Phase dependent advection-diffusion in drift wave - zonal flow turbulence

In magnetically confined plasmas, drift wave turbulence is believed to be responsible for theanomalous transport and the generation of inherent sheared zonal flows providing a self-regulatingmechanism that may control the turbulence itself. The coupled system of drift wave-zonal flow(DW-ZF) is one of the key points in the evasive explanation of the low to high confinement(L-H) transition in fusion plasmas. In plasma turbulence a large number of non-linearly interactingmodes are present which yield an enormous complexity and dynamic richness. However,the nonlinear mode coupling processes allows for coherent structures (sheared flow, acousticmodes, etc.) to be formed which can even grow in the presence of random fields or cascadeto smaller scales, this is a manifestation of the self-organization process with predator-prey dynamics.A powerful and yet simple mathematical framework for describing the self-organisationphenomena was developed by Kuramoto [1]. The Kuramoto model describes the phase dynamicsof a system of stochastic limit-cycle oscillators revolving at arbitrary intrinsic frequenciesand coupled through the sine of their phase differences. Furthermore, under certain conditionsthey spontaneously lock into a common frequency. Most often in modelling, the dynamics ofthe phases is neglected and the so-called random-phase approximation (RPA) is used. However,one could expect that the phase dynamics can play a role in the self-organisation and the formationof coherent structures. In this work we present a study on the importance of a collectivephase dynamic on the characteristic time evolution of the fluctuation energy and the formationof coherent structures. The model is built on the stochastic passive advection-diffusion ofa scalar [2] while using the predator-prey model with stochastic oscillators representing theDW-ZF interaction [3]. Our findings show the occurrence of modulational behaviour prior tothe synchronisation state, and that the synchronization of the phases can result in a significantchange in the dynamics of the fluctuation energy and the saturation state