Traveling-wave State Research Articles

Bistability of Slow and Fast Traveling Waves in Fluid Mixtures

The appearence of a new type of fast nonlinear traveling wave states in binary fluid convection with increasing Soret effect is elucidated and the parameter range of their bistability with the common slower ones is evaluated numerically. The bifurcation behavior and the significantly different spatiotemporal properties of the different wave states - e.g. frequency, flow structure, and concentration distribution - are determined and related to each other and to a convenient measure of their nonlinearity. This allows to derive a limit for the applicability of small amplitude expansions. Additionally an universal scaling behavior of frequencies and mixing properties is found. PACS: 47.20.-k, 47.10.+g, 47.20.Ky

Secondary instability of flow in a curved duct of square cross-section

Experiments and simulations of a travelling wave state of incompressible Newtonian flow in a curved duct of square cross-section are presented. The travelling wave mode develops from the well-documented steady four-cell flow state and is characterized by oscillations of the two Dean vortices near the centre of the outer wall.The oscillations were induced by a carefully positioned pin at 5° from the inlet of the curved section along the symmetry line of the cross-section. It was shown that the travelling wave state is characteristic for curved duct flow and that the pin made it possible to observe the oscillations within the 270° long curved duct. Travelling waves were observed at flow rates above Dn = 170 (Dn = Re/(R/a)1/2, where Re is the Reynolds number, R is the radius of curvature of the duct and a is the duct dimension. The curvature ratio, R/a, is 15.1).If no other disturbances are imposed, the oscillations are the result of the selective amplification of random disturbances in the flow, leading to a broad frequency spectrum. The travelling wave was found to lock in to an imposed periodic disturbance at a selected frequency. The flow structure of the locked state was investigated in detail, using flow visualization and a one-component laser Doppler anemometer to measure streamwise or spanwise velocities. Direct numerical simulations using the package CFDS-FLOW3D are in very good agreement with the experiments and confirm the existence of a fully developed, streamwise-periodic travelling wave state.The inflow region between the two Dean vortices, which transports low-speed fluid away from the outer wall, creates strongly inflectional spanwise profiles of the streamwise velocity. Similarities with twisting vortices in a curved channel and sinuous oscillations of Görtler vortices show that the travelling waves observed here result from a secondary shear instability of these spanwise inflectional profiles.

Chaotic Localized States near the Onset of Electroconvection.

We report on chaotic localized traveling-wave states (pulses) near the onset of electroconvection in a nematic liquid crystal. They coexist with the conduction state, are long lived, appear and die at irregular locations and times, have unique small widths, and vary irregularly in length. In this system there are only four linearly unstable modes. Thus it should be possible to reproduce the dynamics of the pulses by four coupled complex Ginzburg-Landau equations once their coefficients are derived from the equations of motion (the weak-electrolyte model).

Traveling-wave states and secondary instabilities in optical parametric oscillators.

A complex order-parameter description of pattern formation in large aspect ratio optical parametric oscillators (OPOs) with flat end reflectors and uniform pumping is presented starting from the mean-field model of the OPO equations [G.-L. Oppo et al., Phys. Rev. A 49, 2028 (1994)]. It is shown that, in the nondegenerate case, the full OPO equations have an exact continuum family of traveling-wave (TW) solutions, which are preferred to standing-wave (SW) states found in the degenerate case. These solutions correspond to an off-axis emission for both signal and idler fields along two symmetric directions to satisfy momentum conservation in the parametric conversion process. Stability of TW versus SW solutions is investigated by deriving two coupled Newell-Whitehead-Segel equations describing the growth of SW or TW close to threshold. Analytical expressions for long-wavelength phase instabilities of the TW states above threshold are obtained from the coefficients of a Cross-Newell phase equation, and are shown to be the same for OPOs with high or low finesse for the pump field. By direct linear stability analysis of the TW solutions, it is also shown that the appearance of amplitude instabilities may reduce the region of stable TW states in the case of OPOs with a high finesse for the pump field. \textcopyright{} 1996 The American Physical Society.

Absolute and convective instabilities of spatially periodic flows

Stability of monochromatic waves and wave packet evolution are of fundamental importance in the transition to turbulence in open-flow systems. Although the analysis of monochromatic wave stability and the evolution of linear wave packets in spatially homogeneous or parallel flows is generally well understood, such analysis for spatially inhomogeneous flows is not so well understood. In this paper we consider the problem of linear wave packet evolution in a spatially periodic medium. The mathematical formalism of absolute and convective instabilities in spatially homogeneous flows is generalized to the spatially periodic case. The Laplace transform is used to reduce the initial-value problem to a system of ordinary differential equations with periodic coefficients which is then completely analysed using the Floquet theory and the Fourier transform. We define a generalized dispersion relation by $\Delta $($\mu $, $\omega $) = 0, where $\mu \in \text{C}\backslash ${0} is a spatial Floquet multiplier and $\omega \in \text{C}$ is a frequency (and a Laplace transform parameter). We find that a spatially periodic flow is absolutely unstable if and only if $\Delta $($\mu $, $\omega $) has a double root (or more generally a multiple root) in $\mu $ at ($\mu \_{0}$, $\omega \_{0}$) with Im $\omega \_{0}$ > 0 that satisfies the collision criterion: i.e. the double root splits under perturbation in such a way that in the limit, as Im($\omega $ - $\omega \_{0}$) $\rightarrow $ +$\infty $, with Re($\omega $ - $\omega _{0}$) = 0, one of the roots goes interior and the other exterior to the unit circle {$|\mu|$ = 1} in the complex $\mu $-plane. Further results are obtained on the long-time asymptotics of absolutely unstable and absolutely stable flows and the signalling problem for periodic media is introduced. In general the asymptotic states of unstable periodic waves are quasi-periodic in space and/or time. The theory is applied to the finite-amplitude periodic travelling wave states of the real and complex Ginzburg-Landau equation. We find that the Eckhaus instability is always absolutely unstable whereas in the complex case there is an interesting decomposition of the region of unstable finite-amplitude travelling waves into an absolutely unstable region and a convectively unstable but absolutely stable region. The problem of unstable wave packets for the Navier-Stokes equations, linearized about a spatially periodic state, is also formulated.

Traveling wave–standing wave transition in the coupled complex Ginzburg–Landau equations

The coupled Ginzburg-Landau equations are studied numerically. The instability of a chaotic traveling wave state is characterized by means of a stability exponent. When the traveling wave state is unstable, several types of coexistent states of left and right traveling waves appear. Stationary and propagating soliton lattice states are numerically found as a stable coexistent state.

Bound pairs of fronts in a real Ginzburg-Landau equation coupled to a mean field

Motivated by the observation of localized traveling-wave states (`pulses') in convection in binary liquid mixtures, the interaction of fronts is investigated in a real Ginzburg-Landau equation which is coupled to a mean field. In that system the Ginzburg-Landau equation describes the traveling-wave amplitude and the mean field corrsponds to a concentration mode which arises due to the slowness of mass diffusion. For single fronts the mean field can lead to a hysteretic transition between slow and fast fronts. Its contribution to the interaction between fronts can be attractive as well as repulsive and depends strongly on their direction of propagation. Thus, the concentration mode leads to a new localization mechanism, which does not require any dispersion in contrast to that operating in the nonlinear Schr\"odinger equation. Based on this mechanism alone, pairs of fronts in binary-mixture convection are expected to form {\it stable} pulses if they travel {\it backward}, i.e. opposite to the phase velocity. For positive velocities the interaction becomes attractive and destabilizes the pulses. These results are in qualitative agreement with recent experiments. Since the new mechanism is very robust it is expected to be relevant in other systems as well in which a wave is coupled to a mean field.

Convection in binary fluid mixtures. I. Extended traveling-wave and stationary states.

Nonlinear, spatially extended structures of convection rolls in horizontal layers of binary fluids heated from below are investigated in quantitative detail as a function of Rayleigh number for several negative and positive Soret coupling strengths (separation ratios) and different Lewis and Prandtl numbers characterizing different mixtures. A finite-difference method was used to solve the full hydrodynamic field equations in a vertical cross section perpendicular to the roll axes, subject to realistic horizontal and laterally periodic boundary conditions in a range of experimentally accessible parameters. We elucidate the important role that the concentration field plays in the structural dynamics of the nonlinear states of stationary overturning convection (SOC) and of traveling-wave (TW) convection investigated here. Structural differences in the concentration boundary layers and of the concentration plumes in TW's and SOC's and their physical consequences are discussed. These properties show that the states considered here are indeed strongly nonlinear, as expected from the magnitude of advection and diffusion in the concentration balance. The bifurcation behavior of the states is analyzed using different order parameters such as flow intensity, Nusselt number, a mixing parameter characterized by the variance of the concentration field, and the TW frequency. For further comparison with experiments, light intensity distributions are determined that can be observed in side-view shadowgraphs done with horizontal light along the roll axes.Furthermore, detailed structural analyses of all fields are made using color-coded isoplots, vertical and lateral field profiles, and lateral Fourier decompositions. They reveal, among other things, that the mirror-glide operation of lateral translation by one-half a wavelength combined with vertical reflection at the horizontal midplane of the layer is the longest persistent symmetry of TW and SOC states. Transport properties of the TW state are also discussed, in particular the mean lateral concentration current that is caused by the phase difference between concentration wave and velocity wave and that is roughly proportional to the TW frequency. This current plays an important role in the structural dynamics and stability of the spatially localized traveling-wave convection investigated in an accompanying paper.

Universal description of laser dynamics near threshold

Abstract Complex order parameter descriptions of large aspect ratio, single longitudinal mode, two-level lasers with flat end reflectors, valid near onset of lasing and for small detunings of the laser from the peak gain, are given in terms of a complex Swift-Hohenberg equation for Class A and C lasers and by a complex Swift-Hohenberg equation coupled to a mean flow for the case of a Class B laser. The latter coupled system is a physically consistent generalized rate equation model for wide aperture stiff laser systems. These universal order parameter equations provide a connection between spatially homogeneous oscillating states of the complex Ginzburg-Landau equation description of the laser system valid for finite negative detunings, and traveling wave states, described by coupled Newell-Whitehead-Segel equations valid for finite positive detunings. One of the main conclusions of the present paper is that the usual Eckhaus instability boundary associated with a long wavelength phase instability, and which delineates the region of the stable traveling wave solutions for Class A and C lasers, no longer defines the stability boundary for the mathematically stiff Class B laser. Instead a short wavelength phase instability appears causing the stability domain to shrink as a function of increasing stiffness of the system. This prediction is consistent with the strong spatiotemporal filamentation instabilities experimentally observed in a borad area semiconductor laser, a Class B system.

Magnetoconvection dynamics in a stratified layer. 1: Two-dimensional simulations and visualization

view Abstract Citations (15) References (50) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Magnetoconvection Dynamics in a Stratified Layer. I. Two-dimensional Simulations and Visualization Lantz, Steven R. ; Sudan, R. N. Abstract To gain insight in the problem of fluid convection below solar photosphere, time-dependent magnetohydrodynamic convection is studied by numerical simulation to the magneto-anelastic equations, a model appropiate for low Mach numbers. Numerical solutions to the equations are generated on a two-dimensional Cartesian mesh by a finite-difference, predictor-corrector algorithm. The thermodynamic properties of the fluid are held constant at the rigid, stress-free top and bottom boundaries of the computational box, while lateral boundaries are treated as periodic. In most runs the background polytropic fluid configuration is held fixed at Rayleigh number R = 5.44 times the critical value, Prandtl number P = 1.8, and aspect ratio a = 1, while the magnetic parameters are allowed to vary. The resulting dynamical behavior is shown to be strongly influenced by a horizontal magnetic field which is imposed at the bottom boundary. As the field strength increases from zero, an initially unsteady 'single-roll' state, featuring complex time dependence is replaced by a steady 'traveling-wave tilted state; then, an oscillatory or 'sloshing' state; then, a steady two-poll state with no tilting; and finally, a stationary state. Because the magnetic field is matched onto a potential field at the top boundary, it can penetrate into the nonconducting region above. By varying a magnetic diffusivity, the concentrations of weak magnetic fields at the top of these flows can be shown to be explainable in terms of an advection-diffusion balance. Publication: The Astrophysical Journal Pub Date: March 1995 DOI: 10.1086/175411 Bibcode: 1995ApJ...441..903L Keywords: Magnetohydrodynamics; Solar Convection (Astronomy); Solar Interior; Solar Magnetic Field; Two Dimensional Models; Advection; Computerized Simulation; Diffusion; Finite Difference Theory; Magnetostriction; Predictor-Corrector Methods; Solar Physics; CONVECTION; MAGNETOHYDRODYNAMICS: MHD; SUN: INTERIOR; SUN: MAGNETIC FIELDS; VIDEOTAPES full text sources ADS | Related Materials (1) Part 2: 1995ApJ...441..925L

Photonic superguiding state in nonlinear polar crystals.

The present paper establishes the photonic superguiding theory in polar crystals with a high nonlinearity. In quantum theory it is shown that photons can sense an attractive effective interaction by exchange of virtual optical phonons. Such an interaction leads to the superguiding state. In the standing-wave superguiding state, the photons with opposite wave vectors and spins are bound into pairs. In the traveling-wave superguiding state, a propagating photon pair consists of a combination of two photons with opposite transverse wave vectors and spins. We study the particle properties of the photonic superguiding state, the most important property being that the system of photon pairs evolves without scattering attenuations. Quantum fluctuations of the standing-wave superguiding state exceed the vacuum fluctuations while the traveling-wave superguiding state has the squeezing property. We also investigate the wave properties of the photonic superguiding state. It is found that the polar crystals with a high nonlinearity are self-defocusing media. In the standing-wave superguiding state, the system of photon pairs exists in the form of quantized vortices. In the traveling-wave superguiding state, the system of photon pairs exists in the form of quantized temporal solitons.

The influence of thermal noise on the onset of travelling-wave convection in binary fluid mixtures: an experimental investigation

When dealing with systems showing a Hopf bifurcation as the first instability from a conductive state leading to travelling waves, the distinction between convective and absolute instability becomes significant. The convectively unstable regime is characterized by the fact that a homogeneous disturbance may have a positive growth rate, while a single localized perturbation cannot trigger the onset of nonlinear convection. In this paper the convective instability occurring in binary fluid mixtures for a negative separation ratio is utilized for amplifying intrinsic thermal fluctuations, which in this way become accessible to quantitative measurements. The experiments are performed in a quasi-one-dimensional convection channel which, by means of subcritical ramps, effectively prevents the reflection of the travelling waves from the sidewalls. Thus, that range of the convective instability within which linear waves can be observed is strongly enhanced. The temperature variations involved in the observed travelling-wave states are quantified by using the shadowgraph method. By resonantly stimulating the system with its linear Hopf frequency, the reflection ability and some coefficients of the amplitude equation appropriate for describing the convection features near onset can be determined. Without stimulation, travelling-wave states of very small amplitudes showing an erratic spatio-temporal behaviour occur spontaneously inside the convectively unstable regime. The temporal correlation function calculated from the measured light intensity caused by these states is compared with a theoretical expression obtained from a Ginzburg—Landau equation containing a noise term. A very good agreement is found for the amplitude if thermal noise is assumed to be the reason for these fluctuating convection rolls, thus supporting the idea that the response of the system to thermal fluctuations is observed.

CHAOS OF TRAVELING WAVES IN A DISCRETE CHAIN OF DIFFUSIVELY COUPLED MAPS

An infinite set of stationary traveling waves are found in a discrete analog of the nonlinear diffusion Huxley equation. Their profiles are determined by the trajectories of Bernoulli scheme containing two symbols. It is proved that these waves are stable with respect to perturbations moving with the same speed. Thus it is shown that chaos of traveling waves is realized in this system.

Evolution equations for counterpropagating edge waves

Asymptotically exact evolution equations for counterpropagating shallow-water edge waves are derived. The structure of the equations depends only on the symmetries of the problem and on the fact that the group velocity of the edge waves is of order one. As a result the equations take the form of parametrically forced Davey–Stewartson equations with mean-field coupling. The calculations extend existing work on parametric excitation of edge waves by normally incident waves to arbitrary beach profiles with asymptotically constant depth, and include coupling to wave-generated mean longshore currents. Dissipation arises generically from radiation damping, but we also consider heuristically the effects of linear boundary-layer damping. Spatially modulated waves do not couple to the parametric forcing due to the non-locality of the evolution equations and are damped. Thus only spatially uniform wavetrains are expected as stable solutions. If linear dissipation is included the parametric coupling selects standing waves, but in the undamped case travelling wave states are possible. Both classes of solutions are examined for modulational instabilities, and stability conditions for the generic evolution equations are presented. However, modulational instability is found to be excluded in the shallow-water formulation through the effects of the mean flow. Explicit numerical results for two experimentally relevant beach profiles, exponentially decaying and piecewise linear, are presented.

An efficient microwave power source: The free-electron laser afterburner

A kind of microwave power source, called a free-electron laser (FEL) afterburner, that consists of a free-electron laser buncher and a slow-wave output structure sharing a magnetic wiggler field with the buncher is proposed. The buncher and the slow-wave structure can operate in either a traveling-wave state or a standing-wave state. In the buncher, the wiggler field together with the radiation field makes an electron beam bunched, and in the slow-wave structure the wiggler field keeps the beam bunched while the bunched beam interacts strongly with the slow-wave structure and thus produces rf power. The bunching process comes from the free-electron laser mechanism, and the generating process of rf power is in a slow-wave structure. A three-dimensional, time-dependent code is used to simulate a particular standing-wave FEL afterburner and it is shown that rf power of up to 1.4 GW at 17.12 GHz, can be obtained from a 1 kA, 5 MeV electron beam with an energy spread of less than 1% and an emittance of less than 0.5×10−3 π rad m.

Traveling-wave wall states in rotating Rayleigh-Bénard convection.

We show that the wall-localized convection states observed in Rayleigh-Benard convection in rotating cylindrical cells can be explained in terms of a geometry-independent traveling-wave wall state. We calculate the onset Rayleigh number, frequency, and wave number of such a state, as well as its amplitude equation. We also study the large-rotation-rate asymptotic behavior and the small-rotation-rate limit.

Convection in binary mixtures: the role of the concentration field

Structure and dynamics of different roll states appearing close to onset of convection in binary mixtures and the influence of the concentration field on the properties of these states are elucidated. Numerical solutions of the hydrodynamic field equations in a vertical cross section perpendicular to the roll axes are reviewed for room temperature ethanol-water parameters. Extended states of stationary overturning convection (SOC) and of traveling wave (TW) convective rolls are investigated as well as spatially confined states of localized traveling wave (LTW) rolls. The field structure and symmetries of these states are discussed. The appearance of a concentration contrast in TW's between rolls of different turning sense is explained. A mean lateral concentration current that is generated in the TW state by the phase difference between the concentration wave and the velocity wave is reviewed. A comparison of LTW's arbitrary and selected width obtained for different fluid parameters reveals the same field structure under the leading and trailing fronts of these states with only the extensions of the center parts being different.The presence of a large-scale concentration current, the associated large-scale concentration redistribution, and the resulting hindering of the drift of the LTW is reviewed.

Extended states of nonlinear traveling-wave convection. I. The Eckhaus instability.

I describe experiments on one-dimensional states of nonlinear traveling-wave convection in a narrow annular cell. Spatially uniform states are found to be stable within a band of wave numbers whose width grows approximately as the square root of the distance above a saddle-node Rayleigh number. Inside the band, I have measured the static properties of the traveling waves, including their response to spatial inhomogeneities in the Rayleigh number. Outside the stability band, traveling-wave states become unstable to temporally growing modulations of the spatial wave-number profile that propagate through the system at the group velocity of the underlying traveling waves. If allowed to grow, this Eckhaus instability leads, via a subcritical bifurcation, to spatiotemporal defects in which pairs of rolls are created or annihilated. In contrast with the action of the Eckhaus instability in stationary pattern-forming systems, the recurrent appearance and propagation of defects leads to long, erratic transients in which the system may or may not be driven back into the stable wave-number band.

Novel cross-roll state in a cylindrical convection cell with thermally conducting sidewalls.

We describe a new traveling-wave cross-roll state in a water-filled cylindrical convection cell with thermally conducting sidewalls (\ensuremath{\Gamma}=D/2d=4.94). Just beyond convective onset, four axisymmetric toroidal rolls form. For \ensuremath{\epsilon}=(R-${\mathit{R}}_{\mathit{c}}$)/${\mathit{R}}_{\mathit{c}}$>0.15 this pattern loses stability to either a nonaxisymmetric state with rolls normal to the sidewall, or (by skipping from onset to a value of 0.31.3) a pattern consisting of three toroidal rolls with cross rolls in the outermost toroidal roll. The cross rolls are time dependent and typically form modulated traveling waves.

Extended states of nonlinear traveling-wave convection. II. Fronts and spatiotemporal defects

This paper continues a description of experiments on one-dimensional, nonlinear, traveling-wave convection in a binary fluid with separation ratio \ensuremath{\psi}=-0.127 in a narrow annular cell. It is possible to create and manipulate steady-state sources and sinks of traveling waves in this system, as well as to produce stable fronts that separate convecting and quiescent regions. Source defects tend to drift at constant velocity, emitting Doppler-shifted traveling waves whose wave number lies outside the Eckhaus boundary measured for spatially uniform traveling-wave states. I present both qualitative descriptions of such phenomena and quantitative measurements of the amplitude and wave-number profiles of sources, sinks, and fronts.