Dual structures of chaos and turbulence, and their dynamic scaling laws

The decay form of the time correlation function Un(t) of a state variable un(t) with a small wave number kn has been shown to take the algebraic decay 1/{1+(gammanat)2} in the initial regime ttaun(gamma), where taun(gamma) denotes the decay time of the memory function Gamman(t). This dual structure of Un(t) is generated by the deterministic short orbits in the initial regime and the stochastic long orbits in the final regime, thus giving the outstanding features of chaos and turbulence. The kn dependence of gammana, alphane, and gammane is obtained for the chaotic Kuramoto-Sivashinsky equation, and it is shown that if kn is sufficiently small, then the dual structure of Un(t) obeys a hydrodynamic scaling law in the final regime t>tau(gamma) with scaling exponent z=2 and a dynamic scaling law in the initial regime t