ON THE ROLE OF FJÖRTOFT’S SPECTRAL NUMBER IN THE
LINEAR INSTABILITY OF IDEAL FLOWS ON A SPHERE
The normal mode instability of steady solutions to the vorticity equation governing the motion of an ideal incompressible fluid on a rotating sphere is considered. All the types of known solutions are considered: the Legendre-polynomial (LP) flows, Rossby-Haurwitz waves, Wu-Verkley waves and modons. A conservation law for disturbances to each solution is derived and used to obtain a necessary condition for its instability. These conditions specify Fjörtoft’s  spectral number of the amplitude of unstable modes. For the LP (zonal) flows, it complements the well-known Rayleigh-Kuo and Fjörtoft conditions. The maximum growth rate of modes is also estimated, and the orthogonality of any non-neutral or non-stationary mode to basic flow is shown in the energy inner product.
The analytical instability results obtained are especially useful for testing the computational programs and algorithms in the normal mode stability study. Note that Fjörtoft’s spectral number plays a vital part in the linear instability problem.