Study of high-dimensional attractors with scaling properties

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Statistical properties of a high-dimensional strange attractor with scaling properties is studied numerically. We adopted the shell model which was proposed by us to study fluid turbulence. This model was confirmed to have the Kolmogorov scaling properties in its chaotic state. When the Lyapunov dimention of the attractor is around 50, the probability distribution function (pdf) of a velocity variable becomes apart from Gaussian form as the wavenumber increases in the inertial range. In the dissipation range, the form of the pdf is found to deviate from a stretched exponential form and converges asymptotically to P (x) -|x|^<-1.6>. We compared this result with that of Navier-Stokes turbulence obtained from a direct numerical simulation of the equation. In order to interpret the result for the shell model, we employed the orthonormal wavelet analysis to construct the pdf of spatial distribution of Fourier components. This procedure shows that the functional form of the pdf for the Navier-Stokes turbulence also shows the convergence to the same power form in the deep dissipation range. This suggests that the pdf is, at least in the first approximation, determined only by basic properties of the equation. We also applied the wavelet method to observational data of atmospheric turbulence to see its usefulness.