Friday, May 4, 2018

Feedback, climate, algebra and circuitry.

I've been arguing again at WUWT ( (more here)). It is the fourth of a series by Lord Monckton, claiming to have found a grave error in climate science, so it is now game over. My summary after three posts is here.

The claim is, of course, nonsense, and based on bad interpretation of notions of feedback. But I want to deal here with the general use of feedback theory in climate, and the mystery that electrical engineers who comment on this stuff like to make of it. The maths of feedback is trivial; just simple linear equations. And it is best to keep it that way.

A point I often make in commentary is that climate science really doesn't make much use of feedback theory at all. Critics invoke it a lot more. I continually encounter people who think that feedback is the basis of GCMs. I have to explain that, no, they do not form any part of the structure of GCMs, and cannot. A GCM is a solver for partial differential equations. That means it creates for each step a huge array of linear equations relating variables from neighboring cells. That isn't always obvious in the explicit methods they tend to use, but there is still an underlying matrix of coefficients. And because each row just related a few neighboring values, the matrix is sparse. This is an essential feature, because of the number of cells. But global averages, such as would come from a feedback expression, are not sparse. They connect everything. So they cannot fit within the discretised pde framework.

Linear equations and feedback

Problems described as feedback are really just linear equations, or systems of a few linear equations; usually one less equations than unknowns, so on elimination, one variable is expressed as a multiple of another. I described here how a feedback circuit could be analysed simply by writing linear current balance (Kirchhoff rule) equations at a few nodes. In climate, the same is done by balancing global and time average heat fluxes, usually at TOA.

The paper of Roe 2009 is often cited as the most completely feedback oriented analysis. I'll show its presentation table here:

It gives the appearance that ΔR is both input and output, because it is a flux that is conserrved. But the more conventional feedback view is that ΔT is the output. If we take the multi-feedback version of (c)
ΔT = λ₀(ΔR + ΣciΔT )
which I can rewrite setting c₀=-1/λ₀ as just
ΔR + c₀ΔT + ΣciΔT = 0

This is just the equilibrium heat flux balance at TOA since each of the ciΔT is a temperature-responsive flux. I have given the c₀ΔT special status, because it is the Planck term, representing radiation guaranteed by the Stefan-Boltzmann law (c₀ = -4ΔT).

Feedback reasoning and linear equations

Just resolving a linear equation is not a mathematical difficulty. So what is all the feedback talk about? Mainly, it is trying to see the equation as built up in parts. There is no math reason to do that, but people seem to want to do it. The process can be described thus:
  • Select (as in Roe above) a subset to refer to as the reference system. A logical set is the forcing and the necessary Planck response.
    ΔR + c₀ΔT + ΣcₖΔT = 0
    This is like a finite gain amplifier (c₀)
  • Express the other terms as feedbacks relative to c₀:
    ΔR + c₀ΔT *(1 - Σfₖ) = 0, fₖ = -cₖ/c₀
    The f's are then called the feedback coefficients. For stability (see next) they should sum to less than 1. Negative values make this more likely, and so are stabilising. As the coefficient of ΔT, diminishes, it increases the amount by which ΔT would have to change to keep balance. That is said to increase the gain, and creates a singular situation (of high gain) approaching zero.


If the singularity is passed (Σfₖ>1), and the coefficient of ΔT becomes positive, the system is unstable. The reason involves an extra bit of physics. Suppose total flux is out of balance. Then the region into which it flows will cool or heat. The coefficient here is, for a uniform material, called the heat capacity H, and is positive. For a complex region like the Earth surface, that is hard to quantify, but will still be positive. That is, heat added will make it warmer, not cooler. So the equation for temperature change following imbalance is
ΔR + cΔT = H*dΔT/dt
If c is positive, this has exponentially growing solutions, and so is unstable. For c negative, the solutions decay, and lead toward equilibrium.

It's often said that positive feedback is impossible, because it would mean instability. But in the above algebra, that is not true; the requirement is that Σfₖ>1. It is true if you choose a different reference system - just the forcing. That can only work in conjunction with a c₀ΔT where c is negative. Electrically, the reference system is then like an operational amplifier.

Summary so far

Systems often described using feedback terminology are really just linear equations (or systems). Feedback arguments do not yield anything beyond what elementary linear solving can do, including a stability criterion. But with linear algebra, you can identify the various steps of feedback reasoning if you want to.

Systems are not exactly linear

Roe points out that linear feedback is just the use of a first order Taylor Series expansion of a nonlinear relation. This is very direct seen as a linear system. If the forcing R is to be balanced by a flux F which is a function of T and variables u,v which depend on T, then to first order

dR = (∂F/∂T) dT + (∂F/∂u du/dT) dT + (∂F/∂v dv/dT) dT

each partial holding the other variables (from T,u,v) fixed. This gives the required linear relation with the bracketed terms becoming the c coefficients (but negative).

More advanced

There is a lot of approximation here. Not only linearity (usually OK) but also in the use of global averaging. But that doesn't mean linear analysis has to be discarded if you want to take account of these things. You can extend using an inexact Newton's method. Suppose we have the base system

R = F(u,v,T)

where again u and v are variables (like humidity) that depend on T. Suppose we have an initial state subscripted 0, and a perturbed state subscripted 1, of which R₁ is known. Then to first order

F(u₁,v₁,T₁) - R₁ = F(u₀,v₀,T₀) - R₁ + (∂F/∂T)₁ dT + (∂F/∂u du/dT)₁ dT + (∂F/∂v dv/dT)₁ dT = 0

This can be solved as before as a linear equation in dT. Then updating

T = T + dt, u = u + du/dT)₁ dT etc, we can solve again

F(u,v,T) - R₁ + (∂F/∂T)₁ dT + (∂F/∂u du/dT)₁ dT + (∂F/∂v dv/dT)₁ dT = 0

and iterating until F(u,v,T) - R₁. Note that I have not updated the partial derivatives, which are the feedback coefficients. That is what makes it an inexact Newton; convergence is a bit slower, but we probably don't have information to do that update.

So non-linearity is not a show-stopper; it just takes a little longer. This also allows you to work out a more complicated version of F, with, say, latitude variation. You can still use the simpler global feedback coefficients, so the extra trouble is only in the evaluation of F. The penalty will again be slower convergence, and it may even fail. But it gives a way to progress.


  1. The significant point is that it is in fact positive feedback but it's a kind of weak positive feedback that doesn't lead to a runaway condition. This will happen when the feedback is in the form of an Arrhenius rate law, as the highly non-linear exp(-E/kT) does not blow up. Not very difficult to calculate the end points on this, and it likely determines the +33C setpoint in our climate system. This feedback is well known in the material science world but the viscount probably has never set foot in a lab.

    Another point to consider is the nonlinear positive feedback that seems to occur in the ocean oscillations. The annual cycle appears to force the ensuing year to overcompensate and cause a metastable biennial oscillation to sustain year after year as a kind of limit cycle. I believe Michael Faraday and Lord Rayleigh (the smart viscount) first identified this behavior, known as period doubling, in wave experiments and when analyzing the nonlinear Mathieu equation. More recently this has been a significant focus in the research literature on hydrodynamic sloshing models. Not sure why this is not given more thought in ocean models, maybe you know why Nick? All I have seen is one paper that I can no longer seem to find that says the Earth's spherical geometry prohibits it.

    1. Web,
      "the highly non-linear exp(-E/kT) does not blow up"
      It can do, literally. It is the basis of ignition or explosion in chemical reaction. But certainly your ocean sloshing example is likely to be expressible in terms of feedback, with non-linearity leading to clipping which limits the amplitude.

      The main reason why GCM type models don't incorporate a sloshing model is that they can't. Structurally they solve relations between grid cells, which gives huge numbers of equations made tolerable by sparsity (many variables not directly related). Introducing a large scale model destroys the sparsity.

    2. Nick, I suppose it can blow up, depending on the value of E and the temperature. But the region of interest is the heat of vaporization of H2O. I worked it out here several years ago:

      This obviously does not blow up for the low E and modest temperature range and so the feedback end-points can be determined.

      "The main reason why GCM type models don't incorporate a sloshing model is that they can't. Structurally they solve relations between grid cells, which gives huge numbers of equations made tolerable by sparsity (many variables not directly related). Introducing a large scale model destroys the sparsity."

      I would suggest that possibly not along the topologically insulated equatorial region. The lower-dimensionality here allows many simplifying assumptions to be made. This research (google Delplace,Marston) will explode in the next few years I suspect. For the part I worked out, the tidal equations used in the GCMs simplify elegantly and the cyclic climate variability starts to make sense. All I require is a strong period doubling impulse, which is an alternating sign from year to year modulating the forcing.

  2. Nice but your formatting is off, losing part of the right hand side.

    1. Phil,
      I don't have much control of the view window. You could try shrinking the text (Ctrl -).

    2. Phil, apologies, there was something I had done that could have caused the effect you described. Thanks for letting me know. I hope it is fixed now.

  3. I've often said on this issue that net feedback is actually negative, quite strongly so, but it's negative with respect to the forcing input and TOA imbalance output. Temperature change is simply the mechanism by which this negative feedback occurs. As a side effect of that temperature change mechanism other processes are triggered, such as an increase in water vapour and decrease in ice, which also affect the TOA imbalance with the result that the net negative feedback caused by temperature change is damped to some degree.

    1. The base negative feedback is usually described as Planck feedback. Just a kind of Newton's law of Cooling - temperature rise increases radiative outflow. The feedbacks add, so positive feedbacks will eat into this buffer. It's only when the sum, including Planck, approaches zero that runaway becomes an issue.

    2. See my comment above. The feedback for H2O contributions has to be positive, egged on by non-condensing CO2, to avoid the Snowball-Earth possibility.

    3. The Planck feedback is the reason why all of this about positive forcing is nonsense. As you point out it's the net feedback that counts

  4. Nick, Thanks for your very clear presentation on the artificiality of these "feedback" arguments. There were some further devastating comments, particularly from Harry Palmes on the ridiculousness of claiming that the system temperature itself might induce a feedback affecting .... the system temperature. At this point the comments came abruptly to an end, with the moderators announcing "comments now closed" after "Monckton of Brenchley" delivered some particularly trenchent insults to Palmes: this rather surprised me,since I wasn't aware that Watts did that sort of thing, but then I hardly ever visit his site these days

    I was at first sight quite impressed by "Monckton of Brenchley"'s grasp of his brief and his apparent feat of endurance, before realising that "Monckton of Brenchley" was a cover for more than one author. I noticed for instance that "MoB" kept using the American spelling "vapor" and used "favor" on more than one occasion. There again, it's also the Australian spelling, so it became clear that the posts were probably a joint effort with an American/Australian, who it also became clear was taking over when things got technical, for instance over the differentiation of T^4, a subject which Monckton, for all his sneering at others' lack of understanding of the subject was very coy about.

    Also my memory was jogged of Monckton's "coming out" as a revolutionary climate scientist about ten years ago with something very similar. See . The spin was of "his lordship's brilliant mathematical falsification of CAGW", which involved among other things an extremely long-winded and hand-wavy derivation of the derivative of T^4, suggesting his mathematical ability might not be quite as advertised. The same attempt to derive a feedback by fiddling about with the Stefan's Law formula was there as well. I checked the acknowledgements: there was Australian David Evans being thanked for his contributions concerning feedback. So that's who was ghost-writing much of Monckton's stuff, but carelessly leaving in the Aussie spellings, and using a quite different literary style. He seemed to make some sort of effort to ape Monckton's obsessive sneering and abuse of people who suggested that his work might be in any way flawed, but it was far from the real aristocratic rudeness of My Noble Lord.

    1. Bill,
      I think he gets help with the posts. When it comes to discussion, I think he's on his own, and it shows. I actually think David Evans is a bit too knowledgeable (about electronics etc) to be responsible for the emission temperature feedback nonsense. But maybe he just scrubs up the maths.

      On spelling, as you note below, Australian is generally Queen's English (the Labor party are mostly republicans). I write vapor myself, basically because I get sick of fighting the spell checker.

    2. Nick, I'm pretty certain Monckton is getting a huge amount of help on the discussions from someone who uses U.S. spelling: not just "vapor", but also "favor", and who has a quite different literary style. I remember years ago learning something about text analysis of the books of the Bible using style and voabulary: for instance, there are two distinct authors of the book of Isaiah: "Isaiah" and "deutero-Isaiah. I think Evans is being deutero-Monckton here, and writing the lion's share of the comments. He makes an effort to sound like Monckton, using the latter's stilted fashion of addressing others in the third person: "Mr. Stokes is clearly confused", etc., but the mask keeps slipping.

      It's interesting that Evans isn't given as one of the (many) authors of Monckton's forthcoming paper: recognition that he's a toxic presence after the force X nonsense, maybe. It seems he's now reduced to having to impersonate an eccentric, self-obsessed British aristocrat.

  5. Nick, I realise I've been labouring (laboring?) under a misapprehension all these years: Australians actually use British spellings for words like vapour. I was thrown by the anomaly of the "Australian Labor Party.

    That said, David Evans, of force X fame, uses US spellings, so my thesis may stand up after all. See for instance .

    1. David Evans had a significant part of his education in the US, so maybe that's why he's using US spelling so much?