One-dimensional Turbulence Research Articles

Longitudinal structure formation in a nonlinear resonator

A fiber ring resonator, synchronously driven by picosecond pulses, exhibits a spatiotemporal instability. A rapid modulation in the pulse envelope is spontaneously formed and evolves dynamically. This structure may be called one-dimensional optical turbulence. Single-shot detection of these structures is not viable. We report how to obtain experimental information which, when combined with numerical simulation, provides an adequate picture of the processes at work. Intrapulse correlations are studied; they reveal clear differences between dispersion of either sign. Also, they reveal the formation of soliton-like structures in the turbulent regime. A criterion is given that allows the prediction of the degree of complexity.

Turbulent diffusion as a stochastic lagrangian process

AbstractThe present status of stochastic Lagrangian models of turbulent diffusion is reviewed, beginning with a brief historical introduction. The mathematical and physical foundations of these models are described. The derivation of the model for the Lagrangian trajectory of a single particle is illustrated for one‐dimensional inhomogeneous Gaussian turbulence. Single‐particle models for non‐Gaussian turbulence in the convective boundary layer are described. The mean concentration in fixed co‐ordinates is calculated from the statistics of thousands of single‐particle trajectories. Another class of stochastic Lagrangian models gives the relative dispersion of particle pairs and the concentration variance. A list of references is provided for further study.

Modeling energy flow in turbulent beam–plasma experiments

Further analysis of recent experiments improves agreement between a one-dimensional kinematic turbulence model and observations. The model includes both analytic and phenomenological elements. Time behavior, magnitude of turbulent electric fields, and radiated microwaves accord with data. Scaling a three-dimensional radiation rate improves agreement. Introducing ionization from the oscillating turbulent electric field does not improve agreement, arguing that this effect does not appear in typical beam–plasma systems. The model is simple and easy to apply to existing experiments.

The space and scale dependencies of the self-similar structure of turbulence

Two new techniques for analysing the space and scale dependencies of the self-similar structure of turbulence are introduced. Both methods are based on the wavelet transform, but use different wavelets. First, the concept of a ‘turbulence eddy’ is defined in terms of the Mexican hat wavelet transform of the velocity signal in such a way that the ‘eddy’ has a size and a location in space. A new scaling exponent, the eddy capacity D E (0 ≼ D E ≼ 1), is defined in terms of the zero crossings of this wavelet transform. D E is a measure of the space-fillingness of eddies and of their scaling in real space whereas the Hausdorff dimension D' H and the Kolmogorov capacity D' K are scaling exponents that are sensitive to the scaling and space-fillingness in wavenumber space. D E differs from both D' K and D' H in that it is particularly sensitive to phase correlations. For any random phase signal, D E = 1, whereas D' H and D' K depend on the energy spectrum. For well defined spiral signals, and for signals with a fractal intermittency in space, D E = D' K . Secondly, we propose a practical test for discriminating between H-fractals and K-fractals (i. e. between a dense and a sparse distribution of singularities). This method is based on successive averages of wavelet transforms, and the Morlet wavelet is used. The technique is shown to be robust with respect to large amounts of phase noise. Both methods are applied to a one-dimensional (1D) turbulence velocity signal of very high Reynolds number Re λ = 2720. It is found that D E = 1 for the inertial scales larger than the Taylor microscale λ, thus indicating some degree of phase scrambling of the 1D turbulence velocity signal at the larger inertial scales. There is some indication of a spiral-type structure in this signal at scales below λ. Finally, we measure the spatial fluctuation of wavelet energy, comparing an experimental turbulence signal with random phase and spiral-like signals. The magnitude of spatial fluctuations is set by the scrambled phase part of the turbulence, but the increase of energy fluctuation at small scales may be due to locally self-similar small-scale structures.

Numerical prediction of wave boundary layer over a bed with a change in roughness

The behaviour of the turbulent oscillatory wave boundary layer over a sudden change in the bed roughness has been studied using a one-dimensional two-equation turbulence model. The 1D model is being used to describe the 2D flow by making a Lagrangian use of the 1D model, making a sudden shift in time of the bed roughness when the fluid particle passes from a region with one roughness to a region with another roughness. Attention was concentrated on two quantities: the period-averaged velocity and the bed shear stress. The results indicated that the near-bed period averaged velocities are different from zero and directed towards the bed section with the larger roughness. This near-bed streaming was found to increase with increasing difference between the roughnesses of the two bed sections and with decreasing values of amplitude-to-roughness ratio. Streaming velocities as high as 10% of the maximum orbital velocity were predicted. The bed shear stress, on the other hand, responded with a strong peak near the step change in roughness; this amplification in the bed shear stress was found to increase with increasing difference in roughness and with increasing amplitude to roughness ratio. Amplifications experienced in the bed shear stress as large as 3–3.5 were predicted.

Roles of Drizzle in a One-Dimensional Third-Order Turbulence Closure Model of the Nocturnal Stratus-Topped Marine Boundary Layer

This study focuses on the effects of drizzle in a one-dimensional third-order turbulence closure model of the nocturnal stratus-topped marine boundary layer. When the simulated drizzle rate is relatively small (maximum approximately equal to 0.6 mm/day), steady-state solutions are obtained. The boundary layer stabilizes essentially because drizzle causes evaporative cooling of the subcloud layer. This stabilization considerably reduces the buoyancy flux and turbulence kinetic energy below the stratus cloud. Thus, drizzle tends to decouple the cloud from the subcloud layer in the model, as suggested by many observational studies. In addition, the evaporation of drizzle in the subcloud layer creates small scattered clouds, which are likely to represent cumulus clouds, below the solid stratus cloud in the model. The sensitivity experiments show that these scattered clouds help maintain a coupled boundary layer. When the drizzle rate is relatively large (maximum approximately equal to 0.9 mm/day), the response of the model becomes transient with bursts in turbulent fluxes. This phenomenon is related to the formation of the scattered cloud layer below the solid stratus cloud. It appears that the model is inadequate to represent the heat and moisture transport by strong updrafts covering a small fractional area in cumulus convection.

Inertial Ranges and Small-Scale Intermittency in One-Dimensional Turbulence Models

Abstract A set of one-dimensional turbulence models can be constructed by applying severe D-1 directional Fourier truncation to the D-dimensional fluid equations, allowing for numerical calculation over an extremely wide range of scales. At low resolution a reduced 2D model displayed both inviscid energy-enstrophy equipartition and a spectrum consistent with the enstrophy cascade phenomenology in the decay problem. In the present note, these results are extended by using reduced models to examine both the inverse (D = 2) and direct (D = 3) energy cascades. In addition, higher numerical resolution (up to 4096 grid points) is employed to demonstrate unequivocal adherence to the 2D phenomenologies. Small-scale intermittency in the form of spatially intermittent vorticity gradients is also observed. Simulations based on the reduced 3D model were less successful. Although the truncated inviscid equilibrium agreed with an energy equipratition spectrum, forced-viscous simulations failed to show a clear Kolmogoro...

Reply by the Authors to Bernard Etkin

P ROFESSOR Etkin's Technical Comment is very insightful. It 1) identifies an ambiguous statement we made in the Introduction to our paper; 2) puts into sharper focus the potential usefulness of the SDG method; and 3) goes on to discuss some recent turbulence-simulation work at the University of Toronto. This reply addresses the first two items and then ties in some related work in the area of time-correlated gust loads recently conducted at NASA Langley. In the second paragraph of his Technical Comment, Professor Etkin states that our judgment regarding the PSD method (conventional random process analysis) is not strictly true. The judgment he is referring to is contained in the second paragraph of our Introduction and states that the PSD method is not capable of computing a set of physically-realizeable loads. Then, in his third and fourth paragraphs, Professor Etkin proves that, for linear systems subjected to a Gaussian input, such physically realizeable loads may, indeed, be computed. Professor Etkin is correct. Our statement is true, however, when interpreted in the following manner: For an airplane modeled as a linear system subjected to one-dimensional Gaussian atmospheric turbulence, assume that root-mean-square (RMS) values of shear force (SF) and bending moment (BM) are computed (using conventional random process analysis) at many locations along the span of the wing. The integral of this RMS SF distribution along the span from the wing tip to any inboard location yt in general, will not be equal to the RMS value of BM at y(. In this sense, the loads from a conventional random process analysis are not physically realizeable. Also in his second paragraph, Professor Etkin states that the main advantage SDG has over PSD is the ability of the former to use non-Gaussian turbulence inputs. J. G. Jones, the developer of the SDG method, has expressed the same thought when he says indirectly A principal objective of the SDG method is to cover, in a unified approach, the cases of encounters with relatively isolated severe to extreme gusts and flight in more extensive patches of continuous turbulence of generally less intensity. The patches of continuous turbulence referred to here are Gaussian, and it is for this condition that the SDG-PSD overlap has been shown to exist. For this condition, specific values of gust intensity parameter, UQ [Eq. (1) of Ref. 1], fractional exponent, k [Eq. (1) of Ref. 1], and amplitude reduction factors, p^ [Eqs. (4) and (5) of Ref. 1] are required. The Venn diagram in the following sketch depicts schematically the SDG-

The peculiarities of diffusion of the galactic cosmic rays at low energies

I give an interpretation of a result of Simpsonet al. (1988) on the variation with kinetic energyT i of the mean pathlengthX m (T i ) of the galactic cosmic rays (CGRs) in the range 0.1≲T i ≲10.0 GeV nucl−1. I argue that the data onX m (T i ) may be interpreted in terms of a model of GCR diffusion on the one-dimensional Alfven-wave turbulence, having a cutoff in the spectrum at frequencies σ≲σ h , where σ h is the proton gyrofrequency. The cutoff results in changing of the character of variation of the GCR diffusion coefficientD(T i )∼T a in the rangeT i ≫1 GeV nucl−1 towards some more complicated variation at 0.1≲T i ≲1.0 GeV nucl−1 due to the peculiarities of the pitch-angle scattering at θ≈900.

Hydrodynamic equations for a plasma in the region of the distribution-function plateau

A hydrodynamic theory is presented for plasma particles whose distribution function is close to a plateau in some phase-space region for a finite one-dimensional phase-spase interval. In the framework of the methods of moments, a series expansion of the particle distribution function in an appropriate system of orthogonal (Legendre) polynomials near the state with a plateau is carried out. Deviations from the plateau are due to the presence of a heat flux. The expansion obtained for the distribution function is used to truncate the chain of hydrodynamic equations and to calculate the additional terms arising from the finiteness of the phase interval. The case of one-dimensional Langmuir turbulence is considered as an example. Expressions are presented for the moments of the quasilinear integral of collisions, which describe the variations of the macroscopic characteristics of resonant particles in the respective hydrodynamic equations. Criteria for applicability of the equations obtained are presented. Approximate invariants of motion are obtained in the extreme cases of narrow and broad plateaux.

Numerical study of dispersion in the convective boundary layer

A simple one-dimensional turbulence model developed by Sun and Chang (1986a, J. clim. appl. Met. 25,1445–1453) is modified by including γ c (a reference potential temperature gradient) in the diffusion term of the heat equation to study the moist convective boundary layer. The results show that the stratification becomes neutral to slightly unstable in the middle and lower mixed layer but slightly stable in the upper mixed layer as observed, while the wind, moisture and eddy fluxes are very similar to those discussed by Sun and Chang. The results from the one-dimensional model are then applied to a two-dimensional dispersion model. The new concentrations are better than the previous results (Sun, 1986, Atmospheric Environment 20, 1877–1886). The ground level concentration, the mean height of concentration, and vertical speed simulated from this higher order closure model are also in good agreement with field experiments, water tank simulations and results obtained from the statistical models.

One-dimensional turbulence in a viscous polytropic gas

One-dimensional turbulence in a viscous polytropic gas

A one-dimensional turbulence model; Enstrophy cascades and small-scale intermittency in decaying turbulence

Abstract Severe unidirectional Fourier truncation of the equations for 2-D incompressible flow leads to a system of three coupled PDEs in one space dimension with the same quadratic invariants as the original set (i.e. energy and enstrophy). Numerically generated equilibria for inviscid, truncated versions of the reduced system are well approximated by Kraichnan's energy-enstrophy equipartition spectra. Viscous calculations for decaying turbulence at moderate resolution (1024 degrees of freedom) also appear to be consistent with a direct, k −3, enstrophy cascading inertial range when the dissipation is small. Dissipation range intermittency in the form of spatially intermittent enstrophy dissipation with occasional strong bursts producing linear phase locking is also observed. In contrast to full 2-D simulations, no tendency towards the emergence of isolated, coherent vorticity structures is observed. The model consequently mimics some, but not all, of the properties of the full 2-D set.

A least squares formulation of one-dimensional turbulence models

A least squares formulation of one-dimensional turbulence models

A Nocturnal Atmospheric Drainage Flow Simulation Investigating the Application of One-Dimensional Modeling and Current Turbulence Schemes

Abstract We developed a one-dimensional boundary layer model to simulate nocturnal atmospheric drainage flow on a simple forest-covered slope using canopy, soil and radiation parameterizations from previous studies along with turbulence simulation (from Mellor-Yamada) level three. A simulation of the drainage at Unit 19 of the ASCOT Geysers Project, 24 September 1980, is presented, from which we find that whatever horizontal flow-structure assumptions are made significantly affect the results through implied vertical advections. We recommend use of vertical-motion patterns obtained in relatively coarse two-dimensional simulations as a possible solution. Turbulence as simulated develops significant anisotrophy. Careful simulation of turbulence structure will require some reexamination of present turbulence modeling techniques. Nevertheless, using level three and assuming horizontal homogeneity, we were able to simulate well the mean profiles of wind and temperature at Unit 19 for one time when the ambient ...

Nonequilibrium statistical mechanics of one-dimensional turbulence

The method of statistical mechanics is applied to the study of the one-dimensional model of turbulence proposed in an earlier paper. The closure problem is solved by the variational approach which has been developed for the three-dimensional case, yielding two integral equations for two unknown functions. By solving the two integral equations, the Kolmogorov k−5/3 law is derived and the (one-dimensional) Kolmogorov constant Ko is evaluated, obtaining Ko=0.55, which is in good agreement with the result of numerical experiments on one-dimensional turbulence.

Numerical experiments on one-dimensional model of turbulence

The initial-value problem of a forced Burgers equation is numerically solved by the Fourier expansion method. It is found that its solutions finally reach a steady state of ‘‘laminar flow’’ which has no randomness and is stable to disturbances. Hence, strictly speaking, the so-called Burgers turbulence is not a turbulence. A new one-dimensional model is proposed to simulate the Navier–Stokes turbulence. A series of numerical experiments on this one-dimensional turbulence is made and is successful in obtaining Kolmogorov’s k−5/3 inertial-range spectrum. The (one-dimensional) Kolmogorov constant ranges from 0.5 to 0.65.

Nonlinear waves and one-dimensional turbulence in nondispersive media

The main results of the theory based on the solution of Burgers' equation for large Reynolds numbers are reviewed. The basic properties of the arising stochastic regime, which is an example of strong turbulence, and its relation to hydrodynamic turbulence are discussed. Different stages in the evolution of a nonlinear wave are interpreted from the point of view of a flow of noninteracting particles. The statistical properties of Riemannian waves are analyzed for the stage of single-stream propagation. Methods for describing and the characteristics of the turbulence of sawtooth waves, forming at the many-stream stage, are examined. The self-preserving nature of this regime is demonstrated. The coupling of regular and random waves at different stages of propagation is examined. The possibility of describing the evolution of the average velocity with the help of turbulent viscosity is analyzed. Possible generalizations of the theory to related problems are discussed.

Numerical experiments with a one-dimensional higher order turbulence model: Simulation of the Wangara Day 33 case

A one-dimensional stratocumulus model is developed and incorporated into a cloud/mesoscale model to simulate the evolution of the marine stratocumulus-capped mixed layer.

Dynamics of one-dimensional upper-hybrid turbulence in a magnetized plasma

It is shown that the modulational instability of upper-hybrid oscillations in a magnetized plasma gives rise to a one-dimensional collapse of the field distribution accompanied by significant absorption of the high-frequency wave energy.