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

After a brief introduction to the concepts of energy stability, global stability and linear stability three theoretical approaches to different cases of hydrodynamic turbulence under stationary external conditions are discussed. Phase turbulence may occur in the weakly nonlinear limit of the Navier-Stokes-equations of motion when the instability of the basic primary state occurs in the form of a highly degenerate bifurcation as, for example, in the case of convection in a fluid layer heated from below and rotating about a vertical axis. Another quite generally applicable approach towards understanding turbulent fluid flow and its coherent structures in particular is the sequence-of-bifurcations approach. Discrete transitions from simple to complex fluid flows can be analyzed in the form of successive bifurcations when the system exhibits a maximum of symmetries based on the assumption of homogeneity in two spatial dimensions. The spatially periodic solutions generated through the sequence-of-bifurcations approach may not be realized in experimental situations where inhomogeneous onsets of disturbances may lead to chaotic fluid flows long before all spatially periodic solutions become unstable. But the tertiary and quaternary solutions exhibit in the clearest way the dynamic mechanisms that operate in turbulent states of flow. Finally the theory of bounds for turbulent transports in the asymptotic range of high Reynolds and Rayleigh numbers is outlined.

The Problem of Turbulence and the Manifold of Asymptotic Solutions of the Navier-Stokes Equations

The geodynamo, i.e. the process by which the Earth’s magnetic field is generated in the liquid outer core of the Earth, is generally regarded as one of the fundamental problems of geophysics. It is the prevailing opinion among geophysicists that thermal and chemical buoyancy originating in connection with the solidification of the inner core and the general cooling of the Earth cause convection flows in the outer core which consists of liquid iron together with some lighter elements. The mechanical energy of the convection flow provides the energy for the sustenance of the magnetic field against Ohmic losses. The mechanism by which this transfer of energy is accomplished is called a homogeneous dynamo in order to distinguish it from the common inhomogeneous dynamos used in technical applications.

Computations of Convection Driven Spherical Dynamos

Many fluid systems exhibit transitions from simple to complex forms of motion through subsequent bifurcations. Rayleigh-Benard convection in a layer heated from below and the Taylor-Couette system are the best known examples. Typically the homogeneous primary state is replaced by roll like motions in the first bifurcation which is followed by bifurcations to three-dimensional patterns in the form of wavy rolls, knot convection or bimodal cells. Secondary or tertiary bifurcations can also introduce time dependent processes such as oscillatory blob convection. The tertiary and quarternary states in the Rayleigh-Benard case exhibit typical features of turbulent convection such as thermal plumes and time dependent eruptions from the thermal boundary layers. Numerical computations based on Galerkin expansions of the dependent variables have been performed for steady and time periodic three-dimensional flows in dependence on the Rayleigh number, the Prandtl number, and the horizontal periodicity interval. The stability of the steady solutions has been analyzed with respect to disturbances which do not change the horizontal periodicity interval of the steady solutions. The comparison of the numerical computations with the experimental observations shows good agreement.

Higher Order Bifurcations in Fluid Systems and Coherent Structures in Turbulence

The onset of convection in a horizontal layer of an electrically conducting fluid heated from below is studied in the presence of a horizontal magnetic field rotating about a vertical axis. Two different ranges of the rotation frequency are considered: (1) the case of high frequencies of rotation for which the averaging approach is used and (2) the case of finite frequencies for which the averaging approach might be implemented. It is shown that in general the magnetic field stabilizes the static state of pure thermal conduction. However, there exists a parameter range where the inhomogeneity of the magnetic field is strong enough to provide energy for the growth of disturbances even if the fluid layer is stably stratified.

Rotating magnetic field effect on an onset of convection in a horizontal layer of conducting fluid

The stability of steady equilibrium amplitudes of magnetic fields generated by convection flows is investigated. Two cases are considered in detail, for which steady solutions are available from the previous work of Busse (1973) and Busse (1977). In the first case instability occurs primarily because of magnetic flux expulsion at high magnetic Reynolds numbers. In the second case the change in the velocity field caused by the Lorentz force enhances dynamo action. This subcritical finite amplitude dynamo is potentially unstable. In typical cases the nonlinear dynamo oscillations that replace the steady equilibrium solutions are investigated by numerical integration.

Friedrich H. Busse

Papers

The Problem of Turbulence and the Manifold of Asymptotic Solutions of the Navier-Stokes Equations

Computations of Convection Driven Spherical Dynamos

Higher Order Bifurcations in Fluid Systems and Coherent Structures in Turbulence

Rotating magnetic field effect on an onset of convection in a horizontal layer of conducting fluid

Nonlinear Dynamo Oscillations