On the Rayleigh-Kuo criterion for the tertiary instability of zonal flows

This paper reports the stability conditions for intense zonal flows (ZFs) and the growth rate γ_TI of the corresponding "tertiary" instability (TI) within the generalized Hasegawa-Mima plasma model. The analytical calculation extends and revises Kuo's analysis of the mathematically similar barotropic vorticity equation for incompressible neutral fluids on a rotating sphere [H.-L. Kuo, J. Meteor. 6, 105 (1949)]; then, the results are applied to the plasma case. An error in Kuo's original result is pointed out. An explicit analytical formula for γ_TI is derived and compared with numerical calculations. It is shown that, within the generalized Hasegawa-Mima model, a sinusoidal ZF is TI-unstable if and only if it satisfies the Rayleigh-Kuo criterion (known from geophysics) and that the ZF wave number exceeds the inverse ion sound radius. For non-sinusoidal ZFs, the results are qualitatively similar. As a corollary, there is no TI in the geometrical-optics limit, i.e., when the perturbation wavelength is small compared to the ZF scale. This also means that the traditional wave kinetic equation, which is derived under the geometrical-optics assumption, cannot adequately describe the ZF stability.