily the feedback equation, simple though it is, can be misinterpreted.

2. Background

First we’ll use the following plot to place Mr. Stokes’ post in context.

Lord Monckton views the climate system’s equilibrium temperature as a function of the value it would have if there were no feedback. By way of accepting for the sake of argument “all of official climatology except what we can demonstrate to be false,” Lord Monckton has adopted values we’ll call as the equilibrium temperature corresponding to 1850’s carbon-dioxide concentration and as what it would have been had there been no feedback. The point labeled “Pre-Industrial” in the plot above represents those values. The point labeled “IPCC Prediction” results from increasing those values by the “official climatology” ECS value, , and the value it would have had without feedback. (Equilibrium climate sensitivity (“ECS”) is the increment by which doubling carbon-dioxide concentration would increase the equilibrium global-mean surface temperature.)

Observational studies like Lindzen & Choi 2011 have led many of us to believe that ECS is actually much lower than that—if there really is such a thing as ECS. In a video that introduced his theory as a “mathematical proof” that ECS is low, though, Lord Monckton said of previous ECS-value arguments that they had “largely been a competition between conjectures.” He may agree with researchers like Lindzen & Choi, he said, but “they can’t absolutely prove that they’re right.” In contrast, “we think that what we’ve done here is to absolutely prove that we are right.”

By in essence projecting those points to the no-feedback, line, he eventually came to describe climatology’s error as failing to recognize that some of the feedback is response to the emission temperature. And he came to express his proof in the form of the above-mentioned entire-signal rule, of which he said:

“Once that point—which is well established in control theory but has, as far as we can discover, hitherto entirely escaped the attention of climatology—is conceded, as it must be, then it follows that equilibrium sensitivity to doubled CO2 must be low.”

In the passage quoted above, Mr. Stokes’ post contested that theory. So we’ve added a hypothetical curve to the plot to illustrate what high-ECS proponents might think. But to represent substantial feedback to the emission temperature the curve passes to the left of the emission-temperature point on the no-feedback, line in compliance with the entire-signal rule.

3. “Underlying Mathematics”

In a reply to Mr. Stokes’ post Lord Monckton diagrammed his version of the equilibrium scalar feedback equation. That equation seems simple enough, but he wrote of its block diagram that “one can only really understand how it works if one also has a grasp of the underlying mathematics.”

However, by treating the (counterfactual) no-feedback temperature as the input, Lord Monckton’s equation hides the underlying relationship between the output and forcing. To avoid the resultant loss of insight we’ll therefore deal primarily with the forcing relationship, but we’ll map the results to Lord Monckton’s counterfactual-temperature relationship.

For this purpose we’ll simply adopt the more-general notation that the system produces a response to a stimulus ; we won’t try to straddle his temperature-input notation and the conventional forcing-input notation . If we were simply to replace the conventional perturbation values with entire values, we’d thereby have:

That is, the output temperature would simply be the product of a gain and the sum of the input forcing and output-dependent feedback . (The notation for the ratio of feedback to output will be to distinguish it from Lord Monckton’s feedback fraction .) The following explicit expression for the equilibrium-temperature output would seem to follow from elementary algebra:

This is all seemingly simple. But even seemingly simple equations can be hard to interpret. Moreover, Lord Monckton’s theory requires that we deal with entire quantities rather than just small perturbations, so we can no longer ignore nonlinearities.

Therefore, since the formulation may suggest that on the contrary and are constants, we’ll so rewrite the scalar system’s equilibrium equation as explicitly to allow for nonlinearity. Specifically, we’ll replace with and with :

To map this forcing-input formulation to Lord Monckton’s temperature-input formulation , we will also calculate a without-feedback temperature , where denotes ‘s inverse function: . Lord Monckton’s and respectively correspond to and .

4. Example-System Functions

Now we reach specifics: we’ll define the functions and in our system equation .

In doing so we won’t attempt to match the actual climate relationship between equilibrium temperature and forcing. For one thing, no one really knows what that relationship is throughout the entire domain that Lord Monckton would have us acknowledge. To the extent that an equilibrium relationship does exist, moreover, temperature almost certainly isn’t a single-valued function unless that function’s argument is a vector of forcing components instead of the scalar total thereof we’re assuming here. (Those complications are among the reasons why focusing on perturbations is preferable.)

But the question that Lord Monckton’s purported mathematical proof raises isn’t whether we know the relationship; it’s whether, without knowing what that relationship is, high ECS values can be ruled out mathematically. So we’ll merely choose simple relationships that exhibit a high ECS value and watch for any contradictions of what Lord Monckton called “the mathematics of feedback in all dynamical systems, including the climate.”

4.1 Open-Loop Function

For our open-loop function we adopt a fractional-power relationship:

Note that with = 0.25 this function would be a fourth-root relationship reminiscent of the inverse Stefan-Boltzmann equation. As a nod toward the real-world difference between the surface temperature and the effective radiation temperature, however, we’ve instead adopted , with .

As we see, ‘s slope falls off with input :

Since relates an output to an input, the plot above refers to ‘s slope as a “gain.” Of particular importance to this discussion is that the plot shows both the average slope (large-signal gain) and the local slope (small-signal gain). Failure to distinguish between those quantities consistently has bedeviled discussions of Lord Monckton’s theory.

Note also that we refer to both quantities as “open-loop gain”: each is a gain that the system would exhibit if there were no feedback to “close the loop.” Perhaps confusingly—but I think logically—the discussion below uses similar expressions to refer to different quantities.

Specifically, closed-loop gain will refer to the gain that results when feedback does indeed “close the loop.” Lord Monckton sometimes calls this the “system-gain factor.” And just plain loop gain will be the internal gain encountered in traversing the loop: what Lord Monckton occasionally calls the “feedback fraction.” The loop gain results from combining the open-loop gain with the feedback ratio, which we will presently introduce in connection with the feedback function.

Again, the quantity to which our open-loop function relates the output temperature is forcing. If as Lord Monckton does we instead take the input to be the no-feedback temperature, the resultant open-loop function is just the input itself: ; if there were no feedback, the output would be the same as the input.

4.2 Feedback Function

Climatologists sometimes get media attention by speaking of a “tipping point.” But the particular feedback function we chose for the Fig. 1 curve wouldn’t cause one. Since the behavior it results in thereby lacks one of feedback’s more-interesting features, we’ll instead adopt the following feedback function, which causes a tipping point not far beyond the doubled-CO2 equilibrium temperature:

where and

Note that in our feedback-function choice we differ with Lord Monckton’s critics who object that feedback can only be a response to perturbations. Like the Fig. 1 curve’s feedback function, our example, tipping-point-causing function is responsive to the entire output. To be sure, the response seems to become significant in both cases only as the temperature approaches ice’s 273 K melting point; the response approaches zero as the output temperature does. But actually working through the resultant example-system behavior near absolute zero reveals that because of the high forward gain we saw in Fig. 3 the feedback is great enough to cause instability.

Furthermore, although the example function’s value at the doubled-CO~2~ temperature approximates that of the feedback function responsible for Fig. 1, it exceeds it at temperatures very much above or below that temperature. In particular, our chosen function’s feedback to the emission temperature will be even greater than the Fig. 1 function’s.

Now, I don’t really think either of those feedback functions is like the actual climate’s feedback function. I personally don’t think the actual climate has much net-positive feedback at all.

But that’s not the point. Lord Monckton claims to have developed a mathematical proof. That means showing that accepting a high ECS value for the sake of argument would lead to a contradiction with “the mathematics of feedback in all dynamical systems, including the climate.” So the point isn’t whether we believe that premise. It’s whether accepting the premise leads us to a contradiction. And we will search in vain for contradictions among the implications of a system’s exhibiting not only a high ECS value but also a tipping point.

The plot above shows the resultant feedback ratio, i.e., the quantity that multiplies the output or increment thereof to yield our feedback function’s corresponding feedback quantity. It’s the feedback function’s average (large-signal) slope or local (small-signal) slope , where the prime represents differentiation. Again we see that the large- and small-signal versions differ markedly; one does not approximate the other.

Different feedback-ratio functions, plotted above, result if we instead take Lord Monckton’s temperature-input view of the system. Those functions are average and local slopes of a different feedback function, of the function implied by .

The different views’ feedback ratios are somewhat similar at the higher temperatures we’re interested in, but their low-temperature behaviors are quite different. That doesn’t mean that the temperature-input view is wrong. In fact, although I believe the forcing-input view is usually preferable, the temperature-input view may be the more-informative in the case of feedback ratio, which in the temperature-input view happens to equal that view’s loop gain (Lord Monckton’s “feedback fraction”).

5. Resultant Behavior

5.1 Closed-Loop Function

Having now defined our system’s open-loop and feedback functions, we turn to the resulting closed-loop function, i.e., to the function such that our feedback equation implies :

This plot illustrates the tipping point we so chose our feedback function as to cause. No (equilibrium) output values correspond to input values that exceed about 273 W/m². That’s because any higher input value would cause the output to increase without limit: the system would never reach equilibrium. (If pressed, tipping-point partisans would presumably admit to some limit, but let’s just assume their limits are off the chart.) As we will see in due course, inputs that exceed the tipping-point input correspond to a small-signal loop gain that necessarily exceeds unity.

Although the output increases without limit for those values, Lord Monckton says instead that a (large-signal) “feedback fraction” greater than unity would imply cooling. One can see that it wouldn’t, though, by using Lord Monckton’s own and for the sake of simplicity assuming a constant = 1.1 so that the large- and small-signal values are equal. Starting with and , iteratively evaluate . You’ll find that blows up.

Unlike the input, the output has equilibrium values that exceed its tipping-point value, which for the output is about 301 K. The curve’s dotted portion represents them. The dots indicate that the corresponding states are unstable.

You can get an idea of what unstable means by supposing that negative temperatures have meaning in Lord Monckton’s linear system. Again assume a constant = 1.1. If iterations start at = (1, –10) instead of (1, 0), the output will remain at –10: (1,–10) is indeed an equilibrium state. Nudge the input one way or the other, though, and in accordance with the direction of the nudge the output will take off toward positive or negative infinity. Although (0, –10) is an equilibrium state, that is, it’s unstable.

The example system will similarly tend to flee the unstable states and possibly blow up. The example system is nonlinear, though, and the direction of the nudge determines whether it actually does blow up or instead falls to a stable value, i.e., from the dotted curve to the solid one.

Having seen the output behavior from the forcing-input view, let’s turn to Lord Monckton’s temperature-input view. That is, let’s consider the function such that implies :

This view suppresses the nonlinearity in the relationship between forcing and temperature. Since the input is simply what the output would have been without feedback—whose ratio to output is very low throughout most of the function’s illustrated domain—the output over much of the curve nearly equals the input. Toward the right, though, the output pulls away. And, just as in the previous plot, there’s a tipping point.

5.2 “Feedback Fraction”

Now we come to what is perhaps the most-consequential quantity: the loop gain, or, in Lord Monckton’s terminology, the “feedback fraction.” Here again we will see the importance of distinguishing between large- and small-signal versions.

The plot above confirms what we may have surmised from the previous plot: the loop gain is near zero over most of the function domain. But above-unity loop gains on the plot’s right impose the limit we observed on equilibrium-state input values.

Note in particular that it’s the small-signal version of the loop gain whose unity value imposes the limit; the large-signal loop gain is quite modest right up to the tipping point. So it’s important to keep track of which quantity Lord Monckton intends when he discusses the “feedback fraction.”

(Obscure technical note for feedback-theory types: Because of the high small-signal open-loop gain near absolute zero, the system is unstable in that neighborhood even though the feedback approaches zero. For the loop-gain plots we therefore used negative inputs to obtain some of the near-zero behavior, and to that end and were extended as odd functions into the third quadrant.)

Recall that loop gain is the gain encountered in traversing the loop. For the forcing view the large-signal loop-gain version is therefore the ratio of the output-temperature-caused feedback to the total forcing that caused the output temperature. The small-signal version is the corresponding incremental quantity .

Since a unity value of this dimensionless quantity’s small-scale version represents the stability limit, one might think it would be the same thing in both views. If we actually work it out, though, we see there’s a difference.

For the temperature-input view the large-signal loop-gain quantity is the ratio that the feedback temperature bears to the output temperature : it’s , which maps to Lord Monckton’s feedback fraction . The corresponding small-signal version is . In this view the loop gain is the same as the feedback ratio we saw in Fig. 6.

But a comparison of the two views’ small-signal loop gains is instructive:

Although their small-signal tipping-point values are the same in both views, the different views’ loop gains otherwise differ.

5.3 “System-Gain Factor”

We finally come to closed-loop gain. This time we’ll start with Lord Monckton’s temperature-input view. In that view the large-signal version is . That version corresponds to Lord Monckton’s “system-gain factor” . To calculate ECS’s value , he advocates multiplying the “reference sensitivity” by that quantity.

But by definition the quantity whose multiplication by really does yield ECS’s correct value is instead the “secant slope” . Even Lord Monckton seems to admit that the secant slope is the right multiplier. (Well, what he actually says is, “The disadvantage of [deriving the “system-gain factor” as a secant slope] is not, repeat not, that it is wrong, but that it is not useful. . . .”) This view’s small-signal gain corresponds to rather than and therefore equals that secant slope at some point in the relevant interval 265.00 K < R < 266.05 K. As the plot above shows, that small-signal gain is about 3 times the large-signal gain in that interval: it’s about 3 times the “system-gain factor” Lord Monckton uses for his ECS calculation. So if the “secant slope” is correct—as by definition it is—then Lord Monckton’s approach greatly underestimates ECS when ECS is high.

Now, in actuality his approach probably would not result in a serious underestimate if, as many of us believe, ECS is low. That’s because a low value would not result in the great between-version difference that the plot depicts. But that fact doesn’t support Lord Monckton’s theory.

The problem is that his theory’s targets aren’t people who already believe ECS is low. He characterized his theory as a “way to compel the assent” of those who would otherwise believe ECS is high. It would compel assent, he said, because, unlike previous ECS arguments, his theory isn’t a mere conjecture; it’s a proof.

But a proof of low ECS can’t be based on assuming low ECS to begin with; that would be begging the question. Nor would the assent of someone who thinks ECS is high be compelled by an approach that greatly underestimates ECS when it is high. What could arguably compel assent is for the high-ECS assumption to result in contradictions of “the mathematics of feedback in all dynamical systems, including the climate.” That’s why we took a high-ECS system as our example: to expose any such contradictions. But we found none.

Now a point of clarification about the plot. The dotted curves mostly represent unstable equilibrium states as they did in previous plots. But here there’s an exception: the dotted black curve’s vertical segment on the right, at the maximum equilibrium-input value. That segment is merely the line between positive- and negative-infinity values: its abscissa is the value at which the closed-loop gain switches abruptly from positive to negative infinity. So no equilibrium states actually occur on that vertical segment.

It might therefore have been less distracting to omit that segment from the plot. But it provides another opportunity to point out how hard it can be to interpret even simple algebraic equations properly. The corresponding discontinuity in the hyperbola of linear-system gain ratio as a function of loop gain is the basis of Lord Monckton’s above-mentioned belief that loop gains greater than unity imply cooling:

That interpretation is wrong, of course; Fig. 8 showed us that equilibrium output temperature continues to increase beyond the transition to instability.

In the electronic-circuit context Lord Monckton has analogously interpreted that discontinuity is as being the point “where the voltage transits from the positive to the negative rail.” That interpretation is beguiling because of the audience’s experience with audio-system feedback. After all, unity loop gain is the basis for squeal when sound systems suffer from excessive feedback, and that oscillation certainly involves a lot of voltage “transiting.”

One problem with such interpretations is that the hyperbola is valid only for constant open-loop gain. More important, they ignore that the relationship represented by the hyperbola is an equilibrium relationship: the hyperbola doesn’t apply to dynamic effects like oscillation. (Well, the equation on which the hyperbola is based actually can be used to characterize steady-state oscillation. But that would require complex values: the geometric representation would require four dimensions instead of the hyperbola’s two.)

So attempts like Mr. Stokes’ to demystify the feedback equation itself are all well and good. But it’s also important to recognize that the equation’s very simplicity can be misleading, even for someone who “was given training in the mathematics of what are called conic sections.”

Now let’s complete our study of the system’s behavior with the other view of closed-loop gain.

Instead of slightly increasing as the temperature-input view’s large-signal gain did, the forcing-input view’s actually continues to decline right up to the tipping point. But the overall effect is the same: the small-signal gain rises dramatically as the tipping point is approached, whereas the large-signal gain does not.

In short, we’ve worked through a counterexample to the proposition that high ECS values are inconsistent with a system whose feedback responds to emission temperature. In doing so we’ve detected no contradictions of feedback mathematics. But by juxtaposing small- and large-signal versions we’ve seen how important it is to distinguish between them consistently.

6. “Near-Invariant”

Before we conclude, we’ll use one of Lord Monckton’s reactions to such counterexamples to illustrate why it’s important to “work it out” and not just “scope it out.”

Ordinarily Lord Monckton’s reaction to such counterexamples is merely to express his disbelief that the function could be so nonlinear. Or he makes the physical argument that the quasi-exponential response of evaporation to temperature somehow conspires with the quasi-logarithmic response of forcing to concentration to make the entire sum of water-vapor, albedo, lapse-rate, cloud, and other feedbacks linear. Again, though, such arguments are irrelevant. The point isn’t whether we think the function is nearly linear. It’s whether that’s what feedback math requires: it’s whether Lord Monckton has as he claimed achieved an actual proof rather than a mere conjecture.

But this time he argued as follows that it’s “official climatology,” not feedback theory, that imposes the near-linearity requirement. (Presumably he meant “near-invariant” instead of “near-linear” in writing that “official climatology’s view” is “that the climate-sensitivity parameter . . . is ‘a typically near-linear parameter’.”)

“[The counterexample is] spectacularly contrary not only to all that we know of feedbacks in the climate but also to official climatology’s view that the climate-sensitivity parameter, which embodies the entire action of feedback on temperature, is ‘a typically near-linear parameter’.”

“It is only if one assumes that there is no feedback response to emission temperature that climatology’s system-gain factor gives a near-linear feedback response. . . .

“It is only when one realizes that feedbacks in fact respond to the entire reference temperature and that, therefore, even in the absence of the naturally-occurring greenhouse gases the 255 K emission temperature itself induced a feedback that it becomes possible to realize that, though official climatology thinks it is treating feedback response as approximately linear it is in fact treating it – inadvertently – as so wildly nonlinear as to give rise to a readily-demonstrable contradiction whenever one assumes that any point on its interval of equilibrium sensitivities is correct.”

(As an aside we note that Lord Monckton left unspecified the standard by which a system like that of Fig. 1 can be said to be “wildly nonlinear”. Nor did he “readily [demonstrate]” a contradiction that would arise even from a sytem like that of Fig. 8, which we so designed as to provide an imminent tipping point.)

For the sake of convenience we’ll use Lord Monckton’s notation to unpack a couple of those assertions.

First, although he often criticizes “official climatology” for focusing on perturbations in its ECS calculations, he apparently chose in this context not to interpret climatology’s use of “near-invariant” or “near-linear” as limited to the ECS calculation’s perturbation range; he interprets the near-linearity as applying to the function more generally or at least to its entire portion above the emission temperature.

Second, the projection line in Fig. 1 above illustrates what he seems to mean by “It is only if one assumes that there is no feedback response to emission temperature that climatology’s system-gain factor gives a near-linear feedback response.” If you’re considering the stimulus to be only the portion of that exceeds the emission temperature, then the response could be the projection-line portion to the right of the emission temperature. Such a response would have only a single, relatively high slope. If the stimulus is taken as the entire value, though, then the response has a lower-slope portion as well, and that slope change contradicts what he says is “official climatology’s view” that the response is nearly linear.

That his interpretation of “official climatology’s view” thus results in a contradiction isn’t a very compelling argument by itself. His interpretation is almost certainly a misreading of the literature. And, if you choose a contradictory interpretation over the more-probable non-contradictory one, you’re bound to find, well, a contradiction.

But in a further comment he seems to say that climate-model results confirm his interpretation of “official climatology’s view”:

“[W]e are not doing calculations in vacuo. The head posting demonstrates that official climatology regards—and treats—the climate-sensitivity parameter as near-invariant: calculations done on the basis of its error show that the system-gain factor in 1850 was 3.25 and the mean system-gain factor in response to doubled CO2 compared with today, as imagined by the CMIP5 ensemble (Andrews+ 2012), is 3.2. Looks pretty darn near-linear to me.”

Climatology must have intended a nearly linear function, that is, if its slope exhibits so little variation in that interval. And, if climatology intended it to be nearly linear, then feedback would reach zero at the emission temperature: climatology’s position is that there’s no feedback to the emission temperature.

Before we show that this standard for “pretty darn near-linear” is too loose to rule out every counterexample, let’s recognize that drawing an inference from differences so dependent on ensemble selection is a parlous undertaking. For instance, a polynomial fit to the combination of those “system-gain factors” with the and values Lord Monckton attributed to “official climatology” in the same thread would imply a cubic function that intersects at . That’s not very linear. Also, climatology could still be right about ECS even if it’s wrong about lower-temperature behavior.

But let’s nonetheless assume “official climatology’s view” to be that the closed-loop gain won’t vary by more than 3.25 – 3.20 = 0.05. As the plot below shows, this assumption still doesn’t support the inference that “official climatology” has “made the grave error of not realizing that emission temperature (= 255 K) itself induces a substantial feedback.”

For the interval over which Lord Monckton reports the variation in “system-gain factor” the plot displays the temperature-input view’s closed-loop-gain functions not only from the example system but also from that of Fig. 1. As to the large-signal versions, the different feedback functions’ results are virtually indistinguishable, and they vary only negligibly over the interval.

As to the small-signal versions, it’s true that the variation caused by the example, tipping-point-causing feedback function greatly exceeds the arbitrary “near-linear” limit, 0.05. But that limit, which made the “official climatology” closed-loop function look “pretty darn near-linear” to Lord Monckton, is actually 9 times the gain variation 0.0053 caused by Fig. 1’s feedback function—which, again, responds to the emission temperature.

Although a closed-loop function may look “pretty darn near-linear” when we just scope it out, that is, it can look quite a bit different when we actually work it out.

7. Conclusion

The equilibrium scalar feedback equation is the most rudimentary of feedback topics; the algebra is trivial. Yet, as we saw in connection with Fig. 12’s hyperbola, its interpretation isn’t straightforward even when the system is linear. And for nonlinear systems it provides a good occasion to recall that simple rules can result in complex behaviors. So any feedback question calls for following that professor’s advice: Don’t just scope it out; work it out.