Localised states in natural doubly diffusive convection

Fluids subject to both thermal and compositional variations can undergo doubly diffusive convection when these properties both affect the fluid density and diffuse at different rates. This phenomenon can lead to the formation of a variety of patterns, including salt fingers and thermohaline staircases, which have been identified throughout the world’s oceans. In this thesis, we consider natural doubly diffusive convection driven by opposing thermal and solutal gradients in the horizontal direction and aim to determine how states in this system are affected by the physical parameters that characterise the strength of the thermal gradients, the balance between thermal and solutal gradients, and ratios between thermal, solutal and viscous diffusivities. In the particular case when the imposed thermal and solutal gradients balance, a motionless conduction state exists but destabilises when the gradients are sufficiently large. We determine the nature of the associated primary bifurcation using a weakly nonlinear analysis and extend the resulting primary convection branches using numerical continuation to find that large-amplitude steady convection states can coexist with the stable conduction state for both sub- and supercritical bifurcations. We proceed by considering vertically extended domains where spatially localised states, known as convectons, have been found to lie on a pair of secondary branches that intertwine when the onset of convection is subcritical. This process is known as homoclinic snaking and is usually associated with bistability. Here, we show that convectons persist into parameter regimes where the primary bifurcation is supercritical and there is no bistability. We finally consider how the system changes when the imposed thermal and solutal gradients do not balance and the motionless conduction state does not exist. We focus on how the form of convectons change with increasing imbalance and how these localised states cease to exist in sufficiently thermally dominated flows

Complex Patterns in Extended Oscillatory Systems

Ausgedehnte dissipative Systeme können fernab vom thermodynamischen Gleichgewicht instabil gegenüber Oszillationen bzw. Wellen oder raumzeitlichem Chaos werden. Die komplexe Ginzburg-Landau Gleichung (CGLE) stellt ein universelles Modell zur Beschreibung dieser raumzeitlichen Strukturen dar. Diese Arbeit ist der theoretischen Analyse komplexer Muster gewidmet. Mittels numerischer Bifurkations- und Stabilitätsanalyse werden Instabilitäten einfacher Muster identifiziert und neuartige Lösungen der CGLE bestimmt. Modulierte Amplitudenwellen (MAW) und Super-Spiralwellen sind Beispiele solcher komplexer Muster. MAWs können in hydrodynamischen Experimenten und Super-Spiralwellen in der Belousov-Zhabotinsky-Reaktion beobachtet werden. Der Grenzübergang von Phasen- zu Defektchaos wird durch den Existenzbereich der MAWs erklärt. Mittels der selben numerischen Methoden wird Bursting vom Fold-Hopf-Typ in einem Modell der Kalziumsignalübertragung in Zellen identifiziert

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

Parametric Forcing of Confined and Stratified Flows

abstract: A continuously and stably stratified fluid contained in a square cavity subjected to harmonic body forcing is studied numerically by solving the Navier-Stokes equations under the Boussinesq approximation. Complex dynamics are observed near the onset of instability of the basic state, which is a flow configuration that is always an exact analytical solution of the governing equations. The instability of the basic state to perturbations is first studied with linear stability analysis (Floquet analysis), revealing a multitude of intersecting synchronous and subharmonic resonance tongues in parameter space. A modal reduction method for determining the locus of basic state instability is also shown, greatly simplifying the computational overhead normally required by a Floquet study. Then, a study of the nonlinear governing equations determines the criticality of the basic state's instability, and ultimately characterizes the dynamics of the lowest order spatial mode by the three discovered codimension-two bifurcation points within the resonance tongue. The rich dynamics include a homoclinic doubling cascade that resembles the logistic map and a multitude of gluing bifurcations. The numerical techniques and methodologies are first demonstrated on a homogeneous fluid contained within a three-dimensional lid-driven cavity. The edge state technique and linear stability analysis through Arnoldi iteration are used to resolve the complex dynamics of the canonical shear-driven benchmark problem. The techniques here lead to a dynamical description of an instability mechanism, and the work serves as a basis for the remainder of the dissertation.Dissertation/ThesisSupplemental Materials Description Filezip file containing 10 mp4 formatted video animations, as well as a text readme and the previously submitted Supplemental Materials Description FileDoctoral Dissertation Mathematics 201

Introductory Lectures on Turbulence: Physics, Mathematics and Modeling

From Chapter 1: The understanding of turbulent behavior in flowing fluids is one of the most intriguing, frustrating— and important—problems in all of classical physics. The problem of turbulence has been studied by many of the greatest physicists and engineers of the 19th and 20th Centuries, and yet we do not understand in complete detail how or why turbulence occurs, nor can we predict turbulent behavior with any degree of reliability, even in very simple (from an engineering perspective) flow situations. Thus, study of turbulence is motivated both by its inherent intellectual challenge and by the practical utility of a thorough understanding of its nature.https://uknowledge.uky.edu/me_textbooks/1001/thumbnail.jp

1991 Summer Study Program in Geophysical Fluid Dynamics : patterns in fluid flow

The GFD program in 1991 focused on pattern forming processes in physics and geophysics. The pricipallecturer, Stephan Fauve, discussed a variety of systems, including our old favorite, Rayleigh-Bénard convection, but passing on to exotic examples such as vertically vibrated granular layers. Fauve's lectures emphasize a unified theoretical viewpoint based on symmetry arguments. Patterns produced by instabilties can be described by amplitude equations, whose form can be deduced by symmetry arguments, rather than the asymptotic expansions that have been the staple of past Summer GFD Programs. The amplitude equations are far simpler than the complete equations of motion, and symetry arguments are easier than asymptotic expansions. Symmetry arguments also explain why diverse systems are often described by the same amplitude equation. Even for granular layers, where there is not a universaly accepted continuum description, the appropnate amplitude equation can often be found using symmetry arguments and then compared with experiment. Our second speaker, Daniel Rothan, surveyed the state of the art in lattice gas computations. His lectures illustrate the great utility of these methods in simulating the flow of complex multiphase fluids, particularly at low Reynolds numbers. The lattice gas simulations reveal a complicated phenomenology much of which awaits analytic exploration. The fellowship lectures cover broad ground and reflect the interests of the staff members associated with the program. They range from the formation of sand dunes, though the theory of lattice gases, and on to two dimensional-turbulence and convection on planetary scales. Readers desiring to quote from these report should seek the permission of the authors (a partial list of electronic mail addresses is included on page v). As in previous years, these reports are extensively reworked for publication or appear as chapters in doctoral theses. The task of assembling the volume in 1991 was at first faciltated by our newly acquired computers, only to be complicated by hurricane Bob which severed electric power to Walsh Cottage in the final hectic days of the Summer.Funding was provided by the National Science Foundation through Grant No. OCE 8901012