Cold Stove Lids and Convolution II

The Math Error at the Heart of an “Irreducibly Simple Climate Model”

Introduction

In the previous post we saw what the right way is to calculate a linear time-invariant system’s output from its step response. In this one we’ll compare the correct output thus calculated with the estimate that would result from the approach described in Monckton et al.’s 2015 paper, “Why Models Run Hot: Results from an Irreducibly Simple Climate Model.” Since Monckton et al. dealt with feedback systems, though, we’ll first use a change of variables to couch our previous post’s demonstration system in feedback terms and to introduce some nomenclature we’ll employ in discussing Monckton et al.’s paper.

A Simple Feedback System

Fig. 6 is a conceptual diagram of a type of feedback that’s often thought to operate in the climate system. Consider in that connection an increase in the atmospheric content of a non-condensing greenhouse gas such as carbon dioxide. Input x is the resultant forcing, i.e., the resultant initial radiation imbalance, which tends to cause the earth to heat up, and output y is the resultant temperature. For the sake of simplicity the diagram’s top block models the earth as a single lumped heat capacity inversely proportional to coefficient a: it performs time integration of a quantity proportional to the algebraic sum of that forcing and the feedback produced by mechanisms that the bottom two blocks represent.

Specifically, the middle block represents how much the temperature increase would cause outward radiation to increase if the only result of the added carbon dioxide were the infrared-opacity increase caused directly by the carbon-dioxide increase itself. And the bottom block represents the outward radiation’s reduction that other temperature effects such as increased water vapor might cause.

In summary, the forcing tends to cause the earth’s temperature to increase, but this temperature increase also causes an increase in the rate at which the earth radiates energy back out to space, and this outward-radiation increase tends to redress the radiative imbalance to such an extent that when the temperature has increased enough the system reaches equilibrium: the warming stops.

Inspection of that diagram reveals that it implements the following differential equation:

If we make the substitution b = a · (d – f) we find that this feedback system is indeed defined by our original demonstration-model equation, i.e., by:

Nomenclature

We referred to the middle and bottom blocks as both representing feedback mechanisms. That was because both of those blocks add quantities back to the input x that depend on the system output y. But in climate circles it’s more typical to think of only the bottom block as representing a feedback mechanism; the middle-block mechanism is looked upon as a direct effect of the forcing.

For that reason we’ll use the phrase open-loop gain to refer to the equilibrium value 1 / d of the ratio that the output would bear to the input in the absence of the bottom block: it’s the gain that would have prevailed if we hadn’t “closed the loop” by including the bottom-block-implemented feedback path. Monckton et al. assign the symbol λ₀ to that quantity (which they instead call the “Planck sensitivity parameter”).

We’ll call the equilibrium gain 1 / (d – f) that prevails because we do include the bottom-block-including feedback path the “closed-loop” gain. Monckton et al. assign the symbol λ_∞ to that quantity (which they instead call the “equilibrium climate-sensitivity parameter” ).

Finally, we’ll call the product f / d of the feedback coefficient and the open-loop gain simply “loop gain.” Monckton et al. assign the symbol g to that quantity (which they instead call the “closed-loop gain”). Unlike (what we call) the open- and closed-loop gains, which are ratios of total-system output to total-system input, the system’s loop gain is the gain that results from one traversal of its feedback loop.

To emphasize that these terms refer to equilibrium quantities we’ll ignore Fig. 6’s dynamic aspect and thereby reduce it to Fig. 7. In that diagram the blocks within Fig. 6’s dashed rectangle have been replaced with a single block, which applies the open-loop gain λ₀ = 1 / d.

Inspection of that diagram gives us the following (algebraic) equation

and solving for y gives us the relationships we saw above:

We will encounter those relationships again when we consider Monckton et al.’s equation in detail.

Monckton et al.’s Approach

We’re finally ready to see why Monckton et al.’s approach was erroneous: instead of convolving the step response u(t) with the input x(t)’s time derivative, Monckton et al.’s Equation 1 would simply multiply it by the input itself:

Fig. 8’s bottom plot shows what the result would be in the case of our demonstration model if the stimulus is the two-pulse function illustrated by the top plot. Comparison of the bottom plot with the model’s actual response shown by the middle plot indicates that the Monckton et al. approximation can differ so greatly from the actual response that it’s hard to imagine what useful purpose it could serve.

Note that at any given time the true value of the demonstration system’s output depends not only on the input value at that time but also on the input’s prior history. But Monckton et al. would treat the system as having no memory: Monckton et al.’s response estimate would immediately return to zero when the stimulus does. Although our example system is time-invariant, moreover, Monckton et al.’s approach would treat its behavior as changing over time: although the stimulus’s two pulses are equal in magnitude Monckton et al. would approximate the response to the later pulse as being greater in magnitude than the response to the earlier one.

And that brings up another of that equation’s embarrassments: its results depend on where in the stimulus record we place the time origin. Physical systems don’t depend on where we think the time origin should be, and the convolution integral reflects this independence: each given time t’s particular output value y(t) depends on all step-response values and on the stimulus’s derivative values for all previous times. Where we place the time origin doesn’t matter in that calculation.

But it does in the Monckton et al. scheme. We assigned t = +1 to the beginning of the first stimulus pulse, but there’s no reason why we couldn’t have assigned t = 0 to it, in which case Monckton et al.’s estimate of the corresponding response would have been much smaller. And if we had assigned t = –1 to it the estimate would have been zero.

In short, Monckton et al.’s approach just doesn’t make sense.

But probably because it was expressed in a rather elaborate-looking equation, i.e.,

few readers seemed to recognize the problem.

A careful perusal of the equation’s first line would have revealed that what the equation calls for is indeed just the multiplication of those two functions. Specifically, that line expresses the time-t stimulus value x(t) as the product of two factors q_t^-1 and ΔF_t, it expresses the time-t step response u(t) as the product of two other factors r_t and λ_∞, and it expresses the time-t response y(t) simply as ΔT_t. (Recall the previous post’s observation that for dimensional consistency in convolution the step response’s dimensions are the ratio of response units to stimulus units.)

In the expression for the stimulus x(t) the ΔF_t factor is the direct forcing that results from added carbon dioxide alone, whereas q_t is the ratio that the carbon-dioxide component ΔF_t of forcing bears to the total direct forcing, which includes not only that component but also direct forcing from, e.g., other non-condensing greenhouse gases. That is, those quantities’ quotient q_t^-1ΔF_t is simply the direct-forcing total. And the equation’s last line simply expresses this total’s carbon-dioxide portion in terms of concentration, in accordance with the generally accepted logarithmic relationship between concentration and forcing.

As to the first line’s expression for the step response u(t) the best interpretation is probably that the λ_∞ factor is the limit approached by the step response as time approaches infinity—i.e., it’s the equilibrium value of closed-loop gain—and that r_t is the ratio that the step response u(t) bears to that limit at time t. Now, some confusion as to just how the step response’s time evolution is allocated between those factors is introduced by the third and fourth lines’ placement of a subscript t in the notation f_t for the feedback coefficient. Despite the authors’ description of f_t as “the sum of all temperature feedbacks acting on the climate over some period t” the least incoherent interpretation of the paper is that f_t is really a constant and thereby leaves λ_∞ as the constant we’ve interpreted it to be.

The reason for treating f_t as a constant despite the subscript isn’t that there wouldn’t be any storage in the climate’s feedback mechanisms but rather that Monckton et al. have shown no way of separating the resultant feedback component of the response’s time evolution from that evolution’s forward-path component; those components are both included in the “transience fraction” r_t. Indeed, the authors’ transience-fraction description includes the statement 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 r_t.”

Now, it must be admitted that such seemingly inconsistent statements tend to prevent confidence in interpretation. But it’s fairly clear that the quantity r_t λ_∞ in the equation’s first row is intended to represent the system step response. And in any event there’s no question that what Monckton et al.’s Equation 1 expresses is simple multiplication, not convolution.

Stay Tuned . . .

It is inconceivable in light of their credentials that none of the authors had ever been exposed to the elementary differential-equations results we reviewed in the previous post. But their paper contains no evidence that they’d taken such results into account. If they had, why wouldn’t their paper have included a comparison of the response estimate with the actual response?

Still, we can’t completely rule out the possibility that they did take them into account but believed as they’ve since contended that in the climate context their approach produces less error than theory might seem to suggest. So our next post will apply their formula to stimuli and step responses that are more similar than those above to the types normally encountered in climate calculations. We’ll thereby show that even in a climate context the error can be quite significant.