Linear instability of ideal flows on a sphere

Abstract

AbstractA unified approach to the normal mode instability study of steady solutions to the vorticity equation governing the motion of an ideal incompressible fluid on a rotating sphere is considered. The four types of well‐known solutions are considered, namely, the Legendre‐polynomial (LP) flows, Rossby–Haurwitz (RH) waves, Wu–Verkley (WV) waves and modons. A conservation law for disturbances to each solution is derived and used to obtain a necessary condition for its exponential instability. By these conditions, Fjörtoft's (Tellus 1953; 5:225–230) average spectral number of the amplitude of an unstable mode must be equal to a special value. In the case of LP flows or RH waves, this value is related only with the basic flow degree. For the WV waves and modons, it depends both on the basic flow degree and on the spectral distribution of the mode energy in the inner and outer regions of the flow. Peculiarities of the instability conditions for different types of modons are discussed. The new instability conditions specify the spectral structure of growing disturbances localizing them in the phase space. For the LP flows, this condition complements the well‐known Rayleigh–Kuo and Fjörtoft conditions related to the zonal flow profile. Some analytical and numerical examples are considered. The maximum growth rate of unstable modes is also estimated, and the orthogonality of any unstable, decaying and non‐stationary mode to the basic flow is shown in the energy inner product.The analytical instability results obtained here can also be applied for testing the accuracy of computational programs and algorithms used for the numerical stability study. It should be stressed that Fjörtoft's spectral number appearing both in the instability conditions and in the maximum growth rate estimates is the parameter of paramount importance in the linear instability problem of ideal flows on a sphere. Copyright © 2008 John Wiley & Sons, Ltd.