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...

A detailed study of complex-space singularities of the two-dimensional incompressible Euler equation is performed in the short-time asymptotic r\'egime when such singularities are very far from the real domain; this allows an exact recursive determination of arbitrarily many spatial Fourier coefficients. Using high-precision arithmetic we find that the Fourier coefficients of the stream function are given over more than two decades of wavenumbers by $\hat F(\k) = C(\theta) k^{-\alpha} \ue ^ {-k \delta(\theta)}$, where $\k = k(\cos \theta, \sin \theta)$. The prefactor exponent $\alpha$, typically between 5/2 and 8/3, is determined with an accuracy better than 0.01. It depends on the initial condition but not on $\theta$. The vorticity diverges as $s^{-\beta}$, where $\alpha+\beta= 7/2$ and $s$ is the distance to the (complex) singular manifold. This new type of non-universal singularity is permitted by the strong reduction of nonlinearity (depletion) which is associated to incompressibility. Spectral calculations show that the scaling reported above persists well beyond the time of validity of the short-time asymptotics. A simple model in which the vorticity is treated as a passive scalar is shown analytically to have universal singularities with exponent $\alpha =5/2$.Comment: 22 pages, 24 figures, published version; a version of the paper with higher-quality figures is available at http://www.obs-nice.fr/etc7/euler.pd

Given a Taylor series with a finite radius of convergence, its Borel transform defines an entire function. A theorem of Pólya relates the large distance behavior of the Borel transform in different directions to singularities of the original function. With the help of the new asymptotic interpolation method of van der Hoeven, we show that from the knowledge of a large number of Taylor coefficients we can identify precisely the location of such singularities, as well as their type when they are isolated. There is no risk of getting artefacts with this method, which also gives us access to some of the singularities beyond the convergence disk. The method can also be applied to Fourier series of analytic periodic functions and is here tested on various instances constructed from solutions to the Burgers equation. Large precision on scaling exponents (up to twenty accurate digits) can be achieved.