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.

Rotating convection is analysed numerically in a cylinder of aspect ratio one, for Prandtl number about 7. Traditionally, the problem has been studied within the Boussinesq approximation with density variation only incorporated in the gravitational buoyancy term and not in the centrifugal buoyancy term. In that limit, the governing equations admit a trivial conduction solution. However, the centrifugal buoyancy changes the problem in a fundamental manner, driving a large-scale circulation in which cool denser fluid is centrifuged radially outward and warm less-dense fluid is centrifuged radially inward, and so there is no trivial conduction state. For small Froude numbers, the transition to three-dimensional flow occurs for Rayleigh number R ≈ 7.5 × 103. For Froude numbers larger than 0.4, the centrifugal buoyancy stabilizes the axisymmetric large-scale circulation flow in the parameter range explored (up to R = 3.5 × 104). At intermediate Froude numbers, the transition to three-dimensional flow is via four different Hopf bifurcations, resulting in different coexisting branches of three-dimensional solutions. How the centrifugal and the gravitational buoyancies interact and compete, and the manner in which the flow becomes three-dimensional is different along each branch. The centrifugal buoyancy, even for relatively small Froude numbers, leads to quantitative and qualitative changes in the flow dynamics.

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 using realistic boundary conditions and the parameter values used in the experiments of Kolodner [Phys. Rev. E 47, 1038 (1993)]. Particular attention is paid to the multiplicity of states, and their dependence on the applied Rayleigh number and the aspect ratio of the container. Quantitative agreement with the experiments is obtained, and a mechanism explaining the origin and properties of the repeated transients observed in the experiments is proposed.

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A numerical study of the onset of thermal convection in a rotating circular cylinder of radius-to-depth ratio equal to four is considered in a regime dominated by the Coriolis force where the onset is to so-called wall modes. The wall modes consist of hot and cold pairs of thermal plumes rising and descending in the cylinder wall boundary layer, forming an essentially one-dimensional pattern characterized by the number of hot/cold plume pairs, m. In the limit of zero centrifugal force, this onset of convection at a critical temperature difference across the depth of the cylinder is via a symmetry-breaking supercritical Hopf bifurcation which leads to retrograde precession of the pattern with respect to the rotation of the cylinder. For temperature differences greater than critical, a number of distinct wall modes, distinguished by m, coexist and are stable. Their dynamics are controlled by an Eckhaus-Benjamin-Feir instability, the most basic features of which had been captured by a complex Ginzburg-Landau equation model. Here, we analyse this instability in rotating convection using direct numerical simulations of the Navier-Stokes equations in the Boussinesq approximation. Several properties of the wall modes are computed, extending the results to far beyond the onset of convection. Extensive favourable comparisons between our numerical results and previous experimental observations and complex Ginzburg-Landau model results are made.

Simulations of convection in 3He-4He mixtures with a negative separation ratio in two-dimensional containers with realistic boundary conditions and moderately large aspect ratio reveal, at supercritical Rayleigh numbers, the existence of "convectons," i.e., localized states of stationary convection, separated by regions of no convection. The origin and properties of these states are described.