SPECTRAL APPROXIMATION IN THE NUMERICAL STABILITY STUDY OF NONDIVERGENT VISCOUS FLOWS ON A SPHERE

The accuracy of calculating the normal modes in the numerical linear s tability study of two-dimensional nondivergent viscous flows on a rota ting sphere is analyzed. Discrete spectral problems are obtained by tr uncating Fourier's series of the spherical harmonics for both the basi c flow and the disturbances to spherical polynomials of degrees K and N, respectively. The spectral theory for the closed operators [1], and embedding theorems for the Hilbert and Banach spaces of smooth functi ons on a sphere are used to estimate the rate of convergence of the ei genvalues and eigenvectors. It is shown that the convergence takes pla ce if the basic state is sufficiently smooth,and the truncation number s It and N of Fourier's series for the basic flow and disturbances ten d to infinity keeping the ratio NIX fixed. The convergence rate increa ses with the smoothness of the basic flow and with the powers of the L aplace operator in the vorticity equation diffusion term. (C) 1998 Joh n Wiley & Sons, Inc.