Dynamics Days 2002 Abstracts

Oral Presentations (alphabetical by author)

Some Provocative Unresolved Problems in Rayleigh-Benard Convection: Guenter Ahlers, University of California, Santa Barbara

This talk will address a number of experimental observations or measurements pertaining to pattern formation in Rayleigh-B\'enard convection (RBC) which fall into one or both of the following categories:
1.) They seem interesting but have never been addressed theoretically
2.) They disagree with theoretical predictions.
Included are:
a.) the re-emergence of relatively simple patterns in cylindrical samples at large Rayleigh numbers
b.) Unorthodox correlation-length scaling and defect formation in domain chaos, i.e. in RBC with rotation
c.) Formation of "squares" where K\"uppers-Lortz-unstable rolls are predicted in RBC with rotation
d.) A subcritical bifurcation without hysteresis in RBC with rotation
e.) Absence of convection above a predicted Hopf bifurcation in RBC with rotation

Stress Propagation in Granular Materials: Robert Behringer, Duke University

Granular materials present a host of challenging questions at the most basic level. In dense granular materials, complex force structures, known as force chains dominate the transmission of force. Competing models to describe force propagation include purely diffusive, elastic, and wave-like behavior. One way to address the issue of how stresses propagate is to carry out experimental determinations of the Green's function, i.e. the response of a material to a local force perturbation. We have recently carried out such measurements using 2D photoelastic particles that allow us to determine the local force at the particle scale. These measurements indicate that the ensemble-averaged response depends significantly of the amount of order in the packing. In highly ordered packings, the response is consistent with a wave-like propagation of forces, whereas in disordered packings, the response is elastic. Notably, any given realization is typically complex, with large deviations from the ensemble-averaged response. When dense materials are deformed, the stress chains break and reform, leading to large-scale fluctuations.  In Couette shear experiments, we have found that there is a novel transition with second-order-like properties as the density of the sample is varied. Conventional Coulomb models for stresses in shearing materials indicate that the forces should be independent of shear rate. However, we have recently found a slow dependence on shear rate that is not accounted for by conventional models.

Limits on Turbulent Transport: Charles Doering, University of Michigan

Mathematical bounds on mass, momentum and heat transport for solutions of the incompressible Navier-Stokes and Boussinesq equations are reviewed. In some cases the scaling indicated by the bounds is (nearly) saturated by turbulent flows, but in some cases---most notably turbulent Rayleigh-Benard convection---there remains a large gap between experiments and the rigorous heat transport limits. We discuss some current research exploring these issues further.

Dynamics of Turbidity Currents: Gravity-Driven Erosion and Deposition at the Edges of Continents: S. Julio Friedmann, Univ. of Maryland

Turbidity currents are the primary agents of coarse-grained sediment transport across the continental slope and abyss. These are gravity currents which flow when sediment suspended in water is more dense than the ambient fluid (usually sea or lake water). As such, their behavior is very sensitive to changes in mass and momentum, which are functions of sediment concentration, grain-size distribution, current height, and gradient. The critical threshold of erosion is called "ignition", and produces rapid acceleration and bulking up to a maximum 10% sediment concentration at which point the primary suspension mechanism, turbulence, is damped. Below the ignition threshold, turbidity currents will decelerate and dampen to predicable termination. These strong positive and negative feedbacks can be indirectly measured in geological and experimental systems and the system dynamics qualitatively characterized. The relevant parameters can be collapsed to produce simple phase diagrams that help predict grain-size distributions, bed characteristics, pore volume connectivity, and behavioral response. Numerical, experimental, theoretical, and field-based studies will help to place quantitative constraints on system thresholds and response.

Nonlinear Time Series Analysis: P. Grassberger, John von Neumann Institute for Computing, Juelich, Germany

This will be a general introduction, with emphasis on biomedical applications.
After an introduction where the two paradigms of stochasticity and deterministic chaos are opposed, and several physiological examples are sketched, I'll start the more technical part by dicsussing time delay embeddings and choices of parameters for them.
A first application will deal with noise reduction and signal separation based on the geometry of embeddings. Fetal heart beat extraction from a univariant ECG signal is discussed as a special case.
We then discuss classical invariants (metric entropy, attractor dimension, Lyapunov exponents) and argue why using them as indicators for chaotic determinism is not very useful. The same should be true also for alternatives like false nearest neighbors or forecasting errors. In contrast we shall argue that strict determinism is not needed for the arsenal of nonlinear time series analysis to be useful. In contrast, I shall present evidence that effective "attractor" dimensions can be useful for predicting epileptic seizures and localizing epileptic foci.
Finally we shall discuss various methods to study interdependencies between different time series. This includes cross correlation and coherence, mutual information, phase synchronization, and other interdependence measures. We shall discuss their usefulness in EEG analysis, in particular for epilepsy patients. Among these measures, of particular interest are asymmetric measures because they could, independent of time delays, indicate causal connections. Again this is illustrated with epileptic EEGs.

Classical and Quantum Flow Over Hilly Terrain: Branching and Coherence: Eric Heller, Harvard University

Recent experimental and theoretical work on the branching of electron flow under the influence of soft potential energy hills and valleys in nanostructures will be presented (Nature, March 8 2001). The work is applicable to a variety of situations involving propagation through random media, and represents perhaps a new regime in chaos theory. The cumulative effect of long range travel over many correlation lengths of the potential surprisingly leaves strong, preferred branches of flow intact. The combination of such flow with billiard walls will be discussed. Theoretical foundations and quantum implications will be presented.

Integrative Modeling of the Heart: Studying the Dynamics of Atrial Fibrillation: Craig S. Henriquez, Duke University

The promise of computational biology is that it provides an enhanced ability to relate changes at the molecular level to changes in macroscopic function. But as with all techniques, computational modeling and simulation involve making significant tradeoffs,  usually compromising realism to maintain tractability. In this talk, I will discuss the challenges of both creating and analyzing large scale, biologically realistic models of atrial fibrillation and explore some of the reasons why simulation and experimentation must come closer together to fully elucidate the dynamics of wavefront conduction in inhomogeneous, three-dimensional domains.

Making a Splash; Breaking a Neck: The Development of Complexity in Physical Systems: Leo Kadanoff, Work done by Michael Brenner, Peter Constantin, Todd Dupont, Leo Kadanoff, Albert Libchaber, Sidney Nagel, Robert Rosner, and many others

We study the motion of fluids, with the aim of developing a fundamental understanding of fluid flow. Our program is characterized by close cooperation among experimenters, theoreticians, and simulators. The world about us exhibits many beautiful and important fluid flows. Consider clouds and waves, storms, and earthquakes, sunspots and mountain-building. What can we learn from all this richness?
Mostly our work involves solving particular problems, e.g. 'how does heat flow in a pot of water heated over a flame'. But, in following these problems we soon get to broader issues: predictability and chaos, the likelihood of very extreme outcomes, and the natural formation of complex 'machines'.
In the end, we try to ask if there is a 'science of complexity' and are there natural 'laws' of complex things. My answer is 'no', but I do see important lessons to be learned from studying such systems.

Biological Applications of Pattern-Formation Physics: Herbert Levine, University of California, San Diego

In the past several decades, physicists have made great strides in understanding how spatial patterns can arise in systems driven far from equilibrium. Of course, many important issues and significant challenges remain. But, with this sense of progress, many researchers began addressing the question of whether the study of pattern formation could help elucidate the formation of structure in biological systems, often called morphogenesis. Of course, living matter is much more complex than non-living. Yet, this talk will hopefully convince you that not only is this physics-based approach possible, but is in fact extremely promising.
There are many processes one could choose to discuss; for definiteness, I will focus on the life cycle of the soil amoeba Dictyostelium discoideum. In this organism, starvation triggers a day-long series of transformations that take solitary amoebae and create a cooperative multicellular organism; the process culminates in a plant-like fruiting body containing spore cells specialized for survival in harsh conditions. Ideas from the physics of pattern formation have been used to help explain the wave field used for cell guidance, the streaming of cells into the aggregate and the collective motions seen in multicellular stages. Currently, several groups are working on the single-cell chemotactic response from a similar perspective.

Statistics of Lagrangian Velocity in Fully Developed Turbulence: Jean-Francois Pinton, Ecole Normale Superieure de Lyon, France

The understanding of the dynamics of turbulent flows has been a major goal for fundamental and applied fluid dynamics research for almost a century now. On the fundamental side, turbulence is the head figure of a non-linear dissipative system with a very large number of degrees of freedom. On the applied side, the properties of turbulent flows govern the dispersion of pollutants, the physics of mixing, etc. In very recent years, analytical and numerical studies have shown that progress can be made by analyzing the flow properties in the reference frame of a moving fluid particle (the Lagrangian viewpoint), instead of considering the velocity field at a fixed point in space (the Eulerian viewpoint).
In order to completely address turbulence in the Lagrangian frame, one needs to describe the dynamics over the entire range of scales of motion. We have developed such technique, based on sonar principles, to measure directly the velocity of individual small tracer particles over long times. We have analyzed the statistics of the Lagrangian velocity of single particles for flows with turbulent Reynolds numbers between 100 and 1100. We observe that the Lagrangian spectrum has a Lorentzian form in agreement with a Kolmogorov-like scaling in the inertial range. The probability density function (PDF) of the velocity time increments displays a change of shape from quasi-Gaussian a integral time scale to stretched exponential tails at the smallest time increments. This intermittency, when measured from relative scaling exponents of structure functions, is more pronounced than in the Eulerian framework.
Another important observation is that in the erratic course of the particle motion, infinitesimal changes of velocity occur with `random' decorrelated directions but with a correlation of magnitude which persists over the longest times of the flow. Using an analogy with the properties of Multifractal Random Walks, we propose that this feature is essential in the development of intermittency in turbulence.

Observation of Chaos Assisted Tunneling of Ultra-Cold Atoms: Mark Raizen, University of Texas at Austin

We study quantum dynamics of ultra-cold cesium atoms in mixed phase space consisting of islands of stability surrounded by chaos. We use a new method to prepare a minimum uncertainty wavepacket located on one island in phase space. We observe coherent tunneling oscillations between this state and a symmetry related island. We show that this system exhibits chaos assisted tunneling as characterized by the participation of the intermediate stochastic sea, the sensitivity to parameters, and the enormous enhancement in the tunneling rate between distant states.

Scaling in Two-Dimensional and Three-Dimensional Rotating Turbulent Flows*: Harry L. Swinney, University of Texas at Austin

In three-dimensional (3D) turbulent flow, vortices stretch axially and fold, but this process cannot occur in two dimensions. While all turbulent flows are 3D on sufficiently small scales, atmospheric and oceanic flows are approximately 2D on large scales. We study turbulence in a rotating tank where the flow becomes 2D for sufficiently rapid rotation rate (by the Taylor-Proudman theorem), while for low rotation rates the flow is 3D [1]. We find that for 2D turbulence the probability distribution function (PDF) for the difference in velocity between two points is independent of the separation r between the two points, i.e., the flow is self-similar. In contrast, the PDFs for 3D turbulence are gaussian for large r and exponential for small r. We further compare the 2D and 3D turbulence flows by determining structure function scaling exponents and by applying the beta and gamma tests of the hierarchical structure model; these quantities will be defined and discussed. The conclusion is that 2D turbulence in a rotating flow is surprisingly intermittent, but the intermittency is a consequence of large coherent vortices rather than the stretching and folding of vortex lines as in 3D.
*Supported by ONR
[1] C.N. Baroud, B.P. Plapp, Z.S. She, and H. L. Swinney, submitted

On a Generalized Approach to Linear Stability of Spatially Dependent Thin Film Flows: Sandra M. Troian, Princeton University

Recent interest in microscale flows, which can sustain exceedingly large surface to volume ratios, has focused attention on the use of normal or shear force actuation to effect liquid migration along a solid surface. For instance, a thin liquid film supported on a differentially heated substrate will spontaneously flow toward the cooler end in a process known as thermocapillary forcing. In another example, a film contacted by a non-uniform distribution of surface active material will rapidly flow toward the uncontaminated end under the action of Marangoni stresses. Such spreading films typically undergo fingering instabilities at the advancing front which resemble either a series of parallel liquid rivulets or highly ramified dendritic structures. During the past several years, we have used a combination of experiment and theoretical modeling in an effort to provide a unified framework describing the stability characteristics of these and related thin film flows. The presence of capillary, thermocapillary, Marangoni or van der Waals stresses in free surface films creates spatially (and temporally) dependent interface shapes. The linearized operators governing disturbances in film thickness or concentration are therefore typically non self-adjoint. A consequence of this feature is that conventional modal analysis can only describe the asymptotic behavior of these
systems. A rigorous description of the early and intermediate behavior requires a transient growth study. This type of analysis not only identifies the growth rate of optimal disturbances but reveals their initial and evolved waveform shape at all times. Strongly non-normal operators can introduce the possibility of severe disturbance amplification and subsequent non-linear coupling.
In this presentation, we outline the transient growth behavior and amplification of optimal disturbances for thermocapillary driven flows. Time permitting, we review Marangoni driven systems as well. We examine the pseudospectral behavior of the associated disturbance operators and illustrate why transient growth studies offer a more suitable probe of the stability of free surface thin film flows.

Unsteady Aerodynamics of Insect Flight: Jane Wang, Cornell University

The myth `bumble-bees can not fly according to conventional aerodynamics' simply reflects our poor understanding of unsteady viscous fluid dynamics. In particular, we lack a theory of vorticity shedding due to dynamic boundaries at the intermediate Reynolds numbers relevant to insect flight, typically between $10^2$ and $10^4$, where both viscous and inertial effects are important. In our study, we compute unsteady viscous flows, governed by the Navier-Stokes equation, about a two dimensional flapping wing which mimics the motion of an insect wing. I will present two main results: the existence of a preferred frequency in forward flight and its physical origin, and 2) the vortex dynamics and forces in hovering dragonfly flight. If time permits, I will show the recent results on comparing our computational results against robotic fruitfly experiments and modeling three dimensional flapping flight driven by muscles.

A Dynamical Systems View of Planetary Turbulence: Jeffrey B. Weiss, University of Colorado

The turbulence of planetary atmospheres and oceans self-organizes into a spatio-temporal pattern of coherent structures such as vortices and jets. These structures provide insight into the long-standing problem of reducing fluid turbulence to a chaotic dynamical system. The attractor of the turbulence is an evolving population of structures, and the structures' degrees of freedom are the reduced coordinate system which describe the attractor. These ideas are explored in a hierarchy of systems with increasing complexity, from Hamiltonian ordinary differential equations to oceanic observations.

Tracers, Coherent Structures, and Fractional Kinetics: George M. Zaslavsky, Department of Physics and Courant Institute of NYU

We consider chaotic dynamics of tracers in a system of point vortices and, in parallel, the phase space topology of the vortices. There exist a specific connection between coherent structures of the vortex system and tracers transport , which is possible to describe in an analytical way and confirm by simulations. The coherent structures occur as clusters of few vortices , which impose the tracers kinetics of the fractional type. On that way a characteristic exponent of the tracers dispersion can be obtained from the first principles.