Stochastic influences on pattern formation in Rayleigh-Bénard convection: Ramping experiments

Simultaneous influence of additive and multiplicative noise on stationary dissipative solitons

We investigate the simultaneous influence of spatially homogeneous multiplicative noise as well as of spatially δ-correlated additive noise on the formation of localized patterns in the framework of the cubic-quintic complex Ginzburg-Landau equation. Depending on the ratio between the strength of additive and multiplicative noise we find a number of distinctly different types of behavior including explosions, collapse, filling in, and spatio-temporal disorder as well as intermittent behavior of all types listed. Techniques used to analyze the results include snapshots, x-t plots and plots of the spatially and temporally averaged amplitude as a function of the strength of multiplicative noise while keeping the strength of additive noise fixed. Typically 50 realizations are used for averaging to obtain the corresponding data points in these diagrams. For the widths of these distribution as a function of additive noise we obtain a linear decrease in the limit of fairly large, but fixed values of the multiplicative noise. To summarize our findings concisely we show three-dimensional plots of the mean pattern amplitude and the generalized susceptibility as a function of the strengths of additive and multiplicative noise. We critically compare the results of our investigations with those obtained in the two limiting cases of purely additive and of purely multiplicative noise.

Ground level air convection produces frost damage patterns in turfgrass

Frost injury patterns are commonly observed on the warm-season turfgrass species bermudagrass (Cynodon species Rich.), zoysiagrass (Zoysia species Willd.), and buffalograss [Bouteloua dactyloides (Nutt.) J.T. Columbus] in cool-temperate and subtropical zones. Qualitative observations of these injury patterns are presented and discussed. A model for the formation of such patterns based on thermal instability and convection of air is presented. The characteristic length scale of the observed frost pattern injury requires a temperature profile that decreases with height from the soil to the turfgrass canopy surface followed by an increase in temperature with height above the turfgrass canopy. This is justified by extending the earth temperature theory to include a turf layer with atmosphere above it. Then the theory for a thermally unstable layer beneath a stable region by Ogura and Kondo is adapted to a turf layer to include different parameter values for pure air, as well as for turf, which is treated as a porous medium. The earlier porous medium model of Thompson and Daniels proposed to explain frost injury patterns is modified to give reasonable agreement with observed patterns.

State selection in the noisy stabilized Kuramoto-Sivashinsky equation

In this work, we study the one-dimensional stabilized Kuramoto Sivashinsky equation with additive uncorrelated stochastic noise. The Eckhaus stable band of the deterministic equation collapses to a narrow region near the center of the band. This is consistent with the behavior of the phase diffusion constants of these states. Some connections to the phenomenon of state selection in driven out of equilibrium systems are made.

Convective roll dynamics in liquid 4He near the onset of convection

We present results of experiments on Rayleigh-Bénard convection in liquid 4He at several temperatures. We show visually that with carefully defined boundary conditions the basic convection state consists of parallel rolls which are aligned in one of two directions, the angle thus defined as being temperature dependent and we attempt to explain this behavior. We also show directly the skew-varicose instability acting on the basic state and correlate it with fluctuations in the temperature difference across the fluid layer.

Loss of chaotic trabecular structure in OPG-deficient juvenile Paget's disease patients indicates a chaogenic role for OPG in nonlinear pattern formation of trabecular bone

The RANK-RANKL-OPG system of osteoclast regulation may play a key role in determining chaotic structure in trabecular bone. Iliac trabecular bone from juvenile Paget's disease patients deficient in functional OPG shows parallel, anisotropic structure instead of normal chaotic structure. Evidence from experimental systems suggests that RANK-RANKL-OPG controls key nonlinear "chaogenic" parameters, such as friction, forcing frequency, feedback, and boundary forcing. The RANK-RANKL-osteoprotegerin (OPG) system of osteoclast regulation may play a key role in determining chaotic structure in trabecular bone. Iliac trabecular bone from juvenile Paget's disease (JPD) patients deficient in functional OPG shows parallel, anisotropic structure instead of normal chaotic structure. Evidence from experimental systems suggests that RANK-RANKL-OPG controls key nonlinear "chaogenic" parameters, such as friction, forcing frequency, feedback, and boundary forcing. The Belousov-Zhabotinsky reaction-diffusion system, the catalytic oxidation of CO on platinum surfaces, and thermal diffusion in liquid helium allow visualization of nonlinear emergent patterns such as labyrinthine structures, turbulence, and cellular structures, all of which bear some resemblance to trabecular bone. In JPD, the gene for OPG (TNFRSF11B) is subject to an inactivating mutation, leading to increased resorption and accelerated remodeling. Histomorphometric images of iliac crest trabecular bone from teenagers suffering from JPD show a highly unusual array of parallel, regular trabecular plates, instead of the typical chaotic, fractal patterns of normal trabecular bone. Loss of OPG function is associated with a change from chaotic to regular structure, suggesting that the RANK-RANKL-OPG system is controlling key nonlinear "chaogenic" parameters. Looking at trabecular bone from the perspective of nonlinear pattern formation may help understand other phenomena, such as the marked dependence of trabecular bone's architectural and mechanical quality on remodeling rate independent of the trabecular bone mass.

Dynamics of pattern coarsening in a two-dimensional smectic system

We have followed the coarsening dynamics of a single layer of cylindrical block copolymer microdomains in a thin film. This system has the symmetry of a two-dimensional smectic. The orientational correlation length of the microdomains was measured by scanning electron microscopy and found to grow with the average spacing between +/-1/2 disclinations, following a power law xi2(t) approximately t(1/4). By tracking disclinations during annealing with time-lapse atomic force microscopy, we observe dominant mechanisms of disclination annihilation involving tripoles and quadrupoles (three and four disclinations, respectively). We describe how annihilation events involving multiple disclinations result in similarly reduced kinetic exponents as observed here. These results map onto a wide variety of physical systems that exhibit similarly striped patterns.

Vibrational resonance in a noise-induced structure

We report on the effect of vibrational resonance in a spatially extended system of coupled noisy oscillators under the action of two periodic forces, a low-frequency one (signal) and a high-frequency one (carrier). Vibrational resonance manifests itself in the fact that for optimally selected values of high-frequency force amplitude, the response of the system to a low-frequency signal is optimal. This phenomenon is a synthesis of two effects, a noise-induced phase transition leading to bistability, and a conventional vibrational resonance, resulting in the optimization of signal processing. Numerical simulations, which demonstrate this effect for an extended system, can be understood by means of a zero-dimensional "effective" model. The behavior of this "effective" model is also confirmed by an experimental realization of an electronic circuit.

Noise induced propagation in monostable media

We show that external fluctuations are able to induce propagation of harmonic signals through monostable media. This property is based on the phenomenon of doubly stochastic resonance, where the joint action of multiplicative noise and spatial coupling induces bistability in an otherwise monostable extended medium, and additive noise resonantly enhances the response of the system to a harmonic forcing. Under these conditions, propagation of the harmonic signal through the unforced medium is observed for optimal intensities of the two noises. This noise-induced propagation is studied and quantified in a simple model of coupled nonlinear electronic circuits.

Additive noise in noise-induced nonequilibrium transitions

We study different nonlinear systems which possess noise-induced nonequlibrium transitions and shed light on the role of additive noise in these effects. We find that the influence of additive noise can be very nontrivial: it can induce first- and second-order phase transitions, can change properties of on-off intermittency, or stabilize oscillations. For the Swift-Hohenberg coupling, that is a paradigm in the study of pattern formation, we show that additive noise can cause the formation of ordered spatial patterns in distributed systems. We show also the effect of doubly stochastic resonance, which differs from stochastic resonance, because the influence of noise is twofold: multiplicative noise and coupling induce a bistability of a system, and additive noise changes a response of this noise-induced structure to the periodic driving. Despite the close similarity, we point out several important distinctions between conventional stochastic resonance and doubly stochastic resonance. Finally, we discuss open questions and possible experimental implementations. (c) 2001 American Institute of Physics.

Domain coarsening in electroconvection

We report on experimental measurements of the growth of regular domains evolving from an irregular pattern in electroconvection. The late-time growth of the domains is consistent with the size of the domains scaling as t(n). We use two isotropic measurements of the domain size: the structure factor and the domain wall length. Measurements using the structure factor are consistent with t(1/5) growth. Measurements using the domain wall length are consistent with t(1/4) growth. One source of this discrepancy is the fact that the distribution of local wave numbers is approximately independent of the domain size. In addition, we measure the anisotropy of the growing domains.

Heat transfer and convection onset in a compressible fluid: 3He near the critical point

Heat transport in 3He above its critical temperature Tc was studied along the critical isochore in a flat Rayleigh-Bénard cell (height h=1 mm, diameter D=57 mm). The range of the reduced temperature epsilon was 5 x 10(-4)< or = epsilon < or =2 x 10(-1). The temperature difference deltaT(t) across the fluid layer as a function of the time t was measured for different values of the heat current q until steady state was reached. The crossover was observed from the regime dominated by the Rayleigh criterion for the convection onset to that controlled by the adiabatic temperature gradient (ATG), or "Schwarzschild criterion," in good quantitative agreement with predictions. The slope of the convective heat current versus the reduced Rayleigh number was found to be independent of compressibility and the same as for still less compressible fluids. Plots of Nu versus Ra, both corrected for the ATG effect, are presented for early-stage convective turbulence (1 x 10(5)<Ra<5 x 10(8)), with unexpected results for the highest values of Pr approximately 590. The evolution of the transients deltaT(t) upon turning q on and off are described. In the nonconvective regime, the observed transient relaxation curve agrees quantitatively with predictions. In the convective regime, the shape of deltaT(t) changes qualitatively with increasing q and with epsilon. In the Appendix, new data for the thermal conductivity are presented, the impact of the "piston effect" on the temperature profiles inside the fluid is described, and the derivation for deltaT(t) in the diffusive regime is outlined.

Influence of inlet and bulk noise on Rayleigh-Bénard convection with lateral flow

Spatiotemporal properties of convective fluctuations and of their correlations are investigated theoretically in the vicinity of the threshold for onset of convection in the presence of a lateral through-flow using the full linearized equations of fluctuating hydrodynamics. The effect of external forcing by inlet boundary conditions on the downstream evolution of convective fields is separated from the effect of internal bulk thermal forcing with the use of spatial Laplace transformations. They show how the spatial variation of fluctuations and of their correlations are governed by the six spatial characteristic exponents of the field equations.

Simple electronic circuit model for doubly stochastic resonance

We have recently reported the phenomenon of doubly stochastic resonance [Phys. Rev. Lett. 85, 227 (2000)], a synthesis of noise-induced transition and stochastic resonance. The essential feature of this phenomenon is that multiplicative noise induces a bimodality and additive noise causes stochastic resonance behavior in the induced structure. In the present paper we outline possible applications of this effect and design a simple lattice of electronic circuits for the experimental realization of doubly stochastic resonance.

Doubly stochastic resonance

We report the effect of doubly stochastic resonance which appears in nonlinear extended systems if the influence of noise is twofold: A multiplicative noise induces bimodality of the mean field of the coupled network and an independent additive noise governs the dynamic behavior in response to small periodic driving. For optimally selected values of the additive noise intensity stochastic resonance is observed, which is manifested by a maximal coherence between the dynamics of the mean field and the periodic input. Numerical simulations of the signal-to-noise ratio and theoretical results from an effective two state model are in good quantitative agreement.

Nonequilibrium first-order phase transition induced by additive noise

We show that a nonequilibrium first-order phase transition can be induced by additive noise. As a model system to study this phenomenon, we consider a nonlinear lattice of overdamped oscillators with both additive and multiplicative noise terms. Predictions from mean field theory are successfully confirmed by numerical simulations. A physical explanation for the mechanism of the transition is given.