The stability of planetary waves on a sphere

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Abstract
The stability of individual inviscid barotropic planetary waves and zonal flow on a sphere to small disturbances is examined by means of numerical solution of the algebraic eigenvalue problem arising from the spectral form of the governing equations. It is shown that waves with total wavenumber n (the lower index of the Legendre function Pmn which describes the waves’ meridional structure) less than 3 are stable for all amplitudes, whereas those with n ≥ 3 are unstable if their amplitudes are sufficiently large. For travelling waves (m ≠ 0) with n = 3 and 4 and with disturbances comprised of 30 modes, the amplitudes required for instability are approximated by those obtained from triad interactions, and are smaller than those given by Hoskins (1973). For the zonal-flow modes (m = 0) the critical amplitudes are smaller than those predicted by triad interactions, and are close to those obtained from Rayleigh's classical criterion.