Energy flux enhancement, intermittency and turbulence via Fourier triad phase dynamics in the 1-D Burgers equation

Abstract

We present a theoretical and numerical study of Fourier space triad phase dynamics in one-dimensional stochastically forced Burgers equation at Reynolds number Re ≈ 2.7 × 10 4 . We demonstrate that Fourier triad phases over the inertial range display a collective behaviour characterised by intermittent periods of synchronisation and alignment, reminiscent of Kuramoto model (Kuramoto 1984) and directly related to collisions of shocks in physical space. These periods of synchronisation favour efficient energy fluxes across the inertial range towards small scales, resulting in strong bursts of dissipation and enhanced coherence of Fourier energy spectrum. The fast time scale of the onset of synchronisation relegates energy dynamics to a passive role: this is further examined using a reduced system with the Fourier amplitudes fixed in time -a phase-only model. We show that intermittent triad phase dynamics persists without amplitude evolution and we broadly recover many of the characteristics of the full Burgers system. In addition, for both full Burgers and phase-only systems the physical space velocity statistics reveals that triad phase alignment is directly related to the non-Gaussian statistics typically associated with structure-function intermittency in turbulent systems.Turbulence phenomenology provides equations for the energy budget over spatial scales, based on a statistical balance between the injected energy at large scales, its flux across spatial scales and its subsequent dissipation at small scales. Theoretical and numerical approaches to turbulence share a popular setup in terms of Fourier-space variables that assumes periodic boundary conditions in the spatial coordinates. Seminal results started by Kolmogorov in 1941 (see Frisch (1995) provide statistical predictions for the energy spectrum: the modulus squared of the complex Fourier amplitudes E k = |û k | 2 , while nothing is stated about its conjugate variable, the Fourier phase φ k = arg(û k ). A similar situation occurs in wave turbulence theory, which provides coupled evolution equations for the energy spectrum variables only, admitting solutions representing flux of energy-like quantities across spatial scales (see Nazarenko (2011)).Fourier phases are useful variables in Galerkin-truncated discrete models in fluids and nonlinear optics based on triad and/or quartet interactions. There, due to the nonlinear coupling between Fourier energies and the so-called triad or quartet phases (defined as † Email address for correspondence: miguel.bustamante@ucd.ie arXiv:1705.08960v2 [physics.flu-dyn]

Geophysical turbulence dominated by inertia–gravity waves

“…At smaller scales, the effects of rotation would be negligible and strongly nonlinear dynamics would dominate. Given that the shallow water equations are similar to the compressible gas dynamics equations (Whitham 2011), shocks in RSW models at small scales may be expected to generate a wave spectrum close to at these small scales, as is the scenario in compressible flows (Kuznetsov 2004; Falkovich & Kritsuk 2017; Gupta & Scalo 2018; Murray & Bustamante 2018).…”

Existence of positive skewness of velocity gradient in early transition

“…In particular, a triad phase, φ 1 ( p) + φ 1 ( q) − φ 1 ( k), is explicitly represented on the right-hand side, indicating the link between S k and the phase of velocities (see Refs. [11,[38][39][40] for more discussions). This expression explicitly illustrates that both the amplitudes and the phases can affect the evolution of S k .…”

A minimal phase-coupling model for intermittency in turbulent systems

“…While Gaussian random fields feature completely uncorrelated phases, phase correlations can give rise to complex scale-dependent statistics, and it is well known that coherent spatial structures such as shocks require a high level of correlation amongst the phases of the Fourier modes. Notably, so far only very few studies have addressed the connection between the emergence of coherent intermittent structures in real space, non-Gaussian statistics and phase correlations, indicating that bursts of spectral energy fluxes (and dissipation) are produced when Fourier phases become correlated [9][10][11]. Elucidating these connections is important for both fundamental and applied aspects.…”

“…This infinite set of coupled ODEs describes the full Burgers dynamics of the Fourier phases and amplitudes. Recently, it was shown that the dynamics of the Fourier phases φ k (t) determine to a great extent the shock dynamics and the associated non-Gaussian statistics [9,11]. Thus, we take equation ( 3) as a starting point for a minimal model for Fourier phase dynamics in turbulence, the "phase-only" model.…”

“…Panel (b) shows the average phase synchronization, equation(10), as a function of α for various system sizes. Panel (c) shows the ratio between the Kaplan-Yorke dimension, equation(11), and the available degrees of freedom as a function of α for various system sizes.…”