2010-09-04 22:18:05ADVANCED 125: Positive feedback means runaway warming
nealjking

nealjking@gmail...
121.222.93.62

SEE REVISION 3, AT 5 Sep 2010, 9:23 PM, FOR LATEST VERSION


What the science says...

Unlike the simple example of positive feedback we learned in high school, the increase from every round of feedback gets smaller and smaller, in the case of the greenhouse effect. It is a significant factor in the overall warming, but it does NOT lead to a "runaway" trajectory for temperature.


Note: Breaking with custom, we will use some math in this rebuttal, because the argument is about a mathematical topic: the implications of positive feedback. 

 

One of the problems about understanding the extent of global warming is that the total average temperature increase due to CO2 is greater than the first guess: Climatologists 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 2. These second-order effects work as a form of "positive feedback."

An example of this: Imagine the pre-industrial world, with the Earth, land & sea, in rough thermal balance. Then add a dollop of 35% more CO2 to the atmosphere (by burning fossil fuels): Due to the enhanced greenhouse effect, this raises the global average temperature. But the increase in temperature has the effect of increasing EVEN FURTHER the amount of atmospheric CO2, because the heated ocean will hold less CO2 (think of a can of warmed Coke), and the warming Arctic tundras will reveal formerly frozen biomass that will decompose and release more CO2. So you get even more CO2, which gives rise to even more warming, which gives rise to even more CO2...

But this is where the suspicions of the skeptics are aroused: "If more CO2 gives rise to higher temperature, and higher temperature gives rise to more CO2, and this additional CO2 gives rise to higher temperature, doesn't this go on forever? Doesn't that mean that the Earth would turn into an oven? Something is wrong with this picture!"

This line of thought is partly right:

  1. Yes, in a sense this cycle does go on forever; but
  2. No, the Earth will not turn into an oven!

How can these both be true? Well,

  1. The cycle does go on forever, like the reflections between two facing mirrors; but
  2. At each step in the cycle, the incremental increase gets smaller and smaller. After a few cycles, the increase is negligible.

When does positive feedback lead to a "runaway"?

To understand this, consider first the "classic" example of positive feedback: The output of an amplifier is plugged back into its own input. This can lead to an awful squawl. How does this work mathematically? A simple model: Starting at Input level = Io, call the gain factor g:

- Output = g * Input = g*Io

- Input' = Output = g*Io

 and then:

- Output' = g * Input' = g*g*Io

- Input'' = Output' = (g^2)*Io

 and again:

- Output'' = g * Input'' = (g^3)*Io 

- Input''' = Output'' = (g^3)*Io

and so on...

If you work the math, you can see that Output(n) = (g^(n+1))*Io; but another way to figure it out is to use the two equations in a spreadsheet, starting out with Input(1) = Io = 1, with the equations:

Output(n) = g * Input(n)

Input(n+1) = Output(n)

Figure 1 shows the results, for three different values of g: 0.97, 1, and 1.02:

- for g < 1,  the value drops towards 0;

- for g = 1, the value remains unchanged; and

- for g > 1, the value grows larger and larger ("runaway feedback")

Why doesn't the enhanced greenhouse effect run away as well?

However, this simple understanding doesn't apply to the greenhouse-gas situation, because the equation behind the feedback is different.

Let's take a toy model, which is mathematically similar to the case in point:

- C is the atmospheric concentration of CO2: We will begin with initial value = 1

- dC is the incremental increase in concentration of CO2: The first dollop of CO2 will be 0.35

- In general, the radiative forcing due to dC is

RF = a * d(ln(C)) = a*dC/C

(because the radiative forcing depends on the logarithm of the CO2 concentration)

- The incremental increase in temperature, due to the radiative forcing, is

dT = b * RF = b*a*dC/C

- The 2nd-order increase in atmospheric concentration, due to the increase in temperature, is

dC' = e * dT = e*b*a*dC/C

Therefore, defining the feedback parameter f = e*a*b:

dC' = f * dC/C

and

C' = C + dC

Just as with the amplifier case, we can use the two equations above to generate a spreadsheet simulation:

[THE FOLLOWING MAY BE TOO MUCH DETAIL. OPINIONS?]

Just start with the value C = 1, and set the initial kick of extra CO2 to dC = 0.35. If the value of the parameter f is chosen to be 0.1 (for illustration), the second-round increment of CO2 is:

dC' = (0.1) * (0.35)/1 = 0.035 ; the new CO2 value is:

C' = 1 +  0.35 = 1.35

We use the values C' and dC' to kick off the next round of calculation; and so on.

[END OF DETAIL]

The result is shown in Figure 2: Trying three different values of the feedback parameter f (0.1, 1, 2), we see:


- Due to the feedback, the value of C rises quickly;

- However, after several (4 to 10) iterations, there is no further significant change.

- The terminal value of C depends on the value of the feedback parameter f, and on the initial value of dC: in both cases, the bigger, the bigger.

 

CONCLUSIONS

- When you add more CO2 to the system, there is indeed positive feedback, so more CO2 will end up in the atmosphere than you dumped in.

- This does not give rise to a runaway scenario, because the feedback is only logarithmic. Eventually, the impact of that dollop of CO2 comes to an end.

- However, that end will not be an ultimate end until we stop adding more CO2 ourselves: Every time we add another dollop of CO2, we kick off another round of significant feedback cycles. Currently, we are still adding small dollops all the time...
2010-09-04 23:07:29Confirmation
nealjking

nealjking@gmail...
84.151.49.66

John:

- Yes, that is what I was thinking of: Thanks for dealing with the charts.

- I agree that anything that involves this much math is an automatic candidate for Advanced level.

- Even at the Advanced level, I'm not sure if the DETAIL indicated is helpful or not: For someone who is used to using Excel for calculations, it might not be necessary; for someone who is not, it might not be understandable. What does anyone else think?

- It would not be difficult to essentially clip out the equations and restate the message in words. But in that case, I'm not sure there could be any significant difference between Intermediate and Basic. Maybe there should not be a Basic article on this point? It depends on whether someone who would be too blown away by the Intermediate-level explanation is ever going to be concerned with the question of positive feedback anyway.

 

Neal

 

 

2010-09-05 00:22:26On 2nd thought
nealjking

nealjking@gmail...
84.151.49.66

I think we could make reasonable Intermediate and Basic explanations out of this material; once we get agreement on the Advanced version.

Is there anything that should be added to the Advanced version?  Maybe replace the DETAIL by a snapshot of the Excel spreadsheet?

 

Neal

2010-09-05 00:45:55good advanced level
dana1981
Dana Nuccitelli
dana1981@yahoo...
71.140.3.99

 

Actually the reason I thought my 'quantifying AGW' wiki would make for a good advanced rebuttal was because it had some math and physics formulas.  And I agree that this one makes for a very good advanced rebuttal too.

I was going to do something similar for the "it's the Sun" advanced rebuttal - look at the solar radiative forcing formula.  In general when possible, I think it would be good to go into the actual calculations in the Advanced rebuttals.

A few comments on this one:

  • I'd suggest taking advantage of the superscript button for exponentials, and the subscript button for things like CO2.
  • I wouldn't say the feedbacks go on forever.  They go on until the radiative forcing becomes negative and there's a cooling effect.
Overall it's really good though.  Nice job.
2010-09-05 06:16:50Questions
nealjking

nealjking@gmail...
84.151.49.66

dana1981,

 

- I can reference your article when I mention the radiative forcing

- I didn't realize we had subscript capability: CO2 . Cool.

- I don't know what you mean by saying that the RF becomes negative. In step 1a, there will be reduction of IR loss (the RF), leading to warming; in step 1b, the warming leads to out-gassing of CO2; in step 2a, the out-gassing further reduces the IR loss, leading to a llttle more warming; in step 2b, the warming leads to a little more out-gassing. And so on.

Now there is a question as to how I determined the degree of warming due to each "a" step. My understanding is that the addition of the CO2 moves the altitude of the relevant photosphere to upwards, to a colder temperature, and the IR radiative loss decreases in accordance with the photospheric temperature. This creates an imbalance between incoming energy (mostly visible) and outgoing (mostly IR), which leads to the warming; and this will stop when the temperature at the Earth's surface (and therefore the temperature all the way up through the atmosphere back to the new photosphere) has increased to the point that the temperature at the new photosphere is what the temperature at the old photosphere used to be, prior to the dollop of CO2.

So, according to my calculational scheme, in every "a" step, the RF starts out as finite and then drops to 0; in every "b" step, the impact of the temperature increase feeds back to the CO2 increase. In reality, all the dollops have to be differential, because the heating is going on at the same time as the out-gassing. Also, the time scales are not really right, because even if the initial dollop is infinitesimally small, and it takes an infinitesimal amount of time to generate the radiative forcing, it will take a finite amount of time for the resulting increase in surface-level temperature to propagate itself back up to the photosphere; whereas the out-gassing could begin a differential amount of time later. So I guess, in principle, there could be a CO2 overshoot; but it should be pretty small, and could probably be taken into account by increasing the effective "f" parameter.

In any event, the purpose of this article is not necessarily to present the mathematics of the actual enhanced greenhouse effect, but to show that you can have a system with positive feedback that does not run away. This is described as a toy model.

 

Neal

2010-09-05 07:16:27RF
dana1981
Dana Nuccitelli
dana1981@yahoo...
71.140.3.99

Assuming I'm understanding correctly, you're ultimately showing the temperature change for an increasing CO2 radiative forcing, and that it eventually approaches zero slope, but never quite reaches zero (hence the 'forever' comment).

My point was simply that in the real world, where there are other radiative forcings and CO2 won't increase indefinitely, you're eventually going to get a net negative radiative forcing which will cause a cooling and terminate the positive feedbacks you're talking about.

I think the difference is that you're talking about a hypothetical scenario, just isolating the consequences of an ever-increasing amount of atmospheric CO2 (a spherical cow kind of problem), whereas I'm thinking about it in terms of what will happen to the climate in reality.

If you prefer to stick to the hypothetical, I'd suggest making an explicit statement that you're only talking about the response to increasing CO2.

2010-09-05 08:12:42
nealjking

nealjking@gmail...
84.151.49.66

I'm not sure if you're interpreting the graph properly:

1) The horizontal axis is not the amount or concentration of CO2 (that's the vertical axis), but the number of iterations.

- Step 1 is a 35% dollop of CO2 (from a 1-shot burning of fossil fuels).

- Step 2 is the system reaction to that, including the RF and heating, and generating the out-gassing.

- Step 3 is the system reaction to the out-gassing, including RF, heating and more out-gassing. Ad infinitum; but after a few iterations, the increments are unimportant.

2) Fundamentally I don't care about other forcings, because the rebuttal is addressed to the question, "How can you have a system with positive feedback that doesn't blow up?" And my answer is, "Here is a mathematical system with positive feedback, and it obviously isn't blowing up." What's happening is that the system responses to the initial impulse, and the consequential responses, although positive, decreases rapidly so that the sum converges. This happens without any help from anything outside the stated system (e.g., other forcings): The system is self-limiting.

3) I introduce the section on the model by saying: "Let's take a toy model, which is mathematically similar to the case in point:"

- It's a toy in that it does not pretend to be a model for the atmosphere, it has just the features I want to demonstrate a mathematical case of inherently self-limiting positive feedback.

- It's similar in the sense that it uses a forcing that is logarithmic in the concentration.

- But the similarity to the atmospheric RF formula is not really important, in my view, since the issue is conceptual not computational. If the logarithmic example had not worked out, I would have looked for another one. The similarity is a "nice to have".

 

2010-09-05 21:23:27REVISION 3
nealjking

nealjking@gmail...
91.33.109.141

What the science says...

Unlike the simple example of positive feedback we learned in high school, the increase from every round of feedback gets smaller and smaller, in the case of the enhanced greenhouse effect. It is a significant factor in the overall warming, but it does NOT lead to a "runaway" trajectory for temperature.

 

One of the problems about understanding the extent of global warming is that the total average temperature increase due to CO2 is greater than the first guess: 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. These second-order effects work as a form of "positive feedback."

An example of this: Imagine the pre-industrial world, with the Earth, land & sea, in rough thermal balance. Then add a dollop of 35% more CO2 to the atmosphere (by burning fossil fuels): Due to the enhanced greenhouse effect (EGE), the radiation of infrared energy is inhibited, and this reduction in radiative cooling raises the global average temperature. But the increase in temperature has the effect of increasing even further the amount of atmospheric CO2, because the heated ocean will hold less CO2 (think of a can of warmed Coke), and the warming Arctic tundra will reveal formerly frozen biomass that will decompose and release more CO2. So you get even more CO2 in the atmosphere, which gives rise to even more warming, which gives rise to even more CO2...

But this is where the suspicions of the skeptics are aroused: "If more CO2 gives rise to higher temperature, and higher temperature gives rise to more CO2, and this additional CO2 gives rise to even higher temperature, doesn't this go on forever? Doesn't that mean that the Earth would turn into an oven? If the greenhouse effect REALLY has positive feedback, why hasn't this happened already? Something is wrong with this picture!"

This line of thought is partly right and partly wrong:

  1. Yes, in a sense this cycle does go on forever; but
  2. No, the Earth will not turn into an oven!

How can these both be true? Well,

  1. The cycle does go on forever, like the reflections between two facing mirrors; but
  2. At each step in the cycle, the incremental increase gets smaller and smaller. After a few cycles, the increase is negligible.

When does positive feedback lead to a "runaway"?

To understand this, consider first the "classic" example of positive feedback: The output of an amplifier is plugged back into its own input. This can lead to a sustained shriek. How does this work mathematically? A simple model: Start at Input level = Io, and call the gain factor g:

- Output = g * Input = g*Io
- Input' = Output = g*Io

and then:

- Output' = g * Input' = g*g*Io
- Input'' = Output' = (g2)*Io

and again:

- Output'' = g * Input'' = (g3)*Io
- Input''' = Output'' = (g3)*Io

and so on...

If you work the math, you can see that Output(n) = (g(n+1))*Io; but another way to figure it out is to use the two equations in a spreadsheet, starting out with Input(1) = Io = 1, with the equations:

Output(n) = g * Input(n)
Input(n+1) = Output(n)

Figure 1 shows the results, for three different values of g: {0.9, 1, 1.2}.

- for g < 1, the value fades exponentially towards 0;

- for g = 1, the value remains unchanged; and

- for g > 1, the value grows exponentially larger and larger ("runaway feedback")

So if the gain factor is > 1, the system is described as "unstable": Any signal grows rapidly out of bounds.

Why doesn't the enhanced greenhouse effect run away as well?

However, this simple understanding doesn't apply to the greenhouse-gas situation, because the equation behind the feedback is different. Let's illustrate this with a highly simplified model: This is inspired by the case of the enhanced greenhouse effect (EGE), but is simple enough that it can be calculated easily. The purpose of this model is to demonstrate that a mathematical system can have positive feedback without necessarily having runaway behavior.

The model is defined by the assumptions below; note that a, b, e, and f are constant values.

- C is the atmospheric concentration of CO2: We will begin with initial value = 1 (corresponding to 100%).

- dC is the incremental increase in concentration of CO2: The first dollop of CO2 will be 0.35 (corresponding to the 35% of increased CO2 that we have dumped into the atmosphere).

- In general, the radiative forcing due to this dollop dC is

RF = a * d(ln(C)) = a*dC/C

(because the radiative forcing depends on the increase in the logarithm of the CO2 concentration)

- The incremental increase in global temperature, due to the radiative forcing, is assumed to be:

dT = b * RF = b*a*dC/C

- The resulting 2nd-order increase in atmospheric CO2 concentration, due to the increase in temperature and the baking out of CO2, is:

dC' = e * dT = e*b*a*dC/C

(This is just a convenient assumption to make for this toy model.)

Therefore, defining the feedback parameter f = e*b*a:

dC' = f * dC/C

and

C' = C + dC

Just as with the amplifier case, we can use the two equations above to generate a spreadsheet simulation:

The vertical axis shows the increase of temperature, and the horizontal axis shows the number of iterations of feedback.

Trying three different values of the feedback parameter f, {0.1, 1, 1.2}, and two values for the initial dollop, {35%, 105%}, we see:


- Due to the feedback, the temperature does indeed rise beyond the initial jump (due to the dollop of CO2).

- However, after several (3 to 10) iterations, there is no further significant change: It always stabilizes.

- The terminal value of the temperature increase depends on the value of the feedback parameter f: The stronger the feedback, the larger the terminal value; and also the larger the initial dollop, the larger the terminal value.

CONCLUSIONS

- When you add CO2 to this model, there is indeed positive feedback, so even more CO2 will end up in the atmosphere than you dumped in from the combustion of fossil fuels.

- However, this does not give rise to a runaway scenario, because the feedback is only logarithmic. Eventually, the impact of that dollop of CO2 comes to an end.

- Now, in the real world, that end will not be an ultimate end until we stop adding more CO2 ourselves: Every time we add another dollop of CO2 from the combustion of fossil fuels, we kick off another round of significant feedback cycles. Currently, we are still adding small dollops all the time...

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.

2010-09-05 21:31:53Need some help
nealjking

nealjking@gmail...
91.33.109.141

Hi John,

 

I have improved the text, but I might have done something wrong with the formatting. Can you read this properly? On my laptop, it's expanding horizontally beyond the limits of easy readability. I will restart my machine and see if it fixes itself.

If there is a problem, can you see if you can fix it in the Preview?

Also, I will send you a new Figure 0 for insertion: I just want a snapshot of the spreadsheet code, to give a hint as to how the calculation is done.

 

Neal

2010-09-05 21:57:17Problems
nealjking

nealjking@gmail...
91.33.109.141

John,

Something has gone wrong with the text of the article (as linked from the Rebuttal List) and with this thread: the horizontal lines go on too long. If I use command- to shrink the text, the font gets terribly small (for my eyes, anyway). Can you fix this? Maybe the whole article and thread needs to be copied out onto another pair of threads.

 

Neal

2010-09-06 09:38:46Funky formatting
John Cook

john@skepticalscience...
121.222.93.62

Fixed it - there was some style html code inserted into your revised rebuttal that was throwing out the formatting somehow. I just deleted all the style code and it fixed it.

I also added your "Figure 0" - just here in the forum, not in the final rebuttal. I figure we'll do our tweaks here on the forum, then once it's finalised, we can put it into the rebuttal.

2010-09-06 15:14:25OK
nealjking

nealjking@gmail...
91.33.109.141

I'm happy with it now. It's dated: 5 Sep 2010, 9:23 PM.

Comments on the Advanced version?

As soon as it's settled down, I'll abbreviate it for the I & B versions.

 

Neal

 

2010-09-07 10:03:28nice job
dana1981
Dana Nuccitelli
dana1981@yahoo...
71.140.3.99
It looks good to me.  I'd still suggest adding a link to the Advanced 'CO2 effect is weak' rebuttal when you start talking about the CO2 radiative forcing.  And I don't think the initial note about the math is necessary.  But otherwise it looks really good.
2010-09-07 10:57:07
nealjking

nealjking@gmail...
91.33.100.6

dana1981,

 

- The link you suggested was already there; however, now I've underlined it, so it's more visible.

- I removed the initial note for the Advanced article, although I might re-apply it in the Intermediate article, depending on what's included.

 

Neal

2010-09-07 11:55:21Suggested amendment to link text
John Cook

john@skepticalscience...
121.222.93.62

To make the link to the co2 effect page more user friendly, what about this:

(because the radiative forcing depends on the logarithm of the CO2 concentration)

Eg-  make the hyperlink an organic part of the text. Makes it more readable and good from a search engine point of view too - heaps of juicy keywords in that hyperlink means Dana's advanced rebuttal has more chance of ranking highly for words like 'radiative forcing' and 'co2 concentration'.

Thoughts on how we release this rebuttal? Do we release all 3 as blog posts? Nah, overload. What about if we use the basic or intermediate version as the blog post but at the start and finish of the blog post, mention that this is one of 3 different levels of rebuttal, linking to the Basic, Intermediate and Advanced versions. Probably use the intermediate post, a bit of meat for people to chew on but not so technical as to alienate the less advanced readers.

2010-09-07 12:10:07
nealjking

nealjking@gmail...
91.33.100.6

John,

I tried to use the normal HTML for a hyperlink ="URL">text, but it interacted with the editor and the URL information disappeared: very annoying. So I retreated to the "spell it out" approach. Maybe I'll try again later.

Best to start with the Basic or Intermediate: I'm not sure will be better until they're written, which shouldn't take long, once the Advanced is settled.

2010-09-08 00:39:26Feedback
John Cook

john@skepticalscience...
121.222.93.62

"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 2."

Isn't it a factor of 3? Eg - if you double CO2, the direct effect is a warming of around 1 degree C but feedbacks magnify it to 3 degrees C?

Re Figure 2 using two dollops, 35% and 105% - is it necessary to use both? Isn't the case of just using 35% with the different values of f sufficient to illustrate the point? It may be overcomplicating the graph, that's all.

Also, just wondering, for Figure 2, does a higher value of f lead to runaway C?

Note - made a few formatting tweaks to Revision 2. Indented the equations using blockquotes and fixed up your link to the radiative forcing page.

2010-09-08 01:49:28Hhmmm...
nealjking

nealjking@gmail...
91.33.122.182

- factor of 2 or 3: I'm fine with 3, as I recall from one reading that the "most likely" value for a 2X was 3 deg-C, whereas the direct value was 1 deg-C; but I also remember some chatter about 2. In any event, it's an estimate; I've changed it.

- Figure 2: In the Basic and Intermediate versions, I plan to use the simpler version, with dollop = 35% and 3 values of f. However, for the Advanced version, I thought it might be fair to indicate that there are two parameters of interest, dC (dollop) and f (feedback strength); and the final value for CO2 depends on both. Otherwise, the picture looks too simplistic.

- No, for no value of f does Figure 2 give a runaway CO2. 

- The link to the radiative forcing page: "(because the radiative forcing depends on the logarithm of the CO2 concentration)":

should be:

"because the radiative forcing depends on the increase in the logarithm of the CO2 concentration"

I've made this change.

2010-09-08 20:31:33SEE REVISION 3, AT 5 Sep 2010, 9:23 PM, FOR LATEST VERSION
nealjking

nealjking@gmail...
84.151.34.204
The version in the Rebuttal-List Archive got trashed, somehow.
2010-09-10 01:54:34Fedback
jimalakirti

jimalakirti@gmail...
71.34.142.115

In about 10 minutes I will post an edited version of this. I didn't do  much -- it's pretty damned-well written as is. i have tried to shift the emphasis a bit in a coupl of spots and I ask some questions. Things I have changed will be in red and questions and comments will be in green. I am using Macintosh Text Editor so it will e ASCII with rtf coding.

 

Back in a minute. 

2010-09-10 02:04:54Jimalakirti draft of Feedbac
jimalakirti

jimalakirti@gmail...
71.34.142.115

Editorial suggestions are enclosed in square braces and are in red. Comments and questions are in green.

Should all variables and "products", while obvious to scientists, be defined in English to make equations more approachable to lay readers?

 

well, here 'tis.

 

What the science says...

[[ The feedback effect of greenhouse gases is not like the simple example of positive feedback we learned in high school. The increase from every round of greenhouse feedback gets smaller and smaller. Feedback is a significant factor in the overall warming, but it does NOT lead to a "runaway" trajectory for temperature. ]]

 

One of the problems about understanding the extent of global warming is that the total average temperature increase due to CO2 is greater than the first guess: Climatologists must 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. These second-order effects are a result of  a form of "positive feedback."

An example of this: Imagine the pre-industrial world, with the Earth, land & sea in rough thermal balance. Then add a dollop of 35% more CO2 to the atmosphere (by burning fossil fuels): The enhanced greenhouse effect raises the global average temperature. But the increase in temperature has the effect of increasing even further the amount of atmospheric CO2, because the heated ocean will hold less CO2(think of a can of warmed Coke), and the warming Arctic tundra will reveal formerly frozen biomass that will decompose and release more CO2. So you get even more CO2 in the atmosphere, which gives rise to even more warming, which gives rise to even more CO2...

But this is where the suspicions of the skeptics are aroused: "If more CO2 gives rise to higher temperature, and higher temperature gives rise to more CO2, and this additional CO2 gives rise to even higher temperature, doesn't this go on forever? Doesn't that mean that the Earth would turn into an oven? Why hasn't this happened already? Something is wrong with this picture!"

This line of thought is partly right and partly wrong:

  1. Yes, in a sense this cycle does go on forever; but
  2. No, the Earth will not turn into an oven!


How can these both be true? Well,

  1. The cycle does go on forever, like the reflections between two facing mirrors; but
  2. At each step in the cycle, the incremental increase gets smaller and smaller. After a few cycles, the increase is negligible.


When does positive feedback lead to a "runaway"?

To understand this, consider first the "classic" example of positive feedback: The output of an amplifier is plugged back into its own input. This can lead to a sustained shriek. How does this work mathematically? A simple model: [[Start]] at Input level = Io, [[and]] call the gain factor g:

- Output = g * Input = g*Io

- Input' = Output = g*Io

and then:

- Output' = g * Input' = g*g*Io

- Input'' = Output' = (g2)*Io

and again:

- Output'' = g * Input'' = (g3)*Io

- Input''' = Output'' = (g3)*Io

and so on...

If you work the math, you can see that Output(n) = (g(n+1))*Io; but another way to figure it out is to use the two equations in a spreadsheet, starting out with Input(1) = Io = 1, with the equations:

Output(n) = g * Input(n)

Input(n+1) = Output(n)

Figure 1 shows the results, for three different values of g: {0.9, 1, 1.2}. 

runaway1b.gif

- for g < 1, the value fades exponentially towards 0;

- for g = 1, the value remains unchanged; and

- for g > 1, the value grows exponentially larger and larger ("runaway feedback")

So if the gain factor is > 1, the system is described as "unstable": Any signal grows rapidly out of bounds.

Why doesn't the enhanced greenhouse effect run away as well?

However, this simple understanding doesn't [[completely explain]]  the greenhouse-gas situation, because the equation behind the feedback is [[more complex]]. Let's illustrate this with a "toy" [[Is “toy” the best word here? does it sound frivolous to the general reader?]] model: This is inspired by the case of the enhanced greenhouse effect, but is highly simplified, so that it can be calculated easily. The purpose of this model is to demonstrate that a mathematical system can have positive feedback without necessarily having runaway behavior.

Assume the following: 

- C is the atmospheric concentration of CO2: We will begin with initial value = 1 (corresponding to 100%).

- dC is the incremental increase in concentration of CO2: The first dollop of CO2 will be 0.35 (corresponding to the 35% of increased CO2 that we have dumped into the atmosphere).

- In general, the radiative forcing due to this dollop dC is

RF = a * d(ln(C)) = a*dC/C  [[should “a” and other terms introduced later be defined? Or is the reader expected to know what they mean?  I am no longer to be trusted for editing equations.]]

(because the radiative forcing depends on the increase in the logarithm of the CO2 concentration)

- The incremental increase in global temperature, due to the radiative forcing, is assumed to be:

dT = b * RF = b*a*dC/C 

- The resulting 2nd-order increase in atmospheric CO2 concentration, due to the increase in temperature and the baking out of CO2, is:

dC' = e * dT = e*b*a*dC/C

(This is just a convenient assumption to make for this toy [[toy?]] model.) 

Therefore, defining the feedback parameter f = e*b*a:

dC' = f * dC/C

and

C' = C + dC

Just as with the amplifier case, we can use the two equations above to generate a spreadsheet simulation:

runaway3.gif

The vertical axis shows the concentration of CO2, and the horizontal axis shows the number of iterations of feedback. 

Trying three different values of the feedback parameter f, {0.1, 1, 1.2}, and two values for the initial dollop, {35%, 105%}, we see:

runaway2b.gif

- Due to the feedback, the value of C does indeed rise beyond the initial dollop.

- However, after several (3 to 10) iterations, there is no further significant change: It always stabilizes.

- The terminal value of C depends on the value of the feedback parameter f: The stronger the feedback, the larger the terminal value; and also the larger the initial dollop, the larger the terminal value.

CONCLUSIONS

- When you add CO2 to this toy [[toy?]] model, there is indeed positive feedback, so even more CO2 will end up in the atmosphere than you dumped in from the combustion of fossil fuels.

- However, this does not give rise to a runaway scenario, because the feedback is only logarithmic. Eventually, the impact of that dollop of CO2 comes to an end.

- Unfortunately, that end will not be an ultimate end until we stop adding more CO2 ourselves: Every time we add another dollop of CO2 from the combustion of fossil fuels, we kick off another round of significant feedback cycles. Currently, we are still adding small dollops all the time...

Note:This lltoy?]] 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.

 

2010-09-10 02:32:04Jimalakirti draft of Feedback (color)
jimalakirti

jimalakirti@gmail...
71.34.142.115

What the science says...

[[ The feedback effect of greenhouse gases is not like the simple example of positive feedback we learned in high school. The increase from every round of greenhouse feedback gets smaller and smaller. Feedback is a significant factor in the overall warming, but it does NOT lead to a "runaway" trajectory for temperature. ]]

 

One of the problems about understanding the extent of global warming is that the total average temperature increase due to CO2 is greater than the first guess: Climatologists [[must]] 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. These second-order effects are a result of  a form of "positive feedback."

An example of this: Imagine the pre-industrial world, with the Earth, land & sea in rough thermal balance. Then add a dollop of 35% more CO2 to the atmosphere (by burning fossil fuels): The enhanced greenhouse effect raises the global average temperature. But the increase in temperature has the effect of increasing even further the amount of atmospheric CO2, because the heated ocean will hold less CO2 (think of a can of warmed Coke), and the warming Arctic tundra will reveal formerly frozen biomass that will decompose and release more CO2. So you get even more CO2 in the atmosphere, which gives rise to even more warming, which gives rise to even more CO2...

But this is where the suspicions of the skeptics are aroused: "If more CO2 gives rise to higher temperature, and higher temperature gives rise to more CO2, and this additional CO2 gives rise to even higher temperature, doesn't this go on forever? Doesn't that mean that the Earth would turn into an oven? Why hasn't this happened already? Something is wrong with this picture!"

This line of thought is partly right and partly wrong:

  1. Yes, in a sense this cycle does go on forever; but
  2. No, the Earth will not turn into an oven!


How can these both be true? Well,

  1. The cycle does go on forever, like the reflections between two facing mirrors; but
  2. At each step in the cycle, the incremental increase gets smaller and smaller. After a few cycles, the increase is negligible.


When does positive feedback lead to a "runaway"?

To understand this, consider first the "classic" example of positive feedback: The output of an amplifier is plugged back into its own input. This can lead to a sustained shriek. How does this work mathematically? A simple model: [[Start]] at Input level = Io, [[and]] call the gain factor g:

- Output = g * Input = g*Io

- Input' = Output = g*Io

and then:

- Output' = g * Input' = g*g*Io

- Input'' = Output' = (g2)*Io

and again:

- Output'' = g * Input'' = (g3)*Io

- Input''' = Output'' = (g3)*Io

and so on...

If you work the math, you can see that Output(n) = (g(n+1))*Io; but another way to figure it out is to use the two equations in a spreadsheet, starting out with Input(1) = Io = 1, with the equations:

Output(n) = g * Input(n)

Input(n+1) = Output(n)

Figure 1 shows the results, for three different values of g: {0.9, 1, 1.2}. 

runaway1b.gif

- for g < 1, the value fades exponentially towards 0;

- for g = 1, the value remains unchanged; and

- for g > 1, the value grows exponentially larger and larger ("runaway feedback")

So if the gain factor is > 1, the system is described as "unstable": Any signal grows rapidly out of bounds.

Why doesn't the enhanced greenhouse effect run away as well?

However, this simple understanding doesn't [[completely explain]]  the greenhouse-gas situation, because the equation behind the feedback is [[more complex]]. Let's illustrate this with a "toy" [[Is “toy” the best word here? does it sound frivolous to the general reader? Are skeptics likely to jump on this word and trivialize it like they did ‘trick” with the tree rings?;] model: This is inspired by the case of the enhanced greenhouse effect, but is highly simplified, so that it can be calculated easily. The purpose of this model is to demonstrate that a mathematical system can have positive feedback without necessarily having runaway behavior.

Assume the following: 

- C is the atmospheric concentration of CO2: We will begin with initial value = 1 (corresponding to 100%).

- dC is the incremental increase in concentration of CO2: The first dollop of CO2 will be 0.35 (corresponding to the 35% of increased CO2 that we have dumped into the atmosphere).

- In general, the radiative forcing due to this dollop dC is

RF = a * d(ln(C)) = a*dC/C  [[should “a” and other terms introduced later be defined? Or is the reader expected to know what they mean?  I am no longer to be trusted for editing equations.]]

(because the radiative forcing depends on the increase in the logarithm of the CO2 concentration)

- The incremental increase in global temperature, due to the radiative forcing, is assumed to be:

dT = b * RF = b*a*dC/C 

- The resulting 2nd-order increase in atmospheric CO2 concentration, due to the increase in temperature and the baking out of CO2, is:

dC' = e * dT = e*b*a*dC/C

(This is just a convenient assumption to make for this toy [[toy?]] model.) 

Therefore, defining the feedback parameter f = e*b*a:

dC' = f * dC/C

and

C' = C + dC

Just as with the amplifier case, we can use the two equations above to generate a spreadsheet simulation:

runaway3.gif

The vertical axis shows the concentration of CO2, and the horizontal axis shows the number of iterations of feedback. 

Trying three different values of the feedback parameter f, {0.1, 1, 1.2}, and two values for the initial dollop, {35%, 105%}, we see:

runaway2b.gif

- Due to the feedback, the value of C does indeed rise beyond the initial dollop.

- However, after several (3 to 10) iterations, there is no further significant change: It always stabilizes.

- The terminal value of C depends on the value of the feedback parameter f: The stronger the feedback, the larger the terminal value; and also the larger the initial dollop, the larger the terminal value.

CONCLUSIONS

- When you add CO2 to this toy [[toy?]] model, there is indeed positive feedback, so even more CO2 will end up in the atmosphere than you dumped in from the combustion of fossil fuels.

- However, this does not give rise to a runaway scenario, because the feedback is only logarithmic. Eventually, the impact of that dollop of CO2 comes to an end.

- Unfortunately, that end will not be an ultimate end until we stop adding more CO2 ourselves: Every time we add another dollop of CO2 from the combustion of fossil fuels, we kick off another round of significant feedback cycles. Currently, we are still adding small dollops all the time...

Note:This lltoy?]] 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.

2010-09-10 07:35:25Thanks, jimalakirti, John & dana1981.
nealjking

nealjking@gmail...
91.33.118.20

John: I changed the reference regarding the 2nd Figure to dT instead of dC.

jimalakirti: I've adopted many of your changes, but not all: In some cases, your proposals would change from what I meant.

I've propagated these improvements into the Intermediate and Basic versions as well; and vice versa.

Neal

2010-09-10 23:29:49Neal: on comment
jimalakirti

jimalakirti@gmail...
71.34.142.115

I am glad you found something you could use. When I edit a thing I hope to 1) correct any actual errors in spelling or grammar, 2) delete any superfluous or misdirected material (neither of which occurred in this piece), 3) suggest changes for emphasis or clarity, 4) and query anything that doesn't seem clear or that I can't understand.

I am alway glad if some of my thoughts make a difference to the author.  I don't expect everything I suggest to be useful.

 

Glad some of it was useful to you. 

2010-09-11 08:16:01
nealjking

nealjking@gmail...
91.33.124.196

jimalakirti,

 

Thanks again!

 

Neal

2010-10-12 07:36:14Suggestion for an analogy
Peter Hogarth

peter.hogarth@geoacoustics...
81.153.44.253

Neal, one positive feedback analogy I've used that works for (some) engineers is to consider a thermostat set to just above ambient in a closed room.  However this thermostat is wired backwards, ie when the temperature trip point is exceeded (for whatever reason, perhaps someone lights a candle or two) the heater turns on.

What happens next? Oh dear, temperature rises, positive feedback, thermal runaway etc etc.

At this point reality kicks in, most engineers will point out that the temperature will rise until heat loss through the real walls and windows equals energy input from the heater.  A new higher equilibrium temperature is reached.

Might this be also helpful to non-engineers (or non mathematicians)? 

2010-10-12 08:51:16
nealjking

nealjking@gmail...
91.33.105.82

Peter,

I think that's a separate point: the positive feedback situation applies to a particular point (or region) and a certain set of conditions. Based on those conditions, an unstable feedback (e.g., gain >1; or your example of a mis-wired thermostat) drives the system away from equilibrium. However, when the system gets driven into another region, the rules change, and the system stops acting unstable.

But this is true of every unstable situation, otherwise the universe would be continually blowing up. Actually, the only example I know where the constraints don't kick in is in the collapse of a star to form a black hole: There the instability proceeds to the level of a space-time singularity. But every other positive-feedback system hits the wall eventually.

Are you suggesting to use the mis-wired thermostat to explain positive feedback to non-technical people? Or to address the positive feedback of AGW specifically?

2010-10-12 09:50:26
Peter Hogarth

peter.hogarth@geoacoustics...
81.153.44.253

I suspect most peope are familiar with the operation of correctly wired thermostats, and heating rooms. Most also do not dwell on why the universe doesn't continually blow up!  Anyway, it's there as an idea, to be used, modified, discarded etc, I'm not pushing it!