Asymptotic formulas for the Lyapunov spectrum of fully developed shell model turbulence

The scaling behavior of the Lyapunov spectrum of a chaotic shell model for three-dimensional turbulence is studied in detail. First, we characterize the localization property of the Lyapunov vectors in wave-number space by using numerical results. By combining this localization property with Kolmogorov’s dimensional argument, we deduce explicitly the asymptotic scaling law for the Lyapunov spectrum, which in turn is shown to agree well with the numerical results. This shell model is an example of high-dimensional chaotic systems for which an asymptotic scaling law is obtained for the Lyapunov spectrum. Implications of the present results for the Navier-Stokes turbulence are discussed. In particular, we conjecture that the distribution of Lyapunov exponents is not singular at null exponent