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Phase and precession evolution in the Burgers equation.

Michele Buzzicotti1, Brendan P Murray2, Luca Biferale1

  • 1Department of Physics and INFN, University of Rome "Tor Vergata", Via della Ricerca Scientifica 1, 00133, Rome, Italy.

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This study reveals how phase dynamics in the Burgers equation drive energy cascades. Fractal dimensions influence these dynamics, affecting energy flux and correlations between mode amplitudes and frequencies.

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Area of Science:

  • * Fluid dynamics
  • * Nonlinear dynamics
  • * Statistical physics

Background:

  • * The Burgers equation models phenomena like turbulence and wave propagation.
  • * Understanding phase dynamics is crucial for analyzing complex systems.
  • * Fractal sets introduce unique properties to dynamical systems.

Purpose of the Study:

  • * To investigate phase dynamics in the one-dimensional stochastically forced Burgers equation.
  • * To explore the role of Fourier mode reduction on fractal sets.
  • * To connect real-space coherent structures with Fourier-space triad evolution.

Main Methods:

  • * Phenomenological study of the Burgers equation.
  • * Fourier mode reduction on fractal sets.
  • * Analysis of phase and amplitude dynamics of Fourier triads.

Main Results:

  • * In 1D, triad phase alignments drive direct energy cascades to small scales.
  • * Dissipative structures correlate with entangled phase precession and amplitude growth.
  • * Triad precession frequencies exhibit non-Gaussian distributions.
  • * Reducing fractal dimension leads to Gaussian statistics, reduced energy flux, and weaker correlations.

Conclusions:

  • * Phase dynamics significantly influence energy transfer in the Burgers equation.
  • * Fractal properties of the Fourier set modulate these dynamics.
  • * Correlations between phase precession and amplitude growth are key to energy cascade mechanisms.