Ed Lorenz and his equations

Professor Lorenz will celebrate his 90^th birthday on May 23, 2007. Wish him a Happy Birthday!

A piece of our atmosphere

The little white line at the left is a piece of our atmosphere, which we imagine to be enlarged as shown below.

Note: Alas, the enlargement here and on subsequent slides is not a realistic representation of the Earth's atmosphere—see the added (last) slide of this presentation.

How the fluid moves—part 1

This diagram is more appropriate for a shallow layer of water than it is for the Earth's atmosphere.

Cooler fluid is more dense, and therefore tends to fall in the gravitational field. However, viscosity (damping) will inhibit the flow. The result is that if ΔT is less than some critical value there is no movement of the fluid.
For a slightly larger ΔT, the fluid circulates in a steady pattern of rolls, as shown in the diagram.

How the fluid moves—part 2

For an even larger ΔT (greater than a second critical value), the roll circulation pattern becomes unstable. The roll speed increases, which tends to reduce the ΔT gradients, which causes the roll speed to decrease and even reverse—unpredictably! This is the region of deterministic chaos.
For a still larger ΔT, the fluid motion becomes turbulent and the Lorenz equations no longer are sufficiently complex to describe the fluid motion.

The meanings of X, Y and Z

X, Y and Z are quantities that vary with time; they describe how the fluid moves, and how the temperature varies within the fluid layer. τ is proportional to the time.

X is proportional to the roll circulation speed. It can be positive or negative.
Y is a measure of the temperature difference between the rising currents and the falling currents. It can also be positive or negative.
Z is a measure of the deviation from linearity of the temperature gradient in the vertical dimension. It will always be positive, but will increase as the temperature gradient becomes more pronounced near the boundaries at the top and bottom of the layer.

The meanings of σ, r and b

σ, r and b are constant parameters.

σ is proportional to the fluid viscosity.
r is a key parameter, proportional to the temperature difference between the bottom and the top of the layer. It is also proportional to the strength of the gravitational field g.
b is not an important parameter, having simply to do with the shape of the fluid rolls, relating to the ratio of the roll width to the roll height.

How to solve the Lorenz Equations

Choose values for σ, r and b.
Choose initial values for X, Y and Z. Almost any reasonable values will do; these are then the values at τ = 0.
Use the equations and a computation algorithm (recipe) to generate values of X, Y and Z a short time later. This is a simple computation involving only addition and subtraction.
Repeat step 3 a number of times, thus generating values of X, Y and Z for future values of τ.

Table of some calculated values

Graph of X vs time

Graph of X vs Y vs Z — the Lorenz attractor

J. Atmos. Sci., 20, 130 (1963)

Words regarding weather forecasting

“When our results concerning the instability of nonperiodic flow are applied to the atmosphere, which is ostensibly nonperiodic, they indicate that prediction of the sufficiently distant future is impossible by any method, unless the present conditions are known exactly. In view of the inevitable inaccuracy and incompleteness of weather observations, precise very-long-range forecasting would seem to be non-existent.”

— Edward N. Lorenz

The Royal-McBee LGP-30 computer

The computer used by Lorenz was perhaps the machine of choice from 1956 (when the first one was sold) through the 1960s. Here are a few specs:

Components: 113 vacuum tubes, 1450 diodes
Clock speed: 120 kHz
Dimensions: 44" wide, 26" deep, 33" high
Weight: 800 lbs.
Price: in excess of $40,000

The Lorenz waterwheel

The meanings of Ω, η, ζ, σ and r

Ω is proportional to the angular velocity of the wheel.
η is proportional to the horizontal (x) component of the wheel's center of mass.
ζ is proportional to the vertical (z) component of the wheel's center of mass.
The constant σ is proportional to the damping force on the wheel.
The constant r is proportional to the input water flow rate.

The Lorenz waterwheel — an image and a video

According to Professor Lorenz, the idea for this unusual waterwheel apparently originated with Willem Malkus, now an emeritus Professor of Mathematics at M.I.T.

Here is a video of Malkus's waterwheel in operation. Keep your eye on the yellow bucket to see what the wheel does.

Finally, here is a sound file of “Lorenz music”, with X (or Ω) being represented by the organ, Y (or η) by the oboe, and Z (or ζ) by the glockenspiel.

Added slide: Bob Kraft's question

During my talk, Bob Kraft asked a key question: “What time scales are we talking about here?” Although I had worked out the scaling for the waterwheel (since I wanted to make one), I had not checked out the time scaling described in Lorenz's paper, and so did not know the answer, but intuitively I thought it might be on the order of days or maybe a week or two, since that's about how long a weather forecast is good for.

It turns out that if we take H = 10 Km (the scale height of our atmosphere), the natural unit of time (about the time for air to convect through the distance H in the Lorenz model), is approximately 10,000 years! Uh-oh. Something is clearly amiss.

Although Lorenz concludes from his results that “precise very-long-range forecasting would seem to be non-existent” (see slide 11), he also notes, in a comment at the end of his paper, that “There remains the very important question as to how long is ‘very-long-range.’ Our results do not give the answer for the atmosphere; conceivably it could be a few days or a few centuries.”

I've written a few thoughts that may relate to this comment, for those who may be interested. You can peruse them here. I would warmly welcome any thoughts you may have.

— Peter Scott

Emeriti Lunch—March 15, 2007