Tertiary instability of zonal flows within the Wigner--Moyal formulation of drift turbulence

The stability of zonal flows (ZFs) is analyzed within the generalized-Hasegawa--Mima model. The necessary and sufficient condition for a ZF instability, which is also known as the tertiary instability, is identified. The qualitative physics behind the tertiary instability is explained using the recently developed Wigner--Moyal formulation [1] and the corresponding wave kinetic equation (WKE) in the geometrical-optics (GO) limit. By analyzing the drifton phase space trajectories, we find that the corrections proposed in Ref.~[1] to the WKE are critical for capturing the spatial scales characteristic for the tertiary instability. That said, we also find that this instability itself cannot be adequately described within a GO formulation in principle. Using the Wigner--Moyal equations, which capture diffraction, we analytically derive the tertiary-instability growth rate and compare it with numerical simulations. \\ $[1]$ D.~E. Ruiz, J.~B. Parker, E.~L. Shi, and I.~Y. Dodin, {\it Zonal-flow dynamics from a phase-space perspective}, Phys. Plasmas {\bf 23}, 122304 (2016).