Phase and amplitude evolution in the network of triadic interactions of the Hasegawa-Wakatani system

TL;DR

This paper investigates phase and amplitude dynamics in triadic interactions of the Hasegawa-Wakatani system, revealing how resonant and non-resonant triads behave and how a network of triads influences zonal flow dominance and nonlinear phase behavior.

Contribution

It introduces a network formulation of triadic interactions in the Hasegawa-Wakatani system, analyzing the transition to zonal flow dominance and nonlinear phase dynamics.

Findings

Resonant triads can saturate via phase locking with constant amplitude solutions.

Large-scale zonal flows transfer energy to subdominant modes, leading to exponential growth.

Zonal modes become dominant only when many triads are interconnected.

Abstract

Hasegawa-Wakatani system, commonly used as a toy model of dissipative drift waves in fusion devices is revisited with considerations of phase and amplitude dynamics of its triadic interactions. It is observed that a single resonant triad can saturate via three way phase locking where the phase differences between dominant modes converge to constant values as individual phases increase in time. This allows the system to have approximately constant amplitude solutions. Non-resonant triads show similar behavior only when one of its legs is a zonal wave number. However when an additional triad, which is a reflection of the original one with respect to the $y$ axis is included, the behavior of the resulting triad pair is shown to be more complex. In particular, it is found that triads involving small radial wave numbers (large scale zonal flows) end up transferring their energy to the subdominant mode which keeps growing exponentially, while those involving larger radial wave numbers (small scale zonal flows) tend to find steady chaotic or limit cycle states (or decay to zero). In order to study the dynamics in a connected network of triads, a network formulation is considered including a pump mode, and a number of zonal and non-zonal subdominant modes as a dynamical system. It was observed that the zonal modes become clearly dominant only when a large number of triads are connected. When the zonal flow becomes dominant as a 'collective mean field', individual interactions between modes become less important, which is consistent with the inhomogeneous wave-kinetic picture. Finally, the results of direct numerical simulation is discussed for the same parameters and various forms of the order parameter are computed. It is observed that nonlinear phase dynamics results in a flattening of the large scale phase velocity as a function of scale in direct numerical simulations.