Arguments
Does positive feedback necessarily mean runaway warming?
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Positive feedback means runaway warming
"One of the oft-cited predictions of potential warming is that a doubling of atmospheric carbon dioxide levels from pre-industrial levels — from 280 to 560 parts per million — would alone cause average global temperature to increase by about 1.2 °C. Recognizing the ho-hum nature of such a temperature change, the alarmist camp moved on to hypothesize that even this slight warming will cause irreversible changes in the atmosphere that, in turn, will cause more warming. These alleged "positive feedback" cycles supposedly will build upon each other to cause runaway global warming, according to the alarmists." (Junk Science)
Some skeptics ask, "If global warming has a positive feedback effect, then why don't we have runaway warming? The Earth has had high CO2 levels before: Why didn't it turn into an oven at that time?"
Positive feedback happens when the response to some change amplifies that change. For example: The Earth heats up, and some of the sea ice near the poles melts. Now bare water is exposed to the sun's rays, and absorbs more light than did the previous ice cover; so the planet heats up a little more.
Another mechanism for positive feedback: Atmospheric CO2 increases (due to burning of fossil fuels), so the enhanced greenhouse effect heats up the planet. The heating "bakes out" CO2 from the oceans and arctic tundras, so more CO2 is released.
In both of these cases, the "effect" reinforces the "cause", which will increase the "effect", which will reinforce the "cause"... So won't this spin out of control? The answer is, No, it will not, because each subsequent stage of reinforcement & increase will be weaker and weaker. The feedback cycles will go on and on, but there will be a diminishing of returns, so that after just a few cycles, it won't matter anymore.
The plot below shows how the temperature increases, when started off by an initial dollop of CO2, followed by many cycles of feedback. We've plotted this with three values of the strength of the feedback, and you can see that in each case, the temperature levels off after several rounds.
So the climatologists are not crazy to say that the positive feedback in the global-warming dynamic can lead to a factor of 3 in the final increase of temperature: That can be true, even though this feedback wasn't able to cook the Earth during previous periods of high CO2.
Note: A more detailed explanation is provided here.
Last updated on 13 September 2010 by nealjking.
The media (in the UK anyway) really got addicted to talking about 'runaway climate change'. I am not sure where it came from because I've hardly ever seen the term used in the science. I am aware that a lot of climate scientists are very uncomfortable about the use of this phrase.
But there are different ideas bound up in it- if there is a low probability of carbon cycle feedbacks doing something extreme and becoming very nasty even if not becoming self-perpetuating, then is that 'runaway'? How much does something have to run away before it is 'runaway'?
The naysayers get confused by this and cone up with all kinds of wrong arguments based on it.
The ocean is not saturated with CO2.
It would be better to describe the temperature dependance if the distribution coefficient.
But even that would be wrong because I am not sure that the system is in equilibrium.
The earth receives a certain power from the sun. For an earth in equilibrium, all that power must be radiated back out to space. The rate of radiation is proportional to T^4, where T is the temperature in degrees Kelvin (about 290 at the moment).
For a simple analysis of feedbacks, we model the T^4 dependency by a linear one with matching slope at our current temperature. Over a small temperature range (and compared to 300, 1 or 2 degrees is small), the T^4 curve and the linear model will be almost identical.
With the linear model we can now get "runaway" feedback. That is, each iteration of temperature in the spreadsheet analysis above leads to a bigger temperature jump than the one before, so instead of converging on a new value, the temperature just get bigger and bigger.
Of course in this situation, the linear model is not accurate. What happens though is that we jump to a different part of the T^4 curve - a place where the slope is great enough so that the local linear model won't produce runaway feedback.
To summarise. If feedbacks are small (or negative), you will get a multiplier effect - a small temperature change might get (say) doubled by feedback. If feedbacks are big, you may end up jumping to a different part of the T^4 curve. If one looks at the recent ice-age/interglacial temperature movements, it does look possible that feedbacks are kicking us from one temperature regime to another and then back again.
Maybe I should try and post this with graphs and stuff.
I see it as runaway if the system goes to an extreme that is stopped only by a lack of resources of some sort: in the case of the amplifier, limits to power; in the case of the Venusian atmosphere, exhaustion of water.
In this model, the feedback is positive but decreasing, so it just stops adding up. I consider it self-limiting, as opposed to runaway.
- I don't quite get your point: The solubility of CO2 in water declines with increasing temperature. Last I heard, the uptake of CO2 has dropped in recent years, although up til now it has absorbed about half the CO2 produced by fossil fuels.
- I don't understand the issue regarding equilibrium.
The carbon-cycle aspect has a positive feedback, because the increase in T => increase in CO2 => increase in greenhouse effect => increase in T.
The T^4 radiated power has a kind of negative feedback, because the increase in T => increase in cooling => reduction of the increase in T. (But actually, T^4 behavior is not really the way the system works: If it did, we wouldn't be talking about the greenhouse effect.)
First, it ignores the fast feedbacks (water vapour, snow) and goes straight to carbon cycle feedbacks, which is slow and relatively uncertain.
Second, the logarithmic nature of the greenhouse effect is a red herring, at least for small perturbations.
Chris Colose's recent guest post[*] on RealClimate explains the primary reason that positive feedbacks don't *necessarily* cause a runaway:
"Feedback behavior
The ultimate constraint on climate change is the Planck radiative feedback, which mandates that a warmer world will radiate more efficiently and therefore provide a cooling effect. For a blackbody, the emission goes like the fourth power of the temperature. So the question of how the other feedbacks behave is really of how they modify the Planck feedback."
[*] http://www.realclimate.org/index.php/archives/2010/09/introduction-to-feedbacks/
Now, when I stated the same thing on another comment thread on SkepticalScience recently, I was told I was wrong because the Earth is not a black body. This is true, but it is still an object in space that can lose heat only by infrared radiation, and this radiation depends strongly on its temperature. Or, more precisely, on the temperatures of a whole range of different levels in the atmosphere and/or at the surface, each of which affects a different IR frequency band.
Mark
The intent of the post is to address a specific logical question: "How can there be ANY positive feedback if the Earth hasn't done a temperature runaway already?"
To that end, a mathematical model is given as an example of a system displaying both positive feedback and non-runaway behavior.
This post does not propose to describe the full picture of the dynamics of global warming. You might take a look at the "Advanced" version for more perspective.
In the introductory paragraph (Advanced version) you say
"Climatologists must also take into account "second-order" effects which amplify the initial estimate of the warming. It is not easy to calculate these effects, but the general consensus is that, overall, they magnify the temperature increase by about a factor of 3."
This seems to be a reference to fast feedbacks, the ones that increase the warming due to an increase in radiative forcing of 1 W/m2 from about 0.3 C to about 0.8-1.0 C.
The remainder of the article appears to describe a carbon-cycle feedback: CO2 affects temperature; temperature affects CO2.
Forgive me if I have misunderstood, but if I haven't, don't you think it is a little confusing to conflate the two?
"Note: This model incorporates a number of features of the actual feedback mechanism for the enhanced greenhouse effect, in particular the dependence of radiative forcing on the logarithm of CO2. However, it is definitely not intended as a full model for the effect. It's only intended to illustrate the point that there is no contradiction for a system to have positive feedback, while maintaining self-limiting behavior. "
The argument in the intro is: "These alleged 'positive feedback' cycles supposedly will build upon each other to cause runaway global warming, according to the alarmists."
That is what is being controverted.
The first is that a system with positive feedback is by definition unstable. "Alarmists" are always pointing to positive feedbacks, some of them with short time scales like the water vapour feedback and ice-albedo feedback. But the climate is clearly not unstable so the positive feedbacks must not exist, or must be outweighed by negative feedbacks.
The answer to this is that there is a large negative feedback that is fundamental to the Earth system and that is not usually identified as a feedback, namely the Planck feedback as I've discussed earlier (see also the recent article by Chris Colose on Realclimate). That feedback on its own dictates that the Earth's climate sensitivity will be fairly low. The evidence is that there are several fast positive feedbacks that act to increase the sensitivity, but nowhere near enough to make the system unstable.
A second version of the claim might be that "alarmists" are saying the carbon cycle feedbacks will cause runaway warming a la Venus.
The problem with this claim is that no "alarmists" are actually saying this, except Jim Hansen who has suggested it as a very remote possibility, but obviously one with huge consequences. Your discussion of the logarithmic dependence of the greenhouse effect on greenhouse gas concentrations tells us one reason why this runaway warming is not easily triggered. The fact that the Earth has not done so in the past also shows us it's hard to set off. Neither of these things indicate that it's completely impossible.
To be of any help in avoiding confusion, your article needs to be clear about what claim it is addressing. At the moment it's not. Specifically, it conflates fast and slow feedbacks.
Total stable increase for a particular gain (g<1) and forcing (f) is, if you work the math:
V = f / (1-g)
The oft-quoted 3oC increase for a doubling of CO2 (forcing = 1oC) represents a gain of 0.666. This, incidentally, works for negative feedbacks as well - gains with an absolute value < 1.0 are always stable. Differing time constants may cause some oscillation before settling, but systems with |g| < 1.0 are always stable.
Gains > 1 don't tend to exist in natural systems (as they would require infinite energy!); they're pretty common in electronics, amplifying values until you hit the limits of the power supply.
What this set of articles needs to do is clarify that the existence of positive feedbacks in the climate system does not imply |gain| > 1, because the Planck feedback dictates that the the Earth's climate sensitivity is low. It doesn't do that, it goes off onto a tangent about a toy model of a carbon cycle feedback.
The statement you are trying to promote is different from what I am trying to do.
Sorry.
Please explain this.
If the climate did not have equilibrium states formed by the balancing of positive and negative feedbacks, it is unlikely we would be here to see it!
HTH
Respectfully, however, I think that readers might jump to a false or questionable conclusion reading this article, because CO2 is only a piece of the climate change problem.
What Hansen envisions, I think, in his runaway climate change scenario is a series interlinked positive feedback effects which in total might be sufficient to tip the earth into runaway warming. He very specifically mentions destabilization of the methane hydrates as part of his runaway scenario, for example.
I can't speak for Hansen, but what I worry about is the scenario below.
Increases in fossil fuel produced CO2 cause warming, which activates the following positive feedback processes:
The Arctic sea ice/ albedo feedback.
The permafrost decay feedback.
The forest wildfire feedback.
The ocean CO2 release feedback.
The destabilization and release of methane from the shallowest methane hydrates including the Siberian yedoma and thermokarst.
Atmospheric increases in water vapor.
The combined effect of all of these processes destabilizes the oceanic methane hydrate deposits starting with the shallowest ones first. Most methane from this release ends up dissolved in the sea water, and oxidized into CO2. Increasing amounts are able to vent directly to the atmosphere through sudden releases. Destabilization of the hydrates also results in associated deposits of natural gas venting directly into the atmosphere.
Most of this takes place in the Arctic, while the Antarctic continues almost intact. Due to the fact we are coming out of an ice age, large inventories of methane hydrates are available.
As methane releases accelerate, concentrations of the hydroxyl radical in the atmosphere plummet, leading to longer residence times for methane in the atmosphere before it is oxidized into CO2, and increased warming as a result.
About this time, CO2 based warming is approaching a first plateau, as this article predicts, due to diminishing positive feedback returns and saturated absorption bands. So, we get maybe 10 degrees C of temperature increase due to maybe 3000 ppm of atmospheric CO2.
But methane impact on global heating has its own diminishing returns curve, and that curve is in its steepest part.
Water vapor concentrations continue to increase, and the diminishing returns curve of water vapor is also near its steepest point at this time.
The diminishing returns curve of CO2 is at a plateau. When the diminishing returns curve of methane and water vapor plateau, the earth is perhaps another twenty degrees C warmer, on top of the ten degrees C caused by CO2 alone.
At this point many inland lakes lose their water to the atmosphere. Evaporation is greatly increased, from the landmasses.
About this time, the oceans have heated enough to release most of the methane from hydrates. The oceans become anoxic, and hydrogen sulfide starts to evolve from the oceans, killing most land organisms. The hydroxyl radical has been overwhelmed, and atmospheric oxidation times for methane have become hundreds of years.
The oceans begin to boil, and transfer their water into the atmosphere. Soon, most of the water in the oceans has transferred to the atmosphere.
As ground temperatures increase, carbonate begins to convert to CO2. Plate tectonics, driven by the temperature difference between the mantle and the surface slows and stops. So the rock weathering cycle and subduction of carbon containing sediments also stops.
Eventually, with all of the water, CO2, and methane in the atmosphere, the earth becomes another Venus, but hotter at first because we still have all of our water, while Venus has lost its water due to light induced dissociation of water into hydrogen and oxygen, and loss of the hydrogen into space, as happened on Venus.
How likely is all of this?
Well mass extinction events such as the Paleocene-Eocene thermal maximum have apparently taken us part way down this path, according to isotope ratio records.
And the sun is hotter, now.
You've written that very coherently, and it's not by any stretch of the imagination a "crackpot" scenario. But I would say it's very unlikely, even under a BAU emissions scenario.
Lord knows I'm no expert on the PETM, but achieving that kind of CO2 pulse would seem to require burning more fossil carbon than would be projected even under an extreme scenario. And my understanding of the benthic methane hydrate issue is that it's likely to kick in only very slowly (on a global scale, ignoring local exceptions).
The PETM was certainly nasty, and we wouldn't want to subject ourselves to anything like it. But it didn't lead to the kind of Venusian runaway warming you describe, despite involving much higher temperatures than AGW is likely to produce (remember that in addition to the large magnitude of the warming, the PETM was starting from a warmer point). Yes, the sun is hotter now, but the PETM-Holocene difference in TSI isn't that large (a couple of percent?).
Perhaps most importantly, there have been other periods in the past (not just the PETM) when the planet was much hotter than today (think crocodiles and azolla blooms in the Arctic) and no Venusian runaway occurred.
On very long timescales (tens of millions of years), the earth has been gradually cooling. We're going to be unwinding that process by at least a couple of degrees C in a geologically very short time ... but I think the dangers more involve disruption of our agricultural system, expensive and painful impacts from sea level rise, and loss of biodiversity (esp via ocean acidification).
I don't think the Venusian runaway is remotely likely unless our descendants tried really hard to bring it about. (E.g., massive production and release of CFCs).
Again, though, paleoclimate isn't really my area of expertise, so this is just one person's amateur understanding.
Any thoughtful person would be thankful to be wrong about such a scenario, of course.
The fact that we are coming out of an ice age, and starting from a cooler starting point might not save us from such a scenario, though. Our methane hydrate deposits are in equilibrium at ice age temperatures.
The speed at which we are introducing CO2 is absolutely unprecedented, so far as I know. Also, the forcing from fossil fuel use is entirely non-random, unlike most past naturally occurring events. So, our methane hydrates could be particularly susceptible to disruption, and have had no chance to gradually lose methane, and have it safely oxidized into CO2 and sequestered via the rock weathering cycle over many thousands of years.
Yes, there were warmer periods in the past, but we may have gotten to those warmer periods in a safer manner, more gradually, allowing harmless oxidation of methane at reasonable rates.
The permafrost decay positive feedback is a similar concern. If this permafrost loses its frozen plant matter to decay into CO2 and methane gradually, there is no problem. If the accumulated frozen plant matter from thousands of years of ice age conditions decays within a century, though, this might add to warming in an unprecedented manner.
The yedoma and thermokarst of Siberia are a similar concern. These ice age accumulations of methane and methane hydrate could also be susceptible to anomalously rapid dissociation.
The PETM is worrisome, but the event that really worries me is the End Permian. As you point out, the PETM was nasty, but the End Permian mass extinction was the big one, extinguishing on the order of 90 percent of species existing at that time. Direct intrusion of the Siberian Traps volcanism into methane hydrate deposits may have been necessary to cause that one, but we don't know this for sure, so far as I know.
So, I worry that our "clathrate gun" and associated ice age relics might be cocked and loaded, so to speak.
Some things that might save us, as you point out, are the logarithmic nature of the greenhouse effects from the various greenhouse gases, and the diminishing returns positive feedback phenomenon. Also in favor of stability are the endothermic nature of methane hydrate dissociation, and the Planck radiation feedback.
One thing that really worries me is the unpredictable nature of positive feedback phenomena. I frankly doubt the ability of anyone to predict the outcome of such a complex interlocked series of positive and negative feedbacks. If anyone could do it, it would be someone like Hansen- and Hansen is worried, too.
Another thing that worries me is that estimates of the total quantity of methane hydrates differ by at least an order of magnitude.
The sun is a couple of percent hotter than it was during the PETM, but several percent hotter than during the End Permian, I think.
If we take the End Permian event, and add in a more rapid triggering event, a buildup of ice age methane hydrates, and a sun that is five percent or so hotter, what do we end up with?
Recent evidence supporting the clathrate gun hypothesis exists: As reported in Reporting Climate Science .Com
Free copy of the study available here.
I agree with Ned in that, to the level of understanding we have currently, the possibility of a methane clathrate/hydrate release sufficient to trigger a hydrogen sulfide release and/or leading to a Venus-style runaway situation is remote. What is disturbing, however, is that such a possibility even exists. More disturbing is that future conditions may not be a good analog for anything in the paleo record other than the PETM. Without being able to establish an upper bound to the risk, we may find out that we we didn't know was more relevant than what we did.
That should be of concern to all, as this is an experiment to be run once only.
The Yooper
It is, as with many other questions of climate change, a question of rate of change. Thus we do not know if the loaded gun has birdshot or a deer slug.
Archer 2007 is an excellent summary of methane hydrate and their climate change potential.
The hydrate reservoir is so large that if 10% of the methane were released to the atmosphere within a few years, it would have an impact on the Earth’s radiation budget equivalent to a factor of 10 increase in atmospheric CO2. ... Fortunately, most of the hydrate reservoir seems insolated from the climate of the Earth’s surface, so that any melting response will take place on time scales of millennia or longer.
Acoustic images of real-time methane releases as in this example are dramatic evidence that such melting is indeed occurring, albeit in isolated places. As summer Arctic sea ice continues to dwindle in the coming few years, 'science experiments' such as this will no doubt become more frequent and widespread. In my days in the offshore O&G exploration, hydrates were a well-known drilling hazard; punch a hole in one and you cause it to go unstable very quickly. These guys are going looking for them. Combine that plan with another series of avoidable mistakes such as those leading to the BP disaster and you have given your loaded gun to a bunch of drunk teenagers.
Here is a long, but quite thorough 2008 Scripps Institute video on the subject.
I've read one of Archer's papers, but it's been a few months and I'll reread it.
I'd feel more confident in Archer's stuff if he hadn't written several joint papers with ExxonMobil chief scientist Kheshgi:
ExxonMobil Contributed Papers on Climate Science
I can see why an earth scientist might collaborate with ExxonMobil, or it's chief scientist. They undoubtedly have a monumental knowledge of geology, and an immense treasure trove of geological information.
Having said that, though, Archer's estimate of the total amount of methane hydrates is on the low end of current estimates.
It's a really important subject, and I'll get my information about it from sources with no known connection to ExxonMobil.
As I said above, its the rate of release that's critical. Since clathrates are so well-distributed around the world's oceans, their volume is quite significant. But a methane release from an Arctic source may occur independently of one in the Gulf of Mexico.
"information about it from sources with no known connection to ExxonMobil."
Fair point. I note that Maier-Reimer was with the Max Planck Institute when those papers were written. Kheshgi also co-authored a paper with Bert Bolin, who I believe was Chair of the IPCC.
MODELING OF OCEANIC GAS HYDRATE INSTABILITY AND
METHANE RELEASE IN RESPONSE TO CLIMATE CHANGE
This paper seems to show increased levels of hydrate release for shallow hydrate deposits with less than a one degree C temperature increase.
There are chemical reactions that oxidize the methane or transform it into bicarbonate that I was not aware of until recently.
Still, the bigger the reservoir, the smaller the percentage that has to dissociate to cause the climate serious or catastrophic harm.
NETL Methane Hydrates Page
So, the news from these papers appears to me to be bad.
The rate of release from hydrate deposits is limited by the endothermic nature of hydrate dissociation, and by fluid flow limitations, according to the second paper mentioned above, though.
So - a crucial point - whether these deposits will lead to runaway warming may be very dependent indeed on their total quantity.
An order of magnitude difference in estimates of their total quantity, with one of the estimates coming from Archer, who writes papers with ExxonMobil chief scientist Kheshgi, is just unacceptable.
Nothing limited about these numbers: Gas escape features off New Zealand: Evidence of massive release of methane from hydrates
Multibeam swath bathymetry data ... show gas release features over a region of at least 20,000 km^2. Gas escape features, interpreted to be caused by gas hydrate dissociation, include an estimated a) 10 features, 8–11 km in diameter ... If the methane from a single event at one 8–11 km scale pockmark reached the atmosphere, it would be equivalent to ∼3% of the current annual global methane released from natural sources ...
Full paper, with graphics, available here.
Glaciations tend to start gradually, but end abruptly.
Various people, referenced in the your paper and the Wikipedia article on the clathrate gun hypothesis, suspect that lower sea levels during those glaciations can cause pressure induced release of methane from hydrates, rapidly increasing atmospheric methane levels leading to an increase in temperatures and higher sea levels.
If true, this does demonstrate one thing: large scale methane releases from oceanic hydrates can reach the atmosphere.
This is supported by the carbon isotope data from events like the PETM and the End Permian- although direct intrusion of magma from the Siberian Traps volcanism into methane hydrate deposits may have been necessary to trigger the End Permian mass extinction.
These carbon isotope data suggest that two to five or more trillion tons of isotopically light methane from the hydrates entered the active carbon cycle during those events.
Can our greenhouse driven pulse of heat, which is working its way inexorably downward into the oceans, stimulate a similar rapid release of methane?
Climatology's use of "positive" (gain could be either > 1 or < 1) in regard to feedback is not the same as electronics's use of "positive feedback" (gain strictly > 1), IIUC. _That_ is why "positive feedback" has such a strong connotation of "runaway" for many newcomers to climate science.
I think you need to jump on this early and explicitly in any discussion of "runaway", in order to cut down on misunderstandings.
Don't bury the significance of _gain_ halfway down the article at the end of a paragraph ("and call the gain factor g").
Strong atmospheric chemistry feedback to climate warming
from Arctic methane emissions
It's a very important result, IMO, which could provide a bridge from mild CO2 based warming to runaway methane and atmospheric chemistry change based greenhouse heating.
It's a very different atmosphere that they are talking about, with sustained methane release rates of 4 to 13 times those of today. Stratospheric water vapor and stratospheric hydroxyl radical increase, tropospheric hydroxyl radical decreases, and tropospheric ozone increases, leading to indirect warming several times that of the warming from methane itself.
It's particularly worrisome because this appears to be an honest result, resulting from a fair query of a state of the art atmospheric chemistry transport model.
If this work holds up, it may help explain the strong positive feedback of past apparent methane catastrophes including the Paleocene-Eocene Thermal Maximum and the End Permian mass extinction, I think.
There is a theory that the increasing levels of water vapour in the atmosphere due to increased temperatures might greatly increase cloud cover, increasing the earth's reflectance, reducing heat absorbtion into the system. As far as I know, the question of what the impact of increased atmospheric water vapour has on cloud formation and, therefore, albedo is still very open? So I wonder whether it isn't premature to assume that the system is self-limiting due solely to the reason given above? Particularly as the methane release issue casts further doubt on the idea that the system is self-limiting, without some other unconsidered factor coming into play.
Thoughts?
A gain < 1.0 can still provide a several-fold amplification:
Output = Forcing / ( 1 - gain )
A gain of 0.9 will result in an amplification of 10.0, as the sum will be 0.9 + 0.9^2 + 0.9^3 + ..., summing to 10x.
GHG's act as a throttle controlling energy flow out of the Earth climate, all said energy coming from the sun. Input energy inevitably gets dumped to space, the question is in what thresholds are in place in the mid-point of the system, driving internal energy levels so that the throughput can occur.
You seem to be treating this as a limited system, rather than an open system with energy flows. I suggest more reading on your part. I would point you to The Discovery of Global Warming as a starting point.
Not a passive system - a dynamic system. That's a serious error in viewpoint.
-----^v----
ri | |
-^v----*----|+\__| ?
Vin --|-/ |
_|_ |______|
\ /
Consider the circuit above. The op-amp is configured for unity gain. The feedback "gain" = ri/(ri + rf) < 1 and the feedback is positive. Since the forward gain is unity, the open loop gain (referred to as "gain" in #37) = feedback attenuation < 1. The claim in #37 is that this circuit will amplify.
Not so much. The inverting input of the op-amp is wired to the output (Vo). The op-amp is ideal so the non-inverting voltage (v+) is equal to the inverting input voltage (v-) So Vo = v+ = v-. This means the voltage drop across rf = 0 which by Ohm's law means the current through rf (Irf) =0. But Irf = Iri. Thus the voltage drop across ri = 0. Thus Vo = Vi. The gain of this circuit is unity.
Now suppose the op-amp is configured for a gain of 2 and the feedback divide ratio < .5 (again, so that gain as defined in #37 < 1). At first blush it appears capable of amplification. Now add the stability constraint that the input impedance > 0. We find that for stability, ri > rf. The open loop gain for the circuit can be shown to be 2-rf/ri which again must be <= 1 for stability.
This doesn't disprove anything but rather shows that the feedback gain equation is subject to boundary conditions including conservation of energy. I really wonder whether one could devise a physical system capable of doing work using the formulation provided by KR.
KH - I posted before I saw your response. I take your point but am still struggling to understand how feedback amplifies the Milankovitch cycles. The system has to do real work, melting ice, heating the oceans etc. and I thought the argument was the system does more work with feedback than it could do without. That sounds like power amplification to me but I'll come back after I read up some more.
The S-B law indicates that thermal radiation to space scales as T^4 (in Kelvin), and means that excess energy accumulation in the climate, simply by heating/cooling the Earth, will rebalance the input/output levels. That's your negative feedback.
There are fast energy state response elements (water vapor, clouds), slow response elements (ocean CO2, albedo from ice coverage, vegetation), and over and above this the forcings that drive those feedback items.
In the ice age cycles a small amount of insolation variation (orbital changes shifting land/ocean exposure and polar insolation/albedo changes) acted as a forcing, with the temperature sensitive elements responding at various delays.
Currently we are introducing a direct forcing with GHG's, and should expect to see water vapor levels, albedo, vegetation, and in fact CO2 levels from ocean solubility respond to the initial forcing with their own changes.
All of these - forcings and feedbacks - act as throttles on the flow of energy from the sun into the climate and back out again, with rates dependent on the various states, gas concentrations, albedo, etc.
I do think that climatologists use the term feedback in a different way than I am used to thinking about it. In this review, they state "S is proportional to 1/(1 − f)" where S is the sensitivity and f is the "net feedbacks". A control systems guy would call f the open loop gain, as in CL(s) = G(s)/(1-G(s)H(s)) where G(s) is the forward transfer function and H(s) is the feedback transfer function and s is the Laplace operator. I deduce from the above that climatologists model the forward gain as unity (which I find odd - if there were no feedback would all of the energy go into heat flux? Would some get reflected back immediately?) which is why the op-amp is configured for unity gain in my analogy. In any case I'm still trying to formulate a transfer function model for the radiant balance equations so that I can understand how it is that the CO2 could lag the temperature through the entire glaciation cycle, be responsible for the rapid interglacial rise and yet allow the small Milankovitch forcing to turn things around. Seems implausible but I've been fooled by intuition before.
Just the Milankovitch forcings alone should change the average temperature of the Earth by a total of 1-2.5C; the 6-8C swings seen over the ice age cycles are due to the amplification of the forcing change by water vapor, CO2, ice retreat/advancement, etc. The Wiki on this is actually fairly reasonable.
I would suggest caution in reasoning from electronics - that would be an analogy, and while providing an analogy is a useful way to explain something, you cannot reason from an analogy back to the original system, as analogies only resemble the original complex system in part. You will inevitably get tripped up on the differences - there is no substitute for actually studying the real thing. And the real thing includes multiple time lags, chaotic/non-linear variations, and lots of different forcing inputs
Consider a circuit comprised of a voltage source connected to a resistive divider. Assume the divided voltage is our output node. To this output node we connect the control port of a voltage controlled voltage source of gain b whose output is connected to the output node through a series resister. This is our feedback path. Assume all resisters are 1 ohm. If b is zero the output voltage is simply one third the input voltage. As b is increased from zero to 1, the output rises from vin/3 to vin/2. Thus we see a "gain" of 1.5 compared to the no feedback case but in both cases the forcing (vin) is attenuated.
With b=0, the thevenin impedance looking into the output node is 1/3 and therefore the output power density is 4kT/3 W/Hz. With b = 1, the thevenin impedance rises to 1/2 and so too does the power density, to 2kT. This indeed we can see an increase in variance with feedback gain < 1.
Finally note that b in my analogy is not the same as f in #37. In the example, b could actually vary between 0 and 3, (corresponding to 0<=f<1) before the circuit became unstable. However, to get actual gain (in the normal sense of the word, i.e. an output greater than vin), b must be greater than 2. That is, to get actual amplification requires active feedback, as expected.
From the Wiki:
"Each glacial period is subject to positive feedback which makes it more severe and negative feedback which mitigates and (in all cases so far) eventually ends it."
And yet in this tread (see #54), chris accuses me of making an "evidence free assertion" when I tried to make that very point.
Which is it? Is climate sensitivity constant or does it vary as negative feedback(s) kick in at the extremes?
Why does it have to vary? The sensitivity to any particular forcing may remain constant while the forcings themselves vary. For example, as more CO2 enters the atmosphere, that forcing increases with constant sensitivity.
FWIW, I wonder if you've considered a circuit analogy from a simpler time: Hartley and Colpitts oscillators exploit feedback without necessarily running away.
Carbon-cycle modelling still has a lot of uncertainty with many different ways to replicate known data. However, I dont think any of them would give a linear or log-linear response of atmospheric CO2 to temperature.
The Milankovitch forcing are very regular and of essentially constant amplitude. The sensitivity is determined by the sum of all the feedbacks so to the extent that one or more feedbacks depends on temperature or other dynamic variables in the climate, the sensitivity will change. I don't think it is controversial that the assumption of constant sensitivity is only valid over a few degrees.
With regard to your other point, I think perhaps a more apt analogy would be a class of circuits called injection locked amplifiers. In and of themselves they are stable but have positive feedback. They don't oscillate because their complex poles have very low quality factor and are unable to remain on the jw axis for any period of time. However, if we inject a small periodic signal near the eigenvalue frequency, the circuit exhibits behavior very much like a phase-locked loop. Its phase trajectory tracks the input signal inside a bandwidth determined by the injection amplitude. Outside of this bandwidth, the phase trajectory variance falls as f^4. This makes them useful as filters but they are not widely employed because they have a tendency to exhibit chaotic behavior.
The interesting thing about these circuits is that they behave much like oscillators in that each node in the circuit is a delayed version of the previous node (like the CO2 curve in the ice core is kind of a delayed version of the temperature). Another interesting thing is what happens when we add a constant forcing bias. The peak amplitude of the output does not change but rather the duty cycle is modified. It reaches equilibrium by modifying the symmetry of the output waveform. For instance with positive constant forcing the waveform adjusts to spends more time in the positive realm than in the negative realm but the peak amplitude remains unchanged.
What got me thinking about this was a Fourier analysis I did of the Vostok ice core data. The phase noise power spectral density exhibits the same BW tracking fingerprint I saw when analyzing these injection locked amplifiers years ago. This doesn't prove anything but it is an interesting idea.