MATHEMATICAL-MODELS WITH EXACT RENORMALIZATION FOR TURBULENT TRANSPORT .2. FRACTAL INTERFACES, NON-GAUSSIAN STATISTICS AND THE SWEEPING EFFECT

This paper continues the study of a model for turbulent transport with an exact renormalization theory which has recently been proposed and developed by the authors. Three important topics are analyzed with complete mathematical rigor for this model: (1) Renormalized higher order statistics of a passively advected scalar such as the pair distance distribution and the fractal dimension of interfaces, (2) the effect of non-Gaussian turbulent velocity statistics on renormalization theory, (3) the "sweeping" effect of additional large scale mean velocities. A special emphasis is placed on renormalization theory in the vicinity of the value of the analogue of the Kolmogorov-spectrum in the model. In the authors' earlier paper, it was established that the Kolmogorov value is at a phase transition boundary in the exact renormalization theory. It is found here that the qualitative model, despite its simplicity contains, in the vicinity of the Kolmogorov value, a remarkable amount of the qualitative behavior of turbulent transport which has been uncovered in recent experiments and proposed in phenomenological theories. In particular, the Richardson 4/3-law for pair dispersion and interfaces with fractal dimension defect of 2/3 occur in the model rigorously as limits when the Kolmogorov spectrum is approached as a limit from one side of the phase transition boundary; alternative corrections to the Richardson law with the same form as those proposed heuristically in the recent literature and interfaces with fractal dimension defect 1/3, occur in the model when the Kolmogorov spectrum is approached from the other side of the phase transition. It is very interesting that fractal dimension defects of roughly the value either 1/3 or 2/3 for level sets and interfaces of passive scalars have been ubiquitous in recent turbulence experiments. As regards non-Gaussian velocity statistics, a principle of "statistical universality" is established rigorously in the model so that the renormalization theory for eddy diffusivity coincides with the one presented in earlier work in the case of Gaussian velocity statistics. Finally, the authors show that the "sweeping effect" can significantly alter the renormalization theory in the model for suitable infrared divergent velocity statistics with steady or nearly steady velocity fields. However, it is proved here that the renormalization theory in the model in the vicinity of the Kolmogorov spectrum is Galilean invariant and insensitive to this sweeping effect of large scales.