Lorenz System

A simplified system of equations describing the 2-D flow of fluid of uniform depth , with an imposed temperature difference $\Delta T$ , under gravity , with buoyancy $\alpha$ , thermal diffusivity $\kappa$ , and kinematic viscosity $\nu$ . The full equations are

$\begin{displaymath} {\partial\over\partial t}(\nabla^2\phi)={\partial\psi\over\p... ...(\nabla^2\psi)+\nu\nabla^2(\nabla^2\psi) + g\alpha{dT\over dx} \end{displaymath}$

(1)

$\begin{displaymath} {\partial T\over\partial t}={\partial T\over\partial z}{\par... ...pa\nabla^2 T+ {\Delta T\over H}{\partial\psi\over\partial x}. \end{displaymath}$

(2)

Here, $\psi$ is the ``stream function,'' as usual defined such that

$\begin{displaymath} u={\partial\psi\over\partial x}, \qquad v={\partial\psi\over\partial x}. \end{displaymath}$

(3)

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

$\begin{displaymath} \psi=\psi_0\sin\left({\pi a x\over H}\right)\sin\left({\pi z\over H}\right) \end{displaymath}$

(4)

$\begin{displaymath} \theta=\theta_0\cos\left({\pi a x\over H}\right)\sin\left({\pi z\over H}\right) \end{displaymath}$

(5)

grew for Rayleigh numbers

larger than the critical value,

. 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 following terms in his system of equations,

$\displaystyle X$	$\textstyle \equiv$	$\displaystyle \psi_{11} \propto \hbox{ convective intensity}$	(6)
$\displaystyle Y$	$\textstyle \equiv$	$\displaystyle T_{11} \propto \Delta T\hbox{ between descending and ascending currents}$	(7)
$\displaystyle Z$	$\textstyle \equiv$	$\displaystyle T_{02} \propto\Delta\hbox{ vertical temperature profile from linearity},$	(8)

and obtained the simplified equations

$\displaystyle \dot X$	$\textstyle =$	$\displaystyle \sigma(Y-X)$	(9)
$\displaystyle \dot Y$	$\textstyle =$	$\displaystyle -XZ+rX-Y$	(10)
$\displaystyle \dot Z$	$\textstyle =$	$\displaystyle XY-bZ,$	(11)

where

$\displaystyle \sigma$	$\textstyle \equiv$	$\displaystyle {\nu\over\kappa} = \hbox{ Prandtl number}$	(12)
$\displaystyle r$	$\textstyle \equiv$	$\displaystyle {Ra\over Ra_c} = \hbox{ normalized Rayleigh number}$	(13)
$\displaystyle b$	$\textstyle \equiv$	$\displaystyle {4\over 1+a^2} = \hbox{ geometric factor}.$	(14)

Lorenz took $b\equiv{8/3}$ and $\sigma\equiv 10$ .

The Critical Points at (0, 0, 0) correspond to no convection, and the Critical Points at

$\begin{displaymath} ({\sqrt{b(r-1)}, \sqrt{b(r-1)}, r-1}) \end{displaymath}$

(15)

and

$\begin{displaymath} (-\sqrt{b(r-1)}, -\sqrt{b(r-1)}, r-1) \end{displaymath}$

(16)

correspond to steady convection. This pair is stable only if

$\begin{displaymath} r = {\sigma(\sigma+b+3)\over \sigma-b-1}, \end{displaymath}$

(17)

which can hold only for Positive

if $\sigma>b+1$ . The Lorenz attractor has a Correlation Exponent of $2.05\pm 0.01$ and Capacity Dimension $2.06\pm 0.01$ (Grassberger and Procaccia 1983). For more details, see Lichtenberg and Lieberman (1983, p. 65) and Tabor (1989, p. 204).

See also Butterfly Effect, Rössler Model

References

Gleick, J. Chaos: Making a New Science. New York: Penguin Books, pp. 27-31, 1988.

Grassberger, P. and Procaccia, I. ``Measuring the Strangeness of Strange Attractors.'' Physica D 9, 189-208, 1983.

Lichtenberg, A. and Lieberman, M. Regular and Stochastic Motion. New York: Springer-Verlag, 1983.

Lorenz, E. N. ``Deterministic Nonperiodic Flow.'' J. Atmos. Sci. 20, 130-141, 1963.

Peitgen, H.-O.; Jürgens, H.; and Saupe, D. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, pp. 697-708, 1992.

Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989.