Hyperviscosity, Galerkin truncation and bottlenecks in turbulence

TL;DR

This paper investigates how high powers of the Laplacian in hydrodynamical equations cause a transition to truncated conservative dynamics, leading to thermalization at high wavenumbers and viscous behavior at low wavenumbers, with implications for turbulence modeling.

Contribution

It demonstrates that high powers of the Laplacian induce a transition to truncated inviscid dynamics with thermalization, clarifying the origin of the energy bottleneck in turbulence models.

Findings

High powers of Laplacian lead to finite Fourier mode range.

Large wavenumber modes thermalize, small wavenumber modes follow viscous dynamics.

Models with high Laplacian powers exhibit artifacts related to thermalization.

Abstract

It is shown that the use of a high power $\alpha$ of the Laplacian in the dissipative term of hydrodynamical equations leads asymptotically to truncated inviscid \textit{conservative} dynamics with a finite range of spatial Fourier modes. Those at large wavenumbers thermalize, whereas modes at small wavenumbers obey ordinary viscous dynamics [C. Cichowlas et al. Phys. Rev. Lett. 95, 264502 (2005)]. The energy bottleneck observed for finite $\alpha$ may be interpreted as incomplete thermalization. Artifacts arising from models with $\alpha > 1$ are discussed.