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+ (γna t) 2 } in the initial regime t< τn (γ) and the exponential decay αne exp (- γne t) in the final regime t> τn (γ), where τn (γ) denotes the decay time of the memory function Γn (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 γna, αne, and γne 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> τn (γ) with scaling exponent z=2 and a dynamic scaling law in the initial regime t< τn (γ) with scaling exponent z=1. If kn is increased so that the decay time τn (u) of Un (t) becomes equal to the decay time τn (γ), then the decay form of Un (t) becomes the power-law decay t-3/2 in the final regime.