Intermittency in fractal Fourier hydrodynamics: Lessons from the Burgers equation.

We present theoretical and numerical results for the one-dimensional stochastically forced Burgers equation decimated on a fractal Fourier set of dimension D. We investigate the robustness of the energy transfer mechanism and of the small-scale statistical fluctuations by changing D. We find that a very small percentage of mode-reduction (D ≲ 1) is enough to destroy most of the characteristics of the original nondecimated equation. In particular, we observe a suppression of intermittent fluctuations for D < 1 and a quasisingular transition from the fully intermittent (D=1) to the nonintermittent case for D ≲ 1. Our results indicate that the existence of strong localized structures (shocks) in the one-dimensional Burgers equation is the result of highly entangled correlations amongst all Fourier modes.

Real-space manifestations of bottlenecks in turbulence spectra.

An energy-spectrum bottleneck, a bump in the turbulence spectrum between the inertial and dissipation ranges, is shown to occur in the nonturbulent, one-dimensional, hyperviscous Burgers equation and found to be the Fourier-space signature of oscillations in the real-space velocity, which are explained by boundary-layer-expansion techniques. Pseudospectral simulations are used to show that such oscillations occur in velocity correlation functions in one- and three-dimensional hyperviscous hydrodynamical equations that display genuine turbulence.

Nelkin scaling for the Burgers equation and the role of high-precision calculations.

Nelkin scaling, the scaling of moments of velocity gradients in terms of the Reynolds number, is an alternative way of obtaining inertial-range information. It is shown numerically and theoretically for the Burgers equation that this procedure works already for Reynolds numbers of the order of 100 (or even lower when combined with a suitable extended self-similarity technique). At moderate Reynolds numbers, for the accurate determination of scaling exponents, it is crucial to use higher than double precision. Similar issues are likely to arise for three-dimensional Navier-Stokes simulations.

Turbulence in noninteger dimensions by fractal Fourier decimation.

Fractal decimation reduces the effective dimensionality D of a flow by keeping only a (randomly chosen) set of Fourier modes whose number in a ball of radius k is proportional to k(D) for large k. At the critical dimension D(c)=4/3 there is an equilibrium Gibbs state with a k(-5/3) spectrum, as in V. L'vov et al., Phys. Rev. Lett. 89, 064501 (2002). Spectral simulations of fractally decimated two-dimensional turbulence show that the inverse cascade persists below D=2 with a rapidly rising Kolmogorov constant, likely to diverge as (D-4/3)(-2/3).

Resonance phenomenon for the Galerkin-truncated Burgers and Euler equations.

It is shown that the solutions of inviscid hydrodynamical equations with suppression of all spatial Fourier modes having wave numbers in excess of a threshold K(G) exhibit unexpected features. The study is carried out for both the one-dimensional Burgers equation and the two-dimensional incompressible Euler equation. For large K(G) and smooth initial conditions, the first symptom of truncation, a localized short-wavelength oscillation which we call a "tyger," is caused by a resonant interaction between fluid particle motion and truncation waves generated by small-scale features (shocks, layers with strong vorticity gradients, etc.). These tygers appear when complex-space singularities come within one Galerkin wavelength λ(G)=2π/K(G) from the real domain and typically arise far away from preexisting small-scale structures at locations whose velocities match that of such structures. Tygers are weak and strongly localized at first-in the Burgers case at the time of appearance of the first shock their amplitudes and widths are proportional to K(G)(-2/3) and K(G)(-1/3), respectively-but grow and eventually invade the whole flow. They are thus the first manifestations of the thermalization predicted by T. D. Lee [Q. J. Appl. Math. 10, 69 (1952)]. The sudden dissipative anomaly-the presence of a finite dissipation in the limit of vanishing viscosity after a finite time t(⋆)-which is well known for the Burgers equation and sometimes conjectured for the three-dimensional Euler equation, has as counterpart, in the truncated case, the ability of tygers to store a finite amount of energy in the limit K(G)→∞. This leads to Reynolds stresses acting on scales larger than the Galerkin wavelength and thus prevents the flow from converging to the inviscid-limit solution. There are indications that it may eventually be possible to purge the tygers and thereby to recover the correct inviscid-limit behavior.

Hyperviscosity, Galerkin truncation, and bottlenecks in turbulence.

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 conservative dynamics with a finite range of spatial Fourier modes. Those at large wave numbers thermalize, whereas modes at small wave numbers obey ordinary viscous dynamics [C. Cichowlas et al., Phys. Rev. Lett. 95, 264502 (2005)10.1103/Phys. Rev. Lett. 95.264502]. The energy bottleneck observed for finite alpha may be interpreted as incomplete thermalization. Artifacts arising from models with alpha>1 are discussed.

Is multiscaling an artifact in the stochastically forced Burgers equation?

We study turbulence in the one-dimensional Burgers equation with a white-in-time, Gaussian random force that has a Fourier-space spectrum approximately 1/k, where k is the wave number. From very high-resolution numerical simulations, in the limit of vanishing viscosity, we find evidence for multiscaling of velocity structure functions which cannot be falsified by standard tests. We find a new artifact in which logarithmic corrections can appear disguised as anomalous scaling and conclude that bifractal scaling is likely.

A reconstruction of the initial conditions of the universe by optimal mass transportation.

Reconstructing the density fluctuations in the early Universe that evolved into the distribution of galaxies we see today is a challenge to modern cosmology. An accurate reconstruction would allow us to test cosmological models by simulating the evolution starting from the reconstructed primordial state and comparing it to observations. Several reconstruction techniques have been proposed, but they all suffer from lack of uniqueness because the velocities needed to produce a unique reconstruction usually are not known. Here we show that reconstruction can be reduced to a well-determined problem of optimization, and present a specific algorithm that provides excellent agreement when tested against data from N-body simulations. By applying our algorithm to the redshift surveys now under way, we will be able to recover reliably the properties of the primeval fluctuation field of the local Universe, and to determine accurately the peculiar velocities (deviations from the Hubble expansion) and the true positions of many more galaxies than is feasible by any other method.