Convection near threshold for Prandtl numbers near 1

Local properties of patterned vegetation: quantifying endogenous and exogenous effects

Dryland ecosystems commonly exhibit periodic bands of vegetation, thought to form due to competition between individual plants for heterogeneously distributed water. In this paper, we develop a Fourier method for locally identifying the pattern wavenumber and orientation, and apply it to aerial images from a region of vegetation patterning near Fort Stockton, TX, USA. We find that the local pattern wavelength and orientation are typically coherent, but exhibit both rapid and gradual variation driven by changes in hillslope gradient and orientation, the potential for water accumulation, or soil type. Endogenous pattern dynamics, when simulated for spatially homogeneous topographic and vegetation conditions, predict pattern properties that are much less variable than the orientation and wavelength observed in natural systems. Our local pattern analysis, combined with ancillary datasets describing soil and topographic variation, highlights a largely unexplored correlation between soil depth, pattern coherence, vegetation cover and pattern wavelength. It also, surprisingly, suggests that downslope accumulation of water may play a role in changing vegetation pattern properties.

Domain walls and vortices in linearly coupled systems

We investigate one- and two-dimensional radial domain-wall (DW) states in the system of two nonlinear-Schrödinger (NLS) or Gross-Pitaevskii (GP) equations, which are couple by linear mixing and by nonlinear XPM (cross-phase-modulation). The system has straightforward applications to two-component Bose-Einstein condensates, and to bimodal light propagation in nonlinear optics. In the former case the two components represent different hyperfine atomic states, while in the latter setting they correspond to orthogonal polarizations of light. Conditions guaranteeing the stability of flat continuous wave (CW) asymmetric bimodal states are established, followed by the study of families of the corresponding DW patterns. Approximate analytical solutions for the DWs are found near the point of the symmetry-breaking bifurcation of the CW states. An exact DW solution is produced for ratio 3:1 of the XPM and SPM (self-phase modulation) coefficients. The DWs between flat asymmetric states, which are mirror images of each other, are completely stable, and all other species of the DWs, with zero crossings in one or two components, are fully unstable. Interactions between two DWs are considered too, and an effective potential accounting for the attraction between them is derived analytically. Direct simulations demonstrate merger and annihilation of the interacting DWs. The analysis is extended for the system including single- and double-peak external potentials. Generic solutions for trapped DWs are obtained in a numerical form, and their stability is investigated. An exact stable solution is found for the DW trapped by a single-peak potential. In the 2D geometry, stable two-component vortices are found, with topological charges s=1,2,3. Radial oscillations of annular DW-shaped pulsons, with s=0,1,2, are studied too. A linear relation between the period of the oscillations and the mean radius of the DW ring is derived analytically.

Extensive scaling from computational homology and Karhunen-Loève decomposition analysis of Rayleigh-Bénard convection experiments

Spatiotemporally chaotic dynamics in laboratory experiments on convection are characterized using a new dimension, D(CH), determined from computational homology. Over a large range of system sizes, D(CH) scales in the same manner as D(KLD), a dimension determined from experimental data using Karhuenen-Loéve decomposition. Moreover, finite-size effects (the presence of boundaries in the experiment) lead to deviations from scaling that are similar for both D(CH) and D(KLD). In the absence of symmetry, D(CH) can be determined more rapidly than D(KLD).

Stripe domain coarsening in geographical small-world networks on a Euclidean lattice

We study phase ordering dynamics of spatially periodic striped patterns on the small-world network that is derived from a two-dimensional regular lattice with distance-dependent random connections. It is demonstrated numerically that addition of spatial disorder in the form of shortcuts makes the growth of domains much slower or even frozen at late times.

Surface-element analysis of spatiotemporal stripe patterns formed by Ag and Sb coelectrodeposition

Various spatiotemporal patterns of light and dark stripes are formed in the Ag and Sb coelectrodeposition system. In this research, we report the results of the element analysis of three spatiotemporal stripe patterns using an electron probe (x-ray) microanalyzer. The results indicate that all the patterns have an O distribution reflecting the color configuration of the patterns, in addition to Ag and Sb distributions, which have been advocated as forming the patterns. It is suggested that O, as well as Ag and Sb, contributes to the formation of the patterns.

Wave-number selection by target patterns and sidewalls in Rayleigh-Bénard convection

We present experimental results for patterns of Rayleigh-Bénard convection in a cylindrical container with static sidewall forcing. The fluid used was methanol, with a Prandlt number sigma=7.17 , and the aspect ratio was Gamma identical withR/d approximately 19 ( R is the radius and d the thickness of the fluid layer). In the presence of a small heat input along the sidewall, a sudden jump of the temperature difference DeltaT from below to slightly above a critical value Delta T(c) produced a stable pattern of concentric rolls (a target pattern) with the central roll (the umbilicus) at the center of the cell. A quasistatic increase of epsilon identical withDeltaT/Delta T(c) -1 beyond epsilon(1,c) approximately 0.8 caused the umbilicus of the pattern to move off center. As observed by others, a further quasistatic increase of epsilon up to epsilon=15.6 caused a sequence of transitions at epsilon(i,b) ,i=1,...,8 , each associated with the loss of one convection roll at the umbilicus. Each loss of a roll was preceded by the displacement of the umbilicus away from the center of the cell. After each transition the umbilicus moved back toward but never quite reached the center. With decreasing epsilon new rolls formed at the umbilicus when epsilon was reduced below epsilon(i,a) < epsilon(i,b) . When decreasing epsilon , large umbilicus displacements did not occur. In addition to quantitative measurements of the umbilicus displacement, we determined and analyzed the entire wave-director field of each image. The wave numbers varied in the axial direction, with minima at the umbilicus and at the cell wall and a maximum at a radial position close to 2Gamma/3 . The wave numbers at the maximum showed hysteretic jumps at epsilon(i,b) and epsilon(i,a) , but on average agreed well with the theoretical predictions for the wave numbers selected in the far field of an infinitely extended target pattern. To our knowledge there is as yet no prediction for the wave number selected by the umbilicus itself, or by the cell wall of the finite experimental system.

Rayleigh-Bénard convection with a radial ramp in plate separation

Pattern formation in Rayleigh-Bénard convection in a large-aspect-ratio cylinder with a radial ramp in the plate separation is studied analytically and numerically by performing numerical simulations of the Boussinesq equations. A horizontal mean flow and a vertical large scale counterflow are quantified and used to understand the pattern wave number. Our results suggest that the mean flow, generated by amplitude gradients, plays an important role in the roll compression observed as the control parameter is increased. Near threshold, the mean flow has a quadrupole dependence with a single vortex in each quadrant while away from threshold the mean flow exhibits an octupole dependence with a counterrotating pair of vortices in each quadrant. This is confirmed analytically using the amplitude equation and Cross-Newell mean flow equation. By performing numerical experiments, the large scale counterflow is also found to aid in the roll compression away from threshold but to a much lesser degree. Our results yield an understanding of the pattern wave numbers observed in experiment away from threshold and suggest that near threshold the mean flow and large scale counterflow are not responsible for the observed shift to smaller than critical wave numbers.

Rayleigh-Bénard convection in elliptic and stadium-shaped containers

We report on defect formation in convection patterns of stadium-shaped and elliptical horizontal layers of fluid heated from below (Rayleigh-Bénard convection). The fluid was ethanol with a Prandtl number sigma=14.2. The outermost convection roll was forced to be parallel to the sidewall by a supplementary wall heater. The major- and minor-axis aspect ratios Gamma(i)=D(i)/2d, i=1, 2 (D(i) are the major and minor diameter and d the thickness) were 19.4 and 13.0, respectively. For the stadium shape, we found a stable pattern that was reflection-symmetric about the major diameter and had a downflow roll of length L(s) along a large part of this diameter. This roll terminated in two convex disclinations, as expected from theory. No other patterns with the outermost roll parallel to the sidewall were found. The wave numbers of the rolls in the curved sections and L(s) decreased with increasing epsilon identical with DeltaT/DeltaT(c)-1, consistent with a prediction for wave-number selection by curved rolls in an infinite system. At large epsilon, the roll adjacent to the sidewall became unstable due to the cross-roll instability. For the elliptical shape, wave-director frustration yielded a new defect structure predicted by Ercolani et al. Depending on the sample history, three different patterns with the outermost roll parallel to the wall were found. For one, the central downflow roll seen in the stadium was shortened to the point where it resembled a single convection cell. Along much of the major diameter there existed an upflow roll. The new defect structure occurred where the two downflow rolls surrounding the central upflow roll joined. This joint, instead of being smooth as in the stadium case, was angular and created a protuberance pointing outward along the major diameter. We also found a pattern with an upflow roll along the major diameter without the central downflow cell. A third pattern contained a downflow cell, but this cell was displaced by a roll width from the center along a minor diameter. As epsilon increased, the length L(e) between the two protuberances and the wave numbers along the outer parts of the major diameter decreased for all three patterns, analogous to what was found for the stadium. The upper stability limit of these patterns was also set by the cross-roll instability.

Dynamics of pattern coarsening in a two-dimensional smectic system

We have followed the coarsening dynamics of a single layer of cylindrical block copolymer microdomains in a thin film. This system has the symmetry of a two-dimensional smectic. The orientational correlation length of the microdomains was measured by scanning electron microscopy and found to grow with the average spacing between +/-1/2 disclinations, following a power law xi2(t) approximately t(1/4). By tracking disclinations during annealing with time-lapse atomic force microscopy, we observe dominant mechanisms of disclination annihilation involving tripoles and quadrupoles (three and four disclinations, respectively). We describe how annihilation events involving multiple disclinations result in similarly reduced kinetic exponents as observed here. These results map onto a wide variety of physical systems that exhibit similarly striped patterns.

Rotating convection in an anisotropic system

We study the stability of patterns arising in rotating convection in weakly anisotropic systems using a modified Swift-Hohenberg equation. The anisotropy, either an endogenous characteristic of the system or induced by external forcing, can stabilize periodic rolls in the Küppers-Lortz chaotic regime. We apply this to the particular case of rotating convection with time-modulated rotation where recently, in experiment, spiral and target patterns have been observed in otherwise Küppers-Lortz-unstable regimes. We show how the underlying base flow breaks the isotropy, thereby affecting the linear growth rate of convection rolls in such a way as to stabilize spirals and targets. Throughout we compare analytical results to numerical simulations of the Swift-Hohenberg equation.

Traveling concentric-roll patterns in Rayleigh-Bénard convection with modulated rotation

We present experimental results for pattern formation in Rayleigh-Bénard convection with modulated rotation about a vertical axis. The dimensionless rotation rate Omega was varied as Omega(m)=Omega[1+delta cos(phi Omega t)] (time is scaled by the vertical viscous diffusion time of the cell). We used a cylindrical cell of aspect ratio (radius/height) Gamma=11.8 and varied Omega, delta, phi, and epsilon identical with R/R(c)(Omega)-1 (R is the Rayleigh number). The fluid was water with a Prandtl number of 4.5. Sufficiently far above onset even a small delta greater than approximately 0.02 stabilized a concentric-roll (target) pattern. Multiarmed spirals were observed close to onset. The rolls of the target patterns traveled radially inward independent of the sense of rotation. The radial speed v was nearly independent of epsilon for fixed Omega, delta, and phi. However, v increased with any one of Omega, delta, and phi when all the other parameters were held fixed.

Spiral patterns in oscillated granular layers

Cell-filling spiral patterns are observed in a vertically oscillated layer of granular material when the oscillation amplitude is suddenly increased from below the onset of pattern formation into the region where stripe patterns appear for quasistatic increases in amplitude. These spirals are transients and decay to stripe patterns with defects. A transient spiral defect chaos state is also observed. We describe the behavior of the spirals, and the way in which they form and decay. Our results are compared with those for similar spiral patterns in Rayleigh-Bénard convection in fluids.

Pattern formation in rayleigh-Benard convection in a cylindrical container

We report on numerical investigations of pattern formation in the classical Rayleigh-Benard convection with cylindrical geometry in the regime of low Prandtl numbers and moderate aspect ratio. Beyond the onset of convection, we found straight and bent rolls as stable patterns. By increasing the Rayleigh number, we studied the generation of defects, their dynamics in the form of climbing and gliding, the existence of stable targets and spirals as well as the occurrence of core instabilities, a variety of pattern types that were also observed in experiments.

Excitation of Spirals and Chiral Symmetry Breaking in Rayleigh-Bénard Convection

Spiral-defect populations in low-Prandtl number Rayleigh-Bénard convection with slow rotation about a vertical axis were measured in carbon dioxide at high pressure. The results indicate that spirals act like "thermally excited" defects and that the winding direction of a spiral is analogous to a magnetic spin. Rotation about a vertical axis, the spiral analog of the magnetic field, breaks the zero-rotation chiral symmetry between clockwise and counterclockwise spiral defects. Many properties of spiral-defect statistics are well described by an effective statistical-mechanical model.