Supplement to Chapter 2, pages 19-24


Definitions and examples:

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, the air becomes
capable of holding more water vapor.  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

The warmer the earth the more infrared radiation it emits to space
               temperature ----------> emitted radiation

If the incoming solar energy remained constant, an increase in
emitted radiation would tend to cool the earth
               emitted radiation ----------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-------------
 

temperature - emitted radiation: If the earth were to warm, it would
emit more radiation, causing it to cool and lose heat, and vice
versa.  This is a negative feedback loop:

                               ------------->
           temperature                                emitted radiation
                               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.