Energy flux enhancement, intermittency and turbulence via Fourier triad phase dynamics in 1D Burgers equation (1705.08960v2)

Abstract: We present a theoretical and numerical study of Fourier space triad phase dynamics in one-dimensional stochastically forced Burgers equation at Reynolds number $\mathrm{Re} \approx 2.7 \times 10^4$. 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.

• The paper characterizes Fourier triad phase dynamics in the 1D Burgers equation, showing intermittent synchronisation promotes energy flux and reinforces spectral coherence.
• Intermittent phase synchronisation events are linked to non-Gaussian velocity statistics, suggesting a mechanism for structure-function intermittency in turbulence.
• A reduced phase-only model retains key characteristics of the full system, underscoring the critical role of phase dynamics in driving intermittency and turbulence.

Summary of Energy Flux Enhancement, Intermittency and Turbulence via Fourier Triad Phase Dynamics in 1D Burgers Equation

The paper conducted by Murray and Bustamante explores the dynamics of Fourier triads in the one-dimensional stochastically forced Burgers equation at high Reynolds numbers, specifically focusing on the phase dynamics of Fourier triads. Their investigation provides a blend of theoretical and numerical insights, revealing that the Fourier triad phases exhibit intermittent synchronisation and alignment that are somewhat reminiscent of the Kuramoto model. Such synchronisation is found to be intricately connected to the formation and collision of shocks in physical space.

Key Findings and Numerical Results:

1. Fourier Triad Phase Synchronisation: The authors characterize the phase dynamics over the inertial range of scales by intermittent synchronisation and alignment. This collective behaviour of phase dynamics was shown to promote efficient energy flux from larger to smaller scales, which further leads to bursts of energy dissipation and a reinforced coherence in the Fourier energy spectrum.
2. Intermittency and Turbulence: The paper links these intermittent synchronisation events to non-Gaussian velocity statistics typically observed in turbulent systems, drawing parallels between this phenomenon and the classical problem of structure-function intermittency.
3. Phase-Only Modeling: To further explore the predominance of phase dynamics, the authors employed a reduced, phase-only model. This model, which fixes Fourier amplitudes in time while allowing phases to evolve, revealed that key characteristics of the full Burgers system are retained even in the absence of amplitude dynamics. This underscores the role of phase dynamics in driving the observed intermittency.
4. Robustness Across Forcing Scales: Murray and Bustamante extended their analysis to scenarios with a broader range of forcing scales. They found that even under these conditions, as multiple shocks formed, the basic framework of triad phase synchronisation and energy dissipation alignment remained robust.
5. Implications for Dynamics and Stability: The paper also explores the stability characteristics of the phase dynamics in the unforced scenario. They numerically demonstrated that the fully-aligned state is a nonlinear stable fixed point in absence of forcing, providing a fundamental insight into the stability properties of phase synchronisation in this simplified model.

Implications and Future Developments:

This paper's examination of phase dynamics within the Fourier space offers pivotal insights into the mechanisms underlying energy transfer and dissipation in turbulent systems. It suggests that triad phase behaviour may provide valuable perspectives in understanding intermittency and could inform more sophisticated models of turbulence that incorporate these dynamics explicitly.

The work also serves as a foundation for future examinations of more complex systems, such as higher-dimensional Navier-Stokes equations. This focus on phase dynamics as a driving mechanism in energy flux has the potential to reshape perspectives on turbulence modeling and could inspire novel approaches that better capture the multi-scale interactions inherent in turbulent flows. As the field continues to evolve, research inspired by the findings presented here will likely further elucidate the intricate balance of forces in turbulent systems, enhancing both theoretical comprehension and practical application in computational fluid dynamics.