The importance of phase dynamics in generation of coherent structures

Paper in proceeding, 2023

In magnetically confined plasmas (MCP), the transport of heat and particles is determined by collisional and anomalous processes caused by turbulence. A collective effort has been put into modelling the turbulent transport in plasmas using various drift wave (DW) models. However, it is evident that large-scale phenomena have a significant impact on overall transport. Heat transport can be mediated by coherent structures such as streamers and blobs through the formation of avalanche-like events that are intermittent in nature, i.e., localized in time but of large amplitude. Furthermore, at the same time, there are structures such as zonal flows (ZF) and GAMs that are non-linearly generated and mitigate turbulent transport by shearing turbulent eddies. A common denominator for these large-scale structures is the synchronization of smaller scale modes or events to a coherent structure, where phases align in a localized region of space and time. Interestingly, phase synchronization is prevalent in many other fields, such as biological clocks, physiological organisms, and chemical reactors. The dynamical evolution of amplitude and phases have been investigated through simplified equations derived from the Hasegawa - Wakatani (HW) system, where effects of synchronization are studied.
Theoretical studies often deal with the amplitudes of the fluctuating quantities and assume that the phases are randomly distributed according to the random phase approximation (RPA) and thus disregard the dynamics of the phases [1,2]. In this approximation, dynamical amplitudes have a slow variation compared to the rapid change of the phases, which are distributed uniformly over a $2 \pi$ interval [3]. There have been a few general approaches to the randomness in turbulence: the RPA, the diagrammatic method by Wyld and the cumulant expansions, with the aim of systematically characterizing intermittent behavior. Unless a specific case is studied, the diagrammatic method has a drawback since there is no consistent small expansion parameter and no normalization procedure available. Moreover, the intuitive picture of the RPA approach is tempting and is thus widely adopted in turbulence theory. The underlying assumption of randomness in the RPA for the phases of Fourier modes in nonlinearly interacting waves cannot be justified since the phases as well as the amplitudes evolve due to non-linear interactions that act on the same time scales for both. Thus, the phases cannot be randomized faster than the amplitudes, see further discussion in Refs. [4,5]. Understanding the generation of coherent structures and the effects of these structures on transport and turbulence is therefore of crucial importance. In regard to plasma dynamics, simplified models are of interest, assuming an expansion of the state in amplitude and phase, i.e., $\phi \sim \phi_0 \exp(i \theta)$, the basic dynamical equations yield one dynamical equation for the amplitude and one for the phase for each field in the model. In previous papers, models using the passive advected scalar [6] and the Burgers equation [1] where it was found that under certain conditions, the RPA assumption can be invalidated using a phase dependent force and the locking of phases may increase the energy transfer to other modes. The assumption of a fully stochastic phase state of the turbulence is more relevant for high values of scale separation with the energy spectrum following a $k^{−7/2}$ decay rate. The dynamic of the three-body interactions between the phases in the non-linear Burgers’ turbulence shows that the phases lock intermittently. This is due to the k dependence of the coupling strength in the non-linear term which reduces strongly for high-k range due to the dampening effect of the dissipation which does not allow locking of the phases of the small scales. For lower scale dependence the asynchronized and synchronized phases differ significantly, and one could expect the formation of coherent modulations in the latter case. Moreover, the HW have been studied [7] and the work on the predator-prey model of DW – ZF dynamics, it is observed that synchronization may be transferred between the two populations [8].


In this work, we investigate the role of phase dynamics for turbulent fluctuations in a set of direct numerical simulation (DNS) of homogeneous Taylor-Green driven turbulence, simple 2D rotating turbulence flow. The model is the forced incompressible magnetohydrodynamic (MHD) equations. It should be noted that in the study of coupled oscillators describing chemical reactors, the Kuramoto model has been established, and it has been shown that synchronization occurs when a certain threshold is exceeded. In this case the system is strongly forced to generate a vortex and where the phase locking between close neighbours can be quantified.