Linear and Nonlinear Stability of Flows

This chapter deals with some useful aspects of the linear and nonlinear instability of flows on a sphere. The loss of stability of a hydrodynamic flow is the first stage of its transition to a turbulent state. This phenomenon is also very important in meteorology. The movement of the atmosphere is extremely irregular. There are not only damped waves, but also unstable waves, the amplitude of which increases, leading to the complete destruction of a zonal circulation and the appearance of a cyclonic circulation. In its turn, the cyclonic circulation is with time destroyed and replaced by a zonal circulation. It should be noted that despite a large number of studies, the necessary conditions obtained by Rayleigh and Fiortoft remain so far the simplest and most constructive for studying the linear stability of shear and zonal flows. In addition, the semicircle theorems of Howard and Thuburn and Haynes set limits on the growth rate of unstable modes and provide information on the time–space structure of unstable disturbances. Nevertheless, the effectiveness of the necessary conditions for instability can be quite scanty. For example, any sufficiently strong LP flow of degree n ≥ 3 satisfies them. In this connection, every new condition for the instability can be a useful addition to the classical ones. Sections 7.1 and 7.2 contain classical results on the linear instability of parallel shear flows (Squire’s theorem, Rayleigh instability condition, Fjörtoft theorem, Howard’s semicircle theorem) and zonal flows (Rayleigh-Kuo instability condition, spherical analog of Fjörtoft’s theorem and semicircle theorems by Thuburn and Haynes). Arnold’s sufficient condition for nonlinear stability and the first and the second (direct) Liapunov methods for the study of nonlinear instability are briefly discussed in Sect. 7.3. Note that the first Liapunov method has already been used in Sects. 5.5 and 6.6 to prove the nonlinear instability of the zonal RH waves and dipole modons in the Liapunov sense. The Liapunov stability in the invariant sets of perturbations is also discussed here. The definition of nonlinear instability by Zubov in a metric space is defined. It should be used in the case when an invariant set of perturbations is not a linear space and is only a metric space. Equation for kinetic energy of perturbations is considered again in Sect. 7.4. Two mechanisms of generation of perturbation energy, previously described in a number of articles using the Eliassen-Palma flux diagnostics, are given in Sects. 7.5 and 7.6. Unfortunately, this diagnostics cannot be applied to arbitrary steady flow on a sphere. As an alternative, we propose in Sect. 7.6 a method for studying the geometric structure of growing perturbations of any stationary flow on a sphere using the energy and/or enstrophy norms. The method is based on the solution of eigenvalue problem for the symmetric part of the operator linearized about the basic flow. Besides, the eigenfunctions corresponding to the positive eigenvalues give a basis system of unstable orthogonal perturbations, while the instant growth of the kinetic energy (or/and enstrophy) of each such perturbation is determined by the corresponding eigenvalue. Thus, the eigenfunction corresponding to the largest positive eigenvalue gives the geometric structure of the most unstable perturbation in the energy norm (or/and in the enstrophy norm). The geometric structure of a set of unstable perturbations is also discussed here. In order to compare this method with the Eliassen-Palm flux diagnostics we applied it to the numerical stability study of the climatic January barotropic flow. The results of this numerical experiment presented in Sect. 7.7 show the ability of the method to construct the orthogonal system of unstable perturbations and, in particular, correctly reproduce both mechanisms of instability established earlier with the help of the Eliassen-Palm flux diagnostics near zonal jets.