Richard Wiener

My research interests lie in the intersection of three fields of physics: chaos, nonlinear pattern dynamics, and fluid mechanics. Chaos is the study of deterministic systems that display irregular, non-repeating behavior. Pattern dynamics involves systems that develop recurring large-scale structures (i.e. patterns) that evolve dynamically. Examples of patterns in fluid systems include arrays of vortices, standing waves, or convection rolls. Chaotic patterns are ones in which the order of the pattern begins to break down either spatially, temporally, or both. My research is about the borderline between order and disorder in fluid flows. This topic, which is a popular area of research at major universities, is also well suited to the small Liberal Arts College. It is exciting, visually striking, and intellectually challenging. It is widely studied both for its intrinsic interest and its many practical applications. Some of the most important, currently unanswered questions are feasible to pursue with the resources available at a small college.

I perform experimental research on pattern-forming fluid systems such as Taylor-Couette flow. Taylor-Couette flow is the flow of fluid between concentric rotating cylinders [1]. At first glance, it might seem that nothing too exciting or difficult to understand would occur by filling an annulus between cylinders with water and rotating one or both of the cylinders. But the amazing fact is that if one rotates the cylinders even at low speeds a distinct pattern of stacked toroidal vortices emerges. At higher speeds the pattern develops time-dependent waves, then chaos, and eventually turbulence. But depending on parameters such as the gap size between the cylinders, whether the cylinders are co-rotating or counter rotating, and so on, a huge variety of patterns are obtainable. These include spiral vortices, barber pole turbulence, braided vortices, wavy vortex flow, modulated wavy vortex flow, bursting turbulence, to name just a few.

The fundamental question in the field of pattern dynamics is, how do persistent large-scale structures arise, evolve, and interact? Taylor-Couette flow is a great system to explore this question in that it is geometrically simple with precisely controllable experimental parameters, and yet capable of producing a host of patterns that are very challenging to understand. Experiments with Taylor-Couette flow are readily accessible to students, who are often attracted by the intriguing dynamics of patterns one can see in a laboratory. Such experiments also present students with the opportunity to learn fundamental ideas in nonlinear dynamics that are applicable to many scientific fields. Moreover, experiments with Taylor-Couette flow and other related fluid systems employ a variety of advanced techniques to measure, characterize, and visualize flow properties. This provides a wonderful opportunity for students to gain important experimental skills.

A key focus of research in pattern dynamics during the last fifteen years has been to investigate how systems in weakly nonlinear regimes can be characterized by relatively simple amplitude and/or phase equations that are rigorously derived from the Navier-Stokes equation. The Navier-Stokes equation is the classical nonlinear partial differential equation that governs fluid flows. The idea is that patterns can serve as the dynamical entities to which the reduced equations refer. In this way the evolution and interactions of many fluid flows can be predicted and understood, and a link can be established between the Navier-Stokes equation and chaos theory. My primary research interest is to contribute to this effort. I have concentrated on one-dimensional patterns, i.e. patterns that only vary along one direction, since these are the simplest patterns and yet much insight can be gained from studying them. My most recent efforts have explored the control of chaotic pattern dynamics.

The first undergraduate student that I involved in my research, Dan McAlister, has gone on to earn a Ph.D. in physics at the University of Oregon. Dan and I completed a beautiful experiment in which we discovered and investigated solitary waves in Taylor-Couette flow. Solitary waves are particle-like nonlinear waves that hold together and travel indefinitely without dispersing. These solitary waves were predicted to exist by Hermann Riecke of Northwestern University and Hans-Georg Paap of Universitat Bayreuth in Germany [2]. Our experiment is particularly significant in that solitary waves caused by a parity-breaking bifurcation (the underlying physical mechanism) occur in a number of other pattern-forming systems. However, the Taylor-Couette system offered the advantage that it is possible to quantitatively calculate the parameters at which solitary waves exist. Using amplitude-phase equations that are rigorously derived from the Navier-Stokes equation Riecke and Paap made these calculations. Our experimental results confirmed the calculations, and we published them in Physical Review Letters, the premier physics journal in the world [3].

Over the past five years my students and I have been working on a set of experiments using a variant of the Taylor-Couette system that I created. We have been investigating the effects on Taylor vortex flow of replacing the inner cylinder with an hourglass geometry, since this "spatial ramp" induces a time-varying, but nonetheless one-dimensional, pattern of vortices. We have demonstrated that there is a period-doubling cascade to chaotic pattern dynamics [4]. The dynamics involve the diffusive stretching of the vortex pattern, until a pair of vortices catastrophically breaks and a new pair emerges. These breaks, called "phase slips", can occur periodically in time at a regular location or chaotically in both space and time. Our results are in qualitative agreement with predictions from a reaction-diffusion model (see references in Wiener et al. [4]). The hourglass geometry fulfills the symmetry conditions underlying the phase equation used in the model. The mechanism that generates chaotic pattern dynamics is phase diffusion coupled with a ramp-induced Eckhaus instability. Since phase diffusion and the Eckhaus instability are common features of many pattern-forming systems, this mechanism for chaos should be present in other pattern-forming systems. We have also documented several other interesting regimes in Taylor-Couette flow with hourglass geometry that merit additional study. Among these is an intriguing chaotic regime, dominated by spiral excitations, in which axisymmetry (and thus the one-dimensionality of the pattern) is intermittently broken. There is also a highly turbulent regime that still exhibits persistent pattern dynamics involving phase slips. We published the results of our detailed investigation in Physical Review E, with four undergraduate student researchers as co-authors [4].

Recently we have successfully implemented a control of chaos algorithm on the chaotic pattern dynamics that we previously discovered in Taylor-Couette flow with hourglass geometry. We used a control algorithm based on the pioneering work of Ott, Grebogi, and Yorke (OGY) [5]. The key idea of the OGY method is to stabilize one of the many unstable periodic motions that underlie a dynamical system’s chaotic attractor. Stabilization is achieved by small perturbations to an accessible system parameter. The varying strength of the perturbations is determined by a feedback loop that incorporates real-time sampling of the system’s dynamics.

The particular algorithm that we used, recursive proportional feedback (RPF), was developed by Rollins et al [6]. This method is applicable to highly dissipative systems that are adequately modeled by iterative maps of a single variable. For our single variable we used the time interval between phase slips. However, RPF along with other OGY-type control algorithms was developed for concentrated systems, whereas Taylor vortex flow with hourglass geometry is a spatially extended one-dimensional pattern. Nonetheless, we found that by a simple sorting of the spatial behavior of the pattern, it is possible to untangle the temporal dynamics of the phase slips so that RPF is applicable. We are able to control the time interval between phase slips to within 86.6 ± 0.4 periods of revolution, when our target interval is 86.8, using a feedback loop that involves small proportional perturbations (< 2%) to the reduced Reynolds number calculated from real-time sampling. We have demonstrated control for as many as 103 phase slips (until we voluntarily relinquish control). In the absence of control perturbations, the time interval varies between 75 and 140 periods of revolution. The size of the control perturbations is order thousandths of a Hz when the hourglass is rotating at order 3 Hz. The corresponding spatial pattern during control, which we selected during a pre-control experiment, involves only right-drifting phase slips that occur at a single location. Control is robust and repeatable. We have demonstrated (to our knowledge for the first time) control of both the spatially and temporally chaotic dynamics of an extended pattern, using perturbations of a single system parameter. We recently published a paper in Physical Review Letters, with four undergraduate student co-authors, presenting these significant results [7].

REFERENCES:

[1] For a popular discussion of this fluid system see R. J. Donnelly, Physics Today November 1991, page 32.

[2] H. Riecke and H.-G. Paap, Phys. Rev. A 45, 8605 (1992).

[3] R. J. Wiener and D. F. McAlister, Phys. Rev. Lett. 69, 2915 (1992).

[4] R. J. Wiener, G.L. Snyder, M. P. Prange, D. Frediani, and P. R. Diaz, Phys. Rev. E 55, 5489 (1997).

[5] E. Ott, C. Grebogi, and J. A. Yorke, Phys. Rev. Lett. 64, 1196 (1990).

[6] R. W. Rollins, P. Parmananda, and P. Sherard, Phys. Rev. E 47, R780 (1993).

[7] R. J. Wiener, D. Dolby, G. C. Gibbs, B. Smeby, T. Olsen, and A. M. Smiley, Phys. Rev. Lett. 83, 2340 (1999).