By Albert R. Baswell
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Extra info for Advances in Mathematics Research, Vol. 10
Sci. 30 (1973), 150–153.  S. A. Coaker, The stability of a Rossby wave, Geophysical & Astrophysical Fluid Dynamics. 9 (1977), 1–17.  J. L. Anderson, The instability of finite amplitude Rossby waves on the infinite betaplane. Geophys. Astrophys. Fluid Dynamics 63 (1992), 1–27.  B. J. Hoskins, Stability of the Rossby-Haurwitz wave. Quart. J. R. Met. Soc. 99 (1973), 723–745.  R. S. Haarsma, J. D. Opsteegh, Barotropic instability of planetary-scale flows. J. Atmos. Sci. 45 (1988), 2789–2810.
The mean spectral number χΨ is equal to 30for the unstable modes in full accordance with condition (68) which is also valid for polynomial flows (81) . Example 2. Non-zonal RH wave. 2 . 023. The basic flows for different a are presented in Fig. 3a-d. 014, there appears one unstable mode. 014). 013 is neutral and stationary. The amplitude Ψ(x) of the modes is mainly formed by the spherical harmonics Y2±1 (x), Y4±1 (x) and Y4±3 (x). It is in full agreement with the work  where it is noted that for the wave under consideration, the main nonlinear interaction triad is formed by the basic wave and two spherical harmonics (1,2) y (1,4).
Both the inequalities (64) and (71) and the orthogonality conditions (75) and (80) can also serve for testing the computational program. Theorem 7 asserts that unstable modes may exist only in subset M0n − Hn . Thus, exponential growth (as well as exponential decay) of perturbations is impossible in the sets M−n , M+n and Hn , and hence, all the modes of these three sets are neutral. In practice, the probability of the fulfilment of instability condition (68) is quite small, because the set M0n is only the boundary hypersurface separating the basic sets M−n and M+n .