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

TL;DR

This paper derives and revises the Rayleigh--Kuo criterion for the tertiary instability of zonal flows in plasma, showing its applicability and limitations within the generalized Hasegawa--Mima model and highlighting the inadequacy of wave kinetic equations in certain regimes.

Contribution

The paper provides an explicit analytic formula for the tertiary instability in plasma and corrects a historical error in Kuo's analysis, extending geophysical fluid dynamics results to plasma physics.

Findings

Zonal flows are TI-unstable if they meet the Rayleigh--Kuo criterion.

TI occurs when the ZF wave number exceeds the inverse ion sound radius.

No TI in the geometrical-optics limit, challenging wave kinetic equation assumptions.

Abstract

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. $\textbf{6}$, 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 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.