2010-09-08 20:05:00INTERMEDIATE rebuttal #125: Positive feedback means runaway warming


What the science says...

Positive feedback means that a system reacts to a stimulus by reinforcing that stimulus, so the stimulus builds up, and the output builds up, and the stimulus builds up... However, this only leads to a "runaway" instability if the reinforcement is strong enough. If it's not, as in the case with the enhanced greenhouse effect, the feedback can give rise to a definite, but stable, increase over the original stimulus.

Note: Breaking with custom, we will discuss some math in this rebuttal, because the argument is about a mathematical topic: the implications of positive feedback. But we'll try to keep it light.


One of the problems about understanding the extent of global warming (the question of climate sensitivity) 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 add to 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 it work? Starting at input level Io, the output of the amplifier is Io multiplied by the gain factor, g; this gets plugged back into the input, and comes out multiplied by another factor of g; and so on. If we plot the output as a function of the number of cycles of feedback, it looks like this:


We have used three different values for g:

- for g < 1, the output fades towards 0;

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

- for g > 1, the output grows larger and larger

So if the gain factor is > 1, and only in that case, the system is described as "runaway": Any input leads to an output that grows beyond bounds. (In the real world, the system becomes limited by factors not included in the original mathematics, so the output does not really go to infinity, but saturates at a high value. So when a audio amplifier gets into the range of runaway positive feedback, the sound system doesn't explode, it just produces an ear-shattering shriek.)


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

The nature of the positive feedback is different than for the case of the simple amplifier: In the case of the amplifier, the increase in output is proportional to the input; but in the case of the enhanced greenhouse effect (EGE), the increase in output is less than proportional to the input: It is logarithmic. The result is that when we plot a (highly simplified) model of the temperature increase due to the EGE against the number of feedback cycles, we get:

In these plots, we use three different values of the feedback parameter, f:

- Due to the feedback, the temperature does indeed rise beyond the initial jump (due to the dollop of 35% more CO2).

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

- 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.



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

- However, this does not give rise to a runaway scenario: 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 through the burning of fossil fuels, we kick off another round of significant feedback cycles. Currently, we are still adding small dollops all the time...


Note:  A more detailed explanation is provided here.  

2010-09-08 20:06:35OK, now 3 versions


are posted.

Please critique or approve.

2010-09-10 07:36:31Improvements to the Advanced and Basic versions

have been propagated into this version as well.
2010-09-12 02:21:21Intermediate #125:Runaway -- Thumbs up

Looks good to me.
2010-10-20 13:48:07
Alex C


>>>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...

Replace with:

>>>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...



>>>the radiation of infrared energy is inhibited

Hm, is the totality of radiation of IRE inhibited, or just the outgoing IRE?



>>>So when a audio amplifier gets...

Replace with:

>>>So when an audio amplifier gets...



>>>simple amplifier: In the case of the amplifier...

Replace with:

>>>simple amplifier: in the case of the amplifier 



>>>to the input: It is logarithmic.

Replace with:

>>>to the input: it is logarithmic 



>>>runaway scenario: Eventually, the impact...

Replace with:

>>>runaway scenario: eventually, the impact...



Also, in a logarithmic scenario as with the EGE, the output does not approach a terminal value, rather the derivative does (it approaches zero).

d(logx) = 1/x; lim[x->+∞](1/x) = 0 

The output would continue to infinity if you carried it on far enough; logarithmic graphs do not approach an asymptote.  This detail though really doesn't in any way though counter the points you made, it's just that saying that there is a terminal temperature increase value is not necessarily correct.

2010-10-20 13:49:12
Alex C

*This detail though doesn't really in any way counter the...*
2010-10-20 13:49:50
Alex C

Besides those small points though, good rebuttal.
2010-10-20 20:45:50


Alex C:

- The capitalization after the colon is deliberate: I like it that way. Grammar authorities seem to disagree amongst themselves: http://grammar.ccc.commnet.edu/grammar/marks/colon.htm

- I mention "logarithmic" to highlight the distinction with "linear". How I actually model the behavior is as according to the formula: The whole point is that this particular formula is a counter-example to the claim that positive feedback always leads to a runaway. But the formula is

d(log(X)) = dX/X ; and the point is not that log(X) has a terminal value, but that the model responds to a positive "kick in the pants" by jumping up to a terminal amount, as shown.

- "a audio" => "an audio": Yes, I'll fix it.

- Yes, it's outgoing IR radiation: It seemed to me to be obvious in context, but I'll look at it again.

2010-10-20 21:21:38
Alex C


All right, I must have misinterpreted what you had meant by "terminal value" in context.  In that case, since again it looks good and I have nothing more I could add to it...

2010-10-20 21:27:00


Thanks for looking in...



2010-10-28 09:04:10
It looks great to me, although I would argue for it to be advanced rather than intermediate, unless you had other ideas for the advanced version. 
2010-10-28 09:39:50

Well, the Advanced version has even more detail, as you can see by looking at all three versions in the Rebuttal List.