On the linear stability study of zonal incompressible flows on a sphere

The normal mode (linear) stability of zonal flows of a nondivergent fluid on a rotating sphere is considered. The spherical harmonics are used as the basic functions on the sphere. The stability matrix representing in this basis the vorticity equation operator linearized about a zonal flow is analyzed in detail using the recurrent formula derived for the nonlinear triad interaction coefficients. It is shown that the zonal flow having the form of a Legendre polynomial P-n(mu) of degree n is stable to infinitesimal perturbations of every invariant set I-m with \m\ greater than or equal to n. For each zonal number m, I-m is here the span of all the spherical harmonics Y-k(m)(x), whose degree k is greater than or equal to m. It is also shown that such small-scale perturbations are stable not only exponentially, but also algebraically. (C) 1998 John Wiley & Sons, Inc.