## Curry: Thermodynamic Feedbacks in the Climate System

Judith Curry writes:

I’ve posted the chapter on Thermodynamic Feedbacks in the Climate System from my text “Thermodynamics of Atmospheres and Oceans” on my website, the links can be found at

Text: http://curry.eas.gatech.edu/climate/pdf/Ch13_GalleyC.pdf

Figs: http://curry.eas.gatech.edu/climate/pdf/chapter13_figs.pdf

For my more recent thoughts on the subject of climate feedbacks, I refer you to my previous post on the spencer thread, post #23

I will have some time this week (but not alot) to respond to any comments

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## 126 Comments

Judith, thanks!

A couple of thoughts from a brief look:

* your description of the control-theory analogy is much better than what I was attempting on another thread here, thanks.

* In getting to equation 13.3 you reduce all the climate variables essentially to one – the global mean surface temperature (?) T0. Maybe a sentence or two on how realistic that reduction is, what impact it has on the remaining arguments to retain a larger space of internal climate variables perhaps, would be of interest…

OK, so Judy has referred us to previously published summaries of feedbacks, without actually addressing my main point, which is that the time-averaged co-variation of observed surface temperatures and clouds is NOT a measure of feedback (only), but a combination of feedback PLUS a (potentially large) positive bias due to non-feedback short-time-scale cloud forcing of surface temperature. This has biased our estimates of feedback in the climate system in the positive direction.

Until she addresses this point, which both Held and Forster admit is a valid one, I will assume that she has added nothing new to the discussion that was started back in the ‘Spencer on feedback’ thread.

I apologize for sounding so harsh, but at some point the endless repetition of old science like a mantra gets annoying for those of us who are trying to advance the science beyond IPCC talking points.

Roy #2, remember this is a text book. In my experience (Australian high schools) they take a year to produce, and are conservative rather than leading edge. It also takes a while to get the mistakes corrected; for some books this process is not finished before the next book comes out.

Judith, I am looking forward to reading this work. I, along with probably many others, have a lot of catching up to do.

Peter

Roy, in my post over on the spencer thread, I clearly stated that feedbacks were frequency dependent, and that they varied regionally, and that in a nonlinear system these things did not add up in a simple way.

Here is what I stated previously over on your thread (I am reproducing the main points from the previous post, since they are more relevant here)

Evaluating feedbacks using models

Some causation processes are represented explicitly in models; others are indirect results of nonlinear processes.

In a chaotic, nonlinear climate model with 10**7 10**9 degrees of freedom, we do not know how to evaluate the feedbacks.

It is not possible to unambiguously separate individual feedback loops.

Estimating feedback through equilibrium simulations of GCMs, linear analysis, or analysis of vastly simplified models can be misleading.

It is not possible to identify the most important feedbacks.

Example: Cloud feedback

Cloud feedback is regarded as a very important climate feedback.

This evaluation is tied to the magnitude of cloud radiative forcing

In a complex nonlinear system, a large forcing does not necessarily translate into a large and important feedback.

Evaluation of cloud feedback in GCMs using a simple linear analysis shows model disagreement in both magnitude and sign

If plausible projections can be made with different signs of the cloud feedback, it is possible that cloud feedback is not important.

LESSON: Do not confuse forcing with feedback.

Example: Snow/ice albedo feedback

Snow/ice albedo feedback is regarded as a positive feedback.

The sign depends on the time scale under consideration

On glacial time scales, there is a period that follows the onset of warming where snow/ice extent increases, owing to an increase in snowfall LESSON: The magnitude and sign of a feedback can be frequency dependent or associated with a substantial time lag

Example: Water vapor feedback

Water vapor feedback is generally regarded as being positive

However, one study of tropical convection suggests a negative water vapor feedback.

One study in the Arctic suggests a strong positive feedback

LESSON:

The sign and strength of a feedback can vary regionally

——–

So I don’t disagree at all with your point. There is no way we can use simple linear control theory to diagnose true climate feedback; but the kinds of diagnoses used in climate models serve to assess discrepancies among models and with observations, that is their true utility

And yes, pjm is correct, this is a climate textbook, feedback 101 for beginning graduate students, based upon linear control theory at a junior level (undergraduate) engineering curriculum. Few climate scientists have taken in a course in elementary systems and controls. My chapter was designed to help bridge this gap.

Re using surface temperature, any variable could be used in such an analysis. I once did such an analysis where i used sea ice thickness instead.

Dr Curry,

Thank you.

I just started reading this and it is excellent material (despite the fact that I’ll have to review my Calculus to be able to understand it, not using it in my profession – IT Security). I look forward to working my way through it. I’ve become what I would classify myself as a ‘well read AGW skeptic’ via following this site and others (still waiting for that check from Exxon). It has become something of a ‘hobby’ for me and my discussions with some of the general public (and petition waving AGW advocates at local malls) have become very entertaining.

yes, Peter, I know that. Of course, for those who are still learning about feedbacks, her posts and references are certainly useful. I’m referring to Judy not addressing the original point regarding feedbacks, and instead simply referring everyone to what is just the current status quo in the diagnosis of feedbacks.

I’ve suggested before that water, in all its phases, may be either a positive or negative feedback, depending upon need. No hints for mechanisms from me, sorry.

=============================

http://www.phys.unsw.edu.au/RESEARCH/ATMOSPHERIC/atmospheric_research.html

Feedbacks or forcings? …. is there is such thing as an “indirect forcing” or is that really a feedback? And if so, what is its sign, what would be the PDEs to describe its behavior?

One of you said the climate is just the extension of the ocean by other means. Water is the agent by which the ‘English’ of the cosmo-magneto-solar effect is placed on the dynamo of the heat engine that is the earth. I regret that I can only approach the science metaphorically.

============================

Do we know how to bracket the uncertainty? Policymakers might be interested in that – if they knew it existed, or understood its implications.

Water is at various times a forcing, a positive feedback and a negative feedback. It’s all kinda mixed up.

Here’s part of what the TAR sez:

http://www.grida.no/climate/ipcc_tar/wg1/231.htm

Also see “The radiatve forcing due to clouds and water vapor” V. Ramanathan and Anand Inamdar

Frontiers of Climate ModelingBy definition, a forcing has to be an influence from outside of the climate system, so water can never be a forcing, unless you want to suggest that anthropogenic water vapor can force.

It would also be nice if we could simply narrow it down to the primary influences and bracket their individual contributions. Even in extremely complex feedback systems, there are generally only a few poles and zeros that dominate the final output.

Mark

All I know is the IPCC said changes in water vapour are a forcing when they are a result of CH4

oxidation, and they said that Forster and Shine (1999) have estimated a radiative forcing of 0.2 Wm-2 since 1980. I’ll take that as not “never”.

14, I think they’re alluding to contrails.

Bender, re uncertainty, here is excerpts from a presentation I made a few years ago to the NAS Climate Research Committee:

CCSP (the President’s Climate Change Science Program) emphasizes reducing uncertainty.

Reducing uncertainty is probably not the appropriate goal;

we should instead focus on increasing credibility.

FUNDAMENTAL QUESTION:

Is the CCCSP torquing climate science in a direction that is fundamentally less useful for both science and policy?

The answer to this question is probably yes, and both the root of the problem and its eventual solution lies in how scientists and decision makers deal with the issue of uncertainty.

Scientists and funding agencies are better rewarded for proposing and funding large new observational systems, rather than for careful analysis of existing data sets, and preferred data sets become a political issue as the stakes for publicity and funding is accelerated.

Climate modelers (and the agencies that fund climate modelers) infer pressure from policy makers to reduce uncertainties in climate models implied by the spread among predictions made by different models.

This pressure may be torquing climate science in the wrong direction:

By adding new degrees of freedom to individual climate models (e.g. increasing the number of prognostic variables, model resolution, etc.), disagreement among model projections is likely to increase. Such disagreement is a sign that progress is being made in understanding the importance of previously neglected processes.

Given that climate models cannot presently be evaluated using out-of-sample observations, focusing solely on reducing the range of model projections will mislead and there will be no motivation to uncover common flaws among the models.

Evaluation of model errors and prediction uncertainty is essential to establish the credibility of climate projections for decisionmaking

Model errors:

Errors in functional relationships

Numerical errors (coding, model resolution)

Errors in treatment of unresolved degrees of freedom (subgridscale)

Neglect of important processes (e.g aerosol processes)

Prediction uncertainty:

Imperfect models

Chaotic behavior of system

Imperfect initial, boundary conditions

Ensemble simulations are a MUST, and frequentist distributions must be generated for each model, otherwise we are faced with a single unequivocal prediction from each model that may be very far from reality and no understanding of prediction uncertainty using that model.

Simply because it is difficult to evaluate climate model predictions using observations does not imply that we cannot conduct rigorous error analysis on climate models and conduct a rigorous uncertainty analysis on climate model projections.

Basic statistical good practice requires that we do the error analysis and Monte Carlo runs, before we revert to the use of subjective probabilities to combine the conflicting model distributions.

By introducing subjectivity too early, the hard stuff is avoided but we significantly reduce the value of the uncertainty analysis to policy makers.

[deleted a bunch of stuff about how to communicate uncertainty to policy makers and other decision makers]

Credibility = rigorous analysis of observation/model errors

+ uncertainty in projections from monte carlo simulations

+ expert judgment + common sense

Judith Curry says:

Why can’t the policy makers be simply be told that there will always a great deal of uncertainty with these kinds of models and that it is possible that the consensus is completely wrong about the negative impacts of CO2 induced warming. It should be possible to craft sensible policy options that do something to reduce CO2 but take this uncertainty into account.

Contrails create a positive forcing, yes, the values given are about 3-17 mW/m² But I believe they’re talking about chemical oxidation of methane creating water would make the water a forcing.

Regardless, I’m simply saying the IPCC, at least in the TAR, considers water vapor a forcing under certain circumstances. But I’m certainly not saying it usually is or that it’s its primary role. Just that it’s not “never” a forcing.

Although according to the definition, I’m not totally sure if they count the oxidation of methane in the upper stratosphere (which is the main source of water vapor starting at 10 kPa).

TAR WG1 611

wiki sez:

Whatever “no changes in stratospheric dynamics” means… :)

This is the chart on their page for radiative forcing, which includes water vapor as one.

Forcing are external to the system, and feedbacks are internal to the system. Part of the challenge is how you define the system. If the system is the atmosphere, then sources of aerosols say from smokestacks are a forcing. But if you have an atmospheric chemistry parameterization in your model that actually creates aerosol particles, or you have an interactive land surface, then aerosols are not a forcing.

Cloud feedbacks are so interesting because they represent phase boundaries in the system, which are notoriously difficult to to treat in a model of a fluid, and yes their feedbacks can fluctuate in terms of sign and magnitude (depending on the local thermodynamic and dynamic state). Cloud parameterizations represent a key challeng to climate modelers, but diagnosing cloud feedback in these models and inferring much from these diagnoses may not be very fruitful. Some of the diagnoses of cloud feedback (mostly based on linear control theory and the partial derivatives arising therefrom) have some utility as a metric in comparing differernt models and in comparing models with observations.

#16

Oh my.

What is the provenance of this section you’ve cited Dr. Curry? Is this something your team put together?

I’m curious to hear the reactions of other statistician types regarding Dr. Curry’s #16.

As I said elsewhere, anyway; almost all of the anomaly trend is since 1980ish. Except for a few lower anomaly years here and there.

Since around 1995

noneof the monthly figures (GHCN 1880-11/2007 + SST: 1880-11/1981 HadISST1 12/1981-11/2007 Reynolds v2) are negative, or even under 10 or so.Somethings changed drastically, and it’s too quick and abnormal to attribute to some unknown unprovable lag, IMHO. And it certainly doesn’t seem like anything logarithmic could cause it

So what forcings/feedbacks account could account for this drastic change? Or is it non-climate system?

Re #16 Re provenance, this is my own presentation that I made to NAS Climate Research Committee, in the process of planning for an “uncertainty workshop”. i am going to get this posted on my web site. I sort of got eased out of the eventual workshop, and the agenda of the workshop that eventuated can be found at

http://dels.nas.edu/basc/crc_10-04_public_agenda.pdf (the actual presentations dont seem to be online anywhere)

Although I haven’t looked at the data myself, I’ve interpreted some of Pielke Sr.’s literature showing that the increases are due to a rise in Tmin, not Tmax (recalling that the “average temperature” for a day is the average of the daily high and the daily low).

I don’t have a link, but my recollection is that Pielke suspects surface effects due to land cover changes.

Mike B: I would attribute it to mostly land-use, with an unknown amount of the GMTA rise due to the way we measure and process the data and/or AGHG. Coupled with the thermodynamic feedbacks.

Kudos to Judith for dealing with the many uncertainties. That will help gather a better understanding of what is actually going on.

Judith – re the last line of #4 on using surface temperature T0 as the variable – two points:

1) My actual question was more whether the text ought to include some more discussion about why (or whether) it’s justifiable to reduce the internal climate variables down to 1 at that point, and what the consequences would be for the following analysis if you kept several (or a large number of) internal variables.

2) Another idea I had though, from another thread – the most meaningful internal variable is not the mean surface temperature (your T0), but Earth’s internal energy. Changes in Earth’s energy are governed almost entirely by the radiative budget; imbalance leads to heating, an increase in internal energy, and a temperature increase. But instead of a temperature increase, you may get melting ice or something else that corresponds to an increase in internal energy and those makes up for the imbalance for a while. Wouldn’t it be more meaningful to work from that basis in all the analyses?

Thirty years after DIF-EQ I will leave most of that chapter alone, but the final section on thermohalene feedbacks was quite interesting.

#22

Thank you very much for the link, as you know uncertainty estimation is my favorite topic at CA. After reading that abstract I was enthusaistically looking forward to asking about hardcopy follow-up. But upon re-reading your comment I was disappointed to see the last sentence:

Any chance of bringing that material together? Or has some of it now been published? I wonder what fraction of this material was included vs ignored in AR4.

re Dr. Curry #19

What exactly is an interactive land surface?

Sounds difficult to draw the line:

– Wind storm dust is internal to the ‘atmosphere system’, dust from an open pit mine is a forcing, what about a wind storm by an open pit mine?

– Lightning induced forest fire would be part of the ‘system’, but a careless campfire run amok would be a forcing?

I have a naive question: After a quick skim of the chapter, I noticed that the only mention of wind was in the context of surface heat flux. Thinking about Arthur’s notion of using the internal heat energy as the variable of interest, is it possible that a radiative forcing gets partly fed into increases in wind energy? Obviously, such an effect would lead to changes in the variance and distribution of winds over time and space, but is it possible to abstract from that and come up with an estimate of the radiative energy diverted into this increased kinetic energy? Or do we already know that this is trivial?

Not likely trivial, srp; it certainly may be the mechanism by which signs and magnitudes as referenced by JC in #19 can change locally, and rapidly.

Thank you for #19. But what about my suggestion that the signs of water feedback may be dependent upon need? Surely, the climate self-regulates; why haven’t we spiraled to either too hot or too cold? I know, the question is too large or too ignorant.

===============================

Judith Curry (page 27):

Does this mean that absolute humidity over warm ocean will decrease when temperature of the ocean will increase?

Re: Judy’s comment #4, (one more try..)

Yes, Judy, we are in agreement regarding the complexity of feedbacks. But I still get the feeling that my central point is being missed…possibly by many of the participants here as well…so please permit me one more chance, and I will try to be as clear as possible:

In the theoretical treatment of the components of forcing and feedback discussed by Forster and Gregory (2006 J. Climate), they included a term which represented ‘internal variability’. They said that the term could be neglected because, to the extent that it wasn’t correlated with surface temperature variability, it’s effect on the observational estimate of feedback would be zero.

We pointed out that this was incorrect, because ANY heating term MUST be correlated with temperature, since ANY heating term causes a temperature change! (and Forster has admitted this error to us).

Now, here’s the consequence of this for the observational estimation of feedbacks:

**Any internal variability that is not the result of a feedback will cause the observational estimation of feedbacks to be biased in the positive direction.**

In other words, when estimating a cloud feedback, any non-feedback source of cloud variability (e.g. stochastic variability) will cause temperature variability which then leads to a positive bias component in the feedback estimate. (note that it is an energetic necessity that the bias is always positive).

Now, it turns out that there is a larger significance of this “partial misinterpretation” when estimating feedbacks, which is: In our simplified methodology to estimate feedbacks, we assumed that the climate system is MORE SIMPLE than it really is. In other words (at least in this case)….

***…neglecting complexity does NOT lead to a random error in feedback estimates, it leads to a positive bias, which then results in us believing that the climate system is more sensitive than it really is.***

This is significant because there have been some climate experts who say, basically, “…given the uncertainties, cloud feedback could just as easily be positive as negative”. In response, I would say, “No, cloud feedback ‘is what it is’, and to the extent we do not understand all of the forcings on temperature that occur in the real climate system, that lack of knowledge causes us to think the climate system is more sensitive than it really is.”

Could someone please tell me whether all of this makes sense to them?? :)

Thus, it’s NOT the complexity of feedbacks per se that I’m referring to here…it’s that there is more going on in the climate system than feedbacks, which then biases our interpretation of climate sensitivity in the positive direction.

-Roy

re: Larry (#12)..”forcing” = “external”??

This use of the term “forcing” has really irritated me for years. The term “forcing” is too general and useful to allow it to be commandeered for such an ‘anthropocentric’ use!!

Besides, where did this idea that human emissions of CO2 are “external”?? The SUN is external! Any changes generated WITHIN the system are INTERNAL! If you argue, “well, extra CO2 causes a radiative imbalance at the top of the atmosphere, so it’s external”….NO!…ANY internal variability (e.g. cloud cover) causes a radiative imbalance at the top of the atmosphere!! Sheesh!!

-Roy

Roy (#32):

You ask:

My short answer is no — or at least I’m not sure I understand what you’re trying to say. In particular, it makes no sense to me that:

In fact, it seems easy to come up with a counter-example using a simple linear model with an AR(1) structure to account for the feedback: Suppose the effect of the cloud variability amounts to nothing more than iid noise. In the context of this linear model, it is not clear to me how the omission of a variable will

biasthe estimated feedback. So my initial thought is that the statement above is not correct for the general case implied by the statement.However, there may be situations where omission of a variable (such as cloud variability), particular if it has a non-trivial time-series structure (such as LTP, which seems to be ubiquitous in this context),

willresult in over-estimation of the magnitude of feedback because some component of an observed pattern in the data will be incorrectly attributed to feedback when it should be attributed to the omitted variable. The precise conditions for this to occur can likely be worked out.Does this make sense to you? Possibly I have misunderstood your original point. ;-)

Dr. Freeman Dyson, someone I greatly respect, criticizes climate scientists, expecially the modelers, for not putting on their coats, going outside, and seeing what it is that is actually happening up there in the sky and down there in the ocean.

So I ask this question: if we were to consider the actual atmosphere, the actual ocean, the actual sun, and the actual space surrounding earth as the “climate model” to be studied and characterized, what kinds of experiments would be conducted; what kinds of data would be collected and how; and what tools and techniques would be used to analyze that data?

bender #27

It is also worth noting that this workshop was held over 3 years ago, prior to the Wegman or North reports. Also, as nearly as I can tell from te n-line CV’s of the participants, there isn’t a single professional statistician in the group.

I wonder how a panel convened to study uncertainty in GCM’s convened by Dr. Wegman that included only statisticians, econometricians,

and simulation experts would be received by the climate science community?

Perhaps Dr. Curry can shed some light on what I perceive to be a reluctance among climate scientists to collaborate with real statisticians. Trust me, it’s not like the statisticians are swimming in grant money.

TAC (#34):

Yes, TAC, this seems to be everyone’s intuition…but it is incorrect. Please look at our (conditionally accepted) paper to see the details:

Or, since the reviewers of that paper point out that the issue can be demonstrated with a much simpler version of our energy balance model, you can use the following equation (which is about the most fundamental form of a climate model you can have — I believe any climate model in its global average could be reduced to this simple form — to which we have added a “Noise” term):

d[delT]/dt = (H + fb*delT + Noise)/cp

In this equation, the time rate of change (d /dt) of the temperature departure from equilibrium [delT] is the sum of heating terms (“H”..could include anthropogenic forcing or anything else you want), the system feedback (“fb” is the feedback parameter in W m-2 K-1), and the “Noise” we added to illustrate the point…(cp is the heat capacity of the system).

Conceptually, “H” and “Noise” drive the temperature variations, and the feedback term feeds back on the temperature. To illustrate the feedback contamination, we assume “fb” is the true cloud feedback, and “Noise” represents noisy variations in the clouds.

If you put the above equation in an Excel spreadsheet and integrate it in time, average the output to, say, yearly time resolution, then plot [delT] versus [fb*delT + Noise], the resulting regression slope will be a positively-biased estimate of the true feedback, fb.

sorry..the link for our paper didn’t appear for some reason..here’s the URL:

http://www.weatherquestions.com/Potential_Biases_In_Cloud_Feedback_Diagnosis.doc

Roy #32

Could you say why that is so?

Follow on question to #29 ( srp says:January 8th, 2008 at 8:33 pm) – Held, in most of his work, has depicted an expanding Hadley Cell. The tropopause is raised, and, the mean latitude of greatest subsidence is increased. Is this physical and well put? Or, would an increase in heat flow in the ITCZ perhaps mean an increased velocity of Hadley Cells (and, with conservation of energy / classic Newtownian considerations, the more poleward cell types) without a significant change in geometry? Has this been modeled along side the current orthodox GCMs and both been compared with actuals?

Roy, I echo Mark’s question #39

Bias being positive? People have a tendency to overestimate and overstate things. Aside from the propensity to state opinion as fact.

Which are the more factual statements:

“The Earth is warming.” or “The global mean temperature anomaly is trending up.”

“The Earth is .8C hotter now!” or “The anomaly is trending up .8C over the last 125 years.”

“Human produced CO2 is absorbing IR and causing warming. Immediate action is required.” or “Land-use changes and fossil fuel burning, and the side effects of both, are thought primarily responsible for any warming that is occuring, although there are a lot of unknowns and processes that are not well understood at this time. This does not of preclude wise actions for the most dangerous and most probable side effects.”

“Global warming is a fact, caused by humans, and will result in such things as floods, draughts, loss of fresh water, dangerous rises in sea levels, and loss of habitat for both flora and fauna, and we must being mitigation now before we hit the point of no return.” or “If the observed trend of +.8 C in the global mean temperature anomaly reflects warming, and the trend continues to accelerate, there could be potentially dangerous side-effects upon the planet’s eco-systems that we wouldn’t be able to fix by adapting to or reversing/outgrowing by advances in technology. We should wisely use our time and resources to study the repercussions of such side-effects and develop cost/benefit based strategies for mitigating such effects, rather than wasting it focusing on the unknown specifics.”

That said, I disagree (or don’t understand the explanation/reasoning) that any bias

mustbe positive. Any more than I agree that additional CO2mustcause an increase in the anomaly. Maybe, maybe not.#27 Bender, I seem to recall that i posted the link to these presentations previously on climateaudit. i have no idea how to wade through the CA archives to find something like this (perhaps John A or you can figure out how to find this link, but a google search does not show “forum on characterizing and communicating climate change uncertainties”)

Catherine Jacobs presentation is online at (third from the top)

http://www.ag.arizona.edu/AZWATER/presentations/jacobs-p&p.html

The bottom line is that the presentations weren’t really about characterizing uncertainty, but in terms of how to communicate uncertainty and decision making under uncertainty (Catherine Jacob’s presentation is good)

I’ve posted my entire uncertainty presentation on my website

http://curry.eas.gatech.edu/climate/pdf/crc-102103.pdf

(roy, check it out, i use tropospheric temperature trends as an example)

Mark (#39) & Judy (#41):

If a non-zero time-average of daily random cloud variations causes an *increase* in absorbed solar, then *only warming* can result from it (not cooling)…it’s an ‘energetic necessity’.

39, 41, there was an entire thread on that in the past week.

#36 Mike B:

The issue of climate scientists collaborating with statisticians is an interesting one. I am certainly open to doing this. But my experiences with mathematicians and statisticians is that most of the issues that we bring to them are sort of at the level of undergraduate statistics major, and not worthy of Ph.D. level statistics research, so it is difficult to engage high level statisticians in our problems.

The solution seems to lie in two directions:

1) getting climate researchers better educated in statistics and data analysis (note in the past two years we have introduced two new courses at Georgia Tech:

http://www.o3d.org/eas-8803/

http://apollo.eas.gatech.edu/EAS4803/

2) bringing applied statisticians into laboratories to work with climate scientists for a few years. The National Center for Atmospheric Sciences (NCAR) is starting to do this, see

http://www.image.ucar.edu/GSP/

Climateaudit is playing an important role in “forcing” climate researchers to pay more attention to how they are doing their statistical analyses.

bender, i finally found the site with the uncertainty presentations

http://dels.nas.edu/basc/ccu/

#37 you cannot simply do regression the equation you presented because there is too much noise in the system to estimate the derivative from either the forward difference, backward difference or central difference. You first either need a clever way to separate the signal from the noise or you need a more fundamental way to measure the derivative like integrating all the fluxes coming in and out of the earth.

Additionally your constant fb is likely frequency and amplitude dependent. As to weather fb is the feed back, that depends if the cause of it is considered internal or external to the system. Usually feedbacks are considered external but we could create the terms, internal feed back, external feed back and total feedback to avoid confusion.

#29 “We pointed out that this was incorrect, because ANY heating term MUST be correlated with temperature, since ANY heating term causes a temperature change! (and Forster has admitted this error to us).”

Not true. Where there is a phase change there may be no temperature change.

OK, Gary (#48), if the Earth was covered with ice or liquid water, right at 0 deg. C, then I would be wrong. :) (The model I presented is for the “whole system”…globally averaged conditions.)

Judith, 46,

From what I’ve seen here, it isn’t the kind of things that requires PhD level expertise that are the issue. I’m not a statistician myself, and don’t have a good feel for when a problem is in shallow water or when it’s in deep water, but the fact that a mining consultant and an economist were able to poke holes in the work of a physicist who specializes in climate work should tell you something. The problems that have been uncovered to date aren’t ultra esoteric problems.

Larry #45 Do you have the name or link so we can look it up? Thanks.

(36) Judith

The study of error is so incredibly under taught at universities. We had one small exercise in my entire physics degree. I found a very cool, pamphlet sized book that focuses on the study of error. Should be in every upper class and graduate course.

52, I guess it was two weeks ago:

http://www.climateaudit.org/?p=2543

#53 Dennis, agreed. Can you tell me the name of the small book you like (I came across one such book a long time ago, forgot the title).

#51 Larry, this is the whole point, the statistical expertise needed is not very high. so where do we find such people who are interested in working on these problems that have the necessary expertise? until climateaudit came along with a significant level of effort made by retirees and others apparently with a lot of time on their hands, i would have said this population was pretty much zero. At this point, there is some distrust between many climate researchers and climateauditors. I am happy to interact with anyone that emails me personally, but i’ve decided that active collaboration via the blogosphere probably isn’t a good idea for me anyways.

re: #44 and lots o’ others.

Roy said:

Only an increase in the energy content can be assured and then only if there are no ways for the energy to leave the system of interest. The ‘average’ temperature is a whole different matter. What is dT/dE for the entire system?

Judith Curry says:

The government probably needs to step in with legislation that requires independent audits of scientific research used to justify major public policy initiatives. It would cost more but the public would be better served.

As a regular flyer, I can testify that atmospheric conditions at the boundary layer are consistently chaotic. As phase change propagates at the boundary layer, I would think that water vapor experiences a significant decline in thermal energy as it transforms from liquid to solid-state and is cooling at a much higher rate than the surrounding atmosphere. The decline in temperature with altitude may generally be described as a linear process, but there are some important thermal handoffs going on at the same time. Thus within the linear system there exists chaotic thermal perturbation at the boundary layer which can be described by the equation…

If you can capture the thermodynamics of cloud formation, you might could answer the feedback/forcing question. Has anyone done it? (Sorry about using the T word).

55, Judith, if math professionals aren’t interested, because the problems aren’t abstract and challenging enough, maybe that’s not the place to look. As I said, the people who did challenge the HS statistics were a business consultant and an economists. Perhaps you’re been looking in the wrong place, and should be looking at the business schools for the kind of auditing expertise needed. And if it isn’t PhD level, it would certainly be MS level.

Or maybe it’s an applied math thing. I don’t know. I’m sure the expertise is out there, just probably not where you expect to find it.

Judith, re your comment in #55,

Based on my long time lurking, there are many of us aging/retired engineers and scientists that do spend a lot of time on this site. I think that we enjoy the participation in the science and the exchange of experiences and ideas. Perhaps as you say, this will have a positive influence on the science going forward.

RE # 60 Mike Rankin

As a retired scientist, the reason I spend so much time here is because the present generation of green-related scientists is, as a group, improperly trained or not clever enough. They bypass tradition and philosophy and invent faster ways to intellectual suicide. I like to think we add a little wisdom.

RE # 43 Judith Curry

You state –

I am amazed that one has to have lessons in communicating uncertainty in decision making. The whole of Life is one long exercise in making decisions based on imperfect information. Some people fail, some pass, some shine. Some grow poorer, some grow richer. Leady Luck plays a big part – how do you communicate that?

Typically, the ones that fail invent creatures like the “Precautionary Principle”.

Have you ever heard of a workshop that turns out people better able to systematically play the Stock Market than others? This is a classic example of decision making in a climate of uncertainty. When all else fails, you guess or depart. What more can be taught?

(A little humour is meant to shine through this).

All the discussion on atmospheric moisture seems to be focused on cloud formation and “greenhouse effect”. There is another effect that moisture has which CO2 does not have and that is the lapse rate. It has a significant effect whereby the lapse rate decreases as RH at ground increases. This has a significant effect on the radiation balance to the point that it (from my calcs) modifies the absorption effects of water to create higher ground temperatures as the RH falls even though the moisture content is falling ie humidities greenhouse effect is more than nullified by alteration of the lapse rate. So are deserts dry because it is hot or hot because it is dry. My calcs point ot the latter. Also in the tropics I have always wondered when the GHG effect is added why the tropics are still inhabitable. This again supports my calcs that it is the moisture content that is putting a lid on temperature at the tropics. Doesn’t this lead to moisture as being a negative feedback? My intuition tells me that effects such as clouds, contrails, vegetation etc only impact at the edges and determine local climate not average global climate. Surely when 70% of the earth is covered by water land effects are therefore so much less.

Roy

I just spent some time doing a bit of calculus based on the differential equation in #37.

1. Solve assuming delT and t are the only variables.

2. Look at changes in behaviour of the resulting equation if only fb and N are allowed to change.

3. If N varies equally both sides of 0, fb does not vary equally both sides of zero, since the graph of fb against N is curved. If the differences in N add to zero the differences in fb are negative.

4. This means changes to N around an average cause a lowering of fb. If you don’t allow for these changes you overestimate the value of fb.

Just like you said, and all with pencil and paper!

Peter

Judith, thank you for the reference to your book, which I’ve ordered a copy of. (I’ve been looking for something just like this, not just the global climate part.)

I would have to say that it is my experience that climatologists have some fairly uncommon challenges, ones that aren’t covered in undergraduate text books. For example, short-term variability in signals such as global mean temperature present a challenge for people trying to estimate the decadal trend. For other quantities such as pressure or wind velocity fluctuations, you have a problem (over a large range of window sizes), where increasing the integration time correspondingly increases the computed variance of the measured quantity.

These are both problems that can be handled routinely even by non-statisticians, I just disagree that either of these are likely to be discussed at less than say a masters level. And I’ve seen enough abuses of statistics by people writing papers in this field (not all are climatologists) to say that some type of standardization of the analysis techniques is needed.

After a first quick reading of Dr. Currys exposition of thermodynamic feedbacks in the climate system (Chapter 13), I would recommend its handling of the uncertain and controversial parts, as at least a qualitative standard, by which I would like to see the IPCC proceed with the subject of climate modeling. I do not think that the IPCC is motivated to provide such an exposition, but in my view Dr. Curry shows how it might be approached.

Dr. Curry states that block diagrams, signal flow graphs and differential equations and other mathematical relations can be utilized to put the specifications or description of the system configuration for climate feedback into amenable form for analysis and evaluation. She goes on to use linear ordinary differential equations (in approximating climate processes) to present conceptual understandings of how the climate system operates and further to indicate where the major questions arise and how the lack of a clear understanding of some of the physical processes can effect the feedback calculations. One quickly gets a feel for the complexity of these feedback processes on reading this document and the need for further study and better understanding.

A general question I have for Dr. Curry deals with the concerns and reservation that she presents in her thermodynamic rendition of the climate feedback and processes and how those concerns translate to climate models using numerical methods and parameterization. I get the general feeling that in many cases they do when Curry uses a standardized language at the end of a process analysis with her caveats: Given the deficiencies in the climate model parameterization of these processes, the xxxxx feedback determined by these models must be questioned.

If we go to Judith’s book, she writes the gain as G = 1/(1-f), where f is the feedback.

Assuming f is from a random process with = 0, |f| > 0, then

= 1 + + + + … > 1

regardless of what other assumptions you makes about f.

Isn’t this the crux of Roy Spencer’s argument?

Sorry, let me try that again using html code:

Assuming f is a random process with <f> =0, |<f^2>| > 0, then

<G> = 1 + <f^2> + <f^4> + … > 0

Which is to say, it’s positivity biased. [Looks like a real effect to me though.]

Re: #65

Dr. Curry has previously addressed the essence of my query to her in my above post in her Post #27 on the Spencer thread. I would hope that it might be repeated here and perhaps with more specific details.

http://www.climateaudit.org/?p=2543

Judith, “Climateaudit is playing an important role in forcing climate researchers to pay more attention to how they are doing their statistical analyses.”

Are you sure that isn’t a positive feedback rather than a forcing?? :D

LOL! Love it, Sam!

Dr Curry, Still working my way through this and forgive my ignorance, but I’m a little confused by the difference in the feedback parameter equation shown on page 5 of the galley and the one on page 7, they seem to be different. I’ve had some exposure to feedback systems many years ago while starting work on a Phd (never finished) in Operations Research and I think this may be a typo with the equation on page 5 being correct and i being substituted for j in the one on page 7.

Carrick @ 66 and 67

I have had a quick look at Judith’s chapter, but have not had time to go into detail.

On an inital (stress very brief) browse, I do not get a clear sense of whether the equations are for continuous or discrete systems. On balance, it looks like the continuous world. If so, a geometric series on ‘f’ would be invalid.

Sorry to keep hammering this point, but it does seem to be an area of potential confusion.

Will have more time to reply this weekend. Very pleased to see the contemplation that this is receiving!

Carrick, thanks much for ordering the book! A reminder, the errata is posted at

http://curry.eas.gatech.edu/Courses/6140/errata.pdf

Bill in DC you are correct, subscripts should be j, thanks for catching this!

Judith – I should say that my comment wasn’t aimed at you. Some amiguity in the maths is something which (I feel) generally floats around the subject of AGW and modelling. IMO, it would have been far better if climatology had stuck with the sound basis of …. well, you could say “control theory”, but “complex analysis” would do just as well.

Regards

Jordan, thanks for the comment, but I’m not sure I follow your argument. Why don’t you think you can expand a continuous variable in a series expansion?

I imagine f(t) to be a continuous function that describes a band-limited process (e.g., a physical process such as clouds that yields a continuously changing f(t)). As long as |f(t)|

… drats, apologies. The < hit me again.

As long as |f(t)| < 1, you can formally expand 1/(1-f(t)).

After taking the expectation value (averaging over a number of realizations of f(t)), because |f(t)| < 1, the resulting series will be absolutely convergent.

I have a feeling we’re talking in different lingos, so I apologize in advance for any misunderstandings here…

The ability of two (or more people) to have a conversation (which quite often becomes heated!) on the blogs when not even discussing the same thing, SO often… It’s good to see some civility for a change!

:)

Hi Carrik (@75 and @76).

You’ll find a number of posts on this back in thread “IPCC on Radiative Forcing #1: AR1(1990)”, but allow me to summarise my concerns.

When we talk about modelling of dynamic systems we could be dealing with continuous equations or the equations of a discrete system (like difference equations). It is important to state what you are using because the mathematics are not the same for each.

Take an equation like G/(1-f) to represent the gain of a system in feedback (gain=sensitivity). Series expansion works for a discrete model, but not continuous. For example, the unit circle (for ‘f’) is the stability boundary in discrete systems. Not so for continuous systems (some more detail in the above thread).

In the discussion I see around AGW (in the blogosphere and the more public discourse), there are many references to feedback systems and (closed loop) amplification. I know that there are non-linearities and multiple feedback loops in the climate system, but my question is at a much more fundamental level.

I ask myself where the gain is coming from. You cannot have gain (greater than unity) from a stable closed loop system without at least the same level of open loop gain. It is worth saying again … the effect of (practical) feedback is attenuation of the open loop gain. It begs the question: why all the talk about feedback in AGW discussion? The gain (sensitivity) should be evident in the underlying physics – before we even start to think about feedback.

Another little hobby-horse of mine is the term “positive feedback” when we really mean “negative feedback in a stable closed loop system with amplification”. It is an important distinction and I feel the misnomer introduces scope for error and misunderstanding.

That said, climate/temperature is a very interesting example and I keep an open mind about the question of whether it is an active system (open loop gain in excess of unity) or a passive system. You need an active system for the proposed closed loop gain from climate feedback (whether that is stable amplification or the more extreme run-away variety).

Back to your question. For a continuous system, it is better to think of the equation for stable negative feedback as G/(1+GH), and series expansion of the denominatior does not apply. Please do not feel tempted to change the sign in the denominator to (1-GH) because that will represent an unstable closed loop system (with a trivial exception).

I don’t have access to the scientific literature – although I agree with Steve that the starting assumption should always be that the experts are right. That makes it (for me) a little bit stressful to ask these types of questions. I hope that my posts are not immoderate or OT – I always write in good spirit. And I always prefer to express opinion in the form of questions, so please treat the above as a question and invitation for discussion.

I hope this is interesting and thought-provoking.

Re: #78

Gain does seem to be a more descriptive term than feedback.

Nitpick: You mean raises the question not begs. Begging the question means that a deduction contains a proposition that is used to prove itself true as in: Suppose Paul is not lying when he speaking. Paul is speaking. Therefore Paul is telling the truth.

Several proposed climate “feedbacks” have quite reasonable underlying physics. Ice/albedo is a good example. Snow and ice have high albedo. If snow coverage increases, planetary albedo increases. That means more incident solar radiation is reflected and less is absorbed causing the system energy to fall faster. That looks like gain to me and it works in both directions.

Jordan, you can have positive feedback in a physical system as long as you have a stabilizing nonlinearity. And you don’t even need a power supply (a simple delay line will do), unless you want the total gain to be greater than 1.

An example of such a system is this delay differential equation:

x”(t) + x^2(t) x'(t) + w0^2 x(t) + w0^2 kappa x(t-tau) = 0

where kappa > 0, and tau = 1/4 (2*pi/w0) [that is 1/4 period delay]. Take the initial condition (e.g., x(0) = 1, x'(0) = 0) and this will progress to a limit cycle oscillator with an angular frequency near w0…

BTW, it is possible to (at least in the Krylov-Bogoliubov framework) demonstrate stability of the above differential equation.

Obviously it is done in a very different way than you signal processing guys do it. but a rigorous mathematical theory does exist for this type of system (which by the way does appear in nature, e.g., in ecological systems).

I would put the chief limitation of my simple example to be that I was treating the gain function as if the system were memoryless (taking the expectation of the the gain function directly rather than infer it from e.g. the corresponding 1-d differential equation).

DeWitt Payne says:

January 11th, 2008 at 5:47 pm

What if we get more snow due to more precipitation due to more atmospheric moisture due to warmer SST?

Jan,

If the snow still melts during the spring and summer, it’s not really a problem. The snow-ice/albedo effect doesn’t really get rolling until the year round ice/snow extent expands or contracts.

Dewitt,

I was considering the year round condition as I don’t think the yearly cycle particularly relevant since albedo at the poles during the polar winter is meaningless.

4 Judith Curry

I find this interesting. I wonder if you could provide the references for these two studies.

Thanks.

Jan,

Rain melts snow very fast, so the spring and summer temperature would have to drop drastically to avoid this. Something would have to block heat transport from the tropics if the global average temperature has increased. A complete shutdown of the Meridional Overturning Circulation might work, but Wunsch (I think) has said the probability of this happening is effectively zero.

In fact, limiting latitudinal heat transfer would act to lower the global average temperature as the decrease in temperature at the poles would more than make up for any increase in the tropics.

62 Gary

The lapse-rate is only significantly affected where the water-vapor is saturated or near-saturated.

The answer is both — it’s a chicken and egg situation.

The deserts are hot because there is hardly any cloudiness, so insolation is a great deal higher than in more humid regions. But, they are also dry because they are hot. The high temperatures mean that a great deal of water is needed to get saturation, so there is very little precipitation — any clouds entering the area shrink rather than drop precipitation.

DeWitt,

What you are saying does not make sense in the context I asked a simple question based on what has been shown to actually happen:

We can ask Dr Curry without even disturbing her.

Sometimes the the snow/ice “feedback” is also negative. Thing is I’m not all that sure it happens only on glacial time scales I’m even less sure that we can actually regard it as a feedback as opposed to a change in the control law as it does not really change (add to or subtract from) the input but changes what the planet does with in different ways depending on circumstances.

Thank you for you reply Carrick et al.

Could we agree that a limit cycle falls into the class of “not interesting” responses from positive feedback? I would consider it to be instability.

A delay adds phase lag, but you need gain for closed loop instability. Could we also agree that an oscillating system with zero energy input would be a perpetual motion machine of the first kind?

I agree that you can have positive feedback in a physical system as long as you have a stabilizing nonlinearity. If we put oscillators aside for now, positive feedback in a practial system will saturate. It is in this sense that I also suggest positive feedback is not very intersting. When saturation sets in, you will tend to arrive at a self-corecting negative feedback system around the saturation point. The only interesting thing going on at that point is the relationship between the variables in negative feedback. (The positive feedback element will probably just look like a simlpe relationship.)

DeWitt. I’m happy to accept that positive feedback (positive roots) can occur in the climate. I don’t know whether that describes things like albedo and water vapour, but there are credible arguments there (proved?). My last paragraph suggests that the positive feedbacks should not be active in any meaningful sense. At any point in time, the positive feedback elements will be saturated and we should be observing relationships in terms of climate variables and the properties of the system which are controlling the saturation.

Re: #89

Jordan:

I think you’re right regarding misunderstandings (#78). If I understand correctly, what you’re calling gain/+feedback many climate scientists are calling +feedback/runaway +feedback. If so, your saturation argument implies two meta-stable states: one where the variable is wandering around based on other factors with some gain exaggerating its motion, the other where the variable is clamped to the vicinity of the saturation point.

This has interesting implications in a system with many such gain/+feedback situations each with a “tipping point” where the gain switches to +feedback. Such a system could have many different states where some variables are clamped to the saturation point and others are wandering. Internal variation could occasionally cause a variable to cross its “tipping point” and switch the system to a new state. Given the presence of delays in gain in many loops, the result could be a climate that switches pseudo-randomly from one state to another on, say, a decadal time-scale.

Adding in the fact that the weather, which carries the variation, is itself a chaotic system with a continual source of random perturbation percolating upscale (

e.g.a butterfly flapping its wings, a tractor’s location in a field when a T-storm cell is developing), the result would be a climate that switchesrandomlyfrom one state to another on a decadal scale.It seems quite plausible to me that the CO2 forcing could have a different effect on different states, indeed it could have a different effect on

eachdifferent variable ineachdifferent state. This would have serious implications on predicting the effect of CO2 forcing.Very interesting discussion. Quick reply to Pat Keating #84

Curry, J.A., J.L. Schramm, MC. Serreze, and E.E. Ebert, 1995: Water vapor feedback over the Arctic Ocean. J. Geophys. Res., 100, 14223-14229.

http://curry.eas.gatech.edu/onlinepapers.html (should be downloadable from my web site, but it doesn’t seem to be working at the moment)

Lindzen, R.S. Does the Earth have an Adaptive Iris?

http://langley.atmos.colostate.edu/at722/references/Lindzen_etal01.pdf

Held, I.M. and B.J. Soden, Water Vapor Feedback and Global Warming 2000

http://web.gps.caltech.edu/~drf/ge152/homework/solutions/HW1/Held_Soden00%20copy.pdf

There might be a misunderstanding about climate feedbacks that might be causing confusion here.

In the discussion of “feedbacks”, climate people tend to not mention the most important feedback on temperature change – the so-called Planck response…the 3.3 W m-2 K-1 increase in emitted IR with temperature. When this negative feedback is included with all of the positive feedbacks from a highly sensitive climate model, the TOTAL feedback is still negative (although maybe by only 1 W m-2 K-1). If the sum of ALL feedbacks were positive, then the system would be inherently unstable to perturbations.

So when you hear that the sum of the feedbacks in a climate model is positive, remember, that does not include the negative 3.3 W m-2 K-1 Planck response component.

91 Judith Curry

Many thanks for your helpful references. I’ll get your paper when the site is up again.

AK – Can we stick around in the simple world for now. We need to estblish whether there is a fundamental truth – that feedback cannot deliver more gain than the system actually has.

If that is true, we could have as many tipping points and such like as we wish, but the gain (sensitivity) would need to be exaplained in terms of the underlying system properties – you could not say there is gain “because there is feedback”.

A small aside (not inviting discussion). I understand the butterfly effect relates to the ability to forecast the future in a chaotic system (i.e. the accumulation of modelling errors). It would be unphysical to suggest that a butterfly flapping its wings could be the root cause of a hurricane. I would imagine friction in the real world would cause ripples from a butterfly to quickly disspate as ambient heat. But if a perfect model misses even this tiny point of detail, there will be a limit to the value of its output.

Jordan, I think we’re having problems with differences in how we use language here.

I would consider any phase trajectory that converges to a limit-cycle to be an example of stable behavior, not unstable. [If you apply a perturbation to the system in its asymptotic state, it will tend to return to it. Why is that not an example of “stability”?]

Also, a 1/4 period delay circuit does provide energy:

x(t-tau) ~= x(t) – tau x'(t)

So -w0^2 tau kappa is the approximate negative damping associated with the feedback term in the delay equation above. This system requires a battery (it’s “active”), because the total linear damping is < 0….

If I had added a positive, passive damping gamma0:

x(t) + gamma0 x(t) + x^2(t) x(t) + w0^2 x(t) + w0^2 kappa x(t-tau) = 0

such that gamma0 – w0^2 tau kappa > 0, the system can be passive (doesn’t require a power source), even with a feedback term present… In this case, the limit is x=x’=0; but the feedback acts as an “undamping”, namely it sharpens the resonance response of the system.

A simple 1-d cavity (two nearly perfect reflecting surfaces at either end of a box, with a constant propagation delay from one end of the box to the other) is a physical example of this (for example, your speaker cabinet). Not surprisingly, you end up with net gain from the cabinet :

G = 1/(1 – R1 R2)

where R1 and R2 are the complex-valued reflection coefficients associated with the reflection from both ends of the box. The feedback gains in this system are R1 and R2, which for a system without a battery we have the physical requirement |R1|, |R2| < 1.

Whether positive feedback is interesting is somewhat of a subjective issue. It certainly is

importantto have in many physical systems, in order to explain that system’s behavior! Granted in my example, as the system approaches its equilibrium state, the positive feedback gets matched with a positive damping from the saturating nonlinearity. I would never use the term “feedback” to describe a damping term however, because that is never a “feedback” of the system response from the nonlinear damping. [In my parlance, feedback occurs when a portion of the output of a system gets fed back into the system, and these always occur in a physically realizable system with a net positive delay] Positive damping is not a “feedback” in that sense, but rather represents an source of energy loss.Finally, limit cycle oscillators share key mathematical characteristics with more complex phenomena. For example the limit cycle oscillator above is a 1-d approximation of a spontaneous otoacoustic emission (that is sounds produced by the ear in the absence of stimulation; these tend to be narrow band quasi-stable oscillations, some between associated with tonal tetanus). One can write a full model of the ear including a detailed middle ear and cochlea that includes fluid effects, variation in stiffness over the membrane and so forth. In the end, many of the properties of the spontaneous emission produced in the more complex model are recovered with the simple 1-d model.

Jordan:

I’m not sure what you mean here, but if you are saying that the total gain of the system G can not be greater than the feedback gain(s), I believe this assertion is wrong.

The resonance chamber is a counter example of that.

The 1/(1 – R1 R2) expands 1 + R1 R2 + R1^2 R2^2 in the limit that the signal is still present when the delayed response from the R1 * R2 term arrives and so forth. It’s true that if R1*R2 = 1, that you get an infinite gain in a passive system, but if you turn the signal on at t=0, that just means that it continues to grow as t approaches infinity. (In other words, the feedback mechanism “stores” part of the energy that it receives and coherently emits it later; the longer it stores it before emitting it, the greater the emitted amplitude.)

In my comment #95 above, I should have said the gain is “proportional to” 1/(1-R1 R2) not equal to it. If the sound is being transmitted on the same side as the driver (the normal case, use “1” for the side with the driver), then more carefully:

G = T1/(1-R1 R2)

Here T1 is the “transmittance”, with |R1 + T1| <= 1 as well as |R1| < 1 and |T1| < 1. If you take the simplistic example that T1 = 1 – R1 and R2 = R1, this simplifies to,

G = 1/(1 + R1)

For arg(R1) > pi/2 [idealized hard surface R1 = -1 so this is not unrealistic], you end up with |G| > 1 even though |R1| <=1.

I should note that I was sloppy in my language above (#80); you can have a gain > 1 in a passive system with external forcing. The conditions for needing a battery are when you get net gain to a transient forcing (e.g., an impulse). Passive systems get gains >1 by the “build up” over time of the feedback mechanism to external forcings.

Roy, thanks for your clarification.

In my parlance the Planck radiation represents a passive radiation loss, rather than a negative feedback, but it is a good point. [As I use the words, negative feedback would be when a portion of the prior system response is fed back into the system such that it is out of phase with the current system response.]

Jordan (#94)…

I’m not sure what you mean by “gain” then. Using a simple analogy of a microphone, amplifier, and speaker, suppose the microphone is very close to the speaker with some insulation between. The gain of the amplifier remains constant, but the system gain could be larger (for more than very transient sounds) due to feedback. Reducing the amount of insulation increases the gain until the system crosses a “tipping point” and becomes liable for a “runaway feedback” that only saturates when the power requirements cause the amplifier’s gain to become reduced.

If you define “gain” as the ratio of speaker output to input sound (from outside the system), it seems to me the gain of the system become infinite when you cross the tipping point. Is that correct? There are limits to output power (overall and at any frequency) coming from amplifier non-linearity, but I don’t see any limit to gain.

As for your aside, since you don’t want to discuss it, I’ll just mention that my understanding is very different.

Jordan:

I usually do not interchange gain and sensitivity.

Gain is defined as the ratio between output and input for a given physical quantity (e.g., power, voltage, current, pressure, etc).

Sensitivity is defined as the rate of change of the output to the change of one of the parameters of the system. In my 1-d limit cycle model, dx/dkappa could be thought of as a “gain sensitivity.”

As I understand the two words, they have very different meaning and interpretation.

AK:

If you define the gain to be the ratio of output to input, then because the system has a finite power source, the closed-loop gain can’t become infinite. Instead it saturates as the amount of feedback increases.

On the other hand, one could envision reducing the stimulus level to the microphone as we decrease the insulation, in such a way as to keep the output power constant. Then I suppose you could achieve very large gains [at least until you reach the noise floor of the environment.]

94 Jordan

Of course — that’s a fundamental

untruth. It can.It is a waste of time to argue that it’s not possible to get more gain from positive feedback. It certainly can in electronic systems.

The issue is whether the feedback from water vapor is positive or negative, not whether feedback can increase the gain if it is positive.

Carrick – Thanks again. Maybe another time, another place for the butterfly. Not that I don’t want to talk about it – blog management problems. (Price of success – awfully quite on RC in recent times, don’t ya think?)

Conservation of energy is an important part of real-world applcations and I’m trying to steer clear of perpetual motion machines (such as marginally stable with no energy supply). If you need a supply of energy to sustain an output, it suggests there is something in the open loop which has gain greater than 1, at least to account for losses.

For example, I cannot believe that my speaker cabinet can amplify or even sustain a sound without a source of energy. I can accept the cabinet as a passive resonator which (without an input) will eventually decay to silence.

I can see where you are coming to on the phase plane. Our differences in language are not so great. Is it fair to say that some designs are trying to achieve stable oscillation, whereas others are striving for stable non-oscillation? Given that AGW does not postulate an oscillator, I was trying to keep discussion to the latter.

You are absolutely right to say that the assertion would be wrong. Turn it around … I suggest the open loop gain must exceed the closed loop gain for stable, practical closed loop systems.

AK – I see what you mean, although you seem to be playing about with the gain. Why not turn down the amp until the volume of the sound coming from speaker is less than the sound going into the mike. Can you get it to squeal?

Carrick

Good point, and point taken. However, it is hepful to measure your signals at a point which compares apples with apples. It is particularly important where you are considering the combination of outputs and inputs by addition or subtraction in a feedback loop.

Are these points of detail running the risk of drifting from my question?

Pat. Try to get G/(1+GH) to exceed G in a stable negative feedback system.

Oops. Pat – that’s a continuous transfer function (“guilty M’Lud!”)

Jordan:

Actually that’s pretty much what I’ve already said: You need a continuous input to get a net gain > 1.

Again though, we are referring to the gain of the steady-state output to input here. If you are continuously generating a tone through the speaker at the opening of a resonance tube, the output will be much louder than the input from the speaker.

This effect is easily demonstrated by whistling over the top of a beer bottle. If you remove the bottle, the sound is dramatically reduced.

This works precisely because the resonance tube (beer bottle) is storing a portion of the input energy via it’s internal resonance and emitting it later.

Again, gain refers to a ratio of

output to inputnot output to nothing. For a passive system, you get a greater than unity gain because of the continuous supply of energy to the system from the input, so an internal battery is not needed. Nor does the open circuit gain need to be greater than unity, in this example. The open circuit gain of course is |R1 R2| < 1.Also, I get the point about perpetual motion machines. Nobody’s discussing anything remotely related to that here.

I appreciate that, but if the words are to have meaning, they need to be applicable to all systems, not just some. My example provided a simple case where what we individually mean by feedback in a physical system can be explored.

In terms of an AGW system, as with my delay differential equation example above, you can have positive feedback e.g. through CO2 feedback, but when you combine it with the passive losses such as described by Roy, you still end up with a stable system:

Positive feedbacks in a climate system don’t imply that your system is necessarily unstable. They may simply be increasing the sensitivity of the system to a parameter such as CO2 concentration.

Carrick (#100):

OK, the gain becomes asymptotic to infinity as it approaches the tipping point. My point was that the system gain can be made arbitrarily large based on any particular amplifier gain.

Jordan (#102):

No, but I can still get more gain from the system than the amp provides.

I’m not sure if it adds more complexity than you want right now, but if you replace the insulation with an attenuating amp (with new mike and speaker), the volume setting on that amp would be the input variable with a “tipping point.” If the original amp is turned down “

until the volume of the sound coming from speaker is less than the sound going into the mike” the second amp would have to be turned up until it had a gain greater than 1.BTW, it was me with the butterfly. I could start a thread on the new BB and you could get to it at your leisure?

AK:

I agree with that….

And this too.

Of course, your signal must be continuous, a point you were careful to make.

This also happens in cases like the beer bottle example, where you have a loop gain less than one, but a total gain more than 1.

104 Jordan

The standard equation is G/(1-F*G), where G is the OL gain and F is the feedback coefficient, which is your expression with the sign in the denominator corrected. (yours probably assumes the feedback is negative).

If F*G = 0.5, say, the CL gain is 2*G. Which I believe you’ll agree is greater than G.

Obviously, you can’t afford much positive feedback if the OL gain is high.

In the old days, you could ‘goose’ crappy gain this way. Nowadays, gains are usually high, and negative feedback is the general rule.

Carrick – having come from different directions in terms of language, we seem to be edging towards each other.

Language gap – a passive system (to me) means one with no energy input. Covers simple R-L-C circuits, gears and levers, and so forth. There are lots of passive systems which appear to give a form of gain, but you have to be careful about conservation of energy. We can get mechanical advantage from levers and gears. We can get voltage gain from electrical transformers. But in each case, the gain comes at the cost of net energy input and the systems cannot sustain themselves. Maybe there is an analogy with blowing across the top of a bottle – it’s not too important, the key point is that the useful outcome depends on suppy of energy.

I’ll just pause to say again … if you are claiming a closed loop gain of “x”, you must have an open loop gain of more than “x”. There appears (to me) to be a big question mark over the proposition that a climate system has a gain of less than “x”, but we can somehow introduce feedback to make it up to “x”.

On the basis that it is stable, the (continuous) G/(1+GH) does not amplify G. The goal is then to account for the properties of the climate system which lead to “G” – that is, a sensitivity greater than the closed loop sensitivity.

That suggests the (same) G/(1+GH) is amplifying “G” whilst remaining stable. It would need an explanation (gain=sensitivty).

Jordan a passive system is one with no internal power supply. Standard definition.

Certainly agree with this. The equivalent source of energy (main one anyway) for the Earth’s climate is the sun.

I’m claiming exactly the following:

1) gain is the ratio of the output of a system to its input

2) the steady-state total gain (ratio of output to input for a steady stated signal) can be much larger than the loop gain

3) there are many examples of physical systems for which this occurs, a beer bottle is one example

4) In addition to physical examples, this assertion #2 can be proven on mathematical physics grounds. Resonance systems or any other than have positive feedback components behave like this with respect to their feedback parameters.

So I disagree with you on this point, and it is more substantive than just language choice, unless you have some very different definition of what you are calling “open loop gain”. Loop gain in the beer bottle analogy is “R1*R2” to me. If you agree with that statement, then you must agree that upon multiple internal reflections you have

1 + R1 R2 + (R1 R2)^2 + (R1 R2)^3 … = 1/(1 – R1 R2)

[Each higher power of (R1 R2) corresponds to the next higher order internal reflection.]

For arg(R1 R2) < pi/2, 1/(1-R1 R2) > 1, even if |R1 R2| < 1.

This is on pretty solid grounds both experimentally and theoretically, so unless you are using some other definition of loop gain, you have a pretty big hurdle here [this result goes back to Helmholtz, I believe]

I still squirm when you write gain=sensitivity. In my opinion, two are very different concepts.

Hi Carrick.

I certainly don’t want to make you squirm – it is another example of different language across disciplines. Like my problem with “positive feedback” in climatology. We just need to be careful and open with interpretation.

Our differences are lurking around in points 2) to 4) above – but I am very happy to keep looking at examples.

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

Returning to Pat’s amplifier. It is a good example, although I don’t think it answers my question. The idea is to use the forward amplification to get a loud volume from the speaker, then a lot of attenuation before you re-combine in the additive (positive feedback) position of the loop. The key point is that the closed loop gain is less than unity.

To illustrate, let’s say pat has his amplifier setting at 5, and then he inserts screening between the speaker and the mike (mic) which introduces a “gain” (attenuation) with a factor of less than 1/5. Answer: loud noise, not squealing. Point taken.

To help illustrate the nature of my question, you could split this system into two components: a closed loop which produceds a (bounded) output, followed by another amplifier which listens into the closed loop and has a speaker in another room. The second amplifier can be given a setting of 5. The closed loop element can be adjusted to an amplifier setting of 1 and screening less than 0.999. Same result.

Coming back to my question. The second amplifier explains the apparent gain, not the closed loop. That’s what I am trying to say about the climate models.

Avoiding the word “gain”, I have serious doubts about the proposition that such-and-such a climate sensitivity (small-ish) can be made to increase as a result of feedback (and only the addition of feedback). I think it is true to say that the closed loop climate sensitivity cannot exceed the properties of the underlying physical processes (which I would call the open loop sensitivity) just by the presence of feedback. Just like Pat’s experiment – it gives a gain of 5 because that is a property of the amplifier, not the feedback.

Good discussion.

It doesn’t really matter. You get the same answer either way:

The error is given by:

e=U-HY

U is the input

H is the feedback

The output is given by:

Y=e*G

G is the open loop transfer function

Substuting for e we get:

Y=(U-HY)G

Rearanging

(1+HG)Y=u*G

Therefore the closed loop gain is given by

Y/u=G/(1+H*G)

Note that I derived this using negative feedback because I was thinking along the lines of a control system but the positive feedback case:

Y/u=G/(1-H*G)

Follows from the same logic.

I can see where you are coming from John, but the devil is in the detail. It is important to realise that “G” has different form in the discrete and continuous worlds. Wiki has quite a good summary of the bilinear transform which we use map between the two.

Here are two different first order systems which have a similar appearance, except that one is stable and the other not:

G(z) = z/(z-0.5) (OK to use geometric series within stability boundary)

G(s) = 1/(s-0.5) (geometric series invalid regardless of stability boundary)

Oh … I should clarify the above references to “G”. Please take it to be a simple transfer function for the closed loop system.

Jordan:

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.

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

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.

Does anybody know if there is one recognised formula for determining warming that includes TSI, El Nino, La Nina….. (all factors). And why have I never seen an explaination for said formula being converted into temperature via exact formula. Are these graphs and formulas in fact no better then the climate models.

Judith

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

Hello Carrick

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

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 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.

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.

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

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)

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 would like to come back for a second bite at the question of whether series expansion is valid for continuous representations. Sorry to be so technical.

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).

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|

(This is a repost with corrected use of < symbols. Unfortunately, it shows up “correct” in preview, but gets mangled upon posting.)

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.

Hello Carrick

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).

Judith Curry,

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