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 Un (t) of the chaotic modes of interest in terms of the memory function Γn (t). On the assumption of similarity between Un (t) and Γn (t), this equation leads to a closed equation for Un (t), which yields the asymptotic behavior of the time correlation function Un (t) and the corresponding power spectrum In (ω) analytically. Thus it turns out that the time correlation function takes the algebraic form 1/ (1+ t2) for t→0 as predicted previously, and can be classified into three decay forms for t→ according to the wave number kn: the exponential decay e-t, the oscillatory exponential decay e-t cost, and the oscillatory power-law decay t-3/2 cost. All the corresponding power spectra form a dual structure which is Lorentzian as ω→0 and decays exponentially as ω→. In the entire domain 0≤t<, solutions to the closed equation are quite consistent with the numerical results for small kn, while they are consistent with those for large kn, except for the phase. In the case that the integral time scale of Un (t) is equal to that of Γn (t), the closed equation is identical to the direct interaction approximation equation for fluid turbulence in the limit kn →.