Fourier Phase Dynamics in Turbulent Non-Linear Systems

The research presented in this thesis examines in detail the role of triad Fourier phase dynamics across a range of turbulent fluid systems. In 1D Burgers, we see a clear link between the Fourier space triad phases and real-space shocks, the key dissipative structures of the dynamics. This link is evident also in the intermittency statistics, where time periods of high phase synchronisation contribute the majority of the extreme events that characterise intermittent behaviour. The reduction of degrees of freedom is also explored, with Fractal Fourier decimation used to remove modes across all scales of the system. We find that the phase synchronisation mechanism is extremely sensitive to such changes, and coherence is quickly lost as degrees of freedom are suppressed. We further extend these phase dynamics concepts by examining the forward enstrophy cascade in 2D Navier Stokes. Again the importance of the triad Fourier phases is clear, with strong preference for values that contribute to the forward cascade. We will see that at a snapshot in time, only a subset of the Fourier modes are responsible for the formation of small-scale vorticity filament structures that govern the total enstrophy dissipation of the ow. The final stage is to expand the definition of the triad Fourier phase to a non-scalar field in 3D Navier-Stokes. Utilising helical mode decomposition, we show the differing behaviour of the helical triad interaction classes and once again how helical triad phases play a vital role in the efficiency and directionality of energy flux in 3D turbulence. In a similar fashion to the 2D Navier-Stokes enstrophy cascade, we again find only a small energetic subset of the Fourier modes are important contributors to the flux toward small scales, and thus to the intermittent bursts of dissipation that characterise these chaotic flows. Finally we discuss how these exciting new results could be applied to other turbulent systems and how such coherent phase dynamics may lead to a better understanding of the mechanism behind Intermittency in Turbulence.