Lorenz's Simple Model of Convection

In 1964, Edward Lorenz, a humble and mild manored professor of atmospheric science at MIT, published a seminal work which contributed to the founding of a new science: CHAOS.  In this work he studied a very simplified model of convection, the three coupled ordinary differential equations you see below.  In fact this is about as simple as you can make a model of convection and still get something interesting out of it.

Today, we too shall study these equations a little bit.    This may seem a bit overwhelming so please keep in mind that our main objective is to get a sense for how scientists use models to understand nature.

The left hand side of these equations indicates that some variable (i.e. X,Y and Z) is changing in time. The right side of these equations describe what is causing this change. From the right hand side of these equations we see that they are coupled: which simply means that if we want to know how X changes we need to know how Y changes. And if we were to know how Y is chaning in time then we need to know how X and Z are changing in time. In other words they are talking to each other; they all depend on one another and thus a change in one will lead to changes in the other.

And this is why these equations can only be solved on a computer.

The Equations:

X is proportional to the intensity of convection. The change of X in time is governed by

Y is proportional to the temperature differnce between the ascending and descending convection currents. The change of Y in time is governed by

Z is proportional to the deviation of the temperature profile from linear. The change of Z in time is governed by:

NOTES:

a is the Prandtl #;  ; i.e. the ratio of the kinematic viscosity divided by the thermal diffusivity. It is a nondimensional number.
b is a measure of the stability
c is a parameter which depends upon the wave #.
F(t) are forcing terms.

More definitions are needed.

• forcing
• Define viscosity, etc.

Lorenz's Butterfly

If you integrate these equations in time (i.e. solve the three differential equations for each time step) and plot a coloured dot at each point in time representing the coordinates of X, Y, Z, in``phase space'', you get the famous Lorenzian butterfly:

• Talk about predictability.

Some Experiments in Chaos.

Here we can learn about sensitivity to initial conditions and how the behaviour of the solutions of the equations changes under different conditions.

Change the initial conditions:

Set y(0) = 0

Change the property of the fluid:

You can think of changing a as changing the viscosity of the fluid.

Set a = 5.0
Set a = 5.64
Set a = 18.5

References: