A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.

/pdf/pattern-formation-outside-of-equilibrium-1601b9znum.pdf

Pattern formation outside of equilibrium

This paper presents a physicist's approach to Rayleigh-Benard convection widely illustrated with experimental results. The basis of the mechanism of the instability is simply presented with physical reasons for the existence of a critical threshold. A detailed examination of the spatial organization is made with discussion of ordered and disordered structures. Furthermore it is shown that the measurements of the local velocity give a good quantitative description of the convective state. A complete parallel between the Rayleigh-Benard convection near onset and a critical phenomenon is given in the framework of a mean field approach, including both spatial as well as temporal effects. Non-Boussinesq convection is presented as symmetry breaking, changing the second order transition into a (partially) first order transition. The last section is devoted to the ever present but still not completely understood question of the dynamics of the convective pattern; the importance of the existence and motio...

Rayleigh-bénard convection

The study of low-dimensional dissipative dynamical systems has provided a reasonable understanding of the transition to temporal chaos in strongly confined systems for which the spatial structure can be considered as frozen. The situation is still less advanced for weakly confined systems where chaos has both a spatial and a temporal meaning. In order to approach the specificities of the latter, we have chosen to study first a partial differential equation (PDE) displaying a convective-type nonlinear term, steady cellular solutions as in convection, and a transition to spatiotemporal chaos, namely the damped Kuramoto-Sivashinsky (KS) equation: 

$${\partial _t}\phi + \eta \phi + {\partial _{{x^2}}}\phi + {\partial _{{x^4}}}\phi + 2\phi {\partial _x}\phi = 0.$$

(1)

Transition to turbulence via spatio-temporal intermittency

Patterns of an almost periodic nature appear all over the place. One sees them in cloud streets, in sand ripples on flat beaches and desert dunes, in the morphology of plants and animals, in chemically reacting media, in boundary layers, on weather maps, in geological formations, in interacting laser beams in wide gainband lasers, on the surface of thin buckling shells, and in the grid scale instabilities of numerical algorithms. This review deals with the class of problems into which these examples fall, namely with pattern formation in spatially extended, continuous, dissipative systems which are driven far from equilibrium by an external stress. Under the influence of this stress, the system can undergo a series of symmetry breaking bifurcations or phase transitions and the resulting patterns become more and more complicated, both temporally and spatially, as the stress is increased. Figures 1 through 3 show examples of patterns in lasers, binary and ordinary fluids, and liquid crystals. The goal of theory is to provide a means of understanding and explaining these patterns from a macroscopic viewpoint that both simplifies and unifies classes of problems which are seemingly unrelated at the microscopic level. Convection in a large aspect ratio horizontal layer of fluid heated from below is the granddaddy of canonical examples used to study pattern formation and behavior in spatially extended systems. For low values of the vertical temperature difference, which is the external stress parameter in this case and whose non-dimensional measure is called the Rayleigh

Order parameter equations for patterns

The theoretical description of the threshold and near-threshold behavior of electrohydrodynamic convection in nematic liquid crystals in the (low-frequency) conduction regime is reconsidered. We present essentially the full three-dimensional linear stability analysis of the basic state and a major part of the weakly-dependence are treated rigorously but the flexoelectric effect is neglected. Related transitions like the periodic splay-twist instability, which is relevant in polymer materials with positive dielectric anisotropy, are also considered. We give criteria for finding the oblique-roll state and for the competition between different instabilities. Comparison with experiments is made wherever possible. Good qualitative and sometimes quantitative agreement is found Analyse lineaire tridimensionnelle complete de l'etat de base et theorie faiblement non lineaire de l'etat convectif pour decrire le comportement au seuil, et pres du seuil, de la convection electrohydrodynamique de cristaux liquides nematiques dans le regime de conduction (basse frequence). Traitement rigoureux des conditions aux limites aux plaques superieure et inferieure et de la variation temporelle. Extension de l'analyse a diverses transitions, comme l'instabilite de «splay-twist» periodique observee dans des polymeres a anisotropie dielectrique positive. Criteres pour l'observation de l'etat des rouleaux obliques et la competition entre differentes instabilites. Comparaison aux donnees experimentales

On electrically driven pattern-forming instabilities in planar nematics

An analysis is presented of the steady states of two-dimensional convection near threshold in a laterally finite container with aspect ratio 2L [Gt ] 1. It is shown that the allowed wavevectors which can occur in the bulk of the container are reduced from a band |q| ∼ [(R − R0)/R0]½ in the laterally infinite system to a band |q| ∼ (R − R0)/R0 in a system with sidewalls (R is the Rayleigh number and R0 its critical value in the infinite system). The analysis involves an expansion of the hydrodynamic equations in the small parameter [(R − R0)/R0]½, and leads to amplitude equations with boundary conditions, which generalize to higher order those previously obtained by Newell & Whitehead and Segel. The precise values of the allowed wavevectors depend on the Prandtl number of the fluid and the thermal properties of the sidewalls. For certain values of these parameters all the allowed wavevectors are less than the critical value q0. The applicability of the results to convection in a rectangular container is briefly discussed.

http://uqbar.rockefeller.edu/siggia/Publications/1980-9_files/crossJFluMec1981.pdf

Phase-winding solutions in a finite container above the convective threshold

An analysis is presented of the steady states of two-dimensional convection in a laterally finite rectangular container near threshold. It is shown that the presence of sidewalls severely restricts the allowed wave vectors which can occur in the bulk of the container. This effect provides a possible mechanism to explain the observed wavelength increase of convective rolls with increasing Rayleigh numbers.

/pdf/effect-of-distant-sidewalls-on-wave-number-selection-in-2cbkh9pyh0.pdf

Effect of Distant Sidewalls on Wave-Number Selection in Rayleigh-Bénard Convection

In this paper we describe flow through a porous medium in a two-dimensional rectangular cavity driven by differential heating of the impermeable lower surface. The upper surface is held at constant pressure and at a constant temperature equal to the minimum temperature of the lower surface, while the sidewalls are impermeable and thermally insulated. Numerical results for general values of the Darcy–Rayleigh number and the cavity aspect ratio are compared with theoretical predictions for the small Darcy–Rayleigh number limit where the temperature field is conduction-dominated, and with a boundary-layer theory for the large Darcy–Rayleigh number limit where convection is significant. In the latter case a horizontal boundary layer near the lower surface conveys fluid to the hot end of the cavity where it rises to the upper surface in a narrow plume. Predictions are made of the vertical heat transfer through the cavity.

P. G. Daniels

Papers

Phase-winding solutions in a finite container above the convective threshold

Effect of Distant Sidewalls on Wave-Number Selection in Rayleigh-Bénard Convection

Plume flows in porous media driven by horizontal differential heating