  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. A quantitative relation between real-space structure,
statistics, and phase synchronization is currently missing. Here, we address
this problem in the framework of a minimal phase-coupling model, which enables
a detailed investigation by means of dynamical systems theory and multi-scale
high-resolution simulations. We identify the spectral power-law steepness,
which controls the phase coupling, as the control parameter for tuning the
non-Gaussian properties of the system. Whereas both very steep and very shallow
spectra exhibit close-to-Gaussian statistics, the strongest departures are
observed for intermediate slopes comparable to the ones in hydrodynamic and
Burgers turbulence. We show that the non-Gaussian regime of the model coincides
with a collapse of the dynamical system to a lower-dimensional attractor and
the emergence of phase synchronization, thereby establishing a
dynamical-systems perspective on turbulent intermittency.
