In the summary below, the numbers in brackets refer to the group publication list Publications . Our results have been presented also in numerous contributed and invited talks, seminars, and colloquia. A number of Invited Talks were given by our students and postdocs.

Pattern-forming Nonlinear Systems

Another part of our work was done using electro-convection (EC) in a nematic liquid crystal (NLC). NLC molecules are long, rod-like objects which are orientationally ordered relative to their neighbors, but whose centers of mass have no positional order. The axis parallel to the average orientation is called the director n. By confining the NLC between two properly treated parallel glass plates, one obtains a sample with uniform planar (parallel to the surfaces) alignment of n. An ac voltage of amplitude V and frequency f is applied perpendicular to the sample, using transparent conducting films on the inner surfaces of the glass plates. At a critical value V_c, there is a transition from a spatially-uniform quiescent state to convection. The study of EC is a relatively new experimental field for us. During the last four years we have been able to make important contributions to the understanding of the mechanism of EC in NLC. We have also been able to use this system to study STC under conditions which are particularly well suited for a parallel theoretical treatment.

A provocative recent discovery by our group [164] in RBC with Pr~1 is spiral-defect chaos (SDC), a state consisting of small rotating spirals which appear, interact with each other and with other defects, and disappear, irregularly both in space and in time [164,175,182,187, 190]. It came as a complete surprise that SDC with its self-sustained chaotic dynamics and spatial complexity occurs in a parameter range where straight convection rolls are also stable. Our experimental discovery stimulated several numerical studies of solutions of generalized Swift-Hohenberg equations (which are models for RBC) and of the Navier-Stokes equations. This numerical work also led to the observation of SDC. By now SDC has become one of the main examples of spatio-temporal chaos; a number of theoretical groups are studying it intensely.

Motivated in part by the desire to understand more about the system in which SDC occurs, we carried out a lengthy detailed and systematic investigation of RBC for Pr~ 1 [169,170,173,178]. This was done in collaboration with Bob Ecke at Los Alamos. Yu-Chou Hu did his thesis experiments in residence at LANL. The results of this project provide more detailed knowledge about RBC for Pr~1 than is available for any other range of Pr.

A recent extension of our work on SDC has been its study over the range 0.3< Pr<1. (Liu and Ahlers, in print) There are no known pure fluids with Pr < 0.7. However, we became aware of the fact that, on the basis of kinetic theory, this important parameter range of smaller Pr should become accessible in mixtures of gases provided the components have very different masses. The smallest Pr achieved so far by us is 0.15 for a mixture of H_2 and Xe. We studied the SDC onset for 0.15 An extension along a different parameter axis has been our work on RBC in the presence of rotation about a vertical axis. [156, 180,182] Theoretically this has been studied for a long time, but there were no experiments in the theoretically more tractable range near the onset of convection. The addition of a Coriolis force to the equations of motion (the Navier-Stokes equations) renders the system non-potential even near onset, so that persistent time dependence (and chaos) is possible arbitrarily close to the convective onset where the flow amplitudes are arbitrarily small and where thus perturbative theoretical techniques should apply. We found a supercritical transition directly from the conduction state into spatio-temporal chaos (the Kuppers-Lortz state). We found that the characteristic time and length scales of the chaotic state were both finite at onset, whereas the qualitative structure of the usual perturbation theory for problems of this type (the Ginzburg-Landau equations) requires these scales to diverge at onset. The reason for this gross disagreement with theory is not understood; resolution of this disagreement would be a significant step forward in the elucidation of spatio-temporal chaos. We suspect that it will be found in the fact that, even at onset, this system involves an infinite number of modes because its rotational invariance causes convection rolls of all orientations to acquire a positive growth rate simultaneously.

In an effort to find an example of Spatio-temporal chaos which is even simpler that the Kuppers-Lortz case, we turned to EC. Here the spatial anisotropy causes the number of critical modes to be finite. It remained to find a region in a very large parameter space where there was a supercritical bifurcation to a chaotic state, if such a phenomenon indeed existed. It also remained to understand the mechanism and to establish the equations of motion of EC. The prevailing model (the Lesley-Erickson equations) had been shown to be inadequate since they could not explain the experimentally observed Hopf bifurcation. We found that the electrical conductivity of our samples was non-ohmic. [165,174] This suggested that the dynamics of ion recombination and dissociation may play a significant role in this system. Theorists in Bayreuth, Germany developed the Weak Electrolyte Model (WEM) which includes the dissociation-recombination process. We then carried out quantitative measurements of the Hopf frequency and wavevectors and compared them to the prediction of the WEM. [183] Our results could be explained quantitatively. This provided a solid theoretical starting point for an understanding of nonlinear phenomena, including spatio-temporal chaos, in this system.

After some searching, we found that spatio-temporal chaos occurs near the onset of electro-convection in a certain nematic liquid crystal (I52) for sample conductivities above 1 \times 10^-8\ \rm \Omega^-1\, m^-1. [174,186] The chaos evolves via a supercritical Hopf bifurcation from the uniform conduction state. It is the result of the nonlinear interactions between only four modes. Thus, there exists a realistic opportunity of understanding the observed phenomena in terms of a weakly-nonlinear theory in the form of four coupled complex Ginzburg-Landau equations derived from the WEM. This theoretical derivation is now under way in a number of theoretical groups.

To our complete surprise, we found for smaller \Pr (Pr \ltwid 0.7 \times 10^-8\ \rm \Omega^-1\ m^-1) that the pattern immediately above onset consists of localized pulses of convection which co-exist with the conduction state. ([186,189] The pulses have a unique width in the direction perpendicular to \hat n, and have much larger and varying lengths parallel to \hat n. Pulses have been of great interest in the field of nonlinear pattern formation. In one dimension they may be regarded as the equivalent, in a dissipative system, of the solitons of the nonlinear Schr\'odinger equation. The anisotropic localized states which we found are two dimensional, and are quite different from any observed before. It will be interesting to see whether they are contained within the WEM.