Ball-well diagrams, feedback, and equilibria

The ideas of equilibrium and feedback are of fundamental importance, not only in ecology and earth science, but in just about any field. They are simply useful tools for describing certain classes of phenomenon that pop up in just about any subject you can think of. This page is a conceptual explanation of these two ideas used together.

Equilibrium is a point at which a system is balanced in some way. It is not significantly changing over the timescale you are considering. Everything is "normal."

A stable equilibrium is one that has some resilience to minor changes. The system will tend to return to its equilibrium point after some time. Recall Prof. Currie's example from class: a ball sitting at the bottom of a well. If you push the ball up the side of the well (as long as you don't push it too far), the ball will roll back down and eventually settle at the bottom again, right where it started.

The process that returns the system to its stable equilibrium point is negative feedback. A good example of negative feedback is a thermostat and an air conditioner. When the room temperature rises to a certain point, the thermostat triggers the air conditioning to turn on. This then cools the temperature of the room. Then, when the temperature has dropped below a certain point, the thermostat triggers the air to turn off again. Thus, the temperature always stays within certain bounds as a result of the negative feedback mechanism (the thermostat). In the ball-well case, gravity keeps the ball in the bottom of the well (or, to be more technical, the exchange of kinetic and potential energy).

An unstable equilibrium is one in which a minor change would cause the system to drastically alter. The system won't be able to return to its equilibrium point, or it will establish some new equilibrium point different than the original. Picture a ball precariously sitting on top of a hill. If you leave it alone, it stays there, but if you give it a small push in either direction, it will fall off and roll away.

The process that causes such drastic changes in the system is positive feedback. A push toward the left sends the ball even farther toward the left. Or, suppose you had a severely deranged thermostat that operated on positive feedback. When the room heated up to a certain temperature, the thermostat would turn on the heat, continuing to warm the room. To learn more about positive feedbacks and the environment, click this link.

A good example of positive feedback is the ice-albedo situation. Sea ice reflects a lot of light back into space, while darker water underneath it absorbs a lot of light. Now that there is a lot less sea ice around, there is more water to absorb more light as heat, and the warmer oceans melt a lot more sea ice. (in the lecture notes, this was cited from the [Climate.nasa.gov/uncertainties] site, and Blair mentions this in many of his "Climate Denial Crock of the Week" videos—check them out if you haven't already!) [http://www.youtube.com/user/greenman3610]

Consider a situation like the one in the following image. A ball sits atop a hill that is surrounded by two wells. If you leave it alone, it sits in equilibrium, but it if you push it, it will fall into one well or the other depending the direction you push it. It will eventually establish a new equilibrium in one of those wells. Because disturbing this unstable equilibrium could yield more than one very distinct results, it is called a tipping point.

What if those two wells were drastically different climatic conditions? It would be nice to changes in the natural world (or man-made world) could cause the existing unstable equilibrium to get "pushed" in either direction and what the feedback mechanisms were that would determine the eventual destination of the system.

Finally, a word on phase space diagrams. Picture the ball in the bottom of the well again. The plot below shows velocity on the y-axis and position on the x-axis. If the ball is rolling around at the bottom of the well, it has the highest velocity when the position is close to zero (in the middle of the well) and the lowest velocity when the position is at its extremes (up on the sides of the well). So, if you think about the position over time, you get a graph like this one.

Note that the previous diagram assumed that the ball continues to roll without slowing. If there is friction in the system and the ball eventually slows down and comes to a stop, you get something like the next diagram. Every time the ball rolls from one side to the other, it loses some speed, and consequently, it can't roll as high up the side of the well. So the phase space diagram spirals inward. The central point (the equilibrium) is called an attractor.

page revision: 6, last edited: 11 Nov 2009 01:24