Tom Solomon

Chaotic advection in a two-dimensional flow: Le´vy flights and anomalous diffusion

Long-term particle tracking is used to study chaotic transport experimentally in laminar, chaotic, and turbulent flows in an annular tank that rotates sufficiently rapidly to insure two-dimensionality of the flow. For the laminar and chaotic velocity fields, the flow consists of v flow regimes, tracer particles stick for long times to remnants of invariant surfaces around the vortices, then make long excursions (“flights”) in the jet regions. The probability distributions for the flight time durations exhibit power-law rather than exponential decays, indicating that the parrticle trajectories are described mathematically as Levy flights (i.e. the trajectories have infinite mean square displacement per flight). Sticking time probability distributions are also characterized by power laws, as found in previous numerical studies. The mixing of an ensemble of tracer particles is superdiffusive: the variance of the displacement grows with time as tλ with 1<λ<2. The dependence of the diffusion exponent λ and the scaling of the probability distributions are investigated for periodic and chaotic flow regimes, and the results are found to be consistent with theoretical predictions relating Levy flights and anomalous diffusion. For a turbulent flow, the Levy flight description no longer applies, and mixing no longer appears superdiffusive.

Uniform resonant chaotic mixing in fluid flows.

Laminar flows can produce particle trajectories that are chaotic, with nearby tracers separating exponentially in time. For time-periodic, two-dimensional flows and steady three-dimensional (3D) flows, enhancements in mixing due to chaotic advection are typically limited by impenetrable transport barriers that form at the boundaries between ordered and chaotic mixing regions. However, for time-dependent 3D flows, it has been proposed theoretically that completely uniform mixing is possible through a resonant mechanism called singularity-induced diffusion; this is thought to be the case even if the time-dependent and 3D perturbations are infinitesimally small. It is important to establish the conditions for which uniform mixing is possible and whether or not those conditions are met in flows that typically occur in nature. Here we report experimental and numerical studies of mixing in a laminar vortex flow that is weakly 3D and weakly time-periodic. The system is an oscillating horizontal vortex chain (produced by a magnetohydrodynamic technique) with a weak vertical secondary flow that is forced spontaneously by Ekman pumping—a mechanism common in vortical flows with rigid boundaries, occurring in many geophysical, industrial and biophysical flows. We observe completely uniform mixing, as predicted by singularity-induced diffusion, but only for oscillation periods close to typical circulation times.

Passive transport in steady Rayleigh-Bénard convection

Enhancement of the diffusive transport of impurities by two‐dimensional, time‐independent Rayleigh–Benard convection is studied experimentally. Two impurities are used: a molecular dye (methylene blue) and a particulate impurity (latex spheres). The convective flow is characterized by laser Doppler velocimetry, and the transport is monitored by optical absorption techniques. It is found that the transport can be modeled as a diffusive process on long space and time scales, with an effective diffusion coefficient D* whose absolute magnitude and variation with the velocity amplitude W of the flow are in good agreement with recent theoretical predictions. The enhancement factor D*/D scales with the Peclet number approximately as Pe1/2≡(Wd/πD)1/2, where D is the diffusion coefficient and d the layer depth. Several subtle problems that complicate the study of transport phenomena in cellular hydrodynamic flows are discussed.

Shear flow instabilities and Rossby waves in barotropic flow in a rotating annulus

The primary instability of an azimuthal jet is studied experimentally in a rotating annulus with a rigid upper lid (infinite Rossby deformation radius) and a sloping bottom (topographical beta effect). An azimuthal jet is produced by the action of the Coriolis force on fluid pumped between concentric rings of sources and sinks in the bottom of the annulus. The flow is essentially two dimensional by the Taylor–Proudman theorem. Velocity measurements are made with hot‐film probes and particle streak photography. For small forcing flux F, the jet is axisymmetric and has a 1/r velocity profile bounded by narrow free shear layers on each side. At a critical value of F, the inner shear layer becomes unstable to the formation of a propagating chain of vortices; at a larger value of F the outer shear layer also becomes unstable. The critical values of the mode numbers, wave speeds, and F at different annulus rotation rates are in good accord with a linear stability analysis by Lee and Marcus. At onset of instabil...

Invariant barriers to reactive front propagation in fluid flows

We present theory and experiments on the dynamics of reaction fronts in two-dimensional, vortex-dominated flows, for both time-independent and periodically driven cases. We find that the front propagation process is controlled by one-sided barriers that are either fixed in the laboratory frame (time-independent flows) or oscillate periodically (periodically driven flows). We call these barriers burning invariant manifolds (BIMs), since their role in front propagation is analogous to that of invariant manifolds in the transport and mixing of passive impurities under advection. Theoretically, the BIMs emerge from a dynamical systems approach when the advection-reaction-diffusion dynamics is recast as an ODE for front element dynamics. Experimentally, we measure the location of BIMs for several laboratory flows and confirm their role as barriers to front propagation.

Observation of anomalous diffusion and Lévy flights

Chaotic transport is studied experimentally in two-dimensional flow in a rapidly rotating annular tank. The flow consists of a chain of vortices sandwiched between two azimuthal jets. Automated image processing techniques are used to track the motions of neutrally buoyant tracer particles suspended in the flow. If the flow has periodic time dependence, the tracers typically follow chaotic trajectories, alternately sticking in vortices and flying for long distances in the jets. Probability distribution functions (PDFs) are measured for sticking and flight times. The flight PDFs are power laws, indicating in some cases that the particle motion can be characterized as Levy flights (with a divergent second moment for flight times). The variance of an ensemble of particles is found to vary in time as σ2 ∼ tγ, with γ > 1 (superdiffusion). The dependence of the variance exponent γ on the flight and sticking PDFs is studied and found to be consistent with calculations relating Levy flights and anomalous diffusion (γ ≄ 1). In a turbulent flow, Levy flights no longer are present and the mixing appears to be normally diffusive (γ = 1). A review of previous experiments on anomalous diffusion is included.

Chaotic mixing of immiscible impurities in a two-dimensional flow

Experiments compare the chaotic mixing of miscible and immiscible impurities in a two-dimensional flow composed of a chain of alternating vortices. Periodic time dependence is imposed on the system by sloshing the fluid slowly across the stationary vortices, mimicking the even oscillatory instability of Rayleigh–Benard convection. The transport of a miscible impurity is diffusive with an enhanced diffusion coefficient D* that depends on the size of “lobes” which are, in turn, dependent on the oscillation amplitude. The lobes play an important role in the transport of immiscible impurities as well. In this case, the impurity is broken into a distribution of droplets, whose areas determine the nature of the transport. If the characteristic long-term droplet areas are appreciably smaller than the lobe areas, then there is long-range transport with D* equal to that for the miscible case with the same flow conditions. If the droplet areas remain larger than the lobe areas, then there is no long-range transport.

Measurements of the temperature field of mushy and liquid regions during solidification of aqueous ammonium chloride

Experiments are conducted to study the solidification from below of aqueous ammonium chloride. Thermochromic liquid crystal paints are used to visualize the temperature field simultaneously in both the liquid and the mushy layers. In a quasitwo-dimensional cell (thickness 10 mm), mushy-layer and boundary-layer convection are revealed as bumps in isotherms within and above the mushy layer, respectively. The onset, growth and decay of these convective modes are measured by monitoring the progression of the bumps during an experiment. The small-wavelength boundary-layer mode is short-lived (approximately 20-30 min), whereas the larger-wavelength mushy-layer mode survives for several hours, dominating the flow even long after the growth has stopped. Experiments in a Hele-Shaw cell (thickness 2.0 mm) enable simultaneous visualization of both the temperature field and the solid fraction. A coarsening mechanism is observed in which the flow spontaneously changes, reducing the strength of plume convection in one of the channels, and leading to growth of dendrites into the channel. An oscillatory convective mode is also observed, perhaps an indication of one of the oscillatory modes recently predicted by Chen, Lu & Yang and by Anderson & Worster

Barriers to front propagation in ordered and disordered vortex flows

We present experiments on reactive front propagation in a two-dimensional (2D) vortex chain flow (both time-independent and time-periodic) and a 2D spatially disordered (time-independent) vortex-dominated flow. The flows are generated using magnetohydrodynamic forcing techniques, and the fronts are produced using the excitable, ferroin-catalyzed Belousov-Zhabotinsky chemical reaction. In both of these flows, front propagation is dominated by the presence of burning invariant manifolds (BIMs) that act as barriers, similar to invariant manifolds that dominate the transport of passive impurities. Convergence of the fronts onto these BIMs is shown experimentally for all of the flows studied. The BIMs are also shown to collapse onto the invariant manifolds for passive transport in the limit of large flow velocities. For the disordered flow, the measured BIMs are compared to those predicted using a measured velocity field and a three-dimensional set of ordinary differential equations that describe the dynamics of front propagation in advection-reaction-diffusion systems.

Resonant flights and transient superdiffusion in a time-periodic, two-dimensional flow

Abstract Enhanced, passive transport is studied numerically in an oscillating vortex chain with stress-free boundary conditions. The long-range transport is found to be diffusive in the long-time limit with an effective diffusion coefficient D ∗ that peaks dramatically in the vicinity of a few, well-defined resonant frequencies. Superdiffusive transients are also observed for frequencies near these resonant frequencies, with the duration of the transients diverging at the resonant frequencies. Standard analytical techniques based on the Melnikov approximation and on lobe dynamics fail to explain the behavior in the vicinity of these resonant peaks. An alternate explanation is provided, based on flights that have power-law scaling up to a maximum length that also diverges at the resonant frequencies. The long flights for frequencies near the resonant peaks occur because tracers in a lobe return (after an integer number of oscillation periods) to almost precisely the same location in the lobe of another vortex. These periodic orbits correspond to the formation — only at the resonant frequencies — of “tangle islands” within the chaotic region.