Notes on Systems and Feedbacks (Supplement to Chapter 2, pages 19-24)

system: a set or arrangement of things so related or 'coupled' as to form a unity or organic whole. 

state of a system: a description of a system at a fixed point in time.  Usually involves a set of numerical values of a prescribed set of indices.  For example, the present state of the U.S. economy can be described in terms of indices like rate of expansion, unemployment rate, inflation rate, national debt, trade surplus or deficit.  In a similar manner, the current state of the earth system can be described in terms of indices like global mean temperature and sea level, the concentration of carbon dioxide in the atmosphere, total global biomass, mass of ice stored in glaciers and continental ice sheets...  The descriptions of the state of complex systems like the earth system are only partial descriptions at best. 

couplings: one way linkages in which one component of a system affects another component.  In positive coupling between components A and B, denoted by A ---> B in the text, a change in A causes a change of the same sign in B. In negative coupling, denoted by A ----o B,  a change in A causes a change of the opposing sign in B. 
 

Examples of couplings: 

As the temperature of the atmosphere increases, evaporation increases and the amount of water vapor increases.  This coupling, which is positive, can be represented schematically by 
              temperature ---------> water vapor 

Like carbon dioxide, water vapor is a greenhouse (heat trapping) gas. Other things being equal, the more water vapor in the atmosphere, the warmer the temperature of the earth's surface 
               water vapor ----------> temperature 

If the earth were to become warmer, the polar ice caps would tend to melt back and shrink in area 
               temperature ----------o ice cover 

The larger the fraction of the earth that is covered by ice, the whiter the planet and the larger the fraction of incoming solar radiation that is reflected back to space without warming the surface of the planet 
                 ice cover ----------o temperature 

Definition: 

feedback loop: two or more couplings acting in sequence to form a closed loop.  In a positive feedback loop a positive or negative change in one of the components of the system is amplified by the sequence of couplings whereas in a negative feedback loop the change is reduced in amplitude. 
 
Examples of feedback loops: 

temperature-water vapor:  If the temperature warmed for any reason, atmospheric water vapor would increase, which would increase the greenhouse effect and thereby further raise the temperature.  In a similar manner, if the earth cooled, atmospheric water vapor would decrease, causing further cooling.  This is a positive feedback loop.  In symbolic form: 

                               -------------> 
           temperature                                water vapor 
                               <------------- 
 

temperature-ice extent: If the earth were to warm, the ice caps would melt back, making the planet less reflective so that it would warm further.  In a similar manner, if the earth cooled, the ice caps would expand, causing further cooling.  This is also a positive feedback loop. 

                               -------------o 
           temperature                                ice extent 
                               o------------- 
 

Rule for inferring whether a feedback is positive or negative:

Add the number of negative couplings (indicated by (o) in the diagrams).  If it's even (0, 2, 4..) the feedback is positive: if it's odd (1, 3..) the feedback is negative. 
 

Definitions: 

equilibrium state: a state of balance between opposing effects. Once in an equilibrium state a system will remain there so long as it is undisturbed. 

stable equilibrium: a small disturbance will provoke a reaction that brings the system back to equilibrium. 

unstable equilibrium: a small disturbance will provoke a reaction that pushes the system farther away from equilibrium. 

neutral equilibrium: a small disturbance will not provoke a response. 
 

Examples: 

A cone resting on its base is in a stable equilibrium.  If displaced slightly and released, it will return to its equilibrium position. An inverted cone is in an unstable equilibrium: if displaced slightly and released, it will fall over.  A cone resting on its side is in neutral equilbrium: if displaced slightly and released it will remain where it is. 

A marble placed in a bowl with a round bottom is in a stable equilibrium.   A marble placed on top of the same bowl, inverted, is in an unstable equilibrium. These situations are illustrated by Fig. 2-3 in the text.  A marble placed on a flat surface is in neutral equilibrium. 

A small water balloon placed at mid-depth of a large tank of water in which the temperature increases with height is in a stable equilibrium.  If the balloon is raised slightly and then released, it will find itself surrounded by warmer, less dense water.  The water in the balloon will be denser (and therefore heavier) than the water that it displaces so the balloon will tend to sink back toward its original level in the tank.  In a similar manner, if the balloon is lowered and then released, it will find itself surrounded by colder, denser water and it will be lighter than the water it displaces and therefore rise toward its original level. 

The same balloon, placed in a tank of water in which the temperature decreases with height would be in an unstable equilibrium.  (I say 'would be' because such a situation is could not be maintained: the water in such a tank would overturn almost instantly, making the temperature uniform)   If the balloon is raised slightly and then released, it would find itself surrounded by colder, denser water. The water in the balloon will be lighter than the water that it displaces so the balloon will tend to rise farther.  In a similar manner, if the balloon is lowered and then released, it would find itself surrounded by lighter, less dense water and will sink.

 Last Updated:
10/03/2001