Elastohydrodynamic Synchronization of Adjacent Beating Flagella.

It is now well established that nearby beating pairs of eukaryotic flagella or cilia typically synchronize in phase. A substantial body of evidence supports the hypothesis that hydrodynamic coupling between the active filaments, combined with waveform compliance, provides a robust mechanism for synchrony. This elastohydrodynamic mechanism has been incorporated into 'bead-spring' models in which the beating flagella are represented by microspheres tethered by radial springs as they are driven about orbits by internal forces. While these low-dimensional models reproduce the phenomenon of synchrony, their parameters are not readily relatable to those of the filaments they represent. More realistic models which reflect the underlying elasticity of the axonemes and the active force generation, take the form of fourth-order nonlinear PDEs. While computational studies have shown the occurrence of synchrony, the effects of hydrodynamic coupling between nearby filaments governed by such continuum models have been theoretically examined only in the regime of interflagellar distances d large compared to flagellar length L. Yet, in many biological situations d/L ≪ 1. Here, we first present an asymptotic analysis of the hydrodynamic coupling between two extended filaments in the regime d/L ≪ 1, and find that the form of the coupling is independent of the microscopic details of the internal forces that govern the motion of the individual filaments. The analysis is analogous to that yielding the localized induction approximation for vortex filament motion, extended to the case of mutual induction. In order to understand how the elastohydrodynamic coupling mechanism leads to synchrony of extended objects, we introduce a heuristic model of flagellar beating. The model takes the form of a single fourth-order nonlinear PDE whose form is derived from symmetry considerations, the physics of elasticity, and the overdamped nature of the dynamics. Analytical and numerical studies of this model illustrate how synchrony between a pair of filaments is achieved through the asymptotic coupling.

Supertransient magnetohydrodynamic turbulence in Keplerian shear flows.

A subcritical transition to turbulence in magnetized Keplerian shear flows is investigated by using a statistical approach. Three-dimensional numerical simulations of the shearing box equations with zero net magnetic flux are employed to determine the transition from decaying to sustained turbulence as a function of the magnetic Reynolds number R{m}. The results reveal no clear transition to sustained turbulence as the average lifetime of the transients grows as an exponential function of R{m}, in accordance with a type-II supertransient law.

Turbulent magnetic diffusivity tensor for time-dependent mean fields.

We reexamine the nature of the turbulent magnetic diffusivity tensor of mean field electrodynamics and show that the predicted growth rate of the mean field is, in general, incorrect if the tensor is calculated via consideration of time-independent mean magnetic fields. We describe how the traditional expansion procedure for the mean electromotive force should be extended to include time derivatives of the mean magnetic field, and illustrate the consistency of this approach by means of a perturbation analysis for a mean field varying on long spatial scales. Finally, we examine the magnitude of this new contribution to the magnetic diffusion for a particular flow.

Large-scale dynamo action driven by velocity shear and rotating convection.

By incorporating a large-scale shear flow into turbulent rotating convection, we show that a sufficiently strong shear can promote dynamo action in flows that are otherwise nondynamos. Our results are consistent with a dynamo driven either by the shear-current effect or by a fluctuating alpha effect interacting with the shear, but not with either a classical alpha(2) or alpha omega dynamo.

Theoretical study of pattern formation during the catalytic oxidation of CO on Pt{100} at low pressures.

Theoretical studies have thus far been unable to model pattern formation during the reaction in this system on physically feasible length and time scales. In this paper, we derive a computational reaction-diffusion model for this system in which most of the input parameters have been determined experimentally. We model the surface on a mesoscopic scale intermediate between the microscopic size of CO islands and the macroscopic length scale of pattern formation. In agreement with experimental investigations [M. Eiswirth et al., Z. Phys. Chem., Neue Folge 144, 59 (1985)], the results from our model divide the CO and O(2) partial pressure parameter space into three regions defined by the level of CO coverage or the presence of sustained oscillations. We see CO fronts moving into oxygen-covered regions, with the 1 x 1 to hex phase change occurring at the leading edge. There are also traveling waves consisting of successive oxygen and CO fronts that move into areas of relatively high CO coverage, and in this case, the phase change is more gradual and of lower amplitude. The propagation speed of these reaction waves is similar to those observed experimentally for CO and oxygen fronts [H. H. Rotermund et al., J. Chem. Phys. 91, 4942 (1989); H. H. Rotermund et al., Nature (London) 343, 355 (1990); J. Lauterbach and H. H. Rotermund, Surf. Sci. 311, 231 (1994)]. In the two-dimensional version of our model, the traveling waves take the form of target patterns emitted from surface inhomogeneities.

Pattern formation during the oxidation of CO on Pt{100}: a mesoscopic model.

Constantly changing irregular patterns of carbon monoxide (CO) and oxygen are seen during CO oxidation on platinum crystals in the [100] orientation. Ours is the first reaction-diffusion model to reproduce this pattern formation on physically feasible length and time scales, faithfully incorporating the available experimental data. Numerical simulations show patterns made up of CO and oxygen fronts moving at similar speeds to those seen in experiments.