Pattern formation in large domains

Pattern formation is a phenomenon that arises in a wide variety of physical, chemical and biological situations. A great deal of theoretical progress has been made in understanding the universal aspects of pattern formation in terms of amplitudes of the modes that make up the pattern. Much of the theory has sound mathematical justification, but experiments and numerical simulations over the last decade have revealed complex two-dimensional patterns that do not have a satisfactory theoretical explanation. This paper focuses on quasi-patterns, where the appearance of small divisors causes the standard theoretical method to fail, and ends with a discussion of other outstanding problems in the theory of two-dimensional pattern formation in large domains

Secondary instabilities of hexagons: a bifurcation analysis of experimentally observed Faraday wave patterns

We examine three experimental observations of Faraday waves generated by two-frequency forcing, in which a primary hexagonal pattern becomes unstable to three different superlattice patterns. We use the symmetry-based approach developed by Tse et al. to analyse the bifurcations involved in creating the three new patterns. Each of the three examples reveals a different situation that can arise in the theoretical analysis.Comment: 14 pages LaTeX, Birkhauser style, 5 figures, submitted to the proceedings of the conference on Bifurcations, Symmetry and Patterns, held in Porto, June 200

Numerical simulations of sunspots

The origin, structure and evolution of sunspots are investigated using a numerical model. The compressible MHD equations are solved with physical parameter values that approximate the top layer of the solar convection zone. A three dimensional (3D) numerical code is used to solve the set of equations in cylindrical geometry, with the numerical domain in the form of a wedge. The linear evolution of the 3D solution is studied by perturbing an axisymmetric solution in the azimuthal direction. Steady and oscillating linear modes are obtained

The effect of symmetry breaking on the dynamics near a structurally stable heteroclinic cycle between equilibria and a periodic orbit

The effect of small forced symmetry breaking on the dynamics near a structurally stable heteroclinic cycle connecting two equilibria and a periodic orbit is investigated. This type of system is known to exhibit complicated, possibly chaotic dynamics including irregular switching of sign of various phase space variables, but details of the mechanisms underlying the complicated dynamics have not previously been investigated. We identify global bifurcations that induce the onset of chaotic dynamics and switching near a heteroclinic cycle of this type, and by construction and analysis of approximate return maps, locate the global bifurcations in parameter space. We find there is a threshold in the size of certain symmetry-breaking terms below which there can be no persistent switching. Our results are illustrated by a numerical example

Bifurcations of periodic orbits with spatio-temporal symmetries

Motivated by recent analytical and numerical work on two- and three-dimensional convection with imposed spatial periodicity, we analyse three examples of bifurcations from a continuous group orbit of spatio-temporally symmetric periodic solutions of partial differential equations. Our approach is based on centre manifold reduction for maps, and is in the spirit of earlier work by Iooss (1986) on bifurcations of group orbits of spatially symmetric equilibria. Two examples, two-dimensional pulsating waves (PW) and three-dimensional alternating pulsating waves (APW), have discrete spatio-temporal symmetries characterized by the cyclic groups Z_n, n=2 (PW) and n=4 (APW). These symmetries force the Poincare' return map M to be the nth iterate of a map G: M=G^n. The group orbits of PW and APW are generated by translations in the horizontal directions and correspond to a circle and a two-torus, respectively. An instability of pulsating waves can lead to solutions that drift along the group orbit, while bifurcations with Floquet multiplier +1 of alternating pulsating waves do not lead to drifting solutions. The third example we consider, alternating rolls, has the spatio-temporal symmetry of alternating pulsating waves as well as being invariant under reflections in two vertical planes. This leads to the possibility of a doubling of the marginal Floquet multiplier and of bifurcation to two distinct types of drifting solutions. We conclude by proposing a systematic way of analysing steady-state bifurcations of periodic orbits with discrete spatio-temporal symmetries, based on applying the equivariant branching lemma to the irreducible representations of the spatio-temporal symmetry group of the periodic orbit, and on the normal form results of Lamb (1996). This general approach is relevant to other pattern formation problems, and contributes to our understanding of the transition from ordered to disordered behaviour in pattern-forming systems

Towards Convectons in the Supercritical Regime: Homoclinic Snaking 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. A variety of patterns can arise from these buoyancy-driven flows, including spatially localised states known as convectons, which consist of convective fluid motion localised within a background of quiescent fluid. We consider these states in a vertical slot with the horizontal temperature and solutal gradients providing competing effects to the fluid density while allowing the existence of a conduction state. In this configuration, convectons have been studied with specific parameter values where the onset of convection is subcritical, and the states have been found to lie on a pair of secondary branches that undergo homoclinic snaking in a parameter regime below the onset of linear instability. In this paper, we show that convectons persist into parameter regimes in which the primary bifurcation is supercritical and there is no bistability, despite coexistence between the stable conduction state and large-amplitude convection. We detail this transition by considering spatial dynamics and observe how the structure of the secondary branches becomes increasingly complex owing to the increased role of inertia at low Prandtl numbers

Tertiary peralkaline rhyolite dikes from the Skærgaard area, Kangerdlugssuaq, East Greenland.

Tertiary peralkaline acid rocks (comendites) are described from East Greenland. These are dike rocks, which may be aphyric, carry sparse phenocrysts of alkali feldspar, or more abundant quartz, alkali feldspar and katophorite phenocrysts. Regarded as being closely related, are a group of peralkaline granitic rocks associated with the nordmarkitic syenites of the area, which contain quartz, alkali feldspars, arfvedsonite, pyroxenes ranging from hedenbergitic to acmitic, and assessories including zircon, Mn-rich ilmenite, astrophyllite and chevkinite. Analyses are presented for most of these phases and the chemistry of the rocks and that of the crystallization process in the dike centres is discussed. These rocks have an origin clearly distinct from that of the nearby Skaergaard granophyres, which show no peralkaline tendencies, and it is postulated that the comendites have arisen from magmas of nordmarkitic composition which, in turn, arose at depth from a transitional basalt. This suite of rocks seems to reflect closely the tectonic setting, occurring in many areas of crustal attenuation, a situation in accordance with that believed to be present here.Tertiary peralkaline acid rocks (comendites) are described from East Greenland. These are dike rocks, which may be aphyric, carry sparse phenocrysts of alkali feldspar, or more abundant quartz, alkali feldspar and katophorite phenocrysts. Regarded as being closely related, are a group of peralkaline granitic rocks associated with the nordmarkitic syenites of the area, which contain quartz, alkali feldspars, arfvedsonite, pyroxenes ranging from hedenbergitic to acmitic, and assessories including zircon, Mn-rich ilmenite, astrophyllite and chevkinite. Analyses are presented for most of these phases and the chemistry of the rocks and that of the crystallization process in the dike centres is discussed. These rocks have an origin clearly distinct from that of the nearby Skaergaard granophyres, which show no peralkaline tendencies, and it is postulated that the comendites have arisen from magmas of nordmarkitic composition which, in turn, arose at depth from a transitional basalt. This suite of rocks seems to reflect closely the tectonic setting, occurring in many areas of crustal attenuation, a situation in accordance with that believed to be present here.Tertiary peralkaline acid rocks (comendites) are described from East Greenland. These are dike rocks, which may be aphyric, carry sparse phenocrysts of alkali feldspar, or more abundant quartz, alkali feldspar and katophorite phenocrysts. Regarded as being closely related, are a group of peralkaline granitic rocks associated with the nordmarkitic syenites of the area, which contain quartz, alkali feldspars, arfvedsonite, pyroxenes ranging from hedenbergitic to acmitic, and assessories including zircon, Mn-rich ilmenite, astrophyllite and chevkinite. Analyses are presented for most of these phases and the chemistry of the rocks and that of the crystallization process in the dike centres is discussed. These rocks have an origin clearly distinct from that of the nearby Skaergaard granophyres, which show no peralkaline tendencies, and it is postulated that the comendites have arisen from magmas of nordmarkitic composition which, in turn, arose at depth from a transitional basalt. This suite of rocks seems to reflect closely the tectonic setting, occurring in many areas of crustal attenuation, a situation in accordance with that believed to be present here

Analysis of the shearing instability in nonlinear convection and magnetoconvection

Numerical experiments on two-dimensional convection with or without a vertical magnetic field reveal a bewildering variety of periodic and aperiodic oscillations. Steady rolls can develop a shearing instability, in which rolls turning over in one direction grow at the expense of rolls turning over in the other, resulting in a net shear across the layer. As the temperature difference across the fluid is increased, two-dimensional pulsating waves occur, in which the direction of shear alternates. We analyse the nonlinear dynamics of this behaviour by first constructing appropriate low-order sets of ordinary differential equations, which show the same behaviour, and then analysing the global bifurcations that lead to these oscillations by constructing one-dimensional return maps. We compare the behaviour of the partial differential equations, the models and the maps in systematic two-parameter studies of both the magnetic and the non-magnetic cases, emphasising how the symmetries of periodic solutions change as a result of global bifurcations. Much of the interesting behaviour is associated with a discontinuous change in the leading direction of a fixed point at a global bifurcation; this change occurs when the magnetic field is introduced

Near-onset dynamics in natural doubly diffusive convection

Doubly diffusive convection is considered in a vertical slot where horizontal temperature and solutal variations provide competing effects to the fluid density while allowing the existence of a conduction state. In this configuration, the linear stability of the conductive state is known, but the convection patterns arising from the primary instability have only been studied for specific parameter values. We have extended this by determining the nature of the primary bifurcation for all values of the Lewis and Prandtl numbers using a weakly nonlinear analysis. The resulting convection branches are extended using numerical continuation and we find large-amplitude steady convection states can coexist with the stable conduction state for sub- and supercritical primary bifurcations. The stability of the convection states is investigated and attracting travelling waves and periodic orbits are identified using time-stepping when these steady states are unstable