Correlations between the leading Lyapunov vector and pattern defects for chaotic Rayleigh-Bénard convection.

We probe the effectiveness of using topological defects to characterize the leading Lyapunov vector for a high-dimensional chaotic convective flow field. This is accomplished using large-scale parallel numerical simulations of Rayleigh-Bénard convection for experimentally accessible conditions. We quantify the statistical correlations between the spatiotemporal dynamics of the leading Lyapunov vector and different measures of the flow field pattern's topology and dynamics. We use a range of pattern diagnostics to describe the flow field structures which includes many of the traditional diagnostics used to describe convection as well as some diagnostics tailored to capture the dynamics of the patterns. We use the ideas of precision and recall to build a statistical description of each pattern diagnostic's ability to describe the spatial variation of the leading Lyapunov vector. The precision of a diagnostic indicates the probability that it will locate a region where the Lyapunov vector is larger than a threshold value. The recall of a diagnostic indicates its ability to locate all of the possible spatial regions where the Lyapunov vector is above threshold. By varying the threshold used for the Lyapunov vector magnitude, we generate precision-recall curves which we use to quantify the complex relationship between the pattern diagnostics and the spatiotemporally varying magnitude of the leading Lyapunov vector. We find that pattern diagnostics which include information regarding the flow history outperform pattern diagnostics that do not. In particular, an emerging target defect has the highest precision of all of the pattern diagnostics we have explored.

Selection of superior lentil (Lens esculenta M.) genotypes by assessing character association and genetic diversity.

Lentil is one of the most important pulse crops in the world as well as in Bangladesh. It is now considered a main component for training and body building practising in first world countries. Yield varies tremendously from year to year and location to location. Therefore, it is very important to find genotypes that perform consistently well even in ecological farming systems without any intercultural operations. Twenty lentil genotypes were tested during the period from November 2010 to March 2011 and from December 2011 to March 2012 with three replicates in each season to determine genetic variability, diversity, characters association, and selection indices for better grain yield. The experiment was conducted at the breeding field of the Department of Genetics and Plant Breeding, Bangladesh Agricultural University, Mymensingh. This study revealed that all the genotypes possess a high amount of genetic diversity. Plant height and 100-grain weight showed significant positive correlation with grain yield plant(-1) that was also confirmed by path analysis as the highest direct effect on grain yield. The genotypes BM-513 and BM-941 were found to be the best performer in both the seasons and were considered as consistent genotype. The genotypes were grouped into four clusters based on Euclidean distance following Ward's method and RAPD analysis. However, discriminant function analysis revealed a progressive increase in the efficiency of selection and BM-70 ranked as the best followed by the genotypes BM-739, BM-680, BM-185, and BM-513. These genotypes might be recommended for farmers' cultivation in ecological farming in Bangladesh.

Using covariant Lyapunov vectors to quantify high-dimensional chaos with a conservation law.

We explore the high-dimensional chaos of a one-dimensional lattice of diffusively coupled tent maps using the covariant Lyapunov vectors (CLVs). We investigate the connection between the dynamics of the maps in the physical space and the dynamics of the covariant Lyapunov vectors and covariant Lyapunov exponents that describe the direction and growth (or decay) of small perturbations in the tangent space. We explore the tangent space splitting into physical and transient modes and find that the splitting persists for all of the conditions we explore. In general, the leading CLVs are highly localized in space and the CLVs become less localized with increasing Lyapunov index. We consider the dynamics with a conservation law whose strength is controlled by a parameter that can be continuously varied. Our results indicate that a conservation law delocalizes the spatial variation of the CLVs. We find that when a conservation law is present, the leading CLVs are entangled with fewer of their neighboring CLVs than in the absence of a conservation law.

Extensive chaos in the Lorenz-96 model.

We explore the high-dimensional chaotic dynamics of the Lorenz-96 model by computing the variation of the fractal dimension with system parameters. The Lorenz-96 model is a continuous in time and discrete in space model first proposed by Lorenz to study fundamental issues regarding the forecasting of spatially extended chaotic systems such as the atmosphere. First, we explore the spatiotemporal chaos limit by increasing the system size while holding the magnitude of the external forcing constant. Second, we explore the strong driving limit by increasing the external forcing while holding the system size fixed. As the system size is increased for small values of the forcing we find dynamical states that alternate between periodic and chaotic dynamics. The windows of chaos are extensive, on average, with relative deviations from extensivity on the order of 20%. For intermediate values of the forcing we find chaotic dynamics for all system sizes past a critical value. The fractal dimension exhibits a maximum deviation from extensivity on the order of 5% for small changes in system size and the deviation from extensivity decreases nonmonotonically with increasing system size. The length scale describing the deviations from extensivity is consistent with the natural chaotic length scale in support of the suggestion that deviations from extensivity are due to the addition of chaotic degrees of freedom as the system size is increased. We find that each wavelength of the deviation from extensive chaos contains on the order of two chaotic degrees of freedom. As the forcing is increased, at constant system size, the dimension density grows monotonically and saturates at a value less than unity. We use this to quantify the decreasing size of chaotic degrees of freedom with increased forcing which we compare with spatial features of the patterns.

Propagating fronts in fluids with solutal feedback.

We numerically study the propagation of reacting fronts in a shallow and horizontal layer of fluid with solutal feedback and in the presence of a thermally driven flow field of counterrotating convection rolls. We solve the Boussinesq equations along with a reaction-convection-diffusion equation for the concentration field where the products of the nonlinear autocatalytic reaction are less dense than the reactants. For small values of the solutal Rayleigh number the characteristic fluid velocity scales linearly, and the front velocity and mixing length scale quadratically, with increasing solutal Rayleigh number. For small solutal Rayleigh numbers the front geometry is described by a curve that is nearly antisymmetric about the horizontal midplane. For large values of the solutal Rayleigh number the characteristic fluid velocity, the front velocity, and the mixing length exhibit square-root scaling and the front shape collapses onto an asymmetric self-similar curve. In the presence of counterrotating convection rolls, the mixing length decreases while the front velocity increases. The complexity of the front geometry increases when both the solutal and convective contributions are significant and the dynamics can exhibit chemical oscillations in time for certain parameter values. Last, we discuss the spatiotemporal features of the complex fronts that form over a range of solutal and thermal driving.

Stochastic dynamics of micron-scale doubly clamped beams in a viscous fluid.

We study the stochastic dynamics of doubly clamped micron-scale beams in a viscous fluid driven by Brownian motion. We use a thermodynamic approach to compute the equilibrium fluctuations in beam displacement that requires only deterministic calculations. From calculations of the autocorrelations and noise spectra we quantify the beam dynamics by the quality factor and resonant frequency of the fundamental flexural mode over a wide range of experimentally accessible geometries. We consider beams with uniform rectangular cross section and explore the increased quality factor and resonant frequency as a baseline geometry is varied by increasing the width, increasing the thickness, and decreasing the length. The quality factor is nearly doubled by tripling either the width or the height of the beam. Much larger improvements are found by decreasing the beam length, however this is limited by the appearance of additional modes of fluid dissipation. Overall, the stochastic dynamics of the wider and thicker beams are well predicted by a two-dimensional approximate theory beyond what may be expected based upon the underlying assumptions, whereas the shorter beams require a more detailed analysis.

Velocity and geometry of propagating fronts in complex convective flow fields.

We numerically study the propagation of reacting fronts through three-dimensional flow fields composed of convection rolls that include time-independent cellular flow, spatiotemporally chaotic flow, and weakly turbulent flow. We quantify the asymptotic front velocity and determine its scaling with system parameters including the local angle of the convection rolls relative to the direction of front propagation. For cellular flow fields, the orientation of the convection rolls has a significant effect upon the front velocity and the front geometry remains relatively smooth. However, for chaotic and weakly turbulent flow fields, the front velocity depends upon the geometric complexity of the wrinkled front interface and does not depend significantly upon the local orientation of the convection rolls. Using the box counting dimension we find that the front interface is fractal for chaotic and weakly turbulent flows with a dimension that increases with flow complexity.

Erratum: Exploring spiral defect chaos in generalized Swift-Hohenberg models with mean flow [Phys. Rev. E 84, 046215 (2011)].

This corrects the article DOI: 10.1103/PhysRevE.84.046215.

Chaotic Rayleigh-Bénard convection with finite sidewalls.

We explore the role of finite sidewalls on chaotic Rayleigh-Bénard convection. We use large-scale parallel spectral-element numerical simulations for the precise conditions of experiment for cylindrical convection domains. We solve the Boussinesq equations for thermal convection and the conjugate heat transfer problem for the energy transfer at the solid sidewalls of the cylindrical domain. The solid sidewall of the convection domain has finite values of thickness, thermal conductivity, and thermal diffusivity. We compute the Lyapunov vectors and exponents for the entire fluid-solid coupled problem. We quantify the chaotic dynamics of convection over a range of thermal sidewall boundary conditions. We find that the dynamics become less chaotic as the thermal conductivity of the sidewalls increases as measured by the value of the fractal dimension of the dynamics. The thermal conductivity of the sidewall is a stabilizing influence; the heat transfer between the fluid and solid regions is always in the direction to reduce the fluid motion near the sidewalls. Although the heat interaction for strongly conducting sidewalls is only about 1% of the heat transfer through the fluid layer, it is sufficient to reduce the fractal dimension of the dynamics by approximately 25% in our computations.

Spatiotemporal dynamics of the covariant Lyapunov vectors of chaotic convection.

We explore the spatiotemporal dynamics of the spectrum of covariant Lyapunov vectors for chaotic Rayleigh-Bénard convection. We use the inverse participation ratio to quantify the amount of spatial localization of the covariant Lyapunov vectors. The covariant Lyapunov vectors are found to be spatially localized at times when the instantaneous covariant Lyapunov exponents are large. The spatial localization of the Lyapunov vectors often occurs near defect structures in the fluid flow field. There is an overall trend of decreasing spatial localization of the Lyapunov vectors with increasing index of the vector. The spatial localization of the covariant Lyapunov vectors with positive Lyapunov exponents decreases an order of magnitude faster with increasing vector index than all of the remaining vectors that we have computed. We find that a weighted covariant Lyapunov vector is useful for the visualization and interpretation of the significant connections between the Lyapunov vectors and the flow field patterns.

Extensive chaos in Rayleigh-Bénard convection.

Using large-scale numerical calculations we explore spatiotemporal chaos in Rayleigh-Bénard convection for experimentally relevant conditions. We calculate the spectrum of Lyapunov exponents and the Lyapunov dimension describing the chaotic dynamics of the convective fluid layer at constant thermal driving over a range of finite system sizes. Our results reveal that the dynamics of fluid convection is truly chaotic for experimental conditions as illustrated by a positive leading-order Lyapunov exponent. We also find the chaos to be extensive over the range of finite-sized systems investigated as indicated by a linear scaling between the Lyapunov dimension of the chaotic attractor and the system size.

Covariant Lyapunov vectors of chaotic Rayleigh-Bénard convection.

We explore numerically the high-dimensional spatiotemporal chaos of Rayleigh-Bénard convection using covariant Lyapunov vectors. We integrate the three-dimensional and time-dependent Boussinesq equations for a convection layer in a shallow square box geometry with an aspect ratio of 16 for very long times and for a range of Rayleigh numbers. We simultaneously integrate many copies of the tangent space equations in order to compute the covariant Lyapunov vectors. The dynamics explored has fractal dimensions of 20≲D_{λ}≲50, and we compute on the order of 150 covariant Lyapunov vectors. We use the covariant Lyapunov vectors to quantify the degree of hyperbolicity of the dynamics and the degree of Oseledets splitting and to explore the temporal and spatial dynamics of the Lyapunov vectors. Our results indicate that the chaotic dynamics of Rayleigh-Bénard convection is nonhyperbolic for all of the Rayleigh numbers we have explored. Our results yield that the entire spectrum of covariant Lyapunov vectors that we have computed are tangled as indicated by near tangencies with neighboring vectors. A closer look at the spatiotemporal features of the Lyapunov vectors suggests contributions from structures at two different length scales with differing amounts of localization.

Local reduction of organ size in transgenic mice expressing a soluble insulin-like growth factor II/mannose-6-phosphate receptor.

Genetic evidence suggests that the insulin-like growth factor II (IGF-II)/mannose-6-phosphate receptor (IGF2R) slows growth. A soluble form of IGF2R (sIGF2R) is produced by proteolytic cleavage of the intact cellular receptor and is found at high levels in fetal and neonatal plasma. To test the hypothesis that sIGF2R modulates organ size in vivo, we generated transgenic mice expressing a mouse Igf2r complementary DNA in which the transmembrane domain sequence was deleted. The transgene was driven by the keratin-10 promoter and was expressed at the highest levels in the skin and alimentary canal. Transgenics showed disproportionately reduced size of the alimentary canal, where the wet weight was decreased by 9-20% and the dry weight was decreased by 20-30%, whereas the water content per unit dry weight was not significantly changed. In addition, the circulating levels of IGF-II and the latent form of transforming growth factor-beta1 were increased by 58-77% and 56-140%, respectively, whereas plasma epidermal growth factor levels showed a 24-35% reduction. The serum and tissue activities of four lysosomal enzymes were not affected, with the exception of the colon in the line expressing the transgene at highest levels, where enzyme activities were decreased compared with control values. These results support a significant role for the sIGF2R in local modulation of organ size in vivo.

Quantifying mosquito biting patterns on humans by DNA fingerprinting of bloodmeals.

A major debate in infectious disease epidemiology concerns the relative importance of exposure and host factors, such as sex and acquired immunity, in determining observed age patterns of parasitic infection in endemic communities. Nonhomogeneous contact between hosts and vectors is also expected to increase the reproductive rate, and hence transmission, of mosquito-borne infections. Resolution of these questions for human parasitic diseases has been frustrated by the lack of a quantitative tool for quantifying the exposure rate of people in communities. Here, we show that the polymerase chain reaction (PCR) technique for amplifying and fingerprinting human DNA from mosquito bloodmeals can address this problem for mosquito-borne diseases. Analysis of parallel human and mosquito (resting Culex quinquefasciatus) samples from the same households in an urban endemic focus for bancroftian filariasis in South India demonstrates that a 9-locus radioactive short-tandem repeat system is able to identify the source of human DNA within the bloodmeals of nearly 80% of mosquitoes. The results show that a person's exposure rate, and hence the age and sex patterns of exposure to bites in an endemic community, can be successfully quantified by this method. Out of 276 bloodmeal PCR fingerprints, we also found that on average, 27% of the mosquitoes caught resting within individual households had fed on people outside the household. Additionally, 13% of mosquitoes biting within households contained blood from at least 2 people, with the rate of multiple feeding depending on the density of humans in the household. These complex vector feeding behaviors may partly account for the discrepancies in estimates of the infection rates of mosquito-borne diseases calculated parasitologically and entomologically, and they underline the potential of this tool for investigating the transmission dynamics of infection.

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.

Mean flow and spiral defect chaos in Rayleigh-Bénard convection.

We describe a numerical procedure to construct a modified velocity field that does not have any mean flow. Using this procedure, we present two results. First, we show that, in the absence of the mean flow, spiral defect chaos collapses to a stationary pattern comprising textures of stripes with angular bends. The quenched patterns are characterized by mean wave numbers that approach those uniquely selected by focus-type singularities, which, in the absence of the mean flow, lie at the zigzag instability boundary. The quenched patterns also have larger correlation lengths and are comprised of rolls with less curvature. Secondly, we describe how the mean flow can contribute to the commonly observed phenomenon of rolls terminating perpendicularly into lateral walls. We show that, in the absence of the mean flow, rolls begin to terminate into lateral walls at an oblique angle. This obliqueness increases with the Rayleigh number.

Traveling waves in rotating Rayleigh-Bénard convection: analysis of modes and mean flow.

Numerical simulations of the Boussinesq equations with rotation for realistic no-slip boundary conditions and a finite annular domain are presented. These simulations reproduce traveling waves observed experimentally. Traveling waves are studied near threshold by using the complex Ginzburg-Landau equation (CGLE): a mode analysis enables the CGLE coefficients to be determined. The CGLE coefficients are compared with previous experimental and theoretical results. Mean flows are also computed and found to be more significant as the Prandtl number decreases (from sigma=6.4 to sigma=1). In addition, the mean flow around the outer radius of the annulus appears to be correlated with the mean flow around the inner radius.

Front propagation in a chaotic flow field.

We investigate numerically the dynamics of a propagating front in the presence of a spatiotemporally chaotic flow field. The flow field is the three-dimensional time-dependent state of spiral defect chaos generated by Rayleigh-Bénard convection in a spatially extended domain. Using large-scale parallel numerical simulations, we simultaneously solve the Boussinesq equations and a reaction-advection-diffusion equation with a Fischer-Kolmogorov-Petrovskii-Piskunov reaction for the transport of the scalar species in a large-aspect-ratio cylindrical domain for experimentally accessible conditions. We explore the front dynamics and geometry in the low-Damköhler-number regime, where the effect of the flow field is significant. Our results show that the chaotic flow field enhances the front propagation when compared with a purely cellular flow field. We quantify this enhancement by computing the spreading rate of the reaction products for a range of parameters. We use our results to quantify the complexity of the three-dimensional front geometry for a range of chaotic flow conditions.

Nanomechanical motion of Escherichia coli adhered to a surface.

Nanomechanical motion of bacteria adhered to a chemically functionalized silicon surface is studied by means of a microcantilever. A non-specific binding agent is used to attach Escherichia coli (E. coli) to the surface of a silicon microcantilever. The microcantilever is kept in a liquid medium, and its nanomechanical fluctuations are monitored using an optical displacement transducer. The motion of the bacteria couples efficiently to the microcantilever well below its resonance frequency, causing a measurable increase in the microcantilever fluctuations. In the time domain, the fluctuations exhibit large-amplitude low-frequency oscillations. In corresponding frequency-domain measurements, it is observed that the mechanical energy is focused at low frequencies with a 1/fα -type power law. A basic physical model is used for explaining the observed spectral distribution of the mechanical energy. These results lay the groundwork for understanding the motion of microorganisms adhered to surfaces and for developing micromechanical sensors for bacteria.

Bioconvection in spatially extended domains.

We numerically explore gyrotactic bioconvection in large spatially extended domains of finite depth using parameter values from available experiments with the unicellular alga Chlamydomonas nivalis. We numerically integrate the three-dimensional, time-dependent continuum model of Pedley et al. [J. Fluid Mech. 195, 223 (1988)] using a high-order, parallel, spectral-element approach. We explore the long-time nonlinear patterns and dynamics found for layers with an aspect ratio of 10 over a range of Rayleigh numbers. Our results yield the pattern wavelength and pattern dynamics which we compare with available theory and experimental measurement. There is good agreement for the pattern wavelength at short times between numerics, experiment, and a linear stability analysis. At long times we find that the general sequence of patterns given by the nonlinear evolution of the governing equations correspond qualitatively to what has been described experimentally. However, at long times the patterns in numerics grow to larger wavelengths, in contrast to what is observed in experiment where the wavelength is found to decrease with time.