On the normal mode instability of harmonic waves on a sphere

Abstract

The normal mode instability of harmonic waves in an ideal incompressible fluid on a rotating sphere is analytically studied. By the harmonic wave is meant a Legendrepolynomial flow αPn(μ) (n ≥ 1) and steady Rossby-Haurwitz wave of set F 1 ⊕ Hn where Hn is the subspace of homogeneous spherical polynomials of the degree n(n ≥ 2), and F 1 is the one-dimensional subspace generated by the Legendre-polynomial P1(μ). A necessary condition for the normal mode instability of the harmonic wave is obtained. By this condition, Fjörtoft's (1953) average spectral number of the amplitude of each unstable mode must be equal to . It is noted that flow αPn (μ) is Liapunov (and hence, exponentially and algebraically) stable to all the disturbances whose zonal wavenumber m satisfies condition |m| ≥ n. The bounds of the growth rate of unstable normal modes are estimated as well. It is also shown that the amplitude of each unstable, decaying or non-stationary mode is orthogonal to the harmonic wave. The new instability condition can be useful in the search of unstable perturbations to a harmonic wave and on trials of numerical stability study algorithms. For a Legendre-polynomial flow, it complements Kuo's (1949) condition in the sense that while the latter is related to the basic flow structure; the former characterizes the structure of a growing perturbation.