ON THE LINEAR-STABILITY STUDY OF ZONAL INCOMPRESSIBLE FLOWS ON A SPHERE

The normal mode (linear) stability of zonal flows of a nondivergent fl uid on a rotating sphere is considered. The spherical harmonics are us ed as the basic functions on the sphere. The stability matrix represen ting in this basis the vorticity equation operator linearized about a zonal flow is analyzed in detail using the recurrent formula derived f or 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 h ere 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 algebraicall y. (C) 1998 John Wiley & Sons, Inc.