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arXiv1994

A closure model for intermittency in three dimensional incompressible turbulence

A simplified Lagrangean closure for the Navier-Stokes equation is used to study the production of intermittency in the inertial range of three dimensional turbulence. This is done using localized wavepackets following the fluid rather than a standard Fourier basis. In this formulation, the equation for the energy transfer acquires a noise term coming from the fluctuations in the energy content of the different wavepackets. Assuming smallness of the intermittency correction to scaling allows the adoption of a quasi-gaussian approximation for the velocity field, provided a cutoff on small scales is imposed and a finite region of space is considered. In this approximations, the amplitude of the local energy transfer fluctuations, can be calculated self consistently in the model. Definite predictions are obtained on the scaling of the wavepacket energy moments.

chao-dyn↗

arXiv1996

Energy-Conserving Truncations for Convection with Shear Flow

A method is presented for making finite Fourier mode truncations of the Rayleigh--Benard convection system that preserve invariants of the full partial differential equations in the dissipationless limit. These truncations are shown to have no unbounded solutions and provide a description of the thermal flux that has the correct limiting behavior in a steady-state. A particular low-order truncation (containing 7 modes) is selected and compared with the 6 mode truncation of Howard and Krishnamurti (1986), which does not conserve the total energy in the dissipationless limit. A numerical example is presented to compare the two truncations and study the effect of shear flow on thermal transport.

chao-dyn↗

arXiv1997

Langevin approach to generate synthetic turbulence

We present an analytical scheme, easily implemented numerically, to generate synthetic Gaussian turbulent flows by using a linear Langevin equation, where the noise term acts as a stochastic stirring force. The characteristic parameters of the velocity field are well introduced, in particular the kinematic viscosity and the spectrum of energy. As an application, the diffusion of a passive scalar is studied for two different energy spectra. Numerical results are compared favorably with analytical calculations.

chao-dyn↗

arXiv1998

Cascades of energy and helicity in the GOY shell model of turbulence

The effect of extreme hyperviscous damping, $νk_n^p, p=\infty$ is studied numerically in the GOY shell model of turbulence. It has resently been demonstrated [Leveque and She, Phys. Rev. Lett, 75,2690 (1995)] that the inertial range scaling in the GOY model is non-universal and depending on the viscous damping. The present study shows that the deviation from Kolmogorov scaling is due to the cascade of the second inviscid invariant. This invariant is non-positive definite and in this sense analogous to the helicity of 3D turbulent flow.

chao-dyn↗

arXiv1997

Dispersion of passive tracers in closed basins: beyond the diffusion coefficient

We investigate the spreading of passive tracers in closed basins. If the characteristic length scale of the Eulerian velocities is not very small compared with the size of the basin the usual diffusion coefficient does not give any relevant information about the mechanism of spreading. We introduce a finite size characteristic time $τ(δ)$ which describes the diffusive process at scale $δ$. When $δ$ is small compared with the typical length of the velocity field one has $τ(δ) \sim λ^{-1}$, where $λ$ is the maximum Lyapunov exponent of the Lagrangian motion. At large $δ$ the behavior of $τ(δ)$ depends on the details of the system, in particular the presence of boundaries, and in this limit we have found a universal behavior for a large class of system under rather general hypothesis. The method of working at fixed scale $δ$ makes more physical sense than the traditional way of looking at the relative diffusion at fixed delay times. This technique is displayed in a series of numerical experiments in simple flows.

chao-dyn↗

arXiv1997

Different intermittency for longitudinal and transversal turbulent fluctuations

Scaling exponents of the longitudinal and transversal velocity structure functions in numerical Navier-Stokes turbulence simulations with Taylor-Reynolds numbers up to $\rel = 110$ are determined by the extended self similarity method. We find significant differences in the degree of intermittency: For the sixth moments the scaling corrections to the classical Kolmogorov expectations are $δξ_6^L= -0.21 \pm 0.01$ and $\dx_6^T= -0.43 \pm 0.01$, respectively, independent of $\rel$. Also the generalized extended self similarity exponents $ρ_{p,q} = \dx_p/\dx_q$ differ significantly for the longitudinal and transversal structure functions. Within the She-Leveque model this means that longitudinal and transversal fluctuations obey different types of hierarchies of the moments. Moreover, the She-Leveque model hierarchy parameters $β^L $ and $β^T$ show small but significant dependences on the order of the moment.

chao-dyn↗

arXiv1998

Hamilton's principle for quasigeostrophic motion

We show that the equation of quasigeostrophic (QG) potential vorticity conservation in geophysical fluid dynamics follows from Hamilton's principle for stationary variations of an action for geodesic motion in the f-plane case or its prolongation in the beta-plane case. This implies a new momentum equation and an associated Kelvin circulation theorem for QG motion. We treat the barotropic and two-layer baroclinic cases, as well as the continuously stratified case.

chao-dyn↗

arXiv1998

Burgers Turbulence with Large-scale Forcing

Burgers turbulence supported by white-in-time random forcing at low wavenumbers is studied analytically and by computer simulation. It is concluded that the probability density Q of velocity gradient displays four asymptotic regimes at very large Reynolds number: (A) a region of large positive gradient where Q decays rapidly (reduction of gradient by stretching); (B) an intermediate region of negative gradient where Q falls off as the inverse third power of gradient (transient inviscid steepening of negative gradient); (C) an outer power-law region of negative gradient where Q falls off as the reciprocal of gradient (shoulders of mature shocks); (D) a final region of large gradient where Q decays very rapidly (interior of mature shocks). The probability density of velocity difference across an interval r, divided by r, lies on Q throughout regions A and B and into the middle of C, for small enough inertial-range r.

chao-dyn↗

arXiv1999

Large-scale properties of passive scalar advection

We consider statistics of the passive scalar on distances much larger than the pumping scale. Such statistics is determined by statistics of Lagrangian contraction that is by probabilities of initially distant fluid particles to come close. At the Batchelor limit of spatially smooth velocity, the breakdown of scale invariance is established for scalar statistics.

chao-dyn↗

arXiv1999

Note on Forced Burgers Turbulence

A putative powerlaw range of the probability density of velocity gradient in high-Reynolds-number forced Burgers turbulence is studied. In the absence of information about shock locations, elementary conservation and stationarity relations imply that the exponent $-α$ in this range satisfies $α\ge3$, if dissipation within the powerlaw range is due to isolated shocks. A generalized model of shock birth and growth implies $α=7/2$ if initial data and forcing are spatially homogeneous and obey Gaussian statistics. Arbitrary values $α\ge3$ can be realized by suitably constructed homogeneous, non-Gaussian initial data and forcing.

chao-dyn↗

arXiv1999

Free Decay of Turbulence and Breakdown of Self-Similarity

It has been generally assumed, since the work of von Karman and Howarth in 1938, that free decay of fully-developed turbulence is self-similar. We present here a simple phenomenological model of the decay of 3D incompressible turbulence, which predicts breakdown of self-similarity for low-wavenumber spectral exponents $n$ in the range $n_c<n<4$, where $n_c$ is some threshold wavenumber. Calculations with the eddy-damped quasi-normal Markovian approximation give the value as $n_c\approx 3.45$. The energy spectrum for this range of exponents develops two length-scales, separating three distinct wavenumber ranges.

chao-dyn↗

arXiv1996

Iso-topological relaxation, coherent structures, and Gaussian turbulence in two dimensional magnetohydrodynamics

The long-time relaxation of ideal two dimensional magnetohydrodynamic turbulence subject to the conservation of two infinite families of constants of motion---the magnetic and the "cross" topology invariants--is examined. The analysis of the Gibbs ensemble, where all integrals of motion are respected, predicts the initial state to evolve into an equilibrium, stable coherent structure (the most probable state) and decaying Gaussian turbulence (fluctuations) with a vanishing, but always positive temperature. The non-dissipative turbulence decay is accompanied by decrease in both the amplitude and the length scale of the fluctuations, so that the fluctuation energy remains finite. The coherent structure represents a set of singular magnetic islands with plasma flow whose magnetic topology is identical to that of the initial state, while the energy and the cross topology invariants are shared between the coherent structure and the Gaussian turbulence. These conservation laws suggest the variational principle of iso-topological relaxation which allows us to predict the appearance of the final state from a given initial state. For a generic initial condition having X points in the magnetic field, the coherent structure has universal types of singularities: current sheets terminating at Y points. These structures, which are similar to those resulting from the 2D relaxation of magnetic field frozen into an ideally conducting viscous fluid, are observed in the numerical experiment of Biskamp and Welter. The Gibbs ensemble method developed in this work admits extension to other Hamiltonian systems with invariants not higher than quadratic in the highest-order-derivative variables. The turbulence in two dimensional Euler fluid is of a different nature: there the coherent structures are also formed, but the fluctuations about these structures are non-Gaussian.

plasm-ph↗

arXiv1999

Magnetic field correlations in kinematic two-dimensional magnetohydrodynamic turbulence

The scaling properties of the second order magnetic structure function D_2^{(B)}(r) and the corresponding magnetic correlation function C_2^{(B)}(r) are derived for two-dimensional magnetohydrodynamic turbulence in the kinematic regime where the ratio of kinetic energy to magnetic energy is much larger than one. In this regime the magnetic flux function $ψ$ can be treated as a passive scalar advected in a two-dimensional turbulent flow. Its structure function $D_2^{(ψ)}(r)$ and the one for the magnetic field D_2^{(B)}(r) are connected by an exact relation. We calculate D_2^{(ψ)}(r) and thus D_2^{(B)}(r) within geometric measure theory over a wide range of scales r and magnetic Prandtl numbers Pr_m. The magnetic field correlations follow a r^{-4/3} - scaling law and show an anticorrelation at the beginning of the Batchelor regime indicative of the formation of strongly filamented current sheets. Differences to the full dynamic regime, where the ratio of kinetic to magnetic energies is smaller than in the kinematic case, are discussed.

chao-dyn↗

arXiv1999

How Events Come Into Being: EEQT, Particle Tracks, Quantum Chaos, and Tunneling Time

In sections 1 and 2 we review Event Enhanced Quantum Theory (EEQT). In section 3 we discuss applications of EEQT to tunneling time, and compare its quantitative predictions with other approaches, in particular with Büttiker-Larmor and Bohm trajectory approach. In section 4 we discuss quantum chaos and quantum fractals resulting from simultaneous continuous monitoring of several non-commuting observables. In particular we show self-similar, non-linear, iterated function system-type, patterns arising from quantum jumps and from the associated Markov operator. Concluding remarks pointing to possible future development of EEQT are given in section 5.

quant-ph↗

arXiv1997

Nonlinear Analysis of the Lightcurve of the Variable Star R Scuti

It is first shown that the observational light curve data of R Scuti, a star of the RV Tau type, is not multi-periodic, and that it cannot have been generated by a linear stochastic (AR) process. By default, the signal must be a manifestation of deterministic chaos. We use a novel nonlinear time-series analysis, the global flow reconstruction technique, to probe the properties of the irregular pulsation cycles. We show in particular that the chaotic dynamics of this star's complicated lightcurve is captured by a simple 4D polynomial map or flow (4 first order ODEs). Importantly also, the method allows us to quantify an irregular signal which has the potential benefit for extracting novel stellar constraints from an irregular light-curve. Finally, from the low dimensionality 4 of the flow we can infer a simple physical picture of the pulsations, and arguments are presented that the pulsations of R Sct are the result of the nonlinear interaction of two vibrational normal modes of the star.

astro-ph↗

arXiv1999

Resonant Orbits in Triaxial Galaxies

Box orbits in triaxial potentials are generically thin, that is, they lie close in phase space to a resonant orbit satisfying a relation of the form lω_1 +mω_2+nω_3=0 between the three fundamental frequencies. Resonant orbits are confined to a membrane; they play roughly the same role, in three dimensions, that closed orbits play in two. Stable resonant orbits avoid the center of the potential; orbits that are thick enough to pass near the center are typically stochastic. Very near the center, where the gravitational potential is dominated by the black hole, resonant orbits continue to exist, including at least one family whose elongation is parallel to the long axes of the triaxial figure.

astro-ph↗

arXiv1994

Chaos and Noise in Galactic Potentials

ABBREVIATED ABSTRACT: This paper summarises an investigation of the effects of weak friction and noise in time-independent, nonintegrable potentials which admit both regular and stochastic orbits. The aim is to understand the qualitative effects of internal and external irregularities associated, e.g., with discreteness effects or couplings to an external environment, which stars in any real galaxy must experience. The two principal conclusions are: (1) These irregularities can be important on time scales much shorter than the natural relaxation time scale t_R associated with the friction and noise. For stochastic orbits friction and noise induce an average exponential divergence from the unperturbed Hamiltonian trajectory at a rate set by the value of the local Lyapunov exponent. Even weak noise can make a pointwise interpretation of orbits suspect already on time scales much shorter than t_R. (2) The friction and noise can also have significant effects on the statistical properties of ensembles of stochastic orbits, these also occurring on time scales much shorter than t_R. Potential implications for galactic dynamics are discussed, including the problem of shadowing.

astro-ph↗

arXiv1996

Chaos in Static Axisymmetric Spacetimes I : Vacuum Case

We study the motion of test particle in static axisymmetric vacuum spacetimes and discuss two criteria for strong chaos to occur: (1) a local instability measured by the Weyl curvature, and (2) a tangle of a homoclinic orbit, which is closely related to an unstable periodic orbit in general relativity. We analyze several static axisymmetric spacetimes and find that the first criterion is a sufficient condition for chaos, at least qualitatively. Although some test particles which do not satisfy the first criterion show chaotic behavior in some spacetimes, these can be accounted for the second criterion.

gr-qc↗