The Lorenz Equations
Nonlinear Geoscience

The Lorenz Equations

Besides describing the mountains and the clouds and the rhythms of the geomagnetic field, fractals describe the geometry of chaos. Fractals are in many ways the scaffolding or the skeleton of deterministic chaos, so that in great measure, the reason why dynamic systems go chaotic is because they evolve in fractal and not in Euclidean geometry. To explore this idea, let's go back to the archetypal Earth dynamic system: thermal convection. Recall the Benard experiment of a thin layer of liquid heated from below. We know that as the temperature difference between the bottom and top of the layer reaches a critical value, convection (bulk motion of the water) begins and an organized system of convection cells appears that more effectively than conduction transport heat from the bottom to the top of the liquid.

It turns out that it is possible to simulate thermal convection in the atmosphere following the mathematical equations that describe the Benard experiment. Edward Lorenz, the "discoverer" of chaos, worked out a drastically simplified form of the mathematical equations and ended up evaluating a system of three nonlinear differential equations with three unknown time-dependent variables, X, Y, and Z. The variable X is proportional to the velocity of fluid flow in convection. If X>0 the fluid circulates clockwise and if X<0 it circulates counterclockwise. The variable Y is proportional to the temperature difference between ascending and descending fluid, and Z is a measure of the distortion of the temperature gradient away from linearity (remember that in the Benard experiment the temperature gradient was linear during conduction; during convection it changes to become very strong near the upper and lower boundaries and very weak in between.)

At any given instant of time the three variables X, Y, Z can be thought of as the coordinates of a point in three dimension space (this space we shall call phase space). An instant later all three variables will have changed their values -- since they describe the time rate of change of fluid velocity and temperature -- and thus the original point moves to another location described by the new values of X, Y, and Z. Thus, as time passes, the initial point moves smoothly in phase space, describing a trajectory that indicates how the system (the convection process) is evolving in time. For a given set of conditions of the fluid viscosity, temperature gradient, etc., there will be several types of trajectories depending on whether the model represents vigorous convection or steady conduction. In a state that describes vigorous convection, the trajectory looks like this.


The trajectory in the picture starts from the arbitrary point of coordinates (10, 5, 0) and then evolves first by circling one fixed point in phase space, then circling another. What is remarkable about the trajectory is that it will jump from one circuitry to the other at random instants, that is at times unpredictable and devoid of a pattern. Moreover, two slightly different initial points will eventually evolve along totally different trajectories (although always within the attractor), the telltale sign of chaos.

The Lorenz equations are completely deterministic, that is, there is no random component in them and all the variables are uniquesly defined and will evolve in a unique manner. But the outcome will be chaotic, that is stochastic or random. The plot above is arguably the most famous plot in the annals of chaos. It is called a strange attractor, an object in phase space that describes the state of convection at any one time by the values of X, Y, Z. This object is called an attractor because it defines a region in phase space (coincidentally similar to the wings of a butterfly) towards which the system is attracted to and will evolve in forever. That is, no matter for how long we let the system evolve, we can be assured that it will remain forever somewhere within the wings of the butterfly. It is called strange not only because of its shape, but because it is a fractal object, with dimension 2.05! Here is a formidable idea. The intimate geometry of a chaotic system is a fractal object.

To summarize, if chaos and fractals appear in such simplified models of convection as the Lorenz's equations, it is not too risky to think that chaos and fractals should therefore be ubiquitous, as they seem to be.