Noise and Dynamical Pattern Selection in Solidification

The overall goal of this project was to understand in more detail how a pattern-forming system can adjust its spacing. "Pattern-forming systems," in this context, are nonequilibrium contina whose state is determined by experimentally adjustable control parameter. Below some critical value of the control system then has available to it a range of linearly stable, spatially periodic steady states, each characterized by a spacing which can lie anywhere within some band of values. These systems like directional solidification, where the solidification front is planar when the ratio of growth velocity to thermal gradient is below its critical value, but takes on a cellular shape above critical. They also include systems without interfaces, such as Benard convection, where it is the fluid velocity field which changes from zero to something spatially periodic as the control parameter is increased through its critical value. The basic question to be addressed was that of how the system chooses one of its myriad possible spacings when the control parameter is above critical, and in particular the role of noise in the selection process. Previous work on explosive crystallization had suggested that one spacing in the range should be preferred, in the sense that weak noise should eventually drive the system to that spacing. That work had also suggested a heuristic argument for identifying the preferred spacing. The project had three main objectives: to understand in more detail how a pattern-forming system can adjust its spacing; to investigate how noise drives a system to its preferred spacing; and to extend the heuristic argument for a preferred spacing in explosive crystallization to other pattern-forming systems

Solitons and kinks in a general car-following model

We study a car-following model of traffic flow which assumes only that a car's acceleration depends on its own speed, the headway ahead of it, and the rate of change of headway, with only minimal assumptions about the functional form of that dependence. The velocity of uniform steady flow is found implicitly from the acceleration function, and its linear stability criterion can be expressed simply in terms of it. Crucially, unlike in previously analyzed car-following models, the threshold of absolute stability does not generally coincide with an inflection point in the steady velocity function. The Burgers and KdV equations can be derived under the usual assumptions, but the mKdV equation arises only when absolute stability does coincide with an inflection point. Otherwise, the KdV equation applies near absolute stability, while near the inflection point one obtains the mKdV equation plus an extra, quadratic term. Corrections to the KdV equation "select" a single member of the one-parameter set of soliton solutions. In previous models this has always marked the threshold of a finite- amplitude instability of steady flow, but here it can alternatively be a stable, small-amplitude jam. That is, there can be a forward bifurcation from steady flow. The new, augmented mKdV equation which holds near an inflection point admits a continuous family of kink solutions, like the mKdV equation, and we derive the selection criterion arising from the corrections to this equation.Comment: 25 page

Surface Instability in Windblown Sand

We investigate the formation of ripples on the surface of windblown sand based on the one-dimensional model of Nishimori and Ouchi [Phys. Rev. Lett. 71, 197 (1993)], which contains the processes of saltation and grain relaxation. We carry out a nonlinear analysis to determine the propagation speed of the restabilized ripple patterns, and the amplitudes and phases of their first, second, and third harmonics. The agreement between the theory and our numerical simulations is excellent near the onset of instability. We also determine the Eckhaus boundary, outside which the steady ripple patterns are unstable.Comment: 23 pages, 8 figure

Noise and dynamical pattern selection

In pattern forming systems such as Rayleigh-Benard convection or directional solidification, a large number of linearly stable, patterned steady states exist when the basic, simple steady state is unstable. Which of these steady states will be realized in a given experiment appears to depend on unobservable details of the system's initial conditions. We show, however, that weak, Gaussian white noise drives such a system toward a preferred wave number which depends only on the system parameters and is independent of initial conditions. We give a prescription for calculating this wave number, analytically near the onset of instability and numerically otherwise.Comment: 12 pages, REVTEX, no figures. Submitted to Phys. Rev. Let

Traffic Jams, Granular Flow, and Soliton Selection

The flow of traffic on a long section of road without entrances or exits can be modelled by continuum equations similar to those describing fluid flow. In a certain range of traffic density, steady flow becomes unstable against the growth of a cluster, or "phantom" traffic jam, which moves at a slower speed than the otherwise homogeneous flow. We show that near the onset of this instability, traffic flow is described by a perturbed Korteweg-deVries equation. The traffic jam can be identified with a soliton solution of the KdV equation. The perturbation terms select a unique member of the continuous family of KdV solitons. These results may also apply to the dynamics of granular relaxation. PACS numbers: 05.40.+j, 47.54.+r, 81.35.+k, 89.40.+k I. INTRODUCTION The flow of traffic along a limited-access highway is similar in many respects to the flow of a classical fluid [1-3]. For instance, in the absence of entrances and exits, the total number of vehicles on the road is conserved, so t..

THE GENESIS OF WASHBOARD ROADS

We consider a model of cars driving along a compressible or erodible road. Each car is modeled as a mass supported above a wheel by a damped spring. We find conditions under which a flat road surface is unstable and gives way to a regular pattern of corrugations. The instability is mitigated by the damping of the cars' springs and also by the tendency of the road surface to harden as it is compressed. Similar phenomena occur in railroad rails and in calendaring of paper