Spectral approximation in the numerical stability study of nondivergent viscous flows on a sphere

✍ Scribed by Yuri N. Skiba

• Publisher: John Wiley and Sons
• Year: 1998
• Tongue: English
• Weight: 351 KB
• Volume: 14
• Category: Article
• ISSN: 0749-159X
• DOI: 10.1002/(sici)1098-2426(199803)14:2<143::aid-num1>3.0.co;2-o

✦ Synopsis

The accuracy of calculating the normal modes in the numerical linear stability study of two-dimensional nondivergent viscous flows on a rotating sphere is analyzed. Discrete spectral problems are obtained by truncating Fourier's series of the spherical harmonics for both the basic 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 functions on a sphere are used to estimate the rate of convergence of the eigenvalues and eigenvectors. It is shown that the convergence takes place if the basic state is sufficiently smooth, and the truncation numbers K and N of Fourier's series for the basic flow and disturbances tend to infinity keeping the ratio N/K fixed. The convergence rate increases with the smoothness of the basic flow and with the power s of the Laplace operator in the vorticity equation diffusion term.