Climate feedbacks and circuits

I've been arguing again at WUWT. Lord Monckton has been writing a series of articles (see also here, here and here), with more promised, on feedbacks and what he calls the "official equation". Mentioning feedbacks brings out all the engineers talking about feedback and instability. In the course of arguing, I think I see some general confusions arising from inadequate specifications of what circuitry is envisaged, and other general unsoundness. So I thought I could try to clear that up here, and in the process show an actual circuit which would implement the "official equation".

The first thing I try to emphasise here is that climate is not a circuit, and feedbacks are not used in GCM's. They are not the basis of climate science; in fact climate scientists talk about them far less than people imagine. Feedbacks are diagnostic tools - inferred from model output (or climate data) to help understanding. And you are free to imagine any kind of circuit or other apparatus that you think helps. That is a starting point - people are not always imagining the same thing, but they use the same vocabulary.

The basic concept here is climate sensitivity (CS). If you add a warming heat flux, usually from greenhouse effect, how much will the temperature rise? To make this more definite, it is often expressed as equilibrium CS (ECS). If you add a flux and then keep it constant, how much will temperature have risen by the time it has settled to steady state?

A starting point is what can be called Planck sensitivity. We know from the Stefan-Boltzmann law that a warmer planet will radiate more heat, and that will give a relation between flux and temperature change. For a black body, flux F=σT⁴ (S-B, with σ as the S-B constant). I propose to make F analogous to current and T to voltage, so this gives CS = dT/dF = 1/(4σT³) K/(W/m2) This gives it the units of resistance (analogy). With T=255 K, the effective radiating temperature of Earth, that would be 0.26. Earth is not a simple body - the atmosphere has an effect, and GCMs say that the right figure is about 0.31 (Soden and Held, 2006).

But ECS is generally reckoned to be a lot higher, because of the effects of positive feedbacks, especially water vapor feedback. Discussions of these are in S&H just cited, or Roe 2009. So then come the claims that positive feedback is necessarily unstable. It isn't, because it in effect adds to the negative Planck feedback. But if it outweighed that (and any other negative feedbacks) then it would be. That is the basis for talk of tipping points and thermal runaway.

Lord M's posts have started with a statement of the "official equation" which is from Roe 2009. The first version was this:



It is correctly stated there, but was misused, being applied to recent data far from equilibrium. I protested there, so it appeared in later forms (noted here) that could be used to justify the misuse. λ₀ is the Planck feedback described above (0.31 K/(W/m2)). The equation can also be written
ΔT_eq = ΔF/( 1/λ₀ – Σc_i)
where the c's are feedback coefficients, with the convention that positive feedback c is positive. For consistency with λ₀ I will reverse that sign convention and write c₀=1/λ₀:
ΔT_eq = ΔF/( c₀ + Σc_i)

This is easier to interpret. The c's are conductances (but negative for positive feedbacks), and the Planck c₀ is added equally to the others. In fact, the equation is formally just Ohm's law, with combined conductance c = c₀ + Σc_i

That makes it clear that the Planck c₀ has the same status as the other feedbacks, and so is properly called a feedback. It also suggests a simpler interpretation. The "official equation" is just Ohm's law for a combined resistance made up of the feedbacks connected in parallel (where conductances add), and the stability criterion is simply that the resistance is positive. The catch, of course, is that negative c's, associated with positive feedback, are not an everyday circuit element, so some active feedback loop is required to implement them.

Operational amplifiers

These were much discussed at WUWT. An op-amp is a common circuit component, being a high-gain DC amplifier, depicted as on right (Wiki), though the power pins VS are often not shown. Idealised, it has the following properties Output V responds with infinite gain to the difference between V+ and V-. Inputs have infinite input impedance - ie no current can enter Output has zero impedance - any current can be withdrawn without affecting voltage.

One commenter at WUWT who wrote well about op-amps and feedback was Bernie Hutchins. He provided a good set of notes. Another commenter linked to these. I'm going to use one of Bernie's circuits; the basic element is an inverting amplifier:


This works as a see-saw; the input node is held at V=0, and the current entering the input must be V_in/R_in+V_out/R_f = 0 (input impedance infinite). If the R's are equal, V_out=-V_in. Bernie showed a circuit from his notes which uses two such stages to produce a non-inverting output, which can then be fed back as positive feedback:



You can see the experimental readings in light green - the circuit is clearly realisable and stable. The positive feedback increases the gain (ratio of V_out to V_in) from 1 to 3. Coincidentally, this is about the change effected by positive feedback in climate.

I promised a circuit which would implement the "official equation". I'm going to riff on Bernie's circuit. V_in and the initial R determine the entry current i, which corresponds to flux. I've let the first stage have general resistance R₁ and the red feedback be R₂, giving



The second stage operates as a simple voltage inverter no matter what the value of R. The reason is that the R's connect two outputs of zero impedance - ie they can supply any current without affecting voltage, and hence the rest of the circuit. So the relation between V and i is just got by summing currents at the input:
-V/R₁+V/R₂+i=0
or
V = i/(c₁-c₂) where c=1/R
And the loop gain is R₁/R₂ and is stable as long as that is less than 1, ie positive feedback c₂<c₁.

If c₁ is now the sum of negative feedback conductances (connected in parallel) and c₂ is the sum of positives, and ΔT_eq~V, ΔF~i, we have the circuit emulating the "official equation" of equilibrium climate sensitivity with fedback.

So to clarify the issue of positive feedback and stability, it is true that an op-amp, say, with just positive feedback, of any amount, will be unstable. But what is usually meant by positive feedback in climate is feedback which is net positive when then added to the Planck feedback, giving negative overall. That addition increases the climate sensitivity (gain) without instability. But of course it makes it easier for a further perturbation to reach instability. Note that this special treatment of Planck feedback raises the question of whether it should then be regarded as a feedback at all. As I've shown, yes, it can with consistency. You can use terminology in which it is regarded as something else if you want.

Perhaps the simplest way of seeing stability is this. I've noted that the "official equation" has the form of Ohm's law, for a resistor. In math terms, it's isomorphic to an ohmic resistor. It must have the same properties. In particular, a resistor with positive resistance is quite stable. So then, with positive denominator, is the climate system described by the "official equation".

Equilibrium sensitivity

I spent a lot of time at WUWT trying to explain that ECS is an equilibrium analysis. The equation has no time information - it just says what happens when you get from one steady state to another. So it tells you nothing about states along the way, and can't be used to calculate them (as it was in Lord M's first post). Nor can you start drawing Bode plots (much demanded at WUWT) or whatever - that is frequency analysis and out of place in what is, in electrical terms, a DC (direct current) analysis. There is no frequency information to put on a Bode plot.

Update 1 Sept In the interests of relating to other ways in which people might express feedback, I should note two things.

1. I have formulated the circuit as current in - voltage out, because that corrsponds to climate sensitivity thinking. Feedback is more often discussed for a voltage-voltage, or maybe current-current, amplifier. The general way of thinking about this is via two=port networks. The simplification in this circuit is that both input and output have zero impedance (at input because the circuit must ensure that input voltage is zero for any current). So the only non-trivial ratio is z₂₁, or y₂₁ if you're thinking of the denominator.
2. I have used infinite gain amplifiers (op-amps). Feedback theory is often written in terms of finite gain A, as in:
   V = Ai/(1 - A f)
   You could regard the "official equation" as being that with A = λ₀. So then it might be natural to regard the Planck term as the "pre-feedback gain" rather than a feedback. But then you can do the same algebra as above, dividing top and bottom by A, so 1/A becomes something added to the feedback terms, and I think this is more systematic. It reflects the fact that a finite gain amplifier and an op-amp with appropriate feedback are functionally equivalent. It really comes back to what you mean by "pre-feedback". And as I've said, these circuits are just analogues. You can choose.