Bulletin of the American Physical Society
2005 APS March Meeting
Monday–Friday, March 21–25, 2005; Los Angeles, CA
Session L23: Coherent Structures, Pattern Formation and Spatio-Temporal Chaos |
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Sponsoring Units: GSNP Chair: Bruno Eckhardt, University of Marburg Room: LACC 410 |
Tuesday, March 22, 2005 2:30PM - 2:42PM |
L23.00001: Effect of weak dissipative disorder on front formation in pattern forming systems Avner Peleg We study the effect of weak disorder in the linear gain coefficient in pattern forming systems described by the cubic-quintic nonlinear Schr\"{o}dinger equation. We calculate the distribution functions of the front position and amplitude and find that both distribution functions are strongly different from Gaussian. We show that the distribution of the front amplitude is singular at the maximum value of the amplitude, while the distribution of the front position has a lognormal form. These predictions are in very good agreement with our numerical simulations. Implications of the results for other types of dissipative disorder are discussed. [Preview Abstract] |
Tuesday, March 22, 2005 2:42PM - 2:54PM |
L23.00002: Counting intrinsic localized modes in an antif\-erromagnet M. Sato, A.J. Sievers Intrinsic localized modes (ILMs), also called discrete breathers or lattice solitons, are responsible for energy localization in the dynamics of discrete nonlinear lattices [1]. Here we report on the observation of countable ILMs in an atomic lattice by means of a novel nonlinear energy magnetometer [2]. The instrument first produces frequency locked ILMs in the spinwave spectrum of an antiferromagnet and then measures the four wave mixing signal emitted by the sample versus time. This technique makes observable in nonlinear emission the small number of ILMs that remain locked to the driver in steady state. The disappearance of each ILM registers as a step in the time dependent emission power with the surprising result that the energy staircase of ILM deexcitation is uniquely defined. These experiments identify a new direction where future applications may lead to smart materials and directed energy transfer. 1. A. J. Sievers and S. Takeno, Phys. Rev. Lett. \textbf{61}, 970 (1988). 2. M. Sato and A. J. Sievers, Nature \textbf{431}, Nov. 25 (2004). [Preview Abstract] |
Tuesday, March 22, 2005 2:54PM - 3:06PM |
L23.00003: Correlation of intrinsic localized mode properties with sample temperature J.P. Wrubel, M. Sato, A.J. Sievers Intrinsic localized modes (ILMs) in a quasi-1D antiferromagnetic lattice may be eternalized with a moderate powered microwave driver at a locking frequency below the AFMR. These locked ILMs are dynamical sources of nonlinearity in the sample and can therefore be detected in emission by four wave mixing [1]. The emission signal decays in steps at reproducible times as individual ILMs are unlocked from a driver. We have discovered that an unlocked ILM may be recaptured by increasing the amplitude of the driver. To examine the role of the sample temperature on this locking phenomenon the frequency of the driver has been amplitude modulated from 100 Hz to 50 kHz. Our experimental results show that the ILMs are not able to lock to the driver if the sample temperature is unable to follow the modulation frequency. 1. M. Sato and A. J. Sievers, Nature \textbf{431}, Nov. 25 (2004). [Preview Abstract] |
Tuesday, March 22, 2005 3:06PM - 3:18PM |
L23.00004: Collective excitations of charged dust grains in dusty plasma lattices Ioannis Kourakis, Padma Kant Shukla \emph{Dusty Plasmas} (or \emph{Complex Plasmas}) are large ensembles of interacting particles, consisting of electrons, ions and massive, heavily charged, micron-sized dust particulates. The presence of the latter modifies the plasma properties substantially and allows for new charged matter configurations, including liquid-like and solid (quasi-\textit{crystalline}) states (Debye crystals). One- dimensional (1d) dust crystals are formed in discharge experiments, where the electrode \emph {sheath} electric fields and electrostatic interactions constitute a highly nonlinear environment. The nonlinear aspects of horizontal (longitudinal, acoustic mode) as well as vertical (transverse, optical mode) motion of charged dust grains in a (1d) dust crystal are discussed. Different types of localized excitations, predicted by nonlinear wave theories, are reviewed and conditions for their occurrence (and characteristics) in DP crystals are discussed, in a continuum approximation. Dust crystals are shown to support nonlinear \emph{kink-}shaped supersonic solitary excitations, related to longitudinal dust grain displacement, as well as modulated \emph{envelope localized modes} associated with either longitudinal or transverse oscillations. Furthermore, the possibility for highly localized \emph{discrete breather}-type excitations to occur is investigated, for first principles. The relation to known results on atomic chains and on strongly- coupled dust layers in gas discharge plasma experiments is discussed. [Preview Abstract] |
Tuesday, March 22, 2005 3:18PM - 3:30PM |
L23.00005: Formation of Suncups on Snowfields Exposed to Solar Radiation T. Tiedje, Bayo Lau, A. Ballestad, Eric Nodwell A mathematical model is proposed to explain the ablation hollows (suncups) that are observed on snowfields exposed to intense solar radiation. The model is derived by first expressing the distribution of scattered sunlight in the snow in terms of the local slope and curvature of the surface. From this expression, we use a perturbation method valid in the limit of weak surface topography to obtain a differential equation for the snow surface morphology. The resulting non-linear equation is the Kuramoto Sivashinsky equation except with the addition of a conservative non-linear term. In simplified form the equation is: $\partial _t h=cF\left( {\left\langle \ell \right\rangle \nabla ^2h-\left\langle {\ell ^3} \right\rangle \nabla ^4h+\left( {\nabla h} \right)^2+\left\langle {\ell ^2} \right\rangle \nabla ^2\left( {\nabla h} \right)^2} \right)$ where $\left\langle {\ell ^n} \right\rangle $ is the spectral average of the $n^{th}$ power of the photon diffusion length. Multiple scattering from one concave part of the surface to another is treated self consistently. Numerical solutions of this equation with published values for the optical properties of snow yield spontaneous ordered patterns with a characteristic length of 25-50 cm and characteristic formation time under full solar illumination of 5-15 days, depending on the microstructure of the snow. The spontaneous pattern consists of a hexagonal array of parabolic valleys separated by sharp ridges that closely resemble suncups in size, shape and growth time. [Preview Abstract] |
Tuesday, March 22, 2005 3:30PM - 3:42PM |
L23.00006: Pattern formation through alternation of dynamics in a nonlinear optical system K. Saunders, N. Sungar, P.L. Ramazza, J.P. Sharpe We will discuss the experimental observation of a new mechanism for pattern formation in spatially extended nonlinear systems, namely the alternation of dynamics. In this experiment we employ a nonlinear optical device known as a liquid crystal light valve (LCLV) that has been placed in an optical feedback loop. The LCLV arrangement has been studied for over a decade and exhibits a broad range of spatiotemporal behavior. We have found that by alternating one of the parameters of the system (the light intensity) between two values we can generate strong patterns. There is no patterning observed in either of the two states corresponding to each value of the parameter. Thus, alternation of the parameter is crucial. We will discuss the conditions under which patterning occurs and relate our experimental observations to recently proposed theory. [Preview Abstract] |
Tuesday, March 22, 2005 3:42PM - 3:54PM |
L23.00007: Weakly vs Highly nonlinear front dynamics Olivier Pierre-Louis We analyse the nonlinear dynamics of one-dimensional unstable fronts. Our main finding is the existence of two types of dynamics weakly, or highly nonlinear, which respectively lead to continuous or discontinuous morphological transitions. Based on a multi-scale analysis, we list the possible weakly nonlinear equations and determine some of the most relevant ones. We then show that dynamics is not weakly nonlinear, but highly nonlinear in many cases relevant to specific systems (e.g. when an instability occurs in the vicinity of thermodynamic equilibrium in a conserved system). Highly nonlinear equations are explicitly derived, and exhibit unexpected symmetries. The resulting dynamics is discussed. Explicit applications to pattern formation during Molecular Beam expitaxy are presented. [Preview Abstract] |
Tuesday, March 22, 2005 3:54PM - 4:06PM |
L23.00008: The Platonic Ideal of Stalactite Growth Martin Short, James Baygents, Warren Beck, David Stone, Raymond Goldstein, Rickard Toomey The chemical mechanisms underlying the growth of cave formations such as stalactites are well-known, yet no theory has yet been proposed which successfully accounts for the dynamic evolution of their shapes. Here we consider the interplay of thin-film fluid dynamics, calcium carbonate chemistry, and CO$_2$ transport in the cave to show that stalactites evolve according to a novel local geometric growth law which exhibits extreme amplification at the tip as a consequence of the locally-varying fluid layer thickness. Studies of this model show that a broad class of initial conditions is attracted to an ideal shape which is strikingly close to a statistical average of natural stalactites. A linear stability analysis shows is used to explain the instability of this state to the formation of centimeter-scale ripples, as commonly seen on a wide range of speleothem surfaces. [Preview Abstract] |
Tuesday, March 22, 2005 4:06PM - 4:18PM |
L23.00009: Transient evolution of dendritic crystal tips and sidebranch structures Andrew Dougherty Dendritic crystal growth is an inherently non-local problem in pattern formation. Although the theory for steady state tip growth is fairly well established, real dendrites also contain a rich sidebranching structure that interacts with the diffusion field. I will report on experimental studies of the emergence of dendritic crystals that explore how the crystal growth speed, tip size, and sidebranch spacing interact as they evolve towards their steady state values. Starting with a small, nearly spherical seed of NH$_4$Cl held in unstable equilibrium in supersaturated aqueous solution, the temperature is dropped an amount $\Delta T$ ranging from 0.1$^{\circ}$C to 1.5$^{\circ}$C and the subsequent growth is recorded. Even before the steady state tip shape is established, the beginnings of the first sidebranches can be observed, and the dendrite quickly approaches steady state behavior. Once the steady state has been established, the temperature is again changed, and the evolution of the growth speed, tip size, and sidebranch spacing towards the new steady state values will be discussed. [Preview Abstract] |
Tuesday, March 22, 2005 4:18PM - 4:30PM |
L23.00010: Instabilities and motion of tilt grain boundaries in three dimensional stripe patterns Zhi-Feng Huang, Jorge Vinals Unlike two dimensional tilt boundaries in stripe phases for which stationary solutions are known to exist, the three dimensional case is generally unstable. We study the appropriate amplitude equations in the weakly nonlinear regime close to onset, and find a finite wavenumber, anisotropic instability with wavevector along the grain boundary plane. The characteristic wavelength is larger than that of the base stripe pattern. Our study reveals that this new three dimensional instability originates from a phase perturbation of the base periodic modes, as well as from the cross coupling between orthogonal base modes around the grain boundary region. Our results are in agreement with experimental findings in three dimensional lamellar diblock copolymers. [Preview Abstract] |
Tuesday, March 22, 2005 4:30PM - 4:42PM |
L23.00011: Length Scales of Chaotic patterns near the onset of of Electroconvection in the Nematic Liquid Crystal I52 Xiaochao Xu, Guenter Ahlers We report experimental results for Electroconvection of the nematic Liquid Crystal I52 with planar alignment and a conductivity of $1.0\times10^{-8}\, (\Omega\,{\rm m})^{-1}$. The cell spacing was $19.4\, \mu {\rm m}$ and the driving frequency was 25.0 Hz. Spatio-temporal chaos consisting of a superposition of zig and zag oblique rolls evolved by means of a supercritical Hopf bifurcation from the uniform conduction state.\footnote{M. Dennin, G. Ahlers and D. S. Cannell, Science, {\bf 272}, 388 (1996).} For small $\epsilon \equiv V^2/ V_c^2 -1$, we measured the correlation lengths of the envelopes of both zig and zag patterns. These lengths could be fit to a power law in $\epsilon$ with an exponent smaller than that predicted from amplitude equations. The disagreement with theory is similar to that found previously for domain chaos in rotating Rayleigh-Benard convection.\footnote{Y. Hu, R. E. Ecke and G. Ahlers, Phys. Rev. Lett. {\bf 74}, 5040 (1995).} [Preview Abstract] |
Tuesday, March 22, 2005 4:42PM - 4:54PM |
L23.00012: An Anomaly in the Domain Chaos State of Rayleigh-B\'enard Convection with Large Aspect Ratio Nathan Becker, Guenter Ahlers Rayleigh-B\'enard convection-patterns exhibit a type of spatio-temporal chaos known as domain chaos (DC) at the onset of convection when the sample rotates fast enough about the vertical axis. DC is characterized by domains of straight rolls that chaotically change their orientation and size due to the K\"uppers-Lortz instability.\footnote{G. K\"uppers and D. Lortz, J. Fluid Mech. {\bf 35}, 609 (1969).} However, in a large aspect ratio $\Gamma\equiv r/d=82$ cylindrical sample, where $r$ is the radius of the cell and $d$ is the cell thickness, we observed DC in the sample center, surrounded by an annulus of radial rolls populated by occasional defects reminiscent of undulation chaos.\footnote{K. E. Daniels, B.B. Plapp, and E. Bodenschatz, Phys. Rev. Lett. {\bf 84}, 5320 (2000).} This was unexpected because smaller samples do exhibit domain chaos throughout and the weakly-nonlinear theory that describes the supercritical bifurcation to chaos is expected to be more applicable as $\Gamma$ increases. One possible explanation is that the centrifugal force, which is neglected in the theory, plays an important role.\footnote{A. Jayaraman and H. Greenside (private communication).} [Preview Abstract] |
Tuesday, March 22, 2005 4:54PM - 5:06PM |
L23.00013: Meandering of the large-scale circulation of turbulent convection in a cylindrical cell Eric Brown, Denis Funfschilling, Alexei Nikolaenko, Guenter Ahlers The large-scale circulation (LSC) in cylindrical cells of aspect ratio $\Gamma \equiv D/L = 1$ ($D =$ diameter, $L = $ height) filled with water at a mean temperature of 40$^\circ$C and heated from below was studied for Rayleigh numbers $R$ in the range $10^9$ to $10^{11}$. We measured the temperatures of the cell side-wall as a function of time $t$ at eight azimuthal locations on the horizontal mid- plane and from them deduced the azimuthal orientation $\theta(t)$ of the LSC. We found that $\theta(t)$ varied irregularly in time. Although it had a preferred value, on average there was a long-term continuous rotation of the LSC. From the data for $\theta(t)$ we derived $\dot\theta \equiv \Delta\theta/\Delta t$ ($\Delta t$ is the time interval between measurements). The time averages of $\dot \theta(\theta)$ gave a deterministic force $-\partial V/\partial \theta$ corresponding to a potential of the form $V = V_0 [-\cos(\theta - \theta_0) + v_1 \theta]$, and its probability distribution-function $P_ {\dot \theta}(\dot\theta)$ yielded a Langevin force $f(t)$. Integrations of the corresponding stochastic model equation $\partial \theta /\partial t = - \partial V / \partial \theta + f(t)$ produced time series $\theta(t)$ and distribution functions $P_{\theta}(\theta)$ remarkably similar to the experimental data. We attribute $f(t)$ to the action of the turbulent background fluctuations on the LSC, and found that its intensity depended on $R$. [Preview Abstract] |
Tuesday, March 22, 2005 5:06PM - 5:18PM |
L23.00014: Lyapunov exponents for small aspect ratio Rayleigh-Benard convection Janet Scheel, Michael Cross, Mark Paul Positive Lyapunov exponents and their corresponding eigenvectors have been computed numerically for small aspect ratio, 3-D rotating Rayleigh-Benard convection cells with no-slip boundary conditions. The parameters are the same as those used by Ahlers and Behringer (PRL 40, 1978) in their seminal work on aperiodic time dependence in Rayleigh-Benard convection cells. Our work confirms that the dynamics in these cells truly is chaotic as defined by a positive Lyapunov exponent. The time evolution of the Lyapunov eigenvector in the chaotic regime will also be discussed. [Preview Abstract] |
Tuesday, March 22, 2005 5:18PM - 5:30PM |
L23.00015: Patters in particles on free surfaces Bruno Eckhardt, Joerg Schumacher, Jahanshah Davoudi, Guido Boffetta Particles floating on a free surface above a turbulent flow tend to cluster because of the upwelling and downwelling motions in the fluid underneath. The types of patterns that form depend on the stretching and contraction rates of the flow as well as on the correlation times of the flow. With increasing divergence of the flow a transition from elongated structures to point like ones can be induced. Unexpectedly, we find that the presence of correlations can weaken the clustering of particles. [Preview Abstract] |
Tuesday, March 22, 2005 5:30PM - 5:42PM |
L23.00016: Dynamic Buckling and Fragmentation in Brittle Rods Joseph Gladden, Nestor Handzy, Andrew Belmonte, Emmanuel Villermaux We present experiments on the dynamic buckling and fragmentation of slender rods axially impacted by a projectile. By combining the results of Saint-Venant and elastic beam theory, we derive a preferred wavelength $\lambda$ for the buckling instability, and experimentally verify the resulting scaling law for a range of materials including teflon, dry pasta, glass, and steel. For brittle materials, buckling leads to the fragmentation of the rod. Measured fragment length distributions show two peaks near $\lambda/2$ and $\lambda/4$. The non-monotonic nature of the distributions reflect the influence of the deterministic buckling process on the more random fragmentation processes. [Preview Abstract] |
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L23.00017: Strongly non-Gaussian statistics of optical soliton parameters in multichannel transmission due to delayed Raman response Avner Peleg We study the effects of delayed Raman response on soliton dynamics in optical fiber transmission systems with multiple frequency channels. Taking into account the quasi-random nature of pulse sequences in different channels and the collision induced energy exchange we show that soliton propagation in a given channel under many collisions with solitons from other channels is described by a perturbed stochastic nonlinear Schr\"odinger equation with weak disorder in the linear gain coefficient. As a result, the soliton amplitude becomes a random variable with a lognormal distribution. The cross frequency shift is also lognormally distributed and the self frequency shift is a random variable that is not self-averaging. We conclude that this disorder is potentially more harmful than other types of disorder that are usually considered in optical fiber transmission. Our predictions are in very good agreement with extensive numerical simulations employing importance sampling. [Preview Abstract] |
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