Tuesday, February 2, 2010

Lorenz Attractor

The Lorenz attractor is an attractor that arises in a simplified system of equations describing the two-dimensional flow of fluid of uniform depth H, with an imposed temperature difference DeltaT, under gravity g, with buoyancy alpha, thermal diffusivity kappa, and kinematic viscosity nu. The full equations are

partial/(partialt)(del ^2phi)=(partialpsi)/(partialz)partial/(partialx)(del ^2psi)-(partialpsi)/(partialx)partial/(partialz)(del ^2psi)+nudel ^2(del ^2psi)+galpha(dT)/(dx)
(1)
(partialT)/(partialt)=(partialT)/(partialz)(partialpsi)/(partialx)-(partialtheta)/(partialx)(partialpsi)/(partialz)+kappadel ^2T+(DeltaT)/H(partialpsi)/(partialx).
(2)

Here, psi is a stream function, defined such that the velocity components u=(u,w) of the fluid motion are

u=(partialpsi)/(partialz)
(3)
w=-(partialpsi)/(partialx)
(4)

(Tabor 1989, p. 205).

In the early 1960s, Lorenz accidentally discovered the chaotic behavior of this system when he found that, for a simplified system, periodic solutions of the form

 psi=psi_0sin((piax)/H)sin((piz)/H)
(5)
 theta=theta_0cos((piax)/H)sin((piz)/H)
(6)

grew for Rayleigh numbers larger than the critical value, Ra_c" border="0" height="16" width="58">. Furthermore, vastly different results were obtained for very small changes in the initial values, representing one of the earliest discoveries of the so-called butterfly effect.

Lorenz included the terms

X=psi_(11)
(7)
Y=T_(11)
(8)
Z=T_(02),
(9)

where X is proportional to convective intensity, Y to the temperature difference between descending and ascending currents, and Z to the difference in vertical temperature profile from linearity in his system of equations. From these, he obtained the simplified equations

X^.=sigma(Y-X)
(10)
Y^.=-XZ+rX-Y
(11)
Z^.=XY-bZ,
(12)

now known as the Lorenz equations. Here, X^.=dX/dt, Y^.=dY/dt, Z^.=dZ/dt, and

sigma=nu/kappa
(13)
r=(Ra)/(Ra_c)
(14)
b=4/(1+a^2).
(15)

where sigma is the Prandtl number, Ra is the Rayleigh number, Ra_c is the critical Rayleigh number, and b is a geometric factor (Tabor 1989, p. 206). Lorenz took b=8/3 and sigma=10.

The Lorenz attractor has a correlation exponent of 2.05+/-0.01 and capacity dimension 2.06+/-0.01 (Grassberger and Procaccia 1983). For more details, see Lichtenberg and Lieberman (1983, p. 65) and Tabor (1989, p. 204). As one of his list of challenging problems for mathematics (Smale's problems), Smale (1998, 2000) posed the open question of whether the Lorenz attractor is a strange attractor. This question was answered in the affirmative by Tucker (2002), whose technical proof makes use of a combination of normal form theory and validated interval arithmetic.

The critical points at (0, 0, 0) correspond to no convection, and the critical points at

 (sqrt(b(r-1)),sqrt(b(r-1)),r-1)
(16)

and

 (-sqrt(b(r-1)),-sqrt(b(r-1)),r-1)
(17)

correspond to steady convection. This pair is stable only if

 r=(sigma(sigma+b+3))/(sigma-b-1),
(18)

which can hold only for positive r if b+1" border="0" height="14" width="54">.

Lorenz attractor laser-etched crystal (Bathsheba Grossman)
The image above shows a Lorenz attractor laser-etched into glass by digital sculptor Bathsheba Grossman

The Lorenz attractor is a set of differential equations which are popular in the field of Chaos. The equations describe the flow of fluid in a box which is heated along the bottom. This model was intended to simulate medium-scale atmospheric convection. Lorenz simplified some of the Navier-Stokes equations in the area of fluid dynamics and obtained three ordinary differential equations

[Graphics:Images/LorenzAttractorMod_gr_1.gif],

[Graphics:Images/LorenzAttractorMod_gr_2.gif],

[Graphics:Images/LorenzAttractorMod_gr_3.gif].

The parameter p is the Prandtl number, [Graphics:Images/LorenzAttractorMod_gr_4.gif] is the quotient of the Rayleigh number and critical Rayleigh number and b is a geometric factor. Lorenz is attributed to using the values [Graphics:Images/LorenzAttractorMod_gr_5.gif].


There are three critical points (0,0,0) corresponds to no convection, and the two points

[Graphics:Images/LorenzAttractorMod_gr_6.gif] and [Graphics:Images/LorenzAttractorMod_gr_7.gif] correspond to steady convection.

The latter two points are to be stable, only if the following equation holds

[Graphics:Images/LorenzAttractorMod_gr_8.gif].

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Introductory Methods of Numerical Analysis