Phase and precession evolution in the Burgers equation

Abstract

Abstract. We present a phenomenological study of the phase dynamics of the one-dimensional stochastically forced Burgers equation, and of the same equation under a Fourier mode reduction on a fractal set. We study the connection between coherent structures in real space and the evolution of triads in Fourier space. Concerning the one-dimensional case, we find that triad phases show alignments and synchronisations that favour energy fluxes towards small scales -a direct cascade. In addition, strongly dissipative real-space structures are associated with entangled correlations amongst the phase precession frequencies and the amplitude evolution of Fourier triads. As a result, triad precession frequencies show a non-Gaussian distribution with multiple peaks and fat tails, and there is a significant correlation between triad precession frequencies and amplitude growth. Links with dynamical systems approach are briefly discussed, such as the role of unstable critical points in state space. On the other hand, by reducing the fractal dimension D of the underlying Fourier set, we observe: i) a tendency toward a more Gaussian statistics, ii) a loss of alignment of triad phases leading to a depletion of the energy flux, and iii) the simultaneous reduction of the correlation between the growth of Fourier mode amplitudes and the precession frequencies of triad phases.

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

“…is the energy input rate (forcing for small wavenumbers only), D(k, t) ≡ 2ν k k =1 (k ) 2 E k is the dissipation rate and Π(k, t) is the flux across wavenumber k towards large wavenumbers. We derived in (Buzzicotti et al 2016b) an explicit expression for Π(k) in terms of the variables a k and ϕ k3 k1, k2 :…”

“…appropriate linear combinations of the Fourier phases), energy transfers can be enhanced by changing the initial phases, in both chaotic and integrable regimes (Craik 1988;Kim & West 1997;Bustamante & Kartashova 2009;Harris et al 2012; Thompson & Roy 1991;Trillo et al 1994). Despite their apparent background role in turbulence, Fourier phases play a direct role in energy fluxes (Buzzicotti et al 2016b) and high-order correlation functions, so their study could shed light on the problem of intermittency. In continuous fluid models described by partial differential equations, the collection of Fourier phases over a range of spatial scales has been shown to be relevant to the dynamics, via amplitude-phase synchronisation mechanism leading to spatiotemporal chaos in channel flows and magnetised Keplerian shear flows (Chian et al 2010;Miranda et al 2015), and via evolution of phase entropy in cosmological density perturbation solutions (Chiang & Coles 2000).…”

“…In continuous fluid models described by partial differential equations, the collection of Fourier phases over a range of spatial scales has been shown to be relevant to the dynamics, via amplitude-phase synchronisation mechanism leading to spatiotemporal chaos in channel flows and magnetised Keplerian shear flows (Chian et al 2010;Miranda et al 2015), and via evolution of phase entropy in cosmological density perturbation solutions (Chiang & Coles 2000). Motivated by our recent results on triad phase synchronisation in the 2D barotropic vorticity equation and the 1D Burgers equation (Bustamante et al 2014;Buzzicotti et al 2016b), our aim is to develop a theory for triad phase dynamics in the context of classical fluid turbulence. This paper provides the first steps in that direction by looking at the behaviour of triad phases in the stochastically-forced 1D Burgers equation…”

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 .…”

Topical issue on Multi-scale phenomena in complex flows and flowing matter

“…colliding particles, polymers, active matter, micro-swimmers) by several authors [6][7][8][9][10][11]. Few contributions focus on classical problems related to non-linear systems or the turbulent behaviour of Newtonian fluids, but focusing on new observables or novel approaches to anomalous scaling: these include the discussion of dissipated energy in diffusive systems [12], turbulence under rotation or with strong helical properties [13][14][15][16], with waves [17], or with modified non-linear interactions [18,19].The papers in this issue showcase examples of active mutual scientific fertilization across different disciplines and approaches. Among these, the application of techniques inherited from statistical mechanics, both in analyzing largescale dynamics of chaotic systems, as well as mesoscale statistics of glass-forming liquids, but also energy-landscape analysis of mechanical deformation and the study of flowing properties in jammed systems, just to name a few.…”