Phase Transition From Turbulence To Zonal Flows In The Hasegawa-Wakatani System

The transition between two-dimensional hydrodynamic turbulence and quasi-one-dimensional zonostrophic turbulence is examined in the modified Hasegawa-Wakatani system, which is considered as a minimal model of $\beta$-plane-like drift-wave turbulence with an intrinsic instability. Extensive parameter scans were performed across a wide range of values for the adiabaticity parameter $C$ describing the strength of coupling between the two equations. A sharp transition from 2D isotropic turbulence to a quasi-1D system, dominated by zonal flows, is observed using the fraction of the kinetic energy of the zonal modes as the order parameter, at $C\approx0.1$. It is shown that this transition exhibits a hysteresis loop around the transition point, where the adiabaticity parameter plays the role of the control parameter of its non-linear self-organisation. It was also observed that the radial particle flux scales with the adiabaticity parameter following two different power law dependencies in the two regimes. A simple quasi-linear saturation rule which accounts for the presence of zonal flows is proposed, and is shown to agree very well with the observed nonlinear fluxes. Motivated by the phenomenon of quasi-one dimensionalisation of the system at high $C$, a number of reduction schemes based on a limited number of modes were investigated and the results were compared to direct numerical simulations. In particular, it was observed that a minimal reduced model consisting of $2$ poloidal and $2$ radial modes was able to replicate the phase transition behaviour, while any further reduction failed to capture it.