Singular behaviors of two-dimensional numerical turbulence and chaos in a shell model.

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Inviscid two-dimensional quasi-geostroph (QG) flow has been considered to have a finite-time singularity. Some mathematical result and numerical simulations on the quasi-geostrophic system suggested that temperature gradient of a solution to QG equation blows up in a finite time. In this research we performed a numerical simulation by spectral method with more Fourier modes than in previous ones, and re-examined the numerical data which was interpreted to indicate the appearance of the singularity. Our numerical result shows that the previous numerical simulations did not have a sufficient number of modes to resolve the singular behavior of the solution, and a change of variable even suggests that the solution does not blow up in a finite time. We also performed a numerical simulation on viscous QG system, which indicates that there is no cascade phenomenon connected to the blow up of the inviscid solution. These results does not support the appearance of a finite-time singularity, but suggests the regularity of the invisid solution. We also investitated a phenomenological theory of chaos in shell model of turbulence, and obtained an asymptotic formula for Lyapunov spectrum. This theory is based on the fact that the support of Lyapunov vectors in this system in sharply localized in Fourier space, which permits us to relate the Lyapunovspectrum to Kolmogorov similarity law. This formula agrees with numerical result better in the case of larger attractor dimension. This result shows that the shell-model is a rare example of high-dimensional chaos in which the asymptotic formula for Lyapunov spectrum can be obtained. We applied this method also to Navier-Stokes turbulence and obtanied an asymptotic formula for Lyapunov spectrum in the inviscid limit.