On The Rayleigh–Kuo Criterion For The Tertiary Instability Of Zonal Flows

These are the supplementary Matlab codes that can be used to generate the data in figure 3, 4, and 6 in the manuscript "On the Rayleigh--Kuo criterion for the tertiary instability of zonal flows". The abstract of this manuscript reads as follows: This paper reports the stability conditions for intense zonal flows (ZFs) and the growth rate $\gamma_{\rm TI}$ of the corresponding "tertiary" instability (TI) within the generalized Hasegawa--Mima plasma model. The analytic 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 analytic formula for $\gamma_{\rm TI}$ is derived and compared with numerical calculations. It is shown that a ZF is TI-unstable under the Rayleigh--Kuo criterion known from geophysics plus the condition that the ZF wave number must exceed the inverse ion sound radius. For a sinusoidal ZF, these two necessary conditions combined together are also sufficient for the TI. For non-sinusoidal ZFs, the results are qualitatively consistent. As a corollary, there is no TI in the geometrical-optics limit, i.e., when the DW wavelength is small compared to the ZF scale. This also means that the traditional wave kinetic equation under the geometrical-optics assumption cannot adequately describe the ZF stability.