Phase and precession evolution in the Burgers equation

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.

References in corpus (7)

• Burgers Turbulence
• Multifractal statistics of Lagrangian velocity and acceleration in turbulence
• Turbulence on a Fractal Fourier set
• Intermittency in fractal Fourier hydrodynamics: Lessons from the Burgers Equation
• Robust energy transfer mechanism via precession resonance in nonlinear turbulent wave systems
• A new assessment of the second order moment of Lagrangian velocity increments in turbulence
• 3D Euler equations and ideal MHD mapped to regular systems: probing the finite-time blowup hypothesis

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• Inertial range statistics of the Entropic Lattice Boltzmann in 3D turbulence
• A minimal phase-coupling model for intermittency in turbulent systems
• Nonlinear phase synchronization and the role of spacing in shell models
• Precession resonance in water waves
• Unraveling Self-Similar Energy Transfer Dynamics: a Case Study for 1D Burgers System
• Alignments of Triad Phases in 1D Burgers and 3D Navier-Stokes Flows