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Intermittency, turbulence, coupled oscillators, triads
Abstract:
Scale-dependent statistics, i.e. intermittency, are a hallmark of fully developed turbulence. In hydrodynamical turbulence this means a transition from Gaussian statistics on large scales to non-Gaussian statistics on small scales. The Fourier modes of a Gaussian random field are statistically independent. Conversely, it has been shown that, under quite general conditions, Fourier modes with random phases produce approximately Gaussian real-space statistics. This motivates the study of intermittency in turbulent fluids as a scale-dependent coherence phenomenon of the Fourier phases.
To better understand this relation between real-space intermittency and spectral-space coherence, a simple but novel coupled oscillator model is proposed. It is reminiscent of the phase-phase coupling present in the spectral space formulation of the Navier-Stokes equations, in which sets of three phases are coupled in so-called triads.
By studying this model we show that the three-oscillator PDFs can be completely identified with their triad PDFs. A convenient parametrization allows for a very good description of each triad's PDF using only one parameter. Using this parameter we can isolate each triad's contribution to real-space skewness. This establishes a relation between three-oscillator coherence phenomena and real-space intermittency.
