Drift Turbulence Research Articles

Passive scalar in a random wave field: the weak turbulence approach

We consider the evolution of a passive scalar advected by a velocity field which is a superposition of random linear waves. An equation for the average concentration of the passive scalar is derived (in the limit of small molecular diffusion) using the weak turbulence methodology. In addition to the enhanced diffusion, this equation contains the correction to the (Stokes) drift. Both of these terms have the fourth order with respect to wave amplitudes. The formulas for the coefficients of turbulent diffusion and turbulent drift are derived.

Slowly decaying drift turbulence with wave effects

The effect of linear waves on the Charney–Hasegawa–Mima model of drift and geostrophic turbulence is studied numerically for a slowly decaying field. It is shown that wave effects can reduce the efficiency of nonlinear mode coupling, effectively ‘freezing’ the spectrum to a narrow band in wavenumber space. Selective nonlinear interactions tend to favour velocities parallel to the direction of propagation of the waves. Accordingly, anisotropic spectra are observed, steeper in the direction of wave propagation, and shallower in the perpendicular direction.

Turbulent Plasma Transport and State of Equipartition

Turbulent transport may drive a plasma towards a state of Turbulent EquiPartition (TEP). It can be regarded as an attractor with a uniform distribution of Lagrangian invariants respected by the turbulent perturbations in the medium, notably a tokamak plasma. An attractor (the canonical profiles) has been described by invariants, while adequate hydrodynamics has not been derived. In the present paper the model of two dimensional hydrodynamics is suggested assuming low-frequency electrostatic perturbations and the hypothesis that the poloidal, but not toroidal component of the frozen-in low holds in tokamaks. It is shown that instability which drives plasma to the state of TEP is drift-flute type instability and the driving force of the instability is the pressure gradient combining with the shear of the magnetic field. The analysis of the condition for the instability to be developed leads to the important conclusion that the negative magnetic shear stabilizes the instability and thus can suppress the source of drift turbulence in large scale tokamaks, as has already been seen in many recent experiments and in gyro-particle simulations.

Warm-ion drift Alfvén turbulence and the L-H transition

Computations of fluid drift turbulence treating ions and electrons on equal footing, including both temperatures, are conducted in a model toroidal geometry. The resulting `ion mixing mode' turbulence bears features of both electron drift-Alfvén and ion temperature gradient turbulence, and nonlinear sensitivity to the relative strengths of the density and temperature gradients provides a possible route to the bifurcation needed for the L-H transition.

Experiments on Preplanetary Dust Aggregation

For the study of low-velocity dust interactions in the early solar nebula, we have performed two sets of experiments. In the first type of experiments, we studied the grain-mass evolution of a dust cloud embedded in a rarefied turbulent gas environment in which the initially monodisperse spherical SiO2grains (1.9 μm diameter) rapidly aggregate. The analysis of the resulting aggregate structures revealed that the clusters were formed by ballistic cluster–cluster aggregation without restructuring and follow a mass–size relation of the formm∝aDfg, whereDf≈ 1.91 is the fractal dimension andagis the radius of gyration of an aggregate with the massm. A comparison with model calculations shows that the mean collisional velocityvcfalls into the interval 0.07 m s−1≲vc≲ 0.5 m s−1. By extraction of the fractal aggregates from the turbomolecular pump and injection into a levitation tube in the second set of experiments, we were able to observe individual collisions between the aggregates. These types of simulations were performed in the laboratory so that the dominant source of the collisions was relative sedimentation. We investigated 28 collisions between aggregates with monomer numbers betweeni= 1 andi≈ 100 in the collision velocity range 0.001 m s−1≲vc≲ 0.01 m s−1. Our observations show a sticking efficiency of βc= 1 for the above-mentioned aggregate masses and collision velocities with no signs of grain restructuring.As we have attached importance to the similarity between our laboratory experiments and the situation in the solar nebula, e.g., grain size and composition, collision velocities, and friction regime, the results of our investigations are directly applicable to the solar nebula modeling and may be used for time-scale estimations of the aggregate growth in the early Solar System. Our experiments suggest that an ensemble of dust grains which is collisionally self-interacting caused by gas drag effects, such as sedimentation, radial drift, or gas turbulence, will adopt a bell-shaped mass distribution. This evolution results in fractal aggregates with fractal dimensions below or close toDf= 2.

Alfvén-drift turbulence suppression as triggering mechanism for LH transition

The Alfven drift turbulence suppression at the plasma edge is suggested as a triggering mechanism for the L to H transition. The stability theory of Alfven drift-waves shows that with increasing plasma pressure the Alfven waves get coupled to electron drift waves and as a consequence the unstable long wavelength perturbations (most important for transport) are suppressed. The instability can be characterised by two significant parameters, i.e. the normalised plasma beta, β n , and the normalised collision frequency, v n . The resulting turbulent transport coefficient is suppressed when the normalised beta is greater than a critical value, i.e. β n >1+v 2/3 , which depends on the normalised collision frequency v n . The transport coefficients change their dependence on plasma parameters at this threshold. Therefore, the possible scenario for the development of the H-mode could be associated with the stabilisation of the electron fluctuation at the plasma edge. The Alfven drift-wave model predicts the experimental trend of a roughly linear dependence of threshold temperature on magnetic field, with a weak dependence on density at high densities and a strong dependence on density at lower densities.

Finite Beta Electro‐Magnetic Effects at the Edge and the Role of the Scrape‐off Layer in the L‐H Transition

AbstractMagnetic induction becomes important when the skin‐depth equals the wavelength of the electrostatic drift turbulence: \documentclass{article}\pagestyle{empty}\begin{document}$ C/\sqrt 4 \pi \sigma \omega = K_ \bot ^{ - 1} $ \end{document}. It is predicted that this suppresses the longwave perturbations responsible for the bulk of the turbulent transport, providing an explanation for the L‐H transition. Since k⟂ρs ≪ 1 at the edge (ρs ‐ ion Larmor radius), this may occur at rather low β, typical of the separatrix parameters. The above equality is used to derive thresholds for the L‐H transition: β ≈ (k⟂ρs)2(me/mi)in the collisionless skin‐depth limit of very low density plasmas and \documentclass{article}\pagestyle{empty}\begin{document}$ T^3 _e \sim B^2 L_ \bot Z_{eff} /\sqrt m $\end{document} (L⟂‐perpendicular decay length) in the collisional skin‐depth limit of medium to high density plasmas. The latter is in a broad agreement with experiment, pointing to a possible role of low temperature collisional plasma (which is in the SOL) in the physics of the L‐H transition.

Three-dimensional computation of drift Alfvén turbulence

A transcollisional, electromagnetic fluid model, incorporating the parallel heat flux as a dependent variable, is constructed to treat electron drift turbulence in the regime of tokamak edge plasmas at the L - H transition. The resulting turbulence is very sensitive to the plasma beta throughout this regime, with the scaling with rising beta produced by the effect of magnetic induction to slow the Alfvénic parallel electron dynamics and thereby leave the turbulence in a more robust, non-adiabatic state. Magnetic flutter and curvature have a minor qualitative effect on the turbulence mode structure and on the beta scaling, even when their quantitative effect is strong. Transport by magnetic flutter is small compared to that by the flow eddies. Fluctuation statistics show that while the turbulence shows no coherent structure, it is coupled strongly enough so that neither density nor temperature fluctuations behave as passive scalars. Both profile gradients drive the turbulence, with the total thermal energy transport varying only weakly with the gradient ratio, . Scaling with magnetic shear is pronounced, with stronger shear leading to lower drive levels. Scaling with either collision frequency or magnetic curvature is weak, consistent with their weak qualitative effect. The result is that electron drift turbulence at L - H transition edge parameters is drift Alfvén turbulence, with both ballooning and resistivity in a clear secondary role. The contents of the drift Alfvén model will form a significant part of any useful first-principles computation of tokamak edge turbulence.

Critical parameters for turbulent transport in the SOL: mechanism for the L - H transition and its impact on the H-mode power threshold and density limits

An explanation for the L - H transition through the electromagnetic mechanism (suppression of drift wave turbulence by the skin effect) is offered. In dimensional space, the bifurcation is attributed to the involvement of two parameters in L - H transition physics: and (the latter scales as in physical parameters). The maximum of the diffusion coefficient, corresponding to the L - H transition, is reached when the collisional skin depth ( is the drift frequency) equals the characteristic radial displacement of the drift turbulence. The same criterion can also be presented as: - the condition for equal rates of the plasma diffusion into the magnetic field and the diffusion of the magnetic field into the plasma. The analysis yields the combination (scales as in dimensionless parameters) as a critical parameter for the L - H transition, for the case of the scaling for the wavevector of the drift turbulence. This threshold parameter should be applied near the separatrix position. At low densities, the requirement that the collisionless skin depth must be smaller than the radial displacement of the drift fluctuations in the L-mode, which is necessary for turbulence suppression, determines the threshold for the L - H transition. The proposed mechanism for the L - H transition clarifies the R dependence of the H-mode power threshold: scaling is predicted, with for in ITER. The critical parameter for the L - H transition, together with dimensionless parameters characterizing pressure gradient and resistivity, create the set of similarity parameters describing ELM behaviour. The scaling for the separatrix density, normalized to the Greenwald density limit with the machine size and toroidal field which ensures `similar' ELM behaviour, can thus be obtained. For the fixed similarity parameters, the analysis yields weak but favourable dependence of on the major radius. In recent experiments on JET and other machines, the degradation in the edge confinement associated with increased ELM frequency was found to be responsible for the density limit in high-power H-modes. Owing to the approximately dependence, an excess over the Greenwald limit, , by about 30% higher in ITER compared with JET for `similar' conditions (q, , separatrix and the ratio, wall conditions, the use of pellets etc) in ELMy H-modes is predicted. This is with the provision that a limit on the central density, related to mechanisms in the plasma core, is not encountered.

Transport properties of energetic particles in a turbulent electrostatic field

The diffusion of test particles in a turbulent electrostatic field is investigated numerically. The field is obtained by solving the Hasegawa–Mima model for two-dimensional drift turbulence. It is shown that nonlinear coupling significantly reduces the level of transport compared to the linear regime, and the physical mechanisms leading to this effect are analyzed in detail. Finite Larmor radius effects, relevant to alpha-particle transport in tokamaks, also reduce the diffusion rate. The scaling of the diffusion coefficient with Larmor radius is derived from a closure theory, and the predictions are compared to results from computer experiments. It is suggested that measured diffusion rates of particles with different Larmor radii can be used to obtain information about the turbulence.

Test-particle transport in strong electrostatic drift turbulence with finite Larmor radius effects.

Test-particle transport arising from $\mathbf{E}\ifmmode\times\else\texttimes\fi{}\mathbf{B}$ motion in a turbulent plasma is investigated numerically. The electrostatic field is determined by solving the Hasegawa-Mima model for two-dimensional drift turbulence. In the linear regime the particles experience stochastic diffusion, but in the fully nonlinear, strongly turbulent regime the diffusion rate is greatly reduced. Finite Larmor radius effects, relevant to alpha-particle transport in tokamaks, are also shown to strongly inhibit the level of transport.

Thermodynamic description of the relaxation of two-dimensional turbulence using Tsallis statistics.

Two-dimensional Euler turbulence and drift turbulence in a pure-electron plasma column have been experimentally observed to relax to metaequilibrium states that do not maximize the Boltzmann entropy, but rather seem to minimize enstrophy. We show that a recent generalization of thermodynamics and statistics due to Tsallis [Phys. Lett. A 195, 329 (1994); J. Stat. Phys. 52, 479 (1988)] is capable of explaining this phenomenon in a natural way. In particular, the maximization of the generalized entropy ${S}_{q}$ with $q=\frac{1}{2}$ for the pure-electron plasma column leads to precisely the same profiles predicted by the restricted minimum enstrophy theory of Huang and Driscoll [Phys. Rev. Lett. 72, 2187 (1994)]. These observations make possible the construction of a comprehensive thermodynamic description of two-dimensional turbulence.

A PROPOSAL FOR TREATMENT OF TURBULENT MIXING IN A TWO-PHASE SUBCHANNEL FLOW

Abstract We propose a practical method for the treatment of turbulent mixing rate in a two-phase subchannel flow in a hydrodynamic non-equilibrium state. Based on the assumption that the fundamental modes of the inter-subchannel fluid transfer in such a state are turbulent mixing, void drift, and diversion cross flow, the turbulent mixing rate is considered to be equal to that in the hydrodynamic equilibrium state that the flow will attain. The applicability of the method is examined by experiments concerning the axial variation in tracer concentration in a non-equilibrium flow without diversion cross flow. A good agreement is seen between the calculations and the measurements.

The effect of turbulent diffusion on transit-time determination by cross correlation of temperature fluctuations

At high Reynolds numbers heat transport by turbulent diffusion can have a significant effect on the correlation of temperature fluctuations in a flowing fluid. In such a case the conventional methods for transit-time estimation by noise correlation, which are based on the CCF or on the phase of the CPSD, give too low values. This paper provides a theoretical basis for the CCF and the CPSD of temperature fluctuations in the case of turbulent diffusion and drift. Theoretical findings are supported by experimental results. It is shown that more sophisticated methods for transit-time determination give accurate results in the case with and without diffusion. The differences found in estimated transit times are in the range of 5–8%.

Ergodic mixing for turbulent drift motion in an inhomogeneous magnetic field

The turbulent E×B drift of a test particle in an inhomogeneous magnetic field is not reducible to a simple diffusion, but rather leads to a biased diffusion producing an inhomogeneous density distribution (pinch effect). The statistical properties of the long-time chaotic two-dimensional drift motion of a charged particle in the magnetic field B(x,y) and the time-dependent electrostatic potential φ(x,y,t) are studied by numerical symplectic integration. For a conditionally periodic potential with two or more incommensurate frequencies, an ergodic behavior is demonstrated in which the probability density of the particle position is proportional to the magnetic field B. The accuracy of this prediction is found to be independent of the number Nω of the incommensurate frequencies for Nω≥2. The relation of this result with the Kolmogorov-Arnold-Moser theory is discussed.

The influence of drift flow turbulence on surface gravity wave propagation

The theory of surface gravity waves scattering at vortex flows in the ocean is developed in this paper. A scattering amplitude is found in the Born approximation as a function of vorticity which appears very convenient for investigation of scattering at simple localized flows. It is shown that the wave scattering cross-section is determined by the vertical component of vorticity. For a random (turbulent) vortex field the scattering cross-section per unit voume is determined by a vorticity correlation function. The damping of the coherent wave component and the angular spectrum widening are calculated for multiple scattering by vortex turbulence of drift flows. The spectrum angular width evolution for waves scattered at self-similar vortices of the logarithmic boundary layer is determined only by its dynamical speed and the wave vector. The latter result may be used for a remote sensing of oceanic turbulent drift flows based on observations of surface waves.

Linear dynamics of wind waves in coupled turbulent air—water flow. Part 1. Theory

When air blows over water the wind exerts a stress at the interface thereby inducing in the water a sheared turbulent drift current. We present scaling arguments showing that, if a wind suddenly starts blowing, then the sheared drift current grows in depth on a timescale that is larger than the wave period, but smaller than a timescale for wave growth. This argument suggests that the drift current can influence growth of waves of wavelength λ that travel parallel to the wind at speedc.In narrow ‘inner’ regions either side of the interface, turbulence in the air and water flows is close to local equilibrium; whereas above and below, in ‘outer’ regions, the wave alters the turbulence through rapid distortion. The depth scale,la, of the inner region in the air flow increases withc/u*a(u*ais the unperturbed friction velocity in the wind). And so we classify the flow into different regimes according to the ratiola/λ. We show that different turbulence models are appropriate for the different flow regimes.When (u*a+c)/UB(λ) [Lt ] 1 (UB(z) is the unperturbed wind speed)lais much smaller than λ. In this limit, asymptotic solutions are constructed for the fully coupled turbulent flows in the air and water, thereby extending previous analyses of flow over irrotational water waves. The solutions show that, as in calculations of flow over irrotational waves, the air flow is asymmetrically displaced around the wave by a non-separated sheltering effect, which tends to make the waves grow. But coupling the air flow perturbations to the turbulent flow in the water reduces the growth rate of the waves by a factor of about two. This reduction is caused by two distinct mechanisms. Firstly, wave growth is inhibited because the turbulent water flow is also asymmetrically displaced around the wave by non-separated sheltering. According to our model, this first effect is numerically small, but much larger erroneous values can be obtained if the rapid-distortion mechanism is not accounted for in the outer region of the water flow. (For example, we show that if the mixing-length model is used in the outer region all waves decay!) Secondly, non-separated sheltering in the air flow (and hence the wave growth rate) is reduced by the additional perturbations needed to satisfy the boundary condition that shear stress is continuous across the interface.

Nonlinear electrostatic turbulence

A variational method expressing the self-consistency of an electrostatic turbulence in a confined plasma is presented. It properly takes into account all types of resonances if the Kolmogorov broadening due the diffusion is small compared to the spectral width of the turbulent electric potential. The method is applied to study the possibility of a new type of drift turbulence in tokamaks where the various modes are strongly localized by nonlinear effects around their resonant surface. The passing electrons play a major role in driving such a turbulence, a result compatible with the observed increase of transport at the edge.

Transport inferred from consideration of particle orbits in drift turbulence

Particle orbits in the 2D potential hills/valley ('blobs') of low-frequency drift turbulence are analysed. For most of a given blob's lifetime a particle initially on the blob circulates on an orbit locked to that blob even though the blob shape and amplitude are changing. This locking corresponds to an adiabatic invariant being preserved. However, when the blob amplitude decreases to a critical value, the adiabatic invariant is destroyed, the locking ceases, and the particle detaches from the dying blob and attaches to a newly forming blob. This gives a detailed physical picture of the well known gyro-Bohm transport scaling D approximately gamma /k2 where gamma is the turbulence autocorrelation time (corresponding to blob lifetime) and k is the perpendicular wavenumber (corresponding to blob diameter). It is further shown that gyro-Bohm transport is consistent with global measurements in a wide variety of experiments, in addition to the large tokamaks to which it has traditionally been applied. Finally, it is shown that ion orbit behavior becomes qualitatively different if the blobs become sufficiently sharp and steep.

Nonlinear three-wave interaction description for a global drift wave turbulence

A nonlinear description of Drift and Rossby waves turbulence in a dissipative media with a driven force is built, using a statistic based on a three-modes interaction model. The statistical dynamics of the system is investigated on the basis of the existence of a most physically important triad (MPI triad). Then, the neglected modes are modeled as a white-noise contribution leading to a Langevin equation. A probability density function is obtained for the associated Fokker-Planck equation of the turbulent field. The spectrum cascades and the most physically important triad (MPI triad) are analyzed.