Comments on the definition of turbulence

Zusammenfassung Es werden Gründe erläutert, die zur Äusserung einer auf dem Begriff der Ergodizität beruhenden Definition der Turbulenz führen. Wir gehen von der Theorie der Unstabilität der Strömung bei Wirkung einer Störung der endlichen Amplitude aus. Es wird das Ergodenproblem im Landauschen Modell der entwickelten Turbulenz behandelt und einige mit der Ergodenhypothese zusammenhängende Folgen erörtert. Als ein Anfangselement gilt für uns der Begriff der Transitivität. Wir beachten auch die Beziehung zwischen der metrischen Transitivität und der Eigenschaft des Durchmischens der Bewegungen sowie die Relation zwischen dem energetischen Storungsspektrum und der Ergodenhypothese.     

On the large-scale atmospheric turbulence

Summary The dynamical equations of the velocity correlation and spectrum functions for three-dimensional anisotropic turbulence in a stratified, rotating atmosphere are derived and discussed. The characteristics of the horizontal two-dimensional isotropic turbulence are analyzed.National Center for Atmospheric Research, sponsored by the National Science Foundation. On leave from the University of Utah, Salt LakeCity, Utah.     

Local turbulence in the Earth's core

Abstract

It is shown that in the Earth's core, where the geodynamo is at work (and is supplied with energy by the prevailing unstable density stratification), a buoyancy instability of a local character exists which is highly supercritical. This instability results in fully developed turbulence dominated by small scale vortices. The influence of the Earth's rotation and of the magnetic field produced by the geodynamo makes this small scale turbulence highly anisotropic. A qualitative picture of this local anisotropic turbulence is devised and the main parameters characterizing it are estimated. Expressions for the turbulent diffusivity are developed and discussed.     

Modelling of convective turbulence with a two-equation k-ɛ turbulence closure scheme

 In this study we aim at comparing turbulence parameters from field observations and model simulations under convective conditions. The comparison is focused on the depth dependence and temporal dynamics of viscous and diffusive dissipation rates ɛ and χ. The near-surface observations were obtained by using a quasi-free-rising profiler which measured small-scale shear and temperature fluctuations to within the vicinity of the water surface. Convective conditions during the experiment are characterized by low wind speeds (between 0 and 4 m s⁻¹) and a typical heat loss of about 150 Wm⁻². We applied a state-of-the-art two-equation k-ɛ turbulence model with an algebraic second-moment closure scheme. The qualitative agreement of the turbulence quantities resulting from observations and simulations is rather good. The temporal dynamics of the temperature field is simulated correctly, whereas in the spatial dynamics some deficiencies of the model due to its local character can be seen. It is concluded that such models realistically reproduce convective turbulence and therefore represent a reasonable compromise between complexity and simplicity, so that they can be used with acceptable costs in large-scale models. Received: 31 October 2001 / Accepted: 20 September 2002 Acknowledgements The whole project was intitiated by Peter Schlittenhardt, who strongly supported the development of the observational technique in uprising mode and encouraged us to undertake the experiment in Lake Maggiore; for which we will thank him most. Performing the measurement campaign was only possible with the help of several colleagues from the Marine Environment Unit at the Joint Research Centre. Thanks to all of them, but especially to Dirk van d. Linde, Ulisse Devisioni, Bjarke Rasmussen and Hartmut Prandke. The Istituto Italiano di Idrobiologia, Verbania Pallanza, provided the boat Livia for the installation and removal of the measurement system; thanks to the master Stefano Maurizio for his engagement. Ute Tschesche provided the data evaluation and presentation software and Judith Challis helped to polish our English. Part of the study was␣supported by the PROVESS project (MAS3-CT97-0159); thanks to our sponsors at the European Commission. We are further grateful to two anonymous referees for their constructive comments. Responsible Editor: Charitha Pattiaratchi     

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.     

Modeling the effect of saltation on surface layer turbulence

Particle–turbulence interaction has been a research focus in the field of pneumatic transport, especially in aeolian environments. However, knowledge regarding the effect of saltating particles on the turbulence characteristics is very limited. In this article, a process of sand-laden flow from forming sand streamers to stability is investigated via a coupled mathematical model of wind-blown sand that includes the spatiotemporal development. The variations in the turbulence characteristics, such as the mean velocity and turbulence intensity in clean air or sand-laden flow field, are analyzed. The results show that the splash process of sand grains near the wall decrease the wind speed in the saltation layer and destroy the low-speed streaks. Moreover, the profiles of streamwise turbulence intensity exhibit a transition from ‘decreasing’ to ‘increasing’ and approximately intersect at an ‘intensity focus’, which is presented for the first time. Furthermore, it was found that saltating particles could enhance the Reynolds stress. Meanwhile, it was also noticed that the shear stress at the wall surface is greater than the impact threshold and that there is a tendency towards the impact threshold. Therefore, saltation makes the particle Reynolds number of sand-laden flow higher than that under non-saltation conditions, thus changing the particles’ effect on the turbulence intensity. Gravity-dominated saltation is probably the most essential difference between wind-blown sand and other traditional two-phase flows. © 2020 John Wiley & Sons, Ltd.     

Velocity and turbulence variations at the edge of saltmarshes

Saltmarsh vegetation significantly influences tidal currents and sediment deposition by decelerating the water velocity in the canopy. In order to complement previous field results, detailed profiles of velocity and turbulence were measured in a laboratory flume. Natural Spartina anglica plants were installed in a 3 m length test section in a straight, recirculating flume. Different vegetation densities, water depths and surface velocities were investigated. The logarithmic velocity profile, which existed in front of the vegetation, was altered gradually to a skimming-flow profile, typical for submerged saltmarsh vegetation. The flow reduction in the denser part of the canopy also induced an upward flow (the current was partially deflected by the canopy). The skimming flow was accompanied by a zone of high turbulence co-located with the strongest velocity gradient. This gradient moved upward and the turbulence increased with distance from the edge of the vegetation. Below the skimming flow, the velocity and the turbulence were low. The structure of the flow in the canopy was relatively stable 2 m into the vegetation. The roughness length (z₀) of the vegetation depends only on the vegetation characteristics, and is not sensitive to the current velocity or the water depth. Both the reduced turbulence in the dense canopy and the high turbulence at the top of the canopy should increase sediment deposition. On the other hand, the high turbulence zone just beyond the vegetation edge and the oblique upward flow may produce reduced sedimentation; a phenomenon that was observed near the vegetation edge in the field.     

Properties of electric turbulence in the polar cap ionosphere

Small-scale (scales of ∼0.5–256 km) electric fields in the polar cap ionosphere are studied on the basis of measurements of the Dynamics Explorer 2 (DE-2) low-altitude satellite with a polar orbit. Nineteen DE-2 passes through the high-latitude ionosphere from the morning side to the evening side are considered when the IMF z component was southward. A rather extensive polar cap, which could be identified using the ɛ-t spectrograms of precipitating particles with auroral energies, was formed during the analyzed events. It is shown that the logarithmic diagrams (LDs), constructed using the discrete wavelet transform of electric fields in the polar cap, are power law (μ ∼ s ^α). Here, μ is the variance of the detail coefficients of the signal discrete wavelet transform, s is the wavelet scale, and index α characterizes the LD slope. The probability density functions P(δE, s) of the electric field fluctuations δE observed on different scales s are non-Gaussian and have intensified wings. When the probability density functions are renormalized, that is constructed of δE/s ^γ, where γ is the scaling exponent, they lie near a single curve, which indicates that the studied fields are statistically self-similar. In spite of the fact that the amplitude of electric fluctuations in the polar cap is much smaller than in the auroral zone, the quantitative characteristics of field scaling in the two regions are similar. Two possible causes of the observed turbulent structure of the electric field in the polar cap are considered: (1) the structure is transferred from the solar wind, which is known to have turbulent properties, and (2) the structure is generated by convection velocity shears in the region of open magnetic field lines. The detected dependence of the characteristic distribution of turbulent electric fields over the polar cap region on IMF B _y and the correlation of the rms amplitudes of δE fluctuations with IMF B _z and the solar wind transfer function (B _y ² + B _z ²)^1/2sin(θ/2), where θ is the angle between the geomagnetic field and IMF reconnecting on the dayside magnetopause when IMF B _z < 0, together with the absence of dependence on the IMF variability are arguments for the second mechanism.     

Structure of triads in geostrophic turbulence

The interaction between Fourier waves, which results in energy transfer over the spectrum, has been considered using the Boussinesq model in a plane layer during rapid rotation as an example. It has been indicated that the wave triangle spectrum strongly differs on the scales shorter (longer) than the scale of a cyclonic turbulence leading mode.     

Broken ergodicity in magnetohydrodynamic turbulence

Turbulent magnetofluids appear in various geophysical and astrophysical contexts, in phenomena associated with planets, stars, galaxies and the universe itself. In many cases, large-scale magnetic fields are observed, though a better knowledge of magnetofluid turbulence is needed to more fully understand the dynamo processes that produce them. One approach is to develop the statistical mechanics of ideal (i.e. non-dissipative), incompressible, homogeneous magnetohydrodynamic (MHD) turbulence, known as “absolute equilibrium ensemble” theory, as far as possible by studying model systems with the goal of finding those aspects that survive the introduction of viscosity and resistivity. Here, we review the progress that has been made in this direction. We examine both three-dimensional (3-D) and two-dimensional (2-D) model systems based on discrete Fourier representations. The basic equations are those of incompressible MHD and may include the effects of rotation and/or a mean magnetic field B _o. Statistical predictions are that Fourier coefficients of the velocity and magnetic field are zero-mean random variables. However, this is not the case, in general, for we observe non-ergodic behavior in very long time computer simulations of ideal turbulence: low wavenumber Fourier modes that have relatively large means and small standard deviations, i.e. coherent structure. In particular, ergodicity appears strongly broken when B _o?=?0 and weakly broken when B _o?≠?0. Broken ergodicity in MHD turbulence is explained by an eigenanalysis of modal covariance matrices. This produces a set of modal eigenvalues inversely proportional to the expected energy of their associated eigenvariables. A large disparity in eigenvalues within the same mode (identified by wavevector k ) can occur at low values of wavenumber k?=?| k |, especially when B _o?=?0. This disparity breaks the ergodicity of eigenvariables with smallest eigenvalues (largest energies). This leads to coherent structure in models of ideal homogeneous MHD turbulence, which can occur at lowest values of wavenumber k for 3-D cases, and at either lowest or highest k for ideal 2-D magnetofluids. These ideal results appear relevant for unforced, decaying MHD turbulence, so that broken ergodicity effects in MHD turbulence survive dissipation. In comparison, we will also examine ideal hydrodynamic (HD) turbulence, which, in the 3-D case, will be seen to differ fundamentally from ideal MHD turbulence in that coherent structure due to broken ergodicity can only occur at maximum k in numerical simulations. However, a nonzero viscosity eliminates this ideal 3-D HD structure, so that unforced, decaying 3-D HD turbulence is expected to be ergodic. In summary, broken ergodicity in MHD turbulence leads to energetic, large-scale, quasistationary magnetic fields (coherent structures) in numerical models of bounded, turbulent magnetofluids. Thus, broken ergodicity provides a large-scale dynamo mechanism within computer models of homogeneous MHD turbulence. These results may help us to better understand the origin of global magnetic fields in astrophysical and geophysical objects.     

Stochastic similarities between the microscale of turbulence and hydro-meteorological processes

ABSTRACT

Turbulence is considered to generate and drive most geophysical processes. The simplest case is isotropic turbulence. In this paper, the most common three-dimensional power-spectrum-based models of isotropic turbulence are studied in terms of their stochastic properties. Such models often have a high order of complexity, lack stochastic interpretation and violate basic stochastic asymptotic properties, such as the theoretical limits of the Hurst coefficient, when Hurst-Kolmogorov behaviour is observed. A simpler and robust model (which incorporates self-similarity structures, e.g. fractal dimension and Hurst coefficient) is proposed using a climacogram-based stochastic framework and tested over high-resolution observational data of laboratory scale as well as hydro-meteorological observations of wind speed and precipitation intensities. Expressions of other stochastic tools such as the autocovariance and power spectrum are also produced from the model and show agreement with data. Finally, uncertainty, discretization and bias related errors are estimated for each stochastic tool, showing lower errors for the climacogram-based ones and larger for power spectrum ones.     

Formation of strong stationary vortex turbulence in the terrestrial magnetosheath

The microscopic consequences of the presence of nonlinear vortex structures in the near-Earth plasma dispersive medium are studied in this work. In dispersive media, strongly localized vortex structures contain trapped particles, cause pronounced density fluctuations, and intensify transfer processes, mixing in a medium; i.e., they can form strong vortex turbulence. Turbulence is represented as a gas in the ensemble of strongly localized (therefore, weakly interacting) identical vortices composing the ground state. Vortices with different amplitudes are randomly located in space (since they interact with one another) and are described statistically. It is assumed that the steady turbulent state is formed through a balance of mutually competing effects: spontaneous generation of vortices due to nonlinear steepening of the disturbance front, ^noise transfer to small scales, and collisional or collisionless damping of disturbances in the HF region. Noise scaling in the inertial interval takes place since structures merge during their collision. A magnetized plasma medium in the magnetosheath is considered. A new type of turbulent fluctuation spectra with respect to wavenumbers k ^−8/3, which is in satisfactory agreement with satellite observations in space plasma, has been determined. The medium particle diffusion on an ensemble of vortices has also been studied. It has been established that the interaction between structures themselves and between structures and medium particles causes anomalous diffusion in the medium. The effective diffusion coefficient square roothly depends on the noise stationary level.     

On the application of turbulence theory to the determination of marine evaporation

Summary Simultaneous measurements of wind, air temperature, and humidity profiles as well as of evaporation, presented byTakahashi (1958), were compared with relevant formulae suggested by turbulence theory. Satisfactory agreement could be achieved if, near the sea surface, the influence of molecular diffusion and, at higher levels, the effect of thermal stratification on the profiles were taken into account in addition to a suitable reduction of the value of evaporation obtained by means of a floating pan.The chief results of this report have been presented on 27th April 1962 to the Tenth Yearly Meeting of theSocietà Italiana di Geofisica e Meteorologia (Genova: 26–28 April 1962).     

Investigation of the broadband ELF turbulence by observations of the FAST satellite

Scaling properties of variable electric fields in the topside ionosphere have been investigated on scales s from ∼30 m to 2 km by FAST electric field observations with sample rate of 512 s⁻¹, in sixteen events of the broadband ELF turbulence. It is shown that down to scales of a few hundred meters, the power of turbulent electric fluctuations is a power law, ∼s ^α. Scaling index α derived from the slope of logarithmic diagrams (LD) constructed by the discrete wavelet transform of data can be estimated as α = 2.2 ± 0.3, which is close to α estimate earlier reported for scales 1–30 km by electric field observations of the Dynamics Explorer 2 satellite. The behavior of α index is analyzed near the scale of the order of electron inertial length λ_e = c/ω₀ (ω₀ being the electron plasma frequency). At altitudes considered (700–2500 km), λ_e makes 100–900 m. We demonstrate that at scales ≤λ_e, a decrease of LD slope and deviation from the power law are typically observed. As pointed out in the discussion, this feature cannot be identified as a transition to the diffusion range, where dissipation of the turbulence occurs.     

Characteristics of the vertical structure of the atmospheric turbulence in the Tibetan Plateau

Science China Earth Sciences - The Tibetan Plateau (TP) has unique atmospheric dynamics and thermal structures that originate from its giant terrain and complex climate. High vertical-resolution...     

A Reynolds-averaged turbulence modelling approach to the maintenance of the Venus superrotation

Abstract

A maintenance mechanism of an approximately linear velocity profile of the Venus zonal flow or superrotation is explored, with the aid of a Reynolds-averaged turbulence modelling approach. The basic framework is similar to that of Gierasch (Meridional circulation and maintenance of the Venus atmospheric rotation. J. Atmos. Sci. 1975, 32, 1038–1044) in the sense that the mechanism is examined under a given meridional circulation. The profile mimicking the observations of the flow is initially assumed, and its maintenance mechanism in the presence of turbulence effects is investigated from a viewpoint of the suppression of energy cascade. In the present work, the turbulent viscosity is regarded as an indicator of the intensity of the cascade. A novelty of this formalism is the use of the isotropic turbulent viscosity based on a non-local time scale linked to a large-scale flow structure. The mechanism is first discussed qualitatively. On the basis of these discussions, the two-dimensional numerical simulation of the proposed model is performed, with an initially assumed superrotation, and the fast zonal flow is shown to be maintained, compared with the turbulent viscosity lacking the non-local time scale. The relationship of the present model with the current general circulation model simulation is discussed in light of a crucial role of the vertical viscosity.     

Theorem of turbulent intensity and macroscopic mechanism of the turbulence development

Turbulence is one of the most common nature phenomena in everyday experience, but that is not adequately understood yet. This article reviews the history and present state of development of the turbulence theory and indicates the necessity to probe into the turbulent features and mechanism with the different methods at different levels. Therefore this article proves a theorem of turbulent transpor- tation and a theorem of turbulent intensity by using the theory of the nonequilibrium thermodynamics, and that the Reynolds turbulence and the Rayleigh-Bénard turbulence are united in the theorems of the turbulent intensity and the turbulent transportation. The macroscopic cause of the development of fluid turbulence is a result from shearing effect of the velocity together with the temperature, which is also the macroscopic cause of the stretch and fold of trajectory in the phase space of turbulent field. And it is proved by the observed data of atmosphere that the phenomenological coefficient of turbulent in- tensity is not only a function of the velocity shear but also a function of temperature shear, viz the sta- bility of temperature stratification, in the atmosphere. Accordingly, authenticity of the theorem, which is proved by the theory of nonequilibrium thermodynamics, of turbulent intensity is testified by the facts of observational experiment.