Normal Mode Approach to the Stability Analysis of Rossby-Haurwitz Wave

The stability of the steady Rossby-Haurwitz wave (R-H wave) in the nondivergent barotropic model (NBM) on the sphere was investigated with the normal mode method. The linearized NBM equation with respect to the R-H wave was formulated into the eigenvalue-eigenvector problem consisting of the huge sparse matrix by expanding the variables with the spherical harmonic functions. It was shown that the definite threshold R-H wave amplitude for instability could be obtained by the normal mode method. It was revealed that some unstable modes were stationary, which tend to amplify without the time change of the spatial structure. The maximum growth rate of the most unstable mode turned out to be in almost linear proportion to the R-H wave amplitude. As a whole, the growth rate of the unstable mode was found to increase with the zonal- and total-wavenumber. The most unstable mode turned out to consist of more-than-one zonal wavenumber, and in some cases, the mode exhibited a discontinuity over the local domain of weak or vanishing flow. The normal mode method developed here could be readily extended to the basic state comprised of multiple zonalwavenumber components as far as the same total wavenumber is given.