Application of a projection operator method to turbulence

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We investigate the structure of the time correlation function for one-dimensional turbulence with a projection operator method. A closed model equation for the time correlation function is derived assuming the similarity between the time correlation function and the memory function. We obtain the following asymptotic forms of the time correlation functions both by solving the model equation and by carrying out the direct numerical simulation of the Kuramoto-Sivashinsky equation, which is a typical governing equation for one-dimensional turbulence : 1) the time correlation function indicates the algebraic form for the initial regime ; 2) the time correlation function decays exponentially with or without oscillation for the final regime depending on the wavenumber.