## Boulders on Hills

Posted by Jeff Id on May 25, 2009

Another guest post by Dr. Weinstein, I found this one to be interesting as well. He makes a good point about the possibility of strong positive feedbacks. Before we get into his post, there is a subtlety of positive feedbacks which is often missed on blogs. Positive feedback does not guarantee instability. You can have some positive feedback in a stable complex system as long as the net positive is less than 1. If the net feedback is greater than 1 then the system becomes unstable and jumps to extremes. As you approach 1 the system becomes gradually less stable and would exhibit stronger variations in response to everyday weather disturbances.

Personally I think it is unlikely that strong feedbacks of the nature the IPCC requires could exist, Dr. Weinstein points out some of the logic problems with a strong positive feedback climate below. I don’t believe his post precludes a weaker yet positive response which would not cross the threshold to create an unstable climate. IMHO, I’m still putting my $4 on a negative response, it explains why our temps are as stable as they are.

I read a quote from someone on AGW who’s name I can’t remember right now. It was in reference to the stability of our current climate and I think it is appropriate.

You don’t find round boulders perched on the sides of steep hills.

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

## Debunking the CO_{2} Positive Feedback Myth

Leonard Weinstein, ScD

May 11, 2009

**Introduction:**

The climate models used in the IPCC reports have made predictions of increases in global temperature of 2C to 6C over present levels by 2100. These predictions were based on two assumptions:

- The atmospheric CO
_{2}level would go from 290ppm in about 1850 to at least 580ppm in 2100. - The increase in CO
_{2}would have both a direct greenhouse gas effect, and trigger a positive feedback effect. The result would be a small direct increase in temperature from the CO_{2}, but the increased temperature would also result in an increase in water evaporation. The water vapor is the major greenhouse gas in the atmosphere, and its increase would further increase temperature, resulting in a positive feedback until the process self-limited at a significantly higher level.

The predicted direct CO_{2} effect is estimated to be about 1C for the doubling of CO_{2} level. The present level of about 388ppm would have already caused about half of the direct rise, since the effect is nonlinear. The global temperature has apparently increased by about 0.7C in the last 150 years, which is slightly more than the estimated direct CO_{2} effect, but far short of the expected feedback imposed value. In addition, the temperature level up to about 1850 was significantly lower than typical levels during the last several thousand years. Much of the period between about1200 to 1850 has in fact been called the “Little Ice Age”. The abnormal low temperature starting point for the change makes the distinction between natural temperature rise due to a recovery from the abnormally low temperature to the present difficult to separate from CO_{2} and positive feedback induced increases. The CO_{2} increase was small until about 1940, so the positive contribution from the CO_{2} is based on an even smaller maximum temperature increase (about 0.3C) and a shorter time. All of these facts indicate that calculations of any CO_{2} effects and positive feedback additions would have badly missed the actual present temperature if we did not already know it.

The proposed solution to the discrepancy by the IPCC is that sulfate gas and particulate pollution from burning fossil fuels, have caused atmospheric “Global Dimming”, which greatly inhibited the correct level of warming. While this cannot be totally refuted, it is not specifically supportable either. Since the need for a strong positive feedback is needed to support the projections for the temperature rise to 2100, the mechanism for such a rise is examined.

**Proposed mechanism for glacial to interglacial temperature increases:**

It is very likely that over the last several hundred thousand years, axial tilt and precession in the Earth’s orbital motion (Milankovitch cycles). have triggered the transitions from glacial conditions (lasting about 100,000 years) to the significantly warmer inter-glacial periods (lasting 10,000 to 20,000 years) During the glacial cycles massive spreads of glaciers and ocean ice formed over large regions at higher latitudes, and the average Earth temperature was significantly lower than for the inter-glacial periods. The increase of average Solar insolation that occurred due to these orbital variations is not nearly sufficient to directly explain the rapid increase in global temperature, and the rapid melting of much of that ice. The variation has to have been triggered more by the change in distribution of the Solar insolation on the surface. However, once the transition was triggered, it has been hypothesized that some forms of positive feedback amplified the increase in temperature.

The most likely form of the initial trigger was due to local increases in ocean temperature in higher latitudes causing some of the marginal ocean ice to melt, and the increased absorption from the increased ocean area resulting in additional ocean absorption of Solar energy. This positive feedback may have been limited due to the small level of direct increased warming. However, the increased area of ocean and slight increased temperature also caused the water vapor pressure to slightly increase. Since water vapor is a strong greenhouse gas, this led to further temperature increases, resulting in some positive feedback. The increasing ocean temperature eventually resulted in large amounts of CO_{2} to be released, since the solubility of the water to CO_{2} is lower at higher temperature, and the oceans hold the vast majority of ocean+air CO_{2}. It appears that the large increase in CO_{2} lagged the overall increasing surface temperature by about 800 years or so. It is then proposed by some (but it is not likely) that the increased greenhouse effect, due to the increasing CO_{2} level, supercharged the positive feedback by resulting in more heating. This additional heating then released more water vapor, and the positive feedback then took off until some mechanism stopped the process (possibly cloud formation).

**Present conditions and additional heating:**

The present total greenhouse effect from water vapor, CO_{2}, Methane, and other greenhouse gases is estimated to make the surface 33C warmer than a surface without greenhouse gases. The direct contribution from the CO_{2} is estimated to be about 2C. All greenhouse gases other than water vapor are estimated to be about 3C. Why would a small amount of CO_{2}, or all other gases than water vapor cause more of a positive feedback for heating that that due to the water vapor itself? If there is positive feedback from just the initial forcing, it would not require CO_{2} to do something strange, the water vapor would do it (i.e., heating causes more water vapor, which causes more heating, etc.).

Also, since water vapor is the main greenhouse gas, positive feedback would make an unstable system unless there was some mechanism that halted runaway conditions. It is likely that increased cloud formation, or even a haze condition caused by the water vapor, would decrease the effective Solar insolation so that a stable temperature is reached. Any increase in any greenhouse gas with changes small compared to the existing total would be self-limited by the water vapor limiting properties.

There is one way the small direct temperature increase from the CO_{2} could trigger a positive feedback (of limited scope). That would be if the small direct increase were world wide, and significantly decreased total global ice cover on the oceans. The total yearly average of the extent of ocean ice has only been measured for a few decades, and even though it has recently decreased somewhat, the present net effect is that less than 0.3 percent of the Earth’s surface has been exposed to a lower albedo (and thus higher absorption of energy), and this is at locations of very low Solar insolation. Most of even this ice area change is probably due to the natural variation, but the net change of absorbed energy is not sufficient to make a significant difference even if most of it was due to the CO2 increase. In addition, the melted area is now decreasing as we go into a multi-year cooling period. Historical records indicate this slight variation is not unusual over several decade long cycles.

**Conclusion:**

Since water vapor is by far the largest greenhouse gas on the Earth, and since the Earth is mostly water covered, it is easy to see why the response of water and ice to perturbations in the level and distribution of Solar insolation would be most important in the shift from glacial ages to interglacial periods. Reasonable arguments can be made for some positive feedback of water vapor to explain the initial rapid temperature increase during the transition. It is clear that such water vapor feedback would have to be self-limiting, since the increase stops. The claim that the much smaller CO_{2} contribution can then cause an even stronger positive feedback, using water vapor as the main additional feedback mechanism, defies logic.

A recent significant increase of CO_{2}, possibly with a large anthropogenic input, still only contains a total of less than 7% of the atmospheric greenhouse gas content. Only about 1/3 of that is larger than the claimed “natural” levels. It is posited that this can somehow override the water vapor self-limiting mechanism by triggering a small increase in temperature to thus release more water vapor and supercharge a temperature rise to several times the direct effect of the CO_{2} itself. Since the self-limiting mechanism for the water vapor is present, this does not follow logically. If just a small temperature change from any cause would cause a large feedback, then changes from day to night, or winter to summer, which are orders of magnitude larger than the small direct CO2 induced level change, would cause large temperature overshoot to new levels. They don’t (the change can be determined by the change in insolation), and thus there is no significant positive feedback at the present levels. Actual temperature changes have many drivers, but CO_{2} does not appear to be a significant driver beyond it’s direct contribution, at the levels or variations in levels present, and certainly can’t have the amplifying effect claimed. In fact the direct contribution is likely somewhat reduced by the water vapor self-limiting mechanism.

## Carrick said

You can have a stable system with a net positive feedback as long as you have a stabilizing nonlinearity. One important stabilizing nonlinearity for the climate is long-wave radiation into space, which depends on temperature T to the fourth power.

Here’s an example for a one-dimensional model:

T”(t) + alpha T'(t) + beta T(t) + gamma T(t-tau) + delta T^4(t) T'(t) = F(t)

Here alpha > 0 represents a passive “drag” on temperature changes,” beta > 0 is the resistance to change in temperature, gamma > 0 a measure of the positive feedback, and delta a measure of the strength of the radiative losses into space. Finally F(t) is the forcing on temperature from external sources (primarily solar forcing).

[It may not be obvious but “delta T^4(t)” represents a nonlinear loss term when multiplied by T'(t)… See “Van der Pol oscillator equation” for a simpler version of this equation.]

We can expand T(t-tau) in “small tau” as long as the feedback is rapid enough:

T(t-tau) = T(t) – tau T'(t) + …

So approximately we have:

T”(t) + (alpha – gamma tau) T'(t) + beta T(t) + delta T^4(t) T'(t) = F(t)

Note that if eps = alpha – gamma tau 1 because you can’t radiate more energy into space than you are providing. (Again without starting with a more realistic model, exact interpretation of the parameters isn’t possible, but I would guess at first blush that E is the inverse of the transmitted energy into space.]

In any case, for E large (e.g, >~ 2) and r small, you end up with a steady state solution of T = E (the self-oscillations encountered for E=0 are suppressed by the nonlinearity, this is easy to show by a substitution of a simple trial solution, but I’ll leave that out for the sake of “brevity”).

Anyway, here’s a mathematica code to solve this equation numerically (hopefully nothing gets eaten by WordPress), plus a link to the output:

s1 = NDSolve[{Temp”[x] – 0.1 Temp'[x] + Temp[x] +

Temp[x]^4 Temp'[x] == 2,

Temp[0] == 0, Temp'[0] == 0}, Temp, {x, 0, 100}];

Plot[Temp[x] /. s1, {x, 0, 100}, PlotRange -> All]

## Carrick said

Sorry here’s the link again.

## Carrick said

Also word press ate this sentence. Sorry, without a preview it’s hard for me to catch this stuff (that’s not a bitch):

Note that if eps = alpha – gamma tau < 0, we end up with a system with net positive feedback.

## Carrick said

Sigh.. More was missed than I though.

Try again (feel free to delete the previous comment):

Note that if eps = alpha – gamma tau < 0, we end up with a system with net positive feedback.

What is interesting about such a system is you end up with oscillatory motion (in the case of no forcing), with a larger parameter eps giving a more rapid return to the limit cycle oscillation of T. Whether this is physical or not is a big question. You need a power supply to maintain limit cycle oscillations in a system with dissipation like this, but since this is just a 1-D version of a climate model, solar forcings may get introduced through the feedback term as well as through a direct forcing of the system.

A dimensionless version of this system is:

T”(x) – r T'(x) + T(x) + T^4(x) T'(x) = E,

where for sake of simplicity we assume a constant forcing E . Probably E > 1 ….

## Jeff Id said

Radiative Plank emission T^4 is the great stabilizer.

You make a good point that a system can bounce around inside non-linear limits with a strong negative feedback component. I’ve also referred to T^4 gray/blackbody emission here and on other blogs because it is such a powerful limiting factor.

Regarding humidity, it seems to me that this type of strong positive feedback doesn’t fit well with observations. After all, it doesn’t take very long for warm air to pick up the humidity postulated by AGW on a 70% water planet.

It doesn’t mean positive feedback is impossible. If a 1C change caused humidity based driving that created an additional 3C (some are talking 4C), that’s a sign of a pretty unstable system (at least over a local range), T^4 cooling would keep it from launching to a million but any small perturbation would result in some ugly shifts. I don’t believe that our climate is that sensitive because we don’t see large regular shifts of any magnitude at the time-frames we’re discussing.

Maybe they’ll find the ideal thermometer tree and determine that climate is that active, until then it’s hard to believe.

## Carrick said

Jeff, in the example I gave the solution to the model (for constant driving) was a monotonically varying function that asymptoted to a constant solution (it resembles an overdamped exponentially stable solution). I’ve written the notes up from above and added more detail, I apologize for any typos.

In terms of the water-vapor feedback model, I believe it can be derived from the standard linear fluid dynamics equations in a stratified layer planer model of the atmosphere. I have a reference to this someplace which if there is enough interest I coudl dig up, in my memory it dates back to the late ’70s. Anyway these models are the scientific basis for high-sensitivity of anthropogenic CO2 emissions used in global climate models.

There are at least 2 caveats. The first is of course this model doesn’t include the full dynamics of a global model, and a second ones it assumes the water vapor remains a gas once it enters the atmosphere (e.g., it ignores the influence of the extra water vapor on cloud formation). In these simple 1-d models, not including the water vapor feedback gives a climate sensitivity around 1.5C/doubling of CO2, including the water vapor feedback increases this to about 3 C/doubling of CO2.

Ignoring cloud formations is problematic because thin clouds can further enhance the climate feedback (6 C/doubling is possible) and heavy clouds could produce a net cooling relative to models with no water vapor feedback. [I believe this latter model is the basis of Lidzen’s Iris Hypothesis.] Anyway, there have been some attempts

In terms of “it seems to me that this type of strong positive feedback doesn’t fit well with observations”, are you referring to the recent net cooling of the Earth or something else? I believe that climate measurements tend to support the existence of this positive feedback [namely more CO2=increased water vapor in the surface boundary layer.] Reference here, but I’m not surprised since it’s essentially a first-principles result.

That being said, I of course think the debate on this is far from over.

As I understand it(and I’m definitely looking for feedback or errors on this understanding, since I’m not a climate scientist) there are several unresolved issues. One of these global climate models don’t model the water vapor feedback effect at all. Instead they assume a global CO2 sensitivity constant, which gets tweaked to match model with data. A second is they certainly ignore cloud formation assumed with the additional CO2, other than by adjusting the global climate sensitivity to correspond to a “low sensitivity 1-d model” (e.g. 1.5 C/doubling CO2) a moderate sensitivity model (3 C/doubling) or a “high sensitivity model” (6 C/doubling). [I can think of others, but that’s enough for illustration purposes.]The idea that the climate sensitivity would be constant across the Earth is an extremely gross approximation. Because the Earths climate is represented by a nonlinear 3-d model, the global average of the response to forcings of the full model is not in general equal to the response of the full model to the global average of the forcings.

## Jeff Id said

#6 My statement is too simple to be covered in nonlinear control math, I have been considering writing up something myself along the lines of what you did. The key for people to understand it is to keep it simple and that has held me back a bit. Control systems and non-linear differential equations are a bit to swallow.

The current low rate of warming is not described by the models so the feedback is not supported by observation. It’s my guess that this is the reason climate models aren’t more stable in their trend. I’m not experienced enough or knowledgeable enough on the models to say much more.

I’m interested in why you have the term written as dT/dt T^4 rather than cT^4 which is what I would expect from Plank graybody radiative cooling. I would make it a post also and let the readers suffer through the math but I’m not understanding the origins of this equation.

## Carrick said

Jeff ID:

Well suffice it to say I think the current generation of global warming models admit to not handlign the physics correctly, and that is the real source for their disagreement with observation. 1-D layered models (with periodic BCs) are just toy models used to argue the plausibility of a result. They have no place by themselves in making major shifts in national economic policy.

Since the c T^4 represents a radiative loss, in a generalized harmonic oscillator equation, it plays the role of a nonlinear damping constant. The damping term in this system is of the form:

R(T) T’ = [-eps + delta T^4] T’

A straight “T^4” term in the oscillator equation is of the form T^3 * T, and you would end up with a stiffness term of the form

S(T) T = (beta + delta T^3) T

With negative feedback (eps > 0) nonlinear stiffness does not stabilize the equations (no energy is dissipated), and you end up with an exponentially-runaway solution. This particular model is even worse T'(0) = and T(0) large negative, the system is unstable even with no net positive feedback in the system. [You end up with a negative stiffness, which is every bit as bad as negative resistance for stability of the equations.]

## Carrick said

Jeff ID:

I appreciate that this is pretty heavy material, but right now I’m trying to get some type of consensus about what is and isn’t true for nonlinear systems with net positive feedback. Hopefully we can produce some simple computer models that people can play with so they can get an intuitive feel for how these systems behave without having to dive into arcane methods like Krylov Bogoliubov averaging/

## Fluffy Clouds (Tim L) said

Carrick,

If we could get a reasonably simple math model of climate, that would be very interesting to play with. I would be interested in beta testing out the program, If it can be broken I can brake it, if a flaw can be found I can find it! LOL

my two cents… I would like to see a 1 deviation 2 deviation and then a 2% fanning out from there. I don’t see this 95% confidence B.S. ya right like any one can have any confidence as regards to weather/climate!!!!

TX TIM

## Neil Fisher said

Perhaps one of you can answer a question that’s been bugging me re: feedbacks. I notice that there are plenty of people who are saying that positive feedback can be stable as long as the feedback is less than unity, but it becomes unstable once feedback gain exceeds unity. This is fine, and I have no problem with that. Where I have the problem is this: 2 x CO2 sensativity is around 1.3C without feedbacks – let’s call it 1.5C to give us a margin and make the math simple. So am I right in thinking that any model that shows greater than 3.0C for 2 x CO2 due to feedbacks is using a feedback gain of greater than unity? If that’s true (and it seems to be) then all models that show greater than 3.0C per CO2 doubling would be unstable, wouldn’t they? If not, why not? They’d be unphysical because of that instability, wouldn’t they? If not, why not?

## Carrick said

#11 Neil Fisher:

What I was pointing out above is that this statement is in fact false.

As I argue in these posted notes, a forced system with net positive feedback (i.e., a feedback parameter > 1) and a sufficiently strong nonlinear damping will in fact be stable.

Sample system response for the turning on of solar forcing at t = 0.

Take the same system and either weakly force it or increase the feedback suffiiently, and you get a “periodic ringing” that persists indefinitely.

In no cases do you get an exponentially runway system.

## Carrick said

Also in response to this:

For a

linearsystem, the net gainGfrom the feedback parameterfis given byG= 1/(1 –f)As the feedback parameter parameter

fapproaches unity, the gainGapproaches infinity.To get

G= 2 [3C/doubling of CO2], you needf= 1/2.To get

G= 4 [6C/doubling of CO2], you needf= 3/4.I can provide a simple derivation of that feedback formula if that would help.

## Carrick said

Fluffy Clouds:

There are actually some reasonably simple models out there already. See for example this.

I’ll see if I can dig up some other ones.

These models are basically toys, but they let you get a feeling for how changing parameters of the model (such as anthropogenic CO2 forcing) affect climate.

Here for example I assume that the anthropogenic CO2 levels suddenly double after 50 years.

## Jeff Id said

What I was pointing out above is that this statement is in fact false.As I argue in these posted notes, a forced system with net positive feedback (i.e., a feedback parameter > 1) and a sufficiently strong nonlinear damping will in fact be stable.This is correct, my linear example was far too simple and linearity was not specified. I apologize for the confusion it may have created as this is a very complicated subject which unlike most of the stuff done here, I have had opportunity to study in college. The reason for the simplicity of the comment is that there is a lot of detail here and Dr. Weinstein wrote an interesting post.

Consider a unity linear feedback where 1C rise results in an additional 1C positive feedback. The second degree results in an additional 1C so the system goes on until infinity. In reality there is an outgoing plank blackbody/graybody radiation component which dumps energy from earth at T^4. So each 1C rise is substantially more difficult than the last. We come to a limit where it doesn’t go any further.

To cover the whole thing we need to consider that the way the models are built an increase in temp of 1C from CO2 results in a strong positive feedback which gives a 2C or 4C rise or something. Since the CO2 is not the mechanism for feedback (i.e. no cloud nucleation from chemicals or anything) it is only the temperature creating the feedback. My simple example is demonstrating a high (2-4X) gain level which is probably limited mostly by outgoing radiation calculated as cT^4 (c is a constant the term is outgoing so it is negative).

Carrick also points out that cT^4 is negatively unstable, also true but at the same time it is known to be correct and is the value which results in the net outgoing energy balance. There isn’t anywhere else for the heat to go. Considering that CO2 is apparently a low forcing GHG with 3C model total today there isn’t too much downside even by the models which shows the weakness of the gas.

Anyway, I wouldn’t mind doing a post on it sometime with some graphics and other items.

Applying a very high gain to temperature defies logic in my mind as the climate system experiences variances which should create strong water feedbacks. Imagine an ocean event that warm water heats increases the air temperature by 1C evaporation occurs in positive feedback within a short time jumping temps 4C. It would take only a small change in ocean patterns to then drop it right back down.

Of course all this is supposed to be worked out in the models but the reality is that these gain factors are guessed at and despite some claims there is no solid knowledge of the magnitudes of some of these very sensitive parameters. Things such as cloud formation and humidity change are often estimated.

## Carrick said

By the way, Jeff, here is what happens if you use a T^4(t) term rather than a T^4(t) T'(t) term:

s2 = NDSolve[{Temp”[x] – 0.1 Temp'[x] + Temp[x] + Temp[x]^4 == 2,

Temp[0] == 0, Temp'[0] == 0}, Temp, {x, 0, 100}]

In[126]:= Plot[Temp[x] /. s2, {x, 0, 100}, PlotRange -> All]

Result:

As you can see the solution zooms off to – infinity.

## Jeff Id said

#16

I don’t think you’re applying the Stefan-Boltzmann law correctly in this equation. That’s why I was asking before.

http://topex.ucsd.edu/rs/radiation.pdf

For Earth,

Energy in = Energy out = cT^4

first derivative

dEi/dt= cT^3 T’

We get a big energy change for a small positive shift in temps. I’ve never checked how climate models handle this but I’m sure it will be a near future adventure.

## Carrick said

Jeff ID, check your notes: the c T^4 term already refers to the amount of energy radiated by the Earth per unit time.

However if I use your T^3 T’ model

I get similar behavior to what I say before.

(In fact if you replace the original T^4 with any monotonically increasing function F(T), T> 0, you’re going to get qualitatively similar results.)

## Jeff Id said

#18, The outgoing energy flux relationship to Temp is well known and not in question. Therefore, there is a problem is in the way the equation is being set up. I will try an make some time for this on the weekend because what you’re doing is interesting. Lately even 5 minutes is hard to come by. ;)

I was running long algorithm last night and left the computer on. Ann closed the top for me after I went to sleep so it’s going to be another day before I see the result.

## Carrick said

#19, see the last page of your linked document:

Total flux has units of watts (joule/s). Or as I stated “amount of energy radiated by the Earth per unit time.”

If you look up the definition of flux, you get

“energy flux” then being a measure of a

rate.OK?

## Jeff Id said

That’s right, so what are you saying?

## Carrick said

That for the Stephan-Boltzmann equationa black-body radiator, I = c T^4 equation, the quantity “I” refers to energy loss

per unit time,so there is no need to differentiate it to obtain a rate as you did in #17:That said, how I treated the losses is probably not right. I admit I was just going for a conceptual model, really didn’t give too much thought about how one would obtain such an equation from first principles.

I’d be curious to see what you work up when you find a break from the “real world”.

## lweinstein said

Carrick and Jeff,

I still think you are both missing my point. The direct CO2 increase and the possibility of negative feedback from water vapor are not unreasonable even if not presently conclusively proven. However if it is the temperature increase from CO2 causing an increase in water vapor and that increase is the basis for any positive feedback, why is CO2 necessary for the positive feedback? Any thing that causes a temperature increase would be an equal source causing the feedback, because it is stated that it is a water vapor driven positive feedback. Seasonal temperature variations should result in a significant temperature overshoot from a straight insolation driven temperature increase if that were true unless by some strange fact that only a small temperature increase only caused by the CO2 will result in positive feedback.

## Jeff Id said

#23, I got that point from your post. It’s implied in some of my answers. The seasonal variation is much larger than a few C of course so I’m not sure the amplification isn’t there and just driven by the large delta flux (caused by axis tilt of course). Also, the timeframe of the feedback is important, i.e. does it take 1 month or 6 to build the humidity. These points make the conclusion less clear to me.

Something you might find interesting that I ran into from Tamino’s blog – AGW advocate/scientist, was the annual signal in satellite data. I did some FFT analysis and some phase analysis of the temperature data. Anomaly is calculated by averaging like months over and subtracting right. So changes in atmospheric response would cause the Anomaly over 30 years to demonstrate a different signal in our higher CO2 atmosphere compared to before. I found the presence of 1 year and 1/2 year signal – matches Tamino’s work. It is possible that a proper analysis by a well informed individual could extract a ‘atmospheric response’ from the orbital 4% variation in solar flux from distance to the sun. This could quantify the feedback level, although the timeframe is only 6 months.

https://noconsensus.wordpress.com/2008/10/26/half-year-cyclic-variaition-in-rssuah-and-giss-anomaly/

https://noconsensus.wordpress.com/2008/10/25/an-orbital-heating-signal-from-solar-input/

I never posted my hours of phase extraction work on these posts but it seems like there is something here. Either way it relates to the feedback from temperature only point you make and may give a method for quantification.