It is shown that the use of a high power α of the Laplacian in the dissipative term of hydrodynamical equations leads asymptotically to truncated inviscid 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 α may be interpreted as incomplete thermalization. Artifacts arising from models with α > 1 are discussed. PACS numbers: 47.27 Gs, 05.20.Jj A single Maxwell daemon embedded in a turbulent flow would hardly notice that the fluid is not exactly in thermal equilibrium because incompressible turbulence, even at very high Reynolds numbers, constitutes a tiny perturbation on thermal molecular motion. Dissipation in real fluids is just the transfer of macroscopically organized (hydrodynamic) energy to molecular thermal energy. Artificial microscopic systems can act just like the real one as far as the emergence of hydrodynamics is concerned; for instance, in lattice gases the "molecules" are discrete Boolean entities [1] and thermalization is easily observed at high wavenumbers [2]. Another example has been found recently by Cichowlas et al. [3] wherein the Euler equations of ideal non-dissipative flow are (Galerkin) truncated by keeping only a finite -but largenumber of spatial Fourier harmonics. The modes with the highest wavenumbers k then rapidly thermalize through a mechanism discovered by T.D. Lee [4] and studied further by R.H. Kraichnan [5], leading in three dimensions (3D) to an equipartition energy spectrum ∝ k 2 . The thermalized modes act as a fictitious microworld on modes with smaller wavenumbers in such a way that the usual dissipative NavierStokes dynamics is recovered at large scales [25].All the known systems presenting thermalization are conservative. As we shall show themalization may be present in dissipative hydrodynamic systems when the dissipation rate increases so fast with the wavenumber that it mimics ideal hydrodynamics with a Galerkin truncation. This is best understood by considering hydrodynamics with hyperviscosity: the usual momentum diffusion operator (a Laplacian) is replaced by the αth power of the Laplacian, where α > 1 is the dissipativity. Hyperviscosity is frequently used in turbulence modeling to avoid wasting numerical resolution by reducing the range of scales over which dissipation is effective [6].The unforced hyperviscous 1D Burgers and multidimensional incompressible Navier-Stokes (NS) equations are:The equations must be supplemented with suitable initial and boundary conditions. We employ 2π-periodic boundary conditions in space, so that we can use Fourier decompositions such as v(x) = kv k e i k·x . Note that minus the Laplacian is a positive operator, with Fourier transform k 2 , which can be raised to an arbitrary power α. The coefficient µ is taken positive to make the hyperviscous operator dissipative. The Galerkin wavenumber k G > 0 is chos...
Purely helical absolute equilibria of incompressible neutral fluids and plasmas (electron, singlefluid and two-fluid magnetohydrodyanmics) are systematically studied with the help of helical (wave) representation and truncation, for genericities and specificities about helicity. A unique chirality selection and amplification mechanism and relevant insights, such as the one-chiralsector-dominated states, among others, about (magneto)fluid turbulence follow.
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