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

▪ Abstract This review summarizes results for Rayleigh-Benard convection that have been obtained over the past decade or so. It concentrates on convection in compressed gases and gas mixtures with Prandtl numbers near one and smaller. In addition to the classical problem of a horizontal stationary fluid layer heated from below, it also briefly covers convection in such a layer with rotation about a vertical axis, with inclination, and with modulation of the vertical acceleration.

Recent Developments in Rayleigh-Bénard Convection

Thermal convection in a layer heated form below is an exemplary case for the study of non-linear fluid dynamics and the transition to turbulence. An outline is given of the present knowledge of the simplest realisation of convection in a layer of fluid satisfying the Oberbeck-Boussinesq approximation. Non-linear properties such as the dependence of the heat transport on Rayleigh and Prandtl numbers and the stability properties of convection rolls are emphasised in the discussion. Whenever possible, theoretical results are compared with experimental observations. A section on convection in rotating systems has been included, but the influence of other additional physical effects such as magnetic fields, side wall geometry, etc., has not been considered.

https://iopscience.iop.org/article/10.1088/0034-4885/41/12/003/pdf

Non-linear properties of thermal convection

https://scholarlypublications.universiteitleiden.nl/access/item%3A2733176/view

Front propagation into unstable states

A model based on magnetohydrodynamic (MHD) stability of the tokamak plasma edge region is presented, which describes characteristics of edge localized modes (ELMs) and the pedestal. The model emphasizes the dual role played by large bootstrap currents driven by the sharp pressure gradients in the pedestal region. Pedestal currents reduce the edge magnetic shear, stabilizing high toroidal mode number (n) ballooning modes, while at the same time providing drive for intermediate to low n peeling modes. The result is that coupled peeling–ballooning modes at intermediate n (3<n<20) are often the limiting instability which constrains the pedestal and triggers ELMs. These modes are characterized in shaped tokamak equilibria using an efficient new numerical code, and simplified models are developed for pedestal limits and the ELM cycle. Results are compared to several experiments, and nonideal MHD effects are briefly discussed.

/pdf/edge-localized-modes-and-the-pedestal-a-model-based-on-4j2hslubm4.pdf

Edge localized modes and the pedestal: A model based on coupled peeling–ballooning modes

The convective flow in an infinite horizontal fluid layer rotating rigidly about a normal axis is investigated for the special case of infinite Prandtl number and free boundary conditions. For slightly supercritical Rayleigh numbers the solutions of the non-linear steady-state equations are derived approximately by an amplitude expansion. A stability calculation shows that no stable steady-state convective flow exists if the Taylor number exceeds the critical value 2285.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112069001327

Transition from laminar convection to thermal turbulence in a rotating fluid layer

Ballooning stability boundaries for the large-aspect-ratio tokamak

The stability of a general toroidal MHD equilibrium with a continuous pressure profile is investigated for the case where the fluid is surrounded by vacuum. It is found that a disturbance which is localized near the free boundary grows exponentially in time unless a certain necessary criterion is satisfied. Because of the weaker boundary condition, this criterion imposes a more stringent restriction on the configuration than does Mercier's criterion near the boundary. For the cylindrically symmetric case the criterion requires a decrease of the rotational transform and yields a critical relation between the shear and the pressure gradient.

https://iopscience.iop.org/article/10.1088/0029-5515/15/1/007/pdf

The general “peeling” instability

Exact solutions of the steady-state hydromagnetic equations with a solenoidal velocity field are derived for appropriate boundary conditions in a helical geometry.

Exact solutions of the hydromagnetic dynamo problem

Es wurde die existenz von Losungen der Gleichungen des magneto-hydrostatischen Gleichgewichts unter folgenden Bedingungen untersucht. Das MagnetfeldB ist auf einer dreifach zusammenhangenden analytischen toroidalen FlacheT tangential, die zu einer Meridionalebene spiegelsymmetrisch liegt. Der Druck als hinreichend glatte Funktion vonq=ϕ|B|−1dl und der azimutale magnetische FlussF sind vorgegeben. Es zeigte sich, dass ein Gleichgewicht existiert, falls der Druckgradient nicht zu gross ist. Diese Gleichgewichtskonfiguration liegt symmetrisch zur Ebene der Spiegelsymmetrie und besitzt nur geschlossene Magnetfeldlinien. Die in den Anwendungen ausserdem erwunschte Forderung, dass die Flachenq=const dreifachen toroidalen Zusammenhang haben, ist erfullbar, indem die FlacheT so gewahlt wird, dass das durchT undF bestimmte Vakuummagnetfeld diese Eigenschaft hat.

/pdf/uber-die-existenz-toroidaler-magnetohydrostatischer-2jvhd7qomo.pdf

D. Lortz

Papers

Transition from laminar convection to thermal turbulence in a rotating fluid layer

Ballooning stability boundaries for the large-aspect-ratio tokamak

The general “peeling” instability

Exact solutions of the hydromagnetic dynamo problem

Über die Existenz toroidaler magnetohydrostatischer Gleichgewichte ohne Rotationstransformation