Etats localisés dans les systèmes fluides : application à la double diffusion

Les états spatialement localisés sont des solutions physiques possédant une structure spatiale particulière en une région bien définie d'un domaine structuré différemment. Nous nous intéressons aux états spatialement localisés susceptibles de se former lorsqu'une convection d'origine thermique est couplée à une convection d'origine solutale ou induite par la rotation du système. Trois configurations physiques différentes sont abordées : la convection de double diffusion induite par des gradients thermiques et solutaux verticaux dans des couches fluides bidimensionnelles, celle induite par des gradients horizontaux dans des cavités tridimensionnelles et la convection de Rayleigh-Bénard en présence de rotation. Dans chacun des cas, des solutions spatialement localisées sont obtenues et analysées en utilisant la théorie des systèmes dynamiques. Les résultats obtenus dans ce travail révèlent différents scénarios d'un même mécanisme baptisé snaking, observé et analysé è l'aide d'équations modèles.Spatially localized states are physical solutions with a particular structure in a well-defined region in space that is embedded in a different background. We focus here on such states that are formed when thermal convection is coupled to solutal or Coriolis forcing. Three different physical configurations are studied: doubly diffusive convection with vertical gradients of temperature and concentration in two-dimensional fluid layers, doubly diffusive convection with horizontal gradients in three-dimensional fluid layers and Rayleigh-Bénard convection in the presence of rotation. In each of these cases, spatially localized solutions are computed and analyzed using dynamical systems theory. Our results reveal different variations of snaking, a mechanism observed and analyzed using model equations

États spatialement localisés dans la convection de double diffusion

Une méthode numérique de continuation est utilisée pour étudier les états stationnaires spatialement localisés dans la convection de double diffusion au sein d'une couche bi-dimensionnelle d'un fluide binaire confiné entre deux parois horizontales munies de conditions aux limites de non-glissement. La concentration du composant le plus lourd est maintenue supérieure à la paroi inférieure et la convection induite par une différence de température imposée entre le haut et le bas. Dans une certaine gamme de paramètres, des états spatialement localisés appelés convectons se forment. L'origine et les propriétés de ces états sont étudiées et reliées au phénomène d'homoclinic snaking

Homoclinic snaking of localized states in doubly diffusive convection

Numerical continuation is used to investigate stationary spatially localized states in two-dimensional thermosolutal convection in a plane horizontal layer with no-slip boundary conditions at top and bottom. Convectons in the form of 1-pulse and 2-pulse states of both odd and even parity exhibit homoclinic snaking in a common Rayleigh number regime. In contrast to similar states in binary fluid convection, odd parity convectons do not pump concentration horizontally. Stable but time-dependent localized structures are present for Rayleigh numbers below the snaking region for stationary convectons. The computations are carried out for (inverse) Lewis number \tau = 1/15 and Prandtl numbers Pr = 1 and Pr >> 1

Convectons and secondary snaking in three-dimensional natural doubly diffusive convection.

Natural doubly diffusive convection in a three-dimensional vertical enclosure with square cross-section in the horizontal is studied. Convection is driven by imposed temperature and concentration differences between two opposite vertical walls. These are chosen such that a pure conduction state exists. No-flux boundary conditions are imposed on the remaining four walls, with no-slip boundary conditions on all six walls. Numerical continuation is used to compute branches of spatially localized convection. Such states are referred to as convectons. Two branches of three-dimensional convectons with full symmetry bifurcate simultaneously from the conduction state and undergo homoclinic snaking. Secondary bifurcations on the primary snaking branches generate secondary snaking branches of convectons with reduced symmetry. The results are complemented with direct numerical simulations of the three-dimensional equations

Nonsnaking doubly diffusive convectons and the twist instability

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

Localized States in Periodically Forced Systems

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 are 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

Convectons in a rotating fluid layer.

Two-dimensional convection in a plane layer bounded by stress-free perfectly conducting horizontal boundaries and rotating uniformly about the vertical is considered. Time independent spatially localized structures, called convectons, of even and odd parity are computed. The convectons are embedded within a self-generated shear layer with a compensating shear flow outside the structure. These states are organized within a bifurcation structure called slanted snaking and may be present even when periodic convection sets in supercritically. These interesting properties are traced to the presence of a conserved quantity and hence to the use of stress-free boundary conditions

Spatial localization in heterogeneous systems

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

Dynamics of spatially localized states in transitional plane Couette flow

Unsteady spatially localized states such as puffs, slugs or spots play an important role in transition to turbulence. In plane Couette flow, steady versions of these states are found on two intertwined solution branches describing homoclinic snaking (Schneider et al., Phys. Rev. Lett., vol. 104, 2010, 104501). These branches can be used to generate a number of spatially localized initial conditions whose transition can be investigated. From the low Reynolds numbers where homoclinic snaking is first observed (Re < 175) to transitional ones (Re ≈ 325), these spatially localized states traverse various regimes where their relaminarization time and dynamics are affected by the dynamical structure of phase space. These regimes are reported and characterized in this paper for a 4π-periodic domain in the streamwise direction as a function of the two remaining variables: the Reynolds number and the width of the localized pattern. Close to the snaking, localized states are attracted by spatially localized periodic orbits before relaminarizing. At larger values of the Reynolds number, the flow enters a chaotic transient of variable duration before relaminarizing. Very long chaotic transients (t > 104) can be observed without difficulty for relatively low values of the Reynolds number (Re ≈ 250)

Localized rotating convection with no-slip boundary conditions

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