Dimensionality analysis of chaotic density fluctuations in tokamaks

Anomalous transport in tokamak plasmas is thought to be caused by turbulence. Density fluctuations detected with collective scattering or reflectometry indeed revealed the existence of a broad-band turbulence with Δf/f∼Δk/k∼1. The underlying physics mechanisms are difficult to determine because the pertinent equations are highly nonlinear and bound to lead to deterministic chaos as observed. Quasi-linear approximations to the equations yield the limits for the onset of various instability types like drift waves, microtearing, etc, and their typical frequencies and wavelengths at the onset. However, in the experiments the instabilities have grown to a full turbulence at a saturation level which is so high that the various quasi-linear approximations have become invalid. Many plasma quantities in theory assumned to be constant are in reality dragged into fluctuating behaviour, thereby influencing the saturation level, the frequency and wavelength spectra. Notwithstanding the chaotic nature of the plasma transport one can still try to investigate which plasma quantities are involved in the turbulence, which of those are driving the turbulence and which are damping. The theory of deterministic chaos gives some tools for this. The fractal dimension of the attractor yields the lower bound to the number of fluctuating quantities. The sign of the Lyapunov exponents indicates the driving-damping nature of these quantities, whilst the Kolmogorov entropy being the sum of the positive Lyapunov exponents indicates the strength of the driving terms. In a long term programme one could try to identify which plasma quantities belong to which Lyapunov exponents. In this contribution only the issue of the dimensionality of the attractor will be addressed following the method of Grassberger et al. (author) 7 refs., 1 fig., 1 tab