The Lorenz equations are not much harder to write down and solve than
the systems we have already looked at. The biggest difficulty is
actually understanding where they come from. To get there, we first
have to understand a bit about **Thermal Convection** which is
just a fancy name for the idea that hot-air rises and cold air sinks.
In physics there is a classic problem of convection in a thin layer of fluid that is heated
from below (for example a pan with a thin layer of water on a stove).
This problem can show a remarkable range of behavior depending on how
hard you heat the layer. In general, a parcel of fluid that is hotter
than its surroundings will try to do two things. Because it is less
dense, it will try to rise like a hot air balloon; however, because it
is hot it will also lose heat by cooling. So, if it cools faster than
it can move it will just sit there. If it has more heat, or loses it
more slowly it will **convect**. When convection is confined to
a thin layer, it can form all kinds of patterns from no flow at all
(if you don't heat it enough) to simple **rolls** to highly
turbulent chaotic flow (look at a glass tea kettle some time).

Anyway, the Lorenz equations are just a simplified model of thermal
convection in a thin sheet. The principal simplification is that
Lorenz, pretended that the fluid velocity could be described by a
single roll and that the temperature in the layer could be described
by a steady state solution and two time dependent **modes**. In
pictures it looks like

Where *W*(*T*) controls the direction and speed of the roll (if *W* is
positive the roll spins clockwise, if *W* is negative it spins the
other direction, the bigger the absolute value of W, the faster it
spins). controls the horizontal temperature structure. If
is positive the left side of the box is hot, the right side cold (and
vice-versa). Finally controls the vertical temperature field.
It is always positive but higher values imply that the top of the box
is hotter (see the movies below). If both and are zero
the the temperature is just layered with hot material at the bottom
and cold material on top. The important point of this description is
that the spatial structure is assumed to be known everywhere and we
only need to solve for the values of through time.

So finally, the Lorenz equations arise from taking the simple roll solution, substituting into the more general equations of convection and coming up with a dynamical system for the three time dependent variables. They're not much to look at but here they are.

This problem has three adjustable parameters ( ) of which
we will only worry about the **Rayleigh Number** *r* and set the
other two to constants.. The Rayleigh
number is just a measure of how hard the layer is being heated. If
*r* is less than one, the layer is being heated too gently to
convect. Large Rayleigh numbers imply very vigorous convection. The
following section will explore the behavior of the Lorenz equations
for different values of the Rayleigh number.

Mon Sep 22 21:30:22 EDT 1997