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Center for Nonlinear Studies |
It is conjectured that for many equations of hydrodynamical type (including the 3D Navier--Stokes equations, the Burgers equation and various models of turbulence) the use of hyperviscous dissipation with a high power $\alpha$ (dissipativity) of the Laplacian and suitable rescaling of the hyperviscosity becomes asymptotically equivalent to using a Galerkin truncation with zero dissipation and suppression of all Fourier modes whose wavenumber exceeds a cutoff $k_d$. From recent work of Cichowlas et al (\textit{Phys.\ Rev. Lett.} {\bf 95} (2005) 264502), it is known that when $k_d$ is large enough the solution of Galerkin-truncated equations develop a thermalized range at high wavenumbers (with a $k2$ spectrum in 3D). It is proposed to interpret the phenomenon of bottleneck, which becomes stronger when increasing $\alpha$, as an aborted thermalization. In this first talk, I will discuss the cases of eddy-damped-quasi-normal-Markovian (EDQNM) type models. Artifacts which can appear when using hyperviscosity are also discussed along with the numerical verifications.