I will post this comment from a different thread so I can be sure you have read it.

I would like you to post a review of the recent benchmark tests of four different dynamical cores.

Jablonowski, C. and D. Williamson, 2006.
A baroclinic instability test case for atmospheric model dynamical cores.
Q. J. R. Meteorol. Soc,
132, pp 2943-2975.

I would also like to see a review of the article by Roger Pielke Jr. and Sr. (the latter is an “expert” in meteorological mesoscale models).
I will add my comments to these reviews so the general reader can judge
these models based on the reviews.
Jerry

I understand what you are saying, and I know that there is more we agree upon than disagree. Our main differences have been use of language.

In climate discussions, I come across the closed form 1/(1-f) and the explanation of a fraction ‘f’ being fed back from the output to the input. On another thread, you can see this expanded to the form: 1, f, f*f, f*f*f, etc. It clearly refers to a sequence 1 +(f/z) + (f/z)^2 + ….. and the closed form 1/(1-(f/z)).

This is a difference equation of the form x(k) = y(k) + f.x(k-1). This seems to lead to supposition such as: “if ‘f’ could be 0.5 or 0.6, amplification will be 2 to 3 times” (is that not what James Annan suggested?).

Anyway, it looks like we only need to add-back some output to the input and we get amplification. I have my doubts.

Take a simple R-C low-pass filter. This is a passive system and has no steady-state amplification. In this case, the discrete transfer function is (1-b)/(z-b) where b=exp(-T/RC) and T is the sample interval. It doesn’t matter how much we fiddle about with ‘b’, the steady-state amplification is unity (for valid b). The equation accords with how we expect the circuit to behave.

Here’s another example. From any table of z-transforms, we can see that exp(-at) (unit impulse response) maps onto z/(z-b) where b=exp(-aT) and T is the sample interval. It looks innocent enough, but this represents a process with amplification. These equations do not readily reveal the fact that they represent a system with steady-state “gain” = 1/(1-b).

Jordan, note the function I was considering, G = 1/(1 – u), does not employ the variable s. “u” is the forcing from clouds, and in principle could be a constant (in which case the climate forcing from clouds G = constant). Physical arguments require |u| < 1, because cloud forcing is a passive mechanism.

Now if u were varying slowly and in a smooth fashion with |u(t)| < 1, it is easy to show that 1 + u + u^2 + … uniformly converges to 1/(1-u). If one replaces “u” with “s”, where s is not bounded, it’s easy to see why this expansion will fail.

I agree with your points with regard to the construction of digital filters from an analog filter. However, the point is the equation being considered has nothing to do with continuous (analog) versus discrete filter functions. It is as I’ve said an issue of continuum mathematics, not one of filter design.

Let’s take a function F(S) = 1/(s+a). The inverse Laplace Transform converts this to the time-equation f(t)= exp(-at). (Quite a good wikki on Inverse LT.)

There seems to be an assumption that series expansion works for the above function. That is (with a minor adjustment), it can be expressed in the form:

F(s) = 1 – bs + (bs)^2 – …..

This doesn’t work.

The Inverse Laplace Transform only exists for functions which are analytical over the entire complex space “s” with the exception of a finite number of singularities (aka poles).

Series expansion does not meet this criterion because the series only converges for a limited space within s. The Inverse Laplace Transform does not therefore exist for the above series expansion.

This is easily demonstrated with an example. The Inverse Laplace Transform is linear, so we can take the elements of the above series in turn. The Inverse Laplace transform of 1 is a dirac delta function in time. The higher powers are derivatives of the delta function.

If you try to add these together in the time domain, you will get nothing like exp(-at).

I can see part of the confusion. I said the following:

if you are claiming a closed loop gain of “x”, you must have an open loop gain of more than “x”.

It was a slip of the tongue. I meant the feedforward gain (G) must be greater than “x”. Please treat that as a correction.

I’m puzzled why you don’t think

1 + R1 R2 + (R1 R2)^2 + …

is equivalent to

1/(1-R1 R2)

I agree these are equivalent within the radius of convergence. But that’s not my point. I said that geometric expansion is relevant and very useful in the z-plane. But not “s”. When you offered a geometric expansion to explain something, I asked you whether you had slipped back into discrete. You have not answered that question.

This is a completely different issue than whether one can expand a continuous function into a Taylor series…

Sorry if I seem to be picking you up point-for-point. Is the above expansion in R1R2 a Taylor series in the s-plane? On face value, it looks like a straightforward geometric series.

The bilinear transformation is used to convert a continuous time transfer function into a discrete one.

Agreed. I believe that’s what I said in the previous post.

it’s apparent to me that you are mixing concepts from signal processing using continuous versus discrete variables, in with continuum mathematics.

I don’t think I am Carrick. My examples were offered to show that a transfer function in “s” can have a very similar appearance to another transfer function in “z”, even though the two represent completely different systems.

The following text is in the recent thread “James Annan on 2.5 deg C” (the equation doesn’t paste, so I have copied it in by hand)

If you want to look at things in the framework of feedback analysis, there’s a pretty clear explanation in the supplementary information to Roe and Baker’s recent Science paper. Briefly, if we have a blackbody sensitivity S0 (~1C) when everything else apart from CO2 is held fixed, then we can write the true sensitivity S as

S = S0/(1-Sum(f))

where the f_i are the individual feedback factors arising from the other processes. If f_1 for water vapour is 0.5, then it only takes a further factor of 0.17 for clouds (f_2, say) to reach the canonical S=3C value. Of course to some extent this may look like an artefact of the way the equation is written, but it’s also a rather natural way for scientists to think about things and explains how even a modest uncertainty in individual feedbacks can cause a large uncertainty in the overall climate sensitivity.

This appears to be stating that the closed loop gain is 3, when the feed-forward gain is 1.

It is not clear whether the equation is discrete or continuous, but it is crucial to know and to have an explanation as to where the amplification is really coming from. That’s all I’m trying to say.

I finally got to reading your chapter – thank you for posting it.

I have a problem regarding equation 13.3
F = L(Ei, T0, Ij)
and the paragraph following it.

Reading the equation I would first assume that the Ij quantities include many other interesting things such as humidity at various heights, albedo, etc.

However you then comment that “Because T0 is the only dependent variable in the energy balance climate model, the internal quantities must be represented as a function of T0.”

This does not follow in any way from the equation and a natural interpretation of the word ‘internal’.

If we are just given the equation F = L(Ei, T0, Ij) and F = 0, we can assume very little about the Ij quantities. They may not even be functions of the Ei’s, T0 and each other unless you mean to include so-called many-valued functions.

We can of course define the internal quantities Ij to be functions of T0. However other quantities of interest in the climate system may not be ‘internal’ in this sense.

A simple example.

F = E1 + E2 – T0 + I1 – I2
I1 = 2T0 and
I2 = 3T0.

If F = 0, then T0 = (E1 + E2)/2

If however we use

I1 = 2T0 + E1 and
I2 = 3T0 + E1

we get precisely the same result for T0.

There is no a priori reason to suppose the interesting quantities are like the first example and not the second.

Roy, if I understand correctly this is part of what you addressed – clouds are a Ij that is not dependent only on T0.

Peter

The bilinear transformation is used to convert a continuous time transfer function into a discrete one.

This is a completely different issue than whether one can expand a continuous function into a Taylor series, which obviously you can.

You have expressed a closed form of a geometric series in R1R2. Have you slipped back into discrete?

I’m puzzled why you don’t think

1 + R1 R2 + (R1 R2)^2 + …

is equivalent to

1/(1-R1 R2)

R1 and R2 are just complex constants.

In any case, hen you solve the problem from first principles, you end up with the 1/(1 – R1 R2) directly, no need to invert a series. You can obtain the harmonic series either by expanding this result, or by writing it out as a series of multiple reflections…

However, even if they were functions of time, as long as the functions are bounded, you could still either expand them in a series, or invert it.

I’d be interested in what concept you are thinking of here, but you’re definitely wrong on this point.