• The Lorenz attractor - flow patterns in a layer of water & the butterfly effect
In the 1960's a beautiful mathematical object was discovered by Edward Lorenz. He was studying a very simple model of a thin layer of fluid heated from its base.
Regularity
In 1916 Lord Rayleigh(1) had also studied the same system and found that under certain conditions it could behave in a regular manner. Small circular convection currents form what are known as Rayleigh-Bénard convection cells. Rayleigh's 1916 paper describes the regular change in temperature through the layer of fluid with the following equation:
θ = θ0 cos(A x) sin(B z)
The equation contains the regular sine and cosine functions. When the fluid follows this pattern of behaviour it is in principle predictable to some given accuracy.
Irregularity
When Edward Lorenz(2) studied the system he used some equations developed by Barry Saltzman(3). These equations provided a more complete description of the fluid behaviour including allowing for a larger temperature difference between the base and the top. They were as simple a description as possible of how 3 properties of the fluid would change in time.
The properties were given symbols X, Y and Z, where X indicated the intensity of convective motion, Y indicated the difference in temperature between ascending and descending currents and Z indicated how much the temperature varied vertically.
The equations can be written as follows:
d X / d t = σ ( Y – X ),
d Y / d t = X ( ρ – Z ),
d Z / d t = X Y – ß Z
Java App N.B. ! Executable download is here
to plot, change and rotate the 3D Lorenz attractor.
[also includes option to plot the Rössler attractor]
The equations (eqs. 25 to 27 in the Lorenz paper(2)) can be solved approximately to draw the Lorenz attractor.
An animation of the Lorenz attractor and the corresponding flow pattern for a 1 cm thick layer of water is shown:
Drawing the attractor
The attractor is drawn by plotting a line that has three coordinates: (X, Y, Z) at a series of time values.
The How to draw page has example code that can be used.
An initial starting point is set*, (X0, Y0, Z0), and an approximation for the next point along the attractor is calculated, (X1, Y1, Z1), for a small time later, dt.
For the Lorenz attractor the simplest calculations (which can be programmed using many computer languages, or put into 3 columns of a spreadsheet), are:
X1 = X0 + dt × σ × (Y0 - X0)
Y1 = Y0 + dt × X0 × (ρ - Z0)
Z1 = Z0 + dt × (X0 × Y0 - ß × Z0)
The next point on the line (X2, Y2, Z2) is calculated using the values (X1, Y1, Z1). The calculations are repeated for as many points as required. This is known as a deterministic system, i.e. the next state of the system is determined precisely by the present state.
The Butterfly Effect *
The starting point eventually has a very significant effect on the path of the attractor. Two slightly different starting points will eventually draw very different paths (though the general shape of the Lorenz attractor remains the same). This effect is famously known as the 'Butterfly effect'. The difference in starting point could represent two layers of fluid, one in which a microscopic butterfly flaps its wings and the other where it does not. The result will be that in the future the circulations of fluid in each layer will, at times, be in opposite directions. Extrapolating this idea to the behaviour of the Earth's atmosphere it could represent a butterfly flap resulting in a tornado - or not - occurring on the other side of the world some weeks (or months) later.
In principle this means that a universe even as simple as a layer of fluid requires infinitely accurate information to be able to predict its future behaviour at any time in its future. Such information does not belong or 'fit' within that universe. It would be reasonable to say that such knowledge belongs to God. The butterfly effect represents an 'edge' to the logic of the Universe.
The Mandelbrot set java app from this site shows how the calculations for the Mandelbrot set also demonstrate the butterfly effect. Also known as hyper-sensitivity to initial conditions.
Proverbs 30:18,19 "There are three things that are too amazing for me,
four that I do not understand:
the way of an eagle in the sky, ..."
Calculations for a 1 cm thick layer of water
The values of σ, ρ and ß used to draw the animation were σ = 6.58, ρ = 157.268 and ß = 8/3. These values were calculated from various physical constants for a 0.01 m thick layer of water at 20°C using the Rayleigh constant, the Prandl number and a critical number, Rc, (eqs. 21, 22(2)). A value of dt = 0.0014 was used.
Drawing the water layer
The blue shading indicating the temperature of the layer of water was calculated using equation 24(2) and the values Y and Z. The lines indicating the flow were calculated using equation 23(2) and the value of X. The cosine and sine functions of the horizontal and vertical position in the layer were also used.
References
(1) Rayleigh, Lord, On convective currents in a horizontal layer of fluid when the higher temperature is on the under side. Philosophical Magazine, 6th series 32 529-546 (1916)
(2) Lorenz, E. N., Deterministic Nonperiodic Flow. Journal of the Atmospheric Sciences 20 130-141 (1963)
(3) Saltzman, B., Finite Amplitude Free Convection as an Initial Value Problem - I. Journal of the Atmospheric Sciences 19 329-341 (1962)