Spatiotemporal chaos at threshold in rotating Rayleigh-Bénard convection
The complex Ginzburg-Landau equation (CGLE) which describes slow modulations of an amplitude or envelope A of a pattern near a Hopf bifurcation to traveling waves in spatially extended systems, has been used extensively both to study nonequilibrium pattern formation and as a model system for spatio-temporal chaos. The qualitative dynamical behavior of solutions of the CGLE depends on two coefficients, c1 and c3, which, for a given system, can be obtained from the underlying equations by laborious calculations. For small values of these parameters, as well as close to the line c1=-c3, the dynamics is close to that found in the relaxational limit c1=c3=0. In this limit, the dynamics of patterns close to threshold is relatively well understood, and no chaos occurs. For |c1| and |c3| large, the CGLE reduces to the nonlinear Schrödinger equation. In recent years, the complicated and often surprising dynamics that occurs away from these limits, has been intensively studied theoretically. In particular, it has become clear that the CGLE shows various regimes of spatio-temporal chaos when c1c3>1. The precise nature of the various chaotic regimes, as well as the existence and nature of the transitions between them, is still under active investigation in the field of spatio-temporal chaos.
In order to be able to investigate these chaotic regimes experimentally, one would like to have a system where the coefficients of the corresponding CGLE can be tuned in a convenient way through the spatio-temporal chaotic regimes -- so far only one experiment appears to be known where there is indirect evidence that these regimes are accessed. When it was discovered that a forward Hopf bifurcation to a quasi-one-dimensional traveling wave wall-mode occurs in rotating Rayleigh-Bénard convection in bounded containers, it was soon realized by several workers that the rotation rate might serve to tune the coefficients of the corresponding CGLE. In Leiden, Martin van Hecke (then a graduate student, now a postdoc at the Niels Bohr Institute) and I therefore investigated convection in rotating annuli for a range of Prandtl numbers P - earlier investigations by Kuo and Cross of the coefficients for a Prandtl number corresponding to water (P= 6.4) showed that for large P the behavior is close to relaxational. In our analysis, we have taken the circumference of the annulus large enough that the curvature can be neglected, so that bulk-modes are quasi one-dimensional and can be described by the same amplitude equations as the wall-modes. The main result of our calculations is that for Prandtl numbers 0.1 < P <0.2 traveling waves in a rotating annulus are predicted to be an experimental realization of a CGLE whose coefficients scan through the spatio-temporal chaotic part of the phase diagram when the rotation rate is changed. One interesting experimental realization of such a system is a rotating annulus with a superfluid 3He-4He mixture.
Reference
M. van Hecke and W. van Saarloos, Convection in rotating annuli: Ginzburg-Landau equations with tunable coefficients, Phys. Rev. E 55 R1259 (1997).Some references on chaos in the CGLE
B. I. Shraiman, A. Pumir, W. van Saarloos, P. C. Hohenberg, H. Chate and M. Holen, Spatiotemporal chaos in the one-dimensional complex Ginzburg-Landau equation, Physica D 57 241-248 (1992).
H. Chate, Spatiotemporal intermitttency regimes of the one-dimensional complex Ginzburg-Landau equation, Nonlinearity 9 185 (1994).
M. van Hecke, Building Blocks of Spatiotemporal Intermittency, Phys. Rev. Lett. 80 1896 (1998).July 15, 1999
[Pattern formation] [Wim van Saarloos] [Instituut-Lorentz]