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The dynamics near the 1:2 spatial resonance in systems with broken O(2) symmetry are described. The system is assumed to remain SO(2) invariant, and attention is focused on the behavior that replaces the structurally stable heteroclinic cycles present in the O(2)-symmetric system. The symmetry-breaking destroys these cycles, and introduces a variety of new bifurcations into the system, some which are associated with complex behavior. The possible scenarios with increasing symmetry-breaking are described and possible applications of the results are discussed.

In the weakly inviscid regime parametrically driven surface gravity-capillary waves generate oscillatory viscous boundary layers along the container walls and the free surface. Through nonlinear rectification these generate Reynolds stresses which drive a streaming flow in the nominally inviscid bulk; this flow in turn advects the waves responsible for the boundary layers. The resulting system is described by amplitude equations coupled to a Navier-Stokes-like equation for the bulk streaming flow, with boundary conditions obtained by matching to the boundary layers, and represents a novel type of pattern-forming system. The coupling to the streaming flow is responsible for various types of drift instabilities of standing waves, and in appropriate regimes can lead to the presence of relaxations oscillations. These are present because in the nearly inviscid regime the streaming flow decays much more slowly than the waves. Two model systems, obtained by projection of the Navier-Stokes-like equation onto the slowest mode of the domain, are examined to clarify the origin of this behavior. In the first the domain is an elliptically distorted cylinder while in the second it is an almost square rectangle. In both cases the forced symmetry breaking results in a nonlinear competition between two nearly degenerate oscillatory modes. This interaction destabilizes standing waves at small amplitudes and amplifies the role played by the streaming flow. In both systems the coupling to the streaming flow triggered by these instabilities leads to slow drifts along slow manifolds of fixed points or periodic orbits of the fast system, and generates behavior that resembles bursting in excitable systems. The results are compared to experiments.

In the nearly inviscid regime parametrically driven surface gravity-capillary waves couple to a streaming flow driven in oscillatory viscous boundary layers. In an elliptical container of small eccentricity this coupling can lead to relaxation oscillations.

The behavior of a thin liquid film on a uniformly heated substrate is considered. When the substrate is horizontal and the Marangoni number sufficiently large the film breaks up into a periodic array of drops. When the substrate is slightly inclined this drop-like state slides down the substrate. The relation between these states is discussed and their stability properties with respect to longitudinal perturbations are determined. The results shed light on the multiplicity of states accessible to systems of this type and on the possible transitions among them.

Three-dimensional simulations of oscillatory convection in binary mixtures driven by the Marangoni effect have been performed. The upper surface of the fluid is heated by a constant heat flux while the bottom is maintained at a constant temperature. Surface deflection is ignored. Oscillations are the result of concentration-induced changes in the surface tension due to the presence of an anomalous Soret effect. In domains with a square horizontal cross section and aspect ratio Γ=1.5 these take the form of either a standing wave with left–right reflection symmetry or a discrete rotating wave, depending on the separation ratio and the Schmidt number. Standing oscillations with reflection symmetry in a diagonal are unstable. When the cross section is slightly rectangular only the former bifurcate from the conduction state, and the transition to stable rotating waves with increasing Marangoni number proceeds via a sequence of secondary local and global bifurcations. The results are interpreted in terms of predictions from equivariant bifurcation theory.

The results of direct numerical simulations of convection in a uniformly rotating vertical cylinder with no-slip boundary conditions are described, and used to study the dynamics associated with transitions between states with adjacent azimuthal wave numbers far from onset. In certain regimes a novel burst-like state is identified and described. A dynamical-systems explanation for this behavior is suggested.

We study the transverse instability of a liquid ridge on horizontal and inclined substrates using a film evolution equation based on a long wave approximation. The equation incorporates an additional pressure term—the disjoining pressure—accounting for the effective interaction of the film with the substrate. On a horizontal substrate the dominant instability mode is varicose, but may turn into a zigzag mode on a slightly inclined substrate depending on the inclination angle and the ridge volume. For larger angles or volumes the instabilities at the front and back decouple. The linear stability properties of a one-dimensional transverse ridgelike state are studied in detail, and an energy analysis is used to demonstrate that the disjoining pressure provides the dominant instability mechanism at both the front and the back, while the body force is responsible for the main differences between these two instabilities. An amplitude equation for the time evolution of perturbations with small transverse wave numbers is derived that predicts correctly the linear crossing of the most dangerous eigenvalues at zero wave number in the inclined case, in contrast to the situation on a horizontal substrate.

Many physical, chemical, and biological systems exhibit traveling waves as a result of either an oscillatory instability or excitability. In the latter case a large multiplicity of stable spatially localized wavetrains consisting of different numbers of traveling pulses may be present. The existence of these states is related here to the presence of homoclinic snaking in the vicinity of a subcritical, finite wavenumber Hopf bifurcation. The pulses are organized in a slanted snaking structure resulting from the presence of a heteroclinic cycle between small and large amplitude traveling waves. Connections of this type require a multivalued dispersion relation. This dispersion relation is computed numerically and used to interpret the profile of the pulse group. The different spatially localized pulse trains can be accessed by appropriately customized initial stimuli, thereby blurring the traditional distinction between oscillatory and excitable systems. The results reveal a new class of phenomena relevant to spatiotemporal dynamics of excitable media, particularly in chemical and biological systems with multiple activators and inhibitors.

Chimera states consisting of domains of coherently and incoherently oscillating nonlocally-coupled phase oscillators in systems with spatial inhomogeneity are studied. The inhomogeneity is introduced through the dependence of the oscillator frequency on its location. Two types of spatial inhomogeneity, localized and spatially periodic, are considered and their effects on the existence and properties of multi-cluster and traveling chimera states are explored. The inhomogeneity is found to break up splay states, to pin the chimera states to specific locations and to trap traveling chimeras. Many of these states can be studied by constructing an evolution equation for a complex order parameter. Solutions of this equation are in good agreement with the results of numerical simulations., Comment: 16 pages, 35 figures

Nearly inviscid parametrically excited surface gravity-capillary waves in two-dimensional periodic domains of finite depth and both small and large aspect ratio are considered. Coupled equations describing the evolution of the amplitudes of resonant left- and right-traveling waves and their interaction with a mean flow in the bulk are derived, and the conditions for their validity established. In general the mean flow consists of an inviscid part together with a viscous streaming flow driven by a tangential stress due to an oscillating viscous boundary layer near the free surface and a tangential velocity due to a bottom boundary layer. These forcing mechanisms are important even in the limit of vanishing viscosity, and provide boundary conditions for the Navier-Stokes equation satisfied by the mean flow in the bulk. The streaming flow is responsible for several instabilities leading to pattern drift.

Numerical continuation is used to study bifurcations in doubly diffusive convection in three-dimensional enclosures driven by opposing horizontal temperature and concentration gradients, and the results are compared with the two-dimensional case. Direct numerical simulation is used to show that in certain regimes the first stable nontrivial state of the three-dimensional system is a finite amplitude nonlinear oscillation. This state may be either periodic or chaotic. The mechanism responsible for these oscillations is identified, and the oscillations shown to be an indirect consequence of the presence of a steady-state bifurcation to fundamentally three-dimensional longitudinal structures that are absent from a two-dimensional formulation. The role of global bifurcations in generating the chaotic oscillations is elucidated.

Faraday waves are described, under appropriate conditions, by a damped nonlocal parametrically driven nonlinear Schrodinger equation. As the strength of the applied forcing increases this equation undergoes a sequence of transitions to chaotic dynamics. The origin of these transitions is explained using a careful study of a two-mode Galerkin truncation and linked to the presence of heteroclinic connections between the trivial state and spatially periodic standing waves. These connections are associated with cascades of gluing and symmetry-switching bifurcations; such bifurcations are located in the partial differential equations as well.

The generalized Swift--Hohenberg equation with a quadratic-cubic nonlinearity is used to study the persistence and decay of localized patterns in the presence of time-periodic parametric forcing. A novel resonance phenomenon between the forcing period and the time required to nucleate one wavelength of the pattern outside the pinning region is identified. The resonance generates distinct regions in parameter space characterized by the net number of wavelengths gained or lost in one forcing cycle. These regions are well described by an asymptotic theory based on the wavelength nucleation/annihilation time near the boundaries of the pinning region. The resulting theory leads to predictions that are qualitatively correct and, in some cases, provide quantitative agreement with numerical simulations., 43 pages, 19 figures

Chimera states consisting of domains of coherently and incoherently oscillating identical oscillators with nonlocal coupling are studied. These states usually coexist with the fully synchronized state and have a small basin of attraction. We propose a nonlocal phase-coupled model in which chimera states develop from random initial conditions. Several classes of chimera states have been found: (a) stationary multicluster states with evenly distributed coherent clusters, (b) stationary multicluster states with unevenly distributed clusters, and (c) a single cluster state traveling with a constant speed across the system. Traveling coherent states are also identified. A self-consistent continuum description of these states is provided and their stability properties analyzed through a combination of linear stability analysis and numerical simulation.

The theory of stationary spatially localized patterns in dissipative systems driven by time-independent forcing is well developed. With time-periodic forcing related but time-dependent structures may result. These may consist of breathing localized patterns, or states that grow for part of the cycle via nucleation of new wavelengths of the pattern followed by wavelength annihilation during the remainder of the cycle. These two competing processes lead to a complex phase diagram whose structure is a consequence of a series of resonances between the nucleation time and the forcing period. The resulting diagram is computed for the periodically forced quadratic-cubic Swift--Hohenberg equation, and its details interpreted in terms of the properties of the depinning transition for the fronts bounding the localized state on either side. The results are expected to shed light on localized states in a large variety of periodically driven systems., Comment: 5 pages, 4 figures

Using dynamical density functional theory we calculate the speed of solidification fronts advancing into a quenched two-dimensional model fluid of soft-core particles. We find that solidification fronts can advance via two different mechanisms, depending on the depth of the quench. For shallow quenches, the front propagation is via a nonlinear mechanism. For deep quenches, front propagation is governed by a linear mechanism and in this regime we are able to determine the front speed via a marginal stability analysis. We find that the density modulations generated behind the advancing front have a characteristic scale that differs from the wavelength of the density modulation in thermodynamic equilibrium, i.e., the spacing between the crystal planes in an equilibrium crystal. This leads to the subsequent development of disorder in the solids that are formed. For the one-component fluid, the particles are able to rearrange to form a well-ordered crystal, with few defects. However, solidification fronts in a binary mixture exhibiting crystalline phases with square and hexagonal ordering generate solids that are unable to rearrange after the passage of the solidification front and a significant amount of disorder remains in the system., 18 pages, 14 figs

Rotating Rayleigh-Bénard convection exhibits, in the limit of rapid rotation, a turbulent state known as geostrophic turbulence. This state is present for sufficiently large Rayleigh numbers representing the thermal forcing of the system, and is characterized by a leading order balance between the Coriolis force and pressure gradient. This turbulent state is itself unstable to the generation of depth-independent or barotropic vortex structures of ever larger scale through a process known as spectral condensation. This process involves an inverse cascade mechanism with a positive feedback loop whereby large-scale barotropic vortices organize small scale convective eddies. In turn, these eddies provide a dynamically evolving energy source for the large-scale barotropic component. Kinetic energy spectra for the barotropic dynamics are consistent with a k-3 downscale enstrophy cascade and an upscale cascade that steepens to k-3 as the box-scale condensate forms. At the same time the flow maintains a baroclinic convective component with an inertial range consistent with a k-5/3 spectrum. The condensation process resembles a similar process in two dimensions but is fully three-dimensional.

The amplitude equations describing the 1:2 spatial resonance in O(2) symmetric systems are re-examined. Complex dynamical behavior associated with a new class of global connections is identified and analyzed. These connections form further from the mode interaction point where the structurally stable heteroclinic cycles described by Armbruster et al. [Physica D 29 (1988) 257] no longer exist. The resulting behavior is characteristic of systems with O(2)×Z2 symmetry in which the symmetry Z2 is weakly broken.

Nearly inviscid parametrically excited surface gravity–capillary waves in two-dimensional domains of finite depth and large aspect ratio are considered. Coupled equations describing the evolution of the amplitudes of resonant left- and right-traveling waves and their interaction with a mean flow in the bulk are derived, and the conditions for their validity established. Under suitable conditions the mean flow consists of an inviscid part together with a viscous mean flow driven by a tangential stress due to an oscillatory viscous boundary layer near the free surface and a tangential velocity due to a bottom boundary layer. These forcing mechanisms are important even in the limit of vanishing viscosity, and provide boundary conditions for the Navier–Stokes equation satisfied by the mean flow in the bulk. For moderately large aspect ratio domains the amplitude equations are nonlocal but decouple from the equations describing the interaction of the slow spatial phase and the viscous mean flow. Two cases are considered in detail, gravity–capillary waves and capillary waves in a microgravity environment.

The effect of resonant temporal forcing on a system undergoing a Hopf bifurcation with D4 (square) symmetry is studied. The forcing breaks the continuous normal form symmetry of the governing amplitude equations, but the D4 symmetry is preserved. For this system, it is shown that bursts with very large dynamic range may occur. The bursts are associated with visits near solutions “at infinity”, and are related to those found previously for the Hopf bifurcation with broken D4 symmetry [Moehlis and Knobloch, Physica D 135 (2000) 263]. The regime in which an attracting quasiperiodic solution that exists in the absence of forcing “wrinkles” into chaos as the amplitude of the forcing increases is investigated in detail. The overall behavior is governed by the approach of the resulting attractor to solutions at infinity. Windows with stable periodic solutions are found and associated with the traversal in parameter space through Arnol’d tongues. Other aspects of the dynamics are related to the presence of a new type of gluing bifurcation (which we call a “supergluing bifurcation”). © 2001 Elsevier Science B.V. All rights reserved.

Direct numerical simulations of chevrons, blinking states, and repeated transients in binary fluid mixtures with a negative separation ratio heated from below are described. The calculations are performed in two-dimensional containers for experimental parameter values and boundary conditions. Quantitative agreement with the experiments of Kolodner [Phys. Rev. E 47, 1038 (1993)] is obtained, and the origin of the blinking and repeated transient states is elucidated.

Numerical continuation is used to follow branches of steady solutions to the three-dimensional Marangoni–Benard problem in a zero gravity environment. The upper surface of the fluid is heated by a constant heat flux while the bottom is maintained at a constant temperature. Instability arises due to temperature-dependent surface tension effects but surface deflection is ignored. Containers with square and nearly square cross sections and no-slip boundary conditions are analyzed, and the results interpreted in terms of predictions from equivariant bifurcation theory.

In large aspect ratio systems with broken reflection symmetry the onset of instability is closely related to the threshold for absolute instability in the corresponding unbounded system. The upstream boundary (with respect to the group velocity) plays a crucial part in fixing the frequency of the nonlinear wavetrain that results, and hence its stability properties. In contrast in the convectively unstable regime all perturbations decay, although persistent structures can be maintained by the addition of small amplitude noise. The upstream boundary, however distant, continues to play an essential role in frequency selection, with the result that the structures induced by noise are of universal form. A general theory is developed that predicts the selected frequency and wave number for both primary and secondary convective instabilities, and the results illustrated using the complex Ginzburg–Landau equation and a mean-field dynamo model of magnetic field generation in the Sun.

The interaction between two steady-state bifurcations with spatial wave numbers in the ratio 1:3 is considered. Periodic boundary conditions are assumed. The resulting O(2) equivariant normal form, truncated at third order, exhibits a number of global bifurcations that may result in complex dynamics. The origin of this behavior is elucidated with the help of careful numerical simulations and analysis of appropriate return maps. The results generalize to other 1: n resonances.

A new mechanism responsible for generating regular and irregular bursts of large dynamic range near onset of an oscillatory instability is identified. The bursts are present in systems with nearly square symmetry and are the result of heteroclinic cycles involving infinite amplitude states created when the square symmetry is broken. All possible cycles of this type are identified and the resulting bursts described. Global connections involving finite amplitude states are also present. The intricate sequence of bifurcations that result is described in several cases. ©2000 Elsevier Science B.V. All rights reserved.

Existing group theoretic analysis of pattern formation in three dimensions [T.K. Callahan, E. Knobloch, Symmetry-breaking bifurcations on cubic lattices, Nonlinearity 10 (1997) 1179‐1216] is used to make specific predictions about the formation of three-dimensional patterns in two models of the Turing instability, the Brusselator model and the Lengyel‐Epstein model. Spatially periodic patterns having the periodicity of the simple cubic (SC), face-centered cubic (FCC) or body-centered cubic (BCC) lattices are considered. An efficient center manifold reduction is described and used to identify parameter regimes permitting stable lamellae, SC, FCC, double-diamond, hexagonal prism, BCC and BCCI states. Both models possess a special wavenumber k at which the normal form coefficients take on fixed model-independent ratios and both are described by identical bifurcation diagrams. This property is generic for two-species chemical reaction‐diffusion models with a single activator and inhibitor. ©1999 Elsevier Science B.V. All rights reserved.

Fully nonlinear three-dimensional convection in a rotating layer is studied for large Taylor numbers. In this regime, the leading order nonlinearity arises from the distortion of the horizontally averaged temperature profile. As a result, steady rolls, squares, hexagons, triangles, and a pattern called patchwork quilt all have identical Nusselt numbers. A similar degeneracy is present in overstable convection with six patterns having identical time-averaged Nusselt numbers and oscillation frequencies. These results are obtained via an asymptotic expansion in the Taylor number that determines, for each Rayleigh number, the time-averaged Nusselt number and oscillation frequency from the solution of a nonlinear eigenvalue problem for the vertical temperature profile. A number of other patterns are determined by a weakly nonlinear analysis that cannot be extended into the fully nonlinear regime by the present methods, but these patterns are necessarily unstable.

Symmetry-increasing bifurcations of strange attractors in systems with O 2 symmetry are shown to produce traveling waves that reverse their direction of propagation in a chaotic fashion. The resulting dynamics are illustrated using the normal form describing the triple zero instability. q 1999 Elsevier Science B.V. All rights reserved.

Convection in a rotating layer was a subject close to Professor Chandrasekhar’s interest, and graces the cover of the Dover edition of his famous book on Hydrodynamic and Hydromagnetic Stability . In this article I will briefly review the results described in this book, which concern primarily linear stability theory for an unbounded layer. I will then discuss in detail the differences between the theory for the unbounded layer and that for a finite cylindrical container heated from below and rotating about its axis. These differences can be traced to the differences in the symmetries of the two systems, and lead one to expect time-dependent convection at onset, regardless of the value of the Prandtl number. In addition the boundaries are responsible for the presence of a new nonaxisymmetric mode of instability, called a wall mode. The theory predicts that these wall modes will precess at onset and set in before the body modes characteristic of the unbounded layer. These predictions have been confirmed in elegant experiments by R. Ecke and colleagues. In the nonlinear regime perhaps the most interesting phenomenon is the Kuppers–Lortz instability. In this instability a pattern of parallel rolls becomes unstable to another set of rolls oriented at an angle with respect to the first, once the rotation rate exceeds a critical value. This new set is itself unstable in the same fashion etc., resulting in complex dynamics right at onset. I will describe the theory behind this instability and identify similar instabilities for oscillatory convection whether in the form of standing or traveling rolls. Such instabilities are triggered by the formation of heteroclinic orbits connecting two sets of standing or traveling rolls with different orientation. Structurally stable heteroclinic cycles involving four traveling roll states are possible, and result in quite unusual dynamical behavior.

Department of Physics, University of California, Berkeley, California 94720~Received 5 February 1998!In systems with midplane reflection symmetry the dominant spatial resonance is the 1:3 resonance. Numeri-cal continuation is used to study this and other 1:2k11 resonances in two-dimensional convection betweenno-slip perfectly conducting horizontal plates. Periodic boundary conditions are imposed in the horizontal.These resonances influence the process of wave number selection at moderate Rayleigh numbers through thegeneration of hybrid solutions, and thereby modify the Eckhaus picture of wave number selection. Unlike thebetter known mixed modes the hybrid solutions have the same symmetry as a pair of primary rolls. Both hybridand symmetry-breaking mixed modes are computed and their linear stability properties with respect to pertur-bations preserving different spatial periods are determined. A complete description of the effects of midplanereflection on wave number selection emerges. Only steady solutions are considered and the Prandtl number isfixed ats510. @S1063-651X~98!05709-2#PACS number~s!: 47.20.Ky, 05.45.1bI. INTRODUCTION

Stellar dynamos are governed by non-linear partial differential equations (PDEs) which admit solutions with dipole, quadrupole or mixed symmetry (i.e. with different parities). These PDEs possess periodic solutions that describe magnetic cycles, and numerical studies reveal two different types of modulation. For modulations of Type 1 there are parity changes without significant changes of amplitude, while for Type 2 there are amplitude changes without significant changes in parity. In stars like the Sun, cyclic magnetic activity is interrupted by grand minima that correspond to Type 2 modulation. Although the Sun’s magnetic field has maintained dipole symmetry for almost 300 yr, there was a significant parity change at the end of the Maunder Minimum. We infer that the solar field may have flipped from dipole to quadrupole polarity (and back) after deep minima in the past and may do so again in the future. Other stars, with different masses or rotation rates, may exhibit cyclic activity with dipole, quadrupole or mixed parity. The origins of such behaviour can be understood by relating the PDE results to solutions of appropriate low-order systems of ordinary differential equations (ODEs). Type 1 modulation is reproduced in a fourth-order system while Type 2 modulation occurs in a third-order system. Here we construct a new sixth-order system that describes both types of modulation and clarifies the interactions between symmetry-breaking and modulation of activity. Solutions of these non-linear ODEs reproduce the qualitative behaviour found for the PDEs, including flipping of polarity after a prolonged grand minimum. Thus we can be confident that these patterns of behaviour are robust, and will apply to stars that are similar to the Sun.

A robust mechanism generating regular or irregular bursts of large dynamic range near threshold is described. The mechanism arises in the interaction between oscillatory modes of odd and even parity in systems of large but finite aspect ratio, and provides an explanation for the bursting behavior observed in binary fluid convection by Sullivan and Ahlers [Phys. Rev. A 38, 3143 (1988)]. [S0031-9007(98)06353-4]

The incompressible Navier–Stokes equation is considered in the limit of rapid rotation (small Ekman number). The analysis is limited to horizontal scales small enough so that both horizontal and vertical velocities are comparable, but the horizontal velocity components are still in geostrophic balance. Asymptotic analysis leads to a pair of nonlinear equations for the vertical velocity and vertical vorticity coupled by vertical stretching. Statistically stationary states are maintained against viscous dissipation by boundary forcing or energy injection at larger scales. For thermal forcing direct numerical simulation of the reduced equations reveals the presence of intense vortical structures spanning the layer depth, in excellent agreement with simulations of the Boussinesq equations for rotating convection by Julien et al. (1996).

Recent experiments on spiral waves in the Belousov-Zhabotinsky reaction indicate that the waves may break up sufficiently far from the core into chemical turbulence. Simulations of target patterns in the FitzHugh-Nagumo and Ginzburg-Landau models indicate that such transitions occur when the wave frequency pushes the selected wave vector into the regime of absolute Eckhaus instability. This frequency in turn depends on the control parameter and boundary conditions through the solution of a nonlinear eigenvalue problem. [S0031-9007(98)06192-4]

Investigation of the linear stability problem for rapidly rotating convection on an f-plane has revealed the existence of two distinct scales in the vertical structure of the critical eigenfunctions: a small length scale whose vertical wavenumber kz is comparable with the large horizontal wavenumber k⊥ selected at onset, and a large-scale modulation which forms an envelope on the order of the layer depth d. The small-scale structure in the vertical results from a geostrophic balance imposed by the Taylor–Proudman constraint. This primary balance forces rotational alignment and confines fluid motions to planes perpendicular to the rotation axis. For convective transport in the vertical this constraint must be relaxed. This is achieved by molecular dissipation which allows weak upward (downward) spiralling of hot (cold) fluid elements across the Taylor–Proudman planes and results in a large-scale vertical modulation of the Taylor columns.In the limit of fast rotation (i.e. large Taylor number) a multiple-scales analysis leads to the determination of a critical Rayleigh number as a function of wavenumber, roll orientation and the tilt angle of the f-plane. The corresponding critical eigenfunction represents the core solution; matching to passive Ekman boundary layers is required for a complete solution satisfying boundary conditions.An extension of this analysis, introduced by Bassom & Zhang (1994), is used to describe strongly nonlinear two-dimensional convection, characterized by significant departures of the mean thermal field from its conduction profile. The analysis requires the solution of a nonlinear eigenvalue problem for the Nusselt number (for steady convection) and the Nusselt number and oscillation frequency (for the overstable problem). The solutions of this problem are used to calculate horizontal and vertical heat fluxes, as well as Reynolds stresses, as functions of both the latitude and roll orientation in the horizontal, and these are used to calculate self-consistently north–south and east–west mean flows. These analytical predictions are in good agreement with the results of three-dimensional simulations reported by Hathaway & Somerville (1983).

Dynamics of linear and nonlinear waves in driven dissipative systems in finite domains are considered. In many cases (for example, due to rotation) the waves travel preferentially in one direction. Such waves cannot be reflected from boundaries. As a consequence in the convectively unstable regime the waves ultimately decay; only when the threshold for absolute instability is exceeded can the waves be maintained against dissipation at the boundary. Secondary absolute instabilities are associated with the break-up of a wave train into adjacent wave trains with different frequencies, wave numbers and amplitudes, separated by a front. The process of frequency selection is discussed in detail, and the selected frequency is shown to determine the wave number and amplitude of the wave trains. The results are described using the complex Ginzburg-Landau equation and illustrated using a mean-field dynamo model of magnetic field generation in the Sun.

The generation of mean flows by trains of traveling, inviscid water waves is investigated. All ambiguities associated with the velocity potential are resolved by treating uniform wave trains as limits of wave packets, and appealing to the conservation of mass and momentum. This formulation leads to a systematic multiple-scales description of weakly nonlinear wave trains and the associated mean flows. The results are compared with the amplitude equation formulation of Davey and Stewartson and radiation stress formulation of Longuet-Higgins and Stewart which do not conserve mass. The momentum of the wave train can be uniquely specified only by an analysis of the wave generation mechanism. The present theory is sufficiently general that mean flows arising from different generation mechanisms can be included, and shows that a recently proposed singularity associated with mean flows is absent.

Two-dimensional thermosolutal convection with no-slip boundary conditions is studied using numerical simulations in a periodic domain. The domain is large enough to follow the evolution of phase instabilities of fully nonlinear traveling waves. In the parameter regime studied these instabilities evolve, without loss of phase or hysteresis, into a series of confined states or pulses characterized by locally enhanced heat and solute transport. The wavelength and phase velocity of the traveling rolls within a pulse differ substantially from those in the background. The pulses drift in the same direction as the convection rolls on which they ride but more slowly, and are characterized by an exponential leading front and an oscillatory trailing end. Multiple, apparently stable, states are found for identical parameter values. The qualitative properties of the pulses are in good agreement with the predictions of a third-order phase equation which accounts for the relation between wave number and phase velocity, the oscillatory tails and the multiplicity of states. These properties of the pulses are shown to be a consequence of Shil'nikov dynamics in the spatial domain.

We study spatial localization in the generalized Swift-Hohenberg equation with either quadratic-cubic or cubic-quintic nonlinearity subject to spatially heterogeneous forcing. Different types of forcing (sinusoidal or Gaussian) with different spatial scales are considered and the corresponding localized snaking structures are computed. The results indicate that spatial heterogeneity exerts a significant influence on the location of spatially localized structures in both parameter space and physical space, and on their stability properties. The results are expected to assist in the interpretation of experiments on localized structures where departures from spatial homogeneity are generally unavoidable.

The complete spectrum of decaying small-amplitude water waves is described. Viscosity is shown to be responsible both for the decay of gravity-capillary waves and for the appearance of a class of viscous modes that are omitted in the standard potential formulation. For sufficiently small viscosity @as measured by the parameter C[n(gh 3 ) 21/2 !1, where n is the kinematic viscosity, g the acceleration due to gravity, and h the undisturbed depth of the fluid # the viscous modes decay more slowly than the gravity-capillary ones and must be included in weakly nonlinear theories. The analysis indicates that for realistic values of C second-order corrections to the decay rate of gravity-capillary waves are important and suggests a straightforward resolution of existing discrepancies between experimentally measured and theoretically calculated damping rates. @S1063-651X~97!11311-3#

Steady-state symmetry-breaking bifurcations on the simple (SC), face-centred (FCC) and body-centred (BCC) cubic lattices are considered, corresponding to the 6-, 8- and 12-dimensional representations of the group . Methods of equivariant bifurcation theory are used to identify all primary solution branches and to determine their stability; branches with submaximal isotropy are generic for both the FCC and BCC lattices. Complete analysis of the local branching behaviour in the SC (three primary branches) and FCC (five primary branches) cases is given. The BCC case is substantially more complex, owing to the presence of a quadratic equivariant. A degeneracy that arises in the presence of an additional reflection symmetry is analysed first using a normal form truncated at third order. This problem, in which no quadratic equivariants are present, yields 10 primary branches with maximal isotropy and five with submaximal isotropy. The unfolding of the degeneracy is used to show that seven primary branches (six maximal and one submaximal) persist in the generic case, and to determine the form and properties of secondary branches. In certain cases higher-order terms are necessary. The study is motivated by the problem of pattern formation in three spatial dimensions, and extends earlier work by De Wit and co-workers.

The steady-state reflection-breaking bifurcation from a circle of nontrivial equilibria in O(2)-equivariant systems results in a pair of travelling waves. When the continuous part of the group O(2) is weakly broken, the corresponding instability may lead to nonsymmetric but steady states. The transition from this state to the travelling wave state with increasing bifurcation parameter is complex, and typically involves sequences of global bifurcations. The possible scenarios are described in detail and the results are related to the dynamics associated with parity-breaking instabilities of spatially periodic patterns in inhomogeneous systems.

Two-dimensional overstable convection in a rotating layer is studied for large Taylor numbers. In this regime, the leading order nonlinearity arises from the distortion of the horizontally averaged temperature profile. Fully nonlinear solutions in the form of traveling and standing waves are obtained via an asymptotic expansion in the Taylor number. The formulation leads to a nonlinear eigenvalue problem for the vertical structure and frequency of each solution type and allows the computation of the Nusselt number and frequency as functions of the applied Rayleigh number.

Mode interaction between odd and even modes in two-dimensional Boussinesq convection in a box is revisited. It is noted that in the large aspect ratio limit the structure of the amplitude equations depends on the boundary conditions applied at the sidewalls, however distant.With no-slip sidewall boundary conditions the equations approach those for an unbounded layer with periodic boundary conditions; this is not the case for free-slip boundary conditions. Thus only in the former case can the large aspect ratio system be considered a small perturbation of the unbounded system. The reasons for the different large aspect ratio limits are traced to the presence of “hidden” symmetries in the stress-free case. Homotopic continuation is used to extend these results to other types of boundary conditions.

In order to model the solar sunspot cycle, Parker introduced the concept of dynamo waves. These waves owe their existence to two processes, the generation of poloidal field from toroidal by the α–effect, and the generation of toroidal field from the poloidal by the Ω–effect. In their simplest realization these waves are plane waves that propagate in latitude. In this paper we show that spatial inhomogeneities, whether due to variations in the driving α–effect, or simply through the imposition of boundaries in latitude to simulate the solar pole and Equator, have an anomalously important effect, and make it necessary to revise the usual notions of weakly nonlinear bifurcation theory. Two distinct cases are considered. In the first, the solution domain is infinite, but the α–effect is antisymmetric about the Equator at x=0. Both dipole and quadrupole states are considered. The linear problem reveals an unusual feature: the frequency spectrum is continuous and the neutral curve Nω , where N is the dynamo number, plays the role of the more usual neutral curve in the spatial domain. However, the critical dynamo number remains equal to that for an unbounded homogeneous layer. In the weakly nonlinear regime the resulting instability takes the form of dynamo waves that are slowly modulated in space, and is described by an evolution equation of Ginzburg—Landau type but with space and time interchanged. The second case, perhaps more appropriate to the Sun, deals with large but finite domains while including possible slow spatial variations of the α–effect. Surprisingly, in this case sustained dynamo action can only occur when the dynamo number exceeds a value somewhat higher than that for an infinite layer. The onset mode is now localized equatorially. At leading order this dynamo number is independent of the domain size and of the boundary conditions imposed at large ∣x∣, i.e. at the poles. This feature of the problem is related to the distinction between convective and absolute instabilities.

International audience; Localized patches of stationary convection embedded in a background conduction state are called convectons. Multiple states of this type have recently been found in two-dimensional Boussinesq convection in a horizontal fluid layer with stress-free boundary conditions at top and bottom, and rotating about the vertical. The convectons differ in their lengths and in the strength of the self-generated shear within which they are embedded, and exhibit slanted snaking. We use homotopic continuation of the boundary conditions to show that similar structures exist in the presence of no-slip boundary conditions at the top and bottom of the layer and show that such structures exhibit standard snaking. The homotopic continuation allows us to study the transformation from slanted snaking characteristic of systems with a conserved quantity, here the zonal momentum, to standard snaking characteristic of systems with no conserved quantity.

International audience; Doubly diffusive convection in a three-dimensional horizontally extended domain with a square cross section in the vertical is considered. The fluid motion is driven by horizontal temperature and concentration differences in the transverse direction. When the buoyancy ratio N = -1 and the Rayleigh number is increased the conduction state loses stability to a subcritical, almost two-dimensional roll structure localized in the longitudinal direction. This structure exhibits abrupt growth in length near a particular value of the Rayleigh number but does not snake. Prior to this filling transition the structure becomes unstable to a secondary twist instability generating a pair of stationary, spatially localized zigzag states. In contrast to the primary branch these states snake as they grow in extent and eventually fill the whole domain. The origin of the twist instability and the properties of the resulting localized structures are investigated for both periodic and no-slip boundary conditions in the extended direction.

Three-dimensional convection in a binary mixture in a porous medium heated from below is studied. For negative separation ratios steady spatially localized convection patterns are expected. Such patterns, spatially localized in two dimensions, are computed and numerical continuation is used to examine their growth and proliferation as parameters are varied. The patterns studied have the form of a core region with four extended side-branches and can be stable. A physical mechanism behind the formation of these unusual structures is suggested.