Turbulence and spatio-temporal chaos. Final Report

Turbulence phenomena encompass many behaviors at many scales, from chaotic behavior at small scales to coherent structures at large scales. The crucial question is how different scales interact. We have sought clues to a general answer by raising the following questions: Can a deterministic chaotic system described by a determininstic partial differential equation (the Kuramoto-Sivashinsky equation) be characterized at large scales by the same static and dynamic scaling exponents that determine a stochastic equation (the Kardar-Parisi-Zhang equation)? Through an extensive numerical study in both one and two dimensions we have concluded that truly, on large scales, the Kuramoto-Sivashinsky and Kardar-Parisi-Zhang equations belong to the same universality class. The procedure (pioneered by Zaleski) involves coarse graining in Fourier space. Through a detailed analysis of its features, and in particular the effective noise-noise correlation, we are able to comprehend the magnitudes of the physical parameters, and in particular estimate the cross-over times from free to interacting behavior