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.

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Pattern formation outside of equilibrium

A finite-difference method for solving the time-dependent Navier- Stokes equations for an incompressible fluid is introduced. This method uses the primitive variables, i.e. the velocities and the pressure, and is equally applicable to problems in two and three space dimensions. Test problems are solved, and an ap- plication to a three-dimensional convection problem is presented.

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Numerical solution of the Navier-Stokes equations

Bond-orientational order in molecular-dynamics simulations of supercooled liquids and in models of metallic glasses is studied. Quadratic and third-order invariants formed from bond spherical harmonics allow quantitative measures of cluster symmetries in these systems. A state with short-range translational order, but extended correlations in the orientations of particle clusters, starts to develop about 10% below the equilibrium melting temperature in a supercooled Lennard-Jones liquid. The order is predominantly icosahedral, although there is also a cubic component which we attribute to the periodic boundary conditions. Results are obtained for liquids cooled in an icosahedral pair potential as well. Only a modest amount of orientational order appears in a relaxed Finney dense-random-packing model. In contrast, we find essentially perfect icosahedral bond correlations in alternative "amorphon" cluster models of glass structure.

Bond-orientational order in liquids and glasses

Evolutionary games on graphs

This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book

The major planets produce heat flux from their interiors that is comparable to the radiative flux they receive from the sun. The dynamics of convection flows carrying the heat flux are discussed, and the dominating effect of the Coriolis force is demonstrated. The characteristic high-velocity jets in the atmospheres of Jupiter and Saturn can be explained on the basis of Reynolds stresses generated by the fluctuating convective motions. A simple annulus model, which elucidates the more complex mathematical analysis of the spherical case given in an earlier paper (Busse, 1983), is considered in detail. Various aspects of the observational evidence are discussed in relation to the model.

Convection-Driven Zonal Flows in the Major Planets

The problem of the origin of planetary magnetism is formulated as a bifurcation problem and some recent theoretical work on the generation of magnetic fields by buoyancy-driven convection in rotating spherical shells is briefly reviewed. Since the lack of laboratory experiments has hampered the theoretical progress, a possible configuration for a laboratory apparatus is proposed.

Dynamic Theory of Planetary Magnetism and Laboratory Experiments

It is demonstrated both experimentally and theoretically that broken translational and vertical symmetries of a horizontal convection layer heated from below can lead to a qualitatively new effect, namely the onset of time-dependent convection. In particular, the behavior of a non-Boussinesq fluid in a layer with sinusoidal variation of the horizontal boundaries is studied near onset.

Time-dependent convection induced by broken spatial symmetries.

Steady three-dimensional convection in the form of bimodal cells in a fluid layer heated from below with rigid boundaries is studied through numerical computations for Prandtl numbers in the range 10 [lsim ] P [lsim ] 100. The stability of the steady solutions with respect to disturbances of various symmetries has been analysed. Typically, the range of stable steady bimodal convection is restricted by the transition to oscillatory bimodal convection. The oscillations preserve the spatial symmetry of the steady bimodal convection pattern in the case of high P and higher wavenumbers, but break it in the case of lower P or lower wavenumbers in the range that has been investigated. Some comparisons are made with experimental observations. The transition from bimodal to knot convection has also been studied.

Steady and oscillatory bimodal convection

Some concepts used in the theory of convection-driven dynamos in rotating spherical fluid shells are discussed. The analogy between imposed magnetic fields and those generated by dynamo action is evaluated and the role of the Elsasser number is considered. Eddy diffusivities are essential ingredients in numerical dynamo simulations, but their effects could be misleading. New aspects of the simultaneous existence of different dynamo states are described.

Friedrich H. Busse

Papers

Convection-Driven Zonal Flows in the Major Planets

Dynamic Theory of Planetary Magnetism and Laboratory Experiments

Time-dependent convection induced by broken spatial symmetries.

Steady and oscillatory bimodal convection

Remarks on some typical assumptions in dynamo theory