B. Ph. van Milligen

Wavelet bicoherence: A new turbulence analysis tool

A recently introduced tool for the analysis of turbulence, wavelet bicoherence [van Milligen, Hidalgo, and Sanchez, Phys. Rev. Lett. 16, 395 (1995)], is investigated. It is capable of detecting phase coupling—nonlinear interactions of the lowest (quadratic) order—with time resolution. To demonstrate its potential, it is applied to numerical models of chaos and turbulence and to real measurements. It detected the coupling interaction between two coupled van der Pol oscillators. When applied to a model of drift wave turbulence relevant to plasma physics, it detected a highly localized coherent structure. Analyzing reflectometry measurements made in fusion plasmas, it detected temporal intermittency and a strong increase in nonlinear phase coupling coinciding with the L/H (low‐to‐high confinement mode) transition.

Self-similarity of the plasma edge fluctuations

The rescaled range analysis techniques are used to investigate long-range dependence in plasma edge fluctuations [Mandelbrot and Wallis, Water Resources Res. 4, 909 (1969)]. This technology has been applied to data from several confinement devices such as tokamaks, stellarators, and reversed-field pinch. The results reveal the self-similar character of the electrostatic fluctuations at the plasma edge with self-similarity parameters ranging from 0.62 to 0.72. These results show that the tail of the autocorrelation function decays as a power law for time lags longer than the decorrelation time and as long as times of the order of the confinement time. In cold plasma devices (Te<1 eV at the core), there is no evidence of algebraic tails in the autocorrelation function. Some other characteristic features of the autocorrelation function and power spectrum have been investigated. All of these features are consistent with plasma transport as characterized by self-organized criticality.

On the applicability of Fick's law to diffusion in inhomogeneous systems

Two alternative expressions exist for the diffusive flux in inhomogeneous systems: Ficks law and the Fokker-Planck law. Here we re-examine the origin of these expressions and perform numerical and physical experiments to shed light on this duality. We conclude that in general the Fokker-Planck expression should be conceded preference, in spite of the fact that Ficks law seems to be more popular.

Probabilistic finite-size transport models for fusion: Anomalous transport and scaling laws

Transport in fusion plasmas in the low confinement mode is characterized by several remarkable properties: the anomalous scaling of transport with system size, stiff (or “canonical”) profiles, power degradation, and rapid transport phenomena. The present article explores the possibilities of constructing a unified transport model, based on the continuous-time random walk, in which all these phenomena are handled adequately. The resulting formalism appears to be sufficiently general to provide a sound starting point for the development of a full-blown plasma transport code, capable of incorporating the relevant microscopic transport mechanisms, and allowing predictions of confinement properties.

Experimental Evidence of Long-Range Correlations and Self-Similarity in Plasma Fluctuations

To better understand long time transport dynamics, techniques to investigate long-range dependences in plasma fluctuations have been applied to data from several confinement devices including tokamaks, stellarators, and reversed field pinch. The results reveal the self-similar character of the edge plasma fluctuations. This implies that the tail of the autocorrelation function decays as a power law and suggests that there is a superdiffusive component of the anomalous transport. Rescaled fluctuation and turbulent flux spectra from different devices also show a strong similarity. For a range of parameters corresponding to the tokamak ohmic regime and equivalent power for other devices, the spectral decay index may show a universal character.

Additional evidence for the universality of the probability distribution of turbulent fluctuations and fluxes in the scrape-off layer region of fusion plasmas

Plasma density fluctuations and electrostatic turbulent fluxes measured at the scrape-off layer of the Alcator C-Mod tokamak [B. LaBombard, R. L. Boivin, M. Greenwald, J. Hughes, B. Lipschultz, D. Mossessian, C. S. Pitcher, J. L. Terry, and S. J. Zweben, Phys. Plasmas 8, 2107 (2001)], the Wendelstein 7-Advanced Stellarator [H. Renner, E. Anabitarte, E. Ascasibar et al., Plasma Phys. Controlled Fusion 31, 1579 (1989)], and the TJ-II stellarator [C. Alejaldre, J. Alonso, J. Botija et al., Fusion Technol. 17, 131 (1990)] are shown to obey a non-Gaussian but apparently universal (i.e., not dependent on device and discharge parameters) probability density distribution (pdf). The fact that a specific shape acts as an attractor for the pdf seems to suggest that emergent behavior and self-regulation are relevant concepts for these fluctuations. This shape is closely similar to the so-called Bramwell, Holdsworth, and Pinton distribution, which does not have any free parameters.

Impact of different confinement regimes on the two-dimensional structure of edge turbulence

This paper reports the impact of different confinement regimes on the 2D structure of edge turbulence. An image analysis method based on two-dimensional continuous wavelet transformation is used to localize structures (blobs) in the images and to extract their geometrical characteristics (position, scale, orientation angle and aspect ratio). We study the impact of edge shear-layers on these geometrical aspects of blobs. Results show a reduction in the angular dispersion of blobs as the shear layer is established in the boundary, as well as an increase in the elongation of these structures. Similar behaviour is found in NSTX image sequences when going from L to H mode plasmas. During improved confinement regimes the number of detected blobs decreases. Some indications are found suggesting that the turbulence reduction could be scale-selective in the biasing-induced improved confinement regime of TJ-II stellarator. Perpendicular flow reversal is visualized with the cameras and the time scales for flow reversal are found to be less than 50 µs. Radially propagating structures are found in the SOL with velocities in the range ~1000 m/s and with a poloidally asymmetric spatial distribution.

Integrated data analysis at TJ-II: the density profile.

An integrated data analysis system based on Bayesian inference has been developed for the TJ-II stellarator. It reconstructs the electron density profile at a single time point, using data from interferometry, reflectometry, Thomson scattering, and the Helium beam, while providing a detailed error analysis. In this work, we present a novel analysis of the ambiguity inherent in profile reconstruction from reflectometry and show how the integrated data analysis approach elegantly resolves it. Several examples of the application of the technique are provided, in both low-density discharges with and without electrode biasing, and in high-density discharges with an (L-H) confinement transition.

Probabilistic transport models for plasma transport in the presence of critical thresholds: Beyond the diffusive paradigm

It is argued that the modeling of plasma transport in tokamaks may benefit greatly from extending the usual local paradigm to accommodate scale-free transport mechanisms. This can be done by combining Levy distributions and a nonlinear threshold condition within the continuous time random walk concept. The advantages of this nonlocal, nonlinear extension are illustrated by constructing a simple particle density transport model that, as a result of these ideas, spontaneously exhibits much of nondiffusive phenomenology routinely observed in tokamaks. The fluid limit of the system shows that the kind of equations that are appropriate to capture these dynamics are based on fractional differential operators. In them, effective diffusivities and pinch velocities are found that are dynamically set by the system in response to the specific characteristics of the fueling source and external perturbations. This fact suggests some dramatic consequences for the extrapolation of these transport properties to larger size systems.

Uphill transport and the probabilistic transport model

Profile consistency is a long-standing mystery of transport in thermonuclear fusion plasmas. This phenomenon is critically tested by studying the system response to off-axis fueling or heating. The present paper investigates the potential of a recently proposed probabilistic transport model to simulate this phenomenon, and it is found that most of the observed phenomenology can be reproduced, at least qualitatively. In particular, the observed differences between tokamaks and stellarators under off-axis heating may find an explanation in this framework.