Multiple states of spatially localized steady convection are found in numerical simulations of water-ethanol mixtures in two dimensions. Realistic boundary conditions at the top and bottom are used, with periodic boundary conditions in the horizontal. The states form by a mechanism similar to the pinning region around a Maxwell point in variational systems, but are located in a parameter regime in which the conduction state is overstable. Despite this the localized states can be stable. The properties of the localized states are described in detail, and the mechanism of their destruction with increasing or decreasing Rayleigh number is elucidated. When the Rayleigh number becomes too large the fronts bounding the state at either end unpin and move apart, allowing steady convection to invade the domain. In contrast, when the Rayleigh number is too small the fronts move inwards, and eliminate the localized state which decays into dispersive chaos. Out of this state spatially localized states re-emerge at irregular times before decaying again. Thus an interval of Rayleigh numbers exists that is characterized by relaxation oscillations between localized convection and dispersive chaos.

Binary mixtures with a negative separation ratio are known to exhibit time-independent spatially localized convection when heated from below. Numerical continuation of such states in a closed two-dimensional container with experimental boundary conditions and parameter values reveals the presence of a pinning region in Rayleigh number with multiple stable localized states but no bistability between the conduction state and an independent container-filling state. An explanation for this unusual behavior is offered. DOI: 10.1103/PhysRevE.80.025201 PACS number͑s͒: 47.54.Ϫr, 47.20.Bp, 47.20.Ky Many physical systems exhibit spatially localized structures or dissipative solitons. Examples arise in nonlinear optics ͓1͔, buckling of slender structures ͓2͔, and reactiondiffusion systems ͓3͔. Spatially localized oscillations or oscillons are time-dependent structures of this type ͓4͔. Similar structures are found in fluid flows, as indicated by the recent discovery of convectons in binary fluid convection ͓5-7͔ and related systems ͓8-10͔. A binary mixture with negative separation ratio heated from below develops a stabilizing concentration gradient via the ͑anomalous͒ Soret effect resulting in the presence of subcritical steady convection ͑Fig. 1͒. This subcritical regime favors the presence of convectons. These come in two types, even and odd under reflection in a vertical plane through their center, and are located in the so-called pinning region ͓11͔. In this interval of Rayleigh numbers multiple stable convectons, of different lengths and either parity, are present. In horizontally unbounded domains these localized structures appear simultaneously with the ͑subcritical͒ primary branch of spatially periodic steady convection. The resulting convectons are spatially extended at small amplitude but become strongly localized when followed numerically to larger amplitude by decreasing the Rayleigh number. Once their amplitude and length is comparable to the amplitude and wavelength of steady spatially periodic convection both convecton branches enter the pinning region and begin to snake back and forth across it ͑Fig. 1͒ as the convectons grow in length by nucleating additional convection rolls at both ends. This process continues until the length of the convecton becomes comparable to the available spatial domain when the convecton branches turn over toward the saddle node of the periodic branch and leave the pinning region ͓8,10,12͔.The conventional explanation of this behavior uses the presence of bistability between the conduction and spatially periodic convecting states together with spatial reversibility and relies on the similarity between the observed behavior ͑Fig. 1͒ and the corresponding behavior of the SwiftHohenberg equation ͑SHE͒ on the real line ͓13,14͔. The SHE is variational and so permits a comparison between the energy of the trivial ͑conduction͒ state and the energy of the spatially periodic state. When these energies are equal a front from the former to the latter ͑and vice versa͒ will be stationary...

Binary fluid mixtures with a negative separation ratio heated from below exhibit steady spatially localized states called convectons for supercritical Rayleigh numbers. Numerical continuation is used to compute such states in the presence of both Neumann boundary conditions and no-slip no-flux boundary conditions in the horizontal. In addition to the previously identified convectons, new states referred to as anticonvectons with a void in the centre of the domain, and wall-attached convectons attached to one or other wall are identified. Bound states of convectons and anticonvectons called multiconvecton states are also computed. All these states are located in the so-called snaking or pinning region in the Rayleigh number and may be stable. The results are compared with existing results with periodic boundary conditions.

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Numerical continuation is used to compute spatially localized convection in a binary fluid with no-slip laterally insulating boundary conditions and the results are compared with the corresponding ones for periodic boundary conditions (PBC). The change in the boundary conditions produces a dramatic change in the snaking bifurcation diagram that describes the organization of localized states with PBC: the snaking branches turn continuously into a large amplitude state that resembles periodic convection with defects at the sidewalls. Odd parity convectons are more affected by the boundary conditions since the sidewalls suppress the horizontal pumping action that accompanies these states in spatially periodic domains.

Two-dimensional nonlinear thermal convection in a cylindrical annulus is numerically analyzed for a Boussinesq fluid of low Prandtl number ϭ0.025. For a fixed value of the radius ratio, ϭ0.3, different types of steady columnar patterns are found. The stability of these convection patterns and the spatial interaction between them, which result in the formation of mixed modes, are investigated by considering the full nonlinear set of Navier-Stokes equations. Special attention is paid to the strong spatial interaction of the initially unstable modes with wavenumbers nϭ2 and nϭ4, which leads, through global bifurcations, to multiple stable quasi-periodic states of the system. A detailed picture of the nonlinear dynamics until temporal chaotic patterns set in is presented and understood in terms of local and global symmetry-breaking bifurcations of the O(2)-symmetric system.

Pattern selection near the onset of convection in a cylindrical container heated from below is investigated numerically for a water-ethanol mixture, with parameter values and boundary conditions relevant to experiments. The Boussinesq three-dimensional equations for binary fluid convection are simulated for cylinders of aspect ratio Gamma=11 and 10.5 (Gamma identical with R/d, where R is the radius of the cell and d its height). The onset of convection occurs via a subcritical Hopf bifurcation in which the critical mode is strongly influenced by small variations of the aspect ratio of the cell. During the linear regime, an m=1 azimuthal mode consisting of radially traveling waves grows in amplitude in the Gamma=11 cell, while an m=0 azimuthal mode is selected in the Gamma =10.5 cylinder. As convection evolves, simulations for subcritical and supercritical Rayleigh numbers reveal differences in the dynamics. Very close to the critical value, convection is erratic and focuses along one or more diameters of the cell; growths and collapses of the convection amplitude take place, but convection eventually dies away for subcritical values and persists for slightly supercritical values. For larger supercritical values, convection grows progressively in amplitude, and patterns consist of traveling-wave regions of convection initially focused near the cell center, though expanding slowly until a large-amplitude state is reached. Depending on the reduced Rayleigh number, the final state can be a nonsteady state filling the cell or a disordered confined state.

Binary fluid mixtures with a negative separation ratio heated from below exhibit steady spatially localized states called convectons for supercritical Rayleigh numbers. With no-slip, fixed-temperature, no-mass-flux boundary conditions at the top and bottom stationary odd-and even-parity convectons fall on a pair of intertwined branches connected by branches of travelling asymmetric states. In appropriate parameter regimes the stationary convectons may be stable. When the boundary condition on the top is changed to Newton's law of cooling the odd-parity convectons start to drift and the branch of odd-parity convectons breaks up and reconnects with the branches of asymmetric states. We explore the dependence of these changes and of the resulting drift speed on the associated Biot number using numerical continuation, and compare and contrast the results with a related study of the Swift-Hohenberg equation by Houghton & Knobloch (Phys. Rev. E, vol. 84, 2011, art. 016204). We use the results to identify stable drifting convectons and employ direct numerical simulations to study collisions between them. The collisions are highly inelastic, and result in convectons whose length exceeds the sum of the lengths of the colliding convectons.