A Lagrangian method for the numerical simulation of the Kraichnan passive scalar model is introduced. The method is based on Monte Carlo simulations of tracer trajectories, supplemented by a point-splitting procedure for coinciding points. Clean scaling behavior for scalar structure functions is observed. The scheme is exploited to investigate the dependence of scalar anomalies on the scaling exponent $\xi$ of the advecting velocity field. The three-dimensional fourth-order structure function is specifically considered.

We present a numerical method which is used to calculate anomalous scaling exponents of structure functions in the Kraichnan passive scalar advection model [R. H. Kraichnan, Phys. Fluids 11, 945 (1968)]. This Monte Carlo method, which is applicable in any space dimension, is based on the Lagrangian path interpretation of passive scalar dynamics, and uses the recently discovered equivalence between scaling exponents of structure functions and relaxation rates in the stochastic shape dynamics of groups of Lagrangian particles. We calculate third and fourth order anomalous exponents for several dimensions, comparing with the predictions of perturbative calculations in large dimensions. We find that Kraichnan's closure appears to give results in close agreement with the numerics. The third order exponents are compatible with our own previous nonperturbative calculations.

Within a microcanonical scenario we numerically study an $N$-sized linear chain classical inertial $\mathrm{XY}$ model including ferromagnetic couplings which decrease with distance as $r^{-\alpha }$ ( $\alpha \ge 0$). We show that for $N\to \infty$ (thermodynamic limit): (i) The energy per particle $E_N/N$ scales like $N^{*}\equiv (N^{1-\alpha }-1\mathrm{})/(1-\alpha )$; (ii) The properly scaled maximum Lyapunov exponent ${\tilde{\lambda }}_N^{\max }$ tends, for $E_N/\left({\mathrm{NN}}^{*}\right)$ above a threshold, to zero for and only for $\alpha \le 1$. These results are analogous to those observed in low-dimensional and in self-organized critical dissipative systems. This entire picture suggests a connection with the nonextensive thermostatistics recently introduced by one of us.

It is reported that the characteristic relations of type-II and type-III intermittencies, with respective local Poincaré maps of $x_{n+1\mathrm{}}=(1+\epsilon )x_n{+ax}_n^3$ and $x_{n+1\mathrm{}}=-(1+\epsilon )x_n{-ax}_n^3$, are both $-\ln \left(\epsilon \right)$ under the assumption of uniform reinjection probability. However, the intermittencies have various characteristic relations such as ${\epsilon }^{-\nu }$ ( $1/2\le \nu \le 1$) depending on the reinjection probability. In this Letter the various characteristic relations are discussed, and the ${\epsilon }^{-1\mathrm{}/2}$ characteristic relation is obtained experimentally in an electronic circuit, with uniform reinjection probability.

A Monte Carlo scheme is proposed to obtain boundary conditions for the photon-diffusion equation. For a turbid medium with interfaces to nonscattering media, we introduce an interpolated boundary, at a distance of a few transport-mean-free paths from the physical boundary, which acts as the secondary source emitting photons in various directions. The ratio of the fluence rate to the flux is evaluated at both the interpolated and the physical boundaries. The derived boundary conditions are compared with other works and are supported by an independent simulation of an effective source term.

In this Letter we present an experimental analysis of the acoustic transmission of a two-dimensional periodic array of rigid cylinders in air with two different geometrical configurations: square and triangular. In both configurations, and above a certain filling fraction, we observe an overlap, in the range of the audible frequencies, between the attenuation peaks measured along the two high-symmetry directions of the Brillouin zone. This effect is considered as the fingerprint of the existence of a full acoustic gap. Nevertheless, the comparison with our calculation of band structures shows that the triangular lattice has band states in that frequency range. We call them deaf bands. This contradictory result is explained by looking at the symmetry of the deaf bands; they cannot be excited by experiments of sound transmission.

A robust mechanism generating regular or irregular bursts of large dynamic range near threshold is described. The mechanism arises in the interaction between oscillatory modes of odd and even parity in systems of large but finite aspect ratio, and provides an explanation for the bursting behavior observed in binary fluid convection by Sullivan and Ahlers [Phys. Rev. A 38, 3143 (1988)].

We have found an analytic expression for the multivariate generating function governing all $n$-point statistics of random multiplicative cascade processes. The variable appropriate for this generating function is the logarithm of the energy density, $\ln \epsilon$, rather than $\epsilon$ itself. All cumulant statistics become sums over derivatives of “branching generating functions” which are Laplace transforms of the splitting functions and completely determine the cascade process. We show that the branching generating function is a generalization of the multifractal mass exponents. Two simple models from fully developed turbulence illustrate the new formalism.

The formation of toroidal Taylor vortices in a fluid contained within the annular region bounded by two concentric cylinders, the inner one of which rotates, has been observed using molecular dynamics simulation. The quantitative nature of the vortices has been examined over a range of supercritical Taylor numbers. The Fourier amplitudes of the fundamental radial velocity mode and its low-order harmonics have been analyzed; despite the microscopic system size their functional dependence on the Taylor number is in excellent agreement with theory and experiment.

We report experimental evidence of novel kinds of patterns—zigzag destabilized spirals and targets. They have been found in a planar dc driven semiconductor–gas discharge system. Comparison of these patterns with conventional spirals and targets that can form in the same system allows one to conclude that the observed phenomena are specific for large amplitude structures of pattern forming media.

We investigate the dynamical behavior of two limit cycle oscillators that interact with each other via time delayed coupling and find that time delay can lead to amplitude death of the oscillators even if they have the same frequency. We demonstrate that this novel regime of amplitude “death” also exists for large collections of coupled identical oscillators and provide quantitative measures of this death region in the parameter space of coupling strength and time delay. Its implication for certain biological and physical applications is also pointed out.

We formulate the theory of incoherent dark spatial solitons in noninstantaneous self-defocusing nonlinear media. We find that the basic modal constituents of these incoherent dark soliton entities are radiation modes as well as bound states. Our results explain for the first time why incoherent dark solitons are in fact gray and why a transverse $\pi$-phase flip can facilitate their observation.

Stability of optical gap solitons is analyzed within a coupled-mode theory. Lower intensity solitons are shown to always possess a vibration mode responsible for their long-lived oscillations. As the intensity of the soliton is increased, the vibration mode falls into resonance with two branches of the long-wavelength radiation producing a cascade of oscillatory instabilities of higher intensity solitons.

Experiments on thin layers of granular material subjected to vertical vibration have revealed a series of subharmonic bifurcations from flat surface to standing wave patterns. From pressure and surface dilation measurements we show that a solid-liquid-type transition precedes surface wave bifurcation, indicating that these waves are associated with the fluidlike behavior of the layer. Also, we show that depending on the excitation frequency two kinds of waves can be distinguished: hydrodynamic waves which involve lateral transfer of grains within the layer, and dilation waves for which the layer slightly dilates and compacts alternatively in time and space.

When a mixture of two viscous liquids flows past a solid wall there is an ambiguity in the use of the no-slip boundary condition. It is not obvious whether the mass-averaged velocity, the volume-averaged velocity, the individual species velocities, all of the above, or none of the above should exhibit no slip. Extensive molecular dynamics simulations of the Poiseuille flow of mixtures of coexisting liquid species past an atomistic wall indicate that the velocity of each individual liquid species satisfies the no-slip condition and, therefore, so do mass and volume averages.

Direct numerical simulations of the decay of 2D turbulent Navier-Stokes flows inside a square container with no-slip boundaries are presented. Several numerical runs have been carried out for three different values of the Reynolds number. A surprising observation in a majority of these runs is the spontaneous spin-up of the flow during the initial stage of the self-organization of the flow, which is due to normal and shear stresses exerted on the fluid by the rigid boundaries.

We present and explain numerical results illustrating the mechanism of a type of discontinuous bifurcation of a chaotic set that occurs in typical dynamical systems. After the bifurcation, the chaotic set acquires new pieces located at a finite distance from its location just before the bifurcation, and these new pieces were not part of a previously existing chaotic set. A scaling law is given describing the creation of unstable periodic orbits following such a bifurcation. We also provide numerical evidence of such a bifurcation for a nonattracting chaotic set of the Hénon map.

The diffusion properties of the Chirikov-Taylor standard map are shown to be nonuniversal in the framework of the wave-particle interaction, because this map corresponds to a spectrum of waves whose initial phases are all correlated. The occurrence of diffusion is shown to be a peculiarity of the standard map when the average is made over the particles' initial positions. The force correlation time is shown to decrease more slowly with the wave amplitudes for the standard map than for waves with noncorrelated phases. Lévy flights are shown to be one more peculiarity of the standard map.

We report observation of polarization fluctuations in vertical-cavity semiconductor lasers, which allows us to demonstrate and quantify the importance of nonlinear polarization anisotropy. We focus on three aspects, which all fit within the same theoretical framework: (i) a nonlinear spectral redshift, (ii) an extra four-wave mixing peak in the optical spectrum, and (iii) correlations between the polarization fluctuations.

Helical waves have been shown by Moffatt to result in the $\alpha$ effect for high-conductivity plasmas. This model makes it possible to calculate exactly both the magnetic-generation coefficient $\alpha$ and turbulent diffusivity, even when back reaction of magnetic field on the velocity is taken into account, that is, in the context of the nonlinear dynamo. We show that both the $\alpha$ effect and turbulent diffusivity are strongly suppressed in the presence of a relatively weak magnetic field. As a result, fast kinematic magnetic generation reaches the postkinematic regime and substantially slows down.