Len Evens commented:
>I have an open mind about this. Can you demonstrate that the
>phenomenon in your example can occur in the situation Michael
>is interested in? Alternately, perhaps Michael can show that
>it can't occur.
Tobis was arguing that it is impossible in general for
feedback to change the sign of a response to forcing. The above
example shows that this is not true. If Tobis wishes to make a new
argument specific to the climate system with CO2 forcing he is free to
do so.
Len Evens added:
>By the way, last year there was an extended debate started by you in
>which you made a roughly similar argument. You pointed out that
>intuitively in a single variable case certain things which Carbon cycle
>modelers claimed was true could not occur. Your argument seemed
>convincing in the same sense Michael's did. Yet the fact is that the
>models employed in Carbon cycle modeling are much more complex than the
>one dimensional model you were basing your argument on and these models
>exhibited behavior you continued to claim couldn't happen. I don't
>remember your backing down.
Tobis's argument is not convincing. He appears to be assuming
a 1dimensional system. In a higher dimension system the forced
variable can return to zero without turning the feedback off since the
feedback can be selfamplifying. The system can still be stable as the
above example shows.
As to the Carbon cycle what behavoir have I "continued to
claim couldn't happen"? I objected to the the following:
> ... For example, the recent update report of the Intergovermental
>Panel on Climate Change stated, "Future atmospheric CO2 concentrations
>resulting from given emissions scenarios may be estimated by assuming
>that the same fraction remained airborne as has been observed during
>the last decade, that is, 46+7%" (38 p. 35) ...
As I recall a consensus was reached that this estimation
procedure makes no sense.
Michael Tobis responded:
>> This repetition of a refuted argument indicates to me that
>> Tobis is not approaching this subject with a scientific attitude.
>
>At least I'm in good company:
>
> AUTHOR Peixoto, Jose Pinto.
> TITLE Physics of climate / Jose P. Peixoto, Abraham H. Oort.  New York
> : American Institute of Physics, c1992.
>
>on p. 26, tersely dismisses the possibility in question:
>
>"For f loop gain > 1, G sub f system gain would become negative, which
>is a physically unrealistic case."
>
>My argument expands on why this is unrealistic in the case of global
>greenhouse warming. Specifically, it asserts that significant feedbacks
>from greenhouse gases operate through temperature dependent mechanisms,
>and argues that such feedbacks can at most cancel the sense of the
>forcing, but cannot reverse it.
>
>Writing down a mathematical system constitutes no refutation of a claim
>that such a system is physically unrealistic.
Peixoto and Oort also state (p 28): "We should note that these
analysis have some limitations because they do not permit interactions
among the various <feedback> processes or nonlinear responses." Hence
unless you are prepared to argue that there are no interactions between
the various feedback processes involved in the climate system this
section does not in fact support your argument.
Furthermore the "physically unrealistic case" is that of
runaway positive feedback producing a negative response. This is just
a restatement of the fact the the formula for the sum of a geometric
series does not apply when the sum does not converge. (Btw this is
also true for loop gain <1 so the statement in Peixoto and Oort
that: "As f becomes larger and larger negative, G<sub f> tends
asymptotically to zero." appears to be wrong.)
The above example shows your claim about feedback being
unable to change the sign of the response is false in general. You
have not given any argument sufficient to show it is true for the
climate system with CO2 forcing in particular.
Michael Tobis added:
>As to my attitude, I have given Mr. Shearer the benefit of the doubt
>and would prefer that he return the favor, but my arguments stand or
>fall on their merit regardless.
Bob Grumbine said (in a different thread):
> ... On the first, 'Life is too short to again slay the slain.'
>T. H. Huxley, I believe. In other words, having made a conclusive argument,
>you move on to other subjects. In the net, this fails because one of the
>'big lie' people will simply repost the same article in a few months.
I am willing to give the benefit of doubt the first time some
one makes an error. However I find this harder to do when someone
repeats an error after having been corrected.
James B. Shearer
I think it is clear from what you quoted that Michael most recently
was specifically talking about the temperature response of
the climate system. So the internal evidence of your own posting
seems to show that you are in error. (Horror of horrors!)
The fact that long ago he made a general statement about general
feedback does not seem very relevant, so I am not sure why you
are harping on it. Also, Michael responded already in some detail
as to why he thought his argument had some merit.
Let's stop this pedantic nonsense, shall we? Your example does
illustrate a possible kind of phenomenon. The question is whether
or not that phenomenon can actually occur in the real systems we
are interested in. It may not be possible to prove or disprove
this in a mathematical sense, but it is certainly possible to
explore the question further and perhaps gain some enlightenment.
So, instead of arguing about who is allowed to meet your standards
for being a true scientist, suppose you devote your considerable
talents to thinking about that matter. Try to invent a physically
relevant model in which the equilibrium actually shifts in an
unexpected direction. It would be prefereable if this system
actually concerned climate and semiplausible, but it need not be
realistic or justified on the basis of current knowledge.
As far as Michael's scientific credentials are concerned, that
will be settled by people who are in a much better postion than
you or I to make such judgements.
Leonard Evens l...@math.nwu.edu 7084915537
Dept. of Mathematics, Northwestern Univ., Evanston, IL 60208
*********************************************************************
It is impossible for a stable linear feedback system to have opposite
signs of openloop and closed loop gain.
*********************************************************************
While I concede that any stable linear feedback system is equivalent to a
linear transformation, I do not agree that every welldefined linear
transformation can be modelled as a linear feedback system.
I do NOT claim that every linear transformation is equivalent to a
linear feedback system.
Mr. Shearer exhibits a plainly physically realizable linear transformation.
He implies but does not show that it is equivalent to a linear feedback system.
He proposes that this contradicts the highlighted proposition, and that I
am intellectually dishonest for failing to concede the point.
My claim is not that his system is physically unrealizable, but that
it is not realizable *as a linear feedback system*.
If he attempts to exhibit an equivalent linear feedback system,
he will find closedloop gain greater than unity, and hence an
unstable system. If he attempts to implement this with analog circuits,
he will find that rather than the sign of the response changing when he
closes the loop, the system will peg at the power supply.
Casting of the feedback system into a linear transformation presumes that
the delay in the subsytems is negligible compared to the time constants
of the inputs and transformations. This assumption fails when loop gain
exceeds unity.
I will expand on this argument in a forthcoming article. For now, I
note that Shearer has compounded his error with at least two others in his
latest contribution.
As for the following:
: I am willing to give the benefit of doubt the first time some
: one makes an error. However I find this harder to do when someone
: repeats an error after having been corrected.
my own position is somewhat more generous. While waiting for my expanded
discussion, I suggest that Mr. Shearer consider how he wants his crow cooked.
mt
In article <19950721....@almaden.ibm.com>, <j...@watson.ibm.com> wrote:
> Michael Tobis posted:
>>Consider also that if the purpose of the modelling experiment is detecting
>>greenhouse gas sensitivity, we know on physical grounds that the first
>>thing that will happen is an excess of incoming over outgoing radiation.
>>Anything else that happens (except for direct chemical responses to the
>>gases  which may have a very small effect on albedo through fertilizat ion 
>>small because most of the planet is not actually green, and the difference
>>between shades of green is small and not systematic) is a response to
>>the temperature forcing. Thus, these responses cannot reverse the sign
>>of the forcing  if they did, they would turn themselves off!
[Stuff omitted]
> Tobis was arguing that it is impossible in general for
>feedback to change the sign of a response to forcing. The above
>example shows that this is not true. If Tobis wishes to make a new
>argument specific to the climate system with CO2 forcing he is free to
>do so.
I think it is clear from what you quoted that Michael most recently
was specifically talking about the temperature response of
the climate system. So the internal evidence of your own posting
seems to show that you are in error. (Horror of horrors!)
...
Let's stop this pedantic nonsense, shall we? Your example does
illustrate a possible kind of phenomenon. The question is whether
or not that phenomenon can actually occur in the real systems we
are interested in. It may not be possible to prove or disprove
A problem is that there is a physically plausible mechanism by
which a positive forcing can force a negative response: namely
if the Atlantic Conveyor shuts off it _may_ conceivably trigger
ice age conditions around the North Atlantic and we get
albedo runaway negative feedback... of course as forced cooling
can also produce similar runaway, this would imply we're
stuck in an ice age and would make the existence of the
interglacials a major puzzle (orbital forcing? natural oceanic
cycling? CO2 feedback? All three?)
What you say is certainly correct, but I think it is necessary
to consider two points. First, Shearer's example was linear,
and I believe Michael's original remarks referred to small changes
and an assumed linear range. The thermohaline shtuoff you discuss
is almost certainly not the result of linear dynamics. Secondly,
I don't know of any evidence of such a phenomenon in the past
leading to a permanent ice age. Do you? The most common example
in the Younger Dryas event in which what you describe seems to have
happened around the North Atlantic and perhaps more globally,
but then normal warming reasserted itself.
Although things like this aren't known for sure, it was my
impression that the general opinion was that the climate
change started, which resulted in additional CO_2 buildup
from the bisophere and this in turn enhanced the warming.
So your dichotomy is probably an oversimplification.
Also, wasn't it about 280 ppm?
>I don't want to get into the head game tobis and jbs
>are playing with linear mappings, the interesting question
>is what will happen physically.
>
> in the Younger Dryas event in which what you describe seems to have
> happened around the North Atlantic and perhaps more globally,
> but then normal warming reasserted itself.
>
>"normal warming"? Say what?
>Current conjecture is that the Dryas' events was
>due to an ocean circulation instability, what triggered
>it and the transition to the current warm period is not known.
>
>We don't even know what caused the widespread regional
>climate changes in the last 1000 years.
>
>It is clear we have warmed in the last 150 years,
>it is very likely that the warming is mostly due
>to anthropic forcing. It is not clear 1850 was a
>particularly "normal" period, nor that the climate
>back then was any better locally or globally then
>the current.
>
>Oh, and I still wonder just how and when humanity
>contemplates dealing with geochemical CO2 drawdown
>and the implications for C3 photosynthetic plants.
>Is that one we should leave to our descendants?
>
>;)
 Questions about the response of a nonlinear system to forcing 
>Let's stop this pedantic nonsense, shall we? Your example does
>illustrate a possible kind of phenomenon. The question is whether
>or not that phenomenon can actually occur in the real systems we
>are interested in. It may not be possible to prove or disprove
>this in a mathematical sense, but it is certainly possible to
>explore the question further and perhaps gain some enlightenment.
Indeed. I also recall Len posting elsewhere something along the lines of
"whenever we argue about stuff here & you go to the literature, you find that it
has all been discussed there". This is true of [a certain perspective on] this
discussion.
I recommend reading "a nonlinear dynamical perspective on climate change",
Tim Palmer, Weather, vol 48 # 10 p314. A short summary:
The article is about the response of a nonlinear dynamic system to forcing with
especial reference to climate change. To some extent, it is an argument against
"linear thinking".
Using the Lozenz model, (in a suitably sheared and rotated system), it is possible to
arrange for a positive forcing along an axis to result in a negative change in the
timeaveraged state of the system along that axis. This occurs because (by carefully
balancing parameters) the system is fairly insensitive to forcing in most parts of
the phase space. Only near the "backbone" of the butterfly is the system sensitive;
in this region a positive forcing has a positive effect, but predisposes the system
to make a "phase change" to the other wing of the butterfly which has more negative
values. If that makes little sense, you need to see the picture.
Whoops! I'm out of time. Tomorrow... Just to finish, although Palmer is in
general somewhat unimpressed by the predicitive capacities of current GCMs, his
first conclusion in "Implications for modelling" is: "Climate prediction is,
in principle, possible!".
 William Connolley w...@bas.ac.uk
>What you say is certainly correct, but I think it is necessary
>to consider two points. First, Shearer's example was linear,
>and I believe Michael's original remarks referred to small changes
>and an assumed linear range. The thermohaline shtuoff you discuss
>is almost certainly not the result of linear dynamics.
If I understand enough of this, you're saying that Shearer was correct
in substance, but not in details.
>Leonard Evens l...@math.nwu.edu 7084915537
snark
Allow me to bring the discussion back to linear feedback models. I stand
accused of intellectual dishonesty for failing to account for a perfectly
invalid counterargument. I hope I can be forgiven for some impatience that
the discussion has shifted to such niceties without actually recognizing
that fact.
The presumptions of the feedback model are as follows:
1) The climate system, defined as a global mean temperature, responds
continuously to a continuous forcing, the concentration of greenhouse
gases, directly through radiative forcing, in a way that is in principle
a functional dependence.
2) Other phenomena are dependent on temperature, and contribute to
the temperature change in a way that is in principle a defined
continuous functional dependence.
Because of the continuity of the system, for small perturbations a
linear system produces the same response, accurate to first order.
Now we consider the response of this system to a positive forcing.
Since we are interested in the mean output temperature, if all the
subsystems have finite zerofrequency response, we can consider only
the (real, finite) zero frequency gains of the ssubsytem. We are left
with the trivial representation:
 F() <
 
V 
C > W() > + <

> T
in the above, C and T represent physical quantities, and W() and F()
represent functional relationships. Everything else is supposed to
look like arrows, representing the sequence of operations.
So we have T = W(C) + F(T). In the zerofrequency limit, W(C) = w C
(with w a constant multiplier) and similarly F(T) = fT.
Then T = wC + fT
T = wC/(1F)
So we find that the sign of wC is opposite to the sign of T only if
F > 1.
Attempts to build such a system, however, always fail. The reason is
that the system hunts in the wrong direction for its solution. Consider
C=0, T=0 at some inital time, (the basic state having been subtracted),
say W linearizes to 1 (1 Degree C per 100 ppm CO2, for instance) and
F linearizes to 2 (an increase of 1 degree C by other phenomena causes
phenomenon F to add *another* 2 degrees).
There is a solution: everything balances out if T goes to 1 for C
going to 1. But this is because of a *very large* positive feedback.
This will not be realized in practice. In practice, the initial warming
will cause an increase in the temperature, and very shortly thereafter,
that increase will be fed back as a still larger increase. So unlike
systems where F < 1, the system is unstable, not homeostatic, and
in practice doesn't find its solution. Accordingly, Shearer's system
cannot be cast as a continuous (smallsignal linearizable) feedback system.
Does this all apply to climate? I think it does, except in situations
where there is no linearization, i.e., where there are discontinuities.
These would account for the basinofattraction shifts Lorenz observed
in simpler systems. (Note that it does not follow that the climate
system must have such behavior).
My sense is that the regime shifts correspond to catastrophic releases
of potential energy. Is this correct?
As for thermohaline oscillations, I am not sure that these are a case
in point at all: it seems to me that continuity, rather than the much
more severe test of linearity, would suffice to support the argument
that mean responses due to feedbacks cannot have sign reversals. Here,
I'm less sure of myself, but I can conceive of the NADW shutdown
as being part of a slow oscillation triggerred by the input, but
settling in the end to a situation where the global mean temperature
would not be reversed (save for changes in other forcings, of couurse.)
Nevertheless, I can't see that this demonstrates any dishonesty on
my part, and I can't imagine what sort of dishonesty might be displayed
by a claim that feedbacks can't reverse the sense of a system. Mr.
Shearer's counterexample misses the point entirely, and his invocation
of nonlinearity is beside the point (unlike other people's) because
his socalled counterexample was linear.
Finally, I leave it as an exercise for the reader to discover why
F < 1 is NOT unstable, contrary to Shearer's claim.
mt
In article <STEINN.95J...@sandy.ast.cam.ac.uk>,
Steinn Sigurdsson <ste...@sandy.ast.cam.ac.uk> wrote:
[Contributing to the ShearerTobis discussion of feedback]
>A problem is that there is a physically plausible mechanism by
>which a positive forcing can force a negative response: namely
>if the Atlantic Conveyor shuts off it _may_ conceivably trigger
>ice age conditions around the North Atlantic and we get
>albedo runaway negative feedback... of course as forced cooling
>can also produce similar runaway, this would imply we're
>stuck in an ice age and would make the existence of the
>interglacials a major puzzle (orbital forcing? natural oceanic
>cycling? CO2 feedback? All three?)
What you say is certainly correct, but I think it is necessary
to consider two points. First, Shearer's example was linear,
and I believe Michael's original remarks referred to small changes
and an assumed linear range. The thermohaline shtuoff you discuss
is almost certainly not the result of linear dynamics. Secondly,
I don't know of any evidence of such a phenomenon in the past
leading to a permanent ice age. Do you? The most common example
Define "permanent". There is a puzzle as to why the
Earth did leave two major glacial eras, in one case,
if I recall correctly, continental drift forcing
circulation changes was a plausible mechanism, in
our more recent case either orbital forcing or
(volcanic?) CO2 injection or both are possibilities.
The fact that we do enter interglacials suggests that
not all forcing will lead to cooling  yet there is
question as to whether CO2 changes from \sim 180 ppm
to \sim 250 ppm were the cause of, or cause by the current
interglacial.
I don't want to get into the head game tobis and jbs
Please take my word for it. I don't mean to pull rank on you,
but your comment indicates that you haven't got a clue about
what the whole discussion was about. Shearer's example had
absolutely nothing to do with possible shutting off of the
Atlantic flow mechanism. It isn't a matter of details.
Also, Michael knows all about that. This was a very specific
question.
One could go from a relatively steady climate with constant NADW
to a highly variable climate, such as is believed to have
prevailed in the Eem interglacial. Wally Broecker raised
concerns of this sort in his talk at IUGG recently.
>Please take my word for it. I don't mean to pull rank on you,
>but your comment indicates that you haven't got a clue about
>what the whole discussion was about. Shearer's example had
>absolutely nothing to do with possible shutting off of the
>Atlantic flow mechanism. It isn't a matter of details.
>Also, Michael knows all about that. This was a very specific
>question.
Ah, well, it's certainly possible that I was missing the point. I had
thought that jbs had given a specific mathematical example to prove his
point, then you had asked him to provide a realistic example that had a
physical basis. Another poster then provided the Atlantic flow example
as demonstrating jbs's broader point about feedback, which, although
decidedly nonlinear and thus not explicitly satisfying the argument
about linear systems, appeared to meet your criterion for an example with
a physical basis. I guess I erred when I thought that you were asking
jbs to focus on a real application, rather than being tied to a
nonphysical exampleapparently the important thing is for him to
provide a *linear* example with a physical basis, even though a "true"
model is likely nonlinear.
My apologies for wasting your time.
snark
: >Please take my word for it. I don't mean to pull rank on you,
: >but your comment indicates that you haven't got a clue about
: >what the whole discussion was about. Shearer's example had
: >absolutely nothing to do with possible shutting off of the
: >Atlantic flow mechanism. It isn't a matter of details.
: >Also, Michael knows all about that. This was a very specific
: >question.
: Ah, well, it's certainly possible that I was missing the point. I had
: thought that jbs had given a specific mathematical example to prove his
: point, then you had asked him to provide a realistic example that had a
: physical basis. Another poster then provided the Atlantic flow example
: as demonstrating jbs's broader point about feedback, which, although
: decidedly nonlinear and thus not explicitly satisfying the argument
: about linear systems, appeared to meet your criterion for an example with
: a physical basis.
No, the example may have contradicted the general sense of my argument
but it offerred no support for Shearer's "counterexample" which is
in no sense a feedback system.
: I guess I erred when I thought that you were asking
: jbs to focus on a real application, rather than being tied to a
: nonphysical exampleapparently the important thing is for him to
: provide a *linear* example with a physical basis, even though a "true"
: model is likely nonlinear.
It is usually true that a continuous nonlinear system has smallsignal
behavior identical with a particular linear system. If not, one should
point out why. In the present case, I point out below (in an unsatisfyingly
lawyerlike way, making two not entirely consistent arguments, ugh) that
firstly, on a long enough time scale this may not actually provide a
counterexample, and that strictly speaking it is not a feedback system.
: My apologies for wasting your time.
Sigh. Why are people so nasty nowadays? Both I and Len on my behalf are
reeling from my being accused of dishonesty, the more so because it comes
from a person that I have been quite complimentary to  Shearer's
objections to the conventional climatological wisdom are the best informed
and the best stated I have seen in this group, I appreciate his
challenges, and I have been at pains to say so. In return, he calls
me a liar, though what conceivable motivational structure he attributes
to me escapes me.
(Many thanks to Dr. Evens for his support.)
It would be a lot easier for me to let down my defenses at this point
if Shearer hadn't accused me of dishonesty. I'd much rather be considered
thick than dishonest.
Dr. Pierrehumbert's point, by the way, is consistent with mine: the paleo
record actually shows an oscillation in temperature due to NADW
shutdown. This doesn't contradict the linear zero frequency feedback
approximation which allows for *oscillatory* cooling and warming in response
to a forcing that ignoring feedbacks is known to cause a warming. All the
argument says is that at some time after the forcing stops changing
the longterm mean change accounting for feedbacks will be of the same sign
as that neglecting feedbacks.
However, this situation brings up a point similar to Shearer's. The
warming causing NADW shutdown is in the Arctic, while the cooling is
centered nearby in the North Atlantic. This is NOT a closed feedback
loop in any theoretical sense. Such a phenomenon is therefore not
limited by the elementary analysis of feedback systems. It is conceivable
in principle (though not, I believe, in the factual real world situation)
that such a system could result in cooling in the global mean. I don't
believe thhis is a wellsupported hypothesis, just that it isn't excluded
on purely mathematical grounds. This is because it ISN'T a feedback
system.
You may respond that my strong claim is now reduced to a semantic
quibble. I might be willing to acknowledge that position to anyone willing
to acknowledge that I take the pursuit of truth seriously.
As a practical matter, the case is very different on the global scale,
though taking the Arctic and North Atlantic as the system, and modelling
the supply of ice as infinite, there may well be a net cooling. On a global
scale, it is first of all unlikely that the floating meltwater would
overspread enough of the ocean to reverse the very large forcing we are
contemplating. Another point often neglected in this regard is that
by continuity, suppression of NADW formation would suppress deep upwelling
somewhere in the ocean, we know not where, which would just end up moving
surface heating from Northern Europe which could use it to the tropics which
could do without.
Regarding flow regime changes a la Lorenz, I still think they are
associated with catastrophic (in the mathematical sense of the word)
energy releases. I have built three parameterized families of simple
(though highly parallel) ocean models in my thesis work and the test
case I have used is the Holland/Lin 1975 case of the onset of baroclinic
instability in a zonal jet. (It turns out that this is an excellent test
case, because the onset of the instability is very sensitive to modelling
or scaling errors.) Under particular conditions, increasing the zonal forcing
enables an abrupt regime shift to a less energetic, but more complicated
state with baroclinic eddies. I wonder if the marginally unstable case
can meaningfully be linearized. However, in my original statements
that started this mess, I specifically excluded the case of large
nonlinear regime shifts.
Again, Shearer has started a very interesting discussion. I wonder if
he could manage to continue to do so in the future, which I would
welcome with enthusiasm if he were to find the grace to do so without
gratuitous rudeness.
mt
>Mr. Shearer exhibits a plainly physically realizable linear transformation.
>He implies but does not show that it is equivalent to a linear feedback
>system. He proposes that this contradicts the highlighted proposition,
>and that I am intellectually dishonest for failing to concede the point.
I think I know what the misunderstanding is here. Go back and look at
Shearer's posting. Where you see the variable:
y'
it is to be understood as
dy

dt
that is, the derivative of y with respect to time. Thus the equation
that Shearer posted was a differential equation, not a linear
transformation from one variable y to another variable y'. I am quite
sure of this; it is standard terminology, and in addition this way his
posting makes perfect sense.
Intellectual honesty means among other things that when one is confused
one admits to it, and asks for help. In this case a simple and short
question (merely asking what his equation meant) would have clarified
things, and would have been easier both to write and to read than the
many lines of misunderstandings and accusations which you in fact wrote.

Norman Yarvin yar...@cs.yale.edu
"I have observed that persons of good sense seldom fall into disputes,
except lawyers, university men, and men of all sorts that have been
bred at Edinborough."  Ben Franklin
Nice discussion but I think you are overlooking one physical
feature of the actual system...
There are a lot of interesting features of nonlinear systems that are
not represented by the tangent linear system. In particular, I was in error
in specifying that a small change in input made the linear analysis
adequate. In rare occasions (strictly with vanishing probability?)
a regime shift can be triggerred, in which case all bets are off. I
wonder whether this can't be considered equivalent to a discontinuity
in the system, and hence a situation in which the tangent linear system
is undefined, in other words, a linear feedback model does not exist
even for very small perturbations under those rare conditions.
...
1) The climate system, defined as a global mean temperature, responds
continuously to a continuous forcing, the concentration of greenhouse
gases, directly through radiative forcing, in a way that is in principle
a functional dependence.
2) Other phenomena are dependent on temperature, and contribute to
the temperature change in a way that is in principle a defined
continuous functional dependence.
Because of the continuity of the system, for small perturbations a
linear system produces the same response, accurate to first order.
...
Then T = wC + fT
T = wC/(1F)
So we find that the sign of wC is opposite to the sign of T only if
F > 1.
Attempts to build such a system, however, always fail. The reason is
that the system hunts in the wrong direction for its solution. Consider
...
There is a solution: everything balances out if T goes to 1 for C
going to 1. But this is because of a *very large* positive feedback.
This will not be realized in practice. In practice, the initial warming
will cause an increase in the temperature, and very shortly thereafter,
You are assuming here the system is noise free. In practise
the system is noisy and there will be some spectrum of finite
frequency, small amplitude perturbations that may provide an
initial swing to a negative response trapping the system on
a cooling trajectory. In principle, ignoring nonlinear
state transitions.
Does this all apply to climate? I think it does, except in situations
where there is no linearization, i.e., where there are discontinuities.
These would account for the basinofattraction shifts Lorenz observed
in simpler systems. (Note that it does not follow that the climate
system must have such behavior).
My sense is that the regime shifts correspond to catastrophic releases
of potential energy. Is this correct?
I don't think this is true in general.
As for thermohaline oscillations, I am not sure that these are a case
in point at all: it seems to me that continuity, rather than the much
Thermohaline oscillations provide several secondary feedbacks;
notably albedo feedback through ice formation/melting; in principle
there may be albedo feedback on cloud coverage as precipitation
patterns change, and there may be a shift in the carbon cycle to
greater CO2 absorption or a net CO2 release depending on what
happens to deep water circulation.
One interesting point is that there appears to have been a
large (fractional) change in CO2 near the onset of the current
interglacial which then drew down, presumably because of the
response of the carbon cycle to the transient.
: Nice discussion but I think you are overlooking one physical
: feature of the actual system...
: ...
: Then T = wC + fT
: T = wC/(1F)
: So we find that the sign of wC is opposite to the sign of T only if
: F > 1.
: Attempts to build such a system, however, always fail. The reason is
: that the system hunts in the wrong direction for its solution. Consider
: ...
: There is a solution: everything balances out if T goes to 1 for C
: going to 1. But this is because of a *very large* positive feedback.
: This will not be realized in practice. In practice, the initial warming
: will cause an increase in the temperature, and very shortly thereafter,
: You are assuming here the system is noise free. In practise
: the system is noisy and there will be some spectrum of finite
: frequency, small amplitude perturbations that may provide an
: initial swing to a negative response trapping the system on
: a cooling trajectory. In principle, ignoring nonlinear
: state transitions.
If you're saying what I think you're saying, this is also incorrect.
I presume you are saying that the system could find its negative solution
because of very small random perturbations. However, the very small
random perturbations would have to outweigh a substantial initial
forcing in the wrong direction, and then would have to converge on
a trivially unstable solution.
Indeed you have the right phenomenon but the wrong analysis. Consider
the linear stability of the solution of the system around solutions
where F>1 and where F<1. You will find that the F>1 solution is about
as stable as balancing a pin on its tip. It's not even close 
the system always diverges to infinity (until, of course, the model breaks).
Anyone who has played with active analog circuits understands this.
In particular, if C=0; t<0, and C>>0; t>=0, even if the system had somehow
been restrained to T=0;t<0 (which would have to be an active constraint
precisely because of the instability) your perturbation noise would
be quite swamped by the input.
This is all beginning undergrad EE stuff. There is probably a more
sophisticated body of knowledge involving nonlinear feedbacks of which
I am ignorant. It would be interesting to get a better grip on whether
and how this analysis (yes, it extends neatly to countable dimensions and
nonzero frequency cases) applies (or why it doesn't) to Lorenzian fluid
regime changes.
mt
: >Mr. Shearer exhibits a plainly physically realizable linear transformation.
: >He implies but does not show that it is equivalent to a linear feedback
: >system. He proposes that this contradicts the highlighted proposition,
: >and that I am intellectually dishonest for failing to concede the point.
: I think I know what the misunderstanding is here. Go back and look at
: Shearer's posting. Where you see the variable:
: y'
: it is to be understood as
: dy
: 
: dt
: that is, the derivative of y with respect to time. Thus the equation
: that Shearer posted was a differential equation, not a linear
: transformation from one variable y to another variable y'. I am quite
: sure of this; it is standard terminology, and in addition this way his
: posting makes perfect sense.
Yes, I was aware of this. His posting does make sense, until he uses
it as if to display a counterexample of a feedback system whose closed loop
gain is opposite in sense from its open loop gain. That is because he does
not and *cannot* display this system as a stable feedback system.
Note that the derivative *is* a linear operator, so linear feedback
theory applies to systems with differentiators.
However, my simplification to zerofrequency doesn't apply directly,
since the equations have to be recast in integral form and the zero
frequency gain of an integrator is infinite. This raises some
subtleties I had hoped to avoid. Perhaps you will take my word for it
that these can be addressed adequately in a Laplace transform argument,
and that the structure of the argument and the general conclusion about
stability hold in that more general case. (The system will have a
pole in the right halfplane of the transform space if the zerofrequency
gain changes sign when the loop is closed, implying instability. I am NOT
about to explain this to a general audience in ASCII five weeks before
I hand out my dissertation, thanks.)
: Intellectual honesty means among other things that when one is confused
: one admits to it, and asks for help. In this case a simple and short
: question (merely asking what his equation meant) would have clarified
: things, and would have been easier both to write and to read than the
: many lines of misunderstandings and accusations which you in fact wrote.
I make no accusations that I know of. To the contrary, I am compelled to
respond to an accusation. If I have had any misunderstandings, I suggest
you have not shown them. I was quite aware that Shearer's example was
a linear differential equation. (There was no ambiguity at all, since
he presented the usual DE solution.) Since it's linear, it cannot be a stable
feedback system, and thus it cannot represent physics in which a phenomenon
which is dependent on temperature in turn changes the direction of the
shift of the same temperature. This is an established result and claims
to the contrary are wrong. Shearer's "counterexample" was presented as
a claim to the contrary, and it is wrong. I acknowledge that the error is
a subtle one, but it's a real one. My main problem is with the corrolary
that since I didn't open this very can of worms a few months ago I must
be dishonest.
mt
It's a differential equation, and it is stable. What do you mean by a
"feedback system", if not a differential equation? Is it a difference
equation? An integral equation? Can it be multivariate or must it be
univariate? All I saw in your last article was a diagram of arrows and
letters, which is a most ambiguous form of definition.

Norman Yarvin yar...@cs.yale.edu
Let me pull rank on you here. I am a mathematics professor who has
taught this material for many years. I've also discussed this
with Michael in private discussions. He knows perfectly well
what he is talking about. His use of language here was perhaps
a bit loose, and I can understand how someone who is not familiar
with the subject may have misunderstood him. However, I can
assure you he is not confused about this. If anyone is confused
it is you, and that is indicated by the nature of your comment.
Michael has expressed uncertainty about the following question
here and in other postings. Let me try to set it straight to
some degree. Given a nonlinear system of ordinary differential
equations, suppose you have an isolated equilibrium point. That
means that if the state of the system puts you at that point exactly
you won't leave it ever afterwards. Part of the study of
such systems proceeds by considering the linear approximation to
the nonlinear system in the immediate neighborhood of the
equilibrium. Picking coordinates which give the displacement
from equilibrium, the picture is something like this. Let
y be a vector function of time with as many components as
necessary to describe the system. The nonlinear system would
have the form
y' = f(y,t)
for some function f. Often, the function doesn't acutally depend on
t, and the form would become
y' = f(y).
Now take the linear terms of the multivariable Taylor series for
f. The above can be written
y' = Ay + higher order terms
where A is a certain square matrix. y > Ay is of course
a linear transformation. You can now study the linear system
y' = Ay.
Such systems are generally much easier to study. Then the
following question arises. Does the behavior of the solutions
of the linear system agree in some sense which can be made
precise with that of the solutions of the original nonlinear
system. Questions like this were answered by mathematicians
quite a long time ago. The answer is `Yes' unless special
conditions hold among the eignevalues of the matrix A.
So you can say that generically the answer is `Yes'. That
is if they don't hold there has to be something special preventing
it. Now special things do include things like conservation
laws, and conservations laws do hold for systems describing
climate. So it would depend on the effect of the conservation
laws in the particular case. This is a question that
Michael would be in a much better position to consider than
I for the application to climate. My initial guess is that
the conservation laws aren't of a nature so as to affect Michael's
belief that the generic situation is that the linear approximation
is a good approximation sufficiently close to the equilibrium
point, but I am ready to be convinced otherwise.
If you have nonisolated equilibrium points, the situation
can be more complicated. I tried to construct a semiplausible
3 dimensional linear `climatological' example which was roughly of
Shearer's type which ended up with a plane of equilibrium points.
But it did not exhibit anything like the kind of phenomenon
Shearer's example did. If you were in the vicinity of one
equilibrium point and started with plausible initial conditions,
you would just cycle about that same point and not move to another.
My colleague also commented that if one had an equilibrium
orbit rather than a point, i.e., the solution exhibited
periodic behavior once started with the appropriate
conditions and the orbit also satisfied another condition
I won't go into here, then one could state similar results
but they are much more compplicated.
I hope this clarifies some of the discussion. If you didn't
understood most of the words in what I just said, you should
first study differential equations at least the junior/senior
level before commenting further.
: It's a differential equation, and it is stable. What do you mean by a
: "feedback system", if not a differential equation? Is it a difference
: equation? An integral equation? Can it be multivariate or must it be
: univariate? All I saw in your last article was a diagram of arrows and
: letters, which is a most ambiguous form of definition.
Good question. I don't recall the standard aswer, so I'll try
to construct one. The essential feature of feedback systems is
that the processes are logically sequential. That is, we limit
functional blocks to represent realizable processes that depend
only on past values of their input, i.e., that are neither instantaneous
nor clairvoyant. This corresponds well to distinct physical processes,
and there is a large and interesting body of theory based on this
approach, one which is wellknown to mechanical and analog electronics
engineers. Anyway, here goes my attempt, and any corrections or
improvements would be appreciated. It may be more clear with reference
to my crude diagram.
***********************************************************************
An openloop system is defined as a system with an input and an output
which has a defined realizable functional dependence on the input history only,
with the possible exception of stochastic additive quantities at the
input. A closed loop (feedback) system is an open loop system with
an addition at the input of a realizable function of the output history.
By a realizable function is meant one whose result is dependent only on past
values of its input, not on instantaneous or future ones.
***********************************************************************
You may quibble (I know there are mathematicians out there :) ) that an
openloop (jargon  actually it is no loop at all) system is a special
case of a closed loop system with a feedback function of zero, under that
definition. Fair enough, but it still doesn't provide an example of a system
whose closedloop zero frequency gain is of opposite sign from the openloop
gain.
Under these definitions, the zero frequency argument isn't formally sound,
but I'm pretty sure there is an equivalent formally sound argument
in the Laplace transform domain. I'm not about to find it or reconstruct
it just now, though. But the basic idea is the same. If you try to reverse
the sign of the zero frequency gain using feedback you'll get an unstable
system (one in which random perturbations form the solution are amplified
rather than corrected).
This is "cybernetics" in the sense that the word was originally coined 
homeostatic feedback systems. (I'm the only person I know who has actually
read through Norbert Wiener's peculiar halftechnical exposition of the
subject, keeping a promise made to myself in my teen years when I read
the nonmathematical parts. However, this was long ago, and I'm not
claiming to be particularly adept with this material right now.)
Yes, of course the variables can be nonscalar. I'm not sure what the
analogous statement would be in that case, though. The formal statement
of instability in the onedimensional case refers to right halfplane
"poles" (points where the value of the function goes to infinity) of
the Laplace transformed transfer function. It's not obvious, offhand, to
me what the extension to a frequency domain vector function is. Since
the original physical discussion referred to feedbacks from temperature,
a scalar, I fortunately didn't have to think about that.
mt
But reasonable jargon. The path from input to output then back to input is
known as the feedback loop. When working with such a system, it's often
desirable (say, when your amplifier starts to oscillate and you want to prevent
damage to your equipment and/or your eardrums) to eliminate the feedback
without shutting down the system. You typically do this by disconnecting part
of the system in the path from output back to input. I.e., you want to open
the circuit somewhere in that part of the path. This is known as breaking, or
opening, the loop. When the feedback path is intact, the loop is referred to
as closed; otherwise as open. In particular, when a system is running with the
feedback path intact, it's described as running in a closedloop configuration.
When the feedback path is interrupted, as running in an openloop
configuration. The latter is indistinguishable from a system without feedback.
Hence the terms openloop and closedloop for systems.

Carl J Lydick  INTERnet: CA...@SOL1.GPS.CALTECH.EDU  NSI/HEPnet: SOL1::CARL
Disclaimer: Hey, I understand VAXen and VMS. That's what I get paid for. My
understanding of astronomy is purely at the amateur level (or below). So
unless what I'm saying is directly related to VAX/VMS, don't hold me or my
organization responsible for it. If it IS related to VAX/VMS, you can try to
hold me responsible for it, but my organization had nothing to do with it.
Ok consider the following the example. We have two
cylindrical containers p1 and p2 containing w1 and w2 units of water.
Let h1 and h2 be the height of the water level in p1 and p2 above the
bottom of the container. Let s1 and s2 be defined so h1=s1*w1 and
h2=s2*w2. Suppose container p2 is sitting on a spring so that adding
1 unit of water to p2 causes container p2 to sink t units. Suppose
we are adding water at rate a to p1. Suppose there is a hole in the
bottom of p2 causing it to lose water at a rate c*h2 (proportional
to the height of the water in p2 above the bottom). Suppose there is
a flexible hose connecting p1 and p2 which transfers water between
p1 and p2 at a rate proportional to the difference in the height of
the water surfaces in p1 and p2 (note this difference is not h1h2).
Suppose finally the spring is damped and responds much faster than
the water is moving. Then w1 and w2 are related by two simple
linear differential equations.
I believe that for certain values of b, c, s1, s2 and
t increasing a will cause the equilibrium amount of water in p1 to
decrease (a response of opposite sign to the forcing). In particular
if I have calculated correctly this will be true if (1+c/b)*s2 < t <
(1+c/b)*s2 + s1 (the upper bound on t is required for stability).
Michael Tobis claimed:
>No, the example may have contradicted the general sense of my argument
>but it offerred no support for Shearer's "counterexample" which is
>in no sense a feedback system.
I have been assuming feedback was meant in the general sense
of interaction with other variables. This is necessary if one is to
consider the climate system a feedback system.
My example is a feedback system in the following sense.
Suppose we have black boxes A and B connected as follows:

forcing> + >A input> Box A >A output>>
 ________ 
 
  
<B output< Box B <B input<
________
The output of A is transformed by box B and then combined with the
forcing and fed back to box A one time step later. If we allow A and
B to have memory (specifically to remember their last output, so that
the current output is a function of the last output and current input)
it is easy to see that we can chose A and B to emulate the example I
posted (more precisely the similar difference equations which will
behave in the same way as the time step goes to zero). Tobis can
deny this is a feedback system but then it is hard to see how he can
argue that the climate system is a feedback system. For example the
ocean temperature will lag changes in air temperature indicating a
process with memory.
I will be off the net for the next 6 weeks so please copy me
by email if anyone has anything to add to this thread that they wish
me to see.
James B. Shearer
No, he's been arguing that that's the case for a linear feedback system with
underunity zerofrequency gain. He's made that quite explicit a number of
times in this thread. Why do you chose to ignore that?
Ouch! Look, people generally have no responsibility to stay in Usenet
conversations, but this is bad manners. You didn't just disagree with mt
about a scientific issue, you impugned his knowledge and possibly his
character. I would think that you owe him a chance to refute your
accusation in an extended discussion of the scientific question at hand,
without simply replying to his post and then going away before he can answer.
I have mailed this to jbs as he requested.
The above definition includes those types of differential and difference
equations which contain a time variable and use it in the normal manner
(evolving forwards in time). Thus Shearer's counterexample would be a
"feedback system", using the above definition. (which is admittedly
not a formal definition, so I cannot say this for sure.)
>***********************************************************************
>An openloop system is defined as a system with an input and an output
>which has a defined realizable functional dependence on the input history only,
>with the possible exception of stochastic additive quantities at the
>input. A closed loop (feedback) system is an open loop system with
>an addition at the input of a realizable function of the output history.
>By a realizable function is meant one whose result is dependent only on past
>values of its input, not on instantaneous or future ones.
>***********************************************************************
By this description Shearer's example, but with the matrix set to zero,
is an openloop system, and his example with the matrix as it was
stated is a closedloop system. Running openloop the equation would
look like:
y' = 1
0
which gives the solution
y = t+c1
c2
i.e. y(1) increasing at a constant rate and y(2) constant.
Running closed loop the solution is as he stated: stable with y at a
negative value.
I suppose strictly speaking, the "open loop system" above isn't stable;
in particular it is metastable (meaning that small deflections added to
the solution neither decay nor grow.)
But this is a strange way to look at the issue. Running open loop,
i.e. setting the matrix to zero, is like subtracting all the atmospheric
dynamics, and leaving only the warming from CO2, whereupon the
atmosphere warms up at a constant rate, heading toward infinity. It
seems more useful to look at the difference between the full
atmospheric dynamics with no warming added, and the full atmospheric
dynamics with a warming input added in. Mathematically, this is the
difference between an equation with no forcing vector and an equation
with a forcing vector. This is the case in which Shearer's
counterexample shows that there can be reversal of sign.
>Yes, of course the variables can be nonscalar. I'm not sure what the
>analogous statement would be in that case, though. The formal statement
>of instability in the onedimensional case refers to right halfplane
>"poles" (points where the value of the function goes to infinity) of
>the Laplace transformed transfer function.
It is only a slight exaggeration to say that in modeling atmospheric
dynamics, everything is nonscalar. Even the smallest portion of the
atmosphere has at least the variables of temperature, pressure, and
velocity to consider; and one needs to consider these variables at
thousands of points in the atmosphere. Of course one can look at the
temperature at every point in the system, and compute an average, but
the "feedback" does not operate through the average temperature.

Norman Yarvin yar...@cs.yale.edu
"This impatience was very foolish, and in after years I have regretted
that I did not proceed far enough at least to understand something of
the great leading principles of mathematics, for men thus endowed
seem to have an extra sense."  Charles Darwin
: T = wC/(1F)
Actually my claim was weaker than this  I was simply noting that
even if a forcing is towards an unstable regime, with an unreachable
stable solution in the opposite "direction" then the presence
of finite amplitude noise may permit you to reach the stable solution
against the direction of the forcing.
Indeed you have the right phenomenon but the wrong analysis. Consider
...
Funny, I was just about to say that.
A second concern is that your model doesn't really approximate
the true system being considered. In particular, you were considering
the climate as a single parameter system  represented by a global
(time averaged) spatially averaged mean temperature.
However, consider a false model of thermohaline circulation.
Imagine the poles were ice free. Now, in principle, there might
be a ocean circulation feedback such that as CO2 causes tropical
forced warming, circulation falls  with the weakening linear in
some perturbative sense. This could in principle cause polar cooling,
such that the mean polar temperature dropped (by a lot if it went
from ice free to permanent ice), you could then find albedo cooling
and weakened convective warming leads to the mean cooling at the
poles being greater than the radiative forcing, averaged globally,
and this could be a linear response over an finite regime.
In this toy model  and this is _not_ the way things are  a
positive forcing can cause average cooling, with a linear response,
because you are averaging two subsystems that are strongly coupled,
with the cooling of one outweighing the warming of the other.
Things ain't so simple.
: Ouch! Look, people generally have no responsibility to stay in Usenet
: conversations, but this is bad manners. You didn't just disagree with mt
: about a scientific issue, you impugned his knowledge and possibly his
: character.
Disagreeing *is* impugning knowledge, and science proceeds by the
embarassing but functional method of calling other people's claims
into question. I don't object to this at all, though I don't relish it
when it's my own claims that are questioned.
I object to characterizations that I don't play the game fairly, of course.
I would think that you owe him a chance to refute your
: accusation in an extended discussion of the scientific question at hand,
: without simply replying to his post and then going away before he can answer.
By asking to remain updated, he has proceeded correctly. You can't expect him
to run his life based on usenet. By keeping in touch, he is actually
being quite scrupulous, perhaps because the whole basis of his attack on
my character is that I failed to address his original effort to provide
a counterexample to my claim. I continue to claim that I was not dishonest
because his counterexample did not satisfy me.
His new counterexample is also wrong, I am sure, but at the moment I could
stand to leave this aside for a few weeks.
Sigurdsson's counterargument addresses the sense of my assertion without
invalidating its letter  strictly speaking it still isn't a feedback.
I acknowledge that in cases where the response is not in the same domain
as the forcing, the contrary responsive physics may remotely plausibly
reverse the sense of the integrated response. Such cases must be argued
casewise, and the physics independent feedback arguments do not apply.
I am confident that (leaving aside the dubious hydraulics) Shearer's latest
example can be shown to be invalid. I will eventually endeavor to do so,
provided that I survive the rest of the summer, get my thesis wrapped up
and signed, and move on to my new position. For now I will appreciate a
respite from this conversation.
mt
: I have mailed this to jbs as he requested.
ditto.