Fourier phase synchronization in minimal models for turbulent systems

Turbulent systems exhibit a remarkable multi-scale complexity, in which spatial structures induce scale-dependent statistics with strong departures from Gaussianity. In Fourier space, this is reflected by pronounced phase synchronization. We study the collective behaviour of the Fourier phases and investigate their role in the nonlinear transfers, fluxes of conserved quantities and intermittency statistics of various models for turbulent systems. We begin by considering the 1D Burgers equation. From the dynamics of the Fourier phases in this setting we build a minimal phase-coupling model for intermittency. The dynamical and statistical properties of the model are tunable and we find a range of values for which the model displays intermittency, increased phase synchronization and low-dimensional behaviour. The phase-coupling provides a link between the statistical and dynamical systems theories of turbulence and sheds light on the relation between real-space structure, intermittency statistics and phase synchronization, for which a quantitative relation is currently missing in the literature. We extend this approach to a shell model for turbulence where in this setting an amplitude-only equation provides the basis for a minimal model of intermittency. We find that there is a delicate relationship between the dynamics of the Fourier phases and the presence of intermittency, not only in the full shell model, but in the minimal amplitude-only shell model as well. We expand on the concepts of Fourier phase synchronization by examining the direct cascade of enstrophy in the 2D Navier-Stokes system where we find a clear correlation between strong phase synchronization, bursting of enstrophy across scales and enhanced intermittency in the vorticity field. Finally we discuss the significance of these results and how they could be extended to other models for turbulence.