Intermittency in fractal Fourier hydrodynamics: Lessons from the Burgers equation

Abstract

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 non-decimated equation. In particular, we observe a suppression of intermittent fluctuations for D < 1 and a quasi-singular transition from the fully intermittent (D = 1) to the non-intermittent 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.

Phase and precession evolution in the Burgers equation

“…As one can see, shocks are present for all fractal dimensions, while the smooth (for D = 1) ramps connecting two shocks develop small-scale fluctuations that become more and more pronounced as the fractal dimension D decreases. It is important to remark that even if the disorder produces a non-trivial change in the spectrum, a power-law behaviour is always present [23]. Similar results have also been observed in the three dimensional NavierStokes equations [30].…”

“…It is important to mention that in fig. 1 the spectra have no gaps because we have further performed an average over different quenched fractal masks [23]. As one can see, shocks are present for all fractal dimensions, while the smooth (for D = 1) ramps connecting two shocks develop small-scale fluctuations that become more and more pronounced as the fractal dimension D decreases.…”

“…This projection method was originally introduced in [28] to study the inverse energy cascade in 2D Navier-Stokes equations (see [29] for a review). Later, it was exploited to study the statistical properties of small-scale fluctuations in the 3D Navier-Stokes equations [30] and 1D Burgers equation [23]. In these two papers, a tendency towards more regular Gaussian statistics for the small-scale velocity field was observed at lower values of fractal dimension D, indicating that the disorder leads to an energy transfer regime characterised by self-similar fluctuations.…”

“…ii) Joint probability distributions between precession frequencies and energy content and also between precession frequencies and growth rate of energy transfers. Second, in order to better understand the connections between these statistical properties and the presence of shock-like structure in the real space, we will study the effect of the introduction of a quenched disorder given by the projection of the dynamics on a fractal Fourier set, where preliminary results show a strong depletion of shocks [23] with reduced system fractal dimension. We will complement these studies with dynamical-system interpretations in terms of correlations and synchronisation within the state space.…”

On the vortex dynamics in fractal Fourier turbulence

“…However at such fractal dimension, about less than 1% of the modes would survive for the present resolutions, almost annihilating the role of the non-linear transfer and making the energy dissipation almost in a direct balance with the energy injection. We note that, recently, the Burgers equation decimated on a fractal Fourier set of dimension D ≤ 1 was numerically studied [30]. Results obtained at fixed D and for larger and larger values of the Reynolds number, suggest that Fourier decimation is a singular perturbation for the spectral scaling properties.…”