Modelling Synchronization of Large-Scale Modes in Fluid Systems
Paper in proceeding, 2021
In electrically neutral fluids or in plasma flows the presence of nonlinear interactions can lead to the development of turbulence. In general, turbulence is characterized by energetic couplings between different scales of a flow. However, in the context of turbulence driven transport, such as the case of magnetically confined fusion plasmas or the diffusion of cosmic rays, typical flow structures are identified by dominant modes and the global turbulent state is approximated by a superposition of linear contributions (waves in general). These theoretical studies consider the amplitudes of the fluctuating quantities, but disregard the dynamics of the phases by using the so-called random-phase approximation (RPA) for which the existence of a Chirikov-like criterion for the onset of wave stochasticity is assumed. In this approximation one assumes that the dynamical amplitudes have a slow variation compared to the rapid change of the phases, which are considered to be distributed uniformly over a 2π interval. It has been observed that the phase dynamic shows significant departure from the well-known RPA assumptions, with phases locking occasionally (but not in the dissipative high-k range). In the well-known Kuramoto non-linear system with two-body interactions of limit cycle oscillators, it was shown that these types of systems are prone to locking if the coupling strength between the two-bodies has passed a threshold. In non-linear turbulent flow however, three-body interactions between the phases of the various modes is of importance. We will consider examples of synchronization in different fluid system such as Burgers and Navier-Stokes turbulence and in more advanced models such as those for Edge Localized Modes (ELMs) in tokamaks which remain a critical issue for plasma stability and the lifetime of fusion reactors such as ITER. The dynamic of the three-body interactions between the phases in the non-linear Burgers’ turbulence differs from the simplified picture of Kuramoto, and the phases lock intermittently and only in the low to mid-k range.