Collisionless magnetohydrodynamic turbulence in two dimensions

In this paper we give a formulation of two-dimensional (2D) collisionless magnetohydrodynamic (MHD) turbulence that includes the effects of both electron inertia and electron pressure (or parallel electron compressibility) and is applicable to strongly magnetized collisionless plasmas. We place particular emphasis on the departures from the 2D classical MHD turbulence results produced by the collisionless MHD effects. We investigate the fractal/multi-fractal aspects of spatial intermittency. The fractal model for intermittent collisionless MHD turbulence appears to be able to describe the observed k⁻¹ spectrum in the solar wind. Multi-fractal scaling behaviors in the inertial range are first deduced, and are then extrapolated down to the dissipative microscales. We then consider a parabolic-profile model for the singularity spectrum f (α), as an explicit example of a multi-fractal scenario. These considerations provide considerable insights into the basic mechanisms underlying spatial intermittency in 2D fully developed collisionless MHD turbulence.     

Dynamics of fully nonlinear drift wave-zonal flow turbulence system in plasmas

We present numerical simulations of fully nonlinear drift wave-zonal flow (DW-ZF) turbulence systems in a nonuniform magnetoplasma. In our model, the drift wave (DW) dynamics is pseudo-three-dimensional (pseudo-3D) and accounts for self-interactions among finite amplitude DWs and their coupling to the two-dimensional (2D) large amplitude zonal flows (ZFs). The dynamics of the 2D ZFs in the presence of the Reynolds stress of the pseudo-3D DWs is governed by the driven Euler equation. Numerical simulations of the fully nonlinear coupled DW-ZF equations reveal that short scale DW turbulence leads to nonlinear saturated dipolar vortices, whereas the ZF sets in spontaneously and is dominated by a monopolar vortex structure. The ZFs are found to suppress the cross-field turbulent particle transport. The present results provide a better model for understanding the coexistence of short and large scale coherent structures, as well as associated subdued cross-field particle transport in magnetically confined fusion plasmas.     

Richardson's pair diffusion and the stagnation point structure of turbulence

DNS and laboratory experiments show that the spatial distribution of straining stagnation points in homogeneous isotropic 3D turbulence has a fractal structure with dimension D(s)=2. In kinematic simulations the exponent gamma in Richardson's law and the fractal dimension D(s) are related by gamma=6/D(s). The Richardson constant is found to be an increasing function of the number density of straining stagnation points in agreement with pair diffusion occurring in bursts when pairs meet such points in the flow.     

Similarity decay of enstrophy in an electron fluid

A similarity decay law is proposed for enstrophy of a one-signed-vorticity fluid in a circular free-slip domain. It excludes the metastable equilibrium enstrophy which cannot drive turbulence, and approaches Batchelor's t(-2) law for strong turbulence. Measurements of the decay of a turbulent electron fluid agree well with the predictions of the decay law for a variety of initial conditions.     

Some specific mathematical constraints on 2D turbulence

We derive upper and lower bounds for ensemble averages of energy, enstrophy, and palinstrophy for the 2D periodic Navier-Stokes equations. This is carried out both in the general case, and in the case where the energy power law for fully developed turbulence holds. In the turbulent case, the bounds are sharp, up to a logarithm, and provide a new lower bound on the Landau-Lifschitz degrees of freedom. We also prove two properties of the inertial term under the turbulence assumption. One is that as the Grashof number is increased, the ensemble average of this term approaches the force. The other is that an estimate of it via the Ladyzhenskaya inequality is sharp on a considerable portion of the global attractor.     

Stretching in a model of a turbulent flow

Using a multi-scaled, chaotic flow known as the KS model of turbulence [J.C.H. Fung, J.C.R. Hunt, A. Malik, R.J. Perkins, Kinematic simulation of homogeneous turbulence by unsteady random fourier modes, J. Fluid Mech. 236 (1992) 281-318], we investigate the dependence of Lyapunov exponents on various characteristics of the flow. We show that the KS model yields a power law relation between the Reynolds number and the maximum Lyapunov exponent, which is similar to that for a turbulent flow with the same energy spectrum. Our results show that the Lyapunov exponents are sensitive to the advection of small eddies by large eddies, which can be explained by considering the Lagrangian correlation time of the smallest scales. We also relate the number of stagnation points within a flow to the maximum Lyapunov exponent, and suggest a linear dependence between the two characteristics.     

Photon-burst correlation techniques for atmospheric crosswind measurements

With a cw visible laser, the method of photon-burst correlation is used measure atmospheric crosswinds. A scaling law, including the effects of atmospheric turbulence, for performance evaluation of both laser Doppler (LDV) and laser time-of-flight (LTV) velocimeters, is introduced theoretically and established experimentally with field experiments. Crosswind measurements in the night at a range of 500 m with a low-power argon-ion laser are reported. The measured signal particle arrival rate is consistent with the predicted arrival rate based on the scaling law. In addition to the use of higher laser power, it is suggested that with proper inclusion of signal photon bursts resulting from the simultaneous arrival of several particles, routine operation of this type of laser velocimeter for long ranges, up to 1000m, should be feasible.     

Adaptive Function Projective Synchronization of Discrete-Time Chaotic Systems

By backstepping control law and the active control method, adaptive function projective synchronization of 2D and 3D discrete-time chaotic systems with Uncertain parameters are investigated. To illustrate the effectiveness of the new scheme, some numerical examples are given.     

Turbulence in a gas laser

We test a recent assertion [A. Muriel, Physica A 388 (4) (2009) 311] that a gas consisting of excited molecules is turbulent, in contrast to the laminar state of a gas of ground state molecules. Since a lasing gas is made up of excited molecules, we examine if a lasing gas system is indeed turbulent. Surprisingly, from a literature search, it appears that turbulence in a lasing gas medium has never been addressed. To test for turbulence, we use a recently proposed criterion for the existence of turbulence, the presence of multivalued steady-state velocity fields [P. Getreur, A. Albano, A. Muriel, Phys. Lett. A 366 (2007) 101]. To study this subject, we improve an old model of a gas of two-level atoms in a one-dimensional model [A. Muriel, M. Dresden, Physica D 94 (1996) 103] by including the effect of a radiation field with the use of Einstein A and B coefficients. A set of coupled equations for the velocity fields in one dimension are derived. The zeroth order implementation of an iterative solution establishes that the steady-state velocity fields are multivalued, given by the Lambert function. We obtain signature characteristics of turbulence such as velocity reversals, infinite gradients, and stagnation points.     

Statistics and structures of pressure and density in compressible isotropic turbulence

We study statistics and structures of pressure and density in the presence of large-scale shock waves in a forced compressible isotropic turbulence using high-resolution numerical simulation. The spectra for pressure and density exhibit a ?2 scaling over an operational definition of the inertial range. Both the numerical simulation and a heuristic PDF model reveal that the PDFs of pressure increment exhibit a ?2 power law region for the separation in the operational definition of inertial range, quantitatively similar to the PDF of pressure gradient, which also displays a ?2 power law region. Moreover, the statistical relation between density increment and pressure increment has been investigated through a shock-relation model. There is a positive correlation between the vorticity magnitude and pressure, which is different from the case of incompressible turbulence. We argue that this difference is due to large-scale shock waves, another type of intermittent structures in addition to vortex structures in incompressible turbulence.     

2D spectral and turbulence length scale estimation with PIV

A new method to apply spatial two-dimensional power spectral density (2D PSD) analysis to the data measured with Particle Image Velocimetry (PIV) has been introduced. Applying the method to a set of the velocity vector fields characteristic turbulence length scales can be estimated. In this method the computation of 2D PSD has been performed to two kinds of pre-processed data. In the first set, the local average has been spatially subtracted (Spatial decomposition) and in the second set the time-average has been subtracted (Reynolds decomposition). In the computation of 2D PSD the 2D FFT with the variance scaling has been used. Besides 2D spectral analysis this paper uses the distribution analysis of the various turbulence quantities and a structure analysis method to estimate the dimensions of coherent structures in the flow. Another method to analyse turbulence length scales is the estimation of the spatial 2D Auto Correlation Coefficient Function (2D ACCF). All these methods applied side by side to the PIV data increase the understanding of the turbulence, its scales and the nature of the coherent structures.     

Truncated Lévy laws and 2D turbulence

We study the probability distribution functions and scaling properties of truncated Lévy processes with sharp cut-offs. We find that they display features analog to those observed in some 2D numerical simulations of turbulence. Received: 29 October 1997 / Revised: 12 February 1998 / Accepted: 10 April 1998     

Energy spectrum of superfluid turbulence with no normal-fluid component

The energy of superfluid turbulence without the normal fluid is studied numerically under the vortex filament model. Time evolution of the Taylor-Green vortex is calculated under the full nonlocal Biot-Savart law. It is shown that for k<2pi/l the energy spectrum is very similar to the Kolmogorov's -5/3 law which is the most important statistical property of the conventional turbulence, where k is the wave number of the Fourier component of the velocity field and l is the average intervortex spacing. The vortex length distribution converges to a scaling property reflecting the self-similarity of the tangle.     

Generalized tensor ABCD law for an elliptical Gaussian beam passing through an astigmatic optical system in turbulent atmosphere

The propagation of an elliptical Gaussian beam (EGB) through an astigmatic ABCD optical system in a turbulent atmosphere is investigated. An analytical formula for the average intensity of an EGB and a generalized tensor ABCD law for the generalized complex curvature tensor are derived. As an application example, we derived an analytical formula for the average intensity of an elliptical flat-topped beam propagating through an astigmatic ABCD optical system in a turbulent atmosphere. As a numerical example, the focusing properties of an EGB focused by a thin lens in a turbulent atmosphere are studied. It is found that the focused beam at the focal plane becomes a circular Gaussian beam when the atmospheric turbulence is strong enough, and the beam width of the circular Gaussian beam is determined by atmospheric turbulence strength, focal length of the thin lens, and wavelength of the initial beam but is independent of the initial beam widths (i.e., initial intensity distribution).     

Angle-of-arrival fluctuations for wave propagation through non-Kolmogorov turbulence

Recently the increasing experimental evidences have shown that atmospheric turbulence statistics does not obey Kolmogorov’s power spectrum model in portions of the troposphere and stratosphere. These experiments have prompted the investigations of optical wave propagation through atmospheric turbulence described by non-classical power spectra. In this paper, using an original approach and considering a non-Kolmogorov power spectrum which uses a generalized power law instead of constant standard power law value 11/3 and a generalized amplitude factor instead of constant value 0.033, the variances of the angle-of-arrival fluctuations of the plane and spherical waves are derived in weak turbulence for a horizontal path. The concise closed-form expressions are obtained and used to analyze the influence of spectral power law variation on the angle-of-arrival fluctuations.     

Quantized infrared-optical triple resonance on a single cold barium ion

A single trapped and cooled Ba⁺ ion is irradiated by resonant visible light (493, 650 nm) alternating with light at 1.76 µm which may excite the ion to its² D _5/2 metastable state. The (absence of) visible resonance scattering probes the excitation, tuning spectra of which show vibrational sidebands that characterize the ion's temperature. Observed values as low as 120 µK, one-eighth the Doppler limit, are ascribed to electronic Raman cooling by the visible light. Tuning spectra of the events of stimulated deexcitation indicate ion heating by the IR interaction. The results demonstrate the feasibility of vibrational spectrometry on a single particle that oscillates in a potential well, forming a quasi-molecule.     

Analysis of the temporal power spectral models of angel of arrival fluctuations for optical waves propagating through weak non-Kolmogorov turbulence

New analytical expressions for the temporal power spectral models of angle of arrival (AOA) fluctuations are derived for optical plane and spherical waves propagating through weak non-Kolmogorov turbulence. They consider the finite turbulence inner and outer scales, and have a general power law value in the range of 3–4 instead of the standard power law value of 11/3. The results derived in this work can reduce correctly to the previously published analytic expressions for the case of plane and spherical waves propagation through Kolmogorov turbulence case. These results are useful for the understanding the potential impact of derivations from the standard Kolmogorov spectrum.     

Turbulence of incompressible liquid flow near local equilibrium and the principle of secondary maximum of entropy

The variational principle of maximum entropy is used to describe the dynamics of weakly nonequilibrium turbulence using the theory of Reynolds stresses for viscous incompressible liquid flow. From this principle, equations closing the theory of Reynolds stresses and also equations describing mean flow-turbulence interaction for 3D turbulent flows are derived. The theory is reduced to 2D flows and weak turbulence. Thermodynamic analogues and an example of Couette flow are considered.     

Fluctuations of spherical waves in a turbulent atmosphere: effect of the axisymmetric approximation in computational methods

The validity of the axisymmetric parabolic-equation (PE) method for line-of-sight sound propagation in a turbulent atmosphere is investigated. The axisymmetric PE method is a finite-difference method for solving a 2D parabolic wave equation, which follows from the 3D wave equation by the assumption of axial symmetry around the vertical axis through the source. It is found that this axisymmetric approximation has a considerable spurious effect on the fluctuations of the sound field. This is concluded from analytical expressions for the log-amplitude and phase variances, derived both for isotropic turbulence and for axisymmetric turbulence. The expressions for axisymmetric turbulence are compared with the results of numerical computations with the PE method.