Sustained turbulence in the three-dimensional Gross-Pitaevskii model

We study the three-dimensional forced-dissipated Gross-Pitaevskii equation. We force at relatively low wave numbers, expecting to observe a direct energy cascade and a consequent power-law spectrum of the form k^−α. Our numerical results show that the exponent α strongly depends on how the inverse particle cascade is attenuated at ks lower than the forcing wave-number. If the inverse cascade is arrested by a friction at low ks, we observe an exponent which is in good agreement with the weak wave turbulence prediction k⁻¹. For a hypo-viscosity, a k⁻² spectrum is observed which we explain using a critical balance argument. In simulations without any low k dissipation, a condensate at k=0 is growing and the system goes through a strongly turbulent transition from a 4-wave to a 3-wave weak turbulence acoustic regime with evidence of k^−3/2 Zakharov-Sagdeev spectrum. In this regime, we also observe a spectrum for the incompressible kinetic energy which formally resembles the Kolmogorov k^−5/3, but whose correct explanation should be in terms of the Kelvin wave turbulence. The probability density functions for the velocities and the densities are also discussed.     

Density fluctuation spectrum in whistler turbulence

We develop a nonlinear two-dimensional fluid model of whistler turbulence that includes effect of electron fluid density perturbations. The latter is coupled nonlinearly with wave magnetic field. This coupling leads essentially to finite compressibility effects in whistler turbulence model. We find from our simulations that despite strong compressibility effects, the density fluctuations follow the evolution of the wave magnetic field fluctuations. In a characteristic regime where large scale whistlers are predominant, the coupled density fluctuations are found to follow a Kolmogorov-like phenomenology in the inertial range turbulence. Consequently, the turbulent energy is dominated by the large scale (compared to electron inertial length) eddies and it follows a Kolmogorov-like k−7/3 spectrum, where k is a characteristic wavenumber.     

Logarithmic correlation functions in two-dimensional turbulence

We consider the correlation functions of two-dimensional turbulence in the presence and absence of a three-dimensional perturbation, by means of conformal field theory. In the presence of three-dimensional perturbation, we show that in the strong coupling limit of a small scale random force, there is some logarithmic factor in the correlation functions of velocity stream functions. We show that the logarithmic conformal field theory c_8,1 describes the 2D-turbulence both in the absence and in the presence of the perturbation. We obtain the energy spectrum E(k) ∼ k^−5.125 ln(k) for perturbed 2D-turbulence and E(k) ∼ k⁻⁵ ln(k) for unperturbed turbulence. Recent numerical simulation and experimental results confirm our prediction.     

Universal power law for the energy spectrum of breaking Riemann waves

The universal power law for the spectrum of one-dimensional breaking Riemann waves is justified for the simple wave equation. The spectrum of spatial amplitudes at the breaking time t = t _b has an asymptotic decay of k ^?4/3, with corresponding energy spectrum decaying as k ^?8/3. This spectrum is formed by the singularity of the form (x ? x _b )^1/3 in the wave shape at the breaking time. This result remains valid for arbitrary nonlinear wave speed. In addition, we demonstrate numerically that the universal power law is observed for long time in the range of small wavenumbers if small dissipation or dispersion is taken into account in the viscous Burgers or Korteweg-de Vries equations.     

Isotropization of two-dimensional hydrodynamic turbulence in the direct cascade

We present results of numerical simulation of the direct cascade in two-dimensional hydrodynamic turbulence (with spatial resolution up to ). If at the earlier stage (at the time of order of the inverse pumping growth rate τ-Γ_max ^?1), the turbulence develops according to the same scenario as in the case of a freely decaying turbulence [1, 2]: quasi-singular distribution of di-vorticity are formed, which in k-space correspond to jets, leading to a strong turbulence anisotropy, then for times of the order of 10τ turbulence becomes almost isotropic. In particular, at these times any significant anisotropy in the angular fluctuations for the energy spectrum (for a fixed k) is not visible, while the probability distribution function of vorticity for large arguments has the exponential tail with the exponent linearly dependent on vorticity, in the agreement with the theoretical prediction [3].     

Turbulence spectra generated by singularities

The problem of turbulence spectra generated by the singularities located on lines and planes is considered. It is shown that the frequency spectrum of fluid-surface displacements due to whitecaps (linear singularities) is scaled like a weakly turbulent Zakharov-Filonenko spectrum. The corresponding wave-vector spectrum may be highly anisotropic with a decrease in maximum, as in the Phillips spectrum. However, in the isotropic situation, the spectrum differs markedly from the Phillips form. For a highly anisotropic two-dimensional turbulence, the vorticity jumps can generate the Kraichnan power-law distribution in the region of maximal angular peak. For the isotropic distribution, the turbulence spectrum coincides with the Saffman spectrum. For the shock-generated acoustic turbulence, the spectrum has the form of the Kadomtsev-Petviashvili spectrum Eω～ ω^?2 for all spatial dimensionalities.     

Developing homogeneous isotropic turbulence

We investigate the self-similar evolution of the transient energy spectrum, which precedes the establishment of the Kolmogorov spectrum in homogeneous isotropic turbulence in three dimensions using the EDQNM closure model. The transient evolution exhibits self-similarity of the second kind and has a non-trivial dynamical scaling exponent, which results in the transient spectrum having a scaling that is steeper than the Kolmogorov k^−5/3 spectrum. Attempts to detect a similar phenomenon in DNS data are inconclusive, owing to the limited range of scales available.     

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.     

Fractality feature in incompressible three-dimensional magnetohydrodynamic turbulence

Direct Numerical Simulation (DNS) of decaying isotropic 3D magnetohydrodynamic (MHD) turbulence based on the 10243-modes in a periodic box is used to study the statistical properties of turbulence. In this paper, the presence of intermittency in MHD turbulence is investigated through the analysis of the Probability Distribution Function (PDF) for Elsässer fields and total energy fluctuations. We observe that the PDFs of the Elsässer fields fluctuations display a strong non-Gaussian behavior at small scale, which can be ascribed to multifractality feature, while the PDFs of the total energy fluctuations have the same shape over all observed scales and are monofractal. The PDFs have stretched exponential tail and satisfy the function P(|δX|) ～ exp(?A|δX| ^μ ). Numerically, we extract the exponent μ and find that it is constant for monofractal behavior as the length scale varies. To check the notion of self-similarity in the respective fluctuation, we apply the compensated structure functions.     

Ion sound wave excitation in a plasma under a Langmuir turbulence regime

Ion sound wave excitation in a warm non-relativistic (W_b ≤ 400 eV) electron beam unmagnetized plasma system is studied experimentally. The spectrum of these waves shows two peaks at frequencies of 70 and 230 kHz respectively. The origin of these waves is connected with modulational instability and cavity collapse. We show that the energy of bulk accelerated electrons can explain the measured value of kr_De for high-frequency sound waves. The energy of ion sound waves is not high enough to have an influence on the Langmuir turbulence dynamics.     

Probability density,diagrammatic technique,and epsilon expansion in the theory of wave turbulence

We apply the methods of field theory to study the turbulent regimes of statistical systems. First we show how one can find their probability densities. For the case of the theory of wave turbulence with four-wave interaction we calculate them explicitly and study their properties. Using those densities we show how one can in principle calculate any correlation function in this theory by means of direct perturbative expansion in powers of the interaction. Then we give the general form of the corrections to the kinetic equation and develop an appropriate diagrammatic technique. This technique, while resembling that of ϕ⁴ theory, has many new distinctive features. The role of the ϵ = d − 4 parameter of ϕ⁴ theory is played here by the parameter κ = β + d − α − γ where β is the dimension of the interaction, d is the space dimension, α is the dimension of the energy spectrum and γ is the “classical” wave density dimension. If κ > 0 then the Kolmogorov index is exact, and if κ < 0 then we expect it to be modified by the interaction. For κ a small negative number, α < 1 and a special form of the interaction we compute this modification explicitly. We neglect the encountered IR divergencies with the intend to study them in a later publication.     

The energy spectrum of fully developed turbulence generated by random stirring

For fully developed turbulence in an incompressible fluid described by the Navier-Stokes equations with Gaussian random forces the relation between the energy spectrum and the stirring mechanism is investigated within a variational approach. Therein, the effect of nonlinear mode coupling is approximated by a wave number dependent eddy viscosity determined via a nonlinear integral equation for the energy spectrum. For various stirring spectra analytic approximations are compared with the solution obtained numerically with a cutoff in the integral kernel which ensures in eddy relaxing processes that the stirring forces exert strain only on scales larger than the eddy size. The results are compared with renormalization group calculations and closure approximations. Random forces injecting energy at a ratek ^–1 into the wave number banddk aroundk lead to a Kolmogorov distribution of energy. The spectrum of small-scale velocity fluctuations is shown to be universal in the sense that it remains unchanged under variations of the long wavelength stirring spectra.     

Wave turbulence buildup in a vibrating plate

We report experimental and numerical results on the buildup of the energy spectrum in wave turbulence of a vibrating thin elastic plate. Three steps are observed: first a short linear stage, then the turbulent spectrum is constructed by the propagation of a front in wave number space and finally a long time saturation due to the action of dissipation. The propagation of a front at the second step is compatible with scaling predictions from the Weak Turbulence Theory.     

Pre-equilibrium light particle emission in heavy ion induced reactions

We derive a master equation to describe the time evolution of the intrinsic state in heavy-ion collisions. It takes into account the energy supply from the relative motion due to the one-body dissipation mechanism, and the equilibration process and the particle emission due to the two-body residual interaction. We then calculate the energy spectrum and the angular distribution of the emitted protons in¹⁴N,¹⁶O andα induced reactions for various targets and incident energies. The angular distribution is calculated based on a simple phase-space consideration. The numerical results agree very well with experimental data.     

Ergodical time evolution of a Gaussian wave packet in quantum dots

The time evolution of a Gaussian wave packet (GWP) confined in a quantum dot is numerically studied. The quantum dots are modelled by a two-dimensional square box and by the potential x⁴ + y⁴. For the case of an incommensurate energy spectrum the time evolution of observables has no global period. As a result this leads to ergodic phase portraits with a finite phase volume. For the spatially wide GWP the distribution function of quantum observables may be approximated as a Gaussian one. For the case of commensurate transition frequencies in the quantum well the time evolution of observables is periodical and the phase portraits have a zero phase volume.     

Differential integral equation for the spectrum of isotropic homogeneous turbulence

The double and triple velocity correlations for isotropic homogeneous turbulence are constructed and used in v. Kármán-Howarth differential equation for isotropic correlations; it is reduced to a differential integral equation for the spectrum of turbulence; it contains only one dependent function (5.2). The equation of energy, which follows from the above equation, can be reduced toHeisenberg's type of equation for the spectrum of turbulence (5.6). Eddies of the same order of magnitudeinteract; they partly generate partly destroy votricity. Small eddies (large wave numbersk)act on large eddies by mainly destroying them.     

A time operator in quantum mechanics

The lower bound on a continuous energy spectrum suffices t⊙ mathematically preclude the construction of a hermitian time operator canonically conjugate to the Hamiltonian. This problem is overcome by enlarging the Hilbert space in such a way as to have either an unbound spectrum or a doubly degenerate positive spectrum. In the enlarged space, the eigenvalue spectrum θ of such an operator ranges from minus to plus infinity and constitutes the conjugate variable of the energy E. On the other hand, the evolution parameter t of the dynamical equations is related to the expectation value of the time operator. Both extensions yield the usual dynamics for state vectors restricted to the physical subspace. Vectors in the complement subspaces describe either negative energy wave packets with forward evolution or positive energy wave packets with backwards evolution. The time energy uncertainty relation is discussed.     

Hot electron photoluminescence in GaAs crystals

The spectrum and the linear polarization of photoluminescence of hot electrons in GaAs crystals were investigated. Oscillations in the hot photoluminescence (HPL) spectrum due to the subsequent emission of LO-phonons were observed. The study of HPL depolarization in an external magnetic field yielded the scattering time due to the emission of a LO-phonon by a hot electron in the Γ-valley (τ_?0 = 1 × 10^?13 sec) as well as the Γ?L intervalley scattering time. The radiative recombination of hot electrons created in the central Γ-valley via the subsidiary L-valley was observed. The distribution function of hot electrons in a wide energy range was evaluated from the spectra.