Drift Wave-zonal Flow Turbulence Research Articles

Scale selection and feedback loops for patterns in drift wave-zonal flow turbulence

The scale selection and feedback loops for the formation and sustainment of a mesoscopic staircase profile structure are investigated for drift wave-zonal flow turbulence. A mean field model derived from the Hasegawa–Wakatani system and including the evolution of mean density, mean vorticity and perturbed potential enstrophy (PE) is used. It is found that a quasi-periodic zonal staircase forms from self-sharpening of modulation. The principle feedback loop is through the nonlinear dependence of mixing length on electron density gradient, which enters by way of the potential vorticity gradient. Counterintuitively, shearing is not effective. Moreover, the number of steps in the staircase is sensitive to both the drive (production rate of PE and initial density gradient) and damping (flow viscosity and collisional diffusivity) factors. The minimal step scale is selected by competition between the initial density gradient and diffusive dissipation. Finite turbulence spreading is necessary to form the staircase, but moderate enhancement of turbulence spreading tends to wash out the pattern. The staircase retains a memory of its initial state. Both the mean shear and zonal shear affect the staircase evolution. A strong mean shear quenches the pattern by suppressing the drift wave turbulence. The implications of these findings are also discussed.

Collisional Scaling of the Energy Transfer in Drift-Wave Zonal Flow Turbulence.

The collisionality scaling of density and potential coupling together with zonal flow energy transfer and spectral power is investigated at the stellarator experiment TJ-K. With a poloidal probe array, consisting of 128 Langmuir probes, density and potential fluctuations are measured on four neighboring flux surfaces simultaneously over the complete poloidal circumference. By analyzing Reynolds stress and pseudo-Reynolds stress, it is found that, for increasing collisionality, the coupling between density and potential decreases which hinders the zonal flow drive. Also, as a consequence, the nonlinear energy transfer, as well as the zonal flow contribution to the complete turbulent spectrum, decreases the same way. This is in line with theoretical expectations and is a first experimental verification of the importance of collisionality for large-scale structure formation in magnetically confined toroidal plasmas.

Predator-prey model for the self-organization of stochastic oscillators in dual populations.

A predator-prey model of dual populations with stochastic oscillators is presented. A linear cross-coupling between the two populations is introduced following the coupling between the motions of a Wilberforce pendulum in two dimensions: one in the longitudinal and the other in torsional plain. Within each population a Kuramoto-type competition between the phases is assumed. Thus, the synchronization state of the whole system is controlled by these two types of competitions. The results of the numerical simulations show that by adding the linear cross-coupling interactions predator-prey oscillations between the two populations appear, which results in self-regulation of the system by a transfer of synchrony between the two populations. The model represents several important features of the dynamical interplay between the drift wave and zonal flow turbulence in magnetically confined plasmas, and a novel interpretation of the coupled dynamics of drift wave-zonal flow turbulence using synchronization of stochastic oscillator is discussed.

Elasticity in drift-wave-zonal-flow turbulence.

We present a theory of turbulent elasticity, a property of drift-wave-zonal-flow (DW-ZF) turbulence, which follows from the time delay in the response of DWs to ZF shears. An emergent dimensionless parameter |〈v〉'|/Δωk is found to be a measure of the degree of Fickian flux-gradient relation breaking, where |〈v〉'| is the ZF shearing rate and Δωk is the turbulence decorrelation rate. For |〈v〉'|/Δωk>1, we show that the ZF evolution equation is converted from a diffusion equation, usually assumed, to a telegraph equation, i.e., the turbulent momentum transport changes from a diffusive process to wavelike propagation. This scenario corresponds to a state very close to the marginal instability of the DW-ZF system, e.g., the Dimits shift regime. The frequency of the ZF wave is ΩZF=±γd1/2γmodu1/2, where γd is the ZF friction coefficient and γmodu is the net ZF growth rate for the case of the Fickian flux-gradient relation. This insight provides a natural framework for understanding temporally periodic ZF structures in the Dimits shift regime and in the transition from low confined mode to high confined mode in confined plasmas.

Wave-number spectral scaling law of drift wave-zonal flow turbulence in gyrofluid simulations

Wave-number spectral characteristics of ion temperature gradient (ITG) turbulence are investigated based on three-dimensional gyrofluid simulation. The focus is on the back reaction of the self-generated zonal flows (ZFs) on the ambient turbulence through nonlinear mode coupling. Detailed spectral analyses evidently show that the radial wave-number spectra of the ITG-ZF turbulence is characterized by an exponential-law scaling, which is deformed from the usual Kolmogorov-like power law by the back reaction of the ZFs. These results are qualitatively in agreement with a theoretical prediction [Gürcan et al., Phys. Rev. Lett. 102, 255002 (2009)] and fairly match with the experimental observation of the density fluctuation spectrum [Hennequin et al., Plasma Phys. Controlled Fusion 46, B121 (2004)]. The generality of such spectral characteristics in drift wave turbulence with large-scale coherent structures is discussed.

Finite Larmor radius effects on test particle transport in drift wave-zonal flow turbulence

The effect of finite Larmor radius on the transport of passive charged test particles moving in turbulent electrostatic fields is investigated. The turbulent field is governed by a flexible model which is able to produce turbulence where zonal flows are damped or free to self-generate. A subtle interplay between trapping in small scale vortices and entrainment in larger scale zonal flows determines the rate, character and Larmor radius dependence of the test particle transport. When zonal flows are damped, the transport is classically diffusive, with Gaussian statistics, and the rate of transport decreases with increasing Larmor radius. Once the Larmor radius is larger than the typical radius of the turbulent vortices, the rate of transport remains roughly constant. When zonal flows are allowed non-Gaussian statistics are observed. Radial transport (across the zones) is subdiffusive and decreases with the Larmor radius at a slower rate. Poloidal transport (along the zones), however, is superdiffusive and increases with small values of the Larmor radius.

Dynamics of fully nonlinear drift wave–zonal flow turbulence system in plasmas

We present numerical simulations of fully nonlinear drift wave–zonal flow (DW–ZF) turbulence systems in a nonuniform magnetoplasma. In our model, the drift wave (DW) dynamics is pseudo-three-dimensional (pseudo-3D) and accounts for self-interactions among finite amplitude DWs and their coupling to the two-dimensional (2D) large amplitude zonal flows (ZFs). The dynamics of the 2D ZFs in the presence of the Reynolds stress of the pseudo-3D DWs is governed by the driven Euler equation. Numerical simulations of the fully nonlinear coupled DW–ZF equations reveal that short scale DW turbulence leads to nonlinear saturated dipolar vortices, whereas the ZF sets in spontaneously and is dominated by a monopolar vortex structure. The ZFs are found to suppress the cross-field turbulent particle transport. The present results provide a better model for understanding the coexistence of short and large scale coherent structures, as well as associated subdued cross-field particle transport in magnetically confined fusion plasmas.

On generation of Alfvénic-like fluctuations by drift wave–zonal flow system in large plasma device experiments

According to recent experiments, magnetically confined fusion plasmas with “drift wave–zonal flow turbulence” (DW-ZF) give rise to broadband electromagnetic waves. Sharapov et al. [Europhysics Conference Abstracts, 35th EPS Conference on Plasma Physics, Hersonissos, 2008, edited by P. Lalousis and S. Moustaizis (European Physical Society, Switzerland, 2008), Vol. 32D, p. 4–071] reported an abrupt change in the magnetic turbulence during L-H transitions in Joint European Torus [P. H. Rebut and B. E. Keen, Fusion Technol. 11, 13 (1987)] plasmas. A broad spectrum of Alfvénic-like (electromagnetic) fluctuations appears from E×B flow driven turbulence in experiments on the large plasma device (LAPD) [W. Gekelman et al., Rev. Sci. Instrum. 62, 2875 (1991)] facility at UCLA. Evidence of the existence of magnetic fluctuations in the shear flow region in the experiments is shown. We present one possible theoretical explanation of the generation of electromagnetic fluctuations in DW-ZF systems for an example of LAPD experiments. The method used is based on generalizing results on shear flow phenomena from the hydrodynamics community. In the 1990s, it was realized that fluctuation modes of spectrally stable nonuniform (sheared) flows are non-normal. That is, the linear operators of the flows modal analysis are non-normal and the corresponding eigenmodes are not orthogonal. The non-normality results in linear transient growth with bursts of the perturbations and the mode coupling, which causes the generation of electromagnetic waves from the drift wave–shear flow system. We consider shear flow that mimics tokamak zonal flow. We show that the transient growth substantially exceeds the growth of the classical dissipative trapped-particle instability of the system.

Physics of zonal flows

Zonal flows, which means azimuthally symmetric band-like shear flows, are ubiquitous phenomena in nature and the laboratory. It is now widely recognized that zonal flows are a key constituent in virtually all cases and regimes of drift wave turbulence, indeed, so much so that this classic problem is now frequently referred to as “drift wave-zonal flow turbulence.” In this review, new viewpoints and unifying concepts are presented, which facilitate understanding of zonal flow physics, via theory, computation and their confrontation with the results of laboratory experiment. Special emphasis is placed on identifying avenues for further progress.

Theoretical Progress of Zonal Flow

An overview of the theory of zonal flows is presented. The contents include description of the mechanism by which zonal flows are generated, there relation between turbulence and zonal flows, and transport in drift wave-zonal flow turbulence. The physics of zonal flows is a prototypical example of a paradigm shift from the linear-local-deterministic view to the nonlinear-nonlocal-statistical view of structure formation in plasmas. Research regarding the turbulence-zonal flow system in laboratory plasma will deepen our understanding of this aspect of nature.

A Simple Model of Intermittency in Drift Wave-Zonal Flow Turbulence

An analogy between sandpile relaxation and the evolution of the drift wave spectrum by zonal flow induced shearing is used to characterize the probability distribution function of fluctuations in the drift wave population density Ñ, and thus address the predictability of intermittency in drift wave-zonal flow turbulence. Basic conservation and symmetry properties are used to derive a generalized Burgers equation for Ñ in the radial wave number kr. Shocks are symptomatic of shearing events, which are analogous to avalanches in the sandpile. The pdf P(ΔÑ) is shown to be intrinsically asymmetric in kr and the asymptotic behaviors of both left and right tails are determined. The implications of these results for transport are discussed.

On the nature of bursting in transport and turbulence in drift wave–zonal flow systems

The predictions of the extended predator–prey model of the coupled spectral dynamics of drift wave–zonal flow turbulence are presented. The model exhibits three possible types of time-dependent solutions, depending on system parameters, which are: (1) quasiperiodic bursting of the transport and turbulence intensity levels; (2) oscillatory relaxation to a stationary state, and in the collisionless limit; (3) an intensity pulse followed by saturation of zonal flow. These solutions are consistent with the time dependent behavior recently observed in the global gyrokinetic simulations.

Bifurcation and scaling of drift wave turbulence intensity with collisional zonal flow damping

Interacting drift wave–zonal flow turbulence is examined at the spectral level of description using an extended “predator–prey” model. Analytic solutions that describe both the linear scaling of transport with ion–ion collisionality as well as the saturation regime are obtained for a simple model of drift wave turbulence. A theory of self-regulation in this system is presented. The possibility of bifurcation to a state with higher turbulence level and transport is demonstrated. This bifurcation is associated with the appearance of a condensate solution at the largest scales. The possible relevance of this phenomenon to the bursting events of turbulence and transport recently observed in gyrokinetic simulations of ITG instability is discussed.