## 02 April 2009

### Feedback

I'm actually referring to an important mathematical and physical concept, rather than the comments you give (which I also value).

Feedback is an extremely common process. Perhaps the most common case that people know is feedback involved public address systems. You have a microphone to pick up sound, which goes to an amplifier, and then out a speaker. If you put the microphone too near the speaker, then the microphone picks up some minor sound (say the sound of someone putting papers on the stand near the microphone), it goes through the amplifier and the louder sound comes out the speaker. The microphone now picks up that sound, it gets amplified even more and comes out the speaker again. Repeat through a number of cycles and the speaker is now as loud as it can be. This is a runaway feedback. Not all feedback is runaway.

A feedback can also be limited. Consider the atmosphere in this case. It has a temperature, carbon dioxide level, and water vapor level. Both carbon dioxide and water vapor are greenhouse gases, so if there is more of them, the temperature goes up. Suppose we put enough carbon dioxide into the atmosphere that its greenhouse effect (alone) raised temperature by 1 degree. If the atmosphere is warmer, we expect it to have more water vapor. That extra water vapor, let's say, produces another 0.5 degrees of warming. But, since temperature has gone up again, there's still more warming -- another half of this, for 0.25 degrees. This feeds back through increasing the water vapor to make another 0.125 degrees warming. Keep cycling through this feedback and it turns out to add another 1 degree of warming (1 = 0.5 + 0.25 + 0.125 + ...) after you've gone through an infinite number of steps. It's pretty close after a limited number of steps (99.9% of the way there after 10 cycles). We can look at these feedbacks in terms of how much of a response (multiplier) there is to a 1 degree kick to the system. I made up the number 0.5 here (every step is 0.5 times the previous). That's actually about the right size, in addition to being simple to work with. If the response multiplier is, say, 0.75 instead, then instead of getting back a total warming of 2 degrees, we would get back 4 degrees. The formula is
total = 1 / (1 - x)
where x is this multiplier, and the multiplier has to be less than 1.

Feedbacks can also produce cycles. A common illustration of this is predators and prey -- sharks and fish, or foxes and hares. Suppose one year, there are more fish than usual. The sharks, then, have a lot to eat and give birth and raise more sharks than usual. But, now that there are more sharks, they eat up even more fish than usual. That decreases the number of fish below the usual level. With fewer than normal fish, the sharks start dying off. That eventually gives us fewer than normal sharks, so the fish population recovers. And so on. The two populations could keep cycling (if the balances are right), or one could crash (depends on whether the extra sharks eat up too many fish, or whether the time of few fish lasts too long for any sharks to survive).

In the climate system, there are definitely some amplifying feedbacks and some cycling feedbacks. There don't appear to be any runaway feedbacks. That's not terribly reassuring, however, because life can get quite unpleasant even if temperatures don't run off to boiling the oceans.

Ian said...

Nice examples - the predator-prey example gives a clear picture of cycling within constraints.

On the topic of feedback, there's lots of current blog discussion on negative climate feedbacks in particular (all prompted by a Dick Lindzen post on Watts Up). A few posters on Watts Up seem confused by the terms *negative* and *positive* feedbacks in climate science, which are apparently different from their favored use of the terms in an engineering field. Specifically, rather than thinking of a positive feedback as one that intensifies or amplifies a forcing (whether cooling or warming), some commenters think that positive feedback should refer to warming only. Do you have any insight into why this difference in the use of *negative* and *positive* exists to describe feedbacks?

Penguindreams said...

Very odd. I started in electrical engineering and was used to that culture's notion of what feedback was before I ever started looking in to the climate literature. When I did start, I never had any difficulties with climatological usages.

In positive feedback, for electrical engineers, it's simply that the signal (sound from the speakers) is fed back in to the system (microphone) directly. In my climate example, it's temperature increase being fed back in.

Electrical engineers' negative feedback is that you feed back the signal inverted. Instead of feeding back a +1 volt, you pass back a -1 volt. I can't offhand think of any climate components that would carry out a negative feedback in this sense.

What is really at hand (I did check out Lindzen's note) is not whether the feedback is positive or negative -- in my engineer's sense of the term.

The important thing is whether the gain is greater or less than one. The gain is a measure of how big a result you get for how big a kick you give the system. In an audio amplifier, the gain is very much greater than 1. In my climate example, it was 2. Lindzen believes that the gain is more like 0.3 -- still positive, just smaller than the kick provided by greenhouse gas forcing.

In the case of the ice-albedo feedback, it is a positive feedback (and with a gain, at least now, that seems to be greater than 1). Whatever kick you give the system gets amplified, in the same direction. If there is more ice, you get colder temperatures, which gives more ice -- change in the same direction as the kick. If there is less ice, you get warmer temperatures, which gives less ice -- again, same direction as the kick.

For the predator-prey situation, we have a negative feedback process. More sharks means fewer fish, means fewer sharks. Or more fish means more sharks, which gives fewer fish. After 1 cycle, the sign of the kick gets changed.

I don't know what the problem is over there. But this usage is perfectly common in either climate or electrical engineering.

Scruffy Dan said...

One of the best examples of biological feedback cycles is the snowshoe hare/lynx cycle. The cycles are roughly 10 years long, have been well studied and are incredibly interesting... at least for those of us interested in population ecology.

@ Ian

Lindzen's ideas are not new, but so far there isn't much support for them

John Mashey said...

I'm looking for good examples of showing feedbacks graphically, in consistent ways, for the general audience. Do you have any good examples?

(I am convinced that the usual description of positive & negative feedbacks, in words, just doesn't communicate strongly enough to many people.)