Time Correlation Functions In A Similarity Approximation For One-Dimensional Turbulence

The projection operator formalism yields a time evolution equation for the time correlation function U-n(t) of the chaotic modes of interest in terms of the memory function Gamma(n)(t). On the assumption of similarity between U-n(t) and Gamma(n)(t), this equation leads to a closed equation for U-n(t), which yields the asymptotic behavior of the time correlation function U-n(t) and the corresponding power spectrum I-n(omega) analytically. Thus it turns out that the time correlation function takes the algebraic form 1/(1+t(2)) for t -> 0 as predicted previously, and can be classified into three decay forms for t ->infinity according to the wave number k(n): the exponential decay e(-t), the oscillatory exponential decay e(-t) cos t, and the oscillatory power-law decay t(-3/2) cos t. All the corresponding power spectra form a dual structure which is Lorentzian as omega -> 0 and decays exponentially as omega ->infinity. In the entire domain 0 <= t infinity.